i
NATIONAL INSTITUTE OF STANDARDS &

TECHNOLOGY

Researdi Information Center

Gaithersburg, MD 20899

HIDE mobflb

' MBS

Plications

NBSIR 82-2560

NBS: Materials Measurements

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

Washington, DC 20234

I

July 1982

Annual Report

For the Period 1 April 1 981 - 31 March 1 982

NASA Government Order H-27954B

Properties of Electronic Materials

Dj

56

2-2560

. 2

U.S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
1
•4nu»AJt t>crjKX«a ow rrAjrDAjat

NBSIR 82-2560

NBS: MATERIALS MEASUREMENTS

J. R. Manning

UDUAST

SEP 1 4 1982

( OC'O

. u“ - L?

"d V (9 ^

/

U S. DEPARTMENT OF COMMERCE

National Bureau of Standards

Washington, DC 20234

July 1982

Annual Report

For the Period 1 April 1981 - 31 March 1982

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

Summary

Task 1

-

Task 2

-

Task 3

-

Page

1-3

SURFACE TENSIONS AND THEIR VARIATIONS WITH TEMPERATURE

AND IMPURITIES

S. C. Hardy and J. Fine

CONVECTION DURING UNIDIRECTIONAL SOLIDIFICATION

R. J. Schaefer, S. R. Coriell, R. G. Rehm and

G. B. McFadden

MEASUREMENT OF HIGH TEMPERATURE THERMOPHYSICAL PROPERTIES

OF TUNGSTEN LIQUID AND SOLID

D. W. Bonnell

X

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, 1981 to March 31, 1982.

This work is directed toward measurement of materials properties important

to the design and interpretation of space processing experiments and investigations of how the space environment may offer a unique opportunity for

performing improved measurements and producing materials with improved

properties.

This 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 density-driven

1
convection, thereby causing liquids to be stirred as they solidify. This

stirring disturbs the quiescent boundary layer at the solidifying interface,

and can be very undesirable if nearly perfect crystals are required. For

example, it creates interface instabilities, introduces segregation of

components and produces crystal defects in the resulting solid material.

These defects and inhomogeneities, which are particularly troublesome in

electronic technology and other advanced technical applications, might

be avoided in materials produced under microgravity conditions.

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.

Even if gravity-driven convection is small. Important convection effects

still may arise from other sources. For example, convection can be produced

by surface tension gradients. The variation of surface tension with temperature

and composition often is not known, even for important technological materials

such as silicon. NBS currently is making surface tension measurements on

materials considered for space flight experiments. In addition, as part of

this work segregation of impurities to the liquid surfaces is being investigated.

Initial measurements were on gallium and gallium-tin alloys, selected as

model materials useful for studying Marangoni convection and liquid metal

surface effects at moderate temperatures.

The surface segregation studies that are being made include one of the

first applications of Auger analysis techniques to liquid metals. These

techniques have been very successful in producing informative results.

Strong . segregation effects were found in measurements on liquid metal

surfaces. This segregation significantly affects surface tensions.

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

and composition gradients, is being investigated both by theoretical calculations

and by experiments. The experiments are designed to test the theories and

provide quantitative data on convection processes on earth. 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 during materials processing

on earth. Since this type of convection depends 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 and limitations. Measurements of the

specific heat of liquid tungsten uncontaminated by reaction with container

walls currently is being pursued in earth's gravity experiments. Several

aspects of the techniques being developed should also be useful for space

flight experiments.

3

Task 1

The Measurement of Surface Tensions and

Their Variation with Temperature and Concentration

S. C. Hardy

Metallurgy Division

Center for Materials Science

and

J. Fine

Surface Science Division

Center for Chemical Physics

Summary

A series of Ga-Sn liquid alloys has been studied by Auger electron

spectroscopy. The Sn was found to be strongly adsorbed at the surface.

Calculations of surface concentrations from the Auger measurements are shown

to be substantially in agreement with surface concentrations deduced from

thermodynamic data. This result supports the monolayer model of surface

adsorption for the Ga-Sn system.

5
Introduction

Components of a bulk liquid phase which have low intrinsic surface

tensions will preferentially segregate at free surfaces. Although the

thermodynamics of this surface adsorption phenomenon was developed by Gibbs

over a hundred years ago, many questions remain unanswered about details of

the process such as the distribution of the adsorbed species in the surface

regxon and the interactions between coadsorbing species. Adsorption can in

some situations result in large surface tension gradients which generate

fluid flow. These "Marangoni" flows are the dominant type in low gravity.

The study of adsorption at liquid surfaces is currently an active field

of research with a long history. There have been almost no applications of

modern surface analysis techniques to liquids, however, although there would

appear to be no fundamental impediment to such studies for low vapor pressure

and melting point materials. Perhaps the oldest and most widely used of the

new surface analysis techniques is Auger electron spectroscopy, yet we know

of only two previous applications to liquids [1,2]. Both of these were

attempts to measure the surface segregation in binary alloy systems. A

related technique, electron energy loss spectroscopy, has recently been used

to study the oxidation of liquid Sn surfaces [3].

During the past contract year, we have been studying surface segregation

in liquid Ga-Sn alloys using Auger spectroscopy. This system was chosen for

study because of the low vapor pressures and melting points of its components

and because surface tension measurements in the literature Indicated a

strong adsorption of Sn at the interface [4]. Furthermore, the combination

of Ga and Sn is interesting from a surface characterization point of view

since Auger electrons cover a wide energy range for this system: Ga has

prominent lines at 55 e.v. and 1070 e.v. and Sn at 430 e.v. The escape

6
depth for the Auger electrons varies from approximately one to six monolayers

over this energy range. Thus the liquid composition can be sampled to

different depths into the bulk. The primary objective of this work is to

see if the surface concentrations of the alloy components deduced from the

Auger measurements are in agreement with those deduced from the surface

tension data. Such agreement would support the results of the Auger determination of surface concentrations in other alloy systems for which good

thermodynamic measurements are unavailable.

Experimental

A detailed description of the equipment we use has been given in a

previous report [5]. Briefly, the Auger spectrometer is a conventional

single-pass cylindrical mirror analyzer system (CMA) with a coaxial electron

gun. The CMA is mounted vertically so that it can be focused on the top

surface of a sessile drop sample. The electron beam used to excite the

Auger electrons has a diameter of about 3 pm and can be rastered to form a

video image of the drop surface. Auger data is acquired in an analog mode

using first derivative spectral display. The base of the sessile drop is

contained in an alumina cup which is clamped to a simple hot stage mounted

on the rotatable platform of a horizontal manipulator. Four drops are

accomodated by the hot stage and can be successively brought into position

for analysis by rotation. Electrical contact to the drops is made with fine

W wire probes which are insulated from the hot stage. The temperature is

monitored by a chromel-alumel thermocouple spot welded to the top of the hot

stage. The drop temperature was compared with the temperature of the hot

stage by inserting a W-Rh thermocouple directly into a drop in a calibration

experiment. In subsequent work drop temperatures were determined from the

reading of the hot stage thermocouple using these calibration measurements.

7
Sessile drops are formed by placing pieces of high purity Sn and Ga of

the desired weight in the cups and melting them after evacuation of the

system. The surfaces of the drops are covered initially by oxide layers

with high carbon contents. This overlayer of impurities is removed from the

liquid surface by sputtering with argon ions [6]. Although a few residual

islands of impurities may persist after sputtering, the surface area under

analysis is clean, at least to the sensitivity of Auger spectroscopy. For

accurate measurements of surface concentrations it is necessary that the

surface be impurity free since the Auger electron intensities can be significantly

reduced by fractional monolayer coverages.

In our initial measurements, pure Ga and pure Sn samples were studied

in order to establish the relative magnitudes of the three prominent Auger

lines of this alloy system as a function of spectrometer variables. In

subsequent work one of the cups always contained a pure sample of either Ga

or Sn. The spectrum of this standard was recorded before and after each

alloy drop spectrum to allow normalization of the Auger line intensities.

The Sn Auger spectrum, in addition to the strong line at 430 e.v.,

contains a broad, weak, line which overlaps the low energy Ga line at 55

e.v. By superposition of spectra from pure Ga and Sn it was estimated that,

in an alloy sample, this low energy feature of the Sn Auger spectrum would

increase the apparent amplitude of the 55 e.v. Ga line by about 1.9% of the

amplitude of the observed 430 e.v. line for that alloy. Thus, as a first

approximation, we have corrected the 55 e.v. line by substracting this

amount, i.e., 1^=1^ - .019 . Here and are the measured

Ga^ Ga^ Sn Ga^ Sn

peak-to-peak line amplitudes at 55 e.v. and 430 e.v. and I is the adjusted

amplitude. The correction to the low energy Ga line amplitude becomes

B B important as the Sn concentration increases. For = .70, where is

8
the bulk atomic fraction, .019 = .43 . At higher Sn concentration,

Sn Ga

the low energy Sn feature obscures the peaks of the Ga line making measurement impossible. Thus, the determination of the amplitude of the 55 e.v. Ga

line becomes increasingly difficult as the bulk Sn concentration increases

and we anticipate a consequent loss of accuracy. A more elaborate correction

procedure which considered the escape depth of the Auger electrons and the

distribution of Sn near the surface gave essentially the same results as

this simple approach.

In Figure 1 we show the results of the Auger measurements. The observed

peak-to-peak line amplitudes have been normalized using the observed amplitudes

of the corresponding lines in a pure unalloyed specimen under identical

conditions. The amplitude of the 55 e.v. Ga line has been corrected using

the procedure described above. This is the basic experimental data which

will be analyzed subsequently.

Results and Discussion

a. Thermodynamic Data

We wish to compare the surface concentrations estimated from the Auger

measurements with the concentrations derived from surface tension measurements. This latter analysis is made basically with the Gibbs adsorption

equation. The relative adsorption of component A with respect to

component B at the interface is given by

( 1 )

where a is the thermodynamic activity, y is the surface tension, T is the

absolute temperature, and R is the gas constant. To calculate the relative

9
adsorption the surface tension is plotted as a function of Ina and the slope A

is measured along the curve. In Figure 2 we show such a plot for Ga-Sn

alloys at T = 623 K. This data is from the Russian work [4] and is in general

agreement with other data in the literature and with measurements in our

laboratory at several concentrations. Although the slope determination is

somewhat uncertain at low concentrations where 1 Ina ] is large, it is well

oTl defined over most of the composition range. There is an anomalous increase in

surface tension near lna„ = - 0.55. This corresponds to a bulk concentration

of Sn of 0.6 atomic fraction. Similar features are seen in surface tension

measurements in other alloy systems and are generally not understood. We will

not attempt to calculate adsorption near this anomaly, but will simply extrapolate through this region. The adsorptions calculated from the slope in

Figure 2 give the number of excess Sn atoms near the liquid vapor interface.

If these are considered to be confined to a single layer so that atomic fractions

S S in the monolayer, X abd X , can be defined, the following expression for

Sn Gs

the concentration of Sn at the surface is obtained [7]:

Sn / ^ 1 <“Ga-“sn>

( 2 )

Here a and a are the atomic areas of pure Ga and Sn respectively and

Gn sn

g

X„ is the bulk concentration of Sn (atomic fraction). The surface conSn

centration of Sn calculated using this monolayer model and the data shown in

Figure 2 is shown in Figure 3. The Sn is seen to be strongly adsorbed attaining

a concentration at the surface of about 0.6 at a bulk concentration of 0.1.

b. Surface Concentrations from Auger Data

The determination of surface concentrations from measurements of Auger

line amplitudes may be carried out at several levels of approximation. In a

10
first approximation this is often done by equating the surface concentration

S 0 / CO with the normalized line amplitudes, i.e., /I^ where is the

peak-to-peak line amplitude measured for pure Sn. Surface concentrations

calculated in this approximation from our measurements are shown in Figure 4

along with the concentrations deduced from thermodynamics shown previously

in Figure 3. We have converted the surface concentrations of Ga calculated

S S from the Ga Auger data to Sn concentrations by computing = 1

- =

1 - /I^ . The data shown in Figure 4 illustrate that surface concentrations

Ga Ga

calculated in this approximation become inaccurate as the electron energy

increases. This deterioration in the approximation comes about because

Auger electrons of higher energy escape the sample from significant depths

and add a bulk component to the detected signal which obscures the fraction

originating in the surface layer. Thus, it is necessary to account for this

bulk contribution in order to measure surface concentrations.

The Auger electron current from a component A of an alloy AB may be

written as:

tAB r n f \ (-z/Acos6)

^ A

' S^pVa "aB

^

0

( 3 )

where g is a geometrical factor, I is the primary beam current, a is the

p A

ionization cross section, y . is a relaxation coefficient, is the back- A AB

3 scattering factor, C (z) is the concentration of A atoms/cm , z is the

Pi. distance from the surface in units of number of monolayers, and X is the

inelastic mean free path for electrons in units of number of monolayers.

The CO30 term is dependent on the geometry of the analyzer and for our instrument is 0.74. If the concentration C (z) is considered to be uniform in

Pi. the bulk up to a surface layer of a different composition, an integration of

the above equation and the analogous equation for pure A gives:

11
1 1

(4)

.AB

R I,

'AB ^AB

R. I,

N

N.AB {X^ [1-exp (- jAB

) ] + exp (- j

)}

AB

where N is the atomic density (atoms/cm3

), £ = Xcos9, and X^SBand X^ are the

atomic fractions of A in the surface layer and in the bulk respectively.

The subscripts A and AB indicate pure material and alloy. Solving for the

surface concentration,

AB. 0° B -1/Z

,.S - V “

A . -l/l,

1 - e 'AB

(5)

where we have written F.

A

^AB '^AE ^AB

\ ^A «A

Values for the atomic density

required to evaluate equation (5) can be calculated from the mass density.

The backscattering factor and Auger electron mean free paths, however, are

not so well known. The backscattering factor R describes the Auger electrons

which are generated by the backscattered primary beam. R is a function of

primary beam energy E^, Auger electron energy E, and the matrix. We have

derived R values appropriate to Sn and Ga from recent Monte Carlo calculations

[8]

.

