Group Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Title: 15.6.3 - A Multicomponent and Multiphase Model of Reactive Wetting
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 Material Information
Title: 15.6.3 - A Multicomponent and Multiphase Model of Reactive Wetting Reactive Multiphase Flows
Series Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Physical Description: Conference Papers
Creator: Villanueva, W.
Boettinger, W.J.
McFadden, G.B.
Warren, J.A.
Publisher: International Conference on Multiphase Flow (ICMF)
Publication Date: June 4, 2010
 Subjects
Subject: reactive wetting
intermetallic formation
multicomponent and multiphase model
Navier-Stokes flow
 Notes
Abstract: A diffuse-interface model of reactive wetting with intermetallic formation is presented. The model incorporates fluid flow, solute diffusion, and phase change that are based on the total molar Gibbs energy of a ternary system with four phases. Numerical simulations were performed successfully revealing the complex behavior of the reactive wetting process that includes nucleation and growth of an intermetallic phase, and initial rapid spreading followed by a slow and progressive spreading. In addition, the nucleation and growth of the intermetallic phase is shown to be directly influenced by its kinetic coefficient or the interface energy associated with it.
General Note: The International Conference on Multiphase Flow (ICMF) first was held in Tsukuba, Japan in 1991 and the second ICMF took place in Kyoto, Japan in 1995. During this conference, it was decided to establish an International Governing Board which oversees the major aspects of the conference and makes decisions about future conference locations. Due to the great importance of the field, it was furthermore decided to hold the conference every three years successively in Asia including Australia, Europe including Africa, Russia and the Near East and America. Hence, ICMF 1998 was held in Lyon, France, ICMF 2001 in New Orleans, USA, ICMF 2004 in Yokohama, Japan, and ICMF 2007 in Leipzig, Germany. ICMF-2010 is devoted to all aspects of Multiphase Flow. Researchers from all over the world gathered in order to introduce their recent advances in the field and thereby promote the exchange of new ideas, results and techniques. The conference is a key event in Multiphase Flow and supports the advancement of science in this very important field. The major research topics relevant for the conference are as follows: Bio-Fluid Dynamics; Boiling; Bubbly Flows; Cavitation; Colloidal and Suspension Dynamics; Collision, Agglomeration and Breakup; Computational Techniques for Multiphase Flows; Droplet Flows; Environmental and Geophysical Flows; Experimental Methods for Multiphase Flows; Fluidized and Circulating Fluidized Beds; Fluid Structure Interactions; Granular Media; Industrial Applications; Instabilities; Interfacial Flows; Micro and Nano-Scale Multiphase Flows; Microgravity in Two-Phase Flow; Multiphase Flows with Heat and Mass Transfer; Non-Newtonian Multiphase Flows; Particle-Laden Flows; Particle, Bubble and Drop Dynamics; Reactive Multiphase Flows
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Volume ID: VID00386
Source Institution: University of Florida
Holding Location: University of Florida
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Resource Identifier: 1563-Villanueva-ICMF2010.pdf

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7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


A Multicomponent and Multiphase Model of Reactive Wetting


W. Villanueva*, W.J. Boettingeri, G.B. McFadden and J.A. Warren

Physics Department, Royal Institute of Technology, Stockholm, SE-10691 Sweden
SMetallurgy Division, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA
i:Mathematical and Computational Sciences Division, National Institute of Standards and Technology,
Gaithersburg, MD 20899, USA

walterv ~kth.se, william.boettinger ~nist.gov, geoffrey.mcfadden ~nist.gov, and j ames.warren ~nist.gov

Keywords: Reactive wetting, intermetallic formation, multicomponent and multiphase model, Navier-Stokes flow




Abstract

A diffuse-interface model of reactive wetting with intermetallic formation is presented. The model incorporates fluid
flow, solute diffusion, and phase change that are based on the total molar Gibbs energy of a ternary system with four
phases. Numerical simulations were performed successfully revealing the complex behavior of the reactive wetting
process that includes nucleation and growth of an intermetallic phase, and initial rapid spreading followed by a slow
and progressive spreading. In addition, the nucleation and growth of the intermetallic phase is shown to be directly
influenced by its kinetic coefficient or the interface energy associated with it.


