Reference

NBS

Publications

NAT'L INST. OF STAND & TECH ilR 81-2295

AlllQb OMSbMO

ivJS: Materials Measurements

U.S. Department of Commerce

National Bureau of Standards

Washington, DC 20234

Phone: (301) 921-3354

June 1981

Annual Report

For the Period 1 April 1 980 - 31 March 1 981

NASA Government Order H-27954B

Properties of Electronic Materials

/

-Q0

]_QQ U.S DEPARTMENT OF COMMERCE

.U56

81-2295

1981

NATIONAL BUREAU OF STANDARDS

NBSIR 81-2295

NBS: MATERIALS MEASUREMENTS

NATIONAL BUKBAV

OF STANDARDS

library

AUG 1 8 1981

' IS £

mo.

J. R. Manning

U.S. Department of Commerce

National Bureau of Standards

Washington, DC 20234

Phone: (301) 921-3354

June 1981

Annual Report

For the Period 1 April 1980 - 31 March 1981

NASA Government Order H-27954B

Properties of Electronic Materials

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

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

TABLE OF CONTENTS

Page

Summary 1-5

Task 1 - SURFACE TENSIONS AND THEIR VARIATIONS WITH TEMPERATURE

AND IMPURITIES by

S. C. Hardy and J. Fine 7-31

Task 2 - CONVECTION DURING UNIDIRECTIONAL SOLIDIFICATION by

S. R. Coriell and R. J. Schaefer 33-61

Task 3 - MEASUREMENT OF HIGH TEMPERATURE THERMOPHYSICAL PROPERTIES OF

TUNGSTEN LIQUID AND SOLID by

D. W. Bonnell 63-90

Task 4 - HIGH TEMPERATURE THERMOPHYSICAL PROPERTIES OF REFRACTORY

MATERIALS FROM FREE COOLING EXPERIMENTS by

J. H. Colwell, A. Cezairliyan and L. A. Schmid 91-138

Task 5 - EXPERIMENTAL AND THEORETICAL STUDIES IN WETTING AND MULTILAYER

ADSORPTION by

M. R. Moldover, J. W. Schmidt and J. W. Cahn 139-148

l

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, 1980 to March 31, 1981.

The work emphasizes measurement of materials properties and thermophysical

properties important in materials processing applications. It has two main

thrusts

:

1) Carrying out precision measurements in space and investigating the

feasibility of improved measurements when the space environment offers a

unique opportunity for performing such measurements. These measurements

would be useful for either space processing or processes on the ground.

2) Obtaining precision measurements on materials properties when these

properties are important to the design and interpretation of space processing

experiments. These measurements would be carried out either in space or on

the ground.

This work has been carried out in five tasks. These tasks have, as two

of their focal points, the role of convection effects and the role of container effects, both of which would differ in space-based experiments from

those found in ground-based experiments. The results obtained for each task

are given in detailed summaries in the body of the report.

Briefly, in Task 1 - Surface Tensions and Their Variations with Temperature

and Impurities - special equipment has been put into operation which, for the

first time, allows simultaneous liquid-vapor surface tension and Auger surface

1
concentration measurements to be made on metal liquids. This apparatus has

been used to analyze surface tensions and reactions on pure gallium and a

gallium- tin alloy, providing information which is important for prediction

and analysis of surface processes on liquids in space flight experiments.

Applications include, for example, Marangoni convection and surface segregation

effects occurring during solidification. In this work, ionic bombardment

sputtering with argon was found to be a very promising method for producing

clean liquid surfaces, since unexpected convection currents appeared which

allow rapid and efficient cleaning of carbon particles from the whole surface,

or indeed the whole volume, of the liquid droplet. For gallium specimens

cleaned by this method, a linear dependence of surface tension on temperature

was found, in contrast to the quadratic dependence found for less clean

samples. For Ga-Sn (1.9 at.%) alloys the Auger measurements showed strong

temperature-dependent surface segregation of Sn to the surface. This result

was consistent with the simultaneous surface tension measurements made on the

alloys

.

In Task 2 - Convection During Unidirectional Solidification - the

succinonitrile-ethanol system is being investigated as a transparent material

id which convective flows produced by simultaneous temperature and composition

gradients during unidirectional solidification can be quantitatively

measured and compared with theoretical predictions. Knowledge of convective

conditions is important in comparing solidification in microgravity with that

on earth and in evaluating the advantages of space processing. In order to

determine the distribution coefficient k, the phase diagram in the low

2
ethanol range was determined, and k was measured as 0.044. Calculations to

predict the range of convective instabilities then were made using this

distribution coefficient value. It was predicted that ethanol concentrations

-3 -2

in the range 10 to 10 weight percent can provide either convective,

interfacial or oscillatory instability depending on the values of temperature

gradients and solidification velocities. An apparatus has been constructed

to measure convective flows in these ranges in this material, and initial

baseline thermal convection measurements have been made. Velocities are

determined by time lapse photography of polyvinyltoluene spheres having

nearly the same density as the solidifying liquid with the spheres acting as

markers in the liquid to map the convective flows.

In Task 3 - Measurement of High Temperature Thermophysical Properties of

Tungsten Liquid and Solid - this task contributes to a joint effort of the

General Electric Advanced Applications Laboratory (GE) , Rice University

(RICE) and the National Bureau of Standards (NBS) to develop levitation/drop

calorimetry techniques. The immediate objective is to measure the specific

heat of liquid tungsten, in order to provide key benchmark measurements on

thermophysical properties. The current phase of this effort has involved

completing the physical connection of the RICE calorimetric system to the GE

levitation facility. In this phase, improvements have been made particularly

in the temperature measurement area. A data logging system has been added to

eliminate the need for a human observer of the calorimeter heat trend. This

automation, along with proposed automation of the calorimeter data system,

which is made necessary by the erratic levitation holding times of the molten

tungsten drops, can also be expected to aid in automatic operation of similar

space flight experiments.

3
In Task 4 - High Temperature Thermophysical Properties of Refractory

Materials from Free Cooling Experiments - a free-cooling method for measuring

specific heats of reactive and refractory materials is being developed. This

method allows use of the positioning-levitation capabilities available in

space to provide thermophysical measurements without contact with a container.

Tests with a millisecond pulse heating system have been performed using pyroelectric radiometer devices to measure the energy radiated from the specimen.

In addition, calculations have been performed to relate the surface temperature

and radiated energy of spherical samples (which can be measured directly with

a pyrometer and the newly developed pyroelectric radiometer devices) to the

specific heat, thermal conductivity, and emissivity of the sample. Under

certain conditions, it should be possible to determine the temperature dependence of these parameters from simultaneous measurements of energy radiated

and temperature change during free-cooling of a levitated sample.

In Task 5 - Experimental and Theoretical Studies in Wetting and Multilayer

Adsorption - ellipsometric parameters of the liquid-vapor interface between a

binary C_,F^-C^H^ mixture and air have been measured near the predicted

transition from high adsorption to low adsorption. Also a method of measuring

and recording the index of refraction of this transparent binary mixture

simultaneously with measurement of the ellipsometric parameters has been

tested. Separate laser light sources are used for the two types of measurements. Since the index of refraction measurement is extremely sensitive to

the appearance of macroscopic amounts of a second liquid phase, it can serve

to precisely locate the miscibility gap. Thus the simultaneous measurement of

both the index of refraction and the ellipsometric parameters will locate the

4
adsorption transition with respect to the miscibility gap in the same sample.

Measurements of ellipsometric and other optical parameters will allow evaluation

of wetting behavior involving immiscible materials, a topic of major significance

in space processing applications.

5

Task 1

Surface Tensions and Their Variations

with Temperature and Impurities

S. C. Hardy

Metallurgy Division

Center for Materials Science

and

J. Fine

Surface Science Division

Center for Chemical Physics

Summary

Sessile drops of pure gallium and a gallium-tin (1.9 at.%) alloy have

been studied in an Auger spectrometer. The surfaces of the drops were cleaned

by heating to 500 °C and by ionic bombardment with argon. The surface tension

of pure gallium cleaned in this way was found to vary linearly with temperature and could be represented by

y = 724.1 - .0721 T (°C)

.

This linear relationship is significantly different from the quadratic

variation found previously for samples cleaned only by heating in vacuum and

hydrogen. The difference in the two results is tentatively attributed to the

presence of small carbon platelets on the surfaces of the unsputtered drops.

The segregation of tin at the surface of gallium-1. 9% tin drop was

measured by comparing the tin Auger line amplitudes with that of a pure tin

sample with the same spectrometer settings. Although there is some uncertainty

in the attenuation length and backscatter ing factors used in the analysis.

7
surface concentrations in excess of two monolayers were found at 30 °C, with a

sharp decline with increasing temperature. Surface concentrations calculated

from surface tension measurements were in general agreement with this conclusion.

8

'
Introduct ion

It has been widely recognized that surface tension measurements of

liquid metals and semiconductors are flawed by the absence of any independent

characterization of the chemical species present on the surfaces. The development of modern surface spectroscopies makes available a host of techniques

which can address this gap in our knowledge of liquid surfaces, at least for

some low vapor pressure materials. In previously reported work, Auger electron

spectroscopy (A.E.S.) was applied to liquid gallium and provided valuable

information about the state of the surface. The configuration of the spectrometer, however, did not allow the surface tension to be measured with the

desired accuracy. This past year has been largely spent in the assembly and

operation of an Auger instrument which does permit accurate surface tension

measurements of the liquid sample.

Our initial studies of liquid gallium last year showed that the primary

[1 2 3 ] surface impurities were oxygen and carbon ’

’

. The oxygen was undoubtedly

in the form of a gallium oxide and could be effectively reduced and removed by

heating; the carbon, however, remained in the form of small platelets floating

on the surface. In an attempt to clean a spot on the surface to obtain the

spectrum of clean gallium, we sputtered a small area with argon ions using a

conventional low power gun. The ion bombardment was found to generate large

scale fluid flow that eventually carried all of the carbon into the ion beam

where it was sputtered away. The result was a gallium surface which appeared

to be perfectly clean within the measurement limits of the Auger spectrometer.

Our first surface tension measurements on gallium sessile drops with surfaces

cleaned by sputtering will be seen to have a different temperature dependence

9
from our previous results with thermally cleaned surfaces. This observation

reinforces our feeling that the sputter cleaning process will be an important

tool in future studies of liquid surfaces. We will present here our current

ideas on the mechanism responsible for the fluid flow during the sputtering

process

.

The study of surface segregation in liquid alloys is an area which seems

particularly suited to A.E.S. The comparison of directly measured surface

concentrations with those deduced from surface tension values should eventually

result in improved models for the interface. We have begun measurements with

dilute alloys of Sn in Ga and will describe some initial results.

Experimental Apparatus and Procedures

The Auger spectrometer which we have assembled is a conventional singlepass cylindrical mirror analyzer system with a coaxial electron gun. The

CMA is mounted vertically so that it can be focussed on the apex of a

sessile drop located beneath it. This arrangement was chosen because it

maximizes the fraction of the drop surface which can be analyzed by the

spectrometer and provides excellent visibility for photography by locating

the drop near the center of two opposing 8" view ports. The shallow cylindrical cup containing the sessile drop is clamped to a simple hot stage

mounted on a horizontal manipulator which translates in three orthogonal

directions. An ion gun mounted in a side port is also focussed at the drop

apex. Fig. 1 shows schematically the experimental arrangement. The stainless steel bell jar in which these components are mounted is equipped with

three pneumatic vibration isolation supports which are adjusted to level

the cup for sessile drop surface tension measurements. The bell jar is

10
evacuated with sorption and ion pumps to minimize hydrocarbon concentrations.

-9

Pressures in the mid 10 torr range are achieved without baking as measured

by a nude ionization gauge. High purity gases can be admitted to the bell

jar from a connecting system for sputtering or oxidation studies.

The hot stage is also shown schematically in Fig. 1. The seven heaters

are 0.6 mil tungsten wire spirals enclosed in alumina tubes and are pressed

to the bottom of the plate which supports the cup by an opposing plate. The

heater ends are shielded to prevent evaporative contamination of the sessile

drop and the CMA. The cups are generally made of 0.6" O.D. silica cylinders

fused to 1" diameter flat silica plates to provide a wide shoulder at the

base. The body of the cup passes through a hole in a thin steel sheet which

presses the cup base against the heated plate. A chromel alumel thermocouple

is mounted on this pressure plate at the edge of the hole through which the

cup passes. The temperatures read by this thermocouple were in agreement

with those measured by a W-Rh thermocouple inserted into a sessile drop of

gallium contained in the cup.

The hot stage, which is constructed of stainless steel, is mounted by

three thin screws to a silica plate. This silica plate is itself attached to

the platform of the manipulator and provides effective thermal isolation for

the hot stage by its length and its low conductivity. Consequently, the

equilibrium temperature of the stage is extremely stable. This thermal

isolation, however, imposes the penalty of a slow cooling rate at low

temperatures where radiation losses are small. Fig. 2 shows the equilibrium

temperature of the hot stage as a function of heater current. The power

source was a small variable transformer.

11
The use of insulating cups to contain the sessile drop requires that an

electrical contact be made with the sample for Auger analysis. If this is

not done, charge builds up on the sample and disrupts the operation of the

CMA. It is also important to be able to measure the electron beam current in

order to compare different Auger spectra. We make electrical contact to the

drop with a fine W wire spot welded to a nickel rod that is rigidly held in

an electrical feedthrough on the manipulator flange. For surface tension

measurements, the manipulator is used to translate the drop and break contact.

This is necessary to obtain an unperturbed sessile drop. This technique was

also used to insert a fine thermocouple in the drop as mentioned previously.

Prior to forming a sessile drop, the bell jar was evacuated and the hot

stage temperature was raised to 500 °C for about an hour. After cooling, the

system was back filled with dry nitrogen gas. A small port was removed and

gas allowed to flow through the system while the cup was partially filled

with liquid gallium using a syringe. The port was then replaced and the

system very slowly evacuated with the sorption pumps. If evacuation is too

rapid the expansion of gases trapped under the gallium in the cup will expel

the liquid charge. After evacuation the process is repeated to form a

sessile drop large enough for surface tension measurements. Fig. 3 is a

photograph of a sessile drop formed in this way.

The instrumentation for our Auger spectrometer is not yet complete. The

electronics used to form a video image of the sample by rastering the electron

beam over the surface were not received and have had to be reordered. The

imaging of the sessile drop is important because our previous work showed that

the major impurities are in the form of solid platelets floating on the surface.

12
Thus without the video picture we cannot tell if the 3 ym diameter spot

analyzed by A.E.S. is typical of the entire surface. We have nevertheless

proceeded with measurements to gain experience and explore the capabilities

of the device.

Experimental Observations

a. The Gallium Surface

As we have previously reported, (1) the surface of a newly formed

sessile drop of gallium is covered by a thin solid oxide layer with a high

carbon content. Fig. 4 shows the Auger spectrum of such a drop. In addition

to oxygen and carbon, this spectrum shows some chlorine which comes from the

HC1 used in the preparation of the drop to dissolve the surface oxide. After

heating to 410 °C, the oxygen and chlorine lines are gone, but the carbon is

undiminished as seen in Fig. 5a. A brief sputtering (15 min.) at a beam

current of 2 x 10 ^ amperes eliminates the carbon, as seen in Fig. 5b. The

drop, however, is not clean yet because carbon reappears on the surface after

a few hours. Several sputtering and equilibration cycles are necessary

before the carbon is effectively eliminated from the drop. Eventually, no

carbon is detected in the spectrum taken at the apex after sitting for many

days. Although we cannot yet examine for particulates with a video imager,

we believe the surface to be essentially clean after this procedure.

Analysis of other areas on the drop support this conclusion.

The reduction of carbon concentration in the drop by sequential

sputtering and equilibration is probably evidence for volume mixing of

particulates by fluid flow during ion bombardment. The impingent ions

13
remove some carbon from the surface and drive some into the gallium where it

is mixed throughout the drop by convection. Carbon is much less dense than

gallium so buoyancy will drive these small particles back to the drop apex.

-4

A calculation using Stokes equation and a particle diameter of 10 cm gives

velocities which would clear the drop volume of carbon in several hours. This

particle diameter is in reasonable accord with measurements from video images

2 of the surface. Since the velocity varies as r , however, it is quite sensitive

to the diameter chosen, so this calculation only establishes the feasibility

of this process as an explanation for the evolution of the surface carbon

concentration.

There are a number of possible mechanisms which may account for the fluid

flow during ion bombardment: 1) temperature gradients at the surface, 2)

electrostatic forces due to surface charging, 3) surface tension gradients due

to concentration variation of impurities, 4) surface tension gradients due to

implanted argon concentration gradients, 5) surface density gradients due to

implanted argon, 6) momentum transfer from the ion beam, The first of these

seems unlikely for at least two reasons: a) an ion beam at the low power used

here does not produce a change in the drop temperature as measured by a

thermocouple immersed in the drop, b) the impact of the electron beam by

itself does not produce any mass flow even though its total power is com-

_3 parable to that of the ion beam (10 watt). The second explanation also

seems unlikely since carbon is conductive, as is, of course, the metal

surface. Surface tension gradients due to impurity concentration gradients

should not be present for a surface which is clean except for a few isolated

particulates. The fourth and fifth mechanisms are ruled out because they

14
require a significant argon concentration in the gallium and we have never

been able to detect any argon in the Auger spectra during sputtering.

The most likely explanation for the fluid flow generated by argon ion

bombardment seems to be momentum transfer from the ion beam to the liquid.

Although the magnitude of this momemtum transfer is small, other fluid flow

phenomena such as thermocapillary convection are also driven by very small

forces. The rigorous calculation of the flow generated in a sessile drop, or

any fluid body, by a localized beam of ions would be a major research effort.

A very approximate calculation, however, yields flow velocities which suggest

this mechanism to be the one of importance. The ion beam exerts a pressure -22 -12

of about 4 x 10 dynes/cm over its impact area of 10 cm , If we consider

the sessile drop surface under the ion beam to be a plane fluid-vapor interface and neglect fluid flow in the vapor, the Navier-Stokes equation for an

incompressible fluid gives

2 \

9V

z 9z ( 1 )

for the velocity gradient normal to the surface. Here and P^ are the

pressures in the beam and in the liquid at the surface, is the velocity

v -2

normal to the surface and nT is the viscosity. For gallium, nT = 2 x 10

J-i i-i poise, and using the approximate ion beam pressure calculated above gives a

velocity gradient of 1 s \ Assuming a linear velocity function normal to

the surface and a liquid depth of 1 cm yields a normal velocity at the top

surface of 1 cm/sec. Although this is admittedly a very crude approximation,

the velocity calculated is large enough to suggest the beam pressure is

responsible for the fluid flow.

15
Carbon is normally found along with oxygen in the Auger spectra of all

surfaces before cleaning. Thus, it is not surprising that we find carbon on

our gallium surfaces. Gallium drops contained in cups of high density

graphite show intolerable carbon concentrations and it has proved impossible

to eliminate carbon when using them. We presume that the surface of the cup

is eroding at a rate which keeps the gallium contaminated with carbon

particles. The vitreous graphite cups used in the Auger work reported last

year had a much harder surface than the present cups and the gallium could be

cleaned by repeated sputtering. Even with quartz cups, however, carbon is

found on the surface of a freshly formed drop as shown in the spectrum of

Fig. 4. We suspected that the carbon might be originating in the alcohol

which is used in the preparation of the gallium before filling the syringe

for cup loading. However, similar carbon concentrations were found when

distilled water was used as the carrier fluid. In these experiments, a fresh

gallium ingot was used and all tools and glassware were degreased with

acetone and rinsed with distilled water prior to use.

In a further attempt to isolate the source of the carbon, a gallium drop

which had been repeatedly sputtered and showed no indications of carbon after

equilibrating for three days was exposed to air by opening a valve. On reevacuation, the Auger spectrum showed a very large oxygen line but no carbon.