This was done by plotting the R values found in the calculation as a

function of reduced energy, U = Ep/E> Z = 31 (Ga) and Z = 50 (Zn)

.

The

R values appropriate to our U could then be interpolated from the graphs.

In Table 1 we give the values of R for Ga and Sn obtained in this way. For

the alloys, the backscattering factor was assumed to be directly proportional

to the bulk concentration, i.e..

•R = R

Ga,Sn Ga

+ X

Sn

[R,_Sn

- R^

Ga-J ( 6 )
The critical factor in equation 5 is the exponential function of the

inelastic mean free path. Although there have been several measurements of

this quantity for Sn [9,10], we do not know of any measurements for Ga. The

measurement of inelastic mean free paths is difficult and published values

are often of uncertain accuracy. We will, therefore, calculate inelastic

mean free paths for Ga and Sn using the recently developed theory of Szajman

et al. [11] which seems to predict X accurately for aluminum, a metal electronically similar to Ga and Sn. In Figure 5 we show the predicted X

values for the case of Ga and Sn using this theory; we also show the prediction

of the Seah and Dench [12] empirical equation for Sn and the values measured

experimentally at two energies [9,10]. The data point at 430 e.v. was taken

from Seah's work in which he deposited Sn on an Fe substrate and measured

the change in Auger signal with film thickness. We have adjusted his data

using backscattering factors for Sn and Fe calculated as discussed earlier.

This reduced his estimate of X by about 40% and brings his measurement into

good agreement with the predictions of theory. The Szajman theory must

become inaccurate at low energies because it does not predict the minimum in

the X vs. E curve at around 100 e.v. with the strong rise at very low energies

(< 10 e.v.) shown in the Seah and Dench compilation [12]. Lacking any

rigorous method for correcting the Szajman theory in this region, we will

simply add the empirically determined 538/E^ term used by Seah and Dench.

In Table 2 we list the calculated X values which we will use subsequently to

evaluate the Auger data. The X values for Ga and Sn are nearly the same so

we will use their average for all alloy compositions.

c. Comparison of Auger and Thermodynamic Data

In Figure 6 we compare the surface concentrations calculated from

thermodynamics (curve) with the concentrations of Sn determined using

13
the normalized Auger amplitudes, the backscattering factors, and the inelastic mean free paths of Tables 1 and 2 as inputs to equation (5) . The

filled circles are surface concentration obtained for the 430 e.v. Sn line.

S The Ga surface concentration X was also calculated using the 55 e.v. Auger

LrSi s data. This was then converted to a surface concentration of Sn, i.e. , X„ = bn

S

1 - X . This data is shown in Figure 6 as open circles. In Figure 7 we

Gs. have expanded the low concentration range of Figure 6 to better display the

data. The one point which lies significantly off the curve determined from

thermodynamics is believed to be in error due to an impurity island momentarily

drifting into the electron beam during analysis. The deviation of the two

sets of data at high concentrations may indicate an inaccuracy in the

thermodynamic analysis rather than the Auger data. This deviation occurs in

the region containing the anomalous rise in the surface tension which made

the determination of the relative adsorption difficult. A loss in accuracy

in the calculation of the surface concentration using the Sn data at 430

e.v. is to be expected at high bulk concentrations of Sn because the Auger

signal is predominantly from the bulk in this case. The soft Auger electron •

data at 55 e.v. is Increasingly uncertain at high Sn concentrations because

the correction discussed earlier becomes large. The standard deviation of

S S Xg^ determined by Auger from the thermodynamically determined Xg^ is about

7% for the low energy electrons; this increases to 11% for the 430 e.v.

data. The disagreement is surprisingly small considering the uncertainty of

the factors entering into the evaluation. Better agreement with the thermoS dynamic X„ could be obtained for the 430 e.v. measurement if a smaller X Sn

were assumed.

14
Summary

The analysis of the Auger line amplitudes for Ga-Sn alloys at 350 °C

shows that the Sn is strongly adsorbed at the surface. Using calculated

values for the inelastic mean free path and the backscatter ing factors for

the Auger electrons permits the determination of surface concentrations

which are in good agreement with the concentrations derived thermodynamically

from surface tension measurements. This agreement supports the validity of

the monolayer modeJ. which considers all the surface segregated Sn atoms to

be in the top layer. The approach used here to calculate surface concentrations

from Auger measurements is generally useful for metals which have a relatively

soft Auger electron.

15
Table 1. The backscattering factors for Ga and Sn interpolated from the

calculations of Schimura and Shimizu [8],

E (e.v.) ;

i ^Ga

1 ^Sn

1

1

i 1.94 1 2.15

430 i

> 1.60

j 1.74

1070 i 1.34 I 1.44

i

Table 2. The inelastic mean free paths for Ga and Sn in monolayers

calculated using the equation of Szajman and Leckey [11].

E

(e.v)

ka 1 (monolayers) j

^Sn 1 (monolayers) i

538/E^

(monolayers

)

A avg.

(monolayers

)

55

1 .92 ; .0 1i .18 1.09

430

! 4.27 4.20 ; 0 4.23

1070 00 8.87 i 0 8.93

16
References

1. S. Berglund and G. A. Somorjai, J. Chem. Phys. 5537 (1973).

2. P. Laty, J. C. Joud and P. Desre, Surface Sci. 104 , 105 (1981).

3. J. D. Verhoeven, M. A. Noack and A. J. Bevolo, in the Proceedings of

the Materials Processing Research in the Reduced Gravity Environment

of Space Symposium, Guy E. Rindone, Chairman, Boston, 1981.

4. A. G. Nalgiev and K. I. Ibragimov, Zh. Fiz. Kim. ^, 1289 (1974).

5. NBS: Materials Measurements, NBSIR 81-2295, June 1981.

6. J. Fine, S. C. Hardy and T. D. Andreadis, J. of Vacuum Sci. & Tech.

18, 1310 (1981).

7. M. Brunet, J. C. Joud, N. Eustathopoulos and P. Desre, J. Less-Common

Metals 5^, 69 (1977).

8. S. Ichimura and R. Shimizu, Surface Sci. 112 , 386 (1981).

9. M. P. Seah, Surface Sci. 595 (1973).

10. P. Cadman and G. M. Gossedge, J. of Elect. Spect. and Related Phenom.

161 (1980).

11. J. Szajman and R. C. G. Leckey, J. of Elect. Spect. and Related Phenom.

23 , 83 (1981).

12. M. P. Seah and W. A. Dench, Surface Interface Anal. 1, 2 (1979).

17
o

00

00 o O

00 CD OO

O

XI

0)>4

cn

CO

OJB

Cfl 0)

CO 42

CD

CO^/j 0

-*-

CU X33ae & cO c B o CO cu 42

CU 4J oo d

•rH cn 1—1 *H *4— i-O o

(U

M

bO3 £

<0 • 1 o 4^ C

cO

X

CU o a.

d 1 o GQ M 0 *H

4_) 4_) 1 Cd

X

u

n3 4J

cu

d

CL cu

CJ "d

OJ I—I d

cO CUD g 4^ O r- d 3

40

CU 42 >4- H

ao H

ptH

18
700

T=623K

bbO

600

550

500

-4 -3 -2 0

n a sn

Fig. 2. f41 The surface tension of Ga-Sn alloys y as a function of the log of

the thermodynamic activity a .

19
(atomic

fraction)

0

Xsn(atomic fraction)

s Fig. 3. The surface congentration of tin Xg as a function of the bulk

concentration determined from tRe surface tension data and

*- • o Sn equation 2.

20
Xsn

(atomic

fraction)

0 .2 .4 .6 .8 1.0

xfn(otornic fraction)

Fig. 4. S S Comparison of shown in figure 3 with X yalues_calculated from

the Auger measurements in the approximation /I°° . Sn Sn Sn

21
12

E(e.v.)

Fig. 5. The inelastic mean free path for electrons A as a function of electron

energy E.

22
0

Xgpiatomic fraction)

s Fig. 6. The surface concentration X calculated using equation 5 as a function of the bulk concentration The curve is the

thermodynamically determined concentration variation shown in

figure 3.

23
czo

o

E

o

o

CO w

X

.7

.6

.5

A

.3

2

p

Xsn(atomic fraction)

Fig. 7. An expanded display of the surface concentration at low values

of . Sn

Sn

24
Task 2

Convection During Unidirectional Solidification

R. J. Schaefer and S. R. Coriell

Metallurgy Division

Center for Materials Science

and

R. G. Rehm and G. B. McFadden

Mathematical Analysis Division

Center for Applied Mathematics

Summary

A theoretical model is presented of double diffusive convection induced

in a binary liquid during crystal growth. The interface between the liquid

and solid is assumed to remain flat and the crystal is taken to grow at constant

velocity. The nonlinear partial differential equations of fluid flow together

with equations for temperature and concentration are solved numerically. The

numerical methods and some checks used to ascertain the validity of the methods

are briefly described, and sample solutions are presented and discussed.

During directional solidification of pure or ethanol-doped succinonitrile,

it was found that some convection of purely thermal origin was always present.

In the ethanol-doped sample this convection leads to the development of a

macroscopic depression in the solid-liquid interface, which subsequently

becomes the site of a localized cellular structure. An overall cellular

interface appears at conditions close to those predicted for interfacial

instability.

In collaboration with M. E. Glicksman and J. I. Mickalonis of Rensselaer

Polytechnic Institute, the stability of the flow between two vertical, infinite,

rigid, coaxial cylinders held at different temperatures is analyzed by linear

stability theory. When the rigid outer cylinder-fluid interface is replaced

by a crystal-melt interface which can change shape, new "coupled modes" of

25
instability occur at lower values of the Grashof number. These calculations

were motivated by and are in general agreement with recent experiments at

Rensselaer Polytechnic Institute on succinonitrile.
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 discussion of the theoretical work first and then a

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

The theoretical section describes numerical methods for solving the fluid

dynamical, temperature, and concentration equations appropriate to directional

solidification of a binary alloy. Some example results of these calculations

are given. The experimental section describes the fluid flow and crystalmelt 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 Polytechnic 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.

27
THEORETICAL

The Mathematical Model of Solutal Convection

The problem considered is that of natural convection induced during unidirectional solidification of a binary alloy. The crystal is grown vertically

upwards at constant velocity V from a binary liquid with a concentration C of

solute [1]. Figure la shows a schematic diagram of the configuration.

The initial state is one in which the solid rejects or preferentially

incorporates solute at the interface giving rise to an exponential

concentration gradient of solute ahead of the interface with a decay distance

T)/V as shown in Figure lb. Also, an exponential temperature gradient exists

in the liquid with a decay distance </V and in the solid with characteristic

distance Kg/V. Here, k and Xg are the thermal dif fusivities in the liquid and

solid respectively and D is the diffusion coefficient for solute.

A large solute concentration gradient will induce natural, or buoyant convection, often called double diffusive convection [2], if the solute rejected

is lighter than the solvent or a preferentially incorporated solute is heavier

than the solvent. The temperature gradient, on the other hand, is a stabilizing influence since the colder fluid will stably reside near the interface.

It is the double diffusive convection which we wish to calculate.

To examine the natural convection during crystal growth, many simplifications are necessary. The simplifications will be enumerated and discussed

briefly

.

During crystal growth, a spontaneous change in the interface shape, known

as morphological instability and convective instability will be coupled.

Linear stability analysis [3] shows that the wavelength associated with the

most unstable mode of convective instability is much larger than that associ28
ated with the most unstable morphological mode and, therefore, that the coupling between these instabilities is weak in specific parametric ranges. Because

the linear coupling is weak and because we have little expectation at this

stage that we can calculate the nonlinear coupling, we assume that the interface remains planar. The conditions for morphological stability in the

absence of fluid flow are well known; we restrict the parameter range in our

calculations to satisfy these conditions. In addition, we are neglecting

deformation of the interface caused by the flow at present; the validity of

this approximation can be estimated from the numerical results.

The convective flow field is assumed to be two-dimensional and periodic

in the horizontal direction. The task of calculating the convective flow

field is ultimately computational, and a two-dimensional computation is much

more manageable than a three-dimensional one. The horizontal wavelengths for

unstable convective modes calculated from linear stability analysis [1] are

found to be much smaller than horizontal dimensions characteristic of experiments on unidirectional solidification except for very small growth velocities.

Therefore, it is expected that the solutal convection above an interface will

be periodic in the horizontal direction with a wavelength of the order of the

most unstable mode determined from linear stability analysis. The model

selected for computation is one horizontal cell of this periodic, cellular

convection pattern: the length L in Figure la is taken to be one period.

The liquid is taken to be a Boussinesq fluid with constant properties

(density p, kinematic viscosity v, liquid thermal diffuslvity <, solute diffusivity D, etc.) satisfying the Navier Stokes equations. In the Boussinesq

approximation the density of the fluid is assumed to be constant everywhere

except in the buoyancy terms where density variations are multiplied by the

i

29
acceleration of gravity g. This is a very conunon approximation which is

excellent in the present application [2].

In order to compute the flow field, a finite region must be defined. As

discussed above, one cell of a periodic convection pattern in the horizontal

direction is selected. In the vertical direction, an artificial upper boundary is Imposed. As noted earlier, the temperature gradient will tend to stabilize the convection. Therefore, the temperature field can be expected to

limit the height of each convection cell, and if the artificial upper boundary

is far enough from the interface, it should not influence the convection.

(This assumption can be tested by changing the location of the artificial

boundary.

)

Between the liquid and the solid phase of the material, there will, in

general, be a density change. This density change is us.ually small, and in

this study we have neglected it.

The Equations

A recent review of Interfacial and hydrodynamic instability relevant to

crystal growth has been given by Coriell and Sekerka [3]. The equations and

boundary conditions governing double diffusive convection are presented as

Eqs. (2) - (13) in that review. The equations are the Navier-Stokes equations

for a buoyant fluid of constant properties in the Boussinesq approximation,

conservation of energy in the liquid (a convection-diffusion equation for

temperature), conservation of energy in the solid (a diffusion equation for

the temperature in the solid) and conservation of solute (a diffusionconvection equation for the solute concentration). The equations are written

in a reference frame which moves vertically upward at the constant Interface

30
velocity. The equation of state for the fluid relates variations in the

density from a reference state linearly to changes in temperature and in

concentration.

