A111D7 3W14

NBSIR 83-2772

NBS: Materials Measurements

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

Washington, DC 20234

August 1 983

Annual Report

For the Period 1 April 1982 - 31 March 1983

NASA Government Order H-27954B

Properties of Electronic Materials

100

,1156

33-2772

1933

U.S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

AtiUHAL *V*.*-*.*> 09 VTANDAXXMI U>KA*1

NBSIR 83-2772

ft- NBS: MATERIALS MEASUREMENTS

Circ,

CiCtoo

C-*-

J. R. Manning

U S. DEPARTMENT OF COMMERCE

National Bureau of Standards

Washington, DC 20234

August 1 983

Annual Report

For the Period 1 April 1982 - 31 March 1983

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

Page

Summary 1

Task 1 - SURFACE TENSIONS AND THEIR VARIATIONS WITH TEMPERATURE

AND IMPURITIES

S. C. Hardy and J. Fine 5

Task 2 - CONVECTION DURING UNIDIRECTIONAL SOLIDIFICATION

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

R. G. Rehm 19

Task 3 - MEASUREMENT OF THE HIGH TEMPERATURE THERMOPHYSICAL

PROPERTIES OF TUNGSTEN GROUP LIQUIDS AND SOLIDS

D. W. Bonne11 113

i

National Bureau of Standards

Materials Measurements

Summary

This report describes NBS work for NASA in support of NASA’s Materials

Processing in Space Program under NASA Government Order H-27954B (Properties

of Electronic Materials) covering the period April 1, 1982 to March 31, 1983.

This work is directed toward measurement of materials properties important

to the design and interpretation of space processing experiments and

determinations of how the space environment may offer a unique opportunity

for performing Improved measurements and producing materials with improved

properties.

The work is being 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 the High Temperature Thermophysical Properties

of Tungsten Group Liquids and Solids

The results obtained for each task are given in detailed summaries in the

body of the report. Emphasis in Tasks 1 and 2 is on how the reduced

gravity obtained in space flight can affect convection and solidification

processes. Emphasis on Task 3 is toward development of techniques for

thermodynamic measurements on reactive materials, requiring levitation

and containerless processing.

With the advent of the Space Shuttle, it may become feasible to

exploit the unique microgravity environment of space flight to produce

improved materials and improved measurements of important materials

properties. In materials processing on earth, gravity frequently produces

1
density-driven 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 inhoraogeneities , which are

particularly troublesome in electronic technology and other advanced

technical applications, might be avoided in materials produced under

raicrogravity conditions.

In Task 1, surface tension and surface segregation effects are being

investigated to determine the possible influence of these surface effects

on convection and solidification processes. In particular, measurements

are being made of the surface tension of liquid silicon. Silicon is one

of the most important materials used in electronic technology. Because

of the need for fully defect-free material, not readily obtainable under

earth-bound conditions where gravity-driven convection can be important,

silicon is a major candidate material for space flight processing. Under

raicrogravity conditions, Marangoni convection arising from surface tension

gradients could be the main source of fluid flows affecting solidification

processes. Thus surface tension information will be important in planning

materials processing in space. Because of the high reactivity of silicon,

its surface tension is poorly known. Moreover, despite the technological

importance of silicon, the dependence of its surface tension on temperature

and impurities was really not known at all prior to the current work.

This work is designed to provide these surface tension measurements.

2
In Task 2, solutal convection, which results from simultaneous

temperature and composition gradients, and other convective phenomena

arising during directional solidification are being investigated both by

theoretical calculations and by experiments. The experiments are designed

to test the theories and provide quantitative data on convection processes.

Direct measurements of convection are being made during unidirectional

solidification of transparent materials, particularly succinonitrile

containing small amounts of ethanol. The theoretical calculations

specifically include determinations of the effects a reduction in

gravitational force will have on convection.

Interactions between temperature gradients and composition gradients

can make it impossible to avoid solutal convection and other densitydriven convection effects during materials processing on earth. Since

this type of convection depen’ds on gravity, its effects will be much less

pronounced in space. NBS work is directed toward provision of measurements

and development of models to define the conditions under which this type

of convection will be important and determine how it can be avoided or

controlled.

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

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

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

facility is being applied to develop levitation/drop calorimetry techniques

and determine their possibilities for use in space flight experiments.

The critical advantage of such levitation experiments is the avoidance of

specimen contamination since in highly reactive materials even slight

contact with a container during heating and melting could completely

3
invalidate the experimental results. Measurements of the specific heat

of liquid tungsten uncontarainated by reaction with container walls

currently are being pursued in earth's gravity experiments to look at the

limitations imposed by gravity in such work. For example, in these

experiments, the large electromagnetic forces required to counteract

gravity tend to produce instabilities in the liquid droplets. A number

of the automated techniques being developed in this work should also be

useful for space flight experiments.

4
Task 1

Surface Tensions and Their Variations

with Temperature and Impurities

S. C. Hardy

Metallurgy Division

Center for Materials Science

Introduction

Surface tension gradients produced by temperature or concentration

variations on free liquid surfaces result in shear stresses which can in

many situations set the fluid into motion. In low gravity materials

processing the primary fluid flows are expected to be of this type because

buoyancy is greatly reduced. These thermocapillary flows can influence

the temperature and solute fields in the bulk liquid and the morphology,

microstructure and homogeneity of a solid produced from this liquid.

Thus, knowledge of surface tension gradients is essential to the understanding of materials produced from the processes involving free liquid

surfaces in low gravity. We are measuring the surface tension of liquid

silicon in support of space experiments with this material which are

being developed. The results of these measurements will also be of

interest for silicon crystal growth research and production on earth.

The temperature and chemical concentration dependences of the surface

tensions of many materials are poorly known; for highly reactive, high

melting point elements like silicon often only estimates of these parameters

can be found in the literature. Thus in a recent computer study of the

fluid flow in Czochralski growth of silicon, calculations were performed

for values from -0.1 mJ/m^K to -0.4 mJ/m^K for K, the rate of change of surface

tension with temperature [1]. The flow was found to vary significantly

both in magnitude and distribution for this variation in K. These results

5
show that accurate surface tension data is essential for meaningful calculations or quantitative interpretation of observed flow phenomena in

floating zones of silicon.

The uncertainty in K for silicon stems basically from the lack of

systematic measurements of the surface tension over a temperature range.

Various workers have measured the surface tension at different temperatures

[2, 3, 4], However, K values calculated from their data are highly

inaccurate because systematic errors in the different experiments are

sufficiently large to obscure the surface tension differences due to

temperature. It is essential in determinations of K to measure the

surface tension over a temperature range in the same experiment so that

the effects of systematic errors are minimized. This is especially true

for silicon because of its extreme chemical reactivity. Liquid silicon

dissolves almost all materials to some extent thereby picking up impurities

which can surface segregate and significantly lower the surface tension.

Thus it must be assumed that most experiments will have unique impurity

concentrations.

The only systematic study of the surface tension of silicon over a

temperature range that we know of is that of Lukin, et al [5]. The

value they found for K is -0. 1 mJ/m^K. Although this is a reasonable

value, no details or data are presented. Moreover the density variation

with temperature used in their work is significantly different from that

generally accepted. Thus it is not possible to make any judgment as to

the accuracy of their measurements.

Experimental

We are using the sessile drop technique to measure the surface

tension of liquid silicon. This method is based on a comparison of the

profile of a liquid drop with the profile calculated by solving the

6
Young-Laplace equation. The comparison can be made in several ways; we

are using the traditional Bashforth-Adams [6] procedure with modern drop

shape tables we have calculated which virtually eliminate interpolation

errors.

Although we have used the sessile drop technique previously to

measure the surface tension of gallium [7], the present experiments

require essentially new apparatus because of the high temperatures

involved. The melting point of silicon is 1410°C and it is desirable to

make measurements to at least 1600°C in order to determine the temperature

dependence of the surface tension.

Figure 1 is a schematic diagram of the experimental apparatus. The

silicon is contained in a cup which stabilizes the liquid drop and imposes

a circular cross section on the drop base, thus reducing errors arising

from anisotropic wetting often encountered using flat drop supports. The

outside diameter of the cup is used as the length standard in the

calculation of the surface tension. The cup and drop are at the center

of a cylindrical susceptor which is heated inductively. The quartz jacket

enclosing the susceptor and drop connects through an O-ring coupling to a

vacuum system consisting of a water cooled baffle and diffusion pump,

gauges, valves for admitting gases to the system, and a forepump. Quartz

windows at both ends of the tube allow simultaneous photography and

temperature measurement of the drop with an automatic pyrometer.

Figure 2 shows the central region of the heating tube in more detail.

The cup containing the silicon sits on a tungsten platform which in turn

rests on two tungsten rods. These rods extend well out of the susceptor

into a much cooler region of the heating tube. They are clamped to a

I 7
Young-Laplace equation. The comparison can be made in several ways; we

are using the traditional Bashf orth-Adams [6] procedure with modern drop

shape tables we have calculated which virtually eliminate interpolation

errors.

Although we have used the sessile drop technique previously to

measure the surface tension of gallium [7], the present experiments

require essentially new apparatus because of the high temperatures

involved. The melting point of silicon is 1410°C and it is desirable to

make measurements to at least 1600°C in order to determine the temperature

dependence of the surface tension.

Figure 1 is a schematic diagram of the experimental apparatus. The

silicon is contained in a cup which stabilizes the liquid drop and imposes

a circular cross section on the drop base, thus reducing errors arising

from anisotropic wetting often encountered using flat drop supports. The

outside diameter of the cup is used as the length standard in the

calculation of the surface tension. The cup and drop are at the center

of a cylindrical susceptor which is heated inductively. The quartz jacket

enclosing the susceptor and drop connects through an O-ring coupling to a

vacuum system consisting of a water cooled baffle and diffusion pump,

gauges, valves for admitting gases to the system, and a forepump. Quartz

windows at both ends of the tube allow simultaneous photography and

temperature measurement of the drop with an automatic pyrometer.

Figure 2 shows the central region of the heating tube in more detail.

The cup containing the silicon sits on a tungsten platform which in turn

rests on two tungsten rods. These rods extend well out of the susceptor

into a much cooler region of the heating tube. They are clamped to a

8
stainless steel block at one end and rest freely on an identical block at

the other end. These steel blocks rest on the walls of the quartz tube.

The platform and cup can be moved along the rods and the entire assembly

translates inside the quartz tube. This device permits convenient and

rapid sample insertion and removal.

The induction heater operates at 450 kHz and has a power rating

of 2.5 kW. This has proved adequate for these experiments if the

susceptor is insulated sufficiently. The most promising insulation we

have tried is zirconia, Zr02 , because of its low thermal conductivity and

outgassing rate. However, the stability of the insulation in conjunction

with the susceptor material is critical. In our initial measurements we

used zirconia wrapped about a tantalum susceptor. At operating temperature

the tantalum reduced the zirconia and generated volatile species according

to the following reaction [8]:

Ta(s) + Zr02(s) = TaO(g) + ZrO(g)

Similar reactions can occur with other combinations of insulator and

susceptors (9). At present we are using zirconia insulation with molybdenum

susceptors. This combination is reported to be non-reacting up to 2200°

[9, 10]. Metallic heat shields cannot be used because the skin depth at

450 kHz is only 0.06 cm. Since the field decays exponentially as the

thickness over the penetration depth, a heat shield 0.006 cm thick reduces

the field by 10%, a power loss which would be intolerable with our equipment.

The susceptor is constructed of two concentric cylinders of molybdenum

with the third, inner cylinder of tantalum because of its lower vapor

pressure. The total thickness is about 0.11 cm, which is sufficient to

reduce the field to about 14% of the maximum value. Thus the interior of

9
the susceptor is not field free. This may result in net forces on the

sessile drop which will cause distortions. Although we do not have

evidence for such distortions, it is desirable to decrease this field

penetration.

Because of the very high reactivity of liquid silicon, the choice of

cup material is a critical feature of these measurements. Extensive

studies of the compatibility of liquid silicon with various materials

have been made recently in conjunction with the solar cell program of the

DOE. This research suggests that boron nitride is the most promising of

a very limited group of materials [11]. Liquid silicon has a contact

angle with boron nitride of about 130° [11]. For our purposes a high

contact angle is desirable because it minimizes the wetting of the top

surface of the cup by the silicon with the attendent distortion of shape.

More importantly, silicon does not dissolve boron nitride. Thus the

silicon is not grossly contaminated by cup material. The bonding between

the silicon and the boron nitride is sufficiently weak to preserve the

cup on cooling. Most materials bond strongly to liquid silicon and are

cracked by the differential thermal contraction on cooling. Thus the

boron nitride cups can be used for many experiments. Although the DOE

studies showed that silicon did acquire some free boron when in contact

with boron nitride, we do not believe this will have a major effect on

our measurements because boron has a higher surface tension than silicon

and consequently will not surface segregate. The development of a bulk

boron concentration will only increase the surface tension. Studies of

the boron silicon system indicate that 1 atomic percent boron will raise

10
the surface tension of silicon about 25 raJ/m^, i.e., approximately 3% (5).

This would he a very high bulk impurity concentration.

The silicon used in these early experiments is polycrystalline

material of six nines purity. The samples are cut from a large boule

using a carbide cut-off wheel. After rinsing in water and alcohol, the

silicon parallepiped is melted and solidified on a flat boron nitride

block to form a button with roughly the shape of a sessile drop. This

button is then placed in a boron nitride cup and a sessile drop is formed.

The temperature of the silicon is raised to the maximum and the drop is

photographed as it is cooled to various temperatures by reducing the

power setting of the induction heater.

