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PUBLICATIONS

NBSIR 79-1767

411104 030005

NBS: Properties of Electronic

Materials

J. R. Manning

National Bureau of Standards

Washington, D.C. 20234

Phone: (301) 921-3354

June 1979

Annual Report

For the Period 2 April 1978 - 1 April 1979

NASA Government Order H-27954B

-QG

S. DEPARTMENT OF COMMERCE

TIONAL BUREAU OF STANDARDS

MHV I 3 1979

tvfcrf A Cc - Cl rc bdXOO

NBSIR 79-1767

• t* NBS: PROPERTIES OF ELECTRONIC

MATERIALS

J. R. Manning

National Bureau of Standards

Washington, D.C. 20234

Phone: (301) 921-3354

June 1979

Annual Report

For the Period 2 April 1978 - 1 April 1979

NASA Government Order H-27954B

U.S. DEPARTMENT OF COMMERCE, Juanita M. Kreps, Secretary

Jordan J. Baruch, Assistant Secretary for Science and Technology

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

TABLE OF CONTENTS

Page

Summary

Task 1 - SURFACE TENSIONS AND THEIR VARIATIONS WITH TEMPERATURE

AND IMPURITIES by

S. C. Hardy and S. R. Coriell

Task 2 - SOLUTAL CONVECTION DURING DIRECTIONAL SOLIDIFICATION by

W. J. Boettinger, S. R. Coriell, F. S. Biancaniello

and M. R. Cordes 25-81

Task 3 - A THERMOCHEMICAL STUDY OF CORROSIVE REACTIONS IN

OXIDE MATERIALS by

H. S. Parker, R. S. Roth, C. D. Olson, and

E. R. Plante 82-102

Task 4 - THERMODYNAMIC PROPERTIES OF REFRACTORY INORGANIC

MATERIALS AT HIGH TEMPERATURES by

J. H. Colwell 103-131

Distribution List 132-133

National Bureau of Standards

Properties of Electronic Materials

Summary

This report describes NBS work for NASA in support of NASA's

Materials Processing in Space Program covering the period April 2, 1978

to April 1, 1979. The NBS program has two main thrusts:

1) Carrying out precision measurements in space and investigating

the feasibility of improved measurements when the space environment

offers a unique opportunity for performing such measurements. These

measurements would be useful for either space processing or processes

on the ground.

2) Obtaining precision measurements on material properties when

these properties are important to the design and interpretation of

space processing experiments. These measurements would be carried out

either in space or on the ground.

The NBS work is carried out in four tasks. These tasks have, as

two of their focal points, the role of convection effects and the role

of container effects, both of which would differ in space-based experiments from those found in ground-based experiments. The results

obtained for each task are given in detailed summaries in the body of

the report. Briefly, in Task I - Surface Tensions and their Variations

with Temperature and Impurities - measurements of the surface tension,

Y, of liquid gallium were made in vacuum by the sessile drop technique

as a function of temperature T. In the range 150 °C to 750 °C, it

1
2 was determined that dy/dT = -0.068 mJ/m °C. In Task 2 - Solutal

Convection during Directional Solidification - samples of off-eutectic

Pb-rich Pb-Sn were directionally solidified and macrosegregation along

the length of the sample attributed to solutal convection was measured.

The segregation is often sufficient to prevent plane front solidification, thus causing dendritic instead of composite microstructures. In

Task 3 - A Thermo chemical Study of Corrosion Reactions in Oxide Materials

- phase relationships in the ^0^6202 system were investigated for the

80 to 100 percent portion of the system up to 1600 °C with major

corrosion of the platinum containers being avoided, perhaps because of

+2

the small amount of Fe in the specimens. In Task 4 - Thermodynamic

Properties of Refractory Materials at High Temperatures - methods of

determining heat capacities in reactive liquid samples levitated at high

temperatures, especially by use of freely-cooling spherical samples,

were analyzed.

i

2
Task 1

Surface Tensions and Their Variations

with Temperature and Impurities

S. C. Hardy and S. R. Coriell

Metal Science and Standards Division

Center for Materials Science

Summary

Instrumentation and techniques have been developed to measure

the surface tension of liquid metals using the sessile drop method

under high vacuum conditions. The drop profiles are analyzed with the

classic Bashforth and Adams method and by fitting the entire profile

to solutions of the Young-Laplace equation using a computer routine.

The results of the two analyses are in agreement. The surface tension

of gallium was measured from 30 °C to 750 °C in a vacuum of at least

— 8

10 torr. The surface tension values show some adsorption related

temperature, the surface tension

2 -0.068 mJ/m °C. These results are

any previous measurements

curvature below 150 °C; above this

decreases linearly with a slope of

not in quantitative agreement with

3
Introduction

It has been recognized for many years that surface tension

gradients produced by temperature or chemical concentration variations

on free liquid surfaces can generate significant fluid flows. In the

low gravity levels of space vehicles, flows of this kind will be the

dominant ones. A quantitative understanding of the resulting complex

phenomena often requires that the dependence of the surface tension on

chemical concentration and temperature be known with some accuracy.

These thermodynamic properties, however, are not known well enough for

many metals and semiconductors of scientific and commercial interest

to permit meaningful analysis of observed effects or adequate design

of future experiments. Thus accurate measurements of the temperature

and chemical concentration dependence of the surface tension of

materials are necessary before a basic understanding of the fluid

flow behavior is possible. It is the objective of this program to

make such surface tension measurements.

Gallium was selected for initial study because it is a convenient

model fluid for thermocapillary flow experiments and because there is

considerable uncertainty about its surface tension and, more critically,

the temperature variation of its surface tension. Previous measurements are not in agreement, probably due to the sensitivity of the

liquid gallium surface to contamination by impurities.

The evidence for the role of impurities in surface tension

measurements is of two types. The first effect is the observation of

4
a nonlinear temperature dependence of the surface tension. Typically

what is found is that the tension decreases with increasing temperature

as expected, but the values at low temperatures lie below the linear

extrapolation of the high temperature data. This is interpreted as

evidence of enhanced adsorption and consequent surface tension depression at lower temperatures. Several previous measurements have

found such a nonlinear temperature dependence for the gallium surface

[i 21 tension . It must be noted, however, that there does not seem to

be any compelling theoretical reason to expect the surface tension to

decrease linearly with increasing temperature. Recent careful measure-

[31 ments with zinc have found a different dependence .

The second type of effect which suggests the presence of impurities is the observation of a time dependent surface tension. A

freshly formed surface will be reasonably clean. As the concentration

of impurities on the surface builds up to equilibrium, the surface

tension drops to a constant value. Early measurements of the gallium

surface tension depended on forming a new pendant drop and recording

[41 its shape quickly before significant adsorption could occur . The

value found for the surface tension at the melting point was 735 +

2

20 mJ/m ; most subsequent measurements have been slightly lower than

this range.

Some recent measurements of the gallium surface tension obtained

using the pendant drop technique and displaying a typical decrease with

2 time is important because the observed initial values over 800 mJ/m

5
are much higher than any previously reported^. This suggests that

all existing surface tension values for gallium were affected by

impurities. We performed similar experiments last year^ with somewhat different results: although we did not observe high initial

values, occasionally the surface tension would gradually rise from

about 700 mJ/m to around 800 mJ/m . The effect was not reproducible,

which suggested an uncontrolled impurity, probably in the inert

atmosphere in which the measurements were made. We therefore thought

it desirable to make future measurements in a vacuum system. This is

feasible because gallium has a very low vapor pressure. By using the

sessile drop technique we could also conveniently study the surface

tension over a wide temperature range, a primary goal of this program.

Experimental Procedures

The sessile drop technique is considered to be the most accurate

[71 method for measuring surface tensions . It is based on a comparison

of the observed profile of a drop sitting on a horizontal plane with

the profile calculated by solving the Young-Laplace equation. No

approximations are necessary in this comparison, although several

treatments which simplify the surface tension calculation have been

developed. We are using an experimental modification of the sessile

drop technique in which the drop sits in a cup rather than on a

r s i plane 1

• The cup stabilizes the drop and imposes a circular cross

section on the drop base, thus eliminating errors arising from the

anisotropic wetting which sometimes is encountered when the drop rests

on a flat surface.

6
Figure 1 is a schematic diagram of the experimental apparatus

as seen from the top. The drop cup is contained in a vycor tube

which connects to a vacuum system consisting of a 30 liter/sec.

ion pump, two crosses, a nude ionization gauge, a bakeable

isolation valve, and a mechanical pump and cold trap which are not

shown. The cup rests loosely in a flat bottomed depression machined

in a steel block. The depression keeps the cup near the center of the

tube and also catches any gallium which might spill during the

loading procedure or the experiments. The block rests in a semicircular section cut from a stainless steel tube which fits loosely

inside the vycor tube. The steel tube and the block and cup which it

contains can be rotated within the outer vycor tube by using a magnet.

This permits changes and adjustments in the cup orientation. A

chromel-alumel thermocouple is spot welded to the steel block and

to the nickel leads of a pressed seal located near the flange in which

the vycor tube is mounted. The vycor tube has an optical grade quartz

window on the end through which the drop is photographed. A tube

furnace or a heating tape is used to establish the drop temperature.

Before the gallium is placed in the cup, the system is evacuated

and the block and cup assembly baked with a heating tape at about

500 °C for an hour. After cooling, the ion pump is turned off and the

system filled with helium gas. The gallium is inserted in the cup

using a long glass pipette connected to a micrometer syringe. This

3 device is filled with absolute alcohol and about 10 cm of gallium

are drawn into it. The flange containing the window is removed from

7
the vacuum system and quickly replaced with a tightly fitting stopper

through which pass two vacuum couplings. Helium gas is flowed through

a tube which extends into the vacuum system to the area of the cup

through one of these couplings. The pipette containing the gallium is

inserted through the second coupling and pushed down the vycor tube

until it is located directly over the cup. After flushing the system

with helium for about ten minutes, gallium is forced from the pipette

into the cup by moving the plunger of the micrometer syringe. When

the drop has been formed in the cup, the pipette is first withdrawn;

the helium feed tube and stopper are then removed and the window

flange replaced using a new copper gasket. The system is now slowly

evacuated by gradually opening the valve to the cold trap and mechanical pump. This is a delicate part of the procedure because if the

pressure is reduced too quickly, gas generated in the cup can expel

some of the gallium and reduce the drop volume to an unusable size.

After evacuation with the mechanical pump, the valve is closed

and the ion pump started. When it is clear that this is pumping

properly, the valve is tightened with a torque wrench to establish a

vacuum tight seal and the mechanical pump is turned off. When the

— 6

system pressure is about 10 torr, the gallium drop and the cup

assembly are baked at about 400 °C for about an hour. This cleans

the gallium surface and raises the surface tension producing a drop

with a high profile. The ion pump is able to reduce the pressure to

— 8

the low 10 torr range overnight after this treatment. A subsequent

8
bake of the vacuum system at about 250 °C is necessary to get to the

_9

10 torr level.

The sessile drop technique can be used to make highly accurate

surface tension measurements. In order to do so, however, the dimensions of the sessile drop must be measured to an accuracy of a few

4

parts in 10 . This is a nontrival measurement requirement; all

elements of the optical system must be carefully aligned and focussed.

The primary consideration is the lens used to image the drop. The

numerical aperture of this lens must be large in order to obtain

sufficient resolution, which rules out the use of the standard type

of small laboratory telescope. In our system we use a Wollensak

f/5.6 enlarging lens with a 190 mm focal length. The image of the

drop is focussed on the ground glass plate of a 4"x5" camera whose

lens has been removed. The diaphram of this camera is located at the

second focal point of the enlarging lens and a small aperture is used

to eliminate nonparaxial rays. The lens is mounted in a movable

stage taken from a zone refiner. This stage can be precisely translated along the axis of the optical system by turning the screw drive

manually. The image of the drop on the ground glass plate is

observed with a low power telescope as the lens is moved forward and

backward. In this way the drop can be precisely focussed in the film

plane. The drop is illuminated by a high intensity projector located

about 20 feet away, i.e., near infinity for this lens. The entire

optical system is aligned using techniques which will not be described

here.

9
The drop is photographed on a high contrast, fine grained

film designed for graphic arts work. After development the

negative is placed between glass plates and analyzed on a

measuring microscope. The micrometers of this instrument have a -4

least count of 10 inch with a traverse of one inch, which is greater

than the largest dimension to be measured from the photograph. The

image of the drop is kept to this size because the use of a gauge

block with the microscope introduces errors which are intolerably large

The negative must be oriented on the microscope stage so that the

two orthogonal directions of stage translation are parallel to the

horizontal and vertical axes of the drop. This orientation is a

critical preliminary step in order to obtain consistent surface tension

values at different points of the profile. This is accomplished by

rotating the microscope stage until the sum of the coordinates of the

[91 drop diameter are the same at different distances from the drop apex

Thus at Z^, the left hand and right hand coordinates are X^ and X^.

If the stage is now translated to a different Z^, the coordinates will

be X^-AX and X^+AX if the drop is symmetrical about the Z axis, i.e.,

if the microscope Z translation is along the direction of gravity

in the drop photograph. Thus the sum is X^+X^ for each set of coordinates. In practice these sums can be made to agree within a few

4

parts in 10 with no difficulty.

The gallium used in these measurements was nominally of six

nines purity and in ingot form. It was prepared for use by melting

the entire 100 gram ingot under pure alcohol containing a few drops of

10
HC1 to dissolve the surface oxide. After rinsing with pure alcohol

several times, some gallium was drawn into the pipette. The remaining

gallium was then resolidified by cooling and stored under alcohol.

Experimental Results

A preliminary test of the resolution and aberrations of the optics

was made by photographing a precision ball bearing at the position of

the drop. The negative was analyzed on the microscope at 100X by

measuring the distance across the image of the bearing, 2X, as a

function of Z, the distance from the apex. The X dimension can also

be calculated as a function of Z using r, the known radius of the

bearing, i.e., X = v

/ Z(2r-Z) . The quantity X-X^.^ had a

average value of .00005 inch for eight measurements over the profile

of the bearing with no systematic variations detectable. We therefore

concluded that the optical resolution was adequate and the lens was

free of distortions at this level of measurement.

Figure 2 shows a typical drop profile. The cup diameter is

known and is used to provide an absolute dimensional calibration. In

Figure 3 the edge of the drop profile on the negative is shown at 100X

magnification; this is what is seen during measurement. The profile is

not absolutely sharp at this magnification. The intensity changes

over a significant distance, i.e., there is a penumbra. This picture

also shows a dark band and an adjacent light band which is much fainter

and narrower. These are similar to Fresnel fringes and arise from the

diffraction of light by the drop. Their observation means that a

basic physical limitation on the measurements has been reached.

11
The images of the sessile drops were measured as described previously and the surface tension calculated at a number of locations on

the surface using the Bashforth and Adams technique and tables .

We also developed a computer program to calculate a best value of the

surface tension from measurements of the coordinates of many points

on the drop profile. This program repeatedly solves the Young-Laplace

equation for the drop shape while adjusting the surface tension and

several other parameters using a simplex method of function minimiza-

[ 12 ] tion to provide a best fit to the measured coordinates. In Table 1

we list the dimensions measured, the surface tensions calculated by

the Bashforth and Adams technique, the dimensions computed by the

program which give a best fit to the entire profile, and the value

of the surface tension the minimization procedure selected to give this

best fit. The average of the surface tension values calculated by the

2

Bashforth and Adams technique differs from this by only 0.6 mJ/m ,

f\j i.e., ^ 0.1%. In the data we present subsequently surface tensions

given are all calculated using the Bashforth and Adams tables. The

computer program will be used only to process data which we consider

to be in final form.

The surface tension of gallium was measured from the melting

point to about 750 °C at several vacuum levels. Figure 4 shows some

early data taken using a pyrex cup whose circularity and stability

were inferior to the quartz cups used later. The vacuum levels were

not measured for these experiments because the ionization gauge had

not yet been received. From subsequent work we estimate the pressure

12
to have been in the 10 ^ torr range. The data clearly shows curvature

below about 250 °C. This is probably due to adsorption from the

residual atmosphere with surface tension depression as discussed in

the introduction.

Measurements of the gallium surface tension obtained using a

quartz cup are shown in Figure 5. The vacuum in these experiments

_ g ranged from about 5 X 10 torr at the high temperature end to about

_9

1 X 10 torr from 300 °C to room temperature. This data also shows

curvature, but only below about 150 °C. The measurements were made

by raising the temperature to at least 400 °C to clean the surface

and then cooling to the desired temperature. Because the sample is

in vacuum and does not make good thermal contact with the vycor tube,

it is not possible to change the temperature of the drop quickly;

with the heating source removed, three hours are required to cool the

drop to 30 °C from 150 °C. Thus the gallium drop has time to approach

adsorption-desorption equilibrium with the residual atmosphere in the

vacuum system. The depression of the surface tension at low temperature is almost certainly due to adsorption. A regression analysis of

2 the linear data gives dy/dT = - 0.068 mJ/m °C.

