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NBSIR 74-611 (^)

NBS Space Processing Research

E. Passaglia, R. L. Parker

Institute for Materials Research

National Bureau of Standards

Washington, D. C. 20234

Phone: (301) 921-2961

Novemb

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.U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

NBSIR 74-611

NBS SPACE PROCESSING RESEARCH

E. Passaglia, R. L. Parker

Institute for Materials Research

National Bureau of Standards

Washington, D. C. 20234

Phone: (301) 921-2961

November 1974

Annual Report

For the Period 1 November 1973 - 31 October 1974

NASA Contract W-13,475, Mod. 3

This reports was prepared by the National Bureau of Standards under Contract W-13,475, Mod. 3 - NBS Space Processing Research, for the Advanced Missions

Program of the National Aeronautics and Space Administration.

U. S. DEPARTMENT OF COMMERCE, Frederick B. Dent, Secretary

NATIONAL BUREAU OF STANDARDS, Richard W. Roberts. Director

Table of Contents

Page

Summary 2.-2

Task 1 - CRYSTAL PERPECTION IN CZOCHRALSKI GROWTH by M. Kuriyama, 3-23

W.J. Boettinger, H.E. Burdette and R.M. Eaton

Task 2 - EVAPORATIVE PREPARATION OF ULTRA-HIGH PURITY MATERIALS by 24-38

R.C. Paule, W.J. Boettinger and F.S. Biancaniello

Task 3 - VACUUM EFFECTS IN THE PREPARATION OF COMPOSITE MATERIALS 39-59

by H. Yakowitz

Task 4 - MELT SHAPE IN WEIGHTLESS CRYSTAL GROWTH by S.C. Hardy 60-68

and S.R. Coriell

Task 5 - VAPOR TRANSPORT SYNTHESIS AM) CRYSTAL GROWTH OF OXIDES 69-78

by H.S. Parker

Task 6 - SURFACE TRACTION AND OTHER SURFACE PHENOMENA by 79-88

A.L. Dragoo

Task 7 - CONSULTING SUPPORT , 89

Distribution List 90-91

i

National Bureau of Standards

Space Processing Research

Summary

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

Processing Program covering the period November 1, 1973 to October 31, 1974.

The objectives of the NBS program are to perform ground-based studies (and,

where appropriate, space-based studies) of those aspects of space that could

possibly provide a unique environment for making materials more perfect or

more pure. The approach taken deals primarily with experimental and

theoretical studies of the possible effects of the absence of gravitational

forces on those materials preparation processes where the presence of gravity

may be important in reducing perfection or purity. The materials preparation

processes studied comprise six tasks in the areas of crystal growth, purification and chemical processing, and the preparation of composites. There is an

additional seventh task on consulting support.

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

the body of the report. Briefly, in Task 1 - Crystal Perfection in Czochralski

Growth - several crystallographic effects have been examined and found to be

crucial factors in determining the ultimate level of perfection in copper

crystals; very highly perfect crystals have been grown and evidence presented

that gravity is the origin of possible slip dislocations during crystal growth

on earth. In Task 2 - Evaporative Preparation of Ultra-High Purity Materials -

experimental tests of the evaporative purification calculations have been

started, initially on levitated liquid Mo, with some significant reduction of

metallic impurities. In Task 3 - Vacuum Effects in the Preparation of

Composite Materials - the characterization of composites consisting of

reinforcements of silicon carbide, tungsten, bare pyrolytic graphite and

- 1 -
sapphire, each in an aluminum matrix, was carried out by optical, and scanning

electron microscopy. In Task 4 - Melt Shape in Weightless Crystal Growth -

simulated low-g studies of liquid water zones on ice cylinders have shown

that in the absence of a temperature gradient, the stable configuration is

one in which the ice cylinder passes entirely through the liquid water zone.

In Task 5 - Vapor Transport Synthesis and Crystal Growth of Oxides - mercurous

chloride was transported at very high rates and this system may be suitable

for the Apollo-Soyuz Test program. In Task 6 - Surface Traction and other

Surface Phenomena - the enhancement of evaporative purification rates by

capillary convection was studied by a Green's function method. In Task 7 -

Consulting Support - a rather wide range of meetings, consultation, talks

and other interaction with NASA and NASA contractors has taken place.

From these results at NBS, as well as from the Sky lab results on space

processing, the benefits of the space environment on a wide range of materials

preparation and processing techniques are becoming apparent. Better crystals

are, or can be, obtained in certain cases. The possibility of higher purities

looks quite good. While more work is clearly needed, the program is

beginning to bear fruit.
Task 1

Crystal Perfection in Czochralski Growth

M. Kuriyama, W. J. Boettinger

H. E. Burdette and R. M. Eaton

Metallurgy Division

Institute for Materials Research

Summary

As a continuation and expansion of our previous work (NASA Contract

W-13,475 #3 task 1), an intensive study has been performed to seek optimum

solidification parameters for the production of sizable, highly perfect

copper crystals by Czochralski growth. In this report, the objective has

been widened to include crystallographic effects as opposed to the

fluid flow effect, per se. The crystal perfection has been assessed as in

the previous report by nondestructive methods of x-ray diffraction, such

as Borrmann anomalous transmission and rocking curve measurements by double

crystal spectrometry.

This report completes the work on the preparation of sizable, highly

perfect single crystals of copper . Many interesting results about the

crystal growth processes and the resultant crystal perfection have been

obtained. From these results, the following conclusion has been drawn.

With other factors held constant, an optimum range of growth conditions,

characterized by the rotational speeds of the seed and the melt, can be

definitely chosen, which improves the crystal perfection to a certain level.

It has been found, however, that more crucial factors in determining the

ultimate level of perfection are:

1. the crystal perfection in seed crystal and thus the Importance of

the bottle-necking.

- 3 -
2. a particular crystallographlc choice of the growth direction (the

<100> direction is most preferable as the growth direction), and

3. the immobilization of possible dislocations in the growth process.

In particular, the last factor is quite interesting in view of possible

space experiments. This factor can, perhaps, be altered or eliminated in

the growth process of copper in space. Discussions have been made in

detail on this subject, based upon our observations made on the earth.

Further study of the crystallographlc effects of factors 2. and 3. on the

resultant crystalline perfection are strongly recommended.

- 4 -
Introduction

Motivated by man's access to space, one conceived an idea that metal

crystals can be melt-grown in space free of dislocations. One reasoned

this idea, based on the lack of gravity in space and, in turn, the lack of

thermal convection in the melt. Thermal convection, induced by gravity

acting on density differences in the melt, would be believed to influence

the quality of resultant crystals grown on earth.

However, no one could assert with a certainty that the quality of

crystals, or better the degree of crystal perfection, was related to crystal

growth conditions. One lacked a systematic study on their relationship.

The documentation of the growth conditions for the preparation of dislocationfree metal crystals was earnestly needed before one could even attempt the

possibility of manufacturing perfect metal crystals in space.

In previous reports, we documented the crystal imperfections in sizable

copper single crystals grown under various growth conditions by the

Czochralski technique. Emphasis was placed on the following aspects:

a. reproducibility of crystal perfection for given growth conditions,

b. "bottle-necking" effects in the process of crystal growth,

c. annealing effects after sample crystals were prepared in a disc

f orm

.

Results obtained in these previous works seem to indicate that there is a

general trend towards a reproducible relationship between crystal perfection

and growth conditions. For a practical reason, the growth conditions are

described in terms of directly controllable process parameters, such as the

The previous reports will, hereafter, be referred to as Report I (Government

Order H-84832A, National Aeronautics and Space Administration; NBS Report

10873) and Report II (Contract W-13,475 No. 1 - NBS Materials Science and

Manufacturing in Space Research, for the Advanced Mission Program of the

N.A.S.A.; NBS report NBSIR 73-402)
rotational speed of the seed and of the melt.

As a continuation and an expansion of our previous works, optimum

solidification parameters for the production of highly perfect copper

crystals by Czochralski growth are sought in this report, including an

additional aspect

d. the effect of seed orientations,

along with the documentation of crystal imperfections under various growth

conditions. Aspect d is added as an independent parameter so that the

genuine fluid flow conditions in the melt may be separated from other

physical conditions.

In this report, a vital part of the research is, again, the nondestructive assessment of crystal perfection by x-ray dynamical diffraction.

Further development has, therefore, been continued on high resolution x-ray

diffraction methods for the characterization of sizable metal crystals.

At the end of this report, a conclusion will be drawn not only from the

present work, but also from the previous works documented in Reports I and II.

Therefore, this report can be considered as a concluding report on the growth

conditions for the production of large, highly perfect copper crystals by

Czochralski growth.

Experimental Procedures

The details of the crystal-growth procedures and the assessment of crystal

perfection by x-ray dynamical diffraction have been described in Report I and

a previous paper"!" In this report, only changes and improvements are discussed,

along with a brief description of the x-ray techniques.

Large copper single crystals were grown by the Czochralski technique from

single crystal seeds in a vacuum furnace. The pulling speed was 0.10 cm minT''-
In Reports I and II, we have determined that no improvement of crystal

perfection is made by reducing the speed to values as low as 0.01 cm minT^

The melt and seed were rotated around a common axis with various speeds as

described later. Copper of 99.999% purity was used as the melt charge.

Crystal boules were grown with initial narrowing of the boule diameter at

one region, thus forming a "bottle-neck". The grown crystals were usually

about 6 cm long with diameters between 1.5 and 3.0 cm and with a bottleneck diameter of less than 1 mm.

On removing the grown boules from the furnace, the waxing technique was

used as described in Report II. The boules were sliced into discs with an

acid saw. At least three slices were obtained from each boule. We designed

and constructed an improved version of an acid saw (Fig. 1). This saw is

capable of slicing seven discs simultaneously in a much shorter period of time

(approximately less than 8 hours) than the time required by a previous saw

(approximately more than 78 hours). The effect of the new saw on the

resultant perfection of crystal discs was confirmed by an x-ray test to be nil.

The discs were polished on an acid polishing wheel as described in Report I.

The thickness of the copper discs was determined by the x-ray anomalous

transmission effect as described in Report I, II and a previous paper For

the assessment of the perfection of sample crystals, two different x-ray

optical arrangments were employed, namely, an asymmetrical crystallographic

topographic camera (ACT) and a high-resolution double-crystal scanning

diff ractometer (SCAD)^" The quality of the beam in SCAD was the same as that

1 2 given in Report II and previous papers.'

An x-ray image intensifying tube has been introduced in the ACT system

with the collaboration of Professor R. E. Green, Jr. of The Johns Hopkins
University. This image tube has made it possible to assess the quality of

perfection of the sample crystals almost instantaneously. Not only the

surface diffraction topograph, but also the transmission topograph of samples

are visible on the image tube. This greatly facilitates the alignment of

crystals for the maximum coverage.

Experimental Results

1. Fluid Flow Conditions

As in the previous reports, the fluid flow conditions in the melt during

crystal growth are classified into groups by different sets of values of the

principle variables, angular velocity of crucible and seed rotation, as

listed in Table I. Although individual values of the variables differ from

those in previous reports, the group number represents, as a rule, almost the

identical physical condition in the melt. By interchanging the values of y

(seed rotation) and z (crucible rotation), we obtain A = y - z with opposite

signs. Based on the results in Reports I and II and the results in the

present work, there appears to be no significant difference in the resultant

crystal perfection due to the difference in the sign of A. Therefore, we do

not discriminate fluid flow conditions from each other when the sign of A is

changed by interchanging the values of y and z.

In Reports I and II, the orientation of seed crystals was selected so

that the growth axis was approximately 20° from both <111> and <110>. One

of several reasons for this particular selection was to minimize crystallographic effects, if any, on the study of the crystal growth processes from

the melt. This was desirable because the primary question raised had been

the effects of fluid flow on the quality of crystalline perfection. In the

lowest approximation, fluid flow effects would be resulted from a continuum

- 8 -
medium rather than a crystalline medium that might play a

critical role in the real solidification processes.

Since Reports I and II contain sufficient data on the growth conditions,

per se, it is desirable in this report to study crystallographic effects on

crystal growth, and to determine which of the two factors - fluid flow or

crystallography - is more critical to the resultant crystalline perfection.

For this purpose, we have grown crystals from well-oriented single crystal

seeds; the crystals grown in the <111> direction of the seed crystals are

specified by the number 2 as the first digit of their sample number, like

2^54301, and the crystals grown in the <100> direction of the seed crystals

by the digit 4. The crystals grown under the conditions outlined in previous

reports have sample numbers starting with 0.

2. Spectroscopic Data

As in Report II, spectroscopic data - namely, rocking curve widths -

alone supply adequate information regarding the degree of crystal perfection

in sample crystals. The value of each rocking curve width was measured as

the full width of half maximum (FWHM) of each rocking curve, or spectral

profile. The profiles were obtained in both transmission and surface

reflection geometries. The crystals were considered to be more perfect, the

smaller the observed width. Typical rocking curves obtained in both

geometries are shown in Fig. 2.

In Table II, we list the values of rocking curve widths for all the

samples of the 0-series, 2-series (111 - orientation) and 4-series (100 -

orientation) examined in this year's work, along with the values of y, z

and A for the growth conditions. Rocking curve widths observed in the
transmission geometry for crystals of the same degree of perfection vary

with the crystal thickness. In most crystals, the perfection is not uniform

throughout the crystal. In the 0-series crystals, the local variation of

rocking curves was as much as that reported in Report II. In contrast, the

crystals of the 2- and 4-series did not display as large a variation as did

those of the 0-series. The rocking curve widths listed in Table II are the

values averaged over the crystals. In this table, some of annealed crystals

are included for completeness; their sample numbers are distinguished from

the sample numbers for as-grown crystals by a letter A.

It was found in Report II on the 0-series that growth conditions I and

III are equally good for the production of highly perfect crystals. This

has led us to speculate that the value of A is more critical to the resultant

perfection than the individual values of y and z. To check this speculation,

growth condition VI has been added, in which the value of A is kept almost

the same as I and III, but the values of y and z are greatly changed. As

seen in table II, this speculation appears to be proved for the crystals of

the 0-series. In general, the present data are in good agreement with the

results obtained in Report I and II not only for the 0-series, but also for

the other series of crystals. Growth condition V, in which the seed is

rotated in the opposite direction to the crucible, again produced crystals of

low quality from which the rocking curve showed a multi-peak structure as in

Report II.

As far as the rocking curve widths are concerned, it is apparent that

growth condition III should be chosen as an optimum growth condition for

highly perfect crystals, regardless of the seed orientations. However, growth

conditions VI and I are almost as favorable. This is most easily seen in the

- 10 -
values of the surface reflection rocking curve widths, but is also seen in the

transmission rocking curve widths when the effect of crystal thickness is

considered. The rocking curve width in the surface reflection geometry for

sample #458302 under growth condition III is larger than those widths under

other growth conditions. However, this contradiction can be accounted for by

the thinness of the crystal which makes the crystal vulnerable to strains.

In fact, the narrow width observed in the transmission geometry supports the

fact that the crystal is, indeed, more perfect than the others.

Another interesting result can be found in Table II, when the rocking

curve widths are compared with each other within the same growth condition.

As a rule, the crystals grown in <100> direction (4-series) show narrower

widths than the crystals of other series. The exception (#461302) found in

growth condition VI can be explained by the thinness of the crystal, using

the same argument as described above for #458302. If comparisons are made

only among the rocking curve widths from the 4 series crystals under different

growth conditions, it appears that growth conditions I, II, III and VI are

equally effective in producing highly perfect crystals of the <100> orientation.

3. X-ray Topographical Results

The rocking curve measurements, by themselves, doubtless provide adequate

information on the degree of crystal perfection as a functional of growth

condition, as shown above. These measurements, however, are not convincing

to many due to the lack of a visual image of the locations, distributions

and shapes of crystalline imperfections. We will therefore present here the

results of topographic observations made by the ACT system. In view of their

higher resolution, topographs taken by the SCAD system will be used, whenever

necessary, to confirm or resolve subtle differences among the ACT results.

- 11 -
Since, for lack of space, we cannot present all the topographs here, we will

describe the qualities of the crystals as-grown under various growth conditions, based on their topographs.

Examples of the topographs are shown in Figs. 3a and b in both the 220

surface reflection and the 111 transmission geometry from an as-grown crystal

As seen in these topographs, the entire volume of the crystals usually

participates in diffraction simultaneously to form a diffraction topograph,

regardless of the diffraction geometry. This condition implies in the

present ACT system that the possible misorientation over the entire area of

the crystal should be much less than 30 second of arc.

A. The crystals of the 0-series

The topographs obtained from the 0-series crystals reproduces the

identical details of imperfections observed in Report II. As we speculated

and confirmed by the rocking curve measurements, the crystals grown under

growth condition VI also showed the Lomer-Cottrell locking structure as

clearly as the crystals grown under growth condition III, as described in

Report II. The observation of Lomer-Cottrell dislocations in as-grown

2

copper single crystals has been reported in detail in a previous paper.

Furthermore, it is confirmed in this report that the crystals grown under

growth condition II also showed evidence of Lomer-Cottrell dislocations,

although they were, in most cases, less distinct.

B. The crystals of the 2-series

Next, we turn to the x-ray diffraction topographs obtained from the

crystals grown in the <111> direction. The crystals grown under growth

The topograph enlargements are composites as a result of microscope limita

tions during enlargement, and partly as a result of successful growth of

large crystals surpassing the anticipated size of the beam.

- 12 -
conditions I, II, III and VI all showed a distinct Lomer-Cottrell locking

arrangement, although the image-visibility of these dislocations appears to

be dependent on the crystal thickness. Fig. 3b shows the topograph obtained

from a rather thick crystal, in contrast with the typical transmission

topograph of Lomer-Cottrell dislocations in a thin crystal which was shown

2

in Report II and in a previous paper. Fig. 3c shows the SCAD transmission

topograph at a higher resolution where Lomer-Cottrell dislocations are well

demonstrated. The growth conditions in which the value of A is large

produced the crystals of poor quality. The topographs of such crystals

clearly show a mosaic structure, or more accurately, coherent domains

surrounded by extremely low angle grain boundaries. These topographs have

almost the same features as those topographs shown in Figs. 4a and b.

C. The crystals of the 4- series

Here we describe the topographs obtained from the crystals grown in the

<100> direction. All the surface diffraction topographs of the crystals

grown under growth conditions I, III and VI are similar to the surface

topograph shown in Fig. 3a. The transmission topographs from these crystals

clearly show the Lomer-Cottrell dislocation arrangements as seen in the other

series of crystals. The crystals grown under growth condition II show a

somewhat different feature in their diffraction topographs, as shown in Fig. 5a

and b. The origin of these scattered dotted structures has not been explained

3 4 satisfactorily. These structures have been reported by Fehmer and Uelhoff '

in a copper crystal grown by the Czochralski technique. A detailed study of

this structure is currently in progress. Fig. 5c shows this structure in the

SCAD surface topograph with a higher resolution. The growth conditions, IV

and V, produced the crystals having multi-coherent domains, as seen in the

- 13 -
other series of crystals. However, there remains a definite distinction

between their topographs and topographs of other series grown under these

growth conditions. Unlike the other series of crystals in which the anomalous

transmission effect is disrupted so heavily, the 4-series crystals grown under

conditions IV and V show the anomalous transmission effect, as shown in

Fig. 4b. In the transmission topograph, the crystals even show the formation

of Lomer-Cottrell locks in individual domains. This fact implies that the

crystals of the 4-series have much higher degrees of perfection, even under

growth conditions IV and V, than the crystals of the other series.

Typical surface and transmission topographs taken by the SCAD system

are shown in Figs. 6a and b, where scratches, sometimes unavoidable during

the chemical polishing process, are also seen along with dislocations and

other defects. Examples of surface and transmission topographs taken by the

ACT system from annealed crystals are shown in Figs. 7a and b. Fig. 7c

shows the part of the surface topograph, taken by the SCAD system, where no

dislocations are visible.

Conclusion and Discussion

During the past three years, we have performed an extensive study on

the crystal growth of copper single crystals by the Czochralski technique,

with emphasis on the documentation of the resultant crystal perfection. The

total number of crystal boules grown, including those used for seed crystals,

reached 102. From them, 117 discs were examined by the two sophisticated

two-crystal x-ray spectrometer systems which were developed uniquely in our

laboratory. To perform this task in the most effective manner, an extensive

knowledge of x-ray dynamical diffraction was utilized. The development of

our ACT (asymmetrical crystal topography) system has made it possible to

- 14 -
collect the photographic (topographic) data from a large number of sample

discs in both the surface reflection and the transmission modes with

maximum efficiency and precision. Quantitative data, such as the rocking

curve widths, have also been collected at various locations of each sample

with great accuracy. One shortcoming, however, is found in the intensity

of x-rays generated by our x-ray tubes and generator. Although an image

intensifying system has been added to the ACT system, the weak intensity

problems from our x-ray generator have not been eliminated.

In this report, we intend to draw some conclusions on the effect of

growth conditions on the crystalline perfection in copper single crystals

grown by the Czochralski technique. First of all, we have to emphasize

the fact that the same degree of crystal perfection can be reproduced

repeatedly under given growth conditions as shown in Reports I and II as

well as in this report. This reproducibility, at least, makes it possible

and even sensible to have conducted some experiments determining whether

or not there is indeed a correlation between the crystalline perfection and

the controllable growth parameters for the melt. In other words, were it

not for this reproducibility, our experiments would have been meaningless.

As we demonstrated in this report, we had predicted from the previous results

in Reports I and II that a new growth condition, namely VI, should produce as

good a crystal as the other conditions I and III did, and we confirmed

successfully that prediction. This implies that our collected data are

meaningful enough to analyse the validity of the relationship between those

two quantities. It can be concluded, as far as the growth condition is

concerned, that an optimum growth condition for the preparation of highly

perfect copper crystals is, as a rule, characterized by the small value of A

- 15 -
(the difference between the speeds of the seed and the melt). The directions

of the seed and melt rotations should be in the same direction for this

optimum condition. It has been proven that growth conditions with large A's

produce crystals of poor quality that exhibit several coherent but independent

(judged from their well defined boundaries) domains.

The optimum growth condition certainly is significant in the growth of

the 0-series crystals. In the growth of the 2-series crystals (grown in the

111 direction), this optimum condition still has some significance, although

the difference in the crystal perfection becomes more subtle when growth

condition II is compared with, for example, VI. It can even be said that,

for the crystals of the 4-series grown in the <100> direction, the growth

conditions do not have great influence on the resultant perfection, as seen

in Table II. It is, however, fair to conclude that there is a general range

of growth conditions which improves crystal perfection to a certain level,

regardless of their growth directions - the range being specified by the

small value of A.

A more crucial factor influencing the ultimate level of crystal perfection

has been found in the bottle-necking effect, or rather the original perfection

in seed crystals. It is indispensable in the production of sizable perfect

crystals to use a highly perfect crystal as a seed and to reduce the chances

of introducing further dislocations with initial narrowing the boule diameter

at one region.

Annealing does not improve crystal perfection to a large extent, unless

as-grown crystals have been already highly perfect, or have had particular

arrangements of crystal imperfections such as Lomer-Cottrell locks discussed

in Report II and in a previous paper. In general, prolonged annealing tends to

- 16 -
introduce dislocations from the crystal edge and develop pile-ups of dislocations, thus creating extremely low angle boundaries.

A significant result in this report is the observation of crystallographic

effects. Dominance of the growth direction on the ultimate level of crystalline

perfection obviously cannot be ignored, as described in a previous section.

Another effect related to crystallography is slightly more delicate; as

seen in Figs. 3 and 5, some particular sets of dislocation arrangements are

usually observed in the best crystals, and these arrangements are strongly

dependent on the crystal structure of materials, as discussed in a previous

2

paper. The Lomer-Cottrell locks are perhaps unlikely to form, for example,

in the body-centered cubic crystal. The structure observed in Fig. 5 is

very likely related to the types of dislocations originally developed in the

process of solidification, although careful studies have not as yet been

completed

.

2

As we discussed in Report II and a previous paper, the formation of

dislocation structures, such as Lomer-Cottrell locking, plays a decisive role

in the preparation of sizable, highly perfect copper crystals. The crystals

grown under optimum growth conditions have the overall high degree of

perfection, as judged from their display of the prominent Borrman anomalous

transmission effect. All the regions in the interior of the crystals,

therefore, are likely to have similar properties with respect to each other.

If the formation of a Lomer-Cottrell dislocation is energetically favorable

at one place, then there is no doubt that dislocations of a similar type

would occur through the same process everywhere inside the bulk crystal.

In fact, when a slice of the as-grown crystal showed the arrangements

of Lomer-Cottrell dislocations over its entire volume, as shown in Fig. 3,

- 17 - ,
the degree of crystal perfection was equally high for any part of the

crystal boule of about 6 cm long. In contrast, some as-grown crystals

which did not show the Loraer-Cottrell locks showed an extremely high degree

of perfection in one portion of the boule, but did not show the same high

quality of perfection in other parts of the boule. Most of these crystals

tended to develop low-angle grain boundaries somewhere along the length of

the boule. This tendency indicates that the formation of particular sets

of interacting dislocations makes dislocations immobile, thus aids to

produce a sizable, highly perfect crystal. If wide-spread slip occurs or

if many dislocations form randomly, then the locking would not take place

and the dislocations would still be mobile, and would affect the subsequently

grown part of the crystal, resulting in a less perfect crystal.

It is likely that dislocations are generated in the growth process.

They will either disappear by slip through the crystal, or remain randomly

in the crystal at obstructions. During crystal growth on the earth,

crystals are grown in the field of gravity. Gravity exerts a body force on

crystals and can produce slip in the crystals at high temperatures during

their growth process. Especially when the crystal is being pulled, the weight

of the crystal may produce a sufficient shear stress, resulting in slip.

Unless the slip goes through the entire crystal, dislocations are likely to

remain in the crystal and be subject to further motion. If they are sessile,

they won't move, and they will prevent further slip from occurring. On the

earth, it is easier to make dislocations form a certain arrangment, as we

demonstrated, than to make all the dislocations move out of the crystals.

Compared with the crystals grown in other directions, the crystals of

the 4-series grown in the <100> direction have the largest number (8) of slip
systems ({111} planes, <110> directions) available and the maximum shear stress

is produced by the crystal weight on these slip systems. It is thought that

this physical condition will lead quite easily to the formation of LomerCottrell locks and hence crystals of high perfection. These crystallographic

arguments, including the fact that 4-series crystals show the best degree of

perfection, support the hypothesis that gravity is the origin of possible slip

during crystal growth.

In conclusion, it cannot be denied in the preparation of sizable, highly

perfect copper sizable single crystals that a certain range of growth

condition offers an optimum set of directly controllable parameters for the

fluid flow in the melt. This range is more critically limited by the value

of the difference of the rotation speeds of the seed and the melt than by the

individual values of speeds. It is, however, found that more crucial factors

in determining the ultimate level of perfection are 1) the crystal perfection

in seed crystals and the control of inheritance of seed imperfections (the

bottle-necking) , 2) the crystallographic orientation of the growth direction,

and 3) the immobilization of possible dislocations in the growth process. The

<100> direction appears to be most favorable for perfection in crystals. The

crystals could probably have been made perfect by the choice of the proper

growth direction, if no gravity had acted on the crystals. The observed

dislocation numbers in the copper crystals that were grown in the past are

as low as 5 dislocation per cm^. In some annealed crystals, major parts of

the crystal volume display no dislocations at all, although a few dislocations

can be seen near the edge of the crystals.

Based upon the results and conclusions of this report, it is extremely

desirable to further study the crystallographic effects, in particular, by

- 19 -
use of a face-centered cubic metal other than copper. These crystallographic

effects should also be studied in the absence of gravity. From an academic

point of view, many interesting observations, such as the structure shown in

Fig. 5, and the annealing effect, should be studied in detail.

; I.

- 20 -
References

1. M. Kuriyama, J.G. Early and H.E. Burdette, Proc . of AIAA 12th

Aerospace Sciences Meeting (1974) Paper No. 74-204. This paper

Is attached to this report as Appendix I.

2. M. Kuriyama, J.G. Early and H.E. Burdette, Journal Appl.

Crystallography (1974) 7. This paper is also attached to this

report as Appendix II.

3. H. Fehmer and Uelhoff , Journal Cryst. Growth 13/14 (1972) 257.

4. E. Kappler, W. Uelhoff, H. Fehmer and F. Abbink, "Herstellung von

Kupf ereinkristallen Kleiner Verzetungsdichte" Westdeutscher Verlag,

Opladen (1971).

- 21 -
Table I Fluid Flow Conditions

Group Seed Crucible Difference

Number Rotation (y) Rotation (x) (A = y - x)

I - 5.0 - 4.4 - 0.6

+ 5.0 + n 6

II +10.0 + 6.0 4.0

+10.0 + 5.0 + 5.0

-inn -in - s n

III -20.0 -19.6 - 4.0

+90 n + A n +20.0 +20.6 - 0.6

TV +90 n +1 n n +1 n n +90 0 +90 6 + n

V + 5.0 - 5.0 +10.0

- 5.0 + 5.0 -10.0

VI +10.8 +10.0 + 0.8

-10.0 -10.0 + 0.8

-10.8 -10.0 - 0.8

Rotation speed is in units of rotation per minute.

- 22 -
TO Q

O

iH,0

1-13. 0 CM

1 VO ^

1 < vO

LPl

18.16 50.78 50.78 34.55 14.10

CN v£> 00 00 r-( rH ON m >.D m vo p-i vD 0 v£> 00 m rH CM CO cN ri

27.56 45.49 27.56 <3-ONONCMvOCXDCMO<rv3-OCO ocNo^or^cMOOOininrHvo

cooooovocMCMr^ONvOvo«*r*»

COrHrH~3-CT>COm^<-»J-rHlH

(Ill)

Transmission

FWHMsec.

of

arc

8.5

11.0 9.0 0 m 0 0 m 0

CO 0 00 fo 00 CN rH i-l m

29.5

12.5

27.0

10.25

ACT

topographs

only

29.0

13.0

ACT

topographs

only

25.0

9.5

31,5

10.0

not

observed

not

observed

51.5

15.5

ACT

topographs

only

multigrain

multigrain

ACT

topographs

only

10.0 11.0 12.0 10.0

multiple

peaked

15.5 11.0 9.0

11.0 9.310.5 10.5

(220)

Surface

FWHMsec.

of

arc

33.0 29.0 33.0 000000

vO CO 0 ON 00

fO ro CM "B" CN CN

40.5 58.0 oooinmominooom

oooo^^ocoooNC3^NO^^cncn

cncocMcnvO<-cMCMCMCMcocn

Thickness (L)

mm

unobtainable 0.8092 1.0235

0.3294

^

0.9212 0.9212 unobtainable 0.6269 0.2558 0.6557 0.3070 0.50.5566 0.50.6654 0.1471 unobtainable 0.50.8253 0.5

not

observed OOONOONOr^rHO-^J-oOON

<rONcrNr~~ocMrHCTN-<r-d'inON

rHfoco<j-cncor^oo<t<3-inrH NOcocooomin^ooooooocMco

000000000000

Difference

(A

=y

- x)

vO vO 000

1 + + 000000

<r LTl m m

+ + + + 1 1

vO >d- NO vD <r <3- 0 0 0 0 0 c 0

1 + + 1 1 1 1

+10.0 +10.0 -10.0 -10.0 +10.0 +10.0 oooooooooooooocooooo<-<l"oooooooooooo

+ + + + 1 1 I+ + +I 1

Crucible

Rotation

(x)

<r <r

•>3- <r

1 + + 000000

vo vo vO m lh m

+ + + + 1 1

+20.6 +19.6 +19.6 +20.6 +20.6 -19.6 -19 6

+10.0 +10.0 oooo

in U-1 LTl LTl + + 1 1

oooooooooooooo>*»^-

oooooooooooo

rHrHrHrHrHrHrHrHrHrHrHrH + + +I 1 1 IIII+ +

Seed

Rotation

(y) 000

m m u-i

1 + + 000000

000000 <-i r-i <-i <-{ ^ l-i + + + + 1 1 00000000000000 CM CnI CnI CM CM CM CN + + + + + 1 1

+20.0 +20.0 oooom in in m

1 1 + +

oooooooooooooooooo

oooooooooooo

J-\r-{r-i^^,-ir-{,-iT-tr-ir-\<-^ + + +I 1 1 IIII+ +

Crystal Slice

Number

252301 453301 453302 040401 040402

040402A

254301 456301 456302 042301 255301 255302 269301 269302 458301 458302 257301 459302 260302 262302 463402 463402

< <

rHCOCOCMrHCMCOrHCMCvJrHCM oooooooooooo ncorocM mrococorocomco cOcOCOcJNNOvONDOOOrHrH

<r-j-<r<r<-<r<-inininNOvo OOOOCNCMCMCslCMCMvfsr

Group

Number M M M

M M M M M M

M I-l 1—1 1—1 l-H M

M M M M M M M

M hH hH M 1—1 M M

M !-H HH 1—1 M M M

> > M M > > > >

1—IMMt-HMMMI-HMI—IMM >>>>>>>>>>>>

- 23 -



o

rom o

in

<o ^ 2 -

</)— < aoo

UJCO T

COo

roOm

CVJ o

UJ

o: o <

u.

UJoo

rjM

.p^W

1) CO c 0) •H •H

CO

4-1 ^

(1) o i

CO 60 > o o •H

CO

CO

c: •H

•H

B

,^ CO oo CO

>-l 4J

r-lCO T) O C3 •H CO ^§•Ho

0) iH

•H QJU o

lO oom

Om

CSJ aN003S d3d SlNflOO A1ISN31NI

Fig. 3a ACT 220 reflection topograph

of as-grown cropper crystal #255301.
II ; I I r ' I I I
I—

i mm

Fig. 3b ACT 111 transmission topograph

of as-grown copper crystal #250302-

I mm

Fig. 3c SCAD 111 transmission topograph

of as-grown copper crystal #255301.

Fig. 4a ACT 220 reflection topograph

of as-grown copper crystal #459302.

Fig. 4b ACT 111 transmission topograph

of as-grown copper crystal #459302.