The boundary conditions are obtained from conservation of mass, energy

and solute concentration together with a no slip condition for the tangential

component of velocity at the interface. Since we do not allow the Interface

to deform, we can not satisfy all of the usual solidification boundary conditions. In particular, we do not require that the interface temperature

equal the equilibrium temperature. Linear stability calculations have indicated that this does not have a great effect on the conditions for the onset

of convection, A simplified set of these equations, obtained by invoking the

approximations enumerated above, are used to compute double diffusive convection in the present study.

When no convection occurs and the Interface remains planar, a quiescent

steady state exists in a reference frame moving with the constant interface

velocity. This steady state is discussed in some detail, and mathematical

expressions for the temperature and solute concentration fields are given in

References 1 and 3. The fields are shown schematically in Figure 1,

For the computation, a set of dimensionless variables has been used. The

dimensionless concentration is referenced to the equilibrium-state concentration Coo at Infinity and the temperature gradients are made dimensionless using

the equilibrium-state temperature gradient in the liquid at the interface.

All lengths are made dimensionless with respect to the height H of the enclosure used for computation and the time scale is based on the kinematic viscosity

V of the fluid and the height of the enclosure.

When the equations are made dimensionless, several nondlmenslonal parameters result:

31
Ra

gaGLH^

VK

Pr = v/k

Sc = v/D

gcx^GooH^

Ra^

vD

Prg = v/<s

( 1 )

Here g Is the acceleration of gravity. Ra Is the Rayleigh number, a being the

coefficient of thermal expansion and Ra^, is the solutal Rayleigh number,

being the coefficient of expansion for solute. The density in the liquid is

given by the expression p = pQ[l-a(T-To) - ac(C-Co)] where Pq, Tq and Cq are

all reference quantities. Pr is the Prandtl number, Prg = Pr </<g and Sc the

Schmidt number. Additional dimensionless parameters are the geometrical ratio

R = H/L, representing the height to the width of the computational enclosure.

VH

U = — representing the dimensionless interface velocity, and two additional

V

parameters related to material properties; these are k, the partition coefficient (the ratio of the solute concentration in the solid to that in the

A liquid at the interface), and k, the ratio of the thermal conductivity in

the solid to that in the liquid.

In terms of dimensionless variables and parameters, the quiescent

equilibrium solute concentration in the liquid and the temperature in the

liquid and solid are

1-k

Co(y) = 1 + — [exp(-yUSc) - exp(-USc)]

l-( 1-k) [ l-exp(-USc)

]

A 1 To(y) exp (-yUPr)

GlH UPr

( 2 )

A 1 1 *^s Tso(y) = + [l-exp(-UPrgy)

GlH UPr UPrg Gl

32
Here, A is a reference temperature and Gq/Gl is the ratio of the solid to

liquid temperature gradients at the interface. (For more discussion of the

dimensional parameters required to specify the problem or of the equilibrium,

quiescent steady state, see References 1 and 3.) This solute concentration

field and temperature field, together with a zero fluid velocity field, are

the initial conditions for the convection computations described below.

The equations in dimensionless form which describe the double diffusive

convection are —

3u)9a)9 9 _ 9T 9c

- U— + —(uoo) + —(vo)) = + (Ra/Pr) — + (Ra^/Sc) —

9t 9y 9x 9y 9x 9x

(3a)

—

9 c 9 c 9 9 „ - U— + —(uc) + —(vc) = (1/Sc) V^c

9t 9y 9x 9y

(3b)

9T 9T dT.

U-- + --(uT) + --(vT) + V = (1/Pr) v2t

9t 9y 9x 9y. dy

(3c)

= - (3d)

9ij; 9i|; u = — , V = - —

9y 9x

(3e)

in 0 < X < 1/R, 0 < y < 1, and

9Tg 9Tg

U = (1/Prg) v2Tc

9t 9y

s . (3f)

in 0 < X < 1/R, -1 < y < 0.

The boundary conditions are

33
at the top, y

= 1,

3T u=v=0 , c=l and — = 0

3y

at the bottom, y

= - 1,

3Tg

_ 3y

at the sides, x = 0 and x = 1/R, periodic boundary conditions:

ijj(x=0,y) = i(;(x=l/R,y)

w(x=0,y) = to(x=l/R,y)

c(x=o,y) = c(x=l/R,y)

3c 3 c —(x=o,y) = —(x=l/R,y)

3x 3 X

T(x=o,y) = T(x=l/R,y)

3T 3T —(x=o,y) = —(x=l/R,y)

3x 3x

Tg(x=o,-y) = Tg(x=l/R,-y)

3Tg 3Tg

(x=o,-y) = (x=l/R,-y)

3x 3x

for all o < y < 1. Finally, at the Interface y

= o.

u = V = o

3c

- USc(l-k)c

3y

^ 3Tg 3T

T = Tg and k = —

3y 3y

(4a)

(4b)

(4c)

(4d)

34
Here u and v are the horizontal and vertical components of the velocity vector

in the laboratory (stationary) reference frame, \p is the stream function, and

3v 3u

u) = — - — is the vorticity, T and Tg are the variations of the temperatures

3x 3y

from the initial steady-state temperature fields given in Eq. (2) within the

liquid and solid respectively. v2 is the two-dimensional Laplaclan, and all

other symbols have been defined previously.

A feature of the double diffusive convection for a binary alloy which

makes it particularly challenging is the range of the parameters required to

describe it. Equations (1) and the discussion following them demonstrate that

the process is determined by nine dimensionless parameters, even with all the

simplifications in modeling the process discussed earlier. The range of these

parameters which one can hope to investigate must be severely limited therefore; examination of Reference 1 confirms this observation.

A second feature which makes the calculations challenging is the size of

some of these parameters. In a liquid metal the Prandtl number is very small,

of the order 10~2; hence the conduction of heat occurs much more rapidly than

does the diffusion of vorticity or of solute. In the limit when the Prandtl

number vanishes the temperature variation from the equilibrium value would

also vanish.

These observations can be used formally to simplify the temperature

equations as follows. Note first that Prg and Pr are approximately the same

size. Therefore let

T = Pr Ti + Pr2 T2 + ...

(5)

Te = Pr To + Pr2 T,, + . .

.

® ®1

Simultaneously, change the time scale to reflect the faster time for thermal

conduction:

35
Kt K Vt 1 let T = — = - — = — t (6)

h2 V h2 Pr

where t is the dimensional time. Now, formally substitute expansions (5) into

the temperature equations in Eqs. (3), and change to the new time variable t

dTo

in these equations. Note that = exp(-yUPr) s 1 for small Pr, provided

dy

U is not too large, and equate all terms Independent of Pr to zero. The

resulting approximate temperature equations are, for o < x < 1/R,

9^1

V^T]^ = - V for o < y < 1 3t

9t

0 for -1 < y < 0

(7)

The corresponding boundary conditions on Tj and Tg^ are:

9Ti

at y

= 1 = o 9y

at y

= - 1 = o 9y

at X = o and x = 1/R ( 8 )

Ti(x=o,y) = Ti(x=l/R,y)

9Ti 9Ti

(x=o,y) = (x=l/R,y)

9x 9 X

and at the interface, y

= o,

- 3Tg^

T 1 = Tq and k =

1 9y 9y

36
This formal procedure takes advantage of the small Prandtl number, reducing the nonlinear convection-diffusion equation for the temperature variation from the initial state in the liquid to a linear diffusion equation as

a first approximation. The solution for Tj^ is then coupled into the vorticity

equation in Eqs. (3) by the substitution T = PrTj.

The Numerical Method

The equations presented above have been solved by finite difference

methods. A rectangular grid is defined over one period in the horizontal

direction and between the interface and the artificially imposed boundary at

the top for the convection calculation. A similar grid is imposed over a

horizontal period and between the interface and a second artificial boundary

in the solid at y

= - H to calculate the temperature in the solid. At the

nodes or intersection of this grid all dependent variables are defined.

The equations solved numerically are the vorticity equation, (3a), the

solute concentration equation, (3b), the stream function equation, (3d), and

the two approximate temperature equations, (7), for the deviation of the

temperature from the initial state. Note that the vorticity, concentration

and temperature equations are all parabolic in character: they determine the

time evolution of these dependent variables. The vorticity and concentration

equations are coupled, nonlinear, convection-diffusion equations. The

temperature equations are linear to the first approximation for small Prandtl

number.

In contrast, the equation for the stream function is elliptic and linear.

A very brief description of the numerical techniques used to solve these equations and the checks used to ascertain the validity of the numerical solutions

will be presented.

37
For the two coupled nonlinear equations for concentration and vortlcity,

Fromm's method [4] was used. In this method, the convective terms, written in

conservation form (as presented in Eqs. (3a) and (3b)), are approximated by

central differences. The diffusion terms and the time derivatives are

approximated according to the method of DuFor t-Frankel [5]. Fromm's method is

characterized as second order accurate in both space and time, explicit and a

conservative scheme.

The solution to the coupled concentration and vorticity equations when

the temperature field was suppressed (for example by setting Ra = o) was

checked by solving these equations using a different finite difference scheme,

the alternating direction implicit (ADI) scheme [6]. A typical mesh used for

solution of the equations is 16 mesh cells in horizontal direction and 31 mesh

cells in the vertical direction, and for such a mesh, the solutions obtained

by the two different methods agreed to a few percent. The ADI scheme was

found to take about three times longer to compute a transient solution, however, and therefore was not pursued.

The two linear temperature equations were also solved by two different

methods, DuFort-Frankel and ADI, and agreement was found to a few percent.

In this case, however, the ADI was found to be about a factor of ten times

faster for small Prandtl number; the implicit nature of the scheme allowed

time steps for the temperature computation of the same size as those for the

vortlcity computation without much accuracy degradation. Therefore, the ADI

is presently being used to compute the temperature.

The elliptic equation for the stream function, given the vorticity field,

was solved using a package developed by Swarztrauber and Sweet [7]. This

package uses fast direct methods, fast Fourier transforms and cyclic reduc38
tion, and has been found in this and other applications to be both very

accurate and efficient.

The algorithms for calculating the coupled concentration and vorticity

equations are described in detail in Reference 8. They were implemented and

checked by Mr. Winston Chuck during the summer of 1979 after he had graduated

from Cornell University and before he entered a Ph.D. program at the

University of Minnesota. The graphics display, using the NCAR graphics

package [9], as shown in this report and initial attempts to compute the temperature field coupling to the vorticity and concentration distributions were

carried out by Mrs. Kathleen Morrish during the summer of 1980, after she had

completed her Masters degree at the University of Wisconsin and before she

embarked on her Ph.D. at the University of Maryland. Her work is reported in

Reference 10.

Computational Results

Some computational results are given in Figures 2-7. Dimensionless

parameters used in the computations are given in Table 1. The dimensional

parameters used to obtain the values in Table 1 correspond to a cell height of

.125 cm, a solutal diffusion coefficient of 3. x 10"^ cm^/s, an Interface

velocity of 2. x 10"3 cm/s, and temperature gradients of 500 K/ cm and 200

K/ cm for the first and second columns, respectively.

Figures 2 through 5 show contours of constant concentration and contours

of the stream function at two different times. In the steady state, the

stream function determines the flow magnitude and direction; the flow velocity

is tangent to the stream function contour. The calculation is initiated by

adding a small perturbation to the concentration profile and allowing the

disturbance to develop from rest. In this case the bulk concentration is

39
25% larger than its critical value as predicted by linearized analysis [1].

In the early part of the calculation (Figures 2 and 3), a slight distortion of

the concentration profile gradually appears. The cellular structure of the

stream function contours shows good qualitative agreement with that of the

linearized treatment. This Initial period of slow growth is followed by a

rapid, nonlinear growth over a relatively short time Interval, leading to a

steady state flow pattern with significant distortions of the concentration

profile.

Figures 4 and 5 show the steady state pattern which persists after the

transient period has ended. A large distortion of the concentration field is

evident, with an accompanying upwelling of the streamlines near the center

of the figure. Note also that the flow velocity remains orders of magnitude

smaller near the upper boundary of the figure, demonstrating that the stablliing temperature gradient has effectively "capped" the vertical extent of the

flow. Therefore, the boundary conditions applied to the upper surface may

not have too serious an influence on the flow.

Figures 6 and 7 correspond to a flow with larger Schmidt number, Sc =

10. In this case, a longer time is required to achieve a steady state, and

the Influence of the upper boundary determines the height of the convection

cell. The vertical velocity near the center of the cell is 1.0 x 10“2 cm/s,

which is about five times the speed of the interface. If the concentration

far from the Interface is normalized to unity, the maximum concentration at

the Interface is 3.72 and the minimum concentration is 2.23. The maximum and

minimum temperatures along the Interface differ by .013K.

For the two calculations shown above, it is valuable to check the static

stability of the initial state; the fluid is statically stable if the density

40
dPo

decreases with height above the Interface. For static stability < 0, or dy

in dimensionless form

Ra *^^o ^^c *^^o + > 0

Pr dy Sc dy

At the interface, where the concentration gradients are largest,

Ra (l-k)USc

> 0

Pr l-( 1-k) [ l-exp(-USc) ] Sc

Using the values given in Table 1, we find that the fluid is statically stable

initially for both cases.