Heating to 500 °C eliminated the oxygen and the spectrum looked as clean as

that of a sputtered surface. After equilibrating for two days, however, a

strong carbon line was measured. It does not seem likely that this carbon

came from the bulk of the gallium. Rather we suspect that it may have

originated in the hot filaments of the ionization gauge or the heaters. The

16
residual oxygen pressure due to opening the vacuum system was much higher

-9

than normal for several days as the system pumped down to the 10 torr

range. This oxygen may have reacted with the carbon in the hot filaments to

generate CO which accumulated on the gallium surface. However, the surface

showed no oxygen after this procedure so we would have to argue that the CO

decomposed on the cool gallium surface. This does occur on other metal

surfaces. The origin of the carbon found on the surfaces of gallium is still

unknown

.

b. The Surface Tension of Pure Gallium

The surface tension of a gallium drop which had been cleaned by sputtering is shown in Fig. 6. A number of sputtering cycles reduced the carbon

concentration to the degree that none could be detected twelve hours after

sputtering. The surface tension values were calculated from photographs of

the drop profile using the Bashforth and Adams technique. A linear regression analysis of the data gave the following relationship:

y = 724.1 - .0721 T (°C) (2)

We also show in Fig. 6 the nearly quadratic temperature dependence which we

found previously for gallium which was cleaned by heating in vacuum and in

hydrogen atmospheres with graphite and quartz cups. This nonlinear relationship is in good agreement with three other measurements

~>i ^ .

Our present data is in fair agreement with one of the two linear relationships found previously for gallium^ and shown in Fig. 6. Since the Auger

spectrum shows the surfaces used in the present measurements are clean, we

believe the nonlinear surface tension temperature dependence found in our

17
previous work and also in other laboratories may be due to solid carbon

platelets distorting the sessile drops. These platelets accumulate at the

apex of the drop, a uniquely important area for surface tension calculation

because it is the origin from which all the altitudes are measured. Only

very small distortions would be required to produce the maximum low tempera2 ture discrepancy between our two measurements of lOmJ/m at 30 °C, a

difference of less than 1.5%. This explanation can only be proved by

measuring a quadratically varying surface tension for a thermally cleaned

drop, characterizing the carbon concentration, sputtering to eliminate the

carbon, then remeasuring the surface tension. We have not done this.

c. Gallium-Tin

The first liquid Ga-Sn sample studied in the Auger spectrometer was a

gallium drop containing 1.9 atomic percent tin. This alloy was chosen

because measurements had shown a large reduction in its surface tension below

that of pure gallium suggesting that tin was strongly adsorbed at the metalvacuum interface. The sample, however, was contained in a graphite cup and

we were unable to completely eliminate carbon from the drop by sputtering, as

discussed earlier. The amplitudes of the gallium and tin Auger electron

lines are sensitive to the presence of carbon so that the estimation of

surface concentration is inaccurate if the surface is not clean. This

problem was circumvented by recording the Auger spectrum after sputtering

when the surface was free of carbon. Fig. 7 shows the Auger spectrum of the

alloy drop at 30 °C and 450 °C. The amplitude of the gallium line varies

little while that of the tin line decreases strongly as temperature increases.

Since the amplitude of the line is roughly proportional to the concentration

18
of the species, this implies adsorption weakens with increasing temperature.

This is in accord with observations of surface adsorption in other liquid

alloys

.

The concentration of Sn at the surface of the drop was estimated by

comparing the amplitude of the dominant tin line at 430 e.v. with that from

a pure tin sample using the same spectrometer settings. This procedure is

not straightforward, however, because the detected Auger electrons originate

from significant depths and not solely from the surface layer. The primary

electron beam (200 e.v. for our measurements) penetrates deeply into the

sample. The Auger electrons generated by the beam are exponentially

attenuated with distance from their origins. Thus when Auger line amplitudes

from surface layers are compared with those from bulk standards, this

r q I attenuation must be taken into account 1

. For an incoherent scattering

model of a uniformly adsorbed layer of Sn, the intensity of the 430 e.v.

Auger electrons, I^ n

, is given by

I Sn

1 + r_ (430 e.v.) Ga

1 + r n (430 e.v.)

Sn

[i exp (3)

Here 1^ is the intensity of a pure bulk tin sample, m is the layer thickness

and L is the attenuation length for 430 e.v. electrons in tin. The r terms

take into account the Auger electrons which arise from the backscattered

primary beam; they vary with energy and atomic species. In the filled circles

in Fig. 8 we show the surface coverage of tin as a function of temperature

[9] calculated with equation 3 using an attenuation length of 4 monolayers

The r values for Sn and Ga at 430 e.v. were estimated to be 0.62 and 0.46

19
r q

"| respectively . Thus the term — is approximately 0.9.

Sn

We have also measured the surface tension for the gallium-1. 9% Sn drop.

The data shown in Fig. 9 (filled circles on line 1) were taken after sputtering

and before carbon redeveloped at the surface. Although carbon may have been

located on the surface outside of the area analyzed by the electron beam, the

error so introduced into the surface tension values should be small compared

to the large effects of the tin adsorbed at the surface. We also show in this

figure the surface tension relationship (line 4) we find for pure gallium

surfaces cleaned by sputtering. For comparison some surface tension values

for gallium and a gallium-2% tin alloy measured in another laboratory (ref. 6)

are also presented. The results for the alloys are in fairly good agreement

except below 100 °C.

The depression of the surface tension of the alloy below that of pure

gallium is produced by the adsorption of tin at the liquid-vacuum interface.

This adsorption is strongly temperature dependent and produces a temperature

dependent surface concentration of tin, as indicated in Fig. 8. This surface

(Ga) concentration of tin can be estimated from the relative adsorption T Sn

For a dilute solution this can be written as

X

(Ga) Sn dy

Sn RT dX (4)

Sn

where X^^ is the bulk concentration. The coverage in monolayers is found by

(Ga) multiplying T by the area of a tin molecule. In Fig. 8 we show (open

circle) the surface coverage of Sn for the gallium alloy drop calculated by

taking

dy _ ^Ga ^alloy

dX (5)

Sn Sn

20
These results can be compared with the surface coverages estimated from the

Auger line intensities (filled circles in Fig. 8) . The surface coverages are

in good agreement, particularly at the lower temperatures. Given the uncertainties

in attenuation lengths and backscattering factors, the agreement may be

fortuitous. No effort has been made to improve the agreement of the two sets

of data by adjusting these parameters. The surprising aspect of this data is

that it suggests coverages in excess of a monolayer below 125 °C. It was

anticipated that the coverage might reach a monolayer but would then saturate

and remain constant. Large adjustments to the attenuation length, backscattering factors, and the variation of surface tension with concentration

would be necessary to reduce the coverage to below a monolayer.

In conclusion, our initial studies have shown that Auger spectroscopy can

be successfully applied to liquid metals and alloys. Ionic bombardment

sputtering seems to be a promising way of producing clean liquid surfaces

although we have not tested the technique yet for impurities which are surface

active but remain in solution. The study of segregation at liquid alloy

surfaces by Auger spectroscopy combined with surface tension measurements is a

promising area for future work even though attenuation lengths and backscattering factors may not be sufficiently well known for some materials.

21
References

1. S. C. Hardy and J. Fine, Materials Measurements , NBSIR 80-2082, 1

(1980).

2. J. Fine, S. C. Hardy, and T. D. Andreadis, "Symposium on the Physics

of Ionized Gases," Dubrovnik, Yugoslavia, 294 (1980).

3. J. Fine, S. C. Hardy, and T. D. Andreadis, Journal of Vacuum Science and

Technology, in press.

4. 0. A. Timofeevicheva and P. P. Pugachevich, Dokl. A. Kad. Nan K. SSR,

134 . 840 (1960).

5. V. I. Nizhenko, L. P. Sklyarenko, and B. N. Evemenko, Ukrain. Khim.

Zhur. 6^, 550 (1965).

6. A. G. Nalgiev and Kh. I. Ibragimov, Zh. Fiz. Khim. 48^, 1289 (1974).

7. Kh. B. Khokonov, S. N. Zadnmkin, and B. B. Alchagirov, Elektrokhimiya

10, 865 (1974).

8. M. P. Seah and W. A. Dench, Surface and Interface Analysis _1, 2 (1979).

9. M. P. Seah, Surface Science 4£, 595 (1973).

10.

D. M. Smith and T. E. Gallon, J. Phys. D 7_, 151 (1974).

22
1

n iFFT

ia5o.: if n

L+ —non mn i 7

1-1 8

3 9

10

Fie . 1. Schematic diagram of experimental arrangement. 1) CMA

2) electron gun, 3) sessile drop, 4) cup, 5) pressure plate,

6) hot stage, 7) heaters, 8) pressure plate, 9) quartz plate,

10) manipulator, 11) ion gun.

23
I

o o o o

o o o o

ID CO CM

o

o

(0o ) 0jnjBJ0duj0i

24
Fig. 3 Photograph (negative) of sessile drop in Auger spectrometer.
A.E.S.

of

Ga

r

26

Fie.

4.

Auger

spectrum

of

uncleaned

Ga

drnn
27

Auger

spectrum

of

Ga

drop

(a)

after

heating

to

450

°C

(b)

after

sputtering.
Surface

Tension

(mJ/m

2)

Temperature (°C)

Fig. 6. The surface tension of Ga as a function of temperature. 1) present

work, 2) ref. 7, 3) ref. 1.

28
29

Fig.

7.

The

Auger

spectrum

of

Ga-Sn

1.9

at.
M

(monolayers)

Temperature (°C)

Fig. 8. The surface coverage in monolayers of Sn as a function of temperature

as measured from line amplitudes (filled circles) and surface

tension changes (open circles).

30
Surface

Tension

(mJ/m

Temperature (°C)

Fig. 9. The surface tension of Ga-Sn 1.9 at.% as a function of temperature.

T e filled circles along with curve (1) give measurements from the current work. The X's which form curve (2) are data reported in reference 6. Curves (3) and (4) are results for pure gallium, the

X s which form curve (3) being data given in reference 6 and the straight line (4) being results from the present work.

31

Task 2

Convection During Unidirectional Solidification

S. R. Coriell and R. J. Schaefer

Metallurgy Division

Center for Materials Science

Summary

The succinonitrile-ethanol system has been selected for the experimental

study of solute-induced convective and morphological instabilities. The phase

diagram of this system has been determined and the measured distribution

coefficient of 0.044 was used to perform stability calculations. Several

-3 -2

modes of instability are predicted in the composition range of 10 to 10

weight percent ethanol and the solidification velocity range of 0.5 to 5 ym/sec.

Experimental measurements of convective flow have used the tracking of small

neutrally buoyant particles, and have to this point been devoted to measuring

the background of thermally induced convection in pure succinonitrile. Flow

rates of 1 to 5 ym/sec were produced in the present apparatus from thermal

convection alone without solidification whereas convective flows induced by

solidification processes were 15 ym/sec or more.

33
Introduction

During the solidification of a material containing more than one

component, density differences due to chemical inhomogeneities of the

liquid phase are among the several driving forces for convective flow,

in the presence of a gravitational field. The onset of coupled convective and constitutional interfacial instabilities during the directional

solidification of a single phase binary alloy at constant velocity

vertically upwards was previously treated by a linear stability analysis

and specific calculations were made for physical properties appropriate

to the solidification of lead containing tin [1]. For experimental

verification of this type of calculation, and to reveal the nature of

the convective flow once it is initiated, it is necessary to study the

solidification of transparent materials. Such materials have physical

properties which differ greatly from those of metals, but similar

instabilities should occur. We report here calculations and experiments

on the solidification of succinonitrile containing ethanol. For the

lead-tin system, we give some additional results in which the convective

and interfacial instabilities are decoupled and compare these with the

coupled results.

Theory

We recall that the dependence of the perturbed temperature, concentration, fluid velocity, and interface shape on the horizontal

coordinates x and y and the time t is of the form exp [at + iCw^x + w y)],

where and are spatial frequencies. In general, the time constant

a is complex, i.e., a = a +ia.. If a >0 for any values of w and w ,

34
the system is unstable. If a <0 for all values of w and oj J , the

r x y

system is stable. We refer to instabilities corresponding to large

2 2

values of co = (eo^ + co^)

2 (small wavelengths, A = 2tt/(jo) as morphological

(or interfacial) instabilities and to instabilities corresponding to

small values of w (large wavelengths) as convective instabilities.

The properties of the succinonitrile-ethanol system are given in

Table I. The solute diffusion coefficient, D, in the liquid and the

variation of liquid density, p, with solute concentration, c, have not

been measured. Estimated values of these have been used in the calculations, viz., D = 1.0 (10 ~*) cm^/s and e - (3p/3c)/p = 3.07 (10 ^)

(wt.%) \ For a temperature gradient G in the liquid of 10 K/cm, the

critical concentrations of ethanol in succinonitrile above which instability occurs are shown in Fig. 1. The results are similar to the

previous calculations on the lead-tin system, and instability occurs at

-3

concentrations of the order of 10 wt.%. For small velocities the

convective instability occurs at lower concentrations than the morphological instability. The concentration for convective instability

increases with velocity while the concentration for morphological

instability decreases with velocity. The two curves cross at a velocity

of about 3.5 ym/s. In the vicinity of this crossover, i.e., where

convective and morphological occur at about the same critical concentrations, instability occurs by an oscillatory mode for which ^ 0

when = 0. The onset of instability by an oscillatory mode is the

most striking difference between the results for lead-tin and succinonitrileethanol. Fig. 1 also shows the dimensionless ratio of the perturbation

35
wavelength A at the onset of instability to the diffusion boundary layer

thickness D/V. As previously noted, the wavelength corresponding to

convective instability is considerably larger than the wavelength corresponding to interfacial instability. At small velocities, the convective

wavelengths are quite large, e.g., A = 5.7 cm for V = 0.5 ym/s.

The effect of the temperature gradient in the liquid on the

critical concentration for instability is shown in Fig. 2. As long as

the mode of instability does not change, the critical concentration

increases with increasing temperature gradient as would be expected

since the temperature gradient is a stabilizing influence. However, as

illustrated by the V = 3.0 ym/s data in Fig. 2, the critical concentration

at = 10 K/cm is larger than the critical concentration at 20 K/cm;

the mode of instability changes from oscillatory convective (ck i- 0) to

convective (c_^ = 0) as the gradient increases. For V = 4.0 ym/s, the

mode of instability changes from morphological to oscillatory convective

to convective as the temperature gradient increases from 10 to 40 K/cm.

These changes in the mode of instability are illustrated in detail

in Figs. 3-6, which give the concentration at which = 0 as a function

of the spatial frequency oj of the sinusoidal perturbation for G^ = 10

K/cm and V = 2.5, 3.0, 3.5, and 4.0 ym/s, respectively. The solid lines

correspond to o, = 0, while the dashed lines correspond to o. ^ 0; the

values of are indicated by the lower curves in the figures. At V =

2.5 ym/s, the smallest concentration corresponds to a non-oscillatory

convective mode whereas at V = 3.0 and 3.5 ym/s an oscillatory convective

-2 -3 -1

mode occurs with a_^ in the range 10 to 10 s . At V = 4.0 ym/s,

-3

instability first occurs by a morphological mode with c m

= 4.66 (10 ) wt.%

36
and to = 140 cm ^ (not shown in the figure).

Since the liquid diffusion coefficient D has not been measured, some

calculations were carried out with a different value of D. For D = 5

*" 6 2 (10 ) cm /s and G = 10.0 K/cm, the critical concentrations are 2.09

l-i

(10 ^), 5.06 (10 ^), and 3.46 (10 wt.% for V = 0.5, 1.0, and 2.0 pm/s,

respectively. The highest velocity corresponds to an oscillatory convective

instability with non-oscillatory convective instability occurring at the

-5 2 lower velocities. For comparison, for D = 1.0 (10 ) cm /s and = -4 -4

10.0 K/cm, the critical concentrations are 2.63 (10 ), 4.58 (10 ),

and 1.20 (10 wt.% for V = 0.5, 1.0, and 2.0 pm/s, respectively.

Thus, the critical concentration curves (as function of velocity) for

the two different diffusion coefficients cross each other. The critical

concentration for morphological instability is proportional to D for

small velocities, and hence decreases with decreasing D.

In order to obtain an estimate of the rate at which instabilities

develop, linear stability calculations of a for a fixed concentration of

-5 2 ethanol were carried out. For D = 1 (10 ) cm /s, G = 10 K/cm, and

i-j -2

c^ = 10 wt.%, and crystal growth velocites of 1.0, 2.0, and 4.0 pm/s,

-2

the maximum values of o^_ for convective instability are 4.9 (10 ),

3.7 (10 2), and 6.6 (10 ^) s ^ at spatial frequencies of 22, 24, and

28 cm \ respectively. For V = 2.0 and 4.0 pm/s, the maximum values of

-2 -1

for morphological instability are 1.1 (10 ) and 0.12 s at spatial

frequencies of 440 and 950 cm \ respectively. For V = 1.0 pm/s, the

system is morphologically stable, i.e., the maximum value of c^ for

morphological instability is negative. For example, at V = 1.0 pm/s.

37
the amplitude of the convective instability will increase by a factor of

ten in 50 s; during this time the crystal will grow 0.005 cm.

In order to gain some insight into the interaction between convective

and morphological instabilities, we have modified certain parameters in

the calculations so that only one type of instability can occur. Physical

parameters appropriate to the solidification of lead containing tin were

used with V = 40.0 ym/ s and = 200 K/cm. If the gravitational acceleration

g is identically zero, then convection does not occur and only morphological

instabilities are possible. The critical concentration of tin as a function

of the spatial frequency u of a sinusoidal perturbation with g = 0 is

shown in Fig. 7; the minimum value of c^ is 0.215 wt.% at w = 690 cm \ By

requiring that the solid-liquid interface not deform, we can eliminate

the morphological instability and allow only convective instabilities.

The boundary conditions at the solid-liquid interface are based on conservation of enthalpy and solute and a relationship between freezing

point and solute concentration. In order to simultaneously satisfy

these three boundary conditions, the solid-liquid interface must be free

to deform. If we require a planar interface, one of the boundary conditions can not be satisfied. We have required that the interface be

rigid (planar) and satisfied the conservation laws but ignored the

relationship between freezing point and solute concentration. The critical

concentration as a function of w for this rigid interface model is shown

in Fig. 8; the minimum value of c is 0.15 wt.% at u = 45 cm \ The

2 results for the coupled problem, i.e., g = 980 cm/s and interface free

to deform is shown in Fig. 9. The minimum value of c is 0.186 wt.% at ° oo

38
a) = 45 cm for convective instability while the minimum for the morphological

instability is essentially identical to that computed with g = 0.

While there is a 25% difference in the critical concentrations for

convective instability between the rigid and free interface, the most

striking difference between Fig. 9 and Figs. 7 and 8 are the appearance

of oscillatory modes (a^ ^ 0) in the coupled problem. These oscillatory

modes are apparently the result of the coupling between the convective

and morphological instabilities, and would not be expected in the absence

of this coupling. Oscillatory modes do arise in double diffusive convection, but in the simple models which can be treated analytically,

only when the temperature field is destabilizing and the solute field

stabilizing, which is not the case treated here. The calculations for

the succinonitrile-ethanol system have indicated that the onset of

instability can occur by an oscillatory mode.

The main theoretical effort of this task is now directed toward

developing mathematical models of the fluid flow and resultant solute

segregation under conditions for which the linear analysis predicts

convective instability; this research is being carried out in collaboration

with R. G. Rehm of the Mathematical Analysis Division. Numerical

algorithms are being developed and implemented to solve the time dependent

partial differential equations for fluid flow and heat and mass transfer

in two spatial dimensions with a planar solid-liquid interface.

During the past year, a paper "Effect of Gravity on Coupled Convective

and Interfacial Instabilities During Directional Solidification," by

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

39
presented at the XXIII COSPAR meeting and will be published in the

proceedings. An article, "Solutal Convection Induced Macrosegregation

and the Dendrite to Composite Transition in Off-Eutectic Alloys," by W.

J. Boettinger, F. S. Biancaniello , and S. R. Coriell, which describes

previous work of this project, has been published in Metallurgical

Transactions [2].

Experimental Procedures

Because no methods are available for the detailed measurement of

fluid flow patterns in liquid metals, a transparent system is used.

Succinonitrile is chosen as the major constituent because of its metallike solidification behavior and well-characterized physical properties

(Table I). Extremely high purity can be attained in this substance by

multiple distillation and zone refining [3]. Unfortunately, it has a

rather low density, so the choice of solutes having lower density is

extremely limited. Ethanol has been selected for this purpose on the

basis of its low density, relatively high boiling point compared to

other low density materials, and similarity of molecular structure to

succinonitrile.

For input to the stability calculations we need to know the

succinonitrile-rich end of the succinonitrile-ethanol phase diagram.

This was determined by observations of samples immersed in a stirred

water thermostatic bath. Temperatures were measured with precision

thermometers on which temperature differences as small as 0.002 K can be

read. The temperatures are not known with comparable precision on an

40
absolute scale, but only the accurate temperature differences are important

for this experiment. The succinonitrile-ethanol system has a liquidphase miscibility gap, as does succinonitrile with several other alcohols.

