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Electrochemical measurements for the determination of dynamic states in the Bridgman crystal growth configuration

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Electrochemical measurements for the determination of dynamic states in the Bridgman crystal growth configuration
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Sears, Brian R., 1964-
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ix, 264 leaves : ill., photos. ; 28 cm.

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Convection ( jstor )
Diffusion coefficient ( jstor )
Electrodes ( jstor )
Electrolytes ( jstor )
Electromotive forces ( jstor )
Fluids ( jstor )
Liquids ( jstor )
Oxygen ( jstor )
Rayleigh number ( jstor )
Tin ( jstor )
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bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

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Thesis:
Thesis (Ph. D.)--University of Florida, 1990.
Bibliography:
Includes bibliographical references (leaves 259-263).
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Typescript.
General Note:
Vita.
Statement of Responsibility:
by Brian R. Sears.

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University of Florida
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Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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ELECTROCHEMICAL MEASUREMENTS FOR THE DETERMINATION OF
DYNAMIC STATES IN THE BRIDGMAN CRYSTAL GROWTH CONFIGURATION













BY

BRIAN R. SEARS


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


1990












ACKNOWLEDGEMENTS


I would like to acknowledge the support of family, friends, and coworkers

whose help and encouragement have contributed to the timely completion of this work.

First, I would like to thank my faculty advisors, Dr. Tim Anderson and Dr. Ranga

Narayanan, for their guidance and encouragement in my research endeavors. At the

same time, I thank Dr. Archie Fripp, also for his guidance and encouragement, but

additionally for his willingness and enthusiasm to act as research advisor during my

two years of residence at the NASA Langley Research Center. I am grateful to the

National Aeronautics and Space Administration for monetary support through a

university grant.

I would like to thank my coworkers at NASA for the technical and intellectual

support which helped me to get past many problems. In particular, I thank Glenn

Woodell and Bill Debnam for their invaluable technical support. Also, I thank Ivan

Clark, Dave Knuteson, Jim Hurst, and Wayne Gerdes for their support as well as for

many enlightening discussions. These people, and others, have freely offered their

professional guidance as well as their friendship during my stay at NASA.

I am grateful to my parents for their support in my education. I am also

deeply grateful to my wife, Paula, for her continued support and especially for her

sacrifices during the final stages of my dissertation preparation.












TABLE OF CONTENTS

Rae

ACKNOWLEDGMENTS ................................. ii

LIST OF SYMBOLS .................................... v

ABSTRACT .......................................... ix

CHAPTERS

1 GENERAL BACKGROUND .................... 1

Introduction ................................ 1
Vertical Bridgman Crystal Growth ............... 2
Growth of Compound and Doped Semiconductors ...... 4
Buoyancy-driven Convection ..................... 8
Literature Survey ............................ 13
Experimental Approach ........................ 24

2 SOLID-STATE ELECTROCHEMICAL MEASUREMENTS 28

Introduction ................................ 28
Yttria-stabilized Zirconia ....................... 29
Oxygen Concentration Cell ...................... 36
Summary ............... ....... ........... 40

3 OXYGEN DIFFUSIVITY IN LIQUID TIN ........... 42

Introduction ................................ 42
Experimental ............................... 44
Results .................................... 58
Discussion ................................. 77
Summary ...... ....................... ... 81

4 FLOW VISUALIZATION ....................... 83

Introduction ........... .... ..... .... ........ 83
Experimental ............ .... .. ............ 85
Numerical Simulations ....................... 94
Results .................... ...... .... .... 99
Discussion ................................. 118







5 MULTIPLE DETECTOR FLOW VISUALIZATION ..... 121

Introduction ................................ 121
Experimental .............................. 122
Results ............... 126
Discussion ................................. 135

6 SUMMARY AND CONCLUSIONS ................ 138

APPENDICES

A NUMERICAL OUTPUT ........................... 142

B EXPERIMENTAL ELECTROMOTIVE FORCE DATA ... 161

REFERENCES ..................... ......... ............... 259

BIOGRAPHICAL SKETCH ............................... 264












LIST OF SYMBOLS


ao thermodynamic activity of oxygen

A cross-sectional area of the experimental cell

C concentration

C dimensionless concentration, (C C2)/(C1 C2)

Co initial concentration

C, concentration at the lower surface of the fluid cell

C, concentration at the upper surface of the fluid cell

CL concentration of the liquid phase in equilibrium with a solid phase

C, concentration of the solid phase in equilibrium with a liquid phase

C degree Celsius

cm centimeter

Do molecular diffusivity of oxygen

Do' effective diffusivity of oxygen

E potential or electromotive force

E0 initial potential

E, activation energy

EMF electromotive force

F Faraday constant

F dimensionless body force, g/g

g magnitude of gravitational acceleration
g gravity vector







G Gibb's free energy

H height of the fluid cell

I electric current

ID inside diameter

J joule

k Boltzmann constant

K degree Kelvin

kg kilogram

m meter

m index

mm millimeter

mV millivolt

n number of charge equivalents

NO oxygen flux

OD outside diameter

P pressure

VP defined as (VP pog)H2/v2

Pr Prandtl number

r radial coordinate

R radius of the fluid cell

R gas constant

Ra Rayleigh number

Ras solutal Rayleigh number

RaT thermal Rayleigh number

Ra,, first critical Rayleigh number







Ra, second critical Rayleigh number

Sc Schmidt number

sec second

t time

t dimensionless time, tv/IH

T temperature

T dimensionless temperature, (T Ts)/(Ti Ts)

T1 temperature at the lower surface of the fluid cell

T, temperature at the upper surface of the fluid cell

ti ionic transference number

AT temperature difference (between top and bottom of fluid cell)

V volt
v velocity vector

V dimensionless velocity vector, vH/v

W watt

X mole fraction

z axial coordinate

a Seebeck coefficient

P aspect ratio, H/R

Ps solutal expansion coefficient

pT thermal expansion coefficient

y activity coefficient

K thermal diffusivity

JA dynamic viscosity

juo chemical potential of oxygen

v cinematic viscosity, W/p







p density

o, electron conductivity

Oh hole conductivity

ao ionic conductivity












Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements of the Degree of Doctor of Philosophy

ELECTROCHEMICAL MEASUREMENTS FOR THE DETERMINATION OF
DYNAMIC STATES IN THE BRIDGMAN CRYSTAL GROWTH CONFIGURATION

By

Brian R. Sears

December, 1990

Chairman: Dr. Timothy J. Anderson
Major Department: Chemical Engineering

An electrochemical flow visualization technique for characterizing natural

convection in liquid metals and semiconductors in the vertical Bridgman melt-growth

configuration was developed and tested. The ceramic electrolyte yttria-stabilized

zirconia was incorporated into the boundaries of the fluid container to act as a window

through which the dilute oxygen tracer could be injected, extracted, or measured at

the surfaces of the fluid volume. An experimental cell was designed and used to

measure the effective diffusivity of oxygen across tin melts in geometries characteristic

of Bridgman cells. The technique was able to discern transcritical points in the

dynamic state of the melt as a function of imposed temperature gradient. The

electrochemical technique was modified and shown to be capable of describing the

orientation of flow in Bridgman simulations. An improved method for measuring the

binary diffusion coefficient of oxygen in the absence of thermally-driven convection in

liquid metals was designed. The oxygen diffusivity in liquid tin was then studied

experimentally as a function of temperature, and the results were compared to less

well-controlled experimental studies.












CHAPTER 1
GENERAL BACKGROUND


Introduction


Vertical Bridgman melt-growth is a proven method for production of bulk

compound semiconductors. However, inherent compositional inhomogeneities and

extended defects have limited the electronic and optoelectronic applications of these

materials. The electronics industry would greatly benefit from minimization of these

imperfections, but a better understanding of their origin must be obtained. Since

convection in the melt during solidification is known to be responsible for segregation

of the component elements as well as for crystallographic defects, an increased under-

standing of convection is needed. Previously, the nature of convection was inferred

from post-growth analysis of crystals and also through temperature measurement on

growth ampoule surfaces during growth. The opacity of liquid metals and

semiconductors preclude optical visualization techniques and high growth tempera-

tures limit the applicability of other visualization methods. This dissertation presents

a novel flow visualization technique capable of characterizing convection in Bridgman

growth simulations. The present technique involves the application of a solid state

electrochemical cell to introduce, extract, and monitor trace quantities of oxygen

across surfaces of the growth ampoule. Methods by which this flow visualization

technique can be applied to actual crystal growth experiments are discussed.






2
Vertical Bridgman Crystal Growth


Vertical Bridgman crystal growth is one of several methods used to produce

bulk semiconducting materials. It is a preferred method for production of semiconduc-

tors containing volatile or toxic elements since the materials can be easily sealed from

the environment. A Bridgman cell is generally composed of a vertical cylindrical

ampoule containing the semiconductor melt and is housed within a series of heaters

which maintain a thermal gradient along the axis of the cylinder. Directional

solidification of the melt is achieved by one of two commonly used procedures:

mechanical translation of the ampoule relative to an established thermal gradient

(Bridgman method) or translation of the thermal gradient relative to the ampoule by

continuously varying the power input to the heaters (gradient freeze method). The

rate at which the solid-liquid interface moves along the length of the sample is

controlled by, but not necessarily the same as, the rate at which the sample moves

relative to the furnace. A single crystalline sample can be produced by seeding the

end at which solidification will begin. Single crystalline semiconductor materials can

be grown in the Bridgman configuration with either top or bottom seeding, although

most commonly with bottom seeding. Figure 1-1 is a schematic of the basic Bridgman

growth configuration.

Bridgman, Czochralski, and float zone growth of semiconductors are the most

commonly used methods ofbulk production of semiconductor materials from the melt.

Czochralski growth, in which the crystal is pulled from a molten pool, is the preferred

method of growth for the group IV elements, silicon and germanium, although the

float zone technique is also used. The group III-V compound semiconductors are

commonly grown by a liquid encapsulated Czochralski technique in order to minimize






3









Hot Hot
Zone Zone





Insulation // Insulation
Zone \ Zone
DSolid 10z"



Cold / Cold
Zone Zone











Figure 1-1. Schematic of the vertical Bridgman crystal-growth configuration with
a partially solidified sample.






4
evaporative losses of high vapor pressure components. Likewise, group II-VI

compound semiconductors often contain one or more elements which have relatively

high vapor pressures at growth temperatures. This poses two problems, depletion of

one component from the melt as it condenses on cool spots of enclosing walls and

possible leakage of highly toxic elements into the environment. Bridgman growth

offers improved containment of volatile components and is easily modified for growth

of crystals over a wide range of pressures. Also, the Bridgman method appears to be

the best suited technique for growth of solid solution semiconductor materials since

minimization of convection is desired. Forced convection is inherent in Czochralski

growth as the boule and/or crucible are rotated. In Bridgman growth, however,

natural convection is predominant and can be controlled by application of a magnetic

field or reduction of gravity during growth. More detailed discussions of the Bridgman

crystal growth technique are available in the literature [1,2].


Growth of Compound and Doped Semiconductors


The compositional homogeneity of doped crystals and of compound semicon-

ductors is of great importance in applications of the materials. In general, com-

positional homogeneity is desired in semiconductor materials. This is especially true

for alloys of various compound semiconductors when the composition controls

electronic properties. Compositional homogeneity, however, is not easily obtained in

directional solidification of a multicomponent melt. This difficulty results from the

nature of the phase equilibrium established between the liquid and the solid solution

at the corresponding melting temperature. Figure 1-2 shows the solid-liquid phase

diagram [3,4,5] of a pseudo-binary mixture, PbxSn.xTe, which forms a completely







5












920-



900-




BBO C-







CL.
1-880







820



800
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
SnTe Mole fraction PbTe PbTe













Figure 1-2. Phase diagram of the pseudo-binary mixture, PbxSnl.xTe [3,4,5].
C, and CL identify the equilibrium solid and liquid compositions at
a given temperature.






6

miscible solid solution. The symbol Cs indicates the composition of the solid solution

in equilibrium with the liquid of composition CL. As solidification of the melt

proceeds, solute (SnTe) is rejected at the interface and a diffusion boundary layer is

developed in the melt adjacent to the solid-liquid interface. The resulting crystal will

show a compositional trend like that of curve A in Figure 1-3 if the only method of

mass transport is diffusion, i.e. diffusion-limited growth. A quantitative analysis of

solute redistribution for cases in which the distribution coefficient, Ca/C, is constant

(e.g., dilute solutes) is given by Smith, Tiller, and Rutter [6].

As shown in the diffusion-controlled growth curve in Figure 1-3, after the

initial transient, a period of steady-state growth is achieved during which com-

positional homogeneity is predicted. Such a crystal would be considered to be very

high yield. Conditions which eliminate convective mass transfer and thus allow

diffusion-controlled growth are very difficult to realize, however. Application of

magnetic fields to stabilize the melt has been successful in reducing convection in

some materials, but other materials with low magnetic susceptibilities resist

stabilization. The microgravity environment of space, then, offers the only viable

location to conduct convection-free experiments for these materials. Witt et al., in two

separate space experiments, were able to grow doped InSb [7] and Ge [8] crystals.

Analysis of the crystals indicated that ideal, exclusively diffusion-controlled growth

occurred.

On earth, the gravitational field introduces a body force which acts upon

variations in liquid density to cause natural convection in fluids. The convection

disrupts the diffusion boundary layer and enhances the distribution of solute

throughout the melt. Solute rejected at the growth interface is swept away by

























































0.1 0.2 0.3 0.4 0.5
Fraction of length


0.8 0.7 0.8 0.9
along crystal


Figure 1-3.


Characteristic axial compositions of binary solid solutions grown
under diffusion-controlled and fully-mixed conditions. The
distribution coefficient is 0.7, and the initial melt composition is Co.


2.0







1.5







) 1.0
U


0.5


0.0
0.0


Diffusion-controlled -
-.- Fully-mixed -

I A
-
I


I



Curve A ,' -




Cure B
--- Curve B










mm, alm a. i.m nl mum a1m.. nnnu 111n n.1111.n11111111


^^






8
currents and mixed with the bulk melt. Crystals grown under the influence of steady

convection in the melt show a monotonically increasing composition approaching that

shown for complete mixing in Figure 1-3 (curve B) [6]. Under certain conditions,

convection in the melt can be oscillatory or turbulent in nature. This has been shown

to cause transient back-melting of the crystal which results in compositional striations

throughout the final crystal [7,9,10]. The compositional striations resulting from

oscillatory flow are much more damaging to the properties of a semiconductor than

are the monotonic variations in composition resulting from purely steady flow.