A few measurements have been made of the time dependence of the

surface tension. In the linear region it appears to be constant; at

lower temperatures there is some evidence for a slight decrease of

the surface tension over several hours at 100 °C even at a pressure

-9

of 10 torr. Additional work in this area is needed.

13
Conclusions and Discussion

The data presented In Figure 5 are replotted in Figure 6 with

the results of three previous measurements of the gallium surface

tension. Our results clearly do not agree with these earlier measurements. The two nonlinear curves shown in the figure are in approximate

[1 2 ] agreement. The temperature dependences suggested by these authors *

are

y = 708 - .0031(T-29.87) - . 00067 (T-29 . 87)

2

y

= 706.7 - .000647 (T-29. 87)- .0000965 (T-29. 87)

2

.

with T in °C and y in mJ/m2

. [131 The linear curve is described by

Y m = 718 - .101(T-29.87).

The temperature dependence of y we find resembles the relationship

of curve 3 if we ignore the points below 120 °C which are clearly

showing an adsorption effect. However, our value of dy/dT is signifcantly smaller in absolute value than the slope of curve 3. In

rontrast, the present data at high temperature (> 400 °C) is in fair

agreement with the values of y and dy/dT of curves 1 and 2.

The disparate experimental results shown in Figure 6 pose a

dilemma. This results because y and dy/dT give contradictory evidence

about the role impurities are playing in the measurements. Surface

active impurities will lower the surface tension as described by the

Gibbs adsorption equation. Although it is not as clearly defined,

adsorption should also make dy/dT less negative. This is the effect

14
shown in the data of Figure 4 and in the low temperature data of Figure

5. Thus, as the surface gets cleaner, y should increase and dy/dT

should assume a larger negative value.

By the first of the above criteria, the surfaces in our measurements are cleaner than those in the data of curves 1, 2, and 3 because

our y values are the highest over the entire temperature range.

However, the dy/dT criteria would reach the opposite conclusion. The

measurements represented by curve 3 would be identified as the

cleanest at low temperature, and of curve 1 at high temperature. This

is not likely. We conclude that arguments of this kind cannot resolve

the contradictions in the various measurements.

The problems of interpretation of surface tension data found here

are not unique; similar contradictions are encountered in comparisons

[3] of some recent measurements on zinc referred to earlier with

previous results. It seems probable that the different y and dy/dT

values found by different workers is due to the presence of surface

active impurities whose nature and concentration vary with temperature

in an unknown and uncontrolled manner. The measurements shown in

curves 1 and 3 of Figure 6, as an example, were made in evacuated

cells which had been sealed off from the pumps at a pressure no better

than 10 ^ torr. Despite rigorous outgassing prior to seal off, such

systems must have vastly different vacuum levels at low and high

temperatures. The adsorbed chemical species on the liquid surface

are probably coupled to some extent to these variations in cell

atmosphere. Thus one might interpret a curvature such as that shown

15
in curves 1 and 2 of Figure 6 as surface tension depression at high

temperature due to adsorption on the liquid gallium of species being

desorbed from the cell walls.

There is no support in the measurements shown in Figure 6 for

the anomalously high values (y ^ 800 mJ/m^) found by Abbaschian^

and seen occasionally in our work last year^. It is interesting

to note, however, that a linear extrapolation of the high temperature

data of curve 1 in Figure 6 gives such a value at the melting point.

In addition, the old work of Hack et al. contains one anomalously high

value. Thus it is possible that all the measurements summarized in

Figure 6 have been affected by an impurity. A better characterization

of the surface and gaseous impurities is essential before these complex

observations can be interpreted.

16
References

[1] 0. A. Timof eevicheva and P. P. Pugachevich, Dokl. Akad, Nank

SSR, 1J4, 840 (1960).

[2] V. I. Nizhenko, L. P. Sklyarenko and B. N. Eremenko, Ukrain.

Khim. Zhur. j>, 559 (1965).

[3] W. L. Falke, A. E. Schwaneke and R. W. Nash, Met. Trans. 8B ,

301 (1977).

[4] G. L. Mack, J. K. Davis and F. E. Bartell, J. Phys. Chem. 45 ,

846 (1941).

[5] G. J. Abbaschian, Journal of Less-Common Metals 4C, 329 (1975).

[6] S. C. Hardy and S. R. Coriell, Properties of Electronic Materials ,

NBSIR 78-1483, 3 (1978).

[7] D. W. G. White, Trans. ASM 55, 757 (1962).

[8] J. V. Najdich and V. N. Eremenko, Fiz. Metal, i Metalloved 11 ,

883 (1961).

[9] C. A. Smolders and E. M. Duyvis, Recueil ^0, 635 (1961).

[10] F. Bashforth and J. C. Adams, The Theories of Capillary Action ,

University Press, Cambridge, p. 1883.

[11] C. Maze and G. Burnet, Surface Science ^4_, 335 (1971).

[12] L. F. Shampine, SC-RR-72 0657, Sandia Laboratories, Albuquerque,

New Mexico, Sept. 1972.

[131 A. A. Karashaev, S. N. Zadumkin and A. I. Kukhno, Russ. J. of

Phys. Chem. 41, 329 (1967).

17
Table 1. Comparison of a Sessile Drop as Analyzed by the Bashforth

and Adams Technique and the Computer Profile Fitting Routine

Z Z * r r y

(measured) cm

(calculated) cm

(measured)

cm (calculated)

cm

B. & A.

mJ/nr

.08922 .08909 .35736 .35730 691.9

.13930 .13917 .42944 .42940 690.8

.16076 .16063 .45332 .45339 687.0

.19653 .19641 .48650 .48657 688.1

.23230 .23218 .51264 .51273 687.2

.26807 .26795 .53285 .53292 690.2

.30385 .30372 .54781 .54779 689.2

.33962 .33950 .55781 .55768 682.8

.39263 .39251 .56357 .56340 Max. r

.44694 .44681 .55753 .55755 687.1

.48271 .48258 .54598 .54607 686.7

Average y 688.1

Best fit y 688.7

AThe difference between Z (measured) and Z (calculated) is constant;

this constant is one of the parameters fitted in the computer code.

18
Fig. 1. Schematic diagram of sessile drop apparatus: (1) ion

pump (2) cross (3) vycor tube (4) cup containing gallium

(5) thermocouple leads (6) ionization gauge (7) isolation

valve.

19
Fig. 2. Picture (negative) of sessile drop.

20
Fig. 3. Edge of drop as seen through microscope at 100)

21
( 2

ui/puj) N0ISN31 30VddnS

O' 3 3 u u

3

4J cn 3 M

t-i 3

0) rH a to 0

0)

4-i H

3

4-i >

O 3

3

O

•H

4-1 a§

3

3 4J3

m *H 3 -33

s a

•H 3

r-l 4J rH 3

3 *r4 60 3

4-1 3

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U

3

3

0

3

3 i-i 42 O. H

I

<r

60

•r4 P4

22
730

720

710

700

690

680

670

200 400 600 800

TEMPERATURE (°C)

Fig. 5. The surface tension of gallium as a function of temperature

(high vacuum level; quartz cup).

23
740

720

700

680

660

640

620

600

580

560

Hardy & Coriell

J L 200 400 600 800 1000 1200 1400

TEMPERATURE (°C)

Fig. 6. Comparison of the various measurements of the surface tension of gal Hum.

24
Task 2

Solutal Convection during Directional Solidification

W. J. Boettinger, S. R. Coriell, and F. S. Biancaniello

Metal Science and Standards Division

Center for Materials Science

and

M. R. Cordes

Mathematical Analysis Division

Center for Applied Mathematics

Summary

This research is directed towards an understanding of solutal convection caused by the simultaneous concentration and temperature

gradients present during the directional solidification of alloys and

is a continuation of research previously reported under NASA

Government Order H-27954B (NBSIR 78-1483) . Included in this study is

investigation of the effects of convection on the microstructure and

chemical homogeneity of directionally solidified alloys and of the

conditions under which undesirable effects can be eliminated.

The onset of convective and interfacial instabilities during the

vertical directional solidification of a binary alloy is determined

theoretically by means of a linear stability analysis. For values of

the physical constants appropriate to the solidification of lead containing tin, the critical bulk solute concentration delineating the

demarcation between stability and instability is numerically calculated

as a function of growth velocity. For concentrations less than the

critical concentration, the system is stable. For velocities in the

25
range 1-40 ym/s, a convective- like long wavelength instability occurs

at a critical concentration which increases with velocity. For

velocities > 40 ym/s, the concentration at which instability occurs

decreases as velocity is increased and the values of concentration

and wavelength at the onset of instability correspond to the predictions of previous morphological stability theory which neglected

density changes and convection. Application of a vertical static

magnetic field increases the critical concentration for convective

instabilities but a field of a tesla is needed to cause an order of

magnitude change. For a given velocity, the increase in critical

-4

concentration for the onset of convective instability due to a 10

reduction in gravitational acceleration is a strong function of

velocity, ranging from a factor of 20 to 3000.

The occurrence of solutal convection and its effects on the microstructure and macrosegregation of directionally solidified alloys,

particularly off-eutectics, also has been investigated. During plane

front solidification the concentration of less dense solute is

increased in the liquid near the solid-liquid interface and there is

the possibility of solutal convection. In this year's research, the

effect of solute gradient induced convection on the macrosegregation

of Pb-rich Pb-Sn off-eutectic alloys during vertical solidification

is determined experimentally as a function of composition and growth

rate. In many cases macrosegregation is sufficiently great to prevent

the plane front solidification of the alloy. The transition from

26
dendritic to composite structure is found to occur when the composition of the solid is close enough to the eutectic composition

to satisfy a stability criterion based on G/V (temperature gradient/

growth rate)

27
A. THEORY

Introduction

In recent years there have been extensive research and development

efforts on the effect of fluid flow on solidification and on the prop-

[ 1-21

erties of the resulting solids . In general, one attempts to

control fluid flow in such a way as to obtain optimum materials properties. To minimize fluid flow, it is well known that horizontal

temperature and concentration gradients must be avoided. If the fluid

density gradient is parallel to the gravitational field, a motionless

solution of the fluid flow equations is possible; in the remaining

discussion we will assume that this is the case, i.e., there are no

horizontal density gradients.

In the plane front constant velocity directional solidification

of all alloys other than congruent melting compounds and invariant

eutectics, there exists in the liquid an exponential concentration

gradient which extends ahead of the liquid-solid interface with a decay

distance of D/V where D is the liquid interdiffusion coefficient and V

is the interface velocity. There also exists a temperature gradient

which is essentially linear in the region in which the concentration

gradient is exponential. The density, being a function of temperature

and concentration, can have various profiles depending on the solidification conditions and the properties of the fluid.

. In the theoretical section of this report, the onset of convective

and interfacial instabilities during the directional solidification of

a binary alloy at constant velocity vertically upwards (positive zdirection) is treated by a linear stability analysis. We assume that the

28
initial state of the system corresponds to a planar solid-liquid interface of infinite extent in the x and y directions and that there are no

horizontal temperature or solute gradients. Further we assume that the

liquid density p is a linear function of temperature T and solute concentration c. We consider a positive temperature gradient and a liquid

which expands on heating, i.e., (3p/3T) (3T/3z) < 0 and the situation

where (3p/3c) (3c/3z) > 0. Thus, the temperature gradient alone would

cause a negative density gradient and the solute gradient alone would

cause a positive density gradient, e.g., this is the case when a solute

less dense than the solvent is rejected at the solid-liquid interface.

In this model, in the absence of convection, the only fluid flow would

be a uniform flow arising from the density difference between solid and

liquid. Because the usual solidification boundary conditions are used

at the solid-liquid interface, the hydrodynamic and interfacial stability

processes are coupled. In the limit that the liquid density does not

depend on temperature or solute concentration, and the density change on

solidification is negligible, the present treatment reduces to a

morphological stability analysis.

In the unperturbed state in a frame moving with the solid-liquid

interface, the fluid velocity, temperature and density are functions of

z alone; perturbations whose x, y, and time t dependence are of the form

cos to x cos to y exp (at) are introduced, where cj and to are spatial

x y x y

r frequencies and a determines the growth rate of the perturbation. The

onset of convection corresponds to conditions for which the real part of

2 2 2 a vanishes for some value of to(to =w +w ) and is negative for all other

x y

values of to.

29
Theoretical Formulation

The basic theory was described in our previous report 1 We

have now incorporated into the theory the effect of a static magnetic

field. For completeness and for the reader’s convenience, the basic

theory is repeated in this section.

We consider unidirectional vertical solidification of a binary

alloy at constant velocity. We choose an (x^y'^) laboratory coordinate system such that the z' axis is . perpendicular to the planar

solid-liquid interface. The gravitational field is of the form g =

(0,0,-g), i.e., positive g implies a field anti-parallel to the z' axis.

The following differential equations govern the fluid velocity u = — *

(u ^,u ,) , temperature T, magnetic field H = (H ^,H „,H and conx y z x y z centration c [4-6]

V" u = 0, ( 1 )

V' H = 0, (2)

(3u/30 + (uV')u = -(l/p

0)\T (p+H

2

/8tt) + vV^^u - g[a(T-$) +

a

c (c-c)] + (HV')H/4.irp

0 , ( 3 ) (9H/30 + (uV " ) H - (HV')u = nV"

2H, ( 4 )

(3T/9t") + uV"T = kV'

2T, (5)

(3c/3t") + uV"c = DV'

2 c, (6)

where f is the time, p is a pressure, v is the kinematic viscosity, k

Equations (2) and (4) correspond to four equations for the three com- ponents of the magnetic field; however, these four equations are not

independent and actually correspond to three independent equations.

30
is the liquid thermal diffusivity, q is the liquid magnetic diffusivity,

and D is the liquid diffusion coefficient. The prime on the operator

V indicates that it is with respect to the (x' ,y' ,z') coordinate

system. We have assumed that the fluid density p is given by p =

'u % % %

Pq [l-a(T-T) - a^Cc-c)], where p^ is the density at T = T and c = c.

The above equations are based on the Oberbeck-Boussinesq approximation

[4-5] and the coefficients v, k, q, and D are assumed constant.

We transform to a moving coordinate system (x,y,z) attached to

the unperturbed planar solid-liquid interface which is moving with

constant velocity V = (0,0, V) in the z - direction, i.e., x = x",

y = y' , z = z

' —V t , and t = t'

We obtain

V u = 0, (7)

V H = 0, (8)

Ou/at) - (VV)u + (uV)u = - (l/p

Q

) V (p+H

2

/

tt) + vV

2 u - g[a(T-^) +

a

c (c-c)] + (HV)H/Attp

0

, (9)

(3H/3t) - (VV)H + (uV)H - (HV)u = qV

2H, (10)

(3T/3t) - VVT + uVT = kV

2 T, (11)

(9c/at) - VVc + uVc = DV

2

c. (12)

Note that u is still measured in the laboratory coordinate system

(x',y",z')

31
7* The boundary conditions at the solid-liquid interface are that

the tangential components of the fluid velocity vanish (no slip condition) and that the normal component satisfy

v«n(p -p ) » (u*n)p , (13)

Li S L

where v is the solidification velocity, n is the unit normal to the

solid-liquid interface, and p and p are liquid and solid densities,

Li 6

respectively, at the solid-liquid interface. The temperature and concentration satisfy

(v-n)Lv

= (-kjVT + k

s VTs )-n, (14)

(v*n) (c -c ) = (p L

/p

g

)DVc*n, (15)

where L is the latent heat of fusion per unit volume of solid, k^ and

k are thermal conductivities of liquid and solid, respectively, c

S SI

and c are solid and liquid concentrations at the solid-liquid interface, and all quantities are evaluated at the interface. The temperature T^ in the solid satisfies the differential equation

(3T /3t) - (VV)T = k V

2T ,

• (16)

s s s s

where K

g

is the thermal diffusivity of the solid. Diffusion in the

solid has been neglected. In addition, we have

, _ T

si I' (17)

si * kc r

(18)

* The authors are indebted to R. F. Sekerka for detailed discussions

and derivations of the boundary conditions at the solid-liquid

interface

32
(v*n) = p{T -T }, e I

( 19 )

T = T + me - T rK. (20)

e M I M

Here, we have used a subscript I to denote evaluation of the quantity

at the solid-liquid interface; k is the partition coefficient, p is a

linear kinetic coefficient, is the equilibrium temperature of a

curved interface in the presence of solute, 1^ is the melting point of

the pure substance with a planar interface, m is the slope of the

liquidus line, T is the capillary constant, i.e., the solid-liquid

surface tension y

= L^T, and K is the interface curvature. Boundary

conditions for the magnetic field will be discussed subsequently.

