Electrochemical visualization of convection in liquid metal

MISSING IMAGE

Material Information

Title:
Electrochemical visualization of convection in liquid metal
Physical Description:
ix, 183 leaves : ill. ; 29 cm.
Language:
English
Creator:
Hurst, James H., 1959-
Publication Date:

Subjects

Genre:
bibliography   ( marcgt )
theses   ( marcgt )
non-fiction   ( marcgt )

Notes

Thesis:
Thesis (Ph. D.)--University of Florida, 1990.
Bibliography:
Includes bibliographical references (leaves 176-182).
Statement of Responsibility:
by James H. Hurst.
General Note:
Typescript.
General Note:
Vita.

Record Information

Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 001589719
notis - AHL3694
oclc - 23067765
System ID:
AA00003328:00001

Full Text











ELECTROCHEMICAL VISUALIZATION OF
CONVECTION IN LIQUID METAL

















By
JAMES H. HURST


















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 thank the researchers at the Microgravity

Sciences Laboratory of the NASA Langley Research Center. Without their

help and financial support, this research could not have been done. I

would especially like to thank Dr. Archie Fripp for his support, his

patience, and his enthusiasm for the work. I would like to thank David

Knuteson, Brian Sears, Bill Debnam, and Glenn Woodell for their help in

the laboratory and for many enlightening discussions. I appreciate Dr.

Ivan Clark for helpful discussions on an amazing variety of topics and

Wayne Gerdes for his help with the operation of the computers. I would

like to thank my advisor, Dr. Tim Anderson, for always having a sense

of the direction that the research should go and a vision of what is

possible.

I am grateful to my parents for the confidence that they have

always had in me, and to my wife's family for their material support

and their acceptance. More than anyone else, I would like to thank

Becky, my wife, for her encouragement, for her faith in me, for her

understanding, and for giving me a reason to succeed.














TABLE OF CONTENTS


ACKNOWLEDGEMENTS ............................................. ii

KEY TO SYMBOLS AND ABBREVIATIONS ............................ v

ABSTRACT ...... ........................................... viii

CHAPTERS

1 CRYSTAL GROWTH AND CONVECTION .......................... 1

Introduction ................................................ 1
Lead Tin Telluride Properties and Bridgman Growth
Considerations ................... ....... ........... 2
A Bridgman Crystal Growth Experiment ................. 6
Modelling and Measuring Convection ................... 9
Oxygen Tracers for Measuring Convection ............. 12
Research Plan ........................................ 13

2 GENERAL EXPERIMENTAL CONSIDERATIONS .................. 21

Introduction .......................................... 21
Electrochemical Cell Operation ....................... 22
Electrochemical Cells for Radial Diffusion
Measurements ................................. 23
Electrochemical Cells for Axial Diffusion and
Convection Experiments ......................... 25
Diffusivity Measurement .............................. 27
Convection Visualization Technique .................. 30
Non-isothermal Electrochemical Cells ................. 33
Materials and Construction of Apparatus .............. 35
Summary ................. ..................... ......... 42

3 THE IMPORTANCE OF CONVECTION IN LIQUID METAL
ELECTROCHEMICAL DIFFUSION MEASUREMENTS .............. 53

Introduction .................. ........... ..... ....... 53
Sources of Convection in Diffusion Measurements ...... 54
Oxygen Diffusion in Liquid Tin ....................... 57
Axial Diffusion Experiments .......................... 60
Radial Diffusion Experiments ........................... 71
Modelling Procedure and Results ...................... 80
Summary and Conclusions ............................... 91







4 CONVECTION MEASUREMENT AND MODELLING ................ 111
Introduction .................. .... .................. 111
Convection Modelling ................................ 112
Convection as a Function of the
Vertical Temperature Gradient ............... 116
Convection as a Function of the
Side Wall Thermal Conductivity .............. 123
Additional Models ............................ 124
Minimizing Convection in Crystal Growth ............. 126
Summary .................. ............ .............. 130

5 SUMMARY AND CONCLUSIONS .............................. 172

REFERENCES .......... ................... .................... 176

BIOGRAPHICAL SKETCH ......................................... 183














KEY TO SYMBOLS AND ABBREVIATIONS


A2-a an experimental run, where A represents an axial diffusion
experiment, 2 represents an individual electrochemical cell
apparatus, and a represents an individual experimental run.

a sample radius

aO thermodynamic activity of oxygen

B defined in Equation 2-21

b intercept of line

C solute concentration, mole fraction

CO initial bulk concentration

D solute diffusivity

DOeff effective diffusivity of oxygen

E, EMF potential or electromotive force

F body force vector

F Faraday constant

g fraction of sample solidified

g acceleration of gravity

I electric current

JO0 J1 Bessel functions of zero and first order, respectively

k segregation coefficient

k thermal conductivity

L characteristic length

I sample length

M metal

M quantity of ions conducted by electrolyte







M molecular weight of sample

m summation index

n mobile ion valence

2M oxygen dissolved in metal

P pressure

P(02) partial pressure of oxygen

Pr Prandtl number

R resistance

R gas constant

RaT thermal Rayleigh number

RaS solutal Rayleigh number

r growth rate

r radial coordinate

Sc Schmidt number

T temperature

t time coordinate

v velocity vector

W weight of sample

x composition coordinate for PblxSnxTe

x axial coordinate

YSZ yttria-stabilized zirconia

a Seebeck coefficient

pT thermal expansion coefficient

OS solutal expansion coefficient

7 activity coefficient

AGf Gibbs energy of formation

K thermal diffusivity







Am mth root of the Bessel function of first order

AO chemical potential of oxygen

v momentum diffusivity or kinematic viscosity

7 defined in Equation 2-21














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 VISUALIZATION OF
CONVECTION IN LIQUID METAL

By
James H. Hurst

May, 1990


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

An electrochemical technique for the visualization of buoyancy-

driven convection in liquid metals and semiconductors has been

developed and tested. The electrochemical apparatus is designed to

model the conditions in a vertical Bridgman crystal growth furnace.

Electrochemical cells that employ the ceramic electrolyte yttria-

stabilized zirconia are used to titrate oxygen as a tracer element to

and from the liquid metal. The electrochemical cells detect the motion

of oxygen in the liquid metal by diffusion and by natural convection.

Tin is used as a model fluid for semiconductor melts such as lead tin

telluride. Preliminary measurements of the diffusivity of oxygen in

tin have been made. It has been shown that the electrochemical cell

technique can detect the presence of convection in the liquid. The

mass transfer of the tracer element through the sample under various

temperature gradients has been measured and compared with the results

of finite difference computer models for natural convection in the

liquid. The experiments and computer models show that the uncertainty


viii








in electrochemical diffusion measurements is at least partly due to

convection in the liquid sample. Techniques for minimizing natural

convection in liquid metals and semiconductors in diffusion

measurements and in Bridgman crystal growth experiments are

recommended.

















CHAPTER 1
CRYSTAL GROWTH AND CONVECTION



Introduction



There are many difficulties in the production of single crystals

of compound semiconductor solid solutions such as lead tin telluride

(PblxSnxTe) for the manufacture of electronic devices, including

segregation of the main elements lead and tin, and the formation of

extended defects in the solid. Both of the difficulties mentioned are

affected by convection in the melt from which the crystals are grown.

If convection in semiconductor liquids were better understood, it could

be controlled to minimize its effect on the solid. Such an

understanding, however, is lacking because of limits on experimental

techniques for observing convection in liquid metal solutions at

elevated temperatures and pressures. This dissertation presents

research on a new technique that combines crystal growth procedures

with electrochemical cells to observe and measure convection in liquid

semiconductor materials.











Lead Tin Telluride Properties and Bridgman Growth Considerations



Crystal growth of the elemental semiconductor material silicon is

relatively well understood and allows growth of zero dislocation

density material with uniform electrical properties. Silicon based

devices dominate the solid state electronics market in part because of

the availability of a cheap and high quality substrate. The elemental

semiconductor silicon suffers from certain inherent limitations,

however, chiefly because its elemental nature does not allow much

freedom of choice in physical and electrical properties. An obvious

step toward greater choice is the use of compound semiconductor

materials, where the desired properties can be attained by adjusting

the composition of the material. One such compound semiconductor is

PblxSnxTe, a direct bandgap material that is useful in long wavelength

optoelectronic devices (laser, photodetector, or LED). The desirable

feature of Pb_1 SnxTe is its narrow and direct bandgap, which through

wavelength tunability can produce high efficiency photonic devices

operating at long wavelengths (2.5 to 28 micrometers) [1]. The

tunability is achieved either by varying the composition (the value of

x) or by varying the external temperature, pressure, or applied

magnetic field. Both features are important in high resolution

spectroscopy (specifically, atmospheric monitoring), which is one

application of this compound semiconductor material.

Many semiconductor crystals, including PblxSnxTe, are grown in a

variation of the Bridgman-Stockbarger technique [2] in which a molten

sample in a cylindrical ampoule is directionally solidified by moving








the ampoule in its axial direction through a temperature gradient. The

direction of motion is vertical (see Figure 1-1) which allows the

choice of placing the hot end of the temperature gradient at the top or

at the bottom of the sample. The temperature gradient is produced with

a three-zone furnace which has a hot zone (heated above the melting

point of the semiconductor) and a cold zone (kept below the melting

point) separated by an insulated zone that helps to make the

temperature gradient between the two heated zones linear.

It is convenient to consider PblxSnxTe as a pseudobinary solid

solution of the compounds lead telluride (PbTe) and tin telluride

(SnTe) [3,4,5]. As shown in Figure 1-2, the pseudobinary solid-liquid

phase diagram is isomorphous. The phase diagram shows that at all

temperatures and compositions the equilibrium liquid has a higher

concentration of SnTe than the solid. During directional

solidification the liquid will gradually become more rich in SnTe as

the PbTe freezes preferentially into the solid. The concentration

profile in the solid along the direction of solidification will be

somewhere between two limiting cases. The more common case is when the

rate of solute redistribution in the melt, due chiefly to buoyant

convection, occurs faster than solidification. The liquid is

considered to be fully mixed while no mixing occurs in the solid. The

variation in the solid composition in the growth direction for this

case is expressed by the normal freezing equation [2]

C = kCg(l-g)k-1 (1-1)

where C is the solute (SnTe) concentration in the solid, CO is the

concentration in the sample before freezing begins, g is the fraction

of the sample solidified, and k is the ratio of the concentration in








the solid to the concentration in the liquid at the interface (the

segregation coefficient). In the derivation of Equation 1-1, k is

assumed to be concentration and temperature independent, and the solid

and the liquid phases are assumed to have the same density. The other,

less common limiting case occurs when convection in the liquid is

negligible compared with the rate of solidification and solute

redistribution in the liquid occurs primarily by diffusion. This

diffusion-controlled growth has been examined by Tiller et al. [6], and

the solid concentration is described (except for the last-to-freeze

portion, where the diffusion boundary layer is thicker than the

remaining liquid extent) by the equation


C = CG 1-(1-k)exp [-kg]] (1-2)



where r is the growth rate of the solid and D is the concentration

independent diffusion coefficient for the solute in the liquid. In the

derivation of Equation 1-2 the difference between the liquid and solid

phase densities was neglected. The convection- and diffusion-

controlled solid concentration curves are both shown graphically in

Figure 1-3. In the convection-controlled case the solid composition

varies continuously over the length of the sample, while in the

diffusion-controlled case the resulting composition in the solid is

nearly constant over a large section of the ingot, except the first-

and last-to-freeze regions. Most directionally solidified samples of

PblxSnxTe follow closely the convection-controlled case, but it would

be preferable if the diffusion-controlled case could be achieved. Much

of the resulting solid would then be of uniform composition and could

be used for mass production of substrates for semiconductor devices.








To achieve the diffusion-controlled case of ingot composition

distribution, convection in the liquid must be minimized. It would

also be useful if convection could be reduced in the growth of

elemental semiconductor materials, where motion in the liquid can cause

extended defects in the solid [7,8,9]. These defects may be caused by

microscopic changes in the growth rate of the solid as convection

alters the temperature and the liquid composition at the solid-liquid

interface. Successful control of convection in either elemental or

compound semiconductor melts requires an understanding of the forces

that cause convection. The strongest driving force for convection is

gravity acting on density gradients in the molten material. Density

gradients come about both thermally and solutally. Thermal density

gradients occur in a temperature gradient when the density of the

liquid is a function of temperature. A material with a negative

thermal expansion coefficient, such as PbTe (PT -7.0x10-5 K-1) or

SnTe (PT = -1.7x10-4 K-1) [10], will be less dense at higher

temperatures. With these materials, if the bottom of the melt is

hotter than the top, gravity opposes the density gradient. The liquid

is then thermally unstable and convection will occur if the magnitude

of the temperature gradient is sufficiently large. Solutal density

gradients occur in PblxSnxTe because the density of the liquid

decreases with increasing SnTe concentration. At 1000C the density of

PbTe is 7.33 g/cm3, while the density of SnTe is only 5.73 g/cm3 [10].

As the melt at the liquid-solid interface becomes richer in SnTe, the

liquid density decreases. If the interface is at the bottom of the

liquid, gravity opposes the density gradient. The liquid is then

solutally unstable and convection will occur if the magnitude of the








concentration gradient is sufficiently large. In PblxSnxTe the

directions of thermal and solutal instability oppose one another, so

convection occurs in either direction of crystal growth [11,12]. With

these driving forces for convection opposed to one another, PblxSnxTe

is a particularly useful model material for the study of convection in

Bridgman crystal growth.

Convection will also occur whenever horizontal temperature

gradients exist in the liquid, regardless of their magnitude [13,14].

Also, Marangoni convection, driven by surface tension gradients along a

free surface, can result [15]. Gravity-driven convective forces should

be greatly diminished when crystals are grown in the microgravity

environment of space. To this end, researchers are performing crystal

growth experiments both on the ground and in space. Ground-based

experiments have exhibited normal convectivee) freezing, while space-

based experiments have so far shown both convection- and diffusion-

controlled growth [16].