Fig. 5a ACT 220 reflection topograph

of as-grown copper crystal #456302 .

[110] [0 01]

-[HI]

I mm

Fig. 5b ACT 111 transmission topograph

of as-grown copper crystal #456301.

I—

I mm

Fig. 5c SCAD 220 reflection topograph

of as-grown copper crystal #456301.

Fig. 6a SCAD 220 reflection topograph

of as-grown copper crystal #458301,
I

i

-.-....-vrf

"7
Fig. 6b SCAD 111 transmission topograph

of as-grown copper crystal #458301.


\

j

I

i

i

i
I mm

Fig. 7b ACT 111 transmission topograph

of annealed copper crystal #250302A.
!

i

1
I—

i mm

Fig. 7c SCAD 220 reflection topograph

of annealed copper crystal y/040402A

,

Appendix I

FLUID FLOW EFFECTS ON CRYSTALLINE PERFECTION'*'

M. Kuriyama, J. G. Early and H. E. Burdette

Institute for Materials Research

National Bureau o£ Standards

Washington, D. C. 20234

Abstract

In the absence of gravity, thermal convection,

i.e., convection induced by gravity acting on density differences in the melt, would be expected

to be negligible. Fluid flow in the melt, including thermal convection, probably affects the

perfection of crystals grown from the melt. At

present, the relationship between crystal growth

conditions, in particular, fluid flow conditions,

and the degree of crystal perfection has not been

well established for metals. It is, therefore,

highly desirable to document the perfection of

crystals grown fran the melt in terms of directly

controllable process parameters, before one even

begins to analyse the fluid flow conditions in the

melt in terms of thermodynamical variables. In this paper, optimum solidification parameters for

the production of highly perfect copper crystals

by Czochralski growth are sought along with the

documentation of crystal inperfections under

various growth conditions. A vital part of research

of this type is the assessment of crystal perfection. X-ray techniques which do not in their

application produce defects and which allow the

characterization of imperfections in single crys- tals are chosen to assess crystal perfection. The

properties of crystals grown from the melt are anticipated to vary over a large range, since the

growth conditions, especially the fluid flow

conditions, are deliberately changed. The x-ray

techniques employed ranged from ordinary Laue photography through Borrmann topography to doublecrystal scanning diffractometry, thus allowing

crystals with a wide variation in perfection to be

studied. As a set of controllable solidification

parameters, the rotation of the seed and of the

melt and the diameter of the bottle-neck are

chosen. X-ray diffraction topographs are analysed

along with the data obtained from rocking curve

measurements. Tables of growth conditions and

quantitative data of rocking curve widths are

presented.

I . Introduction

Other investigators have reported the growing

of metal single crystals containing regions which

are totally dislocation- free. (1) - (6) However, it

has not generally been possible to grow sizable

metal crystals free of dislocations. In contrast,

semiconducting materials, such as Si and Ge, have

been grown from the melt phase, free of dislocations.

The principle mechanism for the dislocation genera- tion in these crystals has been studied extensively- (7) -(11) The difficulty in growing highly

perfect metal crystals may be attributed to their

low dislocation energy and high dislocation

mobility, compared with those factors for semiconducting crystals. Imperfections can be pro- duced by inpurity particles present in the melt

which may originate in the container used to hold

the melt, or the inperfections may already exist in

the seed crystal and thus easily propagate into the

growing crystal.

In addition to these defect sources, one certainly cannot neglect sources related to thermal

conditions. Among these, the presence of thermal

convection and other fluid flow phenomena in the

melt may affect the solidification process

resulting in the build- ip of inhomogeneous strain

fields sufficiently large to cause dislocations to multiply. Since thermal convection is induced

predominantly by gravity acting on density differ- ences in the melt, it is usually impossible to completely control fliiid flow, especially convection, during crystal growth on the earth's surface.

At the present time, the relationship between

the crystal growth conditions and the degree of

crystal perfection has not been systematically

studied for metals. The primary objective of this paper, therefore, deals with a documentation of the

resultant crystal perfection under various

growth conditions with emphasis being placed on

the following aspects: a. reproducibility of

crystal perfection for given growth conditions,

b. "bottle-necking" effects in the processes of

crystal growth, c. annealing effects after sanple

crystals are prepared in a disc form. In studying crystal perfection as a functional of

fluid flow, it is necessary first to establish the

fact that there is, indeed, such a relation. It must be determined whether or not the variations

of crystal perfection are caused by fluid flow

conditions, per se, or by other factors, such as

the perfection of seed crystals and the procedures

of handling grown crystals. For this reason, aspects b and c are included in this study,

although analysing the results in aspect a alone

seems sufficient to define the optimum fluid flow

conditions in the melt for the present purpose.

In this research, one must examine the crystal

inperfections in the interior of crystals in

sufficient detail without disturbing the already

existing imperfections. X-ray dynamical diffraction by a crystal is known to be very sensitive to inhomogeneous strain distributions inside the

crystal. In nearly perfect single crystals, the

dynamical diffraction effect often produces well

resolved contrasts in intensity distributions

around single dislocations: this technique is known as x-ray diffraction topography. When the

crystal is less perfect, the diffraction topographs usually become more conplex. However, this

tedmique is still useful to detect and analyse

bulk strains. One shortcoming of x-ray topography

lies in its inability to describe the crystal

inperfections quantitatively. When x-rays are diffracted by a crystal, all the information on

irregular arrangements of atoms is contained in

the diffracted x-rays: the intensity distributions

and the angular broadening of the diffracted beams can provide quantitative data on the crystal

imperfections. Double crystal diffractometry can

Sipported by the Advanced Missions Program of the National Aeronautics and Space Administration
be lised to extract quantitative data fron the

diffracted x-ray beams along with x-ray topography.

We, therefore, utilize the dynamical diffraction

effect for the characterization of the perfection

of the crystals.

II. Crystal Growth Procedures

Since Young and Savage (3) succeeded in

preparing copper single crystals of very low

dislocation density, investigations of metal

single crystal growth have been carried out with

copper more often than with other metals. We,

therefore, chose to study copper so that the

previous information on the grcwing conditions and

the degree of the crystal perfection could be

utilized. Unlike previous workers, we grow crys- tals of reasonably large size (1.5 cm - 3.0 cm in

diameter compared with preAdous crystal sizes of

0.1 cm ~ 1.5 cm). The reasons for this are that

1) historically, it has been difficult to grow

such large crystals free of dislocations, 2) in our experiments we substantially change the conditions

of fluid flow, deviating from the conditions in

which most previous crystals of low dislocation

density have been grown, and 3) sizable single

crystals are more useful for industrial purpose.

In order to grow perfect crystals, it is necessary to minimize any possibility of introducing imperfections during the process of crystal

growth. If one can grow crystals without containers, one would expect less perturbation to the

crystals. This has been verified by Kappler et

al.(5) A technique which enables single crystals to be grown without touching a crucible is known as the

Czochralski technique. (12) The material in the crucible is heated in a vacuum slightly above the

melting temperature. Then a small single crystal

(seed) is immersed in the melt and subsequently

withdrawn at a slow rate. The melt adhering to the seed solidifies immediately above the melt

surface as it is being pulled iq^ward. A sche- matic diagram of the Czochralski apparatus is shown in Fig. 1, where the principle variables

defining the growth conditions are denoted by x (pulling speed) , y (seed rotation) and z (crucible

rotation) . The pulling speed is generally between

0.013 cm/min. and 0.1 cm/min. , ;<^iich is slow

enough to minimize the rate of the latent heat

input at the interface. The pulling speed does not

cause as significant a change in diameter, except

for crystals having a very small bottle-neck

diameter (~ 0.05 cm), as does the tenperature of

the melt. When the crystal initially grows with a very small bottle-neck diameter, the pulling speed

affects the diameter of the growing part significantly until the bottle-neck region conpletely

clears the top of the furnace.

Crystals are grown from single crystal seeds,

and copper of 99.9991 purity is used as the melt

charge. The orientation of the seed crystal is selected so that the growth axis lies in the

neighborhood of the direction deviated about 20° from both <111> and <110>. The reason for this

particular selection is based on the considerations

of diffraction conditions with respect to the

possible directions of the Burgers vector in the

copper crystal. A disc shape crystal having (110)

planes as its faces is most convenient, so that the

three strongest diffracting planes, (111), (002) and

Fig. 1 Schematic of Czochralski apparatus A- seed; B-bottle-neck; C-solid/liquid interface region;

D-melt; E-crucible; F-heater; X-pulling direction;

Y-seed rotation; Z-crucible rotation

(220) , can be used in transmission geometry. If the crystal is grown in the <110> direction,

diffraction using these two planes cannot reveal

dislocations associated with the growth direction.

(The growth direction lies in a slip plane.) Since

the growth direction is a unique direction, it is unwise to lose any information concerning those

dislocations (or a slip system (111) <110>)

.

Crystal boules are grown by initially narrowing the diameter at one region of the boule.

Narrowing the crystal diameter (the bottle-neck) is produced by increasing the temperature of the melt.

The boules are sliced into discs by an acid saw, (13)

and the discs are polished on an acid polishing

v^ieel using a saturated solution of CuCJla in

HCj!,.(13) At least three slices of sanple crystals were cut from different regions of each boule. One

of the slices after being polished was annealed at 1000 °C for three days in a hydrogen (dew point = -90°F) flow furnace.

III. Assessment of Crystal Perfection

Theory of X-ray diffraction in inperfect crystals

One of the limiting factors for research of

this type is the ability to determine the properties of crystal imperfections in sufficient detail.

Careful chemical etching can reveal defect

structures in high resolution, but these defects

are revealed only on the crystal surface. By

contrast, x-ray diffraction provides a unique and

powerful tool for the investigation of these

defects in the interiors of crystals. It is also
known that ordinary x-rays are non-destructive to most materials. Although the relatively sin^jle

kinematic diffraction theory is almost adequate

for the investigations of defects in ordinary

imperfect crystals, a canplete and detailed

description of diffraction phenomena requires the more rigorous theory - dynamical theory of diffraction. This is particularly inportant in the study

of nearly perfect crystals where dynamical effects

provide extremely sensitive probes for investigating the imperfections. There have been theoretical

difficulties in the application of the well-known

Ewald and Laue dynamical theoryC14) (15) to inperfect crystals, since this dynamical theory is valid only for perfect crystals. However, a generalized dynamical theory of diffraction has

been formiiLated to be readily applicable to real

crystals containing imperfections(16) (17)

.

The basic idea of employing x-ray diffraction

techniques to a study of crystal imperfections lies

in the fact that an x-ray Laue spot has a fine structure in it and this fine structure is closely

related to inperfections in crystals. The recent advanced techniques, such as x-ray topography, can magnify at high resolution a Laue spot from a large portion of a crystal. In the x-ray topographs

obtained from an inperfect crystal, one observes

black and white images siperijiposed on a background. For a thin crystal where the product of the linear

absorption coefficient y and the crystal thickness

L is less than 1, defects may appear as black (in

the photographic negative; stronger intensity)

images accompanied by a faint anomalous transmission

(or diffraction) beam trace. For intermediate

thickness where 1 < pL < 10, the topographs show

primarily white images. Black images sometimes

appear as well as black-white contrast images.

Usually one should ejqject ccmplicated patterns for

these cases. For a thick crystal (pL > 10), the Borrmann (anomalous transmission) effect (18)

becomes dcminant and is accompanied by v^iite images

of good contrast. Black images or non-uniform

intensity distribution may still be observed in

the topographs. When a crystal is nearly perfect,

the topograph clearly shows the images of individual dislocations. The above qualitative description of images is based on casual observation of

topographs. Mare conplicated pattenns including

interference effects, etc. are also observed.

According to a generalized dynamical theory,(17)

the scattering anplitude of an imperfect crystal

is given by

<k'R' I S I kR> = z F^^^ e3q)[i{a^ + - U(H)}- L]

+ II W(H) F^ „ exp[i(6- + K ) L], (3-1)

where k and k' are the incoming and the outgoing momentun (wave -vector) of x-rays, R and R' are the

positions on the crystal surfaces where the x-rays

are coming in and out, respectively, L is the

crystal thickness, and H is the reciprocal lattice vector perpendicular to the diffracting plane.

This expression is derived for a single Bragg

diffraction condition where |k' | - |k+H| - |k| . The

quantity F is the dynamical amplitude of the trans- mitted (K=0) or the Bragg- diffracted (K=H)

x-ray beam obtained in the usual dynamical theory

for a perfect crystal, where the polarizability

should be corrected by the distortion factor. The

solutions of the dispersion equations for the

incoming beam and the outgoing beams are given by

a and e, respectively. The detailed descriptions

of physical quantities involved are not discussed

here since it is beyond the scope of this paper. However, Bq. (3-1) is helpful to understand

topographic images.

The quantities, U(H) and W(H) , are related

to the crystal inperfections under a certain

region irradiated by the incident beam. When the

crystal is perfect, these quantities become zero, giving ideal dynamical diffraction

<kR |S| k'R'> = Z f[^^ exp[i(a^ + K^) L]. (3-2)

When the crystal becomes really inperfect, the

scattering amplitude reduces to

<kR Is I k'R'> = Z W(H) F^^i e3q)[i(0.+K ], (3-3)

(i^j) ^ ' giving the kinematical scattering amplitude. In this case, W(H) F becomes equal to the x-ray

structure factor. The first term of Eq. (3-1) is,

therefore, considered as the dynamical diffraction

term caused mainly by the diffraction mechanism

similar to the dynamical diffraction in a perfect

crystal. The second term is a new additional

diffraction mechanism due to the presence of

crystal imperfections. This effect is also the

result of dynamical diffraction, although in the

inperfect crystal limit it yields the kinematical

result given by Eq. (3-3).

In the first term of Eq. (3-1), the imaginary

part of a - U(H) can be thought of as the effective absorption coefficient under the diffraction

condition. This quantity is given by

eff = Im [a^ - U(H)

]

- m[1+(-1) 7^ lm[v(0)] ® J' "--^

where e is the parameter indicating a deviation

from the Bragg condition, v(Hj and v(0) are the

0-th and H-th Fourier transform of the polarizability. The ratio, Im v(H)/Im v(0) , generally

is very close to one. M(H) is the distortion

correction factor relating to the second moment of

the strain distribution in the crystal portion

irradiated by the x-ray beam. In a perfect

crystal, M(H) is equal to zero, making y eff

(corresponding the i=l mode) almost zero. This is the phenomena known as the Borrmann anomalous

transmission effect(18) in ^^^lich the transmitted

and the diffracted beams suffer negligible

absorption. When the crystal is not perfect,

M(H) increases, creating higher effective

absorption. Since M(H) is also a function of

position of inperfections, the effective

absorption coefficient becomes larger at the

imperfection sites. This is the primary reason why one can see the image of inperfections clearly

in the Borrmann topographs.

The second term is usually responsible for

black-white contrasts of inperfections seen in

topographs. This term contains almost conplete

information on the properties of imperfections,

and describes the phenomenon previously known as secondary extinction. (17) Unlike the first tern,

vrtiere the contrast is independent of incident

angle, this second term shows that the contrast of

dislocation images changes frcm a black-white one

to a white-black one, depending on the incident

angle or, in other words, the sign of the strain
field gradient. TMs is also experimentally

kncwn as a ccnplementary relation of images: the

topographic contrast formed by the (hkl) and

(HcF) reflections is reversed or conplementary.

Another inportant fact concerning the diffraction

effects in topography is that a local strain

distribution, such as a dislocation line and

isolated imperfections, appear in topographs as a

local image at the sxpposedly corresponding

region, although the image is sometimes clearly

resolved or sometimes slightly blurred. The reason why most imperfections form as a local

image is due mainly to the Borrmann diffraction

mechanism. In topography, a beam of a limited

size, or an extranely well collimated beam is used

as the incident beam. According to x-ray diffraction optics, (16) (17) the diffracted (or transmitted)

beam given by the first tern of Eq. (3-1) cones out

of the exit face of the crystal at the place vrfiere the diffracting plane meets the exist surface, as shown in Fig. 2. If an inperfeetion is located in

this path, the disruption of the Borrmann effect

takes place to produce the image of the imperfection,

due to the change of the effective absorption co- efficient. The scattering mechanism for the second term in Bq. (3-1) is, however, slightly different

from the above optics, thus creating blurred images

of inperfections

.

The schematic illustration of the image

formation is shown in Fig. 3, where the incident

beam sweeps the surface of the crystal. In this

figure, the diffracted intensity is plotted:

(a) corresponds to the intensity due to the first

terra of Eq. (3-1), and (b) to that due to the

second term. When tlie first tern is draninant, one can have a typical Borrmann topograph due to the

nechanism (a) in .this figure. Otherwise one usually

observes the stperposition of the cases (a) and (b)

.

Experimental apparatus for diffraction

For the characterization of the perfection of

sanple crystals, we use two different x-ray

optical alignments, namely an asymnetrical topographic camera and a high resolution doifcle crystal

spectrcmieter. The former, which will be called ATC

hereafter, was designed in order to enable us to survey a large nurber of sanple crytals effectively

in a shorter time period than that required for

ordinary topography. Although this was accom- plished at the expense of high resolution, the

quality of topographs from the ATC remained just

as good as in ordinary x-ray topography. (It was

one of the requirements for this design to maintain

such high quality in the topographs.) In the ATC,

the first crystal is a silicon crystal whose

surface makes an angle of 13.5° with the (111)

diffracting plane. The incident x-ray beam falls on the crystal surface almost parallel to it, and

the (111) diffracted beam appears with a size of

1.7 cm X 2.5 cm, being sufficiently large enough

to cover the entire area of sanple crystals. Care must be taken with the first crystal so that it

would not stperinpose its own surface structure on topographs of the structure of the sample

crystal. This requirement often induces an unavoidable lack of ideal high resolution in the

ATC. The schematic diagram of the ATC is shown in

Fig. 4. Topographs are taken frcsn the sanple

crystal (Cu) both in transmission geometry, lAere

the (111) diffraction of copper is used, and in

reflection geometry, where the (220) diffraction is used.

Fig. 2 Schematic of Borrmann diffraction gecanetry

OtYSTAL 1 ±^ ANOMALOUS

TRANSMISSION

^XDISRUPTION OF THE BORRMANN BEAM

r

ANOMALOUS

TRANSMISSION y BACKGROUND

Fig. 3 Schematic of image formation mechanism ana origin of contrast effects

Fig. 4 Schematic of the asymnetrical topographic

camera (ATC) illustrating both surface reflection

and transmission modes
To obtain diffraction rocking curves and high

resolution topographs at various locations on the

sample crystals, the high resolution double crystal

spectrometer (hereafter called the spectrometer)

has been used with a scanning stage mounted on it. The schematic diagram of the spectrometer is shown

in Fig. 5 vrtiere the first crystal is only shown in

reflection geometry. A silicon crystal of disc

shape vhose surface is a (110) crystallographic

plane has been chosen as the first crystal to obtain a well collimated monochronatic beam from

the (220) Bragg diffraction both in reflection and

transmission geometry. Between the x-ray source

of a point focus x-ray tdbe and the first crystal,

a horizontal slit of 6 mm and a vertical slit of

0.3 inn have been inserted. When the first crystal

is in reflection geometry, two vertical slits of

0.15 nri and 0.10 ram, respectively, are placed a distance of 3 cm apart from each other before the

second crystal. When the first crystal is in

transmission geanetry, no -slits are inserted.

curve width in transmission becomes narrower for

the same degree of perfection, vAien the crystal

disc becomes thicker. Also, the thickness data

are used in the determination of dislocation

densities to estimate the volvme of the crystal

portion where the dislocations are counted on the

transmission topographs. We measure the crystal

thickness by use of the Borrmann (anomalous trans- mission) effect of x-ray dynamical diffraction.

The thickness is calculated from the following

equation:

D = (Lq - Lg)/tan 9, (3-5)

where 9 is the Bragg angle, and L„ and are the

positions of the beam traces. In the present

work, the slit placed in the x-ray beam before the

crystal is 0.01 cm wide. A photograph of a nuclear

plate used for the determination of thickness is

shown in Fig. 6, where the (111) Bragg diffraction

was used with Cu Kai radiation.

TRANWrrTEO

f

I

mm

If

Fig. 5 Schematic of high resolution double crystal

spectrometer with scanning capability illustrating

both surface reflection and transmission modes

The quality of the beam was checked by

measuring the rocking curves of a dislocation free

germaniun crystal which was placed at the second

crystal position. The full width at half maximim

(FWW) of the rocking curve from the (220) geimaniun

diffraction was determined to be 12.8 seconds of arc in the Si (220) surface/Ge (220) surface mode

and 15.5 seconds of arc in the Si (220) transmission/Ge (220) surface mode, using Cu Ka radiation. For topography, exposure times range between 24 hours and 100 hours of transmission, and between

2 and 24 hours for reflection. The ATC requires

mLich shorter exposure times than the scanning

spectrometer. These topographs are recorded on

Ilford L-4 nuclear plates with an emulsion

thickness of 50 um.

In this work, it is also necessary to measure with sufficient accuracy the thickness of crystal

discs without touching them since the rocking

Fig. 6 Photograph used for thickness determination

Table I Fluid Flow Conditions

Group

Nmber

Crucible Rotation (Y) Seed Rotation Xz) Difference

- Y - Z2

I -5.4 rpm -6.0 rpm +0.6 rpn

II +6.0 rpm +10.0 rpm -4.0 rpm

III -20.6 rpm -20.0 rpm -0.6 rpa

IV +30.0 rpm +20.0 rpm +10.0 rpm

V -6.6 rpm +6.0 rpm -12.6 rpm

VI Experimental Results

Fluid Flow Conditions

The fluid flow conditions in the melt during

crystal growth are classified into groups by

different sets of values of the principle variables,

angular velocity of crucible and seed rotation, as
Table II X-ray spectroscopic data

Growth Condition

Neck

Diameter mm

Crystal

Slice Number

Post Growth

Treatment

Thickness

mm

(220) Surface Reflection

FWHM

(111) Transmission

FWHM

Seconds of Arc

Perfection

Ranking uL Seconds of Arc Surface Transmission

I l.A 023302 As Grown 0.40 70 cip not resolved 7 22.05

I 1.4 023303 Annealed 0.40 70 (1) not resolved 7A - 22.05

I <1 024 301 As Grown 0.6580 35 18 2 3 36. 27

I 2 025303 As Grown 0.5883 35 14 2 1 32.43

I 2 025302 Annealed 0.3488 29 14 lA 4A 19.23

^ As Grown 0 . 50 4 7 Not Observed 5 7 27 . 56U JJ Jul Annealed 0. 7549 36 11 5A 2A 41 . 60

I ~ 1 036 301 As Grown 0 . 2463 35 20 2 4 13. 58

1 036303 Annealed 0.3838 32 12 3A 3A 21.15

UZD JUI As Grown 0 . 5502 46 17 4 2 30 . 33

I I I Annealed 0 . 5629 33 11 4A 2A 31 . 03

II A UZ / JU

J

As Grown 0 . 2550 50 Multiple Peaks 6 5 14.06

I I n9 7 m1 Annea led 0. 1 791 40 15 6A 5A 9.87

III <i 028303 As Grown 0.8934 32 14 1 1 49.24

III <i 028301 Annealed 0. 7989 37 10 2A lA 44.03

III >2 030302 As Grown 0.40 43 Multiple Peaks 3 5 22.05

III >2 030303 Annealed 0.4638 31 12 2A 3A 25.56

III -1 037303 As Grown 0. 1439 35 17 2 2 7.93

III ,1 037302 Annealed 0.7101 31 11 2A 2A 39.14

IV Polycrystalline Boule

V 3 034303 As Grown 0.25 Multiple Grains

V 3 034301 Annedled 0.20 Multiple Grains

listed in Table I. By interchanging the values of y and z, we obtain A with opposite signs. There

appears to be no significant difference in the

resultant crystal perfection due to the difference

in the sign of A. Therefore, we do not discriminate fluid flow conditions from each other

when the sign of A is changed by interchanging the

values of y and z. In addition, we have introduced another parameter indicating the diameter of

the bottle-necks grown during the crystal growth

process. There are always at least two different

bottle-neck diameters chosen in each group; one is

very narrow, usually less than 1 inn and the other

between 2 mm and 4 mm.

Spectroscopic Data

In Table II, we list the values of rocking

curve widths (FWHM) observed in both transmission

and surface reflection geometry for all the growth

conditions, including annealing. The crystal is considered to be more perfect, the smaller the

observed width. The crystals are ranked, as shown

in the eighth and ninth columns, in increasing

order of their widths. The annealed crystals are ranked separately with the letter A acconpanied by

their rank. The ranks based on the transmission

data should be judged along with the value of pL,

because, even with the same degree of perfection,

thicker crystals show narrower widths.

In most crystals, the perfection is not uniform throughout the crystal. Rocking curves taken at different locations fron the same crystal

are shown in Fig. 7, where the (111) diffraction

is set in transmission geometry (yL ~ 25) . In this figure, the horizontal axis represents a decreasing glancing angle to the right. The little

hunps on the right side of the individual profiles

correspond to the Bragg diffraction due to Cu Ka2

radiation. The separation of the and 02 peaks

in the Si (220) /Cu (111) non-parallel (11) setting

is -20.8 second of arc, which is the difference

between the separation of aj and a 2 due to Si (220)

and that due to the Cu (111) diffraction.

01 1 1 1 1 1 1 1 1 • 1 1 L_ _J 1 ; , INCIDENT ANGLE

0 t., c d

Fig. 7 Diffraction rocking curves taken in trans- mission geometry for crystal #030303 at various

locations on the crystal

Typical rocking curve profiles are shown in

Fig. 8 and 9 for various crystals. In Fig. 8, the curves were obtained in both transmission and

surface reflection geometry of the copper crystal

with the first crystal, Si (220), in transmission

geometry. The horizontal axis represents an increasing glancing angle to the right. The

snaller peaks correspond to the a2 diffraction:

the separation in the reflection geometry is calculated to be +163.0 seconds of arc. Fig. 9 shows the rocking curve profiles from a crystal of

good quality (#037303) and a typical rocking curve

from a crystal of poor perfection (#035302)

.

The latter crystal did not produce observable

anomalous transmission. These profiles were

obtained with the first crystal, Si (220), in

surface reflection geometry.

X-Ray Topographs of the Grown Crystals

As mentioned previously, diffraction topographs were taken by two different x-ray
TMNSNISnON (III) SURFlCe 1220)

> —10 S«c.

10 Sac.

Fig. 8 Diffraction rocking curves in both surface

reflection and transmission geometry for crystal

#026302. In this alignment, both copper Kaj and

Ka2 radiation conponents are present with the

smaller peak due to Ka2

alignments, the ATC and the scanning spectrometer.

The topographs taken by the ATC usually gave the

entire view of the sanple crystals, unless the

crystal size was larger than the size of the x-ray

beam. Examples of the topographs are shown in

Figs. 10a and b in both the reflection and the

transmission geometry from an as -grown crystal.

Since for lack of space we cannot show all the

topographs here, we shall describe the qualities

of the crystals as- grown under various growth

conditions, based on their topographs.

The crystals grown under Growth Condition I with a narrow neck generally produced surface

topographs (obtained by a surface Bragg diffraction) almost as good as those shown in Fig. 10a,

but with a continuous distribution of darkness

(excess intensity) areas which we call "strain

contours". The transmission topographs (taken in

transmission geometry) indicated, however, much

poorer quality in those crystals, showing the

anomalous transmission only in about 50% of the

total area.

In contrast, the crystals grown in Growth

Condition I with a wider neck showed much poorer

quality in their surface topographs, in which not

only the strain contours, but many lines of

inperfections appeared. These crystals usually did

not produce good transmission topographs.

The crystals grown in Growth Condition II were

of a quality similar to those grown in Growth

Condition I with larger necks. The surface topographs showed many strain contours and often the

topograph images did not cover the entire exposed

area of the crystals. The transmission topographs

were broken ip into many areas, indicating that the

perfect crystal areas were very much limited to small local regions. These results were consistent

in this growth condition, regardless of their

bottle-neck diameters.

The crystals grown in Growth Condition III with a narrow neck generally displayed the highest

quality of perfection. Fig. 10a and b are examples

of the topographs from the crystals grown in this

condition. The surface topographs showed far

fewer crater- type images than those taken from

crystals grown under other growth conditions. The

number of visible dislocations in the surface topoCu # 037303

SURFACE REFLECTION TRANSMISSION

. (220) (III)

10 SECONDS

.Xa. *

inzi 4Q0

^ 200

INaOENT ANGLE

Cu. * 035302 SURFACE REFLECTION (220)

-t^ [^10 SECONDS

INCIDENT ANGLE Fig. 9 Diffraction rocking curves taken from a typically good crystal (#037303) in both surface

reflection and transmission geometry and a typically poor crystal (#035302) in surface reflection

graphs are found to range between 11 and 100 dislocations per cm^ for the crystals as- grown in

this condition. The transmission topographs, however, reveal more inperfeet ions in the interior of

the crystals, as shown in Fig. 10b. One of the common features in the transmission topograplis

from these crystals is shown in Fig. 11, although

this crystal happened to show this feature in a somewhat exaggerated fashion. In this topograph,

many lines run normal to the <111> (direction. The

Bragg diffraction in this case is (111). Also,

additional lines, though less visible, run normal

to the <111> direction. In addition, there appear

black and white bands parallel to the <110> direction. Those lines are caused by extended dislocations associated with a stacking fault, running

almost parallel to the <10T> direction. The lines

probably represent Lomer-Cottrell locking of the

interacting dislocations. (19) (20)

The crystals grown in this condition but with

a large neck diameter generally produced topographs

of almost the same quality of perfection as the

crystals grown in Growth Condition I with a large

bottle-neck. The transmission topographs consisted

of several regions, indicating that there were few

highly perfect regions throughout the crystals. The

crystals grown in Growth Condition IV could not be

investigated by x-ray topography, as they were not

single crystals.

Under the Growth Condition V, regardless of

the bottle-neck diameter, the crystals displayed

typical mosaic structures, as shown in Fig. 12. No transmission topographs were produced.

Now we turn to the annealing effects. As
a b

Fig. 10 ATC topographs of as-grown crystal #028303, growth condition III

a. (220) sijrface reflection canposite topograph b. (Ill) transmission topograph

Many of the topograph enlargements are composite photographs resulting from microscope limitations during

enlargement

.

<J\o> .<oo\>

*\Z-*<iii>

1mm

Fig. 11 ATC transmission topograph of as -grown

crystal #037303, growth condition III

indicated in Table II, at least one crystal from

the boiiles in each growth condition was annealed.

The surface topograph frc»n an annealed crystal of

Growth Condition V is just as bad as Fig. 12,

indicating that the mosaic structure was not affected by annealing. The annealed crystals of

Growth Condition II showed polygonization, which

usually resulted in the formation of several

subgrains

.

All the topographs taken from the annealed

crystals in Growth Conditions I and III showed an

equally high degree of perfection in both surface

and transmission geometry. The resultant degree

of perfection was almost independent of the

bottle-neck diameters. Examples of the surface

and transmission topographs are shown in Fig. 13a

and b of Growth Condition III. As seen in the

transmission topographs, the Lomer-Cottrell locks

of dislocations disappeared and dislocation net works appeared rather randomly.

The nimber of dislocations counted on the

surface topographs for these annealed crystals was

less than 23 dislocations per cm^, the smallest

number observed was 7 disl./cm^. The dislocation

illil

lil>

Fig. 12 ATC surface reflection coirposite topograph

of as- grown crystal #034303, growth condition V

densities can be obtained frcrni the transmission

topographs which show the dislocations in the

interior of the crystals. In this case, the

dislocation density is equal to the nunber of

dislocations multiplied by the length of each

dislocation divided by the volume vihere the dislocations are counted. The average number of

dislocations thus obtained was 15 dislocations per

cm^. The smallest number was 12.8 dislocations

per cm^. In these annealed crystals, there were

always subgrain boundaries observed, which were, of course, considered to be piled- dislocations,

but were not included in the count of individual

dislocations. The siisgrain boundaries would have

eventually been driven out of the crystals, if we had annealed the crystals for a longer period of

time. In the transmission topograph taken on the

scanning spectrometer, the details of the dislocation networks are clearly visible. A few examples

of enlarged topographs taken by the spectrometer

are shown in Fig. 14 a and b, representing a highly
a. b. Fig. 13 Scanning spectrometer topographs of annealed crystal #025302

(220] surface reflection composite topograph b. (Ill) transmission conposite topograph

a. Fighighly perfect region

14 Enlarged regions of scanning spectrometer topographs

b. highly dislocated region

perfect region and a highly dislocated region,

respectively.

V. Conclusion and Discussion

Before concluding this paper, we would like

to emphasize one of our principles followed in

conducting the present work. At present, many

people believe that growing perfect crystals is still an art vAere personal skills play significant

roles. Although we do not object to this view,

and often agree with it based on our experience, we have attenpted to eliminate such art from

the present work. This attitude probably

resulted in producing crystals less perfect than

they could be. However, we see significance in

establishing reproducibility of the resultant

crystal perfection as a functional of growth

conditions, rather than a functional of the "art" part of crystal growth technology. In the present

work, we grew crystals as routinely as possible,

with the least amount of human control. Once the

variables were set, we did not attenpt to modify

these variables in response to local perturbations

which occurred from time to time during growth.

During the growth process, we were tenpted more

than once to correct the diameter by changing

tenperatures or to increase the pulling or rota- tion speed when the melt surface started vibrating

occassionally. However, we made none of these

corrections

.

The spectroscopic data indicated that

Growth Condition III generally produced crystals

of higher perfection, along with crystals

produced under Growth Condition I with

narrow necks. These results are in good

agreement with the observations of the x-ray

topographs. The topographs obtained by the

scanning spectrometer further helped to distinquish

the subtle differences in crystals grown under

conditions I and III. Those topographs

indicated that Grovrth Condition III with narrow
bottle-necks usimlly produced crystals of better

quality than Condition I. The other growth

conditions are clearly inferior to Conditions I and III. It is, therefore, concluded that Growth

Condition III is most optimum for growing a single

crystal of high perfection, when the bottle-neck

is made less than one millimeter in diameter.

