NBSIR 76-980

NBS Space Processing Research

E. Passaglia, R. L.

Institute for Materia

National Bureau of

Washington, D. C.

Phone: (301) 921

February 1976

Parker, R. J... Spae. : ,-o>ss i nr: prorram of

the National Bureau of Standards, Paper

m Materials Sciences in Space with Application to Space Processing Vol. • Progreso

in Astronautics and Aeronautics, L. Steg,

•;,3_ll3 5 (American Institute of

nutir.s, Ne-w York, NY

Ed., pp.

Aeronautics and Astrona

1977)

SI 2

Annual Report

For the Period 1 November 1974 - 31 December 1975

NASA Contract W-13,475, Mod. 4

OF

zV

v.

US. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

NBSIR 76

NBS S|

E. Passaglia, R. L. Parker

Institute for Materials Research

National Bureau of Standards

Washington, D. C. 20234

Phone: (301) 921-2811

February 1976

Annual Report

For the Period 1 November 1974 - 31 December 1975

NASA Contract W-1 3,475, Mod. 4

U S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

NBSIR 76-980

NBS SPACE PROCESSING RESEARCH

E. Passaglia, R. L. Parker

Institute for Materials Research

National Bureau of Standards

Washington, D. C. 20234

Phone: (301) 921-2811

February 1976

Annual Report

For the Period 1 November 1974 - 31 December 1975

NASA Contract W- 13,475, Mod. 4

This report was prepared by the National Bureau of Standards

under Contract W- 13,475, Mod 4 - NBS Space Processing Research,

for the Advanced Missions Program of the National Aeronautics

and Space Administration.

U.S. DEPARTMENT OF COMMERCE, Elliot L. Richardson, Secretary

James A. Baker, III. Under Secretary

Dr. Betsy Ancker-Johnson, Assistant Secretary for Science and Technology

NATIONAL BUREAU OF STANDARDS. Ernest Ambler, Acting Director

TABLE OF CONTENTS

Page

Summary 1-2

Task 1 - CRYSTAL PERFECTION IN CZOCHRALSKI GROWTH by

M. Kuriyama, W. J. Boettinger and H. E. Burdette 3-£i

Task 2 - EVAPORATIVE PURIFICATION OF ULTRA-HIGH PURITY MATERIALS by

R. C. Paule, W. J. Boettinger and F. S. Biancaniello 22-55

Task 3 - VACUUM EFFECTS IN THE PREPARATION OF COMPOSITE MATERIALS by

H. Yakowitz 56-98

Task 4 - MELT SHAPE IN WEIGHTLESS CRYSTAL GROWTH by

S. R. Coriell, S. C. Hardy and M. R. Cordes 99-119

Task 5 - VAPOR TRANSPORT SYNTHESIS AND CRYSTAL GROWTH by

H. S. Parker 120-130

Task 6 - SURFACE TRACTION AND OTHER SURFACE PHENOMENA by

A. L. Dragoo 131-141

Task 7 - CONSULTING SUPPORT 142

Distribution List 143-144

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, 1974 to December 31,

1975. 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 - sizable nickel single crystals have been grown and sliced and the

crystal perfection has been assessed by x-ray dynamical diffraction techniques.

Some preliminary evidence of the effect of growth parameters on crystal perfection has been obtained. Magnetic domain configurations in the nickel slices

have been observed with the x-ray technique. In Task 2 - Evaporative Preparation

of Ultra-High Purity Materials - in anticipation of possible evaporative

purification experiments with A120 3 in space, we have measured the relative

- 1 -
abundances of the various vapor species from molten A1 20 3 and have studied

the effects of non-equilibrium radiative heating of the specimen. We have

also determined the purification rate of Fe impurity in molten levitated

Mo. The Fe impurity was rapidly reduced from 50 ppm to 50 ppb. In Task

3 - Vaccum Effects in the Preparation of Composite Materials - a simple

analytical expression is given of the distribution in depth of electronbeam produced x-rays, and the expression's use to analyze layer structures

or film composites of particles embedded in a matrix is given. In Task 4 -

Melt Shape in Weightless Crystal Growth - experimental and theoretical

investigations of the floating zone process of crystal growth and purification have been carried out with particular emphasis on microgravity