A Bridgman Crystal Growth Experiment



A ground-based experiment in which PblxSnxTe was grown in our

laboratory by the vertical Bridgman technique has confirmed the

existence of convection and the accuracy of the fully-mixed liquid

model from which Equation 1-1 is derived. The first step in preparing

the sample was to weigh the elements in the proportions 50 atomic %

tellurium, 40 at. % lead, and 10 at. % tin (Cominco American, Inc.,

Spokane, WA) for a sample composition of Pb0.8Sn0.2Te. The elements

were placed in a fused silica test tube (General Electric Co., Quartz









Product Div., Cleveland, OH) with an inside diameter of 16 mm which was

then sealed under a vacuum (about 10-6 torr). The tube was then placed

in a furnace where it was heated to 1000C and rocked for 24 hours to

mix the molten solution. Upon removal from the rocking furnace the

sample was immediately quenched in liquid nitrogen to insure complete

solidification before long range compositional segregation could occur.

The sample was removed from the casting silica tube and sealed a second

time under vacuum inside another fused silica test tube for crystal

growth.

The test tube containing the sample was instrumented with six

type K thermocouples (Omega Engineering, Inc., Stamford, CN) mounted on

the outside of the test tube at 1.0 cm intervals along a line parallel

to the axis of the test tube. The thermocouple potentials were

measured and recorded with a scanning digital thermometer (John Fluke

Mfg. Co., Inc., Mountlake Terrace, WA). The growth test tube,

instrumented and containing the sample, was placed in a three-zone

furnace with a 30 mm insulated zone separating the upper zone, heated

to 980C, from the lower zone, heated to 500C. The sample remained

stationary while the furnace was moved to a position where the sample

was entirely in the hot zone. After the measured temperatures showed

that the sample was completely melted, the furnace was moved up until

the top of the insulated zone was at the bottom of the sample. The

furnace was then moved at a constant rate of 0.61 cm/hr until the

entire sample was solidified. The furnace temperature profile as

measured by the thermocouples on the sample is shown in Figure 1-4.

The temperatures measured by thermocouple 0 are noticeably different

from the other temperatures because this thermocouple was located at









the rounded bottom end of the sample container. The differences among

the other temperature measurements show the calibration errors in the

thermocouples. After the furnace had cooled overnight to room

temperature, the tube was removed from the furnace. The ingot did not

adhere to the walls of the fused silica tube and was easily removed

from the container. After directional solidification, the ingot was 16

mm in diameter and 60 mm in length.

The directionally solidified ingot was cut in half through the

axis with a diamond-impregnated wire saw working with a silicon carbide

slurry. The cut face of each half-ingot was polished flat and

electrochemically etched [17]. Etching caused various shades of gray

to appear on the ingot face, as seen in photographs of the cut ingot.

The composite photograph shown in Figure 1-5 was assembled from six

microscope photographs of the ingot. The lines that can be seen on the

ingot within areas of a uniform gray color are cracks that probably

developed during cooling. X-ray diffraction studies have revealed that

the portions of the ingot on either side of the cracks have the same

crystal orientation, while the portions that appeared as different

shades of gray had different orientations. The different gray areas

were therefore the single crystals that made up the ingot. (The

boundaries between different shades of gray that are perpendicular to

the long dimension of the sample are due to different lighting in the

individual photographs.) It can be seen in Figure 1-5 that the etch

has revealed the bulk of the sample to be one single crystal, which

grew out of several small single crystals at the rounded, or first to

freeze, end of the ingot. The composition of the ingot was determined

by electron microprobe analysis performed at many points on the









polished face of the ingot. The microprobe is capable of performing

quantitative analyses accurate to within 5% of the amount of material

present [18]. The microprobe sample points were arranged in linear

arrays across the diameter and along the axis of the ingot. The

tellurium concentration was, within the microprobe measurement error,

0.50 mole fraction at every sample point on the ingot, implying that

the ingot was a solid solution of PbTe and SnTe and that the lead and

tin concentrations reveal the solution composition. The tin

concentration across each diameter of the ingot is constant within the

accuracy of the electron microprobe, while the tin concentration along

the axial or growth direction varies and is shown in Figure 1-6. The

local maximum in the tin concentration in the first to freeze part of

the sample is caused by supercooling prior to nucleation [19]. The

axial tin concentration after this initial anomaly follows the normal

freezing equation (Equation 1-1, which is shown as a solid line in

Figure 1-6) using a segregation coefficient (k) of 0.65. Adherence to

the normal freezing equation indicates that the liquid was fully mixed,

presumably by natural convection, during the growth of the ingot. As

the growth furnace was configured with the hot zone above the cold

zone, the convection was most likely driven by gravity acting on

solutal density gradients generated by the solid-liquid interface as

PbTe froze preferentially to SnTe.



Modelling and Measuring Convection



The description of convection in the Bridgman-Stockbarger crystal

growth configuration as presented to this point has been qualitative.









Better results in crystal growth experiments could be obtained if this

qualitative description were replaced with a more quantitative

understanding of convection in liquid metals and semiconductors. The

factors that determine the driving forces for natural convection may be

understood by application of theory and by careful experimental

measurement.

The theories describing natural convection are expressed in the

differential equations that describe fluid flow. These equations of

motion are well known [20] and cases of liquid convection similar to

Bridgman crystal growth have been discussed in the literature.

Analytic solutions to the fluid flow equations are in many cases not

possible without excessively simplifying assumptions. As new computer

programs are developed, numerical solutions to the fluid flow equations

may be obtained, but the numerical models require accurate physical

property data that is often not available. Current research includes

both analytical and numerical approaches to mathematical modelling of

natural convection in the liquid [21,22,23].

The accuracy of an analytical or numerical model is judged by

comparison to experimental results. Such a comparison is often

lacking, however, because of the difficulties involved in performing

fluid flow experiments with metal and semiconductor systems. With

these materials, much of the experimental work is limited to post-

growth analysis of the resulting solid sample, which includes chemical

etching to reveal the shape and position of the solid-liquid interface

during growth, measurement of the solid composition by electron

microprobe analysis (for comparison to the theoretical curves in Figure

1-3), and X-ray diffraction studies to determine the orientation and








quality of the crystals. Measurement of convection during growth of

metals and semiconductors has until now been limited to the use of

thermocouples to follow changes and oscillations in the temperature at

various points in the liquid [12,24,25].

The most common techniques for observing convection in liquids

are optical. Some techniques, such as Schlieren or differential

interferometry, rely on the pressure and density gradients that

accompany convection to produce visible changes in the refractive index

of the liquid [26]. Another common technique is the observation of

tracer particles or dyes suspended in the liquid [27,28]. These

techniques can only be applied to transparent or semi-transparent

liquids (usually aqueous solutions or oils) and cannot be used with

liquid metals or semiconductors, which are completely opaque. Optical

techniques are also limited in experiments with metals and

semiconductors by the need for a high-temperature furnace around the

test fluid. Convection in geometries similar to crystal growth

configurations have been modelled with transparent liquids so that

optical techniques may be used, but transparent liquids typically have

a Prandtl number (the ratio of the momentum diffusivity to the thermal

diffusivity in the liquid, or v/c) greater than 1, while the Prandtl

number for liquid metals and semiconductors is typically near 10-2.

The accuracy of these models is questionable because the extent of

convection driven by thermal gradients is strongly dependent on the

Prandtl number [29]. Even when natural convection is caused by solutal

rather than by thermal density gradients, crystal growth from a melt

requires a temperature gradient, so experiments in model ssteii,- with a

high Prandtl number will always have limited applicability to the








directional solidification of metals and semiconductors. New methods

for observing convection in liquid metals and semiconductors are needed

in order to avoid the limitations of optical visualization techniques,

and to gather data for comparison to analytical and numerical models.



Oxygen Tracers for Measuring Convection



The development of solid electrolyte electrochemical cells

introduces new opportunities for measuring convection in liquid metals

and semiconductors in crystal growth configurations. The ionic

conductivity of certain ceramic solids was recognized as a useful

property by Kiukkola and Wagner [30], who used these solid electrolytes

to measure thermodynamic properties of metals, semiconductors, and

their oxides. There are several excellent reviews that discuss the

applications of solid electrolytes to thermodynamic measurements

[31,32,33,34]. Solid electrolytes are also widely used in kinetic

experiments, including the measurement of the diffusivity of various

elements through solid and liquid metals and alloys [35,36,37].

Convection is also a kinetic process, and the adaptation of

electrochemical cell techniques to the measurement of convection in

liquid metals and semiconductors provides a valuable tool to study

convection during crystal growth.

The most common solid ceramic electrolytes are the tetravalent

transition metal oxides zirconia (ZrO2) and thoria (ThO2). When doped

with tri- or di-valent metal oxides such as yttria (Y203), calcia

(CaO), or magnesia (MgO), the resulting increase in the oxygen vacancy

concentration gives significant oxygen ion conductivity [38]. The









negligible porosity, inertness, and ionic conductivity of stabilized

zirconia make the ceramic useful as part of a Bridgman crystal growth

ampoule. The ceramic functions as an oxygen-permeable wall through

which oxygen may be electrochemically introduced as a tracer element in

the liquid phase. Oxygen is a useful tracer element because it is

detected by electrochemical cells in concentrations low enough (10-3

mole fraction and below) [39,40,41] that the presence of the tracer

does not affect the fluid properties that determine the nature of the

convection in the liquid. Also, several studies using zirconia

electrolytes in electrochemical cells to study the thermodynamic

properties and the diffusivity of oxygen in metals and semiconductors

provide an extensive data base for oxygen tracer studies of liquid

convection. By building multiple electrochemical cells into a single

Bridgman crystal growth ampoule, the motion of oxygen in liquid metals

and semiconductors can be detected and measured.



Research Plan



Natural convection is a cause of solute segregation and extended

defects in the crystal growth of semiconductors and metals. A

technique for visualizing convection and measuring its extent in liquid

semiconductors and metals would help in developing crystal growth

procedures that would minimize the unwanted effects of natural

convection. Experiments have been designed and performed (Chapter 2)

to test a convection visualization technique which uses a ceramic

electrolyte to follow the motion of an oxygen tracer in a liquid metal.

The model material that was chosen for the experiments was liquid tin.








In preliminary experiments the diffusivity of oxygen in the model

material has been measured (Chapter 3). The diffusivity experiments

and computer models of diffusivity experiments have shown that these

measurements are vulnerable to natural convection caused by temperature

gradients. Experiments have also shown that the electrochemical cell

oxygen tracer technique is able to detect convection in the liquid

metal. The diffusion and convection experiments and the computer

models have demonstrated some of the basic causes of natural convection

in the crystal growth of metals and semiconductors and how it may be

minimized (Chapter 4).







































Sorrpe in
Seoled Arrpok


Ikiotion


Figure 1-1. A Bridgman-Stockbarger crystal growth furnace.
























924



900 Liquid


o




a,850
E
Q) Solid
H--




80
800 '
0.0 0.5 1.0
PbTe Mole Fraction SnTe SnTe














Figure 1-2. Solid-liquid phase diagram of the PbTe-SnTe pseudobinary
solution (from References 3, 4, and 5).





























0.50




S0.40
0
in
(--
c

c 0.30
0


L
0
U-
0.20




I- 0.10

C0.00
cn



0.00


solidified, g


Figure 1-3. Composition distribution in directionally-solidified
Pbl.xSnxTe by fully-mixed and diffusion-controlled crystal gr wth.
Equations 1-1 and 1-2, with CO = 0.20, k 0.7, r/D 2.4 cm .


0.5
Fraction of sample


I -
Diffusion-controlled growth
- Convection-controlled growth









I / I
-|







i i i | -


From





























1000

950

900

850

800


750

700

650

600

550
3C


40
Position (cm)


Figure 1-4. Temperature profile in the Bridgman crystal growth furnace,
and thermocouple placement on the ampoule.


35
Furnace


I I o Io I I I I
000 0c0O










+ +


Distance above *"t
- ampoule bottom .,
0000ooooo 6 cm *'
000= 5 cm *
Aq




AAA 4 cm cm
*oo+o 3 cm
+++++ 2 cm t
.,.xx 1 cm .-
** ** 0 cm I ,*..,
I I I I I I I I I I I


)
































































Figure 1-5. Composite photograph of the directionally- solidified
Pbl-xSnxTe ingot, showing the single crystals in the ingot. The rounded
end of the ingot was the first to freeze.





































+

0 +
CM




+
+
+



o Initial Bulk Sample Concentration
-! + ++





Sample 85-4B Axial Microprobe Scan



o l i l l I I I Il l l i l i l a t Il al l
0 10 20 30 40 50 60
Distance from first-to-freeze (mm)


















Figure 1-6. Composition distribution in the directionally-solidified
Pbl-xSnxTe ingot. (- ) represents the calculated curve for fully-
mixed growth, k = 0.65.

















CHAPTER 2
GENERAL EXPERIMENTAL CONSIDERATIONS



Introduction



This chapter discusses the design, construction and operation of

the electrochemical cells that were used for diffusion and convection

measurements with liquid tin. Within a wide rangg of temperatures and

oxygen partial pressures, the ceramic material stabilized zirconia acts

as a solid electrolyte and provides a means of adding, removing, and

detecting an oxygen tracer in a liquid metal or semiconductor when it

is used as part of an electrochemical cell. Under an electrical or
2-
chemical potential gradient, oxygen anions (02-) can migrate through

stabilized zirconia, which has a transference number of unity for

oxygen anions over a wide range of oxygen activities. When stabilized

zirconia is used as the electrolyte in an electrochemical cell that

also contains a liquid metal sample, an applied voltage will titrate

the tracer element oxygen into and out of the liquid. The cell also

allows measurement of the oxygen concentration in the part of the

liquid that is in contact with the zirconia. When two electrochemical

cells are in contact with the liquid, a response at one cell to a

titration at the other reveals motion of the tracer element in the

liquid. Thus the electrochemical cells provide a method for detecting

21








and measuring the motion of the oxygen tracer in the liquid. Oxygen

moves in the sample both by diffusion and by convection of the liquid.