Results and Discussion

Figure 3 shows the surface tension of liquid silicon as a function

of temperature for three different samples. Two sets of data are in good

agreement with each other: the third set lies about 40 mJ/m^ higher than

these points. The slopes, however, are nearly the same for the two

groupings of data. Linear regression analysis gives slopes of

-0.245 ± 0.012 mJ and -0.237 ± 0.025 mJ

m^K m^K

for the lower and higher data respectively where the range is the estimated

probable error. Thus our early measurements indicate the variation with

temperature of the surface tension of silicon is approximately -0.24 mJ/m^K

This data was acquired before the construction of the vacuum system; the

measurements were made in flowing argon which had been purified by passage

over hot titanium. The 40 mJ/m^ difference in the two sets of data is

probably due to a different partial pressure of oxygen in the flowing argon

11
It is possible, of course, that the surface tension difference arises

from a drop asymmetry which we cannot observe or to bulk contamination

with boron. These values of surface tension and the temperature variation

of the surface tension should be regarded as preliminary and as a

developmental step rather than a quantitative measurement.

We also show in Figure 3 as a dashed line the results of the Russian

measurements referred to previously which have only been reported in

summary form [5]. This curve is in fair agreement in absolute value of

the surface tension with our lower data. However, the temperature

variation is -0.1044 mJ/m4K, a value less than half that which we found.

Part of the discrepancy is due to the use of different density data.

The authors of reference [5] claim to have measured the density and

report their results by giving a value for the coefficient K

p which

occurs in the expression

P Vio - VT- mo>-

The value of K

p for pure silicon they find is 0.1291 x 10

_ ^gm/cm^K.

We have used in our calculations of surface tension the density

measurements of Lucas [12] et al which are reported in detail, are

systematic, and in good agreement with less extensive measurements by

others. Lucas finds K

p = 0.35 x 10 gm/cm K, a value significantly

different from the Russian work. If the lower K

p is used to calculate

surface tensions with our drop measurements, the temperature variation of

the surface tension is reduced to -0.185 mJ/m^, a result still significantly

higher than that reported in the Russian work.

In summary, we have developed the instrumentation required to measure

the temperature variation of the surface tension of silicon. Preliminary

12
measurements in flowing argon using boron nitride cups to contain the

drops suggest that this temperature variation will be near -0.24 mJ/m^K.

In future work we will be making measurements in vacuum and in atmospheres

containing controlled concentrations of hydrogen and oxygen in order to

determine the sensitivity of the silicon surface tension to oxygen partial

pressure. In addition, the effects of other container materials will be

investigated. Of particular interest is Si02 because of its wide use in

industrial silicon crystal growth. Some determination of the impurity

content of the silicon after the surface tension measurements is highly

desirable to assess the extent to which the silicon is contaminated by

contact with the cup. We will characterize our samples using Auger

spectroscopy.

13
References

1. W. E. Langlois, J. Crystal Growth 56, 15 (1982).

2. P. H. Keck and W. Van Horn, Phys. Rev. 9J_, 512 ( 1953).

3. Eljutin, Kostikov and Levin, Izv. Vys. Uch. Sav. , Tsvetn. Met., J2, 131

(1970).

4. Dshemiler, Popel and Zarevski, Fiz. Metri Met., J_8, 83 (1964).

5. S. V. Lukin, V. I. Zhuchkov and N. A. Vatolin, J. of the Less-Common

Metals, 67, 399 (1979).

6. F. Bashforth and J. C. Adams, The Theories of Capillary Action , University Press, Cambridge, 1883.

7. S. C. Hardy and J. Fine, Materials Measurements, NBSIR 80-2082,

1 (1980).

8. W. A. Chupka, J. Berkowitz and M. G. Inghram, J. Chem. Phys. 26 , 1207 (1957).

9. W. D. Kingery, Property Measurements At High Temperatures , John Wiley

and Sons, New York, 1959.

10. The Oxides Handbook , edi’ted by G. V. Samsanov, IFI/Plenum, New York,

1973.

11. T. O'Donnell, M. Leipold and M. Hagan, Compatibility Studies of

Various Refractory Materials in Contact with Molten Silicon,

DOE/ JPL-1012-77 16, (1978).

12. L. D. Lucas, Mem. Sci. Rev. Met., 61, 1 (1964).

14
CO

Q.

2

ZD

CL

15

Fig.

1.

Schematic

diagram

of

experimental

apparatus:

(1)

quartz

tube

(2)

induction

coil,

(3)

O-ring

seal,

(4)

window.
Fig.

2.

Schematic

diagram

of

high

temperature

zone

of

experimental

apparatus:

(1)

quartz

tube,

(2)

zirconia

insulation,

(3)

molybdenum

susceptor,

(4) W

rods,

(5) W

tray,

(6)

BN

cup.
780

1400 1500

TEMPERATURE (° C)

1600

Fig. 3. The surface tension of silicon as a function of temperature

(preliminary results).

17

Task 2

Convection During Unidirectional Solidification

R. J. Schaefer and S. R. Coriell

Metallurgy Division

Center for Materials Science

and

G. B. McFadden and R. G. Rehm

Mathematical Analysis Division

Center for Applied Mathematics

SUMMARY

During solidification of a binary alloy at constant velocity vertically

upwards, thermosolutal convection can occur if the solute rejected at

the crystal-melt interface decreases the density of the melt. We assume

that the crystal-melt interface remains planar and that the flow field is

periodic in the horizontal direction. The time-dependent nonlinear

differential equations for fluid flow, concentration, and temperature

are solved numerically in two spatial dimensions for small Prandtl

numbers and moderately large Schmidt numbers. 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. For fixed velocity as the concentration increases,

the horizontal wavelength decreases rapidly; a phenomenon also indicated

by linear stability analysis. The lateral inhomogeneity in solute concentration due to convection is obtained from the calculations. For a narrow

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

time.

Experimental observations of unidirectional solidification of succinonitrile

containing ethanol have revealed an interaction between convective

flow, interface shape, and the localized development of a cellular interface.

19
The effect is attributed to thermally induced convection and can be modified

by changing the thermal field in the interface region.

In collaboration with M. E. Glicksman and Q. T. Fang of Rensselaer Polytechnic

Institute, and R. F. Boisvert of the Scientific Computing Division, we have

analyzed the stability of the parallel flow between a vertical crystal-melt

interface and a vertical wall held at a temperature above the melting point

of the crystal. Three modes of instability occur: (1) a buoyant mode,

(2) a shear mode, and (3) a coupled crystal-melt mode. For Prandtl numbers

greater than approximately two, the coupled crystal-melt mode occurs at a

lower Grashof number than the other two modes. These calculations and similar

calculations for a cylindrical geometry were motivated by and are in general

agreement with recent experiments on succinonitrile at Rensselaer Polytechnic

Institute.

20
Introduction

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

of the fluid flow and solute segregation which occur during solidification.

The nature of the fluid flow, its effect on the shape of the crystal-melt

interface and on the resulting distribution of solutes is examined. In

particular, the role of solutal convection (gravity driven flow due to solute

gradients) during the vertical directional solidification of a binary alloy

is considered both theoretically and experimentally.

We present a disscussion of the numerical calculations first and then a

description of the experiments: each Section can be read independently.

The results of a number of calculations of the fluid flow and concentration

field during directional solidification of an alloy with Prandtl number of

0.01 and Schmidt number of 10 are described in the Numerical Results section.

The experimental section describes the fluid flow and crystal-melt interface shapes observed during the directional solidification of succinonitrile

containing small additions of ethanol.

In an Appendix, we describe a linear stability analysis relevant to

experimental work being conducted at Rensselaer Poytechnic Institute. As

described in the Appendix an interface instability occurs under conditions

for which in the absence of fluid flow, the crystal-melt interface would be

morphologically stable.

21
NUMERICAL RESULTS

The basic equations and methods for calculating the fluid velocity,

solute, and temperature fields during directional solidification of a binary

alloy vertically upwards have previously been reported [1]. In this report

we present the results of a number of numerical computations for an alloy

of Prandtl number P = 10“^ and Schmidt number Sc = 10. In general we will

use dimensionless variables; Tables I and II define dimensional and dimensionless quantities, respectively.

The solutal Rayleigh number Rs* at the onset of convective instability

can be determined by linear stability calculations. For convective modes of

instability, the results of the linear theory are approximately the same

whether the planar crystal-melt interface is allowed to deform or is maintained

rigid. The critical solutal Rayleigh number as a function of V/D is plotted in

Fig. 1. In these calculations we have kept the imposed temperature gradient

t . in the liquid constant so that Ra(V/D)^ is constant; thus Ra decreases rapidly

as (V/D) increases. Previous calculations [2-4] (see e.g. Table 3 of

ref. 2) have indicated that as V/D increases the thermal field does not affect

the critical solutal Rayleigh number. In fact, Hurle, Jakeman, and Wheeler [4]

showed that for sufficiently large V/D the dependence of density on temperature

could be completely neglected, i.e., they set the Ra = 0, and obtained results

in agreement with those of ref. 2, which corresponded to a liquid temperature

gradient of 200K/cm. Thus, linear stability theory indicates that for

small values of V/D, the thermal field is important while for large values

of V/D the thermal field is unimportant.

The different behavior in these two regimes is further illustrated in

Figs. 2 and 3. The critical bulk solute concentration C for the onset of

instability is plotted as a function of the horizontal wavelength X for

D/V = 0.015 and 0.15 cm in Figs. 2 and 3, respectively. The concentration

22
has been normalized by the minimum concentration C* as a function of wavelength;

thus C/C* = Rs/Rs*. Linear stability theory predicts that the amplitude of

a perturbation varies in time as exp [or (X)t]. For a given value of

C/C* > 1, there is a positive maximum value of a r as a function of X. The

wavelength corresponding to the maximum value of a r for a given value of C/C* is

indicated by the dashed line in Figs. 2 and 3. For D/V = 0.015, the wavelength

corresponding to the maximum growth rate, slowly decreases as C/C* increases.

In contrast, for D/V = 0.15 cm, the wavelength corresponding to the maximum

growth rate rapidly decreases as C/C* increases.

We note that the unperturbed melt density decreases with height if

Rs < kScRa/ (P(k-l) ) or Rs < 429 Ra for the values given in Table II. This

inequality holds for all of the non-linear numerical calculations presented

in this report. Most of our numerical calculations have been carried out for

D/V = 0.15 cm, where linear theory indicates that the thermal field is important and that the wavelength decreases rapidly with increasing solutal Rayleigh

number. For Rs/Rs*=l we choose the horizontal wavelength as that

corresponding to the onset of instability as determined by linear theory.

As the solutal Rayleigh number increases, we find that even though we start

with a given wavelength, the actual numerical solution has a smaller wavelength, usually one half the original wavelength. We then use this new

wavelength as the starting wavelength for calculations at larger solutal

Rayleigh numbers. This behavior is illustrated in Fig. 4. For D/V = 0.15 cm,

the wavelength at the onset of convection XQ

= 8.06 D/V. At Rs/Rs* = 1.62

and 1.65, the wavelength decreases to X0 /2 and X0/3, respectively. At

Rs/Rs* = 3.5, the wavelength decreases again from X0

/3 to X 0 /6. As will

be discussed, the numerical solution may depend on the initial conditions.

Over a narrow range of Rs/Rs* we have found time dependent solutions which

are indicated by the dashed line in Fig. 4.

23
Some tests of the numerical code have previously been described [1], As a

further test, we carried out calculations at X = X Q and Rs/Rs* = 0.977 and

1.020 on a (12, 36) grid and found no flow and flow, respectively, in agreement

with linear stability analysis.

Figures 5-9 illustrate numerical results for Rs/Rs* = 1.29. In our numerical

calculations, we use a dimensionless time, = tv/H . We also define a dimensionless time T

c based on the crystallization velocity, viz., T

c = V^t/D, and

note that T /T.. = (VH/D)^/Sc. In the absence of convection, the time for the

C N

exponential concentration profile in front of the crystal-melt interface to

develop is of the order of Tc /k for k<<l. Specifically, for k = 0.3 the

interface concentration reaches 95% of its final value in a time corresponding

to T c = 10.2 or Tfj = 5.9 for (VH/D) * 4.17 as in Figs. 5-9. For Rs/Rs* = 1 . 29 , the

time to approach steady state in the flow, solute, and temperature fields corresponds to = 14 as shown iniFig. 5 in which the maximum value of the stream

function is plotted against time. In Figs. 6-7 we show the steady state stream

function and concentration respectively as a function of position. At the

top boundary, we have assumed that the fluid velocity and perturbed temperature

gradient vanish, and have used a mixed boundary condition on the solute

concentration, which ensures solute conservation [5]. The top boundary is an

artificial boundary, and we would like to place it sufficiently far from the

interface so that it does not influence the flow and solute segregation near the

interface. From Fig. 6-7 it is evident that the top boundary is having little

effect on the flow and solute concentration near the interface. As the solutal

Rayleigh number increases, the flow is less confined to the interface and eventually it is not feasible in terms of computational times to move the upper

boundary far enough from the interface. As illustrated in Fig. 8, the solute

concentration at the crystal-melt interface varies by 60%. The average solute

24
concentration at the interface deviates by about 3% from the correct value

due to the finite mesh (9 x 36) used in the calculation. In Fig. 9 we show

the average solute concentration in the liquid as a function of distance from

the crystal-melt interface. For comparison, the exponential concentration profile which would hold in the absence of convection is also shown in Fig. 9.

Over a narrow range of Rs/Rs* and X = X0/3, we have found

time-dependent solutions. These are shown in Figs. 10-21; the calculations

were carried out on a (12 x 72) mesh with (VH/D) = 4.17 and Rs/Rs* = 3.3.