Convective transport in melt-growth configurations has become an area of intense

research because the performance of most electronic devices depends largely upon

spatial variations in the composition of the material [11,12].


Buoyancy-driven Convection


Buoyancy-driven convection, or natural convection, in fluids results from the

effect of the gravitational field on density variations within the fluid. These density

gradients can be attributed to thermal or compositional variations. The stability and

corresponding dynamic state of a fluid with respect to the density variation depend

on the orientation of the density gradient with respect to the lines of gravity, the

magnitude of gravitational acceleration, the magnitude of the density gradient, the

fluid properties, and the properties of the boundaries of the fluid volume. This study

is restricted to physical geometries associated with the vertical Bridgman

configuration, i.e. right circular cylinders of fluid bounded by rigid walls on the sides

and bottom. The top surface may be either free or bounded by a rigid wall. Flow due






9
to gradients in surface tension, termed Marangoni convection, can be eliminated in the

case of a rigid boundary.

Some of the parameters which affect the dynamic state of the melt during

directional solidification in the vertical Bridgman configuration are defined in the non-

dimensional forms of the equations of change. The equations of continuity of mass,

momentum, energy, and species are, respectively, as follows:


V = 0 (1-1)



+ (+-V)= -VP + CF + TF + 2 (1-2)
at Sc Pr


Pr- + (VV) = V2T (1-3)
t( aI


Sc + ( = 92C (1-4)
(t )



In this mathematical model, the Oberbeck-Boussinesq [13,14] approximation has

been applied. This states that the density of the fluid can be assumed constant in all

terms except the body force terms in the equations of motion. Because the density,

temperature, solutal, and velocity fields are interdependent, retention of a variable

density in all terms creates an analytically intractable differential problem. The

equations have been cast into simple, non-dimensional form by assuming a linear

dependence of density on temperature and concentration (for a two component

mixture) in the body force terms only. For fluids in which these dependencies are






10
small, such as most fluids used in the Bridgman system, these approximations are

valid [15,16,17]. So the equations of change appear in the simple forms shown

above where dimensionless parameters scale the various terms. The Prandtl number,

Pr, is defined as the ratio of momentum diffusivity to thermal diffusivity of the fluid.

This parameter is a measure of the rate of diffusion of vorticity relative to the rate of

diffusion of heat in the fluid. The Schmidt number, Sc, appears in the presence of

solutal gradients and is defined as the ratio of momentum diffusivity to mass

diffusivity. Similar to the Prandtl number, the Schmidt number is a measure of the

rate of diffusion of vorticity relative to the rate of diffusion of component species in the

fluid.

The solutal and thermal Rayleigh numbers, Ras and RaT, are parameters

which, in general, will change throughout the course of a crystal growth. These

parameters are defined as


= (C,-C2)gH3 (15)
vDy



S= PT(r,- T)gH' (1-6)
VK



where P, and PT are the coefficients of solutal and thermal expansion, respectively;

C, are the concentrations at the lower (1) and upper (2) surfaces of the fluid cell; T, are

the temperatures at the lower (1) and upper (2) surfaces of the fluid cell; g is

gravitational acceleration; H is the fluid cell height; and v, K, and Dg are the

momentum, thermal, and molecular diffusivities, respectively.






11
The Rayleigh number can be thought of as a parameter which characterizes the

driving force for convection resulting from exclusively vertically oriented density

gradients. Considerable attention has been given to the study of hydrodynamic

stability and the dependence of dynamic state on Rayleigh number for fluid layers

having vertically oriented density gradients. A short review of this work as it relates

to vertical Bridgman melt-growth is given in the following section; however, a simple

description of the Rayleigh number as presented by Busse [18] offers an intuitive

description of its relevance. In Busse's description, a fluid layer which is heated on

the bottom and cooled on top will exist in different dynamic states depending on the

magnitude of the imposed temperature gradient. Consider an initially motionless

fluid layer subjected to a thermal gradient as stated; the fluid layer will have a certain

potential energy associated with it as a result of gravity acting on layers of dense fluid

existing on top of layers of less-dense fluid. The fluid also has a viscous nature to it

which serves to dissipate any potential energy which is released. The Rayleigh

number is a parameter proportional to the ratio of the rate of release of potential

energy to the rate of dissipation of energy by viscous forces. The onset of convection

in the stagnant fluid requires that the Rayleigh number exceed a certain threshold

value at which the potential energy is released more rapidly than it can be dissipated

by static viscous forces. Convection is simply the mechanism by which excess

potential energy is released. Further changes in the dynamic state will occur as the

Rayleigh number exceeds additional threshold values, e.g., oscillatory convection and

turbulence.

One might expect that the geometry of the fluid cell would also influence the

dynamic state in a Bridgman cell. Indeed, another dimensionless parameter emerges






12
when the equations of motion are written in component form for cylindrical

geometries. This parameter is the aspect ratio, P, and is defined as the ratio of the

height of the fluid cell to the radius.

To complete this general discussion of buoyancy-driven convection, attention

must be given to the effects of horizontally oriented density gradients in the fluid.

Unlike the conditional stability of fluids having vertical density gradients, fluids

having horizontal density gradients are unconditionally subject to natural convection

[19,20]. Muller et al. [21] presented a two-Rayleigh-number model of buoyancy-

driven convection to account for flow in melt-growth configurations resulting from two-

dimensional density gradients. In actual crystal growth, horizontal thermal gradients

are inherent due to mismatching of material thermal conductivities and also to

furnace imperfections. These horizontal gradients provide a driving force for

convection in addition to any driving forces due to vertical gradients. The resulting

dynamic state is then a non-linear interaction of the independent flows driven by

horizontal and vertical gradients.

In summary, the parameters which determine the dynamic state in ideal,

vertical, Bridgman melt-growth are the dimensionless quantities Pr, Sc, Ras, RaT, and

P. Non-idealities are introduced, however, through the boundaries of the fluid cell and

also by latent heat effects during solidification. Since growth materials must be

contained in an ampoule during growth, lateral heat fluxes are inevitable. These may

be reduced, though, by using better insulating ampoule materials or by better

matching ampoule and growth material thermal conductivities. Non-uniform heating

by heater elements also introduces imperfections in thermal boundary conditions, but

these may be reduced by improved furnace designs.






13
For the sake of completeness, the other modes of convection in Bridgman

growth will be mentioned. Marangoni convection results from gradients in surface

tension, generally due to thermal gradients, on a free surface. This type of convection

can be completely eliminated by replacing any free surfaces with a rigid boundary to

impose a no-slip condition at that boundary. Additionally, forced convection can result

during solidification when volumetric expansion or contraction occurs during the

phase transformation at the growth interface. Accounting for these flows is important

when buoyancy-driven convection is not important, such as in low-gravity melt-

growth. In space, the gravitational field can be decreased to 10"3 10"6 g, decreasing

the resulting buoyancy forces accordingly. Under conditions of melt-growth on earth,

however, buoyancy-driven convection is by far the most dominant source of fluid flow.


Literature Survey


This section encompasses a review of studies on hydrodynamic stability and

natural convection phenomena in fluid layers as they relate to the Bridgman cell.

This discussion will begin with an overview of the classic Rayleigh-Benard problem,

which serves as the foundation for subsequent studies on the hydrodynamic stability

of fluids in various geometries. Extensions of this work which include solutal effects

and geometries characteristic of Bridgman melt-growth will then be discussed.

The earliest work in this area focused on the stability of horizontal fluid layers

heated from below and cooled from above. In this orientation, strata of increasing

density exist on top of one another. One might intuitively conclude that this

arrangement would be statically unstable and would break down into a convective

motion. It has been shown, however, through experiment and mathematical






14

treatments that the fluid layer heated from below will remain stable for temperature

gradients up to some threshold value. Beyond this critical value the fluid layer will

break down into a convective motion. It should be noted that fluid layers having

decreasing density with increasing height (e.g. horizontal fluid layers heated from

above) are statically stable for all magnitudes of density gradient.

The earliest reported experiments which spawned an interest in hydrodynamic

stability were performed by Thomson [22] and Benard [23]. Benard gave a

detailed account of experimental observations of the flow patterns developed as a

result of heating from below a fluid layer with a free upper surface. The cellular flow

patterns observed in shallow layers of fluid are consequently referred to as Benard

cells.

Lord Rayleigh [24] was the first to give an analytical description of the

Benard flow. In his analysis, Rayleigh determined the conditions under which a fluid

layer heated from below would break down into a convective roll pattern. Rayleigh's

approach was a linearized perturbation analysis of the momentum and energy

equations. Although Rayleigh did not consider surface tension driven convection at

the free surface [25,26] (which is quite substantial in the Benard experiments)

in his treatment, his work inspired more general analyses over a broader range of

boundary conditions. Jeffreys [27,28], Low [29], and Pellew and Southwell

[30], in particular, extended the linear analysis to include both free and rigid

boundaries at the upper and lower surfaces. They assumed that these boundaries had

infinite thermal conductivity and heat capacity (i.e. fixed-temperature boundary

conditions), which is a very idealistic approach. Two books which give detailed






15
descriptions of the linear convection problem are by Chandrasekhar [311 and

Gershuni and Zhukovitskii [32].

The effects of imposing thermal boundary conditions other than fixed

temperatures at the surfaces were introduced by Sparrow, Goldstein, and Jonsson

[331. These results are important to the experimentalist, who is constrained in the

use of boundary materials of finite thermal conductivity and heat capacity. Sparrow

et al. extended the linearized analysis to include a constant heat flux condition at a

rigid boundary, as well as a Newton's law of cooling condition at both free and rigid

upper boundaries. The critical Rayleigh number for the onset of convection, hereafter

referred to as the first critical Rayleigh number, for a constant heat flux condition was

found to be significantly lower than for the constant temperature condition. This

result is intuitively correct as an isothermal boundary would damp thermal

perturbations, and thus, stabilize the stagnant fluid. For both free and rigid surfaces,

the first critical Rayleigh number was found to increase monotonically with increasing

heat transfer coefficient at the boundary and asymptotically approached the values

predicted for the constant temperature boundary conditions. Again, this is intuitively

correct as an increased heat transfer coefficient at the surface would tend to damp

thermal perturbations more efficiently.

Sparrow et al. also noted that the critical Rayleigh numbers were higher for

an upper rigid surface than for an upper free surface by approximately 600 for all

values of heat transfer coefficient. For reference, the critical Rayleigh number for a

laterally unbounded fluid layer between upper and lower rigid walls at constant

temperatures is 1707.8. For a free upper surface at constant temperature, the critical

Rayleigh number is 1100.7.






16
The preceding results are particularly applicable to the Bridgman melt-growth

system. Typically, Bridgman crystals are grown with a free upper surface. By

introducing a rigid surface at the top of the melt, by floating a thin fused silica disk

on the surface for example, the fluid would not only be more resistant to transitions

to higher dynamic states (e.g., oscillatory and turbulent regimes) but would also be

subject to additional viscous slowing near that surface. The effect would not be as

dramatic as in the case of infinite horizontal fluid layers, however, since the upper

surface in the Bridgman cell makes up a rather small portion of the total surface area.

A plethora of additional work has been completed in characterizing flows in

infinite horizontal fluid layers. Non-linear perturbation analyses have been used to

test the stability of various flow patterns as well as to test for the existence of other

critical and sub-critical dynamic transitions. These studies are quite relevant in

atmospheric sciences, but I shall diverge here and examine some of the subsequent

work relating to thermally driven flows in laterally bounded fluid layers (primarily

right circular cylinders). Solutal effects will then be discussed in relation to Bridgman

melt-growth.

Early experimentalists soon began to find that approximating infinite

horizontal fluid layers in finite structures presented interesting problems. The

cellular structure expected was often dominated by flow patterns characteristic of the

shape of the bounding side walls. Koschmieder [34], in approximating an infinite

fluid layer in both rectangular and circular dishes, observed rolls of rectangular and

circular shape, respectively. Stork and Miller [35,36], in well-controlled

experiments, made similar observations. The results of theoretical analyses on the

effects of side walls on the preferred flow plan agree well with these experimental






17
observations. Davis [37] performed a linear stability analysis taking into account

the side walls in a rectangular geometry and found that the patterns observed by

Koschmieder were indeed stable near the first critical Rayleigh number. Davis'

analysis was not well-posed, however, in that slip was assumed at two of the four side

walls to facilitate the solution of the problem. This assumption was proven by Davies-

Jones [38] in later work to be valid only for certain rectangular aspect ratios, those

which were studied by Davis. Davies-Jones numerical results agreed very well with

the reported results of Davis. Segel [391 used a modified perturbation analysis,

based on non-linear predictions, for a rectangular field to support the predictions of

Davis.

Charlson and Sani [40] later performed a thorough analysis of flows in

shallow fluid layers heated from below in cylindrical containers. The flow patterns

in these cylindrical geometries, as observed by Koschmieder [34] and Stork and Miiller

[36], consist of concentric toroidal rings, the number of which depends on the aspect

ratio of the fluid domain. Charlson and Sani analyzed the perturbation equations by

recasting them in a variational formulation and applied the Rayleigh-Ritz method to

approximate the solution. They were able to calculate upper and lower bounds to the

first critical Rayleigh number as well as determine the number of toroidal rolls for

aspect ratios, 0 = H/R where R is the radius of the cylindrical container, ranging from

0.1 to 2. Pellew and Southwell [30] and Zierep [41] had previously attempted linear

analyses of the cylindrical case, but assumed "slip" walls in order that the method of

separation of variables could be used to solve the differential equations. The solutions

consequently violated the continuity equation for realistic cases in which no-slip

boundaries are present. Ostrach and Pneuli [42] solved the corresponding linear






18
differential equation for the vertical component of velocity to obtain an upper bound

to the first critical Rayleigh number, but an incorrectly specified boundary condition

instead lead to a predicted lower bound. Sherman and Ostrach [43] subsequently

published an analysis intended to establish a lower bound to the first critical Rayleigh

number. The method of Charlson and Sani, however, has given the most reliable

predictions of the first critical Rayleigh number thus far. They performed the

calculations assuming fixed-temperature upper and lower boundaries for two cases,

insulating side walls (dT/dr = 0) and conducting side walls (T(at r=R) = Tw). The

predicted critical Rayleigh numbers for a given aspect ratio are, in general, lower in

the case of insulating side walls than conducting side walls. This, again, is due to the

damping effect of a highly conducting surface on the thermal perturbation field in the

fluid. Also, for decreasing aspect ratio, the critical Rayleigh number can be seen to

approach the well established value of 1707.8 for a laterally unbounded fluid layer.