By detecting the motion of the tracer, the electrochemical cells can

detect the presence of convection in the liquid metal sample.



Electrochemical Cell Operation



Zirconia (zirconium (IV) oxide or ZrO2) is stabilized in the

fluorite structure by the addition of yttria (Y203) [42,43]. To

preserve electrical neutrality in the presence of the trivalent yttrium

ions, oxygen vacancies are formed in the zirconia lattice. Oxygen

anions migrate through the solid zirconia by an oxygen vacancy

mechanism [44,45,46,47,48,49], and the low solid state diffusivity of

neutral ions in zirconia prevents the migration of other species [50].

The oxygen ion conductivity in zirconia depends on the impurity and

stabilizer concentrations, temperature, and oxygen activity (or the

equivalent oxygen partial pressure). Conduction by oxygen anions is

dominant over a large range of operating temperatures and oxygen

partial pressures [51]. If the oxygen partial pressure is too high or

too low, conduction by holes or electrons increases and the electrolyte

behavior becomes complex [52,53,54,55,56,57,58]. The limits of

temperature and oxygen partial pressure for which the electronic

conductivity of yttria-stabilized zirconia (YSZ) is insignificant, or

where the transference number is greater than 0.99 (called the

electrolytic domain), depends on the specific material. An estimated

lower electrolytic domain limit for YSZ is shown in Figure 2-1. As a

general rule YSZ may be effectively used as a solid electrolyte between








600C and 2000C in oxygen partial pressures above the lower limit that

is shown in Figure 2-1 [59,60,61,62,63,64].

Figure 2-2 gives the Gibbs energy of formation and the

equilibrium oxygen partial pressures for the oxides used in the

experiments described here [65,66]. Based on conservative limits of

the electrolytic domain of YSZ (Figure 2-1) and the information in

Figure 2-2 it is assumed that the measurements made with YSZ for these

oxides are all in the electrolytic domain.


Electrochemical Cells for Radial Diffusion Experiments

The basic electrochemical cell consists of a sample electrode and

a reference electrode separated by the YSZ electrolyte. When oxygen is

used as a tracer element in a molten metal, the sample electrode is the

liquid metal containing dissolved oxygen. The reference electrode must

be a chemical system that has a known equilibrium oxygen partial

pressure. Gas reference electrodes (air, pure oxygen, or gas mixtures)

are commonly used and are simple to construct, when the design of the

experimental apparatus allows. The reference electrode in the radial

diffusion experiments was room air, with an oxygen partial pressure,

P(02), of 0.209 atm [65]. The potential or electromotive force (EL:-1),

E, of the electrochemical cell

Pt I air II YSZ sample, [0], (2-1)

where the dissolved oxygen is designated as [O], depends on the Gibbs

energy of the overall cell reaction
1
202(g) = 2M' (2-2)

where OM represents oxygen dissolved in the sample metal. The

dissociation of 02 occurs at the electrolyte-air interface and is








catalyzed by the porous platinum catalyst on the YSZ surface. The EMF

of the cell is related to the thermodynamic activity of the oxygen

dissolved in the liquid sample, ao(sample) by the Nernst equation

RT P(02)1/2
E = ln (2-3)
nF ao(sample)


which includes the gas constant (R = 1.9872 cal/mole-K), the absolute

temperature (T), the valence of the mobile oxide ion (n = 2

equivalents/mole 0), and the Faraday constant (F = 23061 cal/volt-

equivalent). Equation 2-3 shows that under isothermal conditions the

oxygen concentration at the metal-electrolyte interface can be adjusted

by changing the applied potential, if no overpotentials exist in the

reference electrode. The activity of oxygen in the metal is related to

the concentration of oxygen, C (expressed as a mole fraction), by the

activity coefficient

7 = ao(sample)/C. (2-4)

The value of the activity coefficient can be measured by operating the

cell in a galvanic mode. If the solubility of oxygen follows Henry's

law, the activity coefficient is not required to determine the ratio

between the two corresponding oxygen mole fractions by the relationship


RT C1
E2 E= In -. (2-5)
nF C2

In a diffusion experiment, the applied potential titrates or "pumps"

oxygen electrochemically through the YSZ electrolyte into or out of the

sample. The magnitudes of the initial and final applied potentials

give the ratio between the initial (1) and final (2) oxygen

concentrations at the electrode-electrolyte interfaces.









Electrochemical Cells for Axial Diffusion and Convection Experiments

The electrochemical cells for the axial diffusion experiments

consist of a sample electrode separated by two YSZ solid electrolyte

crucibles from two reference electrodes. The sample electrode is the

liquid metal containing dissolved oxygen. The reference electrode

atmosphere must be isolated from that of the sample electrode [67], and

isoation can be a problem if gas reference electrodes are used. The

isolation problem is reduced with a solid reference electrode, which in

these experiments consists of a pure metal and its stable oxide (M and

MmO) [33]. The EMF (E) of the electrochemical cell

M, MmO II YSZ 1 sample, [0] (2-6)

is related to the oxygen activity (ao), in each electrode by the

relationship [30,68,69]
RT P(02)(reference)1/2
E = In (2-7)
nF ao(sample)


which includes the valence of the mobile ion (n = 2 equivalents/mole
2_
for 02-), the Faraday constant (F = 23061 cal/volt-equivalent), the gas

constant (R = 1.9872 cal/mole-K), and the absolute temperature, T. The

oxygen activity in the solid electrode is determined from the Gibbs

energy of the reversible formation reaction, AGf,
1
mM(s) +02(g) =Mm(s), (2-8)

through the equilibrium relationship [70]

AGf(MmO) = RT In P(02)(reference)1/2, (2-9)

and the potential of the cell is

1
E = -[AGf(MmO) RT In ao(sample)]. (2-10)
nF









This equation is valid when oxygen is present in the sample as

dissolved elemental oxygen. If the oxygen concentration is high enough

for a separate oxide phase to form in equilibrium with the liquid

sample, the oxygen potential is constant and the measured cell

potential is therefore constant.

Equation 2-10 (shown graphically in Figure 2-3 for oxygen in tin

at 700*C) relates the electrochemical cell potential to the activity of

oxygen in the sample and is useful in two ways. First, the measured

cell potential is directly related to the oxygen activity and therefore

the concentration of oxygen dissolved in the sample. Second, an

applied potential will produce a specific oxygen activity (or

concentration) at the electrolyte-sample interface by transporting

oxygen from the reference electrode through the electrolyte.

Electrochemical cells function both as oxygen detectors and as oxygen

pumps in detecting flow in liquid samples. Equation 2-10 is simplified

when applied to the second electrochemical cell, used to detect the

oxygen concentration, provided other parameters (such as the

temperature) remain constant. Taking EO as the EMF corresponding to an

initial oxygen activity ao(0), later values of the EMF, E, can be

compared to the initial EMF of the same electrochemical cell by

RT aO
E E0 = In (2-11)
nF ao(0)

Since oxygen dissolves in liquid metals in only very dilute amounts,

the ratio between oxygen activity and oxygen concentration, C

(expressed as a mole fraction), is a constant (Henry's law) [40]. If

Henry's law applies, Equation 2-11 is expressed as







RT C
E E = In (2-12)
nF CO

where CO represents the oxygen concentration that corresponds to the

initial potential, E0. It is important to note that in Equation 2-12

the measured electric potentials reflect oxygen activities or

concentrations at the sample/electrolyte interface, not in the bulk

sample [71]. Fluid flow in a sample is measured by using multiple

electrochemical cells to detect oxygen concentrations or establish

oxygen concentration boundary conditions at different points in the

sample.



Diffusivity Measurement



The value of the diffusivity of oxygen in the liquid metal is

required for the convection visualization experiments in order to

separate the part of the oxygen mass transfer in the metal that is a

result of convection from the part that is the result of molecular

diffusion. The diffusivity measurements were performed by diffusing

the oxygen in the radial direction in a long, cylindrical sample. The

radial motion of oxygen by molecular diffusion in a cylinder is

described by the radial form of Fick's second law of diffusion [72],


ac 1 a 8c
= r (2-13)
at rOr r r



which relates oxygen concentration in the liquid, C (expressed as the

mole fraction), to time, t, and radial position, r. Equation 2-13

assumes that the oxygen diffusivity is constant. This assumption is

reasonable for the dilute oxygen concentrations that are examined here,








and disproving the assumption would require better diffusivity data

than are currently available (see Chapter 3). The initial condition

for Equation 2-13 is a uniform oxygen concentration, Cl, in the sample

which is maintained by the applied voltage E1. At time t=O, a voltage,

E2, applied between the sample and the air reference electrode fixes a

new concentration, C2, at the YSZ-sample interface (r=a, the radius of

the sample). The axis of symmetry on the center line of the

cylindrical sample (r=0) provides a zero concentration gradient as the

second boundary condition. These initial and boundary conditions are

expressed by

C = C1, t = 0, 0 r r < a, (2-14a)

C = C2, t > 0, r = a, (2-14b)
ac
= 0, t 0, r = 0. (2-14c)
ar


When these conditions are applied to Equation 2-13, the solution is

[73,36]

C -C1 2 -DOA tJO(r,Am)
= 1 m exp (2-15)
C2-C1 a mJl(a,Am)
m=0


where JO and J1 are the Bessel functions of zero and first order,

respectively, and Am is the mth root of the Bessel function of zero

order. If Mt represents the total quantity of oxygen that has passed

through the electrolyte after time t, and M. represents Mt after

infinite time, their ratio is

M, 4 -DOm2t

m=l m









MO is given by
W
M = --(C1-C2), (2-17)

where W is the weight and Mw is the molecular weight of the sample

metal. The quantity Mt is given by integrating the ionic current

through the cell, Iionic' over the time of the experiment,
t

Mt n Iionicdt. (2-18)

0

For large times, the second and all subsequent terms of the sum

in Equation 2-16 are negligibly small, and the sum may be approximated

by

Mt 4 -DO_ 2
= exp (2-19)
M, 1 a .


The first root of the Bessel function of zero order, A1, equals

approximately 2.405 [74]. By substituting Equations 2-17 and 2-18 into

Equation 2-19 and differentiating,

4nFDO W -Do 2t
lionic = --(C1-C2)exp (2-20)
a M[ a



is obtained, which relates the ionic current through the cell as a

function of time to the diffusivity of oxygen through the liquid. By

taking the logarithms of both sides of Equation 2-20, the equation

-ln (lionic) = t/7 ln B (2-21)

is obtained, where
a2
T 2
D0A1


and







4nFDO W
B 2 -(C1-C2)'
a2 Mw


According to Equation 2-21, a plot of -In (lionic) as a function of t

at long times should give a straight line, and the diffusivity (or the

effective diffusivity, if convection is present in the liquid) of

oxygen in the sample can be calculated from the slope of the line.

The value of B is obtained from the intercept of the line of Equation

2-21, and from this value the difference of the initial and final

concentrations, C1-C2, may be calculated. When C1-C2 is combined with

the ratio C1/C2, calculated from the initial and final electrochemical

cell potentials according to Equation 2-5, the initial and final

concentrations of oxygen in the sample may be calculated.



Convection Visualization Technique



Convection in a metal or semiconductor liquid is visualized by

measuring the rate of mass transfer of the tracer element oxygen

through the molten material. In a stagnant liquid, oxygen moves by

diffusion only, and the basic parameter for the mass transfer is the

diffusivity of oxygen through the liquid (DO). Motion of oxygen by

diffusion in a stagnant liquid between two flat, parallel planes is

described by the one-dimensional linear diffusion equation (Fick's

second law of diffusion) [72],

ac a2c
aC 82C
-- = D (2-22)
at ax


which relates oxygen concentration in the liquid, C, to time, t, and

linear position, x. The initial and boundary conditions required for a









solution to the diffusion equation are fixed by the construction of the

sample chamber in the experimental apparatus shown in Figure 2-4. The

concentration of oxygen at the top of the cylindrical sample chamber,

C(x=O), is measured with the detector reference electrode. The

concentration of oxygen at the bottom of the chamber, C(x=-, where 2 is

the length of the sample chamber), is fixed by the pump reference

electrode. Initially, the concentration of oxygen in the sample is

uniform at the value CO. At time t=0, an electrical potential applied

between the sample and the pump reference electrode fixes C(x=-) at C1.

There is no significant flux of oxygen through the top of the sample

chamber (at x=O). This experimental procedure gives the initial and

boundary conditions

C = CO, t = 0, 0 x < 1, (2-23a)

C = Cl, t > 0, x = (2-23b)
ac
0, t > 0, x = 0, (2-23c)
ax


to be applied to the one-dimensional diffusion equation. The solution

is [73,75]

C -CO 4 (-1)m -DO(2m+l)22t (2m+l)rx
= I m exp c2 os 2m (2-24)
CI-C0 7 (2m+l) 42 2S .
m =0


Evaluating the oxygen concentration in the sample at x=0, and choosing

C1 to be much smaller than CO results in the following expression for

the concentration at the detector interface [37]

C 4 (-1)m -DO(2m+l)2 2t
=- X l exp 42 (2-25)
Co m (2m+l) 4=
m=0









This equation converges very slowly when t is small, but when t is

large the summation converges very quickly and all but the first term

may be neglected. With this assumption a reasonable approximation to

Equation 2-25 is

C 4 -D O2t
exp j 2t. (2-26)
CO 7r 4



When Equation 2-26 is combined with Equation 2-12, which was derived

for the electrochemical cell, the ratio of the oxygen concentrations is

replaced by a difference between the potential measured at the detector

reference electrode over time, E, and that potential initially, E0,

RT DO72 RT 4
E E 2 t In (2-27)
nF 4 nF rT.


