NBSIR 85-3217

NBS: Materials Measurements

Reference

NBS

PUBLICATIONS

AlllDb

J. R. Manning

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

Gaithersburg, MD 20899

July 1985

Annual Report

For the Period 1 April 1984 - 31 March 1985

NASA Government Order H-27954B

Properties of Electronic Materials

U.S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

•QC

100

.056

85-3217 I 1985 i

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% NBSIR 85-3217

NBS: MATERIALS MEASUREMENTS

J. R. Manning

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

Gaithersburg, MD 20899

July 1985

Annual Report

For the Period 1 April 1984 - 31 March 1985

NASA Government Order H-27954B

Properties of Electronic Materials

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

TABLE OF CONTENTS

Sumnia ry 1

Task 1 - SURFACE TENSIONS AND THEIR VARIATIONS WITH TEMPERATURE

AND IMPURITIES

S. C. Hardy 5

Task 2 - CONVECTION DURING UNIDIRECTIONAL SOLIDIFICATION

S. R. Coriell, R. J. Schaefer, and G. B, McFadden 18

APPENDIX A: Convection and Interface Stability

during Sol idification 28

APPENDIX B: Double-Diffusive Convection with

Sidewalls 44

Task 3 - MEASUREMENT OF HIGH TEMPERATURE THERMODYNAMIC

PROPERTIES

D. W. Bonnell 71

NATIONAL BUREAU OF STANDARDS

MATERIALS MEASUREMENTS

Summary

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

Science and Applications Program under NASA Government Order H-27954B

(Properties of Electronic Materials) covering the period April 1, 1984 to

March 31, 1985. This work was carried out in three independent tasks:

Task 1. Surface Tensions and Their Variations with Temperature and

Impuri ti es

Task 2. Convection During Unidirectional Solidification

Task 3. Measurement of High Temperature Thermodynamic Properties

The results for each task are given separately in the body of the report.

With the advent of the Space Shuttle and the accompanying availability

of space flight opportunities, it may become feasible to exploit the unique

microgravity environment of space flight to produce improved materials and

improved measurements of important materials properties. Reduction of

convection and enhanced capabilities for containerless processing are two

advantages frequently cited for materials processing in space. Both of these

topics are under investigation in the current work. Tasks 1 and 2 have been

concerned with determining how the reduced gravity obtained in space flight

can affect convection and solidification processes. Emphasis in Task 3 is on

development of levitation and containerless processing techniques which can

be applied in space flight to provide thermodynamic measurements of reactive

material s

In materials processing on earth, gravity frequently produces densitydriven convection, thereby causing liquids to be stirred as they solidify.

This stirring disturbs the quiescent boundary layer at the solidifying
interface and can be very undesirable if nearly perfect crystals are required.

For example, it creates interface instabilities, introduces segregation of

components, and produces crystal defects in the resulting solid material.

These defects and inhomogeneities, which are particularly troublesome in

electronic technology and other advanced technical applications, might be

avoided in materials produced under microgravity conditions. Different

aspects of these convection processes have been of concern in the first two

tasks. Surface tension gradients which may be the main source of convection

in space flight experiments have been the measurement focus in Task 1; whereas

interactions between solidification processes, convection effects, and

interface stability, as influenced by gravity and the lack thereof, are under

investigation in Task 2.

In Task 1, which has now achieved its major goals, measurements were made

of the surface tension of liquid silicon and other materials in order to provide

reliable values for the effects that might be produced by Marangoni convection.

This completes work on this task. Silicon is one of the most important

materials used in electronic technology. Because of the need for a fully

defect-free silicon, not readily obtainable under earth-bound conditions

where gravity driven convection can be important, silicon is a major candidate

material for processing in space. In planning such space processing, it is

important to know the dependence of surface tension on temperature and

impurities, particularly oxygen. Surface tension gradients produce Marangoni

convection and may be the main source of fluid flows which affect solidification

processes in space. During the current reporting period, final sessile drop

experiments were performed to establish the temperature (T) dependence of the

surface tension (y) of clean liquid silicon and of clean liquid gallium.

2
a low-melting point material suggested for use in model space flight experiments, in the temperature ranges above their melting points. Reliable values

of dy/dT for these materials were obtained.

In Task 2, convective phenomena which arise during directional solidification are being investigated by both theoretical calculations and experiments.

Direct measurements of convection and associated solidification effects have

been made during directional solidification of succinonitrile containing

small amounts of ethanol; and calculations are being made of convection,

interface stability and impurity segregation that should be expected during

directional solidification. Measurements were made of the effect of a second

fluid phase present as droplets or bubbles near a solidification interface.

In cooperation with Centre d' Etudes Nucleaire de Grenoble (CENG) of Grenoble,

France, studies have been begun which could lead to experiments with the

MEPHISTO space-fl ight directional solidification system. Numerical calculations

were made of fluid flows in directional solidification systems with rigid

walls. In addition, linear stability calculations have been carried out for

Al-Mg and Pb-Sn alloys of interest for experiments being carried out by other

investigators and related to materials processing in space.

In Task 3, assistance is being provided to a joint project involving

investigators from Rice University (Professor J. Margrave) and General Electric

Co. (Dr. R. T. Frost) in which a General Electric electromagnetic levitation

device is being used to develop levitation/drop calorimetry techniques having

space flight applications. The critical advantage of such containerless

processing experiments is the avoidance of specimen contamination that would

occur upon contact of a reactive material with container walls during heating

and melting. Automated techniques are being developed in the current work both

3
to provide support to the on-going ground-based experimental effort in

levitation calorimetry and also to act as a prototype for flight experiments.

As a major part of the current report an analysis is given of considerations

needed in selecting space flight experiments in this topic area.

4
Task 1

Surface Tensions and Their Variations with Temperature and Impurities

S. C. Hardy

SUMMARY

Measurements of the surface tension of liquid silicon as a function of

hydrogen pressure were performed to determine if reductions in oxygen activity

would alter the surface tension results found previously in measurements done

in purified argon. The result supports the belief that oxygen partial

pressures were insignificant in our measurements. Thus for clean silicon it

is concluded that the surface tension as a function of temperature can be

represented by

V = 885 - 0.28(T - 1410)

where y is the surface tension in mJ/m^ and T is the temperature in Celsius

degrees.

The surface tension of liquid gallium has been measured using the sessile

drop technique in an Auger spectrometer. The samples were cleaned by argon

ion sputtering and were free of impurities to the sensitivity of Auger

spectroscopy. The surface tension of liquid gallium is found to decrease

linearly with increasing temperature and may be represented as

Y

= 708 - .066 (T - 29.8)

where y is the surface tension in mJ/m^ and T is the temperature in Celsius

degrees.

5
Introduction

Research activities during the past year involved studies of the surface

tensions of silicon and gallium. The work with silicon was a continuation of the

program initiated last year; the gallium research had been started several years

ago but had been put aside to undertake the silicon studies. The final six

months of funding for this project was used to complete these two investigations.

Surface Tension of Liquid Silicon

In the work described in the last report we measured the surface tension

of liquid silicon as a function of temperature in purified argon atmospheres.

Our measurements indicated that the surface tension of clean silicon as a

function of temperature could be represented by the following expression in the

temperature range from 141 0°C to 1600°C:

Y

= 885 - 0.28(T - 1410) (1 )

where y is the surface tension in mj/m2 and T is the temperature in Celsius

degrees. These results were in good agreement with a previous measurement at

one temperature in pure hydrogen which is evidence that the oxygen partial

pressures were low in our work. However, other measurements showed that the

surface tension was very sensitive to low concentrations of oxygen. Thus it

is possible the results given by equation 1 were affected somewhat by the

residual pressure of oxygen.

The major instrumentation development necessary for quantitative study

of the variation of silicon surface tension with oxygen partial pressure was

not feasible with the funding and time available in this year's contract. We

attempted, therefore, to reduce oxygen activity in our system and thus obtain

surface tensions higher than those of equation 1 by performing measurements

in argon containing 2.% hydrogen. We found no increase in the surface tension

6
over measurements in pure argon. This result supports our belief that the

oxygen partial pressures are insignificant in our measurements. High

concentrations of hydrogen could not be used because increased heat losses

due to the large thermal conductivity of hydrogen made it impossible to reach

the silicon melting temperature with the available equipment. A manuscript

entitled "The Surface Tension of Liquid Silicon" was prepared and has been

accepted for publication in the Journal of Crystal Growth.

Surface Tension of Gallium

The major effort this year has been directed towards completion of our

studies of liquid gallium. Although the surface tension of gallium had been

studied extensively, there remained significant uncertainties as to its

magnitude and temperature dependence. These data are of interest because

gallium has been suggested as a model fluid for Marangoni flow experiments.

In addition the surface tension is of technological significance in the

processing of compound semiconductors involving gallium.

The scatter in the gallium surface tension data is widely attributed to

the presence of trace concentrations of surface active impurities in the

different measurements. However, no determination of the surface composition

was possible in the earlier experiments; the presence of impurities was

inferred from the surface tension values. In the present work we have measured

the surface tension of liquid gallium using the sessile drop technique in an

Auger spectrometer. This instrument permits a direct analysis of the surface

composition concurrently with the surface tension measurement and thus provides

the surface characterization lacking in earlier work.

There have been seven(^"^) previously reported studies of the surface tension

of gallium over a wide temperature range. Three of these studies found nearly

7
the same non-linear decrease of surface tension with increasing temperature(^ »2»3)

,

Furthermore, some measurements in our laboratory (unpublished) using similar

techniques are in approximate agreement with these other observations of nonlinear behavior. These data are shown in Figure 1 along with three linear

determi nations^^’^’®) . (A fourth linear result is significantly lower and

cannot be shown on the scale of Figure l(^K) Although a non-linear temperature

variation is often associated with surface active impurities, the agreement

of four independent measurements of non-linear behavior suggests some other

factor may be causing the curvature in the temperature variation. In addition,

most of the non-linear data lies above the linear data in Figure 1. This

seems to be inconsistent with a model in which surface active impurities are

causing the non-linearity. Thus the data taken as a whole are anomalous.

Experimental

The Auger spectrometer used in this work is a conventional single-pass

cylindrical mirror analyzer (CMA) with a coaxial electron gun. The CMA is

mounted vertically so that it can be focussed on the surface of a sessile

drop located beneath it. Auger data are acquired in an analog mode using

first derivative spectral display. The electron beam can be rastered to form

a video image of the drop surface. The system is evacuated with sorption and

ion pumps to minimize hydrocarbon concentrations. Pressures in the low 10"^

torr range are achieved without baking as measured by a nude ionization gauge.

An ion gun and gas supply system permit sample cleaning by argon ion sputtering.

The sessile drop is contained in a shallow cylindrical silica cup which

stabilizes the drop and imposes a circular cross-section on the drop base.

The cup has an inside diameter of 1.27 cm. Its top has been beveled to form

a sharp edge at the inside diameter in order to reduce perturbations of the

8
drop base due to anisotropic wetting of the cup top(®). The cup is clamped to

a simple hot stage mounted on a horizontal manipulator. The pneumatic

vibration isolation supports of the stainless steel bell jar are adjusted to

level the cup for the surface tension measurements. A stationary W-Rh

thermocouple is inserted directly into the sessile drop to measure the

temperature and to provide electrical contact during Auger analysis. The

thermocouple is withdrawn from the drop prior to surface tension measurements

by translating the hot stage. The drop is viewed through two opposing 8"

view ports and is photographed in projected light using an f/5.6 enlarging

lens with a 1 90 mm focal length in conjunction with a 4" x 5" camera body. A

fine grained film is used which permits analysis of the negatives at lOOX.

The gallium used in these measurements is of 99.99999% purity. The

samples are prepared in the form of cylindrical slugs by solidification in a

plastic tube under alcohol containing some HCl to dissolve surface oxides.

After loading of the sample into the cup and evacuation of the system, the

temperature of the hot stage is raised slowly to maintain good vacuum conditions.

The gallium melts to form a sessile drop and some impurities desorb from the

surface as the temperature is increased to about 400°C. The drop is then

cooled and examined by rastering the electron beam over the surface and

performing an Auger spectral analysis. After this initial heating the surface

is found to be covered by a thin oxide layer with high carbon content. This

overlayer of impurities is removed from the liquid surface by sputtering with

argon ions(^). Several cycles in which the sample is heated, cooled and

sputtered are necessary before the surface is clean. Even after extensive

sputtering an occasional small solid impurity island will be seen. After

cleaning, the temperature is adjusted to the desired value, the stage translated

9
to break contact with the thermocouple, and the drop photographed.

Results and Discussion

The photographs of the sessile drops were analyzed on a measuring microscope and surface tensions calculated using the traditional Bashforth-Adams 0)

procedure in conjunction with modern drop shape tables. The diameter of the

cup provided a convenient length scale and the density data of Koster et al(^^)

were used in the analysis. Figure 2 shows the surface tension computed for

two drops of different mass in a number of experiments. The drops were

heated, cooled and sputtered prior to the surface tension measurements in

order to obtain a clean surface. Except for the few small solid impurity

islands which probably remained after this procedure, the drop surfaces were

free of impurities to the sensitivity of Auger spectroscopy («1%). The

pressure during the surface tension measurements was less than 10“^ torr at

the highest temperature (»500°C) and less than lO”^ torr at temperatures

below 400°C. Linear regression analysis of the data of Figure 2 yields the

following relationship for the surface tension of gallium in mJ/m2;

Y = 708 - .066 (T - 29.8) (2)

Here T is the temperature in Celsius degrees.

In comparison with the previous measurements summarized in Figure 1,

this result is closest to that of Yatsenko(^) et al who found

Y

= 71 2 - .0606 (T - 29.8) (3)

using density data which differs slightly from that used in our calculations.

An adjustment of their relationship to the density data we have chosen yields

Y = 709 - .066 (T - 29.8) (4)

Thus the surface tension at the melting point and the slope they find are

essentially identical to our results. A density adjustment of the other

10
measurempnts shown in Figure 1 does not significantly improve the agreement

with oiir values.

The range in melting point surface tensions of the various measurements,

excluding the very low result of Konig and Keck(^), is less than 2%. This is

reasonable agreement for surface tension measurements using different techniques

and conditions. However, the range in temperature coefficients is much

larger, the result of Karashaev(^) being about 50% higher than our determination.

The melting point surface tension, temperature coefficient and conditions of

the various measurements are summarized in Table 1. We cannot identify the

cause of the large discrepancy in temperature coefficient found in the

measurements.

As discussed above, the non-linear surface tension data of Figure 1 lie

above the linear data at high temperatures. We have found a similar behavior

for impurity covered surfaces in our measurements. In Figure 3 we show the

surface tension as a function of temperature for a drop with a high surface

coverage of impurity islands composed for the most part of carbon. The

profile of the impurity covered sessile drop appeared perfectly smooth when

examined at 1 OOX magnification which suggests the islands were thin. After

sputtering to obtain a clean surface the linear data of Figure 2 were obtained.