We now solve the equations for a planar solid-liquid interface

located at z = 0; all quantities are functions of z alone. We assume

that in the moving frame there is no dependence on time. The fluid

velocity, magnetic field, temperature, and concentration fields are

"o

= (0,0, -eV), (21)

«0 = (H 0xH

0y’ H

0z>’

(22)

T

o

= d

l - (icG /V*) expLj (-V*z/k), (23)

T

s0

== d

2 - (k G /V exps s (-Vz/k ), s (24)

c o = c

CD ( Cco /k) (k-1) exp (-V*z/D) , (25)

where e = (p /p ) -1, H. , Hn , , d. and d„ are constants, V* =

s L Ox Oy* Oz 1 2 V(p /p ) , G and G are temperature gradients in the liquid and solid,

S L L S respectively, at the solid-liquid interface, c^ is the solute concentration far from the interface, and the subscript 0 indicates that the

33
solutions are for an unperturbed planar interface. From the boundary

conditions we also have the relationships

VL = k G - k Gt , v s s L, L

V* (c /k) (k-1) = DG ,

d-. - (KG /V*) = d_ - (k G /V),

1 L z s s

c = c , si 00

V = + (mc^/k) - d^ + (kG

l /V*)}>

(26)

(27)

(28)

(29)

(30)

where G^ is the concentration gradient in the liquid at the solidliquid interface. These relationships may be used to find explicit

expressions for the constants d^ and d^.

We now consider whether the above solutions, for which there is

only a trivial fluid flow corresponding to the density change on

solidification, are stable with respect to small fluctuations. We

write

U = U

Q + W(x,y, z, t), (31)

H = Hq + H(x,y,z,t), (32)

T = T

Q + T(x,y,z,t), (33)

T

s = Ts0 + rs

( X,y ’ Z,t) > (34)

c = c 0 + C(x,y,z,t), (35)

p = P Q + P

1 (x,y,z,t) > (36)

z = Z sl/X ’ y,t ^’ (37)

34
and will retain only linear terms in the perturbation quantities, W,

H, T, T , C, p. , and z . The function z (x,y,t) represents the shape

of the solid-liquid interface which in general is not planar and the

function Pq(z) satisfies the unperturbed equations. We obtain the

following differential equations for the fluctuations

V W = 0, ( 38 )

V H = 0, 09)

(3W/3t) - V*(3W/3z) = - (Vph

/p Q

) + vV^W - gCotT+c^C) +

(H

0

V)(h/47rp

0

), (40)

dH/dt - v*(3tf/3z) - (h

q

v)w = n v

2H, (41)

(3T/3t) - V*(3T/3 z) 4- W

z (3Tq/3z) = kV

2 T, (42)

(3C/3t) - V*(3C/3z) + W

z (3c0 /3z) = DV

2C, (43)

(3 T / 3t) - V(3T/3z) = k V

2 T , (44)

s s s

where p^ = p^ + Hq*H/4tt.

Using the identities for a function <p and a vector a

curl grad (J> = 0 (45)

and

curl curl a = V(Va) - V

2 a . (46)

the zrcomponent of equation (40) can be written

o o / 9 9

(3V W /at) - V* ( 3V W / 3z) = vV W + g(aV, l+a V, C) +

z z z 1 c 1

(H

0

V)(V

2 H

z /4ttp 0

), (47)

35
0

where e (9^/3x^) + (3^/9y^).

The boundary conditions on the perturbed quantities at z =

follow from the general boundary conditions and are

W = -e(3z _ / 3 1) , (48)

Z Si-i

W = eV(3z / 3x) » (49) x sL

W = eV(3z /3y), (50)

y sL

(3z

sL /8t)Lv

" -k

L

[”(V*/K)G

L

z sL+3T/3z] + k

g

[-(V/ k^G^ +

3T

g /3z], (51)

( 3z / 3 1 ) (1-k) (c /k) = kVG z _-(l-k)VC-(p_ /p )D(3C/3z), (52)

sL 00 c sL L s

G

s

Z sL+ T

s = G

L

Z sL + T '

(3z

sL

/3t) * y{mG

c

Z sL + mC + T^ ZgL - G^-T}

(53)

(54)

From the continuity equation, eqs. (49) and (50) can be combined to

give

(3W / 3z) - -eVV. z _ z I sL

(55)

at z = 0.

The equations and the boundary conditions are homogeneous and

linear. It is convenient to represent the time dependence and the x

and y dependence in the form exp (at+iio x+iu> y) . The quantities id x y x

and to represent spatial frequencies and the general solution of the

equations will involve integration over all values of id^ and to . In

general o is complex, i.e., a => a^+ia^. If is positive for any

values of (id ,id ), then the solutions grow exponentially in time and

x y

36
are unstable. If a^ is negative for all values of (w^w^), the system

is stable. The criterion for the onset of instability is determined

by finding the conditions for which a^ is not positive for any (w^w^)

and is equal to zero for some value of (co ,oj ). We write

x y

W = W (z) exp (at+iw x+iw y)

z z

r x y

H = H (z) exp (at+iw x+iw y)

z z

r x y

T = T(z) exp (at+iw x+iw y)

x y

C = C(z) exp (at+iw^x+iw^y)

T = T (z) exp (at+iw x+iw y)

s s x y

z _ = z exp (at+iw x+iw y)

sL x y

(56)

(57)

(58)

(59)

(60)

(61)

where W , H , T, C , T are functions of z alone, and z is a constant,

z z s Substituting these expressions in the differential equations yields

L{vL+V* (3/ 3z)-a}W = gw^(aT+a C) - L (i(w H +w H n ) +

z 0c x Ox y Oy

H

Qz (3/3z) (62)

{nl+V*(3/3z)-a}H = - (HnV)W , z U z (63)

{Kl+V*(3/3z)-a}T = W Gt exp (-V*z/k),

z L

(64)

(D/.+V*(3/3z)-a}C = W G exp (-V*z/D)

z c (65)

{k L+V( 3/3z)-a}T = 0,

s s

_ ? ? ? ? 2 ?

(66)

where L = [(3 /3z )-w ] and w = w^ +w^ . Equations (62-65) form a

set of coupled ordinary differential equations for the perturbed

37
functions W , H , T, and C in the liquid,

z z When Hq = 0, the above equations reduce to those given in our

r 3

previous report 1 We are primarily interested in the suppression of

the onset of convection by a static magnetic field. It is clear that the

horizontal components of Hq will not suppress the onset of convection

since there will always exist perturbations [characterized by values of

(oj , w )] for which the term w Hn + to H_ =0, viz. for fixed H_. , H„ , x y x Ox y Oy Ox Oy

O r\ i 9 9 1c and w, choose to = -ooHA / (H 0 +H ) and to = toH n / (H- +H„ ) . When

x Oy Ox Oy y Ox Ox Oy

* * Wx^0x+Wy^0y

= equation (62) for W

z is independent of Hqx

and Hq^ .

In the remainder of this report we will take = H„ =0 since finite

Ox Oy

values would not suppress the onset of convection. We will make two

further assumptions concerning the magnetic field equation which greatly

4 2 simplify the calculations. For good conductors n ^ 10 cm /s and it

should be an excellent approximation to neglect the V*(dH^/dz) term in

equation (63). At the onset of convection the real part of a vanishes;

if one further assumes that the imaginary part of a vanishes, equation

(62) can be written as

UHv[L] + V* ( 9/ Sz) }W = gw

2 (af+a C) + (H n

2

/4Trp n

r>) (9^ /9z

2 ), (67)

Z C UZ l) z

which is independent of the perturbed magnetic field. Thus, with these

approximations it is not necessary to specify boundary conditions on

the perturbed magnetic field or to calculate the perturbed magnetic

field^ Based on these assumptions, the effect of a magnetic field

on the onset of convection will be discussed in the Numerical Results

section.

* In general, a constant magnetic field in the x. direction does not

affect a flow with fluid velocities which are Independent of the x coordinate.

38
A A A Since the equation for T does not depend on W , T, or C, it

s z

A be solved analytically. Using the boundary condition T^(z

we find

can

-°°) = 0 ,

T = d_ exp (\z)

s 3

( 68 )

where d^ is a constant and

A = (— V2k ) + { (V/ 2k ) 2 + u) 2 + (o/k )} h

. s s s (69)

We have three coupled differential equations for the functions W , A A A T, and C. The boundary conditions, eqs. (48, 51-55), for W , T, and C at

z

z = 0 also contain the unknown constant z and d^. However, we can

eliminate these constants, and obtain boundary conditions for W^, T,

and C at z = 0, viz.

W + (a/Vw )(9W / 9z) = 0, z z

(9W /9z) + (eVki^k. /a~) (977 9z) - (eVw

2 k A/a )T = 0,

z L 3 s 3

(a

2+a

2 kg A)T - ^k^) (3T/9z) - aynC = 0,

(70)

(71)

(72)

a^k

s AT - k^a^OT/Sz) + (l-k)Va

3

C + (p L

/p

s )Da

3

(9C/3z) = 0, (73)

where

- (k

L

G

L

Vi/K) - (ks°SV/<

S

)> (74)

= (a/y) - mG + G, + T„rw

2

, c L M

* (75)

= aL - a, - k A(GT -G ) v 1 s L s

’ (76)

= a(l-k)(c /k) - kVG . (77)

39
Thus we have four linear homogeneous boundary conditions at z = 0. We

need four additional conditions in the liquid away from the interface

since our differential equations are equivalent to a single eighth

order equation. We consider a rigid planar isothermal boundary at

z = z (a constant). The boundary conditions are

W = (3W /3z) = T = (3C/3 z) = 0. (78)

z z

However, in order to be able to treat a more general class of problems,

we will write the boundary conditions as

W = 0,

z

’

b

;[ T + b

7 (al/3z) = 0,

b^C + b. OC/3z) = 0,

3 4

b c (3W / 3z) + b,(3

2W /3z

2

) - 0,

5 z 6 z

(79)

(80)

(81)

(82)

where b, , b„, b„, b., b r , and b, are constants. Clearly b. = b. = 12 3 4 5 6 14

b = 1 and b» = b„ = b, = 0 corresponds to eq . (78).

j z 3 b

The boundary conditions at z = z^ are not rigorous. In many

solidification experiments, there is a rigid boundary at z' = z (a

1_) constant), which implies that in our moving frame the boundary conditions should be applied at z = z -Vt. In addition, if p i- p , the

unperturbed flow is not zero at z . To take account of these effects

would greatly complicate the calculation. We are basically assuming

that as long as z is sufficiently far from the solid-liquid interface,

L the results will be relatively independent of the precise boundary conditions. The validity of this assumption has been shown by numerical

3] calculations

40
Numerical Methods

The linear homogeneous differential equations (62-67) for

W^(z), T(z), and C(z) are equivalent to eight first order equations

with linear homogeneous boundary conditions at z = 0 [equations (70-73)]

and z = z [equations (79-82)]. The basic method for solving this

L

eigenvalue problem as an initial value problem was discussed in our

[3] previous report . Here, we will describe three important modifications of the previous numerical algorithm, which permit calculations

for a wider range of parameters.

The system of differential equations is solved using the SUPORT

computer code developed by Scott and Watts [see references (7-11) for a

detailed description of this code]. Basically, linearly independent

solutions of the system of differential equations are calculated and

then superimposed to satisfy the boundary conditions. The linear independence of the solutions is maintained by orthonormalizing the

solutions whenever an impending loss of independence is detected^.

For ck ^ 0, the functions W

z (z), T( z) , and C(z) are complex, and

it is convenient to write them in terms of their real and imaginary

parts to obtain sixteen first order real equations with eight boundary

conditions at z = 0 and eight boundary conditions at z = z . For

specified values of ck and c^ and fixed values of the remaining

parameters in the differential equations and boundary conditions,

one calculates eight linearly independent solutions all of which

satisfy the boundary conditions at one endpoint. For particular

values of a. and c (i.e., eigenvalues) a linear combination of these

x 00

41
eight solutions will satisfy the boundary conditions at the other endpoint. In general this will not be possible for our specified values

of a_j and and it is necessary to iterate to find the eigenvalues.

[3] As previously discussed , a linear combination of the eight linearly

independent solutions will satisfy the boundary conditions when the

determinant of the coefficients vanish. In general this determinant

is complex so that we want to find values of cm and such that the

real and imaginary part of this determinant vanishes, i.e., we need

to find the roots of two non-linear equations of two variables.

[12 ] Keller has suggested an alternative method for solving

eigenvalue problems which does not require the evaluation of a determinant. We have included Keller's suggestion as an option in our

numerical code. Specifically, we temporarily replace the homogeneous

boundary condition given by eq. (70) with the inhomogeneous boundary

condition

0

3

W / 9z

3

) = 1* (83)

z

The SUPORT code provides a solution to this modified linear problem

with inhomogeneous boundary conditions. In general, the homogeneous

boundary condition, equation (70) will not be satisfied. We now vary

c^ and a

^ until equation (70) is satisfied. Since equation (70) is

complex, it corresponds to two real equations and we again need to find

the roots of two non-linear equations of two variables. At an eigenvalue clearly the determinant method and Keller's method are equivalent;

however, they may have quite different convergence properties. For

42
the solidification problem treated here with ck ^ 0,. Keller's method

appears significantly better.

Our previous numerical code used the iteration method developed

by Brown as implemented by Gay

In our present code, we are

using the QN code developed by Shampine and Gordon

The QN code

appears to require fewer function evaluations to find the eigenvalues.

In using the SUPORT code we choose an initial estimate for and

and solve the differential equations. The SUPORT code determines

the orthogonalization points for this inital estimate for c^ and o^;

during succeeding iterations the orthogonalization points are not

changed. This fixing of the orthogonalization is necessary when the

determinant method is used to find the eigenvalues and is also used

with Keller's method. The integration of the initial value problem

r I c I is carried out using the Shampine-Gordon code 1

.

We have also written a code which assumes that o. =0; in this

l case W(z), T( z) , and C(z) are real and we have only eight first order

equations. In addition, only cm

is varied to find an eigenvalue.

This code requires considerably less computer time.

Numerical Results

A number of calculations have been carried out using parameters

appropriate to the solidification of lead containing small quantities

of tin. The specific values are given in Table 1. We have neglected

the density change on solidification, i.e., e = 0, and kinetic effects,

i. e. y

-y °°

43
In order to determine the accuracy of our numerical code, we have

[31 previously described a number of tests. In addition, we have compared magnetic field results with results given in Table XIV of

r (S i Chandrasekhar 1 and have compared oscillatory results with the

analytic oscillatory convection results for free boundary conditions'^ .

The calculations correspond to undirectional solidification

upwards of a binary alloy which is rejecting (k<l) a solute whose

density is less than that of the solvent (a >0) . We expect that for

c sufficiently small solute concentration there will be no convection,

and the calculation will determine the bulk solute concentration

above which convection occurs. This value c* depends on the spatial

frequency w of the perturbation and convection occurs if the actual

concentration c^ exceeds c* for any value of oo, i.e., denoting the

minimum value of c* as a function of co by c**; convection occurs if

c > c**

In all the calculations to date, the mode of instability at

c = c** is not oscillatory, i.e., = 0. As will be discussed, we

have found oscillatory modes but these do not correspond to the

minimum value of c*.OO The values of z were chosen such that z >>D/V and ooz >>1. Under

Li L L these conditions the results for c* are independent of z^; they are also

[31 independent of the precise boundary conditions at z = z . In Table

2 and Figure 1, we show c** as a function of growth velocity. As the

growth velocity increases from 0.5 ym/s, there is a shallow minimum in

c** followed by a rapid rise. At velocities slightly larger than 40 ym/s,

44
the minimum value of c* occurs at small wavelengths A = 2 tt/u). We will

denote this small wavelength instability as a morphological instability

since the value of c** and w at c** are not influenced by fluid flow,

00 CO

i . e - , calculations taking account of fluid flow give the same results

as the usual morphological stability theory

^

which neglects fluid

flow. The critical concentration and wavelength for morphological

instability are denoted by c „„ and A^,, respectively, and are given

in Table 2. Neglecting surface tension, the morphological stability

criterion for interface stability is approximately c^ < 7.7/V with V in

ym/s.

Table 2 and Figure 1 also give the results of calculations for

_ ^ 2 which g = 10 g p - 0.098 cm/s'. The long wavelength instability at

velocities less than 4 ym/s is strongly effected by this reduction in

gravitational field. We will denote this long wavelength instability

as a convective instability. However, it should be recognized that

this convective instability gives rise to interfacial instability and

that the morphological instability gives rise to fluid flow. The

critical concentration for morphological instability is unaffected

by the reduction in gravitational field. At velocities slightly larger

than 4 ym/s, the minimum value of c* corresponds to morphological

instability. We note that although the critical concentration is

substantially increased as the constant vertical gravitational field

=4

decreases from g^ to 10 g^, the critical concentration is still rather

small for slow growth, rates (less than 0.01 w/o at V = 0.5 ym/s).
In Table 3, for velocities of 1 and 20 ym/s, the effect of the

temperature gradient in the liquid on the critical concentration for

the onset of convective instability is indicated. For the larger

velocity, the critical concentration is relatively insensitive to the

magnitude of the temperature gradient. At the smaller velocity a

change in temperature gradient from 50 to 1000 K/cm increases the

3/4 critical concentration by an order of magnitude; the quantity c**/G

is nearly constant. The frequency w at the onset of convection is

rather insensitive to the value of G for a given velocity.

l_i Based on the assumptions previously discussed, the effect of

a vertical magnetic field on the onset of convection is shown in

Table 4 for velocities of 1, 5, and 20 ym/s. Magnetic fields of the

order of a tesla are required to give an order of magnitude increase

in the critical concentration. For the dimensionless parameter Q

fixed at 9.0, the effect of magnetic field on c** increases with

velocity. The frequency to at the onset of convection becomes larger

as the magnetic field is increased.