Introduction


The joining of solid metals with molten solders or filler
metals involves wetting of the solid by the liquid, during
which the solid dissolves into the liquid phase and/or an
intermetallic phase or phases form between the spread-
ing liquid and the solid substrate. The strength of the
joint depends on many factors such as wettability, sol-
ubility, and properties of the formed intermetallic com-
pound (see Boettinger et al. (1993) and Eustathopoulos
et al. (1999) for general discussion).
The mechanisms of reactive wetting involve the in-
terplay of fluid flow, heat and mass transport, capillary
phenomena, and phase transformations. To our knowl-
edge, a comprehensive model that incorporates all these
effects is still non-existent. Previous attempts to model
reactive wetting in high-temperature metallic systems
involved fitting experimental dynamic contact angle or
base radius curves by different functions, see for exam-
ple Ambrose et al. (1992, 1993); Eustathopoulos (1998);
Kim et al. (2008); Dezellus et al. (2003). Such empir-
ical formulas provide an easy and straightforward cal-
culation of the extent of spreading or general shape of
the drop at any given time. However, these formulas do
not give much insight into the mechanisms involved in
the reactive spreading process, i.e., intermetallic forma-
tion. In addition, detailed study of certain effects such


as phase change and dissolution can not be discerned
from these formulas. On the other hand, the use of com-
prehensive models requires a more involved numerical
calculations and its attendant challenges, such as insta-
bilities, limitations in parameter ranges, and restrictions
on the choice of length and time scales.
A different analysis of the kinetics of reactive wetting
has been proposed by Saiz et al. (2000). They argued
that the substrate cannot be described as rigid and in-
soluble and, with a sufficient nucleation barrier, a time
regime exists in which intermetallic formation lags the
liquid front. In this regime, the contact angles are then
dictated by adsorption, and the spreading kinetics are
controlled by the movement of a ridge formed at the
liquid-solid-vapor (L-S -V) triple junction. Although this
step can happen only in the early stages of the spreading
process, it can play a critical role in the succeeding steps
and they proposed that it should be taken into account
when modeling a specific system.
Our goal in this paper is to present a diffuse-interface
model of reactive wetting with intermetallic formation.
The multiphase and multicomponent approach is similar
to previous work on the modeling of dissolutive wetting,
see Villanueva et al. (2008, 2009). The model incorpo-
rates fluid flow, phase change, and solute diffusion. In
the next section, the mathematical model and input pa-
rameters are presented, followed by a brief discussion












t = to


t = t' onY

liquid vapor



intermetallic

LL; solid substrate


7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


of the numerical approach. In the results and discussion
section, we first present results for a base set of param-
eters. Then we focus on the factors that can affect the
nucleation and growth of the intermetallic phase such as
interface energies between the intermetallic and liquid,
and kinetic coefficient of the intermetallic phase. Next,
flow patterns and concentration profiles are discussed.

Mathematical Model and Parameters

A ternary system of substitutional elements A, B, and
C with four phases, spreading liquid (denoted by L or
sometimes 1 for convenience), solid substrate (S or 2),
intermetallic (I or 3), and vapor (V or 4) is considered.
See Figure 1. We begin by setting the total molar Gibbs
energy G given by


(G m(xA, B, C, L, S, I, #v, T)

iV+- V4|d (1)


where T is temperature, Vm is the molar volume, as-
sumed constant, and xA,B,c are the mole fractions of
A, B, C-atoms with xA B z + zc 1 The phase-
field variables Ai's vary nas selllli1 between 0 and 1 and
we set the condition that nL + #s + #< + #v 1. Our
approach is similar to Villanueva et al. (2008) with the
main difference that the gradient energies have an anti-
symmetric form. The molar Gibbs energy is postulated
as'
4 4
Gm = CP(#i)Gm + W .'2 ,


W .'2. .' + WLsyv (2)


with the smoothed step function P( i) = ?(10 -
15i + r. .'.). The coefficients eyj's and W's are related
to the thicknesses and interfacial energies. An ideal so-
lution for G'Ls~v and a regular solution for C'I are as-
sumed, and they take the form

G'm = xAo' sA BgoC sB ( ZA Bg)ogc
+RT(xA I A B B~I ~+( zA B~) -
In(1 xA B z)), i = L, S, V, (3)

and

G' = xAoG A B~oC B (1 _A Bg)o I
+ RT(xA I A + B I B +1- zA zB) .
In(1 xA B~) RA B, (4)


Figure 1: Schematic diagram of axisymmetric reactive
wetting with intermetallic formation.