Figs. 10 and 11 show the maximum value of the stream function and the

average interface concentration in the melt, respectively, as a function

of time. The period of oscillation is approximately AT^ = 0.56

or ATC = 0.97, i.e., the interface has moved a distance 0.97 (D/V)

during one cycle. The magnitude of the average interface composition

oscillation is small, viz. about 1.6%, which is considerably less than

the transverse segregation of about 18%. The magnitude of the stream

function oscillation is considerably greater. Figures 12-21 show the

stream function and concentration as a function of position at various

times during one period of the oscillation. It is clear from these

figures that the second vertical cell is undergoing the largest oscillation

with the stream function varying in magnitude from 0.119 to 0.234. Note

that the flow does not reverse in direction; rather these are oscillations

superimposed on a base flow. This may explain why linear stability

calculations, assuming a motionless base state, have not revealed any

time dependent phenomena in this parameter range. We have also observed

that the steady state solution may depend on the initial conditions,

i.e., there may be multiple steady states. As initial conditions, we

have (1) perturbed a uniform concentration field, (2) perturbed a steady

state (exponential) concentration field, and (3) used a solution for a

25
particular value of Rs as the initial condition for a different value of Rs.

In Figs. 22-23, we show the steady state stream function and concentration

field, respectively, for Rs/Rs* = 3.6 using the oscillatory solution at

Rs/Rs* = 3.3 (see Figs. 10-21) as the initial condition. In contrast,

Figs. 24-25 show the steady state stream function and concentration

field, respectively, for exactly the same conditions as Figs. 22-23

except that the initial conditions consisted of a perturbed exponential

concentration field. For this case, the final solution has one half the

wavelength of the initial condition.

In Figs. 26-27, the stream function and concentration field,

respectively, for Rs/Rs* = 5.03 and (VH/D) = 4.17 are shown. Figs. 28-29

are identical to Figs. 26-27, respectively, except that (VH/D) and the

number of mesh points in the vertical direction have been doubled. It is

evident in Figs. 26 and 27 that the effect of the upper boundary is not

* . negligible. As can be seen in Fig. 28 with (VH/D) = 8.33, the stream

function vanishes at a distance of approximately 5.4 (D/V) from the

interface. The solute segregation at the interface is about the same in

Fig. 27 and 29, viz., 15%.

Figures 3-29 were for D/V = 0.15 cm; in Figs. 30-33 we show the results

of a calculation for D/V = 0.015 cm, Rs/Rs* = 1.325 and Ra = 0.127, corresponding to the same temperature gradient as in Figs. 3-29. As previously

discussed, we do not expect the temperature field to stabilize the flow

since the Rayleigh number is very small. As shown in Figs. 30-31, even

though we have taken H = 20 (D/V), the flow is influenced by the upper

boundary, and is certainly not confined to the vicinity of the crystal-melt

interface. There are clearly fairly large lateral solute gradients far

from the interface. Fig. 32 shows the solute concentration at the crystal-melt

26
interface; the solute segregation is about 60%. The average solute

concentration as a function of distance from the interface is compared

with the solute profile in the absence of convection in Fig. 33. The

average interface concentration and the average concentration gradient at

the interface are determined by the boundary conditions, and are independent

of the fluid flow as can be seen from Fig. 33. However, at about 2 D/V

from the interface, the average solute concentration with convection is

about 40% higher than the unperturbed (convectionless) concentration.

Since the Rayleigh number was small for the results of Figs. 30-33, we

carried out an identical calculation with Ra = 0 to determine whether the

thermal field can be completely neglected. The maximum value of the

stream function was 80% larger for Ra = 0 than for Ra = 0.127 indicating

that the thermal field plays a role even for Ra as small as 0.127.

The numerical results presented here provide insight into the solutal

convection which can occur during directional solidification. The range

of parameters which can be studied is severely limited by the computational

times required for these calculations. Results for different values of

the distribution coefficient and temperature gradient in the liquid would

be desirable. We have limited the calculations to a Schmidt number of

ten, which is appropriate for semiconductors. The time to reach steady

state appears to be proportional to the Schmidt number for large Schmidt

numbers so that calculations for metallic melts with Schmidt numbers of

the order of 100 will take ten times as much computer time. The interaction

of the solutal convection, which is driven by the adverse concentration

gradient near the crystal-melt interface, with the thermal convection due

to horizontal temperature gradients in a real furnace is an important

area for further research.

27
EXPERIMENTAL

Background

Succinonitrile containing ethanol provides a useful system for the

investigation of thermo-solutal effects during solidification. The

density of ethanol being significantly less than that of succinonitrile,

unidirectional upward solidification in this system leads to solute

distributions which are predicted to be unstable with respect to convective

perturbations. We previously determined the succinonitrile-rich end of

the phase diagram for this system and, on the basis of the measured phase

diagram parameters and the published physical properties of succinonitrile,

predicted the critical concentration of ethanol in succinonitrile above

which instability occurs, as a function of the velocity of upward

solidification. The predicted instabilities are either interfacial, in

which case the wavelength is short and the shape of the solid-liquid

> . interface deviates from planarity, or convective, in which case the

wavelength is longer and the liquid above the interface starts to flow.

At a fixed temperature gradient, the concentration of ethanol required to

produce interfacial instability decreases with increasing solidification

velocity, while the concentration required to produce convective instability

increases with increasing solidification velocity. At some compositions,

there therefore exists a range of solidification velocities in which

unidirectional upward solidification is predicted to be stable; above

this velocity range interfacial instability is predicted and below this

velocity range convective instability is predicted.

Our experimental observations of solidification processes in the

region of these transitions have used both pure succinonitrile and

succinonitrile doped with an appropriate concentration of ethanol to

28
investigate the behavior when a finite range of stable velocities is

predicted. Because of the low solute distribution coefficient for ethanol

in succinonitrile (k = 0.044), concentrations of interest are very low

(2 - 5 x 10“3 wt.% ethanol) and the samples were thus made from ultrapurified succinonitrile and sealed under vacuum.

Perfectly uniform upward solidification of a substance with a normal

thermal expansion coefficient (i.e., one which expands with increasing

temperature) is convectively stable if no solutes are present. The

convective instability predicted for the succinonitrile-ethanol system

arises from the rejection of the lower-density solute by the advancing

interface. In a real experiment, however, there are sources of convection

which may be described as purely thermal and which interfere with

observation of the effects predicted for truly unidirectional solidification.

Any factor which causes the isotherms in the liquid to deviate from

horizontal planes will be a driving force for convective flow.

There are two important causes for thermally induced fluid flow in

the liquid above a solidifying interface when unidirectional solidification

of a material in a tubular container is attempted by drawing the container

downward through a temperature gradient. First, heat losses or gains

through the container walls to the furnace or outside environment

necessarily imply the presence of a radial component in the temperature

gradient. These must be minimized by design of the crystal growth

apparatus and by control of the interface position. The second origin

for thermally induced convective flow is the distortion of the isotherms

by the container walls. When no solidification is taking place, there is

a gradient discontinuity at the solid-liquid interface due to the different

29
thermal conductivity of the solid and liquid. Because this gradient

discontinuity is not present in the container walls, the isotherms in the

region where the interfce meets the walls must be distorted from planarity.

Succinonitrile is unusual in that the thermal conductivity of solid and

liquid are, within experimental uncertainty, equal, so that at zero

solidification velocity this effect is not important. When solidification

starts, however, a gradient discontinuity is produced at the solid-liquid

interface by the emission of latent heat of fusion. The heat emitted per

unit area of the interface is vL where v is the velocity and L is the

latent heat (4.7 x 10^ J/m^ for succinonitrile). Because in a typical

experiment the discontinuity produced by this latent heat is not negligible

compared to the externally imposed gradients, a significant thermal field

distortion is produced by the container walls.

For the unidirectional solidification of metals it is often possible

to use containers having much lower thermal conductivity than the metal,

in which case the thermal distortions caused by the container walls will

be small. However, lateral heat losses to the environment can be large in

metallic systems due to the high melting temperatures. For an organic

material such as succinonitrile, however, the thermal conductivity of the

container cannot be much lower than that of the material and container

wall effects are unavoidable. Radial heat losses to the environment,

however, can be small because of the low melting temperature. Thus

distortion of the isotherms may be similar in metallic and organic systems,

although the most important cause of the distortion may differ.

In our experiments we have investigated the effects of radial

gradients on "unidirectional" upward solidification under conditions

close to those predicted to result in solute-induced interfacial or

30
convective instability, and have investigated methods to reduce the

effects of radial gradients. We have also measured densities of

succinonitrile-ethanol mixtures because earlier calculations were based

on estimated values.

Experimental Methods

a. Density Measurements

The density of pure liquid succinonitrile and succinonitrile containing

four different concentrations of ethanol (up to 6.67 wt.%) was measured

by determining the temperatures at which calibrated standards with

densities of 0.9700, 0.9800, and 0.9850 had neutral bouyancy. Distilled

succinonitrile was used and measurements were carried out under air at

atmospheric pressure. The temperature of neutral bouyancy was determined

to within 0.1°C.

b. Convective Flow During Unidirectional Solidification

Pure succinonitrile and succinonitrile-ethanol samples are contained

within borosilicate glass tubes 19 mm in diameter and approximately 50 cm

long. The residual impurity content of the succinonitrile samples, before

doping with ethanol, was found to be equivalent to approximately 10“^

wt.% (lppm) of ethanol, in terms of its effect on the solidus and liquidus

temperatures. Most experiments were carried out on a sample containing

2.6 x 10”3 wt.% ethanol.

To produce solidification, the sample tubes are drawn downward through

a temperature gradient established by a thermostatically controlled lower

cold water jacket and an upper heater which is either a similar hot water

jacket or an electric heater. Growth velocities can be from 0.4 to 10 pm/s.

31
An important constraint for these experiments is the requirement of

good visibility in the interface region. This can be attained only if

the cylindrical tube is surrounded in the interface region by a medium

with a refractive index similar to that of the succinonitrile , but which

itself has a flat outer wall for microscopic observation. The best image

is obtained if the index-matching medium is a fluid contained in a flatwalled outer container, but such a system could also undergo convective

processes which would tend to perturb the temperature field within the

solidification tube.

We have therefore surrounded the interface region of the solidification

tube with close-fitting transparent solid pieces with flat outer windows.

Heaters have been incorporated into these pieces in the region of the

solid-liquid interface to provide control over the isotherm shape within

the sample tube. The heaters are in the form of a flat (pancake-like)

coil so that they can duplicate in the transparent solid the temperature

discontinuity existing in the sample at the solid-liquid interface. The

required heater power is proportional to the velocity of motion of the

sample: for a velocity of 1 pm/s, a heater power of about 0.5 W is

appropriate.

The sample tube is photographed from the side by a horizontally mounted

microscope. To measure convective flow, small neutrally bouyant latex

particles are incorporated into the sample. These are photographed using

a triple flash exposure with unequal intervals between the flashes, to

allow determination of the direction of particle motion. The particles

are seen only in dark field illumination, but the interface shape is seen

better in bright field illumination.

32
Results

a. Density

The measured densities of succinonitrile-ethanol solutions are shown

in Figure 34. The measured densities are close to those which are calculated

under the assumption that no change of total volume takes place when the

two components are mixed, although a deviation from this behavior might

have been expected on the basis of the liquid-phase miscibility gap in

the succinonitrile-ethanol system. The lines of constant density lie

almost exactly parallel to the liquidus line in the composition and

temperature range which was measured. The density of the liquid at the

liquidus is thus constant, 0.9880 g/cm^. This leads to the unusual result

that the liquid at the surface of a succinonitrile crystal growing into a

succinonitrile-ethanol mixture has a constant density, no matter what the

shape of the crystal, so long as the ethanol concentration does not exceed

approximately 7 wt.%. Liquid not immediately at the surface, however, is

not constrained to be at the liquidus and therefore has a variable density.

The measurements indicate a concentration dependence of liquid

density a

c = -(3p/3c)/p = 2.7 x 10-

'* (wt.%)

-

* which is close to the

value used previously (3.07 x 10

-

^ (wt.%)

-

*.

b. Convective Flow During Unidirectional Solidification

When the succinonitrile sample is held stationary in the directional

solidification apparatus, the solid-liquid interface is horizontal and

very slightly concave upwards. The amount of this concavity, which

probably results from a small radial heat loss, depends upon the position

of the interface between the hot and cold jackets, the nature of the

surrounding medium, and other details of the growth apparatus design. A

33
concave interface indicates that the liquid in the center of the tube is

warmer than that near the wall at the same level. Therefore, as expected,

there is an upward flow of liquid in the center and a downward flow closer

to the walls, with the liquid immediately above the solid-liquid interface

moving radially inward. The upward flow velocity at the tube center is

typically 10 pra/s. The only microstructural features present at the

interface in either pure or ethanol-doped samples are faint grooves due

to grain boundaries or sub-boundaries.

When solidification is produced by drawing the sample tube downward,

the solid-liquid interface becomes strongly concave in either pure or

ethanol-doped succinonitrile. Simultaneously there is a substantial

increase in the velocity of upward flow at the tube center (to about

40 pm/s for a growth velocity of 2 pm/s). The pattern of the flow

remains unchanged.

» . These flow velocities are sufficiently large that they can be expected

to seriously distort the solute distribution in the liquid above the

solid-liquid interface. It was found, however, that the pancake-shaped

auxiliary heater surrounding the interface region could be effective at

altering this flow pattern.

When the auxiliary heater is not used, the convective flow can result

in a transient macroscopic interface shape change which then leads to a

localization of interfacial instabilities. The sequence of events which

constitute this phenomenon is as follows:

a. The interface is equilibrated in its almost planar configuration

at zero growth velocity.