As an overall check on the computational scheme, the solutal Rayleigh

number for convective instability was determined for prescribed values of all

other parameters, and this value was compared with that determined by linear

stability analysis as described in Reference 1. For a Rayleigh number of 0

and an aspect ratio of 0.5 and all other parameters as given in the first case

shown in Table 1 , the convection code was run repeatedly with Increasing values

of the solutal Rayleigh number until the Initial, small perturbation either

clearly decayed or grew. In the former case the flow was judged stable for

that solutal Rayleigh number and in the latter case unstable. Linear stability

analysis [1] determines the critical solutal Rayleigh number to be 4251. Using

Increments of 250 in Ra^, on a 16 by 31 mesh, Ra^ = 4500 was determined to be

stable whereas Ra^, = 4750 was found to be unstable. The discrepancy between

the critical solutal Rayleigh number determined from the convection code and

that determiined by linear stability analysis is between 6 and 12%. No attempt

was made to determine the critical solutal Rayleigh using the convection code

more precisely because near neutral stability, decay or growth of a perturbation was found to be very slow. Subsequent calculations with the convection

41
code have used estimates made by the linear stability analysis (with a nominal

10% variation) as a guide for selection of parameters.

The double-diffusive convection code described here offers the

opportunity to examine nonlinear features produced by solutal convection

during unidirectional solidification of a binary alloy. Although the range of

parameters over which the code has been run Is very limited at this time, we

hope to extend this range In the future and to explore such nonlinear features

as the magnitude of the convection and the distance above the Interface to

which convection extends as a function of Rayleigh number, solutal Rayleigh

number and Schmidt number for example.

42

1
EXPERIMENTAL

Background

To grow from the melt a crystal containing a uniform concentration of a

solute generally requires solidification at steady state with a planar

solid-liquid interface. Convective instabilities of the liquid phase or

morphological instabilities of the solid-liquid interface lead to a breakdown

of these idealized solidification conditions, with resultant inhomogeneitie&

of the crystal. In a previous article [11] we described a stability analysis

of the succinonitrile-ethanol system which predicted the conditions of

solute concentration and interface velocity which lead to convective or

morphological instabilities. The analysis assumed an initially planar

horizontal interface (unidirectional heat flow) of infinite extent.

Succinonitrile was chosen as the primary constituent because it is a transparent material which solidifies with an unfaceted solid-liquid interface,

which implies that deviations of the interface from its equilibrium temperature

are negligible. The relevant physical properties of succinonitrile are well

known. Ethanol is used as the solute because of its significantly lower

density compared to succinonitrile. The relevant portions of the phase

diagram have been reported previously. The liquidus slope for ethanol in

succinonitrile is 3.6 K/wt.% and the solidus slope is 81 K/wt.%, giving a

distribution coefficient of k = 0.044. Because of the low value of the

distribution coefficient, the convective and interfacial instabilities of

interest occur at low concentrations of ethanol, and very high purity starting

materials are therefore required. Fortunately, it has already been demonstrated

[12] that sufficient purity can be attained in succinonitrile by vacuum

distillation and zone refining, and these methods have been used in the

present experiments.

43
The stability analysis predicts that when succinonitrile containing

ethanol solidifies unidirectlonally upward, there exists for a given temperature

gradient in the liquid a critical concentration above which plane-front

convectionless solidification is unstable. This concentration is a function

of the solidification velocity, and at low velocities the instability appears

first in a convective mode whereas at higher velocities the instability

appears first in an interfacial mode (Figure 8). The Instability wavelength

for the convective mode is considerably larger than that for the interfacial

mode. For temperature gradients which are appropriate for Bridgman-type

crystal growth in this system (about 10 K/cm) the transition from the convective to the interfacial instability mode occurs at velocities of a few

micrometers per second. Moreover, there exists a range of concentrations

-3 -3

between approximately 2 x 10 to 5 x 10 wt . % ethanol for which it is predicted that there is a range of solidification velocities in which plane

front convectionless solidification is stable, but above which interfacial

instabilities occur and below which convective instabilities occur. A sample

was therefore prepared to be within this concentration range so that both

types of instability could be sought.

Experimental Procedures

Samples of pure succinonitrile and succinonitrile doped with ethanol are

sealed under vacuum in round pyrex tubes 19 mm in diameter. Composition is

determined by measurement of the melting range, i.e., the difference between

the liquidus temperature (where the last fragment of solid is equilibrated

with the melt) and the solidus temperature (where liquid first appears at

grain boundaries and in the form of droplets within the solid). The samples

primarily used in these studies were determined thus to have melting ranges

44
-2 -4 „ of 1.2 X 10 °C, corresponding to 1.5 x 10 wt./^ ethanol or equivalent

(designated as the high purity sample), and 0.215 °C, corresponding to 2.6 x -3

10 wt.% ethanol. The samples are innocuated with microspheres of polyvinyltoluene to serve as markers for convective flow in the liquid.

Solidification is carried out by drawing the sample downv/ard through a

temperature gradient device consisting of a lower, low- temperature water

jacket which is maintained below the melting temperature and an upper high

temperature jacket which in some experiments consisted of a water jacket and

in others consisted of an electric heater. A temperature gradient was thus

established, typically in the range of 5-10 °C per centimeter.

The solid-liquid interface was observed and photographed through a

stereomicroscope, directed horizontally at the region in which the temiperature

gradient was established. Bright field illumination was used for best

observation of interface shapes but dark field illumination is most effective

for observations of the particles which delineate the fluid flow. High resolution viewing of the material within the cylindrical tube requires that

the tube be surrounded by a medium which is appropriately matched in refractive

index to the tube contents. A mixture of water and ethylene glycol can be

used for this purpose but more recently it has been found desirable to use a

solid material as described below.

Succinonitrile is a somewhat unusual material in that the thermal conductivity of the solid and liquid are almost identical. Thus, V7hen no solidification

is taking place the gradients in the solid and liquid on either side of the

interface are also identical. UTien the sample tube is held in a vertical

temperature gradient, the linear gradients in the sample and in the container

walls are thus matched and the isotherms, including that which coincides with

45
the solid-liquid interface, are almost planar and horizontal. When solidification

is Induced by drawing the sample downward, the emission of latent heat by

the solid-liquid interface results in a discontinuity of temperature gradient

across the interface, and because this discontinuity is not matched in the

container walls and refractive index matching medium, the isotherms, especially

that delineating the solid-liquid interface, become non-planar. The temperature

gradient in the region near the interface thus has radial components. An

inevitable consequence of this non-planar interface and the associated

radial gradients is the presence of a thermally- induced convective flow in

the liquid phase. This takes the form of an upward flow along the central

axis of the sample tube and a downward flow along the walls. The presence

of this thermally- induced flow is disruptive to the development of the

steady state solute fields as assumed in the analysis of convective and

Interfacial instabilities, and therefore, seriously interferes with the

comparison of theory with experiment. The effect is particularly severe in

the case of materials such as succinonltrile which have a low thermal conductivity compared to that of any useful container.

One way to significantly reduce the radial gradients in the vicinity of

the solid-liquid interface is to add an auxiliary heater surrounding the

sample tube at the location of the solid-liquid interface. Such a heater, if

properly adjusted, can establish a gradient change in the container walls and

the index matching medium which approximates that in the solidying material.

Several designs for such heaters have been investigated, and the most effective

found to this point has been a single loop of small-diameter resistance wire

embedded in a cast block of transparent metallurgical epoxy which fits closely

around the sample tube (Figure 9) . The epoxy block holds the wire in position

46
and also serves as an approximate refractive index matching medium. During

crystal growth the sample tube slides through the epoxy block and the small

space between the block and the tube is filled with a fluid (ethylene glycol)

which is held in place by surface tension and which serves both as a heat

transfer medium and an optical matching medium. The most difficult aspect of

using such a device is that it changes the vertical position of the solidliquid interface as well as the radial gradient, so that both the position

and power of the ring heater must be adjusted to give the flattest solidliquid interface.

Results

Without the use of a ring heater, it is found that convection is present

during solidification of even pure samples of succinonitrile. Because of

radial heat losses, some convection is present even when the growth velocity

is zero.

Figure 10 is an example of a double-exposure photograph of the liquid

above the solid-liquid interface showing the double images of several marker

particles in the liquid, from which convective flow can be measured. The

dense stream of particles results from the break-up of an agglomeration of

particles and does not indicate a region of unusual flow conditions. Figure

11 shows flow velocities measured from this photograph (with the long, line

indicating the dense stream of particles). The crystallization velocity in

these samples was zero, and when crystallization starts the fluid flow field

remains similar in shape but the velocities increase. Thus, Figure 12 gives,

as a function of growth velocity, an example of the upward flow velocity at

the center of the tube 2 mm‘ above the solid-liquid interface for a particular

fixed set of crystallization conditions (temperature difference between hot

and cold bath, configuration of refractive index matching medium, insulation.

47
etc.)* The fluid flow in this case (pure material) is due entirely to the

radial gradients resulting from lateral heat losses and from the flow of heat

between the sample and the chamber walls. The velocities of flow are small

enough that the motion of the particles is not easily seen when viewed in

real time through the microscope, but are large enough to seriously disrupt

the development of any steady state one-dimensional distribution of solute as

assumed in the theoretical analysis. Therefore, it can be expected that the

crystallization conditions used for this set of experiments will not lead to

the onset of interfacial and convective instability at quite the same concentrations and velocities predicted by the theory.

In high purity succinonitrile, the radial gradients and fluid flow

patterns result in a solid-liquid interface which is concave upward to an

extent which increases with increasing crystallization velocity, but which

has no microstructural features except for grooves where grain boundaries or

sub-boundaries Intersect the solid-liquid interface. Any marker particles

which impinge upon the interface are pushed by it but tend to accumulate in

grain boundary grooves or trij unctions where they may become trapped by the ,

solid when they accumulate in sufficient numbers or the velocity becomes high

enough.

-3

In the sample containing 2.6 x 10 wt . % ethanol, the theory predicts

Interfacial instability at solidification velocities above about 10 ym/sec

and convective instability at velocities below about 2.5 ym/sec, with stability

predicted between these velocities. The experiments with this sample are

complicated by the presence of a bowl-shaped interface and thermally induced

convective flow similar to that seen in the pure sample. At steady state

with a growth velocity of 10 ym/s it is found that this sample solidifies

with a cellular solid/liquid Interface, i.e. , interfacial instability has

48
occurred. At a growth velocity of 6 um/sec it was found that approximately

one half of the solid-liquid interface, in the central part of the sample

tube, was in the form of a cellular interface while the outer part of the

sample was non-cellular . At 4 ym/sec, one fourth or less of the interface

was cellular and at slower velocities no cellular structure was observed.

Figure 13 contrasts the smooth but bowl-shaped interface of a high purity

sample with a cellular interface in an ethanol-doped sample. Thus, the

onset of interfacial instability occurred at or somewhat below the velocity

at which it was predicted. At low velocities, where the convective instability is predicted, no flow patterns clearly distinguishable from those

of the pure material could be distinguished among the background of purely

thermal convection.

The localization of interfacial instabilities in the central portion of

the sample tube was found to be a result of macroscopic shape change of the

interface. These shape changes are consistent with what would be expected

from the observed thermal convective flow patterns. Whereas the radial heat

flow and associated thermal convection produce a bowl-shaped (i.e., concave

upward) solid-liquid interface in both the pure and the ethanol-doped sample,

in the latter sample the interface developed an additional central depression

in the center which grew to a depth comparable to its width and then broke

down into a classical cellular structure (Figure 14). No such central

depression was ever seen in the pure material.

The development of the central depression in the interface only in the

sample doped with ethanol suggests that it may be attributed to the accumulation

of solute ahead of the interface directly under the point where the thermallyinduced convective flow converges and turns upward.

49
In the sample containing 2.6 x 10 wt.% ethanol, the solidus and

liquidus temperatures differed by 0.2 K. Interface temperature variations

due to the lateral distribution of solute caused by the thermal convection

can be expected to fall within this range. In a sample solidifying at 2 ym/s in

a temperature gradient of about 6 K/cm, it was found that the axial depression

achieved a maximum depth of about 0.5 mm before the cellular structure

developed. Thus, the interface temperature would vary by 0. 3 K in a depression

of this depth if we neglect the influence of the latent heat, in fair agreement

with the melting point range.

When the same experiment was repeated but with the ring heater surrounding

the interface activated to a power of 0.6 watts, the solid-liquid interface

was greatly flattened and the central depression did not develop even after

three hours of growth. As a result, the cellular interface did not develop.

Although convection measurements were not made in this experiment due to

particle viewing difficulties through the ring heater holder, the flatter

shape of the interface clearly indicated a lower driving force for thermal

convection.

Conclusions and Discussion

The onset of convective and interfacial instabilities during the unidirectional upward solidification of succinonitrile containing ethanol has

been predicted as a function of solute concentration and solidification

velocity. At some concentrations it is predicted that there is a range of

growth velocities in which solidification is stable, with interfacial instability predicted above this range and convective instability predicted

below this range.

50
Experimental measurements of solidification in samples of succinonitrile

solidified upward in a temperature gradient have shown that convection is

present even in high purity samples, where it must be attributed to radial

thermal gradients. The pattern of thermally-induced convective flow is consistent with the observed distortion of the solid-liquid interface due to

heat exchange between the solidifying sample and the container walls. The

thermally- induced convection interferes with any direct observation of the

transition from stability to convective instability because solidification

is not . unidirectional as assumed.

Interfacial instability is never observed in a high purity sample, as -3

predicted, and in a sample doped with 2.6 x 10 wt.% ethanol interfacial

instability is found in the form of a cellular solidification front extending

over the entire interface at velocities very close to that which is predicted.

However, at slower growth velocities the cellular solidification front is

also seen but covers only a restricted area at the center of the interface

and at still slower velocities no cellular structure is seen.

The localization of the cellular structure in the central part of the

interface was found to be the result of a macroscopic but transient depression

which develops in that part of the interface. This depression, which is

never seen in high purity samples, is attributed to solute redistribution

by the thermally-induced convective flow. Activation of a ring heater

surrounding the solid-liquid interface results in a flattening of the interface

and thus a reduction in the driving force for thermal convection. Under

these conditions, the formation of the central depression and of the cellular

interface is suppressed.

51
Table 1. Values Used in Numerical Calculations

Rac

Ra

Sc

Pr

P^^s

U

R

k

k

Figures 2-5

55,000.

