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# Fundamental Approach to Practical Corrosion Problems

surface treatments under both anodic and cathodic delamination conditions. The results

were interpreted in terms of the differing chemical stability of the conversion lIn-
OH- ions generated by oxygen reduction.

3.2 Mathematical Models

It is difficult to demonstrate that a proposed mechanism -II_a-r-- -1. based on

experimental work does indeed give the observed experimental results. Interpretation

is often limited to qualitative and subjective observations. Therefore, a mathematical

model is necessary to identify the phenomena and mechanisms that contribute to the

delamination process.

To date, there are not many mathematical models that simulate explicitly the

propagation of front along the metal-coating interface during the cathodic delamination of

a coated metal. Yet, the occluded environment underneath the paint have been modeled

extensively in terms of crevice corrosion models [58-61] or dishonded coating models.

[62, 6:3] These models are often used to calculate the spatial distributions of es, 4, and the

current density along the metal surface.

3.2.1 Crevice and Disbonded Coating Models

The crevice system, given in Figure :3-3, has been approximated by a geometry that

consists of a dishonded region .Il11 Il-ent to a defect. The boundary located at the plane of

symmetry represents an impermeable wall. The thickness of the dishondment, termed the

gap, is assumed uniform along the metal-coating interface. In dishonded coating systems,

the length of the system in the direction parallel to the substrate is often assumed to

be large as compared to the gap. This facilitates the assumption that variations in the

direction normal to the metal surface are negfligfible.

Chin and Sabde [62] developed a steady-state model for a dishondment system under

cathodic protection. The model accounted for oxygen reduction but neglected corrosion

and hydrogen evolution under the cathodic protection. The ionic species considered were

Na+, Cl-, and OH-, and no other homogeneous chemical reactions were included. The

diffusion and migration of species were considered; however, the diffusion coefficients were

treated as Di = 10-5 cm2S-1 foT all 101110 Species.

Following the development presented by Chin and Sabde [62], Allahar [22] established

a two-dimensional dishondment model in the presence of multiple electrochemical

reactions and one homogeneous reaction. In his work, the assumption that Di was

equal for all species was relaxed. [22] A quasipotential transformation was introduced

Defect~- Disbondment
Plane of itoce
Syrnretry.

Bulk condhi inCotn

Figure 3-3. Schematic diagram of a disbonded coating system on a coated metal.

to solve the system of coupled equations that described the transport of species and the

electroneutrality condition. The application of the transformation increased computational

accuracy and overcame the numerical difficulties arose from the inclusion of homogeneous

reactions .

Sridhar [63] developed a transient model for disbonded coating system to simulate

the evolution of the disbondedment. The species considered in the model were separated

into primary and secondary species. The concentration distributions of the primary

species were calculated using given potential distributions and conservation relationships.

The concentration distributions of the secondary species were calculated using the

concentration of the primary species and the equilibrium relationships associated with

homogeneous reactions. The computational results showed that the pH and potential

gradients in the disbonded region were the consequences of the competition between

anodic dissolution and cathodic reactions. Exposure time, applied potential, solution

conductivity, and crevice geometry were the factors that influenced the pH and potential

3.2.2 Cathodic Delamination Model

Based upon the experimental observations reported by Stratmann et al. [9-18],

Allahar [21, 22] developed the first mathematical model for the cathodic delamination

of coated metal. The model accounted for the coupling of mass transfer of species,

electrochemical kinetics, bond-breakage phenomena, and the propagation of the front

along the metal-coating interface. The key to the development of the model involved

applying the concept that the porosity and the polarization kinetics at the interface were

pH dependent. These pH-dependent hypotheses provided an approach to simulate the

growth of the delamination within a fixed domain. The development of Allahar's work

served to guide that of the present work.

3.2.2.1 pH-Dependent Porosity

Furbeth and Stratmann [9] reported that, during the cathodic delamination process,

the coating degraded into a gel-type medium and the adhesive bond between the coating

and the metal was weakened. The OH- ions produced in the oxygen reduction underneath

mathematical model developed by Allahar, [21, 22] the transition from the degraded region

to the intact region was characterized by the change in porosity. Under the assumption

that the oxidation of the coating was due to the attack of OH- ions, the interfacial

porosity was treated as a function of pH.

The transition of the porosity in the delaminated zone was constructed based upon

the de-adhesion experiments performed by Furbeth and Stratmann. [9] From the tensile

force distributions, Allhar [22] proposed that the porosity was somewhat larger in the

delaminated region due to the interfacial bonds being partially broken. In the intact region

where the delamination has not occurred, the porosity was somewhat smaller. Thus,

the porosity made a transition from the value associated with the delaminated region to

that associated with the intact region. A mathematical formula was then constructed to

describe the the distribution of the porosity in the delaminated domain.

Experimental results revealed that in the delaminated and front regions the

concentrations of the two 1!! r ~ species, OH- and Na+, were approximately equal.

[9] Therefore, the pH distribution was constructed according to the observed CNa+

distribution. The expression for the porosity-pH relation,

be~i be,4
E = ++bs,7 (3-2)
1 +exp [be,2 (pH be,3)] 1 +exp [bs,s (pH be,6)

was obtained by coupling the porosity and the pH distributions, where be,i to be,7 were

fitting parameters.

The use of the pH-dependent interfacial porosity represented a novel approach to

account implicitly for the bond-breaking process during the cathodic delamination. As

proven by Allahar, no propagation was observed when the pH-porosity relation was not

incorporated in the model. [22]

3.2.2.2 pH-Dependent Polarization Kinetics

Due to the presence of the coating, the polarization behavior of electrochemical

reactions at the metal-coating interface is different from that at the metal-electrolyte

interface. Allahar [22] derived expressions for the polarization kinetics at the metal-coating

interface based upon expressions applicable to a bare metal surface.

The current density due to zinc dissolution and oxygen reduction at the zinc-coating

interface were calculated using [22]

(otV-EZn( 3
igt WA WZn o,zn l0 Pzn(3)

and

21im,2 -T'I1~0 211n~aC2,oo1.519i1.5
ii" = -W O na oto (3-4)
Eg Yc Ec 9m

The surface coverage wA represented the surface area available for electrochemical

reactions. The poisoning parameter wzn considered the influence of surface on the

exchange current density of zinc dissolution. The effect of the coating or other formed

deposits on the transport of oxygen through the gel medium was included by a blocking

parameter wo,. As the effect of these parameters varied according to the local pH, the

parameters were linked to pH with the same manner described in the porosity-pH relation.

The use of porosity to represent the bond-breaking process occurring during the

cathodic delamination process provided a mathematical framework for the development

of advanced models. The numerical approach emploi- a by Allhar was not able to include

homogeneous or other chemical reactions due to the ill-conditioned coefficient matrix

generated from governing equations. In the present work, a series of homogeneous

reactions and formation of corrosion products were considered. A different numerical

approach using ?-. \--n! Ion's BAND algorithm [46] was emploi-. I1 in order to incorporate

these chemical reactions into the model.

3.3 Ob jective

The objective of the work was to develop a mathematical model that simulates

the cathodic delamination of a coated metal in the presence of multiple electrochemical

reactions, homogeneous reactions and formation of corrosion products. The electrochemical

reactions considered along the metal-coating interface were zinc dissolution and oxygen

reduction. Water dissociation and a series of reactions associated with Zn2+ hydrolysis

were treated as homogeneous reactions in the model. The corrosion product ZnOH2 WaS

assumed to precipitate along the metal-coating interface. The concepts of pH-dependent

porosity and pH-dependent polarization kinetics remained in the model, but modifications

were made to reduce the number of the fitting parameters used in the program.

The second objective was the use of the developed model to predict the delamination

rate and the delamination kinetics for non-pigmented coated samples. As addressed

earlier, the rate and the mechanism of the delamination system depend strongly on the

application and the property of inhibitors and/or pigments. The chemical variations

associated with these surface treatments, however, are not the major focus of the present

work.

CHAPTER 4
THEORETICAL DEVELOPMENT OF DELAMINATION MODEL

The propagation model simulated the evolution of the delamination process from

given initial conditions. The propagation of front and the bond-breakage reactions

accompanying the delamination process were modeled through the hypotheses that

porosity and the polarization kinetics at the coating-metal interface were pH dependent.

Mathematical constructions for the initial conditions and the pH-dependent hypotheses

followed the development by Allhar. Modifications were made to reduce the number of the

fitting parameters used in the program.

4.1 Porosity-pH Relation

In the present model, the delaminated zone was considered to be a porous medium.

The flux in a dilute, porous electrochemical system Ng* was expressed as [46]

D*
N,* = -zec F V@ D5Vc4 (4-1)
i RT

where ci is the concentration in the aqueous phase. The effective diffusion coefficient Df in

a porous medium was related to the porosity E by [64]

DJ = E1.5Di (4-2)

where Di is the diffusivity in an aqueous medium. Equation (4-1) was recast in terms of

Di using equation (4-2) as

*~Di
N,* = E.5 _Sici i V DiVci) (4-3)
RT

The conservation of a species i in a porous medium was expressed in terms of Nsi as

8(~)= -V (E1.51Vi) ERi 44

where Neiwas the flux in the solution phase.

Position Position

(a) (b)

Figure 4-1. Schematic diagrams for interfacial porosity E and pH as functions of position
in the delaminated zone. The dashed lines separate the domain into the
delaminated, front, and fully-intact regions: a) interfacial porosity, and b)
local pH.

Following the approach taken by Allahar, [22] interfacial porosity was used to

characterize the transition from a degraded region to an intact region in the delaminated

zone. An assumed porosity transition in the delaminated domain, shown in Figure 4-1(a),

was constructed according to the de-adhesion tests conducted by Stratmann et al. [9].

Three transition regions are observed in Figure 4-1(a). The porosity decreases gradually

with position in the delaminated region. In the front region where the delamination is

ongoing, the porosity shows an abrupt decrease with position. The porosity remains

unchanged in the intact region where the delamination has not yet occurred.

Experimental results revealed that in the delaminated and front regions the

concentrations of the two ill r ~ species, Na+ and OH-, were approximately equal. [9]

Thus, the pH distribution in the delaminated domain was constructed based upon the

observed CNa+ distributions. [9] An assumed pH distribution is given in Figure 4-1(b)

where the pH value decreases with position in the delaminated and front regions. As

the OH- ions produced in the oxygen reduction underneath the paint were linked to the

0.012

0.010

,0.008

0.006

a, 0.00)4

0.002

0.000 I
7 8 9 10 11 12 13 14 15
pH

Figure 4-2. Schematic diagram for distribution of interfacial porosity E aS a funCtiOn Of
local pH.

coating degradation and the loss of adhesion, [9-18] the porosity used to represent the

adhesion between the metal and the coating was treated as a function of pH.

The porosity-pH relation was obtained by combining Figures 4-1(a) and 4-1(b) to

yield the E distribution as a function of pH shown in Figure 4-2. The mathematical

expression for the E-pH relationship was obtained by fitting an equation of the form

E(pH) =b'+ be,4 (4-5)
1 + exp(be,2(pH be,3s

to the plot in Figure 4-2 where be,i to be,4 were fitting parameters. The porosity was

assumed to reach the value given by equation (4-5) instantaneously; thus the E-pH

relationship represents an equilibrium condition between E and pH.

4.2 Polarization Kinetics

The electrochemical reactions of interest in the cathodic delamination model involved

zinc dissolution

Zn Zn2+ + 2e- (4-6)

and oxygen reduction

02 + 2H20 + 4e- 40H- (4-7)

The polarization kinetic at the metal-coating interface was derived from that applicable to

the metal-electrolyte interface.

4.2.1 Zinc Dissolution

In an aqueous environment with a bare metal surface, the current density due to zinc

dissolution izn follows the Bulter-Volmer equation [48]

izn = io,znl0 zn (48)

where Pza, Ez,, io,zn are the Tafel slope, equilibrium potential, and exchange current

density, respectively, for the zinc dissolution. A poisoning factor ( was emploi-. I to

calculate the current density due to the zinc dissolution at the metal-coating interface, i.e.

[22]

Lijot CIo,zal0 znI (4-9)

The poisoning factor ( considered the effect of (e Ir filr_ the availability of surface area

to zinc dissolution during the delamination. The factor also accounted implicitly for the

presence of passive films formed on the metal surface. The poisoning factor ( was assumed

to be a function of pH, and the construction of the (-pH relationship was performed in a

manner similar to the construction of the E-pH relationship.

4.2.2 Poisoning-pH Relation

Experimental observations indicated that, during the cathodic delamination process,

anodic dissolution along the metal-coating interface is poisoned due to the presence of the

coating. [9, 16] Thus, the zinc dissolution was considered unfavorable in the delaminated

zone by assigning (
is presented in Figure 4-3(a). In the delaminated region, the poisoning parameter ( is

approximately a constant, indicating that the surface availability to the zinc dissolution is

independent of position in this region. The poisoning parameter decreases exponentially

with position in the front region, demonstrating that the anodic reaction is unfavorable in

the front. In the fully-intact region the value of ( is held as a constant.

Figure 4-3. Schematic diagram for distribution of poisoning factor (: a) as a function of
position, and b) as a function of local pH-.

The relationship between ( and pH-, given in Figure 4-3(b), was constructed by

coupling the distributions shown in Figures 4-3(a) and 4-1(b). The mathematical

expression for the (-pH relationship was obtained by fitting an equation of the form

((p) =bc,i +b(,4 (4-10)
1 +exp(bC,2(pH b(,3)

to the plot in Figure 4-3(b) where bC~1 to b(,4 were fitting parameters.

4.2.3 Oxygen Reduction

Under the assumption that oxygen reduction is mass-transfer-limited at the

metal-electrolyte interface, the limiting current density is given as [48, 65]

nFDo, cos,,m
si~m,oz = (4-11)

where x is the distance that oxygen diffuses through and comm is the oxygen concentration

in the bulk. In the presence of coating and interfacial oxidized lI e. r, the mass-transfer-limited

current density due to oxygen reduction was calculated using [22]

if0 =-o nos oa 1sa t (4-12)
Eg Yc Ec 9m

Position pH
(a) (b)

8 1 1 2 13 1
o"Position pH"

poito; n b sio a untono lcl H

where gm and go were the thickness of the gel-medium and the coating, respectively, and

Ec and Eg were the porosity of the un-degraded coating and the gel-medium, respectively.

The complex term seen at the right side of equation (4-12) was derived by solving the

concentration distribution of oxygen in the direction normal to the metal surface. The

blocking factor cto, accounted for the influence of the coating and the oxidized l?--;r on

the transport of oxygen to the metal surface. The blocking factor cto, was assumed to be a

function of pH and the construction of the cto,-pH relationship was performed in a manner

similar to the construction of the E-pH relationship.

4.2.4 Blocking-pH Relation

An assumed distribution of cso, as a function of position along the metal-coatingf

interface is presented in Figure 4-4(a). The blocking factor is a constant in the

delaminated region, indicating that the transport of oxygen is independent of position

in this region. A largest electrochemical reactivity for oxygen reduction is expected across

the front region; thus, the blocking factor increases in the front. The blocking factor

decreases to a minimum to represent a smallest reactivity in the intact region.

The cto,-pH relation was constructed by coupling Figures 4-4(a) and 4-1(b) to

yield the cto, distribution as a function of pH given in Figure 4-4(b). The mathematical

Table 4-1. Fitting parameters used in the expressions of pH--dependent interfacial porosity,
blocking, and poisoning parameters.
k. E t o,
b1 0.01 4.50 7.50
b2 -3.00 -3.30 -7.00
b3 10.8 10.4 9.80
b4 0.001 -16.0 -0.50
b5 -50.0
bn6 11.10
b7 -10.0

expression for the cto,-pH relation was obtained by fitting an equation of the form

n~o, (ba ,1 ba,4g +or
topH)= 1 (1exp(ba,2(pH -ba,3)) 1+exp(ba,5(pH -ba,6)) ba,7)

to the plot in Figure 4-4(b) where boyl to ba,7 were fitting parameters.

The values of the fitting parameters used in equations (4-5), (4-10) and (4-13) are

listed in Table 4-1. The choice of the values of these parameters might pi?-, an important

role on the computational results. Thus, a systematic sensitivity analysis was performed

and the results are reported in Appendix E.

4.3 Chemical Reactions

The oxygen reduction taking place underneath the coating results in an increase of

pH- in the interfacial degraded lI e. r. For zinc, a series of chemical reactions associated

with Zn2+ hydrolysis and formation of corrosion product Zn(OH)2(s) are possible in

alkaline solutions. [66, 67] In the presented model, multiple homogeneous reactions,

including water dissociation and a series of reactions associated with Zn2+ hydrolysis,

were considered. The mechanisms and equilibrium conditions of these chemical reactions

are summarized in Table 4-2. [67] All the homogeneous reactions were assumed to be

equilibrated because the time constants for reaching the equilibrium conditions are much

smaller than that for the diffusion of the limiting reactant. [68, 69]

The precipitated corrosion product Zn(OH)2(s) is thermodynamically stable within the

pH- ranging from 8.5 to 11. [66, 67] The reaction mechanism of forming solid Zn(OH)2(s)

Table 4-2. Reaction mechanism and equilibrium condition for homogeneous reactions
included in the model. [66, 67]
Reaction NO. C'I. ...ud 1 Reaction Equilibrium Condition
1 H20 OH- + H+ Kw = cOH-CH+
2 Zn" + OH- ZnOH+ log c~O -9.67 + pH
ZnOH~CZn2
3 ZnH+ +20H- HZnO, H20 log = -17.97 +pH

4 H~OE OH-- ZO -+ H20 log = -13.17 + pH
HZnO2-

was assumed to be

Zn2+ + 20H- Zn1(OH)2(s) ( 4

The rate of production of Zn(OH)2(s) depends strongly on the concentrations of Zn2+ and

OH-, thus, the precipitation rate was related to czn+ and COH- by

rpre = k [c H-CZn2+ K~sp] (4-15)

where k is a rate constant and Ksp IS the standard solubility products of Zn(OH)2(s)

at room temperature. [70] The difference between the two terms in the bracket at

the right side of equation (4-15) represents the driving force for formning Zn(OH)2(s)'

Equation (4-15) provides an approach, in terms of the concentrations of Zn2+ and OH-

ions, to incorporate the solid species Zn(OH)2(s) in the complex model. However, this

approach is different from that emploi-e 4 in thermodynamic calculations in which the total

concentrations of Zn2+ and OH- ions were held as constants. [71, 72]

CHAPTER 5
CATHODIC DELAMINATION MODEL

The development of the mathematical model is presented in this chapter. The model

simulated the propagation of the front along the metal-coating interface during the

cathodic delamination of a coated zinc.

5.1 Governing Equations

Under the assumption that the thickness of the coating is much smaller than the

length of the domain, the mathematical model was developed focusing on a one-dimensional

delaminated zone. The significance of the defect was included implicitly at the boundary

sharing with the defect. The dependent variables considered in the model were potential

4 and concentrations of OH-, Na+, Cl-, H+, Zn2+, ZnOH+, HZnO2 ZnO -, and

Zn(OH)2(s)'

The governing equation for the solution potential was derived from the electroneutrality

condition
i=9

i= 1

The governing equation for ci in a 1-D domain was

8 (EC,) 8 (E1.5Vi)
+Ri + Si (5-2)
iit 8ix

where Si represented the rate of production per unit volume by electrochemical reactions.

The conservation equations for the chemically inert species, Na+ and Cl-, were obtained

by assigning SNa+ = 0, and Set = 0. The governing equations for the species participating

in heterogeneous reactions, Zn2+ and OH-, were formulated as

8 (ECZn2+) 8 (E1.5 Zn2+) jat
+Rzns+ + (5-3)
iit 8ix 2F

and

8 (ECOH-) d (E1.5 OH ) i 02
+ROH- +lm0 (5-4)
dt 8x -F

respectively. The net production of Zn2+ and OH- by the homogeneous reactions at any

position was equal to zero, i.e.

Rzn2+ + RZnOH+ RHZnO +ZnO2 = 0 (5-5)

and

ROH- H Zn2+ +2RHZnO2 f3ZnO2 = (5 6;)

Substitution of Ri into equation (5-5) yielded the governing equations for czn2+ and COH-

respectively, i. e.
coat
Gzn2+ +" GZnOH+ GHn + Gzo = 0 (5-7)

and
icoat -coat
GOH- GH+ Gzn2+ + +2Hn Gn 58

where
8 (EC,) a E1.5Vi)
Gi = -+(5-9)
iit 8ix

The equilibrium conditions listed in Table 4-2 were applied as the governing equations

for H+, ZnOH+, HZnO2 and ZnO -. [66] The rate of formation of the corrosion product

Zn(OH)2(s) WaS aSSociated with czns+ and COH- by

8CZn(OH)z
Bt =k [czn2+ ciH Ksp] (5-10)

where the standard solubility product Ksp has a value of 3 x10-17 (mol/1iter)3 at TOOm

temperature. [10]

The phenomena of bond-breakage and coating degradation involve chemical reactions.

Equation (4-5) governs the equilibrium relationship between interfacial porosity and local

pH. The equilibrium E-pH relation is valid under the assumption that the time constants

of bond-breakage reactions are sufficiently small. When the time constants for these

phenomena are large compared to those for the processes of diffusion and migration, the

equilibrium assumption becomes invalid. A non-equilibrium relationship between E and pH

was assumed to follow
iiE
Bt=- kneq (E Eeq) (5-11)

where the equilibrium porosity Eeg is obtained in equation (4-5) and kneq is the rate

constant for the bond-breakage reactions. In the limit that kneq i OO the value of E attainS

its equilibrated value E = Eeq*

5.2 Boundary Condition

At the boundary with the defect, the solution potential and concentrations of

chemical species were fixed at a bulk condition 4, and ci,,, respectively. The boundary

condition for the solution potential 4 at the fully-intact region remained as the electroneutrality

condition. A zero-flux boundary condition Nsi = 0 was used for each species at the

boundary in the fully-intact region.

5.3 Solution Method

The system of coupled equations consisted of four equations written in the form

of equation (5-2) for OH-, Na+, Cl-, and Zn2+, TOSpectively, equation (5-1) for

the electroneutrality condition, equilibrium conditions for H+, ZnOH+, HZnO2 and

ZnO -, equation (5-10) for the corrosion product, and an equation for the porosity-pH-

relationship. When the porosity was assumed to reach its equilibrium value instantaneously,

the equilibrium E-pH relationship, equation (4-5), was used as the governing equation for

E. When the non-equilibrium E-pH relationship was applied, equation (5-11) was used as

the governing equation for E.

The derivative terms were discretized at each node in the domain using Taylor series

approximation. The first-order temporal derivative was given by

8 (ECi) (ECi)n+1 (ECi)n

where the superscripts in brackets n and n + 1 represented the conditions at a given

time t and a time one time step ahead, t + at. Terms of the order (At) and higher were

neglected in the temporal derivative. The spatial derivatives for a non-boundary node m

were approximated using the central finite difference equations [73], i.e.

8/ f+ -Im1+ O(Ax)2 (5-12)
8x 2Ax

42m + -2fm+Im1+ O(Ax)2 (5-13)
8 2 2a>

where f was a generic variable for ci, 4, and E. Terms of the order (Ax)2 and higher were

neglected in the spatial derivatives. A quarter-point method was used to approximate the

derivative term in the boundary conditions. The approximation of m-j was obtained using

iifm+7 \ 2 /

where

f m + Im-1 (-5
mS,, 2 5j

The resulting system of algebraic equations was accurate to the order of (Ax)2

The system of coupled, non-linear, partial differential equations required an iterative

method to converge on a solution starting from an initial guess. A tri-diagonal method,

BAND algorithm, coupled with time step was chosen to calculate the distribution of

cs, 4, and E in the delaminated domain. The mathematical model was developed using

Microsoft V/isual Fortran, V/ersion 9.0 with double precision accuracy. [74]

CHAPTER 6
RESULTS AND DISCUSSION FOR DELAMINATION MODEL

The present mathematical model simulated the evolution of a cathodic delamination

system from a set of given initial conditions. The resulting distributions satisfied the

coupled phenomena of mass transfer, electroneutrality, and disbondment reactions during

the delamination process. The calculated results and discussions are presented in this

chapter.

6.1 Initial Conditions

The geometric parameters used in the model included a coating thickness g, = 45 pm,

a gel-medium thickness g, = 5 pm, and a net length of 0.8 cm for the delaminated, front

and intact regions. The initial lengths of these regions were 0.1 cm, 0.05 cm, and 0.65 cm,

respectively.

The input parameters for the simulation included the grid size ax = 4 x 10-4 cm ,

the time step at = 0.1 s, and the total time t = 60 min. The potential on the metal was

chosen as V = -0.95 VSHE. The polarization parameters for zinc dissolution included pzn

=0.04 V/decade, io,zn =0.008 A/cm2, and Eoz,z =- 0.763 VSHE. The diffusion coefficients

Di for the chemical species are given in Table 6-1. The concentration of dissolved oxygen

at the surface of the coating was 1.26 x 10-3 M. [46]

Table 6-1. Diffusion coefficients of chemical species[46, 75]
CI.. on .1 Di in bulk
Species electrolyte (cm2 S)
02 1.90 x10-s
OH- 5.25x10-s
Na+ 1.47 x 10-s
Cl- 2.03 x10-s
Zn2+ 0.71 x10-s
H+ 9.32 x10-s
ZnOH+ 1.00 x10-s
HZnO2 1.00 x10-s
ZnO 1.00x10-5

-10 L --- -Zn'
o .,

S10.

10"
0.0 0.1 0.2 0.3 0.4
Position / cm

Figure 6-1. Initial concentration distributions of OH-, Na Cl-, and Zn2+ ioUS along the
metal-coating interface.

6.1.1 Initial Concentration Distributions

The objective of this work was to explore the propagation phase of the delamination

process. Thus, an initial condition was established which reflected a system after the

delamination had begun. The initial concentration distributions presented in Figfure

6-1 were constructed based on the experimental data reported by Leng and Stratmann

[14-16] The initial concentrations of Na+ and OH- ions were equal at any position in the

delaminated domain. The shape of the co- distribution followed that of CNa+ distribution.

The concentrations of Na+, Cl-, and OH- ions decreased monotonically with position

in the delaminated and front regions. In the intact region, the concentrations of all

species reached .l-i-mptotic values. The distribution of czn+ was obtained by satisfying

electroneutrality at a given position. The concentrations of the corrosion product and the

species produced in the homogeneous reactions were assumed to be zero at t=0.

6.1.2 Initial Distribution of Porosity

The construction of the relation between porosity and pH- is given in Section 4.1. The

fitting parameters used in the equilibrium E-pH relationship (equation (4-5)) are given in

Table. The initial porosity distribution is shown in Figure 6-2(a) as a function of position

and in Figure 6-2(b) as a function of pH-. As seen in Figure 6-2(a), the porosity decreases

0.012 0.012

0.010 -1 0.010

0.008 -1 0.008

0.006 1 0.006

0.004 1 0.004

0.002 -1 0.002

0.000 I 0.000 I
0.0 0.1 0.2 0.3 0.4 8 9 10 11 12 13 14
Position / cm pH

(a) (b)

Figure 6-2. Calculated initial porosity distribution: a) as a function of position, and b) as
a function of pH.

non-linearly in the delaminated and the front regions. The porosity maintains a uniform

value in the intact region. The porosity-pH plot (Figure 6-2(b)) presents the concept that

at a high pH the adhesive strength is low and that at a low pH the adhesive strength is

high.

6.1.3 Initial Distribution of Polarization Parameters

The polarization parameters t~o, and ( were used to calculate the current densities

due to the zinc dissolution and oxygen reduction at the metal-coating interface. Equations

(4-10) and (4-13) govern the (-pH and cso,-pH relations. The fitting parameters used

in the equations are given in Table 4-1. The initial distributions of ( are presented in

Figure 6-3(a) as a function of position and in Figure 6-3(b) as a function of pH. Figures

6-4(a) and 6-4(b) present the initial distributions of cso, as a function of position and pH,

respectively.

The current density expressions given in equations (4-9) and (4-12) were applied to

generate the polarization plot at the metal-coating interface as a function of pH presented

in Figure 6-5. The pH value of 8.7 corresponds to the positions in the intact region where

local anodic reactions are balanced by local cathodic activities. The corrosion current

density at the metal-coating interface, therefore, is approximately equal to zero in the

0.0 0.1 0.2 0.3 0 4
Position / cm

8 9 10 11 12 13 14
pH
(b)

Figure 6-3. Calculated initial distribution of poisoning factor: a) as a function of position;
and b) as a function of pH.

102 I I10i nl
10 10 -
10 10 -

o"10" 10 -
on 0 10 -

0.2 0.3 0.4 8 9 10 11 12 13
Position / cm pH

Figure 6-4. Calculated initial distribution of blocking factor: a) as a function of position;
and b) as a function of pH.

us -0.5

a! :

-1.0 1
10 "~ 10-m8 10-18 10 "4 10 "2 10-10

Figure 6-5. Interfacial potential as a function of absolute net current density with local pH-
as a parameter. The distributions associated with the pH- values of 8.7 and 9
are superimposed.

intact region. As seen in Figures 6-:3(b) and 6-4(b)), the magnitudes of ( and coo, do not

change over the pH- range 8 to 9. Thus, the polarization curves associated with pH- 8.7 and

9 superimpose as observed in Figure 6-5.

The pH- ranging from 9 to 11 corresponds to the moving front in which the bond-breaking

reactions are ongoing. As shown in Figures 6-:3(b) and 6-4(b), both ( and coo, show an

increase from pH- 9 to 11; consequently, the anodic and cathodic current densities in the

front region are larger than those in the intact region. The increase in the current densities

reflects physically an enhanced electrochemical reactivity in the front region.

The curve of pH- 12 in Figure 6-5 corresponds to the delaminated region in which the

interfacial bonds are partially broken due to the delamination process. As the polarization

parameters ( and coo, account implicitly for the influence of passive films or deposits

on the electrochemical reactions, both anodic and cathodic current densities in the

delaminated region are smaller than those in the front region.

6.2 Equilibrium Porosity-pH Relationship

The simulation results obtained using the equilibrium E-pH relationship are presented

in this section. The results using the kinetic E-pH relationship are presented in Section 6.:3.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Position / cm

Figure 6-6. Calculated distributions of interfacial potential along the nietal-coating
interface with elapsed time as a parameter.

6.2.1 Interfacial Potential Distribution

The calculated distribution of interfacial potential V is presented in Figure 6-6 with

elapsed time as a parameter. In the graphical presentations in the subsequent sections,

t = 30 s was chosen as the initial condition to avoid the artificiality at t = 0. At a given

time, the interfacial potential increases with distance away front the scratch and reaches

a constant value in the intact region. The constant plateau seen in Figure 6-6 represents

the intact region and is observed to shorten with elapsed time. The shape of the potential

distribution is maintained throughout the simulation, indicating that the phenomena and

the hypotheses considered in the model sustains the profile of V while the delamination

front propagates along the nietal-coating interface. These features are consistent with the

experimental results of coated electrogalvanized reported by Stratniann et al. [10] and

Williams et al.. [50]

Following the analysis reported by Leng and Stratnian, [15] the interfacial potential

distributions were differentiated with respect to position to yield distributions of dV/dxr

as a function of position with elapsed time as a parameter given in Figure 6-7. The

sharp peak marks the deflection point of the sharp increase observed in Figure 6-7. The

position of the peak, identified as the delamination front, propagates away front the

~o-0.76

1i -0.78

m -0.82

~j-0.84

4.0
3.5
-t =30 s
3.0 ~ I- t = 10 min
2.5- t = 20 min
-t = 30 min
x 2.0 C ------ t = 40 min
'C, ------ t = 50 min
1.5 t = 60 min

1.0

-0.5 1
0.0 0.1 0.2 0.3 0.4 0.5
Position I cm

Figure 6-7. Calculated distributions of dV/dx along the metal-coating interface with
elapsed time as a parameter.

defect with increasing time. The peak height corresponding to the magnitude of dV/dx

deceases with increasing delamination time. The decreasing trend is in agreement with

the experimental results reported by Leng and Stratmann. [15] The explanation given

by Leng and Stratmann was that, with time, a more gradual change in electrochemical

potential in the front region. [15] The agreement with the experiments demonstrates that

the hypotheses of the pH-dependent interfacial porosity and pH-dependent polarization

kinetics were reasonable for the front propagation during the delamination process.

The rate of propagation of the potential front, calculated from the maxima peak given

in Figure 6-7, is presented in Figure 6-8. The rate initially is large but exponentially

decreases with elapsed time. After a long-time extrapolation, the delamination rate

determined by the potential front is 1.66 mm/hr, approximately two times larger than the

experimental rate for coated steel observed by Leng and Stratmann. [14] The discrepancy

between the theoretical and experimental work can be attributed to the use of the

equilibrated pH-porosity relation in the model. The phenomena of bond breakage and

coating degradation involve chemical reactions. The application of the equilibrium

pH-porosity relation assumes instantaneously that the time constants associated with

breaking bonds are small. When time constants for bond-breakage phenomena are large

6.5 II

E 5.5 tPotential Front
E Rate extrapolated: 1.66 mm/hr
-e5.0

4.5

t 4.0-

1.5 I

0 20 40 60 80 100 120
Time / min

Figure 6-8. Instantaneous velocity of potential front, calculated from the time-dependent
position of the maxima given in Figure 6-7.

compared to those for the diffusion and migration, the assumption of the equilibrium

pH--porosity relation becomes invalid. An investigation for non-equilibrium pH--porosity

will be addressed in Section 6.3.

6.2.2 Concentration Distributions

The distribution of pH- in the delaminated zone is presented in Figure 6-9 as a

function of position with elapsed time as a parameter. The calculated results show that

the pH- in the delaminated and front regions increases with time and remains unchanged in

the intact region. The increase in pH- in the delaminated and front regions is attributed to

the OH- ions produced by oxygen reduction underneath the coating and by diffusion from

the boundary with the defect. The shape of the pH- distribution is maintained throughout

the simulation, which again, demonstrates that the hypotheses and physical phenomena

considered in the model are able to sustain the profile of pH- while the delamination front

propagates along the interface.

The calculated distributions of CNa+ and co- are presented in Figure 6-10 as a

function of position with elapsed time as a parameter. The trends associated with

the CNa+ and co- distributions are similar with those seen in the pH- distribution. A

slight decrease in the co- distribution is observed in the front region. This decreasing

0.0 0.1 0.2
Position

0.3 0.4 0.5
/ cm

g the metal-coating interface with elapsed

10 11 1

10." t =30 s
--t =10 min
--t =20 min
lor' ---- t =30 min
----- t = 40 min
------ t = 50 min
qq. -t=60min

10 *

10"
0.0 0.1 0.2 0.3 0.4 0.5 0.6
Position / cm

(b)

Figure 6-9. Calculated distributions of pH alon~
time as a parameter.

E

O

10 4

E10 5

o 10 7
E

,10` 0

10-111l l 1
0.0 0.1 0.2 0.3 0.4 0.5 0.6
Position / cm

(a)

Figure 6-10. Calculated concentration distributions along the metal-coating interface with
elapsed time as a parameter, a) Na+ ions; and b) Cl- ions.

S10' 2 i
O I Position I cmi
Figure~ 6-1. alulte conenraio ditrbtin of Zin2+ ln h ea-otn
interface\ with-t elase tieaspaaetr

feature demonstratesthat the Cl ions ar exele frm tefotrgonbcueo h
prouctonof H-ion byoxgenreucton

conditionosto aple inth odl

Thgue 61.Clua concentration distributions of the seciesprodued i the hmoene-ousractions

(ZnH HlO2 ,aned ZntitOn o) arez prsentved in Figure 6-12 as a function of position

with elapsed time as a parameter. For salltre spcis the concentrationsi manndechreasewith

pstointhe delaminated and the front regions.Drn the courseraio of Z2 o iese imulaostion, h

cange in ontentration anddcroass the delamionae and the fonthe ato regions becme or

gradual ashosat e h delamina tion p opagates into th fulyintacbyth region. eurai

6.2.3io aPrcpitaed Cnt orrsin rouc

The prneiptation of corrosions poductheses Zn(OH)() a beosred in dh ooee elamintions

psystemsn h dlmia and ctegcorsnsythems [19t 76]The. disribu tie onr of Csn(OH), alculate

10 11 1

10.sD t = 30 s
r? ---t=10 min
E 1" .'t-- ---t= 20 min
o ---- t=30 min
O---t 12 m: ----- t= 40 min
---t=60~min
I10~1 \

o 10 \ 1

10-18
0.0 0.1 0.2 0.3 0 4 0 5 0.6
Position / cm

(a)

i 0.3 0.4 0.5 0.6
Position / cm

10 12
10"

10~Z

10 -
~10-

0.0 0 1

0.2 0.3 0.4 0.5 0.6
Position / cm

Figure 6-12. Calculated concentration distributions along the metal-coating interface with
elapsed time as a parameter, a) ZnOH+ ions; b) HZnOgions; and c) ZnO -
10nS.

10'" Zt~i~t~- t =10 min
m I t = 20 min
S10-' e. t = 30 min
10 ---- t= 40 min
O~ ---- t = 50 min
S10-' e~ ------ t= 60 min

C)10 -

0.0 0 1 0.2 0.3 0 4 0 5
Position / cmn

Figure 6-13. Calculated concentration distributions of precipitated corrosion product
Zn(OH)2(s) along the metal-coating interface with elapsed time as a
parameter.

from the presented model is given in Figure 6-13 as a function of position with elapsed

time as a parameter. The concentration of Zn(OH)2(s) inCTreSeS With time in the

delaminated, front and intact regions. At a given time, the concentration of Zn(OH)2(s)

decreases with position in the delaminated and front regions, and maintains a constant in

the intact region.

It has been known that the formation of passive 1 0-;-r Zn(OH)2(s) On eleCOtOde

surface protects materials from corrosion and moderates the corrosion rate. [19] This

inhibitive feature, in the model presented here, was included implicitly through the use

of the poisoning factor, but not related quantitatively with the local concentration of

Zn(OH)2(s). The solubility of Zn(OH)2(s), aS indicated in the Purbaix diagram, [66] has

a minimum around pH- 9 and increases with increasing pH-. This transition feature is not

observed in Figure 6-13. The inconsistency with the literature might be due to that the

approach taken in the model to incorporate the solid species Zn(OH)2(s) (equation (4-14))

is different from that emploi-. 4 in the Purbaix diagram in which the total concentrations

of Zn2+ and OH- ions were held as constants. [71, 72].

0.012

0.010 :-- = 30 s
t= 10 min
---- t = 20 min
0.008 -.----t = 30 min
1------ t = 40 min
c0.006 ------- t = 50 min
--t = 60 min
S0.004 -

0.002 1

0.000
0.0 0.1 0 2 0 3 0.4 0.5
Position / cm

Figure 6-14. Calculated distributions of porosity along the metal-coating interface with
elapsed time as a parameter.

6.2.4 Porosity Distribution

The calculated porosity distribution is presented in Figure 6-14 as a function of

position with elapsed time as a parameter. The shape of the porosity distribution remains

the same with the initial distributions throughout the simulation. As the delamination

front propagates into the intact region, the interfacial porosity increases to satisfy the

equilibrium E-pH relationship. At a given position, the increase in the porosity with time

is reflected by the increase in pH- observed in Figure 6-9. The trends associated in the

porosity distribution are similar with those observed in the pH- distributions, confirming

that the destruction of the interfacial adhesions is related to the generation of OH- ions

during the delamination process.

In experiments, the delamination front, where the delamination is ongoing, is often

determined by the potential distributions measured using the scanning K~elvin probe.

In the mathematical model presented here, however, it is more reasonable to define the

delamination front by the porosity gradients, because the porosity is the key to represent

the adhesive strength between the metal and the coating. [21, 22]

The distribution of de/dxr along the metal-coating interface is presented in Figure 6-15

as a function of position with elapsed time as a parameter. The sharp peak characterizes a

S-0.20 -------t = 40min
------ t = 50 min
S-0,15 ,- t = 60 min

-0.05 -

0.05
0.0 0.1 0.2 0.3 0.4 0.5
Position I cm

Figure 6-15. Calculated distributions of de/dxr along the metal-coating interface with
elapsed time as a parameter.

porosity front that moves toward the intact region during the delamination process. The

velocity of the porosity front corresponds to the rate of breaking the interfacial bonds

and is approximately 1.50 mm/hr after extrapolated to a longer time (see Figure 6-16).

This value is slightly smaller than that of the potential front, but still larger than the

experimental result of coated steel. [14] The discrepancy between the simulation and

experimental rates can he attributed to the use of the equilibrium pH-porosity relation,

which yields to the upper limit to the propagation rate.

6.2.5 Delamination Kinetics

Following the approach taken by Leng [15] and William [51], the delamination kinetics

was analyzed by plotting the delaminated distance as a function of elapsed time. The

propagation distances determined by the potential and the porosity fronts are presented

in Figures 6-17(a) and 6-17(b), respectively, as a function of time in a double-logarithmic

plot with cation type as a parameter. The calculated reaction order is approximately 0.56

for the potential front and 0.6 for the porosity front. The slopes seen in Figures 6-17(a)

and 6-17(b) are independent of the cation types, and the values are in close agreement

with the reaction order of 0.52 to 0.59 determined by Stratmann et al. [10] for polymer

coated electrogalvanized steel (Figure 6-17(c)). These results indicate that the overall

4.5

4 Porosity Front
E 3 g Rate extrapolated: 1.50 mm/hr
E .

2.5

S2.0

O~ 1.0
0 20 40 60 80 100 120
Time I min

Figure 6-16. Instantaneous velocity of porosity front, calculated from the time-dependent
position of the maxima given in Figure 6-15.

Table 6-2. Diffusion coefficients of cations [70, 75]
Type of Di in bulk
cation electrolyte (cm2 S)
Li' 1.25 x10-s
Na' 1.47x10-s
K( 1.84x10-s
Cs+ 2.10 x10-s

delamination process is primarily limited by the mass transport of ions from the defect

to the delamination front. Due to the co-existence of the potential and concentration

gradients, the transport of ions represents the contributions by migration as well as

diffusion.

6.2.5.1 Influence of Cation Type on Delamination Rate

The influence of cation type on the delamination rate can be seen in Figure 6-18,

where the delaminated distance calculated based on the potential and porosity fronts are

plotted as a function of square root of time. A linear relation between the delaminated

distance and z/ieis observed for all of the cation types. The rate of the propagation

decreases in the order of Cs+ > K(+ > Na+ > Li+ for both potential and porosity

fronts, and this result is correlated to the mobility of the cations in an aqueous medium

(Table 6-2). The result indicates that, with the chemical and physical assumptions, the

Figure 6-17. Delaminated distance as a function of elapsed time in double-logarithmic
scale with cation type as a parameter. The concentration of the electrolyte at

the defect is 0.5 1\. a) Delaminated distance determined by potential front;

b) Delaminated distance determined by porosity front; and c) Experimental
results obtained from coated electrogalvanized steel samples. Data taken from

Stratmann et al. [11] with permission of Corrosion Science.

2.6 2.8 3 0 3.2 3.4
Iog(time / s)

1 1.4 i.8
Iog (timO I mirl)

______

_

" ~Porosity Front

--

7 ,p -0- Cs' m =
,,F'-O- K' m=O
,* -a- Na m =
- -s-- Li' m =
"

4

0-U.

-.O.

-0 6

-0.7

'- O "

-1.1

-1 2

-1.3
2.1

- ~Potential Front

~' '# s' m =0.56
S' -0- K m=0.56
-:~ -e Na m =0.56

pl -r9-- Li m =0.56

4 2.6

-0.8

-0.9

-1.0

-1.1

-1.2

-1.3

-1.4
2.

1.6
).6

2.8 3.0 3.2 3.4

log(time/ I )

3.6 3.8

3.6 3.8

0 100 200

0.27-

E 0 24 ,

-E0.09 -
d 0.06 -
V
S0.1

S0.03

10

Porosity Front

9 Electrolyte: 0M.6
-0- -Cs+
y~ 0- K+
,,C' --A-- Na+
--- Li+
I I I
20 30 40 50 60 7
(Time / s)'

10 20 30 40 50 60
(Time / s)1x

4000

oscdtr M CsCI/

.*KCI

LiCI

Figure 6-18. Delaminated distance as a function of square root of time with cation type as
a parameter. The concentration of the electrolyte at the defect is 0.5 M. a)
Delaminated distance determined by potential front; b) Delaminated distance
determined by porosity front; and c) Experimental results obtained from
coated steel samples. Data taken from Stratmann et al. [15] with permission
of Corrosion Science.

Table 6-3. Diffusion coefficients of anions [75]
Type of Di in bulk
anion electrolyte (cm2 S)
Br- 1.25 x10-5
Cl- 1.47 x10-5
F- 1.84x10-5
CIO, 2.10 x10-5

delamination predicted from the model is principally controlled by the transport of cations

along the metal-coating interface and the rate of this process scales with the mobility

of the nations. This result is qualitatively consistent with experimental measurements

reported for coated steel and coated zinc systems (Figure 6-18(c)). [10, 15]

6.2.5.2 Influence of Anion Type on Delamination Rate

The influence of anion type on the delamination rate was also examined in the

simulation. The resulting propagation distances determined by both potential and porosity

fronts are presented in Figure 6-19 as a function of square root of time. It is clear that

the delamination rate does not vary with the anion types even though their mobilities

in aqueous electrolyte are different (Table 6-3). The production of OH- ions under the

degraded coating attract the cation at the defect, resulting in movements of the cation

from the defect toward the intact region. Consequently, the cathodic delamination is more

sensitive to the cation type than the anion type. The experimental observations by Leng

and Stratmann [11, 15], given in Figure 6-19(c), shows that the anion types influence

slightly on the delamination rate, but the variations of the propagation rate between the

anion types is much less significant than those between the cation types.

6.2.5.3 Influence of Electrolyte Concentration on Delamination Rate

As reported by Leng and Stratmann [15], the concentration of the electrolyte placed

at the defect is also a factor that influences the delamination rate. Figure 6-20 gives the

delaminated distance as a function of square root of time with electrolyte concentration

as a parameter. The higher concentration at the threshold provides a larger driving

force to couple galvanically the intact and the defect zones; therefore, the propagation

4000 O/ or~ NacI

U
0 100 200

0.40 ,

D, 0.35

O3 0.25

d 0.20 -

0.15 -

S0.10 -

0.05
10

Porosity Front

Electrolyte: 0.5 M
-0-- Cr
-0- Br
-Q-F
-V- 010;

20 30 40 50
(Time / s)l

10 20 30 40 50 60
(Time / s)1x

60 70

3000

2000

1000

NaFC104

Figure 6-19. Delaminated distance as a function of square root of time with anion type as
a parameter. The concentration of the electrolyte at the defect is 0.5 M. a)
Delaminated distance determined by potential front; b) Delaminated distance
determined by porosity front; and c) Experimental results obtained from
coated steel samples. Data taken from Stratmann et al. [15] with permission
of Corrosion Science.

Porosity Front ,a
NaCI O a
--o-- 0.5 M a

-o-.01 M d'

o' a

a ',
vi

5OOD

NeCI mtr

01- -- -~ ----------------------

0.25

0.20 -

0.15 -

0.'10 .

0.05 -

0.00
10

30 40
(Time / s)m Z

20 30 403 50
(Time / s)'"

(b)

60 70

4000

1000

delonised water

100

200

Figure 6-20. Delaminated distance as a function of square root of time with electrolyte
concentration as a parameter, a) Delaminated distance determined by

potential front; b) Delaminated distance determined by porosity front; and c)
Experimental results obtained from coated steel samples. Data taken from
Stratmann et al. [15] with permission of Corrosion Science.

rates determined by both potential and porosity fronts increase with the electrolyte

concentration. From the kinetic analysis presented above, one important conclusion is that

the rate-determining step of the cathodic delamination is driven by the transport of the

nations from the defect to the front region, and that the propagation rate scales with the

ionic strength and the mobility of the nations. This conclusion is in agreement with the

6.3 Kinetic Porosity-pH Relationship

The use of the equilibrium pH-porosity relation presented above assumed spontaneously

that the chemical reactions associated with breaking interfacial bonds occur rapidly. This

assumption becomes invalid when the time constants for the bond-breaking reactions

are large compared to those for diffusion and migration processes. The investigatioon for

non-equilibrium pH-porosity relation is necessary.

6.3.1 Potential Front and Porosity Front

To explore the role of finite rates of hand breakage, the equilibrium relationship

between porosity and pH, given as equation (4-5), was replaced by

where k,z, is a rate constant that reflects the time constants of hond-breakage reactions,

and Ee, is obtained from equation (4-5). Different values of k,z, were examined in the

simulations, but only the results for k,ze, = 0.1 and 0.001 s-l are presented here.

The resulting distributions of interfacial potential for k,ze, = 0.1 and 0.001 s-l are

shown in Figure 6-21 as a function of position with elapsed time as a parameter. The

features seen in Figure 6-21 were similar to those observed in Figure 6-6, which were

obtained using the equilibrium pH-porosity relation. The distributions of the potential

gradient dV/dxr for k,ze, = 0.1 and 0.001 s-l are shown in Figure 6-22 as functions

of position with elapsed time as a parameter. Again, the trends associated with the

dV/dxr plot for non-equilibrium pH-porosity relation are similar to those found using the

-0.74 .

-0.76 -

-0.78 km= 0.001 s -
-t =30 s
-0.80 / --t= 20min-
/ / -- -t= 40 min
to-0.82 / /--- 60 min
t / / ,/----- t= 80 min
------t= 100 min
S-084 --t= 120 min

-0.86
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7r
Position / cm

(b)

0.2 0.3 0.4 0.5
Position / cm

Figure 6-21.

Calculated distribution of interfacial potential along the metal-coatingf
interface a) k,ze, = 0.1 s-l; and b) k,ze, = 0.001 s-l

0.3 0.4 0.5 0.6 0.7
Position I cm

0.3 0.4 0.5 0.6 0.7
Position I cm

Figure 6-22. Calculated distribution of interfacial potential gradient dV/dxr along the
metal-coating interface a) k,ze, 0.1 s-l; and b) k,z,, 0.001 s-l

u'u L

- kn q= 0.001 SI
-t= 30 s
- -- -- t = 20 min
-- t = 40 min
---- t= 60 min
----- t= 80 min
------ t= 100 min
- -t =120 min

-0.4 km~= 0.1 s .
-t= 30 s
t= 20 min
-0.3 C I t= 40 min
---- t= 60 min
----- t= 80 min
-0.2 C- II------ t= 100 min
----t= 120 min

I I I I I_ I

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0 7 0 0 0.1 0.2 0 3 0.4 0.5 0.6 0.7
Position / cm Position / cm

(a) (b)

Figure 6-23. Calculated distribution of porosity gradient dE/dx along the metal-coating
interface a) kneq 0.1 s-l; and b) kneq 0.001 s-]

equilibrium relation. The delamination rate determined by Figures 6;-22(a) and 6;-22(b),

after extrapolated to longer time, were 1.63 mm/hr and 1.55 mm/hr for kneq = 0.1 and

0.001 s-l, respectively. These values are slightly smaller than the equilibrium delamination

rate (1.66 mm/hr), indicating that the use of the kinetic pH-porosity relation within the

model influenced the velocity of the potential front.

The propagation of the potential front is then compared with that of the porosity

front. The resulting distributions of dE/dx for kneq = 0.1 and 0.001 s-l are presented

in Figure 6-23 as functions of position with elapsed time as a parameter. The trends

associated with in Figure 6-23 are similar to those observed in Figure 6-15, which were

obtained using the equilibrium pH-porosity relation. The velocity of the porosity front

evidently decreases from 1.37 mm/hr for kneq=0.1 s-l to 0.93 mm/hr for kneq=0.001

s-1. The propagation rate for kneq=0.001 s-l is much smaller than the equilibrium

porosity front rate (1.5 mm/hr) and the value is in good agreement with the experimental

observation of 0.8 mm/hr for coated galvanized steel. [15]

Fr-om the analysis presented above, it is evident that the rate of breaking interfacial

bonds in the cathodic delamination process is controlled by the rate constant kneq, but

-0.6

-0.5

-0.4

-0.3

-0.2

X
`F
,

-100 a l l l i a-120 l
knq = 0.1 s -100-
-80 -It= 30 s kmo= 0.001 s'
--t =20 min -80 t t= 30 s
----t =40 min --- t =20 min
-60 ---- t= 60 min 0 ---t=40 min
I II .... t = 80 min -60 ----- t = so min
I ~~------t = 100 mln I I I----- t = 80 min
a.-0- 2 -40 -4 -----t= 100 min
-20 ----t= 120min

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.0 0.1 0.2 0.3 0.4 0.6 0.8 0.7
Position / cm Position / cm

(a) (b)

Figure 6-24. Calculated distribution of pH gradient along the metal-coating interface a)
kneq0.1 -1; and b) kneq 0.001 s-]

the velocity of the potential front is not. Thus, the next question to be addressed is;

what properties can be correlated to the propagation of the potential front? Figure 6-24

gives the pH gradient distributions for kneq = 0.1 and 0.001 s-l as functions of position

with elapsed time as parameter. The shape and the features within the pH gradient

distributions are similar to those seen in dV/dx and dE/dx plots. The location of the

deflection point in Figures 6-24(a) and 6-24(b), termed pH front, are approximately equal

to the position of the potential fronts in Figures 6-22(a) and 6-22(b), respectively. The

velocity of the pH front changes from 1.63 mm/hr to 1.55 mm/hr when kneq decreases

from 0.1 to 0.001 s-l. This result -II- -- -; that the change of pH along the metal-coating

interface is an important factor that influences the movement of the potential front.

The influence of the rate constant kneq On the velocity of potential front, porosity

front and pH front is summarized in Table 6-4. When kneq decreased from infinity to

10-4 S-1, the rate of the potential front decreased from 1.66 mm/hr to 1.26 mm/hr, and

the rate of the porosity front decreases from 1.50 mm/hr to 0.73 mm/hr. The comparison

between the potential and pH front rates confirms that the movement of the potential

front depends on the pH front. The change of the front velocities indicates that the values

Table 6-4. Calculated velocities of potential, porosity and pH- front
rate constants velocity of potential front velocity of porosity front velocity of pH- front
(k,z,V / s-l) nin/hr nin/hr nin/hr
oc 1.66 1.50 1.69
10-] 1.6:3 1.37 1.61
10-2 1.60 1.26 1.60
10-1.55 0.9:3 1.55
10-4 1.26 0.7:3 1.25

0.012

0.010 -k=1
t =30s
t = 20 min
0.008 C- 1 -- t = 40 min
-e I B---- t = 60 min
2' --- t = 80 min
0.006 ------- t = 100 min
O I `\~ --- t = 120 min
P 0.004

0.002 -\-

0.000
0.0 0.1 0.2 0.3 0.4 0.5
Position / cm

Figure 6-25. Calculated distributions of de/dxr along the nietal-coating interface with
elapsed time as a parameter.

of k,z, influence the propagation of all fronts, but the influence is much more evident on

the porosity front.

As observed in Table 6-4, the velocity difference between the potential and porosity

fronts increases with decreasing k,2,,. The production of OH- ions in the faster potential

front creates a driving force for the bond-breakage reactions that are limited by the

finite rate constant. As a result, the dishondnient occur in a broad region when the

bond-breakage reactions are sufficiently slow. The distribution of porosity for k,ze, = 10-4

s-l is presented in Figure 6-25 as a function of position with elapsed time as a parameter.

Due to the limitation of the finite rate constant, the well-defined porosity front seen

in Figure 6-14 heconies less distinguishable in Figure 6-25. Instead, the change of the

porosity takes place in a broad region and this region expands with increasing time.