The liquidus curve (Fig. 10) was determined from samples weighed out and

sealed under air. A monotectic point was found at approximately 11 °C

and 23 wt.% ethanol. The miscibility gap is rather low but was not

studied in detail because the region of interest lies at very low concentrations of ethanol. The liquidus curve is essentially linear at

concentrations of ethanol of less than 6 wt.%, and its slope in this

region is 3.6 K/wt.%.

The solidus curve is very steep and its determination could only be

carried out with samples held under vacuum, because dissolved air acts

as a significant impurity. Such samples are difficult to prepare with

accurate compositions because the vapor pressure of ethanol is high at

the melting point of succinonitrile. Therefore, the apparatus shown in

Fig. 11 is employed. Zone-refined succinonitrile is first loaded into

the larger chamber, melted and frozen several times with active pumping

to remove all dissolved air, and this side of the chamber is then

hermetically sealed. The mass of the succinonitrile is determined by

weighing. Ethanol is then loaded into the capillary chamber and part of

it is frozen unidirectionally by lowering the capillary into a bath of

liquid nitrogen. This side of the chamber is then evacuated and sealed.

Additions of ethanol to the succinonitrile are carried out by

opening the teflon valve and gently applying heat to the top of the

ethanol column to distill over a small amount. An estimate of the

41
amount transferred is obtained by noting the drop in the ethanol level in the

capillary. A more accurate measure of the concentration of ethanol is obtained

from the measured liquidus temperature (the temperature at which only a minute

fraction of the sample remains solid, and the solid is neither melting nor

freezing). The concentrations determined by these two methods agreed well.

The solidus temperatures were determined as the lowest temperature at which

melting was first observed (at grain boundaries) after the sample had been

homogenized by rapid freezing. The results of these experiments are shown in

Fig. 12, and the slope of the solidus curve is found to be 81 K/wt.%.

The distribution coefficient is a key parameter in the theory of solutal

convection since this coefficient determines the proportion of solute rejected

at a solidifying interface. With the above data, this parameter can be calculated from the ratio of the liquidus slope to the solidus slope at small

solute concentrations to be k = 0.044.

o To investigate the convective and interfacial instability of succinonitrileethanol solutions, the calculations have indicated that one should grow crystals

in the solidification velocity range 0.5 to 10 pm/s. For these experiments

the sample is sealed in a round pyrex tube, 19 mm in diameter. This tube is

drawn downward at the desired velocity through the apparatus shown in Fig. 13.

This apparatus contains an upper water jacket, maintained at a temperature

above the melting point of succinonitr ile, and a lower jacket maintained below

the melting point. Between these reservoirs, the round sample tube is enclosed

in a fluid-filled jacket with flat glass walls. The fluid in this jacket is a

mixture of water and ethylene glycol having a refractive index such that the

optical distortion of the material within the round sample tube is minimized

[3], The upper and lower water jackets are maintained by circulating controllers

42
at temperatures constant within + 0.001 K. The apparatus is heavily insulated

and the sample is viewed in the region of the solid-liquid interface through

small double-walled glass windows.

Convection is measured by photographic tracking of the motion of small

particles suspended in the liquid. Laser Doppler Velocimetry (LDV) was considered as a measurement technique but rejected because at the extremely slow

flow velocities (down to 1 pm/ s) expected in these experiments, the LDV technique

would be overly sensitive to mechanical vibration and electronic noise, and

because it is more important for these experiments to determine the overall

pattern of the flow than to obtain statistical information on velocities at a

fixed point in space, as one normally does with the LDV method.

The particles used for tracking the fluid flow are polyvinyltoluene

spheres 2.02 pm in diameter. The small difference between the density of

these spheres and that of the liquid succinonitr ile, together with their small

size, results in a rate of gravitational precipitation calculated from Stokes

Law to be 0.05 pm/s, so that the spheres follow almost perfectly the flow of

the liquid. The particles are readily visible through a microscope, when

illuminated by low-angle dark field illumination. Photographs are recorded

with exposure times usually between 30 seconds and 2 minutes, to give adequate

travel distance for flow rate measurements. The particles are illuminated by

a microscope light, with a heat-absorbing solution in the optical path to

prevent radiant heating of the specimen.

Experimental Results

During slow solidification, the particles are pushed by the solid/

liquid interface, often accumulating in the small grooves which are

present where grain boundaries intersect the solid/liquid interface.

43
When the particles accumulate into large clusters, or when the interface

velocity becomes rapid, the particles are incorporated into the solid.

No quantitative measurements of this effect have been recorded, and

it does not interfere with the convection measurements. Convection

measurements to date have used a sample of ultra high purity succinonitrile

in which all convection must be attributed to thermal effects. This

sample was prepared by the same techniques which were previously found

[3] to reduce the level of impurities to approximately 5 x 10 ^ to 5 x

— 8

10 mole fraction. The melting and solidification of this sample shows

no indication of impurities, such as grain boundary melting or morphological instability. The objective of the measurements with the high

purity sample is to establish a baseline of residual thermally- induced

convection to which convection in ethanol-doped samples can be compared.

When the temperature reservoirs of the crystal growth apparatus are

adjusted to give a gradient of about 10 K/cm in the liquid, and the

sample tube is held stationary (no crystal growth), the solid- liquid

interface is planar and horizontal, and convective flow rates in the

liquid are generally found to be in the range of 1-5 gm/s. The flow

pattern is somewhat erratic. An occasional source of higher flow rates

under these conditions was seen when a large void, which had originated

as a shrinkage cavity when the sample was originally solidified, migrated

vertically upward through the solid under the influence of the temperature

gradient. Shortly before this void reached the solid-liquid interface

(where it collapsed) , it was seen to have an influence on the convective

flow field such that there was a visible upward flow of the liquid above

44
the void. The thermal distortion which produced this flow, however, was

not sufficient to produce a measurable distortion of the solid-liquid

interface.

When a stationary sample is suddenly repositioned downward, crystal

growth starts almost immediately, with the most rapid growth along the

tube walls. As expected, convective flow is greatly accelerated.

Several hours are required before the interface returns to its original

position and the convective flow again diminishes to velocities characteristic of stationary samples. When the sample was displaced 1 cm,

the flow rate along the sample axis was 20 ym/s at 40 minutes after the

displacement. The flow pattern was a steady flow upward along the

sample axis and downward along the walls.

When the mechanism for pulling the sample downward at constant

velocity is activated, the solid-liquid interface moves downward and

eventually reaches a new steady-state position. Typically 2-3 hours are

required before the interface position becomes effectively stationary.

In addition, the interface changes from the planar horizontal configuration

to an upwardly concave shape. For a sample which was drawn downward at

a velocity of 2 ym/s, convective flow rates were found to increase as

the interface became more concave, so that one hour after the tracking

mechanism was started, flow rates were 15-20 ym/s.

Although the convective flow during constant velocity growth is too

small to be readily visible to the eye when viewed through the microscope, it is desirable to modify the apparatus such that this purely

thermal effect is minimized. To this end, experiments will be carried
out with auxiliary heaters in the index-matching fluid so that the

discontinuity of temperature gradient induced by the latent heat of

fusion can be approximated in this fluid. Experiments with ethanoldoped samples will be commenced to determine the relative magnitude of

the flow rates in pure and doped samples. Simultaneously improvements

in sample illumination will be sought which can reduce the intensity of

light scattered from extraneous sources such as the chamber walls and

thus allow a much wider range of velocity measurements from a single

exposure.

Acknowledgments

We thank W. J. Boettinger, M. R. Cordes, R. G. Rehm, and R. F.

Sekerka for helpful discussions.

46
References

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

J. Crystal Growth _49_, 13 (1980).

2. W. J. Boettinger, F. S. Biancaniello , and S. R. Coriell, Met. Trans.

12A, 321 (1981).

3. R. J. Schaefer, M. E. Glicksman, and J. D. Ayers, Phil. Mag. _32, 725

(1975).

4. C. A. Wulff and E. F. Westrum, J. Phys. Chem. , 67_, 2376 (1963).

5. M. J. Timmermans and Mme. Hennaut-Roland, Journal de Chimie Physique,

34, 693 (1937).

6. H. Fontaine and M. Bee, Bull. Soc. Franc. Mineral. Crystallogr. , 95 ,

441 (1972).

7. R. M. MacFarlane, E. Courtens, and T. Bischofberger , Mol. Cryst. Liq.

Cryst., 35, 27 (1976).

8. M. E. Glicksman, R. J. Schaefer, and J. D. Ayers, Met. Trans., 7A, 1747,

(1976).

47
Table I. Properties of Succinonitr ile

Property Symbol Value Ref .

Molecular Weight W 80. 092 4

Density of Solid p s 1016kg/m2

**

4*

Density of Liquid 988kg/m

2 ** 5 Thermal Expansivity of Solid a

s -5.6x10 /K 6

Thermal Expansivity of Liquid al

-8.1x10“

4

/K 5*

Shear Viscosity n 2.6x10 ^Pa-s** 5 Kinematic Viscosity V 2.6 xlO / s** JU J-4\ 4\ Surface Tension (Liquid-Vapor) t £v

46. 78mJ/m

2 5 Surface Tension (Solid-Liquid) y sl

8 . 9mJ/m

2

3

Refractive Index (Solid) n

s

1 . 4340** 7 Refractive Index (Liquid) ni

1.4150** 7 Equilibrium Temperature (Triple T

r 331. 23K 8

Point)

Latent Heat of Fusion L 4.70x10 J/kg 4

Entropy of Fusion AS 1.42xl0_ j/kgK 4

Heat Capacity of Solid Ss 1913 j/kgK** 4

Heat Capacity of Liquid C

P l 2000J/kgK** 4

Thermal Conductivity of Solid k

s 0. 225j/mKs** 3

Thermal Conductivity of Liquid k

£

0. 223 j/mKs**

-7 2 ,

3

Thermal Diffusivity of Solid K 1.16x10 m /s** *** Thermal Diffusivity of Liquid

s

K l

1.12xl0

_7 m

2 /s** vtJ. JU

Unit Supercooling L/C

P t

23.5 K A A )> Distribution Coefficient for

Ethanol in Succ inonitr ile

k

o 0.044 j'c * * *

Liquidus slope for Ethanol m» 3 . 6K/ wt . % ****

in Succ inonitr ile

Solidus slope for Ethanol in

in Succinonitr ile m

s 81K/wt .

%

Indicates value derived from properties given in cited reference.

** Indicates value of property at melting point, 331. 23K. *** Indicates property derived from other properties in the table.

****ind icat es properties measured in current work.

48
wt.%

100.0

10.0

1.0

0.1 1.0 10.0

V ^.m/s

Fig. 1. The critical concentration c of ethanol in succinonitrile above oo which instability occurs as a function of the velocity V of

directional solidification for a temperature gradient in the

liquid of 10 K/cm. The ratio of the instability wavelength. A,

to the diffusion boundary layer thickness, D/V, is also shown.

The solid and dashed lines correspond to interfacial and con- vective instabilities, respectively, while the dotted line denotes

an oscillatory convective instability.

49

(V

X./D)
Cqo

wt.%

G L K/cm

Fig. 2. The critical concentration of ethanol in succinonitrile above

which instability occurs for crystallization velocities of 0.5,

1.0, 2.0, 3.0, and 4.0 ym/s, and temperature gradients in the

liquid of 10.0, 20.0, and 40.0 K/cm. The x, circles, and squares

indicate morphological, convective, and oscillatory convective

instabilities, respectively.

50
Fig. 3. The concentration of ethanol in succinonitrile at the onset of

instability during directional solidification at V = 2.5 ym/s as

a function of the spatial frequency u of a sinusoidal perturbation.

The solid curves mark the onset of non-oscillatory instabilities

(o.=0); whereas the dashed curves mark the onset of oscillatory

instabilities (the value of a is given on the right hand side).

51
10°

Coowt.%

CO cm

Fig. 4. The concentration of ethanol in succinonitrile at the onset of

instability during directional solidification at V = 3.0 ym/s as

a function of the spatial frequency u of a sinusoidal perturbation.

The solid curves mark the onset of non-oscillatory instabilities

(a.=0); whereas the dashed curves mark the onset of oscillatory

instabilities (the value of o

is given on the right hand side)

.

52
COb

ro

Fig. 5. The concentration of ethanol in succinonitrile at the onset of

instability during directional solidification at V = 3.5 ym/s as

a function of the spatial frequency w of a sinusoidal perturbation,

The solid curves mark the onset of non-oscillatory instabilities

(o_^=0) ; whereas the dashed curves 'mark the onset of oscillatory

instabilities (the value of c_^ is given on the right hand side)

.

53
Fig. 6. The concentration of ethanol in succinonitrile at the onset of

instability during directional solidification at V = 4.0 ym/s as

a function of the spatial frequency u of a sinusoidal perturbation.

The solid curves mark the onset of non-oscillatory instabilities

(o^=0) ; whereas the dashed curves mark the onset of oscillatory

instabilities (the value of is given on the right hand side).

54

10

CT;
o

o

o

o

o

o

00 % *M

o

3

55

Fig.

7.

The

concentration

of

tin

in

lead

at

the

onset

of

instability

during

directional

solidification

atV

-

40.0

pm/s

as

a

function

of

the

spatial

frequency

u

of

a

sinusoidal

perturbation.

The

gravitational

acceleration

g

=0

so

that

only

morphological

instability

can

occur.
56

Fig.

8.

The

concentration

of

tin

in

lead

at

the

onset

of

instability

during

directional

solidification

at V

=

40.0

ym/s

as

a

function

of

the

spatial

frequency

u

of

a

sinusoidal

perturbation.

The

solid-liquid

interface

is

rigid

(planar)

so

that

only

convective

instability

can

occur.
o Eo 30 CO O

•H 4-1 U O 3

0) 3 3

•HT3 30

00 3 0 3

•3 cr 3 3 3 3

T3 4-4

>-> »H

4-1 (0 •H *3 -H 4-4 •h 3

d=>

a 3 co

4-1 CO 0) 0 rC

•H 4J

4-1 4-1 o o 0 4-4 0) 3

cd o (0 "d 3 c -H

4-4

O CO CO 3

a) co ai x)

•3 0 dd 0

4-4

O 4-1 Cd •H 3 dsi •H -Cd X> 4-4 cd

4-1 d*i CO 3 3 4-1 cd dd g 00

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57
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58

WEIGHT

PERCENT

ETHANOL
Teflon Valve

chamber Capillary

Fig. 11. Apparatus for the preparation of succinonitrile-ethanol mixtures

under vacuum.

59
60

50

0.05

WEIGHT PERCENT ETHANOL

0.10

Fig. 12. Detail of succinonitrile-ethanol phase diagram at very small

concentrations of ethanol, showing data points used to determine

the solidus slope.

60
L

4

]

Index

Hot Water

zz~<

Matching Fluid

Cold Water

Fig. 13. Unidirectional crystal growth apparatus. The sample is drawn

downwards to produce crystal growth. Thermal insulation surrounding

the water jackets and sample tubes is not shown.

61

Task 3

Measurement of High Temperature Thermophysical Properties

of Tungsten Liquid and Solid

D. W. Bonnell

Chemical Stability and Corrosion Division

Center for Materials Science

SUMMARY

The combined General Electric Advanced Applications Laboratory

(GE), Rice University (RICE), National Bureau of Standards (NBS) effort

has reached the end of its second phase. This phase was intended to

complete the physical connection of the RICE calorimetric system to the

GE levitation facility, and establish the limits of the system for the

acquisition of solid and liquid enthalpy increment data for tungsten in

an actual measurement effort. These limits have been established,

indicating that data are obtainable on solid tungsten up to the melting

point with a precision comparable to prior high temperature levitation

studies. It appears, however, that while liquid data are obtainable,

the reliability of drop successes is so low that it will be necessary to

completely automate the calorimeter system in order to accommodate the

extremely unreliable holding characteristics of the levitation system.

This unreliability is exacerbated by the nature of the calorimetric

measurement requirement.

A number of necessary improvements in the general system have been

made, particularly in the temperature measurement area, which allow

calorimetric quality heat content measurements to be related to an

accurate temperature. These improvements are expected to form a model

63
which GE will use in a variety of areas, and will hopefully provide the

basis for a generally available very high temperature imaging pyrometric

device

.

Finally, this second phase effort helped establish in detail the

precise requirements and degree of control necessary (1) to provide a

major improvement in the ability of this experimental effort to yield

accurate data and (2) to act as a model for future efforts in high

temperature calorimetry, such as space based experiments where only a

single operator will be available. Such an operator will, of necessity,

need to depend on the instrumentation to control details and provide

appropriate feedback, in order to be free to pursue the goals of the

experiment, without being lost in the simple achievement of operation of

the experiment.

64
INTRODUCTION

The primary aim of this task has been to evaluate experimental

procedures used in the interaction between the General Electric Advanced

Applications Laboratory (GE) and the Rice University (RICE) to measure

the high temperature enthalpy increments of liquid and solid tungsten.

GE has demonstrated a unique apparatus capable of levitating molten

tungsten [1] and RICE has wide experience in high temperature enthalpy

increment measurements, being particularly successful in the use of

drop-type isoperibol calorimeters with levitation heating [2;3a,b;4].

It seemed a natural cooperative effort for RICE to provide the calorimetric apparatus and GE to provide the levitation facility in order to

measure the high temperature enthalpy function of tungsten.

The results of this research are of great theoretical and engineering

interest. From the scientific viewpoint, the unique location of tungsten

at the upper extreme of the metal and element melting point scale should

provide a key part in any extrapolation/interpolation procedure. Models,

such as the well-known Tamman [5] rule and periodic table correlations [6]

,

are currently used for extrapolation. A definitive value for the heat of

fusion of this element will provide a valuable test of such models, perhaps

indicating a significant deviation in such predictions based on lower

melting refractory metals. Questions concerning the liquid state heat

capacity function, i.e., whether it is constant as would be suggested

from studies of lower melting refractory metals (see, for example, [7])

and whether current estimates [8;9] of the discontinuity from Cp(s) to

Cp(l) are even reasonable, are crucial in testing theoretical models

and may provide clues to establishing new parameters in existing

65
or new models. The engineering applications begin with the fact that,

without knowledge of the heat of fusion and liquid enthalpy function,

melting, forming, and casting equipment must of necessity be overdesigned

to allow a satisfactory safety/performance margin. These engineering

effects pervade all high temperature uses of tungsten-containing materials.

In the course of this work, it has been necessary to be present

during much of the actual experimental work at GE , and both RICE and GE

consult this author concerning the resolution of problems at all phases

of the experimental process, from initial design to final data reduction.

All modifications to the RICE calorimetric system were designed by this

author, and a large amount of available research time has had to be

devoted to the adaptation of the GE observation system to be an accurate

pyrometric device.

Liquid holding capability has proved to be the current unresolved

problem. The solution seems to require a totally automated calorimeter

which can be set to detect a falling sample, actuate its gates in real

time, and acquire the sample, while performing in real time, the preliminary calculations of data reduction necessary to establish the ready

state of the calorimeter, the true heat influx, and when the calorimeter

again reaches steady state. Such automation, when coupled to an automatic

version of the now operational GE pyrometric system, will provide the

basis for future experiment, perhaps in space, where real time data

acquisition is not a luxury, but a necessity.

66
EXPERIMENTAL PROCEDURE

The current experimental configuration is shown schematically in

Figure 1 (a,b). The sample is supported prior to levitation on a movable

pedestal, part of a lava and tantalum platform on rails which can be

laterally moved clear of the coil bottom during levitation. This platform

can be used to load successive spheroidal samples, but has only marginal

ability to select the order of introduction among a group of samples.

The tungsten radiation gate below the pedestal assembly must be manually

rotated clear of the drop path to obtain a successful drop. A microswitch on the radiation gate actuator detects the opening of the gate,

turns off the electron beam, and turns down the RF power to the levitation coil to drop the sample under control.

The work coil has been recently been redesigned. The original

design was a lower-coil-only conical design where the molybdenum "shade"

ring acted as a dock for field closure [ 1 0 ; 1 1

] . On our recommendation,

because of severe instability and the inability to melt, the current

coil design depicted in Figure la. was implemented by GE. This coil

style is a conical modification of the coil type used for liquid platinium

containment at RICE [4], incorporating an extra bottom turn to reduce

coil leakage. The requirement to pass the physically large 1 cm)

spherical samples and the large pedestal results in a bottom coil throat

inside diameter of 'v 1.3 cm. This large bottom opening still exaggerates

the possibility of sample loss due to dripping, as was the case for the

original coil, and it is certain that more work needs to be done by GE

in this area. The incorporation of an upper coil turn does, however,

allow higher coil power to be used to improve stability and lessen the

67
probability of coil dripping. The shade ring over the coil has been

retained to continue to serve as electron beam spill protection for the

work coil.