152,900.

1.0

.01

.01

8.33

1.10

.3

1.0

Figures 6-7

11 , 000 .

611.

A

10.0

.01

.01

.833

.856

.3

1.0

52
References

1. S.R. Corlell, M.R. Cordes, W.J. Boettinger and R.F. Sekerka, Convective

and Interfacial Instabilities During Unidirectional Solidification of a

Binary Alloy, J. Crystal Growth 13 (1980).

2. J.S. Turner, Buoyancy Effects in Fluids, Cambridge University Press,

London (1973), Chapter 8.

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

Stability, PhysicoChemlcal Hydrodynamics 281 (1981).

4. J. Fromm, The Time Dependent Flow of an Incompressible Viscous Fluid,

Methods of Computational Physics, Vol. 3, Fundamental Methods in

Hydrodynamics, B. Alder, S. Fernbach and M. Rotenberg (eds). Academic

Press, New York, (1964), pp. 345-382.

5. E.C. DuFort and S.P. Frankel, Math. Tables and Other Aids to Computation,

3 , 135 (1953).

6. R.E. Richtmyer and K.W. Morton, Difference Methods for Initial Value

Problems, Interscience Publishers, New York (1967), pp. 211-216.

7. P. Swarztrauber and R. Sweet, Efficient FORTRAN Subprograms for the

Solution of Elliptic Partial Differential Equations, NCAR Technical Note

lA-1 (July 1975).

8. W. Chuck, A Model of Solutal Convection, Unpublished NBS Report,

September 1979.

9. NCAR Graphics Software, edited by T. Wright, NCAR Technical Note

(preliminary edition), April 1977.

10. K.A. Morrish, A Model of Double-Diffusive Convection, Unpublished NBS

Report, August 1980.

53
11. R. J. Schaefer and S. R. Coriell, Convective and Interfacial Instabilities

During Solidification of Succinonitrile Containing Ethanol, Materials

Processing in the Reduced Gravity Environment of Space, ed. by

Guy E. Rindone (Elsevier, 1982) p. 479.

12. R. J. Schaefer, M. E. Glicksman, and J. D. Ayers, High-Confidence Measurement of Solid/Liquid Surface Energy in a Pure Material, Phil. Mag. 3^, 725

(1975)

.

54
Fig.

1.

Schematic

diagram

of

the

initial

state

of

the

alloy.
C0NCENTRRTI0N C0NT0UR MRP RT T

T—I—I—I—I—I—I—I—I—I—I

\—I—

r

2.49749-01

OWTRJR rtVN .80000 Tl 3.2000 CMTM INTBIVRL IP .20000 PTI3.3)3 2.tn0

Fig. 2. Level lines of the concentration field at dimensionless time

t = .25. The dimensionless parameters for this case appear in

Table 1. The increment in the value of the concentration from

one line to the next is constant, so that the lines bunch together

where the gradients are large.

56
STREBM FUNCTI0N C0NT0UR MRP RT T = 2.49749-01

opnM mm -.zoooo Tt .aooo cmtm wwm. w .«BOO-ot pri$.s)s

Fig. 3. Level lines of the stream function at time t = .25. The

instantaneous velocity is tangent to the level lines. The

location and magnitude of the stream function's highs and lows are indicated in the figure.

.7«I4>-01

57
C0NCENTRRTI0N C0NT0UR MRP RT T = 1.24974+00

oMmit phih .toora ti s. 2ooo omnuR ipitownl w .«ooo pri3.3i:

Fig. 4. Level lines of the concentration field at steady state.

2.1721

58
STRERM FUNCTI0N C0NT0UR MRP RT T 1.24974*00

PTIS.9IS .4SMS

Fig. 5. Level lines of the stream function at steady state.

59
Fig. 6

C0NCENTRfiTI0N C0NT0UR MfiP AT T = 2.50019+00

ni9.3): 1.S3IS

. Level lines of the concentration field

t = 2.5. The dimensionless parameters

Table 1.

at dimensionless time

for this case appear in

60
5TRERM FUNCTI0N C0NT0UR MRP RT T 2.50019+00

i

t

aWTM fMN .70000 TO .70000 CMT«JR INTEKVRL r 10000»00 fTI9.3l=

Fig. 7. Level lines of the stream function at time t = 2.5.

.41280-01

61
100.0

10.0

1.0

0.1

Fig. 8. The critical concentration c of ethanol in succinonitrile above which

instability occurs as a function of the velocity V of unidirectional

upward solidification for a temperature gradient in the liquid of

10 K/cm. The ratio of the instability wavelength, 1, to the diffusion

boundary layer thickness, D/V, is also shown. The solid and dashed

lines correspond to interfacial and convective instabilities, respectively,

while the dotted line denotes an oscillatory convective instability.

62

(VX/D)
[

Fig. 9. Position of ring-shaped heater wire, embedded in transparent epoxy block

surrounding the solid-liquid interface. A fluid layer between the

epoxy block and the borosilicate glass sample tube is held in place by

surface tension.

I i

' \

63
sample of pure succinonitrile, showing motion in the liquid directly above

the solid-liquid interface near the center of the sample tube.

64
65

Fig.

11.

Measured

flow

velocities

from

Figure

9,

with

the

long

arrow

indicatin;

the

position

of

a

concentrated

stream

of

particles.

The

total

inside

diameter

of

the

sample

tube

is

17

mm;

this

diagram

shows

the

central

20%

of

the

tube

diameter.
Flow

Velocity

(/im/s)

Fig. 12. Upward fluid flow velocity 2 mm above the center of the solid- liquid

Interface during solidification of the pure succlnonitrile sample.

This flow is purely thermal in origin and is due to both heat losses to

the environment and (for non-zero growth velocities) to heat exchanges

with the container walls in the vicinity of the solid-liquid Interface.
Fig. 13. Smooth but bowl-shaped interface as seen during solidification of highpurity succinonitrile, contrasted to cellular interface seen at sufficiently

rapid solidification velocities in ethanol-doped succinonitrile.

67
Fig. 14. The macroscopic depression which develops in the center of the solid-liquid

interface of the ethanol-doped sample. Above, it is seen just at the

point where interfacial instability has appeared, and below, it is seen after a cellular structure has developed.

68

wf:
Advances in Space Research (Proceedings of 24th COSPAR Meeting)

MORPHOLOGICAL AND CONVECTIVE INSTABILITIES DURING SOLIDIFICATION

S. R. Coriell*, R. F. Boisvert,** J. I. Mickalonis*** , and

M. E. Glicksman***

*Metallurgy Division, **Scientific Computing Division

National Bureau of Standards, Washington, D.C. 20234 USA

***Materials Engineering Department

Rensselaer Polytechnic Institute, Troy, N.Y. 12181 USA

ABSTRACT

The stability of the flow between two vertical, infinite, rigid, coaxial

cylinders held at different temperatures is analyzed by linear stability

theory. For a Prandtl number of 22.8 and a radius ratio of 0.02, the flow is

unstable to an axisymmetric perturbation at a critical Grashof number of

2150; the wave speed of the instability is comparable to the maximum velocity

of the unperturbed flow. When the rigid outer cylinder-fluid interface is

replaced by a crystal-melt interface which can change shape, two new modes of

instability occur at lower Grashof numbers. There is an asymmetric instability

with a critical Grashof number of 180 and an axisymmetric instability with a critical Grashof number of 460; for both of these modes the wave speed of the

instability is several orders of magnitude smaller than the unperturbed flow

velocity.

These calculations were motivated by and are in general agreement with our recent experiments on succinonitrile. A long vertical cylindrical sample of

succinonitrile was heated by an electrical current through a coaxial vertical

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

the surrounding crystal-melt interface. Above a critical Grashof number of

circa 200, a helical crystal-melt interface formed which rotated steadily

about the cylinder axis; the wave speed was several orders of magnitude less

than the base flow velocity.

69
INTRODUCTION

There has been extensive research and development on the effect of fluid flow

on solidification and on the properties of the resulting solid [1-3]. Both

hydrodynamic instabilities [4] and morphological instabilities [5] have been

studied and the effect of convective flow on morphological stability has been

recently reviewed [6]. Delves [7] showed theoretically 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 and the effect of gravity on this interaction has also been studied [8-9]. 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.

As illustrated in Figure 1, recent experiments [10-11] have demonstrated an interface instability under conditions for which in the absence of flow, the

solid-melt interface would be morphologically stable . In these experiments,

a long vertical cylindrical sample of high purity succinonitrile was heated

by an electrical current through a long, coaxial, vertical wire, so that a vertical melt annulus formed between the coaxial heater and the surrounding

solid-melt Interface. The outer radius of the crystal was maintained at a constant temperature below the melting point (58.1 °C). With this arrangement

the temperature increases in the liquid and decreases in the solid with

distance from the crystal-melt interface, and consequently the interface

would be morphologically stable in the absence of fluid flow [6].

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.

The instability of a similar flow between two infinite vertical plates has

been reviewed by Gershuni and Zhukhovitskii [12]. Recently, Choi and Korpella

[13] and Shaaban and Ozisik [14] used linear stability analysis to calculate

the critical Grashof number for axisymmetrlc instabilities of the axisymmetric

70
i

I

1

1

1

i

!

i

Fig. 1. A long vertical cylindrical sample of succinonitrile

is heated by an electrical current through a coaxial

vertical wire so that a vertical melt annulus forms

between the wire and the surrounding crystal-melt interface. a) below a critical Grashof number of circa 200,

the crystal-melt interface is cylindrical; b-d) , above

the critical Grashof number, a helical crystal-melt

interface forms; d) multiple exposures show motion of

the helical interface.

flow described above occurring between two vertical, infinite, coaxial cylinders

held at different temperatures. We have analyzed this instability problem by

a numerical technique different from that of references 13-14 for parameters

more appropriate to the experiments described in references 10 and 11. In

this case, the flow between two vertical infinite coaxial cylinders is unstable

71
to an axisymmetric perturbation above a Grashof number of 2150 and the wave

speed of this perturbation is comparable to the unperturbed flow velocity.

In contrast, as illustrated in Figure 1 the experimental observations [10-11]

with a crystal-melt interface indicate an asymmetric instability at a critical

Grashof number of about 200 with a wave speed orders of magnitude less than

the unperturbed flow velocity. Therefore, we have Investigated the effect of a crystal-melt interface, which can change shape, on the stability of the

flow between an infinite vertical cylinder maintained at a constant temperature and a coaxial crystal-melt interface. As will be described, the presence

of the crystal-melt interface leads to additional modes of instability. The

lowest mode is asyimnetric with a critical Grashof number less than 200 and

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

flow.

THEORY

As illustrated in Figure 2 we consider a cylindrical coordinate system (r, 4>,

z) such that the melt is contained in the infinite annular region r. t ^ r

1 S and the crystal is contained in the infinite annular region r +L , where r . is the radius of the inner cylinder (wire) , r is the radius of

X SXi the unperturbed solid-liquid- interf ace , and (r + L ) is the outer radius of

S Li S the crystal. We define L=r^-r.,K=r./r^,L =L/L and a dimensionless

sL x_ X sL s s coordinate system (r, (J), z) = (r/L, ({) , z/L) . Further we measure temperature

in units of AT, the difference between the temperature at r = r^ and the

temperature of the unperturbed solid-liquid interface, time in units of L^/v,

and the fluid flow velocity in units of gaL^AT/v, where g is the acceleration

of gravity, which is in the negative z-direction, a is the coefficient of

thermal expansion, and v is the kinematic viscosity. We define dimensionless

fluid flow velocities (u^, u^ , u^) and dimensionless temperature fields T and

T^ in the melt and crystal, respectively. The unperturbed fluid flow velocity

is in the z-direction and is a function of r alone; similarly the unperturbed

temperature fields are functions of r alone.

72
CRYSTAL

Z

HEATING

Fig. 2. Schematic illustration of the cylindrical geometry.

We decompose the fluid velocities, temperature fields, and crystal-melt

interface shape r^ into an unperturbed and perturbed part, and assume that

the (|), z, t dependences of the perturbed quantities are of the form F((f>, z, t)

exp [incj) + iwz + at], where n is an integer, m is a spatial frequency, and

a = + ia^ is a complex number which determines the temporal behavior of

the system. If a^> 0 for any value of n and oj, the system is unstable. We

write

u = W (r)F,

u = W (r)F,

4

>

u = u (r) + W (r)F,

z zo z

T = T (r) + T(r)F,

o

T = T (r) + T (r)F,

s os s

73
= 1/(1-k) + 6F,

where the subscript o denotes unperturbed quantities, and 5 is the amplitude

of the perturbation of the solid-melt interface. The unperturbed fluid

velocity is given by [13]

u (r) = {ci[p^-l + (l-K^)lnp/lnKj - 4 (p Inp /ln< }/ { 16 (1 -k) , zo

with Cl = { (1-K^) (1-3k^) - 4K^lnK}/{ (1-K^) ^ + (1-K^)ln<} and p = (l-K)r.

The unperturbed velocity vanishes at the inner cylinder, r = </(1-k), and at

the unperturbed crystal-melt interface, r = 1/(1-k). In addition, the net

flow vanishes, i.e. , the integral of ru^^(r) from r = k/(1-k) to r = !/(!-<)

is zero. The unperturbed temperature fields are given by

T (r) = T + Inp/lnK

o e

T (r) = T + (k^ /k )lnp/lnK,

OS G L s

where T^ is the dimensionless temperature of the unperturbed crystal-melt

interface, and k and k are thermal conductivities of liquid and crystal,

ij s respectively (we assume that the unperturbed crystal is neither freezing nor melting)

.