Potential Front Porosity Front
-0.4 --- .
0 k --> 0 m =0.57 -0 k --> co m =0.60 i
-0.5 n =01 05 -0.5 m=06
S k =0.01 m =0.59 .a k =0.01 m =0.67 <1
-0.6 V k =103 m = 059 gB v -0.6 k =109 m = 072
O -0.7 O 0 m=05 -0.7 -< 0 m=07

-12I 1 a-12 I I a
20. 6 2 8O 3. . B .
log(Tirne / s) ogTm / s
c(a) (ob)
Fiur 6-26. Deaiae itnea ucto fdlmnto iei
dobl-ogrthi scl wit caintp saprmeeaeaiae
ditac deemndb h oeta rnt n)Dlmntddsac
detrmne by the pooiyfot

6.. DeaiainKntc

Fgives66 propaationae distancedeemndbthposiyfns as a function of dlmnto time i
~~~~obelgrtmcsaewith kneqo typSa a parameter. The sloe s aprxiatelyeult o h qiiru

pH-poroityreatinc but rinceasb tes to ea 0.74t for knq=0- -1 Te angein thed dslopei

an. idicationtatio th eainationmcaimsit rmamastase otoldt

a mixed-ontrol tmecihanisma when the bond-breakage raeactons are sufficintly slow.Th

transeitin ies conssen wionthae a the eprmna resls eote osa doe byt Statmnnet al. [10

thpats the overalldelaminationg proess is lmted byintiswenth delamination prcs.Fgrat is2(

a ie-oto ehns hntebn-raaeratosaesufficiently small.T

CHAPTER 7
ELECTROCHEMICAL IMPEDANCE SPECTROSCOPY

Electrochemical Impedance Spectroscopy (EIS) is a small-signal technique in which

a sinusoidal current or potential perturbation is imposed on tested systems and the

corresponding potential or current response is measured (see Figure 7-1). Comparison of

the input and output signals provides the impedance at a given perturbation frequency.

The influence of a particular phenomenon on impedance response is, in principle,

determined by the time scale of that process. [27] For example, the time constant for

mass-transfer effects is relatively large because the dICI -iffarit- of ionic species in aqueous

medium is small. Therefore, mass-transfer effects are usually apparent at low frequencies,

whereas kinetic and double-lbw;r effects are more important at high frequencies. An

advantage of EIS is that, with a single experimental procedure encompassing a sufficiently

broad range of frequency, the governing chemical and physical phenomena can be

distinguished at a given potential.

Current

o,+ A~I sined o,

Potential

E,I+ AE sinwt

Figure 7-1. Schematic representation for theory of Electrochemical Impedance
Spectroscopy (EIS) where E is potential and I is area-averaged current.
Impedance is defined as the ratio of potential difference between working and
reference electrodes to surface-averaged current.

Experimental impedance spectra are typically interpreted in terms of circuit models

that consist of combinations of passive circuit elements. While equivalent circuit models

are useful for understanding the physical processes or chemical mechanisms that occur in

electrochemical systems, observed impedance spectra frequently show a dispersion that

cannot be fitted using simple elements. The dispersion typically reflects a distribution of

reactivity that is commonly represented in equivalent electrical circuits as a constant-phase

element (CPE). The distributed reactivity may arise from variation of properties either

along the area of electrode (2-D) or along the axis normal to electrode surface (3-D). A

2-D distribution could be associated with geometry-induced current/potential distributions

or surface heterogeneity such as grain boundaries, crystal faces or other variations in

surface properties. A 3-D distribution may arise from changes in the conductivity of oxide

LI-; r [35] or from porosity or surface roughness. [37, 38]

The recent development of local electrochemical impedance spectroscopy (LEIS)

[54, 77] makes it possible to distinguish CPE behavior that has an origin with a 3-D

distribution from one that arises from a 2-D distribution of properties along the surface

of the electrode. [40] In LEIS, similar with traditional impedance methods, a sinusoidal

current or potential perturbation is imposed on tested systems and the corresponding

potential or current response is measured. The local impedance technique consists of a

probe with two micro-electrodes allowing measurements of potential at two positions.

Under the assumption that the Ohmic impedance between the two probes is given by a

constant, the current density at the probe can be estimated from the measured potential

difference AVprobe by

iprobe Vprobe, 71

where d is the distance between the potential sensing electrodes and a is the conductivity

of the electrolyte. The local impedance can then be calculated from the ratio of the

electrode potential measured relative to a reference electrode far away from the surface to

the local current density iprobe.

(a) (b) (c)

Figure 7-2. Passive elements that serve as components of an electrical circuit. a) Resistor;
b) Capacitor; and c) Inductor.

The subsequent sections provide basic concepts involved in electrochemical impedance

spectroscopy. Detailed discussions of technical and theoretical issues associated with EIS is

available elsewhere. [25, 28, 29, 78]

7. 1 Passive Electrical Circuits

Experimental impedance spectra are typically compared to that of known electrical

circuits. Electrical circuits can he constructed from the passive elements shown in Figure

7-2. [25, 28, 29]

The impedance of a passive circuit element is defined as the ratio of the potential

difference between the element clamps to the current flowingf through the element, i.e.

AV
Z =(7-2)
aI

and has units Chms R. For a pure resistor, equation 7-2 yields

Zresistor = R (7-3)

whereas, for a capacitor

Zeapacitor 74

and for an inductor, the impedance is

Zinductor = jeoL (7-5)

(b)

Figure 7-:3. Combinations of passive elements that serve as components of an electrical
circuit: a) in series b) in parallel.

For two passive elements in series, the same current must flow through the two

elements, and the overall potential difference is the sunt of the potential difference for each

element. Thus, the impedance for the series arrangement shown in Figure 7-:3(a) is given

by

Z = Z1 + Z2 76

For two passive elements in parallel, the overall current is the sunt of the current flowingf

in each element, and the potential difference is the same for each dipole. Therefore, the

overall impedance for the parallel arrangement shown in Figure 7-:3(b) is given by

1 1
Z= [ + ]-' (7 7
Z, Z2

Impedance contributions are additive for elements in series; whereas, the inverse of the

impedance is additive for elements in parallel.

It is important to note that different circuit analog models possessing the same

number of time constants can yield a mathematically equivalent frequency response.

[25, 28, 29] The lack of uniqueness of the circuit models creates ambiguity when

interpreting impedance results. A good fit to experimental data does not guarantee that

the model describes correctly the physics of the given system. Additional experimental

observations are needed to verify a proposed model and to avoid ambiguities when

interpreting impedance data.

7.2 Constant-Phase Element (CPE)

Experimental impedance results for a solid electrode/electrolyte interface often revel

a time-constant dispersion that cannot be described by simple elements. To characterize

this time-constant dispersion, the interfacial capacitance is often expressed in equivalent

circuits in terms of a constant-phase element (CPE). [25, 28, 29]

CPE typically reflects a non-ideal double-l} u. r capacitance and is usually related to a

pure capacitor by

Q = Co(juo)l-a (7-8)

where the parameters a~ and Q are constants. When a~ = 1, Q has units of a capacitor and

represents the capacity of the interface. When a~ / 1, the system shows behavior that can

be attributed to distributed properties on electrode surface. The value of a~ may change

from -1 to +1; in this sense, the CPE is treated as an extremely flexible fitting element

and its meaning in terms of a distribution of time constants is less clear.

7.2.1 Origin of CPE

Numerous research efforts have been made in literature to study the origin of the

CPE behavior. P,- ilr v; [38, 79] modeled rough electrodes by surfaces of fractal geometry

with processes dilatational symmetry. The theoretical calculations yielded CPE behavior

with a fractional exponent depending on the fractal dimension. The experiments by

P,- il:<. 1i- and K~erner [:38, 79] showed that the time-constant dispersion on solid electrode

was due to surface disorder (on the atomic scale) rather than geometric roughness (larger

than atomic scale).

De Levie [80, 81] modeled the impedance of porous electrodes under the assumption

that the concentration was uniform and the pores were ideal cylinders. Lasia [82] later

replaced the double 1 e -< c capacitance on pore walls with a CPE. The results reported by

Lasia [82] showed that mass transfer and pore geometry influenced the shape of impedance

spectrum. The models proposed by de Levie and Lasia considered only a single pore

dimension. Song et al. [8:3] developed a model to predict the effect of pore size distribution

on the impedance response of porous electrodes.

P,- il:<. 1i- et al. [84-87] proposed that the time-constant dispersion arose from the

adsorption of molecules or anions on gold electrodes. The capacitance dispersion observed

in the presence of specific adsorption can he assigned to either a slow diffusion or slow

adsorption processes within the double 1.,-:-r or electrode surface.[87]

N. i.--us! Ia and Nisancioglu [41, 42, 45] studied the influence of nonuniform current

and potential distribution on the impedance response of a disk electrode. Their results

indicated the geometry-induced potential and current distribution induced a high-frequency

dispersion that distorted the impedance response. Ni- 1lia, inglu [45] showed the extent

to which this frequency dispersion leads to an error in the values for charge-transfer

resistance and interfacial capacitance obtained from impedance data.

7.2.2 2-D and 3-D Distributions

The explanations of CPE behavior presented above -II__- -r that two kinds of

distributions can he distinguished. A 2-D distribution could be associated with geometry-induced

current/potential distributions or surface heterogeneity such as grain boundaries, < ti--r I1

faces or other variations in surface properties. CPE behavior may also arise from changes

in the conductivity of oxide 1 ,-c -r [:35] or from porosity or surface roughness. [:37, :38] This

Re Re Re Re R Re Re Re

Co Co Co Co zo zzo z

(a) (b)

Figure 7-4. Schematic representation of an impedance distribution for a blocking disk
electrode where Re represents the Ohmic resistance, Co represents the
interfacial capacitance, and zo represents an interfacial impedance
corresponding to CPE a) 2-dimensional distribution of blocking components in
terms of resistors and capacitors, and b) 3-dimensional distribution of blocking
components in terms of resistors and constant-phase elements.

can be described as being associated with a 3-D distribution, with the third direction

being the direction normal to the electrode surface.

A schematic representation of a 2-D distribution for an ideally-polarized blocking

disk electrode is presented in Figure 7-4(a). For a 2-D distribution, the capacitance and

Ohmic resistance could be a function of radial position along the electrode. Integration

of the admittance associated with these circuit elements would yield a global admittance

with a CPE behavior,

Y x dA (7-9)

where A is the electrode area, Y is the global admittance, Z is the global impedance, and

z is the local impedance. The local impedance, in the case of a 2-D distribution would,

however, show ideal behavior. A 3-D distribution of blocking components in terms of

resistors and constant-phase elements is presented in Figure 7-4(b). Such a system will

yield a local impedance with a CPE behavior, even in the absence of a 2-D distribution of

surface properties. If the 3-D system shown schematically in Figure 7-4(b) is influenced

by a 2-D distribution, the local impedance should reveal a variation of CPE coefficients

along the surface of the electrode. Thus, local impedance measurements can be used to

distinguish whether the origin of the CPE behavior arises from a 2-D distribution, from a

3-D distribution, or from a combined 2-D and 3-D distribution.

7.3 Current and Potential Distributions on Disk Electrode

Current and potential distributions on electrode surface phI i- an essential role in

electrochemical fabrication technologies [88] and in interpretation of electrochemical

processes. [89] The geometry of an electrode often constrains the distributions of

current and potential on the electrode surface in such a way that both cannot he

simultaneously uniform. ?- i.--in! .1, [46] developed analytical solutions for current and

potential distributions on a disk geometry, and the development is reviewed in this section.

In a bulk of a well-stirred electrolytic solution where concentration gradients are

negligible within the electrolyte, potential is governed by Laplace s: equationl, i~e.[46]

V2 = 0 (7-10)

where # is the solution potential. The current density i can then he expressed as

i = -sVQ (7-11)

Under the assumption that concentrations are uniform in the electrolyte, the passage of

current through the interface is limited by Ohmic resistance in the electrolyte and by

charge-transfer resistance associated with reaction kinetics. The primary distribution

applies when the flow of current is dominated by the Ohmic resistance and kinetic

resistance can he neglected. The secondary distribution applies when both Ohmic and

kinetic resistances are controlling.

7.3.1 Primary Current Distribution

In the absence of mass-transfer limitations and Faradaic reactions, the primary

current distribution of a disk electrode requires solution of Laplace's equation with a

charging boundary condition at the electrode surface. The primary resistance can he

expressed in the form of [46]

Re (7-12)

The primary current density distribution associated with a disk electrode surface follows

[46]
i 1
(7-13)
21J

where ro is the radius of the disk and < i > is the area-averaged current density on the

electrode. A graph of i/< i > as a function of dimensionless position r/ro is presented in

Figure 7-5. The normalized current density is fairly well behaved near the center of the

electrode, but it approaches infinity at the edge of the electrode. As a result, the primary

current distribution is highly non-uniform for a disk electrode.

7.3.2 Secondary Current Distribution

The secondary current distribution is a consequence of the balance between electrolyte

resistance and charge-transfer resistance. For this case, the distribution requires solution

of Laplace's equation with a boundary condition at the electrode surface that is associated

with both Faradaic reactions and charge of double-l} u. r capacity. The ratio of these two

contributions is, in general, expressed in terms of a dimensionless parameter J

4 R
J = e(7-14)
xr Rt

Large values of J are seen when the Ohmic resistance dominates over the charge-transfer

resistance, and small values of J are seen when the charge-transfer resistance is more

important. The secondary current density distribution on a disk electrode is presented in

Figure 7-6 as a function of the normalized position with J as a parameter. The current

is uniformly distributed when J is sufficiently small and the distribution becomes more

nonuniform as J increases. The curve for J= co represents the primary distribution seen in

Figure 7-5 in which the current distribution is primarily controlled by Ohm's law.

2.0 i 1-

1.5-

S 1.0

0.5

0.0
0.0 0.2 0.4 0.6 0.8 1.0
r /ro

Figure 7-5. Primary current distribution at a disk electrode.

1.4 ,.
Tafel: J

1.2-

----- 1.0 /

0.8 -

0.6 -

0.4
0.0 0.2 0.4 0.6 0.8 1.0

r/r,

Figure 7-6. Secondary current density distribution at a disk electrode with J as a
parameter.

7.4 Ob jective

The studies by ?;- la.; inglu [45] and by Newman [41, 42] demonstrated that the

geonietry-induced current and potential distributions cause a high-frequency dispersion

that distorts the impedance response on a disk electrode. However, their discussions did

not address the dispersion in terms of CPE. 1\oreover, none of the work developed to date

addresses the coupling of 2-D and 3-D distributions, and none of the previous work relates

global impedance response with local impedance.

The objective of this work was to explore, front first principle, the role of nonunifornt

current and potential distributions on the global and the local impedance response of a

disk electrode. The electrochentical systems under study included an ideally-polarized

blocking electrode, an electrode exhibiting a local CPE behavior, and an electrode

exhibiting a single Faradaic reaction. The theoretical development and calculation results

of the work are presented in Chapter 8, Chapter 9, and ChI Ilpter 10, respectively.

CHAPTER 8
IDEALLY POLARIZED BLOCKING DISK( ELECTRODE

This chapter presents the theoretical development and calculation results for the

impedance response of an ideally-polarized blocking electrode. [32] There are several types

of impedance at phI i-, their definitions and notations are also provided in this chapter.

8.1 Theoretical Development

In the absence of mass-transfer effects, the transient response of a disk electrode

requires Laplace s: equation withl fux conditions at the electrode surface. Following

TX .--~!! on's approach, [42] Laplace's equation in cylindrical coordinates was expressed in

rotational elliptic coordinates, i.e.

y = rogyl (8-1)

and

r = o (1+ (2 2)(8-2)

where 0 < ( < 00 and 0 < rl I 1. The coordinate transformation can be seen more clearly

in Figure 8-1. Within the rotational elliptic coordinate system, the electrode surface at y

= and r < ro can be found at ( = 0 and 0 < rl I 1. The reference electrode and counter

electrode located at y7 00o can be found at ( 00o. The insulating surface of the disk at

y = 0 and r > ro is located at rl = 1 and 0 < ( < 00, and the center line at y > 0 and r =

0 is located at rl = 0 and 0 < ( < 00.

Lapace's equation can be expressed in rotational elliptic coordinates as

(1 +' (2_ ) = (8-3)

The potential was separated into steady and oscillating parts as

# = # + @ exp(ject) (8-4)

Insulator

Insulator 9=0

/ Coordinate
Transformation

Electrode
=0fce~

Bulk 5~=03
~...~

-i--~-r-9--I--(--)-c--~--r--

--i-i-~--~--~--i---i-f---i

L--~--~---t--~
--i---s--i---s--t--i---~--c---~--r--

~:::~::i::T::i::i:::i::i:::i::i:::

Insulator

Figure 8-1. Coordinate transformation from a cylindrical coordinate to a rotational elliptic
coordinate. The gridingf in the rotation epiotic coordinate is not drawn to
scale.

where # is the steady-state solution for potential and # is

potential. Thus, equation (8-3) could be written as

80, a2, r r
21 (1 + 2) 2 2rl (1 -

the complex oscillating

a2 r
r12 2

(8-5)

and
80~~ 82~ a2
2( (1 + (2) 2rl (1-92 (8-6)

where 4, and @yi refer to the real and imaginary parts of the complex potential,

respectively.

For a blocking electrode, the current passes from the electrode to the electrolyte by a

means of charging the double-l} u. r capacity. The flux boundary condition at the electrode

surface (( = 0 and 0 < rl < 1) was expressed as

8(V Go) 80 m 8
i = Co) a(87
iit iiB~y so rorl ii a

....-- p

where Co is the interfacial capacitance and a is the electrolyte conductivity. Equation

(8-7) was written in frequency domain as

(8-8)

and

KIV K4~, = i~ (8-9)

where I, represents the imposed perturbation in the electrode potential referred to an

electrode at infinity and K( is the dintensionless frequency

K wor (8-10)

At if = 0 and if = 1, for all (>0, zero-flux conditions impose that

(8-11)

and

877

(8-12)

At the far boundary condition (( 00o and 0 I if < 1),

Or = 0

(8-13)

and

(8-14)

#; = 0

The equations were solved under the assumption of a uniform capacitance Co using

the collocation package PDE2D developed by Swell. [90] To ensure the accuracy of the

calculations, a series of error analysis was performed to verify that the niesh size used in

the program was sufficiently small and the domain size is sufficiently large. Calculations

were performed for different domain sizes, and the results presented here were obtained by

extrapolation to an infinite domain.

1 84, ~
if 8(

Co Co Co Coy

iB(r) Z(r) ie(r) ia(r)

:--- V~~r

Figure 8-2. The location of current and potential terms that make up definitions of global
and local impedance.

8.2 Definition of Impedance

The calculation results presented in the subsequent sections involve several type

of impedance. The notations and the definitions of the impedance are presented in this

section.

A schematic representation of the electrode-electrolyte interface for an ideally-polarized

blocking electrode is given as Figure 8-2, where the block used to represent the Ohmic

impedance reflects the complex character of the Ohmic contribution to the local

impedance response. The impedance definitions presented in Table 8-1 differ in the

potential and current used to calculate the impedance. To avoid confusion with local

impedance values, the symbol y is used to designate the axial position in cylindrical

coordinates.

8.2.1 Global Impedance

The global impedance is defined to be

Z = (8-15)

where the complex current contribution is given by

I = i~r)2nrdr (8-16)
/70

Table 8-1. Notation proposed for local impedance variables[39]
Symbol meaning units
Z global impedance R or SOcm2
Z, real part of global impedance R or SOcm2
Zj imaginary part of global impedance R or SOcm2
Zo global interfacial impedance R or SOcm2
Zo,r real part of global interfacial impedance R or SOcm2
Zo,j imaginary part of global interfacial impedance R or SOcm2
Ze global Ohmic impedance R or SOcm2
Ze,,real part of global Ohmic impedance R or SOcm2
Ze~yimaginary part of global Ohmic impedance R or SOcm2
z local impedance Rcm2
z, real part of local impedance SOcm2
zj imaginary part of local impedance Rcm2
zo local interfacial impedance SOcm2
zo,r real part of local interfacial impedance Rcm2
zo,j imaginary part of local interfacial impedance SOcm2
ze local Ohmic impedance Rcm2
ze~rreal part of local Ohmic impedance SOcm2
ze~yimaginary part of local Ohmic impedance Rcm2
(0)spatial average of potential V
0 time average or steady-state value of potential V
(i) spatial average of current density A/cm2
2, time average or steady-state value of current density A/cm2
y axial position variable cm

The use of an upper-case letter signifies that Z is a global value. The global impedance

has real and imaginary components designated as Z, and Zj, respectively. The total

current can also be represented by I = wrri < 2(r) > where the brackets signify the

area-average of the current density. [39]

8.2.2 Local Impedance

The term local impedance traditionally involves the potential of the electrode

measured relative to a reference electrode far from the electrode surface. [91, 92] Thus, the

local impedance is given by

z (8-17)

The use of a lower-case letter signifies that z is a local value. The local impedance may

have real and imaginary values designated as z, and zy, respectively. [39]

The global impedance can be expressed in terms of the local impedance as

Z = (8-18)

Equation (8-18) is consistent with the treatment of Brug et al. [30] in which the

admittance of the disk electrode was obtained by integration of a local admittance

over the area of the disk.

8.2.3 Local Interfacial Impedance

The local interfacial impedance involves the potential of the electrode measured

relative to a reference electrode Go(r) located at the outer limit of the diffuse double 1e. r..

Thus, the local interfacial impedance is given by

V Go(r)
zo= (8-19)

The use of a lower-case letter again signifies that zo is a local value, and the subscript 0

signifies that zo represents a value associated only with the surface. The local interfacial

impedance may have real and imaginary values designated as zo,, and zo,j, respectively.

[39]

8.2.4 Local Ohmic Impedance

The local Ohmic impedance involves the potential of a reference electrode Go(r)

located at the outer limit of the diffuse double lhe;r and the potential of a reference

electrode located far from the electrode #(oo) = 0 (see Figure 8-2). Thus, the local Ohmic

impedance is given by

ze = (8-20)

The use of a lower-case letter again signifies that ze is a local value, and the subscript

e signifies that ze represents a value associated only with the Ohmic character of the

electrolyte. The local Ohmic impedance may have real and imaginary values designated as

ze,r and ze,,, respectively. The local impedance

z = zo + ze (8-21)

can be represented by the sum of local interfacial and local Ohmic impedances. [39]

8.2.5 Global Interfacial Impedance

The global interfacial impedance is defined to be

Zo /r Io -1(
Zo 2 r(r)1 (8-22)

Zo = ( )-1 (8-23)
zo(r)
The use of an upper-case letter signifies that Zo is a global value. The global interfacial

impedance may have real and imaginary values designated as Zo,, and Zo,j, respectively.
8.2.6 Global Ohmic Impedance

The global Ohmic impedance is defined to be

Ze = Z Zo (8-24)

0.8 ,

0.7 --

0.5 -

0.4 --

0.3 ~ K=1 l.

0.2 --

0.0
0.0 0.1 0.2 0.3 0.4

Z re/ r_7

4

10

102
10"
190

102

10 -
0 24

0.25 0.26

z, I ro=I

0.27 0.28

Figure 8-3. Calculated Nyquist representation of the impedance response for an ideally
polarized disk electrode. a) linear plot showing effect of dispersion at
frequencies K(>1; and b) logarithmic scale showing agreement with the
calculations of N. i..l!!in .1

The use of an upper-case letter signifies that Z is a global value. The global Ohmic

impedance may have real and imaginary values designated as Z,, and Z,4, respectively.

8.3 Results and Discussion

The calculated results for global, local, local interfacial, and both local and global

Ohmic impedance are presented in this section. The results are believed to be incorrect for

frequencies K( > 100 due to the presence of a singular perturbation problem that arises at

the periphery of the electrode at high frequencies. [42]

8.3.1 Global Impedance

The global impedance response presented in Figure 8-3(a) in Nyquist format shows

the influence of time-constant dispersion at frequencies K( > 1. The impedance was made

dimensionless according to Za/roxr in which the units of impedance Z were scaled by unit

U.Lo I t i l

0.27 -

0.26 --

Present Wobrk
0.25 o Newman

0.24 I r i i s
106 1(#5 10" 10" 10'2 10' 100 10' 102 103 10

10" t l l t
10s
Present W~ork
10 -
o Newyman
10 -
~:10 -
-10 -
rj 100
_Slope = -1

109

108 105 10'' 10"' 10- 10'' 100 10' 10' 103 1(

Figure 8-4. Calculated representation of the impedance response for an ideally polarized
disk electrode. a) real part, and b) imaginary part showing agreement with the
calculations and .I-i-uspind~ i1c formula of N.1. in~l! II

area, Ocm2. The expanded logarithmic representation presented in Figure 8-3(b) shows

good agreement with the numerical solutions presented by N. i.-.I!! 11! [42]

The comparison with ?-. i.-.In! .I1's calculations is seen more clearly in the representation

of the real and imaginary parts of the impedance response shown in Figures 8-4(a) and

8-4(b), respectively. As stated by Orazem et al. [78], the slope of log(Zjs/rovr) with

respect to log(K() gives the exponent of the CPE, -a~. The change in the slope of the lines

presented in Figure 8-4(b) shows that the impedance response transitions from ideal ReCo

behavior at low frequencies to a CPE-like behavior at frequencies K( > 1. A deviation from

N. i.--!!! lII's results is seen for frequencies K( > 100. This error is attributed to a singular

perturbation problem, identified by ?-. i.--in lIs. that arises at the periphery of the electrode

at high frequencies. [42]

The change of -a~ is presented in Figure 8-5 as a function of dimensionless frequency

K(. The system behaves as an ideal capacitor at low frequencies with a~ = 1. At

frequencies K( > 1, the value of a~ changes to roughly a~ = 0.85 at K > 10. As the

-1.1 III
-5 -4 -3 -2 -"1 0 1 2
10rg (K)

Figure 8-5. The slope of log(Zjm/rox)) with respect to log(K() (Figure 8-4(b)) as a function
of log(K(). The results were calculated by the collocation method. The value of
this slope is equal to -ca.

slope is not independent of frequency, the frequency dispersion seen at K > 1 does not

represent true CPE behavior.

The frequency K( = 1 at which the current distribution influences the impedance

response can be expressed as

f = (8-25)
2rC/o ro

As shown in Figure 8-6, this characteristic frequency can be well within the range

of experimental measurements. The value m/Co = 104 cm/s can be obtained for a

capacitance Co = 1 pF/cm2 (COTTOSponding to an oxide 111-;-r) and conductivity n

0.01 S/cm (corresponding to a 0.1 M NaCl solution). The value m/C = 103 cm/s can be

obtained for a capacitance Co = 10 pF/cm2 (COTTOSponding to the double 111-,- r on an

inert metal electrode) and conductivity n = 0.01 S/cm (corresponding to a 0.1 M NaCl

solution). Figure 8-6 can be used to show that, by using an electrode that is sufficiently

small, the experimentalist may be able to avoid the frequency range that is influenced by

current and potential distributions.

8.3.2 Local Interfacial Impedance

The calculated local interfacial impedance is presented in Figure 8-7(a) as a function

1 I I

x/Co=10 cm/s

10

10
P

...1 . ....1 . ....I

K = .01
~- -- K =1.0
----- K= 100

0 0.2 0.4 0.6 0.8 1.0
r/r,

102

101

0.01 0.1 1

ro /cm

Figure 8-6. The frequency K(=1 at which the current distribution influences the impedance
response with m/Co as a parameter.

0.36

0.34

S0.32
N7

0.30

0.28

Figure 8-7. Calculated imaginary part of the local interfacial impedance: a) as a function
of frequency with position as a parameter; and b) as a function of position
with frequency as a parameter.

I I

1.C --r/r. = 096

0. r/r = 0.8
-M r/q = 0.51

0.7 -

0.4

K= 1

0.2

.0100
0.0 0.1 0.2 0.3 0.4 0.5 0.6

Figure 8-8. The local impedance in Nyquist format with radial position as a parameter.

of frequency with position as a parameter and in Figure 8-7(b) as a function of position

with frequency as a parameter. The results presented in Figure 8-7 show that the local

interfacial impedance is purely associated with a capacitive behavior. At all frequencies,

zo~jpEK/rox = 1/xr as is expected for an ideal capacitance. The real part of the local

interfacial impedance, not shown here, was equal to zero within computational accuracy.

8.3.3 Local Impedance

The calculated local impedance response is presented in Figure 8-8 in Nyquist format

with normalized radial position as a parameter. The dimensionless impedance is scaled to

the disk area wrri in order to compare with the .-i-mptotic value of 0.25 for the real part

of the dimensionless global impedance. The impedance is largest at the center of the disk

and smallest at the periphery, reflecting the greater accessibility of the periphery of the

disk electrode. Inductive loops, which are not shown in the global impedance, are seen at

high frequencies in local impedance.

l r /= 0.96 --...
ra

--- r / ra O

--- -----

105 10" 109 10' 10^ 100 10' 102 103 10'

0.55
0.50 -
0.45 .
0.40 .
0.35
0.30
0.25

0.10
10

Calculated local impedance with radial position as a parameter: a) real part;
and b) imaginary part.

Figure 8-9.

The real and imaginary parts of the local impedance are presented in Figures 8-9(a)

and 8-9(b), respectively, with radial position as a parameter. The real part of the local

impedance presented in Figure 8-9(a) reaches .I-i-...ph.l~e values at K( 0 and K( 00.

The imaginary part presented in Figure 8-9(b) shows the change of sign associated with

the inductive features in Figure 8-8. The changes in sign occur at frequencies below

K = 100, showing that the inductive loop cannot he attributed to calculation artifacts.

The radial distribution of the real and imaginary impedance is presented in Figures

8-10(a) and 8-10(b), respectively, with dimensionless frequency K( as a parameter. At high

frequencies, e.g. K( = 100, the calculated radial distribution of the real part of the local

impedance follows the expression

(r) 0.5 1- -
~~( )2

(8-26)

derived from equation (8-27) using the expression for the primary resistance

R,

10 I I I
10'
o=0.96
10 -
10 ~- -r / r = 0.51
10 ----r/r 0
100

103

10'
105 10" 10'3 10'2 10~' 100 10' 102 105

(b)

0.6 a 4
Prirnary Resistance
0.5 .--p K = 0.1 -1 3
-- K = 1.0 K = 0.01
0.4 ------ K = 100 -2 K = 1.0

0 0.2 .4 0. 0.8 .0 0. 0.2-. K= 0.4 06 0 .

8..4 Loca Ohi meac
Foloin eqain(-1,telclO mcipdnez consfrtedfeec
bten th oa nefcaladlclipdnes h acltdloa h i meac
is 2 prsne in Fiur 8-1i yus omtwt ailpsto saprmtr

Fiue8-12(b) respctivtelyalipea as a function of feunywt radial position: as ae paramter The

l.. ocal Ohmic i mpedance hsol elvle tK 0adK 0,bti h rqec

rangein 10-2io <81) K<10zehsbthe real andi imaginary components. Figr e 8-12(a) cleal

showse the ..- in l ic behfavior inth lowa frequncys rane wiulthd v oalue distibutmedarond

1/4.etdi iur -1i yus ora ihrda oito saprmtr

Figure 8-12. Calculated values for local Ohmic impedance as a function of frequency with
radial position as a parameter: a) real part; and b) imaginary part.

N
I

0.1 0.2 0.3 0.4 0.5

Z ~K / F T

Figure 8-11.

The local Ohmic impedance in Nyquist format with radial position as a

parameter.

- -r/r = 0.80

. ----- r/ro =0.0 / -

.

-
'

0.55

0.50

0.45

0.40

0.35

0.30

0.25

0.20

0.15

0.10
10

n f

2 -

4~ -tlro=096

6 - rtr,
-r-tro=0.51
8~ -- r
0 -I

2 -
10s

-0.0

-00O

-0.0

-0.0

-0.1

-0.1

.4~ .s qa 1

10" 105 104 10' 10*

101 102 103

10 a0 10 0

0.0108

0.006

S0.002

-0.002 I
103 10' 1 10" 10' 102 109

Figure 8-13. The imaginary part of the global Ohmic impedance, calculated from equation
(8-28), as a function of dimensionless frequency.

8.3.5 Global Interfacial and Global Ohmic Impedance

The local interfacial impedance is associated with a pure capacitance that is

independent of radial position. Thus, the global interfacial impedance should also be

a pure capacitance Co in units of pF/cm2. The global Ohmic impedance Ze is obtained

from the global impedance Z by the expression

Ze = Z- (8-27)

or, in the dimensionless terms used in the present work,

Ze a Zm1
(8-28)
roxr roxr jxKE

The real part of Ze is equal to the real part of Z as given in Figure 8-4(a). The imaginary

part of Ze is given in Figure 8-13 as a function of dimensionless frequency K(. In the

low frequency range Ze is a pure resistance equal to 1.08Re, and, in the high frequency

range, Ze tends towards Re. The imaginary part of the global Ohmic impedance shows

a non-zero value in the frequency range that is influenced by the current and potential

distributions. Figures 8-4(a) and 8-13 show that all the effect of the current and potential

distribution appears in the global Ohmic impedance.

CHAPTER 9
BLOCKING DISK( ELECTRODE WITH LOCAL CPE

In C'!s Ilter 8, it has been shown that the nonuniform current and potential

distributions influence the global and local impedances of an ideally-blocking disk

electrode. The objective of this chapter is to explore the influence of current and potential

distribution on the impedance responses of a blocking electrode exhibiting a local CPE

behavior. In this sense, the goal is to explore the role of coupled 2D and 3D distributions

on the impedance responses of a disk electrode. This chapter presents the theoretical

development and results for the impedance calculations. [39] Experimental validation

provided by Vivier is also presented in this chapter. [39, 93]

9.1 Theoretical Development

The mathematical development presented in this chapter followed that presented

in C'!s Ilter 8. Laplace s equation in cylindrical coordinates was expressed in rotational

elliptic coordinates as equations (8-5) and (8-6) for real and imaginary parts, respectively.

The modification made here was the substitution of the capacitor at electrode surface

((=0) by a constant phase element CPE, i.e.

8(V Go) 8
i = Q --a (9-1)
iit iiwy a rorl iif e=o

where Q can be related to the interfacial capacitance Co by equation (7-8). The flux

boundary condition at the electrode surface ( = 0 was written in frequeno~l- -domain as

K 0-P)cos + ssm d (9-2)
2 2 l 8( e= o

and

K 0sin scos 4, sin -- ~~ (9-3)
2 2 2 rl 8( e~

where V, represents the imposed perturbation in the electrode potential and K( is the

dimensionless frequency

K = ro(9-4)

As seen in equation (9-4), the dimensionless frequency K( includes the CPE coefficient Q,

the frequency w raised to the power of the CPE exponent a~, the disk radius ro, and the

electrolyte conductivity m.

At rl = 0 and rl = 1, zero-flux conditions impose that

= 0 (9-5)

and

At the far boundary condition ( 00o,

Or = 0 (9-7)

and

4, = 0 (9-8)

The equations were solved under assumption of uniform CPE parameters Q and

a~. The simulations were performed using the collocation package PDE2D developed by

Sewell. [90] The calculations were performed for differing domain sizes and the results

reported here were obtained by extrapolation to an infinite domain size. As discussed in

the previous chapter, the calculated results are believed to be incorrect for frequencies

K( > 100 due to the presence of a singular perturbation problem that arises at the

periphery of the electrode at high frequencies. [42]

9.2 Results and Discussion

The calculated results for global, local, local interfacial, and both local and global

Ohmic impedances are presented in this section. A list of symbols for the impedance used

in the subsequent sections is provided in Table 8-1.

9.2.1 Global Impedance

The calculated dimensionless impedance response is presented in Figure 9-1 in

Nyquist format with a~ as a parameter. The representation given in Figure 9-1 applies for

1.0

a= 1
0.8 0.9

0.8

0.6 --
Y 0.7

0.4 --

K=1
0.2 --

(10
0.0
0.2 0.4 0.6

Figure 9-1. Nyquist representation for the calculated impedance response of a blocking
disk electrode with a local CPE with n~ as a parameter.

0= 0.7
1 a= 0.8
----a=0.9 -

10 I I I I t
105
-a= 0.7
10 a=0.8
10 -----a=09

10"
Sloe -
10"

1010 105 10" lb3 10 10~' 10" 0" 10 10 2

(b)

0

105

10'

103

100

yg-'
1(

106 10' 103 10~ 10' 100 10' 102 103 104
K

(a)

Calculated impedance response for a blocking disk electrode with a local CPE
as a function of dimensionless frequency K(: a) real part, and b) imaginary
part.

Figure 9-2.

all values of electrolyte conductivity n and disk radius ro, but different values are obtained

for different values of a~. The impedance was made dimensionless according to Za/roxr

in which the units of impedance Z are assumed to be scaled by area and having units of

Ocm2

The frequency dependence of the impedance response can be seen more clearly

in Figures 9-2(a) and 9-2(b), where the real and imaginary parts of the impedance,

respectively, are presented as functions of dimensionless frequency K( with a~ as a

parameter. The real part of the dimensionless impedance, plotted in Figure 9-2(a),

approaches the expected theoretical value of 1/4 at high frequency. [42] The low-frequency

behavior depends slightly on the value of a~. When plotted against dimensionless frequency

K(, the values of the dimensionless imaginary impedance in Figure 9-2(b) superpose for all

values of a~. This superposition is made possible by the inclusion of a~ in the definition of

frequency K( in equation (9-4).

Orazem et al. [78] cited the utility of logarithmic plots of imaginary impedance as a

function of frequency to identify CPE behavior. The calculated slope of log(Zjm/rox)) with

-0.85 a =0.7

-105' I I K I
-5 -4 -3 -2 -1 0 1 2
log !a

Figure 9-3. The calculated slope of log(Zjs/rox)) with respect to log(K() (Figure 9-2(b)) as
a function of log(K() with a~ as a parameter.

respect to log(K() (Figure 9-2(b)) is presented in Figure 9-3. with a~ as a parameter. Due

to the definition of K(, the slope at low frequencies of the logarithmic plots of imaginary

impedance as a function of K( is equal to -1. At frequencies K( > 1, the slope increases

to approximately -0.85. When expressed in terms of these dimensionless parameters, the

low-frequency response is independent of a~, but the results obtained at higher frequencies

depend on a~.

The calculation of effective CPE coefficient Qeaf provides further evidence that the

low-frequency behavior is unaffected by the current and potential distribution. The

effective CPE coefficient Qeaf for an electrochemical system can be obtained from the

imaginary part of the impedance by

Qea = sm ll-1 (9-9)

The effective CPE coefficient obtained from equation (9-9) scaled by the input value is

presented in Figure 9-4 as a function of frequency with a~ as a parameter. Equation (9-9)

yields the input value for the CPE coefficient at low frequencies, but this calculation is

influenced by the current distributions at frequencies K > 1.

__ _ __ __ __

0.8

-u= 0.7
=0.8
0.4 ----- u==0.9

0.2 In s
10J 10~ 10. 010 10" 10o 101 102 103

Figure 9-4. Effective scaled CPE coefficient as a function of frequency with a~ as a
parameter.

9.2.2 Local Interfacial Impedance

The calculated local interfacial impedance at a~ = 0.8 is presented in Figure 9-5 in

Nyquist format with normalized position as a parameter. All of the lines are superposed

indicating that the local interfacial is independent of position. The real and imaginary of

the local interfacial impedance at a~ = 0.8 are presented, respectively, in Figures 9-6(a)

and 9-6(b) as a function of frequency with normalized radial position as a parameter. All

the lines are superposed and the slopes seen in both real and imaginary impedance plots

are equal to -1. This idealized character of the local interfacial impedance is seen more

clearly in Figure 9-7 in which the impedance is scaled by the dimensionless frequency and

given as a function of radial position with frequency as a parameter. At all frequencies,

the scaled real part of the local interfacial impedance follows

zo,rsK( 1
Scos (cax/2) (9-10)
r~oiT iT

and the imaginary part of the local interfacial impedance follows

xo,j pK 1
S- sin(a xr/2) (9-11)
r~oiT iT

10" .
10"C = 0.8
10*C r / r = 0.96
103 r / ro = 0.80
10' -----r rro=.51
lol -----r/ ro=.0
10 -

162 Slope = -1
10 -

10 -
106 10.s 10" 10" 10' 10'' 100 10' 10 10' 10'

(b)

a =0.8
-rl ro= 0.96

---rl ro=0.51
-----r/ ro=0.0

10" 10' 103 10" 10~ 10P 101 10t 10" 10'

1.8

1.6

1.4

1.2

S1.0

u_0.8
o
0.6

Local Interfacial
Impedance a= 0.8
-r / ro = 0.96
- r / re = 0.80
- --- r / r = 0.51

0.0

-0.2 '
0.0 0.2 0.4

0g,r

Figure 9-5.

Nyquist representation for the calculated local interfacial impedance response
of a blocking disk electrode with a local CPE with normalized radial position
as a parameter.

10s

10 *

10 -

10 '
10'

Slope = -1

Figure 9-6. Calculated local interfacial impedance as a function of frequency with position
as a parameter: a) imaginary part; and b) real part.

0.6 0.8 1.0

0 7

0.14 0.5
0.13 =08a =0.8
0.12 K=100 0.4 K=100
---K=1 ---K=1
0.11- --K = 0.01 -I I -K = 0.01
0.10 -0.3

0 08 C -1 0.2-
0.07-
0.06 L L 0.1
0.0 0.2 0.4 0.6 0,8 1.0 0.0 0.2 0.4 0.6 0,8 1.0
r/ro r/r,

(a) (b)

Figure 9-7. Calculated local interfacial impedance as a function of position with frequency
as a parameter: a) imaginary part, and b) real part.

The results presented in Figures 9-6 and 9-7 show that the calculated local interfacial

impedance is independent of 2-D distributions.

9.2.3 Local Impedance

The calculated local impedance response for a~ = 0.8 is presented in Figure 9-8 with

normalized radial position as a parameter. The dimensionless impedance is scaled to the

disk area grr, to show the comparison with the high-frequency .I-i sphllcl~ value in Figure

9-1. The impedance is largest at the center of the disk and smallest at the periphery,

reflecting the greater accessibility of the periphery of the disk electrode. Inductive loops,

which are not shown in the global impedance, are seen at high frequencies in local

impedance for all the radial positions.

The real and imaginary parts of the local impedance are presented in Figures 9-9(a)

and 9-9(b), respectively, with radial position as a parameter. The real and imaginary

parts of the local impedance presented in Figure 9-9 show a pure CPE behavior at low

frequencies and a geometry-induced dispersion at high frequencies. The imaginary part

presented in Figure 9-9(b) shows the change of sign associated with the inductive features

IIIII IIII
a =0.8
-r / r = 0.96
r r = 0.80
-- r r = 0.51
----- rI ro = 0 0

III 1IIrIII

1.0

0.8

0.6

I

0.4 /I a = Us .
K= 1
0.2--

100

-0.2
0.0 0.2 0.4 0.6 0.8 1.0

Figure 9-8. The local impedance in Nyquist format with radial position as a parameter.

10e
10 a =0.8
10l r / r = 0.96
103 -- r / re = 0.80
102 -- -- r / re = 0.51
F ,1C----- r /ro=0.0
a10 -
A 100 ,
S10 *

10. ~ y ~
10 *
10"
10" 10-s 10" 101 10' 10' 100 10' 10 10' 1(

10'

N ..1

B

10-1 t
10

106 10~ 103 10'2 101 10o 10' 102 103 10

Figure 9-9. Calculated local impedance: a) real part; and b) imaginary part.

10 k0 100 100 r /r,
Y 10
--0.96

0.51
-0.1 0.
0.1 0.2 0.3 0.4 0.5 0.6
z,,r / ran

Figure 9-10. The local Ohmic impedance in Nyquist format with radial position as a
parameter.

seen in Figure 9-8. The changes in sign occur at frequencies well helow K( = 100, showing

that the inductive loop cannot he ascribed to calculation artifacts.

9.2.4 Local Ohmic Impedance

The local Ohmic impedance x, accounts for the difference between the local interfacial

and the local impedances. The calculated Ohmic impedance for a~ = 0.8 is presented in

Figure 9-10 in Nyquist format with radial position as a parameter. The results obtained

here for the local Ohmic impedance are very similar to those reported for the ideally

polarized electrode. At the periphery of the electrode, two time constants (inductive

and capacitive loops) are seen; whereas, at the electrode center only an inductive loop is

evident. These loops are distributed around the .-i-mptotic real value of 1/4.

9.2.5 Global Interfacial and Global Ohmic Impedance

The local interfacial impedance has shown to be associated with an ideal CPE

behavior and to be independent of radial position. Thus, the global interfacial impedance

is given by

Zo = (9-12)

The global Ohmic impedance Z, is obtained from the global impedance Z hy the

expression

Z, = Z Zo

' 0.26
N

10' 100 10' 102 103 10' 100 101 102 103
K K

(a) (b)

Figure 9-11. Calculated values for global Ohmic impedance as a function of frequency with
a~ as a parameter: a) real part; and b) imaginary part.

or, in the dimensionless terms used here,

As Zu1
(9-13)
roxr roxr j"rK

The real part of Z, is given in Figure 9-11(a), and the imaginary part of Z, is given in

Figure 9-11(b) as functions of dimensionless frequency K( with a~ as a parameter. In the

low frequency range Zs/roxr is a pure resistance equal to 0.27, and, in the high frequency

range, Zs/roxr tends towards 1/4. The imaginary part of the global Ohmic impedance

shows a non-zero value in the frequency range that is influenced by the current and

potential distributions. Figure 9-11 shows that all the effect of the current and potential

distribution appears in the global Ohmic impedance.

9.3 Experiments

The predictions made by the calculations can be compared to experimental

observations. Vivier [39] conducted impedance measurements on a glassy carbon disk

electrode to compare with the calculation results. Local impedance measurements were

as well performed on a stainless steel disk to demonstrate that the inductive features

predicted by the simulations are apparent in experiments.

I 0.6 I I
80 a PoeM 2. a 0.DB a
a~~ aac M no
600 o 1.5 o -
0.021 Hz O 25 7 Hz a
~- 400 4 1 -

0 0 400. 1.0 1. o

FB F

between 1 00 k.z and 100 m1z; an )zoedrgo hoigol

9.3.1 -12 Global -mpedance-ln lt o h epne of G lassy-Carbon Electrod

The global impedance measurements were made at three different concentrations of

K(C1. The results obtained in 0.5 AI, 0.06 31 and 0.0065 31 KGC are presented in Figure

9-12 with concentration as a parameter. The differences among the results are most

apparent at high frequencies, as shown in Figure 9-12(b). The results are consistent with

a blocking, but not ideally polarized, electrode. The agreement also -II---- -R that there is

a local capacity dispersion on the glassy carbon disk electrode. A high-frequency feature

is evident in Figure 9-12(b), and this feature appears at lower frequencies for the smaller

concentration.

The dimensionless imaginary part of the impedance is presented in Figure 9-13(a) as

a function of dimensionless frequency. The superposition of data for the three values of

conductivity is in excellent agreement with the calculations (see Figure 9-2(b)), and the

change in slope from a value of -1 appears at frequencies higher than K = 1.

The derivative of the logarithm of the dimensionless imaginary impedance with

respect to the logarithm of dimensionless frequency is presented in Figure 9-13(b). The

1 0 - .. - -, - .-, - -
-0.6 a 0.5 M
10s1 .P o 0.5 M o0.oe M
o 0.6 M 0.aces M
ly a 0,0065 M
o -0.8
S10 -

Ka K

(a) (b)

Figure 9-1:3. Dimensionless analysis for the impedance response of a graphite disk in K(Cl
electrolytes with concentration as a parameter, a) Dimensionless imaginary
part of the impedance as a function of dimensionless frequency
(corresponding to Figure 9-2(b)); and b) Derivative of the logarithm of the
dimensionless imaginary part of the impedance with respect to the logarithm
of dimensionless frequency (corresponding to Figure 9-:3).

dispersion of the data apparent in Figure 9-1:3(b) can he attributed to the fact that

the derivative calculations were performed on experimental data. The superposition of

data for the three values of conductivity is in excellent agreement with Figure 9-3 with

a~ = 0.9, and the transitional frequency between low and high-frequency response is in

good agreement with the theoretical value of K( = 1.

9.3.2 Local Impedance of Stainless Steel Electrode