conditions. Both numerical and analytical techniques have been used, and

we have experimentally measured liquid zone shape and stability. In

Task 5 - Vapor Transport Synthesis and Crystal Growth - seeded growth

experiments were undertaken and improvements in the transport tube

geometry and technique are described. Growth of mercurous chloride at

temperatures in the 350-390 °C range has resulted in good quality growth

of 0.6 O.D. x 1 cm boules. In Task 6 - Surface Traction and Other Surface

Phenomena - ground-based experiments and apparatus were developed for studi

of convection in drops, including necessary photographic techniques. In

Task 7 - Consulting Support - we have continued our rather wide range of

meetings, consultations and talks with NASA and NASA contractors.
Task 1

Crystal Perfection in Czochralski Growth

M. Kuriyama, W. J. Boettinger and H. E. Burdette

Metallurgy Division

Institute for Materials Research

Summary

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

W-13,475 #3, Task 1) , nickel single crystal growth has been chosen for a

study of the relationship between crystal growth conditions and resultant

crystalline perfection in melt-grown crystals. Several interesting crystallographic effects have been found in the resultant crystalline perfection

of copper single crystals studied previously. However, the previous work

alone is not sufficient to draw the conclusion that the observed crystallographic effects take place during growth of most metal crystals. Nickel

has a face-centered cubic crystal structure like copper, while the material

properties such as dislocation mobility, stacking fault energy and dislocation velocities are considerably different from copper. In addition, nickel

is ferromagnetic. Magnetic domain configurations in the interior of sizable

highly perfect nickel single crystals have not been observed before. Possible

observation of these domain configurations would be extremely interesting, not

only from an academic point of view, but also from an industrial point of view.

No one has previously succeeded in producing sizable, highly perfect

nickel single crystals. No one has established proper procedures of handling

grown crystals chemically and mechanically without destroying the crystalline

perfection in the as-grown state. Therefore, emphasis in the present work

3
is placed on developing the most suitable technique for the preparation of

nickel single crystals. This report should be considered to be a preliminary

report to more systematic and complete future investigations.

In this report, the most suitable chemical processing method has been

established experimentally. These methods will enable us to seek the relationship between crystal growth conditions and resultant crystalline perfection systematically with reasonable reproducibility in future work. The

crystal perfection has been assessed by non-destructive methods of x-ray

dynamical diffraction as in the previous work. Preliminary trends of data

have indicated that the growth condition parameters seem to affect the

resultant crystalline perfection. This observation is discussed in comparison with the previous work on copper, although it would be premature to form

any conclusion. The entire procedures have not yet become routine, nor

perfect

A remarkable achievement of the present work is the observation of

magnetic domain arrangements in the interior of nickel single crystals. If

one can systematically produce nickel single crystals which would be better

in perfection than those obtained in this report, one can study the magnetic

properties related to magnetic domain motions in an idealized condition in

order to improve electromagnetic performance for possible industrial purposes.

In view of the progress made in this preliminary study, it is strongly

recommended that this study should be continued in an effort to find whether

an optimum choice of growth conditions improves the resultant crystal perfection, and, if so, to evaluate the role of space in producing nickel

single crystals of better perfection.

4
Introduction

Space has provided mankind with another dimension, which future industries may find unique and useful for the production of new materials. The

lack of gravity and the availability of a high vacuum will be significant

factors in the progress of technology in many areas. It is learned from

experiences in the past that new technology produces new materials, such as

silicon or germanium single crystals free of crystal imperfections. These

single crystals have shown their great usefulness in semiconductor industries

through the developments of communication devices, computer elements, visual

display elements and many other products.

Although it may not be recognized explicitly, the usefulness of single

crystals or textured materials has been demonstrated, for example, through

the use of Fe-Si alloys for increased performance of transformers and motors.

In the days of energy conservation, this aspect may have great significance.

No matter what methods will be used to produce energy in the future, it will

very likely be distributed to the users in the form of electricity. In this

sense, the production of new materials of superior performance should be

sought in all possible ways.