This behavior is indicated by the line in Figure 3. Thus the presence of a

high surface coverage of solid impurity islands is associated with a curved

surface tension-temperature plot with the island covered surface tension

values exceeding those of the clean surface at high temperatures. This is

the general character of the nonlinear data of Figure 1. Presumably the

surface tensions calculated from a drop with a high density of surface islands

are in error due to distortions of the equilibrium drop shape. If this is

11
corrpct, it is surprising that the measurements shown in Figure 1 are in good

agreement. It suggests similar concentrations of solid island impurities in

four independent experiments.

Another empirical observation of an impurity effect is of interest.

During the course of this work it was necessary to open the vacuum system for

a minor adjustment of a component after a gallium drop had been cleaned by

heating and sputtering. Argon was flushed through the bell jar during this

procedure. After reevacuation the surface of the drop was found to be covered

by a uniform, thin solid oxide layer. The surface tension was measured to be

750 mJ/m^, which is significantly higher than the melting point value for a

clean surface. This observation suggests that previous measurements of

anomalously high surface tensions in gallium^^^) may be due to the presence

of thin solid oxide films which distort the equilibrium shape of the drop.

In conclusion, we have measured the surface tension of gallium sessile

drops which are clean to the sensitivity of Auger spectroscopy. The surface

tension decreases linearly with increasing temperature with slope -.066 mJ/m^K.

This result is in excellent agreement with one previous measurement, but is

significantly smaller in absolute value than some other determinations. High

surface coverages of solid impurities were found to generate a non-linear

temperature variation of surface tensions, calculated from sessile drop profiles,

presumably due to distortions of the equilibrium drop shape. It is also

possible for sessile drops whose surfaces are covered with solid layers to

give anomalously high calculated surface tensions.

A manuscript entitled "The Surface Tension of Gallium" has been accepted

for publication in the Journal of Crystal Growth.

12
Table 1. Summary of Linear Surface Tension Results

Reference Y m.p.

(mj/m2)

dy/dT

(mJ/m^K)

Conditions

Karashaev(^) 718 -.101 static vacuum

Khokonov(S) 715 -.088 static vacuum

Konig( 7) 680 1 • O hydrogen

Yatsenko^ 709 -.066 dynami

vacuum

This work 708 1 . O dynamic vacuum

13
References

1. V, I. Nizhenko, L. I. Sklyarenko, and V. N. Eremenko, Ukrain. Khim.

Zhnr. 6 (1965) 559.

2. A. G. M. Nalgiev and Kh. Ilbragimov, Zh. Fiz. Khim. 48 (1974) 1289.

3. 0. A. Timofeevicheva and P. P. Pugachevich, Dokl. Akad. Nauk SSR, 134

(1960) 840.

4. A. A. Karashaev, S. N. Zadumkin and A. I. Kukhno, Russ. J. of Phys.

Chem. 329 (1 967).

5. Kh. B. Khokonov, S. N. Zadumkin, and B. B. Alchagirov, Elektrokhimiya 10

(1 974) 91 1 .

6. S. P. Yatsenko, V. I. Kononenko and A. L. Sukhman, High Temp. (USSR) 10

(1 975) 55.

7. U. Konig and W. Keck, J. Less-Common Metals, 90 (1983) 299.

8. Yu. V. Naidich, V. N. Eremenko, V. V. Fesenko, M. I. Vasiliu, and L. F.

Kirichenko, in "The Role of Surface Phenomena in Metallurgy," editor V. N.

Eremenko, New York, Consultants Bureau, 1963.

9. J. Fine, S. C. Hardy and T. D. Andreadis, J. Vacuum Sci . Technol. 18

(1981 ) 1310.

10. D. W. G. White, Trans. ASM 55 (1962) 757.

11. Van H. Koster, F. Hensel and E. U. Franck, Berichte Der Bunsen Gesellshaft

Fuer Physikal ische Chemie, 74 (1970) 43.

12. G. J. Abbaschian, J. Less-Common Metals, 40 (1975) 329.

14
Hardy (unpubl ished) .

15
Surface

Tension

(mJ/m

Temperature (“C)

Figure 2. Surface tension of two gallium drops of different mass cleaned by

argon ion sputtering.

16
Surface

Tension

(mJ/m^)

Temperature (®C)

Figure 3. Measured surface tension of a gallium drop, a large fraction of whose

surface is covered with solid impurity islands. The line (same as Figure 2) shows

the behavior found when the islands are removed by sputtering.

17
Task 2

Convection during Unidirectional Solidification

S. R. Coriell, G. B. McFadden and R. J. Schaefer

Introduction

The general aim of this task is the theoretical and experimental study

of the fluid flow, solute segregation, and interface morphology which occur

during solidification. In particular, the role of thermosolutal convection

(gravity driven flow due to temperature and solute gradients) during vertical

directional solidification of a binary alloy is studied both theoretical ly

and experimentally. A number of articles describing this research were

published during the past year [1-8].

A brief review of work on this task and related research was presented

at the Workshop on Microgravity Science and Applications, which was held at

the Jet Propulsion Laboratory, Pasadena, CA on December 3-4, 1984; this review

is reproduced in Appendix A. It appears that cellular growth experiments in

the absence of fluid flow would provide definitive data and guidance in

advancing current theories. Since the avoidance of fluid flow in earth-based

directional solidification experiments appears extremely difficult, we are

planning such experiments in collaboration with J. J. Favier and D. Camel of

the Centre d' Etudes Nucleaires de Grenoble. In order to understand the

differences between earth and space experiments, mathematical modelling of

buoyancy driven convection for various geometries and for different gravitational

levels (including time-dependent accelerations) will be continued.

We present a discussion of current numerical modelling first and then a

description of the experimental work; each section can be read independently.

Numerical Results

Our previous calculations of thermosolutal convection during directional

18
solidification [6] assumed periodic boundary conditions on the vertical

boundaries. This is a reasonable approximation when the horizontal wavelength

is small compared to the lateral dimensions of the sample. When the wavelength

is comparable to the lateral dimensions of the sample, it is clearly better to

use boundary conditions appropriate for rigid walls and we have modified the

numerical algorithm accordingly. The periodic boundary conditions correspond

to u = ^v/sx = aT/sx = ;ic/;ix = 0 on the vertical boundaries; here u

and V are the flow velocities in the x and y directions, respectively, T is

the variation of the temperature from the temperature in the absence of flow,

and c is the solute concentration (see reference 6 for a detailed description

and notation). At a rigid wall the fluid velocity vanishes so that u = v = 0,

and the solute flux vanishes so that 9 c/ax = 0. The boundary condition

on the temperature variation depends on the thermal properties of the boundary,

with two limiting possibilities being perfectly conducting (T = 0) or perfectly

insulating (aT/sx =0). We have assumed perfectly conducting boundaries,

which implies that the temperature distribution on the boundary is independent

of time. The rigid boundary conditions are then u = v = T = (;^c/ax) = 0.

We have also assumed perfectly conducting boundaries at the top and bottom of

the sample so that T = 0 at these boundaries.

Numerical calculations were carried out for the following values of the

dimensionless variables: ks/l<L = </<s = (thermal properties of crystal and

melt are the same), Prandtl number Pr = 0.01, Schmidt number Sc = 10, distribution coefficient k = 0.3, crystal growth velocity U = 0.5, thermal Rayleigh

number Ra = 1268, and aspect ratio (height/width) = 2. The Prandtl and

Schmidt numbers are typical of semiconductor melts. Figures 1 and 2 show the

steady state stream function and concentration fields for a solutal Rayleigh

number Rs of 2000 for two different initial conditions. In Figure 1, the

19
flow is downward near both walls and upward in the middle, giving rise to a

concentration profile which is symmetric about the midplane of the container.

The dimensionless solute concentration in the crystal at the crystal-melt

interface ranges from 0.88 at the wall to 1.24 at the center with an average

value of 1.06. The deviation of the average value from unity is numerical

inaccuracy due to the finite mesh size (16 intervals in the horizontal

direction and 80 intervals in the vertical direction). In Figure 2, in the

vicinity of the crystal-melt interface, the flow is downward near the left

boundary and upward near the right boundary. The interface solute concentration

increases from 0.73 at the left boundary to 1.84 at the right boundary. By

symmetry, there is also a steady state solution in which the flow is the

reverse of that of Figure 2, i.e., upward near the left boundary and downward

near the right boundary. We did not find a steady state solution in which

the flow is upward near the walls and downward in the middle.

For a system that is unbounded in the lateral direction, the critical

solutal Rayleigh number for the onset of convection is 963 for the values of

the dimensionless parameters previously given [6]. The linear stability of

the system with rigid vertical walls was determined using the nonlinear code;

the onset of convection occurred at a solutal Rayleigh number of about 400.

Thus, the system with rigid walls is less stable than the laterally unbounded

system. Since rigid walls should inhibit flow, the result was intriguing and

we have investigated it further. A linear stability analysis for the

directional solidification system with sidewalls is nontrivial since it can

not be reduced to ordinary differential equations. Therefore, we have

considered a model system with simple boundary conditions on the horizontal

boundaries. A manuscript "Double-Diffusive Convection with Sidewalls" by

G. B. McFadden, S. R. Coriell, and R. F. Boisvert, describes results obtained

20
from this model and is reproduced in Appendix B. When the boundary conditions

on temperature and concentration are not identical, rigid walls can actually

destabilize the system for certain values of the governing parameters. An

area of further research is the development of a numerical algorithm for

similar linear stability calculations for a directional solidification system.

A number of linear stability calculations for laterally unbounded

directional solidification have been carried out in collaboration with various

groups interested in particular alloys and growth conditions. The critical

solute concentration for the onset of convective and morphological instabilities

during the directional solidification of aluminum containing magnesium has

been calculated for various gravitational accelerations and temperature

gradients. Thermophysical properties for the Al-Mg alloy were provided by

Stephan Rex and Professor Peter R. Sahm (of the Giesserei-Institut der RWTH

Aachen), who are experimental ly investigating these alloys. Recent research

by Caroli et al [9] have suggested that the onset of convective instability

may be oscillatory in time in the lead-tin system at very low temperature

gradients in the liquid. We have carried out linear stability calculations

at a crystal growth velocity of 10 um/s and a temperature gradient in the

liquid of 0.1 K/cm. The convective instability is oscillatory in time and

the behavior is similar to that in the succinonitrile-ethanol system [5].

Under conditions for which growth vertically upwards is convectively or

morphological ly unstable, it is worthwhile to consider the possibility of

growth vertically downward. In collaboration with J. J. Favier of the Centre

d'Etudes Nucleaires de Grenoble, a number of linear stability calculations

for the growth vertically downward of lead containing tin have been carried

out. The onset of instability is usually oscillatory in time. Further linear

21
stability and nonlinear calculations will be performed to elucidate the

advantages and disadvantages of downward growth.

Expe ri mental

Convective effects induced by the presence of a second fluid phase near

the solid-liquid interface have been further investigated by the use of a

succi noni tri le sample containing a droplet of mercury. This system was chosen

on the basis of assumed negligible mutual solubility and a large difference

in thermal conductivity. Because of its high conductivity relative to

succinonitrile, the surface of the mercury drop was effectively isothermal

even in the presence of large thermal gradients in the surrounding succinonitrile.

As a result, the isotherms in the succinonitrile were severely distorted and

bouyancy-dri ven convection was present in the succinonitrile. Convective

flow was seen even when the mercury drop lay just below the succinonitrile

solid-liquid interface, so that no fluid-fluid interface was present.

Irreproducibil ity in the shape of the mercury drop made quantitative rate

measurements meaningless, but the fluid flow in this case did not differ

noticeably from that which was present when the mercury drop contacted the

liquid succinonitrile, thus implying that Marangoni convection was not playing

an important role.

Earlier experiments on the effect of a gas bubble at the solid-liquid

interface of a succi nonitri le-ethanol alloy had shown that a greatly increased

fluid flow was present in the vicinity of the bubble following slight motions

of the bubble. The effect was attributed to Marangoni convection induced by

transient gradients resulting from the bubble motion. The effect was not

observed in the case of the mercury drop sample, where there should be no thermal

gradients and no composition gradients (due to the absence of the ethanol).

22
The experiments thus suggest that a second fluid phase can induce

convection near the solid-liquid interface both by distortion of the thermal

field and by the Marangoni effect. The first mechanism would of course be

greatly reduced by microgravity, but the second mechanism would continue to

operate. The importance of controlling the free volume generated by

solidification shrinkage during space flight experiments is thus emphasized.

In cooperation with Centre d'Etudes Nucleaire de Grenoble (CENG) of

Grenoble, France, we are studying methods for demarkation of solid-liquid

interfaces during directional solidification of metals. These methods would

be employed during flight experiments on the MEPHISTO apparatus to determine

interface shapes at different temperature gradients and growth velocities.

The shape of the solid-liquid interface during solidification of

semiconductor materials can be marked within the growing solid by Peltier

pulsing. In this method, a pulse of electric current is passed through the

sample, normal to the solid-liquid interface, with the result that Peltier

heat is emitted or absorbed at the interface, depending on the direction of

current flow. This Peltier pulse causes a transient change of the solidification

rate, which causes a corresponding change in the composition of the solid.

Upon subsequent metallographic examination, the shape of the interface at the

time of the pulse is revealed by the compositional transient. The Peltier

effect has the advantage that heat is absorbed or emitted directly at the

solid-liquid interface, but unfortunately Peltier pulse marking of metallic

solid-liquid interfaces is difficult because of the low Peltier coefficients

of metal solid-liquid interfaces.

There are, however, other ways in which marking transients can be

produced. One requirement is that the transient be brief enough that it not

23
cause a significant change in the interface shape. For this reason we are

initiating a series of experiments in which the interface will be perturbed

by very short pulses of electric current. Such pulses can be made very large,

in which case the effect of Joule heating will be greater than that of Peliter

heating or cooling. The effectiveness of the pulses will be measured as a

function of several variables to obtain the best marking with the minimum

disturbance of the interface.

24
References

1. Coupled Convective Instabilities at Crystal-Melt Interfaces, S. R.

Coriell, 6. B. McFadden, R. F. Boisvert, M. E. Glicksman, and Q. T.

Fang, J. Crystal Growth 66^ (1984) 514-24.

2. Asymmetric Instabilities in Buoyancy-Dri ven Flow in a Tall Vertical

Annulus, G. B. McFadden, S. R. Coriell, R. F. Boisvert, and M. E. Glicksman, Phys. Fluids^ (1 984) 1359-61 .