The local impedance measurements were performed on a Fe-17Cr stainless steel disk

electrode in 0.05 31 KGC + 0.005 Al N._S(4 electrolyte. The local impedance and local

interfacial obtained at the center of the disk (r/ro = 0) are presented in Figure 9-14.

[9:3] As predicted from the calculations, the local impedance exhibits inductive loops at

high frequency; whereas, the local interfacial impedance shows expected behavior for a

local CPE within all frequency range. The characteristic transition frequency at which

the geometry phI i- a role locates approximately at K=-0.52, which is consistent with the

theoretical prediction K(=1. The local interfacial impedance exhibits an ideal local CPE

O- Iocal
O Iocal Interfacial
Q local ohmic
1.5 o

L. 33 Hz(K=0.045)

a on

.OO

0. -. (K=0.52) .r

0.0 0.5 1.0 1.5 2.0 2.5

Figure 9-14. Experimental local impedance, local interfacial impedance, and local Ohmic
impedance in Nyquist format of a stainless steel disk electrode at the center
of the electrode (r/ro = 0). [93]

behavior, which agrees with the perdition from Figure 9-5. The local Ohmic impedance,

the difference between the local and local interfacial impedance, is given in the rectangle

box in the figure. The shape of the local impedance at r/ro=0 is consistent with that seen

in Figure 9-10.

The local impedance shows CPE behavior at low frequencies and a change in

sign in the imaginary part of the impedance at high frequencies. This appearance of

high-frequency inductive loops is consistent with the calculated local impedance presented

in Figure 9-9(b). The agreement between the model presented here and the experimental

results obtained from the steel electrode illustrates the utility of the model for describing

features of systems that exhibit CPE behavior over a range of frequency.

CHAPTER 10
DISK( ELECTRODE WITH SINGLE FARADAIC REACTION

The results presented in chapters 8 and 9 illustrate that the current and potential

distributions associated with disk electrodes induce an apparent CPE behavior on

the impedance of blocking electrodes. This chapter explores the influence of current

distribution on impedance response of a disk electrode subject to a single Faradaic

reaction. [31]

10.1 Theoretical Development

The mathematical development presented in this chapter followed those presented

in ChI Ilpters 8 and 9. Lapace's equation written in the rotational elliptic coordinates

(equations (8-5) and (8-6)) remained as the governing equations. The key difference

between the present work and those described in the previous chapters was the boundary

condition applied at the electrode surface.

The problem was solved for two kinetic regimes. Under linear kinetics, following

T. wi.-s! 1a [42] and Ni- .Is, loglu [43, 44], the current density at the electrode surface was

expressed as

8(V Go) (as+ a) ioF -8 8
i~ = Co + (Vi o) = (10-1)
Bt RT y a~ rorl iif e=o

The assumption of linear kinetics applies for 2 << io. Under assumption of Tafel kinetics,

the current density at the electrode surface was expressed as

8 (V Go) acF -~ 80
i = Co -t x V- o a(02
dt RT dy a= rorl df e=o 12

where the current in the Tafel regime was assumed to be cathodic. A similar expression

can be developed under assumption of anodic currents. The results presented here are

general because the impedance results do not depend on whether the current is anodic or
cathodic.

The flux boundary conditions (10-1) or (10-2) apply at the electrode surface (( = 0).

The boundary conditions (10-1) or (10-2) were written in frequency domain as

K~y +J V,- ior) =(10-3)

and

K(C;~,.) ii r i~1 Big
K ,-i~)+Jo,y (10-4)

for real and imaginary components, respectively. Here V, represents the imposed

perturbation in the electrode potential referenced to an electrode at infinity and K( is

the dimensionless frequency, defined as

K = woo(10-5)

Under the assumption of linear kinetics, valid for 2 << io, the parameter J was defined to
be
(as + ac) Fioro~
J =(10-6)

For Tafel kinetics, valid for 2 >> io, the parameter J was defined to be a function of radial

position on the electrode surface as

J(rl) =(10-7)

where 2(rl) was obtained from the steady-state solution as

~zo i exp~ c,-(V Go) (10-8)

The local charge-transfer resistance for linear kinetics can be expressed in terms of

parameters used in equation (10-6) as

RT
Rt = .(10-9)
ioF(cas + c)

The local charge-transfer resistance for Tafel kinetics can be expressed in terms of

parameters used in equation (10-7) as

RT
Rt (10-10)
i(rl) Fac

For linear kinetics, Rt was independent of radial position, but, under Tafel kinetics,

as shown in equation (10-10), Rt depended on radial position. Fr-om a mathematical

perspective, the principle difference between the linear and Tafel cases was that J and

Rt were held constant for the linear polarization, whereas, for the Tafel kinetics, J and

Rt were functions of radial position determined by solution of the nonlinear steady-state

problem.

The relationship between the parameter J and the charge-transfer and Ohmic

resistances can be established using the high-frequency limit for the Ohmic resistance to a

disk electrode obtained by N. ein-~! 1o [94] i.e.

Re (10-11)

where Re has units of Rcm2. The parameter J can therefore be expressed in terms of the

Ohmic resistance Re and charge transfer resistance Rt as

4 R
J = e(10-12)
xr Rt

Large values of J are seen when the Ohmic resistance is much larger than the charge-transfer

resistance, and small values of J are seen when the charge-transfer resistance dominates.

The equations were solved under assumption of a uniform capacitance Co using the

collocation package PDE2D developed by Sewell. [90] Calculations were performed for

differing domain sizes, and the results reported here were obtained by extrapolation to an

infinite domain size.

Q~(l(r)

Figure 10-1. Schematic representation of an impedance distribution for a disk electrode
where x, represents the local Ohmic impedance, Corpeet h nef"a
capacitance, and Rt represents the charge-transfer resistance.

10.2 Results and Discussion

The nature of the metal-electrolyte interface of an electrode exhibiting a Faradaic

reaction can he understood in the schematic representation given as Figure 10-1. Under

Lll~linea knelties,~h both Co and R, were independent of radial position; whereas, for Tafel

kinetics, 1/Rt varied with radial position in accordance with the current distribution

presented in Figure 7-6.

The calculated results for global, local, local interfacial, and local Ohmic impedances

are presented in this section. A list of symbols for local and global impedances used is

provided in Table 8-1.

10.2.1 Global Impedance

The calculated global impedance response is presented in Figure 10-2(a) for .7 = 0.1

and in Figure 10-2(b) for .7 = 1.0 with dimensionless frequency K( as a parameter. The real

and imaginary components are presented in dimensionless form to eliminate the influence

of electrolyte conductivity n and disk radius ro. The impedance results for linear kinetics

at .7 = 0.1 match closely with the impedance response

Zu 1 1 1/
+ -(10-13)
wrro 4 r 1+ jK/J~

2.0 J=0.1
Present work
Linear O Newman, Linear
1.5
I ~-K =0.1
1.0 -Tafel

0.5
K=1 K=0.01

0.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

(a)

J= 1.0
0.2~ Present work
O Newman, Linear
... -ff- No current distribution
K=1
c-0.1 -

K=I0

0.0-

0.2 0.3 0.4 0.5 0.6 0.7

Z /I rp

(b)

Figure 10-2. Calculated Nyqluist representation of the impedance response for a disk
electrode under assumptions of Tafel and linear kinetics. Open symbols

represent the result calculated by T. .. in .il [42] a) J = 0.1; and b) J = 1.0.

----------100 t- J=0.1
J=0.1
10-1 C LL-t Slope= -a
Slope= 1
S1C 10~2 J=1.0

N J=1.0 Tafel
N -- Linarr
10 o A Newman. linear
-Tarel
--- Linear s ,
O A Newman, linear
0.1 I li tl10 Il ll
10" 10' 10'3 10" 10-1 100 10" 102 103 10" 10" 10" 10' 10`' '100 10' '102 103
K K

(a) (b)

Figure 10-:3. Calculated representation of the impedance response for a disk electrode
under assumptions of Tafel and linear kinetics and with with J as a
parameter. Open symbols represent the result calculated by N. i.--1! 11!. [42] a)
real part; and b) imaginary part.

derived in the absence of current distribution effects. The impedance response for Tafel

kinetics differs because the charge-transfer resistance is a function of radial position.

The comparison between the impedance for linear kinetics and equation (10-13) for

J = 1 shows the distortion of the high-frequency impedance response associated with the

influence of current and potential distributions.

The calculated results for linear kinetics in Figure 10-2 show good agreement to

the corresponding numerical values obtained by Newman. [42] The comparison with

TN .~..-! Ito's calculations is seen more clearly in the representation of the real and imaginary

parts of the impedance response shown in Figures 10-:3(a) and 10-:3(b), respectively. At

low frequencies, values for the real part of the impedance differ for impedance calculated

under the assumptions of linear and Tafel kinetics; whereas, the values of the imaginary

impedance calculated under the assumptions of linear and Tafel kinetics are superposed

for all frequencies. The slope of the lines presented in Figure 10-:3(b) are equal to 1 at low

frequencies but differ from -1 at high frequencies. As stated by Orazem et al. [78], the

slope of these lines can he related to the exponent a~ used in models for CPE behavior.

0.0 -9

-0.3
4)o.6~ --Tafel
S---- Linear

-5 -4 -3 -2 -1 0 1 2
log (K)

Figure 10-4. The calculated derivative of log(Zjs/rox)) with respect to log(K() (taken from
Figure 10-3(b)) as a function of K( with J as a parameter.

The calculated derivative of log( m/lrox)) with respect to log(K() is presented in

Figure 10-4 as a function of K( with J as a parameter. At large frequencies, the quantity

dlog( m/lrox)/ddlog(K() can be considered to be equal to -a~ where a~ is the exponent

used for models of CPE behavior. The characteristic frequency at which the value of

slope deviates from unity increases with the dimensionless parameter J. The transition

frequencies correspond to the inverse of the RtCo time constant and overlap when given as

a function of
K wCoRT
= LoRtCo (10-14)
J cF

The functional dependence of dlog(Zjs/rox)/ddlog(K() was independent of assumption of

either linear or Tafel kinetics.

When dlog( m/lrox)/ddlog(K() was plotted as a function of log(K(/J), given in Figure

10-5, all the curves for K( < 1 are superimposed. The characteristic frequency K(/J=1 is

associated with the RtCo time constant for the Faradaic reaction and the characteristic

frequency for the effect of the current and potential distributions at K(=1 is associated

with the capacitance and the Ohmic resistance.

r ~1.0 -- -

S0.5

-g0.5

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5
log (KclJ)

Figure 10-5. The calculated derivative of log(Zjs/rox)) with respect to log(K(/J) (taken
from Figure 10-3(b)) as a function of K( with J as a parameter.

10.2.2 Local Interfacial Impedance

The calculated local interfacial impedance for Tafel kinetics with J = 1 is presented in

Figure 10-6 as a function of frequency with normalized radial position as a parameter. At

low frequencies, the local interfacial impedance, for both real and imaginary, is smallest at

the periphery and largest at the center of the disk. The variation at low frequencies is less

distinguishable for smaller values of J. All the curves in Figures 10-6(a) and 10-6(b) are

superposed at frequencies K > 1.

For the linear kinetics calculation, where J is independent of radial position, the

scaled real part of the local interfacial impedance follows

zo,r a
(10-15)
roxr x(12 + K(2

and the imaginary part of the local interfacial impedance follows

zo,ja -K(
(10-16)
roxr x(12 + K(2

Plots similar to Figure 10-7 were obtained for the local interfacial impedance calculated

under assumption of linear kinetics, but for linear kinetics the local interfacial impedance

I l i l l i a ~10" n l s
----- ---------Tafel: J= 1.0 Tafel: J= 1.0
0.3 -._ _____ : 10 *

10"
0o 0.2-
rI ra = 0 6 o 10 *
-- r r=0 r ro 09
0.1 ------r/ ra=0.51 0 O104 P -rr=.
------ rI r,= ----- r r = 0.51
0.0 10 ------r/r,=0
I I 104
10" 10" 105 ~' 10 10 10 l 1 ga 108 10-* 104 10' 102 10' 10" 10" 102 "103 10"
Kt K

(a) (b)

Figure 10-6. Calculated representation of the local interfacial impedance response for a
disk electrode as a function of dimensionless frequency K( under assumptions
of Tafel kinetics with .7 = 1.0: a) real part; and b) imaginary part.

The local interfacial impedance for Tafel kinetics with .7 = 1 is presented in

Figure 10-7 as a function of normalized radial position with dimensionless frequency

as a parameter. Under the Tafel kinetics assumption that .7 is a function of radial

position, as shown in Figure 10-7, the real and imaginary parts of the local interfacial

impedance change around the values given in equations (10-15) and (10-16) and have

minimum values at the periphery of the disk. The dependence of both real and imaginary

of the local interfacial impedance is more evident at low frequency than high frequency.

10.2.3 Local Impedance

The calculated local impedance for Tafel and linear kinetics with .7 = 1 is presented

in Figure 10-8 in Nyquist format with radial position as a parameter. In both cases, the

impedance is largest at the center of the disk and smallest at the periphery, reflecting

the greater accessibility of the periphery of the disk electrode. Similar results are also

obtained for .7 = 0.1, but the differences between radial positions are much less significant.

Inductive loops are observed at high frequencies and these are seen in both Tafel and

linear calculations for .7 = 0.1 and .7 = 1.0.

0 40
0 35

10'

S0 20 -
I I ITafel: J =1,0
"0 15 C------------- Y K= 0.01
NO Tafel. J =1.0 NO 10~2 .---K=1.0
o 10 .1- -- K =100
0.05 C -- K=1.0
000 ------ -- ----------K 10

-0.05 1 il 10'
0.0 0.2 0.4 0.6 0,8 1.0 0.0 0.2 0 4 0.6 0 8 1.0
rI ro r Ir

(a) (b)

Figure 10-7. Calculated representation of the local interfacial impedance response for a
disk electrode as a function of radial position under assumptions of Tafel
kinetics with J = 1.0: a) real part, and b) imaginary part.

The real and imaginary parts of the local impedance for Tafel kinetics with J = 1.0

are presented in Figures 10-9(a) and 10-9(b), respectively. The real part of the local

impedance presented in Figure 10-9(a) reaches .-i-mptotic values at K( 0 and K( 100.

The absolute value of the imaginary part presented in Figure 10-9(b) shows the change

of sign associated with the inductive features seen in Figure 10-8(a). The changes in sign

occur at frequencies below K(=100, indicating that the inductive loop cannot be attributed

to calculation artifacts.

10.2.4 Local Ohmic Impedance

The local Ohmic impedance ze accounts for the difference between the local interfacial

and the local impedances. The calculated Ohmic impedance for Tafel kinetics with J = 1.0

are presented in Figure 10-10 in Nyquist format with normalized radial position as a

parameter. The results obtained here for the local Ohmic impedance are very similar

to those reported for the ideally polarized electrode and for the blocking electrode with

local CPE behavior. At the periphery of the electrode, two time constants (inductive

r_ 0.1
N

0.0 0.1 0.2 0.3 0.4

0.3

0.2

0.0

0.0 0.1 0.2 0.3 0.4

zlreIrTon

0.5 0.6 0.7

Figure 10-8. Calculated representation of the local impedance response for a disk electrode
as a function of dimensionless frequency K( under assumptions of Tafel
kinetics with J = 1.0. a) Tafel kinetics; and b) linear kinetics.

0.5 0.6 0.7 0.8

I I I

10' 100 10' 102 103 104

*

0.8 l l

0.7 -

0.6

0.4 -
Y.. Talel. J=1.0
N 0.3 rr= .6

0.2 ---rtr,=08

0.0 l l
108 10'5 10" 10' 10 2

10'

102

S10'

10"

10sl I I I I I
105 10 10' 103 10~' 10" 0 10 1 2 3

Figure 10-9.

Calculated representation of the local impedance response for a disk electrode
as a function of dimensionless frequency K( with J = 1.0. a) real part; and b)

imaginary part.

0.1 -

o

S0.0

-0.1
0.0

-0-- r / r = 0.96 Ar /r, = 0.51
--r/ r,= 0.80 Frir, =

K=10

0.1 0.2 0.3

Ze,rK I 10~

0.4 0.5 0.6

Figure 10-10. Calculated representation of the local Ohmic impedance response for a disk
electrode as a function of dimensionless frequency K( under assumptions of
Tafel kinetics with J = 1.0

0.04
Tafel: J= 1.0 Tafel: J= 1.0
-r / r = 0.96
0.6 --- rI ro= 0.8 .-r`-.---- 0.02
----- r r = 0.51 I00
", 0.4 -: --r/o= /
... .... ... ../ -0.02 1 /+= 0.96 / .

0.2 0.6

-0.08
0.1 I III iil
10-s 10" 10" 10" 10' 10a 10' 102 103 105 10" 103 102 10~' 10D 10' 102 10"
K KC

(a) (b)

Figure 10-11. Calculated representation of the local Ohmic impedance response for a disk
electrode as a function of dimensionless frequency K( under assumptions of
Tafel kinetics with J = 1.0. a) real part; and b) imaginary part.

and capacitive loops) are seen; whereas, at the electrode center only an inductive loop is

evident. These loops are distributed around the .-i-mptotic real value of 1/4.

The calculated values for real and imaginary parts of the local Ohmic impedance are

presented in Figures 10-11(a) and 10-11(b), respectively, as a function of frequency with

radial position as a parameter. The local Ohmic impedance has only real values at K( 0

and K( 00o, but in the frequency range 10-2 < K( < 100, ze has both real and imaginary

components. This range of dimensionless frequency was not dependent on the value of J.

The local Ohmic impedance obtained for linear kinetics and for different J were similar to

the results reported here.

The representation of an Ohmic impedance as a complex number represents a

departure from standard practice. As shown in previous sections, the local impedance has

inductive features that are not seen in the local interfacial impedance. These inductive

features are implicit in the local Ohmic impedance. As similar results were obtained for

ideally polarized and blocking electrodes with local CPE behavior, the result cannot be

U.ZID tlII are
0.008 -J= 0.01 and 0.1
J = 0 and 0.01
0.270 --
0.1 1
0.25 1 0.006

S0.260 __ ___10_ _1 0.004 1: 0.

N N/
0.255-- 002-

0.250
0.000
01245 I
10" 104 10" 10' 101 100 10" 102 103 10' 10" 10" 10'2 10' 100 10" 102 103
K K

(a) (b)

Figure 10-12. Calculated global Ohmic impedance response for a disk electrode as a
function of dimensionless frequency for linear kinetics with .7 as a
parameter, a) real part; and b) imaginary part.

attributed to Faradaic reactions and can he attributed only to the Ohmic contribution of

the electrolyte.

10.2.5 Global Interfacial and Global Ohmic Impedance

The global interfacial impedance for linear kinetics is independent of radial position

and is given by
Rt
Zo =(10-17)
1 +jecC/o R

The global Ohmic impedance Z, is obtained from the global impedance Z hy the

expression

Z, = Zo(10-18)

The real part and imaginary parts of Ze, obtained for linear kinetics are given in Figures

10-12(a) and 10-12(b), respectively, as functions of dimensionless frequency K with .7 as a

parameter. In the low frequency range Zs/ro~r is a pure resistance with numerical values

that decreases with increasing J. All curves superimpose in the high frequency range

toward .I-oni-n1'1 1 value of 1/4. The imaginary part of the global Ohmic impedance shows

a nonzero value in the frequency range that is influenced by the current and potential

distributions.

At high and low frequency limits, the global Ohmic impedance defined in the present

work is consistent with the accepted understanding of the Ohmic resistance to current

flow to a disk electrode. The global Ohmic impedance approaches, at high frequencies,

the primary resistance for a disk electrode (equation(10-11)) described by Newman. [94]

This result was obtained as well for ideally polarized (ChI Ilpter 8) and blocking electrodes

with local CPE behavior (C'!s Ilter 9). The complex nature of both global and local Ohmic

impedances is seen at intermediate frequencies. This complex value is the origin of the

inductive features seen in the local impedance and the origin of the CPE-like behavior

found in the global impedance.

10.3 Interpretation of Impedance Results

Ni- 1lia; inglu [43] estimated the error caused by frequency dispersion in evaluating

physical properties such as charge transfer resistance and capacitance. A parallel analysis

is presented here in terms of the commonly used CPE models.

10.3.1 Determination of Charge Transfer Resistance

The impedance response of a disk electrode in the absence of current distribution

effects can be expressed by equation (10-13). The corresponding charge-transfer resistance

evident at low frequencies is given by

Rex 1
(10-19)
wrro x J

The effective global charge-transfer resistance can be estimated from the calculated

impedance according to
Rea s Zr 1
(10-20)
wrro x J K(=0
The value of Res/IRt is presented in Figure 10-13 as a function of J under the assumption

of linear kinetics. The results are full agreement with those presented in different format

by ?'- 1a s, nglu [43]. The influence of the frequency dispersion is greatest when J is large,

2.0 .. .. ... .

S1.8

1.6-

1.2 -

0.01 0.1 1 10 100

Figure~ ~ 101.Teaprent value of Res/IRt obtained from the calculated impedance

response at low frequencies as a function of J.

i.e. when the Ohmic resistance dominates over the charge transfer resistance. At J = 100,

an error of 75 percent is seen in the estimation of the charge-transfer resistance.

10.3.2 Determination of Capacitance

The evaluation of interfacial capacitance is perhaps better done in terms of the CPE.

The values of a~ and 1-a~ obtained from Figure 10-4 are presented in Figure 10-14 as

functions of J. The value of a~ ranges from 0.98 for J = 0.01 to 0.87 for J = 10, which

demonstrates that nonuniform current and potential distributions on a disk electrode can

yield high-frequency CPE-like behavior. As J becomes small, i.e. as the charge-transfer

resistance dominates over the Ohmic resistance, a~ tends toward unity. It is significant

that the calculated value of a~ shown in Figure 10-14 corresponds to a range of a~ that is

frequently observed in experiments.

As shown in equation (9-9), the effective CPE coefficient Qeaf for electrochemical

systems follows

Qea = smrl C~T-1

... ..., . ... ... 1.0

0.0 0.1 1 0 .
IJ

Figue 1-14Thea~pparent value of 1-a~ obtained from the calculated impedance response
at high frequencies as a function of J.

The value of effective CPE coefficient, scaled by the interfacial capacitance, is presented

in Figure 10-15 as a function of J. The frequencies reported in Figure 10-15 are limited

to those that are one decade larger than the characteristic frequency because, in this

frequency range, the value of a~ is well-defined. Figure 10-15 was developed taking

into account the observation, seen in Figure 10-4, that the value of a~ is dependent on

the frequency at which the slope is evaluated. Thus, the value of Qeaf reported is that

corresponding to the value of a~ at a given frequency K(.

While the dimensions are not exactly that of a capacitance, the CPE coefficient

is often assumed to have approximately the same numerical value as the interfacial

capacitance. The value of Qeaf presented in Figure 10-15 is a function of frequency. At

high-frequencies, where frequency dispersion pll li- a significant role, the effective CPE

coefficient Qeaf provides an inaccurate estimate for the interfacial capacitance, even

for small values of J where a~ is close to unity. The errors in estimating the interfacial

capacitance are on the order of 500 percent at K( = 100.

o J=0,01 o J=1.0
0 J=0.02 a J=3.0
6 J=0.1
b v J=0.3 dQ.33

S4-

0.1 1 10 100

Figure 10-15. Effective CPE coefficient scaled by the interfacial capacitance as a function
of J.

A number of researchers have explored the relationship between CPE parameters and

the interfacial capacitance. Hsu and 1\ansfeld [95] proposed

Ceff = Q( omax> o1 (10-21)

where max, (or Kmax,) is the characteristic frequency at which the imaginary part of the

impedance reaches its maximum value and Ceaf is the estimated interfacial capacitance.

Equation (10-21) is tested against the input value of interfacial capacitance in Figure

10-16 where Co is the known interfacial capacitance which was independent of radial

position. As described above, Figure 10-16 was developed using local fra in. n. s --- dependent

values of a~ and Qeaf. The frequencies reported in Figure 10-16 are limited to those that

are one decade larger than the characteristic frequency max,. While equation (10-21)

represents an improvement as compared to direct use of the CPE coefficient Qeaf, the

errors in estimating the interfacial capacitance depend on both J and K and range

between -70 to +100 percent.

0.5 -= Oi

0.1 1 10 100

Figure 10-16. Effective capacitance calculated from equation (10-21) and normalized by
the input interfacial capacitance for a disk electrode as a function of
dimensionless frequency K( with J as a parameter. (See Hsu and Mansfeld
[95])

Brug et al. [30] developed a relationship for a blocking electrode between the

interfacial capacitance and the CPE coefficient Q as

Cea [QR 1-) 1/o (10 22)

A similar relationship between the interfacial capacitance and the CPE coefficient Q was

developed for a Faradaic system as

( 1 1 -) /1 J -) /
Cea = 0 Re Re ] Re 1 + 4 (10-23)

Equations (10-22) and (10-23) are compared to the expected value of interfacial

capacitance in Figures 10-17(a) and 10-17(b), respectively. Figures 10-17(a) and 10-17(b)

were developed using local frequeon.-~ Ii-dependent values of a~ and Qeaf over the same

frequency range as is reported in Figures 10-15 and 10-16. The frequencies reported

in Figure 10-17 are limited to those that are one decade larger than the characteristic

frequency wmax. The error in equation (10-22) is a function of both frequency K( and

L.V
-I'
C,,=[L)R '~''1
C4

1.0

o J=0.01 o J=1.0
0.5C O J=002 4 J=3.0
J=0.1 ~ J=10.0
~ J-03

on

1.5

a J=0 01 J=1 0.

0.1 1 10 100

O

0.5

o.o

O
O

0.1 1

10 100

Nu ll In .1. .1I effective capacitance calculated from relationships presented by
Brug et al. [30] for a disk electrode as a function of dimensionless frequency
K( with J as a parameter, a) with correction for Ohmic resistance Re
(equation (10-22)), and b) with correction for both Ohmic resistance Re and
charge-transfer resistance Rt (equation (10-23)).

Figure 10-17.

J. The dependence on J is reduced significantly when both the Ohmic resistance Re

and charge-transfer resistance Rt are taken into account, and the errors in estimating

interfacial capacitance are less than 20 percent. Of the relationships tested, equation

(10-23) provides the best means for estimating interfacial capacitance when frequency

dispersion is significant. The capacitance analysis presented here shows that, for

determining interfacial capacitance, the influence of current and potential distributions on

the impedance response cannot be neglected, even if the apparent CPE exponent a~ has

values close to unity.

CHAPTER 11
CONCLUSION AND RECOMMENDATION

This dissertation covers two research topics that are important to corrosion of metal.

The conclusion associated with the delamination model is presented in Section 11.1, and

that associated with the impedance calculation is presented in Section 11.2.

11.1 Mathematical Models for Cathodic Delamination of Coated Metal

A one-dimensional, transient mathematical model was developed that simulates the

delamination of polymeric coating from a zinc surface. The model included simultaneously

multiple electrochemical reactions, homogeneous reactions, and formation of corrosion

products. The calculation results are in agreement with the experimental observations

reported by Stratmann et al. [9-11, 14-16] for coated steel and coated zinc. The

consistency with experimental observations supports the hypotheses proposed by Allahar

that the porosity and polarization kinetics can he treated as functions of pH.

The simulated results obtained using the equilibrium E-pH relationship demonstrate

that the overall delamination process is preliminary governed by the transport of the

nations from the defect to the front region. The rate of the delamination depends on the

mobility and the concentration of the nations. The anions, on the other hand, have no

significant influence on the delamination rate.

The computational results obtained using a non-equilibrium E-pH relationship

indicate that, when the bond-breaking reactions take place at a sufficiently slow rate,

the potential front and the porosity front become distinguishable. The movement of the

potential front follows the change of pH along the metal-coating interface; whereas, the

movement of the porosity front is limited by the bond-breaking reactions. The kinetic

analysis of the non-equilibrium results also shows that the delamination mechanism

shifts from a mass-controlled mechanism to a mixed controlled mechanism when the

bond-braking reactions are sufficiently slow.

The mathematical model presented here provides a framework for advanced models

in which complex parameters, such as coating property and surface treatments, can be

included. The expansion of the one-dimensional model to a two-dimensional delaminated

zone is recommended in the future. The influence of the expansion on delamination rate or

delamination kinetics will be interesting.

A coupling of a two-dimensional defect with the delaminated zone is also recommended

for a more sophisticated model. This combination will relax the boundary conditions at

the location shared by the defect and the delaminated zone. The coupling will also

account explicitly for the galvanic couple formed between anodic and cathodic sites. The

development of the jointed model might be difficult because of the geometry of the system

and the complex phenomena in the domain. A commercial program, such as COMSOL

Multiphysics, is recommended for the development.

11.2 Influence of Geometry-Induced Current And Potential Distribution of
Disk Electrodes on Impedance Response

The results presented from Chapters 8 to 10 have shown that the geometry-induced

current and potential distributions induce a high-frequency dispersion that distorts the

impedance response of a disk electrode. In all electrochemical systems under study, the

local interfacial impedance exhibits the expected ideal behavior through all frequencies.

The local impedance only shows ideal behavior at low frequency but non-ideal behavior

at high frequency. The inductive loops observed in local impedance is influenced by local

Ohmic impedance, which has imaginary component about the dimensionless frequency

K(=1. The complex value of the local Ohmic impedance is the origin of the inductive

features in the local impedance and the origin of the CPE-like behavior in the global

impedance.

The calculated results presented in (I Ilpters 8 to 10 are compared with literature

and experiments. For the ideally blocking electrode and electrode exhibiting Faradaic

reactions, the calculated global impedance is in excellent agreement with those obtained

by --n 1,The high-frequency dispersion seen in the global impedance has an

appearance of a constant-phase element (CPE), but it can be considered to be only a

quasi-CPE because the CPE exponent a~ is not independent of frequency. The impedance

experiments performed on a glassy carbon disk and stainless steel disk electrode exhibit

good consistency with the calculation results for a blocking electrode subject to coupled

2-D and 3-D distributions. The characteristic transition frequency at which the geometry

pl .i-4 a role is within experimental range. This geometry effect, however, could possibly be

avoided by changing the size of the electrode or the concentration of the electrolyte.

The work explores the role of the current and potential distributions associated with

a disk electrode on both local and global impedance. This is also the first work to express

the geometric effect on impedance response in terms of CPE. The calculated results

illustrate that the use of local impedance spectroscopy is able to distinguish CPE behavior

that has an origin with a 3-D distribution from one that arises from a 2-D distribution.

The electrochemical systems investigated so far did not consider the influence of

convective diffusion that enhances mass transfer in electrochemical systems. It will be

interesting to explore the effect of current and potential distributions on the impedance

response of a disk electrode in the presence of mass-transfer effects. 1Voreover, multiple

heterogeneous reactions can be included in the calculation in the future. Surface coverage

might need to be incorporated when multiple electrochemical reactions take place

simultaneously. It will be interesting to investigate the relation between global and

local impedances when the surface coverage plI li-< a role in the calculation.

APPENDIX A
PROGRAM LISTING FOR THE CATHODIC DELAMINATION

This appendix presents the program listing for the cathodic delamination model. The

program was developed using M~icrosoft V/isual Fortran, V/ersion 9.0 with double precision

accuracy.

A.1 Main Program Listing
implicit double precision(a-h,o-z)
implicit integer(i-n)
common/ca/height,F,frt,vapp,tbetazn,excurz~hc~e~hcns,
& total_time,t_step,z(9),diff(9),fzd(9),curlmhh201mmatk
common/cb/a(11,11),b(11,11),c(11,2001),d112)g1)x11)y(11)nn
common/cc/ii,conc(9,2001),phi(2001),por(201,~h(01,
& dd_phi(200 1),diff_phi(2001),d_c(9,2001),dc9201,
& d_flux(9,2001),flux(9,2001),curzn(2001),cr(01,
& c_aver(9,2001),g_eq(9,2001),c_ini(9,2001 )da201
common/cd/del_1en,fro_1en,sem_1en,tac_1elehelcndl~~~e,
& c_ohfro,c_nafro,c_clfro,c_ohsem,c_nasem, c~~e~~hn,
& c_naint,c_clint,jdel,jfro,jsem,bl_pro,b2_pobr,
& b4_pro,b5_pro,b6_pro,b7_pro,bl_sur,b2_su r~3srbu,
& b5_sur,b6_sur,b7_sur,bl_blo,b2_blo,b3_bl o~4bobl,
& b6_blo,b7_blo,b8_blo,bl_poi,b2_poi,b3_po i~4pibo,
& b6_poi,b7_poi,ph(2001),por_ini(2001),aa( 201,u~o(01,
& bb(2001),block(200 1),poi(2001),b8,b9,b10 b1
common/ce/flux_mig(9,2001),flux_dif(9,200)cre(01dpo20),
& d_v(2001),total_current

!Read input data and discretize delamination zone
call input
!Initialize the concentration distributions
call initial(j)
call setup(j)
!Time-stepping routine
call cal_conc(j)

stop
end
A.2 Subroutine Program Listing
subroutinee that reads input data from input file
subroutine input
implicit double precision(a-h,o-z)
implicit integer(i-n)
common/ca/height,F,frt,vapp,tbetazn,excurz~hc~e~hcns,

& total_time,t_step,rate_k,z_oh,z_na,z_cl, z~nzhzzo,
& z_hzno2,z_zno2,z_znoh2,diff_oh,diff_na,d ifcdifz,
& diff_h,diff_znoh,diff_hzno2,diff_zno2,diffzo2z9,
& diff(9),fzd(9),curolim,h,hh(2001),mn
common/cb/a(11,11),b(11,11),c(11,2001),d112)g1)x11),

character name*40
open(unit=101,file="'input_cd_2.txt",st ats'nnw'
rewind 101

110 format(a20,i6)
120 format(a20,fi2.10)

# of variables
# of mesh points
length cm
F/(RT) J/mole
metal potential V
Tafel slope of Zn dissolution
exchange current density of Zn
gel thickness
coating thickness
total time
time step
rate constant of forming Zn(0h)2
charge number for OH-
charge number for Na+
charge number for Cl-
charge number for Zn2+
charge number for H+
charge number for ZnOH+
charge number for HZnO2-
charge number for ZnO22-
charge number for Zn(0H)2
charge number for OH-
charge number for Na+
charge number for Cl-
charge number for Zn2+
charge number for H+
charge number for ZnOH+
charge number for HZnO2-
charge number for ZnO22-
charge number for Zn(0H)2

130 format(a20,fi6.6)

!Limiting current density of oxygen reduction in a
!metal-electrolyte medium
curolim=4*F*(1.9d-5)*(1.26d-6)
!Calculate grid size
h=height/(nj-1)
!Calculate number of time loops
mm=total_t ime/t_step

do i= 1,n
fzd(i)= frt*z(i)*diff(i)
enddo
do j=1,nj
hh(j+1)=hh(j)+h
enddo
return
end

subroutinee that calculates initial concentration and potential
!distributions
subroutine initial(j)
implicit double precision(a-h,o-z)
implicit integer(i-n)
common/cb/a(11,11),b(11,11),c(11,2001),d112)g1)x11),
& y(11)nn
common/cd/del_1en,fro_1en,sem_1en,tac_1elehelcndl~~~e,
& c_ohfro,c_nafro,c_clfro,c_ohsem,c_nasem, c~~e~~hn,
& c_naint,c_clint,jdel,jfro,jsem,bl_pro,b2 _pobr,
& b4_pro,b5_pro,b6_pro,b7_pro,bl_sur,b2_su r~3srbu,
& b5_sur,b6_sur,b7_sur,bl_blo,b2_blo,b3_bl o~4bobl,
& b6_blo,b7_blo,b8_blo,bl_poi,b2_poi,b3_po i~4pibo,
& b6_poi,b7_poi,ph(2001),por_ini(2001),aa( 201,u~o(01,
& bb(2001),block(200 1),poi(2001),b8,b9,b10b1

character name*40
open(unit=20,file="initial_cd_0.01M.txt ",tts'nnw'
rewind 20

open(unit=35,file='c_ini.text')
open(unit=45,file='parameters_ini.text')

length fully-intact region

c_0H
cNa
cC1
c_0H
cNa
cC1
cOH
cNa
cOH
c_0H
cNa
cC1

the
the
the
the
the
the
the
the
the
the
the
the

delaminated zone
delaminated zone
delaminated zone
front region
front region
front region
sem-intact region
sem-intact region
sem-intact region
fully-intact region
fully-intact region
fully-intact region

for
for
for
for
for
for
for
for
for
for
for
for
for
for
for

porosity
porosity
porosity
porosity
blocking factor
blocking factor
blocking factor
blocking factor
blocking factor
blocking factor
blocking factor
poisoning factor
poisoning factor
poisoning factor
poisoning factor

parameter
parameter
parameter
parameter
parameter
parameter
parameter
parameter
parameter
parameter
parameter
parameter
parameter
parameter
parameter

210 format(a20,fi6.6)
411 format(1x,i4,1x,ei3.5,5e13.5)
412 format(1x,i4,ei3.5,1x,4e13.5)
413 format(1x,a4,6x,a5,7x,a7,3(6x,a7),8x,a3)
414 format(1x,a4,6x,a5,7x,a7,4x,a10,3x,a9,5x a7

jdel=(nj-1)/4
jfro=(nj-1)*5/32
jsem=(nj-1)*3/16

!set length in the delaminated region
!set length in the front
!set length points in the semi-intact region

cc=1/b4_blo
b8= exp(b2_pro*(17-b3_pro))
b9= b2_pro/2.3026d0
bl0=exp(b5_pro*(17-b6_pro))
bl1=b5_pro/2.3026d0

!set nodal points in the domain
do j=1,nj
if(j.1e.jdel) call cini_del(j)
if((j.gt.jdel).and.(j.1e.jfro)) call cini_fro(j)
if((j.gt.jfro).and.(j.1e.jsem)) call cini_sem(j)
if(j.gt.jsem) call cini_int(j)
enddo

do j=1,nj
!Calculate assumed initial pH distribution
ph(j)= -log10(1.0d-17/c_ini(1,j))
!Calculate assumed initial porosity distribution
por_ini (j)=bl_pro/(1+exp (b2_pro* (ph(j)-b3_pro)))+&
& b4_pro/(1+exp(b5_pro*(ph(j)-b6_pro)))+b 7_r
aa(j)= por_ini(j)**1.5
!Calculate assumed initial blocking factor distribution
bb(j)=bl_blowexp(-b2_blo*(ph(j)-b3_blo))
block(j)=((bb(j)/(1+bb(j)))+b7_blo)*&
& ((1/(cc+exp(-b5_blo*(ph(j)-b 6_blo))))+b8 _bo
!Calculate assumed initial poisoning factor distribution
poi(j)=bl_poi/(1+exp(b2_poi*(ph(j)-b3_poi)+
& b4_poi/(1+exp(b5_poi*(ph(j)-b6_poi)))+b 7_o
enddo
return
end

subroutine cini_del(j)
implicit double precision(a-h,o-z)
implicit integer(i-n)
common/cc/ii,conc(9,2001),phi(2001),por(201,~h(01,
& dd_phi(200 1),diff_phi(2001),d_c(9,2001),dc9201,
& d_flux(9,2001),flux(9,2001),curzn(2001),cr(01,
& c_aver(9,2001),g_eq(9,2001),c_ini(9,2001 )da201
common/cd/del_1en,fro_1en,sem_1en,tac_1elehelcndl~~~e,
& c_ohfro,c_nafro,c_clfro,c_ohsem,c_nasem, c~~e~~hn,
& c_naint,c_clint,jdel,jfro,jsem,bl_pro,b2_pobr,
& b4_pro,b5_pro,b6_pro,b7_pro,bl_sur,b2_su r~3srbu,
& b5_sur,b6_sur,b7_sur,bl_blo,b2_blo,b3_bl o~4bobl,
& b6_blo,b7_blo,b8_blo,bl_poi,b2_poi,b3_po i~4pibo,
& b6_poi,b7_poi,ph(2001),por_ini(2001),aa( 201,u~o(01,
& bb(2001),block(200 1),poi(2001),b8,b9,b10b1
!OH-
c_ini(1,j)= c_0hdel+(c_ohfro-c_ohdel)*(j-1)*h/del_1en
!Na+

!Cl-
c_ini(3,j)= c_cldel+(c_clfro-c_c~del)*(j-l1)h/del_1en
!Zn2+
c_ini(4,j)=0.5*(c_ini(1,j)-c_ini(2,j)+c _in(,)
!H+
c_ini(5,j)= 0.0d0
!ZnOH+
c_ini(6,j)= 0.0d0
!HZnO2-
c_ini(7,j)= 0.0d0
!ZnO22-
c_ini(8,j)= 0.0d0
!Zn(OH)2
c_ini(9,j)= 0.0d0

return
end

subroutine cini_fro(j)
implicit double precision(a-h,o-z)
implicit integer(i-n)
common/cc/ii,conc(9,2001),phi(2001),por(201,~h(01,
& dd_phi(200 1),diff_phi(2001),d_c(9,2001),dc9201,
& d_flux(9,2001),flux(9,2001),curzn(2001),cr(01,
& c_aver(9,2001),g_eq(9,2001),c_ini(9,2001 )da201
common/cd/del_1en,fro_1en,sem_1en,tac_1elehelcndl~~~e,
& c_ohfro,c_nafro,c_clfro,c_ohsem,c_nasem, c~~e~~hn,
& c_naint,c_clint,jdel,jfro,jsem,bl_pro,b2_pobr,
& b4_pro,b5_pro,b6_pro,b7_pro,bl_sur,b2_su r~3srbu,
& b5_sur,b6_sur,b7_sur,bl_blo,b2_blo,b3_bl o~4bobl,
& b6_blo,b7_blo,b8_blo,bl_poi,b2_poi,b3_po i~4pibo,
& b6_poi,b7_poi,ph(2001),por_ini(2001),aa( 201,u~o(01,
& bb(2001),block(200 1),poi(2001),b8,b9,b10b1
!OH-
c_ini(1,j)= c_ohfrotexp((hh(j)-del_1en)*10g(c_ohsem/c~hr)fo1n
!Na+
c_ini(2,j)= c_nafrotexp((hh(j)-del_1en)*10g(c_nasem/c~ar)fo1n
!Cl-
c_ini(3,j)= c_clfrotexp((hh(j)-del_1en)*10g(c_clsem/c~lr)fo1n
!Zn2+
c_ini(4,j)=0.5*(c_ini(1,j)-c_ini(2,j)+c _in(,)
!H+
c_ini(5,j)= 0.0d0
!ZnOH+
c_ini(6,j)= 0.0d0

!HZnO2-
c_ini(7,j)= 0.0d0
!ZnO22-
c_ini(8,j)= 0.0d0
!Zn(OH)2
c_ini(9,j)= 0.0d0
return
end

subroutine cini_sem(j)
implicit double precision(a-h,o-z)
implicit integer(i-n)
common/cc/ii,conc(9,2001),phi(2001),por(201,~h(01,
& dd_phi(200 1),diff_phi(2001),d_c(9,2001),dc9201,
& d_flux(9,2001),flux(9,2001),curzn(2001),cr(01,
& c_aver(9,2001),g_eq(9,2001),c_ini(9,2001 )da201
common/cd/del_1en,fro_1en,sem_1en,tac_1elehelcndl~~~e,
& c_ohfro,c_nafro,c_clfro,c_ohsem,c_nasem, c~~e~~hn,
& c_naint,c_clint,jdel,jfro,jsem,bl_pro,b2_pobr,
& b4_pro,b5_pro,b6_pro,b7_pro,bl_sur,b2_su r~3srbu,
& b5_sur,b6_sur,b7_sur,bl_blo,b2_blo,b3_bl o~4bobl,
& b6_blo,b7_blo,b8_blo,bl_poi,b2_poi,b3_po i~4pibo,
& b6_poi,b7_poi,ph(2001),por_ini(2001),aa( 201,u~o(01,
& bb(2001),block(200 1),poi(2001),b8,b9,b10b1
!OH-
c_ini(1,j)= c_ohsem*exp((hh(j)-del_1en-fro_1en)*&
& 10g(c_ohint/c_ohsem)/sem_1en)
!Na+
c_ini(2,j)= c_nasem*exp((hh(j)-del_1en-fro_1en)*&
& log(c_naint/c_nasem)/sem_1en)
!Cl-
c_ini(3,j)= c_clsem*exp((hh(j)-del_1en-fro_1en)*&
& 10g(c_clint/c_cls)sem)/e_1en)
!Zn2+
c_ini(4,j)=0.5*(c_ini(1,j)-c_ini(2,j)+c _in(,)
!H+
c_ini(5,j)= 0.0d0
!ZnOH+
c_ini(6,j)= 0.0d0
!HZnO2-
c_ini(7,j)= 0.0d0
!ZnO22-
c_ini(8,j)= 0.0d0
!Zn(OH)2
c_ini(9,j)= 0.0d0

return
end

subroutine cini_int(j)
implicit double precision(a-h,o-z)
implicit integer(i-n)
common/cc/ii,conc(9,2001),phi(2001),por(201,~h(01,
& dd_phi(200 1),diff_phi(2001),d_c(9,2001),dc9201,
& d_flux(9,2001),flux(9,2001),curzn(2001),cr(01,
& c_aver(9,2001),g_eq(9,2001),c_ini(9,2001 )da201
common/cd/del_1en,fro_1en,sem_1en,tac_1elehelcndl~~~e,
& c_ohfro,c_nafro,c_clfro,c_ohsem,c_nasem, c~~e~~hn,
& c_naint,c_clint,jdel,jfro,jsem,bl_pro,b2_pobr,
& b4_pro,b5_pro,b6_pro,b7_pro,bl_sur,b2_su r~3srbu,
& b5_sur,b6_sur,b7_sur,bl_blo,b2_blo,b3_bl o~4bobl,
& b6_blo,b7_blo,b8_blo,bl_poi,b2_poi,b3_po i~4pibo,
& b6_poi,b7_poi,ph(2001),por_ini(2001),aa( 201,u~o(01,
& bb(2001),block(200 1),poi(2001),b8,b9,b10b1
!OH-
c_ini(1,j)= c_0hint
!Na+
c_ini(2,j)= c_naint
!Cl-
c_ini(3,j)= c_clint
!Zn2+
c_ini(4,j)=0.5*(c_ini(1,j)-c_ini(2,j)+c _in(,)
!H+
c_ini(5,j)= 0.0d0
!ZnOH+
c_ini(6,j)= 0.0d0
!HZnO2-
c_ini(7,j)= 0.0d0
!ZnO22-
c_ini(8,j)= 0.0d0
!Zn(OH)2
c_ini(9,j)= 0.0d0
return
end

subroutinee that records initial guesses
subroutine setup(j)
implicit double precision(a-h,o-z)
implicit integer(i-n)
common/cc/ii,conc(9,2001),phi(2001),por(201,~h(01,
& dd_phi(200 1),diff_phi(2001),d_c(9,2001),dc9201,

& d_flux(9,2001),flux(9,2001),curzn(2001),cr(01,
& c_aver(9,2001),g_eq(9,2001),c_ini(9,2001 )da201
common/cd/del_1en,fro_1en,sem_1en,tac_1elehelcndl~~~e,
& c_ohfro,c_nafro,c_clfro,c_ohsem,c_nasem, c~~e~~hn,
& c_naint,c_clint,jdel,jfro,jsem,bl_pro,b2 _pobr,
& b4_pro,b5_pro,b6_pro,b7_pro,bl_sur,b2_su r~3srbu,
& b5_sur,b6_sur,b7_sur,bl_blo,b2_blo,b3_bl o~4bobl,
& b6_blo,b7_blo,b8_blo,bl_poi,b2_poi,b3_po i~4pibo,
& b6_poi,b7_poi,ph(2001),por_ini(2001),aa( 201,u~o(01,
& bb(2001),block(200 1),poi(2001),b8,b9,b10b1

do j=1,nj
phi(j)=-0.05
conc(1,j)=1.0d-4
conc(2,j)=1.0d-6
conc(3,j)=1.0d-6
conc(4,j)=0.5d-4
conc(5,j)=1.0d-16
conc(6,j)=1.0d-20
conc(7,j)=1.0d-25
conc(8,j)=1.0d-30
conc(9,j)=1.0d-20
por(j)=0.1d0
enddo
return
end

subroutinee for calculating conc. and solution potential
subroutine cal_conc(j)
implicit double precision(a-h,o-z)
implicit integer(i-n)
common/cb/a(11,11),b(11,11),c(11,2001),d112)g1)x11),
& y(11)nn
common/ca/height,F,frt,vapp,tbetazn,excurz~hc~e~hcns,
& total_time,t_step,rate_k,z_oh,z_na,z_cl, z~nzhzzo,
& z_hzno2,z_zno2,z_znoh2,diff_oh,diff_na,d ifcdifz,
& diff_h,diff_znoh,diff_hzno2,diff_zno2,diffzo2z9,
& diff(9),fzd(9),curolim,h,hh(2001),mn
common/cc/ii,conc(9,2001),phi(2001),por(201,~h(01,
& dd_phi(200 1),diff_phi(2001),d_c(9,2001),dc9201,
& d_flux(9,2001),flux(9,2001),curzn(2001),cr(01,
& c_aver(9,2001),g_eq(9,2001),c_ini(9,2001 )da201
common/cd/del_1en,fro_1en,sem_1en,tac_1elehelcndl~~~e,
& c_ohfro,c_nafro,c_clfro,c_ohsem,c_nasem, c~~e~~hn,
& c_naint,c_clint,jdel,jfro,jsem,bl_pro,b2 _pobr,

& b4_pro,b5_pro,b6_pro,b7_pro,bl_sur,b2_su r~3srbu,
& b5_sur,b6_sur,b7_sur,bl_blo,b2_blo,b3_bl o~4bobl,
& b6_blo,b7_blo,b8_blo,bl_poi,b2_poi,b3_po i~4pibo,
& b6_poi,b7_poi,ph(2001),por_ini(2001),aa( 201,u~o(01,
& bb(2001),block(200 1),poi(2001),b8,b9,b10b1

open(unit=102,file='gi.txt')
200 format(11(el0.2))
300 format(/,a7,i4)

do ii=1,mm
do i=1,n
do k =1,n
x(i,k)=0.0
y(i,k)=0.0
enddo
enddo

j=0
do 1=1,40
if (ii.eq.mm) write(102,300) "jcount=", 1
50 j=j+1
do i=1,n
g(i)=0.0d0
do k=1,n
a(i,k)=0.0d0
b(i,k)=0.0d0
d(i,k)=0.0d0
enddo
enddo

if(j.eq.1) call bc_1(j)
if((ii.eq.mm).and.(j.eq.1)) write(102,200) (g(k),k=1,n)
if((j.gt.1).and.(j.1t.nj)) call body(j)
if((ii.eq.mm).and.(j.eq.2)) write(102,200) (g(k),k=1,n)
if((ii.eq.mm).and.(j.eq.(nj-1)/8)) write(102,200) (g(k),k=1,n)
if((ii.eq.mm).and.(j.eq.(nj-1)/4)) write(102,200) (g(k),k=1,n)
if((ii.eq.mm).and.(j.eq.nj-1)) write(102,200) (g(k),k=1,n)
if(j.eq~nj) call bc_2(j)
if((ii.eq.mm).and.(j.eq.nj)) write(102,200) (g(k),k=1,n)
call band(j)

if (j.ne.nj) go to 50

do 100 j=1,nj

!Set boundaries for values of concentrations and solution
!potential
if (c(1,j).1t.-0.5) c(1,j)=-0.50d0
if (c(1,j).gt.0.5) c(1,j)=0.50d0
if (c(2,j).1t.(-0.999*coconc(1,)) c(2,j)=-0.999*coi
if (c(2,j).gt.(1000.0+conc(1,j))) c(2,j)=1000.0*coi
if (c(3,j).1t.(-0.999*conc(2,j))) c(3,j)=-0.999*coi
if (c(3,j).gt.(1000.0+conc(2,j))) c(3,j)=1000.0*coi
if (c(4,j).1t.(-0.999*conc(3,j))) c(4,j)=-0.999*coi
if (c(4,j).gt.(1000.0+conc(3,j))) c(4,j)=1000.0*coi
if (c(5,j).1t.(-0.999*conc(4,j))) c(5,j)=-0.999*coi
if (c(5,j).gt.(1000.0+conc(4,j))) c(5,j)=1000.0*coi
if (c(6,j).1t.-0.5) c(6,j)=-0.50d0
if (c(6,j).gt.0.5) c(6,j)=0.50d0
if (c(7,j).1t.(-0.999*conc(5,j))) c(7,j)=-0.999*cco
if (c(7,j).gt.(1000.0+conc(5,j))) c(7,j)=1000.0*coi
if (c(8,j).1t.(-0.999*conc(6,j))) c(8,j)=-0.999*coi
if (c(8,j).gt.(1000.0+conc(6,j))) c(8,j)=1000.0*coi
if (c(9,j).1t.(-0.999*conc(7,j))) c(9,j)=-0.999*cco
if (c(9,j).gt.(1000.0+conc(7,j))) c(9,j)=1000.0*coi
if(c(10,j).1t.(-0.999*oconc(8,j)) c(10,j)=-0.999*cl
if(c(10,j).gt.(1000.0+conc(8,j))) c(10,j)=1000.0*cl
if(c(11,j).1t.(-0.999*conc(9,j))) c(11,j)=-0.999*cl
if(c(11,j).gt.(1000.0+conc(9,j))) c(11,j)=1000.0*cl
!update new values
phi(j) = phi(j)+c(1,j)

nc(1,j)
nc(1,j)
nc(2,j)
nc(2,j)
nc(3,j)
nc(3,j)
nc(4,j)
nc(4,j)

nc(5,j)
nc(5,j)
nc(6,j)
nc(6,j)
nc(7i,j)
nc(7i,j)
onc(8,j)
onc(8,j)
onc(9,j)
onc(9,j)

conc(1,j)
conc(2,j)
conc(3,j)
conc(4,j)
por (j)
conc(5,j)
conc(6,j)
conc(7,j)
conc(8,j)
conc(9,j)
100 continue

conc(1,j)+c(2,j)
conc(2,j)+c(3,j)
conc(3,j)+c(4,j)
conc(4,j)+c(5,j)
por(j)+c(6,j)
conc(5,j)+c(7,j)
conc(6,j)+c(8,j)
conc(7,j)+c(9,j)
conc(8,j)+c(10,j)
conc(9,j)+c(11,j)

call cal_porosity(j)

if(1.eq.40) then
call cal_flux(j)
call results(j)
endif

!update the hypothesized parameters

!calculate flux and current values
!output files

j=0

enddo
do i=1,n-2
do j=1,nj
c_ini(i,j)=conc(i,j)
por_ini(j)=por(j)
enddo
enddo

enddo
return
end

subroutinee of BAND(J) algorithm
SUBROUTINE BAND(J)
IMPLICIT DOUBLE PRECISION(A-H,0-Z)
IMPLICIT INTEGER (I-N)
DOUBLE PRECISION E (11, 12, 2001)
common/cb/a(11, 11) ,b (11, 11) c(11, 2001) ,d (1 3 1),x(1 1

SAVE E,NP1
101 FORMAT (/15H DETERM=0 AT J=, I4)
IF((J-2).LT.0) GO TO 1
IF((J-2).EQ.0) GO TO 6
IF((J-2).GT.0) GO TO 8
1 NP1= N + 1
DO 2 I=1,N
D(I,2*N+1)= G(I)
DO 2 L=1,N
LPN= L + N
2 D(I,LPN)= X(I,L)
CALL MATINV (N, 2*N+1, DETERM)
IF (DETERM) 4, 3, 4
3 PRINT 101, J
4 DO 5 K=1,N
E(K,NP1,1)= D(K,2*N+1)
DO 5 L=1,N
E(K,L,1)= D(K,L)
LPN= L + N
5 X(K,L)= D(K,LPN)
RETURN
6 DO 7 I=1,N
DO 7 K=1,N
DO 7 L=1,N
7 D(I,K)= D(I,K) + A(I,L)*X(L,K)

8 IF (J-NJ) 11,9,9
9 DO 10 I=1,N
DO 10 L=1,N
G(I)= G(I) Y(I,L)*E(L,NP1,J-2)
DO 10 M=1,N
10 A(I,L)= A(I,L) + Y(I,M)*E(M,L,J-2)
11 DO 12 I=1,N
D(I,NP1)= G(I)
DO 12 L=1,N
D(I,NP1)= D(I,NP1) + A(I,L)*E(L,NP1,J-1)
DO 12 K=1,N
12 B(I,K)= B(I,K) + A(I,L)*E(L,K,J-1)
CALL MATINV (N,NP1,DETERM)
IF (DETERM) 14,13,14
13 PRINT 101, J
14 DO 15 K=1,N
DO 15 M=1,NP1
15 E(K,M,J)= D(K,M)
IF (J-NJ) 20,16,16
16 DO 17 K=1,N
17 C(K,J)= E(K,NP1,J)
DO 18 JJ=2,NJ
M= NJ JJ + 1
DO 18 K=1,N
C(K,M)= E(K,NP1,M)
DO 18 L=1,N
18 C(K,M)= C(K,M) + E(K,L,M)*C(L,M+1)
DO 19 L=1,N
DO 19 K=1,N
19 C(K,1)= C(K,1) + X(K,L)*C(L,3)
20 RETURN
END
! *********************** MAtrix inverse********************

SUBROUTINE MATINV(N,M,DETERM)
IMPLICIT DOUBLE PRECISION(A-H,0-Z)
IMPLICIT INTEGER (I-N)
common/cb/a(11,11),b(11,11),c(11,2001),d(1,3

INTEGER ID(6)
DETERM=1.0d0
DO 1 I=1,N
1 ID(I)=0
DO 18 NN=1,N
BMAX=1.1

DO 6 I=1,N
IF(ID(I).NE.0) GO TO 6
BNEXT=0.0
BTRY=0.0
DO 5 J=1,N
IF(ID(J).NE.0) GO TO 5
IF(ABS(B(I,J)).LE.BNEXT) GO TO 5
BNEXT=ABS(B(I,J))
IF(BNEXT.LE.BTRY) GO TO 5
BNEXT=BTRY
BTRY=ABS(B(I,J))
JC=J
5 CONTINUE
IF(BNEXT.GE.BMAX*BTRY) GO TO 6
BMAX=BNEXT/BTRY
IROW=I
JCOL=JC
6 CONTINUE
IF(ID(JC).EQ.0) GO TO 8
DETERM=0.0
RETURN
8 ID(JCOL)=1
IF(JCOL.EQ.IROW) GO TO 12
DO 10 J=1,N
SAVE=B(IROW,J)
B(IROW,J)=B(JCOL,J)
10 B(JCOL,J)=SAVE
DO 11 K=1,M
SAVE=D(IROW,K)
D(IROW,K)=D(JCOL,K)
11 D(JCOL,K)=SAVE
12 F=1.0/B(JCOL,JCOL)
DO 13 J=1,N
13 B(JCOL,J)=B(JCOL,J)*F
DO 14 K=1,M
14 D(JCOL,K)=D(JCOL,K)*F
DO 18 I=1,N
IF(I.EQ.JCOL) GO TO 18
F=B(I,JCOL)
DO 16 J=1,N
16 B(I,J)=B(I,J)-F*B(JCOL,J)
DO 17 K=1,M
17 D(I,K)=D(I,K)-F*D(JCOL,K)
18 CONTINUE
RETURN

END

subroutinee for the boundary sharing with defect
subroutine bc_1(j)
implicit double precision(a-h,o-z)
implicit integer(i-n)
common/ca/height,F,frt,vapp,tbetazn,excurz~hc~e~hcns,
& total_time,t_step,rate_k,z_oh,z_na,z_cl, z~nzhzzo,
& z_hzno2,z_zno2,z_znoh2,diff_oh,diff_na,d ifcdifz,