34
b. Growth is started at a velocity somewhat less than that which is

expected to produce interfacial instabilities.

c. The interface becomes concave upwards as a result of the heat

flow effects mentioned above.

d. Following an incubation time which depends on the growth velocity,

an additional deeper depression develops in the center of the concave

interface. The size and shape of this depression also depends on the

growth velocity.

e. The depression deepens rapidly and interfacial instabilities

develop within it.

f. The cellular interface within the depression advances rapidly

and catches up to the general contour of the concave interface.

g. The concave interface then propagates at steady state with a

central cellular region in tl)e area where the depression had been.

The incubation time for development of the depression is longer at

lower growth velocities. At 1 pm/s, the depression has not been

observed, and the interface remains bowl-shaped but smooth (no interfacial

instabilities). At 2 pm/s the conditions for formation of the

depression are apparently marginal and the incubation time in different

runs was found to be between 26 and 125 minutes. At a growth velocity of

3 pm/s the incubation time was 13 to 15 minutes. At 4 pm/s the

incubation time becomes more difficult to define because of shape changes

(see below) which make the pit less distinct from the overall concavity

of the interface.

The transient interface pit takes its most dramatic form under the

most marginal conditions for its formation. Thus in the 2 pm/s run in

which the pit took 125 minutes to develop, the pit was very deep, narrow.

35
and pointed before a cellular structure developed near its bottom and

spread upwards along the walls (Figure 35 a). At faster growth rates, or

when the growth conditions at 2 pm/s were somewhat different and the

pit formed more rapidly, the pit was broader, shallower, and more rounded

when the cellular structure developed within it (Figure 35 b). After the

cellular interface formed within the pit, cells spread rapidly up the

sides of the pit as far as the point where the pit merged into the overall

bowl-shaped interface. The pit region then filled in rapidly with cellular

material until the leading edge of this material was essentially coincident

with the overall bowl-shaped contour of the interface. Figure 36 shows

the development of the pit in a run at 2 pm/s. The position, relative

to a fixed point in the apparatus, of the bottom of the pit is shown:

after the cellular substructure develops, the position of the first solid

(the leading edge of the cells) and the last liquid (the last entrained

* . intercellular liquid) are shown.

After the development of the cellular interface in the central pit,

the convective flow pattern in the liquid above this region is different.

Many particles are seen indicating fluid flow downward into the central

cellular interface. The flow appears to be less regular than the simple

pattern which prevails before the cells develop, but we have not yet

discerned its detailed structure.

When the auxiliary heater surrounding the interface is activated,

the formation of the central pit can be suppressed. Ideally, the heater

would be located exactly at the interface level and would transfer heat

uniformly to the sample container walls from all sides. In practice it is

difficult to arrange that the interface lies exactly in the plane of the

heater because the vertical position of the interface is affected by many

36
factors including the growth velocity and the auxiliary heater power.

Circumferentially uniform transfer of heat to the container walls can

also only be approximated. Nonetheless, auxiliary heaters are very

effective at changing the convective flow pattern and eliminating the

central pit.

As an example, on one run at 2 pm/s a central pit developed 33

minutes after the start of growth and the interface within the pit then

became cellular. The auxiliary heater was then turned on and its power

level adjusted to give, as closely as possible, a flat interface near the

walls of the sample tube. After approximately 30 minutes, the central

cellular region of the interface had disappeared. The interface at this

point was relatively flat but tilted to one side: the convective flow in

the liquid was downward on one side of the tube (where the tilted interface

was higher), across the tube above the interface, and upward on the

opposite side (where the interface was lower). Similarly, if growth is

started when the heater is activated to an appropriate level, the central

pit does not develop.

Discussion

These experiments have demonstrated the importance of a previously

unrecoreded interaction between convective flow, interface shape, and the

development of cellular interfaces. As observed in the succinonitrileethanol system, the effect occurs over a small range of growth velocities,

but this range of growth velocities is an important one because it

represents the conditions which might be chosen for an attempt to grow a

crystal of homogeneous, cell-free material. The predicted velocity for

&

37
interfacial instability in the system used for these experiments is

approximately 10 pra/s (depending on the value of the temperature

gradient, which is close to 10 K/cm), and the interactive effects were

observed at growth velocities of 2 to A pra/s. In addition to being

limited to a narrow velocity range, the effect is transient in nature,

with the result that the evolution of interface shapes by this process

will not be understood if observations are not sufficiently frequent.

Figure 37 summarizes the nature of these interactions in pure and

solute-doped systems. At zero growth velocity, the solid-liquid interface

'V' in either system can be planar and horizontal. When solidification

occurs, the thermal effect of the container walls results in a concave

solid-liquid interface and a convective flow pattern in which the liquid

above the solid-liquid interface sweeps radially inward and then up along

the tube center. ;

In the pure material, the interface remains smooth and bowl-shaped.

If solute is present, the radial inward flow redistributes the solute

rejected by the interface and the concentration at the interface is

highest in the central region. The resultant local depression of the

melting point in this region causes the formation of the pit, which

becomes progressively deeper as the geometry for diffusive and convective

removal of solute becomes less favorable.

The absence of the pit in the pure system, and the fact that the pit

formation can be suppressed by the auxiliary heaters, both support the

interpretation that the pit formation is the result of lateral redistribution

of solute at the solid-liquid interface. Formation of a macroscopic pit

is possible only if the interface does not immediately become morphologically

unstable, and the observed velocity dependence meets this condition.

38
Table 1. Definition of symbols used for physical quantities.

v (kinematic viscosity)

k (thermal diffusivity of melt)

ks (thermal diffusivity of crystal)

D (diffusion coefficient of melt)

g (acceleration of gravity)

a (thermal coefficient of expansion of melt)

ac (solutal coefficient of expansion of melt)

k^ (thermal conductivity of melt)

ks (thermal conductivity of crystal)

k (distribution coefficient)

Coo (bulk concentration of solute)

(bulk concentration corresponding to onset of convection)

(temperature gradient in melt)

H (melt height)

V (crystal growth velocity)

X (horizontal wavelength)

t ( t ime

39
Table 2. Dimensionless quantities used in calculations (unless otherwise

specified)

Ratio of Thermal Conductivities

Ratio of Thermal Dif fusivities

Distribution Coefficient

Schmidt number

Prandtl number

Thermal Rayleigh number

Critical solutal Rayleigh number

Dimensionless Crystal Growth Velocity

Ratio of solutal Rayleigh number,

to critical solutal Rayleigh number

Ratio of melt height to horizontal wavelength

kl/k S = 1*°

</ Kg = 1.0

k = 0.3

Sc = V/D = 10.0

P = V/k = 0.01

Ra = (gaG

L/V<)(D/V) 4 = 1268

Rg = (gacC*/VD)(D/V) 3 = 963

VH/D =4-20

Rs = (ga

cC 00/VD)(D/V) 3

,

Rs/Rs = VARIOUS VALUES

H/X = VARIOUS VALUES

40
References

1. R. J. Schaefer, S. R. Coriell, R. G. Rehra, and G. B. McFadden, NBSIR

82-2560, (ed. by J. R. Manning) p. 25 (1982).

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

J. Crystal Growth 49^, 13 (1980).

3. R. J. Schaefer and S. R. Coriell, in Materials Processing in the Reduced

Gravity Environment of Space, ed. by Guy E. Rindone (Elsevier, 1982)

p. 479.

4. D. T. J. Hurle, E. Jakeraan, and A. A. Wheeler, J. Crystal Growth 58 , 163 (1982).

5. S. R. Coriell, R. F. Boisvert, R. G. Rehm, and R. F. Sekerka, J. Crystal

Growth 54, 167 (1981).

41
RAYLEIGH

NUMBER

Fig. 1. The solutal Rayleigh number, Rs , at the onset of convection as a function of the ratio of interface velocity, V, and the diffusion

constant, D. The temperature gradient in the melt was kept

constant so that the thermal Rayleigh number, Ra , varies as the

inverse fourth power of V/D. The values of the other parameters

are given in Table 2.

42
C/C*

LINEAR THEORY FOR D/V - . 015 CM

O

Fig. 2 The normalized critical bulk solute concentration, C, at the onset

of convection as a function of the horizontal wavelength (in units

of D/V * 0.015 cm.) of a sinusoidal perturbation. The wavelength

corresponding to the maximum growth rate of the perturbation is

indicated by the dashed line.

43
LINEAR THEORY FOR D/V - .15 CM

O

Fig. 3. The normalized critical bulk solute concentration, C, at the onset

of convection as a function of the horizontal wavelength (in units

of D/V = 0.15 cm.) of a sinusoidal perturbation. The wavelength

corresponding to the maximum growth rate of the perturbation is

indicated by the dashed line.

44
C/C*

NONLINEAR CALCULATIONS FOR D/V = .15 CM

Fig. 4. The vertical lines indicate the wavelengths and solute concentrations

for which numerical calculations were carried out. The horizontal

lines indicate values of solute concentrations for which the wavelength decreases. The solid curve is the same as in Fig. 3.

45
r

max

0.10

0.15

0.20

0.25

STREAM FUNCTION

Rs - (1 . 29)Rs*

TIME

Fig. 5. The maximum of the stream function as a function of time (in units

of H

z /v) for Rs/Rs = 1.29, VH/D = 4.17, and H/X = 0.517.

The values of the other parameters are given in Table 2.
STREAM

FUNCTI0N

C0NT0UR

MAP

ATT=

24.32

fa£9393

i*

vO

47

The

steady

state

stream

function

asa

function

of

position

for

the

same

parameters

as

in

Fig.

5.

Each

vertical

and

horizontal

division

indicates

the

mesh

used

in

the

numerical

calculations;

although

the

calculation

is

done

for

one

half

wavelength

in

the

horizontal

direction,

slightly

more

than

one

wavelength

is

shown.
C0NCENTRATI0N

C0NT0UR

MAP

ATT=

24.32

48

Fig.

7.

The

steady

state

concentration

asa

function

of

position

for

the

same

parameters

as

in

Fig.

5.
INTERFRCE CONCENTRATION

O

Fig. 8. The steady state interface concentration in the melt for two

wavelengths in the lateral direction for the same parameters as in Fig. 5.

49
CONCENTRATION

Z

Fig. 9. The average steady state concentration in the melt as a function

of the distance (in units of D/V) from the crystal—melt interface

for the same parameters as in Fig. 5. The dashed line indicates

the concentration in the absence of convection.

•50
STRERM FUNCTION

TIME

Fig. 10. The maximum of the stream function as a function of time (in units

of H

2

/ v) for Rs/Rs = 3.3, VH/D = 4.17, and H/X = 1.551. The

values of the other parameters are given in Table 2. The stream

function is periodic in time; the spatial variation of the stream

function during one cycle is shown in Figs. 12-16.
CONCENTRRTION

O

Fig. 11. The average concentration at the crystal-melt interface as a function

of time for the same parameters as in Fig. 10. The spatial variation

of the concentration during one cycle is shown in Figs. 17-21.

52
STREAM FUNCTION CONTOUR MAP AT T = 16.65

.033/ /"""\ . y/0\

CWTMPMN Tl carnw xntcrvhl w .norz-ooi rrti.u* .mmo-ooz

Fig. 12. The stream function as a function of position at dimensionless time

16.65 for the same parameters as in Fig. 10.

53
STREAM FUNCTI0N C0NT0UR MAP AT T = 16.79

caffiut PMH -.MOM TS .MOM CWTM INTERVAL W .7Z1M-001 PTIS.W- .MM4-00I

Fig. 13. The stream function as a function of position at dimensionless time

16.79 for the same parameters as in Fig. 10.

54
STREAM FUNCTI0N C0NT0UR MAP AT T = 16.93

Fig. 14. The stream function as a function of position at dimensionless

time 16.93 for the same parameters as in Fig. 10.

55
STREAM FUNCT10N C0NT0UR MAP AT T = 17.08

Fig. 15. The stream function as a function of position at dimensionless

time 17.08 for the same parameters as in Fig. 10.

56
STREAM FUNCTI0N C0NT0UR MAP AT T = 17.22

Fig. 16. The stream function as a function of position at dimensionless

time 17.22 for the same parameters as in Fig. 10.

57
CBNCENTRflTIBN CENTBUR MAP AT T = 16.65

Fig. 17. The solute concentration as a function of position at dimensionless

time 16.65 for the same parameters as in Fig. 10.

58
CMUUtlWM l.OMQ Tft S.SStt CWTOUt DnBML W .ZMZ PTtS.SU S.0M

Fig. 18. The solute concentration as a function of position at dimensionless time 16.79 for the same parameters as in Fig. 10.

59
(JKrajR PIW 1.0946 TO 3.7772 CWTOUR INTERVAL W .20824 PTO.OU 3.0834

Fig. 19. The solute concentration as a function of position at dimensionless time 16.93 for the same parameters as in Fig. 10.

60
CONCENTRATION CONTOUR MAP AT T = 17.00

Fig. 20. The solute concentration as a function of position at dimensionless time 17.08 for the same parameters as in Fig. 10.

61
CMTIUR PMH 1.0846 T1 9.8408 CWTftJR INTERVAL W .27481 PTI9.9)> 2.8411

Fig. 21. The solute concentration as a function of position at dimensionless time 17.22 for the same parameters as in Fig. 10.

62
STREAM FUNCTION CONTOUR MAP AT T = 21.96

.88875-002

Fig. 22. The steady state stream function as a function of position for

Rs/Rs — 3.6, VH/D = 4.17, and H/X = 1.552. The values of

the other parameters are given in Table 2. The initial state

for this solution was the solution periodic in time shown in

Figs. 10-21 with Rs/Rs =3.3

63
C0NCENTRRTI0N C0NT0UR MflP RT T = 21.96

OKTMJR FMM 1.1119 T9 9.9597 ONTM INTERVAL WF .28416 PTI9.9)* 2.8970

Fig. 23. The steady state concentration as a function of position for the same parameters as in Fig. 22.