The levitation coil is driven by a 25 kW, 450 kHz General Electric

induction furnace. The vacuum feedthrough to the coil is a GE coaxial

design; the entire transmission line including inductive coupling (matching

transformer) allows power factors > 0.8.

The levitation coil is not designed to provide sufficient heating to

melt the sample. The maximum temperature reached by induction power

alone is approximately 2800 K. A simple calculation for the 1 cm diameter

samples of this work using the Stefan-Boltzman Law, i.e.,

dH/dt = E

tAgA(T

4

)

(where dH/dt is the rate of radiant energy loss for a body of area A,

total hemispheric emissivity E^, at a differential temperature A(T) to

its surroundings, a is the Stefan-Boltzmann constant of proportionality,

5.67032 x 10 ^ W*cm ^*K

4

[12]) gives the 2800 K radiation loss as

about 310 watts when a value for E^ of 0.28 [13] is chosen for tungsten.

At the melting point of tungsten, 3695 K [14], the loss is expected

to be about 1000 watts. This additional power is provided by electron

beam (e-beam) heating. In the levitation environment, e-beam heating is

possible only at temperatures where secondary electron emission is high

enough to maintain approximate charge neutrality. Tungsten's secondary

emission characteristic meets this criterion even for electron power

fluxes much in excess of 1 kW [15]. The electron beam gun shown in

Figure la directs a 35 kV, 150 mA focused electron beam onto the top of

a levitated specimen. This power is clearly sufficient to achieve

68
melting even with considerable loss due to charging. A significant

problem in this form of heating is the localized influx of energy due to

spot focusing. While the sample is solid, the top to bottom temperature

gradient is of the order of 50 K, with some uncertainty due to the fact

that the single point of view of the pyrometer provides only an oblique

view of the top and bottom.

Sample outgassing can release large bursts of gas. In the work

coil region, a pressure burst can raise the pressure to a level where an

RF discharge can be supported. When this happens, safety interlocks

must necessarily shut down power to protect the apparatus. Approximately

two thirds of all fresh samples are lost prior to or near the melting

point from this cause. In addition, a major portion of the lower temperature deposition on the internal optics appears to occur from fresh

samples. It is clearly necessary to preprocess all samples before use

to alleviate these problems.

The calorimeter proper is a chrome-over-gold plated copper cylinder

20.3 cm high and 13 cm in diameter. The center tapered receiving well

is a removable copper sleeve 5-1/4 cm top inside dimension and 11 cm

overall in length. A similar sleeve, wound with a manganin wire heating

element, was used for electrical calibration. The calorimeter receiving

well is lined with 0.010 inch tungsten foil and a variety of shards of

tungsten stand in the well to prevent splash back. The splash protection

has not been tested due to lack of liquid drops, but slightly surface

melted samples have impacted the well with no damage and good heat

transfer. This technique of splash protection has worked well for a

variety of other systems [2;4].

69
The total 25 °C heat capacity of the calorimeter as calibrated was .

7893 ± 7 J/°C. The basic uncertainty in total heat using this system,

exclusive of sample loss errors, radiant or electron beam power influx,

but inclusive of reasonable jacket temperature excursions, thermometer

errors, receiving well and foil liner weight errors, etc., is expected

to be much less than 0.5 percent.

A primary problem in this research effort has been the high temperature measurement capability. Under 35 kV electron bombardment, tungsten

emits copious x-radiation. For this reason, visual access is highly

restricted and a special imaging pyrometry system is needed. A General

Electric silicon charge injection diode array camera is used to provide

a 30 frame/sec oscilloscope image of the sample and work coil area at an

image magnification of about 5x. The levitated sample typically covers

about half of the 10,000 diodes of the 100 x 100 camera area. A GE

constructed address/sample-and-hold unit allows selection of any one of

the diode elements for analogue intensity output. The diodes have a

dynamic range of about 20 (max. signal/avg. noise) and a linearity and

long term stability of better than one part in 10,000. The spectral

response, however, (characteristically shown in figure 2) is quite

non-linear. A major portion of this task effort has been devoted to

implementing techniques necessary for conversion of this device into a

pyrometrically reliable tool. The initial effort by GE was to simply

extrapolate the optical response of a selected diode by an exponential

function from ribbon filament temperatures to the region of temperature

interest. Given the fixed reference point of the melting point of

tungsten, this technique should be good to 2 or 3 percent near the

70
reference point. As will become apparent in the following discussion,

this extrapolation is potentially poor, especially if the fixed reference

point is not well identified. The probable error without considerable

precaution could be of the order of 5 percent, i.e., perhaps as much as

200 K. A reasonable goal in enthalpy determinations is one to two

percent overall. This requires significantly better than one percent

pyrometric accuracy to achieve this error level in the face of emissivity

uncertainties, sample temperature inhomogenieties , and the general

problem of optics coating occuring due to sample outgassing and tungsten

deposition.

The optical path arrangement for pyrometry currently in use is shown

in Figure lb. The consequences of restricted access, the camera characteristics, vapor deposition problems, and the lack of black body conditions to the implementation of accurate pyrometry require consideration

of the theory and practice of optical pyrometry.

Plank's law provides the basis for pyrometric measurements, relating

spectral radiant flux to temperature by

CjA' 5/7t exptc^/AT)

where c^ and c^ are the first and second radiation constants (c^ =

3.741832 x 10 ^ W*m; c^ = 1.438786 x 10 ^ m*K), A is the spectral

wavelength, and T is the absolute temperature. is then the radiant

flux per unit area, per unit solid angle, per unit wavelength. The

basis for the current International Practical Temperature Scale

(IPTS-68) [16] at temperatures above the gold point, 1336.15 K is the

71
ratio of the flux at temperature, T, to that from a body at some reference temperature, T(ref). Figure 3, using the right scale, shows the

temperature behavior of this ratio for a fixed reference temperature of

1600 K and an ideal filter with unit transmission in a pass band 1 nm

wide centered at the wavelengths given. This technique is usually

referred to as brightness pyrometry.

Traditional pyrometry operates at a center wavelength of 650 nm

with a typical band pass (including the observer's sensitivity) of about

350 to 400 nm. Photoelectric pyrometers typically use band pass filters

with a transmission window ^ 35 nm wide. It is clear from Figure 3,

that in the realm of 2500 to 4000 K temperature measurements, significantly more sensitivity in temperature measurement is possible if shorter

wavelengths are used. In this temperature region, there is still,

however, more than a factor of two difference in absolute flux, favoring

the longer wavelengths. At 3700 K, a 25 K change in temperature produces

an absolute flux change of 6 percent at 450 nm, 5 percent at 550 nm and

^ 4 percent at 650 nm.

At lower temperatures, another form of pyrometry has been used.

Called two-color or ratio pyrometry, it uses the ratio of the fluxes at

two wavelengths as the measured function of temperature. For any given

wavelength pair, the sensitivity falls rapidly with increasing temperature (see Figure 3, left scale and ratio curves). The high sensitivity

curves use wavelengths where the flux at the shorter wavelength is very

low at low temperature. However, a multicolor measurement using

appropriate band passes can provide redundant temperature measurement.

72
Further, to the extent that emissivity on optics absorption is just

independent of wavelength the measured temperature is independent of

emissivity and changes in absorbance. With multicolor redundancy, it is

possible to measure emittance directly.

The general spectral response function of a silicon diode, shown in

Figure 2, indicates a rapid fall off in sensitivity at shorter wavelengths,

and the catastrophic termination of response at larger wavelengths,

where silicon becomes transparent. A choice of wavelengths is available

in this region and it is not at all clear that the traditional value of

650 nm is necessarily the best choice.

The primary problems plaguing the spectral flux or brightness

pyroraetry method are the uncertainty in emissivity and absorption effects.

The technique obviously requires complete knowledge about the actual

transmission of all optical train elements, but is relatively less

sensitive to the spectral behavior of the absorbances in the path as

long as they do not change and the pyrometer's spectral response band

pass is sufficiently restricted.

In order to implement the brightness pyrometry technique, it was

necessary to provide a reference source. Traditional pyrometers superimpose an image of an internal standard lamp on the attenuated image to

be measured. The standard lamp flux is then adjusted to match the

attenuated source image and the ratio is accomplished as a null measurement. The range of comparison between internal standard lamp and external

source generally uses a broad band absorber as the attenuator. The

external comparison of a black-body source and a hotter body is accomplished by attenuating the light from the hotter body by a device called

73
a sector wheel; essentially this is a chopping wheel behaving as a

mechanical aperture utilizing the eye as a visual smoothing device.

This method can provide for a direct reading instrument of excellent

accuracy and stability, but has no capability for dealing with changing

absorbance in the optical path, such as occurs when vacuum deposition

occurs. Vacuum deposition on the vacuum pyrometer window and reflection

mirror changes the transmission of the optical train typically by a

factor of two to five in the course of a single experiment. This magnitude of change is equivalent to a temperature change of 300 to 500 K.

The installation of a reference lamp internally within the vacuum

system was thus required. The mirror, Ml, shown in Figure lb, reflects

an image of the sample through the vacuum window which is located so

that no direct line of sight exists between the sample and that window.

This essentially eliminates coating of the window and provides x-radiation

protection. The small mirror M2 superimposes an image of the internal

reference filament at the image plane of the sample, offset to one side

about 1/2 cm visually from the sample limb. M2 shades this filament

from deposition and its orientation shields its own surface. Mirrors Ml

2 and M2 are 1/4 wave flat first surface mirrors, ^ 7 cm as used, obtained

from Edmund Scientific Corp. The installation of a General Electric

18A/6V/T10 strip lamp provides the means to establish an initial temperature scale, by comparison with a Pyro micro optical pyrometer, and the

means to make a running measurement of deposition on mirror Ml.

The procedure which has been developed involves replacing the

levitation coil assembly with a thick glass plate and pumping the system

down when a new lamp must be installed. The lamp is then operated over

74
the region from ^ 1600 to 2450 K, brightness temperature (> 1750 to

2770 K true temperature), while measuring the current drawn by the lamp

as a function of brightness temperature monitored by the pyro optical

pyrometer observing through the glass plate. The current is measured

with a 4% digit voltmeter monitoring the voltage drop across a high

current shunt. Current is supplied to the lamp from a well regulated dc

supply. At the same time, the GE camera also monitors the strip lamp

through its optical train. The emissivity of tungsten selected [17] was

0.425 for this filament. The procedure must be repeated frequently, as

the lamp reference is operated far above its long term stability point [18].

This technique provides a low temperature reference to the IPTS-68 once

the absorbance of the thick optical plate is corrected. This is accomplished in a separate experiment by measuring the temperature change,

as a function of lamp brightness temperature, with and without the glass

plate in the optical path. The sighting path through the plate was

identified by physically marking the plate and carefully measuring and

preserving the angles of the calibration experiment. The preferred

method of performing this correction is to actually measure the

absorbance [4] but in most cases, an actual temperature effect

measurement is adequate. The Wien approximation to the Planck Law

J

\ exp (c^/ AT)

is a good approximation in this temperature wavelength regime. Since

transmittance is just —

— o y = e = transmittance

o

75
and in the Wein approximation, behaves just as an emittance, identifying

the flux change as a transmittance effect gives the relation between

•?V brightness temperature IL observed without the absorber to the observed

D

temperature T ^ with the absorber as apparent

1

Vc A ln(e

aX

) _ Kabsorber

apparent

The glass plate used showed a variation in K of a factor of two over the

temperature range 2300 to 2500 K. This is attributed to a color dependence in the glass transmission, and the K , . values were applied absorber

/V at the T^ values measured. The effect of this absorber was to lower

JJ observed brightness temperature by 95 to 52 K. The error due to uncertainties in the exact wavelength dependence were estimated from the

direct lamp temperature versus current measurements to be of the order

of 15 K. This plate must be replaced in future measurements by a sheet

of pyrex or quartz whose visible spectral characteristics are known and

well behaved.

The imaging array camera uses a relay lens with an effective aperture

of 1 in. to provide the necessary magnification to nearly fill the frame

with the sample image. To convert this device to a pyrometric device,

it was necessary to control in a known way, both the relative sensitivity

and the true spectral wavelength response of the system. As a result of

early trials, a combination filter holder, aperture control device was

designed and put into service. This device fits on the entrance aperture

of the relay lens and has two manually turned detented wheels. One

wheel holds one inch aperture narrow band (> 80 mn FWHM) interference

76
filters blocked to 10 or better outside the band pass. The two filters

actually calibrated in this work had band passes centered at 651 run and

451 run. An auxiliary broad band pass neutral density (ND) filter mounted

on front of the filter/aperture device was used as a range control. The

other wheel has a series of knife edged circular orifices: 1, 5/8, 1/2,

5/16, 3/16, and 1/8 inch in diameter, which were to be prepared to

± 0.0005 tolerance.

Calibration consisted of comparing the output of a specific diode

of the imaging array as a function of temperature, filter aperture and

ND filter with clean freshly mounted mirrors using the internal calibrated lamp. A variety of measurements for cross checking redundance

were taken to allow the calculation of any observation from others. The

one inch aperture was found to be larger than the system aperture, as

expected. The 5/16 aperture consistently showed itself to be 7 percent

too small in area or 0.010 inch smaller than the manufactured 5/16

(0.312 inch) aperture. On subsequent check, the actual diameter was

found to be 0.302 ± 0.001 inch, exactly as predicted. All other apertures,

and the transmission factors of the two filters agreed with actual

temperature measurements with a reproducibility of better than 0.6 percent.

The quality of the fit of data over the lower temperature calibration

range fit the Planck Law expression with a maximum error of 15 K and an

average error of 5 K. When translated by aperture changes to the temperature range 3500 to 4100 K, cumulative errors in aperture and fixed

optical path absorbances are expected to contribute less than an additional

10 K error. With the built-in reference, the remaining problem is

changes in the spectral reflectivity of mirror Ml. In fact, the very

77
effort to measure the problem, taking pyrometric measurements at two or

more wavelengths, allows an independent check on temperature when the

wavelength dependence on reflectivity of Ml is measured. Where possible,

experiments are arranged to attempt this, but the inability of the GE

system to provide a significant time of constant temperature liquid

levitation (or even hold on to the completely liquid sample) has restricted

these measurements to stably levitated solids. In that case, the agreement

between the three temperatures, brightness at 650, brightness at 450,

and color ratio between these two measurements has a precision of better

than ± 20 K, indicating accuracy attained of ^ 0.6 percent.

DISCUSSION AND CONCLUSIONS :

The main goal of this phase of this task has been to help RICE install

a calorimetric subsystem on GE's existing and proven [19] liquid tungsten

levitation system. The necessity of converting an approximate temperature

tool into a precision pyrometer was anticipated, and although a variety

of problems were encountered (such as the need to operate around extreme

radio frequency interference (REI) from adjacent unrelated experiments,

internal problems with the diode select logic, etc.), the procedures

which have been established are workable and meet the initially anticipated needs of the project. The calorimetric system has performed

perfectly. The addition of a data logging system at the beginning of

this phase was made to eliminate the need for an observer for the calorimeter heat trend. As expected this addition allowed the expansion of

effort from two to as many as six drop attempts per day. The 8 kJ heat

capacity of the calorimeter block and the vacuum insulation still restricts

the effort to one sample in the calorimeter in four hours.

78
It had been anticipated that if this phase progressed as in fact it

did, that we would have obtained several successful liquid drops and be

in a position to refine the current value uncertainty in the heat of

fusion, H (tungsten),

m The Tamman Rule [5] (AHmm/T = S ^2.3 cal/mol-K) value of 8.5 m

kcal/mol has been the operational value in all the standard references

[ 9 ; 20 ] . Dikhter and Lebedev [21] have obtained a value of AH^ by an

exploding wire technique. Their direct value, 80 ± 4 cal/g, corresponds

to a value of 14.7 ± 0.7 kcal/mol. This value is remarkable as is

implies a value of AS =4. The largest value of AS known for the

m m transition metals is molybdenum, with AS = 3.08 cal/mol*K [3a]. Even

m though the general trend in AS^ seems to be increasing down each group [4]

and tungsten is below molybdenum, this value still appears to be too high.

Kubachewski, et al. [22] give a value of 12.1 kcal/mol for AH

m (silicon)

which was expected [23] to have the highest heat of fusion of any element.

Wouch, et al. [19] report a value for AH^ of 11 ± 1 kcal/mol obtained by

applying a cooling curve model to experimental data. The model ignores

supercooling, but provides a value of AS^ of 2.98 ± 0.3 cal/mole*K, in

line with expectations from the molybdenum data. The value is still

very uncertain, as Wouch, et al. [19] repeatedly noted, especially in the

weakness of this derivation for non-spherical samples. (Terrestrial

levitation). In the course of this work, to establish the GE ability to

levitate and melt, a sample was allowed to fall onto a copper plate in a

one meter drop tube. The sample splashed as though completely molten,

and all (± 1 percent) fragments were recovered. Since the temperature

was estimated to be just above melting, even if AH^ is 8.5 kcal/mole,

79
significant supercooling must have occurred and the Wouch, et al. [19]

model for the cooling curve type of determination [24] should not apply.

Unfortunately, even though the calorimetry and pyrometry are now

functional, the levitation process itself has been plagued with problems.

Although outside the area of this task, improvement in this area is

essential to the task and efforts to deal with the problems have been

made. The work coil was changed at our suggestion. The problem of the

so-called "hot short", where samples would deform, "melt", or explode at

temperatures hundreds of degrees below the melting point of tungsten was

again encountered. Previously, this problem had been diagnosed as being

caused by hydrocarbon contamination during machining of the specimens.

This recurrence was identified as vapor deposition of volatile metals

from the soft solders used in the coil area. Figure 4 shows the effect

of this problem on a group of samples. In addition to the loss of

samples, the shape change can be mistakenly interpreted as melting,

making the use of visual observation of "melting" as an internal pyrometric calibration point unreliable.

The most serious problem has been the inability of the levitation

system to melt and hold tungsten. More than 30 samples have been levitated and successfully e-beam heated in full dress calorimetric runs.

In no run has it been possible to contain the liquid for more than 1 to

2 seconds before inadvertant pouring from the coil occured. With the

current level of hand operations necessary, the very short, unpredictable containment times give an intolerable failure rate. The consequences

of the levitation containment problem are quite important to the eventual

80
success of this program. The current implication under consideration is

that the surface tension/density ratio of liquid tungsten is perhaps as

low as that for gold or lower. This problem was encountered at RICE in

an effort to measure enthalpy increments for tantalum [4] . The measurement effort for platinum also required special efforts to contain. It

is quite possible that this will be a general property of the entire

third long period of the transition elements. Improving containment

could require a significant additional effort on the part of GE to

improve their levitation facility.

Because of the containment problem, it has been necessary to suspend

experimentation in this effort. At a minimum, it will be necessary to

automate the calorimetric system to improve the system response time to

the point where the ability of the calorimeter gate structure to respond

no longer requires long or predictable containment times. The general

characteristics of such a system have already been examined and will be

the subject of future reports when the various aspects have been discussed

fully with RICE and GE. RICE is expecting a new post-doctoral student

to enter this project and the final form of the plans will, of course,

depend on this.

The advantages of an automated calorimeter system extend beyond the

immediate application. It is expected that calorimetric studies will be

among those selected for microgravity experiments. The current effort

requires three or four researchers to operate the apparatus and take

data. The reduction of data to heat content and temperature requires

some time as well. The effort of automating the calorimetric system and

integrating the pyrometry system into such automation obviously is a

81
step which will be done in the normal progress of this experimentation.

This seems to be the obvious time to implement such automation. It is

recommended in the interest of future efforts, that the pyrometric

system be automated as well. The major item necessary to providing an

automated pyrometric system is some method of measuring the change in

absorbance of the primary mirror, Ml (Figure lb).

Figure 5 shows a schematic of the hardware portion of a proposed

automatic imaging pyrometric system. The motor driven aperture and

filter assembly on the pyrometer is a simple extension for computer

control. The novel feature is the modulated photodiode assembly. With

a chopping wheel modulating the internal filament, a simple photodiode

can be used to extract the change in apparent brightness of the reference

filament by lock-in amplifier techniques. With this information in

real-time as well as the output from the imaging array, a microcomputer

can, with careful calibration as has already been developed and described

in this report, provide an almost real-time display of true temperature,

and report the onset of problems in temperature measurement from the

multiple redundancy inherent in multicolor pyrometry.