& diff_h,diff_znoh,diff_hzno2,diff_zno2,diffzo2z9,
& diff(9),fzd(9),curolim,h,hh(2001),mn
common/cb/a(11,11),b(11,11),c(11,2001),d112)g1)x11),
& y(11)nn
common/cc/ii,conc(9,2001),phi(2001),por(201,~h(01,
& dd_phi(200 1),diff_phi(2001),d_c(9,2001),dc9201,
& d_flux(9,2001),flux(9,2001),curzn(2001),cr(01,
& c_aver(9,2001),g_eq(9,2001),c_ini(9,2001 )da201
common/cd/del_1en,fro_1en,sem_1en,tac_1elehelcndl~~~e,
& c_ohfro,c_nafro,c_clfro,c_ohsem,c_nasem, c~~e~~hn,
& c_naint,c_clint,jdel,jfro,jsem,bl_pro,b2 _pobr,
& b4_pro,b5_pro,b6_pro,b7_pro,bl_sur,b2_su r~3srbu,
& b5_sur,b6_sur,b7_sur,bl_blo,b2_blo,b3_bl o~4bobl,
& b6_blo,b7_blo,b8_blo,bl_poi,b2_poi,b3_po i~4pibo,
& b6_poi,b7_poi,ph(2001),por_ini(2001),aa( 201,u~o(01,
& bb(2001),block(200 1),poi(2001),b8,b9,b10b1
common/ce/flux_mig(9,2001),flux_dif(9,200)cre(01dpo20),
& d_v(2001),total_current

ophi=0.402-tbetao*10gl0(excuro)
znphi=-0.763-tbetazn*10gl0(excurzn)
ffl=(por(j)*b7_pro)**1.5
ff2=(por(j)**1.5)*thickness+(b7_pro**1.5)*hc~e
ff=ffl/ff2
diff_phi(j)=vapp-phi(j)
!current density due to zinc dissolution
curzn (j )= (10* *( (vapp-phi (j )-znphi) /tbet azn) ) *sur_c ov (j )*po i(j )
!current density due to oxygen reduction
curo(j)=-curolim*sur_cVj*lcov(j)*blckj)f

dd=bl_pro*b8*b9
ee=b4_pro*bl0*bl1
!solution potential
g(1)= phi(j)+0.1
b(1,1)= -1.0d0
!OH-

g(2)= conc(1,j)-1.0d-3
b(2,2)= -1.0d0
!Na+
g(3)= conc(2,j)-1.0d-3
b(3,3)= -1.0d0
!Cl-
g(4)= conc(3,j)-5.0d-4
b(4,4)= -1.0d0
!Zn2+
g(5)= conc(4,j)-2.5d-4
b(5,5)= -1.0d0
!porosity
g(6)= por(j)-bl_pro/(1+b8*(conc(1,j)**b9))-&
& b4_pro/(1+bl0*(conc(1,j)**bl1))-b7_pro
b(6,2)=-dd*(conc(1,j)**(b9-1))/((1+b8t*oc(,)*b))*)
& -ee*(conc(1,j)**(bl1-1))/(((1+b10(conc(1,)l1)*2
b(6,6)= -1.0d0
!H+ H20 --> OH+ +OH-
g(7)=conc(1,j)*conc(5,j)-1.0d-20
b(7,2)= -conc(5,j)
b(7,7)= -conc(1,j)
!ZnOH+ Zn2+ + OH- --> ZnOH+
g(8)= conc(6,j)-(10.0**1.33)*conc(1,j)*conc(4,)
b(8,2)= (10.0**1.33)*conc(4,j)
b(8,5)= (10.0**1.33)*conc(1,j)
b(8,8)= -1.0d0
!HZnO2- ZnOH+ +20H- --> HZnO2- + H20
g(9)= conc(7,j)-(10.0**4.03)*conc(6,j)*(conc(1,)*2
b(9,2)= 2*(10.0**4.03)*conc(6,j)*conc(1,j)
b(9,8)= (10.0**4.03)*(conc(1,j)**2)
b(9,9)= -1.0d0
!ZnO22- HZnO2- + OH- --> ZnO22- + H20
g(10)= conc(8,j)-(10.0**(-2.17))*conc(1,j)*conc(7j
b(10,2)= (10.0**(-2.17))*conc(7,j)
b(10,9)= (10.0**(-2.17))*conc(1,j)
b(10,10)= -1.0d0
!Zn(OH)2 Zn2+ + 20H- --> Zn(0H)2
g(11)= (por(j)*conc(9,j)-por_ini(j)*c_ini(9,j)) /tse-
& rate_k*(conc(1,j)*conc(1,j)*conc(4,j)-3.0-6
b(11,2)= 2*rate_k*conc(1,j)*conc(4,j)
b(11,5)= rate_k*conc(1,j)*conc(1,j)
b(11,6)= -conc(9,j)/t_step
b(11,11)= -por(j)/t_step
return
end

subroutinee for non-boundary points
subroutine body(j)
implicit double precision(a-h,o-z)
implicit integer(i-n)
common/ca/height,F,frt,vapp,tbetazn,excurz~hc~e~hcns,
& total_time,t_step,rate_k,z_oh,z_na,z_cl, z~nzhzzo,
& z_hzno2,z_zno2,z_znoh2,diff_oh,diff_na,d ifcdifz,
& diff_h,diff_znoh,diff_hzno2,diff_zno2,diffzo2z9,
& diff(9),fzd(9),curolim,h,hh(2001),mn
common/cb/a(11,11),b(11,11),c(11,2001),d112)g1)x11),

common/cc/ii,conc(9,2001),phi(2001),por(201,~h(01,
& dd_phi(200 1),diff_phi(2001),d_c(9,2001),dc9201,
& d_flux(9,2001),flux(9,2001),curzn(2001),cr(01,
& c_aver(9,2001),g_eq(9,2001),c_ini(9,2001 )da201
common/cd/del_1en,fro_1en,sem_1en,tac_1elehelcndl~~~e,
& c_ohfro,c_nafro,c_clfro,c_ohsem,c_nasem, c~~e~~hn,
& c_naint,c_clint,jdel,jfro,jsem,bl_pro,b2_pobr,
& b4_pro,b5_pro,b6_pro,b7_pro,bl_sur,b2_su r~3srbu,
& b5_sur,b6_sur,b7_sur,bl_blo,b2_blo,b3_bl o~4bobl,
& b6_blo,b7_blo,b8_blo,bl_poi,b2_poi,b3_po i~4pibo,
& b6_poi,b7_poi,ph(2001),por_ini(2001),aa( 201,u~o(01,
& bb(2001),block(200 1),poi(2001),b8,b9,b10b1
common/ce/flux_mig(9,2001),flux_dif(9,200)cre(01dpo20),
& d_v(2001),total_current

dd=bl_pro*b8*b9
ee=b4_pro*bl0*bl1
d_aa(j)= (aa(j+1)-aa(j-1))/(2*h)
d_phi(j)= (phi(j+1)-phi(j-1))/(2*h)
dd_phi(j)= (phi(j+1)-2*phi(j)+phi(j-1))/(hth)
diff_phi(j)= vapp-phi(j)

do i=1,n-2
d_c(i,j)= (conc(i,j+1)-conc(i,j-1))/(2*h)
dd_c(i,j)= (conc(i,j+1)-2*conc(i,j)+conc(i,j-1))/(hth
d_flux(i,j)= (-fzd(i)*conc(i,j)*dd_phi(j)-fzd(i)*d_c(i)&
& d_phi(j)-diff(i)*dd_c(i,j))*aa(j)+&
& d_aa(j)*(-fzd(i)*conc(i,j)*d_phi(j)-diff~ i*~~~)
g_eq(i,j)= (conc(i,j)*por(j)-c_ini(i,j)*por_ini(j))/tse~~lxij
enddo

ophi=0.402-tbetao*10gl0(excuro)
znphi=-0.763-tbetazn*10gl0(excurzn)

ffl=(por(j)*b7_pro)**1.5
ff2=(por(j)**1.5)*thickness+(b7_pro**1.5)*hc~e
ff=ffl/ff2
!calculate current densities due to electrochemical reactions
curzn(j)=(10**((diff_phi(j)-znphi)/tbetaz)sucoj)pi)
curo (j)=-curol im*sur_Cov(j) *block(j)*ff
!solution potential(electroneutrality)
g(1)= z(1)*conc(1,j)+z(2)*conc(2,j)+z(3)*conc(3,)z4*oc4j+
& z(5)*conc(5,j)+z(6)*conc(6,j)+z(7)*conc( 7,)z8*oc8j
b(1,2)= -z(1)
b(1,3)= -z(2)
b(1,4)= -z(3)
b(1,5)= -z(4)
b(1,7)= -z(5)
b(1,8)= -z(6)
b(1,9)= -z(7)
b(1,10)= -z(8)
!OH-
g(2)=g_eq(1,j)+curo~j) /(F~t thickn-gess-~q5j-~q4j+
& curzn(j)/(2*F*thickness)+2*g_eq(7,j)+3*geq8j-
& (por(j)*conc(9,j)-por_ini(j)*c_ini(9,j))/tse

a(2,1)= (fzd(1)*conc(1,j)/(hth)-fzd(1)*d_c(1,j)/2h)aa)-
& d_aa(j)*fzd(1)*conc(1,j)/(2*h)-&
& (fzd(5)*conc(5,j)/(hth)-fzd(5)*d_c(5,j)/ (h)aa)+
& d_aa(j)*fzd(5)*conc(5,j)/(2*h)-&
& (fzd(4)*conc(4,j)/(hth)-fzd(4)*d_c(4,j)/ 2h)aa)+
& d_aa(j)*f zd(4)*c onc(4, j)/(2 *h) +&
& 2*(fzd(7)*conc(7,j)/(hth)-fzd(7)*d_c(7,j )2h)aa)-
& 2*d_aa(j)*fzd(7)*conc(7,j)/(2*h)+&
& 3*(fzd(8)*conc(8,j)/(hth)-fzd(8)*d_c(8,j)2h)aa)-
& 3*d_aa(j)*fzd(8)*conc(8,j)/(2*h)
b(2,1)=-2*fzd(1)*conc(1,j)*aa(j)/(hth)+&
& 2*fzd(5)*conc(5,j)*aa(j)/(hth)+&
& 2*fzd(4)*conc(4,j)*aa(j)/(hth)+&
& log(10.0)*curzn(j)/(2*F*tbetazn*thickness-
& 4*fzd(7)*conc(7,j)*aa(j)/(hth)-&
& 6*fzd(8)*conc(8,j)*aa(j)/(hth)
d(2,1)= (fzd(1)*conc(1,j)/(hth)+fzd(1)*d_c(1,j)/2h)aa)+
& d_aa(j)*fzd(1)*conc(1,j)/(2*h)-&
& (fzd(5)*conc(5,j)/(hth)+fzd(5)*d_c(5,j)/ (h)aa)-
& d_aa(j)*fzd(5)*conc(5,j)/(2*h)-&
& (fzd(4)*conc(4,j)/(hth)+fzd(4)*d_c(4,j)/ 2h)aa)-
& d_aa(j)*fzd(4)*conc(4,j)/(2*h)+&
& 2*(fzd(7)*conc(7,j)/(hth)+fzd(7)*d_c(7,j )2h)aa)+

& 2*d_aa(j)*fzd(7)*conc(7,j)/(2*h)+&
& 3*(fzd(8)*conc(8,j)/(hth)+fzd(8)*d_c(8,j)2h)aa)+
& 3*d_aa(j)*fzd(8)*conc(8,j)/(2*h)
a(2,2)= (-fzd(1)*d_phi(j)/(2*h)+diff(1)/(hth))*aa)-
& d_aa(j)*diff(1)/(2*h)
b(2,2)= -por(j)/t_step+(fzd(1)*dd_phi(j)-2*diff(1/h))aj)
& +d_aa(j)*fzd(1)*d_phi(j)
d(2,2)= (fzd(1)*d_phi(j)/(2*h)+diff(1)/(hth))*aa~j+
& d_aa(j)*diff(1)/(2*h)
a (2 ,5) = (-f zd (4) *d_phi (j )/ (2*h) +dif f(4) /(hth) )*aa (j )+&
& d_aa(j)*diff(4)/(2*h)
b(2,5)= por(j)/t_step-(fzd(4)*dd_phi(j)-2*diff(4)/hh)a~)
& -d_aa(j)*fzd(4)*d_phi(j)
d(2,5)= -(fzd(4)*d_phi(j)/(2*h)+diff(4)/(hth))*aa)-
& d_aa(j)*diff(4)/(2*h)
a(2,6)=(-fzd(1)*conc(1,j)*d_phi(j)-diff(1dc1,)&
& (1.5d0+(por(j-1)**0.5d0))/(2*h)-&
& (-fzd(5)*conc(5,j)*d_phi(j)-diff(5)*d_c( 5,)*
& (1.5d0+(por(j-1)**0.5d0))/(2*h)-&
& (-fzd(4)*conc(4,j)*d_phi(j)-diff(4)*d_c( 4,)*
& (1.5d0+(por(j-1)**0.5d0))/(2*h)+&
& 2*(-fzd(7)*conc(7,j)*d_phi(j)-diff(7)*d _c(,)*
& (1.5d0+(por(j-1)**0.5d0))/(2*h)+&
& 3*(-fzd(8)*conc(8,j)*d_phi(j)-diff(8)*d_c(,)*
& (1.5d0+(por(j-1)**0.5d0))/(2*h)
b(2,6)=-conc(1,j)/t_step-1.5*(por(j)**0.5)(fd1*oc1j*
& dd_phi(j)-fzd(1)*d_c(1,j)*d_phi(j)-diff( 1)d~(,)+
& (1.5d0*(por(j)**0.5)*(b7_pro**1.5)/ff2-&
& ffl*1.5d0*(por(j)**0.5)*thickness/(ff2**2)*
& curolim/(F*thickness)+&
& conc(5,j)/t_step+1.5*(por(j)**0.5)*(-fzd(5*oc5j*
& dd_phi(j)-fzd(5)*d_c(5,j)*d_phi(j)-diff( 5)d~(,)+
& conc(4,j)/t_step+1.5*(por(j)**0.5)*(-fzd(4*oc4j*
& dd_phi(j)-fzd(4)*d_c(4,j)*d_phi(j)-diff( 4)d~(,)-
& 2*conc(7,j)/t_step-3.0*(por(j)**0.5)*(-f zd7*oc7j*
& dd_phi(j)-fzd(7)*d_c(7,j)*d_phi(j)-diff( 7)d~(,)-
& 3*conc(8,j)/t_step-4.5*(por(j)**0.5)*(-fzd8*oc8j*
& dd_phi(j)-fzd(8)*d_c(8,j)*d_phi(j)-diff( 8)d~(,)+
& conc(9,j)/t_step
d(2,6)=(-fzd(1)*conc(1,j)*d_phi(j)-diff( 1dc1,)&
& (-1.5d0+(por(j+1)**0.5d0))/(2*h)-&
& (-fzd(5)*conc(5,j)*d_phi(j)-diff(5)*d_c( 5,)*
& (-1.5d0+(por(j+1)**0.5d0))/(2*h)-&
& (-fzd(4)*conc(4,j)*d_phi(j)-diff(4)*d_c( 4,)*
& (-1.5d0+(por(j+1)**0.5d0))/(2*h)+&

& 2*(-fzd(7)*conc(7,j)*d_phi(j)-diff(7)*d _c(,)*
& (-1.5d0+(por(j+1)**0.5d0))/(2*h)+&
& 3*(-fzd(8)*conc(8,j)*d_phi(j)-diff(8)*d_c(,)*
& (-1.5d0+(por(j+1)**0.5d0))/(2*h)
a (2 ,7) = (-f zd (5) *d_phi (j )/ (2*h) +dif f(5) /(hth) )*aa (j )+&
& d_aa(j)*diff(5)/(2*h)
b(2,7)= por(j)/t_step-(fzd(5)*dd_phi(j)-2*diff(5)/hh)a~)
& -d_aa(j)*fzd(5)*d_phi(j)
d(2,7)= -(fzd(5)*d_phi(j)/(2*h)+diff(5)/(hth))*aa)-
& d_aa(j)*diff(5)/(2*h)
a(2,9)= 2*(-fzd(7)*d_phi(j)/(2*h)+diff(7)/(hth)) aa)-
& 2*d_aa(j)*diff(7)/(2*h)
b(2,9)= -2*por(j)/t_step+2*(fzd(7)*dd_phi(j)-2*dif7/h))aj)
& +2*d_aa(j)*fzd(7)*d_phi(j)
d (2 ,9) = 2* (fzd (7) *d_phi (j )/ (2*h) +dif f(7) /(hth) )*aa (j )+&
& 2*d_aa(j)*diff(7)/(2*h)
a(2,10)= 3*(-fzd(8)*d_phi(j)/(2*h)+diff(8)/(hth))aa)-
& 3*d_aa(j)*diff(8)/(2*h)
b(2,10)= -3*por(j)/t_step+3*(fzd(8)*dd_phi(j)-2*dif8/h))aj)
& +3*d_aa(j)*fzd(8)*d_phi(j)
d(2,10)= 3*(fzd(8)*d_phi(j)/(2*h)+diff(8)/(hth))*aaj+
& 3*d_aa(j)*diff(8)/(2*h)
b(2,11)= por(j)/t_step
!Na+ and Cl-
do i=2,3
g(i+1)= g_eq(i,j)
a(i+1,1)=(fzd(i)*conc(i,j)/(hth)-fzd(i)*d~~~)(*)*aj-
& d_aa(j)*fzd(i)*conc(i,j)/(2*h)
b(i+1,1)=-2*fzd(i)*conc(i,j)*aa(j)/(hth)
d(i+1,1)=(fzd(i)*conc(i,j)/(hth)+fzd(i)*d~~~)(*)*aj+
& d_aa(j)*fzd(i)*conc(i,j)/(2*h)
a(i+1,i+1)=(-fzd(i)*d_phi(j)/(2*h)+diff( i)(t)*aj-
& d_aa(j)*diff(i)/(2*h)
b(i+1,i+1)=-por(j)/t_step+(fzd(i)*dd_phi~j-*ifi/hh)a~)
& +d_aa (j )*fzd (i) *d_phi (j )
d(i+1,i+1)=(fzd(i)*d_phi(j)/(2*h)+diff(i )hh)aa)+
& d_aa(j)*diff(i)/(2*h)
a(i+1,6)=(-fzd(i)*conc(i,j)*d_phi(j)-diffidcij)&
& (1.5d0+(por(j-1)**0.5d0))/(2*h)
b(i+1,6)=-conc(i,j)/t_step-1.5*(por(j)**0.)(fdi*ocij*
& dd_phi(j)-fzd(i)*d_c(i,j)*d_phi(j)-diff( i)d~~~)
d(i+1,6)=(-fzd(i)*conc(i,j)*d_phi(j)-diffidcij)&
& (-1.5d0+(por(j+1)**0.5d0))/(2*h)
enddo
!Zn2+

g(5)=g_eq(4,j-urnj)-crnj/(2*F*thickness)+ g~q6j+ q7j+
& g_eq(8,j)-(por(j)*conc(9,j)-por_ini(j)*c~ ii9j)tse

a(5,1)= (fzd(4)*conc(4,j)/(hth)-fzd(4)*d_c(4,j)/2h)aa)-
& d_aa(j)*fzd(4)*conc(4,j)/(2*h)+&
& (fzd(6)*conc(6,j)/(hth)-fzd(6)*d_c(6,j)/ 2h)aa)-
& d_aa(j)*fzd(6)*conc(6,j)/(2*h)+&
& (fzd(7)*conc(7,j)/(hth)-fzd(7)*d_c(7,j)/ 2h)aa)-
& d_aa(j)*fzd(7)*conc(7,j)/(2*h)+&
& (fzd(8)*conc(8,j)/(hth)-fzd(8)*d_c(8,j)/ 2h)aa)-
& d_aa(j)*fzd(8)*conc(8,j)/(2*h)
b(5,1)= -2*fzd(4)*conc(4,j)*aa(j)/(hth)&
& -log(10.0)*curzn(j)/(2*F*tbetazn*thicknes)
& -2*fzd(6)*conc(6,j)*aa(j)/(hth)&
& -2*fzd(7)*conc(7,j)*aa(j)/(hth)&
& -2*fzd(8)*conc(8,j)*aa(j)/(hth)
d(5,1)= (fzd(4)*conc(4,j)/(hth)+fzd(4)*d_c(4,j)/2h)aa)+
& d_aa(j)*fzd(4)*conc(4,j)/(2*h)+&
& (fzd(6)*conc(6,j)/(hth)+fzd(6)*d_c(6,j)/ 2h)aa)+
& d_aa(j)*fzd(6)*conc(6,j)/(2*h)+&
& (fzd(7)*conc(7,j)/(hth)+fzd(7)*d_c(7,j)/ 2h)aa)+
& d_aa(j)*fzd(7)*conc(7,j)/(2*h)+&
& (fzd(8)*conc(8,j)/(hth)+fzd(8)*d_c(8,j)/ 2h)aa)+
& d_aa(j)*fzd(8)*conc(8,j)/(2*h)
a(5,5)= (-fzd(4)*d_phi(j)/(2*h)+diff(4)/(hth))*aa)-
& d_aa(j)*diff(4)/(2*h)
b(5,5)= -por(j)/t_step+(fzd(4)*dd_phi(j)-2*diff(4/h))aj)
& +d_aa(j)*fzd(4)*d_phi(j)
d(5,5)= (f zd(4)*d_phi (j)/(2*h)+diff(4)/ (hth))* aa(j)+&
& d_aa(j)*diff(4)/(2*h)
a(5,6)=(-fzd(4)*conc(4,j)*d_phi(j)-diff(4dc4,)&
& (1.5d0+(por(j-1)**0.5d0))/(2*h)+&
& (-fzd(6)*conc(6,j)*d_phi(j)-diff(6)*d_c( 6,)*
& (1.5d0+(por(j-1)**0.5d0))/(2*h)+&
& (-fzd(7)*conc(7,j)*d_phi(j)-diff(7)*d_c( 7,)*
& (1.5d0+(por(j-1)**0.5d0))/(2*h)+&
& (-fzd(8)*conc(8,j)*d_phi(j)-diff(8)*d_c( 8,)*
& (1.5d0+(por(j-1)**0.5d0))/(2*h)
b(5,6)=-conc(4,j)/t_step-1.5*(por(j)**0.5)(fd4*oc4j*
& dd_phi(j)-fzd(4)*d_c(4,j)*d_phi(j)-diff( 4)d~(,)-
& conc(6,j)/t_step-1.5*(por(j)**0.5)*(-fzd (6*oc6j*
& dd_phi(j)-fzd(6)*d_c(6,j)*d_phi(j)-diff( 6)d~(,)-
& conc(7,j)/t_step-1.5*(por(j)**0.5)*(-fzd (7*oc7j*
& dd_phi(j)-fzd(7)*d_c(7,j)*d_phi(j)-diff( 7)d~(,)-
& conc(8,j)/t_step-1.5*(por(j)**0.5)*(-fzd (8*oc8j*

& dd_phi(j)-fzd(8)*d_c(8,j)*d_phi(j)-diff( 8)d~(,)+
& conc(9,j)/t_step
d(5,6)=(-fzd(4)*conc(4,j)*d_phi(j)-diff( 4dc4,)&
& (-1.5d0+(por(j+1)**0.5d0))/(2*h)+&
& (-fzd(6)*conc(6,j)*d_phi(j)-diff(6)*d_c( 6,)*
& (-1.5d0+(por(j+1)**0.5d0))/(2*h)+&
& (-fzd(7)*conc(7,j)*d_phi(j)-diff(7)*d_c( 7,)*
& (-1.5d0+(por(j+1)**0.5d0))/(2*h)+&
& (-fzd(8)*conc(8,j)*d_phi(j)-diff(8)*d_c( 8,)*
& (-1.5d0+(por(j+1)**0.5d0))/(2*h)
a(5,8)= (-fzd(6)*d_phi(j)/(2*h)+diff(6)/(hth))*aa)-
& d_aa(j)*diff(6)/(2*h)
b(5,8)= -por(j)/t_step+(fzd(6)*dd_phi(j)-2*diff(6/h))aj)
& +d_aa(j)*fzd(6)*d_phi(j)
d(5,8)= (f zd (6) *d_phi (j )/ (2*h) +dif f(6) /(hth) )*aa (j )+&
& d_aa(j)*diff(6)/(2*h)
a(5,9)= (-fzd(7)*d_phi(j)/(2*h)+diff(7)/(hth))*aa)-
& d_aa(j)*diff(7)/(2*h)
b(5,9)= -por(j)/t_step+(fzd(7)*dd_phi(j)-2*diff(7/h))aj)
& +d_aa(j)*fzd(7)*d_phi(j)
d(5,9)= (f zd (7) *d_phi (j )/ (2*h) +dif f(7) /(hth) )*aa (j )+&
& d_aa(j)*diff(7)/(2*h)

a(5,10)= (-fzd(8)*d_phi(j)/(2*h)+diff(8)/(hth))*aa)-
& d_aa(j)*diff(8)/(2*h)
b(5,10)= -por(j)/t_step+(fzd(8)*dd_phi(j)-2*diff(8/h))aj)
& +d_aa(j)*fzd(8)*d_phi(j)
d(5,10)= (fzd(8)*d_phi(j)/(2*h)+diff(8)/(hth))*aa~j+
& d_aa(j)*diff(8)/(2*h)
b(5,11)= por(j)/t_step
!Porosity
g(6)= por(j)-bl_pro/(1+b8*(conc(1,j)**b9))-&
& b4_pro/(1+bl0*(conc(1,j)**bl1))-b7_pro
b(6,2)= -dd*(conc(1,j)**(b9-1))/((1+b8*(conc(1,j)b9)2)
& -ee*(conc(1,j)**(bl1-1))/((1+bl 0+(conc(1,)l1)*2
b(6,6)= -1.0d0
!H+ H20 --> H+ +OH-
g(7)=conc(1,j)*conc(5,j)-1.0d-20
b(7,2)= -conc(5,j)
b(7,7)= -conc(1,j)
!ZnOH+ Zn2+ +OH- --> ZnOH+
g(8)= conc(6,j)-(10.0**1.33)*conc(1,j)*conc(4,)
b(8,2)= (10.0**1.33)*conc(4,j)
b(8,5)= (10.0**1.33)*conc(1,j)
b(8,8)= -1.0d0

!HZnO2- ZnOH+ +20H- --> HZnO2- + H20
g(9)= conc(7,j)-(10.0**4.03)*conc(6,j)*(conc(1,)*2
b(9,2)= 2*(10.0**4.03)*conc(6,j)*conc(1,j)
b(9,8)= (10.0**4.03)*(conc(1,j)**2)
b(9,9)= -1.0d0
!ZnO22- HZnO2- + OH- --> ZnO22- + H20
g(10)= conc(8,j)-(10.0**(-2.17))*conc(1,j)*conc(7j
b(10,2)= (10.0**(-2.17))*conc(7,j)
b(10,9)= (10.0**(-2.17))*conc(1,j)
b(10,10)= -1.0d0
!Zn(OH)2 Zn2+ + 20H- --> Zn(0H)2
g(11)= (por(j)*conc(9,j)-por_ini(j)*c_ini(9,j)) /tse-
& rate_k*(conc(1,j)*conc(1,j)*conc(4,-,j)30-6
b(11,2)= 2*rate_k*conc(1,j)*conc(4,j)
b(11,5)= rate_k*conc(1,j)*conc(1,j)
b(11,6)= -conc(9,j)/t_step
b(11,11)= -por(j)/t_step
return
end

subroutinee for the boundary in the fully-intact region
subroutine bc_2(j)
implicit double precision(a-h,o-z)
implicit integer(i-n)
common/ca/height,F,frt,vapp,tbetazn,excurz~hc~e~hcns,
& total_time,t_step,rate_k,z_oh,z_na,z_cl, z~nzhzzo,
& z_hzno2,z_zno2,z_znoh2,diff_oh,diff_na,d ifcdifz,
& diff_h,diff_znoh,diff_hzno2,diff_zno2,diffzo2z9,
& diff(9),fzd(9),curolim,h,hh(2001),mn
common/cb/a(11,11),b(11,11),c(11,2001),d112)g1)x11),

common/cc/ii,conc(9,2001),phi(2001),por(201,~h(01,
& dd_phi(200 1),diff_phi(2001),d_c(9,2001),dc9201,
& d_flux(9,2001),flux(9,2001),curzn(2001),cr(01,
& c_aver(9,2001),g_eq(9,2001),c_ini(9,2001 )da201
common/cd/del_1en,fro_1en,sem_1en,tac_1elehelcndl~~~e,
& c_ohfro,c_nafro,c_clfro,c_ohsem,c_nasem, c~~e~~hn,
& c_naint,c_clint,jdel,jfro,jsem,bl_pro,b2_pobr,
& b4_pro,b5_pro,b6_pro,b7_pro,bl_sur,b2_su r~3srbu,
& b5_sur,b6_sur,b7_sur,bl_blo,b2_blo,b3_bl o~4bobl,
& b6_blo,b7_blo,b8_blo,bl_poi,b2_poi,b3_po i~4pibo,
& b6_poi,b7_poi,ph(2001),por_ini(2001),aa( 201,u~o(01,
& bb(2001),block(200 1),poi(2001),b8,b9,b10b1
common/ce/flux_mig(9,2001),flux_dif(9,200)cre(01dpo20),
& d_v(2001),total_current

dd=bl_pro*b8*b9
ee=b4_pro*bl0*bl1
d_aa(j)=(aa(j)-aa(j-1))/h
d_phi(j)= (phi(j)-phi(j-1))/h
diff_phi(j)=vapp-phi(j)

ophi=0.402-tbetao*10gl0(excuro)
znphi=-0.763-tbetazn*10gl0(excurzn)
ffl=(por(j)*b7_pro)**1.5
ff2=(por(j)**1.5)*thickness+(b7_pro**1.5)*hc~e
ff=ffl/ff2
!calculate current densities due to electrochemical reactions
curzn (j )= (10* *( (vapp-phi (j )-znphi) /tbet azn) ) *sur_c ov (j )*po i(j )
curo(j)=-curolim*sur_cVj*lcov(j)*blckj)f

do i=1,n-2
c_aver(i,j)= (conc(i,j-1)+conc(i,j))/2
d_c(i,j)= (conc(i,j)-conc(i,j-1))/h
flux(i,j)= -fzd(i)*c_aver(i,j)*d_phi(j)-diff(i)*d_c~ij
g_eq(i,j)= (conc(i,j)*por(j)-c_ini(i,j)*por_ini(j))/tse-
& aa(j)*2*flux(i,j)/h+flux(i,j)*d_aa(j)/2
enddo
!solution potential (Electroneturality)
g(1)= z(1)*conc(1,j)+z(2)*conc(2,j)+z(3)*conc(3,)z4*oc4j+
& z(5)*conc(5,j)+z(6)*conc(6,j)+z(7)*conc( 7,)z8*oc8j
b(1,2)= -z(1)
b(1,3)= -z(2)
b(1,4)= -z(3)
b(1,5)= -z(4)
b(1,7)= -z(5)
b(1,8)= -z(6)
b(1,9)= -z(7)
b(1,10)= -z(8)
!OH-
g(2)= g_eq(1,j)-g_eq(5,j)-g_eq(4,j)+2*g_eq(7,j )+*~q8j-
& (por(j)*conc(9,j)-por_ini(j)*c_ini(9,j)) /tse
a(2,1)= fzd(1)*c_ aver(1, j)* aa(j)*2/(hth)-&
& fzd(1)*c_aver(1,j)*d_aa(j)/(2*h)-&
& fzd(5)* c_aver(5, j)*aa(j)*2/(hth)+&
& fzd(5)*c_aver(5,j)*d_aa(j)/(2*h)-&
& fzd(4)*c_ aver (4, j)* aa(j)*2/(hth)+&
& fzd(4)*c_aver(4,j)*d_aa(j)/(2*h)+&
& 2*f zd (7) *c_aver (7 ,j )*aa (j )*2/ (hth) -&
& 2*fzd(7)*c_aver(7,j)*d_aa(j)/(2*h)+&
& 3*f zd (8) *c_aver (8 ,j )*aa (j )*2/ (hth) -&

& 3*fzd(8)*c_aver(8,j)*d_aa(j)/(2*h)
b(2,1)= -fzd(1)*c_aver(1,j)*aa(j)*2/(hth)+&
& fzd(1)*c_aver(1,j)*d_aa(j)/(2*h)+&
& fzd(5)*c_aver(5,j)*aa(j)*2/(hth)-&
& fzd(5)*c_aver(5,j)*d_aa(j)/(2*h)+&
& fzd(4)*c_aver(4,j)*aa(j)*2/(hth)-&
& fzd(4)*c_aver(4, j)*d_aa(j)/(2*h)-&
& 2*fzd(7)*c_aver(7,j)*aa(j)*2/(hth)+&
& 2*fzd(7)*c_aver(7,j)*d_aa(j)/(2*h)-&
& 3*fzd(8)*c_aver(8,j)*aa(j)*2/(hth)+&
& 3*fzd(8)*c_aver(8,j)*d_aa(j)/(2*h)
a(2,2)= -(fzd(1)*d_phi(j)/2-diff(1)/h)*2*aa(j)/h+
& (fzd(1)*d_phi(j)/2-diff(1)/h)*d_aa(j)/2
b(2,2)= -por(j)/t_step-(fzd(1)*d_phi(j)/2+diff(1)h*laj/+
& (fzd(1)*d_phi(j)/2+diff(1)/h)*d_aa(j)/2
a(2,5)= (fzd(4)*d_phi(j)/2-diff(4)/h)*2*aa(j)/h-&
& (fzd(4)*d_phi(j)/2-diff(4)/h)*d_aa(j)/2
b(2,5)= por(j)/t_step+(fzd(4)*d_phi(j)/2+diff(4)/h**aj/-
& (fzd(4)*d_phi(j)/2+diff(4)/h)*d_aa(j)/2
a(2,6)= flux(1,j)*1.5d0*(por(j-1)**0.5)/(2*h)-&
& flux(5,j)*1.5d0+(por(j-1)**0.5)/(2*h)-&
& flux(4,j)*1.5d0+(por(j-1)**0.5)/(2*h)+&
& 2*flux(7,j)*1.5d0*(por(j-1)**0.5)/(2*h)+&
& 3*flux(8,j)*1.5d0*(por(j-1)**0.5)/(2*h)
b(2,6)= -conc(1,j)/t_step+2*1.5d0*flux(1,j)/h-&
& flux(1,j)*1.5d0+(por(j)**0.5)/(2*h)+&
& conc(5,j)/t_step-2*1.5d0*flux(5,j)/h+&
& flux(5,j)*1.5d0+(por(j)**0.5)/(2*h)+&
& conc(4,j)/t_step-2*1.5d0*flux(4,j)/h+&
& flux(4,j)*1.5d0+(por(j)**0.5)/(2*h)-&
& 2*conc(7,j)/t_step+4*1.5d0*flux(7,j)/h-&
& 2*flux(7,j)*1.5d0*(por(j)**0.5)/(2*h)-&
& 3*conc(8,j)/t_step+6*1.5d0*flux(8,j)/h-&
& 3*flux(8,j)*1.5d0*(por(j)**0.5)/(2*h)+&
& conc(9,j)/t_step
a(2,7)= (fzd(5)*d_phi(j)/2-diff(5)/h)*2*aa(j)/h-&
& (fzd(5)*d_phi(j)/2-diff(5)/h)*d_aa(j)/2
b(2,7)= por(j)/t_step+(fzd(5)*d_phi(j)/2+diff(5)/h**aj/-
& (fzd(5)*d_phi(j)/2+diff(5)/h)*d_aa(j)/2
a (2 ,9) = -2* (fzd (7) *d_phi (j )/2-dif f(7) /h) *2*aa (j )/h+&
& 2* (fzd (7) *d_phi (j )/2-dif f(7) /h) *d_aa (j )/2
b(2,9)= -2*por(j)/t_step-2*(fzd(7)*d_phi(j)/2+dif7)h2aa)/+
& 2* (fzd (7) *d_phi (j )/2+dif f(7) /h) *d_aa (j )/2
a(2,10)= -3*(fzd(8)*d_phi(j)/2-diff(8)/h)*2*aa(j)/+
& 3*(fzd(8)*d_phi(j)/2-diff(8)/h)*d_aa(j)/2

b(2,10)= -3*por(j)/t_step-2*(fzd(8)*d_phi(j)/2+dif8)h2aa)/+
& 3*(fzd(8)*d_phi(j)/2+diff(8)/h)*d_aa(j)/2
b(2,11)= por(j)/t_step
!Na+ and Cl-
do i=2,3
g(i+1)= g_eq(i,j)
a(i+1,1)= fzd(i)*c_aver(i,j)*aa(j)*2/(hth)-&
& fzd(i)*c_aver(i,j)*d_aa(j)/(2*h)
b(i+1,1)=-fzd(i)*c_aver(i,j)*aa(j)*2/(h*h+
& fzd(i)*c_aver(i,j)*d_aa(j)/(2*h)
a(i+1 ,i+1) =- (fzd(i) *d_phi (j )/2-dif f(i) /h) *2*aa (j )/h+&
& (fzd(i)*d_phi(j)/2-diff(i)/h)*d_aa(j)/2
b(i+1,i+1)=-por(j)/t_step-(fzd(i)*d_phi(j)2df~)h**aj/+
& (fzd(i)*d_phi(j)/2+diff(i)/h)*d_aa(j)/2
a(i+1,6)= flux(i,j)*1.5d0*(por(j-1)**0.5)/(2*h)
b(i+1,6)= -conc(i,j)/t_step+2*1.5d0*flux(i,j)/h-&
& flux(i,j)*1.5d0+(por(j)**0.5)/(2*h)
enddo
!Zn2+
g(5)= g_eq(4,j)+g_eq(6,j)+g_eq(7,j)+g_eq(8,j)-&
& (por(j)*conc(9,j)-por_ini(j)*c_ini(9,j)) /tse
a(5,1)= fzd(4)*c_aver(4,j) *aa(j)*2/(hth)-&
& fzd(4)*c_aver(4,j)*d_aa(j)/(2*h)+&
& fzd(6)* c_aver(6, j)*aa(j)*2/(hth)-&
& fzd(6)*c_aver(6,j)*d_aa(j)/(2*h)+&
& fzd(7)* c_aver (7, j)*aa(j)*2/(hth)-&
& fzd(7)*c_aver(7,j)*d_aa(j)/(2*h)+&
& fzd(8)* c_aver (8, j)*aa(j)*2/(hth)-&
& fzd(8)*c_aver(8,j)*d_aa(j)/(2*h)
b(5,1)= -fzd(4)*c_aver(4,j)*aa(j)*2/(hth)+&
& fzd(4)*c_aver(4, j)*d_aa(j)/(2*h)-&
& fzd(6)*c_aver(6,j)*aa(j)*2/(hth)+&
& fzd(6)*c_aver(6,j)*d_aa(j)/(2*h)-&
& fzd(7)*c_aver(7,j)*aa(j)*2/(hth)+&
& fzd(7)*c_aver(7,j)*d_aa(j)/(2*h)-&
& fzd(8)*c_aver(8,j)*aa(j)*2/(hth)+&
& fzd(8)*c_aver(8,j)*d_aa(j)/(2*h)
a(5,5)= -(fzd(4)*d_phi(j)/2-diff(4)/h)*2*aa(j)/h+
& (fzd(4)*d_phi(j)/2-diff(4)/h)*d_aa(j)/2
b(5,5)= -por(j)/t_step-(fzd(4)*d_phi(j)/2+diff(4)h2aa)/+
& (fzd(4)*d_phi(j)/2+diff(4)/h)*d_aa(j)/2
a(5,6)= flux(4,j)*1.5d0*(por(j-1)**0.5)/(2*h)+&
& flux(6,j)*1.5d0+(por(j-1)**0.5)/(2*h)+&
& flux(7,j)*1.5d0+(por(j-1)**0.5)/(2*h)+&
& flux(8,j)*1.5d0+(por(j-1)**0.5)/(2*h)

b(5,6)= -conc(4,j)/t_step+2*1.5d0*flux(4,j)/h-&
& flux(4,j)*1.5d0+(por(j)**0.5)/(2*h)-&
& conc(6,j)/t_step+2*1.5d0*flux(6,j)/h-&
& flux(6,j)*1.5d0+(por(j)**0.5)/(2*h)-&
& conc(7,j)/t_step+2*1.5d0*flux(7,j)/h-&
& flux(7,j)*1.5d0+(por(j)**0.5)/(2*h)-&
& conc(8,j)/t_step+2*1.5d0*flux(8,j)/h-&
& flux(8,j)*1.5d0+(por(j)**0.5)/(2*h)+&
& conc(9,j)/t_step
a(5,8)= -(fzd(6)*d_phi(j)/2-diff(6)/h)*2*aa(j)/h+
& (fzd(6)*d_phi(j)/2-diff(6)/h)*d_aa(j)/2
b(5,8)= -por(j)/t_step-(fzd(6)*d_phi(j)/2+diff(6)h2aa)/+
& (fzd(6)*d_phi(j)/2+diff(6)/h)*d_aa(j)/2
a(5,9)= -(fzd(7)*d_phi(j)/2-diff(7)/h)*2*aa(j)/h+
& (fzd(7)*d_phi(j)/2-diff(7)/h)*d_aa(j)/2
b(5,9)= -por(j)/t_step-(fzd(7)*d_phi(j)/2+diff(7)h2aa)/+
& (fzd(7)*d_phi(j)/2+diff(7)/h)*d_aa(j)/2
a(5,10)= -(fzd(8)*d_phi(j)/2-diff(8)/h)*2*aa(j)/h+
& (fzd(8)*d_phi(j)/2-diff(8)/h)*d_aa(j)/2
b(5,10)= -por(j)/t_step-(fzd(8)*d_phi(j)/2+diff(8)h2aa)/+
& (fzd(8)*d_phi(j)/2+diff(8)/h)*d_aa(j)/2
b(5,11)= por(j)/t_step
!Porosity
g(6)= por(j)-bl_pro/(1+b8*(conc(1,j)**b9))-&
& b4_pro/(1+bl0*(conc(1,j)**bl1))-b7_pro
b(6,2)= -dd*(conc(1,j)**(b9-1))/((1+b8*(conc(1,j)b9)2)
& -ee*(conc(1,j)**(bl1-1))/(((1+b10(conc(1,)l1)*2
b(6,6)= -1.0d0
!H+
g(7)=conc(1,j)*conc(5,j)-1.0d-20
b(7,2)= -conc(5,j)
b(7,7)= -conc(1,j)
!ZnOH+ Zn2+ + OH- --> ZnOH+
g(8)= conc(6,j)-(10.0**1.33)*conc(1,j)*conc(4,)
b(8,2)= (10.0**1.33)*conc(4,j)
b(8,5)= (10.0**1.33)*conc(1,j)
b(8,8)= -1.0d0
!HZnO2- ZnOH+ +20H- --> HZnO2- + H20
g(9)= conc(7,j)-(10.0**4.03)*conc(6,j)*(conc(1,)*2
b(9,2)= 2*(10.0**4.03)*conc(6,j)*conc(1,j)
b(9,8)= (10.0**4.03)*(conc(1,j)**2)
b(9,9)= -1.0d0
!ZnO22- HZnO2- + OH- --> ZnO22- + H20
g(10)= conc(8,j)-(10.0**(-2.17))*conc(1,j)*conc(7j
b(10,2)= (10.0**(-2.17))*conc(7,j)

b(10,9)= (10.0**(-2.17))*conc(1,j)
b(10,10)= -1.0d0
!Zn(OH)2 Zn2+ + 20H- --> Zn(0H)2
g(11)= (por(j)*conc(9,j)-por_ini(j)*c_ini(9,j)) /tse-
& rate_k*(conc(1,j)*conc(1,j)*conc(4,j)-3.0-6
b(11,2)= 2*rate_k*conc(1,j)*conc(4,j)
b(11,5)= rate_k*conc(1,j)*conc(1,j)
b(11,6)= -conc(9,j)/t_step
b(11,11)= -por(j)/t_step
return
end

APPENDIX B
PROGRAM LISTING FOR IMPEDANCE CALCULATIONS

The program listings for the impedance calculations presented in ('! .pters 8 to 10

are given in this appendix. The theoretical development for the cases of ideally-polarized

electrodes (C'h! Ilter 8), electrodes with local CPE (C'!s Ilter 9), and electrodes exhibiting

Faradaic reactions (C'h! Ilter 10) were similar. The key difference was the boundary

condition applied on the electrode surface. The calculations were performed using the

collocation package PDE2D developed by Sewell.[90]

B.1 Main Program Listing
C PDE2D 8.3 MAIN PROGRAM *
C *** 2D PROBLEM SOLVED COLLOCATIONN METHOD) ***
implicit double precision (a-h,o-z)
parameter (neqnmx= 99)
C NXGRID = number of X-grid lines
PARAMETER (NXGRID= 400)
C NXGRID = number of X-grid lines
PARAMETER (NYGRID = 50)
PARAMETER (NEQN = 2)
parameter (nzgrid = 1)
PARAMETER (IRWK8Z= 1)
PARAMETER (IIWK8Z= 1)
PARAMETER (NXP8Z=201,NYP8Z=201,NZP8Z=2,KDEG8Z=1)
common/parm8z/ pi,a,omega,ck,c,deltay,nprob,iprob,DJ,CPalh
dimension xgrid(nxgrid),ygrid(nygrid),zgrid(nzgrid)xuz(:x0
&ny),yout8z(0:nx,0:ny),zout8z(0:nx,0:ny) ,xrs(0)yrs(0)t
&ut8z(0:nsave),uout8z(0:nx,0:ny,4*neqn,0:naersznpzrs
&z(nyp8z),zres8z(nzp8z),ures8z(neqn,nxp8znpzzp)
allocatable iwrk8z(:),rwrk8z(:)
character*40 title
logical linear,crankn,noupdt,nodist,fillin,evempxdptloqi
&t,fdiff,solid,econ8z,ncon8z,restrt,gridi
common/dtdpi4/ sint8z(20),bint8z(20),slim8z(20),blim8z(20
common/dtdpi5/ evlr8z,ev0r,evli8z,ev0i,evempx
common/dtdp30/ econ8z,ncon8z
common/dtdp45/ perdc(neqnmx)
common/dtdp46/ eps8z,cgtl8z,npmx8z
common/dtdp52/ nxa8z,nya8z,nza8z,kd8z
common/dtdp53/ work8z(nxp8z*nyp8z*nzp8z+9)
common/dtdp64/ amin8z(4*neqnmx),amax8z(4*neqnmx)

pi = 4.0*atan(1.d0)
zr8z = 0.0
nxa8z = nxp8z
nya8z = nyp8z
nza8z = nzp8z
kd8z = kdeg8z

nomega=160
a=0.25
ck=0.05
c=3.0d-5
deltav=0.01
DJ=1.0
CPEalpha=0.8

do 78760 iomega=1,nomega
omega=10**(-5+iprob*0.05)
C###########################################################
C A collocation finite element method is used, with bi-cubic Hermite #
C basis functions on the elements (small rectangles) defined by the grid #
C points: #
C XGRID(1),...,XGRID(NXGRID) #
C YGRID(1),...,YGRID(NYGRID) #
C You will first be prompted for NXGRID, the number of X-grid points, #
C then for XGRID(1),...,XGRID(NXGRID). Any points defaulted will be #
C uniformly spaced between the points you define; the first and last #
C points cannot be defaulted. Then you will be prompted similarly #
C for the number and values of the Y-grid points. The rectangle over #
C which the PDE system is to be solved is then: #
C XGRID(1) < X < XGRID(NXGRID) #
C YGRID(1) < Y < YGRID(NYGRID) #
C #
C###########################################################
call dtdpwx(xgrid,nxgrid,0)
call dtdpwx(ygrid,nygrid,0)
XGRID(1) = 0.0d0
XGRID(NXGRID) = 200
YGRID(1) = 0.0d0
YGRID(NYGRID) = 1.0d0

zgrid(1) = 0
call dtdpwx(xgrid,nxgrid,1)
call dtdpwx(ygrid,nygrid,1)
C###########################################################
C If you don't want to read the FINE PRINT, enter ISOLVE = 1. #

C +++++++++++++++ THE "FINE PRINT" (CAN USUALLY BE IGNORED) ++++++++++++++#
C + The following linear system solvers are available: +#
C + +#
C + 1. Sparse direct method +#
C + Harwell Library routine MA27 (used by permission) is +#
C + used to solve the (positive definite) "normal" +#
C + equations A**T*A*x = A**T*b. The normal equations, +#
C + which are essentially the equations which would result +#
C + if a least squares finite element method were used +#
C + instead of a collocation method, are substantially +#
C + more ill-conditioned than the original system Ax = b, +#
C + so it may be important to use high precision if this +#
C + option is chosen. +#
C###########################################################
ISOLVE = 1
C###########################################################
C Is this a linear problem? ("linear" means all differential equations #
C and all boundary conditions are linear) #
C###########################################################
LINEAR = .true.
C###########################################################
C Give an upper limit on the number of Newton's method iterations #
C (NSTEPS) to be allowed for this nonlinear problem. NSTEPS defaults #
C to 15. #
C###########################################################
NSTEPS = 15
C###########################################################
C The solution is saved on an NX+1 by NY+1 rectangular grid covering #
C the rectangle (XA,XB) x (YA,YB). Enter values for XA,XB,YA,YB. #
C These variables are usually defaulted. #
C #
C The defaults are XA = XGRID(1), XB = XGRID(NXGRID) #
C YA = YGRID(1), YB = YGRID(NYGRID) #
C #
C###########################################################
C defaults for xa,xb,ya,yb
xa = xgrid(1)
xb = xgrid(nxgrid)
ya = ygrid(1)
yb = ygrid(nygrid)
call dtdpx3(nx,ny,0,xa,xb,ya,yb,zr8z,zr8z,hx8 z~yzh~~otzy
&ut8z,zout8z,npts8z)

subroutine tran8z(itrans,x,y,z8z)

implicit double precision (a-h,o-z)
&x2hess(3,3),x3hess(3,3)
common/parm8z/ pi,a,omega,ck,c,deltay,nprob,iprob,DJ,CPalh
C###########################################################
C If your region is rectangular, enter ITRANS=0, and you need not read #
C the FINE PRINT. #
C #
C + If X,Y represent polar or other non-Cartesian coordinates, you can +#
C + reference the Cartesian coordinates X1,X2 and derivatives of your +#
C + unknowns with respect to these coordinates, when you define your +#
C + PDE coefficients, boundary conditions, and volume and boundary +#
C + integrals, if you enter ITRANS .NE. 0. Enter: +#
C + ITRANS = 1, if X,Y are polar coordinates, that is, if +#
C + X=R, Y=Theta, where X1 = R*cos(Theta) +#
C + X2 = R*sin(Theta) +#
C + ITRANS = -1, same as ITRANS=1, but X=Theta, Y=R +#
C + ITRANS = 3, to define your own coordinate transformation. In +#
C + this case, you will be prompted to define X1,X2 and +#
C + their first and second derivatives in terms of X,Y. +#
C + ITRANS = -3, same as ITRANS=3, but you will only be prompted to +#
C + define XCART(*) = (X1,X2); their first and second +#
C + derivatives will be approximated using finite +#
C + differences. +#
C + When ITRANS = -3 or 3, the first derivatives of X1,X2 must all +#
C + be continuous. +#
C###########################################################
ITRANS = -3
XCART(1) = a*sqrt((1+x**2)*abs(1-y**2))
XCART(2) = atxty
xcart(3) = z8z
return
end
B.2 Subroutine Listing
subroutine pdes8z(yd8z,i8z,j8z,kint8z,x,y,z8z,t,uu8z
implicit double precision (a-h,o-z)
parameter (neqnmx= 99)
common /dtdp5x/un8z(10,neqnmx)
common /dtdpi8/normx,normy,nz8z
double precision normx,normy,nz8z,normi,norm2,n38z
dimension uu8z(10,neqnmx)
common/parm8z/ pi,a,omega,ck,c,deltay,nprob,iprob,DJ,CPalh
zr8z = 0.0

pr = uu8z(1, 1)
prx = uu8z(2, 1)
pry = uu8z(3, 1)
prxx= uu8z(5, 1)
pryy= uu8z(6, 1)
prxy= uu8z(8, 1)
pryx= uu8z(8, 1)
pii = uu8z(1, 2)
piix = uu8z(2, 2)
piiy = uu8z(3, 2)
piixx= uu8z(5, 2)
piiyy= uu8z(6, 2)
piixy= uu8z(8, 2)
piiyx= uu8z(8, 2)
call dtdpcd(x,y,z8z)
call dtdpcb(x,y,z8z,normx,normy,nz8z,xi,x2,x38nomnr2n3z)
call dtdpec(x,y,z8z,
& prx,pry,zr8z,prxx,pryy,zr8z,prxy,zr8z,zr~z
& xi,x2,x38z,pri,pr2,u38z,prl,pr22,u338z,pr2u3zu3z
& pr21,u318z,u328z,dvol,darc)
call dtdpec(x,y,z8z,
& piix,piiy,zr8z,piixx,piiyy,zr8z,piixy,zrzzrz
& xi,x2,x38z,piil,pii2,u38z,piill,pii22,u33zpi2u8z28z
& pii21,u318z,u328z,dvol,darc)
if (i8z.eq.0) then
yd8z = 0.0
C###########################################################
C Enter FORTRAN expressions to define the boundary condition functions, #
C which may be functions of #
C X,Y,pr,prx,pry, #
C pii,piix,piiy #
C###########################################################
if (j8z.eq.0) then
yd8z = 0.0
C F1 DEFINED
if (i8z.eq. 1) yd8z=
& 2*xtprx+(1+x**2)*prxx-2*y*pry+(1-y**2)*pry
C F2 DEFINED
if (i8z.eq. 2) yd8z=
& 2*xtpiix+(1+x**2)*piixx-2*y*piiy+(1-y**2)*iy
else
endif
return
end

subroutine gb8z(gd8z,ifac8z,i8z,j8z,x,y,z8z,t,uu8z)
implicit double precision (a-h,o-z)
parameter (neqnmx= 99)
dimension uu8z(10,neqnmx)
C un8z(1,I),un8z(2,I),... hold the (rarely used) values
C of UI,UIx,... from the previous iteration or time step
common /dtdp5x/ un8z(10,neqnmx)
common /dtdpi8/normx,normy,nz8z
double precision none,normx,normy,nz8z,normi,norm2,n38z
common/parm8z/ pi,a,omega,ck,c,deltay,nprob,iprob,DJ,CPalh
none = dtdplx(2)
zr8z = 0.0
pr = uu8z(1, 1)
prx = uu8z(2, 1)
pry = uu8z(3, 1)
pii = uu8z(1, 2)
piix = uu8z(2, 2)
piiy = uu8z(3, 2)
call dtdpcd(x,y,z8z)
call dtdpcb(x,y,z8z,normx,normy,nz8z,xi,x2,x38znrinr2n8,

call dtdpcb(
& x,y,z8z,prx,pry,zr8z,xi,x2,x38z,pri,pr2,u3z2
call dtdpcb(
& x,y,z8z,piix,piiy,zr8z,xi,x2,x38z,piil,pii,3z2
if (j8z.eq.0) gd8z = 0.0
C###########################################################
C Enter FORTRAN expressions to define the boundary condition functions, #
C which may be functions of #
C #
C X,Y,pr,prx,pry, #
C pii,piix,piiy and (if applicable) T #
C###########################################################
if (ifac8z.eq. 1) then
C###########################################################
C #
C First define the boundary conditions on the face X = XGRID(1). #
C###########################################################
if (j8z.eq.0) then
C G1 DEFINED
if (i8z.eq. 1) gd8z=
C For Ideally-Polarized Blocking Electrodes
& omegaspii+prx/y
C For Blocking Electrodes with Local CPE
& omega*((deltav-pr)*cos(CPEalpha*pi/2)+piiti(Papap/)+r/

C For Electrode Subject to a Faradaic Reaction
& prx/y+0megaspii+(deltav-prs)*DJ

C G2 DEFINED
if (i8z.eq. 2) gd8z=
C For Ideally-Polarized Blocking Electrode
& omega*deltav-omegaspr+piix/y
C For Blocking Electrodes with Local CPE
& omega*(deltav*sin(CPEalpha*pi/2)-piitcos(Papai/)
& pr*sin(CPEalpha*pi/2))+piix/y
C For Electrode Subject to a Faradaic Reaction
& piix/y+0mega*(deltav-pr)-pii*DJ

else
endif
endif
if (ifac8z.eq. 2) then
C###########################################################
C Now define the boundary conditions on the face X = XGRID(NXGRID). #
C###########################################################
if (j8z.eq.0) then
C G1 DEFINED
if (i8z.eq. 1) gd8z = pr
C G2 DEFINED
if (i8z.eq. 2) gd8z = pii
else
endif
endif
if (ifac8z.eq. 3) then
C###########################################################
C Now define the boundary conditions on the face Y = YGRID(1). #
C###########################################################
if (j8z.eq.0) then
C G1 DEFINED
if (i8z.eq. 1) gd8z = pry
C G2 DEFINED
if (i8z.eq. 2) gd8z = piiy
else
endif
endif
if (ifac8z.eq. 4) then
C###########################################################
C Now define the boundary conditions on the face Y = YGRID(NYGRID). #
C###########################################################
if (j8z.eq.0) then

C G1 DEFINED
if (i8z.eq. 1) gd8z = pry
C G2 DEFINED
if (i8z.eq. 2) gd8z = piiy
else
endif
endif
return
end

subroutine postpr(tout,nsave,xout,yout,nx,ny,uout,nen
implicit double precision (a-h,o-z)
dimension xout(0:nx,0:ny),yout(0:nx,0:ny),zout(0:nx0n)
&rout(0:nx,0:ny),tout(0:nsave),phi_re(0:nx0n)pii(:x0n)
& dprx(0:nx,0:ny),dpry(0:nx,0:ny),dpiix(0:nx0n)
& dpiiy(0:nx,0:ny),diff(0:nx,0:ny),trans( 0:n,:y
dimension cur_re(0:nx,0:ny),cur_im(0:nx,0:ny),cur( 0:x0n)
& subloc_re(0:nx,0:ny),subloc_im(0:nx,0:ny),clr(:x0n)
& ocal_im(0:nx,0:ny),resis_re(0:nx,0:ny),r essi(:x0n)
& gcur_re(0:ny),gcur_im(0:ny)
dimension uout(0:nx,0:ny,4,neqn,0:nsave)
common/parm8z/ pi,a,omega,ck,c,deltay,nprob,iprob,DJ,CPalh
UOUT(I,J,1,IEQ,L) = U-sub-IEQ
UOUT(I,J,2,IEQ,L) = UX-sub-IEQ
UOUT(I,J,3,IEQ,L) = UY-sub-IEQ

open (unit=33,file='data_1.sav')
open (unit=34,file='loc_1.txt')
open (unit=35,file='global.txt')

9 format (10E17.8)
10 format (12E20.12)
11 format (10E17.8)
13 format (7E17.8)

DO I=0,NX
DO J=0,NY
zout(i,j)= atxout(i,j)*yout(i,j)
trans(i,j)= sqrt((1+xout(i,j)**2)*abs(1-yout(i,j)**2)
rout(i,j)=attrans(i,j)
phi_re(i,j)=uout(i,j,1,1,1)
phi_im(i,j)=uout(i,j,1,2,1)
dprx(i,j)= UOUT(I,J,2,1,1)
dpry(i,j)= UOUT(I,J,3,1,1)
dpiix(i,j)=UOUT(I,J,2,2,1)

dpiiy(i,j)=UOUT(I,J,3,2,1)
z8z = 0.0
call dtdpcd(x,y,z8z)
call dtdpcb(x,y,z8z,zi8z,z28z,z38z,xi,x2,x38z,
& dl8z,d28z,d38z,1)
write(33,9) xout(i,j),yout(i,j),zout(i,j),rout(i,j),
& phi_re(i,j),phi_im(i,j),dprx(i,j),dpry(i~ jpi~~)
& dpiiy(i,j)
END DO
END DO

do i=0,nx
do j=0,ny
diff(i,j)=deltav-phi_re(i,j)
cur_re(i,j)=-ck*dprx(i,j)/(atyout(i,j))
cur_im(i,j)=-ck*dpiix(i,j)/(atyout(i,j))
cur(i,j)=cur_re(i,j)**2+cur_im(i,j)**2
subloc_re(i,j)=deltav*cur_re(i,j)/cur(i,)
subloc_im(i,j)=-deltav*cur_im(i,j)/cur(i~j
ocal_re(i,j)= (cur_re(i,j)*diff(i,j)-phi_im(i,j)*cur_imij/
& cur(i,j)
ocal_im(i,j)=-(cur_im(i,j)*(deltav-phi_rei)+
& phi_im(i,j)*cur_re(i,j))/cur(i,j)
resis_re(i,j)=(phi_re(i,j)*cur_re(i,j)+phii~~)
& cur_im(i,j))/cur(i,j)
resis_im(i,j)=-(phi_re(i,j)*cur_im(i,j)-pimij)
& cur_re(i,j))/cur(i,j)
write(34,10) xout(i,j),yout(i,j),zout(i,j),rout(i,j),
& cur_re(i,j),cur_im(i,j),subloc_re(i,j),sulciij)
& ocal_re(i,j),ocal_im(i,j),resis_re(i,j),rei~mij
enddo
enddo

Ctu e=.d
ttcur re=0.0d0

do j=0,ny
if ((j.eq.0).0r.(j.eq.ny)) then
gcur_re(j)=etacur_re(0,j)
gcur_im(j)=etacur_im(0,j)
else if (mod(j,2).eq.1) then
gcur_re(j)=4*etacur_re(0,j)
gcur_im(j)=4*etacur_im(0,j)
else if (mod(j,2).eq.0) then
gcur_re(j)=2*etacur_re(0,j)
gcur_im(j)=2*etacur_im(0,j)

endif
enddo
C Integrate current density on the electrode surface
do j=0,ny
value_re= gcur_re(j)
value_im= gcur_im(j)
c value_re=(gcur_re(j)+gcur_re(j+ 1))*(rout (j+)ru(j)/
ttcur re=ttcur re+value re
ttcur im=ttcur im+value im
enddo
C Calculate Global Impedance
tcur_re=ttcur_re*0.02/3
tcur_im=ttcur_im*0.02/3
globalim_re=deltav*tcur_re/(tcur_re**2+t uri*2
globalim_im=deltav*tcur_im/(tcur_re**2+t uri*2
dimenim_re=globalim_re*ck*a
dimenim_im=globalim_im*ck*a
Reff=4*dimenim re
c_ceff=dimenim_im*omegaspi

write(35,11) omega,tcur_re,tcur_im,globalim_re,globaliim
& dimenim_re,dimenim_im,Reff,c_ceff
return
end

APPENDIX C
MATHEMATICAL MODEL FOR A DISSOLUTION OF ZINC ROTATING DISK(
ELECTRODE

As a preliminary step towards development of the model for the cathodic delamination,

a one-dimensional, transient model was developed for the dissolution and passivation of a

rotating zinc disk in a dilute solution of NaC1. The model treated explicitly the coupling

of mass transport phenomena, electrochemical reactions and homogeneous reactions.

C.1 Model Development

The advantages of using a rotating disk electrode system are that the hydrodynamics

conditions are well understood and the fluid mechanics associated within the system is

well studied. A schematic illustration of the flow field generated by a rotating disk is

presented in Figure where z is the direction perpendicular to the disk and r is direction

along the disk electrode. The rotation of the disk causes a spiral movement of the fluid

resulting in a net velocity toward the disk and in the radial direction.

C.1.1 Mass Transfer

The mass transfer of a species i in an electrochemical system is governed by equation

(2-1) where the flux of a species i is given in equation (2-2). Combination with the

T. I ad i-Einstein equation (equation (2-4)), under the assumptions of a steady-state

condition and an incompressible electrolyte, the governing equation for ci can be rewritten

Di
0 = zeF c4iV (c4V4) + DiV2Ci U Ci Ri (-1)
RT

where the terms on the right side represent the contribution of migration, diffusion,

convection, and production by homogeneous reactions, respectively.

The steady flow created by an infinite disk rotating at a constant angular velocity

in a fluid with constant physical properties was presented by ?-. \--in! .1, [46] Under the

assumption that the velocities in r and 8 direction are negligible, the velocity normal to

Electrode |Insulator

Figure C-1. Schematic representation of a rotating disk electrode system in which a disk
electrode is embedded in a large insulator.

the disks that brings reactants to the surface is expected to follow

vz= H (C-2)

where v is the kinematic viscosity and R. Near the electrode surface, the dimensionless

velocity H in equation (C-2) can be expressed as a power series

where the coefficients a and b have values of 0.51023 and -0.616, respectively.

C.1.2 Electrode Kinetics

The electrochemical reactions of interest in the present model involved zinc

dissolution (4-6) and oxygen reduction (4-7). Under the assumption that the electrochemical

reaction considered are irreversible, the current densities due to zinc dissolution followed

the Bulter-Volmer expression. The oxygen reduction was assumed to be mass-transfer-limited,

thus, the limiting current density, shown in equation (2-39), depends on the concentration

of oxygen in the bulk. The thickness of the diffusion 1 en rrx seen in equation (2-39), in a

cylindrical coordinate, is a function of rotation speed by

( 1 13 v\ 1 /2 (C-4
aSc e
w hecr e Sc is Sch midt num ber a nld r (1 ) i the gam ma- func- t ion- of 1/3

C.1.3 Homogeneous Reactions

The oxygen reduction taking place underneath the coating results in an increase of pH

in the interfacial degraded lI e. r. For zinc, a series of chemical reactions associated with

Zn2+ hydrolysis is possible in alkaline solutions. [66, 67] In the presented model, multiple

homogeneous reactions, including water dissociation and a series of reactions associated

with Zn2+ hydrolysis, were considered. The mechanisms and equilibrium conditions of

these chemical reactions are summarized in Table 4-2.[67]

C.1.4 Boundary Condition

At the far boundary condition, the concentrations and solution potential were fixed

at bulk conditions cs,,= 1.26 x 10-3 M and @,=0. On the boundary of the metal

surface the zero-flux condition was used for the chemically inert species Na+ and Cl-.

The concentration of oxygen was set as zero because the oxygen reduction reaction was

assumed to be mass-transfer limited. The boundary conditions at the metal surface for

Zn2+ and OH- were obtained by relating their fluxes with the current densities due to the

electrochemical reactions on the metal surface. The equilibrium conditions listed in Table

4-2 were treated as boundary conditions for H+, ZnOH+, HZnO2 and ZnO -.

C.1.5 Solution Method

The system of coupled, non-linear, partial differential equations required an iterative

method to converge on a solution starting from an initial guess. A tri-diagfonal method,

BAND algorithm, was chosen to calculate the distribution of ce and 4. The mathematical

model was developed using M~icrosoft V/isual Fortran, V/ersion 9.0 with double precision

accuracy. [74]

C 10 1 Z
10 ZOH.
c10 -

ZnOH'

I0 10" ZnO

O 2 4 6 8 10 12
Dimensionless Position / (z / 8)

Figure C-2. Calculated concentration distributions of species of OH-, Zn2+, H+, ZnOH+,
HZnO,, and ZnO on electrode surface.

C.2 Results and Discussion

The calculated distributions satisfied the coupled phenomena of species mass

transport and electronutrality. The domain length was 0.2 cm, the rotating speed was R

=50 rad/s, the thickness of the diffusion 1 c;- was x = 0.018 cm, and the metal potential

was chosen to be 9= -0.77 V.

The concentration distributions of the chemical species are presented in Figure C-2

as a function of dimensionless position. Due to the electrochemical reactions occurring

on the electrode surface, the concentrations of OH- and Zn2+ have a maximum near the

surface. The concentrations of the species produced in the homogeneous reactions (Ht,

ZnOH+, HZnO,, and ZnO -) are largest near the surface and decrease with increasing

distance away from the electrode.

At steady state, based on the mass balance equation, the homogenous rate for species

Ri can he written by

Ri = V Nsi (C-5)

Thus, the rate for each of the homogeneous reactions can he expressed as follows:

R =V 1V NH+ (6

2.0x10 I 4 5 10 a a
n %zn" + OH' ++ZnOH'
0.0" E 10 12 C znOH +20H ++ HnOn, + H,O
-207r109 R: H,0 ++H'+0 wi : HZn0; +OH' ++Zn02'+ ~O
O .j O, 10f O10'1 -
-6.0x'10 10~2

-8 0>1 10" R 2

; -1.0x10 *.

-1.4x10 I < 10 's 1
0 2 4 6 8 10 12 0 2 4 6 8 10
Dimension less Position /(z / 8) Dimensioniess Position / (z / 5)

(a) (b)

Figure C-3. Calculated rate of the homogeneous reactions included in the model. a) Water
disassociation reaction; and b) Zinc hydrolysis reactions.

R3 HZnOz ZnO

and

R4 = .1ZnO2- (C-9)

The rate of water dissociation is plotted in Figure C-3(a) as a function of dimensionless

position. The OH- ions generated near the electrode surface combine with H+ ions in the

solution to form water molecular. Due to the constraint of the equilibrium condition, the

concentration of H+ ions is small near the electrode; consequently, the rate of this reaction

is approximately zero near the surface. When z/6 approaches 5.4, the concentrations of

OH- ions becomes close to that of H+ ions, leading to the sharp increase in Figure C-3(a).

The rates of the zinc hydrolysis are presented in Figure C-3(b) as a function of

dimensionless position. The negative value of R2 indicates that ZnOH+ ions dissociate

into Zn2+ and OH- ions. This implies that, after the Zn2+ and OH- ions are formed, the

ZnOH+ ions were immediately produced near the surface. This can also be used to explain

the negative rates of reactions 3 and 4.

C.3 Conclusion

A niathentatical model for one-dintensional, steady-state rotating disk electrode

system was developed. In this model, multiple heterogeneous reactions and equilibrated

homogeneous reactions were coupled with mass transport due to migration, diffusion and

convection. The calculated results demonstrates that after corrosion reaction occurs the

pH- value of the electrolyte has been changed significantly due to the formation of OH-

ions. The results also show the coupling among the multiple homogeneous reactions that

take place simultaneously in the system.

APPENDIX D
MATHEMATICAL MODEL FOR GALVANIC COUPLING IN A 2-D CELL

As a step toward the development of comprehensive model for cathodic delamination,

a two-dimensional model was developed that calculated the distributions of concentrations

and potential associated with cut-edge corrosion. Within the present model, the

Zn electrode serves as the local anode and steel as the local cathode. The purpose

of the model was to understand the set up of the galvanic couple by starting from

uniform distributions of all reactive species. The uniform initial conditions permit the

concentration and potential gradients appearing later due to the electrochemical reactions.

Within the model, multiple homogeneous reactions, including water dissociation and a

series of reactions associated with hydrolysis were assumed to occur simultaneously in the

solution phase.

D.1 Model Development

A schematic representation of the Zn-Fe model is given in Figure D-1 where Zn acts

as an anode, Fe acts as a cathode, and NaCl serves as the electrolyte. An insulator

is inserted between the two electrodes and the two vertical walls are composed of

insulators as well. Zinc dissolution was assumed to take place on the anode, whereas

oxygen reduction and hydrogen evolution were both assumed to occur on the cathode. No

iron dissolution was considered on the steel.

Zn Fe

Figure D-1. Schematic representation of a two electrode cell in which Zn serves a local
anode and Fe as a local cathode.

The development of the present model is similar to that presented in Appendix C.

The key differences were that the governing equations were expanded in a two-dimensional

form and different boundary conditions were applied on the anode, cathode and insulator.

In this model, the anodic and cathodic current expression are applied on the Zn (0<

x <0.016 cm) and Fe (0.024< x <0.04 cm) electrodes, respectively. On the region where

the insulator is inserted (0.016< x <0.024 cm), a zero-flux condition was emploi-x I for all

species.

D.2 Solution Method

The calculations by Allahar show that the accuracy of applied numerical technique

pll li- an important role on determining whether the equilibrium relations that describe

the homogeneous reactions can or cannot be incorporated. [21, 22] As a result, a numerical

technique with high-order accuracy is extremely crucial for this particular development.

In the development of the present model, a commercial program based upon

collocation method, PDE2D, was chosen. The use of the collocation method yields

approximations that are of high-order accuracy even when coupled nonlinear partial

differential equations are solved in a multidimensional domain. With this PDE2D

program, all homogeneous reaction can be included simultaneously and the discontinuity

at the bottom boundary can be handled without numerical difficulties.

D.3 Results and Discussion

The dimension of the domain was set to be 0.04 cm in the direction along the

electrodes and 0.02 cm in the direction away from the electrodes. At the bottom

boundary, 0< x <0.016 cm is the region where anodic reaction is dominant, and

0.024< x <0.04 cm is the region where cathodic reactions are more important. The

potential on the metal was chosen as V = -1.1 VSHE. The polarization parameters for zinc

dissolution included pzn =0.08 V/decade, io,zn =12 mA/cm2, Eoz,z =- 0.763 VSHE, PH2

=0.18 V/decade, i0,Hz = 10-smA/cm2, and EO,H2 =- 0.828 VSHE

0.002
8.42x10 1 1
0.001 Z Fe8.41x10 -
8.4x10 -
0.000
S8.39x10 -
S -0.001 8.38x10 -
-0.002 ,
8.38x10 -
-0.003 1- j )-1 8.35x"10 -
8.34x10 I 1
-0.004 0.000 0.004 0.008 0.012 0.016
0.00 0.01 0.02 0,03 0.04
x /cm
xlcm

(a) (b)

Figure D-2. Calculated current density distributions along the x axis. a) 0< r <0.04 cm;
and b) 0< r <0.016 cm.

The overall current density distribution along the :r axis is given in Figure D-2(a)

where the anodic current density is di;111 li4I within 0< :r <0.016 cm, and the cathodic

current density is within 0.024< :r <0.04 cm. The current density drops to zero between

0.016 cm and 0.024 cm, corresponding to the position of the insulator. The distribution

of the anodic current density on the Zn electrode is presented in Figure D-2(b). It is

clear in Figure D-2(b) that the anodic current density increases when approaching the

edge of the electrode. Both Figures D-2(a) and D-2(b) confirm that the geometry of the

electrodes constrain the distributions of current density in a way such that the periphery

of the disk has a greater accessibility. The distributions given in Figure D-2 also show that

the applied collocation method is able to handle the discontinuous transition from the

insulator to the two electrodes.

The concentration distributions of Zn2+ and OH- ions in units of mole/cm3 are

presented in Figure D-3 in contour format. Due to the local anodic reaction, the

concentration of Zn2+ ioUS has a largest around the Zn electrode and decreases with

increasing distance away from the local anode. A similar behavior was observed in the

concentration distribution of OH- ions.

0.020

0.016 -

0.012 -
E
'-0.008 -

0.004 -

0.000
0.00

0.020

0.016

0.012

.5EE--8

-

1.3E-7

8 9E-8 1.8E 2.2

0.008-

0.004 -

0.000 -
0.00

0.03

0.01 0.02 0.03
x/cm

0.01

0.02
x/cm

0 04

Figure D-3. Calculated distributions of concentration in a unit of mole/cm3. a) Zn2+ ioUS;
and b) OH- ions.

0.020 I
Zn Fe
0.016
3 0E-1

E 0.012 6OE-15
.1E-15
0 .008~ 12E-14
1 5E-14
0.004 1.E1

0.000
0.00 0.01 0.02 0 03 0.04
x /cm

F'igur~e D-4. Calculated distribution of ratio of czo OH-- Where Ksp, IS thle standard
solubility product of Zn(OH)2(s)*

192

The calculated concentrations of Zn2+ and OH- ions can be used to predict the

formation of corrosion product Zn(OH)2(s). The predicted distribution of precipitated

Zn(OH)2(s) is presented in Figulre D-4. The magnitude in Figure D-4 is estimated by
C~2+C
cz2OH- Where Ksp IS the standard solubility product of Zn(OH)2(s) With a value of
S2 C
3x10-"7 (mol/1iter)3 at, TOOm temperatulre.[10] The value of czo2 OH, :Which reflects the

driving force of forming Zn(OH)2(s), iS laTgeSt near the local cathode and decreases with

the distance away from the electrode.

D.4 Conclusion

The presented work serves as a preliminary approach for modeling the two-dimensional

cathodic delamination system. The uniform initial concentration distributions applied in

the model allowed us to simulate the natural establishment of the galvanic element. The

computational results indicate that the numerical difficulties associated with including

homogeneous reactions in a two spatial dimension and with the discontinuous region at the