where, for example, oG~ is the molar Gibbs energy of
pure A in the liquid phase and R is the gas constant.
An isothermal, viscous, and incompressible system is
considered. The governing equations, similar to the ap-
proach in Villanueva et al. (2008) and Villanueva et al.
(2009), are the following:
(i) conservative convective concentration equations,


v1 8xAt+uCIn


V JA and


V Je,


where u is the flow velocity and J7A and JnB are fluxes of
A and B measured with respect to the local flow,
(ii) HOH-COnservative convective Allen-Cahn equa-
tions for the phase-field variables,


aSi
t+ u -V 4


M 6 i


L, S, I, (7)


where My 's are kinetic mobilities and with natural
boundary conditions n -Vi =-0,
(iii) a mass continuity equation for incompressible
flow,


V u = 0,


(iv) the Navier-Stokes equations for incompressible
flow with added surface tension forces,


Bu"(+"Vi


-V5x+ V ( )(Vu +VuT)


i: h0 ,s
i=L,S,I


where 1? is a nonclassical pressure.














Table 1: Base set of parameters.

"Gi = -5.0 x 103 J/mol0


"G= -1/2 "G)

.C' = -1 "C'L

"G~ = -2 "G)
"G0 1/2 "GL

~LAff = Af AfC
= 2.6 x 19-13 11101 112/J

=10 Af
lL = 4.0 x103 Pa s
flS ~I = 16l flL
v- 10" -
PL 13 kg/111
Ps = PI 1.05 pL
Pv -10" pL
WI. Os WI = 25 111/J -s
T 450 K
TYLS = 2.21 X 105 J/1110
WLI -1.01 X 104 J/111
WLT = 1.43 x 105 J/11ol
IWsl = 2.54 x 104 J/Inol
Wsvy = 2.49 x 105 J/Inol

IIjre = IfLSIT- 8.75 x 106 J/11ol
tes = 6.32 x 10-" J/m1
LI 6.71 10 J/11
s, = 7.40 x 10-in J/
Lt- = 5.99 x 10-" J/11
e- = 8.24 x 10-" J/11
tf, = 7.94 x 10-" J/11
Tn L, 1.0 x 10-" 113/1110
R = 8.31 J/11ol K<


7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


To complete the concentration equations (Eqns. 5-6),
we write the interdiffusion flux of solutes J4A and JeB
(with ~JA + ~Jg + JC' = )


Je~ =-L4V
J, 6r(bG


L bB gG \


and (10)

(11)


where the variation in G with respect to the composi-
tions .ry are given by
6G 1 dG,
Sj = 24, B, (12)
,z V z
and the L~ij s are

L.AA (1 .rs) .r gAf4(L, OS, #L, OT
+.r4 rsB (OL, #S, PLO-)
+.r(1 rA rs)~C'(OL S, SmI, ) (1)


LeeB = .r rAg(LOWsP-)
+(1-.rsYI) yrs) (OL*PymlOT

+.r(1 rg -- rn)~C'(9L S, SsI, O),


and
(14)


LAB -(1 rs)ZrgBrnAf.4(L O~mI, OT
rgArs(1 rs)MsB(OL, OS, 9I, ,)
+r zA(1 r4 zr )zr Mc (OL, OS, WI, O")-
(15)
The mobilities of A, B, and C can be different in each
phase and are given by


+~~~~~~~ Af'( L- S-W) A, B, C, (16)
where 1 PL OS WI has been substituted for ( .
In the interior of the phases, the diffusivities, Dij, are
defined through the expressionS,
DAA DA
JA =z --~r -Vr and (17)

J, = VzA- Vzrs. (18)

Input Parameters. The model allows for a ternary
phase diagram such as the one shown as an isothermal
section in Figure 2. The phase diagram is idealized and
the parameters are given in Table 1 along with other in-
put parameters.
Given the parameters in Table 1, we can estimate
more familiar material parameters such as diffusivities
D;, and interface kinetic coefficients koeve with the fol-
lowing formulas (Boettinger et al. (2002)),
DL = Aff RT etc. (19)


Table 2: Calculated interface energies yij 's.
*YLS = 2.55 J/111 YLT = 1.01 J/1112
Ysy- = 2.17 J/111 YLI = 0.29 J/1112
Ysr 0.35 J/111 ylv 1.99 J/111
~Since the liquid-solid interface is unstable with the given base set of
parameters, the liquid-solid interface energy is calculated by
suppressing the formation of the intermetallic phase.