3. The Effect of Anisotropic Crystal-Melt Surface Tension on Grain Boundary

Groove Morphology, P. W. Voorhees, S. R. Coriell, G. B. McFadden and R. F. Sekerka, J. Crystal Growth (1984) 425-40.

4. Effect of a Forced Couette Flow on Coupled Convective and Morphological

Instabilities During Unidirectional Solidification, S. R. Coriell, G. B.

McFadden, R. F. Boisvert, and R. F. Sekerka, J. Crystal Growth (1984)

15-22.

5.

Convection-Induced Distortion of a Solid-Liquid Interface, R. J. Schaefer

and S. R. Coriell, Met. Trans. 1 5A (1984) 2109-15.

6.

Thermosolutal Convection during Directional Solidification, G. B.

McFadden, R. G. Rehm, S. R. Coriell, W. Chuck, and K. A. Morrish, Met.

Trans. 15A (1 984) 21 25-37.

7.

Morphological Stability in the Presence of Fluid Flow in the Melt, G. B.

McFadden, S. R. Coriell, R. F. Boisvert, M. E. Glicksman, and Q. T. Fang,

Met. Trans. 15A (1984) 2117-24.

8.

Convective Influence on the Stability of a Cylindrical Solid-Liquid

Interface, Q. T. Fang, M. E. Glicksman, S. R. Coriell, G. B. McFadden,

and R. F. Boisvert, J. Fluid Mech. 1 51 (1 985) 212-40.

9.

Solutal Convection and Morphological Instability in Directional

Solidification of Binary Alloys, B. Caroli, C. Caroli, C. Misbah, and B.

Roulet, J. Phys. Paris 46 (1985) 401 -13.

25
t

26

22,38

10.22,3.

CONTOUR

INTERVAL

OF

.44277.00,

PT,3.3).

.22,22-00,

,.0389

TO

4.,

320

CONTOUR

INTERVAL

OF

.30962

PT(3.3,

=

2.97,6
oo

lf> Q>oo

II Q.Opd

II O

Ui CM

O)

iC7)

TJ- Oo+oCO

<o

CM

II <

CCoo+oooCM

II to

CC

rrCO

(CI 27
APPENDIX A

Proceedings of Workshop on Microgravity Science and Applications

Pasadena, CA; December 3-4, 1984

Convection and Interface Stability during Solidification

S. R. Coriell

During a first order phase transformation, the motion of the interface

between the two phases depends on the transport of heat and solute. For

alloy solidification or crystal growth from the melt, the parent phase is a

fluid and the transport is by both diffusion and convection. For metallic

alloys, thermal diffusivities , <, are four orders of magnitude greater than

solute diffusivities, D, so that fluid flow has a much greater effect on

solute distribution than on the temperature field. The solute distribution

at the crystal-melt interface directly determines the solute distribution in

the solidified material since diffusion rates in the solid are generally

extremely small.

The general problem of mathematical modelling of fluid flow during

solidification is challenging since it involves the non-linear Navier-Stokes

equations and a Stefan problem, i.e., the boundary conditions on the temperature,

concentration, and fluid flow fields are specified at the solid-liquid

interface whose position is unknown. The temperature, concentration, and

fluid flow fields are not only coupled through the convective and buoyancy

terms in the field equations but also through the boundary conditions at the

solid-liquid interface. These boundary conditions directly couple the

temperature and solute field since the melting point depends on composition,

and indirectly couple the fluid flow field to the temperature and concentration

field, since the position of the interface is determined by these fields.

Further, the role played by the solid-liquid surface energy requires calculation

of the mean curvature of the solid-liquid interface. For metals and semiconductors, the Prandtl number (Pr = v/k where \> is the kinematic viscosity)

28
is of the order of 10"2 and the Schmidt number (Sc = v/D) is the the order

of 1 q 2; these parameters indicate widely different length and time scales for

mass, momentum, and energy transport and may cause additional numerical

problems.

Directional solidification of a binary melt to form a single phase solid

at constant velocity V is most amenable to theoretical analysis and is a good

approximation to directional solidification experiments in which there is

uniform relative motion of the sample and its thermal environment. Since the

solute concentration of a liquid is usually quite different from that of a

solid with which it is in equilibrium, there is either rejection or preferential

i ncorporation of solute by the solid at a moving crystal-melt interface.

This results, for a quiescent liquid, in an exponential concentration gradient

that extends ahead of the interface with a characteristic distance of D/V.

The temperature gradient is also exponential with characteristic distance </Vi

however, </V is usually sufficiently large that the temperature gradient is

essentially constant within each phase.

If the crystal-melt interface is planar and there is no fluid flow in

the melt, then at steady state the solidified material will have a uniform

solute distribution. Even in the absence of convection, it is well known

that for sufficiently high solute concentrations the crystal -melt interface

will be unstable [1], and develop into a cellular or dendritic interface with

lateral solute segregation (microsegregation) . For small solidification

velocities (typically less than 10“^ cm/s), an excellent approximation to the

stability criterion is the modified constitutional supercooling criterion,

i.e., the planar interface is stable if

G /(mG^) > 1

29
with

where ki_ and kg are thermal conductivities of liquid and solid, respectively,

G[_ and 65 are unperturbed temperature gradients at the interface in the liquid

and solid, respectively, m is the change in melting temperature with solute

concentration, and is the unperturbed solute gradient at the interface in

the liquid. The product mG^ is always positive, even though solutes can

either raise or lower the melting temperature. Basically, positive temperature

gradients are stabilizing while the solute gradient is destabilizing. Another

stabilizing influence, not evident in the equations, is the solid-liquid

surface free energy, y» which tends to minimize interfacial area. At small

velocities, the surface energy has a negligible effect on the stability

criterion but plays an important role in determining the wavelength \ at the

onset of instability; typically these wavelengths are in the range 1-100 ym

and decrease with increasing velocity.

Recently, there has been renewed theoretical interest in the nonlinear

aspects of the morphologies that occur during directional solidification.

For interface shapes that do not differ too much from planarity, the free

boundary problem may be handled by means of linear stability analysis [1] and

non-linear expansion techniques [ 2 ]. High speed computers have made possible

numerical calculations of nonplanar interface morphologies that differ significantly from planarity [ 3

- 4 ]. Since cellular spacings are generally orders

of magnitude smaller than sample dimensions, computational domains with

periodic boundary conditions corresponding to wavelengths of anticipated cell

spacings are used. In general, it is possible to find a family of steady

state solutions (interface shapes, temperature and concentration fields) for

30
a range of wavelengths. The fundamental problem of wavelength selection has

been longstanding and is just beginning to be understood [5],

Steady state two-dimensional interface shapes and interface concentrations

are shown in Figure 1. The three calculations [4] correspond to bulk alloy

concentrations that exceed the critical concentration for instability by 0.5,

4.0, and 13.0 percent. The cell groove deepens and accompanying solute

segregation increases rapidly as the bulk concentration is raised. Recent

work by Ungar and Brown [6] successfully follows the evolution of cells to

groove depths of up to fifteen times their wavelength. A nonlinear stability

analysis of three-dimensional cells has recently been carried out by

Sri ranganathan et al . [2] in an attempt to understand the complex morphologies

which have been observed experimentally. For example in lead-antimony alloys [7],

the interface morphology at the onset of instability depended on crystal 1 ographic

orientation; nodes (circular wells of liquid penetrating into the crystal)

developed when the growth direction was near [100] or [111] while elongated

cells developed near [110]. At higher degrees of instability, regular

(hexagonal) cells developed.

It is well known that horizontal temperature and concentration gradients

give rise to buoyancy driven fluid flow [8]. In order to minimize these horizontal

gradients growth should be in the vertical direction. For a normal liquid,

which expands on heating, growth vertically upwards will ensure that in the

absence of solute the liquid density will decrease with height. However,

even for this growth configuration since container walls are not perfectly

adiabatic, the differences in thermal conductivity between crystal , melt, and

container, and the latent heat released on freezing will give rise to horizontal

temperature gradients, resulting in fluid flow. The nature of this flow in

31
prototype crystal growth systems is beginning to be understood through extensive

numerical modelling [9]. For vertical growth in the absence of horizontal

temperature and concentration gradients, a motionless solution will formally

satisfy the fluid flow equations; however, such a solution is not necessarily

stable and convective instability may occur.

When both the temperature and solute fields individually cause the density

to decrease with height, there is no possibility of convective instability.

However, even in this case, there can be undesirable convective effects.

During measurements of cell spacings in Al-6 wt% Cu alloys, McCartney and

Hunt [10] observed that the macroscopic interface was highly curved. Since

the curvature could be eliminated by using the ternary system Al-Mg-Si, which

has a very low value of the temperature derivative of the liquid density along

the liquidus line, convection appears important in the stably stratified Al-Cu

system. Presumably, convection due to horizontal temperature gradients

interacts with the solute gradient to cause a non-uniform solute concentration

at the interface.

If a solute that is rejected at the crystal-melt interface decreases the

density of the melt or a solute that is preferential ly incorporated at the

crystal-melt interface increases the density of the melt, then the solute

field in the absence of a temperature gradient will cause the liquid density

to increase with height during growth vertically upwards, and there is the

possibility of convective instability. The unperturbed static density profile

due to both temperature and concentration variation is of little utility in

predicting the onset of convection since one can have a statically unstable

density profile without convection and a statically stable density profile

with convection. The latter case of convection, even though the liquid density

32
decreases with height, is a feature of double diffusive convection and arises

due to the difference in diffusivities of the two diffusing species, viz.,

solute and heat [11]. The results of linear stability analysis [12] of combined

morphological and double diffusive convective instabilities is shown in Figure

2 for the growth vertically upwards of lead containing tin with a temperature

gradient in the liquid of 200 K/cm for three different constant gravitational

accelerations. The system is unstable for tin concentrations c„ above the

curves. The upper line with negative slope represents the onset of morphological

instability, and is essentially independent of the gravitational acceleration.

The three curves labelled with values of the gravitational acceleration

represent the onset of convective instability, and are very sensitive to the

gravitational acceleration. For tin concentrations below the dashed-dot

curve the static liquid density decreases with height. Extremely small

amounts of solute can cause convective instability in experiments on Earth,

e.g., at a growth velocity of 1 ym/s the critical tin concentration is

3.2(10"^) wt%. Additional calculations for a growth velocity of 1 ym/s show

that a vertical magnetic field of 1 tesla increases the critical concentration

by an order of magnitude.

The behavior of the convective stability curves shown in Figure 2 can be

qualitatively understood by recognizing that the relevant length scale is the

diffusion distance D/V. The thermal Rayleigh number Ra is then proportional

to G[_(D/V)^ and the solutal Rayleigh number is proportional to G^(D/V)^ or

equivalently c^(D/V)^. For fixed G|_ and large growth velocities the thermal

Rayleigh number becomes very small and stability is determined by a balance

of the destabilizing solute field and the stabilizing viscous forces;

instability occurs when the solutal Rayleigh number exceeds about ten [13].

33
For small growth velocities the thermal Rayleigh number is large, and stability

is determined by a balance of the destabilizing solute field and the stabilizing

thermal field; the solutal and thermal Rayleigh numbers are roughly comparable.

The coupling between morphological and convective modes of instability

is rather weak except near the concentrations and velocities at which the

morphological and convective curves cross in Figure 2. In the vicinity of

this crossing instabilities which are oscillatory in time may occur.

In order to determine the extent of solute segregation and the nature of

flow field caused by double diffusive convection during directional solidification, the time dependent nonlinear differential equations for fluid flow,

concentration, and temperature have been solved numerically in two spatial

dimensions for small Prandtl numbers and large Schmidt numbers [14]. For

slow solidification velocities, the thermal field has an important stabilizing

influence: near the onset of instability the flow is confined to the vicinity

of the crystal-melt interface. Further, for slow velocities, as the concentration increases, the horizontal wavelength of the flow decreases rapidly--a

phenomenon also indicated by linear stability analysis. For a narrow range

of solutal Rayleigh numbers and wavelengths, the flow is periodic in time.

The steady state stream function and solute field as a function of

position for a semiconductor melt (Schmidt number of 10 and Prandtl number of

0.01) is shown in Figure 3. The crystal-melt interface, assumed planar in

the calculations, is located at the bottom of the figures, and the interface

concentration varies by about 60 percent along the interface. For this

calculation, the thermal Rayleigh number is large and the solutal Rayleigh

number exceeds the critical solutal Rayleigh number for the onset of convection

by only 30 percent so that the flow is confined to the vicinity of the crystal34
melt interface. The maximum flow velocities are relatively small. For example,

for a lead-tin alloy at a growth velocity of 2.0 ym/s and a liquid temperature

gradient of 200 K/cm, the maximum flow velocity was 17.0 ym/s when the bulk

concentration exceeded the critical concentration for instability by a factor

of 4.3. However, this flow velocity is sufficient to cause a 30 percent

variation in the solute concentration at the crystal-melt interface.

We discuss briefly a few experiments which demonstrate the effects of

convection during directional solidification. Boettinger et al [15] studied

the growth of lead rich lead-tin off-eutectic alloys. For growth vertically

upwards, the solute field is destabilizing and substantial variation in

composition in the growth direction (macrosegregation) was found. A vertical

or horizontal magnetic field of 0.1 Tesla did not reduce the macrosegregation,

but downward solidification in small (3 mm) diameter tubes virtually eliminated

the macrosegregation. Sample and Hellawell [16] have recently investigated

the mechanisms of channel segregation during alloy solidification vertically

upwards. Channels containing high solute concentrations develop from positions

near a dendritic growth front when the solute is less dense than the melt,

and plumes rise to the top of the bulk liquid. Recent experiments by Schaefer

[17] on the directional solidification vertically upward of succinonitrile

containing ethanol have shown an interesting interaction between thermal

convection due to radial temperature gradients and the solute field. For

growth conditions near the onset of morphological instability, a macroscopic

pit develops in an otherwise slightly curved crystal-melt interface. The flow

is basically down near the container walls and upward near the axis of the

tube. The solute concentration is increased near where the flow turns upward,

and the macroscopic pit occurs in this region. Morphological instability

35
then takes place in the pit, and the pit disappears leaving a region of

cellular growth in its place.

Except for a very limited range of parameters, the double diffusive and

morphological instabilities which occur during directional solidification are

rather weakly coupled. In particular, the short wavelength morphological

mode is essentially unaffected by the possibility of convection in the

quiescent liquid. Due to very different length scales for convective motions

and for cellular interfaces, it will be extremely challenging to model the

effect of convection on cellular morphologies. Various simplified models for

the effect of a fully developed flow on morphological instability have been

recently reviewed [18]. Delves [19] has calculated the effect of forced flow

parallel to the crystal-melt interface on the onset of morphological instability.

The effect of such a flow on combined double diffusive and morphological

instabilities has recently been treated for Couette flow [20]. Such a flow

does not affect perturbations with wave vectors perpendicular to the flow.