boundary have been overcome in a steady-state model. The results also showed that the

unique configuration of the delamination system led to an interesting distribution pattern

of the species produced in the homogeneous reactions. The emploi-v I numerical technique

will be tested later for treating precipitation of Zn(OH)2 in a traUSient model.

APPENDIX E
PARAMETER SENSITIVITY ANALYSIS

Due to the use of porosity-pH, poisoning-pH and blocking-pH relations, there are

several fitting parameters used in the cathodic delamination program. It is important

to explore the sensitivities of these parameters on the simulation results. The sensitivity

analysis is presented in this appendix. In the report, each fitting parameter were analyzed

using three different values. The velocity of the moving fronts and the kinetic analysis

were checked to determine the sensitivity.

E.1 Porosity

There are four fitting parameters used in the construction of the equilibrated

porosity-pH- relation (see Figure 4-2). The fitting parameter be,i is associated with the

constant value seen at the high pH region. The fitting parameter be,2 gOVOTrlS the slope of

the curve that increases from low pH to high pH regions. The deflection point located in

the middle of the increasing curve is controlled by the fitting parameter be,3. The constant

value seen at the low pH region is governed by the fitting parameter be,4-

E.1.1 b~

The sensitivity analysis for the fitting parameter be,i is summarized in Table E-1.

When the fitting parameter be,i that governs the porosity at high pH increases from 0.01

to 0.1, the rate of the delamination determined by both potential front and porosity

front increase by approximately 60 percent. It is observed that when be,i is equal 0.1, the

interfacial potential in the intact region increases with delamination time. The change of

be,i from 0.01 to 0.001 decreases the moving velocities by approximately 40 percent. The

reaction order, however, is not influenced by the change to the fitting parameter be,1.

Table E-1. Sensitivity analysis for be,i
be,i = 0.001 be,i = 0.01 be,i = 0.1
Potential Front Velocity 1.33 2.19 3.58
Porosity Fr-ont Velocity 1.09 1.6,3 3.27
Reaction Order for Potential Front 0.54 0.55 0.56
Reaction Order for Porosity Front 0.59 0.6 0.6

Table E-2. Sensitivity analysis for be,2
be,2 -5 be,2 -:3 be,2 -2
Potential Front Velocity 2.263 2.19 1.98
Porosity Fr-ont Velocity 2.20 1.6:3 1.18
Reaction Order for Potential Front 0.56 0.55 0.55
Reaction Order for Porosity Front 0.6:3 0.6; 0.6;

0.012

0.010 ~ =-

0.008 -?

E o~oosC b ~= -3
.t 0.006 -

0.002-

0.000
8 9 10 11 12 13 14
pH

Figure E-1. The sensitivity of the slope of the increasing curve to be,2-

E.1.2 be,2

The sensitivity analysis for the fitting parameter be,2 is Summarized in Table E-2. As

illustrated in Figure E-1, the slope of the increasing curve in Figure 6-6 becomes steeper

when the fitting parameter be,2 changes from -2 to -5. The increase in the slope results

in a sight increase in the velocities of both potential front and porosity front. The other

observed feature is that the rates of the two fronts are approximately equal when the

slope is steep. It is also observed that the shape of the resulting potential distributions

(see Figure 6-6) is also influenced by this parameter. The reaction order, however, is not

influenced by the change in be,2-

E.1.3 bs,;3

The sensitivity analysis for the fitting parameter be,,3 is summarized in Table E-:\$.

The deflection point of the increasing curve seen in Figure 4-2 shifts from left to right

when the fitting parameter be,;3 increases from 9.8 to 11.8. When the deflection point is

Table E-:\$. Sensitivity analysis for bs,:4
bs,:4 9.8 be,:4 10.8 be,:4 11.8
Potential Front Velocity 2.7:3 2.19 1.7:3
Porosity Front Velocity 2.27 1.6:3 0.5
Reaction Order for Potential Front 0.5:3 0.55 0.56
Reaction Order for Porosity Fr-ont 0.54 0.6; 1.1

Table E-4. Sensitivity analysis for bs,4
bs,4 =10-4 be,4 =10-3 be,4 =10-2
Potential Front Velocity 1.94 2.19 :3.6;6
Porosity Front Velocity 1.62 1.6:3 2.98
Reaction Order for Potential Front 0.56 0.55 0.56
Reaction Order for Porosity Fr-ont 0.61 0.6 0.72

pushed to high pH region, the velocities of the fronts decrease. For bs: = 11.8, the velocity

of the porosity front decreases to 0.5 nin/hr and the reaction order analysis shows a

kinetic-controlled niechanisni for the porosity front.

E.1.4 be,4

The sensitivity analysis for the fitting parameter be,4 is sunmmarized in Table E-4.

The fitting parameter be,4 coOtrOls the constant value seen at the low pH region in Figure

4-2. As shown in Table E-4, the decrease in this fitting parameter does not have strong

impact on the rate or the niechanisni of the delamination. However, when be,4 increases

to 0.01, the movements of both potential and porosity fronts increase dramatically. For

be,4=0.01, the niechanisni of the delamination remains nmass-transfer controlled for the

potential front, but changes to a nmixed-controlled niechanisni for the porosity front.

E.2 Poisoning Factor

Four fitting parameters were used in the construction of the poisoning factor-pH

relation (see Figure 4-:3(b)). The fitting parameter bC,1 influences the constant value at the

high pH region. The fitting parameter bC,2 gOVerns the slope of the transition from low pH

to high pH regions. The deflection point seen in the middle of the increasing curve changes

with the fitting parameter bC,s4 The constant value seen at the low pH region is related to

the fitting parameter bC,4-

Table E-5. Sensitivity analysis for bC,1
bc,l 3.5 be,, 4.5 be,l 5.5
Potential Fr-ont Velocity 2.18 2.19 2.16
Porosity Front Velocity 1.6 1.63 1.6
Reaction Order for Potential Front 0.55 0.55 0.56
Reaction Order for Porosity Fr-ont 0.6;2 0.6; 0.61

Table E-6. Sensitivity analysis for b(,2
be,2 = -1.3 be,2 = -3.3 be,2 = -5.3
Potential Front Velocity 2.1 2.19 2.2
Porosity Fr-ont Velocity 1.35 1.63 1.75
Reaction Order for Potential Front 0.55 0.55 0.55
Reaction Order for Porosity Front 0.62 0.6 0.62

E.2.1 bc,i

The sensitivity analysis for the fitting parameter be,1 is summarized in Table E-5.

When the high-pH value in Figure 4-3(b) increases by two orders (bc,l changes from 3.5 to

5.5), the rate of the delamination remains approximately the same. The reaction orders

determined based upon the potential front and the porosity front are not influenced much

by the fitting parameter bC,1.

E.2.2 b(,2

The sensitivity analysis for the fitting parameter b(,2 iS Summarized in Table E-6.

The slope of the increasing curve in Figure 4-3(b) becomes steeper when the fitting

parameter be,2 changes from -1.3 to -5.3. The change in be,2 does not influence the rate

determined by the potential front, but has a slight impact on that determined by the

porosity front. For the values that have been tested, the kinetic analysis indicates that the

change in be,2 has no significant impact on the delamination mechanism.

E.2.3 b(,3

The sensitivity analysis for the fitting parameter be,3 1S Summarized in Table E-7.

The deflection point of the increasing curve seen in Figure 4-3(b) shifts from left to

right when the fitting parameter b(,3 changes from 9.4 to 11.4. The simulation results

do not change much when be,3 changes from 9.4 to 10.4, but the velocities of both fronts

Table E-7. Sensitivity analysis for bC,:4
bc,:4 9.4 bc,:4 10.4 bC,:4 11.4
Potential Front Velocity 2.15 2.19 2.06
Porosity Fr-ont Velocity 1.62 1.6:3 1.3:3
Reaction Order for Potential Front 0.55 0.55 0.55
Reaction Order for Porosity Front 0.6;2 0.6; 0.6;2

Table E-8. Sensitivity analysis for b(,4
b(,4 = -15 b(,4 = -16; bC,4 -17
Potential Fr-ont Velocity 2.20 2.19 2.16
Porosity Front Velocity 1.6i8 1.6:3 1.65
Reaction Order for Potential Front 0.56 0.55 0.56
Reaction Order for Porosity Fr-ont 0.6:3 0.6 0.62

decrease slightly when bC,:4 changes from 10.4 to 11.4. For the values that have been

tested, the kinetic analysis shows that the change in be,:4 has no significant impact on the

delaminationniechanism.

E.2.4 b(,4

The sensitivity analysis for the fitting parameter b(,4 iS Sunmmarized in Table E-8.

The computational results remain approximately the same when b(,4 changes from -15, -16

to -17. For the values that have been tested, the simulation results are insensitive to this

fittingf parameter.

E.3 Blocking Factor

The blocking factor was used in the cathodic current density expression. In order

to characterize an enhanced electrochentical reactivity for oxygen reduction in the front

region, a slight increase was assigned in pH- range 10 to 11 in Figure 4-4(b). Seven fitting

parameters were used in the construction of Figure 4-4(b). The fitting parameter badl

is related to the constant value seen at the low pH- region. The fitting parameter ba,2

is associated with the slope of the curve at the low pH- region. The deflection point at

low-pH- curve is controlled by the fitting parameter bo.:4 The fitting parameter ba,4 is

associated with the constant value at the high pH- region. The fitting parameter be 0 is

related to the slope of the short curve seen at high pH-. The length of the constant region

Table E-9. Sensitivity analysis for bo,i

Potential Front Velocity
Porosity Fr-ont Velocity
Reaction Order for Potential Front
Reaction Order for Porosity Front

Table E-10. Sensitivity analysis for ba,2

2.0
1.60
0.56
0.6;2

ba2 = -10
2.29
1.78
0.55
0.62

bo,i = 7.5
2.19
1.63
0.55
0.6

ba,2 = -7
2.19
1.6,3
0.55
0.6

bo,i = 8.5
2.52
2.29
0.57
0.65

ba 2 -4
2.07
1.45
0.55
0.62

Potential Fr-ont Velocity
Porosity Front Velocity
Reaction Order for Potential Front
Reaction Order for Porosity Fr-ont

in the middle range of pH is characterized by ba,6 and the magnitude of the constant seen

at the front region is governed by the fitting parameter box7.

E.3.1 b~

The sensitivity analysis for the fitting parameter boyl is summarized in Table E-9.

When the fitting parameter boyl changes from 6.5 to 7.5, the computational results in

terms of delamination rate and mechanism do not change much. However, when boyl

changes from 7.5 to 8.5, corresponding to an increase in the blocking factor at low pH,

the rates and kinetic order determined by both potential front and porosity front show

increasing tendency.

E.3.2 ba2

The sensitivity analysis for the fitting parameter ba,2 is Summarized in Table E-10.

The slope of the increasing curve at low pH in Figure 4-4(b) becomes steeper when the

fitting parameter ba2 changes from -4 to -10. The increase in the slope results in an

increase in the velocities of both potential front and porosity front. For the values that

have been tested, the kinetic analysis indicates that the change in ba2 has no significant

impact on the delamination mechanism.

Table E-11. Sensitivity analysis for boss
boss = 8. boss 9.8 boss 10.8
Potential Front Velocity 2.51 2.19 2.0
Porosity Front Velocity 1.71 1.6:3 1.5
Reaction Order for Potential Front 0.58 0.55 0.55
Reaction Order for Porosity Front 0.59 0.6; 0.61

Table E-12. Sensitivity analysis for ba,4
ba,4 = -2.5 ba,4 = -1.5 ba,4 = -0.5
Potential Front Velocity 2.11 2.19 2.26;
Porosity Front Velocity 1.18 1.6:3 1.78
Reaction Order for Potential Fr-ont 0.55 0.55 0.55
Reaction Order for Porosity Front 0.62 0.62 0.6

E.3.3 bss

The sensitivity analysis for the fitting parameter bo,,3 is summarized in Table E-11.

The deflection point of the increasing curve at low pH in Figure 4-:3(b) shifts from left

to right when the fitting parameter bo,:3 changes from 8.8 to 10.8. When the deflection

point is pushed to high pH region, the velocities of the fronts decrease. For boss = 10.8, the

velocity of the porosity front decreases to 1.5 mm/hr, but the reaction order based upon

the porosity front does not change with boss.

E.3.4 ba,4

The sensitivity analysis for the fitting parameter ba,4 is Summarized in Table E-12.

The fitting parameter ba,4 is aSsociated with the constant value seen at the high pH

region in Figure 4-:3(b). When ba,4 changes from -2.5 to -0.5, the blocking factor at high

pH increases from 10-5 to 10- ; thus, the delamination rates increase with ba,4. The

delamination mechanism is not influenced significantly by ba,4-

E.3.5 box~

The sensitivity analysis for the fitting parameter box5 is summarized in Table E-1:3.

The fitting parameter box5 is related to the slope of the short curve seen at high pH in

Figure 4-4(b). When the fitting parameter box5 increases from -60 to -40, the slope of the

curve becomes more gradual. The computational results remain approximately the same

Table E-13. Sensitivity analysis for ba,s
ba,s -6;0 ba~s =-50 ba,s -40
Potential Front Velocity 2.20 2.19 2.23
Porosity Fr-ont Velocity 1.66 1.63 1.63
Reaction Order for Potential Front 0.55 0.55 0.55
Reaction Order for Porosity Front 0.6;2 0.6; 0.6;2

Table E-14. Sensitivity analysis for ba,6
ba,6 =10.1 ba,6 =11.1 ba,6 =12.1
Potential Front Velocity 2.15 2.19 2.15
Porosity Front Velocity 1.6 1.63 1.66
Reaction Order for Potential Front 0.55 0.55 0.55
Reaction Order for Porosity Front 0.61 0.6 0.62

for three values of ba,s. For the values that have been tested, the simulation results are

insensitive to this fittingf parameter.

E.3.6 ba,6

The sensitivity analysis for the fitting parameter ba,6 is Summarized in Table E-14.

In Figure 4-4(b), the length of the constant region in the middle range of pH- increases

with the fitting parameter ba,6. The computational results remain approximately the same

when ba,6 inCTreSeS from 10.1 to 12.1. For the values that have been tested, the simulation

results are insensitive to this fittingf parameter.

E.3.7 bari

The sensitivity analysis for the fitting parameter boy7 is summarized in Table E-15.

In Figure 4-4(b), the magnitude of the constant in the middle range of pH- increases from

10-4 to 10-2 When ba,6 changes from -11.1 to -9.1. The increase in the blocking factor

leads to a increase in the delamination rates. The kinetic analysis for -11.1 and -10.1

Table E-15. Sensitivity analysis for bar
br= -11 ba -=-10 ba = -9
Potential Front Velocity 1.96 2.19 2.363
Porosity Front Velocity 1.47 1.63 1.77
Reaction Order for Potential Front 0.55 0.55 0.57
Reaction Order for Porosity Front 0.59 0.6 0.68

shows the same delamination mechanism, but shows a nmixed-controlled niechanisni for the

porosity front when bo,a is equal to 12.1.

The sensitivity analysis presented above indicates that the parameters used in

the construction of the porosity-pH relation (bs,i to bs,4) are 111st sensitive ones in the

simulations. The fitting parameter be,i that governs the porosity at high pH influences up

to 60 percent of the delamination rate. The shape of the resulting potential distributions

is also influenced by that of the equilibrated porosity-pH relation. The the parameters

used in the construction of the poisoning factor-pH relation (bC,1 to b(,4) and blocking

factor-pH relation (bo,i to bo,7) are, in general, less sensitive. However, the increase in the

cathodic current density at the front region results in an increase in delamination rate and

a shift in delamination kinetics.

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BIOGRAPHICAL SKETCH

Mei-Wen Huang grew up in Tainan, Taiwan. She received her bachelor of science

degree front the Clu! non! I1 Engineering Department at the National Chlungt-Hsiang

University in Taiwan in June, 2002. She began her graduate studies at the University of

Florida in August 2002 to pursue a master of science degree. After the completion of the

degree requirements, she joined Professor Mark E. Orazents electrochentical engineering

research group in 2003 to pursue a doctor of philosophy degree. She graduated in the

suniner of 2007 after spending four years being educated in chemical and electrochentical

engineering.

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page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 9 LISTOFFIGURES .................................... 10 ABSTRACT ........................................ 17 CHAPTER 1INTRODUCTION .................................. 19 1.1MathematicalModelsforCathodicDelaminationofCoatedMetal ..... 19 1.2InuenceofGeometry-InducedCurrentAndPotentialDistributionofDiskElectrodesonImpedanceResponse ...................... 22 2BACKGROUNDELECTROCHEMISTRY ..................... 25 2.1MassTransport ................................. 25 2.2SolutionPotential ................................ 26 2.3ElectrochemicalKinetics ............................ 27 2.3.1KineticControl ............................. 30 2.3.2MassTransferControl ......................... 31 3LITERATUREREVIEWONCATHODICDELAMINATION .......... 33 3.1ExperimentalObservation ........................... 33 3.2MathematicalModels .............................. 37 3.2.1CreviceandDisbondedCoatingModels ................ 38 3.2.2CathodicDelaminationModel ..................... 39 3.2.2.1pH-DependentPorosity ................... 40 3.2.2.2pH-DependentPolarizationKinetics ............ 41 3.3Objective .................................... 42 4THEORETICALDEVELOPMENTOFDELAMINATIONMODEL ...... 43 4.1Porosity-pHRelation .............................. 43 4.2PolarizationKinetics .............................. 45 4.2.1ZincDissolution ............................. 46 4.2.2Poisoning-pHRelation ......................... 46 4.2.3OxygenReduction ............................ 47 4.2.4Blocking-pHRelation .......................... 48 4.3ChemicalReactions ............................... 49 5

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...................... 51 5.1GoverningEquations .............................. 51 5.2BoundaryCondition .............................. 53 5.3SolutionMethod ................................ 53 6RESULTSANDDISCUSSIONFORDELAMINATIONMODEL ........ 55 6.1InitialConditions ................................ 55 6.1.1InitialConcentrationDistributions ................... 56 6.1.2InitialDistributionofPorosity ..................... 56 6.1.3InitialDistributionofPolarizationParameters ............ 57 6.2EquilibriumPorosity-pHRelationship ..................... 59 6.2.1InterfacialPotentialDistribution .................... 60 6.2.2ConcentrationDistributions ...................... 62 6.2.3PrecipitatedCorrosionProduct .................... 64 6.2.4PorosityDistribution .......................... 67 6.2.5DelaminationKinetics ......................... 68 6.2.5.1InuenceofCationTypeonDelaminationRate ...... 69 6.2.5.2InuenceofAnionTypeonDelaminationRate ...... 72 6.2.5.3InuenceofElectrolyteConcentrationonDelaminationRate .............................. 72 6.3KineticPorosity-pHRelationship ....................... 75 6.3.1PotentialFrontandPorosityFront ................... 75 6.3.2DelaminationKinetics ......................... 80 7ELECTROCHEMICALIMPEDANCESPECTROSCOPY ............ 81 7.1PassiveElectricalCircuits ........................... 83 7.2Constant-PhaseElement(CPE) ........................ 85 7.2.1OriginofCPE .............................. 85 7.2.22-Dand3-DDistributions ....................... 86 7.3CurrentandPotentialDistributionsonDiskElectrode ........... 88 7.3.1PrimaryCurrentDistribution ..................... 88 7.3.2SecondaryCurrentDistribution .................... 89 7.4Objective .................................... 91 8IDEALLYPOLARIZEDBLOCKINGDISKELECTRODE ........... 92 8.1TheoreticalDevelopment ............................ 92 8.2DenitionofImpedance ............................ 95 8.2.1GlobalImpedance ............................ 95 8.2.2LocalImpedance ............................ 97 8.2.3LocalInterfacialImpedance ...................... 97 8.2.4LocalOhmicImpedance ........................ 98 8.2.5GlobalInterfacialImpedance ...................... 98 8.2.6GlobalOhmicImpedance ........................ 98 6

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............................. 99 8.3.1GlobalImpedance ............................ 99 8.3.2LocalInterfacialImpedance ...................... 101 8.3.3LocalImpedance ............................ 103 8.3.4LocalOhmicImpedance ........................ 105 8.3.5GlobalInterfacialandGlobalOhmicImpedance ........... 107 9BLOCKINGDISKELECTRODEWITHLOCALCPE .............. 108 9.1TheoreticalDevelopment ............................ 108 9.2ResultsandDiscussion ............................. 109 9.2.1GlobalImpedance ............................ 109 9.2.2LocalInterfacialImpedance ...................... 113 9.2.3LocalImpedance ............................ 115 9.2.4LocalOhmicImpedance ........................ 117 9.2.5GlobalInterfacialandGlobalOhmicImpedance ........... 117 9.3Experiments ................................... 118 9.3.1GlobalImpedanceofGlassy-CarbonElectrode ............ 119 9.3.2LocalImpedanceofStainlessSteelElectrode ............. 120 10DISKELECTRODEWITHSINGLEFARADAICREACTION ......... 122 10.1TheoreticalDevelopment ............................ 122 10.2ResultsandDiscussion ............................. 125 10.2.1GlobalImpedance ............................ 125 10.2.2LocalInterfacialImpedance ...................... 129 10.2.3LocalImpedance ............................ 130 10.2.4LocalOhmicImpedance ........................ 131 10.2.5GlobalInterfacialandGlobalOhmicImpedance ........... 135 10.3InterpretationofImpedanceResults ...................... 136 10.3.1DeterminationofChargeTransferResistance ............. 136 10.3.2DeterminationofCapacitance ..................... 137 11CONCLUSIONANDRECOMMENDATION ................... 142 11.1MathematicalModelsforCathodicDelaminationofCoatedMetal ..... 142 11.2InuenceofGeometry-InducedCurrentAndPotentialDistributionofDiskElectrodesonImpedanceResponse ...................... 143 APPENDIX APROGRAMLISTINGFORTHECATHODICDELAMINATION ........ 145 A.1MainProgramListing ............................. 145 A.2SubroutineProgramListing .......................... 145 7

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.......... 173 B.1MainProgramListing ............................. 173 B.2SubroutineListing ............................... 176 CMATHEMATICALMODELFORADISSOLUTIONOFZINCROTATINGDISKELECTRODE ................................. 183 C.1ModelDevelopment ............................... 183 C.1.1MassTransfer .............................. 183 C.1.2ElectrodeKinetics ............................ 184 C.1.3HomogeneousReactions ........................ 185 C.1.4BoundaryCondition .......................... 185 C.1.5SolutionMethod ............................. 185 C.2ResultsandDiscussion ............................. 186 C.3Conclusion .................................... 188 DMATHEMATICALMODELFORGALVANICCOUPLINGINA2-DCELL .. 189 D.1ModelDevelopment ............................... 189 D.2SolutionMethod ................................ 190 D.3ResultsandDiscussion ............................. 190 D.4Conclusion .................................... 193 EPARAMETERSENSITIVITYANALYSIS ..................... 194 E.1Porosity ..................................... 194 E.1.1b";1 194 E.1.2b";2 195 E.1.3b";3 195 E.1.4b";4 196 E.2PoisoningFactor ................................ 196 E.2.1b;1 197 E.2.2b;2 197 E.2.3b;3 197 E.2.4b;4 198 E.3BlockingFactor ................................. 198 E.3.1b;1 199 E.3.2b;2 199 E.3.3b;3 200 E.3.4b;4 200 E.3.5b;5 200 E.3.6b;6 201 E.3.7b;7 201 REFERENCES ....................................... 203 BIOGRAPHICALSKETCH ................................ 210 8

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Table page 4-1FittingparametersusedintheexpressionsofpH-dependentinterfacialporosity,blocking,andpoisoningparameters. ......................... 49 4-2Reactionmechanismandequilibriumconditionforhomogeneousreactionsincludedinthemodel ...................................... 50 6-1Diusioncoecientsofchemicalspecies ...................... 55 6-2Diusioncoecientsofcations ........................... 69 6-3Diusioncoecientsofanions ............................ 72 6-4Calculatedvelocitiesofpotential,porosityandpHfront ............. 79 8-1Notationproposedforlocalimpedancevariables .................. 96 E-1Sensitivityanalysisforb";1 194 E-2Sensitivityanalysisforb";2 195 E-3Sensitivityanalysisforb";3 196 E-4Sensitivityanalysisforb";4 196 E-5Sensitivityanalysisforb;1 197 E-6Sensitivityanalysisforb;2 197 E-7Sensitivityanalysisforb;3 198 E-8Sensitivityanalysisforb;4 198 E-9Sensitivityanalysisforb;1 199 E-10Sensitivityanalysisforb;2 199 E-11Sensitivityanalysisforb;3 200 E-12Sensitivityanalysisforb;4 200 E-13Sensitivityanalysisforb;5 201 E-14Sensitivityanalysisforb;6 201 E-15Sensitivityanalysisforb;7 201 9

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Figure page 2-1Polarizationplotsforoxygenreduction ....................... 32 3-1Cathodicdelaminationsystem ............................ 35 3-2Experimentalinterfacialpotentialdistribution ................... 36 3-3Disbondedcoatingsystemonacoatedmetal. ................... 39 4-1Interfacialporosity"andpHasfunctionsofpositioninthedelaminatedzone.Thedashedlinesseparatethedomainintothedelaminated,front,andfully-intactregions:a)interfacialporosity;andb)localpH. .................. 44 4-2Distributionofinterfacialporosity"asafunctionoflocalpH. .......... 45 4-3Distributionofpoisoningfactor:a)asafunctionofposition;andb)asafunctionoflocalpH. ...................................... 47 4-4DistributionofblockingfactorO2:a)asafunctionofposition;andb)asafunctionoflocalpH. ................................. 48 6-1InitialconcentrationdistributionsofOH,Na+,Cl,andZn2+ionsalongthemetal-coatinginterface. ................................ 56 6-2Initialporositydistribution:a)asafunctionofposition;andb)asafunctionofpH. ........................................... 57 6-3Initialdistributionofpoisoningfactor:a)asafunctionofposition;andb)asafunctionofpH. .................................... 58 6-4Initialdistributionofblockingfactor:a)asafunctionofposition;andb)asafunctionofpH. .................................... 58 6-5InterfacialpotentialasafunctionofabsolutenetcurrentdensitywithlocalpHasaparameter.ThedistributionsassociatedwiththepHvaluesof8.7and9aresuperimposed. ................................... 59 6-6Distributionsofinterfacialpotentialalongthemetal-coatinginterfacewithelapsedtimeasaparameter. ................................. 60 6-7DistributionsofdV/dxalongthemetal-coatinginterfacewithelapsedtimeasaparameter. ....................................... 61 6-8Instantaneousvelocityofpotentialfront,calculatedfromthetime-dependentpositionofthemaximagiveninFigure 6-7 .................... 62 6-9DistributionsofpHalongthemetal-coatinginterfacewithelapsedtimeasaparameter. ....................................... 63 10

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................... 63 6-11ConcentrationdistributionsofZn2+ionsalongthemetal-coatinginterfacewithelapsedtimeasaparameter. ............................. 64 6-12Concentrationdistributionsalongthemetal-coatinginterfacewithelapsedtimeasaparameter.a)ZnOH+ions;b)HZnO2ions;andc)ZnO22ions. ....... 65 6-13ConcentrationdistributionsofprecipitatedcorrosionproductZn(OH)2(s)alongthemetal-coatinginterfacewithelapsedtimeasaparameter. .......... 66 6-14Distributionsofporosityalongthemetal-coatinginterfacewithelapsedtimeasaparameter. ...................................... 67 6-15Distributionsofd/dxalongthemetal-coatinginterfacewithelapsedtimeasaparameter. ....................................... 68 6-16Instantaneousvelocityofporosityfront,calculatedfromthetime-dependentpositionofthemaximagiveninFigure 6-15 .................... 69 6-17Delaminateddistanceasafunctionofelapsedtimeindouble-logarithmicscalewithcationtypeasaparameter.Theconcentrationoftheelectrolyteatthedefectis0.5M.a)Determinedbypotentialfront;b)Determinedbyporosityfront;andc)Experimentalresultsobtainedfromcoatedelectrogalvanizedsteelsamples. ........................................ 70 6-18Delaminateddistanceasafunctionofsquarerootoftimewithcationtypeasaparameter.Theconcentrationoftheelectrolyteatthedefectis0.5M.a)Determinedbypotentialfront;b)Determinedbyporosityfront;andc)Experimentalresultsobtainedfromcoatedsteelsamples. ......................... 71 6-19Delaminateddistanceasafunctionofsquarerootoftimewithaniontypeasaparameter.Theconcentrationoftheelectrolyteatthedefectis0.5M.a)Determinedbypotentialfront;b)Determinedbyporosityfront;andc)Experimentalresultsobtainedfromcoatedsteelsamples. ......................... 73 6-20Delaminateddistanceasafunctionofsquarerootoftimewithelectrolyteconcentrationasaparameter.a)Delaminateddistancedeterminedbypotentialfront;b)Delaminateddistancedeterminedbyporosityfront;andc)Experimentalresultsobtainedfromcoatedsteelsamples.DatatakenfromStratmannetal.withpermissionofCorrosionScience. .................................. 74 6-21Distributionofinterfacialpotentialalongthemetal-coatinginterfacea)kneq=0.1s1;andb)kneq=0.001s1. .......................... 76 6-22DistributionofinterfacialpotentialgradientdV/dxalongthemetal-coatinginterfacea)kneq=0.1s1;andb)kneq=0.001s1. ..................... 76 11

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......................... 77 6-24DistributionofpHgradientalongthemetal-coatinginterfacea)kneq=0.1s1;andb)kneq=0.001s1. ............................... 78 6-25Distributionsofd/dxalongthemetal-coatinginterfacewithelapsedtimeasaparameter. ....................................... 79 6-26Delaminateddistanceasafunctionofdelaminationtimeindouble-logarithmicscalewithcationtypeasaparameter.a)Delaminateddistancedeterminedbythepotentialfront;andb)Delaminateddistancedeterminedbytheporosityfront. .......................................... 80 7-1Smallsignalanalysisofanelectrochemicalnonlinearsystem ........... 81 7-2Passiveelementsthatserveascomponentsofanelectricalcircuit.a)Resistor;b)Capacitor;andc)Inductor. ............................ 83 7-3Combinationsofpassiveelementsthatserveascomponentsofanelectricalcircuit 84 7-4Schematicrepresentationofanimpedancedistributionforablockingdiskelectrode 87 7-5Primarycurrentdensitydistributionatadiskelectrode. ............. 90 7-6SecondarycurrentdistributionatadiskelectrodewithJasaparameter. .... 90 8-1Coordinatestransformationfromacylindricalcoordinatetoarotationalellipticcoordinate ....................................... 93 8-2Thelocationofcurrentandpotentialtermsthatmakeupdenitionsofglobalandlocalimpedance. ................................. 95 8-3Nyquistrepresentationoftheimpedanceresponseforanideallypolarizeddiskelectrode.a)linearplotshowingeectofdispersionatfrequenciesK>1;andb)logarithmicscaleshowingagreementwiththecalculationsofNewman. ..... 99 8-4Representationoftheimpedanceresponseforanideallypolarizeddiskelectrode.a)realpart;andb)imaginarypartshowingagreementwiththecalculationsandasymptoticformulaofNewman. ........................ 100 8-5Theslopeoflog(Zj=r0)withrespecttolog(K)(Figure 8-4(b) )asafunctionoflog(K).Theresultswerecalculatedbythecollocationmethod.Thevalueofthisslopeisequalto. ............................... 101 8-6ThefrequencyK=1atwhichthecurrentdistributioninuencestheimpedanceresponsewith=C0asaparameter. ......................... 102 12

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...................................... 102 8-8ThelocalimpedanceinNyquistformatwithradialpositionasaparameter. .. 103 8-9Localimpedancewithradialpositionasaparameter:a)realpart;andb)imaginarypart. .......................................... 104 8-10Localimpedanceasafunctionofradialposition:a)realpart;andb)imaginarypartmultipliedbydimensionlessfrequencyK. ................... 105 8-11ThelocalOhmicimpedanceinNyquistformatwithradialpositionasaparameter. 106 8-12ValuesforlocalOhmicimpedanceasafunctionoffrequencywithradialpositionasaparameter:a)realpart;andb)imaginarypart. ................ 106 8-13TheimaginarypartoftheglobalOhmicimpedance,calculatedfromequation( 8{28 ),asafunctionofdimensionlessfrequency. .................. 107 9-1NyquistrepresentationforthecalculatedimpedanceresponseofablockingdiskelectrodewithalocalCPEwithasaparameter. ................ 110 9-2ImpedanceresponseforablockingdiskelectrodewithalocalCPEasafunctionofdimensionlessfrequencyK:a)realpart;andb)imaginarypart. ........ 111 9-3Slopeoflog(Zj=r0)withrespecttolog(K)(Figure 9-2(b) )asafunctionoflog(K)withasaparameter. ............................ 112 9-4EectivescaledCPEcoecientasafunctionoffrequencywithasaparameter. 113 9-5NyquistrepresentationforthecalculatedlocalinterfacialimpedanceresponseofablockingdiskelectrodewithalocalCPEwithnormalizedradialpositionasaparameter. ....................................... 114 9-6Localinterfacialimpedanceasafunctionoffrequencywithpositionasaparameter:a)imaginarypart;andb)realpart. ......................... 114 9-7Localinterfacialimpedanceasafunctionofpositionwithfrequencyasaparameter:a)imaginarypart;andb)realpart. ......................... 115 9-8ThelocalimpedanceinNyquistformatwithradialpositionasaparameter. .. 116 9-9Localimpedance:a)realpart;andb)imaginarypart. ............... 116 9-10ThelocalOhmicimpedanceinNyquistformatwithradialpositionasaparameter. 117 9-11ValuesforglobalOhmicimpedanceasafunctionoffrequencywithasaparameter:a)realpart;andb)imaginarypart. ......................... 118 13

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........................................ 133 10-11RepresentationofthelocalOhmicimpedanceresponseforadiskelectrodeasafunctionofdimensionlessfrequencyKunderassumptionsofTafelkineticswithJ=1:0representationofthelocalOhmicimpedanceresponseforadiskelectrodeasafunctionofdimensionlessfrequencyKunderassumptionsofTafelkineticswithJ=1:0.a)realpart;andb)imaginarypart. ................. 134 10-12GlobalOhmicimpedanceresponseforadiskelectrodeasafunctionofdimensionlessfrequencyforlinearkineticswithJasaparameter.a)realpart;andb)imaginarypart. .......................................... 135 10-13TheapparentvalueofRe=RtobtainedfromthecalculatedimpedanceresponseatlowfrequenciesasafunctionofJ. ........................ 137 10-14Theapparentvalueof1-obtainedfromthecalculatedimpedanceresponseathighfrequenciesasafunctionofJ. ......................... 138 10-15EectiveCPEcoecientscaledbytheinterfacialcapacitanceasafunctionofJ. 139 10-16Eectivecapacitancecalculatedfromequation( 10{21 )andnormalizedbytheinputinterfacialcapacitanceforadiskelectrodeasafunctionofdimensionlessfrequencyKwithJasaparameter. ......................... 140 10-17NormalizedeectivecapacitancecalculatedfromrelationshipspresentedbyBruget.alforadiskelectrodeasafunctionofdimensionlessfrequencyKwithJasaparameter.a)withcorrectionforOhmicresistanceRe(equation( 10{22 ));andb)withcorrectionforbothOhmicresistanceReandcharge-transferresistanceRt(equation( 10{23 )). ................................ 141 C-1Schematicrepresentationofarotatingdiskelectrodesysteminwhichadiskelectrodeisembeddedinalargeinsulator. ..................... 184 C-2CalculatedconcentrationdistributionsofspeciesofOH,Zn2+,H+,ZnOH+,HZnO2,andZnO22onelectrodesurface. ...................... 186 C-3Calculatedrateofthehomogeneousreactionsincludedinthemodel.a)Waterdisassociationreaction;andb)Zinchydrolysisreactions. ............. 187 D-1SchematicrepresentationofatwoelectrodecellinwhichZnservesalocalanodeandFeasalocalcathode. .............................. 189 D-2Calculatedcurrentdensitydistributionsalongthexaxis.a)0
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................................ 192 E-1Thesensitivityoftheslopeoftheincreasingcurvetob";2. ............. 195 16

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Thisdissertationcoverstworesearchtopicsthatareimportanttocorrosionofmetal.Therstresearchtopicinvolvesdevelopingamathematicalmodelfordelaminationofpolymericlmsfromactivemetals.Thedelaminationofpaintfrommetalsurface,alsoknownascathodicdelamination,isamajorproblemforautomotiveandbuildingapplications.Experimentalndingsdemonstratedthatthecathodicdelaminationprocessinvolvedcouplingofmasstransfer,electrochemicalreactions,lossofinterfacialadhesions,andpropagationofthedelaminationalongthemetal-coatinginterface.Theseexperiments,inprinciple,areusefulforidentifyingthephenomenaoccurringinthedelaminationsystems.However,itisdiculttoverifythataproposedmechanismdoesindeedgiveobservedexperimentalresults.Therefore,aquantitativeapproachwastakentosimulatethetransit,propagationphaseofthedelaminationprocessoncoatedmetals. Thesecondprojectpresentedinthisdocumentexplored,bytheoreticalcalculations,theroleofcurrentandpotentialdistributionsassociatedwithdiskelectrodesonimpedanceresponse.Electrochemicalimpedancespectroscopy(EIS)isoftenappliedasatooltoinvestigatetherateofcorrosion.Impedancespectra,however,areofteninuencedbythecurrentandpotentialdistributionsonelectrodesurfaces.Thisworkinvestigatestheinuenceofthegeometriceectassociatedwithdiskelectrodesonimpedanceresponsesanddescribesthiseectintermsofconstantphaseelement(CPE). TheintroductionforthecathodicdelaminationmodelispresentedinSection 1.1 .Section 1.2 presentstheintroductionfortheimpedancecalculation. 1 ]andformsaphysicalbarrierbetweenmetalandatmosphere. 19

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Thelaterstageofthedelaminationbecomescomplicatedwhencorrosionproductsareformedunderneaththepaint.FurbethandStratmann[ 10 ]reportedthattheprecipitationofZnCO3,underahighCO2concentrationinatmosphere,inhibitedtheoxygenreductionandledtoapureanodicdelaminationforacoatedelectrogalvanizedsteel.Ogleetal.[ 19 20 ]studiedthecathodicdelaminationongalvanizedsteelandfoundthatthechemicalstabilityoftheinterfacialoxidelayersplaysacriticalroleondeterminingthepropagationrateandthedelaminationmechanism.[ 19 20 ] Itisdiculttodemonstratethataproposedmechanismsuggestedbasedonexperimentalworkdoesindeedgivetheobservedexperimentalresults.Interpretationisoftenlimitedtoqualitativeandsubjectiveobservations.Therefore,aquantitativeapproachisnecessary.Allahar[ 21 22 ]developedtherstmathematicalmodelforthecathodicdelaminationofcoatedmetal.Thekeytohisworkinvolvedapplyingtheconceptthattheporosityandthepolarizationkineticsatthemetal-coatinginterfacewerepHdependent.ThesimulationresultsprovidedqualitativeagreementswiththeexperimentalobservationsreportedbyStratmannetal.[ 9 { 11 14 { 16 ],whichsupportedthehypothesesemployedinthemodel.ThemodeldevelopedbyAllahar,however,didnotincorporatechemicalreactionsthattakeplaceatthelaterstageofthedelaminationprocess. Theobjectiveoftheworkwastodevelop,fromrstprinciples,amathematicalmodelthatsimulatesthepropagationofdelaminationinthepresenceofelectrochemicalandchemicalreactions.Thechemicalreactionsincludedhomogeneousreactionsintheinterfacialoxidizedlayerandformationofcorrosionproductunderneaththecoating.Asecondobjectivewastoexaminewhetherthedelaminationrateandthedelaminationmechanismpredictedfromthemodelagreewiththeexperimentalresults. 21

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23 24 ]andtodetermineinterfacialcapacitance.[ 25 26 ]Theimpedanceresponseofanelectrodeisgeneratedbymeasuringtheratioofappliedpotentialtosurface-averagedcurrentasafunctionoffrequency.Theinuenceofaparticularphenomenonontheimpedanceresponseisdeterminedbythetimeconstantofthatprocess.[ 27 ]Masstransfereectsareusuallyapparentatlowfrequenciesbecausethediusivityofionicspeciesinaqueousmediumissmall.Kineticanddouble-layereectsaremoreimportantathighfrequencies.AnimportantadvantageofEISisthattheinuenceofgoverningchemicalandphysicalphenomenacanbedistinguishedwithasingleexperimentalprocedureencompassingasucientlybroadrangeoffrequency. ThecriticalissueofEISistheambiguityassociatedwiththeinterpretationofimpedanceresults.[ 25 28 29 ]Acommonapproachofinterpretingimpedancedataistocompareexperimentalspectrawiththatofknownelectricalcircuitelementssuchasresistors,capacitors,andinductors.[ 28 ]Thecircuitanalogmodelsarefoundusefulforunderstandingthephysicalprocessesthatcontributetoimpedanceresponses;however,experimentaldatararelyshowtheidealresponseexpectedforelectrochemicalreactions.Theimpedanceresponsetypicallyreectsadistributionofreactivitythatiscommonlyrepresentedinequivalentcircuitsasaconstant-phaseelement(CPE).[ 29 30 ] ThedispersionleadingtoCPEbehaviorcanbeattributedtodistributionsoftimeconstantsalongeithertheareaoftheelectrode(atwo-dimensionalsurface)oralongtheaxisnormaltotheelectrodesurface(athree-dimensionalaspectoftheelectrode).A2-Ddistributionmightarisefromsurfaceheterogeneitiessuchasgrainboundaries,crystalfacesonapolycrystallineelectrode,variationsinsurfaceproperties,orgeometry-inducedcurrentandpotentialdistributions.[ 31 { 34 ]A3-Ddistributionsmaybeattributedto 22

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Thefundamentalelectrochemicalconceptsrelevanttothecathodicdelaminationsystem,suchaselectrodekineticsandtransportindilutesolutions,arepresentedinChapter 2 .ExperimentalandsimulationworkassociatedwiththecathodicdelaminationsystemarereviewedinChapter 3 .TheconstructionsofthepH-dependentporosityandpH-dependentpolarizationkineticsarealsopresentedinChapter 3 ThemodeldevelopedbyAllahardidnotincludechemicalreactionsthatbecomeimportantatthelaterstageofthedelamination.Inthepresentwork,multiplehomogeneousreactionsandprecipitationofcorrosionproductswereconsideredintheoxidizedlayer.ThetheoreticaldevelopmentofthepresentmodelfollowedtheapproachestakenbyAllahar[ 22 ].ThedevelopmentofthemodelispresentedinChapters 4 and 5 .ThecomputationalresultsarepresentedinChapter 6 Thesecondpartofthisdissertation,presentedinChapters 7 to 10 ,explorestheinuenceofgeometry-inducedcurrentandpotentialdistributionsontheimpedanceresponseofadiskelectrode.Electrochemicalimpedancespectroscopy(EIS)isarapidandconvenienttechniquethatprovideselectrochemicalpropertiesoftestedsystemsoverawiderangeoffrequencies.AbriefintroductiontoEISandissuesencounteredinEISarepresentedinChapter 7 ThecurrentandpotentialdistributionsassociatedwithadiskelectrodeembeddedinaninsulatingplanearereviewedinChapter 7 .Thetheoreticaldevelopmentandcalculationresultsfortheideally-polarizedblockingelectrodearepresentedinChapter 8 ,theresultsfortheblockingelectrodewithalocalCPEbehaviorarepresentedinChapter 9 ,andtheresultsforthediskelectrodeexhibitingaFaradaicreactionarepresentedinChapter 10 24

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Thefundamentalelectrochemicalconceptsrelevanttothecathodicdelaminationsystem,suchasmasstransportofionicspeciesindilutesolutionsandelectrodekinetics,arepresentedinthischapter.AdetailedtreatmentofelectrochemistryinelectrolytefromamathematicalperspectivehasbeenpresentedbyNewman.[ 46 ] 46 ] wherethetermsontherightsiderepresentthenetinputduetotheuxNiandthenetrateofproductionduetohomogeneousreactionsRi,respectively.Indiluteelectrochemicalsystems,NiisgivenbytheNernst-Planckequation[ 46 ] whereisthelocalsolutionpotential,uiisthemobility,Diisthediusioncoecient,ziisthechargenumber,visthemassaveragevelocityoftheelectrolyte,andFisFaraday'sconstant.Thetermsontherightsideofequation( 2{2 )representthecontributionsbymigration,diusion,andconvectiontotheuxofaspecies,respectively. Combinationofequations( 2{1 )and( 2{2 ),undertheassumptionthattheelectrolyteisincompressible(rv=0),yieldsthegoverningequationforciinastagnantelectrochemicalsystem 25

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46 ] isgenerallyapplicabletodiluteelectrochemicalsystemswhereRisthemolargasconstantandTistheabsolutetemperature.Combinationofequations( 2{2 )and( 2{4 )yieldstheuxofaspecies Equations( 2{3 )isrewrittenas byemployingtheNernst-Einsteinequation. 46 ] "Xizici(2{7) whereisthepermittivittyofthemedium.Anexpressionbasedontheconceptofelectroneutralityatapoint,i.e. hasbeenusedasthegoverningequationfor.Newmanhasshownthatequation( 2{8 )providesaverygoodapproximationtoPoisson'sequationoutsidethethindoublechargelayernearelectrodes.[ 46 ]ItisimportanttonotethattheassumptionofelectroneutralitydoesnotimplythatLapalace'sequationholdsforthepotential,becausethisapproximationismadeonthebasisofalargevalueofF=inequation( 2{7 ).[ 46 ] 26

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(2{9) Theioniccurrentdensityduetothemotionofchargedparticlesinanelectrolyticsolutioniscalculatedby[ 46 ] Combinationofequations( 2{5 )and( 2{10 ),intheabsenceofconvection,yields wheretheconductivityisdenedas Theioniccurrentdensityinequation( 2{11 )canbedividedintomigrationanddiusioncontributions.Thedrivingforceforthemigrationanddiusioncurrentdensitiesarepotentialandconcentrationgradients,respectively.[ 46 ] Intheabsenceofconcentrationgradients,equation( 2{11 )reducestoanexpressionofOhm'slaw (2{13) Combinationofequation( 2{13 )withequation( 2{9 )yieldsLaplace'sequation (2{14) forthesolutionpotential. 46 ] 27

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AB++e(2{15) Forthereactiondescribedbyequation( 2{15 ),theforwardreactionisanodicandthebackwardreactioniscathodic.Theoverallrateofthereactionrisgiven whererfandrbaretheforwardandbackwardrates,respectively. Undertheassumptionthateachreactionisrstorder,theoverallratecanbewrittenas Fromactivatedcomplextheory,equation( 2{17 )canberecastas RTVcAkcexpnF RTVcB(2{18) whereVistheinterfacialpotential,(knownasthesymmetryfactor)isthefractionoftheappliedpotentialthatfavorsthecathodicreaction,andnisthenumberofelectronstransferred.Equation( 2{18 )canbewrittenintermsofcurrentdensityias[ 46 ] nF=kaexp(1)nF RTVcAkcexpnF RTVcB(2{19) TheinterfacialpotentialVisdenedas whereisthepotentialofthemetalandisthepotentialinelectrolyticsolutionadjacenttotheelectrode. Whentheanodicandcathodicreactionsinequation( 2{15 )reachthesamerate,azerocurrentisobtainedundertheconditionofreactionequilibrium.AttheequilibriumpotentialV0,thenetrateofthereactioniszero;however,theindividualratesofthe 28

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RTV0cA(2{21) or RTV0cB(2{22) Substitutionofi0intoequation( 2{19 )yieldstheButler-Volmerequation[ 46 ] RTsexpcF RTs(2{23) wherethesurfaceoverpotentialsisgivenbys=VV0,theanodictransfercoecientaisgivenbya=(1)n,andthecathodictransfercoecientisgivenbyc=n.Thesurfaceoverpotentialsrepresentsthedeparturefromanequilibriumpotentialsuchthat,ats=0,thetotalcurrenti=ia-icisequaltozero. TheexponentialbehavioroftheButler-Volmerequationresultsinacharacteristicfeatureofelectrochemicalreactions.InthelimitofaFsRT,equation( 2{23 )canbereducedto RTs(2{24) Solvingforsinequation( 2{24 )gives aFlni i0(2{25) or aFlog10i i0(2{26) TheTafelslopefortheanodicreactionaisgivenbytheexpressioninfrontofthelogterminequation( 2{26 ) aF(2{27) 29

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cF(2{28) TheButler-Volmerequation,equation( 2{23 ),canberecastusingtheseTafelslopesinto whereEaandEcaretheeectiveequilibriumpotentialsgivenby fortheanodicreactionand forthecathodicreaction,respectively.[ 47 ] Zn!Zn+2+2e(2{32) andhydrogenevolutionreaction 2H2O+2e!H2+2OH(2{33) maybothoccurinacorrosionsystem.Undersuchcircumstance,theindividualelectrochemicalreactionscanbetreatedindependently.Therefore,aButler-Volmerequationsuchasequation( 2{23 )canbewrittenforeachofthesereactions. ThecurrentdensityduetothereversiblecorrosionreactioniZn 30

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Inmostcorrosionsystems,metaldissolutionisconsideredtobeanirreversiblereaction.Thus,thecurrentdensityduetothezincdissolutioniZnisgivenby whereEa;Znanda;ZnarereplacedbyEZnandZn,respectively.Similarly,thecurrentdensityiH2duetothehydrogenevolutioninreaction( 2{33 )isgivenby undertheassumptionthatthereactionisirreversible.Thetotalcurrentdensityforthegivensystemisthesumoftheindividualcurrentdensity Thepolarizationbehaviorofthecorrosionandthehydrogenevolutionaretermedactivationpolarizationbecausetheratesoftheelectrochemicalreactionsaredrivenbythesurfaceoverpotentials. 2-1 foranoxygenreductionreaction.Thepolarizationbehavioroftheoxygenreductioncontainsactivationandconcentrationcomponents.Thecurrentdensitiesisafunctionofpotentialintheactivationpolarizationpart,butisindependentofpotentialintheconcentrationpolarizationpart.Thereactionrateintheconcentrationpolarizationregimeislimitedbytherateoftransportofoxygentothemetalsurface.Themass-transfer-limitedcurrentdensity,symbolizedasilim;O2,dependsonsolutionagitation,temperatureandconcentrationofthelimitingspecies.[ 48 ]Thenumericalvalueofthe 31

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Polarizationplotsforoxygenreductionattwovaluesofmass-transfer-limitedcurrentdensitylabelledasi1andi2.Thehorizontaldashedlineseparatestheactivationpotentialandconcentrationpolarizationpartsoftheplots. limitingcurrentdensityisgivenby wherexandcO2arethedistancethatoxygendiusesthroughandtheoxygenconcentrationinthebulk,respectively.ThecurrentdensityduetotheoxygenreductioniO2isgivenbythemathematicalexpression toaccountforbothactivationandconcentrationregimes. 32

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Asdescribedinliterature,thedelaminationofpaintunderhumidandcorrosiveenvironmentinvolvesacouplingofmasstransfer,electrochemistry,lossofadhesionatthemetal-coatinginterface,andpropagationofamovingfrontalongtheinterface.Experimentalandsimulationworkassociatedwiththecathodicdelaminationsystemarereviewedbrieyinthischapter. 9 { 18 49 ]employedascanningKelvinprobetomeasurepotentialdistributionsatburiedpolymer/metalinterfaces.Theyperformedtheexperimentsunderacceleratedcorrosiveconditionsandminimizedsurfacetreatments,thusthedelaminationratepredictedfromtheirsampleswerelargerthanthatobservedfromcommercialtechnicalsamples.[ 19 ]Williametal.[ 50 51 ]alsoemployedthescanningKelvinprobetechniquetostudytheinuenceofinhibitorsonthedelaminationmechanismofcoatedgalvanizedsteel. AnewtechniquebasedonFouriertransforminfrared-multipleinternalreection(FTIR-MIR)allowedin-situmeasurementsofthethicknessofwaterlayeratmetal-coatinginterface.[ 52 53 ]Thistechniqueprovidedameanstodeterminetherateofwatertransportthroughthecoatingandtocalculatethediusioncoecientofwaterthroughthepolymerlm.Jorcinetal.[ 54 ]exploredthedelaminationphenomenaatasteel/epoxy-vinylprimeinterfaceusinglocalelectrochemicalimpedancemapping.Theresultsshowedthatthedelaminatedareameasuredbyvisualobservationsaftertheremovalofthecoatingwereapproximatelythreetimessmallerthanthatdeterminedbythelocalelectrochemicalimpedancemapping. 33