If one anneals crystals. Growth Conditions I and III with narrow bottle-necks result in almost

the same degree of crystal perfection. However,

crystals grown in the other conditions did not

show much improvement in quality as a result of

annealing. It is, therefore, further concluded

that annealing becomes most effective only when

the grown crystals are already highly perfect.

In the optimum growth condition, the density

of dislocations can be as low as 11 dislocations

per cm^ for "as-grown" crystals, and as low as 7 dislocations per cm^ for annealed crystals. The

lowest dislocation density measured from the

transmission topographs was 12.8 dislocations/cm^

which is in good agreement with the dislocation

densities determined from surface topographs.

As an initial approach to the study of

fluid flow effects in the melt on metal crystal

growth, we have placed anphasis on the documentation of the resultant crystal perfection under

various fluid flow conditions. The fluid flow

conditions have been described by several directly

controllable process parameters. It is obvious

at this stage that these parameters are not yet

connected to the more sophisticated basic

physical quantities as the thermodynamical

variables which directly characterize the state of fluid flow. There is, however, no unperturbing

method of directly measuring physical quantities

of fluid flow while crystals of high perfection are being grown. Any disturbance of the melt destroys

or drastically changes the resultant perfection of

grown crystals. Once the relation between crystal

perfection and fluid flow conditions as given by

the process parameters is established, one can characterize by other direct methods the fluid flow

motion under a given set of the process parameters,

regardless of the consequences on the crystal

perfection of grown crystals.

1. S. Howe and C. Elbaun, Phil. Mag. 6 (1961) 1227.

2. S.H. McFairlane and C. Elbaum, Appl. Phys. Letters I (1965) 43.

3. F.W. Young and J.R. Savage, J. Appl. Phys. 35 (1964) 1917.

4. H. Fehmer and W. Uelhoff , J. Sci. Instrim. 1_ (1969) 771.

5. E. Kappler, W. Uelhopp, H. Fehmer and

F. Abbink, "Herstellung von Kupereinkristallen

kleiner Versetungsdichte" Forschungsberichte

des Landes Nordrhein-Westfalen Nr. 2181,

Westdeutcher Verlag, Opladen.

6. C.H. Sworn and T.E. Brown, J. Cryst. Growth 15 (1972) 195.

7. W.C. Dash, J. Appl. Phys. 30 (1959) 459.

8. E. Billing, Proc. Roy. Soc. (London) A235

(1956) 37. 9. P. Penning, Philips Res. Repts. 13 (1958) 79. 10. J.C. Brice, J. Crystal Growth 2 TT968) 395.

11. A. SteinCTiann and V. Zimmerli, J. Phys. Chem.

Solids. Sippl. 1, (1967) 81.

12. J.Z. Czochralski, Phys. Chem. 92 (1917) 219.

13. F.W. Young and T.R. Wilson, Rev. Sci. Instru.

32 (1961) 559.

14. P.P. Ewald, Ann. Physik 49 (1916) 1 and 117; 54 (1917) 519. ~

15. M.v. Laue: Erg. exakt. Naturw. 9 (1931) 133.

16. M. Ashkin and M. Kuriyama, J, Phys. Soc. Japan 21 (1966) 1549.

17. M. Kuriyama, J. Phys. Soc. Japan 23 (1967)

1369, Acta Cryst. A25 (1969) 682; Zs. Naturforschung 28a"Tr973) 622.

18. G. Borrmann, Phys, Z. 42 (1941) 157,

Z, Phys. 127 (1950) 297.

19. W.M. Lomer, Phil. Mag. 42, (1951) 1327.

20. A.H. Cottrell, Phil. Mag. 43, (1952) 645.
Appendix II

J. Appl. Cryst. (1974) Vol. 7

An Immobile Dislocation Arrangement in As-grown

Copper Single Crystals Observed by X-ray Topography

by

Masao Kuriyama, James G. Early and Harold E. Burdette

Institute for Materials Research

National Bureau of Standards

Washington, D. C. 20234, U.S.A.

Abstract

X-ray diffraction topography using transmission geometry has revealed

an interesting array of extremely straight and narrow long line images in

sizable copper single crystals grown under particular growth conditions by

the Czochralski technique. These images are analyzed and elucidated by a

model of Lomer-Cottrell dislocations. The formation of these sessile

dislocations usually aids the growth of large copper crystals of high

perfection. The high-degree of perfection over the entire volume of the

crystals accounts for such macroscopic arrangements of Lomer-Cottrell

dislocations which have not previously been observed by electron microscopic

techniques

.

1. Introduction

There have been several reports on growing metal single crystals

containing regions which are totally dislocation-free (Howe and

Elbaum, 1961; McFarlane and Elbaum, 1965). In particular, copper crystals

of high perfection have been prepared and studied in detail by Young and

Savage (1964) and Fehmer and Uelhoff (1969). More recently. Sworn and

Brown (1972) , followed by Tanner (1972) , have demonstrated th^copper

crystals, although small in diameter, could be grown free of dislocations

by initially narrowing the crystal diameter at one region of the boule.*

Even with these studies, however, the documentation of the growth

conditions for the preparation of dislocation-free metal crystals still

is far from complete, unlike in the case of semiconducting crystals

(Billig, 1956; Penning, 1958; Brice, 1968; Dash, 1959; Steinemann and

Zimmerli, 1967) . Continued efforts are still needed to establish

reproducible growth conditions for the production of large,, highly perfect

metal crystals. As one such effort Kuriyama, Early and Burdette (1974)

have recently studied the relationship between fluid flow conditions and

the degree of crystal perfection during the growth of large, highly

perfect copper crystals by the Czochralski technique. In this study,

x-ray diffraction topography has revealed that one particular growth

condition results in many extremely straight dislocation lines in the

crystals. These lines run across almost the entire diameter of the crystal,

and lie in a crystallographic set of three possible slip planes for the

face-centered cubic crystal. Interestingly, these lines can also be

* The reader should keep in mind always that the term "dislocation free"

Is always dependent on the technique used to detect the dislocations.
observed with good contrast in the topographs when this particular set of

slip planes is chosen as diffracting planes. Consequently, these dislocation

lines cannot be identified simply as ordinary edge or screw dislocations.

The purpose of this paper deals with an identification of those extremely

straight line structures observed in these as-grown copper crystals.

2. Experimental Procedures

The details of the crystal growth procedures and the assessment of

crystal perfection by x-ray dynamical diffraction have been described in

a previous paper (Kuriyama, Early and Burdette, 1974). In the present paper, we

shall only describe the experimental conditions relevant to the present

work. Large copper single crystals were grown from single crystal seeds in

a vacuum furnace by the Czochralski technique. The pulling speed was

between 0.013 cm/min. and 0.100 cm/min. The melt and seed were rotated

around a common axis in the same direction with speeds ranging from 5 rpm

to 20 rpm; the relative speed of rotation was 0.6 rpm. Copper of 99.999%

purity was used as the melt charge. Crystal boules were grown by initially

narrowing the boule diameter at one region, thus forming a bottle-neck.

The grown crystals were about 6 cm long with diameters between 1.5 cm and

3.0 cm and with a bottle-neck diameter of 0.5 mm. The boules were sliced

into discs by an acid saw and the discs were polished on an acid polishing

wheel in the same manner as Young and Wilson (1961) . The discs have (110)

planes as their parallel faces, so that the four strongest diffracting planes,

(111), (111), (002) and (220), can be used in transmission geometry.

For the assessment of the perfection of sample crystals, we employed

two different x-ray optical arrangements, namely an asymmetrical crystal

topographic camera and a high resolution double crystal scanning diffractometer

.
The former, which will be hereafter called asymmetrical crystal topography

(ACT) , is equipped with a monochromator made from a silicon crystal whose

surface makes an angle of 13.5° with the (111) diffracting plane for Cu Ka

radiation. The incident x-ray beam falls on the silicon crystal surface

almost parallel to it, and the (111) diffracted beam appears with a size of

1.7 cm X 2.5 cm, sufficiently large to cover the entire area of the sample

copper crystals. In the latter arrangement, which will in the future be

called the scanning dif f ractometer (SCAD), a scanning device is attached

to a high resolution double crystal diffractometer so that the well

collimated narrow beam can scan with high resolution the entire crystal.

This enables us to obtain the profiles and widths of rocking curves at any

location in the crystal. The quality of the beam in SCAD was checked by

measuring the rocking curves of a dislocation free germanium crystal which

was placed at the second crystal position. The full width at half maximum

(FWHM) of the rocking curve from the (220) germanium diffraction was

determined to be 12.8 seconds of arc in the Si (220) surface/Ge (220)

surface mode and 15.5 seconds of arc in the Si (220) transmission/Ge (220)

surface mode, using Cu Ka radiation. This silicon crystal was used as the

first crystal or monochromator in either transmission or reflection geometry

to obtain the rocking curves and scanning topographs from sample copper

crystals.

The thickness of the copper discs was determined by the anomalous

transmission effect (Kuriyama, Early and Burdette, 1974) as shown in Fig. 1.

The thickness can be geometrically related to the distance between the images

L and L . The thinnest disc was 0.1439 mm thick and the thickest was

o s 0.8934 mm thick. The product of the ordinary linear absorption coefficient

and thickness thus ranged from 7.93 to 49.24 for Cu Ka radiation.
3. Rocking Curve Widths and Topographs

The rocking curves were obtained from the copper discs by the (220)

diffraction in the reflection geometry and the (111), (111), (111), and

(002) in the transmission geometry. For as-grown crystals, the peak

intensity of the rocking curves varied up to 30% at different locations

in the crystal. The width of the curves, however, did not vary as much.

The values (FWHM) of the rocking curve widths, averaged over the crystals,

are 34 seconds of arc for the (220) surface reflection, and 14, 13 and 12

seconds of arc, respectively for the (111) and (111), (111) and (002)

diffraction in the transmission geometry'. Typical profiles of the rocking

curves are shown in Fig. 2 for these diffracting planes.

Fig. 3 shows a topograph taken by the ACT in the reflection geometry

of the (220) diffraction. This represents the usual surface reflection

topograph taken from the crystals grown under the conditions described in

section 2. An example of the transmission topographs also taken by the ACT

is shown in Fig. 4, where the (002) diffraction was used. The crystal used

for this topograph was determined by the anomalous transmission effect to

be 0.1439 mm thick. This topograph illustrates the common features in the

transmission topographs from the crystals grown under the present growth

condition. In this topograph, many lines run normal to the [111] direction,

while another set of lines runs normal to the [111] direction. In addition,

there appear black and white bands parallel to the [110] direction. As

indicated in Fig. 4, the first set of lines is denoted by set I, the second

by set II, and the set of bands by set III.

* The rocking curve width in the transmission geometry varies as the crystal

thickness changes. The values reported here are obtained from a crystal of

yL = 7.93.

** Several of the topograph enlargements are composite photographs resulting

from microscope limitations during enlargement.
The transmission topographs obtained from the (111), (111), (002) and

(220) diffractions are reproduced in Fig. 5a, b, c and d, respectively. It

is observed in these topographs that the image contrast of the sets, I and

II, depends on the diffracting planes used. The experimental results from

these topographs can be summarized as follows:

1) The lines in the sets, I and II, are extremely straight, narrow,

and discontinuous, running almost across the entire crystal.

2) The lines in set I lie in the (111) plane; when the images of these

lines are projected on the (110) plane, the plane of the topographs,

they run parallel to the [112] direction.

3) The lines in set II lie in the (111) plane; when the linages of

these lines are projected on the (110) plane, the plane of the

topographs, they run parallel to the [112] direction.

4) In the (111) diffraction, the lines in set I have better contrast

than those in set II. In the (111) diffraction, the relation of

contrast is reversed. In the (002) and (220) diffraction conditions,

both set I and II appear to have equal contrast.

5) Set III consists of black and white bands running in the [110]

direction.

In this paper, we confine ourselves to the identification of sets I and II.

4. A sessile dislocation arrangement

In dynamical diffraction, the image contrast due to crystal imperfections

is determined by the factor {1 - exp[iH • u(r)] , where H is a reciprocal

lattice vector in a reference perfect crystal and u(r) is an atomic

displacement vector at a position r inside the crystal (Kuriyama, 1967; 1969).

In the first order approximation for almost perfect crystals, this factor

may be replaced by 3 {H • u(r)}/3a , where a is a coordinate along the x-ray
propagation direction (Kuriyama, 1970; 1972; 1973). There are two propagation

directions in a single Bragg diffraction; one is the transmitted direction

(K = 0) and the other the Bragg diffracted direction (K = H) . In x-ray

diffraction topography, the principal concern is not with quantitative

information on the imperfections, but rather, the qualitative visual impact

given by the imperfections through such information as their locations,

shapes and distributions. In a crude approximation which is suitable for

qualitative topography, the local contrast condition may be replaced by a

simpler factor H • uCf) or H ' b, where b is the Burgers vector of a

dislocation. It should be, of course, noted even in this crude approximation

that u(r) is not parallel to b. However, the crude contrast factor, ri • b,

is convenient for practical purposes.

This contrast condition first eliminates the possibility that the lines

in sets I and II are glissile dislocations of a pure screw or pure edge

character. Such dislocations lie in a slip plane, and their Burgers vectors

also lie in this plane. VJhen this slip plane is used as a diffracting plane,

there will not be image contrast for these dislocations. In contrast with

this prediction, the lines are clearly visible in the topographs under these

particular diffraction conditions, as the summarized results indicate. Results

2 and 3 state that these lines lie in the (111) slip planes and the (111) slip

planes, respectively for set II and set I. Let us first assume that these

line images are caused by dislocations. The formation of the clear images

under the particular diffraction conditions implies that the Burgers vectors,

or more correctly those significant atomic displacements associated with the

dislocations, do not lie in the dislocation slip planes. Along with
result 1, this strongly suggests that the dislocations observed in the

topographs could be dislocations interacting with each other to

form immobile lines of dislocations i.e. Lomer-Cottrell locks (Lomer, 1951;

Cottrell, 1952).

In k face-centered cubic crystal, the possible interactions of

dislocations on two intersecting slip planes have been studied extensively

(for example, Cottrell, 1964; Hirth and Lothe, 1968) . Let us consider

set II as an example. Result 3 demands that the lines of dislocations

should be either in the [Oil] direction or in the [101] direction. The

possible lock direction [110] would not be visible in the topographs,

because this direction is perpendicular to the [112] direction in which

the observed line images of set II run in the plane of the topographic images.

In order to produce the locks in the direction of [Oil], the two intersecting

slip planes have to be (111) and (111). For the locks in the direction of

[101], the (111) and (111) planes are involved. There are three possible

directions of the Burgers vectors in each slip plane. These three

dislocations in a slip plane can therefore interact with three possible

dislocations in the other intersecting slip plane. After consideration of

(i) which pairs of dislocations can be energetically stable, (ii) which of

the stable pairs can glide afterwards, and (iii) which form annihilated

screw disloactions, only two pairs are found which can form a sessile dislocation

line that runs parallel to a given locking direction. Both pairs result in

the same Burgers vector which lies in neither of the original slip planes.

For the [Oil] locking direction, the resultant Burgers vector is (1/2)

[Oil]. We denote this Lomer-Cottrell lock as
type II A. For the dislocation locked in the [101] direction, the resultant

Burgers vector is given by (1/2) [101]. This lock is denoted as type II B.

In a similar fashion, we find for set I the following Lomer-Cottrell locks;

lA runs parallel to the [Oil] direction with the Burgers vector (1/2) [Oil],

and IB runs in the [101] direction with the Burgers vector (1/2) [101].

Next we apply the approximate contrast conditions for these possible

locks to the diffraction conditions used. The results are shown in Table I.

As mentioned previously, the contrast condition resulting in H • b = 0

simply means that the image contrast under such a diffraction condition is

inferior to the contrast under different diffraction conditions. These

results are in a good agreement with the observed results 3 and 4. Since

all the dislocation lines of these locks make an angle of about 30° with

the (110) plane, they should terminate on the crystal surface. To confirm

this, topographs were taken in high resolution by the SCAD under the (220)

diffraction condition in the reflection geometry. As shown in Fig. 6, we

observe that the dislocation images on the surface topograph are indeed

aligned, though short in their length, in the exactly same way as predicted.

In Fig. 6, the images of the terminating dislocations of set II run in the

vertical direction perpendicular to the [111] direction.

In addition, the high resolution topographs were taken by the SCAD in

the transmission geometry for the (111), (111), and (002) diffraction. As

shown in Fig. 7, the line images of sets I and II are indeed very narrow.

We thus conclude that the assumption of Lomer-Cottrell locks elucidates all

the results listed in section 3 with the exception of result 5.

5. Discussion

Lomer-Cottrell dislocations have been found in face-centered cubic

crystals by electron transmission microscopy (Whelan, 1958; Mader, Seeger
and Thieringer, 1963), They appear, however, as a rather isolated local

event in the entire crystal. The length of the dislocations are short.

Unlike the observations in electron transmission microscopy, the LomerCottrell dislocations observed in the present paper have occurred almost

throughout the entire crystal, and their length is on a macroscopic scale.

The difference between the electron microscopic and the x-ray topographic

observations can be attributed to the following facts. First, the crystals

used for x-ray topography are bulk (1.5 cm in diameter and almost 1 mm thick)

compared with the thin-films used for electron transmission microscopy.

Next, due to the over-all high degree of perfection in the bulk crystals

(judged from their display of the prominent Borrmann anomalous transmission

effect), all the regions in the interior of the crystals are likely to have

similar properties with respect to each other. If the formation of a

Lomer-Cottrell dislocation is energetically favorable at one place, there is

no doubt that dislocations of a similar type would occur everywhere inside

the crystal. This phenomenon is very unlikely to occur in thin films, since

the perfection of these thin films is not as high over the entire film as

these crystals used for x-ray anomalous transmission topography. The

simultaneous use of x-ray topography with a simple metal crystal of high

perfection has, for the first time, made it possible to observe the

extensive arrays of Lomer-Cottrell dislocations on a macroscopic scale over

the entire volume of the crystals.

There are a few additional comments worthy of note concerning the

growing of sizable perfect copper single crystals. When a slice of the

as-grown crystal show the arrangements of Lomer-Cottrell dislocations over

its entire volume, as shown above, the degree of crystal perfection

is equally high for any part of the crystal boule of about 6 cm long.
In contrast, some as-grown crystals grown under different growth conditions

frequently show a high degree of perfection in one portion of the boule, but

not necessarily the same quality of perfection in other parts of the boule.

Most of these crystals tend to develop low angle grain boundaries somewhere

along the length of the boule. This fact may indicate that a particular growth

condition aids dislocation interactions to form immobile Lomer-Cottrell locks.

When this phenomenon takes place, the resultant crystal becomes highly perfect.

If widespread slip occurs or if many dislocations form randomly, as in the

case of the other growth conditions, then the locking would not take place and

the dislocations would still be mobile and affect the subsequently grown part

of the crystal, resulting in a less perfect crystal.

The work reported herein was partially sponsored by the Advanced Missions

Program of The National Aeronautics and Space Administration under Contract

W-13,475 No. 1. The authors wish to express their thanks to Dr. Roland de Wit

and Dr. A. W. Ruff of the Metallurgy Division, National Bureau of Standards

for their discussion.
References

1. Billig, E, (1956) Proc. Roy. Soc. (London) A 235 37-55.

2. Brice, J.C. (1968) J. Cryst. Growth 2^ 395-401.

3. Cottrell, A.H, (1952) Phil. Mag. 43 645-647.

4. Cottrell, A.H. (1964) "Theory of Crystal Dislocations" Gordon and

Breach, New York.

5. Dash, W.C. (1959) J. Appl. Phys. 30 459-474.

6. Fehmer, H. and Uelhoff, W. (1969) J. Sci. Instrum. 2 771-775.

7. Hirth, J. P. and Lothe, J. (1968) "Theory of Dislocations" McGraw-Hill,

New York.

8. Howe, S. and Elbaum, C. (1961) Phil. Mag. 6^1227-1240.

9. Kuriyama, M. , Early, J.G. and Burdette, H.E. (1974) Proc. of AIAA 12th

Aerospace Sciences Meeting, Paper No. 74-204.

10. Kuriyama, M. (1967) J. Phys. Soc. Japan 23 1369-1379

.

11. Kuriyama, M. (1969) Acta Cryst. A 25^ 682-693.

12. Kuriyama, M. (1970) Acta Cryst. A 2^ 56-59.

13. Kuriyama, M. (1972) Acta Cryst. A 28 588-593.

14. Kuriyama, M. (1973) Zs. f. Naturforsch. 28a 622-626.

15. Lomer, W.M. (1951) Phil. Mag. 42 1327-1331.

16. Mader, S., Seeger, A. and Thieringer, H. (1963) J. Appl. Phys. 34

3376-3386.

17. McFarlane, S.H., and Elbaum, C. (1965) Appl. Phys. Letters _1 43-44.

18. Penning, P. (1958) Philips Res. Repts. 13 79-97.

19. Steinemann, A. and Zimmerli, V. (1967) J. Phys. Chem. Solids, Suppl. 1,

81-87.

20. Sworn, C.H. and Brown, T.E. (1972) J. Cryst. Growth 15 195-203.

21. Tanner, B.K. (1972) J. Cryst. Growth 16 86-87

.

22. Whelan, M.J. (1958) Proc, Roy. Soc. A 249 114-137.

23. Young, F.W, and Savage, J,R. (1964) J. Appl. Phys. 35 1917-1924.
Table I

Contrast Factor for possible Lomer-Cottrell Locks

Directions Contrast Factor

Type of Locks Line

Burgers

Vector (111) (111) (002) (220)

. . I A [Oil] [Oil] 0

1

/T

1

2

IB r 1 m 1 [101

J

(J ^3

1

/T

1

2

II A [Oil] [Oil] 0

1

ri

1

2

II B [101] [101] 4 0

1

rr

1

2
Fig. 1 Photograph illustrating the Borrman anomalous transmission effect

in a copper crystal grown under the conditions described in the text. The

photograph indicates the parameters used to determine the crystal thickness.

Lq and Ljj are the beam images diffracted in the transmitted and the Bragg

diffracted direction, respectively. Lg is the image of a part of the lineshaped incident beam.
Si
CM

O

O

0)

a

•rlo

•H

4JO

•H

LiJ -IO

<

CMO

O

1CO

o

Id

CO

in

UJ

o

o

o

o

Jo

o

o

o

o

(U

4-1

CO • > ^-1

" i

60 01 0 60

O

O

Pi CO

CO

•rl

• PI 60 CO

•i-l u

PL4 JJ

QN003S H3d SlNflOO A1ISN31NI

Fig. 3 ACT topograph of the (220) diffraction in the reflection geometry.

Fig. 4 ACT topograph of the (002) diffraction in transmission geometry.

Three sets of images are indicated with the arrows indicating the direction

parallel to the line images of each set.

Fig. 5a ACT transmission topographs obtained from different diffracting

planes. (Ill) diffraction.

Fig. 5b ACT transmission topographs obtained from different diffracting

planes. (Ill) diffraction.

Fig. 5c ACT transmission topographs obtained from different diffracting

planes. (002) diffraction.

Fig. 5d ACT transmission topographs obtained from different diffracting

planes. (220) diffraction.

<00l>

-<lll>

<IIO>

<lll>

I mm

Fig. 6 SCAD topogra^phs taken in the reflection geometry. This magnified

topograph of the (220) surface diffraction shows the images of dislocations

terminating on the surface. Although short in length, they are aligned in

the same direction as observed in Fig. 4.

Fig. 7 SCAD transmission topograph. This magnified topograph of the (111)

diffraction illustrates that the line images of sets I and II are very narrow.
Task 2

Evaporative Preparation of Ultra-High Purity Materials

R. C. Paula

Inorganic Materials Division

; W. J. Boettinger and F. S. Biancanlello

Metallurgy Division

Institute for Materials Research

Summary

Techniques for the ultrapur if ication of materials by high temperature

evaporation of volatile contaminants have been developed with emphasis on

containerless processing. Calculations of multicomponent evaporative

processes have previously indicated that high degrees of purification can

be obtained for selected systems. The calculations and computer programs

have now been simplified, and diagnostics and internal checks have been

added

.

An experimental examination of the evaporative purification calculations

has been started. An RF induction levitation system has been developed to

generate data on purification rates and initial experiments on Molybdenum

are reported. A modulated beam mass spectrometr ic facility has been built

to continuously measure evaporative purification processes as a function of

time. Both of these facilities will be used for a detailed evaluation of

the purification calculations.

- 24 -
Introduction

This Task involves the development of techniques for the ultrapurification of refractory ceramics and metals by high temperature evaporation of

volatile contaminants. The zero-gravity environment of space will be used

to allow the containerless melting and evaporation of relatively large

samples, and to provide high vacuum. Last year's efforts (NASA Contract

W-13,475 #1)''" were largely devoted to the development of calculations to

predict evaporative purification rates and to indicate promising chemical

systems for further investigation. This has now been followed by more

detailed investigations into the basis of evaporative purifications.

The calculations, which have indicated highly successful evaporative

purifications, are based on a number of assumptions. While these assumptions

are quite reasonable, they still must be critically evaluated. The calculational assumptions are that:

(a) all significant vapor species have been considered,

(b) the thermodynamic constants are correct,

(c) ideal solution behavior is obtained,

(d) the evaporation coefficients are unity or near unity, and

(e) there is adequate mixing of the melt to avoid formation of

concentration gradients during the evaporative purification.

Assumptions a. through d^ are being critically examined under this Task.

Assumption e_ is being examined separately under Task #6 of this contract

(Surface Traction and Other Surface Pehnomena)

.

Real chemical systems are expected to have minor deviations from the

above assumptions. Such minor deviations are relatively unimportant and can

be accounted for. We must, however, be sure that there are no gross

- 25 -
deviations that will be detrimental to the purification process.

We have now embarked on an extensive experimental program to further

examine our calculated results. Our experimentation has taken two paths.

First, we have used RF induction coils to levitate and melt metal samples

(primarily Mo) , and by initial and final chemical analyses to measure

evaporative purification rates. Data collected from such experiments will

be used to test the calculations for overall accuracy for small samples

which were truly processed without a container. Secondly, we have designed

and constructed a modulated beam mass spectrometric facility to continuously

measure species evaporating from a molten sessile drop (initially AI2O3).

Such experiments will provide detailed analysis of the specific assumptions

of the calculations mentioned above.

Work has also continued toward the simplification of the evaporative

purification calculations, and to make the computer programing easier for

others to use.

Our overall goals are to conveniently and accurately predict purification rates and ultimate purifications. This includes a description of the

complex chemistry normally found in real purification systems. It is also

our goal to accurately predict optimum materials and conditions for purification.

Levitation Experiments

Electromagnetic levitation with an RF induction coil is one of the best

techniques available to study the evaporative purification of metals and

alloys. Contamination by containers or unmelted or unheated parts of the

sample is absent. Samples are rapidly heated to temperature. Liquid

samples are stirred by the magnetic field and a relatively large surface

to volume ratio is available for evaporation.

- 26 -
The state of the art of levitation melting has been reviewed by several

2 3 4 authors.' * The principle of electromagnetic levitation Involves the

creation of a spatial minimum of the magnetic field produced by a suitably

designed radio frequency Induction coll. This field Induces eddy currents

In a small metal sample placed In this field. The Interaction of these

eddy currents and the magnetic field cause a force to be exerted on the

sample which will suspend It freely Inside the levitation coll. These

Induced eddy currents also heat the sample. Because both the heating and

the lifting are controlled by the strength of the magnetic field, it is not

usually possible to control the lifting and heating Independently. Hence,

If one establishes a magnetic field which will adequately levitate the sample

there is usually no Independent control over the temperature. The temperature

of the sample will adjust to some value determined by the power input to the

sample and the heat lost by the sample.

Many metals and alloys have been levitation melted successfully in inert

atmospheres with few problems. However in the past, a vacuum environment has

posed severe limitations on the usefulness of levitation melting for quantitative studies of evaporative purification. Under vacuum conditions, many

metals vaporize excessively. Metals with sufficiently high vapor pressures

are often either difficult to melt or once melted tend to drip from the bottom

of the levitated charge. In addition, arcing between the sample and the coil

or between adjacent coll turns has often limited the performance of electromagnetic levitation in vacuum. The purpose of the initial phase of this

research was to determine whether these problems could be overcome at least

for some materials so that meaningful purification experiments can be

performed using electromagnetic levitation.

- 27 -
Levitation System - The basic equipment system used in this research

was a commercial levitation melting facility which is part of the Metallurgy

Division Specimen Preparation Laboratory. Briefly, it consists of the

following: a 450 KHz, 20 kW radio frequency generator with an air core

12:3 transformer to improve generator and levitation coil impedance matching;

a coaxial power feed through a vacuum collar; a levitation coil which will be

described later; and a vacuum chamber with supporting vacuum system capable

of a vacuum of 4 x 10~^ Pa (3 x 10~^ torr). Photographs of the system are

shown in Fig . la and lb

.

Choice of metals to be studied and the design of a suitable levitation

coil were the first tasks undertaken. In previous work (in our laboratory)

using conventional funnel shaped coils (Fig. 2a), it was determined that 4 gm

samples of the metals Mo and Nb could be easily levitated but could not be

easily melted. In addition, these metals were good candidates for evaporative

purification with expected eliminations of many metallic impurities and also

of interstitial oxygen^ Efforts were directed toward designing a levitation

coil which would increase the power input to the sample, and hence permit

these metals to be melted and held at temperatures just over their melting

points.

Many of the conventional levitation coils reported in the literature

have been designed on the basis of alloy preparation. They have large diameter windings on the top so that specimens can be easily dropped into the

coil with tongs without venting the vacuum system. They also have a lot of

space in the interior of the coil compared to the sample size so that

oscillating specimens do not hit the coil. According to a report by Hulsey,

these coils have rather low coil gradients (gradient of the magnetic field

- 28 -
per unit coil current along the axis of the coil). Hulsey's experimental

data on the power (P ) developed in a levitated sample as a function of the

square of the coil current times the square of the coil gradient for various

coil designs are shown in Fig. 3. Hulsey indicates that conventional coils

(low coil gradients) typically operate in the region of the curve with

negative slope and have relatively low power input to the sample. That

conventional coils operate in this region of the curve is supported by the

observation reported in the literature that the temperature of a levitated

sample can be raised by lowering the coil current. It is obvious that much

more power can be developed in a levitated sample in a coil with a high coil

gradient.

Following this suggestion, a levitation coil was designed with as high

a coil gradient as was feasible. This levitation coil has identical

opposing sections (wound in the opposite sense) consisting of three turns

each with as small a diameter as was practical for a 4.5 gm sample (sphere

of radius 5 mm). This coil is shown in Fig. 2b. It is wound with 3.175 ram

diameter copper tubing with 0.5 mm wall thickness. Using this coil it was

possible to levitate and melt Nb and Mo at stable temperatures just over

their melting points for up to five minutes.

It was found to be mandatory to omit a refractory coating on the coils

and to clean them carefully after each levitation run to prevent arcing of

the coils.

The size of the sample was found to be very critical to the success of

the levitation melting. This is not surprising because with this coil design

the magnetic field changes drastically over the diameter of the sample.

Molybdenum samples greater than 5 gm would drip from the bottom during

levitation and samples less than 4 gm would not melt.

- 29 -
It should be noted that with this coil samples could be levitated while

remaining solid and then be melted by increasing the coil current. This

indicates that the coil gradient of this coil is large enough to be operating

on the part of Hulsey's curve (Fig. 3) which has positive slope. As mentioned

earlier, this gives increased power efficiency.

Evaporative purification of Mo - Molybdenum samples nominally 4.5 gm

were cut from a single rod of diameter 8.4 mm. Two samples from different

parts of the rod were retained for chemical analysis. Purification

experiments were performed in both the liquid and solid state using the

electromagnetic levitation system just described and are summarized in

Table 1.

For a solid state purification experiment, a conventional coil design

was used (Fig. 2a). Ai sample was dropped into the coil with the power on

with tongs operated from outside the vacuum system. Samples processed in

this manner were heated to a maximum temperature estimated to be about

2500° +100°C in about one minute. These samples were held at temperature

for the times shown in Table 1 and were dropped into cold copper molds by

cutting the power.