At small velocities the convective and morphological instabilities

occur at quite different concentrations (see Table 2)- In this case

the concentration c* as a function of oo exhibits a rather simple OO behavior as illustrated in Figure 2 for V = 5 ym/s. Also shown are

two higher eigenmodes. The system is unstable to a particular convective mode if the concentration lies above the curve for that mode.

-4

The lowest concentration c** above which convection occurs is 8.39(10 )

OO w/o at oo — 10.

46
When the concentrations corresponding to convection and morphological instabilities are of the same order of magnitude, the

concentration c* as a function of u) becomes rather complicated and

00 oscillatory modes (a^O) occur. This behavior is shown in Figures 3-6

for velocities of 30, 40, 50, and 80 ym/s, respectively. The dashed

curves indicate oscillatory instabilities with the value of + given

in the inset of the graph (note that if a is an eigenvalue, then

the complex conjugate a* is also an eigenvalue) . The solid curve

in the lower right of the figures corresponds to the morphogical instability mode; the minimum is this curve occurs at higher values of

a) than are shown in the figure. For V = 30, 40, 50, and 80 ym/s this

minimum occurs at c* = 0.28, 0.21, 0.18 and 0.12 and w = 620, 700, 750,

CO and 900, respectively. These same values are also obtained if fluid

flow is completely neglected. For V = 30 and 40 ym/s, the minimum

value of c* occurs at small wavelengths and corresponds to a nonoo oscillatory (a^=0) convective instability. For these velocities,

c* (u) consists of two non-oscillatory branches both having a minimum OO with the absolute minimum corresponding to the convective mode. The

two non-oscillatory branches are connected by an oscillatory mode

indicated by the dashed line. For V = 40 ym/s, there is an additional

oscillatory branch at low values of to. The oscillatory instability

occurs at concentrations higher than those required for morphological

instability for the four velocities considered. For V = 50 and 80

ym/s, the absolute minimum corresponds to the morphological instability

47
mode and the convective mode is oscillatory. The values of c* at the

relative minimum in these oscillatory modes were indicated in Figure 1

by the dotted curve; the same procedure was also used for the relative

minima for the morphological modes at V = 30 and 40 ym/s. The values

of for the oscillatory modes is of the order of 0.1 Hz or a period

of the order of 60 s.

For V = 80 ym/s, a few calculations were carried out with the sign

of changed (i.e., a heavier solute rejected at the solid-liquid

interface) . The morphological stability mode extended to low frequencies and is a single-valued function of m (in contrast to a

c

> 0; see

Fig. 6). At low frequencies the values of c* were lower than those obtained by the usual morphological stability theory (which neglects fluid

flow) while at large oo all calculations (a^ positive or negative or

fluid flow neglected) gave the same results.

A 2

A 2A A For simple boundary conditions, viz., = (9 W^/9z ) = T = C = 0

at both boundaries, an analytic solution of a double diffusive convection problem is possible. In fact, this solution was used to check

our numerical code. The neutral stability curves are of the form

R + S = 27tt 4

/4 (84)

for a. = 0 and

i

(r^S/ [ (r+s) (1+r) ] } + (s^R/ [ (r+s) (1+s) ] } = 2

it 4

/ 4 for a^0, where R, S,

r, s, are the Rayleigh, solute Rayleigh, Prandtl, and Schmidt numbers,

respectively. Both of the above expressions are of the form

S + a-jR = a

2

, ( 85 )

48
where and a, depend on the properties of fluid, viz., on the Prandtl

and Schmidt numbers. For the present solidification problem the relevant

length is (D/V) and we define Rayleigh numbers R' and S' based on

(D/V) as

R' = -gaG (D/V)

4

/vk, (86)

L

S' = -ga G (D/V)

4 /vD = (ga /vDk) (1-k) (D/V)

3 c . (87)

c c c

00

We now explore the possibility that the convective instabilities

occurring during solidification can be qualitatively understood by

an equation of the form

S' + a

x R' = a

?

. (88)

To make a more direct comparison with the results of our numerical

computations we rewrite this equation in dimensional form exhibiting

only the V, G^, c^, and g dependence, i.e.,

c = a_V

3

/g + a.G /V, (89)

00 3 4 L

where a^ and a 7 depend on all variables other than V, G^, c^, and g,

but perhaps have only a weak dependence on V, G-^» c^, and g. The

above equation has two limiting cases. For sufficiently large V, we

have c = a „V /g while for sufficiently small V, c = a,GT /V. If

00 3

00 4 L we now compare these two limits with our numerical results (see Figure

1 and Tables 2-3), we find qualitative agreement. For V>10 pm/s,

3

c^ is approximately proportional to V , is insensitive to changes in

-4

G , and reduction in g from g to 10 g eliminates the convective

1j 6 6

instability, i.e., morphological instability governs the onset of

49
instability. For small V, rises with decreasing V, is sensitive to

changes in G and reduction in g is not as effective in eliminating

Li the onset of convection. If one is simply interested in fitting the

numerical results given in Tables 2-3, one can generalize the above

expression in the form

a

5 S' + a

1

(R") = a

2

, (90)

or in dimensional form

C

co = a

3y3 /g + a4 (V

3 /g)(gG

L

/V

4

) 5

. (91)

This expression can fit the 20 convective instability data points given

in Tables 2 and 3 with a maximum error of 18%.

At small velocities, the wavelength at the onset of instability is

very large, e.g., 5 cm at V = 0.5 ym/s. Clearly, in small tubes such

wavelengths are not possible. Our calculation assumes infinite tube

diameters, and it is non-trivial to modify the calculation to take

account of the effects of the tube walls. To obtain some estimate of

these effects one can assume that the tube walls simply prohibit small

a) (large wavelength) convective modes. For example, for V = 0.5 ym/s,

-4

convective instability occurs at c** = 3.60 (10 ) w/o at u = 1.2;

if we allow only perturbations with oj _> 20 cm ^ (A<0.1 tt), then in-

-3

stability occurs at a concentration of 1.62 (10 ) at u = 20.

At present we have not investigated the effects of fluid properties

such as Prandtl and Schmidt numbers on the critical concentration for

the onset of convection; such calculations appear worthwhile. For

50
lead containing tin, the maximum concentrations (regardless of

growth velocity) below which there are no convective or morphological

instabilities are 0.19 and 1.9 w/o for constant gravitational acceler-

-4 2

ations of 980 and 980 (10 ) cm/s , respectively. For a given

velocity, the change in critical concentration for the onset of con-

-4

vective instability due to a 10 reduction in gravitational acceleration

is a strong function of velocity, ranging from a factor of 20 to 3000.

Magnetic fields of the order of a tesla increase the critical concentration for convective instabilities by a factor of 10.

For concentrations such that convective instability has occurred,

it would be of great interest to determine the effect of this convection on interface shape and the resulting solute distribution in

the solidified material. A non-linear analysis is necessary for

understanding the effects of fluid flow on solute distribution.

51
B. EXPERIMENT

Background for Experimental Work

The ability to solidify two-phase aligned structures from a wide

range of compositions near a eutectic composition has importance in

the full utilization of this class of composite materials. The early

work of Mollard & Flemings claimed that the range of compositions

for which plane front composite growth was possible was restricted not

only by constitutional supercooling requirements but also by the

requirement that convection be avoided. In fact, their experimental

work using vertical solidification was limited to the Sn-rich side of

the Sn-Pb eutectic wherein the fluid density influenced by temperature

and composition always decreases with distance from the interface

[ 21- 22 ] (height). Subsequently, others argued that as long as a

stagnant boundary layer exists whose thickness is large compared to

the lamellar spacing, plane front growth should be possible in the

presence of convection. This was in fact verified by experiments by

[231 [21] Holder & Oliver and Davis & Fryzuk who demonstrated the growth

[24] of Pb-rich Pb-Sn composites. Verhoeven, Kingery and Hofer subsequently showed that although composite growth was possible in the

case of convection, that significant macrosegregation (10 w/o) down

the length of solidified samples occurs in samples only 1 w/o on the

Pb-rich side of the Pb-Sn eutectic.

The purpose of this research is to describe the microstructure

and macrosegregation produced during the vertical solidification of offeutectic alloys 3-6 w/o from the eutectic composition when the liquid

52
density near the interface increases with distance from the interface

and to determine if a magnetic field or solidification downward can

eliminate convection and hence macrosegregation.

Experimental Procedures

Alloys were prepared from 99.99% pure Sn and Pb in evacuated and

sealed borosilicate crucibles. The directional solidification furnace

is constructed of non-magnetic materials and has a bifilar wound

cylindrical resistance heating element with upper and lower sections

and a circulating water cooled chill zone. A silica tube containing

the desired alloy is pulled down through the chill sealed by O-rings

at constant velocity to accomplish directional solidification. The

cooling water comes directly in contact with the silica tube. Using

two separate A.C. proportional temperature controllers for the furnace,

and water cooled by a heat exchanger, temperatures are constant to

0.1°C over an hour and 1°C in several days. The vertical drive is

powered by a synchronous motor with a multi-speed gear box. Speeds

from 0.5 ym/s to 0.5 mm/s are possible. The tube is attached to the

pull shaft using a device which permits the sample to be suddenly

pulled down about 3 cm to quench the liquid-solid interface.

Electrically insulated 0.25 mm diameter sheathed thermocouples

were used to measure directly the temperature gradient in the liquid

and the solid. The values of the measured liquid temperature gradient

for all runs was 360 K/cm + 10 K/cm. The furnace is designed so that

a permanent magnet with a pole diameter of 8 cm and a pole separation

53
of 11 cm can be slid around the furnace to apply a 0.1T (1 kilogauss)

horizontal magnetic field. Alternately, an 11 cm diameter solenoid

can also be placed over the furnace-chill to apply up to 0.1T vertical

magnetic field. Additionally the furnace can be inverted for solidification vertically downward. Here a graphite plug is used to

maintain a small upward pressure on the inverted melt to prevent

separation of the liquid from the freezing solid.

A specific directional solidification experiment is conducted in

the following manner. A tube filled with the desired alloy is inserted

through the chill, with the alloy completely in the chill section.

A vacuum is pulled on the tube and the furnace temperature is equilibrated for several hours. The sample is then driven quickly into the

furnace melting about 12 cm of alloy. The tube is backfilled with

argon and temperatures are allowed to stabilize for about 1 minute.

Excessive holding times at this stage are undesirable because a layer

of Pb-rich solid solution will slowly solidify and disrupt the initial

liquid composition uniformity. The tube is then pulled at the desired

rate until about 6 cm have solidified. The remainder of the sample

is then quenched delineating the liquid-solid interface in the microstructure. The silica is removed with HF acid and the specimens are

mounted in a cool-setting epoxy for characterization.

Samples are routinely examined in a longitudinal section (parallel

to the solidification direction) using standard metal lographic. techniques. When necessary, transverse sections are also taken. For each

54
experiment the shape of the liquid-solid interface is examined to be

certain that heat flow was unidirectional. In the present experiments,

the macroscopic liquid-solid interface of two-phase composite growth,

when achieved, was concave outward by only 10-20 pm across a 3 mm

diameter sample. Whether the structure is dendritic, lamellar, or

whether there are any signs of growth rate fluctuations was also

determined.

Measurement of macrosegregation down the length of directionally

solidified samples was performed using two methods, x-ray fluorescence

and wet chemistry. The former method, which was described previously

[3 ]

, uses incident monochromatic (22.2 keV) x-rays collimated to 1/2

mm x 1/2 mm. The use of x-rays permits deeper penetration into and

hence better averaging over the two-phase structure present in the

samples than does the use of electron probe methods. This data was

used to non-destructively evaluate the extent of macrosegregation and

to indicate those samples for which wet chemistry would be used. Wet

[25

chemistry using a titration method J was performed on approximately

2 mm discs cut transverse to the growth direction. Accuracy of the

wet chemistry is approximately + 0.25 w/o Pb

Experimental Results

In all experiments conducted on the Pb-rich side of the eutectic,

solidification begins with the dendritic growth of the Pb-rich phase

despite the fact the G^/V (temperature gradient in the liquid at the

interface/growth velocity) substantially exceeds the requirement of

55
constitutional supercooling for the bulk composition of the alloy.

The volume fraction of dendrites decreases slowly as a function of

distance as shown in Figure 7, until the structure is two-phase

composite. The region of dendritic growth in most instances is very

long compared to D/V (where D is the liquid interdiffusion coefficient)

and does not correspond to an initial transient of the type described

by Flemings ^0]. Table 5 is a summary of experiments showing the bulk

composition (C ), growth rate (V), G /V, D/V, the initial length of

O Li the liquid alloy (L) , the length of the dendritic zone (Z°) , and other

parameters to be discussed later.

Figure 8 shows the composition as a function of fraction solidified

(f ) for two 43.7 w/o Pb alloys solidified at 0.5 ym/s with total

initial melt lengths of 5.2 cm and 11.8 cm respectively. [The eutectic

composition is 38.1 w/o Pb . ] The initial solid composition is very

high in Pb content (the boundary of the two phase field occurs at

81 w/o Pb), and falls to the vicinity of the eutectic composition

as solidification proceeds. The position in the samples where the

dendritic structure ends is at a f of 0.12 and 0.11 respectively*

s where f is fraction solidified. Because of the excellent agreement of

the macrosegregation of these two samples as well as the dendrite disappearance position, it is apparent that fraction solidified is the

proper variable to describe changes in composition. This fact suggests

strongly that the dendritic zone is not caused by an initial transient

but by macrosegregation. Henceforth in this paper we will use fraction

solidified as the length parameter to describe macrosegregation.

56
Figure 9 shows the composition as a function of fraction solidified

for a A3. 7 w/o Pb alloy solidified at 2.6 ym/s. With the increased

velocity, the degree of macrosegregation is less and the transition to

lamellar composite structure occurs at a larger fraction solid (0.23)

than for the sample run at 0.5 ym/s.

Figures 10 and 11 shows data for 40.7 w/o Pb alloys solidified at

5.1 ym/s and 1.3 ym/s along with the position of dendrite disappearance.

Table 5 includes the solid composition at the position of dendrite

disappearance (C and the solid composition at f

g = 0 obtained by

extrapolating the macrosegregation curves.

Conclusions and Discussion

The extensive macrosegregation shown in Figures 8 through 11 occurs

only for alloys on the Pb-rich side of the eutectic when freezing

vertically upward. Experiments performed by the present authors on the

Sn-ric'n side of the eutectic under identical conditions as the Pb-rich

experiments exhibit only a relatively omall amount of macrosegregation

produced by thermal diffusion. The measured macrosegregation on the

[Of.!

Sn-rich side agrees with previous work of Verhoeven, Warner & Gibson

using their equation (17) with a thermal diffusion coefficient of

“8 2 —5 2 2.1x10 cm / s-K (D=0.6xl0 cm /s) and no mixing near the liquid solid

interface (6 large) . This absence of significant macrosegregation on

the Sn-rich side in experiments conducted under identical conditions

with those on the Pb-rich side is strong evidence that solutal convection and not ordinary thermal convection is the cause of the

extensive macrosegregation on the Pb-rich side of the eutectic.

57
A qualitative model for the macrosegregation and resultant changes

in microstructure is apparent. Solutal convection is sufficiently

vigorous to disrupt the normal diffusion controlled (Sn-rich) solute

field near the liquid solid interface regardless of whether it is

planar or dendritic. This solute is transported by the convection to

the bulk liquid far from the interface which is almost always well

[ 26 ] mixed* by the radial temperature gradients which exist there

Hence the bulk liquid, which initially was of composition C

q on the

Pb-rich side of the eutectic, gradually becomes more Sn-rich. This

causes a slow change in the solid to compositions which are also more

Sn-rich as solidification proceeds. Qualitatively the situation is

identical to the case of normal solidification for single phase

materials 1 J or Verhoeven et al. J for plane front two-phase solidification. The transition from dendritic to composite structure occurs

when the solid composition is close enough to the eutectic composition

that plane front composite growth is stable. When the bulk liquid composition approaches the eutectic composition, solutal convection ceases.

Macrosegregation after this point is caused only by thermal diffusion.

A quantitative description of the location of dendrite disappearance

is possible. Figure 12 is a plot of G /V vs. the composition of the solid

J-j at the position where the structure changed from dendritic to composite.

* In the present experiments the composition of the liquid determined

after quenching was constant to within + 0.25 w/o pb . The solute

boundary layer was too thin to be detected.