In an extension of their original work, Charlson and Sani [44] investigated

the conditions for the onset of convection in cylindrical geometries for aspect ratios

greater than unity. This linear stability analysis accounted for the possibility of three-

dimensional, non-axisymmetric flow states. The estimates for the lower bounds to the

first critical Rayleigh number are improved over previous calculations by several

authors [45,46,47,48]. Additionally, a transition is predicted in the initial

dynamic state from an axisymmetric to a non-axisymmetric flow as the aspect ratio

is increased above P = 1.23 for insulating side walls and p = 1.64 for conducting side

walls. Fixed-temperature upper and lower boundaries were assumed. The existence

of the non-axisymmetric flow state for larger aspect ratios has been established

experimentally in several studies. Ostroumov [49], Slavnov [50], and Slavnova






19

[51] observed the non-axisymmetric state in transparent fluids of high Prandtl

number. Miller, Neumann, and Weber [52] reported observing the axisymmetric

flow state at P = 1.0 and a non-axisymmetric pattern at P a 2.0 for both water (Pr =

6.7) and liquid gallium (Pr = 0.02).

Miller, Neumann, and Weber's observations are well-documented in the case

of the transparent fluid, water, as they were able to inject visual tracers in the flow.

Thermocouples attached to the outer surface of the ampoule were employed to infer

the flow pattern in liquid gallium, however, and extremely well-controlled thermal

conditions must have been maintained in order to extract accurate thermal data. In

high-temperature crystal growth, this method has not been used effectively to

determine flow patterns under growth conditions. Miller, Neumann, and Weber were,

however, able to infer the level of the dynamic state (i.e., steady, oscillatory, or

turbulent) from thermal measurements in the vertical Bridgman melt-growth with

top-seeding of GaSb. Steady temperature measurements indicated steady flow, while

periodic and non-periodic temperature fluctuations indicated periodic-oscillatory and

turbulent flows, respectively.

The general dynamic states which occur in a typical vertical Bridgman melt-

growth are steady, periodic-oscillatory, and turbulent flows. The stagnant state is not

included since it has never been realized in ground-based growth. As previously

mentioned, stagnation, and consequently diffusion-controlled growth, is presently

feasible only in a zero-gravity environment. Theoretically, the first critical Rayleigh

number delineates the boundary between stagnant and steady flow for vertically

oriented density gradients. In application, however, steady flow is realized at sub-

critical Rayleigh numbers due to horizontal density gradients. This has been observed






20
even in well-controlled experiments [36,42]. In this case, the existence of the first

critical Rayleigh number is no longer valid since it defines the onset of convection

from a stagnant fluid. A change in dynamic state is still expected in the vicinity of

the first critical Rayleigh number, however, but is defined as a transcritical change.

It can be shown through the Navier-Stokes equations that conditions of fluid

motionlessness require that density gradients be oriented only in the vertical direction

[19]. For zero-velocity, the combined Navier-Stokes equations given in Equation (1-2)

become


-VP + R CF + TF=r = 0 (1-7)
Sc Pr



By taking the curl of Equation (1-7), the pressure field drops out and


Vx R CFS + Vx RaTF O. -8)
Sc Pr



Assuming a conservative body force, Equation (1-8) becomes

Ra -- Ra, --..
s(VC x F) + (VT x F)= 0 (1-9)
Sc Pr


and since there is no fixed relation between the concentration and temperature

gradients, Equation (1-9) can, in general, only be valid for concentration and thermal

gradients oriented in the vertical direction. So, the trivial solution of zero-velocity will

not satisfy the Navier-Stokes equations in the presence of horizontal density

gradients.






21
The second critical Rayleigh number is generally defined as that value of the

Rayleigh number at which the flow changes from steady to periodic-oscillatory. This

point of transition cannot be predicted through linear stability analyses and as yet has

not been predicted through non-linear analyses. Experimental observations, however,

have shown a dependence of the second critical Rayleigh number, Rad, on both the

Prandtl number and aspect ratio for fluids heated from below. Krishnamurti [53]

performed experiments with fluids having Prandtl numbers ranging from 0.71 (air)

to 8500 (silicone oil) in a layer of large lateral extent bounded by rigid surfaces on top

and bottom. Ra. is shown to increase with Pr up to Pr = 50, above which Ra. =

55,000 and is Pr independent. A similar result was reported by Silveston [54]. At

low Prandtl numbers, Rad approaches RaE1, and Ra.L was shown experimentally by

Krishnamurti to be Prandtl number independent.

Krishnamurti also observed higher-order dynamic transitions with increased

Rayleigh number. As Ra was increased above Rad, a critical point was reached at

which period-doubling occurred, that is, the frequency of thermal oscillations was

doubled. As Ra was increased further, a regime of non-periodic turbulent convection

was encountered. These last two transitions show the same trend with increasing

Prandtl number as the second critical Rayleigh number. The experimental results of

Miller, Neumann, and Weber [52] with liquid water (Pr = 6.7) and liquid gallium (Pr

= 0.02) show the dependence of Ra1, Rae, and the onset of turbulence on the aspect

ratio in vertical cylinders. As was predicted by the linear theory of Charlson and Sani

[44], Rae1 is observed to increase with increasing aspect ratio. Ra, and the onset of

turbulence show the same trend. In agreement with the results of Krishnamurti, Rad

occurs at much higher values in the case of water (Pr = 6.7) than in the case of liquid






22
gallium (Pr = 0.02) in vertical cylindrical geometries. For example, at an aspect ratio

of 4, Ra8 in gallium was -5x104 and in water was -107.

Knuteson [55] also performed experiments to measure the onset of oscillatory

flow with surface temperature measurements for vertical cylinders of liquid tin (Pr =

0.01). The values of Ra8 reported for several aspect ratios are slightly higher than

those reported by Miller, Neumann, and Weber [521 for liquid gallium (Pr = 0.02) at

aspect ratios ranging from 3.3 7.0, although uncertainty in the measurement of the

vertical temperature gradient could account for this offset. One would expect Ra, for

tin to be slightly lower than for gallium based on the trend of Rae with Pr from

Krishnamurti's work. Knuteson also studied the frequency of oscillations as a

function of aspect ratio and Rayleigh number. Frequency-doubling and turbulence

were also noted in these cylindrical geometries.

An interesting phenomenon observed for the first time in 1983 by Miller [56]

is the appearance of steady flow at Rayleigh numbers above Rad. In his experiments,

Miller grew a Te-doped InSb crystal in a centrifuge (in the thermally unstable

orientation) to allow for variation in the magnitude of the body force. The Rayleigh

number was varied by changing the angular acceleration of the ampoule during

growth. The results showed that steady convection was present initially as the

angular acceleration was increased. A critical centrifugal acceleration was eventually

reached, however, at which oscillatory flow ensued. This was evident both in

temperature measurements and in compositional striations in that region of the final

crystal. As the centrifugal acceleration was increased further, the thermal oscillations

ceased, and steady flow was again realized. One might be inclined to believe that this

relaminarization, as Miller terms it, is a centrifuge effect, but Muller also reports a






23
similar observation in liquid water at normal earth gravity. These are presently the

only observations of this phenomenon reported in the literature.

Solutal convection in Bridgman crystal growth does not manifest itself in the

same manner as thermal convection. Axially directed thermal gradients are generally

linear, and the resulting density gradients are linear as well. The driving force for

convection is then identical at all points in the fluid. On the other hand, the solutal

gradient appears as a result of solute rejection at the growth interface and decreases

exponentially with distance from the interface due to diffusion into the bulk melt. The

driving force for convection is consequently non-uniform. When the two effects are

combined, as must be the case in multicomponent crystal growth, the conditions for

and the nature of convection become quite complex. This type of convection is termed

double-diffusive convection owing to the difference in diffusion velocities of heat and

solute.

The simpler case of fluid layers having both linear thermal gradients and

linear solutal gradients was examined for conditions of stability by Stern [57]. This

case was modelled after an oceanographic phenomenon in which both the temperature

and saline concentration of the ocean's water increase with height. The water is

considered thermally stable but solutally unstable. Stern recognized that the

condition for the onset of convection was not determined by the density gradient itself,

but rather by the combined Rayleigh number, Rag RaT. That is, convection can

result with a net stabilizing density gradient. This type of motion has been termed

double-diffusive convection because it is enabled by the disparity in the diffusivities

of heat and salt. A detailed linear analysis of this problem is given by Veronis

[58,59].






24

In the case of double-diffusive convection in directional solidification of single-

phase binary mixtures, the effects of convective onset are discussed in several papers

[60,61,62,63,64]. Since the destabilizing solutal gradients are developed

near the solid-liquid interface during solidification, the onset of convection is observed

to strongly affect the morphological stability of the interface. Non-uniformities in

composition and crystallographic defects resulting from thermosolutal (double-

diffusive) convection are, consequently, of primary concern to crystal growers.


Experimental Approach


A significant amount of flow visualization in Bridgman geometries has been

reported, although primarily for transparent fluids such as water and silicone oils.

These fluids have different physical properties than the liquid metals and semicon-

ductors of interest here, typically having Prandtl numbers at least two orders of

magnitude larger than the metals. It is not known whether the flow states observed

in high-Pr fluids model the flow states in low-Pr fluids, and it has already been shown

that the critical transitions in dynamic state occur within different ranges of Rayleigh

number depending on the Prandtl number of the fluid. It is the objective of this

research to develop a flow visualization technique which can be used to examine both

critical transitions in dynamic state and corresponding flow patterns for low-Pr fluids

in the vertical Bridgman melt-growth configuration.

The Microgravity Sciences Group at the NASA Langley Research Center, under

whose auspices this work was completed, is currently interested in vertical Bridgman

melt-growth of PbxSnl.xTe, a material which has applications in optical detection of

wavelengths in the infra-red spectrum as well as in making tunable diode lasers.






25
Directional solidification of this pseudo-binary material results in compositionally

inhomogeneous crystals because of rejection of SnTe at the growth interface.

Consequently, growth of this material with bottom-seeding results in a thermally

stable, but solutally unstable melt due to build-up of the lighter component, SnTe,

near the growth interface. Conversely, top-seeding results in a solutally stable, but

thermally unstable melt. PbxSnl.xTe is, therefore, a good model fluid for the study of

natural convection phenomena in the vertical Bridgman system. A flow visualization

technique which can be applied to the PbxSn-xTe system would, therefore, allow for

a better understanding of the effects of various convective motions on crystal

properties.

An electrochemical technique for introducing, extracting, and monitoring dilute

concentrations of oxygen in liquid metals and semiconductors is presented as a viable

technique for visualizing flow in Bridgman crystal growth. In this technique, the

oxygen anion-conducting properties of a dense, inert ceramic, yttria-stabilized-zirconia

(YSZ), are applied in the construction of multiple electrochemical cells in the growth

sample. The melt serves as one electrode for each of the cells in the structure, while

independent reference electrodes are maintained at the outside surfaces of the

electrolyte. By fashioning sections of the ampoule walls out of the YSZ, oxygen can

be introduced or extracted from surfaces of the fluid volume by imposing an electric

potential of required polarity between the melt and the reference electrode.

Alternatively, the concentration of oxygen at surfaces of the fluid volume can be

monitored by measuring open-circuit EMF's between the melt and reference

electrodes. Chapter 2 is devoted to a discussion of both the properties of YSZ and the

theory describing the operation of a solid-state electrochemical cell.






26
The oxygen serves as a dilute tracer element with limited solubility in liquid

metals. Because of its limited solubility, the oxygen was originally thought to be

completely unobtrusive to the flow. This does appear to be true under certain

circumstances, and a full discussion of its effect on the flow will be given in Chapters

3 and 4.

Since this flow visualization technique is in the formative stages, the simplest

possible flow scenario was chosen for this research. Pure tin was employed as a model

fluid so that purely thermally-driven flow could be established. Lead and tellurium

were not used because of safety considerations in heating the materials to

temperatures well above 500C. The application of the techniques outlined in this

work can be generalized, however, to multicomponent melts with only minor

modifications for safety reasons.

To use oxygen as a tracer in the Bridgman simulations, its molecular

diffusivity in tin must be known. An electrochemical cell specifically designed to

measure the molecular diffusivity of dilute oxygen in liquid tin is presented in

Chapter 3. This design is superior to cell designs used in similar studies in the past

because it minimizes thermal gradients in the melt which cause natural convection.

A general discussion of difficulties and misconceptions in experimental diffusivity

measurements will also be given in Chapter 3.

An approach similar to that used in the oxygen diffusivity measurements is

used to study the dynamics of flow in simplified Bridgman simulations. The method

involves measuring the effective rate of mass transfer of the tracer across the

Bridgman cell for various applied temperature gradients. Here, the trends in mass

transfer rate with changing Rayleigh number are indicative of the dynamic evolution






27
of the melt. This technique is discussed in Chapter 4. An extended approach to flow

visualization involves the design of a containing ampoule which will enable

determination of actual flow orientations within the Bridgman cell. This can be

accomplished by using multiple electrochemical detectors on the surface of the

ampoule to observe the migration of oxygen pulses introduced at a known location to

the surface of the melt. The feasibility of this technique is tested and discussed in

Chapter 5.











CHAPTER 2
SOLID-STATE ELECTROCHEMICAL MEASUREMENTS


Introduction


The flow visualization technique described in this work is based on the

electrolytic properties of stabilized zirconium dioxide. The solid-oxide electrolyte

actually serves a dual purpose in these experiments, structural and electrochemical.

The material's rigidity, chemical inertness, and impermeability to atmospheric gases

make it ideal for containment of high temperature liquid metals and semiconductors.

Stabilized zirconia can be cast into a variety of shapes for virtually any application.

The electrolytic properties of stabilized zirconia are manifested in the high

conductivity for divalent oxygen anions under certain conditions of temperature and

oxygen partial pressure. This makes it an ideal medium through which the oxygen

tracer may be added or removed from the metallic or semi-metallic melt.

This chapter is devoted to a discussion of the stabilized zirconia electrolyte, its

applications and limitations. The thermodynamic theory describing the relationship

between chemical and electrical processes in these electrochemical cells is also

presented. Additionally, attention is given to sources of error in electrochemical

measurements using the solid-oxide electrolytes. The purpose of these discussions is

simply to provide a foundation for the subsequent electrochemical studies in Chapters

3 and 4.