For a liquid state purification experiment the new coil design

previously discussed was used (Fig. 2b). A sample was placed in the

levitation coil and turned so that it rested directly on the lower turns

of the coil. The system was evacuated and with the power control set to

a maximum, the RF generator was turned on. Samples processed in this manner

were immediately levitated, became completely molten in one minute, and were

estimated to be heated to maximum temperatures just over the melting point

(2623°C). The fact that samples were molten was determined by visual

- 30 -
Analysis

(ppm

by

weight)

o 1 m in 00 1 CO in iH O 1 CO -*

iH iH

«o O .H o o

CM CM in o o o

o o o

V V

<0.02 <0.02 <0.02 <0.02 0.02

<0.02 <0.02

0) o

u-1 to o o

o o

V

in

.H <N o o o o

<0.05 <0.05 <0.05 <0.05 <0.05 <0.05 <0.05

>

iH ,H O O

in

tH O iH o o o

<0.02 <0.02 <0.02 0.2 <0.02 <0.02 <0.02

u

t-4

CM CM ° v*CM CM in o o d <0,2 <0.2 <0.2 <0.2 .5

<0.2 <0.2

z

CM CM O O

V V

CM CM CM odd V V V

CM CM CM CM CM O O O CM O O O

V CM V V V V

CM iH O O

in in O O CM o o o

0.05 0.05 0.02 0.05 0.02

<0.02 0.05

c

lA lA d d

CM in

iH o o

<0.02 <0.02 <0.02 0.5

0.05

<0.02 <0.02

m<

u-i CM d d

tH CM

tH O O

<0.02 0.05 0.1 0.2 <0.02 0.05

<0.02

H

to in o o

CM CM m

o o o

CM tH tH iH m tH CM O O O O O O O

V

CB o

in in o d

in CM in o o o

m CM CM

tH .H o in in o o

o o o o o o o

CM tH m CM o m

d d d

m m rH CM o m o o o m o o o CM V

CM CM

in

CM O CM

CM in tH CM CM

tH O O tH O O O

V

3u m CM m

tH O CM CM CM CM O CM iH CM

CM

uo m CM in tH in m m tH iH m o tH o • . . O O O tH O O O

o m rH o o o

tH iH tH o o o o o o m tH rH tH tH m ,H

•H

C/i o o r-f iH m m in

CM CM CM CM CM

tH O O CM O O O

e O O CM tH m m oCM

CM CM H CM iH H

O O O O O O O

tH V

tu o o CM CM o o o

iH tH iH m m

CM O O O in tH iH in

Time at

Temp.

min 1 1 tH m O

CO

iH CM CM CO in m m

tH

fl S

•H 3 U 3

^

u

o

iJ

i 1 O O O

tH tH X X X

CO r^ CM

vD o vo \o r^

1 1 1 1 1 1 1 o o o o o o o rH tH tH tH tH tH tH X X X X X X X

cn tH m tH CM CO 00

•H O C (0 M > to Oh 1 1

in m

o o o

tHXXXiH iH <r c;^ CO

<r m .3- m m

1 1 1 1 1 1 1 o o o o o o o iH iH tH tH tH tH tH X X X X X X X <3" tH tH CO ^ ^

Initial Mass gm

1 1

3.9626

/4.0715 4.0175 4.9322 4.8142 4.5133 4.5075 14.5126

4.5172 4.4745

Specimen No. .H CM M (H IET3TUI

iH CM CO

C/3 CO CO a3B3S PTTOS

tH CM ro m vD

hJ ,J ,J ,J _5 J

33B3S PTn^Ti
observation of the shape change of the samples. Accurate measurements of the

temperature have not yet been made. The samples were held molten for the

times shown in Table 1 and were cast into cold copper molds by cutting the

power

.

Samples were chemically analyzed by a commercial firm using spark source

mass spectroscopy for the metallic impurities and inert gas fusion analysis

for the oxygen impurity. Accuracy claimed by the firm is a factor of 2 for

the metallic impurities and +3 ppm for the oxygen impurity. Data for all

detected metallic impurities are given in Table 1 for the various experiments

performed. Exceptions are the metals Ti, Mg, Na, Cd, and Co whose analyses

were subject to interference. Data for the oxygen content are also given

for all but three experiments for which analysis was not performed.

The data for sample L4 would appear to be completely erroneous and at

the present time, no explanation is available. The other liquid state

purification samples show purifications of 95% of Fe, Sn, Si, and others.

The solid state purification samples did not exhibit such purifications due

to the lack of mixing. Within the different groups of these preliminary

experiments - solid state or liquid state - there appears to be little

correlation between impurity level and time. It is quite possible that the

scatter in the data has obscured this correlation. An important part of

future work on this subject will be to determine the source of this scatter.

Within the experimental accuracy of the chemical analysis, the oxygen

level of processed samples is essentially unchanged. The starting level of

the oxygen impurity is so low that the amount of oxygen present in processed

samples is probably determined by the partial pressures of the residual gases

in the vacuum system. Further experiments will attempt to monitor these

- 32 -
partial pressures and to perform experiments with initially higher levels

of oxygen.

Successful levitation has only been accomplished in the last stages of

this year's work. Hence a detailed analysis of the purifications has not

been performed for this report. Significant purifications of metallic

impurities have been obtained using electromagnetic levitation and further

work will emphasize the generation of reproducible data so that evaporation

rates can be obtained.

A New Mass Spectrometric Facility

A new modulated beam quadrupole mass spectrometric (M.S.) facility is

being developed to continuously observe evaporative purification processes

as a function of time. A high sensitivity modulated beam mass spectrometer

is needed to allow the detailed investigation of the specific assumptions

used in the purification calculations. Our new M.S. facility is now nearly

completed. All designs for the system have been completed, all parts

obtained, and the vast majority of the system assembled. The system is shown

in Figures 4 and 5. A summary listing the current state of the system

components is given below.

Component Designed Parts Obtained Assembled

Vacuum pumps, valves and traps X X X

Vacuum

Electronic

envelope

and vacuum

and frame

safety

XXX

Bake-out

protect

and

system

LN refill system

XXX

X X

M.S.

M.S.

Oscilloscope

Chopper

head

electronic

(beam

(mass

sweep

modulator)

consoles

filter)

drive XXX XXX

X

XXX

X

Furnace

Chopper drive and electronics XXX

X X

- 33 -
Several points regarding the design of the M.S. system are worthy of

note. The vacuum enclosure is designed such that the low pressures of noncondensable background gases will be the same in the furnace and in the

mass filter regions. The conventional diaphragm between these two regions

has been eliminated. This should help minimize background signals when

using the modulated beam detection system. If for any reason this innovation

is not satisfactory, a diaphragm can be easily inserted between the two

flanged regions. The vacuum enclosure can then be differentially pumped

in the conventional manner

.

The large tube stainless steel vacuum enclosure is designed for rapid

and efficient pumping, and is fully bakable. Low pressures of the order of

10"^ Pa (10~^ Torr) will be routinely attainable. The frame for the system

has been kept to a minimum to allow easy access.

Except for the molecular beam aperture, the furnace is surrounded by

an optically dense, but vacuum pumpable, water cooled jacket. The jacket,

plus the use of a very small tungsten filament furnace should minimize

outgassing problems due to heat build-up in the system.

The quadrupole mass filter has been positioned to the uppermost portion

of the vacuum chamber so as to allow a maximum separation between the mass

filter and the molecular beam chopper. This was accomplished by offsetting

the mass filter on its flange and by use of an eccentric conical expansion

section in the vacuum wall. The large separation between the mass filter

and the chopper is needed to obtain an adequate drift time and phase shift

of the chopped molecular beam. Examination of phase shift information allows

one to identify the neutral evaporating species in the molecular beam.

- 34 -
Measurements of phase shifts differentiate between parent and fragment ions

which are formed in the M.S. ion source.

The use of a modulated (chopped) beam detection system is advantageous in

that it allows one to ignore unmodulated signals coming from background gases

in the mass spectrometer. By minimizing the effect of background gases,

modulated beam detection allows a greater sensitivity to the evaporating

species which are of interest.

As indicated by the table above, assembly of the mass spectrometric

facility is nearly complete. We are currently testing the vacuum system

and making preliminary checks on the electronics.

Initial mass spectrometric measurements will be made on the evaporating

species coming from a sessile drop of molten AI2O3. Recent calculations by

A. L. Dragoo, under Task #6 of this contract, have indicated that molten

AI2O3 at 2500 K should exhibit good Marangoni mixing. The melt circulation

time is calculated to be "'lO seconds when using a temperature gradient of

only 1 K/cm. AI2O3 should be one of the candidate materials considered for

future space purifications.

Evaporative Purification Calculations

Our computer programs for the evaporative purification calculations are

considered to be working research tools. We have accordingly modified and

improved the programs as follows: (1) programing and symbolism have been

greatly simplified, (2) extensive program notes have been added as well as

information on requirements of the input data, and (3) checks have been added

for satisfaction of stoichiometric and equilibrium constant relationships,

and diagnostic notices are printed if these relationships are not satisfied.

With the exception of the thermodynamic data, the programs now check the

- 35 -
input data for correctness. A subtle partial redundance in the input data

allows the programs to do this.

We have received and complied with a request for aid from

Dr. Franklin F. Y. Wang, Department of Material Science at the State University

of New York at Stony Brook. Dr. Wang, a NASA subcontractor through the Jet

Propulsion Laboratory, requested that we send him a copy of the computer

program for our vacuum evaporative purification calculations. We have sent

him the program, along with test data and results, and have given assistance

in the establishment of the program at Stony Brook. Dr. Wang plans to

examine the purification of Si02 containing B2O3 impurity. We will be

collaborating in this work.

We have also received a request for assistance from Gerald Wouch of

General Electric Space Sciences Laboratory. General Electric, a NASA

contractor, is interested in the evaporative purification of tungsten.

Wouch will be obtaining analytical chemical analyses of initial and final

impurity concentrations in tungsten samples which have been given different

evaporative purification treatments. We have agreed, when this information

is available, to use our computer programs and calculations to look at the

results. This should be of interest both from the viewpoint of examining

the experimental results and the theoretical calculations.

Reports

A talk and paper entitled, Ultrapure Materials; Container less

Evaporation and the Roles of Diffusion and Marangoni Convection , was

presented at the AIAA 12th Aerospace Sciences Meeting in Washington, D.C.

A preprint of the paper is included in Appendix I of this project

report. The paper is being submitted for the symposium publication.

- 36 -
The paper entitled, Calculation of Complex Equilibria Involving

Vaporization into Vacuum , has been accepted for publication in the

December 1974 issue of High Temperature Science . A preprint of this

paper was included in last year's Annual Report.

Status and Conclusions

An electromagnetic levitation system has been developed to perform

quantative evaporative purification experiments. Significant containerless

purification of Molybdenum has been obtained using this system. A modulated

beam mass spec trome trie facility has been built to examine the specific

assumptions involved in the purification calculations. Both of these

experimental facilities will be used to further examine purification rates

and evolve optimum conditions and materials. Programing of the evaporative

purification calculations has been simplified and improved.

- 37 -
Ref er ences

1. R.C. Paule, NBSIR 73-402, 33 (1973).

2. W.A. Pelfer, J. Metals, 17_, 487 (1965).

3. G. Comenetz and J.C.R. Kelly, Jr., "Containment of Hot Matter", in

High-Temperature Materials and Technology , I.E. Cambell and

E.M. Sherwood, Ed., 600, John Wiley & Sons, New York (1967).

4. R.F. Bunshah, "Melting, Casting, and Distillation Techniques", in

Techniques of Materials Preparation and Handling , R.F. Bunshah, Ed.,

801, John Wiley & Sons, New York (1968).

5. L. Brewer and G.M. Rosenblatt, Trans. Met. Soc. AIME. , 224 , 1268 (1962).

6. W.J. Hulsey, AEC Rept. Y-1413, (1963).

7. B. Harris and A.E. Jenkins, J. Sci. Instr., 36, 238 (1959).
o u

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to 0)

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Appendix i

ultrapuke materials: containerless evaporation and

the roles of diffusion and marangoni

convection'*'

Alan L. Dragoo and Robert C. Paule

Institute for Materials Research

National Buxeau of Standards

Washington, D. C. 20234

Abstract

Contamination from containers is a major

problem in preparing ultrapiire refractory

materials. Space with its zero gravity and its

high vacuum offers an opportunity for containerless purification of these materials. The

evaporation of impurities from a melt will involve

many complex chemical equilibria. Thermodynamic

calculations have been modified to describe these

equilibria when impurities in the melt evaporate

into vacuum. The contributions of diffusion and

MauTeUigoni convection to mass transfer rates in

the bulk liquid have been estimated. Calculations

for the evaporative purification of molten alumina

are given.

I . Introduction

Hot containers used in the production and

purification of refractory materials are a common

source of contamination. Space with its zero gravity and its high vacuum offers an opportunity

for containerless purification of refractories.

The distillation of volatile impurities from

a melt is one means of purification. In this

paper, we will describe a technique for estimating

distillation rates. This technique begins with

a phenomenological expression for the magnitude,

Jj^, of the evaporative flux of component i from

the melt. That is,

= a^c^ , (1)

v^ere aj^ is the rate parameter and c^ is the molar

concentration at the surface of the melt. Eq. (1) can be rewritten in terms of the mole fraction x^ by using the relation Cj^ = ctot'^i' where c^ot

the sum of the concentrations of all components

of the melt. The rate parameter aj. depends upon

the chemical environment in the vicinity of the

surface of the melt.

Complex chemical equilibria occur within an evaporating refractory since numerous minor impurities are present and since the high temperatures employed enhance the reaction rates. The

general formalism for the calcu^tion of complex

equilibria previously was modified by one of us^^' to describe evaporative purification. Four

classes of evaporation problems were investigated.

The evaporating gases were considered under

conditions of (1) constant pressure, (2) constant

volume, (3) vacuum, and (4) partial vacuixn with a low-pressure of an oxidizing or reducing gas. The two conditions (3) and (4) are of particular

interest since they allow for the greatest isolation of the sample and the least chance of sample

contamination by the container.

The purification rate of a melt is mautimized

when the melt is well stirred. The stirring

process brings impurities to the surface where

they can be expelled from the system by vaporization. A well-stirred melt also offers the

mathematically simplest condition for the calculation of the mole fraction of each component in

the liquid at selected times. If a plot of mole

fraction of a component against time is made for

a short duration, we can obtain an effective rate

pareuneter for that coir^xsnent. When the

elapsed time is large, we generally find that the rate parameter is a function of time. The

results of the calculation of the mole fractions

at given times will be the subject of the first

portion of this paper.

An actual melt may not be well-stirred in

which case the rate of purification will be less

than in the "ideal" melt described in the previous

paragraph. In addition to the transport of

material by convection which also occurs in the

well-stirred melt, the nonhomogeneous concentrations in the actual melt will give rise to diffusion. Thus, the mole fraction x^ of

component i at the surface will depend upon its rate of evaporation and upon its rate of transport

to the svirface by both convection and diffusion

in the melt.

Our solution of the convective diffusionevaporation problem for the concentration of any

one of the constituents of the melt will be

discussed in the second part of this paper. In this problem, convection will be assumed to be

driven by surface traction, commonly known as the

Marangoni effect. The effective rate pareimeters

for evaporation which are required in the

convective diffusion problem will be obtained

from the results of the complex equilibria

calculations described in the first part of this

paper. The use of the effective rate parameters

will allow us to treat the melt as a binary

system in which only one of the components is

volatile.

II. Calculations of Complex Equilibria

There are many articles in the literature

dealing with the calculation of complex equilibria. These calculations which involve

computer programming cannot be used to solve our problem without extensive modifications. We have

therefore developed otir own equations emd computer

Financial support was provided by NASA under contract W-13,475 # 1.
programs for solving the problem of vaporization

into vacuum. Modifications and adaptations were

made to Kandiner and Brinkley ' s general formalism for the calculation of complex equilibria.

In simplest terms, theii formalism involves the use of a series of mass balance equations for the

independent species of rhe system and a series of

equilibrium constant equations for the dependent

species. The equilibrium constant equations,

in turn, involve relationships between the masses,

the pressures and the activities of the various

species. To describe the evaporation of a melt

into vacuum we have modified the formalism using

the Knudsen equation end the ideal solution

theory. The calculational approach is quite

general and allows for' relatively easy and direct

modifications to describe a variety of situations

of practical interest,

III. Results

Our calculations show that good evaporative

purification of refractories in vacuum may be

obtained. Table I shows the results of calculations for AI2O3 evaporative purification in vacuum at 2400 K, where 10 ppm (mole basis) FeO

and 10 ppm Si02 impurities are initially present. Row 2 of the table shows that the rate of AI2O3

loss is very low, while rows 3 and 4 show that FeO and Si02 are rapidly lost from the melt. The FeO and Si02 concentrations are approximettly

halved after only one second. The remaining

12 species in the table, under the headinc "Moles

of Species Present," represent the moles o." gases

vaporized during each time interval (0-0.005

seconds, 0.005-0.25 seconds, 0.25-0.50 seconds,

etc.). Finally the instantameous pressures of

all species are reported at the bottom of the table. It is evident that the calculationr yield

considerable information. The computer results,

as indicated by this table, are obtained fx :j.'n the solution of each problem.

High rates of purification can result m a depletion of impurities at the surface and

concentration gradients can occur. Homogeniety

of the final melt, however, is highly desired.

This can be accomplished by slowing the purification rates to allow time for better mixing.

The slower rates of purification also allow time

for better control of the experiment.

Table I. Vacuum Evaporation of Al 0 + 10 ppm FeO and 10 ppm SiO (2400 K)

Moles of Species Present

Elapsed Time,

seconds .005 • .250

Liquids

l.OOOOE+00 l.OOOOE+00

9.9821E-06 9.1660E-06

9.9667E-06 8.5491E-06

Si02

Gases

0 1.8199E-07 8.7991E-06

O2 6.1007E-09 2.9106E-07

Al 8.9146E-08 4.4565E-06

AlO 1.2000E-08 5.9193E-07

AIO2 6.662BE-10 3.2430E-08

AI2O 1.8579E-09 9.3499E-08

AI2O2 1.0292E-11 5.1107E-10

Fe 1.7815E-08 8.1233E-07 FeO 8.2998E-11 3.7345E-09

Si 2.2587E-13 9.7506E-12

SiO 3.3262E-08 1.4168E-06

Si02 2.0138E-11 8.4641E-10

Pressure of Species Present (Atm)

.500 .750 1.000

9 .9999E- 01 9, 9999E-•01 9 .9999E-•01 8 .3941E-06 7. 6801E-•06 7 .0209E-•06 7 .2988E- 06 6. 2214E-•06 5 .2953E-06

8 .8694E-•06 8. 7728E-•06 8 .6875E-06

2 .8981E-07 2. 8354E-•07 2 ,7805E-•07 4 .6317E-•06 4. 7084E-•06 4 .7779E-•06 6 .0772E-•07 6. 1106E-•07 6 .1405E-07

3 .2890E-08 3. 2711E-•08 3 .2551E-•08 9 .7772E-•08 9. 9937E-•08 1.0191E-07

5 .2792E-10 5. 3374E-10 5 .3897E-10

7.6845E-•07 7. 1083E-•07 6 .5620E-07

3.4897E-09 3. 1929E-•09 2 .9189E-•09 8 .7050E-12 7. 5844E-12 6 .5827E-12

1 .2495E-06 1. 0768E-•06 9 .2550E-07

7 .3738E-•10 6. 2853E-•10 5.3498E-10

Gases

0 3.5140E-06 3 .4673E-06 3 .4252E-06 3 .3879E-06 3 .3549E-06

02 1.6659E-07 1 .6220E-07 1 .5828E-07 1 .5485E-07 1 .5186E-07

Al 2.2352E-06 2 .2804E-06 2 .3227E-06 2 .3612E-06 2 .3960E-06

AlO 3.7976E-07 3 .8230E-07 3 .8465E-07 3 .8676E-07 3 .8866E-07

AIO2 2.4700E-08 2 .4536E-08 2 .4386E-08 2 .4253E-08 2 .4135E-08

AI2O 7.5013E-08 7 .7043E-08 7 .8953E-08 8 .0701E-08 8 .2292E-08

AI2O2 4.6061E-10 4 .6680E-10 4 .7255E-10 4 .777SE-10 4 .8244E-10

Fe 6.4267E-07 5 .9806E-07 5 .5444E-07 5 .1287E-07 4 .7346E-07 FeO 3,3961E-09 3 .1185E-09 2 .8559E-09 2 .6129E-09 2 .3887E-09

Si 5.7788E-12 5 .0910E-12 4 .4542E-12 3 .8808E-12 3 .3683E-12

SiO 1.0661E-06 9 .2679E-07 8 .OlOOE-07 6 .9028E-07 5 .9330E-07

Si02 7.5356E-10 6 .4638E-10 5 .5185E-10 4 .7039E-10 4 -0037E-10
Table II. Vacuum Evaporation of Al^O^ + 10 ppm CaO (2400 K)

Moles of Species Present

Elapsed Time, 0.005 20.00 40.00 60.00

seconds

Liquids

AI2O3 1 .OOOOE+00 9,.9975E-01 9 .9950E-01 9.9925E-01

CaO 9 .9998E-06 9,.2986E-06 8 .6465E-06 8.0401E-06

Gases

0 1 .6143E-07 6 ,. 4550E-04 6 .4548E-04 6.4535E-04

O2 4 .8005E-09 1 .9197E-05 1 .9193E-05 1.9189E-05

Ca 1 .8850E-10 7 .0120E-07 6 .5210E-07 6.0643E-07

Al 1 .0670E-07 4..2676E-04 4 .2671E-04 4.2665E-04

AlO 1 .2741E-08 5 ,.0957E-05 5 .0949E-05 5.0941E-05

AIO2 6 .2753E-10 2,.5097E-06 2 .5092E-06 2.5088E-06

AI2O 2 .3611E-09 9,.4435E-06 9.4424E-06 9.4413E-06

AI2O2 1 .1602E-11 4,5402E-08 4 .6396E-08 4 .6389E-08

Pressure of Species Present (Atm) Gases 0 3 .1171E-06 3 , 1170E-06 3 .1169E-06 J . iiDoE-Ob

O2 1 .3109E-07 1 OTA Dtt 1^ "7 1 .3107E-07 1 ^TACtti A"7 Ca 5 .7606E-09 c 0 JoXll U J 4.9838E-09 ^ . D J -) jr(~Ui7 Al 2 .6754E-06 2 .6756E-06 2 .6757E-06 2.6758E-0G

AlO 4 .0321E-07 4 .0322E-07 4.0323E-07 4,0323E-07

AIO2 2 .3264E-08 2 .3264E-08 2 .3263E-08 2.3263E-08

AI2O 9 .5332E-08 9 .5338E-08 9 .5343E-08 9.5348E-08

AI2O2 5 .1926E-10 5 .1927E-10 5 .1929E-10 5.1930E-10

Table III. Evaporation of Al 0 + 10 ppm CaO, Vacuum + 10 Atm Oxygen (2400 K)

Moles of Species Present

Elapsed Time , 0.005 60.00 120.00 180.00

seconds

Liquids

1 .OOOOE+00 9.,9984E-01 9.9967E-01 9.9951E-01

9.9999E-06 9.,3413E-06 8.7259E-06 8,151lE-06

Gases Ca 5.8757E-11 6,,5868E-07 6.1532E-07 5.7482E-07

Al 1.8569E-08 2,,228lE-04 2.2278E-04 2.2276E-04

AlO 7 .1134E-09 8,,5352E-05 8.5343E-05 8.5333E-05

AIO2 1.1240E-09 1,,3486E-05 1.3485E-05 1.3483E-05

AI2O 2.2941E-10 2,.7526E-06 2.7523E-06 2.7520E-06

AI2O2 3.6165E-12 4,.3393E-08 4.3388E-08 4.3384E-08

Pressure of Species Present (Atm)

Gases

0 l.OOOOE-05 1,.OOOOE-05 l.OOOOE-05 l.OOOOE-05

O2 1.3491E-06 1,,3491E-06 1.3491E-06 1.3491E-06

Ca 1.7956E-09 1,.6776E-09 1.5674E-09 1.4644E-09

Al 4.6560E-07 4 .6560E-07 4.6560E-07 4.6560E-07

AlO 2.2512E-07 2..2512E-07 2.2512E-07 2.2512E-07

AIO2 4.1669E-08 4 .1669E-08 4.1669E-08 4.1669E-08

AI2O 9.2626E-09 9,.2626E-09 9.2626E-09 9.2626E-09

AI2O2 1.6186E-10 1 .6186E-10 1.6186E-10 1.6186E-10

T
Better experimental control would be particularly important if one wished to produce doped

materials through selective vaporization processes.

The subject of controlled doping, however, will

not be discussed in this paper.

Slower purification of AI2O3 can be obtained

by exposing the melt to a low ambient oxygen

pressure. The pressure can be kept low enough

to maintain vacuum (Knudsen) flow conditions

and yet can be high enough to shift chemical

equilibria and slow the vaporization rates.

Tcibles II and III list results for the evaporative

purification of AI2O3 containing 10 ppm CaO. Table II is for the "pure" vacuum case and Table

III is for a "vacuum" containing 10~^ atm oxygen. The higher oxygen pressures shown in Table III shift the equilibrium away from the reduced vapor

species Ca(g) by the reaction Ca (g) +0 (g)-»-CaO (1) . This causes slower rates of purification. Figure

1 shows the rate of CaO removal both for the case of "self-developed" oxygen (Table II) and for

the case of Pq equal to 10"^ atm (Table III)

.

10.0

9.5 _

9.0 _ o

8.5 |_

8.0

0 40 80 120 160 200 TIME, SECONDS

Fig. 1 CaO Vaporization from AI2O3 at 2400 K.

In making calculations of complex chemical

equilibria one must be sure to include all significant chemical species involved in the system.

Similarily one must use care in the selection of correct thermodynamic data to represent the

chemical equilibrium constants. Ignoring these

factors may result in grossly incorrect answers. The use of adequate chemical species and correct

input data, however, can yield a wealth of information and can allow the proper choice of oxidizing

or reducing conditions to achieve desired

vaporization (purification) goals. The ultimate

choice of evaporation conditions will include the

combined solutions of the "ideal" complex equilibria evaporation problem and the convective

diffusion problem.

IV. Convective Diffusion and Evaporation

A. Description of the Process

Stirring in a levitated melt must be

accomplished remotely. We will consider here one possible way of doing this—that of convection

driven by surface traction, that is, by surface

tension gradients. An imbalance of forces along

the surface of the liquid can arise, first, from

surface temperature gradients in which case the

resulting fluid flow is called "thermal capillary

convection." Secondly, these imbalanced surface

forces can arise from surface concentration

gradients in which case the flow is called "solutal

capillary convection." In addition, the surface

traction may deform the levitated melt somewhat

away from a spherical shape; however, for the

discussion here, the melt will be thought of as having a nearly spherical configuration.

The relationships between the various aspects

of the convective mass transfer problem are depicted in Figure 2. A non-uniform surface

temperature gives rise directly to thermal

capillary convection and indirectly to solutal

capillary convection through the surface concen- tration gradients produced by the non-uniform

evaporation. The non-uniform evaporation also

causes the diffusion to depend upon direction.

In addition to being non-uniform, the diffusionevaporation phenomenon is complicated in other ways

.

Non-uniform surface

temperature

Non-uniform

evaporation

Non-uniform

diffusion:

Vc?^e^Oc/3r)

Surface

concentration

gradients

,

V c

s

Solutal

capillary

convection

Total effect:

Surface

temperature

gradients

,

V T

Thermal

capillary

convection

Fig.

convective-diffusionevaporation

The Relationship of the Components of the

Convective-Diffusion-Evaporation Process

in a Non-uniformly Heated Drop.
The two convective processes do not necessarily act in concert, but may oppose one another. For example, thermal capillary movement near the

surface most often flows from the vicinity of a hot surface region toward that of a cold surface

region because the surface tension for most

materials decreases with increasing temperature.

Solutal capillary flow will tend either to reinforce or to cancel the thermal capillary

flow depending on the variation of the surface

tension with concentration.

The total circulation within a convection

cell —regardless of whether it is driven by

thermal or solutal forces —must be considered in

estimating the convective contribution to the

evaporation rate. Not only is the convective

mass transport from the interior toward the

surface important, but also the spreading of the

liquid over the surface must be considered.

B. A Convective Diffusion Problem

1. Formulation of the Problem

We have formulated a model of a convective

diffusion process which arises when a linear

temperature gradient is imposed along the z-axis

of a spherical drop. The model is simplified

further by the added assumption of steady, non- convective heat transfer. The temperature distriDution in the drop is

a time-dependence of the form exp(-Bt) as does the

surface concentration gradient. The first

observation suggests that when impurity concentrations are less than a few ppm and the temperature

gradient is large, which can be the case for

molten alumina, solutal capillary convection can be ignored. The second observation suggests

that the solutal convective term in the convective

diffusion equation will decay exponentially, so that it rapidly becomes even less significant as time elapses. Solutal capillary convection has

been ignored, then, in our estimates for molten

alumina.

Having introduced the two convective

processes , we turn now to the third branch of

Figure 2 —the non-uniform evaporation. The

boundary conditions for the dynamical problem are greatly simplified by the incorporation of all

chemical effects into the rate parameter a. However, the temperature dependence of a is

considered explicitly by the approximation

a(T) = a + a,T,R'cose . (4) o 11

Since we restrict the temperature field, the

velocity field and the rate parameter to terms of order cos 6 , or sin 9 , we restrict the con- vective diffusion equation, the boundary conditions

and the concentration to these same functions of

6 . We write the concentration as

T = T + T, r'COs6

o 1

(2) c(r,e,t) = c^(r,t) - C2(r,t)cose (5)

Following the scheme of Figure 2 , we first

-vnaider the convective velocity components. For

'Ve thermal capillaury convection, we adapt the

...v.lutions of Young, et al_^^' who calculated the

i-rculation of a bubble rising in a vertical

t-mperature gradient. The velocity field

components in a drop are

= (uVr^) (R^-r^)cose

T T 2 2 2 u. = -(U /R ) (R -2r )sine ,

(3a)

(3b)

where the radius of the drop is R and the characteristic speed is u'^. This speed is

By choosing this form for the concentration, the

convective diffusion equation can be split into

two equations.

!!i 2_ L. 2 !fi \ , 1

,

3t ~

^2 3r l"^ Sr y

3 r

!!i ( 2 !!2\

3t °

^2 3r 3r /

^=2 2

^=2

3c 2°^2 * ^r 17 ' (6b)

U = -(Y^T^R/3y)

,

where Y,p = (3y/3T)q is the temperature coefficient

of the surface tension and u is the coefficient

of the viscosity.

Since the concentrations and the concentration

gradients are time -dependent, the components of

the solutal capillary velocity field will also be

time -dependent. Formulas for these components

can be derived after the manner of Miller and

Scriven 's ^1*^) treatment of an oscillating droplet.

Their work must be modified to include the surface

traction terms in the boundary conditions. The

derivation is lengthy and neither it nor the

results will be given here, but two observations

will be made. First, the velocity components

are proportional to the surface concentration

gradient. Second, the velocity components possess

where

T 2 2 2

(U /R ) (R -r )

T 2 2 2 f„ = -(U /R ) (R -2r )

(7a)

(7b)

and D is the diffusion coefficient. Likewise, the

evaporation equations become

^=1 1 ° —" "o=l -

3 ^^^=2

^=2 ° —= '*o=2 - ^^l^^l •

(8a)

(8b)

Equations (8a) and (8b) are evaluated at r = R.

TP
2. Description of the Solution

The convective diffusion equations (6a) and (6b) were integrated using the method of Greens fvinctions ^-^^^ to obtain solutions for

and c^. The very complicated results ^-^^^ consist of sums over infinite series of terms

.

Some of the terms contain integrals which cannot

be performed easily or which yield very c\imbersome

expressions. Moreover, the solutions for c and

c^ are coupled as are the differential equations

from which they were obtained. Nevertheless,

certain features of the solutions of c^ and c^ can be distinguished.

First, the result for c^ can be separated

into three parts as

Since the solution for C20 resembles that for

c-^Q at r = R, an approximation can be obtained

for cj^p, which is

C2p(R,t) » (a^T^R^/2lTD) (Dt/R^) c^^(R,t).' (12)

A very approximate estimate of c (R,t) was attempted using the Green's function result.

The result is

|c2y{R,t) I « I (a^^^T^RVerrV^) (Dt/R^) c^j^(R,t) |. (13)

Substituting (12) and (13) into (11) , we obtain

''l = =1D ^ =1D ^lU- (9)

Proceeding from left, Cj^j-, is the solution to the pure diffusion problem which is treated by

Crank, (12) section 6.34. The .next part, c-j^j-, is a correction to the diffusion problem due to the non- uniform evaporation. Since the effect of non- uniform evaporation in this problem is to create

ao asymmetric concentration distribution and since

C2_Q is part of the symmetric tena c-^i it represents a second-order, symmetric correction to the diffusion problem and can be ignored. The

convective flow also tends to make the concentration distribution asymmetric, so that we take

^lU ^° ^® ^ negligible symmetric correction.

v^^^*47rR^a^c^j^(R,t) [l-(a^^T^^RV6Tra^D) (Dt/R^)

+ (a^aj^Y^T^^RVlSTrV^) (Dt/R^) ]. (14)

The choice of sign for condition (13) and, hence,

for the convective term in (14) is based on the

assumption that the most important contribution

of the convective flow lies in its bringing

impurity-enriched liquid from the interior to the

surface. If, instead, its major function were

that of smoothing out the concentration differences along the surface in such a way as to reduce

the surface concentration, it could reduce the rate of evaporation.

Although C2 could also be split into three

parts, the pure-diffusion part vanishes due to the assumption of an uniform concentration at the

initial time t = 0 . Thus

,

^=2 = °2D =2U

(10)

where C2d contains the contribution of the non- uniform evaporation to the diffusion problem and

C2U is the convective diffusion part. Equation

(10) is coupled to (9) through 02^. However,

=1D , the solution for cID -which is known-—can be used to find C2u« Unfortunately,

the result is extremely complicated and, therefore,

is not easily used to estimate the evaporation

rates . Care must be exercised in the development

of further approximations to 02^ to avoid approximations which are non-analytic at the surface.

The total rate of evaporation is

3. Estimates of the Evaporation Rates

The data used in these estimates is given in Table IV. The rate parameters, and

ax , were computed for the vacuum evaporation of

an initial concentration of 10 ppm of CaO in molten alumina by means of the complex equilibria

analysis described in Section III. The density p and the viscosity y have been measured by Bates, et al^l"*' at the two temperatures, 2400 and

2600 K, that we have considered here. "Reasonable"

values were assumed for the temperature coefficient

of surface tension, y^, and the diffusion coefficient, D. Experimentally measured values of

Yt for the molten salts are in the range of

-0.01 to -0.1 dyn/cm/K for many salts. Diffusion

coefficients for many liquid systems are of the

order of 1 x 10~^ cm^/s. Our values for these

two quantities are enclosed in parentheses to indicate that they are order-of-magnitude

estimates

.

V^.^^ = 4iTR a^[cT-(a,T.R/3a^)c,] (11) tot 01 11 o ^

= V + V + V

ID 2D 2U

where Cj and 02 are evaluated at r = R. We compare and V2U to V^j-, by estimating the

ratios V2i3/V2^p and V2u/V2^j-,. That is, we estimate

the relative enhancement of the evaporation rate and not the absolute rate

.

Estimates of the rates are given in Table V, The relative rates have been computed for two temperature gradients. At the lower gradient of

1 K/cm, the convective enhancement of the

diffusion-controlled evaporation rate is small

for evaporation times of practical importance.