~coeff
3R~f6









































































and the values are summarized in Table 2.


Table 3: Recalculated interface energies yij's with
0 LI 1.47 x 105 J/Inol and t17 3.87 x 10-" J/111
while keeping the rest of the parameters the same.
*0 LS = 2.55 J/In YLT = 1.03 J/112
Yss- = 2.17 J/mn YLI = 1.54 J/112
Ysl 0.34 J/In ];1- 2.03 J/n2
~Since the liquid-solid interface is unstable with the given base set of
parameters, the liquid-solid interface energy is calculated by
suppressing the formation of the intermetallic phase.







s V



a S+V




.1-L L+I I I+S

U0 2 0 4 0 6 0 8


Figure 2: Idealized phase diagram of a ternary A-B-C
system at 450 K with four phases; liquid (L), solid (S),
intermetallic (I), and vapor (V).



where Af Af = Af = Af, i L,S, I, V. The
diffusivities are DL 109 112/S, DS 10-12 182/S,
and DI 10-12 m2/S while the kinetic coefficients are
kineer 16 s K(/I where / L, S, I.
Interface energies. Values for the interface energies
were obtained from 1D simulations of each pair of
phases (LS, LI, LV, SI, SV, and IV). The interface en-
ergies are computed from the expression,




m ( (f oc), p (f oc); T))


7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


Numerical Approach

The numerical simulations were carried out using Fem-
Lego (Amberg et al. (1999)), an open source PDE solver
USing an adaptive mesh finite element method. All PDEs
are discretized in space using piecewise linear functions.
Each resulting linear system is solved using Krylov-
type iteration methods either by the conjugate gradient
method (CG) for symmetric and positive-definite sys-
tems or the generalized minimal residual method (GM-
RES) otherwise. The system of Navier-Stokes equa-
tions and continuity equation is solved by an incremen-
tal fractional-step algorithm (Guermond and Quartapelle
(1998)) that belongs to a class of projection methods. A
pressure stabilization term is also added in the projection
step to improve stability.
Adaptive mesh refinement and derefinement is uti-
lized due to the need to spatially resolve the interfaces as
a consequence of the phase-field approach. In Figure 3,
cut-off regions corresponding to two different times (ini-
tial and late stage spreading) are shown illustrating the
effective implementation of the mesh refinement and
derefinement. An ad hoc error criterion is used to ensure
mesh resolution along the vicinity of the interfaces. See
Do-Quang et al. (2007) for details of the mesh adaptive
finite element scheme. For the typical example shown in
Figure 3, the initial radius of the drop is Rn 20 inn
and the domain size is (60 inn x 90 inn). The minimum
mesh size is hmin= 0.13 Inn with nodes between 9000-
11000 and triangular elements between 19000-22000.
The capillary time scale tc PLR0D 2YLT iS COnsid-
ered in this study to be especially important, represent-
ing the natural response of the system to reach mechan-
ical equilibrium.


Results and Discussion

Results with a base set of parameters given in Table 1 are
first presented. The initial configuration is a drop that
barely touches the substrate with a contact angle close
to 180 see Figure 4a. There are two ways to visualize
the configuration. Either the 0.5 contours of the phase-
field variables or the maximum phase-field as shown in
Figure 4 can be plotted, that is, at every discrete point in
the domain we find the phase-field variable that has the
largest value and then designate a corresponding color;
red for pi, green for ps, blue for ps, and white for 94.
The maximum phase-field plot has been chosen since
it can directly identify triple junctions, which facilitates
the measurement of contact angles following our previ-
ous approach in Villanueva et al. (2009).
The early stage of spreading is always rapid as vis-
cous or inertial forces dominate followed by a slow and
progressive spreading as diffusion becomes the domi-


(21)


DG,, (.; ; )


- (.r; (2) .r; (foc))







7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


''scale =o0nm


Figure 3: Adaptive mesh refinement and derefinement
of cut-off regions corresponding to two different times
with superimposed 0.5 contours of the phase-field vari-
ables 4L, #s, and #z.