For perturbations with wave vectors parallel to the flow, the onset of

morphological instability is somewhat suppressed and double diffusive

instability is greatly suppressed. When instabilities occur, they are

oscillatory and correspond to travelling waves. A complicated flow field

could be approximated in the vicinity of the crystal -melt interface by a

simple flow such as a Couette flow. This would simplify the modelling of the

interaction between fluid flow and the crystal -melt interface. However,

Glicksman and colleagues [21-23] have recently demonstrated a coupling between

hydrodynamic and morphological instabilities for which such an approximation

is clearly incorrect.

We conclude with a brief description of this coupled instability. A

36
long vertical cylindrical sample of high purity succi noni tri 1 e was heated by

an electrical current passed through a long coaxial heating wire, so that a

vertical melt annulus formed between the coaxial heating wire and the surrounding

crystal-melt interface. The outer radius of the crystal was maintained at a

constant temperature below the melting point of the material.

This arrangement permits the temperature to decrease monotonical ly from the

melt to the solid across the crystal-melt interface, and consequently the

interface would be morphologically stable in the absence of fluid flow.

The thermal gradients in the melt induce buoyancy forces which cause the

fluid to flow upward near the heating wire on the axis and downward near the

crystal-melt interface. When linear stability analysis is used to calculate

the critical Grashof number for instabilities of the axisymmetric flow occurring

between two vertical infinite rigid coaxial cylinders held at different temperatures, it is found that for a Prandtl number of 23 (corresponding to succinonitrile), the flow is unstable to an axisymmetric perturbation above a Grashof

number of the order of 2000, and the resulting wave speed of this perturbation

is comparable to the maximum in the characteristic unperturbed flow velocity.

In contrast, the experimental observations with a crystal-melt interface

indicate an asymmetric helical instability at a critical Grashof number of

about 150 with a wave speed two orders of magnitude less than the unperturbed

flow velocity. Linear stability analysis reveals that the instability is due

to a coupling between a basic hydrodynamic instability in the buoyant flow

and the deformable crystal-melt interface. The crystal-melt interface lowers

the critical Grashof number of an analogous rigid-walled system by an order

of magnitude; furthermore, the hydrodynamic mode that is actually destabilized

by the interface is not the least stable mode in the rigid-walled system. The

37
calculations show that the instability may be regarded either as a large alteration of a basic hydrodynamic instability by the crystal -melt interface, or as

a significant modification of the morphological stability of the interface by

the presence of the buoyant flow. The instability is sensitive to the form

of the flow field; e.g., it does not occur for Couette or Poiseuille flows.

38
References

1. W. W. Mullins and R. F. Sekerka, Stability of a Planar Interface During

Solidification of a Dilute Binary Alloy, J. Appl. Phys. 3^, 444 (1 964).

2. R. Sri ranganathan , D. J. Wollkind, and D. B. Oulton, A Theoretical Investigation of the Development of Interfacial Cells During the Solidification

of a Dilute Binary Alloy: Comparison with the Experiments of Morris and

Winegard, J. Crystal Growth^, 265 (1 983).

3. L. H. Ungar and R. A. Brown, Cellular Interface Morphologies in Directional

Solidification. The One-Sided Model, Phy. Rev. B29, 1367 (1984).

4.

G. B. McFadden and S. R. Coriell, Nonplanar Interface Morphologies During

Unidirectional Solidification of a Binary Alloy, Physica 12D, 253 (1984).

5. J. S. Langer, Pattern Selection in Solidification, Met. Trans. 15A, 961

(1 984).

6. L. H. Ungar, Directional Solidification from a Bifurcation Viewpoint,

Ph. D. thesis 1984, MIT, Cambridge, MA.

7. L. R. Morris and W. C. Winegard, The Development of Cells During the

Solidification of a Dilute Pb-Sb Alloy, J. Crystal Growth^, 361 (1969).

8. D. D. Joseph, Stability of Fluid Motions II (Springer, Berlin, 1976).

9. C. J. Chang and R. A. Brown, Radial Segregation Induced by Natural

Convection and Melt/Solid Interface Shape in Vertical Bridgman Growth,

J. Crystal Growth^, 343 (1983).

10. D. G. McCartney and J. D. Hunt , Measurements of Cell and Primary Dendrite

Arm Spacings in Di rectional ly Solidified Aluminium Alloys, Acta Met.

1851 (1 981 ).

11. J. S. Turner, Buoyancy Effects in Fluids (Cambridge Univ., Cambridge,

1973).

12. S. R. Coriell, M. R. Cordes, W. J. Boettinger, and R. F. Sekerka, Convective

and Interfacial Instabilities During Unidirectional Solidification of a Binary Alloy, J. Crystal Growth^, 13 (1 980).

13. D. T. J. Hurle, E. Jakeman, and A. A. Wheeler, Hydrodynamic Stability of

the Melt During Solidification of a Binary Alloy, Phys. Fluids 624 (

1 983).

14. G. B. McFadden, R. G. Rehm, S. R. Coriell, W. Chuck, and K. A. Morrish,

Thermosol utal Convection During Directional Solidification, Met. Trans.

15A , 21 25 (1 984).

15.

W. J. Boettinger, F. S. Bi ancani el 1 o , and S. R. Coriell, Solutal Convection

Induced Macrosegregation and the Dendrite to Composite Transition in OffEutectic Alloys, Met. Trans. 1 2A , 321 (1 981 ).

39
16. A. K. Sample and A. Hellawell, The Mechanism of Formation and Elimination

of Channel Segregation During Alloy Solidification, Met. Trans. 1 5A , 2163

(1984).

17. R. J. Schaefer and S. R. Coriell, Convection-Induced Distortion of a Solid-Liquid Interface, Met. Trans. 1 5A, 2109 (1984).

18. S. R. Coriell and R. F. Sekerka, Effect of Convective Flow on Morphological

Stability, PhysicoChemical Hydrodynamics 281 (1981).

19. R. T. Delves, Theory of Interface Stability, in Crystal Growth, ed. B.

R. Pamplin (Pergamon, Oxford, 1974).

20. S. R. Coriell, G. B. McFadden, R. F. Boisvert, and R. F. Sekerka, Effect

of a Forced Couette Flow on Coupled Convective and Morphological Instabilities

During Unidirectional Solidification, J. Crystal Growth 69, 15 (1984).

21. Q. T. Fang, M. E. Glicksman, S. R. Coriell, G. B. McFadden, and R. F. Boisvert, Convection Influence on the Stability of a Cylindrical SolidLiquid Interface, J. Fluid Mech. 1 51 , 21 2 (1984).

22. G. B. McFadden, S. R. Coriell, R. F. Boisvert, M. E. Glicksman, and Q. T. Fang, Morphological Stability in the Presence of Fluid Flow in the Melt,

Met. Trans, m, 2117 (1 984)

.

23. S. R. Coriell, G. B. McFadden, R. F. Boisvert, M. E. Glicksman, and Q.

T. Fang, Coupled Convective Instabilities at Crystal-Melt Intefaces, J. Crystal Growth^, 514 (1 984).

40
IT) (M C„ - 1 .005 C„(X Ln

rv

in

CM Cqo “ 1 ^oo (X)

in

in

CM

- 1.13 C„(X)

LO

Fig. 1. Plots of interface shape and distribution of solute in

the solid for a given wavelength as the bulk concentration is increased. The top curve in each plot is the interface shape, the

bottom curve is the concentration of solute. Two wavelengths

are shown.

41
10

Figure 2. The critical concentration c** of tin in

lead above which instability occurs as a function of the velocity V of directional

solidification for a temperature gradient

in the liquid of 200 K/cm. The solid curves represent gravitational accelerations

g

= 980 cm/s^, 10 g , and 10 g . The

upper line with negative slope represents

the onset of morphological instabilities;

the nearly parallel dashed-dot line labeled

(9p/8z) = 0 represents the neutral density

criterion. The dotted extension of the curve labeled g corresponds to oscillatory

instabilities

.

42
STREAM FUNCTI0N C0NT0UR MAP AT T = 24.32

C0NCENTRATI0N C0NT0UR MAP AT T = 24.32

Figure 3.

The steady state stream function and concentration as a function of position for a bulk concentration 29 % above the critical concentration for the onset of convection.

Parameters used in the calculation are typical of semiconductor melts. In the concen t ra t i on plot, the increment of (dimensionless) concentration between contours

is 0.37. The maximum flow velocity is about 5 times the crystal growth velocity of

typically 20 u^/s .

43
APPENDIX B

On Double-Diffusive Convection with Sidewalls

G. B. McFadden, S. R. Coriell, and R. F. Boisvert

National Bureau of Standards

Gai thersburg , MD 20899

Abstract

The effect of rigid vertical boundaries on the onset of convective

instability is calculated for the salt finger regime of double diffusive

convection. The unperturbed state is a quiescient fluid with constant vertical

gradients of temperature and solute, which are stabilizing and destabilizing,

respectively. The horizontal boundaries are taken to be stress-free and

perfectly conducting. The lateral boundaries are perfectly insulating for

solute. Changing from thermally insulating to thermally conducting sidewalls

results in a strong destabilization of the flow for large thermal Rayleigh

numbers even in the limit that the separation between the sidewalls approaches

infinity. Further, for thermally conducting sidewalls, a decrease in the

separation of the sidewalls may destabilize the system.

47.20

44
1. Introduction

For a fluid in a gravitational field with two diffusing coinponents present,

for example, temperature and solute, it is well known that convective instabilities may occur even though the fluid density is strictly decreasing with

height. This instability, known variously as thermosolutal convection,

thermohaline convection, or double-diffusive convection, was first described

by Stommel et al in 1956 [1] and has since been reviewed extensively [2-4],

This instability is of practical importance in many diverse areas involving

convection and heat transfer; see e.g. [5], In what follows, the instability

will be described in terms of the diffusion of heat and salt in a differentially

heated saline solution. The density of a fluid parcel then depends on its

temperature and salinity: heating the parcel makes it lighter and salting it

makes it heavier. Of primary importance is the feature that the diffusion of

heat occurs much more rapidly than does the diffusion of solute.

Consider a vertical density stratification (uniform in the horizontal

directions) with temperature and solute profiles that have opposite effects on

the vertical density profile; that is, when considered separately, one would

tend to increase, and the other decrease, the density as a function of height.

Two distinct regimes occur, depending on which of the two quantities is

"unstably stratified." A fluid that is heated and salted from above (solute

field unstably stratified) is in the "fingering" regime. The conventional

heuristic description of the instability is as follows [3]. The upward

displacement of a fluid parcel carries it to a region of hotter and saltier

water. The temperature in the parcel is relatively quick to equilibrate with

its surroundings, whereas the solute is significantly less quick to do so.

The parcel then finds itself at about the same temperature, but with less

45
solute, than its neighbors, and so the upward displacement is reinforced by

buoyancy. In the simplest model [21, the instability occurs with a monotonic

time dependence ("exchange of stability").

A fluid heated and salted from below falls within the "diffusive" regime.

In this case, as described by Stern [6], when a fluid parcel is displaced

vertically, the relative lag in diffusion rates may result in an overcompensating

restoring force that causes "overstability" to develop, leading to a timedependent. oscillatory behavior.

The present paper is concerned with the effect of sidewalls on flow in

the fingering regime in the absence of applied horizontal gradients. The work

was motivated by numerical results obtained from a simulation of thermosol utal

convection occurring during the unidirectional solidification of a binary

alloy. In this case, the unperturbed solute field in the liquid ahead of the

solidifying planar interface has an exponential vertical profile due to the

rejection or preferential incorporation of solute by the solid phase. The

linear stability of the unperturbed problem for a system that extends to

infinity in the horizontal directions is then non-trivial due to the vertical

variation of the unperturbed state, and results are obtained numerically [7].

Finite difference methods were subsequently employed to describe the evolution

of the instabilities to finite amplitude [8]. In these calculations the twodimensional flow was computed in a hori zontal ly-peri odi c domain, consistent

with the linear stability results for an infinite domain. When the lateral

boundary conditions were altered to model the case of a finite container with

rigid, perfectly conducting sidewalls that are perfectly insulating to solute,

the instability was observed to set in at significantly lower Rayleigh numbers

than for the infinite system. This result was unexpected; it was anticipated

46
that the addition of sidewalls would stabilize the flow as it does in the

simpler case of a si ngl e -component system.

The linear stability for the directional solidification system with

lateral sidewalls is intractable. For this reason an easier problem is

considered here, consisting of a two-dimensional rectangular box with rigid,

solutally insulating lateral walls, with constant vertical gradients of

temperature and solute. We will mainly consider the sidewalls to be perfectly

conducting for temperature; however we will contrast these results with

previous results for perfectly insulating sidewalls, and will give a few

results for mixed boundary conditions. As was observed by Drazin [9] for

the single-component case, if the horizontal boundaries are taken to be

stress-free and perfectly conducting, the solution is separable and the

vertical dependence is simple. The linear stability is then governed by a

set of constant-coefficient ordinary differential equations in the horizontal

variable, so that the effect of the lateral boundary conditions on the stability

of the base state can be examined in detail. We found that the addition of

the sidewalls to the infinite system may result in a significant destabilization

for a range of parametric values.

In section 2 we describe the governing equations for the system. In

section 3 we describe briefly a fully numerical solution to the ordinary

differential equations. In section 4 we give an independent check of the

numerical results by finding zeroes of the characteri Stic determinant, which

in this case may be expressed in terms of closed-form 1 i nearly-i ndependent

solutions to the governing equations. Results are presented in section 5.

with a concluding discussion in section 6.

47
2. The Governing Equations

We consider a two-dimensional box of height H and width 2W. The sidewalls

of the box are taken to be rigid, perfectly thermally conducting, and perfectly

insulating to solute. Following Drazin [9], the top and bottom boundaries

are taken to be stress-free and perfectly conducting to heat and solute. The

fluid occupying the box is described by the Oberbeck-Boussi nesq approximation

to the Navi er-Stokes equations [3]. The fluid is heated and salted from

above, with values of the temperature and salt concentration (Tq', Sq') on

the bottom boundary and (Tq‘ + aT, Sq' + AS) on the top boundary with

aT and AS positive. The density p* of the fluid is assumed to depend

on the temperature T' and the salt concentration S' as

p' = po'd - a[T' - To'] + 3[S' - So'])

where a and 3 are both taken to be positive constants. The constant po' is the

density at temperature To' and salinity So'.