This equation is linear in time with a slope of

RT DOx2
nF 41


and an intercept of

RT 4
In -
nF Ir.


According to Equation 2-27, a plot of the difference between the

initial and the final potentials as a function of time should, at large

times, give a straight line. The diffusivity (or the effective

diffusivity, if convection is present in the liquid) of oxygen in the

sample can be calculated from the slope of the line.







Non-isothermal Electrochemical Cells



The equations presented so far have been derived with the

assumption that the entire cell apparatus is isothermal. If this were

true, there would be no motion in a pure liquid metal sample, since the

driving force for convection in this system is gravity in the presence

of a density gradient caused by temperature variations. If the liquid

is isothermal, the flow visualization experiment reveals only the

diffusivity of the tracer element in the sample. If the same

experiment is repeated with a temperature gradient applied across the

cell, comparison with the isothermal case shows that the tracer in the

sample moves faster than by diffusion alone, and the motion of the

liquid is revealed. Non-isothermal electrochemical cells require

particular consideration, however. To understand the effects of a

temperature gradient on an electrochemical cell, it is necessary to

review the derivation of Equation 2-7. Neglecting electronic

conductivity in the electrolyte, the potential of the cell depends on

the chemical potential of oxygen, pO, in each electrode [68],

1
E = -[(sample) pg(reference)]. (2-28)
nF


The chemical potential is related to a standard chemical potential

(po0) and the oxygen activity by

AO = O + RT In (aO). (2-29)

The electric potential of the cell depends on electrochemical reactions

at the interface between the electrode and the electrolyte. It is the

temperatures, oxygen chemical potentials, and oxygen activities at

these planes that are of concern. The properties on the sample side








will be denoted by (') and those on the reference side will be denoted

by ("). Equations 2-28 and 2-29 become [34,76,77]

1
E = [0' + RT' In aO' i*O" RT" In ao"] + a(T'-T") (2-30)
nF


where a is the Seebeck coefficient corresponding to the electric

potential generated by contact between dissimilar materials at high

temperature. The experimental procedure allows a simplification, in

that the desired data values are not absolute electric potentials but

rather potential differences over time (E EO). Though the

temperatures are not constant over space they are constant over time

and the only variables in Equation 2-30 that are functions of time are

E and ao'. When the relationships between the activities and the

standard states are considered, the electric potential difference

becomes

RT' aO'
E E = In (2-31)
nF ao'(0)


with ao'(0) representing the oxygen activity in the sample that

corresponds to the initial electric potential. Equation 2-31 is nearly

identical to Equation 2-11, which was derived for an isothermal

detector electrochemical cell, but Equation 2-31 shows that the

temperature used in the calculation must be the temperature at the

interface between the sample and the electrolyte. With this

consideration, Equation 2-27 is equally valid for isothermal and non-

isothermal experiments, and oxygen transfer in the liquid sample may be

measured for both diffusive only isothermall) and diffusive plus

convective (non-isothermal) cases.







Materials and Construction of Apparatus



The design of a working electrochemical cell apparatus requires

the consideration of many factors. One factor is the need for a cell

geometry that lends itself to mathematical analysis. Another factor is

the compatibility of cell materials, especially for containment of the

sample and for electrical contact to the sample. Proper

electrochemical contact between electrode and electrolyte materials is

necessary for correct operation of the cell. Also, in the high

temperatures at which the solid electrolytes operate an inert

atmosphere is required to prevent oxygen contamination of the

electrodes and oxidation of the electrical contacts [78]. Finally, for

the measurement of convection, the cell must be heated by a furnace

that can establish the necessary temperature gradient.

The axial diffusion and flow visualization experiments were

performed in the apparatus shown in Figures 2-5 through 2-8 [37]. Tin

was chosen as the sample material for the electrochemical studies

because it has a low vapor pressure and because its thermophysical

properties are available in the literature. The tin sample was

contained in a fused silica tube (General Electric Co., Quartz Products

Div., Cleveland, OH) with an inside diameter of 8.9 mm and a length

between 10 and 20 mm. All fused silica, alumina, and zirconia pieces

that were used in the apparatus were cleaned and etched in a 12:7:1

solution of water, nitric acid, and hydrofluoric acid. The sample

material was cut from zone-refined bars of 99.9999 per cent tin

(Cominco American, Inc., Spokane, WA). Before being placed in the

apparatus, the tin pieces were cleaned and etched in a 5 volume %








bromine in hydrobromic acid solution. The top of the cylindrical

sample chamber was the bottom of a YSZ crucible (8 weight % Y203 in

ZrO2, composition 1372 from ZIRCOA Products, Corning Glass Works,

Solon, OH) with outside diameter of 13.7 mm and height of 25.4 mm that

rested on top of the chamber tube. Inside the crucible was a mixture

of tin pieces and tin (IV) oxide (SnO2) powder (Morton Thiokol, Inc.,

Alfa Products, Danvers, MA) that formed the detector reference

electrode.

The crucible was capped with a plug of machined graphite (POCO

Graphite, Inc., Decatur, TX) and electrical contact with the electrode

materials was made with a 0.75 mm diameter tungsten wire (Alfa

Products) that passed through the graphite. Experiments were attempted

with chromel wire as the electrical contact to the metal sample and to

the detector reference electrode, but oxidation of the chromel limited

the lifetime of the cell too severely. Tungsten was selected as the

material for electrical contact with the metal based on the solubility

of tungsten in the metal and the oxidation potential of tungsten

compared with that of the tin [79,80,81,65]. The tungsten wire

oxidized more slowly than the chromel wire, so more useful data could

be obtained from each electrochemical cell that was constructed. At a

point just above the graphite the tungsten wire was spot-welded to a

copper wire (Alfa Products) which was led out of the cell through a 3

mm diameter alumina tube (McDanel Refractory Co., Beaver Falls, PA).

The detector reference electrode assembly was contained in a 19

mm inside diameter fused silica separator tube (General Electric Co.),

which was fused to the chamber tube. The chamber tube rested in the

bottom of a YSZ crucible (ZIRCOA Products) of outside diameter 25.4 mm








and height 51 mm which was filled with excess sample material. Slots

measuring 1 mm by 1 mm cut in the bottom of the chamber tube permitted

the flow of liquid sample between the chamber and the overflow space to

ensure that the chamber was always completely filled with sample. The

gap between the crucible and the separator tube was closed by an

annular graphite plug through which another tungsten wire made

electrical contact with the sample electrode material. The tungsten

wire was spot-welded to a pair of copper wires which were led out of

the cell through a two-bore, 3 mm diameter alumina tube.

The larger, outer crucible rested atop a mixture of copper (Cu)

and copper (I) oxide (Cu20) powders of 99% purity (metals basis) (Alfa

Products) that had been molded in an hydraulic press at 100 MPa

pressure and then baked for 1 hour at 900C under an argon atmosphere

to form a 25.4 mm diameter, 4 mm thick pellet. The bottom of the

crucible was sanded flat to ensure good electrochemical contact with

the pellet, which was the pump reference electrode [82]. Reference

electrodes made of a nickel-nickel oxide mixture were tried and proved

to be unsatisfactory because the electrical conductivity of the

electrode mixture was too low. The Cu/Cu20 system was chosen for the

pump reference electrode to minimize error because of electrode

polarization during electrochemical titration [83,64].

The pellet rested on top of a solid machined copper cylinder

(Defense Industrial Supply Center, Philadelphia, Pennsylvania) 25.4 mm

in diameter and 75 mm long. The copper cylinder, or expansion block,

was the electrical contact to the pellet and expanded when heated

relative to the fused silica tubes that made the basic structure of the

apparatus. The expansion of the copper block pressed the cell









components together to make good electrical contacts between the block

and the pellet, and between the pellet and the large YSZ crucible. A

pair of copper wires were attached to the copper block and were led

from the cell through a two-bore, 3 mm outside diameter alumina tube.

Electrical contacts to the sample electrode and to the pump reference

electrode were made with pairs of wires so that on each side the

potential could be measured through one wire while the titration

current flowed through the other wire. By this means the measured

potential remained unaffected by potential drops caused by current flow

in the lead wires (IR wire losses) [84]. The entire assembly was

contained in a fused silica tube (General Electric Co.) with an outside

diameter of 32 mm. The copper expansion block rested on the closed

bottom of the fused silica container tube and the wires in the alumina

tube were led out through a 6.4 mm outside diameter fused silica tube

which was sealed to the bottom of the container tube.

Radial diffusion experiments were performed in YSZ tubes (ZIRCOA

Products) closed at one end. The experiments were performed in two

sizes of tubes, and the experimental apparatus that was built around

the smaller tube is shown in Figure 2-9. The larger tube had an inside

diameter of 16 mm, and the smaller tube had an inside diameter of 4.6

mm. The height of the tin sample was 75 mm in the larger tube and 110

mm in the smaller tube. The porous platinum air reference electrode

was made by painting the outside of the tube at the closed end, over a

length equal to the height of the sample, with four coats of platinum

ink (Engelhard, Edison, NJ) and firing the tube for 30 minutes at

10000C after each coat. A pair of platinum wires was attached to the

YSZ tube with fired platinum ink to ensure good electrical contact to









the air reference electrode. The sample electrode was made by casting

99.9999% pure tin metal (Cominco American) in a fused silica tube,

cleaning and etching the resulting ingot, and placing the ingot into

the YSZ tube. The electrical contact to the sample electrode was a

rhenium wire with one end welded to a pair of platinum wires and the

other end immersed in the molten tin.

The electrochemical cell assemblies for the experiments, with the

exception of the air reference electrode in the radial diffusion

experiments, were isolated from room air by stainless steel cell heads

that used Viton O-ring fittings (Cajon Co., Macedonia, OH) to make air-

tight seals to the fused silica or YSZ tubes outside of the furnace.

The cell heads that were used with the axial diffusion experiments are

shown in Figures 2-7 and 2-8. Similar cell heads were used on the open

ends of the YSZ tubes in the radial diffusion experiments. In the

axial diffusion experiments, argon gas flowed into the apparatus at two

places. An argon gas stream entered the cell at the bottom with the

electrical leads to the pump reference electrode. The argon flowed

past the cell assembly and was exhausted through a port in the lower

cell head that sealed the annulus between the container tube and the

separator tube. Another argon gas flow entered the cell through the

alumina tube containing the lead wire to the detector reference

electrode and was exhausted through a fitting on the upper cell head

that sealed the top of the separator tube. In the radial diffusion

experiments, the argon gas flowed into the YSZ tube through the alumina

tube that protected the platinum lead wires to the sample electrode.

The argon gas was exhausted from the apparatus through the cell head

that sealed the opening of the YSZ tube.









The argon gas (Air Products and Chemicals, Inc., Allentown, PA)

was purified in a three-stage process. First, the argon passed over a

catalyst to react hydrogen with oxygen to form water. Second, the

water was removed as the gas passed through a sodium aluminosilicate

molecular sieve desiccant (both from Matheson Gas Products, East

Rutherford, NJ). Finally, the excess oxygen was removed as the argon

passed through a bed of titanium sponge (Alfa Products) that was heated

to 850C. The argon streams were regulated by needle valves and flow

meters and were exhausted through glycerin bubblers to provide a

positive pressure of argon in the apparatus, thus reducing the backflow

of room air.

Cell temperatures were monitored with chromel-alumel (type K)

thermocouples in Inconel sheaths (Omega Engineering, Inc., Stamford,

CN) that entered the cell through fittings in the cell heads. The

thermocouples were tied to the YSZ crucibles or tubes with Inconel wire

and were fixed in place with ceramic cement (Aremco Products, Inc.,

Ossining, NY). The thermocouples were calibrated before use by cycling

them between room temperature and 1000*C together with a standard

thermocouple. The calibration results provided correction factors that

were used to adjust the thermocouple outputs that were recorded during

the experiments. The positions of the thermocouples were carefully

fixed and noted in order to determine precisely the temperature

gradients in the cell.

Both the axial and the radial cell assemblies were placed in a

two-zone tubular furnace made from resistance heater furnace elements

(Thermcraft, Inc., Winston-Salem, NC). Each furnace zone was powered

by a feedback controller (Eurotherm Corp., Reston, VA). The two zones









were separated by ceramic insulation (Cotronics Corp., Brooklyn, NY) 3

cm thick, and each zone was 25 cm long. Between the furnace windings

and the cell in each zone was an annular heat pipe (Dynatherm Corp.,

Cockeysville, MD) that acted as an isothermal liner. With the heat

pipes in place, the temperature change between each furnace zone and

the insulated zone was more abrupt, and the temperature gradient in the

insulated zone was more linear than it was without the heat pipes.

Figure 2-10 shows the electrical connections for monitoring the

potentials and current flows for the two electrochemical cells in the

axial diffusion experiments. The radial diffusion experiments, with

only one electrochemical cell, had the electrical connections that are

shown in Figure 2-9 between the sample and the pump electrodes, with

the electrometer taking the place of the voltmeter for measuring the

cell potential. The potential between the reference electrode (the

detector reference electrode in the axial diffusion experiments) and

the sample electrode was measured with an electrometer (Keithley model

610C, Keithley Instruments, Inc., Cleveland, OH) having an input

impedance of >1014 ohms. The power supply that provided the applied

voltages to the electrochemical cells was a Fluke model 382A

Voltage/Current Calibrator (John Fluke Mfg. Co., Inc., Everett, WA).

The current through the cell between the reference electrode and the

sample electrode was determined from the potential drop measured across

a standard resistor (R = l.000kohm). The potentials, including the

recorder output potential of the electrometer, were measured and

recorded on a Fluke model 2280B data logger (John Fluke Mfg. Co.). The

thermocouple potentials were also monitored by the data logger, which

compensated for the reference temperature and linearized for the









specific thermocouple type. The data logger was in turn controlled

through a IEEE-488 digital interface bus by a Hewlett-Packard model 85B

microcomputer (Hewlett-Packard Co., Corvallis, OR) which recorded the

experimental data on paper and on magnetic disks.