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(b) Schematicrepresentationofexperimentalinterfacialpotentialdistribution.a)interfacialpotentialdistributionasafunctionofpositionwithelapsedtimeasaparameter;andb)dV/dxdistributionasafunctionofpositionwithelapsedtimeasaparameter. distributionsismaintainedthroughouttheexperiments.Thelengthoftheintactregiondecreaseswithincreasingtime.Williametal.[ 50 51 ]alsoobtainedasimilartransitioninthepotentialmeasurementsforacoatedelectrogalvanizedsteelpigmentedwithCrO24andCe3+ions. TheinterfacialpotentialdistributionwasdierentiatedwithrespecttopositiontoyielddV/dxdistributions,showninFigure 3-2(b) ,asafunctionofpositionwithelapsedtimeasaparameter.ThepeakmarkedthedeectionpointoftheabruptincreaseobservedinFigure 3-2(a) .Thepositionofthepeak,recognizedasthedelaminationfront,representedaregionwherethereactionofbreakingbondsisongoing.ThepeakheightcorrespondingtothevalueofdV/dxwasobservedtodeceasewithdelaminationtime.LengandStratmann[ 15 ]suggestedthat,withincreasingtime,therewasamoregradualchangeintheelectrochemicalpotentialacrossthefrontregion. ThedelaminationkineticscanbedeterminedbyplottingthedelaminateddistancecalculatedfromthedV/dxcurveasafunctionoftime.Leng[ 15 ]andWilliam[ 51 ]employedageneralpowerlaw 36

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11 ]reportedavalueofa=0.55fromcoatedsteelwithminimizedsurfacetreatments,whichconcludedthattheoveralldelaminationprocesswasgovernedprimarilybythetransportofcationsfromthedefecttothedelaminatedzone.Theirresultsalsodemonstratedthat,fornon-pigmentedsamplesandashort-timeexposure,therateofthedelaminationdependedstronglyontheionicstrengthandthemobilityofcations.[ 11 ] Experimentalobservationshaveshownthatthelaterstageofthedelaminationbecomesmorecomplicatedwhenpassivelmsorcorrosionproductsareformedunderneaththecoating.[ 10 19 ]FurbethandStratmann[ 10 ]reportedthattheformationofZnCO3,underahighCO2concentrationintheatmosphereinhibitstheelectrontransferofoxygenreduction,resultinginapureanodicdelaminationforcoatedelectrogalvanizedsteel.TheprecipitationofZn(OH)2orZnOatthedefectpreventscorrosionoftheuncoatedarea,therefore,thecathodicreactionbecomesdominatingonthescratchforcoatedzincsamples.Moreover,afterlonger-timeexposure,asthedelaminationfrontmovesfurtherawaythedefect,bothanodesandcathodescanappearunderthepaintanditbecomesdiculttodistinguishwhetherthemovingfrontisanodicorcathodic. Numerousresearcheortshavebeenmadetostudytheinuenceofinhibitorsandsurfacetreatmentsonthedelaminationrateandthedelaminationkinetics.Hernandezetal.[ 57 ]indicatedthatthezinc-aluminumphosphatepigmentsreducedthedelaminationratebyformingaphosphatelayerunderneaththepaint.Ogleetal.[ 19 ]testedvarioussurfacetreatmentsunderbothanodicandcathodicdelaminationconditions.TheresultswereinterpretedintermsofthedieringchemicalstabilityoftheconversionlayerstowardOHionsgeneratedbyoxygenreduction. 37

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TheuseofthepH-dependentinterfacialporosityrepresentedanovelapproachtoaccountimplicitlyforthebond-breakingprocessduringthecathodicdelamination.AsprovenbyAllahar,nopropagationwasobservedwhenthepH-porosityrelationwasnotincorporatedinthemodel.[ 22 ] 22 ]derivedexpressionsforthepolarizationkineticsatthemetal-coatinginterfacebaseduponexpressionsapplicabletoabaremetalsurface. Thecurrentdensityduetozincdissolutionandoxygenreductionatthezinc-coatinginterfacewerecalculatedusing[ 22 ] and ThesurfacecoveragewArepresentedthesurfaceareaavailableforelectrochemicalreactions.ThepoisoningparameterwZnconsideredtheinuenceofsurfaceontheexchangecurrentdensityofzincdissolution.TheeectofthecoatingorotherformeddepositsonthetransportofoxygenthroughthegelmediumwasincludedbyablockingparameterwO2.AstheeectoftheseparametersvariedaccordingtothelocalpH,theparameterswerelinkedtopHwiththesamemannerdescribedintheporosity-pHrelation. Theuseofporositytorepresentthebond-breakingprocessoccurringduringthecathodicdelaminationprocessprovidedamathematicalframeworkforthedevelopmentofadvancedmodels.ThenumericalapproachemployedbyAllharwasnotabletoinclude 41

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Thepropagationmodelsimulatedtheevolutionofthedelaminationprocessfromgiveninitialconditions.Thepropagationoffrontandthebond-breakagereactionsaccompanyingthedelaminationprocessweremodeledthroughthehypothesesthatporosityandthepolarizationkineticsatthecoating-metalinterfacewerepHdependent.MathematicalconstructionsfortheinitialconditionsandthepH-dependenthypothesesfollowedthedevelopmentbyAllhar.Modicationsweremadetoreducethenumberofthettingparametersusedintheprogram. 46 ] whereciistheconcentrationintheaqueousphase.TheeectivediusioncoecientDiinaporousmediumwasrelatedtotheporosity"by[ 64 ] whereDiisthediusivityinanaqueousmedium.Equation( 4{1 )wasrecastintermsofDiusingequation( 4{2 )as TheconservationofaspeciesiinaporousmediumwasexpressedintermsofNias whereNiwastheuxinthesolutionphase. 43

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Schematicdiagramfordistributionofinterfacialporosity"asafunctionoflocalpH. coatingdegradationandthelossofadhesion,[ 9 { 18 ]theporosityusedtorepresenttheadhesionbetweenthemetalandthecoatingwastreatedasafunctionofpH. Theporosity-pHrelationwasobtainedbycombiningFigures 4-1(a) and 4-1(b) toyieldthe"distributionasafunctionofpHshowninFigure 4-2 .Themathematicalexpressionforthe"-pHrelationshipwasobtainedbyttinganequationoftheform totheplotinFigure 4-2 whereb";1tob";4werettingparameters.Theporositywasassumedtoreachthevaluegivenbyequation( 4{5 )instantaneously;thusthe"-pHrelationshiprepresentsanequilibriumconditionbetween"andpH. Zn!Zn2++2e(4{6) andoxygenreduction O2+2H2O+4e!4OH(4{7) 45

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48 ] whereZn,EZn,io;ZnaretheTafelslope,equilibriumpotential,andexchangecurrentdensity,respectively,forthezincdissolution.Apoisoningfactorwasemployedtocalculatethecurrentdensityduetothezincdissolutionatthemetal-coatinginterface,i.e.[ 22 ] Thepoisoningfactorconsideredtheeectofcoating,theavailabilityofsurfaceareatozincdissolutionduringthedelamination.Thefactoralsoaccountedimplicitlyforthepresenceofpassivelmsformedonthemetalsurface.ThepoisoningfactorwasassumedtobeafunctionofpH,andtheconstructionofthe-pHrelationshipwasperformedinamannersimilartotheconstructionofthe"-pHrelationship. 9 16 ]Thus,thezincdissolutionwasconsideredunfavorableinthedelaminatedzonebyassigning1.AnhypothesizeddistributionofasafunctionofpositionispresentedinFigure 4-3(a) .Inthedelaminatedregion,thepoisoningparameterisapproximatelyaconstant,indicatingthatthesurfaceavailabilitytothezincdissolutionisindependentofpositioninthisregion.Thepoisoningparameterdecreasesexponentiallywithpositioninthefrontregion,demonstratingthattheanodicreactionisunfavorableinthefront.Inthefully-intactregionthevalueofisheldasaconstant. 46

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(b) Schematicdiagramfordistributionofpoisoningfactor:a)asafunctionofposition;andb)asafunctionoflocalpH. TherelationshipbetweenandpH,giveninFigure 4-3(b) ,wasconstructedbycouplingthedistributionsshowninFigures 4-3(a) and 4-1(b) .Themathematicalexpressionforthe-pHrelationshipwasobtainedbyttinganequationoftheform totheplotinFigure 4-3(b) whereb;1tob;4werettingparameters. 48 65 ] wherexisthedistancethatoxygendiusesthroughandcO2;1istheoxygenconcentrationinthebulk.Inthepresenceofcoatingandinterfacialoxidizedlayer,themass-transfer-limitedcurrentdensityduetooxygenreductionwascalculatedusing[ 22 ] 47

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(b) SchematicdiagramfordistributionofblockingfactorO2:a)asafunctionofposition;andb)asafunctionoflocalpH. wheregmandgcwerethethicknessofthegel-mediumandthecoating,respectively,and"cand"gweretheporosityoftheun-degradedcoatingandthegel-medium,respectively.Thecomplextermseenattherightsideofequation( 4{12 )wasderivedbysolvingtheconcentrationdistributionofoxygeninthedirectionnormaltothemetalsurface.TheblockingfactorO2accountedfortheinuenceofthecoatingandtheoxidizedlayeronthetransportofoxygentothemetalsurface.TheblockingfactorO2wasassumedtobeafunctionofpHandtheconstructionoftheO2-pHrelationshipwasperformedinamannersimilartotheconstructionofthe"-pHrelationship. 4-4(a) .Theblockingfactorisaconstantinthedelaminatedregion,indicatingthatthetransportofoxygenisindependentofpositioninthisregion.Alargestelectrochemicalreactivityforoxygenreductionisexpectedacrossthefrontregion;thus,theblockingfactorincreasesinthefront.Theblockingfactordecreasestoaminimumtorepresentasmallestreactivityintheintactregion. TheO2-pHrelationwasconstructedbycouplingFigures 4-4(a) and 4-1(b) toyieldtheO2distributionasafunctionofpHgiveninFigure 4-4(b) .Themathematical 48

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FittingparametersusedintheexpressionsofpH-dependentinterfacialporosity,blocking,andpoisoningparameters. k. O2 0.01 4.50 7.50b2 -3.00 -3.30 -7.00b3 10.8 10.4 9.80b4 0.001 -16.0 -0.50b5 -50.0b6 11.10b7 -10.0 expressionfortheO2-pHrelationwasobtainedbyttinganequationoftheform 1+exp(b;2(pHb;3))+b;4 1+exp(b;5(pHb;6))+b;7)(4{13) totheplotinFigure 4-4(b) whereb;1tob;7werettingparameters. Thevaluesofthettingparametersusedinequations( 4{5 ),( 4{10 )and( 4{13 )arelistedinTable 4-1 .Thechoiceofthevaluesoftheseparametersmightplayanimportantroleonthecomputationalresults.Thus,asystematicsensitivityanalysiswasperformedandtheresultsarereportedinAppendix E 66 67 ]Inthepresentedmodel,multiplehomogeneousreactions,includingwaterdissociationandaseriesofreactionsassociatedwithZn2+hydrolysis,wereconsidered.ThemechanismsandequilibriumconditionsofthesechemicalreactionsaresummarizedinTable 4-2 .[ 67 ]Allthehomogeneousreactionswereassumedtobeequilibratedbecausethetimeconstantsforreachingtheequilibriumconditionsaremuchsmallerthanthatforthediusionofthelimitingreactant.[ 68 69 ] TheprecipitatedcorrosionproductZn(OH)2(s)isthermodynamicallystablewithinthepHrangingfrom8.5to11.[ 66 67 ]ThereactionmechanismofformingsolidZn(OH)2(s)

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Reactionmechanismandequilibriumconditionforhomogeneousreactionsincludedinthemodel.[ 66 67 ] ReactionNO. ChemicalReaction EquilibriumCondition 1 H2O!OH+H+ Zn2++OH!ZnOH+ ZnOH++2OH!HZnO2+H2O logcHZnO2 HZnO2+OH!ZnO22+H2O logcZnO22 Zn2++2OH!Zn(OH)2(s)(4{14) TherateofproductionofZn(OH)2(s)dependsstronglyontheconcentrationsofZn2+andOH;thus,theprecipitationratewasrelatedtocZn2+andcOHby wherekisarateconstantandKspisthestandardsolubilityproductofZn(OH)2(s)atroomtemperature.[ 70 ]Thedierencebetweenthetwotermsinthebracketattherightsideofequation( 4{15 )representsthedrivingforceforformingZn(OH)2(s).Equation( 4{15 )providesanapproach,intermsoftheconcentrationsofZn2+andOHions,toincorporatethesolidspeciesZn(OH)2(s)inthecomplexmodel.However,thisapproachisdierentfromthatemployedinthermodynamiccalculationsinwhichthetotalconcentrationsofZn2+andOHionswereheldasconstants.[ 71 72 ] 50

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Thedevelopmentofthemathematicalmodelispresentedinthischapter.Themodelsimulatedthepropagationofthefrontalongthemetal-coatinginterfaceduringthecathodicdelaminationofacoatedzinc. Thegoverningequationforthesolutionpotentialwasderivedfromtheelectroneutralitycondition (5{1) Thegoverningequationforciina1-Ddomainwas whereSirepresentedtherateofproductionperunitvolumebyelectrochemicalreactions.Theconservationequationsforthechemicallyinertspecies,Na+andCl,wereobtainedbyassigningSNa+=0,andSCl=0.Thegoverningequationsforthespeciesparticipatinginheterogeneousreactions,Zn2+andOH,wereformulatedas and 51

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and SubstitutionofRiintoequation( 5{5 )yieldedthegoverningequationsforcZn2+andcOH,respectively,i.e. and where TheequilibriumconditionslistedinTable 4-2 wereappliedasthegoverningequationsforH+,ZnOH+,HZnO2,andZnO22.[ 66 ]TherateofformationofthecorrosionproductZn(OH)2(s)wasassociatedwithcZn2+andcOHby wherethestandardsolubilityproductKsphasavalueof31017(mol/liter)3atroomtemperature.[ 10 ] Thephenomenaofbond-breakageandcoatingdegradationinvolvechemicalreactions.Equation( 4{5 )governstheequilibriumrelationshipbetweeninterfacialporosityandlocalpH.Theequilibrium"-pHrelationisvalidundertheassumptionthatthetimeconstantsofbond-breakagereactionsaresucientlysmall.Whenthetimeconstantsforthesephenomenaarelargecomparedtothosefortheprocessesofdiusionandmigration,theequilibriumassumptionbecomesinvalid.Anon-equilibriumrelationshipbetween"andpH 52

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@t=kneq(""eq)(5{11) wheretheequilibriumporosity"eqisobtainedinequation( 4{5 )andkneqistherateconstantforthebond-breakagereactions.Inthelimitthatkneq!1thevalueof"attainsitsequilibratedvalue"="eq. 5{2 )forOH,Na+,Cl,andZn2+,respectively,equation( 5{1 )fortheelectroneutralitycondition,equilibriumconditionsforH+,ZnOH+,HZnO2,andZnO22,equation( 5{10 )forthecorrosionproduct,andanequationfortheporosity-pHrelationship.Whentheporositywasassumedtoreachitsequilibriumvalueinstantaneously,theequilibrium"-pHrelationship,equation( 4{5 ),wasusedasthegoverningequationfor".Whenthenon-equilibrium"-pHrelationshipwasapplied,equation( 5{11 )wasusedasthegoverningequationfor". ThederivativetermswerediscretizedateachnodeinthedomainusingTaylorseriesapproximation.Therst-ordertemporalderivativewasgivenby

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73 ],i.e. wherefwasagenericvariableforci,,and".Termsoftheorder(4x)2andhigherwereneglectedinthespatialderivatives.Aquarter-pointmethodwasusedtoapproximatethederivativetermintheboundaryconditions.Theapproximationofm-1 4wasobtainedusing @xm+1 4=fmfm1 2 where 2=fm+fm1 Theresultingsystemofalgebraicequationswasaccuratetotheorderof(4x)2. Thesystemofcoupled,non-linear,partialdierentialequationsrequiredaniterativemethodtoconvergeonasolutionstartingfromaninitialguess.Atri-diagonalmethod,BANDalgorithm,coupledwithtimestepwaschosentocalculatethedistributionofci,,and"inthedelaminateddomain.ThemathematicalmodelwasdevelopedusingMicrosoftVisualFortran,Version9.0withdoubleprecisionaccuracy.[ 74 ] 54

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Thepresentmathematicalmodelsimulatedtheevolutionofacathodicdelaminationsystemfromasetofgiveninitialconditions.Theresultingdistributionssatisedthecoupledphenomenaofmasstransfer,electroneutrality,anddisbondmentreactionsduringthedelaminationprocess.Thecalculatedresultsanddiscussionsarepresentedinthischapter. Theinputparametersforthesimulationincludedthegridsize4x=4104cm,thetimestep4t=0.1s,andthetotaltimet=60min.ThepotentialonthemetalwaschosenasV=-0.95VSHE.ThepolarizationparametersforzincdissolutionincludedZn=0.04V=decade,i0;Zn=0.008A/cm2,andE0;Zn=-0.763VSHE.ThediusioncoecientsDiforthechemicalspeciesaregiveninTable 6-1 .Theconcentrationofdissolvedoxygenatthesurfaceofthecoatingwas1.26103M.[ 46 ] Table6-1. Diusioncoecientsofchemicalspecies[ 46 75 ] ChemicalDiinbulkSpecieselectrolyte(cm2=s) O21.90105OH5.25105Na+1.47105Cl2.03105Zn2+0.71105H+9.32105ZnOH+1.00105HZnO21.00105ZnO221.00105

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InitialconcentrationdistributionsofOH,Na+,Cl,andZn2+ionsalongthemetal-coatinginterface. 6-1 wereconstructedbasedontheexperimentaldatareportedbyLengandStratmann[ 14 { 16 ]TheinitialconcentrationsofNa+andOHionswereequalatanypositioninthedelaminateddomain.TheshapeofthecCldistributionfollowedthatofcNa+distribution.TheconcentrationsofNa+,Cl,andOHionsdecreasedmonotonicallywithpositioninthedelaminatedandfrontregions.Intheintactregion,theconcentrationsofallspeciesreachedasymptoticvalues.ThedistributionofcZn2+wasobtainedbysatisfyingelectroneutralityatagivenposition.Theconcentrationsofthecorrosionproductandthespeciesproducedinthehomogeneousreactionswereassumedtobezeroatt=0. 4.1 .Thettingparametersusedintheequilibrium"-pHrelationship(equation( 4{5 ))aregiveninTable.TheinitialporositydistributionisshowninFigure 6-2(a) asafunctionofpositionandinFigure 6-2(b) asafunctionofpH.AsseeninFigure 6-2(a) ,theporositydecreases 56

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(b) Calculatedinitialporositydistribution:a)asafunctionofposition;andb)asafunctionofpH. non-linearlyinthedelaminatedandthefrontregions.Theporositymaintainsauniformvalueintheintactregion.Theporosity-pHplot(Figure 6-2(b) )presentstheconceptthatatahighpHtheadhesivestrengthislowandthatatalowpHtheadhesivestrengthishigh. 4{10 )and( 4{13 )governthe-pHandO2-pHrelations.ThettingparametersusedintheequationsaregiveninTable 4-1 .TheinitialdistributionsofarepresentedinFigure 6-3(a) asafunctionofpositionandinFigure 6-3(b) asafunctionofpH.Figures 6-4(a) and 6-4(b) presenttheinitialdistributionsofO2asafunctionofpositionandpH,respectively. Thecurrentdensityexpressionsgiveninequations( 4{9 )and( 4{12 )wereappliedtogeneratethepolarizationplotatthemetal-coatinginterfaceasafunctionofpHpresentedinFigure 6-5 .ThepHvalueof8.7correspondstothepositionsintheintactregionwherelocalanodicreactionsarebalancedbylocalcathodicactivities.Thecorrosioncurrentdensityatthemetal-coatinginterface,therefore,isapproximatelyequaltozerointhe 57

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(b) Calculatedinitialdistributionofpoisoningfactor:a)asafunctionofposition;andb)asafunctionofpH. (b) Calculatedinitialdistributionofblockingfactor:a)asafunctionofposition;andb)asafunctionofpH. 58

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InterfacialpotentialasafunctionofabsolutenetcurrentdensitywithlocalpHasaparameter.ThedistributionsassociatedwiththepHvaluesof8.7and9aresuperimposed. intactregion.AsseeninFigures 6-3(b) and 6-4(b) ),themagnitudesofandO2donotchangeoverthepHrange8to9.Thus,thepolarizationcurvesassociatedwithpH8.7and9superimposeasobservedinFigure 6-5 ThepHrangingfrom9to11correspondstothemovingfrontinwhichthebond-breakingreactionsareongoing.AsshowninFigures 6-3(b) and 6-4(b) ,bothandO2showanincreasefrompH9to11;consequently,theanodicandcathodiccurrentdensitiesinthefrontregionarelargerthanthoseintheintactregion.Theincreaseinthecurrentdensitiesreectsphysicallyanenhancedelectrochemicalreactivityinthefrontregion. ThecurveofpH12inFigure 6-5 correspondstothedelaminatedregioninwhichtheinterfacialbondsarepartiallybrokenduetothedelaminationprocess.AsthepolarizationparametersandO2accountimplicitlyfortheinuenceofpassivelmsordepositsontheelectrochemicalreactions,bothanodicandcathodiccurrentdensitiesinthedelaminatedregionaresmallerthanthoseinthefrontregion. 6.3 59

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Calculateddistributionsofinterfacialpotentialalongthemetal-coatinginterfacewithelapsedtimeasaparameter. 6-6 withelapsedtimeasaparameter.Inthegraphicalpresentationsinthesubsequentsections,t=30swaschosenastheinitialconditiontoavoidthearticialityatt=0.Atagiventime,theinterfacialpotentialincreaseswithdistanceawayfromthescratchandreachesaconstantvalueintheintactregion.TheconstantplateauseeninFigure 6-6 representstheintactregionandisobservedtoshortenwithelapsedtime.Theshapeofthepotentialdistributionismaintainedthroughoutthesimulation,indicatingthatthephenomenaandthehypothesesconsideredinthemodelsustainstheproleofVwhilethedelaminationfrontpropagatesalongthemetal-coatinginterface.ThesefeaturesareconsistentwiththeexperimentalresultsofcoatedelectrogalvanizedreportedbyStratmannetal.[ 10 ]andWilliamsetal..[ 50 ] FollowingtheanalysisreportedbyLengandStratman,[ 15 ]theinterfacialpotentialdistributionsweredierentiatedwithrespecttopositiontoyielddistributionsofdV/dxasafunctionofpositionwithelapsedtimeasaparametergiveninFigure 6-7 .ThesharppeakmarksthedeectionpointofthesharpincreaseobservedinFigure 6-7 .Thepositionofthepeak,identiedasthedelaminationfront,propagatesawayfromthe 60

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CalculateddistributionsofdV/dxalongthemetal-coatinginterfacewithelapsedtimeasaparameter. defectwithincreasingtime.ThepeakheightcorrespondingtothemagnitudeofdV/dxdeceaseswithincreasingdelaminationtime.ThedecreasingtrendisinagreementwiththeexperimentalresultsreportedbyLengandStratmann.[ 15 ]TheexplanationgivenbyLengandStratmannwasthat,withtime,amoregradualchangeinelectrochemicalpotentialinthefrontregion.[ 15 ]TheagreementwiththeexperimentsdemonstratesthatthehypothesesofthepH-dependentinterfacialporosityandpH-dependentpolarizationkineticswerereasonableforthefrontpropagationduringthedelaminationprocess. Therateofpropagationofthepotentialfront,calculatedfromthemaximapeakgiveninFigure 6-7 ,ispresentedinFigure 6-8 .Therateinitiallyislargebutexponentiallydecreaseswithelapsedtime.Afteralong-timeextrapolation,thedelaminationratedeterminedbythepotentialfrontis1.66mm/hr,approximatelytwotimeslargerthantheexperimentalrateforcoatedsteelobservedbyLengandStratmann.[ 14 ]ThediscrepancybetweenthetheoreticalandexperimentalworkcanbeattributedtotheuseoftheequilibratedpH-porosityrelationinthemodel.Thephenomenaofbondbreakageandcoatingdegradationinvolvechemicalreactions.TheapplicationoftheequilibriumpH-porosityrelationassumesinstantaneouslythatthetimeconstantsassociatedwithbreakingbondsaresmall.Whentimeconstantsforbond-breakagephenomenaarelarge 61

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Instantaneousvelocityofpotentialfront,calculatedfromthetime-dependentpositionofthemaximagiveninFigure 6-7 comparedtothoseforthediusionandmigration,theassumptionoftheequilibriumpH-porosityrelationbecomesinvalid.Aninvestigationfornon-equilibriumpH-porositywillbeaddressedinSection 6.3 6-9 asafunctionofpositionwithelapsedtimeasaparameter.ThecalculatedresultsshowthatthepHinthedelaminatedandfrontregionsincreaseswithtimeandremainsunchangedintheintactregion.TheincreaseinpHinthedelaminatedandfrontregionsisattributedtotheOHionsproducedbyoxygenreductionunderneaththecoatingandbydiusionfromtheboundarywiththedefect.TheshapeofthepHdistributionismaintainedthroughoutthesimulation,whichagain,demonstratesthatthehypothesesandphysicalphenomenaconsideredinthemodelareabletosustaintheproleofpHwhilethedelaminationfrontpropagatesalongtheinterface. ThecalculateddistributionsofcNa+andcClarepresentedinFigure 6-10 asafunctionofpositionwithelapsedtimeasaparameter.ThetrendsassociatedwiththecNa+andcCldistributionsaresimilarwiththoseseeninthepHdistribution.AslightdecreaseinthecCldistributionisobservedinthefrontregion.Thisdecreasing 62

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CalculateddistributionsofpHalongthemetal-coatinginterfacewithelapsedtimeasaparameter. (b) Calculatedconcentrationdistributionsalongthemetal-coatinginterfacewithelapsedtimeasaparameter.a)Na+ions;andb)Clions. 63

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CalculatedconcentrationdistributionsofZn2+ionsalongthemetal-coatinginterfacewithelapsedtimeasaparameter. featuredemonstratesthattheClionsareexpelledfromthefrontregionbecauseoftheproductionofOHionsbyoxygenreduction. ThecalculateddistributionofcZn2+isgiveninFigure 6-11 asafunctionofpositionwithelapsedtimeasaparameter.Theshapeofthedistributionismaintainedthroughoutthesimulation.InthefrontregiontheconcentrationofZn2+ionsincreaseswithpositionatpartoftheregionanddecreaseswithpositionattheotherpartoftheregion.ThisfeatureshowsthatthedistributionofcZn2+isconstrainedbytheelectroneutralityconditionappliedinthemodel. Theconcentrationdistributionsofthespeciesproducedinthehomogeneousreactions(ZnOH+,HZnO2,andZnO22)arepresentedinFigure 6-12 asafunctionofpositionwithelapsedtimeasaparameter.Forallthreespecies,theconcentrationsdecreasewithpositioninthedelaminatedandthefrontregions.Duringthecourseofsimulation,thechangesinconcentrationacrossthedelaminatedandthefrontregionsbecomemoregradualasthedelaminationpropagatesintothefully-intactregion. 19 76 ]ThedistributionofcZn(OH)2(s)calculated 64

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(b) (c) Calculatedconcentrationdistributionsalongthemetal-coatinginterfacewithelapsedtimeasaparameter.a)ZnOH+ions;b)HZnO2ions;andc)ZnO22ions. 65

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CalculatedconcentrationdistributionsofprecipitatedcorrosionproductZn(OH)2(s)alongthemetal-coatinginterfacewithelapsedtimeasaparameter. fromthepresentedmodelisgiveninFigure 6-13 asafunctionofpositionwithelapsedtimeasaparameter.TheconcentrationofZn(OH)2(s)increaseswithtimeinthedelaminated,frontandintactregions.Atagiventime,theconcentrationofZn(OH)2(s)decreaseswithpositioninthedelaminatedandfrontregions,andmaintainsaconstantintheintactregion. IthasbeenknownthattheformationofpassivelayerZn(OH)2(s)onelectrodesurfaceprotectsmaterialsfromcorrosionandmoderatesthecorrosionrate.[ 19 ]Thisinhibitivefeature,inthemodelpresentedhere,wasincludedimplicitlythroughtheuseofthepoisoningfactor,butnotrelatedquantitativelywiththelocalconcentrationofZn(OH)2(s).ThesolubilityofZn(OH)2(s),asindicatedinthePurbaixdiagram,[ 66 ]hasaminimumaroundpH9andincreaseswithincreasingpH.ThistransitionfeatureisnotobservedinFigure 6-13 .TheinconsistencywiththeliteraturemightbeduetothattheapproachtakeninthemodeltoincorporatethesolidspeciesZn(OH)2(s)(equation( 4{14 ))isdierentfromthatemployedinthePurbaixdiagraminwhichthetotalconcentrationsofZn2+andOHionswereheldasconstants.[ 71 72 ]. 66

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Calculateddistributionsofd/dxalongthemetal-coatinginterfacewithelapsedtimeasaparameter. porosityfrontthatmovestowardtheintactregionduringthedelaminationprocess.Thevelocityoftheporosityfrontcorrespondstotherateofbreakingtheinterfacialbondsandisapproximately1.50mm/hrafterextrapolatedtoalongertime(seeFigure 6-16 ).Thisvalueisslightlysmallerthanthatofthepotentialfront,butstilllargerthantheexperimentalresultofcoatedsteel.[ 14 ]ThediscrepancybetweenthesimulationandexperimentalratescanbeattributedtotheuseoftheequilibriumpH-porosityrelation,whichyieldstotheupperlimittothepropagationrate. 15 ]andWilliam[ 51 ],thedelaminationkineticswasanalyzedbyplottingthedelaminateddistanceasafunctionofelapsedtime.ThepropagationdistancesdeterminedbythepotentialandtheporosityfrontsarepresentedinFigures 6-17(a) and 6-17(b) ,respectively,asafunctionoftimeinadouble-logarithmicplotwithcationtypeasaparameter.Thecalculatedreactionorderisapproximately0.56forthepotentialfrontand0.6fortheporosityfront.TheslopesseeninFigures 6-17(a) and 6-17(b) areindependentofthecationtypes,andthevaluesareincloseagreementwiththereactionorderof0.52to0.59determinedbyStratmannetal.[ 10 ]forpolymercoatedelectrogalvanizedsteel(Figure 6-17(c) ).Theseresultsindicatethattheoverall 68

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Instantaneousvelocityofporosityfront,calculatedfromthetime-dependentpositionofthemaximagiveninFigure 6-15 Table6-2. Diusioncoecientsofcations[ 70 75 ] TypeofDiinbulkcationelectrolyte(cm2=s) Li+1.25105Na+1.47105K+1.84105Cs+2.10105 6-18 ,wherethedelaminateddistancecalculatedbasedonthepotentialandporosityfrontsareplottedasafunctionofsquarerootoftime.Alinearrelationbetweenthedelaminateddistanceandp 6-2 ).Theresultindicatesthat,withthechemicalandphysicalassumptions,the 69

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(b) (c) Delaminateddistanceasafunctionofelapsedtimeindouble-logarithmicscalewithcationtypeasaparameter.Theconcentrationoftheelectrolyteatthedefectis0.5M.a)Delaminateddistancedeterminedbypotentialfront;b)Delaminateddistancedeterminedbyporosityfront;andc)Experimentalresultsobtainedfromcoatedelectrogalvanizedsteelsamples.DatatakenfromStratmannetal.[ 11 ]withpermissionofCorrosionScience. 70

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(b) (c) Delaminateddistanceasafunctionofsquarerootoftimewithcationtypeasaparameter.Theconcentrationoftheelectrolyteatthedefectis0.5M.a)Delaminateddistancedeterminedbypotentialfront;b)Delaminateddistancedeterminedbyporosityfront;andc)Experimentalresultsobtainedfromcoatedsteelsamples.DatatakenfromStratmannetal.[ 15 ]withpermissionofCorrosionScience. 71

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Diusioncoecientsofanions[ 75 ] TypeofDiinbulkanionelectrolyte(cm2=s) Br1.25105Cl1.47105F1.84105ClO42.10105 6-18(c) ).[ 10 15 ] 6-19 asafunctionofsquarerootoftime.Itisclearthatthedelaminationratedoesnotvarywiththeaniontypeseventhoughtheirmobilitiesinaqueouselectrolytearedierent(Table 6-3 ).TheproductionofOHionsunderthedegradedcoatingattractthecationatthedefect,resultinginmovementsofthecationfromthedefecttowardtheintactregion.Consequently,thecathodicdelaminationismoresensitivetothecationtypethantheaniontype.TheexperimentalobservationsbyLengandStratmann[ 11 15 ],giveninFigure 6-19(c) ,showsthattheaniontypesinuenceslightlyonthedelaminationrate,butthevariationsofthepropagationratebetweentheaniontypesismuchlesssignicantthanthosebetweenthecationtypes. 15 ],theconcentrationoftheelectrolyteplacedatthedefectisalsoafactorthatinuencesthedelaminationrate.Figure 6-20 givesthedelaminateddistanceasafunctionofsquarerootoftimewithelectrolyteconcentrationasaparameter.Thehigherconcentrationatthethresholdprovidesalargerdrivingforcetocouplegalvanicallytheintactandthedefectzones;therefore,thepropagation 72

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(b) (c) Delaminateddistanceasafunctionofsquarerootoftimewithaniontypeasaparameter.Theconcentrationoftheelectrolyteatthedefectis0.5M.a)Delaminateddistancedeterminedbypotentialfront;b)Delaminateddistancedeterminedbyporosityfront;andc)Experimentalresultsobtainedfromcoatedsteelsamples.DatatakenfromStratmannetal.[ 15 ]withpermissionofCorrosionScience. 73

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(b) (c) Delaminateddistanceasafunctionofsquarerootoftimewithelectrolyteconcentrationasaparameter.a)Delaminateddistancedeterminedbypotentialfront;b)Delaminateddistancedeterminedbyporosityfront;andc)Experimentalresultsobtainedfromcoatedsteelsamples.DatatakenfromStratmannetal.[ 15 ]withpermissionofCorrosionScience. 74

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15 ] 4{5 ),wasreplacedby @t=kneq(""eq)(6{1) wherekneqisarateconstantthatreectsthetimeconstantsofbond-breakagereactions,and"eqisobtainedfromequation( 4{5 ).Dierentvaluesofkneqwereexaminedinthesimulations,butonlytheresultsforkneq=0.1and0.001s1arepresentedhere. Theresultingdistributionsofinterfacialpotentialforkneq=0.1and0.001s1areshowninFigure 6-21 asafunctionofpositionwithelapsedtimeasaparameter.ThefeaturesseeninFigure 6-21 weresimilartothoseobservedinFigure 6-6 ,whichwereobtainedusingtheequilibriumpH-porosityrelation.ThedistributionsofthepotentialgradientdV/dxforkneq=0.1and0.001s1areshowninFigure 6-22 asfunctionsofpositionwithelapsedtimeasaparameter.Again,thetrendsassociatedwiththedV/dxplotfornon-equilibriumpH-porosityrelationaresimilartothosefoundusingthe 75

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(b) Calculateddistributionofporositygradientd"/dxalongthemetal-coatinginterfacea)kneq=0.1s1;andb)kneq=0.001s1. equilibriumrelation.ThedelaminationratedeterminedbyFigures 6-22(a) and 6-22(b) ,afterextrapolatedtolongertime,were1.63mm/hrand1.55mm/hrforkneq=0.1and0.001s1,respectively.Thesevaluesareslightlysmallerthantheequilibriumdelaminationrate(1.66mm/hr),indicatingthattheuseofthekineticpH-porosityrelationwithinthemodelinuencedthevelocityofthepotentialfront. Thepropagationofthepotentialfrontisthencomparedwiththatoftheporosityfront.Theresultingdistributionsofd"/dxforkneq=0.1and0.001s1arepresentedinFigure 6-23 asfunctionsofpositionwithelapsedtimeasaparameter.ThetrendsassociatedwithinFigure 6-23 aresimilartothoseobservedinFigure 6-15 ,whichwereobtainedusingtheequilibriumpH-porosityrelation.Thevelocityoftheporosityfrontevidentlydecreasesfrom1.37mm/hrforkneq=0.1s1to0.93mm/hrforkneq=0.001s1.Thepropagationrateforkneq=0.001s1ismuchsmallerthantheequilibriumporosityfrontrate(1.5mm/hr)andthevalueisingoodagreementwiththeexperimentalobservationof0.8mm/hrforcoatedgalvanizedsteel.[ 15 ] Fromtheanalysispresentedabove,itisevidentthattherateofbreakinginterfacialbondsinthecathodicdelaminationprocessiscontrolledbytherateconstantkneq,but 77

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Calculatedvelocitiesofpotential,porosityandpHfront rateconstants velocityofpotentialfront velocityofporosityfront velocityofpHfront (kneq/s1) mm/hr mm/hr mm/hr 1.50 1.69 101 1.37 1.61 102 1.26 1.60 103 0.93 1.55 104 0.73 1.25 Figure6-25. Calculateddistributionsofd/dxalongthemetal-coatinginterfacewithelapsedtimeasaparameter. ofkneqinuencethepropagationofallfronts,buttheinuenceismuchmoreevidentontheporosityfront. AsobservedinTable 6-4 ,thevelocitydierencebetweenthepotentialandporosityfrontsincreaseswithdecreasingkneq.TheproductionofOHionsinthefasterpotentialfrontcreatesadrivingforceforthebond-breakagereactionsthatarelimitedbytheniterateconstant.Asaresult,thedisbondmentoccurinabroadregionwhenthebond-breakagereactionsaresucientlyslow.Thedistributionofporosityforkneq=104s1ispresentedinFigure 6-25 asafunctionofpositionwithelapsedtimeasaparameter.Duetothelimitationoftheniterateconstant,thewell-denedporosityfrontseeninFigure 6-14 becomeslessdistinguishableinFigure 6-25 .Instead,thechangeoftheporositytakesplaceinabroadregionandthisregionexpandswithincreasingtime. 79

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(b) Delaminateddistanceasafunctionofdelaminationtimeindouble-logarithmicscalewithcationtypeasaparameter.a)Delaminateddistancedeterminedbythepotentialfront;andb)Delaminateddistancedeterminedbytheporosityfront. 6-26(a) givesthepropagationdistancedeterminedbythepotentialfrontasafunctionoftimewithkneqasaparameter.Whentherateconstantkneqchangesfrominnityto104s1,theslopesofthelineschangeslightlyfrom0.57to0.59.Theslightchangeintheslopedemonstratesthattherateconstantkneqdoesnothavesignicantimpactsontherate-determiningstepoftheoveralldelaminationprocess.Figure 6-26(b) givespropagationdistancedeterminedbytheporosityfrontsasafunctionoftimewithkneqasaparameter.Theslopeisapproximatelyequalto0.6fortheequilibriumpH-porosityrelationbutincreasesto0.74forkneq=104s1.Thechangeintheslopeisanindicationthatthedelaminationmechanismshiftsfromamass-transfercontrolledtoamixed-controlmechanismwhenthebond-breakagereactionsaresucientlyslow.ThetransitionisconsistentwiththeexperimentalresultsreportedbyStratmannetal.[ 10 ]thattheoveralldelaminationprocessislimitedbykineticswhenthedelaminationrateissucientlysmall. 80

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35 ]orfromporosityorsurfaceroughness.[ 37 38 ] Therecentdevelopmentoflocalelectrochemicalimpedancespectroscopy(LEIS)[ 54 77 ]makesitpossibletodistinguishCPEbehaviorthathasanoriginwitha3-Ddistributionfromonethatarisesfroma2-Ddistributionofpropertiesalongthesurfaceoftheelectrode.[ 40 ]InLEIS,similarwithtraditionalimpedancemethods,asinusoidalcurrentorpotentialperturbationisimposedontestedsystemsandthecorrespondingpotentialorcurrentresponseismeasured.Thelocalimpedancetechniqueconsistsofaprobewithtwomicro-electrodesallowingmeasurementsofpotentialattwopositions.UndertheassumptionthattheOhmicimpedancebetweenthetwoprobesisgivenbyaconstant,thecurrentdensityattheprobecanbeestimatedfromthemeasuredpotentialdierence4Vprobeby d(7{1) wheredisthedistancebetweenthepotentialsensingelectrodesandistheconductivityoftheelectrolyte.Thelocalimpedancecanthenbecalculatedfromtheratiooftheelectrodepotentialmeasuredrelativetoareferenceelectrodefarawayfromthesurfacetothelocalcurrentdensityiprobe. 82

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(b) (c) Passiveelementsthatserveascomponentsofanelectricalcircuit.a)Resistor;b)Capacitor;andc)Inductor. Thesubsequentsectionsprovidebasicconceptsinvolvedinelectrochemicalimpedancespectroscopy.DetaileddiscussionsoftechnicalandtheoreticalissuesassociatedwithEISisavailableelsewhere.[ 25 28 29 78 ] 7-2 .[ 25 28 29 ] Theimpedanceofapassivecircuitelementisdenedastheratioofthepotentialdierencebetweentheelementclampstothecurrentowingthroughtheelement,i.e. andhasunitsOhms.Forapureresistor,equation 7{2 yields whereas,foracapacitor andforaninductor,theimpedanceis 83

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(b) Combinationsofpassiveelementsthatserveascomponentsofanelectricalcircuit:a)inseriesb)inparallel. Fortwopassiveelementsinseries,thesamecurrentmustowthroughthetwoelements,andtheoverallpotentialdierenceisthesumofthepotentialdierenceforeachelement.Thus,theimpedancefortheseriesarrangementshowninFigure 7-3(a) isgivenby Fortwopassiveelementsinparallel,theoverallcurrentisthesumofthecurrentowingineachelement,andthepotentialdierenceisthesameforeachdipole.Therefore,theoverallimpedancefortheparallelarrangementshowninFigure 7-3(b) isgivenby 84

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38 79 ]showedthatthetime-constantdispersiononsolidelectrodewasduetosurfacedisorder(ontheatomicscale)ratherthangeometricroughness(largerthanatomicscale). DeLevie[ 80 81 ]modeledtheimpedanceofporouselectrodesundertheassumptionthattheconcentrationwasuniformandtheporeswereidealcylinders.Lasia[ 82 ]laterreplacedthedoublelayercapacitanceonporewallswithaCPE.TheresultsreportedbyLasia[ 82 ]showedthatmasstransferandporegeometryinuencedtheshapeofimpedancespectrum.ThemodelsproposedbydeLevieandLasiaconsideredonlyasingleporedimension.Songetal.[ 83 ]developedamodeltopredicttheeectofporesizedistributionontheimpedanceresponseofporouselectrodes. Pajkossyetal.[ 84 { 87 ]proposedthatthetime-constantdispersionarosefromtheadsorptionofmoleculesoranionsongoldelectrodes.Thecapacitancedispersionobservedinthepresenceofspecicadsorptioncanbeassignedtoeitheraslowdiusionorslowadsorptionprocesseswithinthedoublelayerorelectrodesurface.[ 87 ] NewmanandNisancioglu[ 41 42 45 ]studiedtheinuenceofnonuniformcurrentandpotentialdistributionontheimpedanceresponseofadiskelectrode.Theirresultsindicatedthegeometry-inducedpotentialandcurrentdistributioninducedahigh-frequencydispersionthatdistortedtheimpedanceresponse.Nisancioglu[ 45 ]showedtheextenttowhichthisfrequencydispersionleadstoanerrorinthevaluesforcharge-transferresistanceandinterfacialcapacitanceobtainedfromimpedancedata. 35 ]orfromporosityorsurfaceroughness.[ 37 38 ]This 86

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88 ]andininterpretationofelectrochemicalprocesses.[ 89 ]Thegeometryofanelectrodeoftenconstrainsthedistributionsofcurrentandpotentialontheelectrodesurfaceinsuchawaythatbothcannotbesimultaneouslyuniform.Newman[ 46 ]developedanalyticalsolutionsforcurrentandpotentialdistributionsonadiskgeometry,andthedevelopmentisreviewedinthissection. Inabulkofawell-stirredelectrolyticsolutionwhereconcentrationgradientsarenegligiblewithintheelectrolyte,potentialisgovernedbyLaplace'sequation,i.e.[ 46 ] whereisthesolutionpotential.Thecurrentdensityicanthenbeexpressedas Undertheassumptionthatconcentrationsareuniformintheelectrolyte,thepassageofcurrentthroughtheinterfaceislimitedbyOhmicresistanceintheelectrolyteandbycharge-transferresistanceassociatedwithreactionkinetics.TheprimarydistributionapplieswhentheowofcurrentisdominatedbytheOhmicresistanceandkineticresistancecanbeneglected.ThesecondarydistributionapplieswhenbothOhmicandkineticresistancesarecontrolling. 88

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Thischapterpresentsthetheoreticaldevelopmentandcalculationresultsfortheimpedanceresponseofanideally-polarizedblockingelectrode.[ 32 ]Thereareseveraltypesofimpedanceatplay;theirdenitionsandnotationsarealsoprovidedinthischapter. 42 ]Laplace'sequationincylindricalcoordinateswasexpressedinrotationalellipticcoordinates,i.e. and where01and01.ThecoordinatetransformationcanbeseenmoreclearlyinFigure 8-1 .Withintherotationalellipticcoordinatesystem,theelectrodesurfaceaty=0andrr0canbefoundat=0and01.Thereferenceelectrodeandcounterelectrodelocatedaty!1canbefoundat!1.Theinsulatingsurfaceofthediskaty=0andr>r0islocatedat=1and0<1,andthecenterlineaty>0andr=0islocatedat=0and0<1. Lapace'sequationcanbeexpressedinrotationalellipticcoordinatesas @(1+2)@ @(12)@ Thepotentialwasseparatedintosteadyandoscillatingpartsas =+~exp(j!t)(8{4) 92

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Coordinatetransformationfromacylindricalcoordinatetoarotationalellipticcoordinate.Thegridingintherotationepioticcoordinateisnotdrawntoscale. whereisthesteady-statesolutionforpotentialand~isthecomplexoscillatingpotential.Thus,equation( 8{3 )couldbewrittenas 2@~r and 2@~j where~rand~jrefertotherealandimaginarypartsofthecomplexpotential,respectively. Forablockingelectrode,thecurrentpassesfromtheelectrodetotheelectrolytebyameansofchargingthedouble-layercapacity.Theuxboundaryconditionattheelectrodesurface(=0and01)wasexpressedas r0@ 93

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8{7 )waswritteninfrequencydomainas K~j=1 and K~VrK~r=1 where~VrrepresentstheimposedperturbationintheelectrodepotentialreferredtoanelectrodeatinnityandKisthedimensionlessfrequency K=!C0r0 At=0and=1,forall>0,zero-uxconditionsimposethat and Atthefarboundarycondition(!1and01), ~r=0(8{13) and ~j=0(8{14) TheequationsweresolvedundertheassumptionofauniformcapacitanceC0usingthecollocationpackagePDE2DdevelopedbySwell.[ 90 ]Toensuretheaccuracyofthecalculations,aseriesoferroranalysiswasperformedtoverifythatthemeshsizeusedintheprogramwassucientlysmallandthedomainsizeissucientlylarge.Calculationswereperformedfordierentdomainsizes,andtheresultspresentedherewereobtainedbyextrapolationtoaninnitedomain. 94

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Thelocationofcurrentandpotentialtermsthatmakeupdenitionsofglobalandlocalimpedance. Aschematicrepresentationoftheelectrode-electrolyteinterfaceforanideally-polarizedblockingelectrodeisgivenasFigure 8-2 ,wheretheblockusedtorepresenttheOhmicimpedancereectsthecomplexcharacteroftheOhmiccontributiontothelocalimpedanceresponse.TheimpedancedenitionspresentedinTable 8-1 dierinthepotentialandcurrentusedtocalculatetheimpedance.Toavoidconfusionwithlocalimpedancevalues,thesymbolyisusedtodesignatetheaxialpositionincylindricalcoordinates. wherethecomplexcurrentcontributionisgivenby ~I=Zr00~{(r)2rdr(8{16) 95

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Notationproposedforlocalimpedancevariables[ 39 ] Symbol meaning units orcm2Zr orcm2Zj orcm2Z0 orcm2Z0;r orcm2Z0;j orcm2Ze orcm2Ze;r orcm2Ze;j orcm2z cm2zr cm2zj cm2z0 cm2z0;r cm2z0;j cm2ze cm2ze;r cm2ze;j cm2hi V timeaverageorsteady-statevalueofpotential Vhii A/cm2 A/cm2y cm 96

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39 ] 91 92 ]Thus,thelocalimpedanceisgivenby Theuseofalower-caselettersigniesthatzisalocalvalue.Thelocalimpedancemayhaverealandimaginaryvaluesdesignatedaszrandzj,respectively.[ 39 ] Theglobalimpedancecanbeexpressedintermsofthelocalimpedanceas Equation( 8{18 )isconsistentwiththetreatmentofBrugetal.[ 30 ]inwhichtheadmittanceofthediskelectrodewasobtainedbyintegrationofalocaladmittanceovertheareaofthedisk. ~{(r)(8{19) Theuseofalower-caseletteragainsigniesthatz0isalocalvalue,andthesubscript0signiesthatz0representsavalueassociatedonlywiththesurface.Thelocalinterfacialimpedancemayhaverealandimaginaryvaluesdesignatedasz0;randz0;j,respectively.[ 39 ] 97

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8-2 ).Thus,thelocalOhmicimpedanceisgivenby ~{(r)(8{20) Theuseofalower-caseletteragainsigniesthatzeisalocalvalue,andthesubscriptesigniesthatzerepresentsavalueassociatedonlywiththeOhmiccharacteroftheelectrolyte.ThelocalOhmicimpedancemayhaverealandimaginaryvaluesdesignatedasze;randze;j,respectively.Thelocalimpedance canberepresentedbythesumoflocalinterfacialandlocalOhmicimpedances.[ 39 ] or Theuseofanupper-caselettersigniesthatZ0isaglobalvalue.TheglobalinterfacialimpedancemayhaverealandimaginaryvaluesdesignatedasZ0;randZ0;j,respectively. 98

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(b) CalculatedNyquistrepresentationoftheimpedanceresponseforanideallypolarizeddiskelectrode.a)linearplotshowingeectofdispersionatfrequenciesK>1;andb)logarithmicscaleshowingagreementwiththecalculationsofNewman. Theuseofanupper-caselettersigniesthatZisaglobalvalue.TheglobalOhmicimpedancemayhaverealandimaginaryvaluesdesignatedasZe;randZe;j,respectively. 42 ] 8-3(a) inNyquistformatshowstheinuenceoftime-constantdispersionatfrequenciesK>1.TheimpedancewasmadedimensionlessaccordingtoZ=r0inwhichtheunitsofimpedanceZwerescaledbyunit 99

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(b) Calculatedrepresentationoftheimpedanceresponseforanideallypolarizeddiskelectrode.a)realpart;andb)imaginarypartshowingagreementwiththecalculationsandasymptoticformulaofNewman. area,cm2.TheexpandedlogarithmicrepresentationpresentedinFigure 8-3(b) showsgoodagreementwiththenumericalsolutionspresentedbyNewman.[ 42 ] ThecomparisonwithNewman'scalculationsisseenmoreclearlyintherepresentationoftherealandimaginarypartsoftheimpedanceresponseshowninFigures 8-4(a) and 8-4(b) ,respectively.AsstatedbyOrazemetal.[ 78 ],theslopeoflog(Zj=r0)withrespecttolog(K)givestheexponentoftheCPE,.ThechangeintheslopeofthelinespresentedinFigure 8-4(b) showsthattheimpedanceresponsetransitionsfromidealReC0behavioratlowfrequenciestoaCPE-likebehavioratfrequenciesK>1.AdeviationfromNewman'sresultsisseenforfrequenciesK>100.Thiserrorisattributedtoasingularperturbationproblem,identiedbyNewman,thatarisesattheperipheryoftheelectrodeathighfrequencies.[ 42 ] ThechangeofispresentedinFigure 8-5 asafunctionofdimensionlessfrequencyK.Thesystembehavesasanidealcapacitoratlowfrequencieswith=1.AtfrequenciesK>1,thevalueofchangestoroughly=0:85atK>10.Asthe 100

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Theslopeoflog(Zj=r0)withrespecttolog(K)(Figure 8-4(b) )asafunctionoflog(K).Theresultswerecalculatedbythecollocationmethod.Thevalueofthisslopeisequalto. slopeisnotindependentoffrequency,thefrequencydispersionseenatK>1doesnotrepresenttrueCPEbehavior. ThefrequencyK=1atwhichthecurrentdistributioninuencestheimpedanceresponsecanbeexpressedas AsshowninFigure 8-6 ,thischaracteristicfrequencycanbewellwithintherangeofexperimentalmeasurements.Thevalue=C0=104cm/scanbeobtainedforacapacitanceC0=1F/cm2(correspondingtoanoxidelayer)andconductivity=0:01S/cm(correspondingtoa0.1MNaClsolution).Thevalue=C=103cm/scanbeobtainedforacapacitanceC0=10F/cm2(correspondingtothedoublelayeronaninertmetalelectrode)andconductivity=0:01S/cm(correspondingtoa0.1MNaClsolution).Figure 8-6 canbeusedtoshowthat,byusinganelectrodethatissucientlysmall,theexperimentalistmaybeabletoavoidthefrequencyrangethatisinuencedbycurrentandpotentialdistributions. 8-7(a) asafunction 101

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ThefrequencyK=1atwhichthecurrentdistributioninuencestheimpedanceresponsewith=C0asaparameter. (b) Calculatedimaginarypartofthelocalinterfacialimpedance:a)asafunctionoffrequencywithpositionasaparameter;andb)asafunctionofpositionwithfrequencyasaparameter. 102

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(b) Calculatedlocalimpedancewithradialpositionasaparameter:a)realpart;andb)imaginarypart. TherealandimaginarypartsofthelocalimpedancearepresentedinFigures 8-9(a) and 8-9(b) ,respectively,withradialpositionasaparameter.TherealpartofthelocalimpedancepresentedinFigure 8-9(a) reachesasymptoticvaluesatK!0andK!1.TheimaginarypartpresentedinFigure 8-9(b) showsthechangeofsignassociatedwiththeinductivefeaturesinFigure 8-8 .ThechangesinsignoccuratfrequenciesbelowK=100,showingthattheinductiveloopcannotbeattributedtocalculationartifacts. TheradialdistributionoftherealandimaginaryimpedanceispresentedinFigures 8-10(a) and 8-10(b) ,respectively,withdimensionlessfrequencyKasaparameter.Athighfrequencies,e.g.K=100,thecalculatedradialdistributionoftherealpartofthelocalimpedancefollowstheexpression r0(r)=0:5s r02(8{26) derivedfromequation( 8{27 )usingtheexpressionfortheprimaryresistance

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TheimaginarypartoftheglobalOhmicimpedance,calculatedfromequation( 8{28 ),asafunctionofdimensionlessfrequency. or,inthedimensionlesstermsusedinthepresentwork, r0=Z r01 TherealpartofZeisequaltotherealpartofZasgiveninFigure 8-4(a) .TheimaginarypartofZeisgiveninFigure 8-13 asafunctionofdimensionlessfrequencyK.InthelowfrequencyrangeZeisapureresistanceequalto1:08Re,and,inthehighfrequencyrange,ZetendstowardsRe.TheimaginarypartoftheglobalOhmicimpedanceshowsanon-zerovalueinthefrequencyrangethatisinuencedbythecurrentandpotentialdistributions.Figures 8-4(a) and 8-13 showthatalltheeectofthecurrentandpotentialdistributionappearsintheglobalOhmicimpedance. 107

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InChapter 8 ,ithasbeenshownthatthenonuniformcurrentandpotentialdistributionsinuencetheglobalandlocalimpedancesofanideally-blockingdiskelectrode.TheobjectiveofthischapteristoexploretheinuenceofcurrentandpotentialdistributionontheimpedanceresponsesofablockingelectrodeexhibitingalocalCPEbehavior.Inthissense,thegoalistoexploretheroleofcoupled2Dand3Ddistributionsontheimpedanceresponsesofadiskelectrode.Thischapterpresentsthetheoreticaldevelopmentandresultsfortheimpedancecalculations.[ 39 ]ExperimentalvalidationprovidedbyVivierisalsopresentedinthischapter.[ 39 93 ] 8 .Laplace'sequationincylindricalcoordinateswasexpressedinrotationalellipticcoordinatesasequations( 8{5 )and( 8{6 )forrealandimaginaryparts,respectively.Themodicationmadeherewasthesubstitutionofthecapacitoratelectrodesurface(=0)byaconstantphaseelementCPE,i.e. r0@ whereQcanberelatedtotheinterfacialcapacitanceC0byequation( 7{8 ).Theuxboundaryconditionattheelectrodesurface=0waswritteninfrequency-domainas Kn(~Vrr)cos and Kn(~Vrsin where~VrrepresentstheimposedperturbationintheelectrodepotentialandKisthedimensionlessfrequency K=Q!r0 108

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9{4 ),thedimensionlessfrequencyKincludestheCPEcoecientQ,thefrequency!raisedtothepoweroftheCPEexponent,thediskradiusr0,andtheelectrolyteconductivity. At=0and=1,zero-uxconditionsimposethat and Atthefarboundarycondition!1, ~r=0(9{7) and ~j=0(9{8) TheequationsweresolvedunderassumptionofuniformCPEparametersQand.ThesimulationswereperformedusingthecollocationpackagePDE2DdevelopedbySewell.[ 90 ]Thecalculationswereperformedfordieringdomainsizesandtheresultsreportedherewereobtainedbyextrapolationtoaninnitedomainsize.Asdiscussedinthepreviouschapater,thecalculatedresultsarebelievedtobeincorrectforfrequenciesK>100duetothepresenceofasingularperturbationproblemthatarisesattheperipheryoftheelectrodeathighfrequencies.[ 42 ] 8-1 9-1 inNyquistformatwithasaparameter.TherepresentationgiveninFigure 9-1 appliesfor 109

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NyquistrepresentationforthecalculatedimpedanceresponseofablockingdiskelectrodewithalocalCPEwithasaparameter. 110

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(b) CalculatedimpedanceresponseforablockingdiskelectrodewithalocalCPEasafunctionofdimensionlessfrequencyK:a)realpart;andb)imaginarypart. allvaluesofelectrolyteconductivityanddiskradiusr0,butdierentvaluesareobtainedfordierentvaluesof.TheimpedancewasmadedimensionlessaccordingtoZ=r0inwhichtheunitsofimpedanceZareassumedtobescaledbyareaandhavingunitsofcm2. ThefrequencydependenceoftheimpedanceresponsecanbeseenmoreclearlyinFigures 9-2(a) and 9-2(b) ,wheretherealandimaginarypartsoftheimpedance,respectively,arepresentedasfunctionsofdimensionlessfrequencyKwithasaparameter.Therealpartofthedimensionlessimpedance,plottedinFigure 9-2(a) ,approachestheexpectedtheoreticalvalueof1=4athighfrequency.[ 42 ]Thelow-frequencybehaviordependsslightlyonthevalueof.WhenplottedagainstdimensionlessfrequencyK,thevaluesofthedimensionlessimaginaryimpedanceinFigure 9-2(b) superposeforallvaluesof.ThissuperpositionismadepossiblebytheinclusionofinthedenitionoffrequencyKinequation( 9{4 ). Orazemetal.[ 78 ]citedtheutilityoflogarithmicplotsofimaginaryimpedanceasafunctionoffrequencytoidentifyCPEbehavior.Thecalculatedslopeoflog(Zj=r0)with 111