64
STREAM FUNCTION C0NT0UR HflPRTTs 10.21

camutnoH -.22191 it .22191 onuur inibml r .*012-001 no.s>« -.29*27-001

Fig. 24. The steady state stream function as a function of position for

Rs/Rs - 3.6, VH/D = 4.17, and H/ A = 1.552 (same as Figs. 22-23).

e initial state for this solution was a perturbed exponential concentration field. The steady state solution has one half the

r^h

1 hH

gt:h

^ 0f^ the initial state * Comparison with Fig. 22 indicates that the steady state solution depends on the initial conditions.

65
C0NCENTRATI0N C0NT0UR MAP AT T = 10.21

WTSUR FR8H 1.1188 TS 3. SMS CSfTlUR IMTERVflL W .2058 FT (3.3)*

. 25. The steady state concentration as a function of

the same parameters as in Fig. 24.

3.2721

position
STREAM FUNCTI0N C0NT0UR MAP AT T = 10.15

CamUR FR8H -.29247 TS .29247 CANTOR INTERVAL 9F .58494-001 ^1(3.3)= .24439-001

Fig. 26. sceady st a te s tream Wt^ri as a function of position for Rs/R§ 5.03, VH/D = 4.17, and H/X = 3.104. The values

of the other parameters are given in Table 2.
CONCENTRATION CONTOUR MAP AT T = 10.15

OKT8UR FIWH 1.3084

Fig. 27. The steady state concentration as a function of position for the same parameters as in Fig. 26.

68
CMTftJR FMH -.90890 T8

STREAM FUNCTI0N C0NT0UR MAP AT T = 3.33

.29618-001

Fig. 28. The steady state stream function as a function of position for

Rs/R§ = 5.03, VH/D = 8.33, and H/X = 6.208. The values of

the other parameters are given in Table 2. This calculation

is identical to that of Fig. 26 except that the height has

been doubled.

69
OKTM PMM l.OOM Tl

C0NCENTRRT10N C0NT0UR MAP RT T = 3.33

i. ms

Fig. 29. The steady state concentration as a function of position for the same parameters as in Fig. 28.

"V

-

70
STREAM FUNCTION CONTOUR MAP AT T = 3.41

Gamut mat -.ran tb .rose camut interval w .ibuo rro.m* .uszs-oib

Fig. 30. The steady state stream function as a function of position for Rs/Rs = 1.33, VH/D = 20.0, H/X = 1.547, Ra = 0.127,

Rs - 6.78, and D/V = 0.015 cm. The values of the other

parameters are given in Table 2.

71
ONTaUR FRW 1 .0015 Tl 4.5888 CKHIt INTBQfflL V .99871 PT 13.31= 2.1128

Fig. 31. The steady state concentration as a function of position for the same parameters as in Fig. 26.

72
INTERFACE CONCENTRATION

o

Fig* 32. The steady state interface

wavelengths in the lateral

as in Fig. 31.

concentration in the melt for two

direction for the same parameters

73
CONCENTRATION

Z

Fig. 33. The average steady state concentration in the melt as a function

of the distance (in units of D/V) from the crystal-melt interface

for the same parameters as in Fig. 30. The dashed line

indicates the concentration in the absence of convection.

74
TEMPERATURE

(°C)

ETHANOL CONCENTRATION (wt %)

Fig. 34. Temperature and composition dependence of the density of dilute solutions

of ethanol in succinonitrile.

75
Fig. 35. Central pits formed during the solidification of succinonitrile containing

2.6 x 10~3 wt.% ethanol,

a. at a growth velocity of 2 pm/s.

76
77
TIME (minutes)

Fig. 36. T^.rf^p Dositions during the formation of a central pit, as a

the leading and trailing edges of the cellular structure are then shown.

78
CO

h- 33 3 44 —I cO 3 o o , T5 •H

|4

A CO i-43 3

cx

443 > O 3 •H I TJ X3 T33 1- e3 1 i-4 t4 5

443 3oX

G

(43 •3U

r4 44 3

>44 3 44 >44 • 3 M o

II >

J-I

g 3os co

4-) . 0) CO u G «

>4 >44 3 M

GM

0)

4-1 4-1 COcx

g

4) *H

CO 4-1 3 3 P o

rH e •w

a 3

4-> 3 C O i-l H14-1 0) X

a) 4-1 5 a

4-1 *H 3 g > o g *h

CO o LU —

44 44 3 c

>4

O

CX-H

44y

y 33 A oc CX-H >44 3 *H 3>

a> P S ex 3 3 3 •H 43 G 3 3 > 0. O X i-4 44 •H 3 1 Oy 3 -a

<44

O

O a) r-4

•3

C

i4

O 3 -H

i-4 44

>44 COy

TJ i4 C >4-1 CO "HT3

CO -H 3 i-4 a O

CO CO

CD

Oc 0) o XCO 4-4

>4-1 -H 3

4-1 G • M CO

H COO

<44 i-4 M Q, 'O i—I 0

rl (0 tI 3 -H CD H h

>44 0) CO

i-4 3 X

cO 0 4J G "H T) <11 S

•H >4

co 3 li

a)a o

|4 co a> >44

1-4

X (4

H 44 4)

CO 44 see CO H -H

4= «

44

MC

CUy44e

co co 2 e a s 0) >44 443 o

r4 *4 o

CD <U 44 X G

44 (-4 •H O

13 CD

e'- en

APPENDIX

Coupled Convective Instabilities at Crystal-Melt Interfaces

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

National Bureau of Standards

Washington, D.C. 20234

and

M. E. Glicksman and Q. T. Fang

Materials Engineering Department

Rensselaer Polytechnic Institute

Troy, NY 12181

The interaction between fluid flow and the crystal-melt interface is

of central importance in determining the solute distribution in the

solidified material, which in turn determines its properties. The effect

of convective flow on morphological stability has been recently reviewed

[ 1 ]

Calculations by Delves [2] showed that a flow parallel to the

crystal-melt interface could increase stability and give rise to traveling

waves on the interface. The Interaction of morphological instability

with a melt subject to thermosolutal instability has also been studied

[3J. In both these cases, morphological instability occurs even in the

absence of fluid flow, although in general the critical value of the

parameter for the onset of instability is changed by the flow.

Recent experiments [4-7] have demonstrated an interface instability

under conditions for which in the absence of flow, the crystal-melt

interface would be morphologically stable. In these experiments, a long

vertical cylindrical sample of high purity succinonitrile was heated by

passing an electrical current through a long, coaxial, vertical wire, so

that a vertical melt annulus formed between the coaxial heater and the

surrounding crystal-melt interface. The outer radius of the crystal was

maintained at a constant temperature below the melting point (58.1°C).

81
With this arrangement the temperature increases in the melt and decreases

in the crystal with distance from the crystal-melt interface, and

consequently the interface would be morphologically stable in the absence

of fluid flow.

Buoyancy dictates that the fluid flows upward near the heated wire

and downward near the crystal-melt interface. There are analytic solutions

of the fluid flow and temperature equations in which the flow velocity is

vertical and the flow velocity and temperature are functions of the radial

coordinate alone. Choi and Korpela [8] and Shaaban and Ozisik [9] have

calculated the critical Grashof number for axisymmetric instabilities of

the axisymmetric flow between two vertical, infinite, coaxial, rigid

cylinders held at different temperatures. We [6] have repeated these

calculations by a different numerical technique and for physical properties

appropriate to succinonitrile (Prandtl number of 22.8). The flow between

two vertical infinite coaxial cylinders containing succinonitrile is

unstable to an axisymmetric perturbation above a Grashof number of 2150

and the wave speed of this perturbation is comparable to the unperturbed

flow velocity.

When these calculations are carried out with the outer rigid cylinder

replaced by a crystal-melt interface, quite different results are obtained.

The onset of instability occurs by an asymmetric mode at a Grashof number

of 176, which is in agreement with experiment. The linear stability

calculations indicate that the density change on solidification and the

crystal-melt surface tension are of little importance in determining the

instability. The calculations predict that the period of oscillation

(time for the wave to traverse one wavelength) is proportional to the

fifth power of the gap width (the difference between the radius of the

82
crystal-melt interface and the wire radius). This has been verified

experimentally with the period of oscillation ranging from 300 to 60000 s.

In order to understand better the "coupled, mode" of instability, we

use linear stability theory to analyze a simpler geometry, viz., we

consider the stability of the parallel flow between a vertical crystalmelt interface and a vertical wall held at a temperature above the melting

point of the crystal. Three modes of instability occur: (1) a buoyant

mode, (2) a shear mode, and (3) a coupled crystal-melt mode. The first

two modes are well known from previous research [10] on the instability of

the flow between two vertical walls. The behavior of these modes as the

Prandtl number varies will also be described.

Theory

We consider an (x, y, z) Cartesian coordinate system such that in the

unperturbed state the region x < 0 is crystalline, the region 0 < x < L

is molten, and at x = L there is a rigid isothermal wall, whose temperature

is greater than the melting point of the planar crystal by an amount AT.

We assume that the force of gravity acts in the negative y direction and

that all quantities are independent of the z-coordinate. We measure

lengths in units of L; temperature T in units of AT; time t in units

of L

2 /v, where v is the kinematic viscosity, and velocities in units of a

reference velocity U, which will be specified subsequently. The

dimensionless equations for the fluid velocity u and the temperature are,

in the Boussinesq approximation

V» u = 0,

( 3u/3t) + Re (u»V)u = - Vp + V

2u + (G/Re)Tj,

(3T/3t) + Re u»VT = (V 2T)/P,

( 1 )

( 2 )

(3)

83
where Re = UL/v is the Reynolds number, p is a dimensionless pressure,

G = gaATL^/v 2 is the Grashof number, j is a unit vector in the ydirection, P is the Prandtl number, g is the gravitational acceleration,

and a is the coefficient of thermal expansion. The temperature Tg in

the crystal satisfies

(3T

s /3t) = V

2T/Ps

, (4)

where Pg = P</<g = v/<g and < and Kg are the thermal dif fusivities of

the melt and crystal, respectively.

The boundary conditions at the crystal-melt interface are that the

tangential component of the fluid velocity vanishes and that the normal

component satisfies

u»n = - e(v»ri). (5)

Here n is the normal to the crystal-melt interface, e = -1 + pg/pL,

pg and p^ are the crystal and melt densities, respectively, and v

is the local solidification velocity. At the crystal-melt interface, the

temperature is continuous and equal to the equilibrium temperature, viz.,

TSI - TH - -f K. (6)

where the subscript I denotes evaluation at the crystal-melt interface.

Temperature is measured relative to the dimensionless melting point T^/ AT

of the planar crystal-melt interface, T = Tmy/(LHvAT)

y is the

crystal-melt surface free energy, Hv

is the latent heat of fusion per

unit volume of solid, and K is the mean curvature of the crystal-melt

interface. Conservation of energy at the crystal-melt interface requires

A(v-n) = (-kr VT + VTS

) »n, (7)

where A = vHvRe/k s AT and kr is the ratio of the thermal conductivity

of the melt kL to that of the crystal kg.

84
At the rigid wall y

= 1,

u = 0 (8)

and

T - 1. (9)

In the crystal we require that far from the crystal-melt interface, the

temperature gradient is constant and assumes a value such that the

unperturbed planar interface is stationary.

A solution of the differential equations satisfying the boundary

conditions is

u

yo = [1/4 - (x - 1/2) 2

] [ (G/ 6Re) (x-1/2) + 61], (10a)

T0 = x, (10b)

TSo = krX, (10c)

Po = [G/2Re - 12I]y, (lOd)

where I is the net flow in the y-direction, i.e. , the integral of u

y

from

x = 0 to x = 1. The subscripts x and y denote the x and y components of

u and the subscript o denotes the unperturbed or base flow and temperature

fields.

We examine the stability of the flow with respect to small

perturbations. We decompose the fluid velocity, temperature fields, and

crystal-melt interface shape xj into an unperturbed and perturbed part,

and assume that the y and t dependences of the perturbed quantities are

of the form F(y, t) = exp (iooy + at), where w is a spatial frequency and

a = ar + ia-t is a complex number which determines the temporal

behavior of the system. If o r > 0 for any value of w, the system is

unstable. We write

85
ux

= Wx (x)F, (11a)

u

y

= Uyo + Wy(x)F, (lib)

T = x + T(x)F

,

(11c)

Ts = kj.x + Tg (x)F

,

(lid)

P = P 0 + Pl(x)F, (lie)

x x = 6F, (Ilf)

where 6 is the amplitude of the perturbation of the crystal-melt interface.

The linear ordinary differential equations for the perturbed

quantities, Wx , Wy, pj_, T, and Ts can be written as

DWX + iwWy = 0, (12a)

AWX + iuDWy + Dpj = o. (12b)

-Re(Du

yo

)Wx + (D 2 - A)W

y

- iwpj + (G/Re) T = 0, (12c)

-PReW

x

4- (D 2 - A

T

)T = o. (12d)

(D 2 - w

2 - P

s a)Tg = 0, ( 12e)

where D = (3/3x), A = w

2 + a + iooReu

yo , and A,j, = w

2 + P(a + iwReu

yo

).

The perturbed temperature field in the crystal, Tg , can be found analytically

and is of the form exp [(u)2 + Pg<j)^ 2x].

It is desirable for purposes of numerical computation to prescribe the

boundary conditions at x = 0 rather than at the crystal-melt interface x = xj.