Summary



Experiments for measuring the motion of oxygen in liquid tin by

diffusion and convection were designed and built. The experiments take

advantage of the ionic conductivity of yttria-stabilized zirconia to

measure, establish, and change the concentration of oxygen in the tin

sample. The geometries of the experimental apparatus provide initial

and boundary conditions for the solution of the diffusion equation.

Radial experiments were designed to measure the molecular diffusivity

of oxygen in liquid tin, and axial experiments were designed to measure

convection in tin caused by temperature gradients in the sample. The

materials and construction of the experimental apparatus were

described.





43





















-5

-10

-15

E -20

-25

-30


g -35

-40

-45

-50 IiI l
5.0E-004 7.0E-004 9.OE-Op4 1.1E-003
1/Temperature (K )















Figure 2-1. The lower limit of the electrolytic domain of the solid
electrolyte yttria-stabilized zirconia (from Reference 50).


































Ur-


marks phase transition

---------- -----------
SSaTra tor liquid- -- ----
0-




0- -










.0-
0 2







,


Temperature (C)
Temperature (CC)


900


10


lo P(02)
(atm)
-3

-6

-9

-12

-15

-18

-21

-24

-27

-30

-33


00


Figure 2-2. Gibb's energy of formation of and oxygen partial pressure
over oxides used in the electrochemical cell experiments.


-2




-4




-6




-8




-10


600


-
























700



660



>620
E

S80
580
0

540

is saturation point
500



460 I
0.0 4.0 8.0
Oxygen activity x 10 at 700C













Figure 2-3. Electric potential of the cell Cu20 I YSZ 1 Sn, [Q], as a
function of the oxygen activity in the tin. The saturation point is
Esat = 471.6 mV, asat = 7.75x10-11 (from References 40 and 85).






















xGzO








a~U


YSZ



Fused
silica
wall


YSZ


Copper/copper oxide pump
reference electrode


Geometry of the axial diffusion experiments.


Figure 2-4.




















Copper wires in
alumina tubes


Graphite plugs


Tungsten wires


Yttria-stabilized
zirconia crucibles


.marks thermocouple
positions



Copper expansion
block


Fused silica
separator tube



Fused silica
container tube



Detector Sn/Sn02
reference electrode


Liquid sample


Pump Cu/Cu20
reference electrode


Copper wires in
alumina tube


Figure 2-5. Apparatus for the axial diffusion experiments, including
the details of the electrochemical cells and the materials of
construction.





48



























i







ii


























Figure 2-6. Photograph of the apparatus for the axial diffusion
experiments, showing the larger YSZ crucible, the graphite cap, the
thermocouples, and the fused silica container tube.
thermocouples, and the fused silica container tube.
















Argon gas n


Thermocoup'e leads
S Argon












Upper fumoce block


Isothermol liners <





Lower furnace block


Copper ire to detector reference eectrode

-Alumi tube



Stoinless steel cell heads with 0-ring fittings
Fused silica separator tube
je =-- Copper i'res to sample electrode

- Fused silica continer tube


Argon gos in


- Copper 'es to pump reference electrode


Figure 2-7. Apparatus for the axial diffusion experiments, including
the furnace and the cell heads with argon and electrical connections.





50























-iL- :N- 0 -, --'



































Figure 2-8. Photograph of the apparatus for the axial diffusion
experiments, including the furnace and the cell heads with argon and
electrical connections.
electrical connections.

















Two platinum
wires in an
alumina tube


Porous
platinum
electrode



Thermocouple e
positions N


Alumina
tube

Yttria-
stabilized
zirconia
tube


Two platinum
wires attached
to a rhenium
wire







Liquid
tin
sample


Figure 2-9. Apparatus for the radial diffusion experiments, including
the materials of construction.

















Electro- RU I t[ i Uic iL
meter Electrode
V YSZ
Sample
Electrode

V) YSZ R
Pump
Reference
Electrode Power
Supply Vy










Figure 2-10. Electrical connections for measuring the electrochemical
cell potentials and the titration current in the axial diffusion
experiments.

















CHAPTER 3
THE IMPORTANCE OF CONVECTION IN LIQUID METAL
ELECTROCHEMICAL DIFFUSION MEASUREMENTS



Introduction



The electrochemical visualization of convection in metal and

semiconductor melts depends on the motion of oxygen through the sample,

which occurs by both convective and diffusive processes. Convection is

detected when the motion of the oxygen tracer is greater than what

would be expected from molecular diffusion alone. The magnitude of the

molecular diffusivity of oxygen through the liquid sample must be known

with certainty, since the electrochemical technique which detects

convection cannot distinguish between the two mechanisms for the motion

of oxygen in the liquid. This chapter describes experiments that were

performed with liquid metal samples in electrochemical cells. The

model material for the convection measurement experiments was liquid

tin. The tin samples were cylindrical, and experiments were performed

with oxygen diffusion in the axial and radial directions. The values

of the oxygen diffusivities in liquid tin, both those reported in the

literature and those reported here, are examined. The electrochemical

cells were able to detect convection in the liquid metal samples.

Computer modelling of natural convection in diffusion measurements was









performed to estimate the effect of convection in the experimental

samples on the measured oxygen diffusivities. The examination also

revealed some of the causes of natural convection in oxygen diffusion

measurements.



Sources of Convection in Diffusion Measurements



The greatest challenge in designing experiments to measure solute

diffusivities in liquid metals is the elimination of convection in the

samples. When convection is present, the solute moves through the

sample by bulk flow as well as by diffusion, resulting in a determinate

error in the measured diffusion coefficient. Convection caused by

cooling or solidifying the sample, or by mechanically initiating

convection [86,87], is forced convection and may be avoided by the

appropriate choice of experimental technique. For example, with

techniques that measure the solute concentration in the liquid metal

spectrographically (such as NMR techniques) or electrochemically,

cooling and solidifying the sample is not necessary. Mechanical motion

may be avoided by adding or subtracting solute to or from the sample

through a membrane or through a porous wall. But even techniques that

do not require mechanical motion may be subject to natural, or

buoyancy-driven, convection due to the action of gravity on local

density variations in the sample. The causes of natural convection

must be understood and minimized in order to obtain accurate values for

solute diffusivities.

Density gradients in the liquid sample may be expressed as

vectors with components parallel to and perpendicular to vertical (the









direction of the gravity vector). The case of vertical gradients has

been studied extensively and is reviewed by Chandrasekhar [88]. It has

been shown that if the density gradient is parallel to gravity (with

the less dense liquid above the denser liquid) the configuration is

stable and no convection will occur. If the density gradient is

antiparallel to gravity (with the denser liquid above the less dense

liquid) the configuration is unstable and convection will occur if the

density gradient is large enough to overcome the viscous forces that

try to hold the liquid stagnant.

The balance between buoyancy and viscous forces is expressed by

the dimensionless Rayleigh number, Ra [20], which is given by


-PTgL4 aT
RaT (3-1)
Kv ax


when the density gradient is produced by a temperature gradient, and by

-/3BgL4 a
Ra = (3-2)
,Dc ax


when the density gradient is produced by a concentration gradient. In

Equations 3-1 and 3-2, PT and PS are respectively the thermal and

solutal coefficients of expansion of the sample material, g is the

acceleration of gravity, L is the characteristic length of the geometry

being examined, K is the thermal diffusivity of the sample, DO is the

diffusivity of the solute 0 in the fluid, v is the kinematic viscosity

of the fluid, OT/ax is the vertical temperature gradient, and aC/ax is

the vertical concentration gradient. In a situation with vertical

density gradients only, convection occurs when the Rayleigh number in

the antiparallel direction exceeds a certain critical value. If the









Rayleigh number is less than the critical value or is positive parallel

to the gravity vector, the liquid will be stagnant. If the thermal or

solutal critical Rayleigh number is exceeded, the stagnant liquid is

unstable and the liquid will convect in response to the instability.

From the basic Navier-Stokes equations which describe fluid flow,

Verhoeven [14] has shown that convection will result any time the

horizontal density gradient in the liquid is not zero. Verhoeven terms

this phenomenon "thresholdless" convection to distinguish it from the

"threshold" convection described previously, the threshold being the

critical value of the Rayleigh number that must be exceeded for

convection to occur. In other words, convection will occur whenever

there is a horizontal density gradient in the liquid sample, even if

the vertical density gradient alone would allow the liquid to be

stagnant.

Density gradients which lead to convection in a liquid sample are

usually caused by temperature or solute concentration gradients. To

minimize convection, one must keep the horizontal temperature and

concentration gradients as close to zero as possible, while making sure

that the vertical gradients do not exceed the critical thermal and

solutal Rayleigh numbers. These critical values depend on the geometry

of the sample, especially the aspect ratio, and can be found in several

sources in the literature [14,89,90,91].

Control of the concentration gradients is easily achieved and

involves the use of sample container materials that are impervious to

the diffusing species. Control of the temperature gradients is much

more difficult, since the thermal conductivities of liquid metals and

their container materials often do not differ by much more than one









order of magnitude. In the axial diffusion experiments that are

described in this chapter, for example, the thermal conductivity of tin

is 40 W/m-K, while the thermal conductivity of the fused silica

container is 1.4 W/m-K [92,93]. As is seen in the experiments reviewed

and performed here, lack of control over the temperature gradients in a

diffusion measurement can lead to errors in the experimental results.



Oxygen Diffusion in Liquid Tin



Two reports of the diffusivity of oxygen in liquid tin were found

in the literature. Both measurements were made with the

electrochemical cell technique but different sample geometries were

used. In both cases the diffusivity measurement was reported with the

assumption that there was no convection in the liquid tin.

Ramanarayanan and Rapp [36] measured the diffusion of oxygen in

tin contained in a long, narrow tube of ceramic electrolyte material,

using a geometry similar to the radial diffusion experiments described

in Chapter 2. In these experiments oxygen was diffused in the radial

direction into and out from a cylindrical tin sample with a uniform

initial oxygen concentration. The researchers do not report any checks

made to determine whether there was convection in the liquid, but they

do report that their furnace was isothermal to within 1IC in the 7.6

cm long region where the sample was located. In these experiments the

tin samples had aspect ratios (1/r) of 18. In this geometry, according

to Verhoeven [14], the onset of convection would occur when the

temperature gradient exceeds 13*C/cm, with the lower part of the sample

hotter than the top. The apparatus that contained the sample was very









simple, and the temperature boundary conditions on the sample would be

similar to the conditions that were imposed by the furnace.

While it is unlikely that the temperature gradient in the furnace

exceeded the critical value for the onset of convection, it is also

difficult to state with confidence that the furnace and sample was

isothermal to within l*C at 1400C, as these authors did. The

apparatus was instrumented with one Type S thermocouple mounted on the

outside of the ceramic tube that contained the sample, and it is

impossible for one thermocouple to indicate a temperature gradient.

There are two options for measuring the temperature gradient on the

sample: either multiple thermocouples should be attached to the

apparatus as near to the sample as possible, or the furnace should be

calibrated with a dummy sample carefully instrumented with

thermocouples [94]. Both procedures will detect temperature gradients

created, in an otherwise isothermal furnace, by the presence of the

experimental apparatus. Among other things the apparatus can act as a

heat pipe, carrying heat from the interior of the furnace out to

ambient conditions. In the calibration of the furnace it is also

important to consider the limitations of thermocouples. Type S

thermocouples at 14000C have an inherent error of 3.5C [95] (which

may be reduced if multiple, calibrated thermocouples are used). A

temperature difference less than the error cannot be reported with

confidence.

The analysis of the thermal boundary conditions and whether they

lead to convection in the liquid sample is further complicated by

effects at the ends of the long, narrow tube employed in the

experiments of Ramanarayanan and Rapp [36]. One would expect that even









a small vertical temperature gradient would be translated into a

convection-inducing horizontal temperature gradient at the bottom of

the sample, since the ratio of the thermal conductivities of the liquid

tin and the zirconia container is about 20 [92,96]. At the top of the

sample, the free surface boundary condition between the liquid and the

argon atmosphere in the tube allows convection driven by spatial

variations in surface tension [97]. The impact of the end effects on

the measured diffusivities may be minimized by making the sample as

long and thin as is practical. Verhoeven [14] has shown that for

aspect ratios (1/r) greater than 22, the temperature gradient required

for the onset of convection depends only on the radius of the sample,

and that smaller radii require higher temperature gradients for

convection to occur. Ramanarayanan and Rapp could have reduced the

possible impact of convection on their measurements by using a longer,

narrower sample.

Otsuka, Kozuka, and Chang [98] measured the diffusivity of oxygen

in liquid tin by an electrochemical cell technique that diffused oxygen

along the axis of a cylindrical sample. From a sample initially at a

uniform concentration, oxygen was removed at one face of the cylinder

and diffusion was measured by the rate of oxygen depletion at the

opposite face of the cylinder. The researchers varied the length of

the cylinder and reported an oxygen diffusivity independent of sample

length, which together with the agreement between their results and the

analytical solution to the diffusion equation was cited as evidence

that there was no convection in the liquid tin sample in the

experiments. The authors do not give a description of the sample









geometry sufficient for calculating the vertical temperature gradient

required for the onset of convection.

The chief concern in an axial diffusion experiment of the type

reported by Otsuka et al. is the horizontal temperature gradients that

result from the combination of materials with differing thermal

conductivities, especially the combination of conducting and insulating

materials. These materials can divert the heat flow from an otherwise

vertical-only temperature gradient to produce horizontal temperature

gradients that will induce convection in the sample. The authors of

this report do not discuss how the temperature measurements were made,

but another article by the same authors [37] describe similar

experiments that were instrumented with only one thermocouple. Such an

arrangement can monitor the general furnace temperature but will not

give information on the temperature gradients in the apparatus and in

the sample.