The same computer can easily have the capability to handle the calorimeter automation and data logging requirement. The same data system,

with auxiliary floppy disk storage, can provide complete data reduction

in a few minutes after the experiment ends. In fact, a running monitor

of the final period of the experiment may be able to significantly

shorten the 4 hour per run time period currently necessary for best

work.

82
REFERENCES

1. G. Wouch, "Containerless Melting and Solidification of Metals and

Alloys in the Terrestrial and Space Environment", Ph.D Thesis, Drexel

Univ. (1978).

2. A. K. Chaudhuri , D. W. Bonnell, A. L. Ford, and J. L. Margrave,

High Temp. Sci., 2, 203 (1970).

3. J. A. Treverton and J.L. Margrave,

(a) J. Phys. Chem. 75, 3737 (1971).

(b) J. Chem. Thermodynam. 3, 473 (1971).

4. D. W. Bonnell, "Property Measurements at High Temperatures--Levitation

Calorimetry Studies of Liquid Metals", Ph.D Thesis, Rice Univ. (1972).

5. G. Tamman, Z. Phys. Chem. 85, 273 (1913).

6. J. L. Margrave, High Temp. -High Press. 2, 583 (1971).

7. D. W. Bonnell, A. J. Valerga, and J. L. Margrave, "Radiation and

Conduction Loss Corrections to Free Drop Calorimetry," in preparation

(1981).

8. JANAF Thermochemical Tables , (Joint Army, Navy, Air Force) Second

Edition, D. R. Stull et al . (Eds.), NSRDS-NBS 37, U. S. Government

Printing Office (1971).

9. R. Hultgren, P. D. Desai , D. T. Hawkins, M. Gleiser, K. K. Kelley,

and D. D. Wagman, Selected Val ues of the Thermodynamics Properties

of the Elements , Amer. Soc. for Metals, Metals Park, OH (1973).

10.

R. J. Begley, G. Comenetz, P. A. Flinn, and J. W. Salatka, Rev. Sci.

Instr. 301, 38 (1959).

11. J. C. Lewis, H. R. J. Neumeyer, and R. G. Ward, Rev. Sci. Instr.

39, 569 (1962).

12. E. R. Cohen and B. N. Taylor, J. Phys. Chem. Ref. Data 2, 663 (1973).

13. C. C. Jun, S. Ebrahim, and M. Hoch, High Temp. -High Press. 2, 43

(1970).

14. A. Cezairliyan, High Temp. Sci. 4, 248 (1972).

15. R. T. Frost, Private Communication (1975).

16. The International Practical Temperature Scale, Metrologica 5, 35

(1969).

83
17. R. D. Larrabee, J. Opt. Soc. Am. 49, 619 (1959).

18. H. J. Kostkowski and R. D. Lee, Theory and Methods of Optical

Pyrometry , NBS Monograph 41, U.S. Government Printing Office,

Washington, DC (1962).

19. G. Wouch, E. L. Gray, R. T. Frost, and A. E. Lord, Jr. , High Temp.

Sci. 10, 241 (1978).

20. K. K. Kelley, U. S. Bureau of Mines Bull. No. 584 (1960).

21. I. Ya. Dikhter and S. V. Lebedev, High Temp. -High Press. 2, 55,

(1970).

22. 0. Kubachewski, E. L. Evans, and C. B. Alcock, Metallurgical

Thermochemi stry , Fourth Edition, Pergamon Press, London (1967).

23. J. L. Margrave, Private Communication (1972).

24. J. L. Margrave, D. W. Bonnell, and J. Weingarten, "Thermodynamic

Property Determination in Low Gravity", Final Report, NASA Contract

NAS8-32030 (1977).

* The trade names listed in the descriptions of equipment in this report are provided for identification purposes only.

84
FRONT VIEW

FIGURE la

Fig. la. Front cross sectional veiw of the GE-Rice coupled apparatus

showing the relative orientation of major components.

85
TOP VIEW

FILTER

FIGURE lb

Fig. lb. A top view showing details of the optical arrangement for pyrometry.

86
ABSOLUTE

RESPONSIVITY

TYPICAL

SPECTRAL RESPONSE

for SILICON DIODES

WAVELENGTH nm

Fig. 2. Characteristic spectral response behavior for silicon photodiode

type devices. The GE imaging array elements have a similar response

function.

87
SPECTRAL

WAVELENGTH

RATIO

PYROHETRY TEMPERATURE FUNCTIONS

TEMPERATURE (K)

Fig. 3. Plank law functional behavior for two type of pyrometry. The

dashed (and solid) curves are spectral radiance ratios (right

scale) for the labeled wavelengths for T

^ = 1600 K, typical of

brightness pyrometry. The dot-dash curves show the flux ratios

(left scale) for wavelength pairs, typical for multicolor pyrometry.

88

SPECTRAL

RADIANCE

RATIO

w

(A.

T>

/w

(A.

Tref)

20000

15000

10000

5000
HOT

SHORT"

SAMPLES

LU

ocDg

u.

a>

s-ai

3:

•a

ai P 03-3 c -a jd

•r- ai 3 -P O (0 x: i—

in zsc- (0 • S- </> cn ai . •r- in in s- ai

•t—

U *1“ £= -p

S-o

S- -p

4-> (0

in

aiCLE

ra cn

toai

(0

in

-P

S-o

-po 3 v- o s- XI 3

Q.

C E

03

S- r—

03 (0 P

03 <0 C S

-a

ai

sai>ou

ai

s- T3

CL) P

S-o

in

in03

(0

(0in -O

ai o0

r—

01X

03 ~oc

(0

ino

CL

a> a co

in -C

Q.

03 -P

S- 03 03 E O 1- -P O O M—

x: <u Q. "O

Oai -p

Q) -r- s-3

in

in<u

s-CL S-o

sa O 03 a. >

03> <po

x:

03C

•I- o

JZ •!- p

03NO

•P

00

•H

89

never

heated

above ~

3300

K.

Note

tool

marks

and

degree

of

deforma-
COMPUTER

oc

LU

t—3

Q.£o

90

Schematic

of

optical-mechanical

arrangement

fora

computer

operated

real

time

direct

reading

pyrometric

system.

See

text

for

discussion.
Task 4

High Temperature Thermophysical Properties of Refractory

Materials from Free Cooling Experiments

Jack H. Colwell, Ared Cezairliyan, and Lawrence A. Schmid

Thermophysics Division

Center for Chemical Physics

Summary

Free cooling experiments are to be used to obtain high temperature

(1200-4000 K) , heat capacities, thermal radiative properties, and thermal

dif fusivities of materials while samples are levitated in a space environment.

The data are to be derived from the simultaneous measurement of the total

radiant heat loss and the temperature change of spherical samples as they

freely cool by radiation alone. Pyroelectric detectors are being investigated

for use in making the total radiance measurement. The dynamic response of

these detectors have been tested using a millisecond pulse heating apparatus,

which permits metal strips to be heated at rates up to 4000 K/s and cooled

at rates up to 1500 K/s. In these experiments the pyroelectric detector

measures the total normal radiance from an exposed portion of the strip

while the radiance temperature of the strip is determined by a high speed

pyrometer. A small (~1%) discrepancy between the heating and cooling

portions of the experiments is ascribed to the radiance of the strip, not

to the behavior of the detector. Normal total emittances derived from

these high speed measurements are in excellent agreement with the steady

state results of others. The large temperature gradients that can develop

in rapidly-cooling spherical samples complicates the interpretation of the

free cooling experiments. Mathematical expressions have been derived for

values of the physical parameters of materials from the observed time

dependence of a sample's surface temperature and total radiance.

91
Introduction

The objective of this research is to measure the high temperature (1200-

4000 K) thermophysical properties, viz., heat capacities, thermal emittances,

and possibly thermal dif fusivities , of refractory materials. The experiments

will exploit the microgravity environment of space to maintain the samples,

when hot, without physical contact. RF induction will be used to heat samples,

and subsequently, will be used at low power levels to maintain the position

of the samples. Samples will be spherical, fabricated in that form as solids

and, when liquid, maintained in that form by their surface tension. Experiments

will consist of heating the samples to a high temperature in a cold-walled

vacuum chamber and then allowing the samples to freely cool by radiation. As

the sample cools, its surface temperature will be determined by multicolor

radiation pyrometry and the total radiative heat loss will be determined

simultaneously by a wide-band pyroelectric detector. The spherical geometry

of the sample makes it possible for the total radiance measurement to be made

with a single detector. A unique feature of these experiments is that it

will permit liquid samples to be followed into the supercooled region all the

way to their point of recalesccnce , thus yielding thermophysical data on

supercooled liquids and possibly kinetic data on the solidification process.

Containerless liquid refractory metals undergoing free cooling may be expected

to supercool by as much as 1000 K.

The cooling rates of the samples will be a function of the temperature

and the size of samples

t

1

» 2

] . For smaller samples, less than 5 mm diameter,

at 4000 K, the cooling rate will exceed 1000 K/s. Larger samples, 10 mm

diameter at 1500 K will be cooling at rates of the order of 10 K/s.

92
Because of the dynamic nature of the cooling experiments, temperature gradients

will exist within the samplest 2

]. For metals with a high thermal diffusivity,

the total temperature differential from surface to center will be a maximum

of tens-of-degrees and will have little effect on the measurements. For

materials of lower diffusivity, the temperature differentials can reach

hundreds-of-degrees . The temperature differential is, however, an inverse

function of the sample radius, so by measuring different sized samples the

physical parameters can still be determined. For materials within a particular

range of dif fusivities , the diffusivity itself can be obtained. The mathematical

procedures for determining the values of the material parameters from the

experimental observables is described in detail in Part B of this report.

One of the innovative aspects of this work is the use of pyroelectric

detectors in dynamic measurements of the total radiance. In Part A of this

report we describe the experiments with a pyroelectric detector using the

rapid electrical heating apparatus of the Dynamic Measurements Laboratory

at NBS. In these experiments, metal strips are heated to a given temperature

in less than one second, and then allowed to cool. The temperature is

measured pyrometrically and a portion of one surface is viewed by the pyroelectric

detector. The pyroelectric detector gives a determination of the total

normal emittance of the surface as a function of temperature. The main

advantage of this apparatus is that heating and cooling rates can be generated

which are several times faster than expected in the free-cooling experiments,

thus thoroughly testing the dynamic response of the detector.

The vacuum assembly is nearly complete and the inductive power supply

is now on hand for the ground-based simulation of the free-cooling experiments.

93
This effort is being detained while we wait for the renovation of a new

laboratory which will house the experiments. In addition to the experiment

described in this report, we are proceeding with the development of a

millisecond-resolution two-color pyrometer which will be tested in the

simulation experiments.

94
Part A:

Experimental Procedures

F 3 1 The millisecond pulse heating apparatus 1- J used in these experiments

impresses a nearly constant voltage across a conductive specimen for a

predetermined length of time. The constant voltages are produced by a bank

of batteries. Voltage levels are determined by the number of cells in the

circuit and an adjustable series resister. By the time the sample reaches

the temperature where measurements can begin, T > 1200 K, all switching

transients have died away and the current is constant except for the slow

decrease caused by the increase in the resistance of the specimen as its

temperature rises. The duration of the pulse is determined by the circuit

characteristics and a temperature limit, and generally is less than one

second.

The sample was viewed from one side with a high speed pyrometer which

focused on a target area of 0.2 mm diameter at the center of the specimen.

The pyrometer uses a photomultiplier tube as a detector, which is alternately

exposed to light from the specimen and from a calibrated strip lamp. The

sequence of measurements made by the pyrometer is governed by a 12 aperture

chopper rotating at 200 Hz. Six apertures admit light from the specimen and

are completely open as are two of the apertures admitting light from the

lamp. The other four apertures contain two pairs of light attenuators. With

each rotation of the chopper, six observations of the specimen are made,

interspersed with six reference observations of the lamp. The pairs of

reference attenuators used in the chopper are not perfectly matched so their

signals show up as two closely spaced doublets on the output (see Fig. 1)

.

The specimen temperature is determined only when the signal lies between

the upper and lower reference lines; this constitutes a change of 20-25

percent in the brightness temperature of the specimen depending on

95
the particular temperature range. Temperatures in different ranges are

obtained by placing calibrated neutral-density filters in the light path

from either the lamp or the specimen. To cover a wide range of temperature,

therefore, successive experiments must be run with different filter combinations

to cover each successive temperature range.

The electrode assembly that holds the specimen is enclosed in a cylindrical

vacuum chamber that is sealed by an O-ring to the bottom plate. The pyrometer

views the specimen through a glass window in the side of the vacuum chamber,

with the objective lense 15 cm from the specimen. The pyroelectric detector

is on the opposite side of the specimen, mounted in a side tube of the

vacuum chamber so that no windows need intervene between the specimen and

the detector. The light chopper, which is required for pyroelectric detection,

is a small (30mm diameter) rotating sector device and is operated at a

chopping frequency of 200 Hz. The chopper motor has teflon bearings

and has now been operated for tens of hours in a vacuum with no apparent

overheating or other ill effects.

The portion of the specimen viewed by the pyroelectric detector is

determined by a pair of horizontal knife edges that obscure the top and

bottom portions of the strip, so that the detector views only the central

portion. The knife edges are mounted on the electrode support assembly, 5

cm from the specimen. The detector is 25 cm from the specimen so the

effective sample length is 25% larger than the slit width. The active area

of the detector is 1 mm in diameter so the penumbra region increases the

effective length of the specimen by 0.2 mm. In these experiments, the gaps

between the knife edges were 12 to 15 mm, so any error in the penumbra

correction should have a negligible effect on the measurement. Our first

experiments were with tungsten strips, 6.35 mm wide, 0.25 mm thick, with

96
45 ram of length exposed between the electrodes. The opening between the

knife edges was 14.6 mm. The outputs of the pyrometer and pyroelectric

detector were recorded with two dual-trace oscilloscope-camera combinations.

Numerical values were obtained from the photographs using a traveling microscope.

A subsequent set of measurements was made with niobium strips of the same

size as the tungsten strips and with a knife edge opening of 12.7 mm. For

this second set of measurements, data were recorded using a pair of digital

oscilloscopes from which the data could be transferred directly to the PDP-11

computer for processing. The light chopper for the pyroelectric detector was

synchronized with the pyrometer chopper so that the two sets of data could be

accurately correlated.

The linearity of the response of the pyroelectric detector was established

using the inverse distance law. A DXW 1000 W tungsten halogen lamp was used

as a light source. The detector was moved over distances from 50.0 to

141.4 cm from the lamp. Measurements were made for irradiances covering

more than three orders of magnitude from 32 mW/cm2 to 20 yW/cm2

. After values

were measured from 50 cm out to 141.4 cm, the lamp power was reduced until

the detector signal was the same at 50 cm as it was previously at 141.4 cm.

The process was then repeated. These tests showed that the linearity

of the detector was better than 1% over this wide range. The principal

uncertainty in the calibration procedure comes from the geometric

distribution of the filament in the light source. The filament of the lamp

is a coiled-coil construction, approximately 20 mm long and 5 cm in

diameter. There is some uncertainty in the point taken as the origin

for the distance measurement. Also, the angle of the lamp to the light

path to the detector has an effect, probably due to changes in the amount

that the front coils of the lamp obscure th,e back coils at different detector

97
distances. Care has to be exercised that the lamp had reached thermal

equilibrium after each power change.

Experimental Results

Oscillographic traces of one of the tungsten strip heating experiments

are given in Fig. 1. The two oscillograms are of the same experiment with

the data collected at different sweep rates on separate oscilloscopes. The

upper trace in each oscillogram is the pyrometric temperature signal,

increasing to the point where the current is cut off and then decreasing as

the specimen cools. The horizontal bands are the pyrometer reference

levels, their corresponding radiance temperatures are indicated on the

lower oscillogram. The lower trace on each oscillogram is the AC signal

from the pyroelectric detector, the peak-to-peak distance being a measure

of the total radiant energy. The shape of the pyroelectric signal was

puzzling when first observed. With slowly varying light intensities, the

two halves of the AC signal are found to be symmetrical, so the asymmetric

signal observed in the rapid heating experiments was unexpected. The upper

half of the pyroelectric signal in the oscillograms corresponds to light

falling on the detector, and the lower half to the detector obscured by the

light chopper. A clue to the explanation of the asymmetric signal is found

in Fig. 2 which is the response of the detector when suddenly exposed to a

constant radiation source. The initial response is a large positive signal

during the exposed half of the cycle, and a very small negative signal

during the obscured half of the cycle. With time the positive peaks decay

and the negative peaks increase until they are symmetrical about the zero

signal axis; the relaxation time, t, for this process is approximately 100

ms for this detector. t is the thermal relaxation time of the detector and

is determined by its heat capacity, c, and the thermal conductance, h.

98
between the detector element and its mounting, so that x = c/h. The initial

response in Fig. 2 is when the detector is first heated above ambient, the

temperature increase is large in the exposed cycle and small in the obscured

cycle because little heat is being lost to the heat sink. As time goes on,

the average temperature of the detector increases and more heat is being lost

to the heat sink, so that the temperature increase during the exposed cycle

becomes less and the decrease during the obscured cycle becomes greater,

eventually becoming equal when the steady state is reached. For a steady

light source, as in Fig. 2, this relaxation phenomena introduces no detectable

error (~0.3% resolution in this experiment) between the peak to peak voltages

at the beginning of the relaxation period and those after the steady state

was reached.

A second relaxation phenomena that affects the signal from a pyroelectric

detector is the rise time of the device. For the detector used in these

experiments the time constant was approximately 0.5 ms. As a consequence of

the rise time and the drop in the signal due to the thermal relaxation, the

maximum signal with constant irradiation occurs at approximately 2.0 ms into

each half cycle.

The series of experiments on tungsten strips covered thermodynamic

temperature ranges from 1500 to 3000 K. Heating rates varied from 4000 K/s

at the lowest temperatures to 2000 K/s at the highest temperatures and the

corresponding cooling rates ranged from 500 to 1500 K/s. A direct comparison

of the normal total radiance with the radiance temperature of the specimen

showed that the measured total radiance when the specimen was cooling was

always 1 to 2% lower than when it was being heated, except for the region

near the maximum temperature.

99
Thermodynamic temperatures were calculated from the measured radiance

temperatures using the spectral emittance data of de Vos!-

4

] and Abbott I- 5 -L

The Stef an-Boltzmann law and the total normal radiance data were then used to

calculate the total normal emittance of the tungsten strips as a function of

temperature. The calculated values agree within 1% with the publish values

of Abbott^-

5

] which were obtained in steady-state experiments. This close

agreement on an absolute scale is fortuitous for the intensity calibration of

our detector was not thought to be that precise.

Fig. 3, 4, and 5 are the results of experiments with niobium strips over

three different temperature ranges; these are direct computer plots with no

intermediate treatment of the data. The plot of normal total radiance

(pyroelectric voltage) versus radiance temperature again shows the slight

difference between the heating and cooling curves. The difference increases

the longer the sample cools and is largest in the higher temperature ranges.

Normal total emittance values from these experiments are within 5-10% of the

steady-state values determined by Abbott!-

5 ], the discrepancies resulting from

the lack of precise values for the normal spectral emittance required to

derive chermodynamic temperatures.

Fig. 6 presents the normal total radiance and radiance temperature

measurements during the melting of a niobium strip. The radiance temperature

shows the flat plateau, characteristic of melting curves in these dynamic

experiments^ 5

] ; the radiance temperature drops by only 2-3 K in a period of

200 ms. The normal total radiance, however, has a short plateau and then

drops off steadily (down 3-4% after 100 ms) until it finally drops precipitously

as the specimen melts through. In an earlier melting experiment the timer

switch stopped the current prior to the specimen melting through, the melting

plateau was only about 100 ms wide. The measured width of this partially

melted sample after the experiment was found to be 1-2% narrower

100
than it was originally because of the melting at the edges. This

indicates that the fall off in the normal total radiance in Fig. 6 may

be due to a narrowing of the specimen. The radiance temperature is

unaffected by the narrowing of the specimen because it is determined by

observing a small spot of the center of the strip.

The melting point of niobium is known, 2750 1

, so the normal

total emittance of liquid niobium at the melting point can be evaluated;

it is 0.256.