However, a large gradient, on the order of

100 K/cm, might yield an appreciable enhancement

of the evaporation rate. Such gradients would

probably cause many non-linear effects in the

melt, so that our estimates are less certain for

the larger gradients

.
Table IV. Parameters for the Estimation of Evaporation Rates

Units 2400 K 2600 K Ref

.

30 /R

o 3ai/R

D

s~"'"/K -2 . cm /s

mm^/s

0.00338

4.44xl0~'

(lxl0~^)

(1x10"^)

0.01225 This work

-5 4.44x10

(lxlO~^)

(lxlO~^)

dyn/cn/K

N/m/K

(-0.1)

(-1x10"'*)

(-0.1)

(-1x10""^)

cp

Pa'S

110

.11 62.062 14

g/cm

kg/m^

2.584

2584

2.330

2330

14

R (1 mol AI2O3) cmmm

cm/s^

nm/s

2.1

21

0.064

0.64

2.2

22

0.12

1.2

*-.T U evaluated for a temperature gradient of IK/cm.

Table V. Estimated Magnitudes of Various Contributions to the

Evaporation Rate for CaO from Molten Alumina

id'

Temp

Gradient

K/cm 2400 K 2600 K

1 Sxio"® Ixio"®

100 Sxio"'* Ixio""*

1 0.4 t 3 t

100 4000 t 3xlo'* 1

Times t in seconds. Estimates are for times less than 1x10 s.
Surface traction forces are weak forces

relative to the buoyant force on earth and

convective flows generated by them can be damped

significantly if the viscosity of the liquid is high. Thermal capillary enhanced stirring in

molten alumina will occur if a large temperature

gradient is used to overcome the moderately high

viscosity—60 to 100 cp—of the melt. A liquid

with a viscosity on the order of 1 cp, however,

could have ratios of

^2i!'^^lD

^'^^'^'^ would be

about 100 times greater than those given in

Telble V. Such a low viscosity liquid could

exhibit a noticeable enhancement of the evaporation rate by surface traction driven convection

even at gradients as low as IK/cm.

V. Conclusions and Future Work

The phenomenological description of an evaporation rate as the product of a rate parameter and a surface concentration allows the

separation of the computation of the evaporation

rate into two more mamageable problems:

• the computation of the rate parameter

from the analysis of .the complex chemical equilibria,

• the solution of the convective diffusion

problem.

These computations have been illustrated here by

the evaporative purification of molten alumina.

A very rough estimate of the enhancement of

the evaporation rate by thermal capillary con- vection was calculated for CaO in an alumina

melt, with alumina, significant enhancement by

capillciry convection may be attained when a large

temperature gradient—on the order of 100 K/cm

is used. With liquids having low viscosities of

etbout 1 cp, however, significant enhancement will

occur even for small temperature gradients of

1 K/cm.

To date, we have studied the complex equilibria by means of the calculations described

here. Mass spectrometric observations of the

evaporative process are planned in order to experimentally check the chemical calculations.

The experimental data will also provide information about departures from the solution ideally

assumed in the calculations . In addition to alumina, other materials will be examined mass spectrometrically

.

References

1. Paule, R. C, Calculation of Complex

Equilibria Involving Vaporization into

Vacuum, to be published.

2. Kandiner, H. J. and Brinkley, S. R. , Ind.

Eng. Chem. 42^, 850 (1950).

3. White, W. B., Johnson, S. M., and Dantzig,

G. B., J. Chem. Phys. 2£, 751 (1958).

4. Gordon, S. emdMcBride, B. J., NASA, SP-273

(1971)

.

5. Feldmann, H. F., Simons, W. H. and

Bienstock, D., U. S. Bur. of Mines, RI-7257

(1969) .

6. Zeleznik, F. J. and Gordon, S., Ind. Eng.

Chem. 60, 27 (1968)

.

7 . Van Zeggeren , F . and Storey , S . H . , The

Computation of Chemical Equilibria , Cambridge University Press, 1970.

8. Klein, M. , Physical Chemistry , Vol 1/Thermo

dynamics. Chapter 7, edited by Eyring, H. E Jost, W. and Henderson, D., Academic Press,

New York, 1971.

9. Young, N. D., Goldstein, J. S. and

Block, M. J., J. Fluid Mech. 6, 350 (1959).

10. Miller, C. A. and Scriven, L. E., J. Fluid

Mech. 32^, 417 (1968) .

11. Morse, P. M. and Feshbach, H., Methods of

Theoretical Physics , McGraw-Hill Book Co.,

Inc. (New York, 1953), Chapter 7,

12. Dragoo, A. L., to be published.

13. Crank, J., The Mathematics of Diffusion , Oxford University Press (London, 1956) . 14. Bates, J. L., McNeilly, C. E. and J. J. Rasmussen, "Properties of Molten Ceramics,"

in Ceramics in Severe Environments , ed. by

Kriegel, W. W. and Palmour, H., Ill, Plenum

Press (New York, 1971) .

15. Janz, G. J., Lakshminarayanan, G. R.,

Tomkins, R. P. T. and Wong, J., Section 2. Surface Tension Data in Molten Salts , Vol. 2, NSRDS-NBS 28, Nat. Stand. Ref. Data

Ser., Nat, Bur. SteUid. (U.S.) (August 1969).
Task 3

Vacuum Effects in the Preparation of Composite Materials

H. Yakowitz

Metallurgy Division

Institute for Materials Research

Summary

Characterization of composites consisting of reinforcements of silicon

carbide, tungsten, bare pyrolytic graphite and sapphire each in an aluminum

matrix was carried out by optical and scanning electron microscopy. Electron

probe microanalysis techniques were also utilized. These composites were

prepared in an oxygen-free vacuum under Contract NAS 8-29620. The most

promising composite of the group appears to be the sapphire-aluminum followed

by the graphite-aluminum.

A new means to deduce local strains near the matrix-reinforcement interface

is being explored. The method consists of using optical diffraction of electron

channelling patterns to monitor the effects of strain. This report shows the

validity of the method as well as initial results.

Signal processing equipment for the scanning electron microscope was

fabricated in order to better observe the matrix- reinforcement interface.

This equipment permits us to obtain the first and second time derivatives of

the signal as well as their absolute values. The equipment is relatively

inexpensive; circuit diagrams are included in this report.

- 39 -
Introduction

A major objective of this work Is to measure strains In the vicinity of

the matrlx-relnforcement Interface in composite materials. One possible means

to achieve this end Is to prepare selected area electron channelling patterns

in the scanning electron microscope. Then if these patterns can be correlated

with strains, a map of the strains near the interface can be prepared.

Obtaining the electron channelling patterns is usually straightforward. However,

assessing the amount of strain in the region from which the channelling pattern

originates is not at all straightforward.

Originally, divergent beam (Kossel) mlcrodif fraction was going to be used

to provide quantitative strain correlation with the electron channelling

patterns. This method proved less than satisfactory

for a variety of reasons including the impossibility of obtaining satisfactory

Kossel patterns without tilting the specimen in the scanning electron microscope.

Such tilting seriously degrades the electron channelling pattern. Hence, Kossel

and electron channelling patterns could not be prepared simultaneously, the

result being that the Kossel method could not be used as originally anticipated.

Therefore, a different method of assessing strains from electron channelling

patterns had to be developed. The method chosen is to use a tapered tensile bar

strained a known amount as a calibration source. Electron channelling patterns

can then be prepared and the strain determined. Optical diffraction is being

used to assess the changes in the electron channelling pattern as a function of

strain.

The strain correlation represents the main thrust of the work for the year.

In addition, signal processing equipment for the scanning electron microscope

was fabricated in order to better observe the matrix-reinforcement Interface.

- 40 -
Finally, several composite materials supplied by Mr. I. C. Yates of the

Marshall Space Flight Center were examined and characterized. These consisted

of aluminum matrices with various reinforcers; the specimens were prepared by

Convair Aerospace Division of General Dynamics under Contract NAS 8-29620.

Tapered Tensile Bar Approach

The idea for using the tapered tensile bar for electron channelling

pattern studies was devised by Davidson [1]. The original idea was derived

from a method used to relate etch pit density to macroscopic strain in Fe-3 1/4%

Si transformer steel [2].

The specimen was designed for use with an Instron straining machine and is

shown schematically in Fig. 1. Both aluminum and nickel were selected as

materials to carry out the strain calibration and correlation. Aluminum was

chosen because of its desirable properties as a matrix for composites: (1)

Relatively low processing temperature, (2) Good wetting characteristics, (3)

Susceptibility to controlled oxidation so as to permit density controlled

composites, (4) Availability, and (5) Cost. Nickel was chosen because it

rates well on items (2) , (4) , and (5) above and does not require a prohibitively

high processing temperature. Density control with a nickel matrix may, however,

be a problem. In both cases, nickel and aluminum, commercially pure polycrystalline stock was used to prepare the tapered tensile bars since material of

this purity and type will almost certainly be used for any large-scale

manufacturing of composites.

The aluminum bars were strained to a nominal value of 1.2% after being

annealed for 61 hours at 525° C.

Strain was then determined along the bar, a grid having been placed on

the surface and measured before and after deformation. The shape of the strain

- 41 -
versus position on the bar curve is shown in Fig. 2. Based on Davidson's

determination of the degradation of electron channelling patterns as a function

of strain, this strain characteristic is entirely satisfactory for the desired

correlation [1].

The bar was then annealed, this time for 92 hours at 525°C, to promote

grain growth. A grid was placed on the surface by a photochemical process.

This was necessary since no surface strains can be tolerated from scribing

a grid. The bar was then electropolished to its final form; the grid was not

destroyed by polishing in a solution of 5% perchloric acid in ethanol for

five minutes at 50 volts, 0.2 amps and a temperature of -70°C.

Electron channelling patterns of a number of grains in the bar were then

prepared; these represent the "unstrained condition." These patterns are

shown in Fig. 3. Each is sharp and characteristic of well-annealed unstrained

material.

The way in which changes in these patterns arising from strains can be

assessed must now be considered. Davidson has used the human eye which is

fairly sensitive to comparisons of overall pattern contrast and distortion,

widths of high and low index channelling lines and the absence of high order

lines [1]. Davidson claims a relative accuracy of "at least 1%" for the eye.

This method also utilizes the information from the whole electron channelling

pattern.

One can assess, somewhat more quantitatively, line breadths by monitoring

the signal electronically along a line. Figure 4 taken from last year's

report [3] shows the effect of rolling strain on the line profile. This method

has a disadvantage in that only a portion of the electron channelling pattern

is monitored.

A method which uses the whole electron channelling pattern is the optical

transform method. Here the optical diffraction pattern or transform of the

- 42 -
electron channelling pattern is taken. Changes is the electron channelling

pattern affect the optical transform. Hence, the aim is to obtain and evaluate

optical transforms of electron channelling patterns.

Optical Transforms

The optical bench used to generate optical transforms is shown schematically

in Fig. 5. Since laser light is used, the system exhibits the property

known as spatial coherence, i.e., the illumination consists of a spatial

distribution of complex-valued amplitude [4].

The Fourier transform of a complex function g of two independent variables,

X and y, can be called Fig) and defined by

+00 F{g^ = g(x,y) exp [-1271 (f x + f y)]dxdy

X y

—00

Hence, Fig) is a complex-valued function of two independent variables f^ and f^,

referred to as frequencies. Similarly the inverse Fourier transform of a

function G(f ,f ) can be called F {g} and defined by

X y

'

F ^{Q} =

J_oo

G(f ,f ) exp [i27T(f X + f y)]df df

X y X y X y

Now, the assertion is made that the system represented in Fig. 5 is a linear

system, and its properties can be described by linear systems theory. This

simply means that the response of the sample, in this case a transparency of

an electron channelling pattern, to several stimuli acting simultaneously is

identically equal to the sum of the responses that each of the component

stimuli would produce individually. Thus, if a stimulus is decomposed into

a linear combination of elementary stimuli, each of which produces a known

response of convenient form, then by the property of linearity, the total

response can be determined as a corresponding linear combination of the

- 43 -
the responses to the elementary stimuli. Fourier analysis provides a basic

means of performing one such decomposition. For example, consider the

inverse-transform relationship

+00

g(t) =

I

G(f) exp(i27rft)df

—00 which expresses the time function, g, in tems of its frequency spectrum.

This expression represents a decomposition of the function g(t) into a linear

combination, in this case, an integral, of elementary functions each having

the specific form exp(i2TTft). Therefore, the complex number G(f) is just a

weighting factor which must be applied to the elementary function of frequency,

f, in order to synthesize the desired g(t).

In an entirely similar way, the two-dimensional Fourier transform may be

regarded as a decomposition of a function g(x,y) into a linear combination of

elementary functions having the form exp [i2TT(f ^x + f ^y) ] . Such functions have

a number of interesting properties, e.g., for any particular frequency pair

(f^,f^), the corresponding elementary function has zero phase alone lines

described by

£

y y

where n is any integer. Figure (6) shows that this elementary function can be

regarded as being directed in the xy plane at some angle 9, with respect to the

X-axis, given by

e = tan-l /X

Furthermore, the distance between zero-phase lines, known as the spatial period,

is given by

L =

2 2 1/2

(f ^ + f y'^

X y

- 44
Hence, the Fourier spectrum G of a function g is a description of the

weighting factors that must be applied to each elementary function in order to

synthesize the desired g. However, in the case of the analysis of electron

channelling patterns, the pattern represents the function g. The patterns

possess varying degrees of sharpness as a function of strain in the specimen

or other factors such as a coating on the specimen [3].

Therefore, the form of the optical diffraction pattern is expected to

change as the electron channelling pattern sharpness changes. Loss of sharpness

can lead to loss of high frequency components. In analogy to the simple

example represented in Fig. 6, the frequencies f and f would be altered;

X y

and hence, the values of 9 and L would be expected to change concomitantly.

The strain calibration thus consists of preparing optical transforms from

each of the electron channelling patterns taken from the grains in the annealed,

unstrained tapered tensile bar. Then, after straining, new electron channelling

patterns are prepared and their optical transforms recorded. If the electron

channelling pattern alters due to the applied strain, then the optical transform is also expected to alter. With the aid of Fig. 2, the strain as a

function of position in the tapered tensile bar is known. Hence, the end

result is a set of electron channelling patterns and their corresponding

optical transforms for both the unstrained and strained states. The optical

transform method utilizes the whole electron channelling pattern and is unbiased.

The experiment consists of preparing the electron channelling pattern from the

composite, taking the optical transform and comparing it with the calibration

set in order to determine strain as a function of position in the composite.

To test the response of the system, a set of four photographs of a

pseudo-electron channelling pattern was prepared. The sharpness of each was

successively degraded to produce the set shown in Fig. 7. Then the optical

- 45 -
transform of each was prepared. The pattern produced by Fig. 7a shows sharp

spots due to the periodicity of the stipple and the expected symmetry (Fig. 8a).

If the central portion of this pattern is enlarged by means of a projector

lens, the pattern shown as Fig. 9a results. Figure 7b produced the pattern

shown in Fig. 8b without a projector lens. The spot pattern is absent. Use

of a projector lens yields the pattern shown in Fig. 9b. Clearly some periodicity

remains here. Figures 7c and 7d yielded the patterns shown in Figs. 9c and 9d.

The progressive effects are apparent. This simple test set indicates the kinds

of effects one can expect to see when real electron channelling patterns are

subjected to optical transform techniques.

Figure 10 shows an electron channelling pattern of the <111> pole in

niobium as well as its corresponding optical transform. The symmetry elements

can be easily seen. Next, an aluminum single crystal was bombarded by argon

ions at one end so as to cause local surface damage. Electron channelling

patterns and their corresponding optical transforms are shoim in Figs. 11 and

12 for the undamaged and damaged portions respectively.

Characterization of NASA Supplied Composite Materials

The materials to be described were prepared by Convair Aerospace Division

of General Dynamics under Contract NAS 8-29620. The materials were supplied

by Mr. I. C. Yates of Marshall Space Flight Center, NASA monitor for the

above contract.

All materials had an aluminum matrix. The reinforcements were respectively

(1) SiC in the form of 0.1 mm diameter filaments, (2) Chopped tungsten wires

0.3 mm in diameter, (3) Bare pyrolytic graphite in the form of 0.01 mm diameter

fibers and (4) Sapphire (Al^O^) filaments about 0.25 mm in diameter.

The composites were prepared in an oxygen-free vacuum. The rationale is

that to achieve good bonding, good wetting of the reinforcement by the matrix
is necessary. But wetting truly occurs only in the three-phase, solid- liquidgas, system. Therefore, in a two-phase system consisting of just solid and

liquid, conventional wetting concepts may not apply; and free dispersion (good

bonding) can be achieved with any reinforcement. The oxygen-free vacuum was

obtained by cycling the vacuum chamber containing mixed aluminum chips and the

-3

reinforcement to a pressure of 10 torr pressurizing v/ith argon and evacuating

-3

to 10 torr several times. Then the sample is heated to 600° C and the

evacuation-argon, pressurization-evacuation step repeated once again. The

specimen is then heated to 1000° C and manual mixing is carried out followed

by slow cooling. Empirically, the 1000° C temperature was established as that

which gave the best matrix-reinforcement dispersion and bonding.

The specimens were supplied in their original ingot form. Each specimen

was mechanically polished and examined with the optical microscope and with a

scanning electron microscope equipped with X-ray microanalysis capabilities.

SiC-Al Specimen

Optical examination revealed material in the matrix grain boundaries,

an extensive reaction zone at the SiC-Al interface, the presence of an extraneous phase in the matrix and that the SiC filament from which the reinforcer

had been prepared contained a core (Fig. 13). Scanning electron microscope

examination shows the SiC to be unevenly dispersed in the matrix and that the

extraneous phase seems to have an affinity for the SiC (Fig. 14). The extraneous phase was identified as tungsten bearing by means of the X-ray microanalysis

unit attached to the scanning electron microscope. Furthermore, the core

material of the SiC was shown to consist of tungsten. The tungsten bearing

material in the composite was subjected to quantitative electron microanalysis

procedures [5]. Results showed the material to consist of 38% aluminum.

- 47 -
balance tungsten by weight. This corresponds fairly closely to something

having the formula Al^W.

The reaction zones at the SiC-matrix interface are shown in Fig. 15.

The SiC has cracked in many places. The matrix exhibits shrinkage pores.

Reaction of the SiC with the matrix seems to have proceeded on a non-uniform

basis. The extraneous tungsten phase is probably from the tungsten rod used

to manually mix the components in vacuo at 1000° C. Even without the extraneous

tungsten, it appears that this composite is not entirely satisfactory.

W-Al Specimen

Optical examination shows that the dispersion of the tungsten wires in

the aluminum is fairly good. The wires appear to be randomly oriented. (Fig. 16)

Optical examination also revealed that the matrix consists of dendritic grains

whose boundaries contain a separate phase (Fig. 17).

Scanning electron microscope examination showed the reinforcers to be

very irregular in shape and aspect ratio. The separate phase in the grain

boundaries appears to be eutectic-like and may well be Al^W. (Fig. 18) X-ray

microanalysis indicates that the aluminum and tungsten had reacted during

preparation of the composite. All of the tungsten bearing material contains

aluminum. Quantitative electron microanalysis of this material indicates

about 42 1/2 percent aliaminum by weight, the balance being tungsten. This

corresponds to 83.5 atom percent of aluminum or something having a formula

o Al^W. Al^W is reported to be hexagonal in structure with a = 4.902A and

C = 8.857A [6].

The interface of the reinforcement with the matrix is variable ranging

from straight and sharp to regions where local reaction appears to have taken

place. (Fig. 19) This variability plus the existence of what is likely to

- 48 -
be a brittle phase throughout the grain boundaries indicates that this

particular composite will probably not be suitable for engineering applications.

Graphite-Al Specimen

Both optical and scanning electron microscope examination indicate

reaction between the graphite and the matrix; the dispersion appears to be

reasonably good (Fig. 20). The amount of reaction which occurred in individual

graphite fibers appears to be variable. In some cases a black appearing core

remains while in others this core is absent. The size of the core is variable

as well. (Fig. 20)

The black core and surrounding grey areas were subjected to quantitative

electron microanalysis procedures. The black core contains 29.2% aluminum

by weight, balance carbon. This corresponds to 15.5 atom percent aluminum

or something approximating AlC^. The grey region is variable but contains an

average 72.3% by weight of aluminum, balance carbon. This corresponds to

53.8 atom percent aluminum and is very likely the well-documented compound

Al^C^ 17] 18]. It is likely that a composition gradient exists throughout

the carbon-bearing material.

This composite may be worth additional effort to obtain mechanical

properties tests. Perhaps the time at temperature during fabrication should

be lengthened slightly to attempt to react the fiber completely to Al^C^.

Sapphire-Al Specimen

The grain boundaries are clean. The dispersion of the sapphire is very

poor; all of the reinforcement seems to be at the very edge of the ingot.

Probably it floated up and could not be sunk again during the molten stage of

fabrication. Tungsten is also present again probably due to the stirring rod.

- 49 -
Figure 21 shows the general aspect of this composite in the region containing

the sapphire. Most of the material is sapphire free but has a tungstenbearing impurity.

The tungsten-bearing impurity was shown by quantitative electron microanalysis

techniques to consist of 41.3 percent by weight of aluminum, balance tungsten.

This corresponds to 82.6 atom percent of aluminum. Hence, the reaction of

tungsten with molten aluminum at 1000°C seems to produce material containing

around 80 to 83 atom percent aluminum as shown throughout this study.

In this composite the tungsten bearing material seems to exist both in

nodules and in whisker-like needles, whose aspect ratio is roughly 30 to 1.

Remarkably, the interface between these needles is straight and very clean at

a magnification of 10,000 diameters (Fig. 22). Clearly, this "accidental"

tungsten aluminum composite shows more promise than the composite deliberately

fabricated of tungsten in aluminum.

In regions where the sapphire is present, the matrix-sapphire interface

appears to be excellent. The sapphire filaments appear to exhibit a variable

friability. Some are badly fissured while others nearby are completely smooth

(Fig. 23). Sapphire is not wetted by aluminum under usual fabrication conditions; however, the fabrication conditions used to make this specimen clearly

have resulted in good matrix-reinforcement bonding. More effort to obtain a

better dispersion of the sapphire would seem to be in order. If this can be

achieved, then the sapphire-aluminum composite should be subjected to a full

battery of mechanical properties tests.

In addition, another effort at producing a tungsten-aluminum composite

having the aspects illustrated in Fig. 22 probably is worthwhile.
New Signal Processing Equipment for Scanning Electron Microscopy

Since the characteristics of the matrix-reinforcement interface are of

great Interest, the possibility of enhancing the information obtainable from

the conventional scanning electron microscope (SEM) by signal processing is

attractive. Therefore, some unique signal processing equipment, designed by

Mr. C. E. Florl of NBS, was adapted in prototype form to the SEM. Preliminary

tests were so encouraging that we are having a permanent set of the signal

processing equipment fabricated for adaption to the SEM. What follows is a

description of the equipment, which can be built relatively inexpensively,

and the results of the preliminary tests.

The image from an SEM provides the viewer with a representation of a

specimen surface in varying shades of grey ranging from black to white on

a display cathode ray tube (CRT) or photographic material. The shade of a

particular picture point is related to the signal produced by one of the detector

systems, which monitor the interaction of the primary beam with the specimen.

The detector may respond to primary (back scattered) electrons, secondary

electrons, absorbed electrons (specimen current), X-rays, or long wavelength

light (cathodolumlnescence) . The detector signal is amplified to produce a

signal suitable to intensity modulate a CRT, which is scanned in synchronism

with the primary beam on the specimen to create an image. The characteristics

of the SEM signal have been considered elsewhere [9]. The amplification of

the detector signal, which we shall refer to as signal processing, takes in

its simplest form a direct proportionality between the detector signal and

the output signal. In certain circumstances, such a "direct image" contains

useful Information to an observer: the natural contrast — the signal ratio,

I^/l2, between two points of Interest — is sufficient for the human eye to

discern. Such a condition is realized when primary electrons are used to

form the image from a rough surface; the natural contrast level can be 25%

- 51 -
or more, which can be easily observed by the human eye. However, In many

instances the natural contrast is too weak to be easily observed, or at the

opposite extreme, the contrast between two regions in the image may be so

strong that slight contrast changes within the regions may be invisible to the

eye. For these reasons various operators which functionally transform the

signal have been developed. The major reason for performing such transformations is to increase the amount of useful information which can be extracted

from the specimen. Most SEM s are equipped with two operators — differential

amplification of the signal, usually called the "black level operator," and

nonlinear amplification of the signal, referred to as the Y operator. [10]

There are a variety of other operators which can be applied to the signal.

In this report three of these additional operators, (1) the first time derivative

of the signal, (2) the absolute value of the first time derivative, and (3) the

second time derivative are discussed. These operators can enhance fine

details from specimens which produce widely varying signal levels within the

field of view, can provide apparent vertical or oblique illumination of the

specimen, and can produce apparent sharpening of edge details (crispening) in

the image. -

'

For this work an amplifier has been constructed which can produce the Y

and time dependent signal transformations described above. An additional

feature of the amplifier is that it can mix, in any proportion, the direct

signal with the derivative signal of interest. For certain image transformations, orthogonal scanning, in which two scans at right angles are superimposed,

has been found to be useful. Orthogonal scans with perfect registration are

not generally possible with present SEM's. Obtaining accurate registration

with analog scan generators is difficult. Therefore, we have constructed a

digital scan generator with orthogonal scan capabilities. In addition.

- 52 -
individual coordinates and their respective grey levels are recoverable in

computer compatible format from this generator, making possible the eventual

use of a digital computer for image processing. The schematic diagrams for

this signal processing amplifier and digital scan generator are shown in

Fig. 24.

The SEM Recording Process

The image in an SEM is usually observed on a CRT having a long persistence

phosphor and recorded photographically on a CRT having a short persistence

phosphor. Only photographically recorded images are considered. The brightness range of recorded images is restricted by both the characteristics of

the recording film and by the number of statistically valid grey levels which

can be distinguished [9J. The exposure density of a film is a logarithmic

function of the input signal, progressing from black (no signal or response)

to white (full signal or maximum film response). This progression is divided

by the eye into a number of grey levels in which a range of input signal produces effectively only one film response or shade of grey. The signal transformations discussed in this paper will only be considered with regard to the

photographic image. However, we must recognize that the signal is transformed

by the response of both the CRT phosphor and the recording film.

Time Dependent Operators

The SEM operator can choose between (1) low contrast-sensitivity over

the entire range of signal intensity or (2) using the black level and y

operators to give high contrast-sensitivity within a restricted range of

signal intensity. This choice can be altered by using the first time

derivative of the signal, as illustrated in Fig. 25 to modulate the brightness

- 53 -
of the CRT [11]. The first time derivative gives enhanced sensitivity to

certain contrast changes at all signal levels. Modulation of the brightness

by this derivative signal produces an image in which all areas of constant

signal have the same grey level regardless of constant signal intensity.

Changes of the signal level produce a variation of brightness. However, pure

derivative images produced in SEM's do not give the usual impression of depth,

since the static levels of the signal are no longer distinguishable. Furthermore, the time derivative of the signal is not responsive to changes in signal

intensity along the scan line. Therefore, contrast boundaries in the original

image which do not cross the raster lines do not appear in the derivative

image. The derivative signal from a line-type feature, such as an edge,

oriented at an angle 9 to the scan line will be reduced in intensity by a

factor of sin 9 from that value when the scan line is perpendicular to the

feature. For these reasons, it is usually best to mix in variable proportions,

depending on the subject and on the purpose of the image, the original signal

with the derivative signal. The major reason for utilizing derivative operators mixed with the original signal is to reduce vast grey level differences

and to provide increased contrast in regions of rapidly changing signal level.

A basic characteristic of derivative processing is to apparently sharpen edge

details in the image; this effect is referred to as "crispening" in television

technology [12]. Observers have an overwhelming preference for crisper photographs and thus the derivative processing may be esthetically pleasing [12].

However, crispening must be critically evaluated so that false interpretation

of specimen features does not occur. For example, at some magnification where

the beam diameter encompasses several picture elements producing an unsharp

image, a condition occurs where derivative processing will falsely provide

an impression of improved resolution on the micrograph.

- 54 -
An undesirable feature of the time derivative transformation is that the

signal-to-noise ratio is always less than that obtainable without the transformation. The derivative operator is extremely sensitive to high frequency

components in the total signal spectrum. Since noise in the SEM is always

high frequency, the noise is amplified to a greater degree than the signal.

This difficulty is of minor importance in many SEM applications, but when

the signal-to-noise ratio is inherently small, the derivative operator may

not be useful. Another drawback is the case where one wishes to enhance fine

details in the presence of rapidly changing high contrast levels. The value

of the derivative for sudden signal changes varies widely for the fine details

and large signal changes at edges. Saturation must be conceded at these large

signal changes to enable fine details to be distributed over the full grey

level range. Blooming of the film can be avoided by providing an electronic

"clamp" in the derivative circuit to prevent undesirably high signals.

Fig. 26 shows the effects of time differentiation on the image of a

simple object, a round hole machined into a thin aluminum disk. Fig. 26a shows

the direct image of the hole formed with secondary electrons. The specimen was

normal to the primary beam. Shadowing effects occur because of the geometrical

relationship of the emissive electron detector and the specimen. Fig. 26b

shows the same field imaged with specimen current which provides a uniformly

illuminated image. Figs. 26c to h show various time derivatives of the specimen

current signal. Fig. 26c shows the first time derivative scanned in the

vertical direction only. The anisotropy and information loss caused by this

transformation are apparent. The image depends strongly on the orientation

of the specimen relative to the scan line; detail parallel to the scan lines

is totally lost. The illumination appears to be oblique. Anisotropy in an

image is not necessarily undesirable; shadowing in the emissive signal image

- 55 -

t
enhances the "three-dimensionality" effect. Anisotropy which results from

specimen geometry and/or signal collection effects is an inherent feature of

the SEM imaging process. Special cases exist in which an isotropic image is

produced. However, the artificial anisotropy of the first time derivative

represents an undesirable information loss along a line (arrows). Orthogonal

scanning, in which two scans at right angles to one another are superimposed,

has the effect of rotating the line of information loss in the image.

Fig. 26d arrows, and an orientation dependent response is still obtained.

The absolute value of the first time derivative, Fig. 26c, converts all

negative derivative signals from the blacker end of the grey scale to the

whiter end. Figure 26e shows an image of the hole with the absolute value of

the first derivative unidirectionally scanned in the vertical direction;

information is again lost. If orthogonal scans are used. Fig. 26f, the entire

edge is outlined without information loss. The absolute value of the first

derivative is not isotropic, but it is a closer approximation to isotropy.

We have found the absolute value operator useful to outline major features

when regions with few details are studied. In regions containing much detail,

the absolute value image is extremely complicated, making interpretation

difficult.

For complicated specimens, the second time derivative. Fig. 26d, can

be extremely useful. This operator has the advantage of being isotropic when

orthogonal scanning is used, i.e., specimen orientation relative to the

scan lines is immaterial. Fig. 26g shows the second derivative image of the

hole scanned in one direction — edge details are apparent but information

is again lost parallel to the scan line. Orthogonal scanning with the second

derivative. Fig. 26h, produces a uniformly illuminated edge having sharp

detail. The illumination is effectively vertical.

- 56 -
While images of simple subjects are useful for illustration, it is

informative to show the use of these derivative operators on a typical SEM

specimen, e.g., a fracture surface of iron. Figure 27a shows the direct

image with the secondary electron signal while Figs. 27b, c, and d show the

first time derivative, its absolute value, and the second time derivative

with orthogonal scanning respectively. With the first derivative Fig. 27b,

the impression of depth is lost, and the edges appear to be obliquely illuminated. With the second derivative. Fig. 27d, the impression of depth is

retained due to enhanced sensitivity to fine detail facing the detector, and

the edges again appear to be obliquely illuminated. The orthogonally-scanned

second derivative is expected to give the appearance of vertical illumination

but this condition is not obtained for this specimen with the secondary electron

signal. An isotropic operator will not convert an anisotropic direct image

to an isotropic image. The absolute value of the first derivative produces

a strong outlining effect, Fig. 27c. In Figs. 27e, f, and g the derivatives

are mixed in equal proportion with the direct signal. The crispening of edges

resulting from the use of the derivative operators and the enhancement of

fine details are apparent.

Hopefully, the permanent version of this signal processing equipment will

be installed on the SEM in the very near future. Matrix-reinforcement interface characterization utilizing the equipment will then commence.

Conclusions

1. Of the composites prepared under Contract NAS8-29620, aluminumtungsten, aluminum-sapphire and perhaps aluminum-graphite show promise of

being useful in an engineering sense.
2. Optical transforms of electron channelling patterns can be used to

assess pattern perfection.

3. The tapered tensile bar coupled with the optical transform technique

will provide a means to calibrate electron channelling patterns as a function

of strain.

4. Signal processing equipment added to the scanning electron microscope

will enable better matrix-reinforcement interface assessment.

- 58 -
References

[1] D. C. Davidson, SEM/74 (0. Johari, ed.) IITRI, Chicago, 111. (1974)

p. 927.

[2] G. T. Hahn, P. N. Mincer and A. R. Rosenfield, Exp. Mech. 11, 248 (1971).

[3] H. Yakowitz and D. E. Newbury in NBS Materials Science and Manufacturing

in Space Research (E. Passaglia and R. L. Parker), NBSIR 73-402 (1973)

p. 57.

[4] G. W. Stroke, An Introduction to Coherent Optics and Holography , Academic Press, New York (1966).

[5J H. Yakowitz in Practical Scanning Electron Microscopy (J. I. Goldstein

and H. Yakowitz, eds.) Plenum Press, New York (1975) Ch. IX, X.

[6] J. Adam and J. B. Rich, Acta Cryst. 8^, 349 (1955).

[7] M. von Stackelberg and E. Schnorrenberg , Z. physik. Chem. B27 , 37

(1934).

[8] J. H. Cox and L. M. Pidgeon, Can. J. Chem. 41, 1414 (1963).