We keep only terms linear in the perturbation quantities (W^, W^, W^, T, T^,

and 6); the fluid dynamical and temperature equations are then (in the OberbeckBoussinesq approximation with viscosity and thermal diffusivity assumed

constant)

(D*D-A)W - 5^G(f+g)(Du ) + ;7 - iwp = 0 (la)

z zo

D*(f+g) + (n/r)(f-g) + 21cjoW^ = 0 (lb)

(n/r)D*(f-g) + A(f+g) + 2Dp + 2icoDW^ = 0 (Ic)

(D*D-A^)T - J2PG(f+g)(DT^) = 0 (Id)

74
[D*D-A + (2n-l)/r^]g - Dp - (n/r)p = 0 (le)

(D*D-A^g)Tg =0 (If)

where D = 3/9r, D* = D + (1/r),

A = (n^/r^) + + iu)Gu + a, oz

A^ = (n^/r^) + H- P(io)Gu^^ + a),

Aj^ = (n^/r^) + 0)2 + P^a.

The function p is the radial part of the perturbed dimensionless pressure, f = W + iW , g

= W - iW^ [15], G = gaL^AT/v2 is the Grashof number, P is the

r <j) <P Prandtl number (ratio of kinematic viscosity to thermal diffuslvity k of

i-i the melt), and P = Pk /k with k the thermal diffuslvity of the solid. The

S 1j s s temperature field in the solid can be expressed in terms of Bessel functions

of order n with complex argument. The remaining equations for the fluid

variables are equivalent to a system of eight first-order equations for W^,

DW^, f, p, g, Dg, T, and DT. For n = 0, we have = 0, f = g, and equations

(Ib-c) are equivalent to equation (le) so there are six first-order differential

equations in the variables W^, DW^, f, p, T, and DT.

The boundary conditions at the inner cylinder are the usual no slip and no normal flow; we take the temperature as constant, so that the perturbed

temperature vanishes. Thus, at r = </ (1 -k) we have

= f = g = T = 0 (2)

The boundary conditions at the crystal-melt interface r^ are the usual solidification boundary conditions [6,8]. The no slip condition holds and the

normal flow is such as to balance the density change on solidification. The

temperature is continuous across the crystal-melt interface and is equal to

the equilibrium temperature which depends on interface curvature through the

75
Gibbs-Thomson equation. The net heat flux at the interface balances the

evolution of latent heat due to interface motion.

It is desirable for numerical purposes to set the boundary conditions at the

unperturbed interface rather than at the actual crystal-melt interface. This

is easily done in linear theory since any function, e.g., f(r), evaluated at

r^ = r^ + 6F can be written as f(r^) = f(r_ ) + (3f/9r)5F, where r^ =

I lo I lo lo

1/(1-k) and the partial derivative is evaluated at r = Again, for

numerical purposes, it is desirable to eliminate and 6 from the boundary

conditions and write them in terms of W^, f, g, and T. When this is done, we have the following linear, homogeneous, boundary conditions at the unperturbed

crystal-melt interface

ha.1 (f+g) + zoT = 0

f - g = 0

(3a)

(3b)">-

aiW + (Du )GT = 0

^ z zo

(3c)

-a2T + ai(k /k )(DT) = 0,

s (3d)

where e = (p /p )-l with p and p the density of melt and crystal, respectively,

S Lj s

I

ai = y{ [ (n^-l)/r^^] + co^} - G(DT^),

with Y = , where T^^ is the melting point of a planar interface and

r is the ratio of crystal-melt surface tension and latent heat, H^, per unit

volume,

a 2 = {(DT )/T }{ai + G(DT )[l-(k^/k )]} - Aa,

s s o L s

with A = H gaL^/(k v) . We assume that the outer radius of the crystalline

V s annulus is maintained at a fixed temperature so that the perturbed temperature

vanishes, i.e..

76
T (4)

s + L ) = 0 . s

The ratio (^J)T^)/T^, evaluated at and appearing in the definition of a2 , can be expressed in terms of Bessel functions.

For succinonitrile, it is an excellent approximation to take the thermal

properties of liquid and crystal equal, i.e. , k = k and P = P . Further,

it is reasonable (and numerical calculations have verified) that the density

change upon solidification and the effect of the crystal-melt surface tension

are not important. Thus, setting k = k , e = 0, and y “ the boundary

j_i s conditions, equations (3a-d) , reduce to the following:

f = g = 0, (5a)

- (DT )W + (Du )T = 0, (5b)

o z zo

{[(D^’ )/T ] + Aa/[G(DT ) ] - DT = 0. (5c)

s s o

We note that if the coefficient of T in equation (5c) is sufficiently large,

then the above boundary conditions further reduce tof=g=W^=T=0,

which are identical to those at a rigid boundary (see eq. (2)).

The numerical methods used to solve the linear eigenvalue problem defined by

the differential equations [eqs. (1)] and the boundary conditions [eqs. 2-4] are similar to those previously described [8]. We set = 0, and vary G and

o^, keeping all other parameters constant, until we find a solution of the

differential equations satisfying the boundary conditions. The program

SUPORT [16] is used to solve the differential equations and the program SNSQE

[17] is used for the non-linear iteration procedure. For the case of rigid

boundaries, we have compared our results for G and with n = 0 and P = 15

to those of Table 1 of Choi and Korpela [13], who used a quite different

numerical method; there is agreement to about 1%.
NUMERICAL RESULTS

We present results for physical properties appropriate to succinonitrile for

a radius ratio k = 0.02. We take L = 0.25 cm, and L = 1; for succinonitrile

s [19] we have P = = 22.8, e = 0.0283, y = 4.53 (10"^+), A = 1.017 (lO"^) , and

k^/k^ = 1.0. The Grashof number as a function of the spatial frequency m is

shown in Figure 3. We have found three distinct modes of instability, two

axisymmetric modes (n = 0) and an asymmetric mode (n = 1) . The lowest mode

is asymmetric with a minimum at w = 1.35 with G = 176 and = -0.977 (10 ^).

The lower S3nnmetric mode has a minimum at o) = 1.45 with G = 464 and = -3.65(10 ^) , whereas the higher s3Tnmetric mode has a minimum at w = 2.3 with

G = 2152 and a. = -43.7. If we take y = e = 0, the values of G and o.

1 1

change by less than 0.1%. For the remaining discussion we assume y

= e = 0 so that the boundary conditions at the unperturbed crystal-melt Interface,

equation (3), reduce to equation (5). The higher symmetric mode in Figure 3

is identical to that obtained if the crystal-melt interface is replaced

by a rigid isothemal boundary, i.e., replace the crystal-melt boundary

conditions, equation (5), with equation (2). For this mode, in fact, equation

(5) approximately reduces to equation (2) since the factor multiplying T in

equation (5c) is very large compared to unity, and hence T = 0 at the solidmelt boundary. Since with y = 0, 6 = -r/(DT^), it follows that there is very

little interface deformation in this mode.

In contrast the lower axisymmetric and the asymmetric mode only occur when a crystal-melt boundary is present. For these modes the coefficient of T in

equation (5c) is a complex number of order unity so that T and become the same order of magnitude. In turn, W^ does not vanish at the unperturbed

crystal-melt interface [see eq. (5b)].

For the remaining discussion we call the higher symmetric mode the "convective

mode" since it involves little Interface deformation and occurs when a rigid

boundary is present. The lower symmetric and the asymmetric mode which would

not occur in the absence of either fluid flow or the crystal-melt interface

will be denoted as "coupled modes." We define a dimensionless wave velocity

s = -o^/ (Gw) in units of (vG/L) = 0.104G cm/s so that positive s indicates

78
F=22B fc=0.02

B H3’ *

8i;w^ :

' L H j. in

Fig. 3. The Grashof number at the onset of instability

as a function of the spatial frequency m of a sinusoidal perturbation. The n = 0 modes are axisymmetric while the lowest mode (n = 1) is

asymmetric.

-3,

waves traveling upwards. The maximum unperturbed flow velocity is 8.2 (10 );

“3 “5 “5

the wave speeds are 8.8 (10 ), 5.4 (10 ), and 4.1 (10 ) for the convective,

coupled symmetric, and coupled asymmetric modes, respectively. The wave

speed for the convective mode is approximately the same as the maximum flow

velocity [13] , whereas the wave speed for the coupled modes are two orders of

magnitude smaller than the maximum flow velocity. It is interesting that the

crystal-melt transformation, which involves both melting and freezing with

latent heat absorption and evolution, can only follow the relatively slow

fluid motions of the coupled mode.

79
We have carried out additional computations with a 0 to estimate how

— o rapidly instability develops. With o =2 (10 ), the asymmetric coupled

mode has a minimum at w = 1.4 with G = 205 and o. =-1.06 (10 ). This

1

Grashof number is 16% above the critical Grashof number at the onset of

instability; the perturbation amplitudes would increase by an order of

magnitude in a time of 1.15 (10^) [2.77 (10^) s = 46 min]. Thus, it takes a relatively long time for the instability to develop. Further, calculations

indicate that the dimensional time for an instability to develop is roughly

proportional to L^. Even L = 1 cm would result in such a slow development of

the instability as to be unobservable over several days.

A more detailed study over a wider parametric range will be presented in a future publication. We remark that the coupled mode of instability occurs

for planar solid-melt Interfaces, e.g., let ic 1. In this case, of course,

there is no distinction between symmetric and asymmetric modes.

ACKNOWLEDGMENTS

The authors gratefully acknowledge the support of this investigation by the

Materials Processing in Space Program, National Aeronautics and Space . Administration. We also thank Drs. S. H. Davis, S. Ostrach, R. G. Rehm,

D. A. Saville, R. F. Sekerka, and J. A. Simmons for helpful discussions.
REFERENCES

1. J. R. Carruthers, in: Preparation and Properties of Solid State Materials , Vol. 3, eds. W. R. Wilcox and R. A. Lefever, Dekker, New York, 1977, p. 1.

2. D.T.J. Hurle, in; Current Topics in Materials Science , Vol. 2, eds.

E. Kaldis and H. J. Scheel, North-Holland, Amsterdam, 1977, p. 549.

3. S. M. Pimputkar and S. Ostrach, J. Crystal Growth 55 , 614 (1981).

4. P. G. Drazin and W. H. Reid, Hydrodynamic Stability , Cambridge University

Press, Cambridge, 1981.

5. J. S. Langer, Rev. Mod. Phys . 52 , 1 (1980).

6. S. R. Coriell and R. F. Sekerka, Physicochemical Hydrodynamics 2, 281 (1981).

7. R. T. Delves, in: Crystal Growth , Vol. 1, ed. B. R. Pamplin, Pergamon,

Oxford, 1974, p. 40.

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

J. Crystal Growth 49 , 13 (1980)

.

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

Adv. Space Res 1 ^ 5 (1981)

.

10. J. I. Mickalonis, M. S. Thesis, Materials Engineering Department, Rensselaer

Polytechnic Institute, Troy, New York (1982).

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

Seminar on Thermal Sciences, Miami, Florida, 1982.

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

Fluids , Keter, Jerusalem, 1976.

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

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

Transfer, Munich, Germany, 1982.

15. V. K. Garg, J. Fluid Mech . njO, 209 (1981).

16. M. R. Scott and H. A. Watts, SIAM J. Numerical Anal . 14 , 40 (1977).

17. SLATEC Common Math Library, The Program SNSQE was written by K. L. Hiebert

and is based on an algorithm of Powell [18].

18. M.J.D. Powell, in: Numerical Methods for Nonlinear Algebraic Equations , ed. P. Rabinowitz, Gordon and Breach, New York, 1970.

19. R. J. Schaefer and M. E. Glicksman, in: Modeling of Casting and Welding

Processes , eds. H. D. Brody and D. Apelian (The Metallurgical Society of

AIME, Warrendale, PA, 1981) p. 375.

81
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Task 3

Measurement of High Temperature Thermophysical Properties

of Tungsten Liquid and Solid

D. W. Bonnell

Materials Chemistry Division

Center for Materials Science

SUMMARY

With the end of phase two experiments, it has been necessary to

reassess strategy and consider alternative experimental approaches. The

most promising immediate line has been identified as improving the

mechanics of calorimeter operation. A complete plan for achieving this

improvement has been proposed, and a subset of the proposal implemented.

Actual testing of the new design has been carried out and was quite

successful, with 12 samples being successfully dropped, four in a nearly

molten state. Only one correctable fault in the gate linkage was found

and is being rectified.

A new procedure has been developed for fabricating samples which is

less expensive and more convenient. About 20 samples are now available.

This method of fabrication was developed in MBS' shops and consists of a

grinding process to give clean spheroidal specimens '^1.2 5 cm in diameter.

The larger sample has been tested successfully at GE. The larger samples

are expected to provide Improved stability.

A complete system automation plan has been developed which will act

as a model for future flight based systems as well as being a guide for

improvements to the current system. Details of timing, required hardware and comptater devices are discussed, and a sketch of software requirements is given.

Current plans are to schedule a new experimental series as soon as

possible. Rice is bringing a new student on board who has already been

acquainted with the experimental problems and apparatus.

83
INTRODUCTION

The evaluation of experimental procedures used in the interaction

between the General Electric Advanced Applications Laboratory (GE) and

Rice University (Rice) to measure high temperature enthalpy increments

for liquid and solid tungsten has been the primary aim of this task.

Coupling of GE’s unique apparatus [Wouch, 1978] for levitating and melting tungsten with Rice's expertise in high temperature measurements with

isoperibol calorimeters used in conjunction with levitation heating

[Chaudhuri, et al., 1970; Treverton and Margrave, 1971 a, b; Bonnell,

1972] has been expected to provide the means to measure hertofore estimated thermophysical properties of tungsten.

The location of tungsten at the extreme upper end of the melting

point scale of metals and elements provides a unique point for any correlation. Currently, questions such as the nature of the liquid state

heat capacity function, or even values of liquid heat capacity, are only

answered by estimates. A review of the state of experimental data for

the heat of fusion of tungsten [Bonnell, 1981] indicates the best value

is still uncertain by perhaps 50 percent, not significantly better than

an estimate.

A variety of theoretical, correlative and practical applications

await directly measured reliable data for the fundamental thermodynamic

quantities of heat content and heat and entropy of fusion for tungsten.

The importance of this effort, in spite of experimental difficulty, is

still clear. The difficulty of measurement by any technique is well

recognized. This effort represents the dividing line between experiments which can be performed in the terrestrial environment, and those

which require the advantage of no gravity to accomplish.