29

Yttria-stabilized Zirconia


Zirconia (ZrO,) can be doped with yttria (Y20) to form a stable solid solution

which ranges from approximately 8 to 50 mole percent yttria [65,66]. This solid

solution (YSZ) is arranged in the cubic (fluorite) structure [67]. The conductivity

in zirconia can be attributed to electrons, holes, and oxygen vacancy defects. Heavily

doping zirconia with rare earth metal oxides such as CaO, Y20,, or MgO is known to

increase the concentration of oxygen vacancies. In the case of yttria, these vacancies

are necessary to maintain charge neutrality due to the valency difference between Zr4'

and Y*. The ionic conductivity of YSZ is consequently increased to a level which

significantly dominates any electronic conductivity, at least over a broad range of

temperature and pressure.

The electrolytic domain of solid-oxide electrolytes is generally defined as the

realm in which the ionic transference number, t., is greater than 0.99. The

transference number is given by


S= (2-1)
0m + 0k + im



where ao is the ionic conductivity, ao is the electronic conductivity, and ah is the hole

conductivity. The sum of all transference numbers is unity. For a given dopant

concentration, the electrolytic domain is determined by the temperature and oxygen

partial pressure. The effects of each of these variables will now be discussed in

relation to the electrolytic nature of the material.

The ionic conductivity of YSZ initially increases with increased levels of doping.

A maximum is reached, however, at concentrations of yttria between 8 and 10 mole






30
percent, corresponding to the monoclinic-cubic solid solution phase boundary [681.

Increasing the yttria composition beyond 10 mole percent results in a decrease of ionic

conductivity. The corresponding activation energy for ionic conduction is observed to

reach a sharp minimum at the phase boundary as well. The optimal doping level for

YSZ appears to be the minimum concentration of yttria necessary to produce the

fluorite structure, -8 mole percent.

Oxygen partial pressure has a strong influence on the electron and hole carrier

concentrations. Under conditions of low oxygen partial pressure, oxygen is removed

from the lattice according to the equilibrium equation


Oo Vo + 2 e- + (Po) (2-2)


where Oo is an oxygen atom in its designated lattice site, Vo** is an oxygen vacancy

site, e- is a mobile electron, and O, is gaseous oxygen. Two electrons must be

liberated to accommodate each vacant oxygen site, and consequently, the electronic

conductivity due to free electrons becomes important at low oxygen pressures. At the

other extreme, high oxygen partial pressures will force oxygen into the lattice

according to the equilibrium equation


102(P) + 0 + 2n* (2-3)

where n* is a mobile hole, having an equal but opposite charge of an electron. Here,

the filling of each previously vacant oxygen site must be accompanied by the liberation

of two holes. Thus, at high oxygen pressures, the electronic conductivity due to holes

becomes important. The electrolytic domain of solid-oxide electrolytes must, then,

exist for intermediate oxygen partial pressures. The width of the active pressure






31
window will, in general, be a function of temperature since the electron and hole

carrier concentrations are temperature dependent.

The electrolytic domain of YSZ is defined as the range of temperatures and

oxygen partial pressures over which the ionic transference number is greater than

0.99. That is, the electronic conductivity within the electrolyte is less than 1% in this

range. If the ionic and electronic conductivities are known as functions of temperature

and pressure, then the limits of the electrolytic domain can be determined. At high

oxygen partial pressures, conduction by electrons is insignificant and the upper

pressure boundary of the electrolytic domain is defined by


t = 0.99 = o (2-4)
Oi + oim



Similarly, at low oxygen partial pressures, conduction by holes is insignificant and the

lower pressure boundary of the electrolytic domain is defined by


S= 0.99 =- o (2-5)
O, + o.



The electronic conductivities of YSZ were investigated by Kleitz et al. [69]. The

particular samples were 9 mole % yttria-doped zirconia, and the temperature range

investigated was 1170 1550C. These results will be extrapolated to lower tempera-

tures in order to estimate the electrolytic domain in the temperature range used in

the present work (550 800C). The equations given by Kleitz et al. for the electron

and hole conductivities are, respectively,









I
a, =5.5xlOP10 P 4e) (2-6)


I
.= L4P4eaxp( (2-7)


where o, and oh are in (Qcm)'1, the Boltzmann constant, k, is in eV/K, and Po, is in

atm. The ionic conductivity of YSZ electrolytes was studied by Strickler and Carlson

[70] and Schouler et al. [71], among others. The two references mentioned give

results in reasonable agreement, and the data of Strickler and Carlson, given by the

following equation, shall be used here:


o. = 115exp 78 (2-8)



The temperature dependence of the upper and lower pressure limits (i.e., the

pressures at which Equations (2-4) and (2-5), respectively, are satisfied) of the electro-

lytic domain can be calculated from Equations (2-4) (2-8). The functions describing

these limits are given by the following equations:




InP = 2.88 9.956 (2-9)
kT


n Po- = 52.27 11.76 (2-10)
kT







33
where Po0 is the upper pressure limit at which hole conduction becomes important,

and Po" is the lower pressure limit at which electron conduction becomes important.

A graphical representation of these limits is shown in Figure 2-1. A lower tempera-

ture limit also exists for the YSZ electrolyte. It can be noted from Equation (2-8) that

the ionic conductivity decreases with temperature. This is due to a decrease in the

mobility of oxygen ions as the thermal lattice energy is lowered, that is, the

distribution of ions with sufficient energy to overcome the electrostatic binding forces

becomes less. A critical lower temperature will eventually be reached at which the

ionic carrier density will be insufficient to maintain an appropriate ionic conductivity.

This lower temperature limit is generally observed to be in the range of 550 600*C.

Operating outside of this electrolytic window in experiments involving oxygen

concentration cells will result in uncertainties due to non-equilibrium conditions.

Consequently, it is important to choose oxygen atmospheres which fall within the

electrolytic domain for a given operating temperature. The oxygen concentration cells

used in these studies maintained oxygen atmospheres which fall well within the

electrolytic domain at the chosen operating temperatures (550 800C). Figure 2-2

shows the Gibb's energies of formation [72] of the various oxides used in the

construction of the experimental cells. The equilibrium oxygen partial pressures may

be read from the dashed oxygen isobars. A copper/copper(I) oxide reference system

was used in some of the experiments, while a platinum/air reference system was used

in others. Each of these can be seen to provide oxygen atmospheres which ensure

proper electrolytic behavior of the electrolyte.














T(K)


0.8 0.9 1.0
Temperature-" x 1000


Electrolytic domain (t. > 0.99) of 9 mole % yttria-stabilized zirconia.


Figure 2-1.


















i I I I I I I I I I I I I i i 1 1 11i i I 1 i ( tm )
^- PQ, (otm)

S-25 10-5


.
0 -c CuO(s)

010-




-050

0 --- ---- 10-25






1 100
-150 o- f n '(ll 0- 1
--- SiO (s)
-175 =---- --- -. 10

5 -200 10-1



-225 10-40
10-50 45
-250 1 1 1 I I I f t.. I 1 1
500 600 700 800 900 1000
Temperature (OC)












Figure 2-2. Gibb's energies of formation of the most stable oxides of materials
used in the electrochemical cells [72]. The dashed lines are oxygen
isobars.






36
Oxygen Concentration Cell


The oxygen concentration cell is a thermodynamic system consisting of a solid-

oxide electrolyte separating two electrode compartments having independent oxygen

chemical potentials. Each compartment must have a metallic electrode contacting the

electrolyte interface to physically couple the chemical and electrical processes in the

cell. In operation, a reversible open-circuit electric potential, E, is developed across

the electrolyte which is related to the variation in oxygen chemical potential, go,

across the electrolyte. Wagner [73] derived this relationship which is stated as

Po
E = -1 F tdpo (2-11)
nF,



where n is the valence of oxygen in the electrolyte (n = 2), F is the Faraday constant,

and IO' and Eo" are the oxygen chemical potentials at each electrode (Oo' < Ao"). The

integration can be carried out when the variation of ti with J is known. As shown

earlier, ti is a function of temperature and oxygen pressure, and the variation of t.

with these independent variables must be determined experimentally. In fact, a fair

amount of experimental work has been completed in an attempt to characterize the

conductive properties of solid-oxide electrolytes (Kleitz et al. summarize many of these

studies [69]). The results of the various investigations are often in extreme

disagreement, however, for a number of reasons. First, the stabilized oxides used in

experiments are of varied compositions and have been synthesized by a number of

different processes. The single-crystal grain sizes, in particular, can vary

substantially from one process to another. Ionic and electronic transport mechanisms






37
within and across the grain boundaries will then have a significant effect on the

overall charged carrier conductivities. Second, even for a given composition and grain

structure, the carrier conductivities may not remain constant over time. Extended use

of stabilized oxide electrolytes at high temperatures results in changes in the material

such as (for YSZ in particular) segregation of yttria-rich layers at the grain

boundaries, formation of tetragonal ZrO, [74], and impurity segregation at the grain

boundaries [75]. These temporal changes due to annealing rule out the effective use

of Equation (2-11) in its general form, and the need for mathematical simplification

is noted.

The most obvious simplification of Equation (2-11) is to take advantage of the

invariance of t.. within the electrolytic domain. As long as the electrolyte is

maintained within the previously described range of temperature and oxygen partial

pressure, ti. can be assumed to be constant and equal to unity. For the case t, >

0.99, Equation (2-4) reduces to


AG = ~o Lo" = -2FE (2-12)



In terms of oxygen activity, ao, Equation (2-12) is given by


RTIn a0 -2FE (2-13)
ao"

where R is the universal gas constant, and T is the absolute temperature.

The oxygen concentration cell generally uses a reference electrode and a

working electrode. The oxygen activity in the reference electrode is fixed by using a

metal/metal oxide chemical system or a gas mixture of known oxygen composition.






38
In the case of the diffusion measurements and flow visualization cells used in this

work, the working electrode is the tin melt. Having chosen an appropriate reference

electrode, the activity of oxygen in the tin can be calculated directly from Equation (2-

13) with experimental electromotive force (EMF) measurements.

The EMF measurements in oxygen concentration cells are subject to experi-

mental errors, however. A general discussion of the sources of these errors will be

given here as they relate to the experiments in this study. The first source of

measurement error results from non-isothermal operation of the electrochemical cell.

The previous equations relating the EMF to the oxygen chemical potential difference

were derived under the assumption of isothermal and isobaric conditions. Additional

terms must be included, however, in the case of non-isothermal cells. Goto and

Pluschkell [76] presented the following simplified equation relating the EMF to

oxygen chemical potentials at electrodes of different temperatures:


E = 4'F [o(TP.' ) P po(Tlto + a[ T"]. (2-14)



Goto and Pluschkell define a as a constant parameter which is related to the partial

molar entropy and heat of transfer of oxygen ions in the electrolyte and the partial

molar entropy and heat of transfer of electrons in the electrodes. It is essentially the

overall Seebeck coefficient of the electrochemical cell. This Seebeck coefficient is not

constant, however, but is a function of oxygen partial pressure in the cell. Fischer

[77] measured a(Po,) for 9 mole percent yttria-stabilized zirconia with dual platinum

electrodes in the temperature range 687 1037C and reported the following empirical

equation,









a(P = 0.492 0.02201n5 (2-15)


where the units of a are (mV/C) and the oxygen pressure is in mmHg. In a separate

experiment, Fridman et al. [78] measured the Seebeck coefficient of 10 percent

yttria-stabilized zirconia at 1175*C in air and obtained a value of -0.47 mV/*C,

agreeing very well with the value of-0.492 mV/C from Fischer's results. The order

of magnitude of a from these experiments is characteristic of most stabilized zirconia

electrolyte materials [79]. The contribution of the Seebeck coefficient of the electrode

materials to the overall Seebeck coefficient is very small since the partial molar

entropy and heat of transfer of electrons in metals is small in comparison to the

identical properties of oxygen ions in the electrolyte.

Further errors in EMF measurements may result when currents are passed

through the electrochemical cell by applied voltages different from the equilibrium

open-circuit EMF. Passing currents through the cell can result in electrode and

electrolyte polarization. For large currents, IR drop in the electrode and extension

wires leading between the cell and current source can be important. For this reason,

electrode materials of high electrical conductivity are generally preferred. Polarization

may also occur at the electrode-electrolyte interface due to the build-up of oxygen

resulting from kinetic limitations of the half-cell reactions. Electrode polarization may

also occur in an electrode depleted of oxygen, e.g. when diffusion of oxygen to the

interface is the limiting kinetic factor. IR drop through the electrolyte due to the

resistance to flow of ions in the electrolyte lattice may also become important for large

current densities. In general, large currents in an electrochemical cell lead to the






40
production of an irreversible EMF, and the equations relating the reversible EMF as

given previously will not be valid.

The flow visualization experiments which will be described in Chapter 4 are

especially subject to the EMF measurement errors described for non-isothermal cells.

In these experiments, the effective diffusivity of oxygen across a tin melt is calculated

for various Rayleigh numbers. The temperature gradients which are developed across

the electrolytes range up to 120C, for which the corresponding Seebeck voltage is

approximately 6 mV. However, since the diffusivities are calculated from the slope

of the measured EMF versus time curve, the temporally-constant Seebeck voltages do

not enter into the calculation. Also, EMF measurement errors due to electrode

polarization in the diffusivity experiments do not present a problem because of the

small currents developed in the potentiostatic removal of oxygen.


Summary


Electrochemical measurements involving solid-oxide electrolytes provide an

effective means of determining thermodynamic as well as kinetic properties of oxygen

in liquid metals. The accuracy of these measurements is, however, dependent upon

maintenance of proper experimental conditions. The solid-oxide electrolytes are

known to function properly only under certain conditions of temperature and oxygen

partial pressure. Additionally, errors resulting from non-isothermal operation of the

cells and polarization under high current loads must be accounted for in the analysis

of EMF data.

A mathematical relationship has been presented which relates the reversible

EMF developed across the oxygen concentration cell to the chemical potential






41

difference of oxygen. This relationship will be used in the following chapters to gain

insight into the kinetics of oxygen transport through liquid tin. Chapter 3 focuses on

measurement of the binary diffusion coefficient of dilute oxygen in liquid tin. Chapter

4 then extends the methods used in Chapter 3 to measure the combined diffusive and

convective mass transport of oxygen in Bridgman crystal growth simulations using

liquid tin as a model fluid.