58
It resembles the expected form of the relation between G /V vs.

L

composition for the stability of plane front solidification, i.e., it

falls slightly below the requirement of the simple constitutional

supercooling criterion

G

l

/V = - m/D (C

E

-C

s

) (92)

where m is the liquidus slope of the Pb-rich phase. It also agrees

[ 211 with the limited data of Davis & Fryzuk on zone melted off-eutectics

where macrosegregation is avoided by the fact that solid of constant

composition if fed into the melted zone. Hence, it is a conclusion

of this paper that the transition from dendritic to composite structure

occurs when the composition of the solid satisfied a stability criterion

for plane front growth of a composite.

Quantitative description of the observed macrosegregation is more

r r\ /

r\ s -| difficult. Verhoeven et al. * using a thermotransport-corrected

[271 Burton-Prim-Slichter type analysis of plane front composite growth

has formulated an expression to predict macrosegregation. The 40.7 w/o

Pb sample run at 5.1 ym/s is dendritic for an extremely short distance

(long compared to D/V) and hence this analysis is applicable. Using

a value of the thermotransport coefficient D" mentioned previously,

a value of 90 ym for 6, the thickness of the unmixed zone near the

interface, gave the best fit to the measured data in close agreement

with Verhoeven. This calculated curve of the macrosegregation is

shown superimposed on the data of Figure 10. In this experiment, it

was determined experimentally that the liquid composition remained on

59
the Pb-rich side of the experiment up to the quench point and solutal

convection probably existed throughout this experiment. The calculated

value of the bulk liquid composition using this value for 6 also agreed

to within 0.25 w/o of the measured composition of the quenched liquid.

Samples with large regions of dendritic growth cannot be fit with

this equation. Clearly the existence of the dendritic interface influences both the driving force for solutal convection and the geometry

for the flow. In the data shown in Figure 9 a substantial number of

data points exist in the dendritic zone of the sample and a marked

change in slope occurs as the volume fraction of dendrites decreases.

Quantitative description of these curves is a topic for future research.

It is significant to note two aspects of solutal convection

observed in this research. There is no evidence to suggest that as

the concentration of the bulk liquid approaches the eutectic composition

during solidification (and hence the concentration gradient goes to

zero) that the convection becomes less vigorous. For example, in

Figure 10, there is no systematic deviation of the data from the calculated curve which might suggest that 6, the width of the unmixed

zone, increased as the liquid composition reached the eutectic composition. Additional evidence on this point is seen in Figure 8

where the sharp break in the data at a f

g of about 0.25 marks the

transition from macrosegregation caused by solutal convection to

macrosegregation caused by thermal diffusion. This sharp transition

from regimes of convection to no convection may be reflected in the

extremely low concentrations (i.e., deviation from eutectic composition)

60
required for the onset of convection described in Section A of this

report. Additionally, the fact that containers as small as 3 mm used

for this work do not apparently suppress convection may be reflected

in the rather small change of critical composition for the onset of

convection as a function of to discussed in Section A.

The presence of growth rate fluctuations have been observed in

some cases in the present study. However, as can be seen for example

in Figure 7, many samples have been obtained which contain no fluctuations. Hence, it is the belief of these authors that these fluctuations

are caused by artifacts, such as for example, by nonsteady thermal

r og 1 contractions L “

. Solutal convection seems to be sufficiently steady

as to not cause growth rate fluctuations.

Future work on this contract will involve experiments under

magnetic field and solidification downward to determine if solutal

convection can be suppressed. Also attempts will be made to predict

quantitatively the extent of macrosegregation during the dendritic

growth of these off-eutectic alloys.

Acknowledgements

We acknowledge the substantial contributions of Professor

Robert F. Sekerka of Carnegie-Mellon University and Consultant to the

National Bureau of Standards to the theoretical part of this research.

We would also like to thank Barry I. Diamondstone for the wet chemical

analysis and J. R. Carruthers, M. Kuriyama, and R. G. Rehm for

helpful discussions.

61
References

[1] J. R. Carruthers, Preparation and Properties of Solid State

Materials , Vol. 3, Eds. W. R. Wilcox and R. A. Lefever (Marcel

Dekker, Inc., New York, 1977) p. 1.

[2] D. T. J. Hurle, Current Topics in Materials Science , Vol. 2,

Eds. E. Kaldis and H. J. Scheel (North-Holland, Amsterdam, 1977)

p. 549.

[3] W. J. Boettinger, S. R. Coriell, F. S. Biancaniello , and M. R.

Cordes, NBS: Properties of Electronic Materials , Eds. J. R.

Manning and R. L. Parker (NBSIR 78-1483, National Bureau of

Standards, Washington, D. C.) P. 15.

[4] D. D. Joseph, Stability of Fluid Motions I and II (Springer-Verlag,

Berlin, 1976)

[5] G. Z. Gershuni and E. M. Zhukhovitskii, Convective Stability of

Incompressible Fluids (Keter, Jerusalem, 1976).

[6] S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability

(Oxford, London, 1961).

[7] M. R. Scott and H. A. Watts, SIAM J. Numer. Anal. 14^, 40 (1977).

[8] M. R. Scott and H. A. Watts, SAND 75-0198, Sandia Laboratories,

Albuquerque, New Mexico, 1975.

[9] B. L. Darlow, M. R. Scott, and H. A. Watts, SAND 77-1328, Sandia

Laboratories, Albuquerque, New Mexico, 1977.

[10]

B. L. Darlow, M. R. Scott, and H. A. Watts, SAND 77-1690, Sandia

Laboratories, Albuquerque, New Mexico 1977.

62
[11]

M. E. Lord and H. A. Watts, SAND 78-0522, Sandia Laboratories,

Albuquerque, New Mexico 1978.

[12] H. B. Keller, Numerical Solution of Two Point Boundary Value

Problems (Society for Industrial and Applied Mathematics,

Philadelphia 1976).

[13] K. M. Brown, SIAM J. Numer. Anal. _6, 560 (1969).

[14] D. M. Gay, CNA-109 (Center for Numerical Analysis, The University

of Texas at Austin) Dec. 1975.

[15] L. F. Shampine and M. K. Gordon, SAND 75-0450, Sandia Laboratories,

Albuquerque, New Mexico 1975.

[16] L. F. Shampine and M. K. Gordon, Computer Solution of Ordinary

Differential Equations - The Initial Value Problem (W . H

Freeman, San Francisco, 1975).

[17] J. S. Turner, Buoyancy Effects in Fluids (Cambridge, London,

1973), Chapter 8.

[18] R. F. Sekerka, Crystal Growth: An Introduction , Ed. P. Hartman,

(North-Holland, Amsterdam, 1973) p. 403.

[19] R. T. Delves, Crystal Growth , Vol. 1, Ed. B. R. Pamplin

(Pergamon, Oxford, 1974) p. 40.

[20] F. R. Mollard and M. C. Flemings, Trans. AIME 239 , 1534 (1967).

[21] K. G. Davis and P. Fryzuk, Can. Met. Quart. IT), 273 (1971).

[22] J. D. Verhoeven and R. H. Homer, Met. Trans. 1, 3437 (1970).

[23] J. D. Holder and B. F. Oliver, Met. Trans. J5, 2423 (1974).

63
[24] J. D. Verhoeven, K. Kingery and R. Hofer, Met. Trans. 6B, 647

(1975).

[25] M. J. Tschetter and R. Z. Bachman, Talanta 21 , 106 (1974).

[26] J. D. Verhoeven, J. C. Warner, E. D. Gibson, Met. Trans. 3, 1437

(1972)

[27] J. A. Burton, R. C. Prim, and W. P. Slichter, J. Chem. Phys. 21 ,

1987 (1953).

[28] J. D. Verhoeven and E. D. Gibson, J. Cryst. Growth 1JD, 235 (1971).

64
Table 1. Values Used in Numerical Calculations

(unless otherwise stated)

v (kinematic viscosity) 2.43(10" 3

) 2, cm / s

< (thermal diffusivity) .108

2, cm /s

r\ (magnetic diffusivity) 7 . 6 (10

3

) 2, cm /s

D (diffusion coefficient) 3. 0(10~ 5

) cm

3

/ s

g (acceleration of gravity) 980.0 / 2 cm/ s

a (thermal coefficient of expansion) 1.15(10“ 4) K"

1

a

c (solutal coefficient of expansion) 5 . 2 (10~ 3

) (w/o)

1

Pq (density of Pb at melting point) 10.66 gm/ cm

3

(thermal conductivity) .159 J/ cmKs

k (thermal diffusivity of solid)

s .202 ,2 cm/ s

k^ (thermal conductivity of solid) .297 J / cmKs

(latent heat per unit volume) 256.0 J/cm3

m (liquidus slope) -2.33 K/ (w / o)

T^r (capillary parameter) 1.0(10" 5

) cmK

p (kinetic coefficient) oo cm/ sK

k (distribution coefficient) 0.3

e (fractional density change) 0

r (Prandtl number v/k) 2 . 25 (10~ 2

)

s (Schmidt number v/D) 81.0

k Ik

s 1.87

U

s /kL

1.87

g

l

200.0 K/cm

65
Table 2. Bulk concentrations, c**, at which instability first occurs as a function of velocity V (G =200 K/cm)

Li

g = g

e = 980 cm/s

2

V(ym/s) c**(w/o) CO X (cm) Dw/V c"CO c~MS(w/o) XMS

(cm)

0.5 3.60(10

-4

) 5 0.7 1.14 15.5 0.039

1.0 3.16(10

-4

) 2.4 0.8 0.57 7.76 0.031

2.5 4.10(10“ 4

) 1.0 0.7 0.23 3.13 0.024

5.0 8.39(10" 4

) 0.6 0.6 0.11 1.58 0.019

10.0 3. 10(10~ 3

) 0.4 0.4 0.057 0.804 0.015

20.0 1.89(10" 2

) 0.26 0.36 0.028 0.412 0.012

30.0 6. 40(10~ 2

) 0.18 0.35 0.019 0.281 0.010

40.0 1. 86(10

_1

) 0.14 0.34 0.014 0.215 0.009

50.0 1. 75(10

_1

) 0.008 4.5 0.011 0.175 0.008

80.0 1. 16(10

_1

) 0.007 3.4 0.0071 0.116 0.007

g = 10- 4

g

e

0.5 8.4(10" 3

) 6 0.6 1.14 15.5 0.039

1.0 3.09(10" 2

) 4 0.5 0.57 7.76 0.031

2.5 3. 63 (10

_1

) 2.2 0.34 0.23 3.13 0.024

4.0 1.87 1.4 0.34 0.14 1.97 0.021

See Table 1 for values of physical quantities used in the calculations.

For c

oo <c^", the unperturbed density is a monotonically decreasing

function of height.

66
Table 3. Bulk concentrations, c**, at which instability first occurs as a function of temperature gradient

V(ym/s) G

L

(K/cm) c** (w/o)

OO

1.0 50.0 1.15(10" 4

)

1.0 100.0 1.89(10'4

)

1.0 200.0 3.16(10~ 4

)

1.0 400.0 5 . 37 (10~ 4

)

1.0 1000.0 i.ii(io" 3

)

20.0 50.0 1. 88(10~ 2

)

20.0 100.0 1. 86(10~ 2

)

20.0 200.0 1. 89(10

_2

)

20.0 400.0 1 . 95 (10~ 2

)

20.0 1000.0 2 . 10(10~ 2

)

67
Table 4. Bulk concentrations, c**, at which instability first occurs as a function of magnetic field H

Qz (a^O assumed)

V (ym/ s) c**(w/o) Dco/V H0z (T) Q

1.0 3.16(10~4

) 0.8 0.0 0.0

1 . 0 3.55(10"4

) 0.9 0.050 9.0

1.0 3 . 09(10

-3

) 2.7 1.00 3 . 6(10 3

)

5.0 8.39(10" 4

) 0.6 0.0 0.0

5.0 1.90C10

-3

) 0.8 0.25 9.0

5.0 4 . 16 (10-3

) 1.0 0.50 3.6U01)

5.0 1. 16(10

-2

) 1.1 1.00 1.4(10 2

)

5.0 5.70(10- 2

) 1.5 2.50 9.0(10 2

)

20.0 1.89(10-2) 0.36 0.0 0.0

20.0 2 . 63 (10

-2

) 0.42 0.25 0.56

20.0 4. 19(10

-2

) 0.53 0.50 2.25

20.0 8.84(10-2) 0.68 1. 00 9.0

r &

I 2 The dimensionless number Q

= (Hg

zD/V) / (4TTpr)v) based

on D/V as the relevant length; for the values used in these

calculations Q

= 3.6(10 ) (H n /V) where H n is in T and V is

. , Uz Uz in ym/ s

68
Table

5

-

Summary

of

Experiments

Q

W

rH NO cn CN O o CN rH

• • » • o o o O o

00o

CNno

• • on o

cn

<r o

r-.

oo

oom

m

o

vo

oo

fL|o m

cn

<r cnm

cn cn cn m ^ ^

nt

tH

CNO

o

CN

t-H o

CN

i—

O

CN

B m

eg CO o l . o

m r-i w i—i rH o

vO . cn o

II e> Q W

U

69

38.1

w/o

Pb
LOG

LOG V jjL m /s

Fig. 1. The critical concentration c** above which instability occurs as

a function of the velocity V of directional solidification of lead

containing tin. The solid curves with positive slope represent

the onset of convective instabilities f^r constant gravitational

accelerations of g = 980 cm/s and 10 g . The solid curve with

negative slope represents the onset of morphological instability;

the nearly parallel solid line labeled (9p/9z) = 0 represents the

neutral density criterion.

70
4

V = 5 /xm /s

G L

= 200 K/cm

Fig. 2. The concentration at the onset of instability during

directional solidification at V = 5 ym/s for a dilute

alloy of tin in lead as a function of the spatial

frequence to of a sinusoidal perturbation. Three

eigenmodes are shown; the system is unstable with

respect to a particular convective mode if the bulk

concentration c lies above the curve for that mode. co —Uf For c < 8.4 (10 ) w/o, the system is stable with

respect to all small perturbations.

71
oo

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a)

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X)

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a)

S-4 cd

cn •H

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cn

d

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a)

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43

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4-1

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d

cn o

•H

I—I 4-1 cd c

X! 4- 2) 4-i a

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bO

d

d r— o c>

bi) •rH Pd

76

Solidification

proceeded

from

left

to

right.

At

the

extreme

right

there

no

dendrites.

Solidification

velocity

is

1.3

ym/soriginal

compositio
Fig-. 8. Solid composition as a function of fraction solidified

for two 43.7 w/o Pb samples of different length L,

solidified at 0.5 yn/s. Agreement shows fraction

solidified to be an accurate parameter to describe

these experiments. The regions of dendritic and

composite growth are indicated. The sharp break in

the curve is the demarcation between regions of

macrosegregation induced by solutal convection and

thermal diffusion.

77
IDOo

fOo

CVJ Ooo

co

CO "g 0

0. 4-) •H0 o • X 4-1 CN 0

CD 0

+J

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4-1 o 0

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Li_ i

i < i CD o

4-4 •H4-J ==- 0 G 0 0 Q rQ 0

i—

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COo CD 4-1 00 00 • CD Cd' 0

on !-t

G

O

<r00

CD 4d

130 e

wasX0

i—

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4-> •H

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a. 0 0

or 0 CD -0H 0O

Ljl o

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N

X

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CD 0 X i—l CO X) 0 0 o r0 E

)-l

— 0 0

CJ

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cD

p 0 -0 - 0 4-J

CTN

00

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78
CDOo

(06

tooo

rO« o

0J6do o

Ixl Q

-Jo

coo<

££

Li.

79

Fig.

10.

Macrosegregation

data

fora

40.7

v//o

Pb

alloy

solidified

at

5.1

ym/s.

The

curve

is

the

best

fit

of

the

data

using

only 6

asa

parameter

using

Verhoeven's

equation

(17) L .
(0

o

in

o

o

IO

o

CVJ o

o

o

(qd°/M ) Nomsodwoo anos

CO

"e

cn

i—

4J

cO x)

QJ •H

M-l •HX3

00

a)

!-< 00

a)cno

S-J o

co

oo

•H

|j-i

80

FRACTION

SOLIDIFIED
05O

Q.

LlJ oz<

cr<

Ui0- CL

COQ

UJb

orQz

UJo<9d

05zob

05o0. 2o

CN

bD •H

pH

81

A

plot

of

G^/V

versus

solid

composition

at

the

position

of

dendrite

disappearance

for

all

of

the

samples.

The

data

follows

the

general

trend

of

plane

front

interface

stability

for

off-eutectic

alloys.
Task 3

A Thermochemical Study of Corrosive Reactions in Oxide Materials

H. S. Parker, R. S. Roth and C. D. Olson

Ceramics, Glass and Solid State Science Division

Center for Materials Science

and

E. R. Plante

Chemical Stability and Corrosion Division

Center for Materials Science

Summary

Phase relationships in the K20-"Fe203" system are described for

the 80-100 mole percent E^O^ portion. Two structures analogous

to the Na-Beta alumina structures were found, a 3-type at about

85.2 mole percent Fe 2°3 anc* a high temperature 8'" -type at about

91 mole percent Fe202» Vapor pressure measurements of the 8“type

-4

indicate pressures of the order of 10 atm at 1300° C. Vacuum

melting and low oxygen pressure heatings in open containers result

in excessive sample volatility.