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Thecalculatedslopeoflog(Zj=r0)withrespecttolog(K)(Figure 9-2(b) )asafunctionoflog(K)withasaparameter. respecttolog(K)(Figure 9-2(b) )ispresentedinFigure 9-3 .withasaparameter.DuetothedenitionofK,theslopeatlowfrequenciesofthelogarithmicplotsofimaginaryimpedanceasafunctionofKisequalto1.AtfrequenciesK>1,theslopeincreasestoapproximately0:85.Whenexpressedintermsofthesedimensionlessparameters,thelow-frequencyresponseisindependentof,buttheresultsobtainedathigherfrequenciesdependon. ThecalculationofeectiveCPEcoecientQeprovidesfurtherevidencethatthelow-frequencybehaviorisunaectedbythecurrentandpotentialdistribution.TheeectiveCPEcoecientQeforanelectrochemicalsystemcanbeobtainedfromtheimaginarypartoftheimpedanceby TheeectiveCPEcoecientobtainedfromequation( 9{9 )scaledbytheinputvalueispresentedinFigure 9-4 asafunctionoffrequencywithasaparameter.Equation( 9{9 )yieldstheinputvaluefortheCPEcoecientatlowfrequencies,butthiscalculationisinuencedbythecurrentdistributionsatfrequenciesK>1. 112

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(b) Calculatedlocalinterfacialimpedanceasafunctionofpositionwithfrequencyasaparameter:a)imaginarypart;andb)realpart. TheresultspresentedinFigures 9-6 and 9-7 showthatthecalculatedlocalinterfacialimpedanceisindependentof2-Ddistributions. 9-8 withnormalizedradialpositionasaparameter.Thedimensionlessimpedanceisscaledtothediskarear20toshowthecomparisonwiththehigh-frequencyasymptoticvalueinFigure 9-1 .Theimpedanceislargestatthecenterofthediskandsmallestattheperiphery,reectingthegreateraccessibilityoftheperipheryofthediskelectrode.Inductiveloops,whicharenotshownintheglobalimpedance,areseenathighfrequenciesinlocalimpedanceforalltheradialpositions. TherealandimaginarypartsofthelocalimpedancearepresentedinFigures 9-9(a) and 9-9(b) ,respectively,withradialpositionasaparameter.TherealandimaginarypartsofthelocalimpedancepresentedinFigure 9-9 showapureCPEbehavioratlowfrequenciesandageometry-induceddispersionathighfrequencies.TheimaginarypartpresentedinFigure 9-9(b) showsthechangeofsignassociatedwiththeinductivefeatures 115

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(b) CalculatedvaluesforglobalOhmicimpedanceasafunctionoffrequencywithasaparameter:a)realpart;andb)imaginarypart. or,inthedimensionlesstermsusedhere, r0=Z r01 TherealpartofZeisgiveninFigure 9-11(a) ,andtheimaginarypartofZeisgiveninFigure 9-11(b) asfunctionsofdimensionlessfrequencyKwithasaparameter.InthelowfrequencyrangeZe=r0isapureresistanceequalto0:27,and,inthehighfrequencyrange,Ze=r0tendstowards1=4.TheimaginarypartoftheglobalOhmicimpedanceshowsanon-zerovalueinthefrequencyrangethatisinuencedbythecurrentandpotentialdistributions.Figure 9-11 showsthatalltheeectofthecurrentandpotentialdistributionappearsintheglobalOhmicimpedance. 39 ]conductedimpedancemeasurementsonaglassycarbondiskelectrodetocomparewiththecalculationresults.Localimpedancemeasurementswereaswellperformedonastainlesssteeldisktodemonstratethattheinductivefeaturespredictedbythesimulationsareapparentinexperiments. 118

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(b) Complex-impedance-planeplotsfortheresponseofaglassycarbondiskinKClelectrolyteswithconcentrationasaparameter.a)forfrequencyvaluesbetween100kHzand10mHz;andb)zoomedregionshowingonlyhigh-frequencydata. 9-12 withconcentrationasaparameter.Thedierencesamongtheresultsaremostapparentathighfrequencies,asshowninFigure 9-12(b) .Theresultsareconsistentwithablocking,butnotideallypolarized,electrode.Theagreementalsosuggeststhatthereisalocalcapacitydispersionontheglassycarbondiskelectrode.Ahigh-frequencyfeatureisevidentinFigure 9-12(b) ,andthisfeatureappearsatlowerfrequenciesforthesmallerconcentration. ThedimensionlessimaginarypartoftheimpedanceispresentedinFigure 9-13(a) asafunctionofdimensionlessfrequency.Thesuperpositionofdataforthethreevaluesofconductivityisinexcellentagreementwiththecalculations(seeFigure 9-2(b) ),andthechangeinslopefromavalueof1appearsatfrequencieshigherthanK=1. ThederivativeofthelogarithmofthedimensionlessimaginaryimpedancewithrespecttothelogarithmofdimensionlessfrequencyispresentedinFigure 9-13(b) .The 119

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(b) DimensionlessanalysisfortheimpedanceresponseofagraphitediskinKClelectrolyteswithconcentrationasaparameter.a)Dimensionlessimaginarypartoftheimpedanceasafunctionofdimensionlessfrequency(correspondingtoFigure 9-2(b) );andb)Derivativeofthelogarithmofthedimensionlessimaginarypartoftheimpedancewithrespecttothelogarithmofdimensionlessfrequency(correspondingtoFigure 9-3 ). dispersionofthedataapparentinFigure 9-13(b) canbeattributedtothefactthatthederivativecalculationswereperformedonexperimentaldata.ThesuperpositionofdataforthethreevaluesofconductivityisinexcellentagreementwithFigure 9-3 with=0:9,andthetransitionalfrequencybetweenlowandhigh-frequencyresponseisingoodagreementwiththetheoreticalvalueofK=1. 9-14 .[ 93 ]Aspredictedfromthecalculations,thelocalimpedanceexhibitsinductiveloopsathighfrequency;whereas,thelocalinterfacialimpedanceshowsexpectedbehaviorforalocalCPEwithinallfrequencyrange.ThecharacteristictransitionfrequencyatwhichthegeometryplaysarolelocatesapproximatelyatK=0.52,whichisconsistentwiththetheoreticalpredictionK=1.ThelocalinterfacialimpedanceexhibitsanideallocalCPE 120

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Experimentallocalimpedance,localinterfacialimpedance,andlocalOhmicimpedanceinNyquistformatofastainlesssteeldiskelectrodeatthecenteroftheelectrode(r=r0=0).[ 93 ] behavior,whichagreeswiththeperditionfromFigure 9-5 .ThelocalOhmicimpedance,thedierencebetweenthelocalandlocalinterfacialimpedance,isgivenintherectangleboxinthegure.Theshapeofthelocalimpedanceatr=r0=0isconsistentwiththatseeninFigure 9-10 ThelocalimpedanceshowsCPEbehavioratlowfrequenciesandachangeinsignintheimaginarypartoftheimpedanceathighfrequencies.Thisappearanceofhigh-frequencyinductiveloopsisconsistentwiththecalculatedlocalimpedancepresentedinFigure 9-9(b) .TheagreementbetweenthemodelpresentedhereandtheexperimentalresultsobtainedfromthesteelelectrodeillustratestheutilityofthemodelfordescribingfeaturesofsystemsthatexhibitCPEbehavioroverarangeoffrequency. 121

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Theresultspresentedinchapters 8 and 9 illustratethatthecurrentandpotentialdistributionsassociatedwithdiskelectrodesinduceanapparentCPEbehaviorontheimpedanceofblockingelectrodes.ThischapterexplorestheinuenceofcurrentdistributiononimpedanceresponseofadiskelectrodesubjecttoasingleFaradaicreaction.[ 31 ] 8 and 9 .Lapace'sequationwrittenintherotationalellipticcoordinates(equations( 8{5 )and( 8{6 ))remainedasthegoverningequations.Thekeydierencebetweenthepresentworkandthosedescribedinthepreviouschapterswastheboundaryconditionappliedattheelectrodesurface. Theproblemwassolvedfortwokineticregimes.Underlinearkinetics,followingNewman[ 42 ]andNisancioglu[ 43 44 ],thecurrentdensityattheelectrodesurfacewasexpressedas RTV0=@ r0@ Theassumptionoflinearkineticsappliesfor{<
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10{1 )or( 10{2 )applyattheelectrodesurface(=0).Theboundaryconditions( 10{1 )or( 10{2 )werewritteninfrequencydomainas K~j+J~Vr~0;r=1 and K~Vr~0;r+J~0;j=1 forrealandimaginarycomponents,respectively.Here~VrrepresentstheimposedperturbationintheelectrodepotentialreferencedtoanelectrodeatinnityandKisthedimensionlessfrequency,denedas K=!C0r0 Undertheassumptionoflinearkinetics,validfor{<>i0,theparameterJwasdenedtobeafunctionofradialpositionontheelectrodesurfaceas where{()wasobtainedfromthesteady-statesolutionas {=i0expcF RTV0(10{8) Thelocalcharge-transferresistanceforlinearkineticscanbeexpressedintermsofparametersusedinequation( 10{6 )as i0F(a+c)(10{9) 123

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(b) CalculatedrepresentationoftheimpedanceresponseforadiskelectrodeunderassumptionsofTafelandlinearkineticsandwithwithJasaparameter.OpensymbolsrepresenttheresultcalculatedbyNewman.[ 42 ]a)realpart;andb)imaginarypart. derivedintheabsenceofcurrentdistributioneects.TheimpedanceresponseforTafelkineticsdiersbecausethecharge-transferresistanceisafunctionofradialposition.Thecomparisonbetweentheimpedanceforlinearkineticsandequation( 10{13 )forJ=1showsthedistortionofthehigh-frequencyimpedanceresponseassociatedwiththeinuenceofcurrentandpotentialdistributions. ThecalculatedresultsforlinearkineticsinFigure 10-2 showgoodagreementtothecorrespondingnumericalvaluesobtainedbyNewman.[ 42 ]ThecomparisonwithNewman'scalculationsisseenmoreclearlyintherepresentationoftherealandimaginarypartsoftheimpedanceresponseshowninFigures 10-3(a) and 10-3(b) ,respectively.Atlowfrequencies,valuesfortherealpartoftheimpedancedierforimpedancecalculatedundertheassumptionsoflinearandTafelkinetics;whereas,thevaluesoftheimaginaryimpedancecalculatedundertheassumptionsoflinearandTafelkineticsaresuperposedforallfrequencies.TheslopeofthelinespresentedinFigure 10-3(b) areequalto1atlowfrequenciesbutdierfrom1athighfrequencies.AsstatedbyOrazemetal.[ 78 ],theslopeoftheselinescanberelatedtotheexponentusedinmodelsforCPEbehavior. 127

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Thecalculatedderivativeoflog(Zj=r0)withrespecttolog(K)(takenfromFigure 10-3(b) )asafunctionofKwithJasaparameter. Thecalculatedderivativeoflog(Zj=r0)withrespecttolog(K)ispresentedinFigure 10-4 asafunctionofKwithJasaparameter.Atlargefrequencies,thequantitydlog(Zj=r0)=dlog(K)canbeconsideredtobeequalto-whereistheexponentusedformodelsofCPEbehavior.ThecharacteristicfrequencyatwhichthevalueofslopedeviatesfromunityincreaseswiththedimensionlessparameterJ.ThetransitionfrequenciescorrespondtotheinverseoftheRtC0timeconstantandoverlapwhengivenasafunctionof J=!C0RT Thefunctionaldependenceofdlog(Zj=r0)=dlog(K)wasindependentofassumptionofeitherlinearorTafelkinetics. Whendlog(Zj=r0)=dlog(K)wasplottedasafunctionoflog(K=J),giveninFigure 10-5 ,allthecurvesforK<1aresuperimposed.ThecharacteristicfrequencyK/J=1isassociatedwiththeRtC0timeconstantfortheFaradaicreactionandthecharacteristicfrequencyfortheeectofthecurrentandpotentialdistributionsatK=1isassociatedwiththecapacitanceandtheOhmicresistance. 128

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Thecalculatedderivativeoflog(Zj=r0)withrespecttolog(K=J)(takenfromFigure 10-3(b) )asafunctionofKwithJasaparameter. 10-6 asafunctionoffrequencywithnormalizedradialpositionasaparameter.Atlowfrequencies,thelocalinterfacialimpedance,forbothrealandimaginary,issmallestattheperipheryandlargestatthecenterofthedisk.ThevariationatlowfrequenciesislessdistinguishableforsmallervaluesofJ.AllthecurvesinFigures 10-6(a) and 10-6(b) aresuperposedatfrequenciesK>1. Forthelinearkineticscalculation,whereJisindependentofradialposition,thescaledrealpartofthelocalinterfacialimpedancefollows r0=J (J2+K2)(10{15) andtheimaginarypartofthelocalinterfacialimpedancefollows r0=K PlotssimilartoFigure 10-7 wereobtainedforthelocalinterfacialimpedancecalculatedunderassumptionoflinearkinetics,butforlinearkineticsthelocalinterfacialimpedancewasindependentofradialposition. 129

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Athighandlowfrequencylimits,theglobalOhmicimpedancedenedinthepresentworkisconsistentwiththeacceptedunderstandingoftheOhmicresistancetocurrentowtoadiskelectrode.TheglobalOhmicimpedanceapproaches,athighfrequencies,theprimaryresistanceforadiskelectrode(equation( 10{11 ))describedbyNewman.[ 94 ]Thisresultwasobtainedaswellforideallypolarized(Chapter 8 )andblockingelectrodeswithlocalCPEbehavior(Chapter 9 ).ThecomplexnatureofbothglobalandlocalOhmicimpedancesisseenatintermediatefrequencies.ThiscomplexvalueistheoriginoftheinductivefeaturesseeninthelocalimpedanceandtheoriginoftheCPE-likebehaviorfoundintheglobalimpedance. 43 ]estimatedtheerrorcausedbyfrequencydispersioninevaluatingphysicalpropertiessuchaschargetransferresistanceandcapacitance.AparallelanalysisispresentedhereintermsofthecommonlyusedCPEmodels. 10{13 ).Thecorrespondingcharge-transferresistanceevidentatlowfrequenciesisgivenby r0=1 Theeectiveglobalcharge-transferresistancecanbeestimatedfromthecalculatedimpedanceaccordingto r0=Zr JK=01 4(10{20) ThevalueofRe=RtispresentedinFigure 10-13 asafunctionofJundertheassumptionoflinearkinetics.TheresultsarefullagreementwiththosepresentedindierentformatbyNisancioglu[ 43 ].TheinuenceofthefrequencydispersionisgreatestwhenJislarge, 136

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TheapparentvalueofRe=RtobtainedfromthecalculatedimpedanceresponseatlowfrequenciesasafunctionofJ. 10-4 arepresentedinFigure 10-14 asfunctionsofJ.Thevalueofrangesfrom0.98forJ=0:01to0.87forJ=10,whichdemonstratesthatnonuniformcurrentandpotentialdistributionsonadiskelectrodecanyieldhigh-frequencyCPE-likebehavior.AsJbecomessmall,i.e.asthecharge-transferresistancedominatesovertheOhmicresistance,tendstowardunity.ItissignicantthatthecalculatedvalueofshowninFigure 10-14 correspondstoarangeofthatisfrequentlyobservedinexperiments. Asshowninequation( 9{9 ),theeectiveCPEcoecientQeforelectrochemicalsystemsfollows

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Theapparentvalueof1-obtainedfromthecalculatedimpedanceresponseathighfrequenciesasafunctionofJ. ThevalueofeectiveCPEcoecient,scaledbytheinterfacialcapacitance,ispresentedinFigure 10-15 asafunctionofJ.ThefrequenciesreportedinFigure 10-15 arelimitedtothosethatareonedecadelargerthanthecharacteristicfrequencybecause,inthisfrequencyrange,thevalueofiswell-dened.Figure 10-15 wasdevelopedtakingintoaccounttheobservation,seeninFigure 10-4 ,thatthevalueofisdependentonthefrequencyatwhichtheslopeisevaluated.Thus,thevalueofQereportedisthatcorrespondingtothevalueofatagivenfrequencyK. Whilethedimensionsarenotexactlythatofacapacitance,theCPEcoecientisoftenassumedtohaveapproximatelythesamenumericalvalueastheinterfacialcapacitance.ThevalueofQepresentedinFigure 10-15 isafunctionoffrequency.Athigh-frequencies,wherefrequencydispersionplaysasignicantrole,theeectiveCPEcoecientQeprovidesaninaccurateestimatefortheinterfacialcapacitance,evenforsmallvaluesofJwhereisclosetounity.Theerrorsinestimatingtheinterfacialcapacitanceareontheorderof500percentatK=100. 138

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EectiveCPEcoecientscaledbytheinterfacialcapacitanceasafunctionofJ. AnumberofresearchershaveexploredtherelationshipbetweenCPEparametersandtheinterfacialcapacitance.HsuandMansfeld[ 95 ]proposed where!max(orKmax)isthecharacteristicfrequencyatwhichtheimaginarypartoftheimpedancereachesitsmaximumvalueandCeistheestimatedinterfacialcapacitance.Equation( 10{21 )istestedagainsttheinputvalueofinterfacialcapacitanceinFigure 10-16 whereC0istheknowninterfacialcapacitancewhichwasindependentofradialposition.Asdescribedabove,Figure 10-16 wasdevelopedusinglocalfrequency-dependentvaluesofandQe.ThefrequenciesreportedinFigure 10-16 arelimitedtothosethatareonedecadelargerthanthecharacteristicfrequency!max.Whileequation( 10{21 )representsanimprovementascomparedtodirectuseoftheCPEcoecientQe,theerrorsinestimatingtheinterfacialcapacitancedependonbothJandKandrangebetween70to+100percent. 139

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Eectivecapacitancecalculatedfromequation( 10{21 )andnormalizedbytheinputinterfacialcapacitanceforadiskelectrodeasafunctionofdimensionlessfrequencyKwithJasaparameter.(SeeHsuandMansfeld[ 95 ]) Brugetal.[ 30 ]developedarelationshipforablockingelectrodebetweentheinterfacialcapacitanceandtheCPEcoecientQas AsimilarrelationshipbetweentheinterfacialcapacitanceandtheCPEcoecientQwasdevelopedforaFaradaicsystemas Equations( 10{22 )and( 10{23 )arecomparedtotheexpectedvalueofinterfacialcapacitanceinFigures 10-17(a) and 10-17(b) ,respectively.Figures 10-17(a) and 10-17(b) weredevelopedusinglocalfrequency-dependentvaluesofandQeoverthesamefrequencyrangeasisreportedinFigures 10-15 and 10-16 .ThefrequenciesreportedinFigure 10-17 arelimitedtothosethatareonedecadelargerthanthecharacteristicfrequency!max.Theerrorinequation( 10{22 )isafunctionofbothfrequencyKand 140

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(b) NormalizedeectivecapacitancecalculatedfromrelationshipspresentedbyBrugetal.[ 30 ]foradiskelectrodeasafunctionofdimensionlessfrequencyKwithJasaparameter.a)withcorrectionforOhmicresistanceRe(equation( 10{22 ));andb)withcorrectionforbothOhmicresistanceReandcharge-transferresistanceRt(equation( 10{23 )). 10{23 )providesthebestmeansforestimatinginterfacialcapacitancewhenfrequencydispersionissignicant.Thecapacitanceanalysispresentedhereshowsthat,fordetermininginterfacialcapacitance,theinuenceofcurrentandpotentialdistributionsontheimpedanceresponsecannotbeneglected,eveniftheapparentCPEexponenthasvaluesclosetounity. 141

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Thisdissertationcoverstworesearchtopicsthatareimportanttocorrosionofmetal.TheconclusionassociatedwiththedelaminationmodelispresentedinSection 11.1 ,andthatassociatedwiththeimpedancecalculationispresentedinSection 11.2 9 { 11 14 { 16 ]forcoatedsteelandcoatedzinc.TheconsistencywithexperimentalobservationssupportsthehypothesesproposedbyAllaharthattheporosityandpolarizationkineticscanbetreatedasfunctionsofpH. Thesimulatedresultsobtainedusingtheequilibrium"-pHrelationshipdemonstratethattheoveralldelaminationprocessispreliminarygovernedbythetransportofthecationsfromthedefecttothefrontregion.Therateofthedelaminationdependsonthemobilityandtheconcentrationofthecations.Theanions,ontheotherhand,havenosignicantinuenceonthedelaminationrate. Thecomputationalresultsobtainedusinganon-equilibrium"-pHrelationshipindicatethat,whenthebond-breakingreactionstakeplaceatasucientlyslowrate,thepotentialfrontandtheporosityfrontbecomedistinguishable.ThemovementofthepotentialfrontfollowsthechangeofpHalongthemetal-coatinginterface;whereas,themovementoftheporosityfrontislimitedbythebond-breakingreactions.Thekineticanalysisofthenon-equilibriumresultsalsoshowsthatthedelaminationmechanismshiftsfromamass-controlledmechanismtoamixedcontrolledmechanismwhenthebond-brakingreactionsaresucientlyslow. 142

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Thisappendixpresentstheprogramlistingforthecathodicdelaminationmodel.TheprogramwasdevelopedusingMicrosoftVisualFortran,Version9.0withdoubleprecisionaccuracy.

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!subroutineofBAND(J)algorithmSUBROUTINEBAND(J)IMPLICITDOUBLEPRECISION(A-H,O-Z)IMPLICITINTEGER(I-N)DOUBLEPRECISIONE(11,12,2001)common/cb/a(11,11),b(11,11),c(11,2001),d(11,23),g(11),x(11,11),&&y(11,11),n,njSAVEE,NP1101FORMAT(/15HDETERM=0ATJ=,I4)IF((J-2).LT.0)GOTO1IF((J-2).EQ.0)GOTO6IF((J-2).GT.0)GOTO81NP1=N+1DO2I=1,ND(I,2*N+1)=G(I)DO2L=1,NLPN=L+N2D(I,LPN)=X(I,L)CALLMATINV(N,2*N+1,DETERM)IF(DETERM)4,3,43PRINT101,J4DO5K=1,NE(K,NP1,1)=D(K,2*N+1)DO5L=1,NE(K,L,1)=-D(K,L)LPN=L+N5X(K,L)=-D(K,LPN)RETURN6DO7I=1,NDO7K=1,NDO7L=1,N7D(I,K)=D(I,K)+A(I,L)*X(L,K)

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TheprogramlistingsfortheimpedancecalculationspresentedinChapters 8 to 10 aregiveninthisappendix.Thetheoreticaldevelopmentforthecasesofideally-polarizedelectrodes(Chapter 8 ),electrodeswithlocalCPE(Chapter 9 ),andelectrodesexhibitingFaradaicreactions(Chapter 10 )weresimilar.Thekeydierencewastheboundaryconditionappliedontheelectrodesurface.ThecalculationswereperformedusingthecollocationpackagePDE2DdevelopedbySewell.[ 90 ]

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Schematicrepresentationofarotatingdiskelectrodesysteminwhichadiskelectrodeisembeddedinalargeinsulator. thedisksthatbringsreactantstothesurfaceisexpectedtofollow whereisthekinematicviscosityand.Neartheelectrodesurface,thedimensionlessvelocityHinequation( C{2 )canbeexpressedasapowerseries 3zr wherethecoecientsaandbhavevaluesof0.51023and-0.616,respectively. 4{6 )andoxygenreduction( 4{7 ).Undertheassumptionthattheelectrochemicalreactionconsideredareirreversible,thecurrentdensitiesduetozincdissolutionfollowedtheBulter-Volmerexpression.Theoxygenreductionwasassumedtobemass-transfer-limited;thus,thelimitingcurrentdensity,showninequation( 2{39 ),dependsontheconcentrationofoxygeninthebulk.Thethicknessofthediusionlayerxseeninequation( 2{39 ),ina 184

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3(C{4) whereScisSchmidtnumberand1 3isthegammafunctionof1/3. 66 67 ]Inthepresentedmodel,multiplehomogeneousreactions,includingwaterdissociationandaseriesofreactionsassociatedwithZn2+hydrolysis,wereconsidered.ThemechanismsandequilibriumconditionsofthesechemicalreactionsaresummarizedinTable 4-2 .[ 67 ] 4-2 weretreatedasboundaryconditionsforH+,ZnOH+,HZnO2,andZnO22. 74 ] 185

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CalculatedconcentrationdistributionsofspeciesofOH,Zn2+,H+,ZnOH+,HZnO2,andZnO22onelectrodesurface. TheconcentrationdistributionsofthechemicalspeciesarepresentedinFigure C-2 asafunctionofdimensionlessposition.Duetotheelectrochemicalreactionsoccurringontheelectrodesurface,theconcentrationsofOHandZn2+haveamaximumnearthesurface.Theconcentrationsofthespeciesproducedinthehomogeneousreactions(H+,ZnOH+,HZnO2,andZnO22)arelargestnearthesurfaceanddecreasewithincreasingdistanceawayfromtheelectrode. Atsteadystate,basedonthemassbalanceequation,thehomogenousrateforspeciesRicanbewrittenby Thus,therateforeachofthehomogeneousreactionscanbeexpressedasfollows: 186

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(b) Calculatedrateofthehomogeneousreactionsincludedinthemodel.a)Waterdisassociationreaction;andb)Zinchydrolysisreactions. and TherateofwaterdissociationisplottedinFigure C-3(a) asafunctionofdimensionlessposition.TheOHionsgeneratedneartheelectrodesurfacecombinewithH+ionsinthesolutiontoformwatermolecular.Duetotheconstraintoftheequilibriumcondition,theconcentrationofH+ionsissmallneartheelectrode;consequently,therateofthisreactionisapproximatelyzeronearthesurface.Whenz=approaches5.4,theconcentrationsofOHionsbecomesclosetothatofH+ions,leadingtothesharpincreaseinFigure C-3(a) TheratesofthezinchydrolysisarepresentedinFigure C-3(b) asafunctionofdimensionlessposition.ThenegativevalueofR2indicatesthatZnOH+ionsdissociateintoZn2+andOHions.Thisimpliesthat,aftertheZn2+andOHionsareformed,theZnOH+ionswereimmediatelyproducednearthesurface.Thiscanalsobeusedtoexplainthenegativeratesofreactions3and4. 187

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Asasteptowardthedevelopmentofcomprehensivemodelforcathodicdelamination,atwo-dimensionalmodelwasdevelopedthatcalculatedthedistributionsofconcentrationsandpotentialassociatedwithcut-edgecorrosion.Withinthepresentmodel,theZnelectrodeservesasthelocalanodeandsteelasthelocalcathode.Thepurposeofthemodelwastounderstandthesetupofthegalvaniccouplebystartingfromuniformdistributionsofallreactivespecies.Theuniforminitialconditionspermittheconcentrationandpotentialgradientsappearinglaterduetotheelectrochemicalreactions.Withinthemodel,multiplehomogeneousreactions,includingwaterdissociationandaseriesofreactionsassociatedwithhydrolysiswereassumedtooccursimultaneouslyinthesolutionphase. D-1 whereZnactsasananode,Feactsasacathode,andNaClservesastheelectrolyte.Aninsulatorisinsertedbetweenthetwoelectrodesandthetwoverticalwallsarecomposedofinsulatorsaswell.Zincdissolutionwasassumedtotakeplaceontheanode,whereasoxygenreductionandhydrogenevolutionwerebothassumedtooccuronthecathode.Noirondissolutionwasconsideredonthesteel. FigureD-1. SchematicrepresentationofatwoelectrodecellinwhichZnservesalocalanodeandFeasalocalcathode. 189

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21 22 ]Asaresult,anumericaltechniquewithhigh-orderaccuracyisextremelycrucialforthisparticulardevelopment. Inthedevelopmentofthepresentmodel,acommercialprogrambaseduponcollocationmethod,PDE2D,waschosen.Theuseofthecollocationmethodyieldsapproximationsthatareofhigh-orderaccuracyevenwhencouplednonlinearpartialdierentialequationsaresolvedinamultidimensionaldomain.WiththisPDE2Dprogram,allhomogeneousreactioncanbeincludedsimultaneouslyandthediscontinuityatthebottomboundarycanbehandledwithoutnumericaldiculties. 190

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(b) Calculatedcurrentdensitydistributionsalongthexaxis.a)0
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(b) Calculateddistributionsofconcentrationinaunitofmole/cm3.a)Zn2+ions;andb)OHions. FigureD-4. CalculateddistributionofratioofcZn2+c2OH 192

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D-4 .ThemagnitudeinFigure D-4 isestimatedbycZn2+c2OH 10 ]ThevalueofcZn2+c2OH 193

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Duetotheuseofporosity-pH,poisoning-pHandblocking-pHrelations,thereareseveralttingparametersusedinthecathodicdelaminationprogram.Itisimportanttoexplorethesensitivitiesoftheseparametersonthesimulationresults.Thesensitivityanalysisispresentedinthisappendix.Inthereport,eachttingparameterwereanalyzedusingthreedierentvalues.Thevelocityofthemovingfrontsandthekineticanalysiswerecheckedtodeterminethesensitivity. 4-2 ).Thettingparameterb";1isassociatedwiththeconstantvalueseenatthehighpHregion.Thettingparameterb";2governstheslopeofthecurvethatincreasesfromlowpHtohighpHregions.Thedeectionpointlocatedinthemiddleoftheincreasingcurveiscontrolledbythettingparameterb";3.TheconstantvalueseenatthelowpHregionisgovernedbythettingparameterb";4. E-1 .Whenthettingparameterb";1thatgovernstheporosityathighpHincreasesfrom0.01to0.1,therateofthedelaminationdeterminedbybothpotentialfrontandporosityfrontincreasebyapproximately60percent.Itisobservedthatwhenb";1isequal0.1,theinterfacialpotentialintheintactregionincreaseswithdelaminationtime.Thechangeofb";1from0.01to0.001decreasesthemovingvelocitiesbyapproximately40percent.Thereactionorder,however,isnotinuencedbythechangetothettingparameterb";1. TableE-1. Sensitivityanalysisforb";1 PotentialFrontVelocity 1.33 2.19 3.58PorosityFrontVelocity 1.09 1.63 3.27ReactionOrderforPotentialFront 0.54 0.55 0.56ReactionOrderforPorosityFront 0.59 0.6 0.6 194

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Sensitivityanalysisforb";2 PotentialFrontVelocity 2.26 2.19 1.98PorosityFrontVelocity 2.20 1.63 1.18ReactionOrderforPotentialFront 0.56 0.55 0.55ReactionOrderforPorosityFront 0.63 0.6 0.6 FigureE-1. Thesensitivityoftheslopeoftheincreasingcurvetob";2. E-2 .AsillustratedinFigure E-1 ,theslopeoftheincreasingcurveinFigure 6-6 becomessteeperwhenthettingparameterb";2changesfrom-2to-5.Theincreaseinthesloperesultsinasightincreaseinthevelocitiesofbothpotentialfrontandporosityfront.Theotherobservedfeatureisthattheratesofthetwofrontsareapproximatelyequalwhentheslopeissteep.Itisalsoobservedthattheshapeoftheresultingpotentialdistributions(seeFigure 6-6 )isalsoinuencedbythisparameter.Thereactionorder,however,isnotinuencedbythechangeinb";2. E-3 .ThedeectionpointoftheincreasingcurveseeninFigure 4-2 shiftsfromlefttorightwhenthettingparameterb";3increasesfrom9.8to11.8.Whenthedeectionpointis 195

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Sensitivityanalysisforb";3 PotentialFrontVelocity 2.73 2.19 1.73PorosityFrontVelocity 2.27 1.63 0.5ReactionOrderforPotentialFront 0.53 0.55 0.56ReactionOrderforPorosityFront 0.54 0.6 1.1 TableE-4. Sensitivityanalysisforb";4 1.94 2.19 3.66PorosityFrontVelocity 1.62 1.63 2.98ReactionOrderforPotentialFront 0.56 0.55 0.56ReactionOrderforPorosityFront 0.61 0.6 0.72 pushedtohighpHregion,thevelocitiesofthefrontsdecrease.Forb";3=11.8,thevelocityoftheporosityfrontdecreasesto0.5mm/hrandthereactionorderanalysisshowsakinetic-controlledmechanismfortheporosityfront. E-4 .Thettingparameterb";4controlstheconstantvalueseenatthelowpHregioninFigure 4-2 .AsshowninTable E-4 ,thedecreaseinthisttingparameterdoesnothavestrongimpactontherateorthemechanismofthedelamination.However,whenb";4increasesto0.01,themovementsofbothpotentialandporosityfrontsincreasedramatically.Forb";4=0.01,themechanismofthedelaminationremainsmass-transfercontrolledforthepotentialfront,butchangestoamixed-controlledmechanismfortheporosityfront. 4-3(b) ).Thettingparameterb;1inuencestheconstantvalueatthehighpHregion.Thettingparameterb;2governstheslopeofthetransitionfromlowpHtohighpHregions.Thedeectionpointseeninthemiddleoftheincreasingcurvechangeswiththettingparameterb;3.TheconstantvalueseenatthelowpHregionisrelatedtothettingparameterb;4. 196

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Sensitivityanalysisforb;1 PotentialFrontVelocity 2.18 2.19 2.16PorosityFrontVelocity 1.6 1.63 1.6ReactionOrderforPotentialFront 0.55 0.55 0.56ReactionOrderforPorosityFront 0.62 0.6 0.61 TableE-6. Sensitivityanalysisforb;2 PotentialFrontVelocity 2.1 2.19 2.2PorosityFrontVelocity 1.35 1.63 1.75ReactionOrderforPotentialFront 0.55 0.55 0.55ReactionOrderforPorosityFront 0.62 0.6 0.62 E-5 .Whenthehigh-pHvalueinFigure 4-3(b) increasesbytwoorders(b;1changesfrom3.5to5.5),therateofthedelaminationremainsapproximatelythesame.Thereactionordersdeterminedbaseduponthepotentialfrontandtheporosityfrontarenotinuencedmuchbythettingparameterb;1. E-6 .TheslopeoftheincreasingcurveinFigure 4-3(b) becomessteeperwhenthettingparameterb;2changesfrom-1.3to-5.3.Thechangeinb;2doesnotinuencetheratedeterminedbythepotentialfront,buthasaslightimpactonthatdeterminedbytheporosityfront.Forthevaluesthathavebeentested,thekineticanalysisindicatesthatthechangeinb;2hasnosignicantimpactonthedelaminationmechanism. E-7 .ThedeectionpointoftheincreasingcurveseeninFigure 4-3(b) shiftsfromlefttorightwhenthettingparameterb;3changesfrom9.4to11.4.Thesimulationresultsdonotchangemuchwhenb;3changesfrom9.4to10.4,butthevelocitiesofbothfronts 197

PAGE 198

Sensitivityanalysisforb;3 PotentialFrontVelocity 2.15 2.19 2.06PorosityFrontVelocity 1.62 1.63 1.33ReactionOrderforPotentialFront 0.55 0.55 0.55ReactionOrderforPorosityFront 0.62 0.6 0.62 TableE-8. Sensitivityanalysisforb;4 PotentialFrontVelocity 2.20 2.19 2.16PorosityFrontVelocity 1.68 1.63 1.65ReactionOrderforPotentialFront 0.56 0.55 0.56ReactionOrderforPorosityFront 0.63 0.6 0.62 decreaseslightlywhenb;3changesfrom10.4to11.4.Forthevaluesthathavebeentested,thekineticanalysisshowsthatthechangeinb;3hasnosignicantimpactonthedelaminationmechanism. E-8 .Thecomputationalresultsremainapproximatelythesamewhenb;4changesfrom-15,-16to-17.Forthevaluesthathavebeentested,thesimulationresultsareinsensitivetothisttingparameter. 4-4(b) .SeventtingparameterswereusedintheconstructionofFigure 4-4(b) .Thettingparameterb;1isrelatedtotheconstantvalueseenatthelowpHregion.Thettingparameterb;2isassociatedwiththeslopeofthecurveatthelowpHregion.Thedeectionpointatlow-pHcurveiscontrolledbythettingparameterb;3.Thettingparameterb;4isassociatedwiththeconstantvalueatthehighpHregion.Thettingparameterb;5isrelatedtotheslopeoftheshortcurveseenathighpH.Thelengthoftheconstantregion 198

PAGE 199

Sensitivityanalysisforb;1 PotentialFrontVelocity 2.0 2.19 2.52PorosityFrontVelocity 1.60 1.63 2.29ReactionOrderforPotentialFront 0.56 0.55 0.57ReactionOrderforPorosityFront 0.62 0.6 0.65 TableE-10. Sensitivityanalysisforb;2 PotentialFrontVelocity 2.29 2.19 2.07PorosityFrontVelocity 1.78 1.63 1.45ReactionOrderforPotentialFront 0.55 0.55 0.55ReactionOrderforPorosityFront 0.62 0.6 0.62 inthemiddlerangeofpHischaracterizedbyb;6andthemagnitudeoftheconstantseenatthefrontregionisgovernedbythettingparameterb;7. E-9 .Whenthettingparameterb;1changesfrom6.5to7.5,thecomputationalresultsintermsofdelaminationrateandmechanismdonotchangemuch.However,whenb;1changesfrom7.5to8.5,correspondingtoanincreaseintheblockingfactoratlowpH,theratesandkineticorderdeterminedbybothpotentialfrontandporosityfrontshowincreasingtendency. E-10 .TheslopeoftheincreasingcurveatlowpHinFigure 4-4(b) becomessteeperwhenthettingparameterb;2changesfrom-4to-10.Theincreaseinthesloperesultsinanincreaseinthevelocitiesofbothpotentialfrontandporosityfront.Forthevaluesthathavebeentested,thekineticanalysisindicatesthatthechangeinb;2hasnosignicantimpactonthedelaminationmechanism. 199

PAGE 200

Sensitivityanalysisforb;3 PotentialFrontVelocity 2.51 2.19 2.0PorosityFrontVelocity 1.71 1.63 1.5ReactionOrderforPotentialFront 0.58 0.55 0.55ReactionOrderforPorosityFront 0.59 0.6 0.61 TableE-12. Sensitivityanalysisforb;4 PotentialFrontVelocity 2.11 2.19 2.26PorosityFrontVelocity 1.18 1.63 1.78ReactionOrderforPotentialFront 0.55 0.55 0.55ReactionOrderforPorosityFront 0.62 0.62 0.6 E-11 .ThedeectionpointoftheincreasingcurveatlowpHinFigure 4-3(b) shiftsfromlefttorightwhenthettingparameterb;3changesfrom8.8to10.8.WhenthedeectionpointispushedtohighpHregion,thevelocitiesofthefrontsdecrease.Forb;3=10.8,thevelocityoftheporosityfrontdecreasesto1.5mm/hr,butthereactionorderbasedupontheporosityfrontdoesnotchangewithb;3. E-12 .Thettingparameterb;4isassociatedwiththeconstantvalueseenatthehighpHregioninFigure 4-3(b) .Whenb;4changesfrom-2.5to-0.5,theblockingfactorathighpHincreasesfrom105to103;thus,thedelaminationratesincreasewithb;4.Thedelaminationmechanismisnotinuencedsignicantlybyb;4. E-13 .Thettingparameterb;5isrelatedtotheslopeoftheshortcurveseenathighpHinFigure 4-4(b) .Whenthettingparameterb;5increasesfrom-60to-40,theslopeofthecurvebecomesmoregradual.Thecomputationalresultsremainapproximatelythesame 200

PAGE 201

Sensitivityanalysisforb;5 PotentialFrontVelocity 2.20 2.19 2.23PorosityFrontVelocity 1.66 1.63 1.63ReactionOrderforPotentialFront 0.55 0.55 0.55ReactionOrderforPorosityFront 0.62 0.6 0.62 TableE-14. Sensitivityanalysisforb;6 PotentialFrontVelocity 2.15 2.19 2.15PorosityFrontVelocity 1.6 1.63 1.66ReactionOrderforPotentialFront 0.55 0.55 0.55ReactionOrderforPorosityFront 0.61 0.6 0.62 forthreevaluesofb;5.Forthevaluesthathavebeentested,thesimulationresultsareinsensitivetothisttingparameter. E-14 .InFigure 4-4(b) ,thelengthoftheconstantregioninthemiddlerangeofpHincreaseswiththettingparameterb;6.Thecomputationalresultsremainapproximatelythesamewhenb;6increasesfrom10.1to12.1.Forthevaluesthathavebeentested,thesimulationresultsareinsensitivetothisttingparameter. E-15 .InFigure 4-4(b) ,themagnitudeoftheconstantinthemiddlerangeofpHincreasesfrom104to102whenb;6changesfrom-11.1to-9.1.Theincreaseintheblockingfactorleadstoaincreaseinthedelaminationrates.Thekineticanalysisfor-11.1and-10.1 TableE-15. Sensitivityanalysisforb;7 PotentialFrontVelocity 1.96 2.19 2.36PorosityFrontVelocity 1.47 1.63 1.77ReactionOrderforPotentialFront 0.55 0.55 0.57ReactionOrderforPorosityFront 0.59 0.6 0.68 201

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Thesensitivityanalysispresentedaboveindicatesthattheparametersusedintheconstructionoftheporosity-pHrelation(b";1tob";4)aremostsensitiveonesinthesimulations.Thettingparameterb";1thatgovernstheporosityathighpHinuencesupto60percentofthedelaminationrate.Theshapeoftheresultingpotentialdistributionsisalsoinuencedbythatoftheequilibratedporosity-pHrelation.Thetheparametersusedintheconstructionofthepoisoningfactor-pHrelation(b;1tob;4)andblockingfactor-pHrelation(b;1tob;7)are,ingeneral,lesssensitive.However,theincreaseinthecathodiccurrentdensityatthefrontregionresultsinanincreaseindelaminationrateandashiftindelaminationkinetics. 202

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## Material Information

Title: Fundamental Approach to Practical Corrosion Problems
Physical Description: 1 online resource (210 p.)
Language: english
Creator: Huang, Mei Wen
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

## Subjects

Subjects / Keywords: cathodic, corrosion, delamination, impedance
Chemical Engineering -- Dissertations, Academic -- UF
Genre: Chemical Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

## Notes

Abstract: Our research covers two topics that are important to corrosion of metal. The first project involved developing a mathematical model for delamination of polymeric coating from active materials. The model described the coupled phenomena of mass transfer, electroneutrality, and loss of interfacial adhesion during the delamination. The key to this work was to use pH-dependent porosity and polarization kinetics to simulate the disbondment during the delamination process. The results predicted from the model, under the assumption of equilibrium pH-porosity relation, concluded the delamination process is limited by the transport of cations from defect to delaminated zone, and the delamination rate scales with the mobility and ionic strength of cations. The equilibrium pH-porosity relation becomes invalid when time constants for the bond-breaking reactions are large compared to those for the diffusion and migration processes. The investigation of kinetic pH-porosity relation showed that when the bond-breaking reactions occur at a sufficiently small rate, the delamination mechanism shifts from a mass-transfer-limited mechanism to a mixed-controlled mechanism. The second project explored the role of current and potential distributions associated with disk electrodes on impedance response. It has known that the geometry-induced current and potential distributions lead to a high-frequency dispersion on impedance response of a disk electrode. The contribution of the work was to express the geometric effect on impedance response in terms of constant-phase element (CPE) and to use both global and local impedances to study this geometric effect. The systems studied in this project included an ideally blocking electrode, an electrode exhibiting a local CPE behavior, and an electrode exhibiting a Faradaic reaction. The results showed that the global impedance is influenced by the current and potential distributions at high frequencies. While the local interfacial impedance exhibits the expected behavior for a given system, the local impedance shows inductive behavior at high frequency and ideal behavior at low frequency. The local impedance is influenced by Ohmic impedance, which has complex behavior at intermediate frequencies. This complex character is believed to be the origin of the inductive features seen in the local impedance and the CPE-like behavior found in the global impedance.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Mei Wen Huang.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.

## Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021337:00001

## Material Information

Title: Fundamental Approach to Practical Corrosion Problems
Physical Description: 1 online resource (210 p.)
Language: english
Creator: Huang, Mei Wen
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

## Subjects

Subjects / Keywords: cathodic, corrosion, delamination, impedance
Chemical Engineering -- Dissertations, Academic -- UF
Genre: Chemical Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

## Notes

Abstract: Our research covers two topics that are important to corrosion of metal. The first project involved developing a mathematical model for delamination of polymeric coating from active materials. The model described the coupled phenomena of mass transfer, electroneutrality, and loss of interfacial adhesion during the delamination. The key to this work was to use pH-dependent porosity and polarization kinetics to simulate the disbondment during the delamination process. The results predicted from the model, under the assumption of equilibrium pH-porosity relation, concluded the delamination process is limited by the transport of cations from defect to delaminated zone, and the delamination rate scales with the mobility and ionic strength of cations. The equilibrium pH-porosity relation becomes invalid when time constants for the bond-breaking reactions are large compared to those for the diffusion and migration processes. The investigation of kinetic pH-porosity relation showed that when the bond-breaking reactions occur at a sufficiently small rate, the delamination mechanism shifts from a mass-transfer-limited mechanism to a mixed-controlled mechanism. The second project explored the role of current and potential distributions associated with disk electrodes on impedance response. It has known that the geometry-induced current and potential distributions lead to a high-frequency dispersion on impedance response of a disk electrode. The contribution of the work was to express the geometric effect on impedance response in terms of constant-phase element (CPE) and to use both global and local impedances to study this geometric effect. The systems studied in this project included an ideally blocking electrode, an electrode exhibiting a local CPE behavior, and an electrode exhibiting a Faradaic reaction. The results showed that the global impedance is influenced by the current and potential distributions at high frequencies. While the local interfacial impedance exhibits the expected behavior for a given system, the local impedance shows inductive behavior at high frequency and ideal behavior at low frequency. The local impedance is influenced by Ohmic impedance, which has complex behavior at intermediate frequencies. This complex character is believed to be the origin of the inductive features seen in the local impedance and the CPE-like behavior found in the global impedance.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Mei Wen Huang.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.

## Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021337:00001

Full Text
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FUNDAMENTAL APPROACH TO PRACTICAL CORROSION PROBLEMS

By
MEI-WEN HUANG

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2007

S2007 Mei-Wen Huang

To my parents in Taiwan, who, ah--an- support all my efforts especially for my PhD

ACKENOWLED GMENTS

I sincerely express my gratitude to my advisor, Professor Mark E. Orazem, for his

technical guidance throughout my graduate research. I will ak- .1-< be grateful for his

patience and encouragement. I also thank Professor K~evin Ogle, Dr. C!~!s-I i Ia Alley,

Dr. Bernard Tribollet, Dr. Vincent Vivier, and Dr. Nadine Piibi~re for their expertise in

electrochemistry. I express my appreciation to my committee members, Professor Jason

Butler, Professor K~irk Ziegfler, Professor C'I I.) 1. Martin, Professor Chlungt-Won Park, and

Professor Darryl Butt for their contribution to my proposal presentation and dissertation

defense.

I thank our group members, Patrick McE~inney, Sunil Roy, Shaoling Wu, and Bryan

Hirschorn. Special appreciation goes to the former group members K~erry Allahar and

Nellian Perez-Garcia for their support. I was very fortunate to be a member of this group.

I wish to acknowledge IRSID, Arcelor Innovation in France, for funding the project

and for giving me a chance to perform experiments in their laboratories in 2006. I also

express appreciation to my friends in Arcelor, Yannick Lerous, Taigo Machado, and

Aurelie Felten for their friendship and for their assistance in life and work.

Finally, and most importantly, I express my thanks and gratitude to my parents and

my friends in Taiwan, who have .ll.-- li-- supported me in spirit along this journey. They

gave me the courage to pursue my dreams, and I am ak- .1-< going to be grateful for having

them by my side.

page

ACK(NOWLEDGMENTS .......... . .. .. 4

LIST OF TABLES ......... ..... .. 9

LIST OF FIGURES ......... .... .. 10

ABSTRACT ......... ...... 17

CHAPTER

1 INTRODUCTION ......... ... .. 19

1.1 Mathematical Models for Cathodic Delamination of Coated Metal .. .. 19
1.2 Influence of Geometry-Induced Current And Potential Distribution of Disk
Electrodes on Impedance Response ...... ... 22

2 BACKGROUND ELECTROCHEMISTRY ..... ... 25

2.1 Mass Transport ......... . . 25
2.2 Solution Potential ......... . 26
2.3 Electrochemical K~inetics ........ ... 27
2.3.1 K~inetic Control ........ ... 30
2.3.2 Mass Transfer Control ...... .. .. 31

3 LITERATURE REVIEW ON CATHODIC DELAMINATION .. .. .. .. 33

3.1 Experimental Observation ......... ... 33
3.2 Mathematical Models ........ .. .. 37
3.2.1 Crevice and Disbonded Coating Models ... ... .. 38
3.2.2 Cathodic Delamination Model ...... .. 39
3.2.2.1 pH-Dependent Porosity .... ... .. 40
3.2.2.2 pH-Dependent Polarization K~inetics .. .. .. .. 41
3.3 Objective ........ . .. 42

4 THEORETICAL DEVELOPMENT OF DELAMINATION MODEL .. .. 43

4.1 Porosity-pH Relation ......... .. .. 43
4.2 Polarization K~inetics ......... .. .. 45
4.2.1 Zinc Dissolution ......... .. .. 46
4.2.2 Poisoning-pH Relation . ...... .. 46
4.2.3 Oxygen Reduction ......... .. .. 47
4.2.4 Blocking-pH Relation ....... ... .. 48
4.3 CI..~ ..... I1 Reactions ......... .. .. 49

5 CATHODIC DELAMINATION MODEL ........

5.1 Governing Equations .. ... .. .. 51
5.2 Boundary Condition .. ... .. .. 5:3
5.3 Solution Method .. ... . .. 5:3

6 RESITLTS AND DISCUSSION FOR DELAMINATION MODEL .. .. .. 55

6.1 Initial Conditions ........ .. .. ... .. 55
6.1.1 Initial Concentration Distributions .... .... .. 56
6.1.2 Initial Distribution of Porosity ..... .. . 56
6.1.3 Initial Distribution of Polarization Parameters .. .. .. 57
6.2 Equilibrium Porosity-pH Relationship ..... .. . 59
6.2.1 Interfacial Potential Distribution .... ... .. 60
6.2.2 Concentration Distributions ...... .. .. 62
6.2.3 Precipitated Corrosion Product ..... ... .. 64
6.2.4 Porosity Distribution ........ ... .. 67
6.2.5 Delantination K~inetics ... .. . .. .. 68
6.2.5.1 Influence of Cation Type on Delantination Rate .. .. 69
6.2.5.2 Influence of Anion Type on Delantination Rate .. .. 72
6.2.5.3 Influence of Electrolyte Concentration on Delantination
Rate ............ ........ 72
6.3 K~inetic Porosity-pH Relationship . ..... 75
6.3.1 Potential Front and Porosity Front .... ... .. 75
6.3.2 Delantination K~inetics . ...... .. 80

7 ELECTROCHEMICAL IMPEDANCE SPECTROSCOPY .. .. .. .. .. 81

7. assive Electrilcal Circuits .......... 8:
7.2 Constant-Phase Element (CPE) . ..... .. .. 85
7.2.1 Origin of CPE .. ... .. .. 85
7.2.2 2-D and :3-D Distributions . . . .. 86
7.3 Current and Potential Distributions on Disk Electrode .. .. .. .. 88
7.3.1 Primary Current Distribution ...... .. . 88
7.3.2 Secondary Current Distribution ..... ... .. 89
7.4 Objective ......... .. .. .. 91

8 IDEALLY POLARIZED BLOCKING DISK( ELECTRODE .. .. . 92

8.1 Theoretical Development .... ... .. 92
8.2 Definition of Impedance .. ... ... .. 95
8.2.1 Global Impedance .... ... .. 95
8.2.2 Local Impedance .. ... ... .. 97
8.2.3 Local Interfacial Impedance ..... .. .. 97
8.2.4 Local Ohmic Impedance . .... .. .. 98
8.2.5 Global Interfacial Impedance ...... .... .. 98
8.2.6 Global Ohmic Impedance . ..... .. .. 98

8.3 Results and Discussion ......... .. .. 99
8.3.1 Global Impedance ......... .. .. 99
8.3.2 Local Interfacial Impedance ...... .. . 101
8.3.3 Local Impedance ......... ... .. 103
8.3.4 Local Ohmic Impedance . ... .. 105
8.3.5 Global Interfacial and Global Ohmic Impedance .. .. .. .. 107

9 BLOCKING DISK( ELECTRODE WITH LOCAL CPE .. .. .. 108

9.1 Theoretical Development ......... .. .. 108
9.2 Results and Discussion ......... .. .. 109
9.2.1 Global Impedance ......... .. .. 109
9.2.2 Local Interfacial Impedance ...... .. . 113
9.2.3 Local Impedance ......... ... .. 115
9.2.4 Local Ohmic Impedance . ... .. 117
9.2.5 Global Interfacial and Global Ohmic Impedance .. .. .. .. 117
9.3 Experiments .... .... .. .... .. 118
9.3.1 Global Impedance of Glassy-Carbon Electrode .. .. .. 119
9.3.2 Local Impedance of Stainless Steel Electrode .. .. .. 120

10 DISK( ELECTRODE WITH SINGLE FARADAIC REACTION .. .. .. .. 122

10.1 Theoretical Development ......... .. .. 122
10.2 Results and Discussion ......... .. .. 125
10.2.1 Global Impedance ......... .. .. 125
10.2.2 Local Interfacial Impedance ...... .. .. 129
10.2.3 Local Impedance ......... ... .. 130
10.2.4 Local Ohmic Impedance . .... .. .. 131
10.2.5 Global Interfacial and Global Ohmic Impedance .. .. .. .. 135
10.3 Interpretation of Impedance Results ...... .... .. 136
10.3.1 Determination of C'I I.vge Transfer Resistance .. .. .. .. 136
10.3.2 Determination of Capacitance ...... .. . 137

11 CONCLUSION AND RECOMMENDATION ... .. .. 142

11.1 Mathematical Models for Cathodic Delamination of Coated Metal .. .. 142
11.2 Influence of Geometry-Induced Current And Potential Distribution of Disk
Electrodes on Impedance Response ..... .. .. 143

APPENDIX

A PROGRAM LISTING FOR THE CATHODIC DELAMINATION .. .. .. 145

A.1 Main Program Listing ......... .. .. 145
A.2 Subroutine Program Listing ........ ... .. 145

B PROGRAM LISTING FOR IMPEDANCE CALCULATIONS .. .. .. .. 173

B.1 Main Program Listing ........ .. .. 173
B.2 Subroutine Listing ......... . .. 176

C MATHEMATICAL MODEL FOR A DISSOLUTION OF ZINC ROTATING
DISK( ELECTRODE ......... . .. 183

C.1 Model Development ........ .. .. 183
C.1.1 Mass Transfer ........ .. .. 183
C.1.2 Electrode K~inetics ....... ... .. 184
C.1.3 Homogeneous Reactions . ..... .. .. 185
C.1.4 Boundary Condition ....... ... .. 185
C.1.5 Solution Method ......... .. .. 185
C.2 Results and Discussion ......... .. .. 186
C.3 Conclusion ......... ... .. 188

D MATHEMATICAL MODEL FOR GALVANIC COUPLING IN A 2-D CELL .. 189

D.1 Model Development ........ .. .. 189
D.2 Solution Method ......... ... .. 190
D.3 Results and Discussion ......... .. .. 190
D.4 Conclusion ......... ... .. 193

E PARAMETER SENSITIVITY ANALYSIS ..... .. . 194

E.1 Porosity ........ .... .. 194
E.1.1 b,,i ........... ......... 194
E.1.2 be,2 ............ ........... 195
E.1.3 be,3 ............ ........... 195
E.1.4 be,4 ........... ......... 196
E.2 Poisoning Factor ......... ... .. 196
E.2.1 bc,i ........... ......... 197
E.2.2 b(,2 ............ ........... 197
E .2.3 b(,3 . 197
E .2.4 b(,4 . 198
E.3 Blocking Factor ......... . .. 198
E.3.1 b,,i .......... ......... 199
E.3.2 ba2 ............ ........... 199
E.3.3 ba,3 ......... ... .. 200
E.3.4 ba,4 ........ . .. 200
E.3.5 be,s ........ . .. 200
E.3.6 ba,6 ............ ........... 201
E.3.7 b,,7 ......... ... .. 201

REFERENCES ......._._.. ........_._.. 203

BIOGRAPHICAL SK(ETCH ......... . .. 210

LIST OF TABLES
Table pa

4-1 Fitting parameters used in the expressions of pH-dependent interfacial porosity,
blocking, and poisoning parameters..

4-2 Reaction mechanism and equilibrium condition for homogeneous reactions included
in the model.

ge

6-1 Diffusion coefficients of chemical species .....

6-2 Diffusion coefficients of cations .....

6-3 Diffusion coefficients of anions ......

6-4 Calculated velocities of potential, porosity and pH

8-1 Notation proposed for local impedance variables .

E-1 Sensitivity analysis for be,i ..