[9] D. E. Newbury and D. C. Joy, SEM/73 (0. Johari, ed.) IITRI, Chicago,

111. (1973), p. 151.

[10] C. E. Fiori, H. Yakowitz and D. E. Newbury, SEM/74 (0. Johari, ed.) IITRI,

Chicago, 111. (1974) p. 167.

Ill] K. F. J. Heinrich, C. E. Fiori, and H. Yakowitz, Science 167, 1129 (1970).

112] P. C. Goldmark and J. J. Hollywood, Proc. IRE, 39 (10), 1314 (1951).

- 59 -

-J .J

J /

Fig. 1 Tapered tensile bar shown

actual size.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 I.I 1.2

Position, (Taper to Wide End) Inches

Fig. 2 Actual strain as a function

of position in aluminum tapered

tensile bar nominally strained

1.2%.



Fig. 4a Electron channelling pat- pig_ 45 intensity profile along

tern of annealed electropolished i^^g ^-^ p^g^ Scale: 2 volts/cm.

aluminum. Line indicates region

from which profile information was

taken.

A

Fig. 4c Electron channelling pattern of aluminum crystal rolled to

5% reduction and then electropolished.

Line indicates profile trace position.

Fig. 4d Intensity profile along

line in Fig. (4c). Scale: 2 volts/cm.

o

LiJ (T

Li_ <

CL

C/) 3 CO

•*-> E

ta s- s- o

(O ^-

Ql to Q. c

s- 4- -(-> O

O) o

"I

> »-> Q. O O

4-> O)

rO $- E ra CU CL

OJQ.o

Lf) +-> • -o

CD 0)

•1— if) U- 3

crLU

CO<

_i

+-> tA c: ^ +-> o

O) O s-

•r- 0) +J

1— 4- (_> to 4- (U CU

+-> 0)

13 <U

+-> fO (O

sr-- E s ^ •p— n3 c>—

(/I 0) C +-> CD O -l-> •1- 4-> fO U_ c Q.

to

t/) 3 (tJ CD

•SC U s- C

O 4-> <+- toO 'oJ o +-> c

c

cn+-> O) (tJ •r- 3 ^ Ll_ O -O

D1

(Uc

c

fOo

c

o

s4-> o

Qi

O

-a3

O)

to Qu

(T3 C

(U

• +-> C7)+->

•1- fO

Ll_ Q.

1 1 Bi I I

Q)

S-S- O)

Z3 E

I— o

-Q (/)

+-> 4— U O

O)

> o

o c

1— <u

-— </) J3

• (0 cn

•I— <u

Ll_ CO

to n3o tn -O 4-> <ui~

to Z3

• n3 M CD fO

•1- o 0)

Ll_ (/)

o

E

CD

to

u -a

• s-

Fig. 8a Optical transform of

Fig. (7a) without projector lens.

The spot pattern is due to the

periodicity in the stipple. The

cross is due to the overall pattern

symmetry.

Fig. 8b Optical transform of

Fig. (7b) without projector lens.

The spot pattern is gone since the

stipple pattern is completely

blurred.

Fig. 9a Optical transform of

Fig. (7a) with projector lens. The

cross is seen to consist of a num- ber of small spots and has long

range.

4-

Fig. 9c Optical transform of

Fig. (7c) with projector lens. The

cross is poorly defined. Compare

with Figs. (9a, b).

Fig. 9b Optical transform of

Fig. (7b) with projector lens. The

cross is less well defined than in

Fig. (9a).

Fig. 9d Optical transform of

Fig. (7d) with projector lens. The

cross is virtually absent. Compare

with Figs. (9a, b, c).

Fig. 10a Electron channelling

pattern of the <ni> pole of a niobium single crystal.

Fig. 10b Optical transform of

Fig. (10a) with projector lens. The

symmetry elements are apparent.

Fig. lla Electron channelling pattern Fig. lib Optical transform without

of annealed electropolished aluminum. projector lens of Fig. (lla).

Fig. 12a Electron channelling Fig. 12b Optical transform, without

pattern of deformed aluminum. projector lens, of Fig. (12a). Pattern is considerably less sharp than

Fig. (lib).

Fig. 13 Optical micrograph showing

grain boundary phase, SiC fibers

with tungsten core and tungsten

impurity in Al matrix - SiC

reinforced composite.

Fig. 14 Scanning electron micrograph

(specimen current mode) showing

aspect and dispersion of SiC fibers

in Al matrix. White regions are W

impurity. Note cracks and shrinkage

cavities in the matrix.

Fig. 15a Reaction zones at SiC-Al

Interface. Note cracks in SiC

fibers and nonuniformity of reaction

zone.

Fig. 15b Detail of reaction zone

at SiC-Al interface. White

material is Al^W impurity.

Fig. 16 Aluminum matrix with

tungsten wire reinforcement. Optical micrograph shows dispersion of

reinforcement and randomness of

reinforcer orientation.

Fig. 17 Optical micrograph showing

a grain boundary phase in Al-W

composite.

Fig. 18c Scanning electron micro- Fig. 18d Grain boundary phase

graph showing extent of grain revealed as being eutectic-like

boundary phase. material.

1"

Fig. 19 Optical micrograph showing local reaction at matrixreinforcement interface in Al-W

composite. Reaction product

contributes to formation of grain

boundary phase.

Fig. 20 Scanning electron micrograph of graphite rei nforcer-Al

matrix composite. Reaction of Al

with the graphite has occurred

leaving grey (AljCs) regions and some black core (AlCs) areas.

Fig. 21 Scanning electron micrograph of sapphire reinforced Al

matrix composite showing

agglomeration of sapphire. Black

regions are tungsten bearing

impurity.

Fig. 22a Sapphire free region

of sapphire-Al matrix composite.

White needles are tungsten bearing impurity having composition

41.3 wt. pet. Al , balance W.

Fig. 22b Detail of W-Al needle

in sapphire free region of

sapphire-Al matrix composite.

Note reaction free interface

with matrix.

Fig. 22c Interface of W-Al

needle. Interface is remarkably

clean and sharp.

Fig. 22d Interface of W-Al needle

showing no visible reaction within

500A of boundary. Excellent bonding

would be predicted for this situation.

Fig. 23a Sapphire filaments in

sapphire-Al matrix composite. Note

cracks in case of central feature

and absence of cracks in neighboring

filaments. White material is W

bearing impurity.

Fig. 23b Same as Fig. (23a) but

showing sapphire-Al interface to

reaction zone free.

be

> - • ! f w> * — * 9
i

I
Fig. 25 Effect of derivative

operators on signal from typical

SEM specimen: (a) Direct emissive

signal, (b) First time derivative

of signal, (c) Absolute value of

first time derivative, (d) Second

time derivative (inverted) of signal.

4->

(/) -o O C

c

0) ro 3 O

1— </» (O> Q)>

+J 4-> 3 (d I— >

o •-- 10 s- XI <u

-o

<uc

(do

CO

0)>

no>

>> s- I— <u I— T3

(OC (U O E

a>T- o •-> x:

»-> -o

s_ c

o cvj

-o

0)c

c

fOo

u

•I- s- Q. a> o >

s-

>

1— (d r— >

<0 -r- C J- O <U t- -o

4-> U (U

<U E

s- •-

•I- -l-> -o

•I- T3 C C

3 C\i

c

o

4-> Ol O 40 C (U

s-a. • r— •!— O

-o +J C C (O o <o c

•>- O 0) E

x: cn 3

4J <o r—

S- E r- O T- -rc

o

•f—

+-> -o o

<U 4- C c to

(OO 4- >>

(/) o o.o

0) 0) S- > c -l-> o

-l-> _J to > c:

fO

S- 5^

O) o

T5 -4-> (O <U c (U E o

CD -a

-)-> o

lO +J to

to o

o

I c

o

•r4-> O

0) 0) E s-

•I— "r-" -M T3

-o

14- O) O C

C

2 O) (O I— toO -

S- I—

fO >

>, 0) •.- X> 4-> 4J "O r— > . O) O -I- >^ 4J to S- •—

fd oj I—

(U c: E•1- IT) O

(U •> Q.>

CO ^O• " ro

to •> 'r— E a f3(U I— Si

>. • C J- 4-> •I- (13 C

E 0}

<U -r- 1_ I— S- S- O Q- 3

3: o

o-!-> CM >— _

fO •!-

• E O

C7) S- 0)

•r- O CL

Ll_ C (/)

C

0)cn • (O to E <u

•r- Ol

(tJ C E O •!- s- +J +J o c — J0) s~=3 >> O

sc:

0)

CD -a

O)c

>

(U

O)O

c oE E

O -f- o o

to

CD o

to 'r- E 4->

•r- <0c

u E

<U ^ <U 3

S- I S_ r—

•I- J3 •!- I— Q 'Q .|-

o

C

c

to c

o o

CO •(- -M

O) ro CD E fO S-

<U -r- >

•r— •

M >>

rtj I— > I—

•r— ro

J- C

<u o

T3 -r-

-l-> <u u

E <U

•I- S-

-l-> -t- T3

4-> •!- V) E

o

S- -MO

to

•n-(=<a

CJ)

-o

S- c

O <L) S- S_

S- fO

(T3 CL

> Q.

fO

to (U o s- r— ro

lO U

Fig. 27a Secondary electron image

of fracture surface of pure iron.

No image processing applied, 20kV,

5nA specimen current, specimen

tilted 30°.

Fig. 27b First time derivative of

image in Fig. (27a). Note flattening and apparent oblique

illumination. Scanned orthogonally.

Fig. 27c Absolute value of first

time derivative. Note strong outlining of edges. Scanned

orthogonal ly.

Fig. 27d Second time derivative

of image of Fig. (27a). Note

impression of vertical illumination

and less apparent flattening.

Scanned orthogonally.

Tig. 27e Same as Fig. (27b) but

mixed in equal proportions with

direct signals scanned orthogonally;

note crispening of image.

Fig. 27f Same as Fig. (27c) but

mixed in equal proportions with

direct signal. Scanned orthogonally.

Fig. 27g Same as Fig. (27d) but

mixed in equal proportions with

direct signal. Scanned orthogonally.

Note crispening of image as well as apparent vertical illumination.
Task 4

~ Melt Shape in Weightless Crystal Growth

S. C. Hardy and S. R. Coriell

Metallurgy Division

Institute for Materials Research

Sununary

Experimental investigations of the distribution of water at the end of

an ice cylinder in a simulated low gravitational field have yielded different

results for isothermal and linear gradient thermal geometries. In the

isothermal arrangement, the stable configuration of the ice and water is

one in which the cylinder passes entirely through the water zone. In a

gradient, however, it is possible to have a cap of water on the end of the

cylinder. Theoretical calculations have been carried out on the shape and

stability of a vertical rotating liquid zone subject to a weak vertical

gravitational field. The first order correction to the zero gravity stability

result is obtained.

- 60 -
Introduction

The shape and stability of liquid bodies in zero gravity have been

subjects of experimental investigation since the pioneering work of Plateau^'''^

over a hundred years ago. The quantitative theoretical treatment of these

(2) problems is equally old, the basic variational mathematics having been

successfully formulated prior to 1900. With the development of space

technology, a new interest has arisen in phenomena involving liquids in zero

gravity. The particular concern of this project is the shape and stability

of the melt during containerless crystal growth in space.

The zero gravity condition has been previously simulated by using

immiscible liquids of equal density, soap bubbles, or specimens sufficiently

small to make the surface tension forces much larger than the volume forces

of gravity. Numerous configurations of liquid and solid surfaces have been

(3) studledr (1) zones between solid plates; (2) zones between rings; (3) zones

between spheres and (4) pendant drops, and many more complicated arrangements.

In addition, the effects of rotation on the shape and stability of the liquid

has been investigated^^ not only in the laboratory but in space.

With the exception of the recent space experiments by Walterf^^ most of

this extensive research has not involved solidification. Furthermore, because

of the particular boundary conditions imposed, the analogy with solidification

may not be strong. The boundary conditions differ from those appropriate to

solidification in two significant respects. First, surface coatings were often

used on the solid surfaces to prevent or restrict wetting and eliminate the

phenomena known as "capillary run-off". Since crystals generally are wetted

by their melts, this clearly does not model solidification. Secondly, the

experiments were essentially isothermal whereas solidification generally

involves high thermal gradients. Surface tensions are temperature dependent

- 61 -
and, thus, the stability of liquid zones in high temperature gradients may not

be the same as in isothermal conditions.

Our experimental research in the past year has been directed to

examining the shape of liquid zones in near zero gravity with actual meltcrystal interfaces in contact with the zones. Rather than concentrate on

quantitative measurements, we have attempted a variety of qualitative

experiments with the intention of doing more detailed work on interesting

areas later. We have worked exclusively with water-ice because of convenience

and prior experience.

In addition, we have made a study of the variational mathematics used

in treating liquid zones. A detailed calculation has been completed applying

these techniques to the stability of a cylindrical zone between flat plates

in a weak gravitational field and the first order correction to the zero

gravity stability criterion has been determined. This calculation is important

to our experiments and to experiments in space because zero gravity is never

actually attained; it is approximated. Thus, it is essential to understand

how small volume forces modify the zero gravity stability criteria.

Experimental Procedures

The technique we have used to simulate zero gravity uses a liquid that

is immiscible with water and of nearly the same density as a supporting medium.

After trying a number of liquids, we selected a solution of di-n-butyl

phthalate and xylene because of availability and adequate solubility at

temperatures near 0°C. The solutions are made up in a constant temperature

bath at 0°C, the temperature at which we require density balancing. The

final density is compared with that of water by forming a small water drop

in the solution and watching its motion. This is done using a micrometer

syringe and a thin teflon tube inserted into the solution from the top.

- 62 -
Exact density balancing is impossible to achieve when there is a temperature gradient across the water zone; it is very difficult to achieve in

isothermal experiments. Therefore, we do not attempt it; rather, we perform

identical experiments using solutions with densities higher and lower than

that of the water. If the same result is found in both cases, we consider

the observation to be independent of precise density balancing.

The experiments are performed in cylindrical glass cells similar to

that shown in Figure 1. The curvature of the cell results in a distorted

image i.e. a sphere appears flattened in the vertical dimension. Prior to

an experiment a polycrystalline cylinder of ice, previously formed in a

glass tube, is frozen into the socket at the bottom of the cell. The

solution is cooled to below 0°C and then poured into the cell until the end

of the ice cylinder is covered.

The cells are used in two arrangements which provide different thermal

geometries. In one case, the cell is placed in a constant temperature bath

maintained below 0°C. Heat flow is radial in this arrangement; the ice

cylinder and solution are isothermal. In these isothermal experiments, the

ice cylinders are generally not melted to produce a liquid zone. Instead,

a drop of water is formed in the oil and brought into contact with the ice.

In the other case, the sides of the cell are insulated, its bottom is placed

in a very cold bath, and a heat leak is introduced at the top by inserting a

metal rod into the cell and touching the surface of the solution. The heat

flow is approximately along the axis of the cell and a thermal gradient can

be established. This gradient is about 10°C/cm. in most experiments.

- 63 -
Experimental and Theoretical Results

The major experimental effort this year has used the semi- infinite

cylinder geometry shown in Figure 1. We have examined what happens at the

free end of the ice cylinder when a drop of water is brought into contact

with it or when it melts. We have also made some studies of the freezing

of the liquid zones.

Figure 2 is a sequence of pictures taken during the addition of water

to the end of an ice cylinder in an isothermal immiscible liquid of density

slightly greater than 1. A small drop of water was injected into the solution through a teflon tube as shown in 2a. The tube was then slowly lowered

until the drop touched the end of the ice cylinder. Upon contact, the drop

immediately ran down the cylinder and formed a liquid zone about its end as

seen in 2b. Pictures 2c and 2d show the ice cylinder after adding several

larger water drops. Note the detached drops which have floated upward in

the cell. This demonstrates that the solution density was higher than that

of the water and, therefore, would exert a buoyant force on the water which

would be in the direction opposite to that of the observed motion. Thus, the

distribution of water around the cylinder rather than in a "pendant" drop is

attributed to surface tension forces rather than inexact density balancing.

In Figure 3, we show a similar experiment in which drops of water are

added to the side of the cylinder at two points. Again, the solution density

was slightly greater than that of water. The stability of the liquid zones

suggests that the small buoyant forces acting on the water are insignificant

in comparison with the surface forces.

The distribution of the water about the cylinder seen in Figures 2 and 3

is not a new observation; Plateau obtained the same results in the last
century using olive oil, water, and an iron wire. From the perspective of

containerless crystal growth in space, however, this behavior suggests

possible difficulties in controlling the distribution of the melt when in

contact with crystal because of capillary run-off.

The shapes of liquid zones in temperature gradients can be significantly

different from those foand in isothermal experiments. Figure 4 shows the

successive changes in the distribution of the water when an ice cylinder is

slowly melted from its end in a temperature gradient. The ice and supporting

liquid are initially below the melting point (4a) . The temperature at the

top of the cell is then increased by lowering the heat leak. As the temperature

rises at the cylinder end, the ice melts and a cap shaped water zone develops

(4b). This zone slowly grows as melting continues. However, it retains its

cap shape with a slightly convex liquid-solid interface (4c) . Capillary run

off and rapid freezing occurred shortly after 4c, leaving a smaller liquid

zone and an irregular, distorted ice cylinder (4d)

.

A somewhat different process is seen when an ice cylinder is rapidly

melted at the free end. In this case, the water cannot form a stable cap

at the cylinder end like that seen in (4b, 4c). Rather the large volume of

water moves down the cylinder and stabilizes in a shape like those found in

the isothermal experiments (see Fig. 3). This process is shown in Figure 5.

An originally cylindrical ice cylinder was very rapidly melted at its end by

a thermal impulse. The mass of water produced by this process flowed down

the shank and stabilized at a point where the temperature was significantly

below the melting point as indicated by rapid freezing at the base of the

drop (5a) . Melting of the end of the cylinder and freezing at the base of

the water drop continued simultaneously producing a swelling of the ice
cylinder and a cap of water (5b, 5c). A planar ice-water interface developed

which then moved upward as the cell continued to cool. Eventually, a

spherical cap of ice on a cylindrical shank was produced. This was the

result anticipated for Walter's space experimentf^^

The most striking difference between the isothermal and temperature

gradient observations is the apparently stable cap of water formed at the

cylinder end in the latter experiments. This phenomenon is seen for solution

densities both greater and less than 1 at 0°C indicating that it is not

attributable to buoyancy. The stabilization of the cap may be due to changes

in the shape of the solid or to variations in surface tension because of the

temperature gradient. Quantitative experiments will be necessary to isolate

the stabilizing factor. .

'

. > In addition to the experimental observations described above, calculations have been carried out on the shape and stability of a vertical rotating

liquid zone subject to a weak vertical gravitational field. A discussion and

the details of these calculations are given in the Appendix. Here, we

briefly summarize the most important results. As illustrated in Figure 6,

we consider a liquid zone between two solid disks of radius R separated by a

distance L; in cylindrical coordinates (r,z) the boundary conditions on the

liquid zone are r(-L/2) = r(L/2) = R. The volume of liquid in the zone is

ttR^L so that in the absence of gravity the shape of the zone is cylindrical,

r(z) = R. We assume that in a coordinate system rotating with constant

angular velocity about the z-axis the liquid is stationary. We find that

the zone is stable if

^ L/2a < Tr{l - [(e^/2) h(a/R)]^ + ...}

- 66 -
1

where a = R/[l + pfi^R^/y] , e = Pga^/y,

h(a/R) = (9/8) [1 - (a/R)2 + 3(a/R)'+ + (a/R)^],

p is the density difference between the liquid and surrounding medium, y is

the liquid-medium surface tension, and g is the gravitational field. This

equation gives the leading correction to the zero gravity stability result;

the equation should be valid as long as [(e^/2) h(a/R)]^ < < 1. Since

h(a/R) > 0 for all (a/R) , it is clear that a weak gravitational field always

destabilizes the liquid zone. For no rotation the stability criterion becomes

1

(L/2R) < 7t{1 - [9e2/4]3}

with e = pgR^/y. In order for the correction to the zero gravity result to

be less than 10%, e < 0.02. For water in the earth's gravitational field,

R - 0.04 cm corresponds to e = 0.02; in 10 ^ gravity R can be 100 times larger.

Several lines of research for the future are indicated by the experimental

and theoretical work of the past year. The determination of what stabilizes

a drop of water at the end of an ice cylinder in a thermal gradient is of

high priority. This will require quantitative experiments and possibly

theoretical calculations. In addition, measurements of liquid zone stability

between circular plates in simulated low gravitational fields would be

relevant in order to determine the dynamics of instability and to delineate

the stability criterion for geometries not readily amenable to theoretical

treatment. For example, in general the gravitational field will not be

parallel to the rotation axis of the liquid; one must then solve partial

differential equations instead of the ordinary differential equations described

in the Appendix. The complexities of such calculations suggest that an

experimental simulation of the situation may be a better approach to the

problem.

- 67 -
References

1. Plateau, J.A.F., Experimental and Theoretical Researches on the Figures

of Equilibrium of a liquid mass . .

. , translated in Annual Reports of

the Smithsonian Institution, 1863-1866.

2. Bolza, 0., Lectures on the Calculus of Variations, Dover, New York, 1961.

3. Princen, H.M. , Surface and colloid science, 2^ (eds. Matijevic and Eirich)

Wiley-Interscience, New York, 1969.

4. Carruthers, J.R. and Grasso, M. , J. Crystal Growth 13/14, 611 (1972).

5. Carruthers, J.R. and Grasso, M. , J. Appl. Phys . 43, 436 (1972).

6. Carruthers, J.R. , Proceedings Third Space Processing Sjrmposium Skylab

Results, Volume 2, 837 (1974).

7. Walter, H.U., Proceedings Third Space Processing Symposium Skylab

Results, Volume 1, 257 (1974).

8. Pitts, E., J. Fluid Mech. 63, 487 (1974).

9. Heywang, W. , Z. Naturforsch. 11a, 238 (1956).

10. Padday, J.F., Phil. Trans. Roy. Soc. (London) ^69, 265 (1971).

11. Padday, J.F. and Pitt, A.R. , Phil. Trans. Roy. Soc. (London) 275, 489

(1973).

12. Gillette, R.D. and Dyson, D.C., Chemical Engineering 2^, 44 (1971).

13. Howe, W. , Ph.D. Dissertation, Friedrich-Wilhelms Universitat zu Berlin

(1887).

14. Hocking, L.M. , Mathematika 7_. 1 (I960).
Appendix I

Stability of a Liquid Zone

We apply the calculus of variations to study the stability of a liquid

zone (bridge) between two solid circular surfaces (plates) . We assume that

the two circular plates are perpendicular to a line joining their centers and

to the direction of the gravitational force and that the liquid remains at

the edges of the plates. The distance between the plates is L and the radii

of the upper and lower plates are and R, respectively. We define a

cylindrical coordinate system (r,(|),z) with origin at the center of the plates

and midway between them. This is illustrated in Figure 6 for R = R. We will

u assume that the liquid shape r = r(z) is independent of (|> . The boundary

conditions on r(z) are r(-L/2) = R and r(L/2) = R^. Further, the volume V

of liquid in the zone is fixed (strictly, the mass is fixed, but we assume

the density is constant), i.e.,

L/2 L/2

V = / Trr^dz = / G(r) dz = constant

-L/2 -L/2

In our coordinate system which may be rotating with constant angular velocity

^ about the z-axis, we assume that the liquid is stationary, i.e., we are

carrying out a static calculation in which the energy of the system is

completely determined by r(z), the shape of the liquid-vapor interface

(although we use the word vapor, the calculation is applicable to any fluid).

(2) The calculus of variations enables us to determine r(z) such that the

energy is stationary and further to determine whether the energy is a

minimum and the shape stable. The energy can be written as a sum of surface,

gravitational, and rotational energies, i.e..
L/2

E' = / F(r,z,r ) dz , -L/2

^

where

^ F(r,z,r^) = y2TTr(l + r^)^ + gp-nr^z - -|pi;^2^ ^

Here Y is the liquid-vapor surface energy, the subscript z indicates

differentiation, i.e., r^ = (dr/dz), g is the magnitude of the gravitational

acceleration which is in the negative z direction, and p is the density

difference between the liquid and vapor. Since the volume is fixed, we

introduce the Lagrange multiplier p and define H(r5Z,r ) = F(r,z,r ) -

z z pG(r) . If the integral

L/2

E = / H(r,z,r ) dz

-L/2

is a minimum, then E' is also a minimum since the integral of G(r) is a

constant. We now consider variations in r, i.e. r -> r + 6r and

r r + 6r . Defining

z z z L/2

AE = f {H(r + 6r,z, r + fir ) - H(r,z,r )} dz,

-L/2

expanding H(r + 6r,z,r + ^r^) to second order in 6r and 5r^, and performing

some partial integrations in which the boundary terms vanish since

6r(+L/2) = 0, we obtain

L/2

AE = / dz {[8H/3r - (d/dz) OH/3r )] [6r]}

-L/2

T L/2

+ ^ / dz {[32H/8r2-(d/dz)02H/8r9r ) ] [ (6r)2]+[92H/9r2] [ (5r )2]}

2 -L/2 ^ ^ ^

For E to be stationary requires that AE vanish to first order in 6r, i.e.,

OH/ar) - (d/dz) OH/9r ) = 0.

z
Evaluating this expression yields

_1 .1 '

y {r"^ (1 + r2) 2 - r (1 + r2 ) + _ ^^^2^-2 - p = 0.

z zz z /

This equation is called the Euler equation for the variational problem being

considered. The coefficient of Y is the curvature of the interface and the

above equation can also be derived by considering the pressure balance across

the liquid-vapor interface. The pressures in the liquid and vapor are

P (•if^2r2 - gz) + p' and p (^^v^ - gz) + p', respectively, where p' and p'

are constants. Then, using the Laplace equation which states that the pressure

jump across the interface is the product of Y and the curvature, we obtain the

Euler equation with p = p^ ~ ^v*

An expression for the voltme in the zone can be obtained by writing

the Euler equation in the form

{d/dz}{Yr/(l + r2)2 + (pg/2)zr2 - (p/8)fi2r4 - pr2/2} = (pg/2)r2

and integrating with respect to z from (-L/2) to (L/2). This yields

L

{Yr/(l-h:2) + (pg/2)zr2 - (p/8)j^2r'+ - pr2/2}| ^ = (pg/2TT)V

^ z = -L/2

For = R = r(+L/2), this reduces to

-— L/2

YR {(l-h:2) 2

I } = (pg/2iT)(V - 7rR2L). ^ -L/2

Later, we will discuss solutions of the Euler equation; temporarily

we assume we have found a solution of the Euler equation satisfying the

boundary conditions and the volume constraint (the voltmie constraint specifies

a particular value of p) . Assuming that r(z) satisfies the Euler equation,

our expression for aE can be written

L/2 — —

AE = TT / dz {-[pfi2r2+Y/r(l+r2)2][(6r)2] + [Yr/ (l+r2) 2] [ (gr )2]}

-L/2 ^ z z
For stability, we want E to be a minimum and consequently AE > 0 for all

possible variations 6r(z) . The conditions under which AE > 0 are well known

from the calculus of variations; we will simply state these conditions. First

3

the coefficient, Yr/ (1 + » °f

^"^^z^^

must be non-negative (Legendre's

condition). This condition is obviously satisfied, and the strengthened

Legendre condition (coefficient of (6r^)^ positive) is satisfied except when

r = 0 or r is infinite. On physical grounds, it is obvious that the liquid

z zone is not stable if r = 0; therefore in the remaining treatment, we will

assume that the strengthened Legendre condition holds. It is interesting

(9) that Heywang in his calculation for vertical zone pulling (identical rod

radii) used the condition r^ infinite as a criterion for instability.

The second condition for Ae > 0 is the conjugate point condition and

can be stated as follows. We first define a second differential equation,

the Jacobi equation, which is the Euler equation of the quadratic functional

AE. Since (<Sr) must satisfy the constraint

L/2

/ r(6r) dz = 0

-L/2

we introduce the Lagrange multiplier p and write

L/2

AE = AE - Try

J dz r(6r)

-L/2

The Euler equation for AE (r is fixed and we consider variations in 6r) is

3 1

[d/dz][YrW /(l+r^)2] + [p^V+y/vU+r^)^] W = -yr/2, where W = <Sr and

z z z W = <Sr ; this is the Jacobi equation of the original variational problem

z z for E. Let u(z) and v(z) be solutions of the homogeneous (y = 0) and

inhomogeneous (y 7^ 0) Jacobi equation, respectively, such that u(-L/2) =

v(-L/2) =0.
Define

z m = / r u dz,

-L/2

z n = / r V dz,

-L/2

and

D(z, -'L/T) = mv - nu.

For stability, it is necessary that D(z, -L/2) 0 for -L/2 < z ± L/2.

Obviously D(-L/2, -L/2) = 0; defining z* as the next greater zero of

D(z, -L/2), i.e. D(z*, -L/2) = 0, the stability criterion can also be

written as

L/2 < z*.

The point z* is called the conjugate of the point (-L/2).

We will now give a brief discussion of solutions of the Euler equation.

For g ^ 0, it is not possible to solve the equation analytically, and

approximations or numerical techniques are required. Extensive tables of

numerical solutions of the Euler equation (for fi = 0) are available from

J.F. Padday f''"^''"'''^ When g = 0, the solution of the Euler equation can be

reduced to a simple quadrature. Writing the Euler equation in the form

1

(y/r) (d/dr) [r/(l + r^)^] - — pfi^r^ - p = 0, and integrating yields

1

z Z

r/(l + r^) = (p/8y) f2^r^ + pr^/2Y + = A(r) where is an integration

constant and A(r) is defined by the second equality. Solving for r and

z integrating again gives

z = C_ + /dr {A/(r2 - a2)^},

where is an integration constant. For 9. = Oy the above integral can be

written in terms of elliptic integrals. To show this we define two constants
V and 9 such that

P = 2Y/[v(l + cos e)]

and

= V cos 6/(1 + cos 9).

We then have

z = + fdr{ [r2+v2cos9]/[(r2-v2cos29)(v2-r^)]^}.

2 1

Introducting the substitution r = vA(t,6) = v[l-sin^ 6 sin^ t]^

(provided v is positive), the above equation becomes

z = C2-v/dt {A(t,6) + cos 6/A(t,6)}

or

z = C2 - V {E(t/9) + cos 6 F(t/9)},

where F(t/9) and E(t/9) are elliptic integrals of the first and second kind,

respectively. The above equation and the equation r = vA(t,9) give a

parametric representation (with t as the parameter) for the solution of the Euler

equation in the absence of gravity and rotation. The solution contains three

constants v and 9 which allows one to satisfy the two boundary conditions

and the volume constraint. The properties of the solutions have been discussed

(12) by Gillette and Dyson. These authors, making use of earlier results of

(13) Howe, have determined the stability of the solutions from the conjugate

point criterion for the case R = R. Their basic result is that for a given

u

(L/R) , there is a unique stable solution provided the volume V in the liquid

zone lies in a specified range, i.e., defining V and V as the minimum and

maximiim volume, respectively, for which a stable liquid bridge exists, the

bridge is unstable if V < V, or if V > V„ and is stable if < V < V„ . For " A B A — — B

L < 2TrR, the solution corresponding to is constricted in the middle and the

one corresponding to V bulges out; for L > 2itR, both solutions bulge outward.

B
The values of V and V are given graphically by Gillette and Dyson. As

A B

pointed out by these authors, the extension of the stability calculation to

the case R ^ R should be straightforward,

u We will now consider the stability of a liquid cylinder. Initially, we

consider the zero gravity case. It is well known that the cylinder is

unstable when L > 2ttR; we will derive this result. It is clear that r(z) = R

is a solution of the Euler equation with p = (y/R) - 4 p fl^ R^ and the zone

1

volume V = 7T R^ L. It is convenient to define a = R/ [1 + p

9,'^ R^/y]^ and

n = z/a. In terms of n and for r(z) = R, the Jacobi equation is

W + W = -]ja2/2Y. nn Since we are only interested in when D(z, -L/2) = 0, we are free to choose

any non-zero value for y; we choose y such that (-ya^/2Y) = 1. Solving the

differential equation, we take

u(n) = sin(n + 6)

and

v(r)) = 1 - cos(n + 3)

where 3 = L/2a and both u and v vanish at n = -3(z = -L/2). The integrals m

and n are then

m = R a {1 - cos(n + 3)}

n=Ra{n+3- sinCn + 3)}

The equation D(z*, -L/2) = 0 can be written as sin ^ {sin ? - C cos O = 0,

where ^ = (3 + n*)/2 and ri* = z*/a. The root E, = 0 of this equation corresponds

to z* = -L/2; the next larger root is C = ^ or z* = 2-na - L/2. The stability

condition (L/2 < z*) is then

1

L < (2itR)/(1 + p 9^ R^/y)^.
For Q = 0, this is the well known result L < 2TrR. The above equation has been

(14) derived by Hocking by considering the dynamics of axisjnnmetric perturbations

of an infinite cylindrical column of liquid.

We now consider the effect of a weak gravitational field upon the

preceding stability result by carrying out a perturbation treatment. We take

the liquid volume V = ttR^L so that in the absence of gravity the liquid shape

is cylindrical. We expand all functions in a series in powers of g; the zeroth

order terms are the zero gravity results given in the preceding paragraph. It

is convenient to write the Euler and Jacobi equations in dimensionless form;

we define the dimensionless variables n = z/a, ^ = r/a, $ = W/a, e = pga^/y,

1

to = pn^a^/iy, P = ap/y, and M = (-ya/2Y), where a = R/[l + p^^R^/y]^. The

Euler and Jacobi equations are now

1 3

[^(1 + ^^)'^r^ - m /(I + ^^)^ + en - co^2 - P = 0

n , nn n and

3 1

[H** /(I + f^)^] + {[-I'd + -^2)2^-1 + 2^^>2} $ = "I'M

We expand m, $, P, and M in powers of e, i.e.

= "l'o(n) + e'Fi(n) + £^'^2(11) + •••

* = *o(^) + £^$2^^) •••

P = Pq + ePi + e2P2 + . .