This is easily accomplished in linear theory because any function, e.g., f(x),

evaluated at x = xj = 6F can be written as f(xj) = f(0) + 6F(3f/3x), where

the partial derivative is evaluated at x = 0. Upon eliminating Tg and 5,

we find the following boundary conditions for the fluid variables, Wx ,

Wy, and T at x = 0:

(1 + rw

2 )ReW

x

- eaT = 0, (13a)

(1 + Iw2 )W

y

- (Du

yo

)T = 0, (13b)

[ ( 'jj 2 + P c a) 1/2 (k + Too2

) + aA/Re] T - k (1 + rw

2 )(3T/3x) = 0. (13c)

b r r

86
These boundary conditions are furhter simplified by taking T = e = 0.

It is not surprising and numerical calculations have verified that these

are excellent approximations.

The boundary conditions at the rigid isothermal wall at x = 1 are

Wx

- Wy = T = 0. (14)

In order to understand the physical mechanism of instability, it is

useful to compute the average kinetic energy of the perturbed flow field.

The average kinetic energy in the disturbances is obtained by integrating

the quantity (Real U-,)^ + [Real(u - uWrt )]^, as given by eq. (11), over a

x y yo

basic cell having unit width in the x-direction and one wavelength

(2x/a)) in the y-direction (Real indicates the real part of the complex

functions ux and u

y

) . At the onset of instability (or = 0) the rate

of change of the kinetic energy in the disturbance vanishes, and from

eqs. (12b) and (12c), it follows that

» . “<|DWx|

2 +

| DW

y|

2 + “

2

| Wx| 2 + “

2

J Wyl 2 + (G/Re) ^Real(IW

y

*)>

- Re(Duyo ){Real(WxW

y*)> - Real[W

y

(0)DW

y

*(0) ] = 0, (15)

where the < > indicates integration from 0 to 1 in the x-direction and

the * indicates the complex conjugate. .The sum of the first four terms

represents energy lost to viscous dissipation and is always negative. It

is balanced by the sum of the remaining three terms. We denote the terms

proportional to G and (Duyo ) as buoyant and shear terms, respectively.

The remaining term W

y

DW

y

evaluated at x ** 0 is due to the crystal-melt

interface and would vanish for a rigid interface. In the results to be

presented subsequently, we have found modes of instability for which one

of the terms, i.e., buoyant, shear, or interface, is dominant.

87
The numerical methods used to solve the linear eigenvn ie problem

defined by the differential equations [eqs. 12a-d] and the boundary conditions

[eqs. 13-14] are all similar to those described previously [3]. We

usually vary G and aj., keeping o r and all other parameters constant,

until a solution of the differential equations is found which satisfies

the boundary conditions. The program SUPORT [11] is used to solve the

differential equations, and the program SNSQE [12-13] is used for the

non-linear iteration procedure. Calculations of the critical Grashof

number for the flow between two rigid plates are in agreement (to 4

significant figures) with Table 3 of Korpela, Gozum, and Baxi [14].

Numerical Results

For the calculations reported here, we have taken 1=0 (see eq. 10a)

so that the net flow in the y-direction vanishes, and we choose our

reference velocity U such that Re = G. Further we take T = e = 0

*. . and kr = P/Pg = 1, i.e. , we neglect both the crystal-melt surface energy

and the density change during solidification, and we assume that crystal

and melt have the same thermal properties. In most of our calculations

we set A = vH^Re/kgAT = HvgoL'Vk s v = 1.017(10^), which corresponds to

succinonitrile with L = 0.25 cm. The corresponding value of A for

lead with L = 0.25 cm is A = 625. In general we calculate the onset

of instability viz. a r = 0. For the Prandtl number P and all other

parameters fixed, we calculate the Grashof number G and as functions

of the spatial frequency w for o r = 0. Typical results for succinonitrile

P = 22.8 are shown in Figures 1-4. As previously discussed the instabilities

can be characterized as buoyant, shear, or interface. For P = 22.8, the

mode with the lowest Grashof number shown in Figure 1 is primarily an

interface mode. The next highest branch represents a buoyant mode,

88
followed by a shear mode at higher values of G. The interface mode is

shown in more detail in Figure 2, where values of as a function

of w are also included. Note that < 0, meaning that the interface

deformation propagates upward, whereas the base flow is downward near the

interface. The buoyant mode is shown in Figure 3 and has a value

three orders of magnitude greater than the interface mode. There is also

a shear mode with a minimum Grashof number of 6200 shown in Figure 4.

| The variation of the shear and buoyant modes of instability as a

function of Prandtl number for the flow between two vertical rigid walls

has been described by Gershuni and Zhukhovitskii [10]. In Figures 5-7, we

show the behavior of the various modes as a function of Prandtl number

for the crystal-melt interface. Figure 5 shows the minimum Grashof number

(as a function of w) as the Prandtl number varies. In Figures 6 and 7,

we have plotted a^ and w, respectively , as a function of Prandtl

number; the a± and w curves correspond to minimum values of the

Grashof number as a function of w. The buoyant mode, which has a large

value of 0^ and a large wave speed (comparable to the unperturbed

flow velocity), is hardly affected by the presence of the crystal-melt

interface, and is essentially identical to the buoyant mode for a rigid

boundary. For high values of the Prandtl number the Grashof numbeir for

the interface mode is two orders of magnitude lower than the Grashof

number for the shear mode of instability. As the Prandtl number decreases,

the interface and shear modes approach each other, while the Grashof

number for the buoyant modes rises abruptly and is much larger than the

Grashof number for the shear mode for Prandtl numbers less than 10.

89
At a Prandtl number of 2.5, a plot of Grashof numbei ui, ,t i unction

of co, shown in Figure 8, has two minima, which correspond to the two

Grashof numbers appearing in Figure 3. The size of each of the last

three terms appearing in eq. (15) are shown in Figure 9; that is, the

shear terra (dashed line), the buoyant term (dot-dashed line), and the

interface term (solid line). Positive values indicate a destabilizing

influence since the term tends to increase the kinetic energy of the

disturbance to the main flow, whereas negative values indicate a

stabilizing influence. At wavenumbers less than unity, the interface

term dominates the instability, while for wavenumbers greater than two,

the shear term is driving the disturbance.

As the Prandtl number is decreased. Figures 10-17, the second minimum

in Grashof number, corresponding to the shear mode, disappears (cf. Figure

3). Both shear and interface- terms are significant for the remaining

> . minimum. As P is further reduced, the wavenumber at the minimum Grashof

number increases and the effect of the interface term becomes less and

less significant. For Prandtl number P = .01, Figures 16 and 17, the

crystal-melt interface term is negligible and the shear mode drives the

disturbance.

In Figures 18-20, we show results for lead with L = 0.25 cm, A = 625,

P = 0.0225, P s = 0.012, kr = 0.535, and y

= e = 0, for both the

crystal-melt interface and a rigid boundary. The Grashof number as a

function of w is essentially independent of the type of boundary. For a

rigid boundary = 0, while for the crystal-melt interface = -2.7

at the minimum value of the Grashof number of 7630. This corresponds to a

period of oscillation of 60 s.

90
References

1. S. R. Coriell and R. F. Sekerka, PhysicoChemical Hydrodynamics 2, 281

(1981).

2. R. T. Delves, Crystal Growth (edited by B. R. Pamplin) Vol. 1, 40,

(1974).

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

Crystal Growth 49, 13 (1980).

4. J. I. Mickalonis, M. S. Thesis, Materials Engineering Department,

Rensselaer Polytechnic Institute, Troy, NY (1982).

5. M. E. Glicksman and J. I. Mickalonis, Proceedings of the 16th

Southeastern Seminar on Thermal Sciences, Miami, Florida, 1982.

6. S. R. Coriell, R. F. Boisvert, J. I. Mickalonis, and M. E. Glicksman, Adv.

in Space Research (proceedings of the 24th COSPAR Meeting) 3, 95 (1983).

7. Q. T. Fang, M. S. Thesis-, Materials Engineering Department, Rensselaer

Polytechnic Institute, Troy, NY (1983).

8. I. G. Choi and S. A. Korpela, J. Fluid Mech. 99, 725 (1980).

9. A. H. Shaaban and M. N. Ozisik, 7th International Conference of Heat

Transfer, Munich, Germany (1982).

10. G. Z. Gershuni and E. M. Zhukhovitskii , Convective Stability of

Incompressible Fluids, Keter, Jerusalem, 1976.

11. M. R. Scott and H. A. Watts, SIAM J. Numerical Anal. J4, 40 (1977).

12. SLATEC Common Math Library, The Program SNSQE was written by K. L.

Hiebert and is based on an algorithm of Powell [13].

13. M. J. D. Powell, in: Numerical Methods for Nonlinear Algebraic

Equations, ed. P. Rabinowitz, Gordon and Breach, NY (1970).

14. S. A. Korpela, D. Gozum, C. B. Baxi, Int. J. Heat Mass Transfer 16 ,

1683 (1973).

91
I

Planar Crystal—Melt Interface

Succinonitrile (F*=22.8)

u

Fig. 1. The Grashof number, G, at the onset of instability as a function

of the spatial frequency, co, of a sinusoidal perturbation for

Prandtl number, P, of succinonitrile, viz., P = 22.8. The

three curves correspond to different modes of instability.

92
400.0

800.0

1200.0

1600.0

2000.0

2400.0

o

d.

Planar Crvstal-Melt Interface

Succmonitrile (P=22.8)

o4

'f

COo

CDO

3

8

OO

1 1 1 1 or

0.00 0.25 0.50 0.75 1.00 125

CO

Fig. 2. The Grashof number (solid curve, left ordinate) and the imaginary

part of the time constant, o±, (dashed curve, right ordinate)

as a function of the spatial frequency, u, of a sinusoidal perturbation for Prandtl number of 22.8. The solid curve is

identical to the lower curve of Fig. 1.

93
{

Planar Crvstal-Melt Interface

Fig. 3. The Grashof number (solid curve, left ordinate) and the imaginary

part of the time constant, aj[, (dashed curve, right ordinate)

as a function of the spatial frequency, w, of a sinusoidal

perturbation for Prandtl number of 22.8. The solid curve is

identical to the middle curve of Fig. 1.

94
Planar Crystal—Melt Interface

Succinonitrile (P=22.8)

Fig. 4. The Grashof number (solid curve, left ordinate) and the imaginary

part of the time constant, o±, (dashed curve right ordinate)

as a function of the spatial frequency, oj, of a sinusoidal

perturbation for Prandtl number of 22.8. The solid curve is

identical to the top curve of Fig. 1.

95
PLRNRR CRYSTRL-MELT INTERFRCE

P

Fig. 5. The minimum Grashof number (as a as a function of Prandtl number,

number, three different modes of

function of spatial frequency)

For large values of the Prandtl

instability occur.

96
PLANAR CRYSTAL-MELT INTERFACE

Fig. 6. The imaginary part of the time constant, o±, (corresponding

to the minimum values of the Grashof number as a function

of u>) as a function of Prandtl number.

97
PLRNRR CRYSTRL-MELT INTERFRCE

P

Fig. 7. The spatial frequency, w, (at the minimum value of the Grashof

number) as a function of the Prandtl number.

98
PLRNRR CRYSTAL-MELT INTERFRCE

CD

Fig. 8.

12000 -i

11000-

10000 -

9000 -

8000 -

7000 -

6000 -

5000 -

4000 -

3000 -

2000-

1000-

0

--0

P-2.5

n r 0.5 1 2

1—

2.5

T

3 3.5

CO

The Grashof number, G, at the onset of instability as a function

of the spatial frequency, o>, of a sinusoidal perturbation for

Prandtl number, P, of 2.5.

99
POWER

PLflNRR CRYSTflL-MELT INTERFACE

Fig. 9. The interface (solid curve), shear (dashed curve), and buoyant

(dot-dashed curve) contributions to the rate of change of the

average kinetic energy (power) of an instability as a function

of spatial frequency for Prandtl number of 2.5.

100
PLRNRR CRYSTRL-MELT INTERFRCE

CD

12000

11000 -

10000 -

9000 -

8000 -

7000 -

6000 -

5000 -

4000 -

3000 -

2000 -

1000 -

0

--0

P-2.0

T

1 1 1 1 1 1 1.5 2 2.5 3 3.5

CO

Fig. 10. The Grasilof njunber, G, at the onset of instability as a function

Pran^l

SP

i

^equency, u>, of a sinusoidal perturbation for Prandtl number, P, of 2.0.

101
POHER

PLRNRR CRYSTRL-RELT INTERFRCE

Fig. 11. The interface (solid curve), shear (dashed curve), and buoyant

(dot-dashed curve) contributions to the rate of change of the

average kinetic energy (power) of an instability as a function

of spatial frequency for Prandtl number of 2.0.

102
PLRNRR CRYSTRL-MELT INTERFROE

Fig. 12. The Grashof number. G, at the onset of the Instability as a forPrandU n^r^ofTo?"'

7 ’ "* °f 3 SlnuSOldal Perturbation

103
POWER

PLRNRR CRYSTRL-MELT INTERFRCE

Fig. 13. The interface (solid curve), shear (dashed curve), and buoyant

(dot-dashed curve) contributions to the rate of change of the

average kinetic energy (power) of an instability as a function

of spatial frequency for Prandtl number of 1.0.

104
PLRNRR CRYSTflL-MELT INTERFACE

12000-1

11000 -

10000 -

9000-

8000-

7000-

CJD 6000 -

5000-

4000-

3000-

2000 -

1000 -

0 -

0

“I 1 1 1 *1 11 0.5 1 1.5 2 2.5 3 3.5

CO

Fig. 14. The Grashof number, G, at the onset of instability as a function

of the spatial frequency, uj, of a sinusoidal perturbation for

Prandtl number, P, of 0.5.