Conclusions and Discussions

The small discrepancy between the heating and cooling portions of

the correlation between the pyroelectric voltage and the radiance temperature

is thought to due to changes in the total radiance determinations and

not due to any error in the temperature measurement. We feel that the

change is most likely due to differences in the total radiance of the

strip during heating and cooling. While cooling, the strip will be

losing energy from the edges of the strip (4% of surface area) as well

as the faces so that the average radiance from a face will be less than

the radiance from the center of the face. During the heating period

this tendency is mitigated in two ways. The rate of heating is much

faster than the rate of cooling, so that the temperature gradients will

not be as fully developed. More importantly, however, when heating, the

current will tend to follow the path of lowest resistance, i.e. the

coolest path, and will suppress the development of the temperature

gradients. There is the possibility that the discrepancy is inherent in

the pyroelectric detector. Because of the finite rise time (and the

thermal relaxation) of the device, a different voltage signal could be

obtained for rapidly increasing and decreasing signals. This possibility

101
has not been investigated, but if it were a major contributing factor to

the observed discrepancy, it would be expected to be largest near the

peak of the temperature curve were the signals are changing most rapidly;

this is not the case. In any case, the heating and cooling rates in

these strip heating experiments are several times larger than those that

will be encountered in the free-cooling experiments, so errors from this

source will be minimal.

The principal aim of this stage of our research is to establish that

pyroelectric detectors can be used to make accurate total irradiance

measurements in high speed experiments. These experiments have accomplished

that.

References

1. J.H. Colwell, NBSIR 79-1767, NBS : Properties of Electronic Materials

,

J.R. Manning, Ed., p. 103.

2. J.H. Colwell, NBSIR 80-2082, NBS Materials Measurements , J.R.

Manning, Ed., p. 88

3. Ared Cezairliyan, J. Res. Natl. Bur. Stand., 75C , 7(1971).

4. J.C. DeVos, Physica, 2_0 690 (1954).

5. G.L. Abbott: Total Normal and Total Hemispherical Emittance of

Polished Metals, Part III. WADD Tech. Rep. 61-94, U.S. Air Force,

Sept. 1963, W.J. Parker and G.L. Abbott, NASA SP-55, S. Katzoff,

Ed. , p. 11.

6. Ared Cezairliyan, J. Res. Nat'l. Bur. Stand., 77A , 333 (1973).

7. Ared Cezairliyan, High Temp. -High Pres. 4, 453 (1972).

102
1930 K

1680 K

1530 K

Fig. 1 Tungsten strip heating experiment. Two oscillograms of the same experiment at different sweep rates, 50 ms/cm in the top

oscillogram, 200 ms/cm in the bottom. The top curve in each case

is the pyrometer output, the bottom pair of curves is the ac pyroelectric voltage. The horizontal traces are the reference

levels of the pyrometer and their corresponding radiance temperatures

are indicated on the lower oscillogram. The radiance temperature at

the peak is 1912 K which corresponds to a thermodynamic temperature

of 2056 K.

103
Fig. 2 The pyroelectric response upon instantaneous exposure to a constant light source. Sweep rate is 50 ms/cm, pyroelectric

chopping frequency is 200 Hz. Variation of the two voltage levels

is due to thermal time constant of the detector; the peak-to-peak

voltage is not affected by the relaxation process.

104
PYROELECTRIC

VOLTAGE

(mV)

RADIANCE TEMPERATURE (K)

1500 1600

250

200

150 -

100

50

1700

T

_ 1800

I 1900 2000

~1~

i 2000

P

Q-'-

.

<3

Q''

•

•

•

. 0"'

A

-<—

6

<£•

5'" O'"

o--' ©:

0.-'

1 1 100

J I 200

TIME (ms)

300

1900

1800

H 1700

1600

1500

Fig. 3 Niobium-strip heating experiment to 1900 K. Arrows on curves indicate scales to be used. The pyroelectric voltage (proportional

to normal total radiance) and the pyrometrically determined radiance

temperature are shown as a function of time, and also plotted

against each other. On the latter curve the encircled points are those with the temperature increasing. Near the maximum temperature,

the heating rate is 4200 K/s and the cooling rate is 500 K/s.

105

RADIANCE

TEMPERATURE

(K)
PYROELECTRIC

VOLTAGE

(mV)

1700

RADIANCE TEMPERATURE (K)

1800 1900 2000

250

200

150 -

H 2100

100

- 2000

1900

1800

1700

100 200

TIME (ms)

300

Fig. 4 Niobium-strip heating experiment to 2100 K. Arrows on curves indicate scales to be used. The pyroelectric voltage (proportional

to normal total radiance) and the pyrometrically determined radiance

temperature are shown as a function of time, and also plotted

against each other. On the latter curve the encircled points are those with the temperature increasing. Near the maximum temperature,

the heating rate is 3600 K/s and the cooling rate is 700 K/s.

106

RADIANCE

TEMPERATURE

(K)
PYROELECTRIC

VOLTAGE

(mV)

RADIANCE TEMPERATURE (K)

TIME (ms)

Fig. 5 Niobium-strip heating experiment to 2375 K. Arrows on curves indicatescales to be used. The pyroelectric voltage (proportional

normal total radiance) and the pyrometrically determined radiance

temperature are shown as a function of time, and also plotted

against each other. On the latter curve the encircled points are those with the temperature increasing. Near the maximum temperature,

the heating rate is 200 K/s and the cooling rate is 1200 K/s.

107

RADIANCE

TEMPERATURE

(K)
800

700

600

500

400

300

200

100

0

i

2500

2400

2300

2200

- 2100

100

L . 200

_L

300

TIME (ms)

__ _J

400

2000

500

. 6 Niobium-strip melting experiment. The pyroelectric voltage

(proportional to normal total radiance) and the pyrometrically

determined radiance temperature are shown as a function of time.

The plateaus of the curves indicate the period where the surface of

the specimen is melting. The precipitous drop in the pyroelectric

voltage indicates the point that the specimen melts through and

collapses. The melting point of niobium is 2750 K.

108

RADIANCE

TEMPERATURE

(K)
Part B:

Mathematical Analysis of the Free-Cooling Sphere

1. Overview of the Analysis

The free-cooling calorimetric method discussed in this report aims at the

determination of the temperature dependence of the thermal parameters (heat

capacitance, diffusivity, and emittance) of refractory materials at high

temperatures by optical observations of the time dependence of the surface

temperature T

g (t) and the radiance E(t) (radiated heat flux per unit area) of a

hot solid sphere of the material as it cools down. (The possibility of melting

will not be discussed in what follows. This will be the subject of future

research.

)

The emittance e(T ) of the surface is simply given by

e = E/(oT 4

) (1)

s

where o is the Stefan-Boltzmann constant. Thus the real problem is to express

the diffusivity a and the heat capacity per unit volume c in terms of E(t) and

T

s (t). In the limit a -» 00 (or R -* 0 where R is the radius of the spherical

sample), the temperature throughout the sample can be taken equal to T

g

, and c

can be found from the following simple relation:

- (4tiR 3 / 3) cdT /dt = 4 ttR2 E or c = -3E/RT (2a, b)

s s

where T^ = dT^/dt. For finite a, however, the temperature inside the sphere is

higher than T , and so eq. 2b can only be the leading term in a more complicated expression containing higher-order correction terms involving a. Such an

expression for c will be derived in what follows, first for the case of constant

c and a (cf. eqs. 30 and 37), and then for the case of temperature-dependent

parameters c(T) and a(T) (cf. eqs. 56 and 57). Similar expressions will be

derived for a as a function of the time derivatives of T and E (cf. eqs. 35

s

109
and 55)

.

An essential feature of the free-cooling method described in this report

is that the temperature field within the sample is assumed to be unknown at the

beginning of the experiment. Thus the heater, be it an induction coil, furnace,

or some other device, need not be precisely controlled in order to assure that

a specified quantity of energy be injected into the sample, or that a specified

initial interior temperature field be achieved. The price that must be paid

for this freedom is that the readings of T (t) and E(t) for a time interval

s At~R /a at the beginning of an observational run after the heater is turned off

cannot be used to determine a(T) and c(T), because T

g and E are too sensitive

to the details of the unknown initial interior temperature field. It is only

when the initial transients have died out and the thermal behavior of the

sample enters the "post-transient regime" that the expressions for a and c in

terms of E(t) and T (t) (and their time derivatives) that are derived in this

s report can be applied. For the sample sizes contemplated, however, only a

couple of seconds are required for the transients to die out in the case of a

poor thermal conductor like uranium dioxide, and this is to be compared with a

toLui observation time of nearly 30 seconds. In the case of metal samples the

time required for the transients to die out is a much smaller fraction of the

total observation time. These estimates are discussed at the end of Section 3.

In order to relate the analysis of this report to the existing literature

on the analysis of free-cooling, it should be noted that there are three

different kinds of cooling problem which must be clearly distinguished:

(1) the predictive or initial-value problem in which the initial temperature

field within the body is given, and it is necessary to calculate T (t)

knowing the functional dependence E(T ) of the radiance on the surface

temperature;

1 1.0
(2) the inverse problem, in which E(T

g

) is known, T

s (t) is given, and it is

necessary to calculate the initial interior temperature field;

(3) the calorimetric problem, in which E(t) and T

g (t) are given, and it is

necessary to calculate c(T) and a(T). A knowledge of the initial interior

temperature field is neither given nor desired. (This category of problem

is sometimes also referred to in the literature as the "inverse problem".)

This report will concern itself only with the first and third categories. Only

\j the "weak form" of the predictive problem will be discussed, by which is meant

the prediction of T

g (t) (and the interior temperature field) in the posttransient regime. Inasmuch as T (t) in this regime is determined by the thermal

s parameters alone, it is possible (in the case of constant parameters) to derive

a universal cooling law for this regime. This law has its simplest form when t

is expressed as a function of T , and an explicit expression for t(T

g

) is given

for the case of arbitrary (but constant) thermal parameters (cf. eqs. 26 and

27). In the weak form of the predictive problem, the radiance E(T

g

) may be

regarded as a forcing function that is producing a "forced decay" in T

g (t).

The "free decay modes" arising from an arbitrarily specified interior temperature field have died out, and the expression for the interior temperature

T(r,t) can be characterized by a form-invariant functional dependence on the

forcing function E(T ) or, equivalently, on T . That is,

s s

T(r,t) = T [r , T (t)]. (3)

s

Such an expression for T(r,t) is given, which has the form of the sum of

polynomials in r multiplied by powers of T

g (t) in which all coefficients are

time- independent numbers (cf. eqs. 14 and 23). This solution corresponds to

the "particular solution" in the theory of linear differential equations, but

the present problem is nonlinear because the forcing function E(T

g

) is a

function of the unknown surface temperature'.

Ill
2. Outline of Solution Method Employed

A four-step procedure is employed to solve the weak form of the predictive

problem: (1) First the sphere is imagined to be immersed in a controlled

temperature bath with a given time-dependent temperature T

g (t). This is the

only boundary condition imposed (except for the requirement that the interior

temperature be everywhere finite) . This is a linear problem whose solution can

be expressed as a convolution integral. This integral is then transformed so

as to cast the solution into a simple and novel form in which the temperature

is expressed as an infinite (but subsequently truncated) sum of characteristic

polynomials in r of degree (2n+l) , each of which is multiplied by (-l) n

dn

T

g /dtn

.

Thus the time derivatives of the surface temperature can be regarded as

"driving" the solution for the interior temperature. (2) the next step consists

of imposing the boundary condition that relates (3T/3r)

g to E(Tg

). When the

solution to the linear problem is substituted into this boundary condition, the

boundary condition assumes the form of an ordinary differential equation in

T

g (t) that is linear except for the fact that the forcing function E(Tg

) is a

nonlinear function of the dependent variable T . The differential equation is

of infinite order, but is truncated to fourth order. (3) An iterative procedure is used to find a self-consistent solution to this nonlinear differential

equation in which dT

g /dt and the higher derivatives are expressed as sums of

powers of T . This is the "particular solution" that characterizes the posttransient regime. (4) The equation dT

g /dt = F(T

g

) is integrated to give an

explicit expression for t(T ) , which is the universal cooling curve for a hot

sphere in the post-transient regime.

The same ordinary differential equation that results from the imposition

of the boundary condition involving the radiance E also serves as the basis of

the solution of the calorimetric problem, except that E(t) and T (t)

112
(and hence all of its derivatives in the ordinary differential equation) must

now be regarded as known functions, so that the differential equation becomes

an algebraic equation in which the unknowns are a and c. If a and c are constants,

then it is only necessary to know the values of E and T

g at two moments during the

cooling in order to construct and solve a system of two algebraic equations for

the two unknowns a and c. However, an alternative procedure for solving for

a and c is employed that leads to expressions for them that lend themselves to

an easily automated iterative solution.

The problem of finding the temperature dependence of variable parameters

a(T) and c(T) is solved by introducing new dependent and independent variables

that reduce the nonlinear partial differential equation to the same form as

that for constant parameters except for the presence of a very small effective

heat source density whose effect can be neglected. The same solution procedure

is followed as in the case of constant parameters, and the end result is

expressions for a(T) and c(T) in terms of E(t) and T

g (t) (and their derivatives

that can be solved by a simple iterative procedure.

There appears to be no discussion of the weak form of the predictive

problem in the existing literature, except for the limiting case in which the

interior temperature may be considered to be constant throughout the sample. The

strong form of the predictive problem, i.e. the initial value heat problem,

has of course a superabundant literature . 1

The calorimetric problem has been investigated by a small group of

mathematicians 2-10

, but the investigations have been limited either to two

constant parameters, or else only one variable parameter, and the resulting

expressions are complicated and difficult to solve numerically. One important

cause for this complication is the fact that these authors insisted on

113
maintaining the requirement of uniform initial interior temperature, and the

concept of the post- transient regime was not introduced. Without limiting

the analysis to this regime, and simultaneously rigidly maintaining specified

initial conditions, the problem becomes overdetermined, and in general ill-posed.

3. Predictive Problem for Constant Thermal Parameters

Although the primary purpose of this report is to present the solution of

the calorimetric problem, the necessary analysis must be confronted already in

the simplest predictive problem (i.e. the one for constant parameters). The

solution of this problem also provides estimates of the numerical values of

various quantities that become observational data in the calorimetric problem.

The partial differential equation governing the flow of heat in a body

in the absence of any interior heat sources is

V • (k V T) = c 3T/3t (4)

where k is the thermal conductivity and c is the heat capacity per unit volume.

(Strictly speaking, since c is the product of the mass density and the constantpressure specific heat, and because the mass density is a function of temperature,

c is the heat capacity of the quantity of material that occupies unit volume

at some given reference temperature , and is not strictly the heat capacity per

unit volume at an arbitrary temperature, although this simplified nomenclature

will be used in what follows. If p(t) is the mass density, then the constantpressure specific heat C = c/p. C and p enter into the formalism only as

P P

the product c = pC^.)

The boundary condition that must be applied to eq. 4 is obtained by noting

that the integral of the right side of eq. 4 over the volume occupied by the

hot sphere is equal to the rate of decrease of heat content of the sphere, and

114
so must be equal to 4 ttR

2 E where the radiance E is the radiated heat flux per

unit area. Applying Gauss' Theorem to the volume integral of the left side of

eq. 4, the following boundary condition results:

-k(3T/9r) = E(t) (calorimetric problem) (5a)

s = eaT

1

* (predictive problem) (5b)

s

where the subscript s designates quantities evaluated at the surface, a is the

Stefan-Boltzmann constant, and e is the emittance. This boundary condition

assumes that the sphere is radiating into a "cold vacuum", i.e. a vacuum at

absolute zero of temperature. This assumption is convenient, but in no way

essential for what follows.

With proper scaling, all quantities can be made dimensionless, and all

parameters (including dimensionless ones) can be eliminated from the equations,

reducing them to a universal form that applies to spheres of all sizes with all

possible (constant) thermal parameters. In order to do this, it is necessary

to introduce a characteristic length r^, time interval ti* and temperature Tj.

The first two are self-evident:

rj = R ; tj e R

2 /a. (6a, b)

The definition of the characteristic temperature Tj is not so obvious. Various

definitions can be found in the literature, such as the highest or lowest

temperatures encountered in the problem, or an equipment- related temperature

such as that of a furnace to which the sample is exposed. If, however, one

wishes to remove even dimensionless constants from the equations, only the

following definition will suffice:

Tj E (k/Rea) 1 / 3

. (6c)

This characteristic temperature is determined by the size and thermal properties

of the sample, and is independent of the starting conditions of the experiment

or of the properties of the environment to which the sample is exposed. For a

115
tungsten sphere for which R = 1 cm, Tj = 17,300 K which is nearly five times

the melting temperature (3653 K) , whereas for a sphere of uranium dioxide of

the same size, Tj = 3650 K, which is only 500 K higher than the melting temperature (3151 K)

.

The dimensionless variables (designated by a circumflex) corresponding to

the characteristic magnitudes defined above are

r E r/R ; t = at/R2

; T = T/T 1 ; (7a,b,c)

and the corresponding equation and boundary condition can be written as follows:

3 2 (rT) / 9f 2 - 3(rT)/3t = 0 (8a)

- (3T/3r) = E(t)

s (calorimetric problem) (8b)

= T

4 s (predictive problem) (8c)

E(t) = E (t) [k(T 1 /R)] _1 = RE/kTi. (9)

The first expression for E above shows that it may be regarded as a dimensionless

heat flux for which the scaling flux is [k(T}/R)], i.e. the flux driven by the

temperature gradient t^/R.

As noted in Section 2, the first step in the solution of eq. 8 above

consists of replacing the boundary condition (8c) by the requirement that at

the surface of the sphere T be equal to the (as yet unspecified) temperature

T

g (t) which is regarded as the forcing function that drives the interior

temperature T(r,t). It is more convenient to deal with the temperature

difference (T - T ) t and to this end the dependent variable is defined as

e(£,t) = r[f(r,t) - T

g (t) ]

.

( 10 )

116
Thus the equations to be solved are

3 2

0 / 3r 2 - 30/3t = r dT

g /dt (11a)

0 (0, t) = 0(1, t) = 0 . (lib)

This is a linear inhomogeneous equation with Dirichlet-type boundary conditions

that is driven by the forcing function r dT

g /dt. It is easily confirmed that

the solution can be written in the following form:

GO

0(r,t) =— / j

( sin(miTf) A (t) exp[-(m-rr) 2 t] (12a)

7T m in m-1

where

A (t) = m

t

exp [ (mir ) 2 f*] [dT ( t

' ) / dt ' ] s dt f (12b)

An alternative form results from splitting the integral of (12b) into two parts

as follows:

A (t) = Bm + A (t)

m mm

where is a constant given by

m

o

B = / exp [ (mir) 2 t ’ ] [dT (t’)/dt'] dt'

and

•/ A (t) = / exp [ (mu 2

) t'] [dT (t')/dt'] dt

m / s

(13a)

(13b)

(13c)

If eq» 13a is substituted into eq. 12a, we have the more familiar form of the

solution in which the terms involving B describe the rapidly decaying normal

m

modes, and the summation involving A^(t) describes the inhomogeneous part of

the solution that is driven by the forcing function r dT

g /dt on the right side

117
of eq. 11a. Because A(0) =0, it is evident that only the part of the solution

(12a) that involves the B is non-zero at t = 0, and that I (— 1)

m

B /ml are m m

the Fourier coefficients for a sinusoidal expansion of 0(r,O). The longest-lived

normal mode (m=l) decays like exp (— tt 2 £ ) = exp [- (Tr 2 a/R“) t ] , so when t% R^/a,

this longest-lived mode has a magnitude of order exp(-TT 2

) = 5 x 10“ 5

, so it is

safe to say that by this time the post-transient regime has become established.

When the solution (12a) is written using (12b) rather than (13) , the

entire solution is regarded as "driven", with the effects of the driving force

reaching back to t = - °°. That is, one effectively takes t = - 00 to mark the

beginning of the experiment, and takes the position that for all finite t the

decaying homogeneous solution has long since died away, leaving only the

inhomogeneous driven solution. For a solid sphere whose interior temperatures

can be influenced only by the environment to which its surface is exposed, this

is a physically more natural point of view than that posed by the initial-value

problem, which requires that the initial interior temperature field be

specified, rather than the past history of the surface temperature.