An important question in the present investigation is whether the

values reported by Ramanarayanan and Rapp and by Otsuka et al. for the

molecular diffusivity of oxygen in liquid tin should be accepted as the

true values, or whether convection in the authors' experiments has

introduced error into their measured values. To find an answer to this

question, experiments were performed which examined the effect of

vertical and horizontal temperature gradients on the motion of oxygen

through liquid metals by diffusion and by convection. Computer

calculations were also performed on models of diffusivity measurements

to determine whether convection may have affected the oxygen

diffusivity measurements.








Axial Diffusion Experiments



Solid-electrolyte electrochemical cells were used for a series of

experiments to measure the diffusivity of oxygen in liquid tin and to

examine the effect of convection in the diffusivity measurements. The

electrochemical cell technique has two principle advantages in

diffusivity measurements: (1) the measurement of the diffusivity may

be made without cooling or solidifying the metal sample, and (2) the

oxygen concentration may be accurately measured in very dilute metal

solutions. With dilute solutions, the effect of the oxygen on the

physical properties of the metal is minimal. For example, an estimate

calculated from data for similar systems reported by Verhoeven [14]

shows that at the oxygen concentrations that were used in these

experiments (less than 10-4 mole fraction), the presence of oxygen

would cause a change in the density of tin on the order of 0.001%.

Since the density change is the most important fluid property in a

study of natural convection, this estimate led to the conclusion that

the low concentrations of oxygen in these experiments did not have a

significant effect on the fluid properties of tin.

The axial diffusion experiments were designed to measure the

diffusion of oxygen along the axis of a cylindrical liquid tin sample.

The electrochemical cells that were used for the axial diffusion

experiments are described in Chapter 2, as are the mathematical

procedures for interpreting the data from the experiments. The

electrochemical titration experiments were performed by first applyi!.

a potential, from 460 to 540 mV, between the sample and the pump

reference electrode that corresponded to an oxygen concentration in the








sample of about 10-4 mole fraction [99], compared to the saturation

concentration of 2x104 mole fraction [36]. After allowing several

hours for the oxygen concentration to become uniform in the sample, as

indicated by agreement between the pump and detector electrode

potentials, the applied potential was abruptly switched to 2000 mV

(except in experimental runs A2-a and A2-b, where the applied potential

was switched to 2500 mV), which imposed a vanishingly small oxygen

concentration in the portion of the sample adjacent to the pump

reference electrode. The imposed oxygen concentration for an applied

potential of 2000 mV was about 10-20 mole fraction, several orders of

magnitude lower than the initial concentration, and was considered to

be equal to zero in these experiments. The detector potential was

monitored as a function of time (two to six hours, depending on the

experimental conditions), and the difference between the time-dependent

potential and the initial potential plotted as a function of time

yielded the effective diffusivity for the oxygen tracer mass transfer

through the liquid tin sample, according to Equation 2-27.

As a test of the approximation that led to Equation 2-26, where

only the first term in an infinite sum was retained, the value of the

first term of the sum was compared to the value of the second term.

The values vary with the sample length and the effective diffusivity of

oxygen in the sample, but when average values for both of these

parameters were taken the second term was found to be 1% of the first

term after about 1000 seconds, or 0.3 hours. Since the results that

are tabulated here were taken from data recorded beginning, on average,

at 1.5 hours into the titration, the approximation is considered to be

valid for these experiments.









Figure 3-1 shows two examples of the results from the axial

diffusion experiments. Two artifacts of this plot should be explained.

The first artifact occurred at the beginning of the oxygen titration

(time = 0) when the potential difference rose immediately in an initial

transient and then dropped quickly to the expected curve. This initial

transient appeared in about half of the experiments and was a sudden

increase in the potential of the detector electrochemical cell in

response to the increased potential that is applied across the pump

electrochemical cell. The cause of the initial transient is not known

with certainty, but it appears to be an electronic polarization effect

in the electrolytes or the metal that dissipates in a short time. The

second artifact occurred at the end of the titration (time = 5 hours)

when the potential difference became constant. This artifact occurs

when the oxygen concentration in the tin sample becomes so low that the

electrochemical pumping process is at steady state with the oxygen

leaking into the sample from the argon atmosphere and the materials of

the apparatus. The final potential difference that is plotted in

Figure 3-1 for Run Number A5-a corresponds to an oxygen concentration

of about 10-7 mole fraction in the tin metal.

The dimensions and thermal conditions of the axial diffusion

experiments are listed in Table 3-1. In Tables 3-1 through 3-3, the

run numbers consist of A, signifying an axial diffusion experiment, a

number, signifying the particular electrochemical cell apparatus that

was used for that run, and a lower case letter, signifying the

individual experimental run. The temperature measurements in these

experiments were subject to some uncertainty because the thermocouples

could not be located as close to the sample as desired. The







Table 3-1. Dimensions and thermal conditions from the axial diffusion
experiments.

Sample Sample
Length Radius
Run No. (1) (r) R/r T' AT/Ax AT/Ar
(mm) (mm) (C) (C/cm) (oC/cm)

A2-a 13.06 4.24 3.08 700.0 -0.15 -
b 683.8 7.3 -

A5-a 19.60 4.46 4.39 701.4 -0.07 -
b 712.1 -10.11 -

A6-a 19.50 4.46 4.37 699.6 0.07 -
b 717.4 -9.93 -

A7-a 16.08 4.46 3.61 700.4 -0.71 -
b 692.0 1.60 -
c 702.5 1.00 -

A8-a 12.32 4.43 2.78 700.2 0.0 0.2
b 700.3 -0.21 -0.2
c 700.7 0.43 -

A9-a 14.25 4.43 3.21 702.4 0.43 -0.6
b 700.1 -0.50 -1.1
c 702.4 1.86 -0.3
d 700.8 3.36 0.3

AlO-a 12.26 4.44 2.76 700.8 -0.14 0.8
b 703.0 0.33 -1.9

A12-a 12.32 4.46 2.76 699.8 0.31 1.9
b 699.5 0.33 1.9

A13-a 12.32 4.46 2.76 700.8 0.29 0.9
b 698.9 -0.04 1.1








thermocouples that measured the vertical temperature gradient were

separated from the sample by a zirconia crucible wall, the excess

liquid tin, and the fused silica sample chamber wall. The

thermocouples that measured the horizontal temperature gradient were

located 2 mm and 8 mm radially away from the top of the tin sample and

were also separated from the sample by the various cell materials. The

size of the thermocouples and the tendency of the liquid tin to attack

the thermocouple metals prevented the thermocouples from being located

any closer to the sample. The temperature measurements listed are the

best that could be obtained, but there may be significant differences

between the measurements and the actual temperature gradients that were

in the samples.

The geometry of the axial diffusion experiments made it difficult

to control the temperature gradients in the samples. A major

difficulty was the uncertainty in the Type K thermocouples that were

used. At the temperature of the experiments (7000C) the Type K

thermocouple uncertainty is 5.4C [95], and when this uncertainty is

applied to thermocouples spaced 2.0 cm apart, the uncertainty in the

measured temperature gradient becomes l.4C/cm. Temperature gradients

smaller than this are reported in the experimental results because the

thermocouples that were used were calibrated against one another. The

calibration of the thermocouples improved their precision to within

+1C, but their accuracy was unchanged. With the calibration of the

thermocouples the estimated error in the measured temperature gradients

is 0.3C/cm. Another difficulty was that the horizontal temperature

gradients could not be adjusted with the furnace controllers. They







Table 3-2. Calculated thermal Rayleigh numbers, measured detector
potentials, and calculated oxygen concentrations from the axial
diffusion experiments.

Run No. RaT EO Cxl105
(mV) (mole fraction)

A2-a 115 61.9 4.37
b -5620 11.2 9.83

A5-a 273 6.6 16.9
b 39400 22.5 15.0

A6-a -268 7.7 15.8
b 38000 63.6 6.49

A7-a 1260 19.5 12.1
b -2830 44.4 5.42
c -1770 36.8 8.46

A8-a 0 74.1 3.29
b 128 57.4 4.91
c -262 43.1 6.97

A9-a -469 25.7 11.0
b 545 10.7 14.9
c -2030 23.6 11.6
d -3660 19.1 12.4

AlO-a 84 65.3 4.12
b -197 27.5 10.7

A12-a -189 46.6 6.27
b -201 49.1 5.86

A13-a -177 44.7 6.72
b 24 27.6 9.64








would vary in a non-reproducible fashion with the temperature of the

furnace elements and with the position of the apparatus in the furnace.

Table 3-2 lists certain conditions of the axial diffusion

experiments, including the calculated thermal Rayleigh numbers, the

initial detector potentials, and the initial oxygen concentrations that

were calculated from the potentials. The characteristic length that

was used in the calculation of the Rayleigh number was the vertical

length of the tin sample. Table 3-3 lists the slopes and intercepts

that came from the plots of the experimental data. In Table 3-3,

measured represents the intercept of the line that was drawn through

the data points while theoretical represents the intercept calculated

according to Equation 2-27,

RT 4
theoretical = n -. (3-3)
nF 7


Table 3-3 also lists the effective oxygen diffusivities that were

calculated from the slopes. Figure 3-2 is a plot of the diffusivities

as a function of the ratio of measured to btheoretical. This plot

shows that the value of bmeasured/btheoretical is a good indication of

whether convection in the liquid is affecting the diffusivity

measurement.

Stability theory calculations [88,89,90] predict that the onset

of convection will occur when the critical Rayleigh number is exceeded.

For the axial diffusion experiments, where the sample is tin at 700C,

the critical Rayleigh number depends on the aspect ratio (./r) of the

sample and is between 10,000 and 30,000 for the geometries that were

used in the axial diffusion experiments. However, conventional

stability theory calculations usually assume thermal boundary




68


Table 3-3. Straight line slopes and intercepts calculated from the
data, the ratios of the measured slope to the slope calculated by
Equation 2-27, and the measured effective oxygen diffusivities from the
axial diffusion experiments.


Run No.


A2-a
b

A5-a
b

A6-a
b


slope
(mV/hr)

8.58
51.3

10.1
108.

11.0
238.

24.4
11.0
37.8


-bmeas.
(mV)

-23.6
-78.9

12.9
211.

5.42
35.2

32.5
16.1
70.7

59.0
60.6
8.30

273.
356.
10.7
54.2

5.62
44.7

12.2
12.8

8.15
30.6


9.29
10.1

8.69
24.7


bmeas./btheor.


-2.33
-7.92

1.27
20.5

0.535
3.41

3.21
1.60
6.96

5.82
5.98
0.819

26.8
35.2
1.05
5.35

0.554
4.40


A7-a
b
c

A8-a
b
c

A9-a
b
c
d


AlO-a
b

Al2-a
b

Al3-a
b


D effx105
(cm2/s)

3.93
23.9

10.4
110.

11.2
239.

16.7
7.61
25.8

25.5
25.5
11.0

61.4
111.
12.6
18.0

6.93
30.5


62.7
62.6
27.0

113.
204.
23.2
33.2

17.1
75.9

22.8
24.8

21.4
60.5


1.21
1.27

0.804
3.02









conditions on the sample that cannot be achieved in an experimental

apparatus, such as side walls that are perfectly insulating or

perfectly conducting. The typical consequences of these assumptions is

that the only temperature gradients in the sample are in the vertical

direction. In the axial diffusion experiments, where there are

horizontal temperature gradients in the sample as well as the

externally imposed vertical temperature gradients, convection occurs

when the Rayleigh numbers are much lower than the critical values

calculated from stability theory.

The effective oxygen diffusivities that were measured in the

axial diffusion experiments are plotted with the imposed vertical

temperature gradients in Figure 3-3. The measured effective

diffusivities are plotted with the measured horizontal temperature

gradients in Figure 3-4. In Figure 3-5, the measured effective

diffusivities are plotted with the Rayleigh numbers calculated from the

experimental conditions. Experimental runs with large vertical

temperature gradients were performed to measure the differences in

effective oxygen diffusivity caused by large versus small and by

positive versus negative vertical temperature gradients.

In the first experiments that were performed (Experiments A2, A5,

and A6), the vertical temperature gradient was deliberately switched

from small to large magnitudes to see the difference between a stagnant

and a convecting liquid. After it became apparent that there was

considerable convection even with very small vertical temperature

gradients, experiments were performed at positive and at small negative

vertical temperature gradients, and the measurement of the horizontal

temperature gradients was begun.









In all of the axial diffusion experiments, the reproducibility of

the results was a problem. The measurement of the reproducibility was

limited by the number of runs that could be performed in each apparatus

before oxidation of the cell materials, especially the electrical

contacts to the electrode materials, interrupted the operation of the

cell. Some good examples of the reproducibility of the experiments

were obtained, however. Cell A12 is the best example, where two runs

performed under nearly identical conditions gave nearly identical

results. Runs A8-a and A8-b are another case of good reproducibility,

since the differences in the temperature gradients in the two runs are

small. Examples of poor reproducibility are found in the results from

Cells A10 and A13, where there are small differences in the temperature

gradients but large differences in the effective diffusivities.

With a few exceptions, the results from the different cells are

generally consistent with one another. The cluster of effective

diffusivity values measured with nearly zero vertical temperature

gradients shows the consistency of the experiments. The results are

sufficiently scattered that it is difficult to state quantitative

conclusions from the results of the axial diffusion experiments.

Qualitative conclusions may be drawn from the results taken as a whole.

One conclusion is that small changes in the vertical temperature

gradients have a larger impact on the effective diffusivities than do

small changes in the horizontal temperature gradients. Also, Figure 3-

4 shows that positive horizontal temperature gradients (hotter at the

circumference than at the axis) produce smaller effective

diffusivities. Finally, the results from the two experimental runs

with the highest Rayleigh numbers confirm the stability theory








prediction that large destabilizing temperature gradients will produce

convection. The other experimental runs, with Rayleigh numbers either

near zero or in the stabilizing direction, show that there are other

causes of convection besides destabilizing vertical temperature

gradients.