nant process. However, phase change can also affect the
spreading process in both stages as was found in Vil-
lanueva et al. (2009) for dissolutive wetting. Figure 4b
shows an early time t/te 1 where the base radius has
increased to R/Ro = 1.07 with an apparent contact an-
gle of the drop of 132.3 The growth of the intermetal-
lic phase is also rapid, as expected by a balance of ten-
sions with the set of interface energies in Table 2. In the
magnified view of Figure 4b, we see that the intermetal-
lic phase has grown ahead of the spreading liquid and
forms LIV and SIV triple junctions. The drop profile in-
cluding the structure of the contact line region is qualita-
tively similar to SEM images of a Sn-0.7 Cu solder on a
Cu-substrate performed experimentally by Nogita et al.
(2009). The drop continues to spread further and reaches
R/R<3 = 2.14 at time t/te = 200 as shown in Figure 4c;
the intermetallic phase thickens and remains ahead of
the spreading liquid. In the magnified view of the con-
tact line region, one can observe that the solid substrate
is not planar, nor is the intermetallic layer, which is con-
sistent with SEM images of contact line regions reported
in the literature, e.g. Yin et al. (2008) and Nogita et al.
(2009).
Note that there are four possible triple junctions that
can be formed, namely, LSV, LIV, SIV, and LIS. Only
two stable triple junctions (LIV and SIV) are formed be-
cause the LS interface is always wet by the intermetallic
for the base set parameters. In principle there should
be a parameter set where the other two triple junctions,


4 2


Figure 4: Numerical simulation of the spreading of a
molten metal drop on a solid substrate with the forma-
tion of intermetallic phase between the drop and the sub-
strate. Th iuesossnapshots with a maximum
phase-field plot at times t/te = 0, 1, 200 where to is
a capillary time scale.


LSV and LIS, are stable and co-exist. In this case the
intermetallic phase should remains behind the spread-
ing liquid for a longer time period than the present case.
However we have yet to find the set of interface ener-
gies that will give this outcome. In addition, there is the
possibility of forming a stable LSIV quadrijunction as
proposed by Cahn (1991) but it is a challenge to find
the set of interface energies that will generate this par-
ticular case given our present methodology.
A number of factors can affect the nucleation and
growth of the intermetallic phase such as interface ener-
gies between the intermetallic and other phases, and ki-
netic coefficients of the intermetallic phase. The present
phase field model does not cleanly separate the equiva-
lent interface kinetic coefficients for the LI and SI in-
terfaces. However by decreasing the mobility of the
phase field for the intermetallic phase, one can retard the
growth of the intermetallic between the liquid and solid
phases. Using the approximate formula in Equation 20,
a case is now considered where the mobility of the phase
field is lowered by a factor of 12 from the base state,
while leaving the rest of the parameters the same. The
intermetallic phase becomes more sluggish in the sense







7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010













Figure 6: Comparison of the growth of inter-
metallic phase with different liquid-intermetallic inter-
face energies: (a-c-e) YLI 0.29 J/112 at times
/tet 0.1, 0.3, 1.0, respectively, and (b-d-f) yLI =
1.54 J/111 at times t/te = 0.1, 0.3, 1.0, respectively.



same corresponding snapshot in the fast kinetics case are
also less than an interface thickness in difference. The
contact angles are less than a degree in difference except
for the time t/te 20 where the fast kinetics case is
2.4" lower in 91.
Figure 6 shows the effect of increasing the liquid-
intermetallic interface energy yLI from 0.29 J/m21 to
1.54 J/111 on the nucleation and growth of the inter-
metallic phase. We should note that varying an inter-
face energy y;4 in the model means modifying the cor-
responding parameter W~j and tij which can also al-
ter the other interface energies. However, a 1D recal-
culation of the interface energies with the changes in
the LI-interface, 0 LI .7x12JIo n 1
3.87 x 10-" J/m1 (while keeping the rest of the param-
eters the same), yielded only a slight change in other
interface energies (see Table 3).
Snapshots for yLI 0.29 J/112 at times t/te
0.1, 0.3, 1.0 are shown in Figures 6a-c-e, respectively,
while Figures 6b-d-f correspond tO YLI 1.54 J/111
with the same time sequence, respectively. At t/t,
0.1 in Figures 6a and 6b, both cases show no intermetal-
lic growth and the profiles are almost identical. But then
at t/te 0.3, the growth of the intermetallic phase with
YLI 0.29 J/112 (Figure 6c) has become noticeable as
compared to the case with a higher YLI 1.54 J/112
(Figure 6d) where the intermetallic phase is just about
to nucleate. At a later time t/te 1.0 with the lower
YLI (Figure 6e), the intermetallic phase grows thicker
and is now clearly ahead of the spreading liquid. For
the case with a higher yLI (Figure 6f), the growth of the
intermetallic phase has become noticeable but it is thin-
ner compared to the case with lower yLI. Thus Figure 6
demonstrates that a higher liquid-intermetallic interface
energy can also retard the growth of the intermetallic
phase although the intermetallic phase finally gets ahead
of the spreading liquid.