If the derivative in the integrand of (12b) is expanded in powers of

(t* - t) , the integration can be carried out explicitly and the solution

assumes the form

OO

0 ( t , t) = (-l) n

T

(n)

(t) P (r) (14a) / j s n n=l

where

and

V

n)

(t) E d" T

s /d£n

CO

Pn(r) = 2 ^ ^ (-1) (hitt) (^n+l)

s in ( m7T £)

,

(14b)

(14c)

1

118
The P (r) are polynomials of degree (2n+l) that have the following properties:

P (0) = 0 n for n 0; (15)

Po

(1) “ !> (16a)

P

n

(l) - 0 for n > 0; (16b)

d

2 P /dr 2 =

o

o, for 0< r< 1 (17a)

d

2 P /dr 2 = n

- Vi for n > 0 . (17b)

The first few of these polynomials can be found in handbooks. (See, for

example, p. 87 of reference 11.) Property (17b) can then be used to generate

successively higher-order polynomials by double integration using properties

(15) and (16b) to determine the two constants of integration. The four

lowest-order polynomials are listed below:

P

Q

(r) = r (18a)

Pl(r) = r(l-r 2 )/6 (18b)

P 2 (r) = r (7 - 10 i

2 + 3 r

4 )/360 (18c)

P 3 (r) = r (31 - 49r 2 + 21 r

4 - 3 r6 )/15,120 (18d)

P 4 (r) = r (381-620 r

2+ 294 r4 - 60 r6 + 5 r

8

) /l, 814 ,400. (18e)

Using the property (17b) , it is a very easy matter to show by direct

substitution that the differential equation (11a) is satisfied. The properties

(15) and (16) guarantee that the boundary conditions (lib) are satisfied.

The form of the solution given in (14) represents a different way of

stating that the interior temperature field is determined by the history of

of the surface temperature. Whereas the form of the solution given in (12)

represents this history by an integral reaching into the distant past (with

a kernel that provides "memory loss") , the form given in (14) represents this

history by means of all of the time derivatives of the surface temperature

119
calculated at the present instant of time. Instead of the ever-diminishing

influence of the surface temperature as one reaches deeper into the

past, it is now the ever-diminishing significance of the higher-order

time derivatives of surface temperature as the order of the derivative

increases. This decreasing influence of T

s

^ n ^ as n increases results

from the fact that P decreases as n increases. For this reason, a

truncation of the infinite series in (14a) is feasible in the case of a

numerical calculation. In the calculation described in this report,

only the terms of (14a) up to and including n = 4 were retained.

The infinite set of polynomials P^ does not span the entire class

of continuous functions on the interval 0 <_ r <_ 1. The set is, however,

sufficiently large to include all permissible solutions to (11) . To see

this, it should be noted that since (9/r) = T - T , the set (P /r)

ought to be capable of describing the most general form of T - T

g inside the sphere, and on the surface should vanish. This latter requirement

is satisfied in view of (16b) and the fact that n = 0 is excluded from

the summation in (14a). As to the completeness of the set (P^/r), it

should be noted that (as (18) indicates) it is isomorphic to the set of

all even powers of r, which corresponds to the requirement that V_ f =

0 at the center of the sphere, which in turn follows from the fact that,

in the absence of a point source or sink at the origin, the heat flux

must vanish there.

The second step of the solution method that was outlined in Section

2 above consists of substituting (14) into the boundary condition (5b)

in order to obtain an ordinary differential equation that determines the

time dependence of T

g (t). The dimensionless form of the boundary condition

(5) is

120
- ( 30/ 3r) = E(t) (calorimetric problem) (19a) r=l

~ i = T

4

. (predictive problem) (19b)

Differentiation with respect to r of the solution given in (14) , and

subsequent substitution into (19) yields the following result:

CO ^ v (~l) n

[2^n

B

n

/ (2n) I ] dn T^/dt11 = E(t) (calorimetric problem) (20a)

n=l

= T (predictive problem) (20b)

where are the Bernoulli numbers, which are tabulated in sundry handbooks 12

.

The first 7 of these are

Bi = 1/6, B 2 = 1/30, B 3 = 1/42, B 4 = 1/30, B 5 = 5/66, B 6 = 691/2730,

B 7 = 7/6.

If the series in (20) is truncated at n = 4, the following ordinary

differential equation results:

( 21 )

d

4 T 9 d

3 T

2 s 4725 dtP 945 dt 3 + (22a)

.. d

2 T - df

1 s _ j. s 45 dt z 3 dt

= E(t) (calorimetric problem)

= T

g

4

. (predictive problem) (22b)

In the calorimetric problem, T^n

^(t) and E(t) are given functions of

time, so when (22a) is written in dimensional form it becomes an algebraic

equation in the two unknowns a and c. This case is discussed in Section

4 below. In the predictive problem however, one must solve the ordinary

differential equation (22b) , which would be linear except for the fact

that the forcing function T

4 is a nonlinear function of the dependent

s variable T

g

. Except for this nonlinearity, one could assert that the

solution of (22b) consists of the sum of a homogeneous part that requires

a knowledge of four initial conditions, and a particular solution that

121
is entirely determined by the forcing function and is independent of the

initial conditions. The form of (14a) shows that the initial conditions

of (22b) determine the initial interior temperature field, and vice

versa. This homogeneous solution has completely died away by the time

the system has entered the post-transient regime that is completely

determined by the forcing function T

4

. The "particular solution" that

describes this regime can be found analytically by an iterative process

which leads to a set of self-consistent expressions for the derivatives

T^ as functions of T alone, i.e., there are no arbitrary degrees of

s s freedom that must be specified by initial conditions or any other condition.

These self-consistent expressions are the following:

T ( 1 ) =

s

t(2) =

s

T

4 - 12

5 T

' +

12

25

-10+

432

875

36 f

7 _

s 396

p

1 0 s 1512

25

-13

(23a)

(23b)

p

( 3) =

s 756 T

10 -

s 149045

(23c)

t(4 )=22680 T

13 . (23d)

s s

If these expressions for T are substituted into (14a), 0(r,t)

becomes the sum of- products of powers of r and T

g in which the coefficients

are numbers, rather than functions of t. That is, 0 becomes a forminvariant function of r and f

g (or, equivalently, of the forcing function

T

4

)

,

and its total time dependence is contained in the time dependent of

T (t)

.

This is one way of characterizing the post-transient regime,

s The expressions given in (23) are valid only if the truncation of

the infinite series in (20b) is valid. The range of validity of this

truncation is indicated by Table I below which uses the eqs. 23 to

122
calculate the magnitudes of the various terms of eq. 22b for a range of

values of T

4

. (Powers of 10 are indicated in parentheses following the

numbers, e.g. 3. 9 (-2) = 3. 9x10”

2

.)

Table I

T

s T

4

s (_tO)/3)

s (+t( 2 )/45)

s (_2t( 3 >/945) (+T (4) /4725)

s

1 1.000 0.525 0.384 -4.709 4.800

3/4 0.316 0.223 3. 96(-2) -6. 00(-2) 0.114

1/2 6.25(-2) 5. 64(-2) 4. 70(-3) +7. 92(-4) 5 . 86 (-4)

1/4 3.91(-3) 3. 86(-3) 4. 72 (—5) +1. 43(-6) 7.15 (-8)

1/8 2.44(-4) 2. 44(-4) 3.80C-7) +1. 48(-9) 8. 73(-12)

A lef t-to-right sweep of the first row of Table I shows that the magnitudes of

the higher-order terms in (22b) are definitely not small compared with the

magnitude of the forcing function T

4

. However, this is the case for T <_ 1/ 2

,

s s so it seems safe to conclude that in this range the truncation of the infinite

series is a valid procedure.

It is hoped that it will be possible to extend the free-cooling calorimetric

method up to temperatures that exceed the melting temperatures of refractory

materials by several hundred degrees Kelvin. If the methods and results of

this report (and its intended extension to include melting) are to be used,

it is necessary that the size of the sample sphere be chosen so that the

condition

T = T /T i = T (k/Reo)" 1/3

< 1/2 (24)

s s s —

is satisfied. Referring to the remarks made in Section 1 above, it is evident

that this condition is easily satisfied for a tungsten sphere of radius 1 cm.

In fact, at the melting point of tungsten (3653 K) , T^ = 0.211 for a sphere

123
of this size. However, for a sphere of uranium dioxide of the same size

at its melting point (3151 K) , T =0.86. However, for a sphere of

s radius 1 mm, at the melting point of uranium dioxide T

g = 0.40, so

condition (24) is satisfied. Thus, in applying the free-cooling method,

one of the first design considerations is to choose a sample sphere size

that is small enough to satisfy condition (24). In general, for metals

any radius up to 1 cm would be safe, but for refractory non-metals, 1 mm

should be regarded as the upper limit.

The eq. 23a can be written in the form

dt = - dT /F(T ) (25)

where F(T ) is the expression contained in the brackets on the right

side of (23a). This equation can be integrated to give £(1^). When

this is carried out to an accuracy consistent with that of eq. 23a, the

following equation results:

t = — (T-3 -1) - — £n f

9

Us

' 15 s + — (1 - T

3

) + —-— (1 - T

6

) 75 K s' 1575 ^ s'

(26a)

where

R 1/3

t 5 at/R2 and T e —^T . (26b, c)

s k s

The arbitrary integration constant in (26a) has been chosen so that t =

0 for T = 1. It should be noted, however, that (26a) is reliable only

s for T^ £ 1/2. The first term on the right side of (26a) is the wellknown expression that characterizes the cooling of a very good conductor.

The other terms allow for the presence of a temperature gradient within

the cooling material. The last two terms are not important. If they

are omitted, the error that is introduced at T

g = 1/2 is less than 1%,

and considerably smaller than that for smaller values of T . Eq. 26 is

124
a universal cooling curve that is valid for spherical samples of all

sizes for materials having constant thermal parameters. It is useful

for estimating the cooling time At = t^ - t^ that is required for a

For example, using parameters appropriate to a tungsten sphere of radius

1 cm, one can calculate the time required for the sphere to cool from

3000 K to 1500 K. In this case (27) yields At = (501.2 + 0.62) seconds.

This shows that the second term, that allows for an interior temperature

gradient is negligible, although it would have a relatively greater

significance in the case of a short time interval near the beginning of

the cooling process. This cooling time of 502 seconds is to be compared

with the time R

2 /a =3.3 seconds that is required for the post-transient

regime to become established. The same calculation for a small sphere of

uranium dioxide having a radius of 1 mm yields At = (28.5 + 0.37) = 28.9

seconds for the sphere to cool from 3000 K to 1500 K. This interval is

to be compared with R

2 /a = 2.0 seconds, which is the approximate time

required for the post- transient regime to become established.

4. Calorimetric Problem for Constant Thermal Parameters

In the calorimetric problem, E(t) and T(t) are given functions, and

a and c are unknown constants. A knowledge of the fact that the cooling

obeys the T

4 radiation law is necessary only for a calculation of the

emittance e, which is given by

sample to cool down from an initial temperature T_^ to a final temperature

Tf Dropping the last two terms in (26a) , this becomes

(27)

e = E (t) / a [T (t) ] 4

, (28)

125
where a is the Stef an-Boltzmann constant. This relation is actually the

definition of e, and it is valid even if e is not constant. From a

knowledge of e(t) and T(t), one could find the temperature-dependent

emittance e(T).

The basic equation is now (22a) rather than (22b). When the definitions

contained in (7) are used to convert (22a) into dimensional form, it can

be written as follows:

- T (1-F) = 3E/Rc

s where

and

T E d T /dt

s s

F E R

2 hs

1 VdF

2

21 a dt T

15aT

This yields the following expression for c:

R^

d4T

s 105a z dt* )

(29a)

(29b)

(29c)

c = [3E/R (-T )] (1 - F) _

1

. (30)

s

For a perfect conductor, F = 0 because a = 00 . For finite a, F -* 0

as the sphere cools down because the cooling curve T

g (t), although still

descending, tends to flatten out so that the higher derivatives of T

g (t)

that appear in F tend to vanish. This suggests a simple procedure for

evaluating c and a: First, corresponding readings of E and T

g near the

cold end of the cooling curve are used to calculate c from (30) with F =

0. Then this value of c together with corresponding readings of E and

T

g near the beginning of the post-transient regime are used in (29a) to

calculate a numerical value for F. Using this numerical value of F in

(29c) , together with the known numerical values of the derivatives of

T

s (t), a cubic equation in (1/a) results which can be solved for a.

This value of a could then be used in (30) together with the cold temperature

readings to calculate an improved value for c, and so on.

126
A simple analytical expression for a can be derived by neglecting

and t( 4 ) in (29c). To this approximation.

F = 15aT

" R

z d

s 15a dt

Jin (-T

s

) , (31)

and when this is used in (29a), the following expression for a results:

-1

R

15a

d AnQ-TS

dt

3E

(32)

RcT

This expression for a involves c on the right side, but this can be

eliminated by a substitution. If (29a) is differentiated, and t( 3 ^

and T^ 4

) are ignored as they were in (31), as well as T

2

, the following

s s approximate expression results:

T = - 3 E/Rc.

s (33)

d Jln(-T ) s

-1

d £n E

15a

" dt dt

-l

It can be shown that the following exact expression can be derived

without making any of the above approximations:

(34)

R

2 15a

d£n(-T ) s L dt

- f (a,T

g

)

_1

d£nE

dt

- g(a,E)

-1

(35a)

where

f (a»T ) =

s

g(a,E) =

2 R2

21 a

1 _

R/

35 a

‘ d

2 T

1 s T

s

dt'

d

z E

dt 2

10

1!

a

d

3 T

dt

2_ R^

45 a

s

d

3 1 E

E dt

149 * " 4725

T e dT /dt.

s s

R

4 —

?

d

4 E

dt^

(35c)

(35d)

127
Because this expression involves only the ratios T

^ n

^/T and E^n

^/E, it

s s is insensitive to the calibration of T and E, which is a significant

experimental advantage. Furthermore, because this expression involves

only a and not c, an iterative process can be employed to calculate an

accurate value of a using readings of E(t) and T (t) and their derivatives

s at a single point on the cooling curve. Then these same readings and

the calculated value of a can be used to calculate c from (30) without

the need for further iteration. In fact, the F that appears in (30) can

be written in the following form:

R

z 15a

d£n (-T ) s dt

- f (a, T

g

) (36)

where f(a,T

s

) is defined in (35b).

The factor (1-F) that appears in (30) involves derivatives of T .

It is possible to make use of (35a) to exchange this factor for one that

involves only derivatives of E. The resulting expression for c is

where

c = [3E/R(-T

g )](l+G),

R

2 15a

d&nE

dt g (ot,E) ,

(37a)

(37b)

where g(a,E) is defined in (35c). It seems likely that, as a matter of

experimental technique, T

g and its derivatives will have higher precision

than E and its derivatives, so (30) is likely to give better accuracy

than (37).

Although these expressions for a and c appear to involve readings

taken at only one point on the cooling curve, in actual fact they involve

readings at several neighboring points, because these are necessary to

calculate the various derivatives of T and E which are not measured

s

128
directly in the experiment. However, once these derivatives have been

calculated (and any necessary smoothing of the data carried out) , the

expressions (35) and (30) (or (37)) can be used to march along the

cooling curve and calculate an independent pair of values for a and c at

each point of the curve. Thus, if these thermal parameters are actually

mild functions of temperature rather than constants, this procedure

ought to give reasonably accurate expressions for a(T) and c(T).

There are two sources of error in this procedure, however, which

will be corrected in the next section. One is that the calculated

values of a and c should not be associated with the surface temperature

of the point on the cooling curve at which they were calculated, but

rather with a temperature resulting from some kind of averaging throughout

the volume of the sphere. Second, the temperature dependence of the

thermal parameters makes the basic differential equation nonlinear, and

not just the boundary condition. In spite of these considerations, the

application of the results of this section will provide good first

approximations for insertion into the formulas that are derived in the

next section.

5 . Calorimetric Problem for Temperature-Dependent Parameters

The strategy that will be employed in this section is to invoke

appropriate redefinitions of both the dependent variables in order to

reduce the problem to one which, in its dimensionless form, is identical

to the dimensionless form of the calorimetric problem for constant

parameters. In this way a formal solution of the equation and boundary

conditions is achieved. When the transition is made back to the dimensional

quantities appropriate to the experiment, the effects of the changes in

129
variables that were made at the beginning of the solution manifest

themselves, with the result that terms involving the derivatives of the

parameters with respect to temperature appear in the expressions for a

and c. These expressions are the desired generalizations of those

derived in the preceding section. This approach automatically refers

everything to the surface temperature T

g and to the values of the parameters

at the surface, even though the effects of the higher temperatures of

the interior are implictly taken into account.

The equation and boundary condition are the same as given in eqs. 4

and 5a, except that now k(T) and c(T) are arbitrary and independent

functions of temperature. Instead of working with T as the dependent

variable, the enthalpy density h(T) now assumes this role where

h(T) = I c(T') dT' + h

r

, (38a)

T

r

and

h = h (T ) . (38b)

r r

T^ is any convenient constant reference temperature. Strictly speaking,

h(T) is actually the enthalpy of that quantity of matter which occupies

unit volume at the reference temperature, but which at a different

temperature occupies a slightly different volume because of the temperature

dependence of the mass density. For simplictly, however, h will be

called simply the enthalpy density. In terms of h, the partial differential

equation and the boundary condition become

V • [a(T)V h] = 3h/3t (39a)

and

- a

g (3h/9r) r=R = E(t) (39b)

where

a(T) 5 k(T) / c(T) (40a)

130
and

a = a (T ) (40b)

.

s s

a is a function of t via the time dependence of T (t)

,

and this function

s s a (t) will be used to define a dimensionless time variable t:

s

= t?-2

T E R‘

L /

a (t ' ) dt ? . s (41)

Using (41), the eq. 39a can be put into the following form:

R

2 3h/3x (42)

where the second term on the left is an effective heat source density

arising from the effective heat flux -{(a/a ) - 1] Vh. This effective

flux vanishes at the surface of the sphere where a - a . Because of

this, the volume integral of the effective heat source is zero. If the

effect of this effective source density were taken into account by an

iterative perturbation technique, it would be found that h would have to

be corrected by a small amount 6h such that V6h would be very nearly

equal to the effective flux density -[(a/a ) - l]Vh. Because this vanishes

S

'

at the surface, so would V6h and (3Sh/3r)

s

. Because of this, the correction

5h would produce no change in the boundary condition (39b) which gives

rise to the equation from which the expressions for a and c are derived.

In other words, the second term on the left side of (42) would produce a

slight change in the interior h-field, but it would produce no significant

effect on the equation that is derived from the boundary condition

involving E, and this equation is all that matters for the solution of

the calorimetric problem.

The reference temperature T , and the values of the sundry quantities

at this temperature, are used to make the equation and boundary condition

131
dimensionless. Dimensionless quantities will be designated by a circumflex.

Thus

c = c/c^ = c(T)/c(T^); and a = a/a^ = a(T)/a(T^). (43 a,b)

The characteristic temperature is now defined in terms of the values of

the material constants c and k at the reference temperature:

r r

TjE (k /Re a)

1/3

i

. (44) r r

This is used to scale both h and T:

h E h/c

rT! = / c(T') dT’ + hr /crT!

T

r (45a)

where

T e T/T i

.

(45b)

The dimensionless independent variables are

t

r = r/R and x = a R

2 r /

V") dt'. (46a, b)

In complete analogy with the variable 0 defined in (10) , it is convenient

to introduce the variable q defined by

( r , x ) - h

g

(x) n >]• (r,x) = r

In terms of these variables, the equation becomes

3 2 q/3r 2 + r

jy - d iWdxJ = 3n/3r

(47)

(48a)

where

1 CM <U

III >~ “1

3r

(=--

a

_ s i) <i

^ - ni‘ 3 r

II >"i > 1 N> 3_

3r

-

<

SL -

a L s

N -2 3h

n l) r

2 —

3r_

•

(48b)

(48c)

132
The boundary conditions are

(n) - = 0 ; r=o(n)'' i = n = 0 r=l s

- (3q/9r)~ = E/a

r— _l s

where

/x ~ 1/3

a E a(T ) / a(T ) ; E =RE/k T x ; T 1 = (k /Re a)

x/

s s r r r r

(49)

(50)

(51a, b,c)

Except for the presence of the ry term in (48a) , this equation and the

homogeneous boundary conditions (49) are identical to the system defined by

(10) and (11), and could be solved in the same way if the ry term were ignored.

The inclusion of this term presents no fundamental problem since it can be

handled by an iterative approach that treats it as a prescribed forcing function

like (-dh

s /dx), except that y depends on r as well as t. The formal solution

of (48a) including y(r,x) (regarded as a specified function rather than as a

function of ri) has been carried out, but will not be given here, because

numerical analysis (not yet carried out) will almost certainly show that the

error caused by ignoring the ry term in the case of any material of practical

interest is not greater than the error caused by truncating the infinite series

for q. For this reason, the ry term will be ignored, and the solution for

q(r,x) will be taken to have the same form as that given in (14).