At present, Fe-Si alloys are unique, because of their desirable performance in cyclic magnetization processes, their easiness of being rolled

in a commercial size and their rather good level of crystal perfection. In

contrast, most other fundamental ferromagnetic materials have not so far

satisfied high performance conditions for many reasons. For example, no

one has succeeded in producing sizable , highly perfect nickel single crystals,

which might be the precursor of commercial textures materials. These perfect

magnetic single crystals provide an idealized condition for a study of

5
magnetic domain behavior, which will assist in gaining control over

magnetic properties under cyclic magnetization processes for possible

industrial applications.

It has been established in previous reports* that crystal growth conditions play a significant role in producing sizable copper single crystals

of high perfection from the melt by the Czochralski technique. The crystal

growth conditions have been described in terms of directly controllable

process parameters, such as the rotational speed of the seed crystal and

of the melt. These parameters represent the fluid flow conditions in the

melt during crystal growth. As discussed in Report III, the reproducible

relationship between the crystal growth conditions and the resultant crystalline perfection has considerable practical significance in space manufacturing processes, since the lack of gravity and, in turn, the lack of thermal

convection may make the resultant fluid flow in the melt more controllable.

It has been concluded further in Report III that, although an optimum growth con

dition for the preparation of highly perfect copper crystals is, as a rule,

characterized by a physical quantity controlling the fluid flow in the melt,

more crucial factors influencing the ultimate level of crystal perfection

are found in some crystallographic effects during growth: the growth direction and the formation of a particular set of sessile dislocations. It is

worthwhile mentioning that this second factor may be related directly to

the effect of gravity. Are these crystallographic effects more prominent

or more likely to happen in copper single crystals than in other crystals?

The previous reports will, hereafter, be referred to as Report I (Government Order H-84832A, National Aeronautics and Space Administration, NBS

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

and Manufacturing in Space Research for the Advanced Missions Program of

the National Aeronautics and Space Administration; NBS report NBSIR 73-402)

and Report III (Contract W-13,475 Mod. 3 - NBS Materials Science and Manufacturing in Space Research for the Advanced Missions Program of the

National Aeronautics and Space Administration; NBSIR 74-611)

6

I
The previous work lacks evidence on this aspect. While nickel also has a

face-centered cubic crystal structure, its material properties such as dislocation mobility, stacking fault energy and dislocation velocities are

considerably different from copper. In view of these crystallographic,

mechanical and magnetic properties, we have chosen nickel single crystals

in the present work to study the relationship between the crystal growth

conditions and the resultant crystal perfection. Since there is, at present,

not much previous information available for the growth of sizable, highly

perfect nickel crystals, emphasis in the present report is placed on developing the most suitable technique for the preparation of nickel crystals. This

report is, therefore, considered to be a preliminary report to more systematic

and complete future investigations.

Experimental Procedures

As described in previous reports, a vital part of research of this type

is the non- destructive assessment of crystal perfection in the interior of

crystals by dynamical diffraction. The anomalous Borrmann transmission effect

serves, to lowest degree, as a measure indicating whether crystals are of a

sufficient quality to be investigated further. When a crystal is imperfect,

ordinary x-rays cannot penetrate farther than 10 ym of the crystal even when

a Bragg diffraction condition is met. On the other hand, these x-rays can

pass through a crystal one millimeter thick under Bragg diffraction conditions

(the anomalous transmission effect), when the crystal attains a certain level

of perfection. Above this level of perfection, crystals begin to display

their special characteristics unobtainable in imperfect crystals or polycrystalline materials. It is, therefore, important to find a method which is capable of producing crystals above this level, and to establish a method of

characterizing the crystal perfection without further damaging the crystals

in the evaluation processes.

7
There have been two reports ' in the past, of which we are aware, of

x-ray topographic observation of nickel crystals. One"'" dealt with nickel

crystals in plate form prepared from nickel bromide by chemical reduction,

2 and the other with nickel sheets treated by a strain anneal (secondary

crystallization) method. The first group obtained diffraction topographs

in the transmission geometry from large crystal grains in the nickel plate.

However, the crystal was very thin (in addition, with a hard radiation).

Consequently, the anomalous transmission effect was not prominent. The

latter used a multi-grain sample 0.5 mm thick, and observed the anomalous

transmission effect in one of the grains. However, there have been no

systematic studies on the production of nickel crystals of high perfection.