Figure 5: Comparison of the growth of intermetallic
phase between a fast intermetallic kinetics (a-c-e-g) cor-
responding to times t/t, 1 1.3, 3, 20, respectively,
and a slow intermetallic kinetics (b-d-f-g) with the same
corresponding times, respectively. In both cases the in-
termetallic phase eventually grows ahead of the spread-
ing liquid drop.


that it will respond more slowly to disequilibrium. In
Figure 5, a comparison between the two cases, fast inter-
metallic kinetics (which corresponds to 16 s K(/11) and
slow intermetallic kinetics (192 s K(/m1) is shown. Fig-
ures 5a-c-e-g show snapshots for the fast kinetics case
with a sequence in time, t/te 1 1.3, 3, 20, respec-
tively, while Figures 5b-d-f-g correspond to the slow ki-
netics case with the same time sequence, respectively.
At t/te 1 the intermetallic phase has grown ahead
of the liquid drop with the fast kinetics case (Figure 5a)
while no intermetallic phase has yet grown between the
liquid drop and the substrate with the slow kinetics case
(Figure 5b). The base radii of the liquid drop for both
cases are essentially identical at R/Rn 1.07 for the
fast kinetics case and R/Ro 1.03 for the slow kinet-
ics case. The apparent contact angles 01 for both cases
are less than a degree apart, 132.3" for the fast kinetics
case and 132.1" for the slow kinetics case. The time
sequence t/t, 1.3, 3, 20 for the fast kinetics case
(Figure 5c-e-g) shows the intermetallic phase moves to-
gether but always ahead of the liquid drop. The thick-
ness of the intermetallic phase is also increasing while it
slowly penetrates the substrate. The intermetallic phase
is non-planar and so is the substrate. The time sequence
t/te = 1.3, 3, 20 for the slow kinetics case (Figure 5d-
f-h) shows the nucleation and growth of the intermetallic
phase. The base radii of the liquid drops compared to the







7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


The initial concentrations in each phase can be set ar-
bitrarily in the model. At equilibrium, the liquid phase
is rich in A while the substrate is rich in B (refer to the
phase diagram in Figure 2). Initially, we start with the
substrate having equilibrium values of .rf = 0.05 and
zr = 0.70 while the liquid has lower concentration of
B, reB 0.05, than the equilibrium value of J:r = 0.22
but higher concentration of A, zr = 0.85, than the equi-
librium value of zrL = 0.54. Solute diffusion is much
slower than fluid flow. So for the purpose of demonstrat-
ing that the concentrations in the liquid and intermetal-
lic reach their corresponding stable or equilibrium val-
ues within a feasible computational time frame (see Fig-
ure 7), we increase the effect of diffusion compared to
fluid flow, that is, the ratio between the solute transport
due to convection and solute transport due to diffusion
(also known as the solutal Peclet number) is decreased
by a factor of 500. Figure 7a shows the concentration
profile of B at an early spreading stage t/t, 1 with a
superimposed normalized velocity profile and 0.5 con-
tours of the phase-field variableS PL, Os, and 97. The
concentration of B in the liquid evaluated at the center
of the drop (along the z-axis and midpoint between the
LV and LI-interface) has only increased by 2 10-" from
the initial value of J:r 0.05 while the concentration of
A (not shown) has increased by 3 104 from the initial
value of .r 4 = 0.85. The intermetallic phase has formed
by this time and has spread ahead of the liquid phase.
The flow pattern consists of a vortex with center out-
side of the drop and near the liquid-vapor interface, and
a flow downward from the upper part of the drop then
redirected to the contact line region. And as expected,
there is negligible flow in the solid substrate and in the
intermetallic phase at any given time. The base radius of
the liquid drop is R/Rn = 1.07 and has a contact angle