The same argument that converted the boundary condition (19a) into the

ordinary differential equation (22a) now leads to the following equation:

d

4h

4725 d^ 945 dx

d

3 h 1 d2

fi s 1 s 45 dx 2

i dh

1 s -

xr —j— = E/a

3 dx s (52)

This is identical in form to (22a) except that T

g

, t, and E are replaced by

h , x, and E/a respectively. To convert to the dimensional form it is only

necessary to note that it follows from (45a) and (41) that

133
(53)

dhc dx

dTc dr (s), - ©"-1 dT c (T ) s s R

2 c dT

dt c T a c T-i dt

s r

1 s \ / r Successive differentiation proceeds in the same way, with care being

taken to note that

dcc dx

r dx\

1 1 dc^

dTC dF

dc dT

S £ dT dt (54)

and similarly for da^/dx.

The same procedures employed in Section 4 that led to expressions

for a and c in the case of constant parameters now lead to similar expressions

that include the effects of the variability of the parameters. The expression

for a = a (T ) is

s s

R^

15a

d£n(-T ) s dt £,(T

s’

cs’“s )

-1

where

f ’ (T , c ,a ) =

s s s

d£na d£n c

s _

s dt dt

d£nE , /T, N -ir - 8

-l

(55a)

2_ r£

21 a

d

2 T

1 R

2 1

d

3 T

+

dt 2 10 a • dt 3 s I w

s s d

2 £n c d

2 £n a

s s

+

(

1

dt z

dT

s

• dt

T

s

(d£n a

s

dt 2 d£n c

s dt

d£n c

d£n a - 3 dt

dt

dt

!)(

!

d£n a d£n c

s 1

1

2 s s dt dt )

and g'(E,a )

diln

“s 1_

dt 35 a

s d

2£n a

dt‘

1 d/E _2_ Rf 1 d

3 E

E dt 2 45 a E dt 3

' d to E

ilLvt\ - 3 dt dt + 2

(

cl£n a \

J

(55c)

134
where the terns involving e( 4

) and third and higher derivatives of

£n a and £n c as well as cubics in the first derivatives of these have

s s been neglected.

The two alternative expressions for c = c(T ) are

s s

c =

s -25- jiR(-T ) I

R"15a

d£n(-T ) dt

- f ' (T ,c ,a ) s s s

v-1 (56)

where f * (T , c ,a ) is given by (55b), and

s s s

c =

s 3E

R(-T

s

)

11 + is- nF- (57)

where g* (E,a

g

) is given by (55c).

The point of view to be taken in applying these expressions is that

E(t) and T

g (t) are given, and after suitable smoothing their sundry time

derivatives are calculated. These functions are then substituted into

(35) and (30) to obtain the time-dependent functions a(t) and c(t) which

are then used for a and c on the right sides of (55) and (56) to

s s evaluate more accurate functions a (t) and c (t) which are reinserted

s s into the right sides of (55) and (56) , and so on until the process

converges. From the final functions a (t) and c (t) , together with

s s T (t) , the functions a(T ) and c(T ) can be found. Because a and c are

s s s functions of T alone, and the functional relations are the same in the

interior of the sphere as on the surface, the subscript s can be dropped

since the computed functions are the desired functions a(T) and c(T).

It is probable that a numerical analysis will show that, for the

kinds of functional relations a(T) and c(T) that are encountered in the

case of any real material of practical interest, many of the terms in

(55b) and (55c) are negligible in the sense that they are smaller than

the experimental error of the larger terms.

135
6 . Discussion of Part B

The working formulas for the application of the free-cooling calorimetric

method are (55) and (56) (or eq. 57). It is noteworthy that (55) (the

expression for a) is completely insensitive to the units used to measure

E and T

g since only relative values enter into the formula. Thus, for

the calculation of a, it is not necessary to calibrate the instruments

in absolute terms since arbitrary units can be used. In the case of the

calculation of c, the value of (E/T

s

) is the only quantity that must be

calibrated. If the value of c is known for the sample in question at

any temperature (most likely the coldest temperature in the experimental

range) , then the calibration can be made at that temperature. If c is

completely unknown for the material in question, then an entirely different

material and different sphere size can be used for calibration purposes

if c for the calibration material is known at some temperature.

The derivation of the working formulas made no assumption whatever

about the dependence of E on T

g

. Rather, it was simply assumed that

E(t) and T (t) were independent given functions. Only in the calculation

of the emittance e(T ) was any assumption made about the physical nature of

the cooling. (Cf. eq. 1.) Thus the formulas for a and c apply also to

a sphere that is cooled by some mechanism other than radiation into a cold

vacuum so long as the assumption of spherical symmetry remains valid.

There is no fundamental reason why the methods employed in this report

could not be adapted to cylindrical geometry. This geometry admits the

possibility of controlled electrical heating of refractory materials (an

established experimental technique). Such heating would be represented in

the formalism as a prescribed heat source density. The definition of the

post-transient regime would have to be modified only to the extent of

including the prescribed heat source as one of the determining factors.

136
along with the cooling law and the thermal parameters. Such prescribed heat

sources have already been included in the analysis for spherical geometry.

Although their inclusion complicates the formalism, it introduces no

fundamental difficulties.

Following a numerical test and confirmation of the relations derived

in this report, the proposed next stage of development of the analysis is

the inclusion of the possibility of a solid spherical shell surrounding a

molten core that shrinks in size as the sphere cools. The object of this

analysis would be to determine the heat of fusion, i.e. the jump in enthalpy

density across the solidification front. For this purpose h is a much more

natural variable than T, so the formalism of Section 5 should provide a natural

starting point.

137
References for Part B

1. M.N. Ozisik, Heat Conduction (Wiley, New York, 1980)

2. B.F. Jones, J. Math. Mech. 11 , 907 (1962)

3. J. Douglas and B.F. Jones, J. Math. Mech. 11 ? 919 (1962)

4. B.F. Jones, Comm. Pure Appl. Math. _1<5, 33 (1963)

5. J.R. Cannon and B.F. Jones, Int. J. Engng Sci. 1 , 457 (1963)

6. J.R. Cannon, J. Math. Anal. Appl. 188 (1964)

7. J.R. Cannon, P.C. DuChateau, and D.L. Filmer, Math. Biosciences 9_,

61 (1970)

8. J.R. Cannon and P. DuChateau, SIAM J. Appl. Math. 24-, 298 (1973)

9. J.R. Cannon and P. DuChateau, Int. J. Engng Sci. 11 , 783 (1973)

10. J.R. Cannon and P. DuChateau, SIAM J. Appl. Math. _39, 272 (1980)

11. V. Mangulis, Handbook of Series (Academic Press, New York, 1965)

12. See, for example, p. 40 of reference 11.

138
Task 5

Experimental and Theoretical Studies in Wetting and Multilayer Adsorption.

M. R. Holdover and J. W. Schmidt

Thermophysics Division

Center for Chemical Physics

J. W. Cahn

Center for Materials Science

Summary

Equipment has been assembled and/or fabricated to measure the

ellipsometric parameters of the liquid-vapor interface of binary mixtures

near the predicted transition from high adsorption to low adsorption.

Preliminary measurements on the mixture C_,F. . - C_,H_ . have not yet discovered

7 14 7 14

the predicted transition. A variety of interfering phenomena have been

tracked down. Further measurements are in progress.

139
Introduction

We have assembled a semi-automatic ellipsometer together with

sample cells and a thermostat capable of precisely monitoring the ellipsometric parameters of the liquid-vapor interface above binary liquid

mixtures. We have made a computer model of the ellipsometer in order to

optimize its sensitivity to the appearance of the predicted high adsorption layer. We have tested a method of measuring and recording the

index of refraction of the binary mixture under study simultaneously

with the recording of the ellipsometric parameters. The index of refraction measurement is extremely sensitive to the appearance of macroscopic amounts of a second liquid phase; thus it will serve to locate

the adsorption transition with respect to the miscibility gap in the

specific liquid sample under study. We expect the strategy of simultaneously measuring the index of refraction and the ellipsometric parameters to obviate the necessity of carrying out detailed exacting

chemical analyses of each sample of each mixture to be studied.

1 2 In preliminary measurements we have searched for the predicted ’

high adsorption-low adsorption transition in the liquid-vapor interface

between a C_,F, , - C_,H_, , mixture and air saturated with this mixture’s

7 14 7 14

vapor. We have found no evidence of such a transition in a sample which

undergoes phase separation near 23°C. (The consolute temperature in

this system is near 46° C.) This sample is approximately 7% fluorocarbon

on a mole fraction basis. In these preliminary measurements, the ellipsometric parameters showed a smooth temperature dependence between 23°

C

and 60° C which would be equivalent to the adsorption of a 70A thick

layer of the fluorocarbon at the interface between the vapor and this

140
hydrocarbon-rich liquid. This temperature dependence is probably

spurious. At the moment our highest priority is to determine its

origin. These preliminary measurements could have detected an abrupt

high adsorption-low adsorption transition equivalent to a 20A change in

the thickness of a fluorocarbon-rich layer at the liquid-vapor interface. We expect improved sensitivity in future measurements and to scan

the composition range in order to locate and trace out the expected

interface transition curve on a phase diagram.

The index of refraction measurement revealed a secular drift in the

temperature at which bulk phase separation occurred (roughly 0. 5°C/month)

.

This was traced to a leak in the quartz sample cell which we have now

repaired

.

The measurement system we are developing will be automated. This

is required because of the long equilibration times encountered.

Experimental Procedures

The experimental procedures we have carried out will be described

under the following subheadings; (1) Optical Table, (2) Thermostat, (3)

Equilibration (4) Sample cell, (5) Model for ellipsometer , (6) Optical

components for ellipsometer, (7) Optical components for index of refraction measurement , and (8) Sample preparation. When it is appropriate,

we will indicate the rationale for the procedure followed as well as the

improvements we anticipate in the near future. Figure 1 shows the

experimental arrangement.

(1) Optical Table: The ellipsometric measurement is sensitive to

the angle of the incident light. As the liquid-vapor surface sloshes

about (from ambient vibrations) the angle of incidence varies. Further141
more, the path of the reflected beam through various optical components

is also a function of the instantaneous position of the surface. These

two effects modulate the intensity of the reflected beam. Thus they

appear as noise in the ellipsometry . A commercially manufactured air

suspension table was purchased and installed to use as a low vibration

support for the entire experiment. Even at the comparit ively low

sloshing frequencies the table significantly reduced the liquid-vapor

surface vibrations. The table is held very nearly level by a system of

valves which admit compressed air to the legs. When functioning properly the variations in level are + 1 milliradian. This seems to be

satisfactory at the present precision of our measurements. We have

found that occasionally one of the valves may stick. Then, much larger

excursions in the position of the table level occur. These excursions

are correlated with room temperature variations (which alter the air

pressure in the table suspension) and are corrected periodically as the

valve finally opens. This larger amplitude tilting produced an unacceptably large ellipsometric signal whose origin was tricky to locate.

Ultimately we may have to regulate the table level with a much smaller

dead band than the compressed air servo system tolerates. This is

possible by moving two small weights on the table.

(2) Thermostat: We are now using a two stage thermostat. The

temperature of the outer stage is controlled by circulating water from

an external circulater /thermostat . We have used this circulator between

15 C and 60°C. The temperature of a particular point on the outer shell

is now monitored with a thermistor. Temperature fluctuations of this

point are on the order of + 0.05°C.

142
The thermal relaxation time of the sample is several hours (see

below) . We change the temperature setpoint of the outer stage by switching resistors in parallel with the temperature sensitive element of the

regulator. The switching is done at preselected times. This hardware

based programming scheme will eventually be replaced with a software

based one incorporating feedback.

The inner shell of the thermostat is regulated by a home-made

electronic servo system using a thermistor sensor. The set point

-4

temperature exhibits noise of less than 10 C; however, there are substantial temperature gradients across the inner stage (roughly 30 mK)

.

In the near future two improvements will be made to reduce this gradient

by nearly two orders of magnitude. First, the point heaters on the

inner shell will be replaced by a distributed heater placed symmetrically

around the shell. Second, the temperature of the outer shell’s set

point will be servoed in such a way as to maintain the outer shell’s

temperature much closer to that of the inner shell.

(3) Equilibration: Equilibration of concentration gradients in

binary mixtures via diffusion is a notoriously slow process. Typical

-5 2 diffusion constants in the liquids of interest are 10 cm /s.

Typical dimensions of our samples are 3 cm; thus characteristic times

should be several hours when one liquid phase is present and much longer

when two liquid phases are present. A magnetic stirring bar has been

enclosed in a glass envelope and placed within the sample cell. This

bar is turned periodically (e.g. 2 minutes every 30 minutes) by a second

magnet built into the thermostat which in turn is attached to a motor

outside the thermostat. We are able to directly observe equilibration

143
of the ellipsometric parameters of the liquid-vapor surface after a

temperature change. When one liquid phase is present, equilibration

takes about one hour. When two phases are present, equilibration can be

monitored using the index of refraction measurement because the index of

only the upper phase is measured. We have observed that equilibration

took between two and four hours when the sample was cooled into the two

phase region even though the liquid-vapor interface acts as a nucleation

site for the lower fluorocarbon-rich liquid phase and the sample is

stirred vigourously.

(4) Sample cells: The sample cells are essentially hollow prisms.

They are fabricated from fused quartz windows and quartz tubing held

together with a high melting temperature solder glass. The cells are

filled through a quartz tube which is sealed with a teflon stopper and

0-ring. (It is possible to seal the cells by fusing the tube shut;

however, at this point we have not done so). The prism angle is 60° and

is oriented with its apex upward. The angle of incidence for the

ellipsometer is near 60°. This minimizes the influence of the residual

birefringence of the cell's windows on the ellipsometer performance.

This angle of incidence is also close to Brewster's angle for the

hydrocarbon-rich mixture; thus, it is near optimum for the ellipsometer's sensitivity.

In equilibrium, the liquids under study are in contact with quartz

and vapor (the stopper is in the vapor phase). The hydrocarbon-rich

phase of C^F^-C^H^ mixtures totally wets the quartz. Thus the sample

cells' walls and corners as well as the glass encapsulated stirring bar

do not act as nucleation sites for the appearance of the fluorocarbon

144
rich phase. The fluorocarbon-rich phase can only appear at the liquidvapor interface, the interface we are studying ellipsometrically . The

absense of "traps" for the fluorocarbon-rich phase insures that the

index of refraction measurement will clearly indicate the initial

formation of macroscopic amounts of this phase.

(5) Model for the Ellipsometer : The ellipsometer arrangement is a

conventional one : source-polarizer-compensator-sample-analyzer-detector

.

In order to choose reasonable design compromises, a computer program was

written to calculate the intensity at the detector as a function of the

angle of incidence, the orientations of the polarizer, compensator, and

analyzer, the index of refraction of the substrate fluid, and the

thickness and index of refraction of the multilayer adsorbed film we are

searching for. This program is an essential tool for the design of the

instrument and the interpretation of the measurements. With it we can

explore the sensitivity of the measurements to extraneous factors such

as the tilt of the table and component imperfection.

(6) Optical components for the ellipsometer: A lmW He-Ne laser is

used as a source. The polarizer and analyzer crystals have an extinction

ratio near 10 . A Faraday modulator is inserted between the polarizer

and analyzer. In effect, it rotates the plane of polarization through a

few tenths of a degree. The modulator is excited with an AC signal so

that the photomult iplyer output can be monitored with a lock-in amplifier.

A DC signal is also fed back to the modulator to insure that the intensity

of the detector is near null. The ellipsometer is set up for extinction

when the sample is at a high temperature. Then we expect essentially

zero preferential adsorption at the liquid-vapor surface. As the sample

145
is cooled the DC component of the current fed back to the modulator is

continuously monitored. If a thin dielectric film, i.e. an adsorbed

layer, appears on the liquid surface the extinction will be maintained

by the feedback signal. This method does fail when the layer becomes on

the order of 1/8 of a wavelength of light thick. Then two optical

components must be adjusted to maintain extinction.

An integrator with a 2 minute period is used in the feedback loop

to average over the interface vibrations.

The most serious problem we have encountered with this arrangement

is an ambient temperature dependent residual ellipticity introduced by

the modulator. We have started monitoring this by periodically inserting

a test sample between the modulator and a second detector. This sample

simply a quartz flat whose optical properties are constant. In the

future, we will attempt to use thermostated modulator of our own design.

(7) Optical components for index of refraction measurement: The

index of refraction measurement is based on measuring the deviation that

a collimated beam of light experiences upon passing through the prismshaped sample cell. A second He-Ne laser is used as a light source.

Its beam is spatially filtered and expanded to fully illuminate the

cell. The beam passes through the cell and is reflected back upon

itself. (This doubles the sensitivity of the index of refraction measurement and reduces the number of windows needed for the thermostat). A

beam splitter is used to separate the returning beam from the incident

one. A telescope focuses the returning beam upon a position sensitive

detector. The voltage from the detector provides a continuous monitor

of changes in the index of refraction of the sample.

146
(8) Sample preparation: The C^F^-C^H^ mixtures are prepared from

the pure components as received. A rough measurement indicates the

consolute temperature of our samples are within 1° C of that reported in

the literature for these components. Because we intend to establish the

location of the adsorption transition in relation to the phase separation

temperature it is not necessary to prepare mixtures of a precisely known

composition. Instead we prepare mixtures such that two phases are

present at the temperature at which phase separation is desired. Then

the hydrocarbon-rich upper phase is drawn off and used as the sample.

Experimental Results

As we have indicated, the experimental techniques are still being

actively developed. The tests of the apparatus that have been carried

out to date have not yielded new results of physical significant. We

have not yet observed the predicted phase transition, but we have

scanned only one composition.

Conclusions and Discussion

The measurement technique we have developed can be improved significantly. Substantial progress is expected in the near future. An

automatic data acquisition system has been ordered. As of this writing,

the primary factor limiting the resolution of our present equipment is a

large temperature dependent shift in the apparent etllipsometric parameters of our fluid samples.

References

1. M. R. Moldover and John W. Cahn, Science 207 , 1073 (1980).

2. J. W. Cahn, J. Chem. Phys. 66, 3667 (1977).

147
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148
NBS-114A (REV. 2-80

U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR

REPORT NO.

NBSIR 81-2295

2. Performing Organ. Report No, 3. Publication Date

June 1981

4. TITLE AND SUBTITLE

NBS: Materials Measurements

5. AUTHOR(S)

J. R. Manning

6. PERFORMING ORGANIZATION (If joint or other than NBS. see instructions)

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract/Grant No. NASA Gov't Order

JH- 2 7 954.B8 . Type of Report & Period Covered

Annual Report

Apr. 1 , 1980-March 31, 1081

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. State. ZIP)

Materials Processing in Space Division

National Aeronautics and Space Administration

Washington, DC

10. supplementary notes

Document describes a computer program; SF-185, FlPS Software Summary, is attached.

11. ABSTRACT (A 200-word or less factual summary of most si gnifi cant information. If document includes a significant

bi bl iography or literature survey, mention it here) This report describes NBS work for NASA in support of NASA's Materials Processing

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

Materials) covering the period April 1, 1980 to March 31, 1981. The work emphasizes

measurements of materials properties and thermophysical properties.

This work has been carried out in five tasks. These tasks have, as two of their

focal points, the role of convection effects and the role of container effects, both

of which would differ in space-based experiments from those found in ground-based

experiments. The results obtained for each task are given in detailed summaries in

the body of the report. The five tasks are Task 1 - Surface Tensions and Their

Variations with Temperature and Impurities; Task 2 - Convection During Unidirectional

Solidification; Task 3 - Measurement of High Temperature Thermophysical Properties of

Tungsten Liquid and Solid; Task 4 - Thermodynami c Properties of Refractory Materials at High Temperatures; Task5- Experimental and Theoretical Studies in Wettinq and Multilayer Adsorption.

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolons)

Adsorption; Auger spectroscopy; convection; el 1 ipsometry ; gallium; levitation

calorimetry, specific heat; surface tension; thermophysical properties; tungsten; wetting

13. AVAILABILITY

fxJ Unlimited

[f[] For Official Distribution. Do Not Release to NTIS

Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402.

[XJ Order From National Technical Information Service (NTIS), Springfield, VA. 22161

14. NO. OF

PRINTED PAGES

151

15. Price

$ 14.00

U S COMM- D C 6043-P80