CHAPTER 3
OXYGEN DIFFUSIVITY IN LIQUID TIN


Introduction


The binary diffusion coefficient of dilute oxygen in liquid tin is investigated by

using an oxygen concentration cell. The flow visualization experiments discussed in

Chapter 4 are based upon the measurement of effective diffusivities of oxygen across

a tin melt which is subjected to thermal gradients. The resulting effective diffusivity

is a measure of the total rate of mass transfer of oxygen due to diffusion and

convection. Since it is the overall effect of convection on mass transfer within the melt

that is of interest, the diffusion effects must be subtracted out. This can be done since

the concentration field depends linearly on the velocity field. Hence, the first goal of

this study is to experimentally determine the molecular diffusivity of oxygen in liquid

tin in the absence of convection.

In order to remove convective effects in these diffusivity measurements,

thermal gradients which induce natural convection must be eliminated or at least

substantially minimized. Previous investigations of oxygen diffusivities in liquid tin

[80,81,82,83], as well as in other liquid metals, have generally not taken suffi-

cient care to eliminate thermal gradients. On occasion, researchers have even

imposed small thermal gradients (hot on top) in order to "stabilize" the melt; however,

this technique has adverse effects in the creation of small horizontal thermal

gradients due to mismatching of thermal properties of the melt and its container.






43
Otsuka and Kozuka [84], in particular, presented oxygen diffusivities in liquid lead

under these conditions. They imposed a 1.5C/cm vertical temperature gradient across

the electrochemical cell used in the measurements. Hurst [83] modelled this case

numerically and observed low-level convection which was significant enough to cast

doubt on the reported results. A detailed discussion of the effects of thermal gradients

in diffusion measurements is given by Hurst [83].

In addition to thermally-driven convection, solutal effects must also be

considered in diffusion experiments. In particular, the orientation of the diffusion cell

must be assessed in terms of the direction of the resulting concentration gradient with

respect to the direction of gravity. Ramanarayanan and Rapp [80] and Hurst [83]

each employed radial diffusion schemes, where the resulting oxygen concentration

gradient is perpendicular to the direction of gravity. As discussed in Chapter 1, this

orientation causes unconditional hydrostatic instability, resulting in natural

convection. The other orientation, used previously by Otsuka, Kozuka, and Chang

[82] and in the present work, is axial diffusion. In this case, the resulting oxygen

concentration gradient is aligned with the direction of gravity. The presence of a

driving force for convection is then conditional upon the solutal Rayleigh number and

aspect ratio for a given experiment.

Consequently, a need was recognized for improved design of oxygen concentra-

tion cells for diffusion measurements. An improved cell design is presented here for

making oxygen diffusivity measurements under isothermal conditions. The mass

diffusivity of oxygen in liquid tin is then given as a function of temperature and

compared to the results of the previous investigators.






44
Experimental


A schematic of the experimental cell is given in Figure 3-1. As shown, the cell

is actually composed of two oxygen concentration cells which share a common working

electrode in the tin melt. Copper/copper(I) oxide reference electrodes are used to

establish a known oxygen potential at the reference side of the electrolyte disks.

Copper was chosen for several reasons: availability and low cost, machinability, and

its physical properties. Copper's high electrical conductivity makes it an ideal

electrode material with high resistance to polarization. The high thermal conductivity

of copper helps to minimize thermal gradients which will induce convection in the

melt.

The cylindrical cell was designed for axial diffusion measurements. The

purpose of this is to align the concentration gradients, which are developed over the

course of an experiment, with the gravity vector. Since the side walls of the diffusion

cell are impermeable to oxygen, horizontal concentration gradients are not a

consideration. As described in Chapter 1, vertical density gradients developed as a

result of solutal gradients may or may not be of sufficient magnitude to cause an onset

of convection. The criterion for the onset of convection is the magnitude of the solutal

Rayleigh number. Certainly, conducting the experiment so that the fluid will be less

dense on top (corresponding to removal of oxygen at the bottom) will avoid develop-

ment of solutally-driven convection. However, removing oxygen from the top creates

an unstable density distribution, and convection may ensue depending on the solutal

Rayleigh number. It is not possible to calculate Ras apriori since no data on the

coefficient of solutal expansion for the tin-oxygen system is available in the literature.

Verhoeven [19] reported that the low oxygen concentrations of oxygen in liquid metals
















Rheniun Extension
Wire








Alunina Overflow
Tube


Copper/Copper Oxide
Reference


- Copper Lead
Wires



Fused Silica
Spacer



Copper
Electrodes





Fused Silica



- Tin Melt


Figure 3-1.


Schematic diagram of the experimental cell used for oxygen diffus-
ivity measurements.






46
will not produce density gradients large enough to surpass the first critical Rayleigh

number. However, this statement can be shown to be incorrect. The solutal Rayleigh

number is given by


H ap 1
& p, (3-1)
RaS = &
Do.



where ap/az is the density gradient and p. is the mean density of the fluid. The

required density gradient for the onset of convection can be estimated by inserting

typical parameters: Ra.1 = 2500 (for p = 1), H = 0.5 cm, g = 980 cm/sec2, Do = 6x10"

cm2/sec, and v = 1.65x103 cm2/sec. The value of the minimum density gradient for

instability thus obtained is


( ap 1 -4xlO %Icm (3-2)
az p.)



which is not an unreasonable variation in a liquid metal system [19]. The diffusivities

are calculated in this study for removal of oxygen from both the top and bottom faces

of the fluid cell to test the effect of the direction of diffusion on the experimental

diffusivities.


Cell Construction

The tin sample was contained within a fused silica cylinder (General Electric

Co., Quartz Products Division, Cleveland, OH) sandwiched between two yttria-stabi-

lized zirconia disks (ZIRCOA Products, Solon, OH). The quartz cylinders used in this






47

series of experiments were varied in height from 0.389 cm to 0.750 cm and measured

0.729 cm ID and 0.945 cm OD. The YSZ disks were 0.953 cm diameter by 0.158 cm

thick with a reported composition of 8 wt.% yttria (4.5 mole %). The disks were

cemented to the fused silica cylinder with Aremco 571 magnesia-based ceramic

adhesive (Aremco Products, Inc., Ossining, NY) after placing the tin sample inside.

The tin sample itself was cast from zone-refined bars of 99.9999% purity (Cominco

American, Inc., Spokane, WA), which, after machining, was cleaned and etched with

a 5 volume % bromine in hydrobromic acid solution.

Since the axial diffusion experiment requires that the tin sample be completely

confined and make full contact to the upper and lower electrolyte surfaces, a

modification had to be made to allow for thermal expansion of the liquid sample upon

heating. Consequently, a 1mm diameter hole was ground through the upper

electrolyte disk at its center to allow for escape of excess tin. A 0.318 cm OD alumina

overflow tube (McDanel Refractory Co., Beaver Falls, PA) was cemented, again with

Aremco 571 ceramic adhesive, to the upper surface of the disk to contain the overflow

and keep it electrically isolated from the copper electrode.

The copper electrode components were machined from 2.54 cm diameter bars

of copper (Defense Industrial Supply Center, Philadelphia, PA) of unknown purity.

Electrical contacts were made to the copper electrodes with 0.5 mm diameter copper

wire of 99.9% purity (Johnson Matthey Inc., Seabrook, NH) by drilling a small hole

in the electrode, inserting the end of the wire, and mechanically pressing the junction

to form a press-fit around the wire. The reference electrode system consisted of a 1:1

mole ratio of copper and copper(I) oxide powders (Alfa Products, Danvers, MA). The

copper and copper(I) oxide powders were each 99.95% pure. A 4:1 mole ratio of copper







48
to copper(I) oxide was used initially, but the cell lifetime was quite short due to

reduction of the oxide in the inert environment of the ambient argon stream. The 1:1

mole ratio system exhibited equally good electrical conductivity, however. The powder

mixture was packed loosely into recesses machined into the copper electrode pieces,

and then compressed by inserting the reference side of the electrolyte disks into the

recesses on top of the powder.

The upper electrode consisted of a single piece of copper having a 0.320 cm hole

drilled through it axially to hold the alumina overflow tube. The bottom electrode

consisted of two pieces, an outer cylindrical sheath, 2.54 cm OD, and a smaller

cylinder which recessed the reference system. A fused silica spacer was inserted

between the upper copper electrode and the outer copper sheath to electrically insulate

the two electrodes from one another. The primary purpose of the outer sheath was

to maintain isothermal conditions within the melt.

Electrical contact was made to the tin sample by extending a 0.25 mm diameter

rhenium wire (Johnson Matthey) through the top of the overflow tube. The rhenium

wire was electrically contacted to a copper extension wire by twisting the two together

over a length of approximately 3 cm. The copper extension wire was drawn into a

narrow fused silica capillary tube (General Electric Co.), and the copper-rhenium

contact was forced into the end of the capillary to maintain pressure on the twisted

junction. Resistance measurements of the combined copper-rhenium extension wire

before and after experiments showed no change. The copper-rhenium junction

extended no more than 4 cm above the top of the cell, well within the isothermal

region of the furnace.






49
The temperature of the cell was measured with a type R (platinum/platinum

13% rhodium) thermocouple which was inserted approximately 2 cm into the top of

the outer copper sheath. The accuracy of type R thermocouples is reported to be

0.25% [85].

The cell was contained within a 2.8 cm ID fused silica tube (General Electric

Co.) which was capped at the top by a brass cell head. The brass cell head had four

ports, three of which sealed 0.318 cm diameter feed-throughs with o-ring fittings. The

fourth served as a connection to the vacuum system and also the outlet for the

purified argon stream. Alumina feed-throughs were used to isolate the electrical

connections passing through the cell head. The core of each electrical feed-through

was then plugged with RTV sealant (Dow Corning Corp., Midland, MI). The purified

argon was introduced through one of the alumina feed-throughs. The argon (Air

Products and Chemicals, Inc., Allentown, PA) was purified in two steps. First, the gas

stream was passed over a catalyst to react hydrogen with oxygen to form water, which

was then removed as the gas was passed through a canister containing sodium

aluminosilicate desiccant (Matheson Gas Products, East Rutherford, NJ). Second, the

gas stream was passed through a bed of 800C titanium sponge (Alpha Products). The

final purified argon stream had a measured residual oxygen partial pressure of 1.6 x

10-2 mmHg (see Chapter 5).

The furnace components were Kanthal wound resistance heater blocks

controlled by Eurotherm (Eurotherm Corp., Reston, VA) temperature controllers. An

isothermal liner (Dynatherm Corp., Cockeysville, MD) 30.5 cm in height was inserted

into the furnace core to establish an isothermal region in which the diffusion cell could

be placed. The liner was 3.4 cm ID by 6 cm OD. The cell, 5.6 cm in height, was






50
positioned in the middle of the isothermal liner. The temperature along the liner was

measured by extending a narrow type R thermocouple into the furnace, between the

liner and fused silica tube containing the cell. The temperature did not vary by more

than 0.27*C along the length of the liner. Additionally, another isothermal liner (10.2

cm in height) and heater block were placed on top of the other components to increase

the height above the cell which was heated. It is believed that this may reduce any

conductive heat losses through the connections leading between the cell and the cell

head at the top of the apparatus, as well as limit any radiative heat losses from the

top of the cell. These precautions, as well as the copper sleeve encasing the diffusion

cell, are believed to significantly reduce any thermally-driven convection in the melt

to levels which will not noticeably affect the diffusivity measurements.


Procedure--Transient Diffusion Experiments

The oxygen diffusivity was determined experimentally by a combined

potentiostatic and EMF method. Initially, a uniform oxygen concentration was

established within the melt by pumping oxygen in or out of the tin through an applied

voltage across either or both oxygen concentration cells. Recall, the experimental cell

is a combination of two oxygen concentration cells which share a common tin working

electrode and are represented as follows:



Cu,Cu2O (1) II YSZ II Q in Sn I Re I Cu (I)


Cu I Re I 0 in Sn II YSZ |I Cu,CuaO (2)






51

The apparatus was then left open circuit until the measured EMF's across each cell

were steady and equal. Then, at the start of the diffusion experiment, a zero oxygen

concentration boundary condition was established at one of the tin surfaces (upper or

lower) by applying a large voltage, 1.2 2.0 V, between the two electrodes of the

corresponding oxygen concentration cell. The open-circuit EMF at the other oxygen

concentration cell was then measured over time to yield an oxygen depletion curve for

the surface of the tin sample contacting that electrolyte. Figure 3-2 shows typical

EMF versus time data for three different sample heights. The diffusivity can be

shown to be related to the slope of the EMF versus time curve at long times by solving

the corresponding one-dimensional boundary value problem.

The diffusion equation is given by


aC C12 C
a =Do2C (3-3)
at a z

where C is the oxygen concentration in tin, t is time, z is axial position, and Do is the

binary diffusion coefficient of oxygen in liquid tin. The initial condition and boundary

conditions used in the solution of this problem are the following:

C=Co at t=O, OszsH (3-4)


C=0 at t>0, z= (3-5)


ac 0 at t 0, z=H (3-6)
az
































0 '

I

























* Figure 3-2.


1 30 40
Time, t (min)


Experimental EMF data for axial diffusion of oxygen through liquid
tin. Representative data for three sample heights are given.






53
The general solution to this boundary-value problem is given by


expf Dolm-w2
WC()t) H2\ sin M (3-7)
Co _-o x+ I 1\-W H
2 22

Since the experimental EMF measurements are related to the concentration of oxygen

at the axial position z = H, Equation (3-7) will be expressed as


C(H,t) 2(-1) e D(3-8)
Co m +1 [ H2 2 t



At large times, however, only the first term is meaningful and Equation (3-8) reduces

to


C(H,t) A [4 [(o! i2 (3-9)
C, H2 4



The relationship between the oxygen concentration at z = H and the measured

EMF is obtained from an examination of the half-cell reactions in the electrochemical

cell. For the transfer of oxygen from the copper/copper(I) oxide reference electrode to

the tin melt, the half-cell reactions are as follows:

Cu2O + 2e (in Sn) -* 2Cu + 02" (in YSZ) (III)

O2 (inYSZ) 0 + 2e (inSn) (IV)

where 0 is oxygen dissolved in liquid tin. The overall cell reaction can then be

expressed in terms of chemical and electrical processes.

Cu2O 2Cu + (V)






54
2e" (in Cu) 2e- (in Sn) (VI)

At electrochemical equilibrium, the cumulative Gibb's energy change for the overall

cell reaction must be equal to zero.