At an intermediate stage t/te 20 (Figure 7b), the
drop has spread further, with a base radius R/Ro = 2.30
and a contact angle BL = 85.4 The flow pattern is gen-
erally the same compared to the previous time t/te = 1
except that the vortex moves further away from the solid
substrate. The concentration of A and B in the liquid are
zr = 0.73 and Jrs = 0.10, respectively, while the con-
centrations in the intermetallic phase are zr = 0.44 and
Jrn = 0.37. Since the average concentration of the sys-
tem implies only three-phase SIV co-existence, the liq-
uid drop is expected to disappear at equilibrium. In ad-
dition, the equilibrium concentration of the intermetallic
phase is expected to reach the I-corner of the SIV co-
existence triangle (see phase diagram) which has values
zr' = 0.28 and zr = 0.53. At a later stage t/te 200
(Figure 7c), the liquid drop shrank, the intermetallic
phase expanded, and portion of the substrate has dis-
solved. Furthermore, the liquid drop has spread further


0 123 45 6
cl


0 1 23
b


4 5 6


0 1 23


4 5 6


Figure 7: Concentration profiles of B at (a) early fast-
spreading stage t/te 1 (b) intermediate stage t/te
20, and (c) late slow-spreading stage t/te = 200 with
superimposed normalized velocity profiles and 0.5 con-
tours of the phase-field variableS PL, Os, and PI.







7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


namics of liquid drop spreading in metal-metal systems,
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Dezellus O., Hodaj F., Eustathopoulos N., Progress in
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Amberg G., Parallel adaptive computation of some time-
dependent materials-related microstructural problems,
Bull. Pol. Acad. Sci., Vol. 55, 229-237, 2007

Eustathopoulos N., Dynamics of wetting in reactive
metal/ceramic systems, Acta Mater., Vol. 46, 2319-
2327, 1998

Eustathopoulos N., Nicholas M.G., Drevet B., Wettabil-
ity at high temperatures, Pergamon Materials Series, El-
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1039-1053, 1998

Kim C., Kang S.C., Baldwin D.F., Experimental eval-
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033537, 2008

Nogita K., Gourlay C.M., Nishimura T., Cracking and
phase stability in reaction layers between Sn-Cu-Ni sol-
ders and Cu substrates, JOM, Vol. 61, 45-51, 2009

Saiz E., Cannon R.M., Tomsia A.P., Reactive spread-
ing: adsorption, ridging and compound formation, Acta
Mater., Vol. 48, 4449-4462, 2000

Villanueva W., Griinhagen K., Amberg G., and Agren J.,
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with R/Ro = 3.24 and contact angle BL = 61.2 The
concentration of A and B in the liquid are zr = 0.60
and Zg = 0.20, respectively, and are close to the sta-
ble values zr = 0.54 and zB = 0.22 (corresponding
to the L-comner of the LIV co-existence triangle in the
phase diagram). The concentration of A and B of the
intermetallic layer are zr = 0.36 and zB = 0.48 with
its source of A-atoms coming from the liquid drop and
B-atoms from the substrate. Although the average con-
centration (.7 4 = 0.07, rPs = 0.35) lies inside the SIV
co-existence triangle, these concentrations are closer to
the I-comer of the LIV co-existence triangle, that has
values zr = 0.34 and zB = 0.48, than the I-comner of
the SIV co-existence triangle that has values zr = 0.28
and Zg = 0.53. A check at times t/tc 300, 400, 500
yielded the values (zr = 0.36,2Brs 0.48), (zr, -
0.36,2Brs 0.48), (2r = 0.36,.rs = 0.47), respec-
tively, which are also closer to the I-comner of the LIV
co-existence triangle. However, a validation by examin-
ing a 1D planar interface confirms that a three phase SIV
co-existence yields the right concentrations at the cor-
ners of the SIV co-existence triangle. Thus we conclude
that solute trapping is likely occurring in the rapidly
forming intermetallic. We note that in real solder joints
the intermetallic layer is made of many small grains that
require multiple nucleation events. Treatment of this ef-
fect is beyond the scope of this investigation.


Conclusions

We have presented a multicomponent and multiphase
model of reactive wetting with intermetallic formation.
The model incorporates fluid flow, phase change, and
solute diffusion. Numerical simulations were performed
successfully revealing the complex behavior of the re-
active wetting process that include the nucleation and
growth of an intermetallic phase, initial rapid spread-
ing followed by a slow and progressive spreading. In
addition, we have shown that the formation of the in-
termetallic phase can be controlled by the kinetic coef-
ficient of the intermetallic phase and/or the associated
interface energy.


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7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010


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Yin L., Chauhan A., Singler T.J., Reactive wetting
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