105
POWER

PLANAR CRYSTAL-MELT INTERFACE

Fig. 15. The interface (solid curve), shear (dashed curve), and buoyant

(dot-dashed curve) contributions to the rate of change of the

average kinetic energy (power) of an instability as a function

of spatial frequency for Prandtl number of 0.5.

106
PLANAR CRYSTAL-MELT INTERFACE

Fig. 16. The Grashof number, G, at the onset of instability as a function

of the spatial frequency, u>, of a sinusoidal perturbation for

Prandtl number, P, of 0.01.

i .

107
POWER

PLRNRR CRYSTRL-MELT INTERFRCE

Fig. 17. The interface (solid curve), shear (dashed curve), and buoyant

(dot-dashed curve) contributions to the rate of change of the

average kinetic energy (power) of an instability as a function

of spatial frequency for Prandtl number of 0.01.

108
I

Planar Interface

Lead (FMJ.0225)

O

Fig. 18. The Grashof number, G, at the onset of instability as a function

of the spatial frequency, to, of a sinusoidal perturbation for

Prandtl number, P, of lead, viz., P = 0.0225. The solid curve corresponds to a rigid interface while the dot-dashed curve corresponds to a crystal-melt interface.

109
G

(xlO”

9)

Planar Crystal—Melt Interface

Lead (P=0.0225)

Fig. 19. The Grashof number (solid curve, left ordinate) and the imaginary

part of the time constant, a£ , (dashed curve, right ordinate)

as a function of the spatial frequency, oj, of a sinusoidal

perturbation for Prandtl number of 0.0225. The solid curve is

identical to the dot-dashed curve of Fig. 18.

no
G

(xlO

- *)

Planar Rigid Interface

Lead (P=0.0225)

O

Fig. 20. The Grashof number as a function of the spatial frequency, w,

of a sinusoidal perturbation for Prandtl number of 0.0225. The

solid curve is identical to the solid curve of Fig. 18. The

Imaginary part of the time constant vanishes.

Ill

Task 3

Measurement of The High Temperature Thermophysical

Properties of Tungsten Group Liquids and Solids

D. W. Bonnell

Materials Chemistry Division

Center for Materials Science

SUMMARY

The General Electric Space Systems Division (GE) developed prototype

"gulp" calorimeter has been tested in joint Rice University (RICE) and The

National Bureau of Standards (NBS) experiments by dropping levitated tungsten

solid at 2800 K, and found to function mechanically to perfection. The steps

necessary to establish this device as a working calorimeter for both ground

and potential space applications is discussed. The central problems remaining

are the addition of calorimetric temperature measurement hardware precise to

0.001 K and resistant to electromagnetic interference, the development of

techniques to achieve 0.1 percent enthalpy determination uncertainty with

expected 18 K temperature rises, and direct calibration of the calorimeter.

The concentration of effort to measurements at the melting point of

liquid tungsten have yielded the first reported direct measurement of the heat

of fusion of tungsten. The value obtained was 53.0 ±2.3 kJ/mole (12.7 ±0.5

kcal/mole) and, in comparison with estimates and previous indirect methods, is

considered to be a current best value.

113
INTRODUCTION

The primary objective of this task is to develop, in conjunction with

General Electric Space Systems Division (GE) and Rice University (RICE), the

techniques, methodology, and experimental apparatus for measuring thermophysical properties (e.g. heat of fusion, enthalpy increment, heat capacity) of

materials at extremely high temperatures. Electromagnetic levitation of

conductive samples using auxiliary heating techniques such as electron-beam

heating has been coupled to a drop-type block calorimeter system. Specific

application to tungsten has been a prime area of interest because it is the

highest melting element. Measurements of tungsten are crucial for estimation

of the thermophysical properties of othe elements by correlation. The third

long period transition elements have all proved to be difficult to measure

directly, and little data is available for VIB and VI IB elements in spite of

their importance both scientifically and for engineering purposes. Accurate

data are necessary to encourage further theoretical investigation of the

liquid state at high temperature. Modern computer capacities promise new

models based on detailed evaluation by molecular/atomic dynamics [1], but will

require validation by experimental determination of definitive thermophysical

properties.

The technical difficulty of thiseffort has provided a measure of the

dividing line between experiments which can be performed in the terrestrial

environment, and those which require the absence of gravity for successful

accomplishment. Continued advance of ground based efforts seems at present

the clearest means to successful future experiments in space, as considerable

hardware development is still necessary to take advantage of the low-gravity

envi ronment.

114
GULP CALORIMETRY

In the initial development of the levitation technique, it was recognized

that the technique was a method to gain the advantage of non-material support

in a one gravity field, giving many of the advantages of the space microgravity environment. The technique's application to the measurement of the

heat content of levitated materials added a separate requirement, whose

simplest solution is the "drop" calorimeter technique, in which gravity

provides a rapid and hence approximately isothermal transfer of the sample

from the high temperature state into contact with a calorimetric mass whose

temperature is near the reference state. A significant problem in conducting

similar high temperature calorimetry in space is the transfer of a sample to

the calorimeter without the aid of gravity and without giving up the advantages

of the containerless environment. The problem is not insuperable. Acoustic

positioning techniques are capable of manipulating levitated objects [2-4] but

the acoustic techniques still appear to have severe materials problems to

overcome at the very high temperatures which are of greatest interest, and are

most difficult to achieve in a controlled fashion on the ground. Electrostatic

techniques are possible [5] and can be used in a vacuum, an important advantage

where some methods of auxiliary heating need to be used. Less well conceived

and apparently untested are techniques using an electromagnetic pulse to move

a sample. Such a method would be difficult to develop in a one-g environment

in conjunction with RF levitation. Since the design of ground based levitation

coils is still somewhat of an art, another method was sought. In 1974, this

investigator made the suggestion that it would be more straightforward to

attempt to move the calorimeter instead, and J. L. Margrave of Rice University

coined the name "gulp" calorimeter for this technique.

115
The advent of Space Shuttle experiment opportunities has made this

problem an important one, as high temperature calorimetry is one of the most

obvious kinds of science certain to benefit from the advantages of space for

experimentation. General Electric Space Laboratories has undertaken a ground

pilot development of a "gulp" calorimeter. NBS expertise has been sought in

this connection in an advisory capacity, since use of the device on the ground

would almost certainly occur within the framework of the current GE/RICE/NBS

levitation effort. The final development by GE [6] is a excellent application

of mechanical engineering ingenuity, and closely mimics the behavior a spacebased device must have. Certain compromises were necessary. A primary one is

the physical size of the calorimeter. Traditional drop calorimeters are

designed for total heat rises of the order of one degree to minimize the

temperature difference between calorimeter and surroundings. The mathematical

basis for data analysis of the thermal behavior of an isoperibol (drop-type)

calorimeter [7,8] invokes a linear approximation of. the combined effects of

the exponential Newton's law of cooling (conduction/convection) and the power

law radiation transfer process. This approximation is generally excellent

(better than 0.0005 average percent deviation per 10 second measurement period

over 10-20 minute intervals where calorimeter block/jacket temperature differences are 1-2 degrees [8]). Th*is linear relationship is used to determine

when the calorimeter has equilibrated with the heat input from the sample and,

more importantly, to correct for the heat contribution to/from the calorimeter

during the equilibration time of calorimeter with sample. For typical

equilibration times of ca. 3000 seconds, worst case error from this source is

less than 0.1 percent and thus negligible in measurements of a few tenths

percent overall accuracy.

116
The GE calorimeter prototype was designed to be tested on the ground, and

to be a reasonable model (i.e., physical overall dimensions and system total

mass) for a flight system. In addition, capture times were required to be as

short or shorter than ground experimental arrangements, to minimize radiation

losses from the sample. A pnuematic actuation system was chosen by GE as the

most reasonable system capable of the required speed of operation with

significant masses. This choice limited the moving mass of the calorimeter to

the order of one kg, approximately 5 percent of the current ground

calorimeter. Predicted heat rises for 10 gm samples of freezing liquid

tungsten at 3695 K are about 18 K. No isoperibol calorimeter has apparently

been reported using such large heat increases. By the above analysis, one

would assume that existing data analysis techniques would cause the error from

such large calorimeter/jacket temperature differences to contribute at least

one percent to the uncertainty of the heat measurement. In addition, widespread practice has determined, for calorimeters with 1 atmosphere gas

pressure (at 1 gravity) in the jacket region, that a jacket/calorimeter gap of

the order of 1 cm yields the best uniformity of convection driven temperature

transfer. This was a difficult structural requirement, and was ignored for

prototype purposes, as this system is expected to operate in vacuum. Operated

in atmosphere the gulp calorimeter system would require careful calibration to

establish its true heat capacity behavior and the thermal effects of movement

as a function of differential temperature. Even restriction to operation in

vacuum below the 10 Pa (0.0001 atm) point, where gas thermal conduction

becomes much less significant, will still require attention, as the radiation

transfer, and incidental mechanical connection conduction will not be

negligible. At the moment, there is just too little experimental data

available to make such corrections a priori.

117
There are two possible solutions to this problem: 1) new numerical

techniques for evaluation of the true thermal exchange process; and/or 2)

controlling the jacket temperature to track the block temperature - a form of

adiabatic calorimetry. For the first, both the radiation and conduction

processes have well defined functional forms, and a direct determination of

the net transfer coefficients for the two processes should be relatively easy

to model from experimental calibrations, although separating the two processes

may prove difficult. The hardware solution, using the surface temperature of

the calorimeter as a control signal to heaters controlling all "jacket"

surfaces near the calorimeter at the calorimeter surface temperature is quite

attractive. This adiabatic wall process is used extensively in calorimetry

where large temperature excursions are routine (e.g. low temperature calorimetry

and differential scanning calorimetry) and involves only existing commercially

available technology. Although the gulp calorimeter system is considerably

more massive, and has both a much more complicated physical geometry and more

complicated thermal pathways than the above examples, a combination of the

above two methods should make high accuracy calorimetry possible with the

prototype design.

The GE designed calorimeter uses a pnuematic ram with special high speed

valves to move the calorimeter from its rest position to the capture region.

For ground based testing, this location is just below the work coil, with

clearance for the momentum opened calorimeter gate. Timing and capture trials

with levitation heated samples of tungsten were conducted with the prototype

instrument as a joint GE/RICE/NBS experimental effort. The calorimeter ram

and gates were instrumented by GE to allow oscillograms of the time trajectory

of the calorimeter to be obtained and to detect when, in that trajectory, the

block radiation gates were fully open [6]. A simple time delay relay was

118
inserted in the signal line from the drop detector (originally constructed for

tungsten experiments) which triggers the capture cycle. The delay was adjusted

to retard the pnuematic impulse about 0.02 sec until the timing oscillogram

intersected the falling sample location in the middle of the gate open time.

The block radiation gate is counterweighted to open inertially as the calorimeter accelerates, and to close by momentum as the calorimeter stops at

maximum excursion. The calorimeter system adapter to the levitation chamber

was designed to limit excursion to a location approximately 4 cm below the

levitation point (about 2 cm below the drop detector location), allowing

clearance for the coil and pedestal assembly [for experimental details, see

reference 9]. The calorimeter mount has a pair of overlapping radiation

shutters which are opened by cams moving with the calorimeter. The calorimeter

is just beginning to open the block gates as it passes the point of opening

the main external shutters, and the block gate open time of more than 0.05 sec

allowed several centimeters leeway in sample location at "capture" (defined

for this purpose as the time the sample crossed the gate hinge plane and would

be retained by the closing gates, rather than possibly having some of the

sample ejected). A capture point corresponding to a drop distance of about 7

cm (0.12 sec) was selected. Actual testing with a hot levitated sample of

tungsten (no e-beam heating) was successful. This system halves the drop time

of the current system, and would presumably improve drop success significantly.

It is still to be determined how robust the system is with missed drops, and

how well the timing repeats as the system wears.

The remaining development needed with this new system is the actual block

temperature determination. Discussions with GE led originally to the selection

of a miniaturized commercial 100 ohm platinum resistance thermometer. It

appeared that a custom version of the Hewlett/Packard quartz oscillator based

119
thermometer used for the existing system would have been required to meet

geometry constraints, allow passage through a vacuum wall, and accommodate the

calorimeter movement. The installation used for the existing RICE/GE system

is unique to a nonmoving system, and would have been extremely difficult to

adapt to the gulp calorimeter (requiring a hollow ram and other changes).

Also, the 10 times larger increase in temperature rise anticipated relaxes the

measurement constraint by the same factor. Although no commercial sources of

electronic readout units capable of 0.001 K precision have been located, it

should not be necessary to resort to potentiometric techniques to achieve this

temperature sensitivity. However, a significant problem requiring some

development exists here, as very small voltages will be necessary to avoid

self-heating error and to minimize input energy to the calorimeter. In

addition, DC determinations of the resistance of the thermometer are quite

likely to be seriously disturbed by electromagnetic interference (EMI) for the

RF heater. Thus, AC excitation and detection techniques will be necessary to

determine the resistance of the thermometer. Expected heat inputs for the

highest temperature experiments is of the order of 8000 J. Equilibration time

for the gulp calorimeter will likely be between 1500 and 6000 seconds. For

the upper limit estimate, if the thermometer is to contribute less than two

percent of the total heat to the block, the maximum RMS voltage for excitation

will be 1.6 volts. This is convenient for modern operational amplifiers, and

a low (few 100 Hz) frequency AC self-balancing bridge circuit of this sensitivity should be within the state-of-the-art. The design may be non-trivial

because of the electromagnetic interference rejection requirement. Other

sensors, (e.g. thermocouples or thermistors) have been used, but are generally

not considered to be as suitable (e.g., because of hysteresis). Temperature

sensitive integrated circuits (e.g., Analog Devices AD590 or equivalent) might

120
be attractive temperature sensors, given that specialized apparatus to achieve

the required sensitivity must be constructed.