AG, + AGG, = 0 (3-10)



The Gibb's energy change for the chemical process given in Equation (V) is



AGV,= AGO + RTln[ aOacc.2 (3-11)




where AG* is the standard state Gibb's energy change of Equation (V), and the

activities of the pure phases, ac, and aco, are equal to unity. The activity of oxygen

in tin can be expressed as

a = C (3-12)



where y is a function only of temperature, assuming Henry's law holds for dilute

oxygen in tin. For the electrical process,

AG, = 2FE (3-13)



From Equations (3-10) (3-13), the resulting relationship between E and C is


-2FE = AG + RTIn(yC) .


(3-14)






55
Since Equation (3-14) can be written for any arbitrary time in the diffusion experi-

ment, the following relationship is obtained for the ratio C(H,t)/Co:


-2F(E(t) Eo) = RTIn C(Ht)) (3-15)




where E(t) is the measured EMF at any arbitrary time, t, and Eo is the initial EMF.

Combining Equations (3-9) and (3-15), this relationship is obtained:


2F(E(t) E) = 2D 4 (3-16)
RT 4 H2)



Thus, at large times, the rate of change of the measured EMF becomes constant and

is proportional to the oxygen diffusivity. The experimentalist must be careful to wait

sufficiently long to be certain that the approximation given in Equation (3-9) is valid.

Generally, Equation (3-9) is considered valid when the second term in Equation (3-8)

is less than 1% of the first term. This implies that only data taken after a minimum

length of time, t., should be used to calculate the diffusivity. The value of tm is

given by


H2
t = 0.178 (3-17)
Do



From this, it can be seen that samples of small height are preferred. The time

required for the experiment decreases as the square of the sample height. Addi-

tionally, experiments requiring long times will be more prone to errors resulting from






56

low-level convection in the melt. First, the cumulative amount of oxygen carried by

convection is greater for long duration experiments than short duration experiments.

Second, larger sample sizes are more prone to convection simply due to their size. The

viscous damping of convection afforded by the side walls becomes less effective as the

bulk is removed further from the walls. Maintaining isothermal conditions in the

sample also becomes more difficult as the size is increased, especially in high-

temperature furnaces which often have only small, truly isothermal zones.

A least squares analysis of the EMF data according to Equation (3-16) should

result in an intercept at E(t) Eo = (RT/2F) ln(4/I). This has been used as a measure

of the reliability of the data [82]. An intercept deviating from this predicted value

would indicate non-ideal experimental conditions, i.e. convection or oxygen leakage.

Diffusion experiments were carried out for sample heights ranging from 0.389

to 0.750 cm. Ideally, the oxygen diffusivities calculated from each cell height should

be identical at a given temperature. Should the measured diffusivities show a

variation related to cell height, then the data would be suspect. This, again, would

indicate the presence of convection or oxygen leakage into or out of the cell.


Procedure--Steady-state Diffusion Experiments

The diffusivity of oxygen in tin can be determined from a steady-state

experiment as well as the transient experiment. The same apparatus is used in both.

In this case, however, a steady current was passed through the entire cell, from one

reference copper electrode to the other. Since the current is transferred in the form

of oxygen ions through the electrolytes, a linear oxygen gradient is established axially

across the tin melt in the absence of convection. Since the current in equals the

current out, the total amount of oxygen in the tin sample does not change over time.






57
In order to calculate the diffusivity in the steady-state experiment, the

concentration gradient must be known. The flux of oxygen, No, is proportional to the

concentration gradient of oxygen, and the proportionality factor is the diffusivity:


No = DOC (3-18)



Assuming ionic conduction only, the flux of oxygen is calculated from the cell current,


No (3-19)
2A



where I is the cell current, A is the cross-sectional area of the tin sample, and the 2

originates from the divalency of oxygen anions in YSZ. So, the diffusivity is given by

the equation

I
Do
o- *dC (3-20)




EMF measurements across each of the oxygen concentration cells will give the

relative concentrations of oxygen at the opposing surfaces of the tin melt. The

unfortunate aspect of this method, though, is that the concentration gradient cannot

be determined exclusively from the EMF measurements. Knowledge of the absolute

concentration of oxygen at some reference EMF must be obtained. One reference state

for which experimental data is available in the literature is the saturation point for

oxygen in tin [80,86,87]. The results of the three references are in reasonable

agreement, and the results of Ramanarayanan and Rapp [80] are plotted in Figure 3-3







58
for the saturation mole fraction of oxygen as a function of temperature. The

corresponding saturation potential developed between the copper/copper(I) oxide

reference and the saturated tin in the present cell was measured to be 491(4) mV at

700"C, the temperature at which the steady-state diffusion experiments were carried-

out. With this information, the oxygen diffusivity in steady-state galvanic cells can

be calculated from the measured EMF's between the tin and the individual copper

reference electrodes.

The steady-state experiments were carried-out for both positive and negative

gradients of oxygen. The case where the oxygen concentration increases with height

is expected to be hydrostatically stable since the density of the fluid decreases with

height. Conversely, a decreasing oxygen concentration with height will create a

hydrodynamic stability problem completely analogous to the Rayleigh-Benard problem.

The diffusing component in this case, however, is the oxygen solute rather than

thermal energy. Should the solutal Rayleigh number exceed the critical value for

onset of convection, then the measured diffusivity will reflect the additional mass

transport afforded by the convective flow. This information is critical in assessing the

merit of a flow visualization technique which uses dilute oxygen as a tracer element

in liquid metal samples.


Results


Transient Diffusion Experiments

The experimental results for the transient diffusion experiments will be

presented in two groups: 1.) oxygen depletion from the bottom of the tin sample and

2.) oxygen depletion from the top of the tin sample. A marked difference is observed


















































800
Temperature (C)


Figure 3-3. Oxygen solubility in liquid tin as a function of temperature [80].


0.001









0.0001


0.00001 l
600






60

between these two cases in the general trends of the measured EMF data. Figure 3-4

shows the two general trends of the measured EMFs which were observed in the

experiments. The first group of experiments (bottom-depletion) is characterized

entirely by normal diffusion curves, that is, EMF trends like that shown in the lower

curve of Figure 3-4. This is the expected trend assuming that no convection or oxygen

leakage is present in the liquid sample. The second group (top-depletion), however,

is characterized by both trends in EMF, and the type of trend is seen to be correlated

with the initial oxygen concentration and cell height. The transient diffusion results

for depletion of oxygen from the bottom of the sample will be presented first.

Table 3-1 summarizes the experimental conditions for each of the isothermal

diffusion experiments in which oxygen was removed from the bottom face of the tin

sample by applying a large voltage between the tin and the lower copper reference

electrode. The temperature range investigated was 547C to 827C. The initial

oxygen concentration in the samples was varied from approximately 8.5 x 108 to 5.7

x 10" mole fraction (based on the oxygen saturation data of Ramanarayanan and Rapp

[80] and saturation EMF measurements in the present work).

The diffusivity of oxygen was calculated from linear EMF data like that shown

in Figure 3-2. The linear region of the data was fit by a least-squares method and the

corresponding lines were drawn through the data. A summary of the results from

each run is given in Table 3-2. The diffusivities were calculated from the slopes of the

linear data, and the corresponding ordinate intercepts may be compared to the

predicted intercepts from Equation (3-16). It can be noted that the experimental

intercepts are less negative than the predicted values in each case. This would

indicate that either low-level convection is present in the sample despite precautions










































Time, t (min)


Figure 3-4.


Representative EMF versus time curves for transient diffusion
experiments for removal of oxygen from both the top (upper and
lower curves) and bottom (lower curve only) surfaces of the tin
sample.









Experimental conditions of the bottom-depletion diffusivity studies
including sample height, temperature, initial oxygen mole fraction,
and applied voltage.


3-2
3-7
3-8
3-9
3-15
3-17
3-20
3-22
3-26
3-27
3-28

3-33
3-34
3-35
3-38
3-39

3-41
3-42
3-43
3-44
3-45
3-46
3-47
3-48
3-63
3-64
3-65
3-66
3-67
3-68
3-69
3-70
3-71
3-72
3-73

3-80
3-81


Height(cm)

0.389
0.389
0.389
0.389
0.389
0.389
0.389
0.389
0.389
0.389
0.389

0.750
0.750
0.750
0.750
0.750

0.742
0.742
0.742
0.742
0.742
0.742
0.742
0.742
0.467
0.467
0.467
0.467
0.467
0.467
0.467
0.467
0.467
0.467
0.467

0.691
0.691


Initial Mole
Fraction. X0


Applied Volt.(V)


Temp.(*C)

590
547
547
550
641
723
721
722
720
719
718

730
727
726
729
730

688
690
689
770
771
771
827
827
633
633
633
677
677
677
727
725
724
779
782

705
705


1.5x10-'
6.3x10-'
1.O0x10-
8.5x10-'
5.9x10-5
2.6x10-'
2.3x10-'
2.5x10-4
1.6x10-'
1.8x10-'
2.1x10-'

1.8x10-'
2.0x10-'
1.4x10-'
1.8x10-'
2.0x10-'

9.1x10-s
1.1x10-'
7.8x10-s
2.8x10'-
2.8x10-4
2.8x10'-
5.7x10-'
5.6x10'-
4.8x10-s
4.7x10-s
5.0x10-5
8.6x10-s
8.4x10-'
9.2x10-5
2.0x10-'
1.6x10-'
2. x10-'
3.5x10-'
4.3x10'-

1.0x10-'
1.3x10-


Table 3-1.


1.2
1.2
1.2
2.0
1.5
1.5
1.5
1.5
1.5
2.0
1.5

1.5
1.5
1.5
1.5
1.5

1.5
1.5
1.5
1.5
1.5
1.5
1.5
1.5
1.5
1.2
1.8
1.5
1.5
1.5
1.5
1.5
1.5
1.5
1.5

1.5
1.5









Calculated oxygen diffusivities and corresponding intercepts from a
statistical fit of the linear data for each experimental run in which
oxygen was removed from the bottom of the tin sample.


Exy. #

3-2
3-7
3-8
3-9
3-15
3-17
3-20
3-22
3-26
3-27
3-28

3-33
3-34
3-35
3-38
3-39

3-41
3-42
3-43
3-44
3-45
3-46
3-47
3-48

3-63
3-64
3-65
3-66
3-67
3-68
3-69
3-70
3-71
3-72
3-73

3-80
3-81


Intercept (mV)


Do (cm'/sec)

5.llxlO
2.80x10lO
4.00x10-5
4.30x105-
5.80x10-5
7.54x10-5
7.57x10'"
8.76x10-5
8.12x10-5
8.40x10-5
8.32x10-5

6.37x10-5
6.88x10-5
6.30x10-5
7.67x10-5
7.72x10-"

5.86x10-5
6.23x10-5
5.72x10-5
6.91x10-5
6.96x105-
7.28x10-5
1.14x10-4
1.38x10-

4.80x10-5
4.94x10'-
5.55x10-5
5.95x10-5
6.06x10-5
6.13x10-5
7.63x10-5
7.30x10-5
7.62x10-5
8.44x10-5
1.00x10'-

6.91x10-5
7.16x10"5


Predicted
Intercept (mV)


-4.69
-4.27
-5.34
-4.98
-6.41
-7.96
-7.46
-6.54
-7.97
-8.40
-8.54

-6.70
-7.52
-6.29
-7.68
-7.94

-6.37
-7.85
-6.10
-6.68
-7.08
-7.07
-9.50
-9.78

-5.87
-6.08
-7.28
-6.67
-6.95
-7.10
-8.21
-7.78
-7.96
-8.24
-9.12

-7.32
-8.20


Table 3-2.


-8.98
-8.53
-8.53
-8.57
-9.51
-10.37
-10.34
-10.36
-10.33
-10.32
-10.31

-10.44
-10.41
-10.40
-10.43
-10.44

-10.00
-10.02
-10.01
-10.85
-10.87
-10.87
-11.45
-11.45

-9.43
-9.43
-9.43
-9.89
-9.89
-9.89
-10.41
-10.39
-10.38
-10.95
-10.98

-10.18
-10.18






64

to eliminate thermal gradients or oxygen leakage into the cell is causing a slow drift

in the measured EMF's.

The diffusivities from Table 3-2 are plotted as a function of reciprocal

temperature in Figure 3-5. An Arrhenius dependence on temperature is assumed, and

the line drawn through the data corresponds to a linear least-squares analysis. The

following equation is obtained:


D = 1.65x10-3 exp 6150 (3-21)

where R is 1.987 cal/g-mole K Two interesting trends are observed, however, when

the data from the experiments for each individual sample height are analyzed

separately. Figure 3-6 shows a least-squares analysis for each of the three sample

heights which were studied over reasonable temperature ranges. The first trend is

observed in the activation energies calculated from the Arrhenius relationship. These

are summarized in Table 3-3 for the various data sets. E, is lowest in the case of the

H = 0.389 cm data set and highest in the case of the H = 0.742 cm data set. The

point, however, is not that the activation energy should change with sample

dimensions. On the contrary, the activation energy is dependent only on the "activity"

of oxygen in the solvent and not on the geometry of the sample. The point is that the

temperature ranges studied for each sample height (also shown in Table 3-3) are

different. The activation energy shows an increase with temperature, at least as

calculated from this limited data.

The second trend observed in Figure 3-6 indicates an overall lowering of the

calculated diffusivities as the experimental sample height is increased. This trend

leads to the conclusion that convection is not a likely cause of experimental error, but























































0.9 1.0 1.1
Temperature1 x 1000 (K-1)


Figure 3-5.


Experimental oxygen diffusivities from transient diffusion experi-
ments using five different cell heights. The solid line is a least-
squares fit of the data.


0.001


0.0001


0.00001
0.1


III1 1 1 I IIIIIII I I I I I I IIiI | I I

000o0 H = 0.389 cm
suoms H = 0.467 cm
ooooo H = 0.691 cm
AAAAA H = 0.742 cm
AAAA H = 0.750 cm




A
A




U


,I
50


1.3


""""""""""""""""""""


1


3


1.3

















































0.9 1.0
Temperature 1


1.1
x 1000 (K-)


Figure 3-6.


Linear least-squares analyses of the experimental diffusivities given
for individual sample heights. Analyses were not made for H = 0.691
cm and H = 0.750 cm since the temperature was not varied for these
sample heights.