The chemical corrosion of platinum by compositions in the 80-

100 mole percent F e 2°3 range appears to be minimal at temperatures

up to 1600°C although there is some evidence of diffusion of the

volatile species into the platinum container. The absence of con-

+2 tainer attack may be attributed to the low amount of Fe in the

specimens at the pressures and compositions used in these experiments.

82
Introduction

The objectives of this task are to investigate the nature,

extent and limitations on experimental studies imposed by samplecontainer reaction at high temperature and to evaluate containerless

techniques for melt and solid state investigations. Alkali and

transition elements are among the most reactive and are of great

technological importance because of their application in the fields

of MHD, fuel cells, catalysts and solid state electronic and ionic

conductors. Reaction with the container can severely compromise

the bulk composition of the sample and adversely affect the electrical

and chemical properties. This, together with the inability in general

to electromagnetically levitate ceramic materials, makes low gravity

experiments attractive for containerless study of such materials.

The systems chosen as the initial experimental materials include

the KFeO^-FeO^ system and the CaO-Y^O^-FeO^ system. These systems are

of interest not only because of the reactivity with container materials

+2 (particularly Fe ) and other ceramics but also because of their

interesting electronic and ionic conductivity. Because of the high

temperatures involved, the container choice is limited to the noble

metals and their alloys for oxidizing atmospheres and to molybdenum,

tantalum and tungsten for inert or reducing atmospheres.

The general approach is to : a) determine the maximum temperature

limits for the container material and characterize the reaction products, b) determine the equilibrium phase assemblages and vapor

pressures in selected portions of the systems within the limitations

83
imposed by a), and, c) in collaboration with MSFC provide specimens

for acoustic levitation experiments at MSFC and characterize and

correlate results of levitation experiments with container experiments to evaluate problems associated with low gravity experiments.

The work described in this report on the KFeC^-FeC^ system is

a continuation of that undertaken during the previous contract

year [1]

Experimental Procedures

A detailed description of the preparation of the compositions

used for the quenching experiments described herein is given in the

previous report [1]

together with the characterization methods

following heat treatment of each specimen.

The quench furnaces used were of a standard design, wound with

Pt-Rh alloy wire and utilizing the furnace winding as one arm of an

a.c. bridge for temperature control. For temperatures up to 1500°C,

Pt-90Ptl0Rh thermocouples were used for measurement. Above 1500°C,

a disappearing filament optical pyrometer was utilized. The thermocouples and pyrometer were calibrated in situ against known melting

points and suitable corrections applied.

Equilibrium at a given temperature was considered to have been

reached when continued heating produced no further changes in the

x-ray powder diffraction pattern of the specimen.

84
Experimental Results

A. Phase Equilibria and Specimen-Container Reaction (H. S. Parker,

R. S. Roth and C. D. Olson)

The compositions and temperatures investigated are shown in

Table 1 and Figure 1. Included in Figure 1 are some data from the

previous report [1]. The B n, -alumina structure occurring in this

system at compositions in the 90-92 mole percent "F^O^" range has

not been recognized previously, although the 3-structure has been

reported [2],

Attempts were made to heat specimens of the 3"' composition in

open platinum containers at 1200°C and partial pressures of oxygen

-3 +2 +3

in the 10 atm range, with the hope of increasing the Fe /Fe

ratio and thereby lower the observed minimum temperature of stability.

These attempts were unsuccessful, in every case the only phase

present after heat treatment was Fe^O^. This suggests that the

potassium volatility under these conditions is too great to allow

the attainment of equilibrium in open containers. This is in contrast

to similar specimens (90.9 mole percent "Fe202 M

, Table 1) which

ruptured the container during heating in air at 1450°C for 24 hours

yet still contained only 3'".

As discussed in the previous report [1], little interdiffusion

and minimal reaction was observed between the platinum container

and a specimen of K^O :5 ,75Fe

2 02 (85.2 mole percent Fe20.2) after

heating at 1475°C for 19 hours. Specimens of K

2 0:4Fe20^, K20:llFe2 0

3

,

K20:12Fe202 and K

2 0:16Fe202 (80, 91.7, 92.3 and 94.1 mole percent

Fe202» respectively) were heated at 1450°C for 20 hours in sealed

85
Table 1 . The System KFeO -"Fe 0

2 e

2u

3

Composition

mole percent

Temp

6 C

k

2 o Fe2°3

20 80 1200

1450

16.67 83.33 1200

1446

15.4 84.6 1200

1200

1399

1520

14.3 85.7 1200

1200

1450

12.5 87.5 1400

1520

11.1 88.9 1200

1325

1400

1446

10.0 90.0 1200

1396

1450

9.1 90.9 1200

1398

1450

1450

1520

8.3 91.7 1200

1200

1450

1475

1600

Time Results^

hrs

19 6 + KFeO

2 20 partial melting. 8

19 6 + tr 8"

19 partial melting. 8

19 6

23 8

22 8

20 leaked. 8"' + Fe,0.

3 4

19 8

24 8

16 partial melting. 8

20 small leak. 8 + 8'"

22 leaked. 8"' + Fe^O^

19 8 + Fe 0

16 8 + Felfof

20 small leak. 8" f + 8 +

18

Fe3°A

partial melting. 8+8

19 8 + Fe_0~

20 8 + Fefof

6 + 16 8"^

19 8 + Fe 0-

20 8"’

24 leaked. 8"'

16 leaked. 8"'

20 partial melting. Fe_0

19

6 + 8"'

8 + Fe 0

24 8 + Fett

16 8". ^ J

19 Fe 0 + 8"’

19 melted. 8"' + Fe^O^

86
7.7 92.3 1400

1450

1450

5.9 94.1 1200

1400

1450

1520

1608

4.2 95.8 1200

1396

1520

20 Fe

? 0 + 8

20 8"* 4 FeO

24 8"' + Fe^O,

3 4

24 8 + FeO

20 6 + Fe^

20 leaked! 7e 0, + 8'"

20 8"' + Fe^O, SJ (partial

melting;

19 Fe^O (partial melting)

24 FeO + 8

20 Fefoif + 8 / 20 Fe3°4 + tr (partial

melting)

Si / — All compositions calcined at 625°C for 18 hours, 800°C for 90 hours

and 1000°C for 72 hours with periodic grinding prior to use in

quenching experiments . All quenching experiments done in sealed

platinum capsules. —^ Based on visual observation for detection of melting and x-ray

diffraction powder technique for phase identification. —

c / If partial melting occurred, presence of liquid phase could not be

detected visually.

heavy wall (1.5 mm) platinum tubes and sectioned for SEM examination.

An empty heavy wall tube was given the same heat treatment for comparison.

In the K20:4Fe2C>3 specimen, little sample-container reaction or

interdiffusion was evident, as shown in Figure 2. The three pores

which intersect the platinum surface, however, are rich in potassium

and iron as shown by the maps in Figure 2c and 2d. Examination of all

specimens at higher magnifications revealed a number of small pores,

in a band, remote from the inner surface of the platinum tubes (see

Figure 3, 4 and 5). No such porosity could be found in the empty

87
platinum tube given the same heat treatment. Figure 6 shows the

platinum container wall used for heating a specimen of 1^

0 : 1^6203

and left exposed to laboratory air after SEM examination. These

dendrites are rich in potassium and appear to be associated with pores

and fissures in the platinum, suggesting potassium accumulation in

these areas

It is possible that these pores are the result of diffusion of

the volatile species from the specimen along grain boundaries with

subsequent agglomeration and pore development.

Specimens of ^0^6203 (87.5 mole percent "Fe

2 03 ") were prepared

from hot-pressed material in the form of discs, 1 cm diameter and

0.1 to 0.2 cm thick, spheres 0.2 to 0.6 cm in diameter, and small

bars about 0.2 x 0.1 x 1 cm and supplied to MSFC for use in their

acoustic levitation and drop tube melting experiments. The sample

was melted in the drop tube apparatus, but melting was accompanied

by extreme outgassing of the sample and electrical discharges in the

apparatus, terminating the experiment. As discussed elsewhere in

this section, the potassium and oxygen pressures in these samples are

apparently too high for heating at high temperatures in vacuum or

reduced oxygen pressures in open containers without seriously compromising the bulk composition and introducing difficult problems

in design of experiments.

B. Vapor Pressure Studies (E. R. Plante)

Vapor pressure measurements yield useful data for projected

containerless operations under zero gravity by helping to define

conditions which minimize loss of volatile constituents. During

88
the current project year, vapor pressure measurements on the composition approximated by "K

2 0*6Fe

2

0

3 " were carried out and will be

summarized in this section of the report. These measurements complement the vaporization studies on KFe0

2 -"K2 0.7Fe

2 0

3

" which were

summarized in the previous report [1], The previous vaporization

measurements were interpreted in terms of the reaction

7/6 (K

2 0.Fe

2 0

3

) - 2K(g) + 1/2 0

2 (g) + 1/6 (K^CWFe^) . The composition of the 3 phase, "K

2 0.7Fe

2 03

", is not known precisely. This

phase may contain as little as 5.75 moles of and has a small

solid solution range in which additional Fe

2 0

3 may be dissolved. Under

the conditions of the vaporization experiments the oxygen pressure

is self generated and the K

2 0-Fe

2 03

phase may become oxygen deficient.

In fact, the vapor pressure measurements do indicate slight changes

in the composition of the evaporating phase. However, in the interpretation of the vaporization data, the composition of the residue

phase in the previous experiments, and the evaporating phase in the

current experiments will be assumed to be Ko 0*6Fe

2

03>

Vapor pressure measurements were made using a modulated beam

quadrupole mass spectrometer. A molecular beam issuing from a Pt

Knudsen cell was ionized by 30 V electrons and the major ions, K

+

and 0* were alternately monitored. The mass spectrometer was

calibrated using observed weight losses and integrated ion currents

(I 32

At A and I+g At /T) . Because of the complicated vaporization

process a nominal 0.34 mm diameter effusion hole was employed in the

89
effusion cell to aid in obtaining equilibrium pressures. Temperatures

were measured by thermocouple at the base of the effusion cell and

confirmed by optical pyrometer sighting into the effusion cavity.

The sample used in the current measurements was prepared by a commercial supplier and had a molar ratio of K

2 0/Fe2 0

3 0 f 1/7 . This composition corresponds to a two phase region in the phase diagram at

elevated 0^ pressure (see Figure 1) which is not stable under the

conditions of the vacuum vaporization experiments. Thus, in the early

stages of the measurements, oxygen pressures were high because of the

reaction

6 Fe

2°

3

( c ) = 4Fe

3

0

4 (c) + 0

2 (g)

which competed with the reaction,

K

20-6Fe2 0

3

(c) = 2K(g) + 3/2 0

2 (g) + 4Fe

3

0

4

(c).

Three sets of experiments were carried out using two samples of

molar ratio K

2 0/Fe2

C> 3of 1/7. In the first two experiments, the

initial starting composition were taken as 1/7 and the final composition was taken as 1/6. In the third experiment the initial and

final composition were both assumed to be 1/6. The three independent

sets of calibration constants determined for 0

2 and K were consistent

using these assumptions.

In previous measurements on KFeC>

2 [1], we noted that certain nonequilibrium effects were observed. Measurements of ion currents were

made at constant temperature but the ratio of l

3 2/*39 var lec* with

time depending on whether the temperature was increased or decreased

90
in going from one constant temperature to another. When the temperature

was increased, the K+

intensity increased with time while the 0*

intensity decreased with time. If the temperature were decreased,

then the K+intensity decreased with time and the 0* intensity

increased with time. Although it was possible to obtain and reproduce

a steady state in which the K+

and 0* intensities did not depend on

time, it was expedient to obtain data by varying the temperature in

one direction only and accepting the K

+

and 0* ion currents at the

mean time interval of each temperature because this data is also

required to calculate the mass spectrometer sensitivity constant.

The gradual variation in K+

and 0* intensities with time at constant

temperature is believed to result from a slight variation in composition of one or both of the phase boundaries.

In the measurements on 6Fe20^-Fe20^ the same non-equilibrium

effects as mentioned above were observed; however, the approach to

steady state conditions was much slower than in the KFeO^-I^O • 6Fe2C>2

measurements and it was not possible to demonstrate that equilibrium

conditions could actually be achieved. The observation that the

increase in the K+

and the decrease in the 0* signals with time can

be reversed by decreasing the temperature so that the K

+

signal decreases

and the 0+ signal increases is evidence that the composition of the

evaporating phase has crossed the equilibrium phase boundary and is

tending toward equilibrium.

91
As in the previous measurements, K+

and 0* signals were taken

at the mean of the time interval of each constant temperature point.

Since none of the data is obtained at equilibrium but the system

is trending toward equilibrium it follows that the K pressures are

too low and the oxygen pressures too high for data obtained when the

temperature is increased while the K pressures are too high and the

oxygen pressures too low for data obtained with decreasing temperature.

Some typical data for K and pressures are shown in Figure 7 .

The upper curve labeled A and the lower curve labeled A show the

respective K and 0

^ pressures when the temperature is varied from

lower temperature to higher temperature while the curves labeled

B show the K and pressures when the direction of temperature

change is reversed. In spite of the non-reproducibility of the data,

it is possible to show that the variation in the K20.6Fe20^ composition is not very great. If the vaporization reaction is written

to produce one mole of vapor and the K20.6Fe20^ composition is

essentially constant then the product should vary K

linearly with 1 /T. Figure 8 shows the same data as shown in Figure

7 plotted in this way and it is clear that a linear plot is obtained.

In fact, all of the data obtained in the measurements closely follow

this curve.

As noted previously, the variation in the K+and 0+ intensities

With time at constant temperature show that the composition of

K20.6Fe202 varies somewhat with temperature. Although we assumed

92
that the average composition is K9 0.6Fe20^, it is possible to calculate

shifts in the 0

2 content from the assumed average composition using

the observed ion intensities. At a temperature of 1300°C the composition of "K

2 0.6Fe202

" turns out to be K20.6Fe202 ^ . Thus, the

deviation in 0

2 content from the assumed average composition is not

large.

Finally, a comparison of K pressures over KFeC^-K^O. 6Fe20^ and

K

2 0. 6Fe2 02~Fe20^ is of interest. Although equilibrium pressures were

not obtained a comparison of K pressures for phases of fixed com8 2 position may be made using the data obtained from P* .P* for the

K 0-

57 43

Z former and P* .P* for the latter. These comparisons assume

K

that PA - 1/4 Pv for the KFeO_ vaporization and P = 1.33 P_ for

^2 K ^ jn- t>2 K20.6Fe202. Average potassium pressures for the respective phases

plotted as a function of temperature are shown in Figure 9. Pressures

of K over K20.6Fe

2

C>2 are approximately 1/2 those over KFe09

. The

-4

pressures at 1300°C in the 10 atm range predict that loss of K would

be experienced unless a high ambient pressure of 0

2 was maintained

or experiments were of short duration. This is entirely consistent

with the findings in other parts of the study.

Conclusion and Discussion

Phase assemblages in the system KFe02- MFe2Q2 M have been determined

by the quenching technique in the 1200°-1600°C temperature range for

compositions between 80 and 100 mole percent Fe

202. The existence

of a 8' M -alumina structure type between 1400° and 1520°C at about a

K20:llFe202 composition is reported.

93
The corrosive attack of platinum by specimens in the 80-100 mole

percent F e

2C

3

range appears minimal at temperatures in the 1450°-1600°C

range. However, the development of arrays of small pores in the

platinum suggests diffusion of volatile species from the specimen into

the tube and agglomeration into pores. The absence of container

+2 attack is attributed to the fact that the amount of Fe in the

specimens in a closed environment is too small to cause a serious

reaction problem with platinum. It is known that, in silicate systems,

+2 the presence of Fe does lead to container problems and bulk composition shifts (see e.g. Min. Mag. (12) 4jL, 555-6 (1977)). For this

reason, it is planned to investigate the effect of SiO^ additions

to selected compositions in the KFeC^-FeC^ system and determine the

extent of container reaction at low oxygen pressures.

Vapor pressure measurements on K20:6Fe20^ (85.7 mole percent

^e2^3^ s^ow the potassium pressures to be about one-half those over

-4

KFeC^. However, pressures in the 10 atm range at 1300°C predict

that loss of K would be expected unless a high ambient oxygen pressured was maintained to suppress volatility or experiments were of

short duration. This is confirmed by the excessive outgassing of a

specimen containing 87.5 mole percent during a drop tube melting

experiment performed by MSFC for NBS.

94
References

1. J. R. Manning and R. L. Parker, NBS: Properties of Electronic

Materials, NBSIR 78-1483, pp. 84-100, June 1978.

2. T. Takahasi, K. Kuwabara and Y. Kase, "Formation of K-8Fe20^

and Phase Diagram of the System KFe0

2 -Fe202" , Denki Kagaku 43

[5] 273-7 (1975). In Japanese.

95
CO ©

(0

^ M O

0o

‘±

cvi a>

E

96

MOLE
0.05 mm

Specimen and platinum container of K^OrAFe^O.