E-2 Sensitivity analysis for be,2 ...

E-3 Sensitivity analysis for be,3 ...

E-4 Sensitivity analysis for be,4 ......

E-5 Sensitivity analysis for beC ... ..

E-6 Sensitivity analysis for b(, ... ..

E-7 Sensitivity analysis for be,3 ...

E-8 Sensitivity analysis for b(,4 ...

E-9 Sensitivity analysis for bo,i ...

E-10 Sensitivity analysis for ba,2......

E-11 Sensitivity analysis for ba,3 ...

E-12 Sensitivity analysis for ba,4 ......

E-13 Sensitivity analysis for ba,s ...

E-14 Sensitivity analysis for ba,6 ......

E-15 Sensitivity analysis for bo,~7 .....

55

69

72

79

96

194

195

196

196

197

197

198

198

199

199

200

200

201

201

201

front

LIST OF FIGURES

Figure page

2-1 Polarization plots for oxygen reduction . ... .. :32

:3-1 Cathodic delamination system ......... .. :35

:3-2 Experimental interfacial potential distribution .... .. :36

:3-3 Dishonded coating system on a coated metal. ..... .. :39

4-1 Interfacial porosity E and pH as functions of position in the delaminated zone.
The dashed lines separate the domain into the delaminated, front, and fully-intact
regions: a) interfacial porosity; and b) local pH. ... .. .. 44

4-2 Distribution of interfacial porosity E as a function of local pH. .. .. .. .. 45

4-3 Distribution of poisoning factor (: a) as a function of position; and b) as a function
oflocalpH. ........... ...... ..... 47

4-4 Distribution of blocking factor n~o,: a) as a function of position; and b) as a
function of local pH. ........ ... .. 48

6-1 Initial concentration distributions of OH-, Na', Cl-, and Zn2+ ioUs along the
nietal-coating interface. ......... . 56

6-2 Initial porosity distribution: a) as a function of position; and b) as a function of
pH.......... ........... .... 57

6-3 Initial distribution of poisoning factor: a) as a function of position; and b) as a
function of pH. ........ .. .. 58

6-4 Initial distribution of blocking factor: a) as a function of position; and b) as a
function of pH. ........ .. .. 58

6-5 Interfacial potential as a function of absolute net current density with local pH
as a parameter. The distributions associated with the pH values of 8.7 and 9
are superimposed. ........ ... .. 59

6-6 Distributions of interfacial potential along the nietal-coating interface with elapsed
time as a parameter. ........ ... .. 60

6-7 Distributions of dV/dxr along the nietal-coating interface with elapsed time as a
parameter. ......... ... .. 61

6-8 Instantaneous velocity of potential front, calculated front the tinte-dependent
position of the nmaxinia given in Figure 6-7. ..... .. . 62

6-9 Distributions of pH along the nietal-coating interface with elapsed time as a
parameter. ........ . .. 6:3

6-10 Concentration distributions along the metal-coating interface with elapsed time
as a parameter, a) Na+ ions; and b) Cl- ions. .... ... .. 63

6-11 Concentration distributions of Zn2+ 10nS along the metal-coating interface with
elapsed time as a parameter. ......... .. .. 64

6-12 Concentration distributions along the metal-coating interface with elapsed time
as a parameter, a) ZnOH+ ions; b) HZnO2 ioUS; and c) ZnO ions. .. .. .. 65

6-13 Concentration distributions of precipitated corrosion product Zn(OH)2(s) alOng
the metal-coating interface with elapsed time as a parameter. .. .. .. .. 66

6-14 Distributions of porosity along the metal-coating interface with elapsed time as
a parameter. ......... . .. .. 67

6-15 Distributions of de/dx along the metal-coating interface with elapsed time as a
parameter. ......... ... .. 6;8

6-16 Instantaneous velocity of porosity front, calculated from the time-dependent
position of the maxima given in Figure 6-15. ..... .. . 69

6-17 Delaminated distance as a function of elapsed time in double-logfarithmic scale
with cation type as a parameter. The concentration of the electrolyte at the
defect is 0.5 M. a) Determined by potential front; b) Determined by porosity
front; and c) Experimental results obtained from coated electrogalvanized steel
samples. . .. ........ . 70

6-18 Delaminated distance as a function of square root of time with cation type as a
parameter. The concentration of the electrolyte at the defect is 0.5 M. a) Determined
by potential front; b) Determined by porosity front; and c) Experimental results
obtained from coated steel samples. . ...... . 71

6-19 Delaminated distance as a function of square root of time with anion type as a
parameter. The concentration of the electrolyte at the defect is 0.5 M. a) Determined
by potential front; b) Determined by porosity front; and c) Experimental results
obtained from coated steel samples. . ...... . 73

6-20 Delaminated distance as a function of square root of time with electrolyte concentration
as a parameter, a) Delaminated distance determined by potential front; b) Delaminated
distance determined by porosity front; and c) Experimental results obtained
from coated steel samples. Data taken from Stratmann et al.with permission of
Corrosion Science. ......... . 74

6-21 Distribution of interfacial potential along the metal-coating interface a) kneq =
0.1 s-l; and b) kneq= 0.001 s-l. ... .. 76

6-22 Distribution of interfacial potential gradient dV/dx along the metal-coating interface
a) kneq= 0.1 s-l; and b) kneq= 0.001 s-l. . .. .. 76

6-2:3 Distribution of porosity gradient dE/dxr along the metal-coating interface a) k,z,,
S0.1 s-l; and b) k,z, 0.001 s-l. .. ... .. .. 77

6-24 Distribution of pH gradient along the metal-coating interface a) k,z,, 0.1 s- ;
and b) k,z,, 0.001 s-l. .. ... .. 78

6-25 Distributions of de/dxr along the metal-coating interface with elapsed time as a
parameter. ........ .. .. 79

6-26 Delaminated distance as a function of delamination time in double-logarithmic
scale with cation type as a parameter, a) Delaminated distance determined hv
the potential front; and b) Delaminated distance determined by the porosity
front............... .......... ... 80

7-1 Small signal >.1, llh--;-; of an electrochemical nonlinear system .. .. .. .. 81

7-2 Passive elements that serve as components of an electrical circuit. a) Resistor;
b) Capacitor; and c) Inductor. ......... ... .. 8:3

7-3 Combinations of passive elements that serve as components of an electrical circuit 84

7-4 Schematic representation of an impedance distribution for a blocking disk electrode 87

7-5 Primary current density distribution at a disk electrode. .. .. .. 90

7-6 Secondary current distribution at a disk electrode with J as a parameter. .. 90

8-1 Coordinates transformation from a cylindrical coordinate to a rotational elliptic
coordinate .. .......... ........... 9:3

8-2 The location of current and potential terms that make up definitions of global
and local impedance. ......... .. .. 95

8-3 Nyquist representation of the impedance response for an ideally polarized disk
electrode. a) linear plot showing effect of dispersion at frequencies K(>1; and b)
logarithmic scale showing agreement with the calculations of N. i.us .1 .I .. 99

8-4 Representation of the impedance response for an ideally polarized disk electrode.
a) real part; and b) imaginary part showing agreement with the calculations
and .I-i-usiI Il e formula of Newman. . ...... .. .. 100

8-5 The slope of log(Zjs/ro~r) with respect to log(K() (Figure 8-4(b)) as a function
of log(K(). The results were calculated by the collocation method. The value of
this slope is equal to -c0. . .. ... .. .. 101

8-6 The frequency K(=1 at which the current distribution influences the impedance
response with m/Co as a parameter. . ..... .. 102

8-7 Imaginary part of the local interfacial impedance: a) as a function of frequency
with position as a parameter; and b) as a function of position with frequency as
a parameter. . .... .. 102

8-8 The local impedance in Nyquist format with radial position as a parameter. 103

8-9 Local impedance with radial position as a parameter: a) real part; and b) imaginary
part. ............ ............... 104

8-10 Local impedance as a function of radial position: a) real part; and b) imaginary
part multiplied by dimensionless frequency K. .... ... .. 105

8-11 The local Ohmic impedance in Nyquist format with radial position as a parameter. 106

8-12 Values for local Ohmic impedance as a function of frequency with radial position
as a parameter: a) real part; and b) imaginary part. .. .. .. .. 106

8-13 The imaginary part of the global Ohmic impedance, calculated from equation
(8-28), as a function of dimensionless frequency. .... .. .. 107

9-1 Nyquist representation for the calculated impedance response of a blocking disk
electrode with a local CPE with a~ as a parameter. .. .. .. .. 110

9-2 Impedance response for a blocking disk electrode with a local CPE as a function
of dimensionless frequency K(: a) real part; and b) imaginary part. .. .. .. 111

9-3 Slope of log(Zjs/rovr) with respect to log(K() (Figure 9-2(b)) as a function of
log(K() with a~ as a parameter. ........ ... .. 112

9-4 Effective scaled CPE coefficient as a function of frequency with a~ as a parameter. 113

9-5 Nyquist representation for the calculated local interfacial impedance response of
a blocking disk electrode with a local CPE with normalized radial position as a
parameter. ......... ... .. 114

9-6 Local interfacial impedance as a function of frequency with position as a parameter:
a) imaginary part; and b) real part. . ...... .. 114

9-7 Local interfacial impedance as a function of position with frequency as a parameter:
a) imaginary part; and b) real part. . ...... .. 115

9-8 The local impedance in Nyquist format with radial position as a parameter. 116

9-9 Local impedance: a) real part; and b) imaginary part. ... .. .. .. 116

9-10 The local Ohmic impedance in Nyquist format with radial position as a parameter. 117

9-11 Values for global Ohmic impedance as a function of frequency with a~ as a parameter:
a) real part; and b) imaginary part. . ..... .. 118

9-12 Complex-impedance-plane plots for the response of a glassy carbon disk in K(Cl
electrolytes with concentration as a parameter, a) for frequency values between
100 kHz and 10 mHz; and b) zoomed region showing only high-frequency data. 119

9-1:3 Dimensionless analysis for the impedance response of a graphite disk in K(Cl
electrolytes with concentration as a parameter, a) Dimensionless imaginary part
of the impedance as a function of dimensionless frequency (corresponding to
Figure 9-2(b)); and b) Derivative of the logarithm of the dimensionless imaginary
part of the impedance with respect to the logarithm of dimensionless frequency
(corresponding to Figure 9-:3). . .. ... ... .. 120

9-14 Experimental local impedance, local interfacial impedance, and local Ohmic impedance
in Nyquist format of a stainless steel disk electrode at the center of the electrode
(r/ro =0). ...... ...... ........... 121

10-1 Representation of an impedance distribution for a disk electrode where x, represents
the local Ohmic impedance, Co represents the interfacial capacitance, and Rt
represents the charge-transfer resistance. ...... .. . 125

10-2 Nyquist representation of the impedance response for a disk electrode under
assumptions of Tafel and linear kinetics. Open symbols represent the result calculated
by N. i.--us! .Is a) J = 0.1; and b) J = 1.0. ...... .. .. 126

10-3 Representation of the impedance response for a disk electrode under assumptions
of Tafel and linear kinetics and with with .7 as a parameter. Open symbols represent
the result calculated by N. i.--n! Ias a) real part; and b) imaginary part.. .. .. 127

10-4 Derivative of log(Zja/rox)) with respect to log(K() (taken from Figure 10-:3(b))
as a function of K( with .7 as a parameter. ..... .. . 128

10-5 Derivative of log(Zja/rox)) with respect to log(K(/J) (taken from Figure 10-:3(b))
as a function of K( with .7 as a parameter. ..... .. . 129

10-6 Representation of the local interfacial impedance response for a disk electrode
as a function of dimensionless frequency K( under assumptions of Tafel kinetics
with .7 = 1.0: a) real part; and b) imaginary part. ... ... .. 1:30

10-7 Representation of the local interfacial impedance response for a disk electrode
as a function of radial position under assumptions of Tafel kinetics with .7 =
1.0: a) real part; and b) imaginary part. ..... .. .. 1:31

10-8 Representation of the local impedance response for a disk electrode as a function
of dimensionless frequency K( under assumptions of Tafel kinetics with .7 = 1.0.
a) Tafel kinetics; and b) linear kinetics. . .... .. 1:32

10-9 Representation of the local impedance response for a disk electrode as a function
of dimensionless frequency K( with .7 = 1.0. a) real part; and b) imaginary part. 13:3

10-10Representation of the local Ohmic impedance response for a disk electrode as a
function of dimensionless frequency K( under assumptions of Tafel kinetics with
J=1.0 .. ....... ..... .. 133

10-11Representation of the local Ohmic impedance response for a disk electrode as a
function of dimensionless frequency K( under assumptions of Tafel kinetics with
J = 1.0representation of the local Ohmic impedance response for a disk electrode
as a function of dimensionless frequency K( under assumptions of Tafel kinetics
with J = 1.0. a) real part; and b) imaginary part. ... ... .. 134

10-12Global Ohmic impedance response for a disk electrode as a function of dimensionless
frequency for linear kinetics with J as a parameter, a) real part; and b) imaginary
part. . .. ....... . 135

10-13The apparent value of Res/IRt obtained from the calculated impedance response
at low frequencies as a function of J. . .... .. .. 137

10-14The apparent value of 1-a~ obtained from the calculated impedance response at
high frequencies as a function of J. . ...... .. 138

10-15Effective CPE coefficient scaled by the interfacial capacitance as a function of J. 139

10-16Effective capacitance calculated from equation (10-21) and normalized by the
input interfacial capacitance for a disk electrode as a function of dimensionless
frequency K( with J as a parameter. . ...... .. 140

10-17Normalized effective capacitance calculated from relationships presented by Brug
et.al for a disk electrode as a function of dimensionless frequency K( with J as a
parameter, a) with correction for Ohmic resistance Re (equation (10-22)); and
b) with correction for both Ohmic resistance Re and charge-transfer resistance
Rt (equation (10-23)). . .. ... .. 141

C-1 Schematic representation of a rotating disk electrode system in which a disk
electrode is embedded in a large insulator. .. .. .. 184

C-2 Calculated concentration distributions of species of OH-, Zn2+, H+, ZnOH+,
HZnO2 and ZnO on electrode surface. ... .. .. 186

C-3 Calculated rate of the homogeneous reactions included in the model. a) Water
disassociation reaction; and b) Zinc hydrolysis reactions. .. .. .. 187

D-1 Schematic representation of a two electrode cell in which Zn serves a local anode
and Fe as a local cathode. .. ... .. .. 189

D-2 Calculated current density distributions along the x axis. a) 0< x <0.04 cm;
and b) 0< x <0.016 cm. ......... . .. 191

D-3 Calculated distributions of concentration in a unit of mole/cm3. a) Zn2+ ioUS;
and b) OH- ions. ............ ........... 192

D-4 Calculated distribution of ratio of czna2+*O-KpweeKpI h tnadslblt
product of Zn(OH)2(s). ......... . .. 192

E-1 The sensitivity of the slope of the increasing curve to be,2. .. .. .. .. 195

Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

FUNDAMENTAL APPROACH TO PRACTICAL CORROSION PROBLEMS

By

Mei-Wen Huang

August 2007

Cl.! ny~: Mark E. Orazem
Major: Chemical Engineering

My study covers two important research topics that are relevant to corrosion of

metal. The first project involved developing a mathematical model for delamination of

polymeric coating from active materials. The model described simultaneously the coupled

phenomena of mass transfer, electroneutrality, loss of interfacial adhesion, and propagation

of delamination during the delamination process. The key to this work included the use of

pH-dependent porosity and pH-dependent polarization kinetics to simulate implicitly the

bond-breaking reactions that occurred during the delamination process.

The computational results predicted from the model, under the assumption of

equilibrium pH-porosity relation, concluded the overall delamination process is limited by

the transport of cations from defect to delaminated zone, and the rate of the delamination

process scales with the mobility and ionic strength of cations. This conclusion is in good

agreement with the interpretation obtained from experiments. The consistency with

experiments supported the pH-dependent hypotheses emploi- II in the model.

The equilibrium pH-porosity relation becomes invalid when time constants for the

bond-breaking reactions are large compared to those for the diffusion and migration

processes. The investigation of kinetic pH-porosity relation showed that when the

bond-breaking reactions occur in a sufficiently small rate, the delamination mechanism

shifts from a mass-transfer-limited mechanism to a mixed-controlled mechanism. A similar

mechanism transition was also observed in experiments.

The second project presented in this document explored, by theoretical calculations,

the role of current and potential distributions associated with disk electrodes on

impedance response. It has known that the geonietry-induced current and potential

distributions lead to a high-frequency dispersion on impedance response of a disk

electrode. The contribution of the work was to express the geometric effect on impedance

response in terms of constant-phase element (CPE) and to use both global and local

intpedances to study this geometric effect. A coherent notation was proposed for global

and local intpedances which accounted for global, local, local interfacial, and both global

and local Ohmic intpedances.

The electrochentical systems under study in the second project included an

ideally-polarized blocking electrode, an electrode exhibiting a local CPE behavior, and

an electrode exhibiting a single Faradaic reaction. The calculation results front these

systems showed that the global impedance of a disk electrode is influenced by the current

and potential distributions at high frequencies. While the local interfacial impedance

exhibits the expected behavior for a given system, the local impedance shows inductive

behavior at high frequency and ideal behavior at low frequency. The local impedance

is influenced by the Ohmic impedance, which has complex behavior at intermediate

frequencies. The representation of an Ohmic impedance as a complex number represents

a departure from standard practice. This complex character is believed to be the origin of

the inductive features seen in the local impedance and the origin of the CPE-like behavior

found in the global impedance.

CHAPTER 1
INTRODUCTION

This dissertation covers two research topics that are important to corrosion of metal.

The first research topic involves developing a niathentatical model for delamination

of polynteric films front active metals. The delamination of paint front metal surface,

also known as cathodic delamination, is all in r problem for automotive and building

applications. Experimental findings demonstrated that the cathodic delamination process

involved coupling of mass transfer, electrochentical reactions, loss of interfacial adhesions,

and propagation of the delamination along the nietal-coating interface. These experiments,

in principle, are useful for identifying the phenomena occurring in the delamination

systems. However, it is difficult to verify that a proposed niechanisni does indeed give

observed experimental results. Therefore, a quantitative approach was taken to simulate

the transit, propagation phase of the delamination process on coated metals.

The second project presented in this document explored, by theoretical calculations,

the role of current and potential distributions associated with disk electrodes on

impedance response. Electrochentical impedance spectroscopy (EIS) is often applied as a

tool to investigate the rate of corrosion. Impedance spectra, however, are often influenced

by the current and potential distributions on electrode surfaces. This work investigates the

influence of the geometric effect associated with disk electrodes on impedance responses

and describes this effect in terms of constant phase element (CPE).

The introduction for the cathodic delamination model is presented in Section 1.1.

Section 1.2 presents the introduction for the impedance calculation.

1.1 Mathematical Models for Cathodic Delamination of Coated Metal

Reactive metals in applications such as automotive and architecture are often

protected by covering their surfaces with a nlicron-thick 1.0 c- of organic/polynteric

coatings. The organic coating provides a matrix in which anticorrosive pigments and/or

inhibitors are dispersed [1] and forms a physical barrier between metal and atmosphere.

While additives are introduced into polymeric coatings, the detachment of polymers from

metal surface reduces significantly the efficiency of the corrosion protection.

The adhesive strength between the metal and the coating ]II i-s an essential role on

determining the rate of corrosion. Sugama et al. [2] demonstrated that thermoplastic

polymers such as poly(ethylene) exhibit a poor adherence to the metal surface due to

the lack of functional groups. George et al. [3] and Sugama et al. [2] showed that the

incorporation of functional groups, for example, methacrylic acid (jl A) and -COOH, onto

the thermoplastic polymers improved the bond strength and further reduced the corrosion

rate. Despite the fact that the functional groups on the coating lead to a strong adhesion,

the experimental results by Leidheiser et al. [4, 5] indicated that the bond structure is

vulnerable to alkaline environment created by oxygen reduction underneath the coating.

Yasuda et al. [6, 7] found that small molecules with high permeability, such as

water and oxygen, can penetrate through a defect-free coating and be reduced at the

metal-coating surface. So long as the cathodic reaction began to occur underneath the

paint, the OH- ions produced in the interfacial l er-is promoted the decomposition

of polymers, resulting in a delamination of the polymer film from the metal surface.

Hammond [8] and Sugama [2] verified the bond-breakage phenomenon by a reduction of

the bonding energy for -COOH group and an increase in the bonding energy for -COONa

group.

Stratmann et al. [9-18] investigated experimentally the cathodic delamination for

coated steel and coated electrogfalvanized steel. Following the interpretation reported

by Stratmann et al. [9-11, 13-16], the cathodic delamination system consists of a defect

where the bare metal is exposed to atmosphere, and a delaminated zone where the

interfacial bonding is partially damaged due to the delamination process. Exposure of the

metal surface to atmosphere favors metal dissolution at the defect. Along the delaminated

zone where the anodic reactions are prohibited because of the presence of the (U i fling

oxygen and water penetrate the coating and react at the interface. Due to the nature of

the configuration, the early stage of the cathodic delamination is often described as the

formation of an electrochemical cell with distinct anodic and cathodic zones.

The later stage of the delamination becomes complicated when corrosion products are

formed underneath the paint. Furbeth and Stratmann [10] reported that the precipitation

of ZnC03, under a high CO2 concentration in atmosphere, inhibited the oxygen reduction

and led to a pure anodic delamination for a coated electrogfalvanized steel. Ogle et al.

[19, 20] studied the cathodic delamination on galvanized steel and found that the chemical

stability of the interfacial oxide 111-;- vs pl 1-<~ a critical role on determining the propagation

rate and the delamination mechanism. [19, 20]

It is difficult to demonstrate that a proposed mechanism -II_a-r-- -1. based on

experimental work does indeed give the observed experimental results. Interpretation

is often limited to qualitative and subjective observations. Therefore, a quantitative

approach is necessary. Allahar [21, 22] developed the first mathematical model for the

cathodic delamination of coated metal. The key to his work involved applying the concept

that the porosity and the polarization kinetics at the metal-coating interface were pH-

dependent. The simulation results provided qualitative agreements with the experimental

observations reported by Stratmann et al. [9-11, 14-16], which supported the hypotheses

emploi-, I1 in the model. The model developed by Allahar, however, did not incorporate

chemical reactions that take place at the later stage of the delamination process.

The objective of the work was to develop, from first principles, a mathematical

model that simulates the propagation of delamination in the presence of electrochemical

and chemical reactions. The chemical reactions included homogeneous reactions in the

interfacial oxidized 1.>.;r and formation of corrosion product underneath the coating. A

second objective was to examine whether the delamination rate and the delamination

mechanism predicted from the model agree with the experimental results.

1.2 Influence of Geometry-Induced Current And Potential Distribution of
Disk Electrodes on Impedance Response

Electrochemical impedance spectroscopy (EIS) is a powerful technique that has been

used extensively in interfacial electrochemistry to study electrochemical kinetics [2:3, 24]

and to determine interfacial capacitance. [25, 26] The impedance response of an electrode

is generated by measuring the ratio of applied potential to surface-averaged current as

a function of frequency. The influence of a particular phenomenon on the impedance

response is determined by the time constant of that process. [27] 1\ass transfer effects

are usually apparent at low frequencies because the diffusivity of ionic species in aqueous

medium is small. Kinetic and double-l] w;r effects are more important at high frequencies.

An important advantage of EIS is that the influence of governing chemical and physical

phenomena can he distinguished with a single experimental procedure encompassing a

The critical issue of EIS is the ambiguity associated with the interpretation of

impedance results. [25, 28, 29] A common approach of interpreting impedance data is

to compare experimental spectra with that of known electrical circuit elements such as

resistors, capacitors, and inductors. [28] The circuit analog models are found useful for

understanding the physical processes that contribute to impedance responses; however,

experimental data rarely show the ideal response expected for electrochemical reactions.

The impedance response typically reflects a distribution of reactivity that is commonly

represented in equivalent circuits as a constant-phase element (CPE). [29, :30]

The dispersion leading to CPE behavior can he attributed to distributions of time

constants along either the area of the electrode (a two-dimensional surface) or along the

axis normal to the electrode surface (a three-dimensional aspect of the electrode). A 2-D

distribution might arise from surface heterogeneities such as grain boundaries, crystal

faces on a poli- l i--r I11;11< electrode, variations in surface properties, or geometry-induced

current and potential distributions. [:31-34] A :3-D distributions may be attributed to

changes in the conductivity of oxide 1.v. ris [35, 36] or from porosity or surface roughness.

[37, 38] This CPE behavior can be described as arising from a 3-dimensional distribution,

with the third direction being the direction normal to the electrode surface. [39, 40] Jorcin

et al. [40] demonstrated that the use of local electrochemical impedance spectroscopy

(LEIS) makes it possible to distinguish the CPE behavior that has an origin with a 3-D

distribution from one that arises from a 2-D distribution of properties along the surface of

the electrode.

The disk electrode geometry is well-defined and amenable to numerical calculation

of the impedance response. ?-. i.--us! 1is [41, 42] calculated the current and potential

distributions on disk electrodes and developed both numerical and analytical treatments

for the impedance response of a blocking electrode and an electrode subject to a Faradaic

reaction. The results demonstrated that geometry-induced current and potential

distributions cause a time-constant dispersion that distorts the impedance response.

Nisancioglu [43-45] showed the extent to which this frequency dispersion causes an

error in the values for charge-transfer resistance and interfacial capacitance obtained

from impedance data. The discussion by Nisancioglu and by N. i..--us .II, however, did not

address the common practice of describing non-ideal impedance response in terms of

constant-phase elements.

The first objective of this research topic was to calculate, from first principles,

the influence of non-uniform current and potential distributions associated with a disk

electrode on impedance response. The second objective of the work was to describe

the role of the time-constant dispersion in terms of CPE behavior and to relate global

impedance response with local impedance. The impedance calculations were performed

for an ideally-polarized blocking electrode, a blocking disk electrode with a local CPE

behavior, and a disk electrode subject to a simple Faradaic reaction.

The structure of the dissertation is divided according to the two research topics.

The first part of the dissertation, presented from C'!s Ilter 2 to C'!s Ilter 6, deals with

fundamental electrochemical concepts, theoretical development, simulation results, and

discussions that are associated with the cathodic delamination system. The objective

of this part was to gain an understanding of the phenomena that may contribute to the

delamination process through developing a mathematical model for cathodic delamination.

The fundamental electrochemical concepts relevant to the cathodic delamination

system, such as electrode kinetics and transport in dilute solutions, are presented in

('!s Ilter 2. Experimental and simulation work associated with the cathodic delamination

system are reviewed in ('! .pter 3. The constructions of the pH-dependent porosity and

pH-dependent polarization kinetics are also presented in ('! .pter 3.

The model developed by Allahar did not include chemical reactions that become

important at the later stage of the delamination. In the present work, multiple homogeneous

reactions and precipitation of corrosion products were considered in the oxidized 1.>c.y

The theoretical development of the present model followed the approaches taken by

Allahar [22]. The development of the model is presented in ('! .pters 4 and 5. The

computational results are presented in ('! .pter 6.

The second part of this dissertation, presented in ('! .pters 7 to 10, explores the

influence of geometry-induced current and potential distributions on the impedance

response of a disk electrode. Electrochemical impedance spectroscopy (EIS) is a rapid and

convenient technique that provides electrochemical properties of tested systems over a

wide range of frequencies. A brief introduction to EIS and issues encountered in EIS are

presented in Chapter 7.

The current and potential distributions associated with a disk electrode embedded

in an insulating plane are reviewed in ('! .pter 7. The theoretical development and

calculation results for the ideally-polarized blocking electrode are presented in ('! .pter 8,

the results for the blocking electrode with a local CPE behavior are presented in ('! .pter

9, and the results for the disk electrode exhibiting a Faradaic reaction are presented in

CHAPTER 2
BACKGROUND ELECTROCHEMISTRY

The fundamental electrochemical concepts relevant to the cathodic delamination

system, such as mass transport of ionic species in dilute solutions and electrode kinetics,

are presented in this chapter. A detailed treatment of electrochemistry in electrolyte from

a mathematical perspective has been presented by N~ -.In .11 [46]

2.1 Mass Transport

In an electrochemical system, conservation of mass restricts the governing equation for

the concentration of a species i to [46]

8ci
i=- V Nsi + Ri (2-1)

where the terms on the right side represent the net input due to the flux Nsi and the net

rate of production due to homogeneous reactions Ri, respectively. In dilute electrochemical

systems, NVi is given by the N. IIno-Planck equation [46]

Nsi = -zecsFuiVQ DiVc4 + civ (2-2)

where # is the local solution potential, as is the mobility, Di is the diffusion coefficient, ze

is the charge number, v is the mass average velocity of the electrolyte, and F is Faraday's

constant. The terms on the right side of equation (2-2) represent the contributions by

migration, diffusion, and convection to the flux of a species, respectively.

Combination of equations (2-1) and (2-2), under the assumption that the electrolyte

is incompressible (V v = 0), yields the governing equation for ce in a stagnant

electrochemical system

i= zingFV (c4V4) + DiV2Ci Ri (2-3)

where convective contributions to the flux of a species are negfligfible. The ?-. I t-i-Einstein

equation [46]

ui = D,/RT (2-4)

is generally applicable to dilute electrochemical systems where R is the molar gas constant

and T is the absolute temperature. Combination of equations (2-2) and (2-4) yields the

flux of a species
Di
Nsi = -zeciF V@ DiVci (2-5)
RT

Equations (2-3) is rewritten as

=i De [ziV (ciV ) + V2C4] Ri (2-6)

by employing the N. Its-i-Einstein equation.
2.2 Solution Potential

The governing equation for the solution potential in an electrochemical system is

Poisson's equation [46]

V2 iC 27

where e is the permittivitty of the medium. An expression based on the concept of

electroneutrality at a point, i.e.

ze-ce = (2-8)

has been used as the governing equation for 4. Newman has shown that equation

(2-8) provides a very good approximation to Poisson's equation outside the thin

double charge lw. ;r near electrodes. [46] It is important to note that the assumption
of electroneutrality does not imply that Laplc eq ato hold fo t epoetil

because this approximation is made on the basis of a large value of F/e in equation (2-7).

[46]

The conservation of charge is given by

V -i = (2-9)

The ionic current density due to the motion of charged particles in an electrolytic solution

is calculated by [46]

i =F ~zes (2-10)

Combination of equations (2-5) and (2-10), in the absence of convection, yields

i = -VQ Fl zDVci (2-11)

where the conductivity n is defined as

a = zfC-iecr (2-12)

The ionic current density in equation (2-11) can be divided into migration and diffusion

contributions. The driving force for the migration and diffusion current densities are

potential and concentration gradients, respectively. [46]

In the absence of concentration gradients, equation (2-11) reduces to an expression of

Ohm's law

i = -xV@ (2-13)

Combination of equation (2-13) with equation (2-9) yields Laplace's equation

V2 = 0 (2-14)

for the solution potential 4.

2.3 Electrochemical Kinetics

The rate of charge-transfer or Faradaic reactions taking place at electrode surface

is important in corrosion systems. The reaction rate, characterized by current density,

depends on the composition of the electrolytic solution .Il11 Il:ent to the electrode. [46]

Consider a simple heterogeneous electrochemical conversion of A to B on an electrode

surface

A W B+ + e- (2-15)

For the reaction described by equation (2-15), the forward reaction is anodic and the

backward reaction is cathodic. The overall rate of the reaction r is given

r = rf Tb (2-16)

where rf and Tb are the forward and backward rates, respectively.

Under the assumption that each reaction is first order, the overall rate can be written

r = kfcA kbCB (2-17)

From activated complex theory, equation (2-17) can be recast as

r kx( (1 P)nF V~.Dx( -pnF C)' 2
RT RT

where V is the interfacial potential, p (known as the symmetry factor) is the fraction of

the applied potential that favors the cathodic reaction, and n is the number of electrons

transferred. Equation (2-18) can be written in terms of current density i as [46]

nF=s cl (1 P)nF -pnF 29

The interfacial potential V is defined as

V = W- (2-20)

where W is the potential of the metal and # is the potential in electrolytic solution

.Il1i Il:ent to the electrode.

When the anodic and cathodic reactions in equation (2-15) reach the same rate, a

zero current is obtained under the condition of reaction equilibrium. At the equilibrium

potential Vo, the net rate of the reaction is zero, however, the individual rates of the

reactions are non-zero. The current density at the equilibrium potential is defined as

exchange current density io and is calculated using either

=i kg exp Vo c (2-21)

nFi -nRT 22

Substitution of io into equation (2-19) yields the Butler-Volmer equation [46]
(aaF -acF
zoex p -ex q (2-23)
RT RT

where the surface overpotential rl, is given by rl, = V Vo, the anodic transfer coefficient

as, is given by as, = (1 P)n, and the cathodic transfer coefficient is given by ac, = pn.

The surface overpotential rl, represents the departure from an equilibrium potential such

that, at rl,=0, the total current i = i, ic is equal to zero.

The exponential behavior of the Butler-Volmer equation results in a characteristic

feature of electrochemical reactions. In the limit of ctaFrl > RT, equation (2-23) can be
reduced to

i = zoI~ exp ,FT (2-24)

Solving for rl, in equation (2-24) gives

195 = In~rR (2-25)

rls = 2.303 loglo .~ (2-26)

The Tafel slope for the anodic reaction P, is given by the expression in front of the log

term in equation (2-26)
RT
So = 2.303~ (2-27)
cAF

The corresponding Tafel slope for the cathodic reaction is given as

RT
c = 2.303~ (2-28)
acF

The Butler-Volmer equation, equation (2-23), can be recast using these Tafel slopes into

i = 10(V-Ea)/Pa 10(V-EC)/0C (2-29)

where E, and Ec are the effective equilibrium potentials given by

E, = Vo Pa logo io (2-30)

for the anodic reaction and

Ec = Vo Pc logo io (2-31)

for the cathodic reaction, respectively. [47]

2.3.1 Kinetic Control

In electrochemical systems, multiple Faradaic reactions often occur simultaneously on

an electrode surface. For example, zinc dissolution

Zn Zn+2 + 2e- (2-32)

and hydrogen evolution reaction

2H20 + 2e- H2 + 20H- (2-33)

may both occur in a corrosion system. Under such circumstance, the individual electrochemical

reactions can be treated independently. Therefore, a Butler-Volmer equation such as

equation (2-23) can be written for each of these reactions.

The current density due to the reversible corrosion reaction izn

Zn C Zn2+ + 2e- (2-34)

can be calculated by

izn = 10(V-EaZn)/0aZn 10(V-ECZn)/#C,zn (2-35)

In most corrosion systems, metal dissolution is considered to be an irreversible reaction.

Thus, the current density due to the zinc dissolution izn is given by

izn = 10(V-EZn)/0Zn (2-36)

where Ea,zn and a,~zn are replaced by Ezn and Pzn, respectively. Similarly, the current

density iH, due to the hydrogen evolution in reaction (2-33) is given by

iHz = -10-(V-EH2) 11.* (2-37)

under the assumption that the reaction is irreversible. The total current density for the

given system is the sum of the individual current density

inet = izn + iHz (2-38)

The polarization behavior of the corrosion and the hydrogen evolution are termed

activation polarization because the rates of the electrochemical reactions are driven by

the surface overpotential rl.

2.3.2 Mass Transfer Control

The rate of electrochemical reactions can also be limited by the rate at which

reacting species are carried to the electrode surface. An example of such a case is

illustrated in Figure 2-1 for an oxygen reduction reaction. The polarization behavior

of the oxygen reduction contains activation and concentration components. The current

densities is a function of potential in the activation polarization part, but is independent

of potential in the concentration polarization part. The reaction rate in the concentration

polarization regime is limited by the rate of transport of oxygen to the metal surface. The

mass-transfer-limited current density, symbolized as ilim,oz, depends on solution agitation,

temperature and concentration of the limiting species. [48] The numerical value of the

.. + 2H20 +4e~ 4 40H~

c~Activation
polarization
Concentration
polarization ji 1

log | iq |

Figure 2-1. Polarization plots for oxygen reduction at two values of mass-transfer-limited
current density labelled as ii and i2. The horizontal dashed line separates the
activation potential and concentration polarization parts of the plots.

limiting current density is given by

-nFDo, co,,oo

slim,O,

(2-39)

where x and co, are the distance that oxygen diffuses through and the oxygen concentration

in the bulk, respectively. The current density due to the oxygen reduction io, is given by

the mathematical expression

+ 10(V-Eog)/4

(2-40)

to account for both activation and concentration regimes.

ilim,Oz1

CHAPTER 3
LITERATURE REVIEW ON CATHODIC DELAMINATION

As described in literature, the delamination of paint under humid and corrosive

environment involves a coupling of mass transfer, electrochemistry, loss of adhesion

at the metal-coating interface, and propagation of a moving front along the interface.

Experimental and simulation work associated with the cathodic delamination system are

reviewed briefly in this chapter.

3.1 Experimental Observation

The kinetics and the mechanisms of the delamination process have been investigated

experimentally for coated steel and coated electrogalvanized steel using local electrochemical

and physical techniques. Stratmann et al. [9-18, 49] emploi-l da scanning K~elvin probe

to measure potential distributions at buried polymer/metal interfaces. They performed

the experiments under accelerated corrosive conditions and minimized surface treatments,

thus the delamination rate predicted from their samples were larger than that observed

from commercial technical samples. [19] William et al. [50, 51] also emploi-, a the scanning

Kelvin probe technique to study the influence of inhibitors on the delamination mechanism

of coated galvanized steel.

A new technique based on Fourier transform infrared-multiple internal reflection

(FTIR-MIR) allowed in-situ measurements of the thickness of water lw. ;r at metal-coating

interface. [52, 53] This technique provided a means to determine the rate of water

transport through the coating and to calculate the diffusion coefficient of water through

the polymer film. Jorcin et al. [54] explored the delamination phenomena at a steel/epoxy-vinyl

prime interface using local electrochemical impedance mapping. The results showed that

the delaminated area measured by visual observations after the removal of the coating

were approximately three times smaller than that determined by the local electrochemical

impedance mapping.

The samples used in delamination experiments are typically made by applying a

micro-thick polymer 111-;-r on a metal substrate using a roll coat procedure. [19] After

few d .1-< Of curing, the samples are scribed to create a defect area. This well-defined

defect serves as a reservoir for electrolyte when the coated samples are placed into a

controlled-humidity and subjected to intermittent salt sprays. Depending upon whether

the metal dissolution occurs at the defect or under the paint, the electrochemical

mechanisms of the delamination can be divided into two broad categories: anodic and

cathodic delamination. As reported in the literature, the delamination often begins in

a condition where the uncoated area plI li-< a role of a local anode, and the coated part

represents a local cathode.

Furbeth and Stratmann [9-11] reported that the cathodic delamination process on

a coated metal with a coating defect was the result of a galvanic couple formed between

the defect and the intact zone. Cathodic reactions at the metal-coating interface were

balanced by anodic reactions at the bare metal exposed by the defect. A schematic

diagram of a delamination system, following the interpretation obtained by Stratmann

et al., is presented in Figure 3-1. The system consists of a defect where the bare metal is

exposed to an electrolyte of NaC1, and a delaminated zone where the interfacial adhesive

bonding between the substrate and the paint is partially damaged due to the delamination

process. The exposure of the metal to the electrolyte initiates an anodic reaction at the

defect. On the delaminated zone where the metal dissolution is limited due to the presence

of the (e Ir filr_ small molecules such as water and oxygen diffuse through the polymer film

and are reduced at the interface. [6, 7, 12, 16] The cathodic delamination process is driven

by the formation of an electrochemical cell formed between the anodic and the cathodic

sites.

Leng and Stratmann [15] -11---- -r. I1 that, after the delamination began to occur,

the intact coating-substrate interface was replaced gradually by two new interfaces: a

substrate-electrolyte 1... -r and electrolyte-polymer 1... -r. The electrochemical reactions,

SDefect Delaminated Z~one
O,

Figure :3-1. Schematic representation of cathodic delamination of a coated metal, following
the interpretation obtained by Stratmann et al. [9-11] The system consists of
a defect where the metal is exposed to an electrolyte, and a delaminated zone
where the interface adhesive strength is partially broken.

therefore, expanded from the scratch and propagated away from the defect with a

well-defined delamination front. The OH- ions produced in the cathodic reaction

mechanism promoted coating dishondment through polymer decomposition and hydrolysis

of interfacial bonds. [51] Leng and Stratmann [14] found that the reduction of oxygen

partial pressure decreased significantly the delamination rate, concluding that the oxygen

reduction was important in the delamination process. Other radical peroxide intermediates

generated in the oxygen reduction have also been proposed as possible .I__-oessive species

during the delamination process. For a case of an ultrathin plasma polymer 1 isl, r

Grundmeier [55, 56] -11_0---- -1. that the metal-coating interface was destroyed by the

attack of the intermediates formed in the cathodic reaction mechanism.

The potential distributions obtained from the scanning K~elvin probe were used to

interpret the delamination mechanism for coated zinc and coated steel. [9-18, 50, 51] A

schematic representation of a potential distribution, following the experimental results

reported by Stratmann et al., is given in Figure :3-2(a) as a function of position with

elapsed time as a parameter. The zero position in the figure represents the boundary

between the defect and the delaminated zone. Three different regions are observed in

Figure :3-2(a). The interfacial potential shows a gradual increase with position in the

delaminated region, an abrupt increase in the front region, and approximately a constant

value in the intact region. As observed in Figure :3-2(a), the shape of the potential

PositnPsto
(a b

time as aparameter

thgue potetia Shmeasuremreents for aoate elperognalvanized steel pigmentealdwisth ritOn and

Teinterfacial potential distribution was difernctiatedpoito with rsettoposidtion a
to~~ yielde; ndb dV/dx distributionsshwinFgr3-() as a function of position with lpe
elapsedtime as a parameter.Thpekmrethdeecinpntoteabutnras

cosrrespiond ing tonane hou the vau o V/xwabervmed to. dheleaswth dftelaintatio teime

LeengandStrtan[1]---- ht with increasing time, Wila ta.[0 1 lothierewa a smolre gradualo i

chane itn the electrocemilpoent i ral catelctross the ze fro tee region.wthCr~-a

The deaintration kineticcan dsrbeto a detrinfrted bypltingthe dselaminted distance

caylcltd fromx dsrbthens dV/dx curver 32( as a function of tie en 1] n iliam [51]

emaplo ie da geeal powmeer. law ekmre h dfeto on f h butices

observexde =nFgr32a. h oiin ktheea recgie (3-1)elmnaio ro

to describe the delamination mechanism, where .race k, and taez are delaminated distance,

rate constant, and delamination time, respectively. Furbeth and Stratmann [11] reported

a value of a = 0.55 from coated steel with minimized surface treatments, which concluded

that the overall delamination process was governed primarily by the transport of cations

from the defect to the delaminated zone. Their results also demonstrated that, for

non-pigmented samples and a short-time exposure, the rate of the delamination depended

strongly on the ionic strength and the mobility of cations. [11]

Experimental observations have shown that the later stage of the delamination

becomes more complicated when passive films or corrosion products are formed underneath

the coating. [10, 19] Furbeth and Stratmann [10] reported that the formation of ZnC03,

under a high CO2 concentration in the atmosphere inhibits the electron transfer of oxygen

reduction, resulting in a pure anodic delamination for coated electrogfalvanized steel.

The precipitation of Zn(OH)2 or ZnO at the defect prevents corrosion of the uncoated

area, therefore, the cathodic reaction becomes dominating on the scratch for coated zinc

samples. 1\oreover, after longer-time exposure, as the delamination front moves further

away the defect, both anodes and cathodes can appear under the paint and it becomes

difficult to distinguish whether the moving front is anodic or cathodic.

Numerous research efforts have been made to study the influence of inhibitors and

surface treatments on the delamination rate and the delamination kinetics. Hernandez et

al. [57] indicated that the zinc-aluminum phosphate pigments reduced the delamination

rate by forming a phosphate 1 u-