In the Oberbeck-Boussinesq approximation the dimensionless governing

equations take the form

-D(v24,)/Dt = -v2(v2^) + (Ra/Pr)(8T/3x) - (Rs/Sc ) (3S/3x) (1)

DT/Dt = V^T/Pr (2)

DS/Dt = v2s/Sc (3)

for -L<x<L, 0<y<l, where L = W/H, D/Dt = 3/3t + u3/3x + v3/3y,

u = 3\|;/3y, V

= “3ij^/3x, T = (T' - Tq')/aT, S=(S' - Sq')/aS, Ra = gaATH^/v<

is the thermal 'Raylei gh number, Rs = g3ASH3/vD is the solutal Rayleigh number,

Pr = v/k is the Prandtl number, and Sc = v/D is the Schmidt number. Here

u and V are the x and y components of the fluid velocity, is the stream

function, g is the magnitude of the gravitational acceleration (which is in the

negative y direction), v is the kinematic viscosity, k is the thermal diffusivity.

48
and 0 is the solute diffusivity. Our sign convention is such that both Ra

and Rs are positive for the case of a fluid heated andtsalted from above. In

choosing dimensionless units we have used a length scale H and a time scale

h2/v.

The governing equations admit a quiescent base state ( \p = 0) with

linear temperature and solute fields, T = y and S = y. The boundary conditions

chosen to describe this state are \p = d^\p/dy^ = 0 and T = S = 1 for y

= 1,

\p = d^ip/dy^ = T = S = 0 for y

= 0, and \{; = d\p/dx = 3S/9x = 0 and T = y

for X

= -L and x = L.

The equations describing the linear stability of the base state are then

-a(v2;j;)/8t = -v2(v2J) + (Ra/Pr)(3T/8x) - (Rs/Sc ) (3S/ 3x ) (4)

3T/3t - 3$/3x = v2f/Pr (5)

3S/3t - 3j/3x = V^S/Sc (6)

with = d^^/dy^ = T = S = 0aty = 0 and y

= 1 , and ij; = d1f/dx = 3S/3x =

T = 0 at X

= -L and x = L. Here S, and T are perturbations to the base state.

The boundary conditions at y

= 0 and y

= 1 are consistent with a y-dependence

of the form sin(miy). We will consider the case with the lowest vertical

wavenumber n = 1. Indeed, for the particular boundary conditions chosen

here, higher values of n simply correspond to several copies of n = 1 systems

stacked vertically one upon the other, so that knowledge of the n = 1 system

for all values of the width L therefore includes the behavior for all n as

well. We also confine the discussion to the case of "exchange of stabilities;"

that is, at the onset of instability we assume that the convection is steady

and time-independent. This is known to be true in the simpler case of a

laterally infinite domain, and the assumption was also tested numerically in

a number of cases for the system discussed here, with no observed oscillatory

49
behavior. With these assumptions, the governing equations reduce to a set of

ordinary differential equations: ‘

(d 2 _ ^2)2j = RaDT - RsDS (7)

-DJ = (d 2 - Tr2)T (8)

-D$ = (d2 - tt2)s (9)

.-1 for ij;(x), T(x), and S(x), where now D = d/dx. The boundary conditions are

i); = Dijl = T = DS = 0 at X

= -L and x = L. The Prandtl number Pr and the Schmidt

number Sc have been removed from the linear problem by rescaling the temperature

and solute field, so that the perturbed temperature and solute fields have ~ A ^ ^ the form T(x,y) = PrT(x)sin(Try ) and S(x,y) = ScS(x)sin(Try) , respectively.

The perturbed stream function has the form ij;(x,y) = ij)(x)si n( rry ) . There are

therefore just three parameters appearing in the linear problem for the onset

of steady convection: the dimensionless half width L, the thermal Rayleigh

number Ra, and the solutal Rayleigh number Rs. For given values of L and Ra,

we seek the critical solutal Rayleigh number Rs*, which is the smallest

value of Rs for which the governing equations admit a non-trivial solution.

Note that for the case of a 1 ateral ly-unbounded system, the equations

admit periodic solutions T = S = kcos(kx), = (k^ + Tr^)sin(kx), where the

solutal Rayleigh number is then Rs = Ra + (k^ + Tr2)3/k2. The critical

solutal Rayleigh number then satisfies Rs* - Ra = 27ii^/4 = 657.5, and

corresponds to a critical wavenumber k* = tt/ v/T]

3. A Numerical Approach

The governing equations (7-9) may be rewritten as a set of eight real,

coupled, first order linear ordinary differential equations, with the parameter

Rs considered as an eigenvalue. As in earlier treatments of eigenvalue problems

50
described by ordinary differential equations [7], the numerical solution was

obtained using the linear boundary value problem software package SUPORT [10].

Following a method described by Keller [11], the homogeneous eigenvalue problem

is converted to a nonsingular inhomogeneous problem by altering the form of

the boundary conditions so that the solution satisfies, say, D'^\|;(L) = 1

rather than i|)(L) = 0. Using SUPORT to solve this problem, the desired

solution is obtained by using a nonlinear root solver to vary Rs until the

solution satisfies the proper boundary condition !j)(L) = 0. For most of the

calculations the subroutine SNSOE [12,13] was used to find solutions to this

equation. The nonlinear root solver PITCON [14] was also used to track

solutions by a continuation method to obtain Rs*(L), the critical Rayleigh

number as a function of the half-width L.

4. An Analytical Approach

The set of ordinary differential equations (7-9) has constant coefficients

and so admits solutions whose x-dependence is of the form exp(iax), where the

wavenumber a may be real or complex. With the assumption ij)(x) = i|;oexp( i ax)

,

T(x) = Tq exp(iax), and S(x) = Sq exp(iax), a solution is obtained provided

that a^ is a root of the fourth-degree polynomial

(a2 + 7r2)[(a2 + ,^2)3 . a2AR] = 0 (10)

where aR = Rs - Ra. In terms of a^, there is one root a^ = and three othpr

roots obtained from the relation (a^ + ir2)3 _ a^AR = 0, which is the same

relation that appears for a single-component system [9]. The latter roots a^

are shown in Fig. 1 as a function of AR. There is one root a^ which is

p always real and negative, which we will label -a§ with a 3

> 0. The remaining

two roots a^ may be either real and positive or complex conjugate pairs,

depending on whether AR is greater than or less than 27 tt^/ 4. For aR > 27ti^/4,

51
9 9 d we label the two roots as a[ > a^, with a-| > 0 and a

2

> 0. If aR < 27tt /4

the two roots are denoted by a[ and a^, where a-j = u + iv and 32 = , with y> 0

and V > 0. The superscript denotes the complex conjugate. For aR = 27tt^/4,

9 9 9 the two roots aj^ and a^ become equal, and for AR = 0, all three roots a^',

32 , and -33 become equal to -it . For these special values of aR

the degeneracy of the roots forces one to include secular forms, such as

X exp(iax), to obtain linearly independent solutions.

There are therefore eight linearly independent solutions that may generally

be obtained from the eight roots for a. Because of the symmetry of the

equations and boundary conditions to reflection about the origin x = 0 , these

solutions may be combined into either even or odd solutions about x = 0 [9].

/I We denote even solutions as those for which T(x) and S(x) are even functions

of X and the stream function ijl(x) is odd; the reverse is true for the odd

solutions. Even or odd eigenfunctions are then constructed by expressing the

eigenfunction as a sum of the four even or odd linearly independent solutions

to the linear ordinary differential equations using four undetermined coefficients.

Imposing the boundary conditions ij; = D\j; = T = DS = Oatx = L results in a

homogeneous system of linear equations for the four unknown coefficients.

For non-zero solutions the determinant of the system must vanish, which is

the required equation for the solutal Rayleigh number Rs in terms of L and

Ra. For the even case, this determinant takes the form

irRaT 4 ( 6]Ai + 62A 2 + B3A 3 ) ( 11 )

-Rs(- 6i aiT-|Ai - B2a2T2A2 S3a3T3A3) = 0

and for the odd case it has the form

TrRa(-3iTiB-| - 62T 2

B 2 + & 3

^ 3^ 2 ) ( 12 )

-RsT4( -3-|a-|Bi - 3292^2 + 33^3^3) = ^

52
where

T] = tan(a-| L)

T 2

= tan(a2L)

T 3 = tanh(a 3 l)

14 = tanh ( ttL )

3

^ = (tt^ + af)"^

$2 - ( TT +32)

33 = (tt^ - a^)"^

A] = (T2/32) - (T3/a3)

A 2 = (T3/a3) - (Ti /ai

)

A3 = (Ti/ai) - (T2 /a 2 )

Bi = (T3/a2) + (T2/33)

B2 = -(Ti/a3) - (T3/ai)

B 3

(T] /32) - (l2/a] )

Note that the numbers a], a 2 , and 33 all depend on the difference

aR = Rs - Ra. For given values of Ra and L, solutal Rayleigh numbers were

computed from these relations using the program ZEROIN by Shampine and Watts

[15]. The resulting values of Rs agreed with those values obtained from the

numerical solution of equations (7-9) to within the assigned tolerances

(usually five digits). As discussed below we also examine the asymptotic

behavior for large widths L >> 1 , and aR < 27tiA/ 4, in which case we have

T] = i, T 2

= -i, and T 3 = T 4 = 1, and the two determinants become equal.

5. Results

In Figures 2 and 3 we give marginal stability results as a function of

the width 2L for Ra = 0 and Ra = 3000, respectively. The results are given

in terms of aR in order to facilitate comparison with the 1 ateral ly-unbounded

case, for which aR* = Rs* - Ra = 27 ttA/ 4, independent of the value of Ra.

In both figures several curves are shown, corresponding to the lowest two

even and odd modes in each case. The critical value aR* then corresponds

to the lowest value of aR for each width. For Ra = 0 (Figure 2), the thermal

53
field contributes no driving force to the flow and the flow corresponds to

that for a single component. These curves therefore reproduce results of

Drazin [9], Hall and Walton [16], and Daniels [17], In particular, Rs*

approaches the asymptotic value of 27ir^/4 as L becomes infinite. That

is, for the single component case, as the distance between sidewalls becomes

larger and larger, they have less and less of an effect, and the stability of

the system is approximated more and more closely by that for the laterally

unbounded case, for which Rs* = 27it^/4. Roughly speaking, the variation

in Rs* for finite L is due to the fact that a set of rolls with preferred

wavelength 2ir/k* = 2 sFY cannot exactly fit between the walls and still

satisfy the proper boundary conditions, so that a modulation of the wavenumber

over the length L is required, which also causes an increase in Rs*.

For Ra = 3000 (Figure 3) the asymptotic value of aR* for large L is

about 50, which is well below the value for a laterally unbounded system. As

will be discussed shortly, this destabilization becomes even more pronounced as

Ra increases, with the asymptotic values of aR* tending to minus infinity as

Ra -* “. Thus for the two-component system the sidewalls do not have less

and less of an effect as the width of the system becomes larger and larger.

It can easily be shown that the destabl ization relative to the single

component case is related to the fact that T and S satisfy different boundary

conditions. Indeed, if they satisfied the same boundary conditions, then by

A

subtracting equation (9) from (8) it can be seen that the difference S(x) - T(x)

satisfies (D^ - ir2)(s - T) = 0 with homogeneous boundary conditions, and

.A so must vanish identically. Eliminating T(x) from Eqs. (7-9) then results in

equations describing a system with a single destabilizing component with an

effective Rayleigh number Rs - Ra, for which the asymptotic limit for large

54
L is 21-n^!^. Figure 2 can therefore be interpreted in two ways. It gives

Rs versus width (2L) for the single component system with Ra = 0 as shown,

or, equivalently, it gives aR versus width for a two-component system with

matching boundary conditions on the sidewalls (both perfectly insulating in

this case).

Thus for Ra = 3000, we may consider Figures 2 and 3 to represent the

cases of thermally insulating and thermally conducting sidewalls, respecti vely

.

We may continuously pass from one case to the other by writing the thermal

boundary conditions in the form

cos((j))T(L) + sin( (j))DT(L) = 0

cos((j))T(-L) - sin( (j))DT(-L) = 0,

for 0 < (j) < ir/2. The behavior of the two lowest ("destabilized") modes in

Figure 3 may be observed as the boundary conditions change. In Figure 4 we

show the change in AR with <() for a width 2L = 4.0 for the three lowest odd

modes. All three modes show a decrease in AR.as (j> ranges from tt/ 2

(thermally insulating) to zero (thermally conducting) and the relative order

of the modes is preserved. The lowest mode shows a much larger change as

(j) 0. The lower two modes in Figure 4 correspond to the two odd modes

shown in each of Figures 2 and 3. We see from Figure 4 that aR varies

continuously with (^. Only the first pair of even and odd modes in Figure 2

are destabilized to values aR < 27tt^/ 4 as Ra increases (Figure 3); the

second pair of modes apparently still asymptote to aR = 27tt^/ 4 as L - ».

The lowest curve in Figure 3 does not descend to its asymptotic value

as L increases, but shows a minimum value of aR* = -420 near 2L = 1.5. For

widths larger than this value, therefore, decreasing the separation of the

sidewalls results in a significant destabilization of the system. In Figure

55
5 additional results are given for Rs* as a function of Ra for 2L = 0.5, 1.0,

1.5, and 2.0. These values for 2L are less than the critical wavelength

2.8 for an unbounded single component system. For Ra = 10^ and 2L = 0.5,

Rs* = 4.4(10^) or aR* h -5.5(10^) indicating that the magnitude of aR*

increases with increasing Ra. For large L, this behavior can be described

analytically and will be discussed below. Note that the 2L = 1.5 and 2.0

curves cross at Ra = 10^ so that there is a range of Ra values for which the

2L = 1.5 case is less stable than the 2L = 2.0 case. Clearly for sufficiently

small L, as indicated by the 2L = 0.5 curve, significant stabilization occurs.

It is clear, then, that for T = DS = 0 on the sidewalls, the critical

solutal Rayleigh number does not approach that for an infinite system without

sidewalls as the system becomes larger and larger. Examination of the eigenfunctions shows that, for large thermal Rayleigh numbers, most of the flow

occurs in a neighborhood of the walls, cf. Figure 6, in a manner suggestive

of a buoyancy layer near a heated wall [3]. Thus for large L the interior of

the region is relatively free from convection, whereas in the single-component

system the interior is filled with an array of convection cells having more

or less equal strength, cf. Figure 7. In view of this, an alternative

interpretation of the large L limit can be obtained by fixing the left hand

boundary, for example by setting ”x = x + L, and then letting the right hand

boundary recede to positive infinity while keeping the left hand boundary at

'5T = 0. With this change of variables, both the even solutions and the odd

solutions tend to the same set of solutions as L - ®; namely, those for an

infinite hal f-i nterval with exponential decay as ^ » (for aR < 27ttV4).