84
DISCUSSION

Details of the plan for automation are now in a condition where an

integrated approach is clear and natural. The component areas are: (1)

pyrometer; (2) window correction; (3) calorimeter data acquisition; (4)

calorimeter operation; and (5) RF and e-beam power. The techniques for

accomplishing items (1) and (2) have been developed and reported on

earlier [Bonnell, 1981]. earlier. Although it is probable that some

version of that system will be necessary to obtain precise temperatures

above the melting point of tungsten, we anticipate the majority of data

will be obtained at the melting point in the next series of experiments

and we have adequate techniques for hand correcting individual drops.

Experience has shown, however, that manual multicolor pyrometry is not

feasible with this system, and brightness temperatures will be the prime

mode of temperature determination.

Due to more serious problems, the actual implementation of the

pyrometry automation has been deferred [Frost and Stockoff, 1981]. A

major problem is the inability to control the sample, as it liquifies,

well enough to provide a controlled drop into the calorimeter. In a few

instances, when the calorimeter was not attached, it has been possible

to liquify tungsten. This is a dynamic process, requiring a skilled

operator to balance heating rate (as power input from the e-beam heating

system) and flotation point in the coil. Recovered melted samples were

resolidified as soon as molten. A sample was melted recently which,

on inadvertent release, fell to the bottom of a 1 meter drop tube and

splashed, giving a myriad of tiny spherules. This event establishes the

feasibility of the experiment, and answers the question of whether

liquid drops can be made. The analysis is based on an integrated form

of the radiative loss.

85
Assume the sample is a body of surface area A and total hemispheric

emissivity E^. At a temperature T, the radiative heat loss for the

sample, dH* during time dt is given by the Stephan-Boltzman law as

4

dH* = E^aT dt (1)

TO TO/ TO

where a = 5.67032 x 10 joule sec cm K (1.35524 x 10 calorie

_1 _2 -4 sec cm K ) [Cohen and Taylor, 1973] is the Stephan-Boltzman radiation

constant. The correction to (1) due to the fact that radiation actually

occurs to non-zero temperature surroundings is less than one part in 10^

above 3000 K and will be ignored. Let C* be the total constant pressure

heat capacity for a sample of mass m. Then

dH* = -C*dT

P

Combining (1) and (2) gives, on rearrangement

dt

( 2 )

(3)

Now assume that C* is independent of temperature and that there are no

temperature gradients in the body. The first assumption is consistant

with all experimental evidence for liquid refractory metals. The second

is reasonable for liquids, in view of the electromagnetic stirring of

the melt and good thermal conductivity of the sample. This should not

cause an error of more than a few percent even where (as in levitated

solids) noticable gradients ('^bO K) have been observed. With the initial conditions that at t = 0, T = T , the initial sample temperature,

o

(3) may be integrated from T^ to T corresponding to time from 0 to t

.
Solving for the final temperature gives

T = C*T^

P o 1/3

3E^AatT +

T o C*

P

(A)

Equation (A) can be substituted into (2) to give the total heat loss,

AH*, in time t as

r

Ah* = C*T

r p o (5)

where

3E AatT^

i o C*

P

( 6 )

The expression in brackets in (5) may be expanded by the binomial formula as

1 - ( 1 + y'

1/3 2 3

^ ' = l/3y - 2/9y + 13/81y (7)

Taking only the leading term of this expression, equations (5) and (6)

reduce to

AH* = l/3yC*T E E AatT^

r P o T o

( 8 )

which is the integral of (1) assuming constant temperature.

From equation (A) , we can estimate the degree of undercooling

achievable with the G. E. apparatus. The sample was "^10 gms of 17.6 gm/cc

[Smithells and Brandes, 1976] tungsten. The drop distance is '^1 meter

(t-0.A5 sec). C* is 0.A6 calories assuming the Hultgren, et al [1973]

-1 -1

estimate for C^(l) = 8.5 cal mole *K . This estimate may be in error

by 10-20 percent or more, but should not lead to an erroneous conclusion

in this case. The minimum temperature at drop is the melting point of

87
tungsten. The value of Cezairliyan [1972] of 3695 K is the current best

estimate for T (tungsten) and is considered more reliable than the

m

JANAF [1966] value. This value is in excellent agreement with the

average of investigations since 1960 [Cezairliyan, 1972]. The value of

is estimated by extrapolation from data reviewed by Smithell [Smithell and Brandes, 1974]. With these data T., , is obtained from equation (4) as 3476 K, an under cooling of ^6 percent. This is certainly a

reasonable value, well within accepted limits of supercooling [Perepezko,

1980]. This estimate ignores other loss modes, such as conduction

(negligible in vacuum) and material loss due to vaporization. The

largest error (except for and uncertainty) is an over-estimation

in the assumption of sphericity, which is considered to be negligible.

CALORIMETER OPERATION

The data acquisition system currently attached to the calorimeter

is a reasonable stand-alone system. With its interface, it provides

300 Baud RS232 serial output of the six digit temperature of the calorimeter block at the thermometer's clocked update period. The format was

carefully designed to allow direct acquisition and use by a teletypelike device or to allow interfacing to a computer port.

The calorimeter gate mechanisms have been the subject of development in the past year. When it became clear that manual operation would

not achieve a successful liquid drop except by extreme good fortune,

alternatives were considered. Figure 1 shows a sketch of proposed

modifications to the calorimeter jacket and shield assembly to solve

this problem. The salient features are (1) an optical drop detection

technique; (2) passive optical and electron beam shielding; (3) a poweractuated radiation shield; (4) an extension of the calorimeter jacket to

88
behave as a shield, with a thermometer to detect failed drops; (5) a

power-actuated block gate, (6) and a thermal shunt mechanism.

The optical drop detector is the prime item in the system. The

choice of optical detection was made because it seemed to impose the

least restrictions on future samples while retaining the non-contact

qualities of levitation. The primary problem anticipated was the fact

that a tungsten sample is radiating more than 1000 watts at its melting

point, with a substantial portion in the near IR and visible. The

movement of the sample causes considerable flickering in the chamber.

Even with good collimation, it was unclear how likely false triggering

would be with a simple photo-diode detector. One possibility would be

to provide an optical trap encompassing the field of view of the diode

as suggested in figure 1. A second effort suggestion was the use of an

ordinary interrupted light beam, encoding the light beam by modulating

it at a relatively high frequency (ca. 1-10 KHz). Then detection by a

synchronous detector would provide the necessary isolation. This is,

however, a relatively expensive and complex solution, and the simpler

one would be preferred. The actual implementation at GE utilized a 4 mm

glass light pipe with a small tantalum nose piece shaped to give a

roughly horizontal slit entrance aperture. The opening is located below

the work coil and is shaded from the sample by it. Bench tests implied

that this arrangement would yield a signal to noise trip point of the

order of 10:1. That is to say, if a sample at Tm (tungsten) = 3695 K is

the selected trip point, the trigger would not trip to a light source

from a sample at a temperature below '^2500 K, and the sample would have

to be at a temperature of 5000 K, before stray light would trip the

trigger. To achieve this rejection, it was necessary to shield the

light pipe from grazing light which actually constituted the most serious

89
spurious trigger source. Mr. Brian Stephenson of Rice and myself did

the final testing and adjustment of this device at GE. A direct map of

the apparatus' response to a bench light source was done to set the trip

relative to that lamp. Since the required gain change for the tungsten

experiments required decrease in sensitivity of the order of 1000-fold,

it was necessary to investigate the circuit gain function to center the

response at the desired temperature. The grazing incidence light response problem was solved by an overlapping spiral wrap of copper foil on

the light pipe. A series of tests were carried out on the trip mechanism

in situ with actual tungsten samples. In 12 drops, the mechanism tripped

at each drop and never false triggered.

The original gate suggestions (see figure 1) recommended rotating

gates driven by springs with a solenoid and pawl mechanism to release

the gates. The time between drop detector actuation and arrival of the

sample at the calorimeter is ^0.3 second. The available distance to the

first gate location cuts this time to '^0.1 second. A rotating gate is

the simplest way to achieve such short opening times without large

forces. However, GE, using commercial rotary solenoids, devised an

entirely different operating mechanism, which involved only the single

existing single mechanical feedthrough and operated both gates with a

single linkage. The available torque from the rotary solenoid is high

enough to open both gates in the necessary time. The problems with this

design are (1) a thermal isolation element must be part of the gate

linkage and (2) the drop-leaf operation of the main radiation gate does

not allow the extension of the jacket above the calorimeter as a collimating shield. (See figure 1). The live tests showed that the thermal isolation link was a weak design point. GE is now redesigning this

90
item. The collimating shield is probably not absolutely essential, and

the shield mechanism for the gate provides some protection. Further

testing will be required to determine if heat from this source is a

problem. The additional problems in getting the design of Figure 1

sequencing properly is a good argument for the redesign. Mr. Daniel

Rutecki of GE is to be complemented on his ingenuity and understanding

of the requirements of the situation in converting my initial design

into working hardware.

The entire new system has been tested for functionality at GE with

tungsten samples at the melting point. Four samples were partially

molten, and one, which was held momentarily at liquefaction appeared to

have been more than 80 percent molten on inadvertent loss. This success

is as good as all our previous efforts, and represents a success ratio

very much higher than any previous effot.

The rest of the features of the design of figure 1 have not yet

been implemented. The thermal shunt deserves some mention. A significant problem in high temperature drop calorimetry is cycling the system.

\^ere vacuum insulation is concerned, the drift rate of the calorimeter

must require hours to return to starting conditions. Normally, this is

not a problem, as it is necessary to dismount the calorimeter to weigh

the dropped specimen. It is a simple matter to adjust the block to

starting temperature conditions at this time. However, in this system,

a failed drop requires dismounting as well, just to avoid this several

hour wait. Under the best of conditions, a run requires about two hours

from drop to dismount. Dismount time is about one hour. A high conductance thermal link which could be made between the jacket and block

could shorten this three hour turnaround considerably in the case of

91
failed drops. In the design shown, it also seems feasible to incorporate the post drop weighing process with the thermal shunt. If the

thermal shunt contained a sensitive load cell, it would be possible to

lift just the cup to obtain the weight. Commercial vendors exist who

can supply remote weighing devices with the necessary 1 mg or better

sensitivity for loads of more than the ^200 gm of the sample cup. These

features are considered highly desirable in the fully automated system.

SYSTEM AUTOMATION

Figure 2 shows a block schematic of the suggested automated computer system. This is a straight-forward automation problem which

should be implemented with off-the-shelf hardware. From the hardware

viewpoint, the external input/output (I/O) structure is most important.

Figure 2 breaks this down to the signal level and organizes it in terms

of I/O types. Analog input signals are characterized by resolution and

digitizing speed. The only signal requiring any real speed is OBJECT

DETECTOR INTENSITY and then only if computer software is to replace the

comparator logic currently being used. Assuming the detector is located

1.5 cm below the sample levitation coil, the sample will be in the view

point of the detector for '^0.016 sec while falling. Assuming several

observations are necessary to trigger reliably, a sampling rate of

'^250 Hz is required. The PYROMETER SIGNAL is updated at the camera

frame rate, ^^30 Hz [see Bonnell, 1981 for details]. The maximum resolution required is in JACKET TEMPERATURE and PYROMETER SIGNAL. The

amplifiers and track and hold for the PYROMETER SIGNAL need be stable to

no better than 1:2000, so twelve bit bipolar digitization is more than

adequate. These requirements are now well within the state-of-the-art

for monolithic analog-to-digital converters. In fact, a single multiplexed converter such as the Analog Devices AD363 system [Analog Devices,

92
1980] a 16 channel multiplexed, 30 KHz, 12 bit system, is quite adequate. Assuming a computer with a 32 bit word (and instruction cycle

time) of one MHz, and an interrupt driven I/O routine of 100 instructions and overhead of 12 instructions per interrupt, all seven channels

can be serviced at a 1 KHz rate utilizing only twenty percent of machine

capacity. Having the OBJECT DETECTOR INTENSITY input to, say channel 0

and 4, a sample rate of 1 KHz for eight channels causes the OBJECT

DETECTOR INTENSITY to be sampled at 250 Hz, and all other signals at

125 Hz. This would require a 16 bit (or 2-8 bit) bidirectional parallel port to handle channel addressing, hand shaking and transfer, and a

real-time timer to clock the acquisition. The assumption of 100 instructions for the I/O loop assumes a self triggered acquisition system. A

considerable safety factor is included in these estimates, as this

sampling rate would provide ^40 samples over the falling ball, and four

readings of each sample of the pyrometer.

The Parallel I/O channel is used for reference and control information. Eight aperture sizes and eight band pass filters is quite adequate to provide multicolor pyrometry for the temperature range required.

A simple binary coded switch on the APERTURE SIZE and the FILTER //

shafts (see Bonnell, 1981 for pyrometer system details) provide absolute

position indication with no operator initialization. INDUCTION HEAT STATE

and ELECTRON BEAM STATE are indicators as to the on/off status of the

two heating units, allowing processing code to correct for effects due

to these power sources. The SHIELD TEMPERATURE, INDUCTION POWER, and

ELECTRON BEAM POWER provides for correction of the leakage rate to the

calorimeter by accounting for strong e-beam and induction effects. This

information will be stored with the calorimeter temperature for final

93
data reduction. As these effects have not been negligible in the past,

corrections are necessary.

The timing on processing the parallel port is keyed to the pyrometry filter action, and should involve processing of the order of

25 percent of the A-to-D machine overhead. The Parallel I/O channels

are to be processed as a task syncronous with PYROMETER SIGNAL and would

be scheduled for reading on the completion of an aperture or filter

change.

Stepping the aperture and filter wheels could be done directly, but

the timing required is of the order of the A-to-D sample time. To be

specific, if eight apertures are available and eight filters are in use,

it is desirable to expect to use a different filter to acquire a different color temperature at each reading of the pyrometer signal. A

30 Hz update rate on the pyrometer signal implies a 30 Hz filter (and

aperture, if necessary) change rate. Eight filters/apertures mounted on

a wheel require 25 steps of a nominal 200 step/revolution stepper motor.