.

M = Mg + eMi + e^Mj + . .

.

Recall that "I'gCn) = R/a. Substituting these expressions into the differential

equations, expanding all quantities in powers of e, and then equating

coefficients of e, e^, we obtain

(a/R) - a)(R/a)2 = Pq

¥1 + ^1 = n - Pi
^2 + H'p = (¥2a/2R) [3(a/R)2 - 1] - (a^? /2r) - p

+ $o[(3'l'2/^P - ('l'2/'i'0) -

(31'2/>1'J) + ('^2^/2>f2)]

-

$^ (^i^Z-^o) + $l(^l/*l'o)[(3/l'2) - 1] +M2 , -1 -3

where we have used the Identity 2bi = ^ q - . These equations can be

solved analytically by a straightforward but tedious method. The boundary

conditions on ^ are 1'(+3) = and

V = 4*2 dn = 2B1'o . -3

The conditions on and ^2 then

1'i(±3) = ^2(13) = 0 ,

/^g^i dn = 0 ,

and

/!b

^2>i'o^2 + dn = 0 .

The solutions for m-^ and >1'2 which satisfy the boundary conditions are

^l(n) = n - 3 sin n/sin 6

'i'2(n) = {(a/2R)(3a2/R2 - i)} {^2 _ (32/3) (1 + sin2n/sin23)

- (3n sin n/2 sin 3) + (32/6 sin2B) (1 - cos 3 cos n)

+ [6/sin B] [(n2 - 32) (cos n)/2 + (3/2) (cos n - cos 3)]}

- {a/2R} {32/3 - 3nsinTi/sin3 + [32/3sin23] [2 - cos2n - cos 3 cos nl

- (3/sin 3) (cos n - cos 3)}

We note that "i*! (3) = I'l (-3) and H*? (3) = ^Fo (-3) = 0; these values satisfy

the previously derived equation (for the voltime in the zone) which for V = itR2l

requires that 1'(3) = +'l'(-3).

It is desirable to express the conjugate point equation, D(z*, -L/2) = 0,

as a power series in e. We write
u(n) = uo(n) + euiCn) + e^u2(n) + ...

v(ri) = VqCih) + e vjCn) + V2(n) + ...

n* = nj + e nf + + . •

•

The conjugate point equation requires evaluation of functions at n*; it is

more convenient to evaluate these functions at n*. To do this we expand in

a Taylor series about n*; thus for any function f = fg + efi + £^f2» we have

. f(n*) = fo + e[fi + n* fo^]

+ e2[f2 + n* fi^ + n* fo^ + (n*)2 fo^^/2] +

where all quantities on the right hand side of the equation are evaluated at

n = n*. Expanding the conjugate point equation in powers of e, and equating

coefficients of e and we obtain the following equations for n* and n*

1 2 n* = -(uj/a)

r]* - -(u2/a) - (n*ui^/a) + (vi/ q) [I(uq^> i) + 1(ui^q)].

All quantities on the right are to be evaluated at n = ri* = 2Tr - 3; we have

introduced the notation

1(f) = f dT] . -e In obtaining the above results, we made use of the zeroth order solutions which

we take as (uq/m) = sin(n + 3) and (vg/a) = Mgll - cos(n +3)]; evaluation at

nj yields uq = 0, Uq^ = a, uq^^ = 0, K^FqUq) = 0, vq = 0, Vq^ = 0, Vq^^ = aMg,

and ICVqVq) = 27raMo¥oA lengthy but straightforward calculation yields

n* = (a/R) a2 Tr(7r - 3)

n* = -Tr{(3/2) + ai/4 + aia2 + (27/16) a^a^ - (3/2) aj}

- Tr3 {-2aia2 + 4a2} - 77^3 {3aia2 - 6ap

- 3^71 {(3/8) - (5/24) ai + (3/8) aia2 - (25/24) aia2 + (15/4) ap

- {•n-3cos 3/sin3} {-ai + (7/3) aia2 + (7/12) aia2 + 8a2}
- {TTg^cos^B/sin^B} h(a/R),

2 where = (a/R)2, a2 = [(Sa^/R^) - 1], and h(a/R) = (9/8) [1 - aj + 3ai + a^] .

The stability criterion is

3 < n* = n* + en* + e^n* + . .

.

This can be simplified since we know that instability occurs when 3 = tt.

Letting s = ir - 3 and recognizing that s < < 1, it is clear that the dominant

term in n* is the coefficient of (l/sin^3). The stability criterion can then

2 be written as s^ > (e^/2) tt^ h(a/R) or

1

(L/2a) < TT {1 - [(e2/2) h(a/R)]3}.

This eqxiation gives the leading correction to the zero gravity stability

result; the equation should be valid as long as [(e^/2) h(a/R)]-^ < < 1.

We note that it was necessary to carry out the calculation to second order

in in order to obtain corrections to the zeroth order result, i.e., the term

en* does not affect the stability criterion. Physically, this was expected

since a change in the sign of g (and consequently e) should not change the

stability.

It should be possible to extend these results to arbitrary values of e

by numerically solving the Euler and Jacobi equations. We wish to acknowledge

helpful discussions with R. F. Sekerka.



Fig. 2 The addition of water to the end of an isothermal ice cylinder. The temperature

is approximately -0.2°C

r

Fig.. 4 The melting of an ice cylinder

in a linear temperature gradient

c D

Fig. 5 The rapid melting of an ice cylinder

in a linear temperature gradient

Fig. 6 Rotating zone between circular plates

Task 5

Vapor Transport Synthesis and Crystal Growth of Oxides

H. S. Parker

Inorganic Materials Division

Institute for Materials Research

Summary

The use of closed tube transport techniques is often advantageous

in crystal growth where the desired material has a very high melting

point, a destructive phase inversion or high vapor pressure. Transport

is generally achieved in a temperature gradient by either chemical means

through the addition of a suitable transporting agent or by evaporationcondensation of materials with vapor pressures sufficiently high.

This section of the report discusses further efforts to determine

a suitable transporting agent for the chemical transport of ZxO^ alone

and in combination with Ta20^. No successful agent was found within

the limitations of temperature and pressure imposed by the quartz

reaction tubes.

The results of transport rate measurements on the transport of

mercurous chloride by an evaporation-condensation mechanism are presented. Experimental conditions were chosen to enhance and minimize

the convective effect. Because of the relatively simple transport

mechanism involved and the satisfactorily large rates obtainable, a

proposal was submitted to NASA for participation in the Apollo-Soyuz

Test Program.

- 69 -
Introduction

The major objective of this study is to investigate the effects of

convective and diffusive flow in vapor transport crystal growth utilizing

a closed tube system. Mass transport in the presence of a temperature

gradient can be achieved either by the addition of a suitable transporting

agent, generally a halide, or by a simple evaporation-condensation

mechanism. The general applications of this technique have been

described by Schafer [1] and Laudise [2]. Chemical transport can be

of particular advantage, if a suitable transporting agent can be found,

in cases where a high melting point or destructive phase transitions

make more conventional techniques difficult or impossible. In more

favorable cases, where the material of interest has a vapor pressure

sufficiently high, the choice of an evaporation-condensation technique

has the advantage of a simpler reaction mechanism. In both cases,

however, convective and diffusive effects are intimately interrelated

in a gravity environment. Independent control of parameters such as

temperature, temperature gradient and pressure can seldom be achieved

without influencing the relative contributions of convection and diffusion to the observed transport.

The initial material chosen for investigation by the chemical

transport technique during the previous year was a potentially superior

type of stabilized ZrO^, using la^^ to produce an orthorhombic, oxygenexcess form as detailed in the previous report [3]. These experiments

involved selection of a suitable transport reaction for the simultaneous

transport of ZrO^ and la.^^ in a molar ratios of approximately 6:1 and

8:1.
A second material was also chosen for investigation during the

present year as representative of an evaporation-condensation transport

process. Mercurous chloride was selected both from the standpoint of

its adaptability to this technique as well as its important optical

properties

.

The extremely large birefringence of mercurous chloride, approximately four times that of calcite, was first measured by Dufet [4] in

the late 1890' s. The possibility of utilizing this material for construction of polarizing prisms or other optical components has been

neglected, with the exception of one paper by Barta [5] on the growth

of Hg2Cl2 crystals from the melt and from the vapor although no characterization of the crystals is given. Preliminary work at NBS on

Bridgman vapor growth and optical properties has indicated that the

material is transparent from 0.4 ym to '^^16 ym and the birefringence is

in excess of 0.5 at 10.6 ym.

Mercurous chloride has a high vapor pressure [5] at moderate

temperatures, ranging from about 0.4 atmospheres at 350 °C to 44

atmospheres at the triple point, approximately 540 °C. See Figure 1,

also [5,6]. The vapor species in equilibrium with Hg2Cl2(s) has been

shown to be Hg2Cl2(g) at temperatures below about 375-400° and

Hg(g) + HgCl2(g) at higher temperatures [7,8]. The vapor pressure in

the temperature range of interest, 350-460 °C, ranges from about 0.5

to 10 atmospheres.

Extremely high transport rates in the absence of convection were

observed by Prof. Wiedemeier in his experiments on Skylab involving

chemical transport of GeSe with Gel,. The physical perfection and size

- 71 -
was also markedly better as compared to crystals grown in ground-based

experiments. This suggests that a series of crystal growth experiments

by a simple evaporation-condensation mechanism might be of value in

further understanding the phenomena of mass transport under convectionfree conditions. In this case, the transport agent and the chemical

reaction between transport agent and the material at the source and sink

end are eliminated. However, in the case of evaporation-condensation,

independent control of the total pressure in the system is lost, since

the vapor pressure of the material is fixed solely by the operating

temperature. The supersaturation and growth rate are, in part, determined by the choice of AT and thus Ap.

Experimental Procedures

Work on establishing a chemical transport reaction for the simultaneous transport of ZrO^ and Ta20^ was undertaken during the previous

year of this contract and the experimental details are described in

the first annual report [3].

Efforts to establish a suitable transport reaction for ZrO^ and

8Zr02:Ta20^ or 6Zr02:Ta20^ were continued using halogens and halogens

plus sulfur as transporting agents. An additional problem was the

severe corrosion of the quartz tube wall when simultaneous transport

of Zr02 and Ta20^ was attempted. Transport experiments up to the

practical working limits of temperature and pressure for quartz tubes,

1100-1200 °C and estimated pressures of 7 atmospheres failed to produce

any significant increase in mass transport above the milligram quantities

obtained in 168 hour runs at lower temperatures and pressures. In view

of these extremely low rates, further attempts to transport Zr02 were

abandoned

.

- 72 -
For transport of Hg2Cl2, three different furnace configurations

were used. In the initial experiments, a horizontal three zone furnace

was utilized. In order to provide greater flexibility in adjustment

of temperature gradients and to enable experiments to be done with the

transport tube vertical a multi-tapped furnace was constructed. By the

use of suitable baffles and shunt taps, it was possible to either

maximize the convective effect by having the hotter (source) end of the

transport tube at the bottom, or to minimize the effect by reversing

the gradient. Temperature was monitored by eight thermocouples

positioned along the transport tube. The tube holder was designed

to allow insertion of the ampoule into the furnace at temperature.

Approximately 30 minutes were required for the tube and charge to

establish equilibrium with the temperature of the furnace. A blank

sample tube was used to adjust the thermal profile of the furnace

prior to a run. An isothermal zone at least 5 cm in length was

established at the higher temperature end with a linear gradient to

the colder end where a second isothermal zone was established.

All transport experiments were carried out in fused quartz tubes.

Prior to filling, the tubes were cleaned and evacuated to pressures

—6

lower than 2 x 10 torr and baked out at 500-700 °C while under

vacuum. The initial horizontal transport experiments were done in

tubing of 1.8 cm inside diameter. The vertical transport experiments

with the hotter end at the bottom (maximum convection) were done in

1.1 cm inside diameter tubes. In experiments with the hotter (source)

end at the top, it was necessary to provide a support to hold the

charge. By careful selection, it was possible to choose 1.1 cm outside

- 73 -
diameter (0.8 cm inside diameter) quartz tubes which fit tightly

within a 1.1 cm inside diameter tube. An open fused quartz grid across

the upper end provided a support for the charge. This method also provided a convenient adjustment of the transport path length.

The mercurous chloride was obtained from a commercial source in

powder form and spectroscopic analysis indicated the following impurities

in weight percent: Ti:0. 003-0. 03 percent, Fe:0. 001-0. 01 percent. Si,

Cu, Mg:0. 001-0. 0001 percent. In order to provide a dense charge for

the transport experiments, the as-received powder was isostatically

pressed into cylinders of appropriate diameter. A few experiments

were performed using once-transported material (Bridgman furnace) but

no difference in results were noted. In all cases, the tube and charge

—6

were thoroughly dried at 100 °C at pressures of less than 5 x 10 torr

before sealing off. The presence of water vapor has been shown by

Gucker and Munch [8] to enhance the dissociation of Hg2Cl2 vapor.

Experimental Results

Horizontal transport experiments were primarily qualitative in

nature for the purpose of observing the magnitude of the transport

rates to be expected. At a source temperature of 450 °C and a AT of

100 °C, corresponding to a Ap of 8.4 atm., an entire charge of 54.1

grams was transported in less than 68 hours over a distance of 13 cm.

Over the same transport distance, a source temperature of 390 °C and

a 20° AT, corresponding to a A p of 0.5 atm, resulted in the transport

of 15.2 grams during a 48 hour period. This corresponds to a transport

rate of 0.32 grams/hr.

- 74 -
The results of transport experiments with the ampoule in a vertical

position are summarized in Table 1. With one exception, all runs were

made at temperatures in the region where the dissociation of Eg^Cl^

vapor would be minimized. This imposes a maximum experimental temperature of about 400-410 °C. Because the primary objective was the

observation of transport rates under different conditions, no attempt

was made to obtain large single crystals.

Runs made with the bottom of the transport tube hottest to maximize

the effects of convection were chosen to cover a wide range of temperatures and differential pressures. Unfortunately, the times chosen

for the two highest pressure runs were too long and the entire charge

was transported, making an accurate rate determination impossible.

In the opposite configuration, top of the tube hottest to mimimize

convection, all experiments were done in the 325° - 410 °C range and

the duration of the runs was drastically shortened. As expected,

transport rates were much lower. A marked increase in rate was noted

when the Ap was increased from 0.9 to 1.6 atmospheres by lowering the

temperature at the deposition end of the tube. Reduction of the

transport distance to 7 cm resulted in an increase of the rate to

0.660 g/hr which corresponds to a volume deposition rate of about 0.09

cubic centimeters per hour, based on a density of 7.15 grams per cubic

centimeter for Ylg^Cl^. This would be sufficient to produce crystals

large enough for property measurements and characterization in

relatively short experiments of 6-8 hours duration.

In both sets of experiments, the transported product was coarsely

polycrystalline, with the largest crystals 3-4 mm in size. The largest

- 75 -
Q. <]

00 « • • O O 00

O % % -^J,

COU

•H

4-1 U

0) >

CO

fl: <Uao

(0uH

CN

iH

CN

bO aO

CO H

00 00

m 1 1 1 1 1 1 1 • o o o o o o O

p Csj .H <-{ iH iH rH rH e X X X • o CO 00 as oo 00 CU • • • • • • • J3

CO rH CN rH vO in

<u

A

A 4-1 iH A A O 4JUOa •U COc o a) CO rH '

a\ CO

O

u

m ctj CM LO o vO 4-J C Pi M O ch o VO

CO • • • • • • • IH

>-i o o o o o O o o H M A A A A C

Q)

4-) ^ W)

U

4-»

C S o CO CO o CM o CU to (U o M iH rH rH M rH rH rH T3

CT

H rH U O • • ( . - O • rH > (U g u O CU o 3 u

•X3 o 4J 3

•H CO C)0 4J u 4J •H '

•H (U

O fa 3 O rH CO in CO a O Ou M O o C^ M CN CO 00 CJ^ O e (U 60 • • • \z • • • • CU CU CU < S

CM o CN

M

O CO CM 0) 4-1 4-) C/3 CO eg rg g 'TD CO U u U H 10 <u • C rH u

CO o CO o

0)m

OO in rH 3 a s vO 00 <j\ • • • • 4J CO \o u 4-1 O > H CU CO CO 4J IH u <u •> o o u3 (U c

CO 4-1 o

CO •H

U • 0) CU o u

rH

O <f •<r

au Pu CO • u o o 00 CM >-i CO •H « CO • • • • • • • 3 C a

«» o iH CO o O o o O i-> CO o U CO o CO o- 4-J CO

4-1 CO •H

CO > d) t3

(30 QJ

O

a

U § CO

C \ o o O o o m o o 3 •H

4-1

U rH o in CN in in 4J

U CJ 01)

M 4J 0 en <r CO CO CO fO CO ^ c o U •H CU •H

4-) PQ 0) rH

u

CO •H •H 1 B 3 4-1 CO

(U cr

C CO

H

wW <: § o o o eg in o o O 0 00 rH a\ o iH rH H CM CO <r CO <r ^1 ^1 ^1 - 76 -
crystals were produced in the lowest temperature and lowest Ap run,

at a deposition rate of 0.021 grains per hour. Several attempts at

seeded growth were unsuccessful. In every case, the seed evaporated

before deposition occurred.

Conclusion and Discussion

In view of the extremely low rates obtained in attempts to transport

Zr02 with halogens or halogens plus sulfur, no further efforts to determine a suitable reaction were made. It is suggested that, if transport

of Zr02 were essential, it would be necessary to utilize a flowing

rather than a closed transport system and to utilize suitable refractory

tubes to allow the use of higher temperatures than is possible in fused

quartz ampoules.

Measurements of the transport rate of Hg2Cl2 by evaporationcondensation under conditions of minimum convection have indicated that

0.4 to 0.660 grams per hour can be transported in the 350° - 400 °C

temperature range. These rates are sufficiently high so that specimens

of reasonable size can be obtained in experiments of relatively short

duration. The problem of preventing seed evaporation before steadystate transport and growth occurs is being investigated further.

Because of the high transport rates obtainable and the relatively

simple evaporation-condensation mechanism involved, a proposal for

growth of Rg^Cl^ crystal was submitted to NASA for participation in

the Apollo-Soyuz Test Program.

- 77 -
References

1. a) H. Schafer, "Chemical Transport Reactions", Academic Press,

New York, 1964.

b) H. Schafer, "Preparative Methods in Solid State Chemistry",

P. Hagenmuller, Ed., 251-277, Academic Press, New York, 1972.

2. R. A. Laudise, "The Growth of Single Crystals", 225-256, PrenticeHall Inc., New Jersey, 1970.

3. E. Passaglia and R. L. Parker, "NBS Materials Science and Manufacturing

in Space Research", NBS Report No. NBSIR 73-402, 25-32, November 1973.

4. M. H. Dufet, Bull. Soc. Franc de Mineral. Crist. 21, 90 (1898).

5. C. Barta, Kristall and Technik 5^, 541 (1970).

6. Gmelins Handbuch der anorg. Chemie, System No. 34, Vol. B, 438-440,

Velag Chemie, Bergstrasse (1967).

7. J. Weland, Helv. Phys. Acta 11, 46 (1929); 11, 77 (1929);

Z. Phys ik 77.. 157 (1932).

8. Gucker, Jr. and Munch, J. Am. Chem. Soc. 59, 1275 (1937).

- 78 -
Temperature, °C

1.2 1.4 1.6 1.8

•lO-^/T

Figure 1. Vapor Pressure of Hg CI . Data" from references [5,6].

Task 6

Surface Traction and Other Surface Phenomena

Alan L. Dragoo

Inorganic Materials Division

Institute for Materials Research

Summary

Work was continued on the problem of steady thermo-capillary

convection cells In nearly spherical drops. Computer programs

were completed to plot the streamlines for such cells.

Regarding the problem of the enhancement of evaporation

purification rates by capillary convection, a convectlve

diffusion problem was solved by the Green's function method.

The concentration was found to be expressible as a simple

diffusional term plus a term for the effect of the non-uniform

rate of evaporation upon diffusion plus a term for the effect

of convection upon diffusion. The very complicated forms for

non-uniform evaporation term and for the convectlve term

required the development of a very approximate expression for

the rate of evaporative purification in order to make simple

estimates. An estimate of the convectlve enhancement of the

purification rate was performed for the removal of CaO-trace

impurity from molten alumina.

- 79 -
Introduction

This study of surface traction and other surface phenomena

during the past year has continued both the work on the analysis

of steady thermocapillary convection in liquid, drops and the

work on evaporative purification of the previous year. Most

of the results of these two areas of work are presented in

the two preprints which are attached to this report:

Alan L. Dragoo, "Steady Thermocapillary Convection

Cells in Liquid Drops, " presented at the

International Colloquium on Drops and Subfiles,

Pasadena, California, August 28-30, 1974. (See Appendix III)

Alan L. Dragoo and Robert C. Paule, "Ultrapure

Materials: Containerless Evaporation and the

' Roles of Diffusion and Marangoni Convection,"

presented at the AIAA 12th Aerospace Sciences

Meeting, Washington, D. C, January 30 - February 1,

1974. (See Appendix I, Task 2)

These two papers will be summarized briefly in the next section.

Appendix I to this report contains the details of the solution

of the convective diffusion-evaporation problem, the solution

of which was treated approximately in the paper on "Ultrapure

Materials."

- 80 -
Theoretical Results

The question of how much surface tension-gradient driven

convective flow might enhance mass transport in a nearly

spherical, levitated melt at zero-g can be divided into two

parts

1) How rapid is -the convective flow under steady-state

conditions?

2) For the most-rapid convective mode, how much does

capillary convection enhance the rate of

evaporative purification?

In answering both questions, creeping flow was assumed, so

that all equations could be linearized with respect to the

velocity. More important, thermal gradients along the

surface of the melt were considered to be the sole cause of

the surface tension gradients and the presence of a single

condensed phase was assumed. The effect of these tx^o

assumptions, in the case of most liquids, is probably to

Overestimate the convective enhancement of the purification

rate.

Regarding the assumption of surface thermal gradients

as the sole cause of the capillary convection, surface

concentration gradients were considered to be insignificant.

For many liquids undergoing the loss of an impurity

by non-uniform evaporation, the change in surface tension

- 81 -
due to thermal gradients and the change due to concentration

gradients may tend to cancel each other out. However, since the

surface concentration of an impurity is likely to be reduced

greatly soon after evaporation begins, the concentration effect is

expected to diminish quickly. The thermal gradients then would be

left as the predominant source of the long-term capillary convection.

The second assumption, that of a single condensed phase, is

probably the more critical one in practice. The drag upon the rise

of a bubble due to contaminants at the interface has been recog1 2 nized for many years ' . A similar "problem" would occur if a

solid phase were present on the surface of a melt. However, this

problem is more likely to be troublesome in the case of an oxide

scum on the surface of a molten metal than iri the case of a molten

oxide such as alumina which was considered in "Ultrapure Materials."

In addition, the problem of a surface contaminant requires the consideration of a specific chemical system and of concentration

distributions about which there is little information and which

unnecessarily complicate the answer to our question at this point.

To answer the initial question in as general a way as possible, the

answer can be given by setting an upper limit on what is likely to

be observed in practice. Such an answer can show which materials

require further consideration with respect to evaporative purification and which ones can be ruled out.

The answer to the question of hov/ rapid the thermal capillary

- 82 -
convection might be was worked out mostly during the previous

contract year. This work was re-examined this year in the course

of preparing a paper for the International Colloquium on Drops and

Bubbles. This year, computer programs for plotting the streamlines

were completed; examples are given in the paper on "Steady Thermocapillary

Convection Cells." Th^ circulation time was redefined to be the

reciprocal of the average vorticity of a convection cell. The circulation time provides a means of comparing the rate of circulation

in different liquids under similar thermal conditions and between

different convective modes—different thermal conditions—for the

same liquid.

In addition to capillary convection, a liquid can respond to

non-uniform surface tension by deformation of its surface. In general,

both will occur. In the paper on "Steady Thermocapillary Convection

Cells" both were included—in fact, both must be included to satisfy

both the normal and th^tangential stress balance conditions at the

surface of theliquid. However, in the paper, the emphasis was placed

on the convective phenomenon. For completeness, the surface deformation will be compared here with the rate of convective flow. For

the (n,m,a)-mode, this comparison can be made by calculating the ratio

of the radial strain S

^ to the ^aracteristic speed U ^ , mn ^ mn

where is the coefficient of viscosity and 3^ is the coefficient of

surface tension. The significance of the deformation of the surface

- 83 -
decreases as the order n of the mode increases—that is, as the

number of hot and cold spots on the surface of the drop increases.

The second question—how much does convection enhance the rate

of evaporative purification—was examined in the paper on "ULtrapure

Materials" for the ease of thermocapillary convection and for the

purification of molten alumina.

To find an expression for the evaporation rate, a convective

diffusion problem was solved in which the convection cell was that for

the lowest order mode—the (l,0,e)-mode—obtained from the solution

of the steady flow problem. This mode also has the most rapid circulation rate. The solution of the differential equation for convective

diffusion, the boundary conditions and the initial condition was

3 achieved by the Green's function method. In general, the concentration can be written as

CCr,Q^t) = C^C^.t) C-iT C^L^.t) h>^Cco:>&) , (2)

where P^(cos6) is a Legendre polynomial. The coefficient c^(r,t)

can be expanded as

where c (r,t) is a diffusional term, c' (r,t) contains the effect

o nD

of non-uniform evaporation upon the diffusion process and c „(r,t)

nU

contains the effect of convection on the diffusion process. Due to

the natures of the non-uniform evaporation terms in the boundary

condition and of the convective term in the convective diffusion,

equation, c (r,t) is coupled directly or indirectly to all the

n coefficients in the set {c (r,t)}. An exact solution for (3) requires

m

- 84 -
the solution of an infinite hierarchy of equations—an impossible task.

However, useful approximations can be obtained by considering the

problem as a perturbation problem. Then, after deciding first

what the most important mass transport process is, the integrals

can be evaluated to find the first-order corrections to the concentration terms and to the- expression for the evaporation rate.

A glance at the flow pattern for the (1, 0, e)-mode—Fig. 1

in "Steady Thermocapillary Convection Cells''—will show that

diffusion must be the most important step in bringing the

majority of impurity molecules to the surface in most instances.

In this work, isotropic, radial diffusion was taken as the most

important mass transport process —the solution of which was

4 5 obtained by Berthier and was summarized by Crank . The solution

for diffusion in a sphere of radius R with uniform evaporation from

the surface is

(4)

where

is the Sherwood number, is the evaporation rate parameter,

D is the diffusion coef f icient , ,C is the initial concentration,

o

'

j^(x) is the zero-order spherical Bessel function and the

"wave-numbers" {3 } are given by the root equation

(5)

Using the solution to this diffusion problem, the integral

expressions for c' (r, t) and for cnU

(r, t) were evaluated to

- 85 -
find the first-order corrections due to non-uniform evaporation

and to convection.

Since the evaporation rate parameter a(T) was expanded only

to first-order in the temperature T, as

(6)

where the temperature is given by

T - % + Xr€^o%& J (7)

the rate of evaporation was expressed as

where the integral was over solid angle. Assuming that c^ (R, t)

- Cqp (R, t), approximate solutions were found for c^^ (R, t) and

for c^y (R, t) , namely

C,oCf?,i) c: 4LU, Ca L,

^

±

t ;—TTT?

where L, = T.R /D, (10)

- 86 -
T

U = -(Y^T^R/3p) is the characteristic, speed for the (1, 0, e)-Tnode,

is the temperature coefficient of the surface tension, { 3^}

is given by Eqn. (5) and {3^'''^ is given by

(L^')j,((iS') = /^."V. ^Z^-'") (11)

Eqns. (9) and 0-0) were too complicated for estimating the

evaporation rates, so an attempt was made to identify the most

important terms in the sumjnations to factors m.ultiplying the

isotropic rate (t) = Attq^c^j, (R, t) . The approximation which

was obtained is

where the Peclet num.ber (mass)

A/^ =r u ^R/O - - ^ 77 /? V3/U O)

can also be viewed as a Marangoni number (mass) , N

Mam

- 87 -
REFERENCES

1. V. G. Levich, Physlcochemical Hydrodynamics , Prentice-Hall

Englewood Cliffs, N. J. (1962), Chapters 7 and 8.

2. J. F. Harper and D. W. Moore, J. Fluid Mech. 32., 367 (1968).

3. P. M. Morse and H. Feshbach, Methods of Mathematical Physics , McGraw-Hill Book Co., Inc., New York (1935), Chapter 7.

4v.G. Berthier, J. Chim. Phys . 49, 527 (1952).

5. J. Crank, The Mathematics of Diffusion , Oxford University

Press, London (1956), 6,34.

6. N. D. Young, J. S. Goldstein and M. J. Block, J. Fluid Mech.

6, 350 (1959).

7. G. Arfken, Mathematical Methods for Physicists, Academic

Press, New York (1966), Chapter 12.

8. Handbook of Mathematical Functions , edited by M. Abramowitz

and I. A. Stegun, National Bureau of Standards, Applied

Mathematical Series, No, 55» U. S. Department of Commerce,

Washington, (1964).

- 88 -
Appendix I: Solution of a Convective Diffusion Problem for ThemocapillaryDriven Convection in a Drop and Evaporative Removal of an Impurity

We imagine that a nearly spherical drop of radius R has a temperature

field described by

T =T^ + rcos 6 (I .1)

where is the average temperature <T> of the drop and is the "vertical"

temperature gradient (8T/8z) along the axis of the drop. The thermocapillary

velocity field in the drop according to the result of Young, et. al.^ ,

has the components

u^ = (U^/R^) (R^-r.^) cos 6 (l.2a)

u = -(U^/R^) (R^-2r^) sin 6 (l.2b)

o

u, = 0, (1. 2c)

where the characteristic speed

U*^ = -Y^T^R/3y (1.3)

is a function of the temperature coefficient of the surface tension

Yip = (8y/9T) , of the coefficient of viscosity, y. Further, we assume

that the evaporation coefficient a can be expanded as a power series

in the temperature and, to first-order, is

a(T) = a + -a.T, Rcos 9 (l.A)

o i 1

To find the concentration c as a function of position r in the drop and

of time t, the convective diffusion equation

|| 4- u • Vc = DV^C, " (1.5)

where D is the diffusion coefficient, the boundary conditions

= a(T)c(R,t) (I. 6a)

dr

R

I7I =0 (I. 6b)
and the initial condition c=C^, O^r^R must be solved. Having

-> -> found ; c(r, t) , the concentration distribution c (R, t) can be readily

obtained, and the rate of evaporation from the drop

a(T)c(R, t)di7 (1.7)

where dQ, is an element of solid angle, can be calculated.

Since the problem has axial symmetry—that is, the temperature,

the velocity components and the boundary conditions are independent

of the angle cj) —we anticipate that the solution can expanded in

a series of Legendre polynomials P^(x), x = cos 0, as

c(r, X, t) = c (r, t) (-1)\ (r, t)P (x) (l.8)

o X J n n n=l

Substituting (1.8) into (1.5) and using the integrals evaluated in

Appendix II, we obtain for the coefficient c^(r, t) the inhomogeneous

differential equation

r

where H is the operator

i d_ / 2 d_

2 dr V r

^

The details of the source term v/hich is written formally as the

operator-operand relation ^^c are

^ n n n where

n r ni 8r r L 2n-l dr ^„ 2n+3 9r J

n>2

^(e), . J [-eCbzH. + l£«)l2±2i . ^"1 (Liib)

n e 2n-l r 2n+3 r J
The factors and f appearing In (1. 11a) and (I. lib), respectively,

are given by

= •(U^/R^)(R2-r^) (1. 12a)

= -(u'^/R^)(R^-2r^) (I.12b)

The homogeneous part of equation (1.9) is merely the radial part

of the diffusion equation in which the concentration has a 9-dependence

,

The source term on the righthand side of (1.9) contains the effects

of convection.

Substituting (1.8) into the boundary condition (1. 6a), we

obtain

9c

dr

= a 6 c (R, t) - a,T,R6 c (R t) + a c (r t) I „ o no » ^ 1 1 nl ' on' 'n>l

is. -Wfe -Vl^^' ^>ln.2 ^fSs'^n+l

(1.13)

Assuming the convective effects to have been absent prior to t = 0,

the initial condition becomes

c (r, t) = <5 C , 0 ^ r ^ R (1.14) n no o

Equations (1.9) and (1.13) may be integrated by the method of

3 \ Green's function . The Green's function G (r, tlr , t ) which is n I o o required to find the coefficeint t) must satisfy the inhomogeneous

differential equation

_ n(n+l)_. 1 !|n ^ _ ^ _ .,^^3^ n Z n D ot o o
and the boundary conditions

dr ^n

a + G

R

° ^

= 0 (I. 16a)

R

: .. dG

! O

The required Green's function is

0=1

[3^ + L(L-l)-n(n+l)][j (3 )]^ ^I'^^^

where u(t-t ) is the unit step function—u(t-t ) = 1 if t < t and

=

o 00

0 if t^ > t, j^(x) is a spherical Bessel Function, the coefficient

is defined by the root equation

or

a-h.)j„(B„) =e„j„+i (g^) (i.i8b)

and

L = a R/D = Sc (1.19)

o

is the Sherwood number. The set of coefficients ^3^^ depends on the

order n of the spherical Bessel function; however, to simplify

notation, this dependence will be indicated by a superscript "(n)" only

when a superscript is needed to avoid confusion.
Writing (1.9) and ( .15) in terms of the variables r^, t^,

multiplying (I.9)by the Green's function and (1.15) by c^(r^, '

the two equations are combined and then integrated using Green's

theorem to obtain

t

c (r, t) = c^^ (r, t) 6 + f dt [G

^

C - C

^

G ] n ' OD ' no o n dr n L n dr n _ Jo o 'R o 'R

t .R dt

^ 2 dr r G (r, tlr , t ) o o n ' o o

o

.^c(r , t ) (1.20) n o o where

exp[-3^Dt/R^] r/R)

c^^ (r, t) = 2LC /. 5—^^ "

. (1.21) °° ° ^

[3^ + L(L-l)] Jo^V

is the solution to the problem of diffusion in a sphere without

convection and with uniform surface evaporation^. Formula (1.21)

can be obtained from (1.9), (1.13) and (1.17) by taking n = 0 and

by ignoring the convective and the non-uniform evaporation terms.