These four solutions have spatial variations given by

56
F] (x) = Gxp(i 3] x)

F2(x) = exp(-ia2x)

F3(x) = exp(-33x)

F 4 (x) = exp(-Trx)

and thp characteristic determinant derived from the boundary conditions at

X = 0 is then

TrRa

| 6 i (-i/a2 - 1 / 33 ) + 62 ( 1/33 - i/3]) + 63(i/ai + i/32)} (13)

-Rs{-i 6 i 3 ] (-i /32 - 1 / 33 ) + i 62 a 2 (1 /a 3 " Ua]) + 63a 3 (''/ai + i/az)} = 0

which is the large L limit of both equation (11) and equation (12). For

large but finite L, therefore, the exponential decay of the eigenfunctions

with distance from the wall means that there are basically two isolated

regions of flow, each located near one of the walls, each of which approaches

the limiting case of flow in a semi -i nf i nite slab as the sidewall separation

becomes large.

For L infinite we have also considered the asymptotic dependence of the

critical solutal Rayleigh number Rs* on Ra for large Ra. This involves

deriving an expansion of the roots a^ for -aR >> 1, inserting them into the

determinant (13), and again expanding for Ra >> 1 . The calculations were carried

out using the symbol manipulation software MACSYMA [18], with the result being

Rs = 2TrRa3/4(i + (3/j8)iTRa-1

+

(1 5/1 6) Tr^Ra'I /^ + 0(Ra-3/4)} (14)

The dependence on RaV^ is also characteristic of the width of the long narrow

cells that occur in the fingering regime [3,6]. An intermediate result is

3 1/2 that a 3

7T /Ra ' , and the fact that a

3 tends to zero as Ra ~ causes a

slowly decaying tail of the disturbance to extend to the interior of the

region for large Ra.

For large L, the flow is confined to the neighborhood of the walls as

57
illustrated in Figure 6. Since the flow near one wall is essentially

(independent of the flow near the other wall, the critical value of the solutal

Rayleigh number is nearly the same for even and odd modes (see Figure 3). The

local flow adjacent to each wall is in opposite directions for the odd mode

(see Figure 6) and is in the same direction for the even mode. As the separation

of the walls decreases, the localized flows begin to interact and the critical

Rayleigh numbers for the even and odd modes become increasingly different.

As shown in Figures 8 and 9 for 2L = 3.0 and 1.5, for the odd mode the interaction

tends to weaken the flow in the interior whereas for the even mode the flow

in the interior is reinforced. The even mode consists of two convection

cells separated by a dividing vertical streamline midway between the walls.

The odd mode undergoes a transition from a flow with essentially two cells

(Figure 6) to a mixed flow with a dividing separatrix at the center of the

region (Figure 8) to a single unicell (Figure 9) as L decreases. The difference

between the critical solutal Rayleigh numbers for the even and odd mode is

134 and 1747 for 2L = 3.0 and 1.5, respectively. Thus, for 2L = 1.5, the

unicell is significantly less stable than the two cell structure.

6. Conclusions

In the presence of sidewalls the critical solutal Rayleigh number Rs*

for convective instability can be significantly lower than that for a laterally

unbounded system. The flow in the unbounded system consists of a periodic array

of convection cells of equal strength, whereas the presence of sidewalls allows

a localized flow to develop near the walls at a significantly lower solutal

Rayleigh number. Maximum destabilization occurs for a finite value of L.

The stability of the finite system is sensitive to the particular form

of the boundary conditions imposed on the sidewalls. Changing the thermal

58
boundary conditions from conducting to insulating stabilizes the system (and

considprably alters the nature of the flow in the interior of the system at

the onset of instability). This effect is consistent with the statement that

the change from Neumann to Dirichlet boundary conditions for the temperature

field has the effect of imposing a constraint on the temperature field, which

is therefore less effective in stabilizing the flow field.

7. Acknowledgements

This work was conducted with the support of the Microgravity Sciences

Program, National Aeronautics and Space Administration. A portion of the

work was performed while one of the authors (GBM) was visiting the Institute

for Theoretical Physics, University of California Santa Barbara, whose hospitality

is gratefully acknowledged. We also thank R. F. Sekerka for helpful discussions.

59
References

1. H. Stommel, A. B. Arrons, and D. Blanchard, Deep-Sea Res. 3, 152 (1956).

2. H. E. Huppert and J. S, Turner, J. Fluid Mech. 106, 299 (1 981 ).

3. J. S. Turner, Buoyancy Effects in Fluids, Cambridge University Press,

New York, NY, 1973.

4. J. S. Turner, Ann. Rev. Fluid Mech. 6, 37 (1 974).

5. C. F. Chen and D. H. Johnson, J. Fluid Mech. 138, 405 (1984).

6. M. E. Stern, Tellus 12, 1 72 (1 960).

7. S. R. Coriell, M. R. Cordes, W. J. Boettinger, and R. F. Sekerka, J.

Crystal Growth 49, 13 (1980).

8. G. B. McFadden, R. G. Rehm, S. R. Coriell, W. Chuck, and K. A. Morrish,

Metall. Trans. 15A, 2125 (1984).

9. P. G. Drazin, Z. angew. Math. Phys. 26, 239 (1975).

10. M. R. Scott and H. A. Watts, SIAM J. Numerical Anal. 14, 40 (1977).

11. H. B. Keller, Numerical Solution of Two Point Boundary Value Problems,

Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, PA,

1976, vol. 24.

12. SLATEC Common Math Library, National Energy Software Center, Argonne

National Laboratory, Argonne, IL. SNSQE was written by K. Hiebert

(1 980), based on Powell [13].

13. M. J. D. Powell, Numerical Methods for Nonlinear Algebraic Equations.

P. Rabinowitz, ed., Gordon and Breach, New York, NY, 1970, 87.

14. W. C. Rheinboldt and J. V. Burkardt, ACM Trans. Math. Software 9, 215

(1983).

15. L. Shampine and R. C. Allen, Jr., Numerical Computing: An Introduction,

W. B. Saunders & Co., Philadelphia, PA, 1973.

16. P. Hall and I. C. Walton, Proc. Roy. Soc. Lond. A. 358, 199 (1977).

17. P. G. Daniels, Z. angew. Math. Phys. 28, 577 (1977).

18. MACSYMA Reference Manual, Version 10, 1983. By the Mathlib Group - Room

832, Laboratory for Computer Science, Massachusetts Institute of

Technology, 545 Technology Square, Cambridge, MA 021 39.

60
Fi gure Captions

Figure 1

Figure 2

Figure 3

Figure 4

Figure 5

Figure 6

Figure 7

Figure 8

Figure 9.

. The real and imaginary parts of the roots a^ of the cubic equation

(a^ + TT^)^ - a^AR = 0. The dashed curve indicates the real

part of two complex conjugate roots.

. The values^AR =^Rs - Ra for Ra = 0 as a function of dimensionless

width for T = DS = p

on the lateral walls. The aR curves for any

values of Ra with DT = DS = 0 on the lateral walls are identical to

these curves.

The values^AR =^Rs - Ra for Ra = 3000 as a function of dimensionless

width for T = DS = 0 on the lateral walls.

. Dependence of aR = Rs - Ra on the thermal boundary conditions

for the lowest three odd modes with Ra = 3000 and L = 2.0. Here

the thermal boundary conditions on the jateral walls h^ve the form

cos((}))T(L) + sin(4))DT(L) = 0 and cos( (j>)T{-L) - sin( <j))DT{-L) = 0.

. The critical solutal Rayleigh number Rs* as a function of thermal

Rayleigh number Ra for widths of 0.5, 1.0, 1.5, and 2.0.

. The variation with x of Jhe el^genfuncti ons \p, S, and T for Ra = 3000

and a width of ten with T = DS = 0 on the walls corresponding to a critical value of AR* = Rs* - Ra = 46.4. Also shown is a contour

plot of the stream function \|;(x)sin( -ny ) with contour spacings

of 0.25.

^ -A

. The variation with x of ^he eigenfunctions ij;, S, and T for a width of ten with ot = DS = 0 on the walls corresponding to a critical

value of aR* = Rs* -^Ra = 671.3. Also shown is a contour plot of

the stream function ij;(x )si n( ) with contour spacings of

0.25; solid and dashed curves indicate positive and negative

values, respectively. For these boundary conditions aR is

independent of Ra.

Spatial variatj[on oj the stream function for Ra = 3000 and a width

of three with T = DS = 0 on the walls. The lowest even (upper plot)

and odd (lower plot) modes are shown corresponding to aR = Rs - Ra = 99.3

and -34.7, respectively.

Spatial variation^of the stream function for Ra = 3000 and a width

of 1.5 with T = DS = 0 on the walls. The lowest even (upper plot)

and odd (lower plot) modes are shown correspond! ng to aR = 1331.1

and -415.7, respectively.

61
62
Figure 2

0.0 2.0 4.0 6.0 6.0 10.0

Width

63
(

Figure 3

64
Figure 4

65
Figure 5

66
Figure 6

67
Figure 7

68
<

Figure 8

1

0.5

0 “1.5 -1 -0.5 0 0.5 1 1.5

69
Figure 9

70
Task 3

Measurement of High Temperature Thermodynamic Properties

D. W. Bonnell

SUMMARY

Major efforts in the interaction between Rice University (RICE), the

General Electric Space Facilities Laboratory (GE) and the National Bureau of

Standards (NBS) through this project have concentrated in two areas over the

past year. The first area is the prototyping of software on a portable

computer, suitable for transporting to the experimental site which would

provide both data acquisition and data reduction, independent of the need to

rely on external computers or data transmission to ascertain the current

state of the experiment. This effort is intended to provide support both to

the on-going ground-based experimental effort in levitation calorimetry, and

also to act as the prototype for flight experiments. The second effort has

been the exploration of experimental flight opportunities in the area of high

temperature properties measurements. D. W. Bonnell has been selected as a

member of the "High Temperature Advisory Committee", a NASA sponsored group,

which met at the Jet Propulsion Laboratory (JPL) in February of 1985 to consider

the general requirements of high temperature utilization of mi crogravi ty , and

to provide scientific guidance to the JPL levitation research, particularly

use of acoustic techniques. A visit to RICE in March of 1985 was made to

discuss flight experiment possibilities, to discuss current ground-based

efforts and examine current experimental arrangements, and to discuss automation

access. The results of these efforts suggest that there is potential for

flight experimental work, although the applicability of acoustic positioning

in significant high temperature efforts is yet to be established. Silicon

71
metal seems to be a good candidate, since it is extremely reactive, very

difficult to levitate in a one gravity environment, and apparently possesses

the highest heat of fusion of any element. A variety of experiments, including

volume change on melting, density, and cooling curve heat capacity measurements

appear within the current or projected capabilities of hardware being developed

for flight (e.g., the GE electromagnetic package).

Introduction

For the past several years, an experimental interaction among workers at

General Electric Space Facilities Laboratory (GE, R. T. Frost), Rice University

(RICE, J. L. Margrave), and the National Bureau of Standards (NBS, D. W.

Bonnell) has been carried out with the purpose of exploring the limits of

ground-based high temperature levitation research. This work developed an

ultra-high temperature calorimeter system which yielded the first precision

value for the heat of fusion of tungsten [Bonnell, 1983]. The levitation

system was not really capable of the stable levitation of liquid tungsten

necessary for calorimetry of the liquid, but provided a test bed to examine

techniques needed for high temperature calorimetry, such as automatic multicolor imaging pyrometry [Bonnell, 1984 and Bonnell, 1982], and other aspects

of high temperature calorimetry. This work has culminated in the publication

of a review chapter on the subject. Levitation Calorimetry [Bonnell, et al

1985]. One of the aspects of calorimetry brought out by the need to move

disparate groups and laboratory equipment to a distant site is the fact that

calorimetric studies are generally deliberate, meticulous research, and the

apparatus and techniques which have been developed over the years for

determinations in such an environment are not necessarily the most effective

in a remote environment situation. Since flight experiments will have needs

72
for immediate data evaluation and response similar to those of the RICE-GENBS interaction, this seemed an appropriate time to begin development of a

prototype data acquisition and analysis system which supports the experimentalist

in data acquisition, and provide local data evaluation to reduce lag in

identifying experimental problems. In the past, data has simply been collected

without real-time evaluation, and processed later by remote transmission and

batch calculation. With the advent of self-contained portable computers

supporting high level languages, an evaluation of prototyping languages and

codes has been undertaken.

Because Shuttle-based high temperature microgravity experimental facilities

and opportunities will soon be available, research with scientific goals (as

opposed to methodology development goals) can now consider the microgravity

environment as another viable means of pursuing those goals. In high

temperature research efforts, it is still the case that problems utilizing

microgravity must be very carefully selected, and that the process of developing

experiments must still recognize the infancy of scientific experimentation in

space, and be designed accordingly. In particular, the experimental aspects

should include relatively simple measurements to be made in early experiments,

with only one or two variables to be measured, and have strai ghtforward environmental conditions to be satisfied. The measurements should have high probability

of success even with marginal apparatus performance. A third criterion

should be that an experimental series should be designed to test, in earlier

phases, the apparatus performance which will be needed in later phases. High

temperature flight experiments should be selected, keeping in mind these

requirements and the current restrictions of upper temperature limits of 1500

to 2000 K, the availability of perhaps 1% temperature precision at best, a

73
limited multisample capability, and limited operator interaction.

Flight Experimental Opportunities

Although one important class of experiments that would benefit directly

from microgravity are the determination of basic thermodynamic data such as

fusion heats and heat capacity/enthalpy functions, these experiments require

overall precision better than 1-2 percent to be of real value, and generally

require several parameters of the system to be measured or controlled. An

ideal begining experiment would require only optical access to the sample,

should allow the sample to be reused during the flight for multiple

determinations, and should yield data for which approximately correct estimates

already exist.

Of intrinsic importance, as well as being important parameters in groundbased high temperature determinations, are the density of the liquid, and the

volume change on melting. In many cases above 1500 K, the densities of liquid

metals are known only approximately, and the general best representation of

the temperature dependence is a linear function [Metals Reference Book, 1976].

This representation is heavily weighted by high accuracy low temperature

measurements (e.g, Hg), and by the available precision of the high temperature

data. In microgravi ty, surface tension will quickly force a molten sample

into a spherical shape, and the requisite positioning forces should distort

this sphericity only slightly. Observing a sample as it cools with a

dimensionally-calibrated recording optical system (either video or film

camera) will readily allow determination of the sample volume from its physical

dimensions. This technique has been attempted in ground-based research

[Bonnell and Margrave, 1974] with mixed success. In earth-based experiments,

the sample is strongly distorted into a "pear" shape by the combination of

74
gravity and the electromagnetic support fields. One dimensional imaging,

with the assumption the specimen was a figure of revolution about the vertical

axis limits accuracy to a few percent. As the total change in density upon

melting is typically of the order of 10 percent, it is clear that levitationbased techniques need improvement. The volume change on melting is an

important parameter itself, being generally larger than the volume change

over the useful liquidus temperature range. Measurement of this parameter

requires only a just molten sample. Thus, data for this property is an

excellent minimal apparatus performance target.