(80 mole percent Fe_0_) after heating at 1450°C

for 20 hours.

Platinum map of 2(a).

Iron map of 2 (a)

Potassium map of 2(a).

Figure 2. (a)

(b)

(c)

(d)

97
3 ym

Figure 3. Platinum container used for heating specimen of

K

? 0:4Fe20^ (80 mole percent Fe20^) at 1450°C

for 20 hours.

6 ym

Figure 4. Platinum container used for heating specimen of

K20:12Fe202 (92.3 mole percent ^^O-) at 1450°C

for 20 hours (dark area specimen, light area

platinum)

98
6 ym

Figure 5. Platinum container wall used for heating specimen of

K^OrlSFe^O^ (94.1 mole percent Fe^O^) at 1450°C for 20

hours

1

1 f—H

50 ym 10 ym

Figure 6. Platinum container used for heating specimen of

K20 :llFe

2 03 (83.3 mole percent Fe^CL) at 1475°C

for 19 hours, left exposed to air after SEM

examination.

99
PRESSURE

T.

# C

100
57 A3 Figure 8. Plot of P£ *Pq of same data shown In Figure 7, demonstrating

that composition range of ^O^Fe^O^ is small.

101
PRESSURE

OF

POTASSIUM

(ATM)

T, *C

1300 1200 1100 1000 900

Figure 9. Comparison of K pressures over KFeOj-KjO.fiFe-O- and

K^O^FCjOyFa^O^ for assumed constant compositions.

102
Task 4

Thermodynamic Properties of Refractory Inorganic

Materials at High Temperatures

Jack H. Colwell

Chemical Thermodyamics Division

Center for Thermodynamics and Molecular Science

Summary

The various techniques that have been used for high-temperature

thermodynamic measurements are reviewed, detailing the major problems

involved. The potential for obtaining thermodynamic data from freelycooling spherical samples is presented. The use of optical pyrometry and

arrays of photodiode detectors for the measurement of surface radiance

temperatures are contrasted and the requirements for converting these

measurements to thermodynamic temperatures are discussed. The use of

pyroelectric detectors is proposed for the determination of the total heat

loss from samples. The measurement difficulties presented by several

particular aspects of the experiments, such as the radiance of transparent

materials, sample vaporization, and temperature gradients within spherical

samples, are discussed.

103
Introduction

High temperature thermodynamic data has been largely derived from

drop calorimetry measurements^. In these measurements samples are

heated to some fixed (known) temperature in a furnace and then "dropped"

into a calorimeter. The heat released in the calorimeter is a measure of

the enthalpy change of the sample between the temperatures of the furnace

and the calorimeter. From a series of such determinations at different

furnace temperatures, the heat capacity and other thermodynamic variables

for the sample material can be derived. At moderate temperatures the

sample temperature within the furnace is usually determined by measuring

the furnace temperature with a resistance thermometer or thermocouple. At

sufficiently high temperatures, optical pyrometers are used to measure the

temperature of the sample within the furnace; this arrangement usually

gives a reasonable approximation to a black-body radiation source. Many

types of calorimeters have been used varying from the "block" type, where

a temperature rise of the calorimeter is determined, to calorimeters where

a phase change in a calorimetric material, such as the melting of ice in

the Bunsen ice calorimeter, is determined.

In striving to carry this type of measurement to higher temperatures

(>2000 K) , material problems became a limitation. Furnace design becomes

difficult but this can be circumvented by direct heating of samples by

induction heating or radiation sources. Sample containment, however,

remains as a stumbling block. As temperatures are increased, all materials

become increasingly reactive so that the samples must be kept from any

contact with foreign materials in order to prevent their reaction or

104
contamination. Several ways have been found around this problem. Samples

have been suspended within the induction coils by fine wires of the same

r 2

material as the sample, an approach which works up to the melting point

In these experiments a deep hole bored in the side of the sample served as

a black-body cavity for temperature measurements.

The latest step in this line of development is the use of electrof 3

magnetic levitation to suspend the sample while it is heated . By

properly shaping induction coils and applying sufficient power, an electro

magnetic lift can be imparted to a sample so that it is levitated in the

gravitational field. After the temperature of the sample reaches a steady

state, the radiance temperature of the sample is determined and the levitating field is shut off, dropping the sample into a calorimeter. These

experiments are difficult and tedious because the power required to levitate a sample frequently precluded effective temperature control of the

sample. Coils have to be developed for individual cases dependent on the

density and electrical resistance of the sample and the range of temperatures to be covered. These experiments, on the other hand, are not

limited by the melting of the samples and measurements have been made well

into the liquid range of several metals. It is necessary, however, to

line the calorimeter with a sheath of the sample material to prevent

reactions at that stage of the experiment.

A. parallel development in the measurement of high-temperature heat

capacities has been the use of high-speed resistive heating of metal

[ 4 ] samples, the entire experiment taking less than one second . The heat

capacity is derived from the simultaneous measurement at millisecond

intervals of the power being dissipated in the sample and its rate of

105
temperature rise. Temperatures are determined with a high-speed pyrometer

by observing the black-body radiation within the tubular sample through a

small hole in the wall of the tube. The experiments are nearly "containerless" even through the current and potential leads make contact with

the sample. The current leads are remote from the measurement area and

are cooled so there is little likelihood of impurity diffusion to the

measurement area in the short duration of the experiments. The knife

edges used for potential contacts are made from the same material as the

sample so that no contamination can occur. Experiments can be carried to

the point where the sample is partially molten, but the collapse of the

sample prevents any extension of the measurement further into the liquid

range This technique, as with the use of electromagnetic levitation,

has been limited to metals. Although "non-conductors" are usually sufficiently conductive at high temperatures to permit electrical heating, such

experiments will be very difficult^, The high speed experiments require

that the sample be heated uniformly. This is assured in the case of

metals where their electrical resistance has a positive temperature coefficient, as a result the coolest point in the cross section of the

sample gets the most current and therefore, the most heating. The reverse

is true of nominal insulators, the conductivity is semiconducting in

nature so that the most heating will take place at any hot spot which

accidently occurs.

With the advent of the Space Shuttle Program the possibility was

presented of conducting containerless, high temperature thermodynamic

measurements on materials that are not good metallic conductors. An

106
obvious approach would be an experiment similar to the electromagnetic

levitation experiment except that the low-gravity environment of space

would be used to "suspend" the sample as it is heated and its temperature

determined. Since the sample cannot be dropped, it would be necessary to

inject it into the calorimeter in some way or for the calorimeter to

engulf the sample. The latter prospect led John Margrave to dub such a

device the "gulp calorimeter" 1 *

. Although this approach to the problem

is straightforward and involves no new principles, it would be cumbersome

in practice. Each datum of the enthalpy determinations would probably

require a separate sample and a new calorimeter liner or possibly even a

different calorimeter. Even if the device could be automated the experiments would still be very time consuming and most likely still require

considerable operator participation. An alternative method which we have

been studying is to simply heat a sample to a high temperature and then

allow it to freely cool within an evacuated cold-walled chamber. If the

rate of energy loss and the temperature of the sample can be determined as

a function of time one has a direct measure of the heat capacity as simply

the rate of energy loss divided by the rate of temperature change. It is

the problems likely to be encountered in making these two measurements on

which we have concentrated our attention. There are other aspects of such

an experiment for which we have postponed consideration or which are being

pursued by other workers. Positioning of the sample within the apparatus

and making it hold that position is one such case. In the space environment an apparatus will be subjected to accelerations which a freely

107
floating sample within the apparatus would not experience. To keep the

sample in a fixed position within the apparatus the same accelerations

must be imparted to the sample. Electromagnetic positioning is anticipated for this purpose (see below). Sets of coils, similar to those

used for the ground-based levitation of metals, can be used, but operating

at very low power so that little energy is dissipated to the sample.

Thomas Frost has been studying the application of this technique to the

space environment^^. An alternative is to use acoustic positioning^,

but as this technique requires a gas atmosphere, we are not giving it much

consideration at this time.

Heating of the sample to a high temperature after it is positioned in

the apparatus can be carried out in a variety of ways and should not

present a major difficulty. If electromagnetic positioning is utilized,

the same coils could be used for inductively heating the sample. It is

likely, however, that the positioning coil would not be strongly coupled

with the sample so that the inductive heating would have to be supplemented with some form of radiative heating to achieve the highest temperatures. Nonmetallic samples will have to be heated radiatively to some

high temperature (^2000 K) before they will be sufficiently conductive

even to be positioned electromagnetically , so it is unlikely that inductive

heating can be utilized in this case.

Temperature Measurement

No matter what type of experiment we are going to perform to determine the thermodynamic properties of a sample, we must determine the

temperature of the sample. For the temperatures we are considering

108
—greater than 2000 K, in a vacuum environment, —radiation thermometry is

the only means available. The radiance of a black body is given by

Planck’s distribution law

J

b

(A,T) = C

1

A_5TT" 1 [exp(C

2

/ AT) - 1]

_1

(1)

where and are the radiation constants and A is the wavelength of the

radiation

Temperatures are not determined by measuring in an

absolute sense but by measuring relative to that of another source.

The International Practical Temperature Scale is derived by assigning a

numerical value to the temperature of a black body at the melting point of

gold. In designing high temperature experiments one usually tries to

achieve a black body source in some way —a cavity in the specimen can give

a good approximation. In experiments where the sample is unconfined, this

is not easily accomplished and is not possible for molten samples. One

must then resort to measuring the surface radiance which is given by

J ( A, T, 0) = e(A,T,0)J

b

(A,T) (2)

where e(A,T,0) is the spectral emittance of the sample surface. The

emittance is a function of the condition of the surface, the angle of the

emission, 0, relativeto the normal to the surface, and has a temperature

and wavelength dependence which is characteristic of the material. By

integrating over all angles one obtains the hemispherical spectral radiance

W(A,T) = e(A,T)W

b

(A,T) = e( A,T)iTJb

( A,T) (3)

where e(A,T) is the hemispherical spectral emittance. Further, integrating over all wavelenths gives one the total hemispherical radiance

W(T) = e(T)W (T) = e(:T)aT

4 (4)

109
where e(T) is the total hemispherical emittance and a is the StefanBoltzmann constant. The latter equation defines the total radiative heat

loss per unit surface area as a function of temperature.

There is an extremely large literature on the radiative properties of

materials but the subject is not well understood. Emittance measurements

are not easy to make, particular for wide ranges of angle, temperature,

and wavelength. The sensitivity to surface condition makes the use of

literature values questionable (emissivity is the emittance of an optically flat surface which should also be clean and free of damage), and, in

any case, at high temperatures few values exist. It appears, however,

that whatever technique is used to determine temperature, it will be based

on some assumption about the emittance of the sample so one must be

cognizant of the literature on the subject. In the electromagnetic

levitation experiments of Margrave and coworkers ^the emittance of the

molten sample was determined at the melting point, a temperature that was

known, and it was assumed that it did not change with increasing temperature. The assumption, in this case, although an expedient, is based on

the knowledge that for metals, the emittance is not a strong function of

temperature or wavelength.

The assumption of the emittance being a weak function of temperature

and wavelength is also the basis of the only practical method available

for skirting the problem of an unknown emittance. The typical optical

pyrometer is monochromatic, using filters to define a narrow band of

wavelengths usually centered at 0.653 pn. The temperature is then defined

by (using Wien’s Law as an approximation for Planck’s Law)

110
J(A,T) = e(A,T)C ( 5 ) 1

A

5

tt 1 exp(-C

2 /AT)

By using a two color pyrometer where the radiance is determined simultaneously at two different frequencies, one obtains

£

i V

e

2 A/

exp(C

2

/T)[(l/A

2

) - (1/A

1

)

( 6 )

If one assumes the emittances are equal then the ratio of the radiances

gives the temperature; or better, if the radiances can be determined at

a known temperature, such as a melting point, then one need only assume

that the ratio of the emittances is a constant to obtain any other

temperature. An idea of the validity of these assumptions can be obtained from Fig. 1 which shows the extensive emissivity data for tungsten

[121 obtained by de Vos . The assumptions that the emittances are either

independent of temperature or equal at two different wavelengths can be

seen to fail badly. The assumption that the ratio of emittances is

independent of temperature is rather good at short wavelengths but

begins to break down above 1 ym. If one goes to a three color pyrometer

one can assume that the slope of the e versus X curve is constant or,

with a point of calibration, that the curvature is constant. The threecolor device is probably more useful being used as two separate two-color

devices where the redundancy is likely to show up systematic errors, from

the failure of the basic assumptions or from instrumentation faults.

The instrumentation of optical pyrometry has undergone some recent

development ^ ^^ and, in general, these devices far exceed our demands.

Ill
Automatic pyrometers have been constructed which can record data at

microsecond intervals. The highly stable silicon photodiodes are replacing photomultipliers as detectors so that continuous chopping for

intercomparison with a reference standard is no longer necessary and a

continuous output can be used. Fiber optics can be used to carry light

to a remote detector system if required^.

The use of pyrometers in our experiment could cause some difficulties because we would be viewing a sphere, possibly of small size,

that is not rigidly positioned and will be moving about to some degree.

An optical pyrometer is usually used to view the normal spectral radiance

from a surface, but in viewing a sphere this would only be achieved at

the midpoint of the spherical image. If the sphere behaved as a gray

body, i.e. had a constant emittance, independent of angle as well as

wavelength and temperature, it would not matter what part of the sample

surface was viewed for the sphere would appear as a flat disc of uniform

radiance. A real sphere, however, will not be gray and the emittance

will have a pronounced angular dependence. The general behavior found

for this dependence is shown in Fig. 2 where the emissivity relative to

the normal emissivity is plotted as a function of angle. This angular

dependence is in addition to the Lambert cosine law correction for

projected area. The important point is that all opaque materials do

appear to have an extended region going out to between 30 and 45° from

the normal to the surface where the emittance is constant. Thus, in

observing a sphere with a fixed pryometer, the sphere could undergo

lateral displacements of approximately half the radius of the sphere

112
without affecting the observation as long as the field of view is small

compared with the radius of the sphere. Similar displacements parallel

to the viewing direction, causing the sample surface to be out of focus,

can probably also be tolerated.

The possibility of having to cope with relatively large sample

displacements led us to consider using an array of radiation detectors in

place of an optical pyrometer. The individual detectors could be of the

silicon diode type with lenses or apertures to define a field of view

much larger than the sample at the position of the sample. The signal of

the detector will depend, in addition to the radiance of the sample, on

the size of the sample and its distance from the detector. To establish

the position of the sample at a given instant an array of detectors would

be used. With an appropriate geometric array the magnitude of the instantaneous signals at the detectors would vary as the inverse square of

the distance of the sample from each detector. This information can be

used to establish the position of the sample and then the the signal at

any one detector can then be related to the sample radiance. The arrangement which appears best for this purpose is to have four detectors at the

apices of a regular tetrahedron. This arrangement allows for two sets of

detectors to be symmetrically placed about a sample by having them occupy

alternate corners of a cube as diagrammed in Fig. 3. This arrangement

also allows for three sets of mutually perpendicular positioning coils to

occupy the faces of the cube. The size of the sample would have to be

determined, probably by photographic techniques. The bandwidth at the

individual detectors would be determined by the characteristics of the

detectors and any filters that may be used, just as with a pyrometric system.

113
The detector array could be adapted to two-color determinations. The

best arrangement for this would probably be to use two separate detector

systems but it could be done by switching filters on a single array.

In order to get the most accuracy in the temperature measurements it

will probably be necessary to obtain information on the emittances of the

various materials being studied by conducting separate laboratory experiments. For materials in the solid state, one could construct samples

with cavities to get a black-body temperature while simultaneously

measuring the surface emittance. For liquid state measurements, skull

[14] melting could be used to achieve uncontaminated samples. Although

the temperature of the melt could not be obtained with any accuracy, the

radiance as a function of wavelength at various arbitrary temperatures

would be useful in identifying unexpected large fluctuations in emittances.

In resorting to separate emittance measurements, use of optical

pyrometric measurements will be much simpler than using the silicon

detector system both from an instrumental standpoint and because of the

quantity that must be measured. The optical pyrometer as used measures

the normal spectral radiance as given by Eq. 2 (0=0°). The detector

system, however, in looking at a sphere will be observing a radiance

which is equivalent to the hemispherical spectral radiance given by Eq.

3. This can be seen in Fig. 4 where the top portion is a pictorial

description of the hemispherical radiance, i.e. all the radiation leaving

the elemental flat surface at all angles falls on a hemispherical detector. The bottom portion of Fig. 4 shows the radiation from a sphere

falling on an elemental detector surface at a distance. It can be shown

114
analytically that the radiance as a function of emission angle reaching

the detector is the same in each case, but a symmetry argument will

suffice. If the elemental detector in the bottom picture is considered

as a portion of a larger sphere symmetrically enclosing the radiating

sphere, then all the radiation leaving the inner sphere must fall uniformly on the outer sphere. Since each elemental surface area of the

inner sphere is radiating as the elemental area in the top picture the

elemental detector in the lower picture must be receiving the hemispherical radiance. As a result, if we were to try to determine the required

emittances in separate experiments, the detector system would require

that we observe a sphere with one of the detectors in the same fashion as

it is to be used in the actual experiment, or measure a flat radiating

surface at all angles to the normal and integrating over a hemisphere.