0.001


0.0001


0.00001
0.8


II I I I I I III I I |II II 11111 | 11111 a i !I |I II

oDoP H = 0.389 cm
enmi. H = 0.467 cm
ooooo H = 0.691 cm
AA.AAA H = 0.742 cm
AAAAA H = 0.750 cm




A







''a'a i l a a a a a
-- *






67

Table 3-3. Experimental activation energies for the diffusion of oxygen in liquid
tin. Values are given for the individual data from sample heights of
0.389 cm, 0.467 cm, and 0.742 cm as well as for all data combined.


E, (cal/g-atom) Calculated from Data: Temperature Range

7340 H 0.389 547 7230C
7690 H = 0.467 633 7820C
9930 H 0.742 688 8270C
6150 All data 547 827C





that oxygen leakage is most likely the primary factor in experimental inconsistencies.

If convection were present in the sample, then the experimental diffusivities would

be expected to increase with sample height. Small sample sizes provide a greater

resistance to bulk convection due to the closeness of the bounding walls, while these

viscous effects will be less apparent to the bulk in larger samples. So, the most

probable explanation for these variations is a source of oxygen in the experimental

cell. The larger sample heights require longer times to complete a single diffusion

experiment, and consequently, the oxygen leakage will have a more pronounced effect

on the calculated diffusivities than in the case of smaller sample heights. For this

reason, the diffusivities calculated from the smaller cells are likely the most accurate

of all the reported results.

The most probable source of oxygen in the diffusion cell is the overflow tunnel

through the upper YSZ disk. Although the diameter of the hole is small (1 mm), the

oxygen in the small volume of tin which exists there acts as a virtual oxygen leak.

Since the oxygen concentration cell defined by this upper YSZ disk is used to monitor







68
the oxygen depletion from the upper surface of the tin sample, the virtual leak directly

affects the EMF measurements. Further evidence supporting this theory can be

gathered from the data plots in Figure 3-2. The EMF curves begin to noticeably sag

at long times, indicating an inclusion of oxygen somewhere in the tin sample.

Certainly, however, more than one oxygen source could exist in the cell. Other

possibilities include outgasing from the silica container or YSZ due to chemical

reduction as well as leakage of ambient oxygen through micropores in the cemented

junctions of the container. Outgasing from the ceramics is not likely, however,

considering the stability of those particular oxides in the oxygen atmosphere within

the cell. Also, no visual evidence of degradation on the surfaces of the materials is

apparent. Leakage of oxygen through micropores, on the other hand, is a possibility

since the ceramic cement used to fix the pieces together is porous upon curing. Thus,

two possible sources of oxygen leakage have been identified (overflow channel leakage

and micropore diffusion), but the magnitude of the error introduced into the calculated

diffusivities is uncertain.

The second group of transient diffusion experiments were performed by

applying the voltage across the upper electrolyte to remove oxygen from the top of the

tin sample. The EMF was then measured across the bottom electrolyte to allow for

calculation of the oxygen diffusivity. The two general shapes of the EMF versus time

curve observed in the transient diffusion experiments were shown in Figure 3-4, and

both of these trends were actually observed in the case of oxygen depletion from the

top of the sample. The lower curve is the normal diffusion curve, resulting from

purely diffusive mass transport in the melt. The upper curve, however, exhibits an

interesting phenomenon which is uncharacteristic of a well-controlled diffusion







69
experiment. A sharp rise in the EMF a short time into the experiment indicates that

oxygen is being removed quite rapidly from the tin which is adjacent to the lower

electrolyte. Based on the results from the diffusion experiments under stabilizing

density gradients, this depletion cannot be a result of purely diffusive mass transport.

The apparent cause of this rapid disappearance of oxygen from the bottom surface

must, then, result from convective transport within the melt.

The sudden rise in EMF at short times into the experiment apparently

corresponds to the initial onset of convection in the fluid. As the fluid begins to flow,

oxygen-poor tin from the upper portion of the melt is swept to the bottom, resulting

in a corresponding rise in the equilibrium EMF. However, the slope of the EMF is

then observed to decrease after the initial sharp rise. This can be attributed to the

sudden decrease in the driving force for convection. The initial onset of convection

causes the oxygen to be mixed throughout the melt, and the solutal gradients are

thereby reduced. The convection will then subside and settle into a less energetic

flow. One would expect the flow to slowly decrease as the oxygen concentrations are

reduced from the continued removal of oxygen at the upper surface.

Several of the experiments in this group resulted in normal diffusion curves,

and the diffusivities were calculated in the usual manner. The remaining experi-

ments, however, had to be analyzed differently. An effective diffusivity can be defined

which is a measure of the combined rate of mass transport of oxygen due to both

diffusion and convection. This effective diffusivity is calculated in much the same

manner as the binary diffusion coefficient is calculated, from the slope of the EMF

versus time curve. For these experiments, the effective diffusivity is calculated from

the slope at the point where the rapid rise in EMF is observed. The experimental







70

parameters of each run are detailed in Table 3-4. The resulting diffusivities from this

group of experiments are tabulated along with the experimental and predicted

ordinate intercepts from the linear fit in Table 3-5. The runs which exhibited an

apparent onset of convection are marked as such.

The diffusivities from Table 3-5 are plotted in Figure 3-7 for comparison with

results from the bottom-depletion experiments. Each of the data points which

exhibited a normal diffusion curve lies very close to the linearized data from the

bottom-depletion experiments. The cluster of points which lie well above the

linearized data, on the other hand, all exhibited the sharp rise in EMF due to the

apparent onset of convection. The onset was not observed in the experiments with





Table 3-4. Experimental conditions of the top-depletion diffusivity studies
including sample height, temperature, initial oxygen mole fraction,
and applied voltage.

Initial Mole
Ex. # Height(cm) Temp.(OC) Fraction. X, Applied Volt.(V)

3-1 0.389 589 2.0x10-5 1.2
3-3 0.389 590 1.5x10-5 1.2
3-10 0.389 550 8.7x10' 1.5
3-11 0.389 551 9.2x10-' 1.5
3-12 0.389 639 5.0x10O5 1.5
3-14 0.389 640 4.4x10- 1.5
3-16* 0.389 724 1.6x10'- 1.5
3-19* 0.389 721 2.3x10-4 1.5
3-21* 0.389 721 2.1x104 1.5
3-24* 0.389 721 2.4x10-4 1.5
3-36 0.750 727 2.1x10-4 1.5
3-54* 0.742 867 8.3x10"4 1.5

indicates observation of the sharp change in slope in the EMF versus time
curve






71

Table 3-5. Experimental oxygen diffusivities from top-depletion experiments.
The ordinate intercepts from a least-squares analysis of the data are
also listed for comparison with the predicted intercepts.


Predicted
Exp. # D (cm'/sec) Intercept (mV) Intercept (mV)

3-1 4.32x10-s -5.0 -8.97
3-3 4.73x10-s -6.2 -8.98
3-10 4.06x10- -6.3 -8.57
3-11 4.33x10" -5.7 -8.58
3-12 6.66x10- -9.1 -9.48
3-14 6.23x10' -7.8 -9.50
3-16* 2.12x104 -25.9 -10.38
3-19* 2.70x104 -26.6 -10.34
3-21* 2.69x10- -28.6 -10.34
3-24* 2.26x10- -15.6 -10.34
3-36 8.78x10- -10.5 -10.41
3-54* 3.90x10' -5.2 -11.86

indicates observation of the sharp change in slope in the EMF versus time
curve




bottom-depletion because the resulting density gradient in the melt was oriented such

that the more dense fluid was underneath the less dense fluid. In the case of the top-

depletion experiments, however, the density gradient was oriented in the opposite

direction, and the hydrostatic stability of the fluid was then conditional upon the

magnitude of the density gradient and the geometry of the fluid cell. The stability

problem is analogous to the Rayleigh-Benard problem for vertical thermal gradients,

except that the resulting density gradient in this case is not linear, but decreases

exponentially away from the depletion surface.

Since the hydrostatic stability of the melt is dependent upon the magnitude of

the solutally induced density gradient, one might expect a correlation of the observed

convective onset with the initial oxygen concentration in the melt. This is indeed the

case. In considering only the experiments listed in Table 3-4 which used a sample


















































0.8 0.9 1.0 1.1 1.2
Temperature-' x 1000 (K-1)


Figure 3-7.


Effective diffusivities calculated from transient diffusion experiments
with top-depletion of oxygen. The solid line corresponds to the least-
squares analysis of data from experiments with bottom-depletion of
oxygen.


0.001


0.0001


0.00001 0
0.'


I i i I i I I I I I l I I T i I Ii i I I I I |







4 -- 2 data points
















Bottom-depletion experiments
Q0000 H = 0.389, top-depletion experiments
ooooo H = 0.750, top-depletion experiments
Doo0D H = 0.742, top-depletion experiments
I I I I ""i II "p


7






73
height of 0.389 cm, the onset of convection was observed in only those having an

initial oxygen concentration greater than 10 mole fraction. This is strong evidence

of the ability of very small, solutally-induced density gradients to drive natural

convection in these liquid metal systems.

A final observation in these results involves the influence of the aspect ratio

on the condition for hydrostatic instability. Experiment number 3-36 in Table 3-4

showed no evidence of convection despite an initial oxygen mole fraction of 2.1x104.

This mole fraction is shown to be sufficient to cause convection in the 0.389 cm tall

samples, but appears to be insufficient to initiate convection in this 0.750 cm tall

sample. The diffusivity calculated from experiment number 3-36 agrees well with

measurements from bottom-depletion experiments. Experiment number 3-54, on the

other hand, evidenced a strong onset to convection for an initial oxygen mole fraction

of 8.3x104. The sample dimensions were almost identical to those in experiment

number 3-36. Thus, the initial oxygen concentration and cell dimensions both

influence the hydrostatic stability of the tin sample during a transient diffusion

experiment. This is not surprising since the aspect ratio and Rayleigh number are the

two parameters which were found to define the stability criterion in the Rayleigh-

Benard problem.


Steady-state Experiments

The results of the steady-state experiments carried-out at 700*C do not agree

well with the diffusivities calculated from the transient diffusion experiments; but,

further evidence is found to substantiate the occurrence of convection in the presence

of destabilizing oxygen gradients. Recall, the steady-state experiments are carried-out

by passing a current from one reference electrode to the other, thus creating a linear






74
oxygen gradient across the tin sample. Measurement of the equilibrium EMFs across

each of the electrolytes then allows calculation of the oxygen concentrations from

solubility data in the literature. The calculated diffusivities are tabulated for both

positive (concentrated on top) and negative (concentrated on bottom) oxygen gradients

in Table 3-6 along with the applied cell currents. The height of the sample in each

case is 0.691 cm. The diffusivities are also plotted in Figure 3-8 as a function of the

applied cell current.

The diffusivities calculated for negative oxygen gradients are approximately a

factor of three larger than those calculated for positive oxygen gradients. This

discrepancy leads to the conclusion that convective mass transport must be playing

a role in the negative oxygen gradient runs. Indeed, this orientation does lead to an

increasing fluid density with height, and the hydrostatic stability of the fluid is again

dependent upon the magnitude of the density gradient and aspect ratio. In these

experiments, the aspect ratio is maintained constant and the density gradient varied

by changing the oxygen concentration gradient across the sample. The melt is

statically unstable to all of the applied gradients, however, indicating that the first

critical Rayleigh number is exceeded in each case.

The diffusivities calculated from the positive oxygen gradient runs are

substantially lower than the corresponding values determined from the transient

diffusion study. The results from the transient experiments are more accurate,

however, owing to the minimal experimental error involved in data measurements.

The steady-state results are subject to error both internally and externally. The

internal errors result from uncertainties in the EMF measurements due primarily to

polarization effects as indicated by an increasing effective diffusivity with current.







75

Oxygen diffusivities calculated from steady-state experiments. The
cell current and corresponding oxygen gradient are also listed for
each experiment. (positive gradients indicate increasing concentra-
tion with increasing height)


Cell Current (uA)

5
10
15
20

10
15
20
25
30
35
40
45
50
55
60
65
65
70
75
80
85


Oxygen
Gradient. dX,/dz (cm'1)

12.4x10-s
13.4x10-s
18.8x10-'
21.7x10-s

-3.65x10-s
-5.46x10-s
-7.38x10-s
-9.25x10-s
-11.0x10-
-12.7x10-5
-14.2x10-5
-15.9x10-
-18.0x10-s
-19.7x10-
-20.8x10-5
-22.4x10-s
-20.2x10-s
-22.2x10-5
-24.3x10-
-26.6x10-s
-28.9x10-s


Do (cm/lsec)

0.9x10'
1.6x10-5
1.7x101
2.0x105

6.0x10-5
6.Ox10-5
5.9x10-5
5.9x105
6.0x10-
6.1x105
6.2x10-5
6.2x10-1
6.1x10-
6.1x105
6.3x105
6.4x105
7.1x105
6.9x105
6.8x105
6.6x105
6.4x10-5


The measurements are made across oxygen concentration cells which are subjected

to continuous electrical currents. By the time a steady-state is reached, a significant

layer of oxide can be built-up at the surface of the electrolyte at the most negative

reference electrode. This leads to measurement errors of varied magnitude depending

on the extent of the oxide layer. For example, the saturation potential measured for

oxygen saturated tin before one experiment was 491(2) mV, but after applying a 40

pA current for 3 to 4 hours, the measured saturation potential was 510(4) mV. The

saturation potential eventually drifted back to its original value, but only after being


Table 3-6.






















1.OE-004





.OE-005








0
6.0E-005






S24.0E-005






S2.0E-005





0.OE+000


I I I I I I i I I I I I
- ooooo Decreasing concentration with height
00000 Increasing concentration with height






0 0
-O

O
o oo
o 0 00 00 0 O 0 0












- 0
0
0 0


S,


0 10 20 30 40
Applied


50 60
current


70
(uA)


80 90 100


Figure 3-8. Steady-state diffusivities plotted versus applied current at 700*C.
The cell height was 0.691 cm.


I







77
held open-circuit for several hours. A 1 mV error in the measured EMF results in a

2.5% error in the calculated oxygen mole fraction at the electrolyte-tin interface.

Additional errors in the diffusivity calculations from steady-state data are

external to the experiments. As mentioned previously, oxygen solubility data from

literature sources must be used in the calculation of the diffusivity. The available

sources report oxygen solubilities which are in disagreement by approximately 20%,

although this is considered to be reasonable agreement for this type of study. A 1%

error in the solubility results in a corresponding 1% error in the calculated oxygen

diffusivity.