Current density work at RICE will use a mirror arrangement to give 3

orthogonal views, which should improve volume measurement accuracy to the

order of 1-2 percent, although in earth gravity, this is an ambitious effort.

To improve on this error level in a microgravity experiment should be much

easier. A ^% error level for a 1 cm diameter sample corresponds to 3 axis

radii measurements precise to 0.02 mm for a sphere. As long as the sample

sphericity error is of this order, amounting to about 0.08 eccentricity

,

corrections to the spherical measurements should contribute with less than 1

percent uncertainty. In earth gravity, eccentricities of the order of 0.85

(2 to 1 axes ratio) are common, requiring large non-spherici ty corrections.

In the microgravity environment, one would expect excellent sphericity for a

suspended liquid, so that the limiting parameters are optical resolution and

sample vibrations, both harmonic and rotational. There are still problems

with the sample moving rapidly under the influence of the electromagnetic

field (vibrating, spinning, stirring of the melt, etc.) which can limit total

precision. In the microgravity environment, the field can be reduced to a

level just sufficient for positioning after the sample reaches high temperature.

75
Spinning, which would produce centrifugal distortions, can be controlled by

introducing a deliberate vertical asymmetry in the levitation coil.

This experiment gives directly the temperature dependence of density for

very little additional effort over that necessary to simply test levitation

and temperature measurement capability. It is not highly dependent on accurate

positioning of the specimen, and shows a clear advantage for microgravity

measurements. Assuming a calibrated pyrometric or even a moderately bandwidth

limited radiance detector, the shape of the light output data during cooling

will yield the functional dependence of the high temperature liquid density

with temperature.

Cooling Curve Cp Measurements

The next stage in such an experimental series would be to include a

relatively high speed 1 or 2 color imaging pyrometer system with 0.5 % or

better accuracy, and to monitor the rate of heat loss during cooling to obtain

heat capacity (Cp), Weingarten, et al [1977]. Although a much more demanding

experiment in terms of system control, this technique represents the next phase

of effectively utilizing the microgravity environment. When the completion

of a series of density and cool i ng- function heat capacity experiments have

eliminated experimental problems associated with flight microgravity heating

and positioning, temperature measurement, and specimen changing, it will be

appropriate to consider high precision calorimetric studies, perhaps utilizing

a "gulp" calorimeter system similar to that prototyped by GE and recently

tested [Frost and Stockoff, 1983; Bonnell, 1983].

Levitation Systems

The recent meeting of the High Temperature Advisory Committee at JPL

provided a review of acoustic positioning technology [Doremus, 1985]. These

76
systems are quite attractive for manipulating non-conductors, e.g., ceramic

and glass materials. However, in the ceramics area, materials are generally

less reactive and more tractable for earth-bound measurement up to temperatures

well above 1800 K. This was pointed out to the JPL staff by the committee,

with the recommendation that JPL plan soon to attempt design and development

efforts for a high temperature acoustic positioning system to operate above

2300 K, probably based on the single-mode acoustic levitator described by

Barmatz [1985] using a compact YAG laser to heat samples of less than 1 mm

diameter. It was the committee's opinion that avoiding the problem of heating

an entire levitator to temperatures above 2300 K would eliminate a variety of

serious problems in the implementation, including heat rejection, total power

required, early apparatus failure, hot wall outgas contamination of the

sample, etc. The apparatus failure problem, particularly at launch, and the

likelihood that a hot container's walls could cause more contamination than

levitation would eliminate are powerful arguments in favor of a localized

heating arrangement. There are major problems associated with stable levitation

in the presence of large gas temperature gradients which need to be investigated

and overcome, and the acoustic positioning effect may cause undesirable gas

circulation near the sample, but such problems seem to be development problems,

amenable to engineering solutions. The advantages of using acoustic positioning

extends research opportunities to ceramics, glasses, molten salts, and other

refractory non-conductors.

Although acoustic positioning obviously has significant implications for

the future of high temperature space experimentation, it appears that current

planning should consider electromagnetic levitation the most probable near

term high temperature technique available, thus limiting opportunities to

77
conductors. Frost [1985] of GE indicates that an electromagnetic levitator

system with optical image recording and pyrometry, based in part on the SPAR

levitation system used for beryllium in the late 1970's [Wouch, et al , 1978]

will be available in the near future with capability of temperatures to the

order of 2000 K.

Sample Choice

With this temperature capability, it is practical to consider systems

with melting points above 1500 K. Electromagnetical ly levitatable elements

which have melting points between 1500 K and 2300 K and are difficult to

levitate in a one gravity field include manganese (1^ = 1514 K) , silicon

(1683 K) , and thorium (1 964 K) . Other elements in this melting point range,

which have been studied by levitation, include Be (1551 K) , Co (1768 K) , Fe

(1809 K), Gd (1 585 K) to Yb (1 700 K) rare earths, Ni (1 726 K) , Pd (1825 K)

,

Sc (1812 K) , Ti (1 943 K) , V (1710 K) , and Y (1 768 K) . See Bonnell, et al

[1985] for a recent review of the status of most of these latter elements.

It is interesting to note that liquid data for the gadolinium to ytterbium

higher melting rare earths have not been mentioned widely in the literature

[but see Pankratz et al , 1962, 1963; and Robie, et al , 1979], and thus what

problems might occur with their levitation are uncertain. However, cerium

[Kuntz and Bautista, 1976], praseodymium [Stretz and Bautista, 1976], and

neodymium [Stretz and Bautista, 1975], have been studied by levitation, thus

the ground-based problems should be readily solvable.

Scandium should be similar to titanium and presents significant problems

in atmosphere control due to its ready tendency to form the oxide. Lack of

measurements are probably due to the expense of the metal and the fact that

most current uses are of the oxide. Its future potential as a very light metal

78
with a higher melting point than aluminium could put it into the same class

as titanium as an alloying material. Since no obvious barrier to ground-based

studies seems to exist, this material should be attempted on the ground first.

Of the elements listed above as difficult or without reports of successful

levitation, thorium's melting point is at the upper end of the range available,

which requires complete functioning of the apparatus to insure obtaining

liquid data. It is also somewhat radioactive, increasing the on-board risk

slightly in case of a catastrophic system failure. Since thorium should be

levitatable on earth with precautions, it too should be attempted in a ground

environment first.

Manganese is a strategic material, considered to be the single most

important additive in the steelmaking process, where about 90% of the total

manganese consumed in the U.S. is used [Critical Materials Requirements of

the U.S. Steel Industry, 1983]. The pure metal is prepared by electrodeposition,

and is obtainable in only thin {> 0.2 cm) platelets, which cannot be consolidated

even by arc melting. The material is extremely brittle and has not been

levitated electromagnetically due to its unavailability in reasonable bulk

form. Although some older thermochemical data have been reported [cf. Metals

Handbook, 1976], this element would appear to be a good candidate for a

microgravity investigation.

Silicon is a semiconductor, and does not achieve metallic conductivities

sufficient to levitate until heated near its melting temperature. The obvious

importance of this element to the semiconductor industry, its extreme

reactivity, which would make containerless studies very advantageous, and the

estimates that it has the largest heat of fusion (^a 60 kj/mol ) of any element

make it the most attractive candidate for initial high temperature microgravity

79
experiments. Its melting point is comparable to that of Be, which has already

been levitated by the GE SPAR levitator [Wouch, et al , 1978], which makes a

successful first flight experiment likely. Emissivity data are available to

above the melting point, so temperature measurement should not present abnormal

problems. Combining flight experiments with ground-based studies of the

volume change on melting and liquid density of silicon and/or beryl ium appears

to be an attractive initial effort. The results of the GE-RICE-NBS interaction

in the recent past make it clear that a successful flight experimental program

will need the support of a strong ground-based parallel effort to attract and

support personnel, and to provide the continuity of effort necessary to

conserve expertise during the flux of students typical of academic research.

Studies of alloys, particularly of extreme refractory systems in

composition regions where the liquidus is within the capabilities of the

flight apparatus, are also attractive candidates when multiple sample handling

capability has been demonstrated. Since a number of specimens of varying

composition would be necessary to establish trends, these efforts should

follow the initial demonstration types of experiments. Detailed information

on density and Cp variations should be quite interesting. Our analysis

[Bonnell, et al , 1 985] of the cerium-copper alloy system data of Dokko and

Bautista [1980] points out the failure of the simple mole fraction estimations

generally used for alloys.

During the March 1985 discussions at RICE, the possibility of processing

specimens as part of the experimental program was discussed. Gas/condensed

phase reactions are generally straightforward to accomplish in the containerless

environment, with the general limitation that the product remain a liquid (or

near the liquidus) and be a good conductor during the processing. The liquid

80
limitation is due to the typically strong decrease in gas diffusivity in

solids much below their melting point, which halts the reaction upon

solidification. The usual method of using powdered solid to increase surface

area and minimize diffusion distances requires a container, as to the extent

the powder couples with the RF field of the levitation coil, each individual

particle will be el ectrostatical ly repelled from its neighbors, with the

result that any sample not strongly compacted will be dispersed by the RF

field.

Gas-liquid high temperature reactions have significant advantages in the

electromagnetic containerless environment when compared to reactions in a

containter. Electromagnetic stirring provides a much more rapid mixing

process near the surface than simple diffusion, and the entire surface of the

sample is available to the reactant without stirrer or wall reactions to

consider. The introduced gas can be metered into the reaction environment

without unwanted reactions through cool nozzles, allowing accurate mass

balance determinations, and controlled reaction rates, in contrast to sealed

tube type syntheses. In addition, reaction products can be either volatile

or non-volatile, or both, and can be separated during the reaction process.

Groups of substances to prepare and make density or Cp measurements

similar to those described above for the metallic elements include the oxides,

nitrides, and phosphides of conducting elements. In these systems, the stepwise addition of a reactant gas (e.g., oxygen) to the atmosphere surrounding

a levitated liquid allows the direct reaction product to be formed. Previous

work [Margrave, et al , 1971, Bonnell, et al , 1969] notes that reactions are

generally simple, tending to yield the stable product of lowest oxidation

state due to the effective excess of metal at the reaction interface. At

81
higher temperatures, carbides are also quite interesting, but their exceptionally

high melting points and the fact that many metal carbides are good electrical

conductors makes these syntheses better candidates for a parallel groundbased program. Of the metals which can be melted in expected flight apparatus,

the following compounds should be readily synthesized, with volume change

measurements used to track extent of reaction, and perhaps cooling-curve

thermal measurements possible. Boric oxide (B 2 O3 ) is a possible first

material. Its liquidus properties are reasonably well known [JANAF, 1971;

Robie, et al , 1 979], and it has the useful property that the liquid oxide is

considerably lower melting (723 K) than the metal (2077 K) . Of the metals

suggested as potentials for early flight experiments, the manganese-oxygen

system is very interesting. There are a number of manganese oxides, ranging

from MnO 2058 K) to Mn 304 (1863 K - this transition is dependent on oxygen

pressure, being from 1873 K at 1 atm 8 ( 03 ), to 1828 K at 0.1 atm 8 ( 03 ) - see

Driessens [1967] and the commentary in Phase Diagrams for Ceramists [1975,

fig. 4157]). The tendency of levitation synthesis to lead to the lower oxide

should allow a natural progression in this system. The phase diagram of the

manganese-oxygen system is similar to that of the Fe-0 system in complexity,

and appears to need more work to permit displaying the detail known in the Fe-0

system. Manganese oxides also exhibit compound non-stoicometry around the

MnO composition which containerless preparation could investigate. Parallel

experiments on the better known Fe-0 system (from the Wustite Feo .9470 ( T|y, =

1652 K) to Fe 303 (1895 K)) might also be in order, either as preparation for

studies of the Mn-0 system, or as followup in the event containerless processing

of oxides proves exceptionally fruitful.

Other oxides which could be considered include silica (Tm = 1996 K) and

82
oxides of vanadium, which has stable solids of compositions VO = 2350 K) ,

V 2 O 3 (2240 K), V 3 O 5 (2100 K), VO 2 (1818 K) , V 2 O 5 (943 K) and Y 2 O 3

. The Y-0

system is particularly attractive if the available temperature range becomes

higher, because of the interest in oxide film corrosion protection by yttrium

in superalloys. Also, in general, the sesquioxides of the lanthanides exhibit

solid polymorphism. If these oxides can be prepared, and solidified in the

high temperature polymorph, due to lack of external nucleation sites, new

insight into high temperature behavior could result.

Toropov, et al [1972] note that the first three ionization potentials of

Sc, Y, and La are low so that they can give up two electrons for the formation

of oxygen bonds, while the third electron is a conductivity band electron.

Thus ScO and YO may exhibit metallic conductivity, particularly at elevated

temperatures. This possibility certainly makes these oxides excellent ground

base test case possibilities, and should be excellent choices for determining

the limits of electromagnetic positioning of compounds. This information

will be necessary to determine if alternate heating will be required during

syntheses.

The major result of these experiments, however, would be the actual

processed specimens, including the probability of large crystalline specimens,

for chemical characterization and microstructural comparison of the containerless

prepared material with conventional materials. This pioneering work should

provide important scientific guidance for the use of the microgravity

containerless environment as a tool in the synthetic chemist's arsenal for

tailoring materials.

Liquid Tungsten

Previous work on this project yielded the first levitation measurement

83
of the heat of fusion of tungsten [cf. Bonnell, 1983^ Bonnell, 1984]. The

value, 53.0 ± 2.3 kj/mol , was obtained by analysis of fraction melted versus

heat content data from drop calorimetry of the specimens. Continued efforts

to obtain Cp data on the levitated liquid has proved fruitless, as the

available generator power at GE was apparently insufficient to provide stable

liquid levitation. Recently, German workers [Arpaci and Frohberg, 1984],

utilized a 50 kW, 900 kHz induction heater to obtain enthalpy increment data

for tungsten liquid and solid. Although those authors unfortunately give no

details of the coil design, the use of higher power solved the difficulties

in levitating molten tungsten, as they reported 20 measurements of the liquid

over the temperature span from 3699 K to 4017 K, with an average deviation of

0.42 %, which is indicative of excellent high temperature measurements. Arpaci

and Frohberg obtained the heat of fusion directly from the difference of their

solid data as 50.3 ± 0.4 kJ/mol at 3683 K. They apparently did not directly

determine the melting temperature, but used the commonly referenced literature

value 3683 K to determine emissivities and establish their temperature scale.