Most of the values for the diffusivity of oxygen in liquid tin

that were measured in the axial diffusion experiments are less than the

upper error limits for the values that were reported in the literature.

It should be noted, however, that all of the results are higher than

the literature values. The conclusion to be taken from the comparison

to the literature values for the molecular diffusivity of oxygen in tin

is that convection existed to some degree in all of the axial diffusion

experiments. More definitive conclusions concerning the extent of the

convection require better experimental designs that will improve the

reproducibility of the effective diffusivity measurements. Still, it

is possible to conclude that the geometry of the axial diffusion

experiments is prone to convection and that other geometries may give

better results in diffusivity measurements.



Radial Diffusion Experiments



The radial diffusion experiments were designed to measure the

diffusion of oxygen in the radial direction in a long, cylindrical

sample. The apparatus and the mathematics for interpreting the data

are described in Chapter 2. A radial diffusion experiment was

conducted by first holding the applied potential of the electrochemical

cell constant at a value of E1 for a few hours to insure a uniform









oxygen concentration in the cylindrical sample. The applied potential

was then switched to a new value, E2, and the cell current was

monitored as a function of time as the oxygen diffused into or out from

the sample in the radial direction. The ionic current through the cell

as a function of time gave the rate of the oxygen tracer mass transfer

through the liquid tin sample.

In an ideal electrochemical cell, the current through the cell

will become zero as the concentration of oxygen in the sample

approaches the value set by the applied potential. In an actual

electrochemical cell, there is a electronic current that remains after

the ionic current (the flow of oxygen ions through the electrolyte) has

vanished. The electronic current is a constant depending on the

conductivity of the electrolyte material and is the value of the

current that is reached after sufficiently long times. In the larger

YSZ tube, the electronic current could be measured after 1.5 to 3

hours, while in the smaller tube the electronic current could be

measured after 0.5 to 1.0 hours. The ionic current may be found

according to the relationship

ionic = total electronic' (3-4)

Examples of the ionic current as measured in the radial diffusion

experiments are plotted in Figure 3-6.

The apparatus in the radial diffusion experiments had the

advantage over the axial diffusion experiments in several ways. First,

the assembly of the apparatus offered little chance for oxygen leakage

between the halves of the electrochemical cell. Also, the use of only

three materials (the tin sample, the zirconia tube, and the porous

platinum electrode) in concentric cylinders provided a symmetry and a








simplicity that eliminated many of the uncertainties in the temperature

measurements. Finally, as will be shown later, the long cylindrical

sample was less subject to convection-induced determinate errors in the

measured diffusivity.

Table 3-4 lists the thermal conditions and the applied potentials

in the radial diffusion experiments. Table 3-5 lists the slopes and

intercepts of the plots of

-In Iionic = t/ In B (3-5)

that were made from the experimental data, and the diffusivity values

that were calculated from the slopes. In Tables 3-4 and 3-5, the run

numbers consist of R, signifying a radial diffusion experiment, a 1 or

5, signifying experiments performed in the larger or the smaller YSZ

tube, a number signifying the date of the experiment and a letter

signifying the individual experimental run. Measurements were made

with two inside tube diameters: 1.598 cm and 0.456 cm. Owing to the

thermal expansion of the tin, the aspect ratio of the sample would

change with the temperature. In the larger diameter tube the aspect

ratio (2/r) was 9.68 at 700C and 9.51 at 800*C. In the smaller

diameter tube the aspect ratio was 48.4 at 7000C, 48.6 at 750C, and

48.7 at 8000C.

A visual inspection determined which data fell close enough to a

straight line to be included in the least squares calculation for the

slope and the intercept of the line. To test the approximation that

produced the equation for the line, where only the first term in an

infinite sum was retained, the value of the first term of the sum was

compared to the value of the second term. The values vary with the

sample radius and the effective diffusivity of oxygen in the sample,







Table 3-4. Dimensions, thermal conditions, and applied potentials from
the radial diffusion experiments.


Run No.


AT/Ax
(C/cm)

Tube, Diameter = 1.
0.11
0.12
0.14
0.11

0.20
0.18
0.26
0.35


E1
(mV)

598 cm
995.2
999.4
998.6
998.5

942.9
928.3
918.9
911.9


R1-117A
118A
118B
119A

R1-120A
120B
123A
123B


R5-1023A
1024A
1024B
1024C
1024D
1025A
1025B
1025C
1025D
1025E
1026A
1026B
1208A
1208B
1211A
1211B
1212A

R5-1027A
1030A
1030B
1030C
1031A
1031B
1031C
1101A
1101B
1102A
1102B
1103A
1103C
1106A
1106B
1106C


T
(C)

Large
699.9
700.5
693.5
699.1

798.8
801.6
801.8
799.8

Small
697.8
696.9
697.3
697.4
697.7
697.9
697.7
697.7
697.7
698.0
697.5
697.9
697.5
697.7
697.6
697.5
697.4

748.3
748.7
748.7
748.4
748.6
748.5
748.4
748.6
748.9
749.3
749.4
747.7
748.0
748.0
748.1
747.4


1588.9
1389.8
1589.3
1292.0
1490.8
1687.0
1391.6
1193.1
1391.6
1589.5
1292.3
1490.9
1687.3
1389.0
1586.6
1289.9


Tube, Diameter = 0.456 cm
-0.00 1495.0
0.01 1295.2
0.01 1494.8
0.01 1693.2
0.01 1395.2
-0.01 1593.7
0.03 1295.1
0.03 1494.2
0.03 1195.4
0.03 1394.6
0.01 1593.0
0.07 1294.9
-0.07 1669.5
-0.08 1256.2
-0.07 1688.4
-0.03 1292.8
-0.03 1691.8


-0.03
-0.04
-0.03
-0.04
-0.03
-0.02
-0.04
-0.02
-0.11
-0.02
0.00
-0.03
-0.02
0.00
0.00
0.02


E2
(mV)


1401.6
1401.7
1401.6
1401.9

1396.9
1396.0
1395.7
1395.9


1295.9
1494.8
1693.2
1395.2
1593.9
1295.1
1494.2
1195.4
1394.6
1593.3
1294.9
1593.0
1256.2
1670.8
1292.8
1690.6
1294.1

1391.3
1589.3
1292.0
1490.6
1687.0
1391.6
1193.1
1391.6
1589.7
1292.3
1490.7
1687.2
1391.8
1586.6
1289.9
1487.8








-- continued.


Table 3-4

Run No.


R5-1108A
1108B
1109A
1109B
1109C
1110A
1110B
1110C
1110D
1113A
1113B
1113C
1113D


T
(oC)

797.7
798.6
798.7
798.3
798.8
798.1
798.1
798.1
798.1
798.3
798.1
798.2
798.8


AT/Ax
(C/cm)

-0.04
0.03
0.02
0.02
0.02
0.01
0.01
0.02
0.02
0.03
0.04
0.04
0.07


E1
(mV)

1482.7
1674.8
1383.9
1186.2
1481.7
1764.2
1383.0
1764.6
1383.2
1672.3
1381.4
1672.4
1381.2


E2
(mV)

1674.8
1384.3
1186.2
1481.7
1764.0
1383.0
1764.6
1383.2
1673.6
1381.4
1672.4
1381.2
1672.4






Table 3-5. Slopes and intercepts for Equation 3-5 that were measured
in the radial diffusion experiments, and the calculated effective
oxygen diffusivities. (Points that were discarded from the calculation
for the mean diffusivity are marked with *.)

Run No. 1/r -In B DOeffx05
(min-) (cm2/s)

Large Tube, Diameter = 1.598 cm
R1-117A 0.0620 -2.86 11.9*
118A 0.1020 -3.60 18.8*
118B 0.1009 -3.55 18.6*
119A 0.0553 -2.24 11.6*

R1-120A 0.0426 -2.31 8.42*
120B 0.0764 -2.94 12.5*
123A 0.0341 -2.56 6.71*
123B 0.0237 -1.66 4.82*

Small Tube, Diameter = 0.456 cm
R5-1023A 0.198 0.667 2.96*
1024A 0.0209 0.0749 0.314*
1024B 0.0825 -0.554 1.24
1024C 0.0852 -0.429 1.28
1024D 0.0218 -0.438 0.326*
1025A 0.0791 0.0384 1.19
1025B 0.567 0.696 1.21
1025C 0.113 0.803 1.70
1025D 0.242 1.046 3.63*
1025E 0.0855 0.0322 1.28
1026A 0.0791 0.132 1.18
1026B 0.0892 -0.175 1.34
1208A 0.0383 0.214 0.574*
1208B 0.0452 -0.219 0.677*
1211A 0.0407 0.253 0.610*
1211B 0.0414 -0.0467 0.621*
1212A 0.0381 0.231 0.571*

R5-1027A 0.137 -0.386 2.05*
1030A 0.0833 -0.324 1.25*
1030B 0.0978 -0.515 1.47
1030C 0.0883 0.0606 1.32*
1031A 0.144 -0.937 2.15*
1031B 0.0994 -0.828 1.49
1031C 0.0970 1.066 1.45
1101A 0.114 0.726 1.70
1101B 0.0859 -0.321 1.29*
1102A 0.0883 -0.133 1.32*
1102B 0.0994 0.247 1.49
1103A 0.158 -0.921 2.36*
1103C 0.0102 -0.705 1.53
1106A 0.0969 -0.456 1.45
1106B 0.0990 -0.335 1.48
1106C 0.110 0.364 1.65




\







Table 3-5 -- continued.

Run No. 1/r -In B D effx105
(min ) (cm /s)

R5-1108A 0.175 -0.825 2.62*
1108B 0.101 -0.829 1.51
1109A 0.0612 0.462 0.917*
1109B 0.0667 -0.162 0.999*
1109C 0.147 0.367 2.20
1110A 0.0855 -0.786 1.28*
1110B 0.286 -1.941 4.29*
1110C 0.0984 -0.922 1.47
1110D 0.181 -1.355 2.71*
1113A 0.103 -0.795 1.55
1113B 0.123 -0.972 1.85
1113C 0.102 -0.882 1.52
1113D 0.126 -1.041 1.89








but when average values for both of these parameters were taken the

second term was found to be 1% of the first term after about 5000

seconds, or 1.4 hours, in the larger tube and 400 seconds, or 7

minutes, in the smaller tube. The results that are tabulated here for

the smaller tube were taken from data that was recorded beginning, on

average, at 6 minutes into the titration and continuing for several

minutes. The approximation is considered to be valid for these

experiments. The results that are tabulated for the larger tube were

taken from data that was recorded beginning, on average, at 0.6 hours

into the titration. Even though the lines formed by the data taken

from the larger tube appeared straight, the validity of the

approximation is borderline and so these results are questionable.

Stability theory states that the experiments with the larger tube

diameter and the smaller aspect ratio would be more vulnerable to

convection induced by temperature gradients, and the results of the

experiments confirm this prediction. The effective diffusivities

measured in the larger tube were all higher than the diffusivities

measured in the smaller tube. Even in the smaller tube, with a

geometry that was more resistant to buoyant convection, there was a

degree of scatter in the measurements. Five measurements, numbered R5-

1208A, R5-1208B, R5-1211A, R5-1211B, and R5-1212A, had a suspected

determinate error. The diffusivities that were measured in these runs

were consistently smaller than most of the other measurements and were

taken after the sample had been at the experimental temperature for

about 50 days. It was about this time that the measured potential of

the electrochemical cell began to drift lower, indicating an impurity

or a chemical side reaction in the sample. It was concluded that these









diffusivity measurements were suspect and they were not included in the

calculation of the mean diffusivity. The other values marked in Table

3-4 as not included in the calculation of the mean diffusivity were

excluded by the Student's t test [100] as too scattered. They were,

however, included in the calculation of the confidence limits of the

result.

The mean diffusivity, calculated with 95% confidence limits, from

the radial diffusion measurements is

Doeffective = 2.5(+8.8,-1.3)xlO-4exp[-(57203000)/RT], (3-6)

where R = 1.9872 cal/mole-K and T is the absolute temperature (K).

Figure 3-7 shows the results of the radial diffusion experiments

plotted together with the reported diffusivities from the literature.

A general trend that is evident in the three sets of results is that

the radial diffusion measurements give smaller diffusivities than the

axial measurements, and that the smaller diameter, higher aspect ratio

measurements give the smallest results of all. This trend is in line

with the prediction from stability theory that high aspect ratio

geometries are the least vulnerable to convection. This trend also

hints that even in the lowest measured diffusivity values, convection

may still be adding a determinate error to the measured diffusivities.

Figure 3-8 shows a small but consistent correlation between the

diffusivity and the vertical temperature gradient measured on the

outside of the zirconia tube in the experimental apparatus. The

correlation shows a slightly higher measured diffusivity in the

experiments with positive vertical temperature gradients. Plots of the

measured diffusivity against other experimental variables, such as the

potential applied to the electrochemical cell or whether oxygen was









titrated into or out of the tin, showed no apparent correlation. The

temperature gradient is thus left as the only known source of

determinate error in the diffusivity measurements.



Modelling Procedure and Results



Mathematical models of the oxygen diffusivity experiments were

calculated with the FLUENT computational fluid dynamics software

(FLUENT/CVD versions 2.93 and 2.99) developed by Creare, Inc. (Hanover,

NH). The software ran on a MicroVAX III workstation (Digital Equipment

Corp., Maynard, MA) and on the CRAY-2S supercomputer (Cray Research,

Inc., Mendota Heights, MN) at the National Aeronautics and Space

Administration Langley Research Center in Hampton, Virginia. FLUENT

solves the equations of motion for fluid flow by a finite difference

numerical procedure on a user-defined computational grid.

Dimensionless forms of the equations of motion are given in Table 3-6.