Discussion


The diffusivity of oxygen in liquid tin has been studied by other investigators

previously, and their results are compared to the present results in Figure 3-9. The

studies of Ramanarayanan and Rapp [80] and Hurst [83] used radial diffusion

measurements in cylindrical geometries, while Otsuka and Kosuka [81] and Otsuka,

Kozuka, and Chang [82] used axial diffusion measurements similar to those reported

in this work. The radial diffusion results are in significant disagreement with the

axial diffusion results. A probable cause is the presence of buoyancy-driven convection

in the radial geometries due to the development of horizontal density gradients as the

oxygen is depleted from the lateral boundaries of the fluid volume. At first, this

argument may seem to contradict the physical evidence since the reported diffusivities

in the radial geometries are lower than those reported for axial geometries. This

cannot be easily judged, however, since the two types of experiments are carried-out

differently.






















































0.8 0.9 1.0 1.1
Temperature-1 x 1000 (K1)


1.2 1.3


Figure 3-9.


Comparison of experimental diffusivities from the present work,
Ramanarayanan and Rapp [80], Otsuka and Kozuka [81], Otsuka,
Kozuka, and Chang [82], and Hurst [83].


0.001


0.0001


0.00001 o
0.'


i I I I I I l I I I I I I i I i i I 1 1 1 1 I
- eP9 Present work
S- Raomanaryanan and Rapp
Otsuka, Kozuka, and Chang
-------- Otsuka and Kozuka
Hurst









Ia o




"" ""--- "




I I0


7






79
The presence of convection in the present studies is shown to yield an effective

diffusivity larger than the molecular diffusivity. This can be justified intuitively by

considering the effect of convection on the surface concentration of oxygen at the

surface opposite to the depletion surface. In the absence of convection, the oxygen

diffuses away from this surface into the bulk relatively slowly as the concentration

gradient in the bulk evolves. The corresponding change in the equilibrium EMF

across the oxygen concentration cell will increase at a corresponding rate. In the

presence of convection, however, the oxygen is removed more rapidly from this surface,

and the corresponding change in the equilibrium EMF will reflect this. Since the

diffusivity is proportional to the rate of change of the EMF with time (at long times),

the apparent diffusivity must increase in the presence of convection. However, the

diffusivities are calculated differently in the case of radial diffusion experiments.

Here, a potential is applied to deplete oxygen from the lateral boundaries of the

cylindrical cell, and the ionic current (corresponding to the flux of oxygen out of the

tin) is measured as a function of time. The diffusivity is then found to be proportional

to the negative logarithmic slope of the ionic current over time (at long times). In the

absence of convection, the ionic current decreases relatively quickly as the oxygen is

depleted from the melt adjacent to the lateral wall. The presence of convection,

however, will replenish oxygen at the lateral wall with oxygen rich fluid from the

bulk, altering the evolution of the ionic current with time. The resulting effective

diffusivity will then be in error, but whether it should decrease or increase with

convection is not intuitively obvious.

Diffusion studies of oxygen in liquid metals have been carried-out in the past

under the assumption that the oxygen was present at such small concentrations that






80
any resulting density gradients would be insufficient to drive natural convection. This

is simply not true, and care must be taken in constructing oxygen diffusion cells such

that any density gradients are aligned with the gravity vector. The cell design

presented in this study appears to be superior to designs used in the past primarily

due to the minimization of thermal gradients in the melt The solutoconvective

driving forces can be minimized simply by choosing an orientation which aligns the

solutal gradient with gravity, but elimination of thermal gradients is not as simple

owing to the high temperatures at which these oxygen diffusion studies are carried-

out. The highly conductive copper sheath which encloses the diffusion cell is a simple

modification which greatly increases the isothermal character of the diffusion cell.

The only drawback of the present design is the oxygen leakage which occurs

through the overflow tunnel in the upper YSZ disk. However, this problem is averted

when the cell is run in the top-depletion mode because the EMF measurements are

then taken across the bottom oxygen concentration cell. Care must be taken, however,

to reduce oxygen gradients such that the critical instability for the onset of convection

is not surpassed. This implies that the initial oxygen concentration must be

maintained as low as possible while still remaining within the electrolytic domain of

the electrolyte.

A point of interest not yet mentioned concerns the applied voltage for creation

of the zero-concentration boundary condition in the transient diffusion experiments.

The applied voltages are listed in the tables of experimental parameters for each of

the case studies. Overall, they were varied from 1.2 to 2.0 volts with no noticeable

effect on the calculated diffusivities. In reality, the equilibrium oxygen concentration

at the electrolyte-tin interface is not zero, but several orders of magnitude lower than






81
the bulk oxygen concentration depending on the value of the applied voltage. At the

temperatures studied here, an increase in the applied voltage of 100 mV reduces the

equilibrium oxygen concentration by approximately one order of magnitude. So, at

700C, an applied potential of 1.2 volts establishes an oxygen concentration of 1011

mole fraction at the interface. An interesting observation, however, is that applied

voltages a 1.5 volts caused a rather large infusion of copper into the YSZ disks.

Apparently, the copper migrated into the grain boundaries of the zirconia and was

even observed to diffuse across the entire thickness of the disk in some cases. In all

likelihood, the presence of copper in the grain boundaries will adversely affect the

conductive properties of the electrolyte and should be avoided.


Summary


An improved experimental cell design for measuring the binary diffusion

coefficient of dilute oxygen in liquid metals is presented. The vertically oriented cell

is applied to the tin-oxygen system for determination of the oxygen diffusivity in the

temperature range 547 827C. The results compare favorably with previous

investigations which used similar experimental procedures, although the present

results are considered to be more reliable due to better controlled thermal conditions.

Results from investigations employing radial diffusion techniques show significant

differences from the axial diffusion cases. These differences are explained in terms

of natural convection phenomena resulting from density variations in the radial

diffusion orientation.

Convection resulting from vertical gradients in oxygen concentration is also

shown to be significant when the resulting density gradient is positive and exceeds a






82
threshold value. The threshold value is found to be dependent on the magnitude of

the density gradient and the geometry of the fluid sample. The fluid is found to be

hydrostatically stable for all negative vertical density gradients.












CHAPTER 4
FLOW VISUALIZATION


Introduction


The basic experimental approach used to measure the diffusivity of oxygen in

liquid tin (Chapter 3) is extended to study certain aspects of the dynamic states in a

simplified Bridgman configuration. The vertical diffusion cell is modified and used to

measure the transport of oxygen in tin melts which are subjected to axial thermal

gradients. The measured transport rates reflect the overall ability of the fluid to

transfer oxygen across the fluid cell. This rate is then indicative of the tendency of

the fluid to disperse solute which is introduced at one boundary (as in solute rejection

during melt-growth of multicomponent semiconductors which form solid solutions)

throughout the bulk. By defining an overall mass transfer coefficient (the effective

diffusivity), the relative level of convection in a fluid cell can be studied as a function

of certain external parameters. For example, in the Rayleigh-Benard problem (which

concerns the stability of dynamic states in horizontal fluid layers heated from below),

the dynamic state of a fluid layer is observed to change with the imposed vertical

temperature gradient. Similarly, the dynamic state of a confined fluid volume, such

as in Bridgman melt-growth configurations, changes with imposed vertical

temperature gradient. These flow transitions have been observed in many fluids, but

have only been characterized in detail for high Prandtl number, Pr, fluids since these

fluids can be studied visually. The flow characteristics of low Pr fluids, which are of






84

interest in Bridgman melt-growth, have not been studied in much detail (as outlined

in Chapter 1) owing to the absence of a comprehensive flow visualization technique.

It is for this reason that the applicability of solid-state electrochemical techniques in

flow visualization is investigated.

Hurst [83] initially proposed an experimental technique similar to that used

in this study. The particular design which he adopted was, however, not capable of

maintaining well-controlled thermal conditions across the melt. The conditions must

be controlled to the extent that measurements are reproducible as the thermal

gradient is cycled. Hurst's data show significant scatter due to the inability to

accurately characterize the thermal conditions within the fluid cell. The experimental

design proposed in this work is much simpler and affords more accurate temperature

measurements.

The results obtained from the diffusion studies in Chapter 3 indicate that the

basic diffusion cell design and technique applied in the transient studies can be

applied as a flow visualization tool in low Pr fluid systems. The purely diffusive mass

transfer of oxygen in liquid tin was shown to be substantially lower than the combined

mass transfer in the presence of convection. This low diffusivity is fortuitous in that

oxygen can then be used effectively as a tracer which can be sensed at the boundaries

of the fluid volume. Consequently, the goal of this study is to test the viability of

using electrochemical sensors to trace dilute oxygen in dynamic fluid systems

associated with the Bridgman configuration.

The physical problem chosen for these initial tests is that of a vertically

oriented cylinder of fluid heated from below. This configuration is primarily of

academic interest since Bridgman melt-growth is generally carried out by heating






85
from above and directionally solidifying from the bottom up. The appearance of

dynamic transitions when heating from below, however, provides an ideal medium for

testing the electrochemical technique. Two fluid aspect ratios (p = H/R) are studied

in these initial experiments. A cell having p = 1 is studied first since it can be

modelled numerically as a two-dimensional flow [441, and the numerical results are

compared with experimental observations. The oxygen transport rates, in particular,

are compared for experimental and numerical results for various applied vertical

temperature gradients. The experimental technique is then applied to a fluid cell

having p = 5.3, again, to study the effects of varying the imposed vertical temperature

gradient across the melt. The details of the experimental and numerical approaches

are also presented.


Experimental


The experimental approach is analogous to that used in the oxygen diffusivity

studies. The experimental cell design is modified, however, to allow for the

application of a thermal gradient along the axis of the cell. The cell design is shown

schematically in Figure 4-1. The outer copper sheath used in the diffusivity studies

is removed and the upper and lower copper reference electrodes are extended to

protrude into the upper and lower zones of the Bridgman furnace. In this manner,

a thermal gradient may be imposed simply by maintaining the upper and lower zones

of the furnace at different temperatures. Liquid tin is used as a model fluid for these

studies primarily due to its low vapor pressure and low toxicity, although the

experimental method may be extended to most any metallic or semi-metallic fluids.











Overflow
Tube





Thermocouples




Liquid
Tin





Copper
Electrode


Copper
-Extension
Wires








Fused
Silica

- YSZ


Copper
Copper


+
Oxide


Figure 4-1. Experimental cell design for measuring the effective diffusivity of
oxygen across liquid tin.








Cell Design

The tin melt was held within a fused silica cylinder sandwiched between two

YSZ disks. The ID of the cylinder was 2.2 cm in each case, and the YSZ disks were

2.54 cm in diameter by 1.58 mm thick. The heights of the fluid cells used were 1.1

cm and 5.79 cm to yield aspect ratios of 1 and 5.3, respectively. A 1 mm diameter hole

was ground through the upper YSZ disk to allow excess tin to flow up the 4.75 mm

OD alumina overflow tube which was cemented to the upper surface of the disk with

Aremco 571 ceramic cement.

The reference electrodes were constructed from 2.55 cm diameter copper bars

(Defense Industrial Supply Center). The ends facing the YSZ were hollowed out to

create a recess for the 1:1 mole ratio copper/copper(I) oxide powder reference system.

The upper copper electrode had a 4.8 mm diameter hole drilled along its axis to

receive the overflow tube.

Type R thermocouples were inserted into small holes which were drilled into

the sides of the two copper electrodes at the ends nearest the fluid cell. The tips were

coated with ceramic cement to maintain electrical insulation from the electrodes. The

thermocouples were made from the same wire stock, and calibration relative to one

another showed no measurable difference at cell temperatures.

Electrical connection to each of the copper electrodes was made by inserting a

copper wire into small holes drilled into the electrodes and pressing the walls of the

hole around the wire. Electrical connection to the tin was made by twisting a copper

extension wire to a short length of rhenium wire which could then be fed down the

overflow tube into the tin melt.






88
A cast tin ingot was initially placed within the quartz container, and the cell

was constructed around it. The cell was then placed inside of a fused silica tube

which was capped at the top by a brass cell head. The entire apparatus was then

connected to a vacuum source as well as a purified argon source. A schematic of the

entire experimental setup is shown in Figure 4-2, including the cell, furnace, vacuum,

argon, and electronic instrumentation. A detailed schematic of the furnace itself is

shown in Figure 4-3.

The furnace is a three zone Bridgman furnace which allows heating of the two

copper electrodes to desired temperatures by the upper and lower zones. The central

zone is required to eliminate any lateral heat losses from the sample. The brass heat

pipe located in this zone is designed to create a smooth temperature gradient between

the upper and lower zones, and thus reduce any heat transfer to or from the sides of

the sample. The cell tube shown in Figure 4-3 is the fused silica container used to

isolate the experimental cell from the oxygen-rich atmosphere.


Procedure

The experimental procedure is completely analogous to that used in the

transient diffusion studies in Chapter 3. The boundary value problem, however, is not

so easily solved in the case of a fluid in motion. The convective term must be included

in the species balance for oxygen in order that the problem be fully posed. The oxygen

balance is then given by the following equation:


a + ( .V)C = Do2C (4-1)
W




























































II-
01 I


0


=Ea
.-
6a









Cell Tube


Isotherma I
Liner


'Insulation







Furnace Brass Heat
Blocks Pipe







Insulation


Isothermal I
Liner


Figure 4-3. Schematic of the furnace used in the effective diffusivity studies.






91
The velocity field is not analytically tractable, however, due to the coupling between

the energy and momentum equations as well as the complex thermal character of the

experimental cell. The problem can be solved numerically, though, and is done so by

using the FLUENT computational fluid dynamics code (creare.x Inc., Hanover, NH)

to model the experimental cells.

The experimental approach used here is based on a one-dimensional

approximation to mass transfer of oxygen across the dynamic fluid cell. The one-

dimensional diffusion equation is used to model the transfer of oxygen in a dynamic

diffusion experiment across a convecting melt.


C Do a2 (4-2)
t az2



The effective diffusivity, Do", then accounts for both diffusive and convective mass

transport. Since the experiments are carried out at constant mean temperatures, the

variation of the effective diffusivity is expected to be due solely to changes in the

dynamics of flow in the melt (i.e. the temperature dependence of the binary diffusion

coefficient is less than 0.5% per *C in the temperature range studied here and is not

expected to have a significant effect on the effective diffusivity as the thermal gradient

is changed).

The boundary value problem to be solved for calculating the experimental

effective diffusivity is then identical to the problem solved in Chapter 3 for calculation

of the binary diffusion coefficient from transient diffusion experiments. The effective

diffusivity is related to the measured cell EMF by the following equation at large

times:




Full Text

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