This value and other literature on the melting temperature of tungsten have

been reviewed by Cezairliyan [1972] and a recommended value of 3695 K was

selected in light of improvements in measurement methodology. Using this

temperature (3695 K) , the Arpaci and Frohberg [1984] heat of fusion, aH^,

changes insignificantly (to 50.4 ± 0.4 kJ/mol), relative to the solid value

from their fitted equation of 118.7 (± 0.3) kJ/mol at 3695 K. Our analysis

of our heat of fusion determinations gave the value, 119.5 ± 2 kj, for the

solid independently, which is also in nearly exact agreement with JANAF [1971].

Arpaci and Frohberg considered the literature of solid tungsten carefully,

and noted positive deviations in the recent literature of over 4%. Comparing

84
solid enthalpy values shows our derived solid value to be within 0.6% of the

Arpaci and Frohberg value, and well within the combined error limits. Because

our value of the heat of fusion is independent of literature solid data, and

references a solid value in good agreement with the Arpaci and Frohberg value,

the two studies can be directly compared, and the agreement is quite good,

considering that in both cases, differences in rather large numbers are involved.

Arpaci and Frohberg do not state where they derive their error estimate for

aHji, , but apparently, using the difference fit H(p.,T) = H(p,,Tni) + Cp x (T - T^j,)

they have accepted the error in the intercept term as the error in aH^.

Refitting the data of their Table 2 indicates a standard error for the fit of

888 J, in line with what would be expected from an average deviation of 0.42%

for ^ 170 kj/mol. Combined with the 298 J error estimate from the solid

data gives an error estimate for of at least 600 J/mol. The better

error estimate should be to use the larger uncertainty. Alternatively,

combining the uncertainty of the solid with the intercept error suggests an

uncertainty of 670 J/mol. It is suggested that the uncertainty is better

represented by a value of at least ± 0.8 kj/mol. The agreement between the

calorimetric heats of fusion [Arpaci and Frohberg, 1985, Bonnell, 1984] is

within 1.5% of the total heat, which is excellent agreement for difference

measurements.

Although a aH^^ of 51 kj is higher than the Tamman rule [1913] estimate by

nearly 70 percent, this value is in accord with periodic correlation estimates

which indicate that the entropy of melting, aS^, should be somewhat higher

than the fusion entropy for molybdenum (12.9 J/mol. K). Our work [Bonnell,

1983, 1984] gives aS^ = 14.3 ± 0.6 J/mol. K, in reasonable agreement with the

value of Arpaci and Frohberg [1984], corrected to a 3695 K melting point, of

85
13.6 + 0.2 J/mol.K. Using an error-wei ghted average of the two values for

aS[|, gives a best value of 13.8 ± 0.3 J/mol.K. The entropy of fusion of

tungsten is thus 7% larger than molybdenum.

With two independent calorimetric studies in good agreement, it appears

that the melting thermodynamic properties of tungsten are now reasonably well

known. The coincidence of the heats of fusion also indicate that the emissivity

values used by Arpaci and Frohberg, and their derived heat capacity of the

liquid are rel i abl e.

Prototype Data Acquisition and Analysis

The advent of "personal" computers with the capability of supporting

high level languages, and with a relative ease of being transported suggested

that it was an appropriate time to consider a computer-based acquisition

system which could be brought to the experimental site along with the rest of

the apparatus. Previous experimental efforts had developed a datalogging

system, which could be used via a telephone link to the NBS computer system

to reduce calorimetric data. The relatively slow speed and unrel iabil ity of

transmission for this work has been a general problem area. The large calorimeter

used in this work requires about 1 hour to reach steady state after an impulse

change. In order to provide for more than one run in a day, it was necessary

to wait until the end of the day to process data, so the second and succeeding

run of the day could not be planned on the basis of the previous runs. With

a large number of aborted drops, it would be desirable to be able to analyze

the drift rate of the calorimeter in an ongoing fashion to determine when the

calorimeter was ready.

The data reduction program, CALOR, is written in FORTRAN, and during

this effort, was converted to ANSI FORTRAN 77 running on the NBS mainframe

36
Sperry 1100/82. Among other changes, this involved replacing the character

manipulation performed for report generation originally implemented in integer

data types with true character data types, adding new code to permit CALOR to

recognize and process a variety of input stream formats, and other minor code

cleanups. The program is, however, by nature, a batch program, and requires

multiple runs to determine such data characteristics as the end of the main

period and whether a portion of a drift period is linear, even though the data

stream collected may exhibit curvature. Once these characteristics are determined,

its reporting facilities are excellent. A real-time control and report program

actually has different goals than the final data reduction program, and a

reasonable way to approach the problem was to develop two codes, one for the

run-time acquisition and monitoring process, and one for final data reduction,

both to run from files created on the portable computer system.

For this purpose, NBS has procured a transportable computer system,

compatable with the IBM* Personal Computer type of system. The Corona* PCC-2

portable was chosen because of its ability to be maximally configured (2 5-

1/4" floppy disk drives, 512 kBytes memory, 9 in. internal monitor with ultrahigh resolution text display, serial and parallel ports, full size, enhanced

function keyboard, etc.) and leave expansion slot space for a full size modem

card, a data acquisition card, and a battery backed-up clock card. Although

other vendors (with restricted choices of add-ins) can now meet many of these

requirements, that was not the case when the computer was procured. The

advantage of IBM PC compatabi 1 ity is the large number of languages, e.g.,

FORTRAN 77, BASIC, PASCAL, APL, C. FORTH, etc., available in implementations

* Mention of products by brand name is to provide identification of a commercially obtained or evaluated item, and is not to be construed as Government endorsement or certification of suitability for other applications.

87
likely to be portable across a variety of different machine configurations.

The availability of FORTRAN 77 for the compilation of CALOR has proved to

be more problematical than expected. Until recently, no full language FORTRAN

77 compiler was available for the IBM PC, running under MS-DOS, the overwhelmingly

accepted operating system. Of the subset compilers, Microsoft FORTRAN has

received the best reviews, particularly for both code quality and freedom from

bugs. However, the implementation tested, version 3.20, evidenced a number

of missing features and implementation bugs which required major revision of

portions of the character manipulation code. As examples, only one length

character variable can be typed in a statement; attempts to use *n designators

resulted in undefined objects; character concatenation is not supported, nor

are substring operations; character type variables cannot be intermixed with

numeric types in COMMON. Considering the amount of debugging necessary to

overcome these problems, and the fact that the resulting code would be somewhat

specific to the compiler, it was decided to delay this implementation until

one of the new full languages compilers was released. Two compilers are under

consideration; Digital Research* (DR) FORTRAN 77 and the IBM/Ryan-McFarland*

Professional FORTRAN. The DR FORTRAN, although a full language compiler, was

reviewed by NBS [Kahaner, 1985] for use and found to have so many bugs that

its use was not recommended. Some preliminary testing has been undertaken on

the Professional FORTRAN compiler by the NBS Scientific Computing Division

with highly positive results. They could find no significant problems and so

far have found it to be a complete full language implementation. The processor

fully supports current levels of MS-DOS (i.e., versions 2.1 and 3.0) and can

access files at any tree depth. However, delivery of the processor from IBM

has been slow, and a copy for this project was received too late to complete

the port of CALOR to the PC in this reporting period.

88
For the data control portion of the project, a number of factors were

considered. Primary was the selection of a language which was interactive

and portable, handled serial I/O well, and provided a powerful and flexible

development environment, since the actual algorithms to provide real-time

support to the experimentalist must be developed piecemeal, with considerable

testing against recorded as well as actual datasets. The languages considered

as meeting the interactive requirement and the wide use necessary to assure

portability were BASIC, APL, and FORTH. FORTH still lacks a standard method

for handling floating point and would require extended programming expertise

for its users until the control system was stable. It was felt after

examination of the FORTH language that it was the most powerful and flexible

for control codes, but that algorithm development would be quite difficult

and protracted for this project.

BASIC was rejected, as the defacto standard Microsoft BASIC makes modular

development extremely difficult. Its lack of local parameters, the lack of

support for passed parameters to subroutines, its lack of support for the

8087 math co-processor, and its restriction to the small (< 64 kByte) memory

model all mitigate against the language. In its favor is the fact that much

microcomputer assembly language support code for data acquisition devices is

compatable with the BASIC CALL statement. Also, with restrictions, interpreter

BASIC code can be compiled with the BASIC compiler for major execution improvements, but with the disadvantages of compilation. However the compiler,

and the interpreter until very recently, have been unable to utilize the MSDOS 2.0 level tree structured file system, thus further limiting the capabilities of BASIC. Although it is perfectly possible to write extremely

sophisticated systems in BASIC, maintenance of the code generally requires the

89
author's support. It appears that so much code is written in BASIC because

of its simplicity, its similarity to FORTRAN syntax, the advantages of

interpreter debugging, the universal availability of similar versions of the

language on microcomputers, and until very recently the lack of alternate

1 anguages.

The advent of low cost semiconductor memory and microprocessors which

can address large extents of memory (e.g., 512 kByte or more), has made the

availability of another interactive language possible. Although APL requires

considerable memory for the processor alone (~192 kB) and the manipulations

of APL can be memory intensive, the parallel nature of the language permits

execution speeds often approaching that of compiled language when handling

quantities of data. For example, with an 8087 coprocessor, APL sums vectors

of 1000 double precision values in ~100 ms. Least square fit of a linear

model to 60 data points requires <2 s. The sum in GW BASIC requires over 8

seconds, and is in error in the 10 decimal place due to improper conversion

of decimal numbers to double precision. Calorimetric data reduction requires

linear estimation, numeric integration, and running differences calculations,

all of which require double precision arithmetic and fast execution. These

computations are simple code in APL in contrast to BASIC.

Algorithms in APL are generally totally portable where file I/O is not

involved. The modularity of the language permits that non-portability to be

localized easily. Scientific Time Sharing Corporation* (STSC), one of the

major sources of mainframe APL, has made a strong entry into the microcomputer

market by releasing a complete implementation of their mainframe processor,

APL*PLUS, for IBM and compatable PC's. The supported environment is superior

to mainframe usage, and is comparable in flexibility to the best FORTH

90
environments. The majority of algorithm development required is quite suited

to the vector capabilities of APL, particularly the least square fitting of

the initial and final period processes, and the integration of the main period

data. Although less accepted than BASIC, the clear superiority of APL over

BASIC makes it the preferred choice for the interactive calorimetric system

prototypi ng.

Specific interactive code development will be begun as soon as the

porting of the data reduction code, CALOR, is complete.

91
References

1 .

2 .

3.

4.

5.

6 .

7.

8 .

9.

10 . n.

Arpaci, A. and Frohberg, G., Z. Metal Ikde. 7^, 614-618 (1984).

Bonnell, D. W. , Chauduri , A. K. , and Margrave, J. L., unpublished work

(1 969).

Bonnell, D. W. and Margrave, J. L. , unpublished work. Rice University

(1 974).

Bonnell, D.

82-

2560, p.

Bonnel 1 , 0.

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Bonnel 1 , D.

84-

2882, p.

W., NBS: Materials Measurements, ed. J. R. Manning, NBSIR83-1 03 (1 982).

W., NBS: Materials Measurements, ed. J. R. Manning, NBSIR11 3-1 30 (1 983).

W., NBS: Materials Measurementd , ed. J. R. Manning, NBSIR1 35-1 52 (1 984).

Bonnell, D. W. Montgomery R. L. , Stephenson, B., Sundareswaren, P. C.,

and Margrave, J. L., in Heat Capacity of Solids, ed. A. Cezairliyan,

McGraw-Hill, (1 985 in press).

Critical Materials Requirements of the U.S. Steel Industry, (prepared

under the National Materials and Minerals Policy Research and Development

Act of 1980, Dept, of Commerce, March 1983).

Cezairliyan, A., High Temp. Sci.,^, 248 (1 972).

Dokko, W. and Bautista, R. G. , Met. Trans. B 11 , 51 1-18 (1 980).

Doremus, R. , Informal Report to R. Hal pern, NASA Headquarters , April 5,

1985.

12. Driessens, F. C. M. , Inorg. Chim. Acta,^* ^93 (1 967).

13. Frost, R. T. and Stockoff, E., Final Report, NASA Contract NAS8-34231

,

Task 7 (GE Space Systems Division, Valley Forge, PA, 1983).

14. Frost, R. T. , private communication, February 1 985.

15. JANAF ThermoChemical Tables, 2nd Edition, ed. D. R. Stull, H. Prophet,

et al , NSRDS-NBS 37 (U.S. Govt. Printing Office, 1971), and supplements

1 974-1 982.

16. Kahaner, D. , ( NBS-Scienti fic Computing Division), private communication,

1 985.

17. Kuntz, L. L. and Bautista, R. G. , Met. Trans. B7 , 107-113 (1976).

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18. Margrave, J. L., Treverton, J. A., and Wilson, P. W., High Temp. Sci.,

3, 1 63-1 67 (1 971 ).

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Brandes, ( Butterworths , London and Boston, 1 976, p. 944).

20. Pankratz, L. B., King, E. G. , and Kelley, K. K. , U.S. Bureau Mines Rept.

Inv. 6033 (1 962).

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(1 963).

22. Phase Diagrams for Ceramists, 1975 supplement, ed. Levin and McMurdie,

(Amer. Ceram. Soc., Columbus, OH, 1975).

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,

Handbook of Phase Diagrams of Silicate Systems, Vol . II Metal -Oxygen

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93
NBS-114A (REV. 2-e'') U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

1. PUBLICATION OR

REPORT NO.

2. Performing Organ. Report No. 3. Publication Date

SHEET (See instructions) NBSIR 85-3217 AUGUST 1985

4. TITLE AND SUBTITLE

NBS: Materials Measurements

5. AUTHOR(S)

J. R. Manning

6. PERFORMING ORGANIZATION (If joint or other than NBS, see instructions) 7. Contract/Grant No. national bureau of standards H-27954B

DEPARTMENT OF COMMERCE 8. Type of Report & Period Covered

WASHINGTON, D.C. 20234 Annual Report

April 1, 1984-March 31,

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. Stote, ZIP}

Microgravity Science and Applications Division

National Aeronautics and Space Administration

Washington, DC 20546

10. SUPPLEMENTARY NOTES

f ] Document describes a computer program; SF-185, FIPS Software Summary, Is attached.

11. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bi bliography or literature survey, mention it here) This report describes NBS work for NASA in support of NASA's Microgravity Science

and Applications Program under NASA Government Order H-27954B (Properties of Electronic

Materials) covering the period April 1, 1984 to March 31, 1985. This work has been

carried out in three independent tasks:

Task 1. Surface Tensions and Their Variations with Temperature and Impurities

Task 2. Convection during Unidirectional Solidification

Task 3. Measurement of High Temperature Thermodynamic Properties

The results for each task are given separately in the body of the report.

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key

convection; directional solidification; gallium; levitation calorimetry

surface tensions; thermodynamic measurements

words by semicolon s)

; silicon;

13. AVAILABILITY 14. NO. OF

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