Writing (1.20) as

we can identify, in addition to the diffusional term c^^^, the non-uniform

evaporation term c '^ and the convective term c ... We see from (1.22) that the

nD nU zero order coefficient c (r, t) is not necessarily equal to c (r, t)

— 'J UD

c^^ is only the diffusional part of c . The non-uniform evaporation

CD o term can be v/ritten out using (1.13), (1. 16a) and (1.17) as
. — —

(1.23)

Since the factor c in the integrand of cnU

is complicated,

n the details of c will not be given here.

nU

' • To find the rate of evaporation from the surface, we

first set r = R —but we do not set r = R —to obtain the

concentration at the surface. This introduces a little

simplification into (1.20), (1.21) and (1.23) by removing

the ratios of spherical Bessel functions i (6^r/R)/i (6 ) o 0 o a and j (3 r/R)/i (6 ). The rate of evaporation can be. obtained

n a no

by substituting c (R,t) into (1.8) and then by using (1.4)

and (1.8) in the integrand of (1.7). Ignoring the small

deformations of the surface which sould bring in the higher

order terms, the rate of evaporation is

Although the rate of evaporation only depends explicitly

on the concentration terms C-(R,t) and c^(R,t), we must bear

0 / I in m nd that CQ(R,t) and c^(R,t) are found by solving a

hierarchy of equations which couple all of the concentration

terms out to an indefinite order.

o

n

(1.24)
An apparent weakness of the Green's function method is that

we are ultimately required to know a. priori what we have set

out to find. In this instance, we must know the complete set

of concentration terms to solve the hierarchy of equations.

However, this paradox can be circumvented by making a physical

guess as to which concentration terms are the most important

ones in the set. Here, we will assume that the pure isotropic

diffusion term predominates, that c^^(r,t)- c^(r,t) > c (r,t)|

(JU (J n ' n and we will use CQp(r,t) as given by (1.21) to estimate the

perturbations caused by non-uniform evaporation and by convection. Further, we will only use this approximation to

obtain as approximation for the evaporation rate.

Solving (1.23) for the non-uniform evaporation term, we

find

(1.25)

where

(I. 26a)

(I. 26b)

and ^ (1.27)
Turning to the convective term, we find

'-00

' : (1.28)

The integrals

X

are evaluated in Appendix II.
Appendix n : Some Definite Integrals of Legendre Polynomials

and of Spherical Bessel Functions

Legendre Polynomials

The orthogonality condition for Legendre polynomials

is

(II.L.l)

where x = cos

The integral over the product of three Legendre polynomials

/

(ILL. 2)

can be obtained from the more general integral over the

product of three spherical harmonics ^ . The integral is

found to be

Similarly,

( ILL. 3)

Spherical Bessel Functions

The orthogonality condition for spherical Bessel functions

defined over the range 0$ x <:1 can be obtained from the orthog gonality condition for ordinary Bessel functions since the

spherical Bessel function
From formula (11.45) in the Handbook of Mathemetical Functions,

the orthogonality condition is found to be

j'^^^j(^^^yn(^<r^^ - 6>

, if V ^ a, n > -1;

- if V = a, n > -1, b = 0;

:f V = a, n > -1, b ^ 0;

(Il.S.l)

where 3i,32j'-'3re positive roots of

^^^iO;^^/3o) ^^uj^Ufio) =0 (U.S. 2)

The symbol "j^(x)" denotes the derivative

dj (x)/dx.

n The orthogonality condition (II.S.l) applies when

both spherical Bessel functions have wavenumbers, 6^, which

satisfy the same root equation. We will now evaluate two

integrals in which the wavenumbers satisfy different root .

equations. Let {B} satisfy the equation

=• fij^^iC^^ (U.S. 3a)

and {X} satisfy = A^^c?.) (U.S. 3b)

These two conditions can also be written as

J

. /-NX - -2 - ^ CA^ (U.S. 4b)

respectively.
The integral

^6 can be evaluated by using the differential equation

X^jJ^'m -tX^^W +/7y^-nt*A-^')7^^^x) =-0 (U.S. 6)

.2 , /*

to obtain

This integration can be performed immediately and

ACft^-A^ '^<^"-' (U.S. 7)

The final equality is obtained by using the root equations

(U.S. 4a) and (II. S.4b).

The second integral to be evaluated is

The integrand of (U.S. 8) can be obtained by operating on

the integrand of (U.S. 5), n-^n-1, with

(2) Thus, I^ can be found by applying the same operator to

the result for I (U.S. 7): n

(U.S. 9)
The differentiation of (II. S. 9) is straightforward, but tedious.

The derivative of the spherical Bessel functions,

the recurrence relation

J ^ ' C^l^^i) (II.S.11)

(2) and the root equations are used. The integral I is n

-r- to
Appendix III

STEADY THERMOCAPILLARY CONVECTION CELLS IN LIQUID DROPS

Alan L. Dragoo*

Institute for Materials Research, National Bureau of Standards

Washington, D. C. 20234

ABSTPACT

A nominally spherical drop is used as a model for a theoretical analysis

of thermocapillary convection and for estimates of convective flow rates in

"levitated" melts at zero-g. Since in practice temperature fields and the

resulting convective flov/ can be more complicated than the simple vertical

temperature gradient and the single vortex ring, respectively, the convective

flow arising from a general steady-state temperature field is analyzed. Expressions for the components of a steady velocity field are obtained by

adapting the analytical method of Miller and Scriven. The vortex rings are illustrated by means of typical streamlines for the simpler, more symmetric,

temperature fields. The circulation time is introduced as a measure of the

rate of circulation in a convection cell and typical values are given for

several materials.

INTRODUCTION

IVhen buoyant forces are negligible, such as in a space laboratory,

convective flows may still occur in a liquid as the result of gradients

in the surface—or interfacial tension. These convective flows are commonly

called the Marangoni effect although the Marangoni effect includes both this

phenomenon of convection and the phenomenon of the deformation of a free

liquid surface (Ref . 1) . Among the causes of gradients in the surface tension

are gradients in the concentration and in the temperature along the surface

of a liquid.

Concentration and temperature gradients may not be completely eliminated

in many processes, and in some instances, their presence may be necessary to

produce the convection which is desired. The growth of crystals from a melt

is a process in which the elimination of convection in the melt is desireable

because convection produces non-uniform growth conditions, and, thereby, an increase in the niomber of dislocations in the crystal . The evaporative purification of a levitated melt is a process in which the opposite result is

desired: rapid convection is important here because it increases the rate of

purification by replenishing the impurity concentration at the surface and

because it tends to maintain an uniform composition throughout the melt.

Financial support was provided by NASA under contract W-B ,475 #1.
The question of how much surface tension driven flows might enhance the

rate of purification of a levitated melt at zero-g is the motivation for the

work reported here. As a model which will begin to supply part of the answer

to this question, we will consider thermocapillary convection—that is,

convection resulting from a temperature gradient along the surface (Ref. 2) —in

nearly spherical drops. The temperature field at the surface of the drop

will be treated in a general way by writing it as an expansion in spherical

harmonics. The model will examine one of the convective modes corresponding

to one of the terms in the expansion of the surface temperature field. The

equations of motion—the Navier-Stokes equations--will be solved within the

assumptions of a steady-state and of creeping flow--terms which are nonlinear

in the velocity will be ignored. The mathematical analysis will proceed along

the lines of the method which Miller and Scriven (3) applied to the oscillations

of a fluid droplet although here we will not retain the time-dependence of

their problem. The physical boundary conditions will account for both the

convective flow and for the deformation of the surface so that both aspects

of the Marangoni effect will appear in the problem. The velocity field

components which are obtained from the Miller-Scriven analysis will be used

to derive the circulation time x whose inverse characterizes the rate of

circulation within a convection cell. An expression for x will be worked

out in detail for convection cells having axial symmetry. Estimates of x will

be given for a variety of materials when the convection pattern is a single,

axially symmetric vortex ring. Also, relative circulation times will be

calculated for several higher order, axially symmetric convective modes.

Illustrations of these convective modes will be given.

THE TEMPERATURE FIELD

The temperature field responsible for the convection is considered to be

a general, non-uniform, but steady, field which can be written as: n

T(r,e,({)) = T + ^^t"^ (9, (J)) (a=e,o), (1) o n=l m=0 a mn mn

(2a)

(2b)

are spherical harmonics as defined by Morse and Feshbach (4) . The function

(cos0) is an associated Legendre polynominal. The simple case of a constant

vertical temperature gradient through the drop—Y^^^ = P-j^ (cos9) = cos 6 — is

a special case of the solution of Young, Goldstein and Block (5) who derived

the velocity field in a bubble rising in a vertical temperature gradient.

where, „e , Y = P (cosO) cos m d)

. mn n Y = P (cos0) sin m d) mn n
SOLUTION OF THE NAVIER- STOKES EQUATIONS

To obtain the velocity field u(r,0,(})) and the hydrodynamic pressure field

p(r,9,tj)) within a drop of radius R, we solve the linearized Navier-Stokes

equations 2^

yV u = Vp (3a)

V-u = 0, (3b)

where y is the coefficient of viscosity. We impose the requirements

1) of a finite solution at r = 0;

2) of the kinematic condition u (R) =0, where u is the radial

r r component of the velocity; and

3) of the physical boundary conditions which will be examined in

the next section.

Since Eqns. (3a,b) are linear, a general solution can be written as a superposition of all the modes. Thus, it is sufficient to find a solution

for one of the modes (n,m,a)

.

Eqns. (3a,b) can be integrated according to the method of Miller and

Scriven. This method integrates (3a, b) in terms of u and the radial component

of the vorticity, where the vorticity is defined by

C = V X u. (4)

The results of the Miller-Scriven method which satisfy requirements "1" and

"2" are

u {r,QA) =-A'^^r""-^ (R^-r )y'^ {Q,<^) (5a) r mn mn

C (r,e,(|)) = b'^ r^'-'-Y^ (9,(j)) . (5b) r mn mn

The remaining integration constants A*^ and B*^ will be obtained from the

physical boundary conditions in the next section.

The velocity components—u„ and u^ —can be obtained from u and c by

9 © r r a relation due to Sani (6)

u = e^u^ + [r^/n(n+l)] [^ii^\-^j.xV^^;^] , (6)

where e is the radial unit vector,

r

'ii =

'

- ^ h <^>

is the surface gradient operator and R is the operator
Eqn. (4) and the results for u , , u , can be used to derived i;. and C^. r 0 (p 09

The hydrodynamic part p(r,0,(l)) of the pressure can be found by taking

the divergence of (3a) which yields V2p = 0.

p(r,0,*) = rV (9,*). (9) mn mn

The coefficient F*^ can also be obtained from (3a) by mn

r- V^u = V^(ru^) = (r/y)0p/3r), (10)

from which it can be shown that

f'^ = 2(2n+3)MA'^ /n. (11) mn mn

- PHYSICAL BOUNDARY CONDITIONS

The balance of stresses at the surface requires:

1) that the normal stress on the surface due to the hydrostatic pressure

and to the motion of the fluid is balanced by the surface tension; and

2) that the shear stresses due to the variation of the surface tension

are balanced by the fluid motion.

The deformation of the surface must be included in these conditions. The

deformation is assumed to be small, so that

R + AR = R [1 + £ (e,(j)) ] , (12)

where AR is the displacement of the surface and

£(6,4)) = E*^ Y*^ {QA) (13) mn mn

is a radial strain. The coefficient E*^ also must be obtained from the

boundary conditions

.

The normal stress due to the fluid

-P,, = + ^L^^'^L - 2mOu /9r)/ * (14) rr o mn mn r r+R

is balanced by the surface tension produced stress

Y(^ (15) ^ ^2

*A simplification of the surface conditions is introduced at this point by

neglecting the interfacial dilational elasticity and the interfacial shear

elasticity which contribute an interfacial viscosity term to the normal stress

equations—see Scriven (7) and Miller and Scriven (5) for discussions of these

properties.
where y is the local value of the surface tension and R]^ and R^ are the

principal radii of curvature. For small deformations of the surface of

a spherical drop, Lamb (8) has shown that

^ + - = - [2 + (n-1) (n+2) E° Y° ] . (16)

Rj^ R^ R mn mn

2 The surface tersion coefficient is expanded in terms of the mean value y

and the temperature coefficient as

Y = Y + Y^[T + L E ]R Y , (17)

a T mn mn mn mn

where L*^ is the coefficient obtained in the expansion of T(R+AR) of the

deformeS surface about the temperature T(R) of the undeformed surface and

TRY is the; (n,m,a)-term in the expansion of the surface temperature field, mn mn

Setting the^hydrostatic pressure p = 2y /R, and considering terms to

first order a.n Y in the normal stress°condi?ion, we obtain the first

equation for A and , mn mn

6yR'^"^"^A^ /n = [Y (n-1) (n+2) + 2y r"] E^ = 2y T° r". (18) mn o T mn n\n T mn

Instead of solving the shear stress conditions directly, it is more

convenient to take the surface divergence and the surface curl of the force on an element of surface. The surface divergence equation is

V^^^ Y = (i?u ) - V ^u ] ^ (19) II dr r II r r=R

.

2

Eqn. (19) can be simplified since V u

II r then, yields a second equation for a'^ and E*^ , mn mn

= 0. The divergence condition.

2(2n+l) ur""^"'"A^ /n - (n+1) y R^E^ = (n+Dy T^ r". (20) mn T mn mn T mn

The radial part of the surface curl equation yields the result that B*^ =0,

mn

or Q =0.

rEqns. (18) and (20) can be solved simultaneously for A*^ and E*^ , mn mn

A^ = n(n+l) (n+2)y y T^ /2uR (21) mn o T mn mn

e'' = y T° R^/d'^ , (22) mn T mn mn

where D*^ = (2n+l) (n+2)Y +Y R^. (23)

mn o T mn
THE VELOCITY FIELD AND THE VORTICITY FIELD

Defining a characteristic rate of flow

= -n(n+l) (n+2) Y Y^T^ R^/2mD^ , (24) mn o T mn mn

the components of the velocity u can be written as

u (r,e,(j)) = f''"-'-(l-r^)Y^ (d,<t>) (25a)

r mn mn

' Up, (r,e,(})) = (f""Vn) [l-^)f^] • (8y'^ 736) (25b)

•. o mn n+1 mn

u^(r,e,(j)) = (f''"Vn) [l-(^)r^] (l/sin6 )

• (3 Y^ 736 ) , (25c)

•-. <P mn n+1 mn where f = r/R.

Although c, =0 within the drop, Q and c, generally do not vanish. Using

the definition of the vorticity, Eqn. (4), and Eqns. (25a, b,c),

Cfl = Z^^r^'d/sinG) (SY*^ /3())) (26a)

6 mn mn

^ . = -z'^ r"OY^ /36) , (26b)

<p mn mn

where

= 2(U° /R) (2n+3)/n(n+3) . (27)

. ;. mn mn

The rate of flow, or speed, is

2 2 2 1/2 V = [u +u„+un , (28) r 69

where in general,

n

u = Z, Eu (n,m,a) (29a) r n=l m=0 a r

n

u. = I. Zu„(n,m,o) (29b)

9 n=l m=0 a 9 n

u, = Eu(|) (n,m, a) . (29c)

(fi n=l m=0 a Thus, the general expression for v can be very complicated. Only expressions

for the pure modes will be investigated here.

For the special case of the lowest order mode (n,m,a) = (1,0, e) and where

eLqi =0, we obtain equations for a levitated drop in a constant vertical

temperature gradient. In this case, the temperature in the drop is

T = T + T, Rfcos9

,

o 1

(30)
which results in the velocity components

= -<Y^T^R/3m) (l-r^)cose (31a)

= (y T-R/3M) (l-2r^)sine (31b)

D T 1 = 0.

These results can be obtained from the solutions of Young, et. al (5) , as mentioned in the Introduction. In addition, we find that

= Y T R/9Y , (32) mn T 1 o so that the drop is spherical in the limit ^f^Y^ 0«

SOME CHARACTERISTICS OF AXIALLY SY^L^1ETRIC MODES

For an axially symmetric mode (n,0,e), the expression for the rate of

flow, Eqn. (28) , is

,.n-l. 2.- -2.2. .2 n+3 .2.2_1.2,l/2

V = (r /n) [n (1-r ) (P ) +(1 —— r ) (P ) ] (33) n n+1 n

e 1 where v = v/U , P is the nth order Legendre polynomial and P is the

associated Legendre polynomial of first degree. Eqn. (3 3) can'^ie used to

identify the stagnation points since v = 0 at these points.

Within the drop, n rings of stagnation points about the axis of the drop can be readily identified: take = (n+1/ (n+3) and P_ = 0; since P has n nodes and 0 < ({) < 2it, n rings have been identified. On the surface of the

drop (f = 1) , the rate of flow v = 0 if pl = 0. The associated Legendre

polynomial P^ has (n+1) nodes, including one at each pole. These two

stagnation points at the poles and the (n-1) stagnation rings on the surface

coincide with the hot and cold spots of the temperature field and define

the; boundaries of the convection cells in the drop. If n > 1, v = 0

when r = 0. That is, flow does not occur through the center of the drop

when n > 1. Finally, we must consider the possibility of other internal

stagnation points. That is, are there other points such that

2„ -2,2,^ 2 n+3 ,2 2 ,^1,2 ^ ^ n (1-r ) (P ) + (1-—7 r ) (P ) =07 n n+1 n The answer is no. Since each term is either positive or zero, and since

neither (l-r^) and (1- (n+3/n+l) f 2) nor Pn and P^ vanish at identical values

of r and cos 9, respectively, there are no other internal stagnation points.

For an axially symmetric problem, a stream function ijj(r,6) can be obtained

from

^ g r sin

u, = ^(34b)

9 r sin 6 3r
where u = u /U (a = r,9). For the (n,0,e) -mode, the streamfunction is a a mn

ijj = -r""^-'-(l-r^) sinep-'-(x)/n(n+l) , (35) n where x = cos9. For the mode (l,0,e), the streamfunction is that for Hill's

spherical vortex (9). Stagnation points, indicated by "N" , and typical

streamlines for Hill's spherical vortex are illustrated in Fig. 1 and for

the modes n = 2,3,4,5 in Figs. 2-5, respectively. The drops are seen in

vertical cross-sections. The model rings lie in planes perpendicular to

the axis in each illustration. The straight lines within the circles

represent the boundaries of the convection cells, and here ii = 0. The

positions of the stagnation points are listed in Table 1 for these five

modes.

Table 1. Stagnation "Points" for Axially Symmetric Convection Cells in

,

, , Nominally Spherical Drops

Mode n f Interior Type Surface (r = 1)

e e Type

1 1//2 ring 0° point

180° point

2 0 point 0° point

• /3/5 54° 44' 8" ring 90° ring

125° 15' 52" ring 180° point

3 0 point 0° point

/2/3 39° 13' 54" ring 63° 26' 5" ring

It 90° ring 116° 33' 54" ring

II 140° 46' 6" ring 180° point

4 0 point 0° point

/5/7 30° 331 20" ring 49° 6' 24" ring

II 70° 7' 28" ring 90° ring

It 109° 52' 32" ring 130° 53* 36" ring

II 149° 26' 40" ring 180° point

5 0 point 0° point

1/2/3 25' 1' 2" ring 40° 5' 17" ring

II 57° 25' 14" ring 73° 25' 38" ring

ft 90° ring 106° 34' 22" ring

If 122° 34' 46" ring 139° 54' 43" ring

II 154° 58' 58" ring 180° point

I
AXIS

Figure 1. Streamlines and Stagnation Points ("N") for the (1 , 0 , e) -Mode

Hills' Spherical Vortex.
¥
AXIS

I

Figure 2. Streamlines and Stagnation Points for the (2,0,e)-Mode
1 "-v
AXIS

Figure 3. Streamlines and Stagnation Points for the (3,0,e)-Mode

AXIS

I

Figure 4. Streamlines and Stagnation Points for the

(4, 0, e)-Mode

AXIS

I

Figure 5. Streamlines and Stagnation Points For the (5,0,e) -Mode

THE CIRCULATION TIME

The circulation time t will be introduced in this section as a means of

characterizing the rate of convective mass transfer from the interior of the

convection cell to the vicinity of the surface of the drop. The reciprocal

circulation time t~1 is defined by

x"-*- = ^ u-d£// dS

= j t-ndS/f dS, (36)

where the circulation integral

j> u-d£

is taken around the boundary of the cell formed by the intersection of the

cell with a vertical plane containing the axis of the drop, such as any of

the cell boundaries shown in Figures 1-5. The second equality follows

from Stokes's theorem, where the surface integrals

/t-ndS and / dS

are taken over the region bounded by the circuit of the line integral; n is

the unit normal vector to the surface and here is equal to e^ . According

to Eqn. (36) , the reciprocal circulation time is the average'*' vorticity

of the cell. The circulation time resembles the period of rotation of a rigid body. Indeed, if the fluid circulated about the vortical center

as a rigid body, Eqn. (36) would yield 4tt times the frequency of rotation

and T would be the period of the rotation reduced by 4TT. Unfortunately,

such a simple interpretation of thermocapillary convection cells is not

possible, but t can still be used to characterize the rate of circulation

within the cell.

For the axially symmetric mode (n,0,e), the integrals in Eqn. (36) can be performed yielding

T = (n+2) [P (x')-P (X' -)]/2ze (6- -e-) (37) ns n s n s+1 on s+1 s for the circulation time for the s-cell, where 1 < s $ n+1, and x' and x'

j_ J- s s+1

are roots of

P (x) = 0. (38) n Cell-s has its vortical center at the stagnation point whose angular

position 6 is given by the root x of

s s P^(x) = 0; (39)
6 lies within the range whose lower bound is 9 ' and whose upper bound is 9

s s s+1 The circulation time may be either positive or negative, the sign depending

upon the direction of circulation in the cell. Using Eqns. (1) , (24) and (27)

,

we can show that, apart from the geometric factor, the circulation time t depends

ns only on the material parameters y and and on the temperature gradient

at the surface.

The circulation times for the mode n = 1 are given in Tables 2-5 for a variety of liquids. Here, we assume "unit conditions": L^-^ = 0, and a unit temperature gradient, T^^ = -l°/cm. The assumption of unit conditions

is indicated for the circulation time by t°, where the superscript "o"

designates unit conditions. Table 2 lists circulation times for some liquids

at room temperature; Table 3, for five liquid metals; Table 4, for four

molten oxides; and Table 5 for three molten sodium halide salts.

Table 2. Circulation Times* for Some Liquids at Room Temperature (298.15 K)

Y Material ^ T Viscos:.ty

mN/m/K** mPa.s**

Acetone -0. 112 (11) 0 . 316 (10) 0. 0323

DC 200, 20 CS+ -0. 062 (5) 19 (5) 3. 5

200 cs -0. 065 (5) 193 (5) 34

1000 CS -0. 061 (5) 793 (5) 180

Ethanol -0. 9832 (11) 1 .092 (10) 0. 150

n-Hexadecane -0. 106 (5) 3 .086 (12) 0. 334

Krytox 143 AZ"*""^ -0. 1 (13) 32 .4 (13) 4

Water -0. 1477 (11) 0 .8904 (13) 0. 06908

temperature gradient at the poles has a magnitude of l°/cm.

**1 mN/m = 1 dyn/cm; 1 mPa.s = 1 cp = 10~2 dyn-s/cm^.

"hDow Corning silicone oils of the DC 200 series.

"•"•A perfluoroalkylpolyether

.

T I , S
Table 3. Circulation Times* —Metals

Materials ^T

mN/m/K

Temp . T

K

' Viscosity

mPa. s

T ,s

Aliaminum

Copper

Mercury

Sodium

Tin

-0.356 (10)

-0.06 (10)

-0.2049 - (11)

-0.09833 (14)

-0.0706 (13)

930

1070

1370

1470

290

470

470

670

500

870

4.5 (15)

2.5

4.5 (15)

3.9

1.554 (15)

1.052

0.450 (15)

0.284

1.97 (13)

1.05 (13)

0.14

0.08

0.9

0.7

0.0869

0.0588

0.00524

0.00331

0.320

0.170

*The temperature gradient at the poles has a magnitude of l°/cm,

Table 4. Circulation Times* —Oxides

Material ^T

mN/m/K

Temp

K

Viscosity

Pa. s

AI2O3 (0.1)+ 2400

2600

0.11 (15)

0.062

13

7

0.0354 (18)

at 720 K

1410

1670

5.02 (17)

2.01

1620

651

Geo. 0.056

at 1390 K

1750

1930

12.2 (17)

0.787

2500

160

SiO, 0.031

at 2000 K

2280

2680

2820

717 (17)

102

46.4

8 X 10'

1 X 10^

5 X 10^

*Temperature gradient at the poles has a magnitude of l°/cm.

"''Estimated.
Table 5. Circulation Times*—Sodium Halides

Material ^T (18) Temp . T

TT^ J! J / T ^ \ Vxscosity (17)

mN/m/K K

° mPa . s

NaCl -0.0719 1090 1.38 0.220

1150 1 . 08 0.1/2

NaBr -0.0809 1060 1.28 0.181

1170 1.00^

6

0.142

Nal -0.129 1030 1.15 0.102

1100 0.96 0.085

*Temperature gradient at the poles has a magnitude of l°/cm.

The steady-state temperature gradient is obtained by balancing the

conductivie heat flux through the drop against the radiant heat flux away

from the drop in the cooler hemisphere. Sufficient heat is applied to the

hotter hemisphere to maintain the steady-state gradient. Since the radiant

heat flux '^T'^, liquids such as molten metals which have high thermal conduc-,

tivities will only have steep steady-state gradients at high temperatures.

Although copper has the highest thermal conductivity of the liquid metals

considered in Table 3, it is the only one of the five metals which is

considered at temperatures high enough for a rate of radiant heat loss to

be attained which could produce a steady-state gradient of l°/cm. Thus,

the circulation times given for copper in the table are the only ones which are physically attainable under steady-state conditions. For the

other metals, the physcially attainable value ofT^ will be longer than

T®: the attainable steady convective rates will be less than the rates

produced by a gradient of l°/cm.

The long circulation times estimated for the oxides, ^2*^3' ^^^^2

SiO^ show the importance of the viscosity in determining whether thermocapillary

convection can occur in the liquid. For these three oxides, thermocapillary

convection is expected to be negligible. However, ^1^0^ has a lower viscosity

than these other three oxides and, thus, a shorter circulation time. Since

temperature gradients much larger than l°/cm can be attained in molte" oxides,

significant thermocapillary convection might be observed in molten alumina.

The n = 1 mode has been demonstrated on earth by the work of Young,

Goldstein and Block (5) who observed the balancing of the thermocapillary and

buoyancy forces on small bubbles in a vertical temperature gradient. The low-g

of a space laboratory in addition to facilating the levitation of a drop would

allow the observation of the higher convective modes, in particular, the n = 2 mode. The circulation could be observed by the movement of a dye or of a radiotracer.
Relative circulation times for the axially symmetric modes n=2,3,4,5 are given in Table 6. These times are calculated relative to the single vortex

(n = 1) time t°. The sign of the circulation time for a particular cell indicate

the direction of circulation with respect to that in the single vortex drop.

Since the convective rate decreases as n increases, the higher modes appear

to be less desirable than the n = 1 mode for obtaining rapid purification.

However, the convective rates for some of these higher modes still can be

large enough for the convective pattern to be studied and for these modes

to have some utility for the purification of levitated melts in space.

SUMMARY

Equations describing the convective flow velocities in thermocapillary

convection cells in drops have been derived. When the thermal conditions

approximate a pure, symmetric mode—that is, when the temperature field can

be described by a single Legendre polynomial, the streamlines of the vortex

and the positions of the stagnation points in each cell can be calculated.

The circulation time, which is the reciprocal of the average vorticity of

the cell and which characterizes the rate of circulation, also can be

calculated for these high symmetry cells. An estimate of the circulation

time for a liquid drop is an indicator of the probable significance of

thermocapillary convection in the drop when a temperature gradient is imposed

across it.

Table 6. Relative Circulation Times, t° /'^ff ^or Axially Symmetric Modes

with n > 1 * Mode n Cells s t° /t?

ns 1 2 1 3.968

2 -3.968

3 1 9.981

2 -7.365

3 9.981

4 1 19.994

2 -13.505

3 13.505

4 -19.994

5 1 34.960

2 -22.691

3 20.647

4 -22.691

5 34.960

*T"» is the circulation time for the s-cell of the nth mode,

ns

T* is the circulation time for mode n = 1; a temperature gradient with a magnitude of l°/cm is assumed at the poles. The sign indicates the

direction of circulation with respect to the n = 1 mode.
REFERENCES

1. L. E. Scriven and G. V. Sternling, Nature 187 , 186 (1960)

.

2. V. G. Levich, PhyEicochemical Hydrodynamics , Prentice-Hall, Englewood

Cliffs, N. J., (1962), Chapters 7 and 8.

3. C. A. Miller and L. E. Scriven, J. Fluid Mech. 32^, 417 (1968) .

4. P. M. Morse and H. Feshbach, Methods of Theoretical Physics , McGraw-Hill,

New York (1953), Chapter 10.

5. N. O. Young, J. S. Goldstein and M. J. Block, J. Fluid Mech. 6, 350 (1959).

6. R. L. Sani, "Convective Instability", Ph.D. Thesis, Department of

Chemical Engineering, University of Minnesota, 1963.

7. L. E. Scriven, Chem. Engineering Sci. 12^, 98 (1960).

8. H. Lamb, Hydrodynamics , Sixth Ed., Cambridge University Press, London

(1932) , p. 475.

9. L. M. Milne-Thomson, Theoretical Hydrodynamics , Fifth Ed., Macmillan,

New York (1968), Section 19.51.

10. Handbook of Chemistry and Physics, 47th Ed., R. C. Weast, ed. Chemical

and Rubber Company, Cleveland, Ohio, (1966)

.

11. J. J. Jasper, J. Phys. Chem. Ref. Data 3^, 841 (1972).

12. R. S. Marvin and D. L. Hogenboom, "Viscosity of Liquids," in the

American Institute of Physics Handbook , Third Ed., D. E. Gray, ed.,

McGraw-Hill, New York (1972).

13. P. G. Grodzka and T. C. Bannister, Sci. 176 , 506 (1972).

14. Liquid Metals Handbook, Sodium (NaK) Supplement , C. B. Jackson, ed.

,

U. S. Atomic Energy Commission and the Bureau of Ships, U. S. Department

of the Navy, Washington (1955)

.

15. C. J. Smithells, Metals Reference Book , 4th Ed., Vol. 3 Plenum, New York (1967

16. J. L. Bates, C. E. McNeilly and J. J. Rasmussen, "Properties of Molten

Ceramics", Ceramics in Severe Environments , W. W. Kriegel and H. Palmour

III, ed. , Plenum, New York (1971).

17. G. J. Janz, F. W. Dampier, G. R. Lakshminarayanan, P. K. Lorenz and

R. P. T. Tomkins, Molten Salts , Vol. 1, NSRDS-NBS 15, National Bureau of

Standards, U. S. Department of Commerce, Washington (1968).

18. G. J. Janz, G. R. Lakshminarayanan, R. P. T. Tomkins, and J. Wong, Molten

Salts, Vol. 2, NSRDS-NBS 28.
Task 7

Consulting Support

NBS personnel have consulted with other groups engaged in the Space

Processing Program, including Marshall S.F.C., General Electric (Valley

Forge), State University of New York at Stony Brook, Grumman Aerospace,

Jet Propulsion Laboratory and others. We have shared computer programs

and data with SUNY. We have assisted in the evaluation of proposals. We

have attended technical meetings in San Francisco, California; Huntsville,

Alabama; Washington, D. C; Pasadena, California, and others. We have

provided advice and consultation with NASA Headquarters as required. We

have organized a Space Processing Seminar Series with invited speakers from

Bell Laboratories, J. T. Baker Company, Renssalaer Polytechnic Institute,

General Electric Corporation and others.
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BIBLIOGRAPHIC DATA

SHEET

1. PUbLlCA 1 ION OK RbFORl NO.

NBSIR 74-611

I. (jov t Accession

No.

3. Recipient's Accession No.

4. TITLE AND SUBTITLE

NBS Space Processing Research

5. Publication Date November 1974

6. Performing Organization Code

7. AUTHOR(S)

E. Passaglia, R. L. Parker

8. Performing Organ. Report No. NBSIR 74-611

PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234 ,

10. Project/Task/Work Unit No.

11. Contract/Grant No. W-13,475 Mod 3

12. Sponsoring Organization Name and Complete Address (Street, City, State, ZIP)

NASA - Advanced Missions Program

600 Independence Avenue S.W. \ Washington, D.C. 20546

13. Type of Report & Period

Covered

Annual Report

1 1 h'^K^^ 1 1n),

14. Sponsoring Agency Code

IS.SUPPLEMENTARr NOTES , /

Annual Report

16. ABSTRACT (A 200-word or less factual summary almost si^ilicant information. If document includes a significant

bibliography or literature survey, mention it here.)

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

covering the period Novanber 1, 1973 to October 31, 1974. The objectives of the NBS

i program are to perform ground-based studies (and, where appropriate, space-based

1 studies) of those aspects of space that could possibly provide a unique environment for

making materials more perfect or more pure. The approach taken deals primarily with

experimental and theoretical studies of the possible effects of the absence of gravitaI tional forces on those materials preparation processes where the presence of gravity

may be important in reducing perfection or purity. The materials preparation

processes studied comprise six tasks in the areas of crystal growth, purification and

! chemical processing, and the preparation of composites. There is an additional seventh

j task on consulting support.

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name; separated by semicolons)

1 Crystal growth; crystal perfection; gravitational effects; purification; space

"processing; zero-g.

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