Typical small stepping motors are capable of averaging 3000 steps/sec

with proper ramping. At this rate, a filter can be changed between

pyrometer reading updates. Considering the other processing expected of

the computer, and the inexpensiveness of intelligent stepper drivers

with firmware ramping, it is proposed to use devices such as the Cybernetic Micro System CY512 [CMS 1982] stepper system which is capable of

independent automatic operation of the stepping motors. With this

system each motor can be fully programmed and operated via a standard

eight bit parallel port with handshake lines. An actual movement is

invoked by sending a seven bit code (e.g., ASCII "D") . This is to be

gated by an external enable timer started by the pyrometer frame clock

and open for the dead time allowable until the stepper cannot be started

94
and complete the movement before the pyrometer reading is updated. This

tr ial-and-error set up is not essential, but provides insurance in the

case where changes in coding cause a running system to exceed CPU resources. When set up in this way, the interrupt routine would, on

recognizing the pyrometer track and hold latch, start the task to step a

filter (and aperture, as necessary), and, on checking the APEPvTURE SIZE

and FILTER NUMBER on receipt of PYROMETER SIGNAL, determine that the

filter was in place at data acquisition time. This additional code will

run in interrupt mode and should easily fit in one of the eight A-to-D

time slices, thus using about three percent of available CPU time at

worst

.

The RELAY CONTROL channel is in actuality a bit parallel port with

at least four high speed "relay” closures under computer control. Speed

requirements should be better than the read rate of the OBJECT DETECTOR

INTENSITY, since actions such as gate release, e-beam enable, etc. may

need to be accomplished in this time frame. The contacts should be

capable of 1-3 amps. The functions are self explanatory, with the

relays being in series with mechanical switches for INDUCTION HEAT, EBEAM ENABLE and RADIATION GATES UNLOCK. The SAMPLE RELEASE relay

parallels the manual release and a bit should be set for return by the

manual switch if the sample is manually released.

The THERMAL SHUNT was discussed earlier; the actual actuating

mechanism could be a solenoid, a small pneumatic or hydraulic cylinder

or a mechanical linkage. The maximum throw should not need to be more

than 1/2", as the calorimeter block could be hollowed out to accomodate

a shaft to lift the cup for weighing. A tiny pneumatic cylinder is

probably the most convenient item. Note that weighing could occur

during the thermal shunt operation.

95
The SERIAL I/O channels are simple RS232 protocol ASCII coded

channels. Each channel should be capable of any normal BAUD rate between 110 and 9600, especially to include 300 BAUD, 1200 BAUD and

9600 BAUD. All these channels will be operated Interrupt driven on a

character by character basis. Since input rates are very low (SYSTEM

CONSOLE, 1 or 2 char/sec intermittently; CALORIMETER TEMPERATURE, 10

characters at 300 BAUD every 10 seconds; SAMPLE WEIGHT, 10 characters at

300 BAUD every experiment) the characters can be processed individually

as scheduled tasks.

The above requirements Indicate a minimum CPU of the Z80 class,

operating at 4 MHz. Memory requirements are set by the amount of code

and data, but even if a substantial portion of the background tasks are

BASIC or FORTRAN, 65 K bytes should be sufficient for the system needs.

An existing FORTRAN program for the post processing has been run in less

than 30K bytes. An interrupt timer is required to drive the A-to-D

multiplexer and a real time time-of-day clock should be available against

which to log the data, as well as provide between run cycle timing.

Twin 5-1/4” floppy disks will be required to store and edit data, and

hold system software. A minimal hardcopy unit would be a simple low

speed 80 column impact printer. Graphics capability or a plotter could

be used to provide post run diagnostics, but is not essential for day to

day operation. The display can be a standard alphanumeric CRT with

80 character by 24 line black and white capability. This can be the

system display device under non-automation conditions. The software

should keep the operating parameters indicated in figure 2 as well as a

variety of status displays running, updated at scheduled intervales.

The variety of microprocessor systems meeting these requirements includes

most units on the market. Since extensive software development is

96
The SERIAL I/O channels are simple RS232 protocol ASCII coded

channels. Each channel should be capable of any normal BAUD rate between 110 and 9600, especially to include 300 BAUD, 1200 BAUD and

9600 BAUD. All these channels will be operated interrupt driven on a

character by character basis. Since input rates are very low (SYSTEM

CONSOLE, 1 or 2 char/sec intermittently; CALORIMETER TEMPERATURE, 10

characters at 300 BAUD every 10 seconds; SAMPLE WEIGHT, 10 characters at

300 BAUD every experiment) the characters can be processed individually

as scheduled tasks.

The above requirements indicate a minimum CPU of the Z80 class,

operating at 4 MHz. Memory requirements are set by the amount of code

and data, but even if a substantial portion of the background tasks are

BASIC or FORTRAN, 65 K bytes should be sufficient for the system needs.

An existing FORTRAN program for the part processing has been run in less

than 30K bytes. An interrupt timer is required to drive the A-to-D

multiplexer and a real time time-of-day clock should be available against

which to log the data, as well as provide between run cycle timing.

Twin 5-1/4" floppy disks will be required to store and edit data, and

hold system software. A minimal hardcopy unit would be a simple low

speed 80 col impact printer. Graphics capability or a plotter could be

used to provide post run diagnostics, but is not essential for day to

day operation. The display can be a standard alphanumeric CRT with

80 char by 24 line black and white capability. This can be the system

display device under non-automation conditions. The software should

keep the operating parameters Indicated in figure 2 as well as a variety

of status displays running, updated at scheduled intervales. The variety of microprocessor systems meeting these requirements includes most

units on the market. Since extensive software development is involved
involved and at least six parallel ports and three serial ports are

needed, the choice should be made on a basis of convenience of peripherals and availability of interrupt driven task scheduled software

packages in assembly code.

The software is a non-trivial package and beyond the scope of this

task to develop. However, the basic features can be sketched around the

hardware design. A core interrupt routine to service the A-to-D system,

the stepping motor parallel ports, the serial ports and the clocks is

required. This can be vectored, but a single level polled structure

should be adequate. The foreground program would be a scheduled task

execution routine, which processes tasks scheduled by the interrupt

routines. Typical of such scheduled tasks are (1) Process calorimeter

temperature and report drift status; (2) collect current aperture size

and filter, and report sample temperature(s) ; (3) change experiment mode

to (a) initialize, (b) levitating, (c) melting, sample acceptable, (d)

drop sample, (e) post drop process, (f) end of experiment; (4) Request

sample weight and thermal shunt system; (5) process sample weight;

(6) report system status of (a) shield temperature, (b) coating absorbence, (c) induction and e-beam power levels, (d) filter in use, etc.

As the operating system is developed, a further breakdown of the

task structure will be possible. Changes, such as the current calorimeter gate automation, may significantly alter operation. The current

mechanical tripping of the calorimeter makes it possible for the computer not to monitor the drop detector and removes considerable complexity from the software. The loss in generality is not yet clear, but

is probably not significant.

98
; %

Table 1 presents an operation scenario which details the overall

progress of an automated experiment. With the controls and monitors

described, the system will follow and report on each stage of the experiment without operator intervention. The only operator action required as far as the calorimeter system is concerned, will be a simple

GO/NO GO decision at step 4. Step 5 can be manual or inadvertant and

the result will be the same, and processing without further operator

action will take place.

This automation scheme expects direct operator action to control

sample processing. That is to say, an operator will externally adjust

induction power and e-beam power to achieve desired results. It would

j be possible to include an ability to perform these actions under auto-

^ matic control, but a complete video image (at least 20 x 20 pixel resolui tion) of the coil region should be monitored to determine sample stability,

I and several (at least three) 12 bit digital-to-analog outputs would be

required to control the induction power and e-beam power. This addition0 ,

I

al complexity would perhaps double the size of the operating system

i

I

software, and will require a processor ^five times the instruction cycle

r time and data fetch time of the Z80, i.e., a Motorola 68000 grade GPU or

!

a minicomputer system. This is not a prohibitive problem, simply one

I which I recommend be considered after the current system functions well.

j

t

99
Table 1

OPERATION SCENARIO

(1) CALORIMETER AUTOMATION IN "MONITOR MODE —Dropped object detector disarmed (radiation gates are locked closed) —Follow initial drift —Notify operator when minimum drift data acquired

(2) SAMPLE LEVITATED, STABILIZED —Check for influence, correct drift data, notify operator —Monitor induction power, sample temperature

(3) AUXILIARY HEATING ON (ELECTRON-BEAM) —Check for additional heat input, monitor shields...

(4) OUTGASSING. . .SATISFACTORY—ARM OBJECT DETECTOR (OPERATOR DECISION)

(5) SAMPLE DROPS (or is dropped)

--Object detector signals

--Induction furnace, e-Beam off

--Gates actuate in sequence

--Shield monitored for BAD DROP, GO TO ERROR —Dropped object detector is disarmed

(6) DATA SYSTEM ACQUIRES A0, DETERMINES END OF RUN

(7) GATES RESET, Visual or Other Inspection Satisfactory

--Thermal shunt activated, mass determined —Store data —Hardcopy results... —Diactive thermal shunt

RESTART ^ GO TO 1

(8) ERROR —Calorimeter thermal shunt actuated

--Inspection to determine fault —Gates checked for operation...

RESTART GO TO 7
REFERENCES

1. Analog Devices, Inc. Data Acquisition Components and Subsystems , Norwood, MA (1980).

2. ASCII, American Standard Code for Information Interchange. American National Standards Institute (ANSI) Standard X3.4—1977.

3. D. W. Bonnell, NBSIR-81 ed J. R. Manning (1981).

4. D. W. Bonnell, "Property Measurements at High Temperatures —Levitation Calorimetry Studies of Liquid Metals", Ph.D Thesis, Rice Univ.

(1972).

5. A. Cezairliyan, High Temp. Sci. 248 (1972).

6. A. K. Chaudhuri, D. W. Bonnell, A. L. Ford, and J. L. Margrave,

High Temp. Sci., 203 (1970).

7. E. R. Cohen and B. N. Taylor, J. Phys. Chem. Ref. Data 663 (1973).

8. Cybernetic Microsystems, CY512 Intelligent Stepper Motor Controller

Los Altos, CA 1982.

9. R. T. Frost and E. Stockoff, private communication (1981).

10. R. Hultgren, P. D. Desai, D. T. Hawkins, M. Gleiser, K. K. Kelley,

D. D. Wagman, Selected Values of the Thermodynamic Properties of

the Elements , American Society for Metals, Ohio (1973).

11. JANAF Thermo chemical Tables, 2nd Edition, NSRDS-NBS 37 (1971) and

updates

.

t 12. T. A. Sidorov and M. M. Mezdrogina, Fiz. Khim. Granits Razdela Kontaktiruyuschchikh Faz , ed V. Eremenko, p. 45-7, Naukova Dumka,

Diev, U.S.S.R. (1976)

.

13. C. J. Smithells and E. A. Brandes, Metals Reference Book , 5th

edition, Butterworths , London (1976) (E^, p. 1023) (pm, p. 944).

14. J. A. Treverton and J. L. Margrave (a) J. Phys. Chem. 3737

(1971) . (b) J. Chem. Thermodynam. 473 (1971) .

15. G. Wouch, "Containerless Melting and Solidification of Metals and

Alloys in the Terrestrial and Space Environment", Ph.D Thesis,

Drexel Univ. (1978).

*Trade names referenced in this report are provided as examples for

specificity of state of the art capability in commercial devices. Such

references are not intended to imply uniqueness of these devices, or the

unsuitability of similar devices of another manufacturer to the applications.

101
TOP VIEW

DRIVE SPRING

SOLENOID

RELEASE

ROTATION

ROTATING

RADIATION

GATE

o o

O O

o o

oo oo

LIGHT TRAPhP;

ROTATING

RADIATION

GATE

(ANALOG TEMPERATURE)

PHOTODIODE

OBJECT

DETECTOR

(ANALOG INTENSITY)

- SHIELD

(ANALOG

TEMPERATURE)

TEMPERATURE

CONTROLLED

JACKET

BLOCK

RADIATION

GATE

SPRING DRIVE

SOLENOID & PAWL

RELEASE

THERMAL SHUNT

BLOCK

THERMOMETER

(SERIAL TEMPERATURE)

Fie. 1. Proposed calorimeter modifications to allow full automation.

top view is a detail of the main rotating radxation gate assemb y.

102
Fig.

2.

Block

diagram

of

automation

system

with

a

detailed

breakdown

of

the

necessary

input/output

structure.

NBS-114A (REV. 2-8C) U.S. DEPT. OF COMM. 1. PUBLICATION OR 2. Performing Organ. Report No. 3. Publication Date BIBLIOGRAPHIC DATA

SHEET (See instructions)

REPORT NO. NBSIR 82-2560 July 1982

4. TITLE AND SUBTITLE

NBS: Materials Measurements

5. AUTHOR(S)

J. R. Manning

6. PERFORMING ORGANIZATION (If joint or other than NBS. see instructions)

NATiGNAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract/Grant No. NASA^I^Order

8 . Type of Report & Period Covered

Annual Report

April 1 , 1981 - Mar. 31

,

9.

oPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street, City. State. ZIP)

1982

Office of Materials Processing in Space

National Aeronautics and Space Administration

Washington, D. C. 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 Processinc

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

Materials) covering the period April 1, 1981 to March 31, 1982.

This work is directed toward measurement of materials properties important to

the design and interpretation of space processing experiments and investigations of

how the space environment may offer a unique opportunity for performing improved

measurements and producing materials with improved properties. This 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

^ Emphasis in Tasks 1 and 2 is on how the reduced gravity obtained in space flight

ji|; 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.

iT2. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolons)

Auger spectroscopy; convection; gallium- tin alloys; levitation calorimetry;

: segregation; specific heat; surface tension; thermophysical properties; tungsten.

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