The next steps to complete the development of the "gulp" calorimeter

concept can be summarized:

1) Develop temperature measurement technique to required sensitivity with

adequate EMI rejection.

2) Calibrate calorimeter by existing techniques, with specific emphasis

on establishing validity of data analysis and calorimeter performance.

3) Incorporate changes as dictated by the results from 1) and 2), and

repeat until calorimeter system heat capacity/temperature response is

known over operating temperature range to 0.1 % level.

The unique RICE/GE interaction would seem to be ideal for this task, as

the capabilities of RICE to accomplish the testing required in 2) appear a

natural complement to the development expertise of GE in the completion of

this task.

HEAT OF FUSION OF TUNGSTEN

The primary thrust of this project has been to assist RICE and GE in

developing methodology for high temperature thermophysical property measurement utilizing the GE coupled RF Levitation/Electron-beam heating facility

with a RICE supplied and operated calorimeter system. A prime problem in this

work has been sample temperature determination, as the location of the levitation process in vacuum, the x-ray hazard posed by electron-beam heating and

the high vapor pressure of the target material, tungsten, at its melting point

all impose stringent requirements on optical access, and make commercial

pyrometers impractical. For routine monitoring, GE developed an imaging

system based on a silicon diode video camera, with selection electronics to

allow using individual diodes as intensity detectors. Recent efforts [10]

121
have established the techniques for converting this type of device to a

practical pyrometer system, with tracability to the International Practical

Temperature Scale [11]. The combination of the apparent inability of the

newly redesigned levitation coil to maintain liquid tungsten, the long cycle

time of the calorimeter, the inherent noise characteristics of the existing

camera system, and the difficulty of real time compensation for window obscuration have made actual temperature measurements extremely difficult. For that

reason, and in order to isolate problems in this effort, it was decided to

concentrate efforts on measurements at the melting point, with the specific

aim of obtaining the heat of fusion of liquid tungsten.

Previous reports of this parameter have been primarily based on the

Tamman Rule [12] (AH /T =2.3 cal/mole.K {9.6 J/mole.K}, 1 calorie = 4.184 mm

joule) used in the standard references [e.g., references 13 and 14]. This

gives the value, 36 kJ/raole, for the heat of fusion using the best estimate

value for the fusion temperature, 3695 K, determined by Cezairliyan [15].

Dikhter and Lebedev [16] reported AH

m of 80 ± 4 cal/gm, corresponding to 62 ±

3 kJ/mole. This value is probably too high, as it predicts a value for the

entropy of melting (AS

m

) of 17 J/mole.K, a value almost 25 percent higher than

the currently highest known value for a transition metal (molybdenum, 12.9

J/mole.K. AS

m for tungsten is expected to be somewhat higher than for molybdenum, although too much of the data for the transition metals is estimated to

draw strong conclusions. One would expect, in this region of the transition

metal portion of the periodic table, that the second- to-third long period

change in AS

m would be less than a 25 percent increase. Thus, this value is

viewed as an upper bound. The report by Wouch, et al., [17] is based on the

application of a cooling model to experimental data with rather large temperature error bounds. The model ignores supercooling, and yielded a heat of

122
fusion of 47.7 kJ/mole, with a corresponding entropy of fusion of 12.9 J/mole.K.

From their reported fit to the model, a change in AH

m of 8 kJ/mole occurs with

an increase of only 0.5 times the standard deviation of the best fit and ± 32

kJ/mole at the 2o point is the variance in the fitted AH

m

- Although somewhat

insensitive, this model approach has produced the best estimate of AH^ prior

to the direct measurements of this task.

Results of drops estimated to be 50 percent or more molten out of 25

drops for this work are given in Table 1. The estimates of percent melt for

partially fused samples were based on visual inspection. Sectioning is

planned to refine these values, but some of these samples may be reused in

subsequent efforts, as partially melted samples are generally more successful.

The completely melted sample was weighted 5 times the partially melted data in

recognition of the uncertainty in the partial melt estimates. Two methods

were used to derive essentially identical values of the heat of fusion. In

the first method, the difference between radiation corrected [8] drop heat

(molar basis) and the enthalpy value for the solid, 119.395 kJ/mole, from

JANAF [18] was corrected to a 100 percent melted basis, and a weighted average

taken. In the second method, a two parameter weighted least square analysis

of corrected drop heat verses fraction melted was made. The heat of fusion

was then just the slope of this function, and its intercept is the solid

enthalpy. That intercept was 119.5 ± 2 kJ/mole, in excellent agreement with

the JANAF [18] value. Because of this agreement, both methods are essentially

numerically identical, and the value for the heat of fusion obtained by both

methods was 53.0 ±2.3 kJ/mole. The statistical uncertainty is of the order

of twice the anticipated error from temperature uncertainty, excess e-beam

disturbance of the calorimeter, and normal calorimetric errors.

123
Several factors contributed to the relative success of these efforts: 1)

nearly flawless operation of the new automatic gate system; 2) improved

initial stability and low outgassing of the specially cleaned ground samples

[19], and improved heating and holding characteristics of the slightly smaller

samples (a set of large samples (1.25 cm diameter, 20 gm) were extremely

stable levitators, but proved impossible to melt, even on the surface); 3) the

redesigned coil, which improved levitation stability (only in the case of

sample N did the coil show any evidence of holding a molten sample, and that

sample may have completed melting as it was being lowered in the coil preparatory to dropping in that case).

However, a number of problems, related to the age of the GE equipment,

and the large number of hand operations necessary to ensure successful operation resulted in a significant number of failures (at least 1 in 4) due to

system error, either outright hardware mi sfunctions or human error. The

remainder of failures to melt are attributable to electron-beam adjustment

failure. This appears to be the current primary difficulty to obtaining

successful melting. A number of tests and setup techniques were employed in

attempts to correlate tuning technique with successful melting. A special

fixture to simulate high temperature emission grounding was constructed. One

of the larger balls was mounted on a 0.32 cm diameter tungsten sting, attached

to a baseplate flange, which allowed the placing of a specimen in the coil for

beam focusing. This established that the levitation field did not seriously

disturb beam focus, but pointed up a number of equipment improvements

necessary to make tuning more reproducible.

It appears that successful melting occurs only when the electron beam is

focused so tightly that it can actually "drill" a hole through the sample (see

124
figure la and lb). The local heating is apparently considerable, and significant amounts of material are ejected, often with accompanying arc- induced

shut-off of the RF generator and loss of sample. It would seem that this

tight focus and consequent undesirable local heating should not be necessary

for melting on the basis of estimates of the secondary emission characteristics

at the melting temperature of tungsten. However, inability to observe and

control the beam focus as desired has made systematic testing of focus effects

difficult.

A new molybdenum shade ring, to protect the smaller work coil and to

provide a run-to-run place to tune the electron beam without mounting special

fixtures, will be installed. A number of minor hardware modifications to the

e-beam control apparatus have also been suggested. These improvements should

be completed about the time GE finishes the development of a new two-color

pyrometry camera system, implementing part of the design proposed earlier

[19]. In order to improve performance, a new series of smaller samples will

be prepared with a weight of approximately 6-7 gm. In addition, a new coil

with smaller (1.4 verses 1.6 cm) inside diameter and solenoid geometry to

maximize the field gradient for these smaller samples and improve liquid

holding capability has been designed and recommended to GE.

CONCLUSIONS

A prototype "gulp" calorimeter system has been developed and tested for

mechanical performance. The design seems suitable for a potential flight

experiment series, but a number of developments are still necessary before the

unit operates as a true calorimeter.

The ground-based effort to determine thermophysical properties for

tungsten have yielded a value of 53.0 ± 2.3 kJ/mole for the heat of fusion of

tungsten. This value needs additional experimentation for confirmation, but

125
appears at present to represent a best value for that parameter. Experimental

difficulties are still a serious problem. It is clear that the electron beam

performance is the most important obstacle to successful levitation melting.

Focus, beam shape, and positioning are all quite critical factors, even though

the more than 3000 watts available is about 7 times the necessary power to

melt the tungsten samples if absorbed. It is recommended that GE devote some

effort to improving this aspect of their experimental facility, with a view

toward determining the critical factors in electron beam melting of levitated

specimens. Improvements have been suggested to improve control and allow some

experimental tests to be performed.

It appears that this research still requires some developmental effort to

achieve its goal of obtaining tungsten liquid enthalpy increments. The

problems are understood, but will require laboratory time to resolve.

126
Table 1.

Sample Weight, gm Drop enthalpy,

3

kJ/M % melted (est) AHm,

b kJ/M

A 7.895 158.36 75 51.9

D 9.740 148.62 60 48.5

F 9.473 148.44 50 58.2

G 9.596 158.83 75 52.7

N 9.468 172.58 100 53.18

RiA 9.3249 159.35 75 53.1

53.0 + 2.3

(

a Radiation loss corrected

^computed on 100 % melt basis

Qweighted average/regression - see text

127
REFERENCES

1. R. Munro, Private Communication, (1983).

2. T. Wang, Jet Propulsion Laboratories, Private Communication (1983).

3. P. C. Nordine and R. M. Atkins, Rev. Sci. Instrum., 53, 1456 (1982).

4. W. A. Oran, L. H. Berge, and H. W. Parker, Rev. Sci. Instrum. (1981).

5. C. D. Hendricks in "Materials Processing in the Reduced Gravity

Environment of Space," ed. G. E. Rindone, MRS Symp. Proc 9, (NorthHol land, New York, NY, 1982), p. 59-65.

6. R. T. Frost and E. Stockoff, Final Report, NASA Contract NAS8-34231, Task

7 (General Electric Space Systems Division, Vally Forge, PA, 1983).

7. J. Coops, E. S. Jessup, and K. van Nes, "Experimental Thermochemistry,"

ed. F. D. Rossini, (Interscience Publishers, Inc., New York, 1956).

8. D. W. Bonnell, "Property Measurements at High Temperatures - Levitation

Calorimetry Studies of Liquid Metals," Ph.D Thesis, Rice University

(1972).

9. G. Wouch, "Containerless Melting and Solidification of Metals and Alloys

in the Terrestrial and Space Environment," Ph.D. Thesis, Drexal Univ.

(1978).

10. G. Wouch, R. T. Frost, D. Rutecki , N. Beser, and E. C. Okress,

Final Report, NASA Contract NAS8-34231 (General Electric Space Systems

Division, Valley Force, PA, 1975).

11. D. W. Bonnell, "NBS: Materials Measurements," ed. J. R. Manning, NBSIR

81-2295 (1981).

12. IPTS-68 - The International Practical Temperature Scale of 1968 -

Metrologica, 5, 35 (1969).

13. G. Tamman, Z. Phys. Chem. , 85, 273 (1913).

14. K. K. Kelley, U.S. Bureau of Mines Bull. No. 584 (1960).

128
15. R. Hultgren, P. D. Desai , D. T. Hawkins, M. Gleiser, K. K. Kelley, and

D. D. Wagman, "Selected Values of the Thermodynamic Properties of the

Elements" (American Society for Metals, Metals Park, OH, 1973).

16. A. Cezairliyan, High Temp. Sci., 4, 248, (1972).

17. I. Ya. Dikhter and S. V. Lebedev, High Temp. - High Press., 2, 55,(1970).

18. G. Wouch, E. L. Gray,, R. T. Frost, and A. E. Lord, Jr. - High Temp.

Sci. , 10, 241 (1978).

19. JANAF Thermochemical Tables, NSRDS-NBS 37 (1972).

20. D. W. Bonnell, "NBS: Materials Measurement," ed. J. R. Manning, NBSIR

82*2560 (1982).

129
la. Run P

lb. Run T

Figure 1. Recovered tungsten samples, showing electron beam "drilling"

of stably levitated samples. Note the obvious signs of

ejecta surrounding the site of melting. The specimens are 1 cm

in diameter, and the hole in run T is about 0.7 cm deep.

130
NBS-114A (REV. 2-80) U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR

REPORT NO.

2. Performing Organ. Report NoJ 3. Publication Date

4. TITLE AND SUBTITLE

NBS: Materials Measurements

5. AUTHOR(S)

J. R. Manning

6. PERFORMING ORGANIZATION (If joint or other than NBS. see instructions)

NATIONAL bureau of standards

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract/Grant No. NASA Gov't. Order

H-27Q54B

8. Type of Report & Period Covered

Annual Report

April 1, 1982 -March 31,

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. State, ZIP) TOT

Office of Materials Processing in Space

National Aeronautics and Space Administration

Washington, DC 20546

10.

SUPPLEMENTARY NOTES

| | 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

bibliography or literature survey, mention it here)

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

in Space Program under NASA Government Order H-27954B (Properties of Electronic

Materials) covering the period April 1, 1982 to March 31, 1983. This work is directed

toward measurement of materials properties important to the design and interpretation

of space processing experiments and determinations of how the space environment may

offer a unique opportunity for performing improved measurements and producing materials

with improved properties.

The work is being 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 the High Temperature Thermophysical Properties of

Tungsten Group Liquids and Solids

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

the report. Emphasis in Tasks 1 and 2 is on how the reduced gravity obtained in space

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

toward development of techniques for thermodynamic measurements on reactive materials,

requiring levitation and containerless processing.

12.

KEY WORDS (S/x to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolons)

surface tension, convection, directional solidification, thermophysical properties,

silicon, tungsten, binary alloy

13.

AVAILABILITY

| XI Unlimited

| | For Official Distribution. Do Not Release to NTIS

|~

| Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402.

Order From National Technical Information Service (NTIS), Springfield, VA. 22161

14. NO. OF

PRINTED PAGES

15. Price

USCOMM-DC 8043-P80