In either case the task would be difficult compared to a simple measurement of the normal radiance.

Most of the above discussion concerning emittances is based on the

premise that the radiating samples are opaque. Since our primary interest is in dielectric materials we will surely be encountering materials

that are transparent in at least some wavelength regions. There is very

little information available on the thermal radiation from bulk samples

of transparent materials. The most useful discussion we have found is by

Gardon^'^ who analyzed the radiation from ordinary glass. The principle

point is that the emission of radiation is no longer a surface phenomenon

but a bulk phenomenon. As a result, the thickness of the sample and

absorption coefficient of the material become determining parameters.

115
Figure 5 summarizes some of the results of Gardon.- It is the product of

the absorption coefficient, y, and the sample thickness, X, which determine the effective thickness of a sample, for if the absorption coefficient

is large the material will appear opaque and all the radiation will be

from the surface layer. This thickness dependence is going to have a

pronounced effect on what one observes with spherical samples; the size

of the observation area of constant emittance will now be much smaller

than was determined above for opaque materials.

Free-Cooling Experiment

To gather thermodynamic information from a free-cooling experiment

one must be able to measure, in addition to temperature, the total heat

loss from the sample. In our case, we are considering a spherical

sample at temperatures in the range 2000 to 4000 K. At these temperatures the principal mode of heat loss will be by radiation. Heat loss

through evaporation will be a consideration, but if it becomes large it

will probably defeat the experiment for other reasons. For the range of

temperatures considered, more than 99% of the thermal radiation will be

in the band of wavelengths from 0.4 ym to 12 ym, i.e. from the far blue

region of the visible through most of the near infrared.

In recent years there has been considerable research and commercial

development in pyroelectric detectors. A pyroelectric is a poled ferroelectric material which, when heated, changes its spontaneous polarization and with suitably applied electrodes can be used to generate a

current in an external circuit. The current is proportional to the rate

116
of change of temperature, so, to observe a continuous source, the radiation must be chopped in some way. Pyroelectric materials range from the

plastic film of polyvinyl fluoride, to the organic crystals of triglycine

sulfate, to the inorganic crystals of LiTaO^. LiTaO^ with its high Curie

temperature of 883 K is best suited for a high temperature apparatus and

is used in many commercial devices. Since the pyroelectrics are thermal

sensing devices, their response to different wavelengths is not a function

of the pyroelectric material but of the surface coatings or the device

configuration. Devices using gold-black coatings are effectively black

over the spectral range of interest and tfte devices can be made less

dependent on the quality of the coating by incorporating the pyroelectrics

into wedge shaped light traps . The sensitivity of commercial detectors

appear to exceed our demands but the linearity is questionable.

Pyroelectric detectors could be used in a manner similar to that

proposed above for the use of silicon diode detectors in temperature

measurements, except that radiation choppers would have to be installed

in front of the detectors. In using these detectors the sample size and

the distance from the sample to the detector would have to be determined.

A detailed knowledge of the effective area of the detector would be

needed to derive the total hemispherical radiance at the sample surface,

but this would come out of the calibration procedures.

117
It is contemplated that the detectors would be exposed directly to

the radiation from the sample with only apertures to cut down reflected

radiation. If any evaporation is occurring, the detectors would also be

exposed to the evaporated atoms which would most likely condense out.

Assuming the vacuum is good enough for most of the evaporated atoms to

travel directly to the surface of the detector and that their sticking

coefficient is high, one could get a reasonable measure of the heat loss

from evaporation. There is obviously much room for error on this point,

for instance, the vapor could react with the detector coating. The

coating itself, because of its highly porous nature, would probably have

to adsorb a considerable amount of the evaporated species before its

r is 1 radiation absorption characteristics would be appreciably affected J

;

also a light trapping device could help in this regard. One could use

windows such as KRS-5 (thallous halides) or Irtran-2 (ZnS) which are

transparent over the wavelengths concerned to discriminate against evaporated atoms but the reflection characteristics of these polished

surfaces would be quickly affected by the condensation of the evaporated

atoms. The various possibilities in this regard would have to be studied

in separate laboratory experiments for each substance studied.

r 8 1 Margrave and Weingarten L J proposed, in the study of metals, that

the induction coils be used to heat the sample to a steady state temperature, At that point the power loss from the sample by all mechanisms

would be equal to power dissipated by the inductive heating. The latter

can be calculated if the metal’s resistance is known, but whether it

could be calculated for a semiconducting material where the heating

would not be uniform is uncertain.

118
In conducting a free-cooling experiment one would want to remove the

heat source quickly so that initial perturbations can die away and the

measurements begun before the sample has cooled appreciably from its

highest temperature. This is done simply when radiant heating is being

used by shutting off the source. If the sample is being heated inductively

by the electromagnetic positioning coils, care must be exercised in

reducing the field so that the sample does not go into wide oscillation

about its equilibrium position. In rocket experiments where metallic

samples were heated and positioned in this way it was found that the

r 19

field could usually be reduced to a low level in a few couple seconds

A potentially major problem in the free cooling experiments is the

temperature gradients that may exist in the sample. Since all the heat

must leave from the surface of the sample, heat in the interior of the

sample must be conducted to the surface and if the thermal conductivity

of the material is small or the heat capacity or sample size is large,

large temperature gradients will develop. For most metals the gradients

would be small and not greatly affect a heat capacity determination, but

for non-metals the problem could become severe and the solution intractable.

The rate of cooling of a sphere by radiation when no gradient exists

is obtained by equating the rate of heat loss

(7)

with the thermal capacity

dH = (4/3)7TR

3 cpdT ( 8 )

which yields

119
dT/dt ( 9 ) = -3 ec7T4

/(cpR),

c is the specific heat and p is the density of the material, £ is the

total hemispherical emittance and R the radius of the sphere. When the

thermal conductivity, k, is finite the temperature distribution within an

opaque sphere must satisfy the equation

9 2 T 2 3T = £c _3T (10)

„ 2

r r 3r k 9t

dr

where r is the radius vector. The solution must also satisfy the boundary

condition at the surface

9T 4

k — = EOT at r = R (11) dr

i.e. the energy leaving the surface must equal the energy conducted to

4 the surface from the interior of the sphere. The T term on the right

hand side of Eq. (11) makes the problem nonlinear and it cannot be

solved analytically. If the boundary condition is made linear in temperature

k = hT (12) dr

the problem can be solved (though not simply) Approximate solutions

can be generated by an iterative procedure of substituting

h = goTq

3

(13)

in Eq. 12, with T^ as the starting surface temperature. After generating

a solution for a small temperature change from an initial Tq, a second

determination is made starting from the previous final temperature.

Each time, however, new roots to a transcendental equation must be

found and the starting conditions generated by a Fourier's series involving those roots fit to the preceeding solution; a most laborious procedure. This problem was worked out several years ago by D. L. Ayers

120
using a finite-difference computing technique and a brief summary of the

[ 21 ] results was published . The generalized solutions are in terms of two

dimensionless quantities

N

po

= kt/(cpR2

) (14)

and

N

rc

= k/(eaT

0

3 R) (15)

When N„ exceeds 10 or N exceeds 30 the cooling proceeds in a isoFo rc thermal manner, that is, as given by Eq. 9. If one could obtain sufficiently accurate data there would be the possibility, for a region where

the thermal gradients were moderately large, of deriving a value for the

thermal conductivity as well as the heat capacity. This could be accomplished by varying the size of the sample for this has a marked

effect on the cooling pattern; doubling the size of the sample would have

the same effect as simultaneously doubling the heat capacity and halving

the thermal conductivity. By measuring small samples the gradients would

be small and the heat capacity could be found. That value could then be

used in measurements on a large sample with its large gradients to get a

value of the thermal conductivity.

The nature of the temperature gradient within a spherical sample

using the boundary condition of Eq. 12 is shown in Fig. 6, where the

temperature T as a function of the radial vector r is plotted relative

to the temperature at the center, r = 0. The surface temperature will

be at some point R = u)r where u is a scaling factor which depends on the

parameters affecting the temperature gradient. For example, if k is

121
large, wr would approach zero and the gradient would be small; if k

approached zero, tor would approach it and the surface temperature would

approach zero (we assumed the radiating sphere sees surroundings at 0

K) . The dashed curve is an estimate of the behavior that would be

obtained using the true radiation boundary condition of Eq. 11.

The behavior of transparent materials in free cooling experiments

will probably be radically different from that described for opaque

materials. The radiation from a transparent material does not originate

at the surface but from within, so the description for thermal gradients

given above cannot hold. When the path length for radiation within a

material is very long it probably has the effect of making the effective

thermal conductivity large.

Conclusions and Discussion

At this time the potential of free-cooling experiments for obtaining

high-temperature thermodynamic data looks promising, but there are many

aspects of such experiments that will require considerably more study as

well as extensive laboratory investigation. The success of the freecooling approach, as presently conceived, hinges on the ability to

measure the total energy loss from the samples. We propose using

pyroelectric detectors to measure the radiant energy loss, but their

suitability for this purpose has yet to be established in laboratory

tests.

It is intended that measurements will extend to temperatures where

samples will be evaporating at a considerable rate. The heat loss from

122
evaporation can probably be established, but the sample vaporization

will complicate the radiation determination. Tests should be made on

rapidly evaporating samples to determine the effects on both radiation

loss and temperature measurements, but a suitable experimental arrangement

has not yet been devised.

For temperature determinations it will be highly desirable if an

optical pyrometer type device can be used for radiance measurements.

The methods for this type of measurement using multi-color rapid-response

instruments are reasonably well established and the emittance parameters

for the different materials would be much easier to assess than would be

the case with the use of a detector array. At this time we do not see

how a pyrometer can be kept focused on a sample if the sample is moving

about to any extent. Electromagnetic positioning does not appear to be

a problem for metals but may be for poor conductors. Before any development

can begin, therefore, we must ascertain that samples of the materials of

interest can be positioned within required limits. Other NASA contractors

could be of assistance to us in this regard.

In the event that the free-cooling experiments are not feasible, we

are continuing to give consideration to the various ways that receiver

(drop) type calorimeter experiments can be conducted.

123
References

[1] T. B. Douglas and E. G. King, Experimental Thermodynamics Vol. 1, J.

P. McCollough and D. W. Scott, Eds. (Butterworth, London, 1968) p.

293.

[2] L. Leibowitz, M. G. Chasanov, and L. W. Mishler, Trans. Metall. Soc.

AIME 245, 981 (1969).

[3] A. K. Chandhuri, D. W. Bonnell, A. L. Ford, and J. L. Margrave, High

Temp. Sci. _2, 203 (1970).

[4] A. Cezairliyan, J. Res. Nat. Bur. Stand. (U.S.) 75C (Eng. and

Instr.), 7 (1971).

[5] An evolution of this high speed technique which is under development

is to carry out the measurements at much higher speed (microsecond

intervals between readings) so that the integrity of the sample is

maintained upon melting. See G. R. Gathers, J. W. Shaner and R. L.

Brier, Rev. Sci. Instrum. 47, 471 (1976) or A. Cezairliyan and C. W.

Beckett, Microsecond-Resolution Pulse Calorimetry for Semiconducting

Materials at High Temperatures (A Feasibility Study) , NBS Publication

NBSIR 75-793, October 1975.

[6] A. Cezairliyan, Thermophysics Division, NBS, Washington, DC, private

communication

.

[7] Workshop on an Electromagnetic Positioning System in Space, May 1-2,

1978, Washington, D.C.

[8] J. L. Margrave and J. S. Weingarten, Thermodynamics Property Determination in Low Gravity , Rice University, Contract NAS8-32030,

Final Report, January 31, 1977.

[9] W. A. Oran, D. A. Reiss, L. H. Berge, and H. W. Parker, Preliminary

Characterization of a One-Axis Acoustic System , NASA TM-78213.

124
[10] See for example D. L. McElroy and W. Fulkerson, Techniques of

Materials Preparation and Handling , Vol. 1, pt. 1, Chap. II, R. F.

Bunshah, Ed., Wiley and Sons, New York, 1968.

[11] D. W. Bonnell, J. A. Treverton, A. J. Valerga, and J. L. Margrave,

Temperature, Its Measurement and Control in Science and Industry ,

Vol. 4, pt. 1, p. 483, H. H. Plumb, Ed., Instr. Soc. of America,

1972.

[12] J. C. de Vos, Thesis, Amsterdam, 1953.

[13] See also the references in footnote [5].

[14] In skull melting a block of material is heated locally to melting in

such a way that the unmelted portion forms a crucible for the molten

portions.

[15] R. Gardon, J. Am. Ceram. Soc. 34, 278 (1956); 44, 305 (1961).

[16] W. R. Blevin and J. Geist, Appl. Optics L3, 1171, 2212 (1974).

[17] W. M. Doyle, B. C. McIntosh and R. L. Maxwell, Detectors for

Wavelength Independent Radiometry , Electro-Optics/Laser meeting,

Paper IX-2, New York, N.Y. (1976).

[18] J. Geist, Radiometric Physics Division, NBS, Washington, D.C.,

private communication.

[19] R. T. Frost, Space Science Center, General Electric Corporation,

King of Prussia, Pa., private communication.

[20] L. R. Ingersoll, 0. J. Zobel, and A. C. Ingersoll, Heat Conduction ,

• Univ. of Wisconsin Press, 1954.

[21] D. L. Ayers, J. Heat. Trans., Trans ASME 92_, 180 (1970). Unfortunately, these solutions may now be lost. We have made considerable

effort to locate the author of this paper or someone familiar with

the work, but we have not been successful.

125
Fig. 1. Emissivity of tungsten as a function of temperature

and wavelength after de Vos [12],

126
1.4 Metals

Angle of Emission

Fig. 2. Directional spectral emittance relative to normal emittance

typical of metals and dielectric materials.

127
Fig. 3. Arrangement (not to scale) for positioning photodiode detectors

pyroelectric detectors, and electromagnetic positioning coils

about a spherical sample.

128
Measurement of Hemispherical

Emittance

0

Measuring Emittance of a Sphere

at a Point Detector

Fig. 4. Equivalence of detected hemispherical radiation.

129
Directional

Spectral

Emissivity

Angle of Emission

Fig. 5. Directional spectral emissivity of glass determined by the

product of the absorption coefficient, y, and the sample

thickness, X.

130
H80

131

Fig.

6.

Radial

temperature

distribution

of

a

radiating

opaque

sphere.

Surface

temperature

of

sphere

will

be

at

some

point

R

-

cor

where

w

depends

on

the

heat

capacity,

thermal

conductivity

and

emittance

of

sample.
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NBS-114A (REV. »-7»i

U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO.

NBSIR 79-1767

4. TITLE AND SUBTITLE

NBS: Properties of Electronic Materials

5. Publication Date JUNE 1979

7. AUTHOR! S) J. R. Manning

8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, DC 20234

11. Contract/Grant No. Government Order

H-27954B

12. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street, City, state, ZIP)

George C. Marshall Space Flight Center

National Aeronautics and Space Administration

Marshall Space Flight Center, AL 35812

13. Type of Report & Period Covered Annual

,

Apr. 1978 - Mar. 1979

15. SUPPLEMENTARY NOTES

Document describes a computer program; SF-185, FIPS Software Summary, is attached.

16. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant bibliography or

literature surrey, mention it here.) The report describes NBS work for NASA in support of NASA's Materials Processing

in Space Program covering the period April 2, 1978 to April 1, 1979.

The results obtained are given in detailed summaries in the body of the report.

Briefly, in Task I - Surface Tensions and their Variations with Temperature and Impurities - measurements of the surface tension y of liquid gallium were made in vacuum by

the sessile drop technique as a function of temperature T and, in the range 100 °C to

~50 °C, it was found that dy/dT = -0.068 mJ/iir °C. In Task 2 - Solutal Convection and

Liquid Diffusion Coefficients - samples of off-eutectic Pb-rich Pb-Sn were directionally

solidified, and the macrosegregation and changes in microstructure along the length of

the sample, attributed to solutal convection, were measured. In Task 3 - A Thermochemical Study of Corrosion Reactions in Oxide Materials - phase relationships in the

K20-Fe2 O3 system were investigated up to 1600 °C with major corrosion of the platinum

containers being avoided, perhaps because of the small amount of Fe + ^ in the specimens.

In Task 4 - Thermodynamic Properties of Refractory Inorganic Materials as Functions of

Temperature - methods of determining heat capacities in reactive liquid samples levitated at high temperatures, especially by use of freely-cooling spherical samples, were

analysed.

17. KEY WORDS (six to twelve entries; alphabetical order; capi talize only the first letter of the first key word unless a proper name;

separated by semicolons)

Convection; Heat capacity measurement; Pb-Sn off-eutectic alloys; phase diagram of

K20 -Fe 203 system; Solidification; Surface tension of liquid gallium; Thermodynamic

properties

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