The physical properties of the liquid are included in the calculation

as temperature-dependent functions, and multiple chemical species may

be included with their binary diffusivities. Heat conduction through

solid materials may also be a part of the model when the effect of the

container on convection in the liquid sample is to be considered.

An important question raised by the experiments of Otsuka et al.

[98] is the effect of the several different materials in the

experimental apparatus on the thermal boundary conditions on the

sample. The goal of the computer models is to see how the thermal

boundary conditions on the sample differ from those imposed by the

furnace, and to see if there is a possibility for convection in the





81


Table 3-6. Dimensionless equations of change for modelling fluid flow.

Continuity V v = 0

1 av RaT RaS 2
Motion + v Vv = -VP +- TF +- CF + V v
Pr at Pr Sc

Pr T
Energy + Pr(v VT) = V2T
Sc at

ac
Species + Sc (v VC) = V2C
at

v = velocity vector
Pr = Prandtl number
t = time coordinate
P = pressure
RaT = thermal Rayleigh number
T = temperature
RaS = solutal Rayleigh number
C = species concentration
Sc = Schmidt number (momentum diffusivity/mass diffusivity)
F = body force vector (including gravity)









liquid sample. The article published by Otsuka et al. does not

describe the geometry of their apparatus in sufficient detail for a

realistic model, so experiments on the diffusion of oxygen in liquid

lead, described by Otsuka and Kozuka [37], were substituted. The

geometries of the experiments with tin and lead were very similar (as

seen in the diagrams in the respective articles) and the authors

assumed in both cases, based on the same reproducibility tests, that

there was no convection in the liquids.

The computer models of the axial diffusion experiments were done

on two scales. At the larger scale, called the furnace model, the

entire experimental apparatus was modelled and at the smaller scale,

called fluid flow models, only the lead samples were modelled. The

furnace model was calculated to determine the thermal boundary

conditions that should be applied to the sample in the fluid flow

model. This model was based on the description in the article by

Otsuka and Kozuka and their diagram, reproduced here as Figure 3-9. In

the furnace model the sample material was treated as a conducting solid

along with the zirconia electrolyte materials, the alumina container

materials, and the reference electrode materials of the electrochemical

cells. The argon atmosphere was included as a gas moving by natural

convection in the space above the sample and the surrounding ceramic

materials. The physical properties that were taken from the literature

and used in the FLUENT models are listed in Table 3-7. The furnace

model used cylindrical geometry and, taking advantage of the symmetry

of the apparatus, included only two dimensions on a finite difference

grid with 100 nodes in the axial direction and 40 nodes in the radial

direction. The center line of the apparatus was represented by a








Table 3-7. Physical properties
flow model calculations.

Material Property

Zirconia Thermal Conductivity

Alumina Thermal Conductivity

Nickel Thermal Conductivity

Argon Thermal Conductivity
Viscosity
Specific Heat

Lead Thermal Conductivity
Viscosity
Specific Heat
Density
Oxygen Diffusivity


used in the FLUENT furnace and fluid


Value (T in Kelvin) Reference

1.296 + 7.196x104T W/m-K [96]

13.47 5.71x10-3T W/m-K [101]

50.2 + 0.0216T W/m-K [92]

0.0160 + 2.76x10-5T W/m-K [102]
6.5x10-5 kg/m-s [102]
520 J/kg-K [102]

7.99 + 0.0136T W/m-K [92]
3.44xl03 2.00x10-6T kg/m-s [103]
138.3 J/kg-K [104]
11240 0.98T kg/m3 [104]
2.35xl04 cm2/s [37]









symmetry boundary condition, with no temperature gradients and no

velocities normal to the symmetry boundary. Otsuka and Kozuka built

their experiments with four different sets of length and diameter

dimensions for the sample, and in the furnace model this was

represented by an average of the four sets of dimensions.

A temperature profile of 1.5*C/cm was established in the

experimental furnace by Otsuka and Kozuka in order to prevent

convection in the sample caused by destabilizing temperature gradients.

A common practice in diffusion measurements is to design the furnace so

that there is a slight vertical temperature gradient to make the top of

the sample less dense than the bottom. The sample is given this sort of

temperature gradient to stabilize the liquid and prevent convection.

In the furnace model, the thermal boundary condition on the outside of

the apparatus in the region of the sample was a vertical temperature

gradient of 1.5C/cm, with the higher temperature above the lower

temperature on the sample, in order to duplicate the thermal conditions

that were in the experiments that were being modelled.

The upper portion of the apparatus, well above the sample, was

given a steeper vertical temperature gradient in the furnace model.

This steeper gradient modelled the portion of the apparatus that passed

from the interior of the furnace to the unheated ambient conditions.

At the top of the apparatus, outside of the furnace, the alumina tubes

make contact with 0-ring seals, which have a maximum operating

temperature much lower than the sample temperature. The extension of

the apparatus out of the furnace was modelled to determine whether the

sample was affected by heat that was conducted out of the furnace by

the materials of the experimental apparatus.









The FLUENT calculations on the furnace model showed that the

materials were far enough into the furnace that sufficient heat was not

conducted from the cooler part of the apparatus to affect the

temperature of the sample. The calculations also showed that if the

apparatus had been homogeneous, there would have been no horizontal

temperature gradients in the apparatus near the sample. Since the

apparatus was not homogeneous and there was an imposed vertical

temperature gradient, the differing thermal conductivities of the

materials in the apparatus caused horizontal temperature gradients in

the sample. The plot of the calculated temperature isotherms in Figure

3-10 shows how the alumina and zirconia, with their lower thermal

conductivities, bore most of the vertical temperature gradient. The

steepest horizontal gradients in the sample were in the upper and lower

corners, near the junctions of materials with three different thermal

conductivities.

The fluid flow models, as opposed to the furnace model, consisted

only of the cylindrical liquid metal sample with thermal and velocity

boundary conditions imposed. These models were also calculated on a

two-dimensional grid, shown in Figure 3-11, with 50 nodes in the axial

direction and 20 nodes in the radial direction. The two-dimensional

grid presumes that the flow in the sample is axisymmetric. As a test

of this assumption, two three-dimensional models were evaluated. In

the first model, cylindrical coordinates were used and a symmetry

condition applied along the axis of the cylinder. In the other three-

dimensional model, rectangular coordinates were used and the outer wall

of the fluid volume consisted of stepped computational cells that

approximated the cylindrical wall. The three-dimensional models were








given the same dimensions and thermal boundary conditions as the two-

dimensional Model A. A sample of the velocity vectors that were

calculated for the cylindrical three-dimensional model is plotted in

Figure 3-12. Both three-dimensional models gave axisymmetric results

that were reasonably close to the results of the two-dimensional case.

Muller, Neumann, and Matz [21] also predict that the flow in a cylinder

heated from the top would be axisymmetric. The three-dimensional test

cases established the credibility of the two-dimensional model, and the

reported cases were all done in two dimensions in order to reduce the

computational time.

In the two-dimensional fluid flow model the axis of the

cylindrical sample was modelled by a symmetry boundary condition. The

velocity boundary conditions were that the fluid was not moving at the

walls (the "no-slip" boundary condition). The thermal boundary

conditions on the sample in the fluid flow model were taken from the

results of the furnace model. The top and bottom faces of the sample

were isothermal walls, with the top at 1275.0 K (1001.8C) and the

bottom at 1273.8 K (1000.6C). The thermal boundary condition for the

side wall was a third-order polynomial equation that was calculated

from a least squares fit of the position versus temperature data taken

from the results of the furnace model. The polynomial equation that

was used for the models with a length of 12 mm is

T(K) = 1273.81 + 0.10761x 6.5125x10-3x2 + 4.3585x104x3, (3-7)

where x is the distance in millimeters measured in the axial direction

from the bottom of the sample volume. The data and the fitted curve,

shown in Figure 3-13, show a temperature profile that was nearly linear









except for the steeper gradients at the top and bottom corners of the

sample.

Five fluid flow models were calculated. Four models, Models A,

C, D, and E, modelled the four aspect ratios (the cylinder length

divided by the cylinder radius) that were used by Otsuka and Kozuka in

their experiments. The fifth model, Model B, had the same aspect ratio

as Model A but with an adiabatic side wall instead of the polynomial

temperature profile. The calculations for each model were begun by

first calculating the temperature distribution in the sample owing to

conduction only. After the calculations had converged to a solution

for the temperature field in the sample, the velocity calculations were

begun with a stagnant liquid as the initial condition. The temperature

calculations continued as the velocity calculations proceeded, but the

flow in the liquid had only a small effect on the temperature field.

After the velocity calculations had converged, the stream lines in the

liquid for each model were plotted. The stream lines for Model A, with

an aspect ratio of 2.67, are shown in Figure 3-14, the stream lines for

Model C, with an aspect ratio of 4.80, are shown in Figure 3-15, the

stream lines for Model D, with an aspect ratio of 1.78, are shown in

Figure 3-16, and the stream lines for Model E, with an aspect ratio of

3.20, are shown in Figure 3-14.

The stream function and velocity fields in each case defined the

extent of convection in the sample. After these fields had been

calculated, FLUENT was used to calculate oxygen mass transfer through

the liquid. The initial and boundary conditions for the oxygen were

chosen to match the experimental procedure for the electrochemical

cells. Initially, the oxygen concentration in the lead was uniform at









10-3 mole fraction. The molecular diffusivity of oxygen in lead was

fixed at the value given for the average temperature of the models by

Otsuka and Kozuka in their experimental results. At time t=O, the

oxygen concentration at the lower face was fixed at zero, and FLUENT

would then calculate oxygen mass transfer toward this boundary in five

second time steps. The results of the calculations were interpreted in

the same manner as the data from the electrochemical cell experiments.

The oxygen concentrations across the top face of the sample were

averaged at several large time values, the ratio between each time-

dependent average concentration and the initial concentration was

converted to a difference between a time-dependent and an initial

electrochemical cell potential according to Equation 2-12, and an

effective oxygen diffusivity for each model was calculated from the

time-dependent potential difference according to Equation 2-27. The

averaging of the oxygen concentration across the top face of the sample

was considered to be a valid approximation because the difference

between the highest and the lowest concentration was only about 10% of

the average. The calculated oxygen mass transfer was then compared to

the value for the molecular diffusivity of oxygen that was used in the

calculations.

Model B was calculated with the same dimensions as Model A in

order to provide a basis for comparison. In Model B, the top and

bottom faces of the sample had the same temperatures as in the other

models, but the side walls were treated in the calculation as adiabatic

walls, with no heat exchange between the wall and the fluid. The

temperature field calculated by this model showed no horizontal

gradient, and the subsequent velocity calculation gave a velocity of









zero at all points in the liquid, showing a completely stagnant liquid.

Because the liquid was completely stagnant, no stream lines or velocity

vectors are shown for Model B. The oxygen diffusivity calculated by

the adiabatic wall model was slightly less than the molecular

diffusivity that was used in the calculation owing to the convergence

tolerance that was applied in the diffusion calculation.

Each of the four models that were calculated showed two counter-

rotating cells of fluid, driven by the horizontal temperature gradients

in the corners of the cylinder. (In Figure 3-16, the second, smaller

rotating cell does not appear at the resolution of the stream function

contours that were plotted. The cell is in the lower left corner of

the sample and is made apparent by the shape of the bottom stream

function contour.) The calculated values for the oxygen mass transfer

are tabulated together with other results of the calculations in Table

3-8. It should be noted that the value of the effective diffusivity

that was calculated for Model E is 2% smaller than the molecular

diffusivity that was used in the calculations. This is an inaccuracy

in the FLUENT software that was caused by the choice of grid spacing

and convergence criteria for the calculations. A more accurate value

for the diffusivity could have been obtained from the model by

continuing the calculation on to longer times, but the 2% error was

considered to be small enough for the purposes of this discussion.

Table 3-8 includes the maximum fluid velocity, which typically occurred

in the upper half of the sample along the axis of the cylinder.

In Figure 3-18, the calculated effective diffusivities are

plotted together with the measured effective diffusivities and the 95%

confidence limits from the experiments of Otsuka and Kozuka. The





90

Table 3-8. Dimensions, thermal conditions, maximum fluid velocities,
and calculated diffusivities for the fluid flow models.


Model Length Dia.
(mm) (mm)


AT/Ax
(*G/cm)


2.67
3.20
4.80
2.67
1.78


V
maximum
(mm/s)


1.00 0.0
1.33 0.0536
1.00 0.0446
1.00 0.174
1.33 0.164


B
E
C
A
D


12.0
8.0
12.0
12.0
8.0


9.0
5.0
5.0
9.0
9.0


Deffxl05
(cm2/s)

23.2
23.0
23.5
30.1
34.4









calculated diffusivities and maximum fluid velocities correlate well

with the aspect ratios taken from the sample geometries and show that

the shorter, broader geometries are subject to a greater magnitude of

convection. Figure 3-18 shows that over half of the scatter in the

molecular diffusivity data of Otsuka and Kozuka may be accounted for by

the convection in their sample, and that the horizontal temperature

gradients caused a definite error in the measured molecular

diffusivity. The convection in the liquid increased the mass transfer

of oxygen through the sample, so the actual value of the molecular

diffusivity of oxygen in liquid lead is probably lower than the result

reported by Otsuka and Kozuka, but it is most likely within their 95%

confidence limits.

The model calculations also show that the horizontal temperature

gradients in the sample result from the action of the vertical

temperature gradient on the experimental apparatus. If it were

possible to design an apparatus with side walls that are perfectly

insulating, the stabilizing vertical temperature gradient would ensure

that there would be no convection in the liquid. More realistically,

better values of the molecular diffusivity may be measured in samples

that are isothermal or have high aspect ratios.



Summary and Conclusions



Values of the diffusivity of oxygen in liquid tin from literature

sources and from experiments show that, in a general sense, the value

will be lower for measurements made with higher aspect ratios. This

correlation indicates that in conventional techniques for measuring