In particular, no one has yet succeeded in growing sizable nickel single

crystals of high perfection from the melt.

In view of this, particular efforts were placed during this contract

year on the following aspects:

(I) development of a method of growing nickel crystals from the melt

by the Czochralski technique, and

(II) development of chemical processing methods for obtaining samples

from grown crystal boules for x-ray topographic examination.

Since we had not anticipated obtaining crystals of good quality at the onset

of this work, an additional effort was made on

(III) modification of the asymmetric (double) crystal topographic (ACT)

camera for such imperfect crystals.

I. Crystal Growth

One of the objectives for the present contract year is to grow sizable

nickel single crystals by the Czochralski method using the same equipment

used for the previous copper work. A considerable amount of time has been

8

i
spent to achieve this objective with limited resources. First, the entire

furnace and vacuum system was completely disassembled, so that the system

could be cleaned chemically to remove all deposits and traces of copper.

Wherever possible, parts were etched with various acid solutions. The system was then reassembled with new seals, and all the parts contaminated

heavily by copper were replaced with new ones up to the limit of commercial

availability

.

It was found, after testing the system, that the melting point of

nickel was at the extreme limit of the furnace capability and that the

system could not hold a good vacuum at this high temperature. These facts

necessitated some modifications in thermal shields and for a supply of

various atmospheres. Several atmospheres were tested, but all of them

accentuated heat loss. Finally, it was concluded that an Ar gas could be

tolerated under the present conditions because of its less efficient heat

transfer property than other inert gases.

It is extremely desirable to maintain visual contact with the meniscus

area during crystal growth. This condition imposed severe constraints on

the arrangements of thermal shields and on the choices of crucible shapes.

Several crucible shapes of different materials were tested, including the

dual crucible combination utilizing a graphite outer crucible. Cylindrical

alumina crucibles have been selected as the most convenient at present.

After each use, the crucible was discarded to avoid possible cracking during

subsequent crystal growth.

For this year's purpose, 99.99% commercially available nickel stock

was obtained in the form of 6 mm diameter rods. We have not yet been concerned with the purity of nickel in the present work which is still at the

preliminary stage of establishing a proper procedure of nickel single crystal

growth.

9
Large nickel crystals were grown with this arrangement in a fashion similar

to that used previously for copper single crystal growth. Either a polycrystalline or a single crystal seed was used. Both the seed and the melt were

rotated around a common axis independently with different rotational speeds.

The pulling speed was chosen to be 0.10 cm min-

^ for convenience. During

the crystal growth, attempts were always made to narrow the diameter of a

crystal boule initially at one region ( bottlenecking ) • Bottle-necking was

not successful most of the time, because of the lack of temperature control,

as described later. The grown crystals were about 12 cm long with diameters

between 1.5 and 2.5 cm. When crystals were bottlenecked successfully, the

neck diameter was always slightly less than 1 mm. On removing the grown

boules with a bottleneck from the furnace, the waxing technique was used

as in the copper work described in Report II.

II. Crystal Cutting and Polishing

For the assessment of crystal perfection it is necessary to prepare

smooth single crystal surfaces of desired crystallographic orientation without introducing further damage into the crystals during the preparation. It

is usually desirable to obtain samples of different crystallographic orientation in a disc form from the same grown boule. The conventional methods of

cutting and polishing introduce dislocations to a considerable depth below

3 4 the crystal surface. There have been two reports ' in the past concerning

the chemical cutting and polishing of nickel crystals. However, these

reported techniques were found to be unsatisfactory when the crystal perfection in original boules was improved. During this contract year, extensive

work has been carried out to develop the chemical procedures for cutting and

polishing nickel single crystals.

10
We used the same cutter which was developed for copper single crystals

in Report III. The following chemical compositions were found to he most

effective for the strain-free cutting of samples from the grown nickel boules:

nitric acid 45 volume %

hydrochloric acid 33 volume %

acetic acid 20 volume %

ethyl alcohol 2 volume %

Among these, ethyl alcohol plays an important role in maintaining the solution

in a highly volatile state. A freshly prepared solution should be fed into

the cutting baths (acid trays) at regular intervals. It is also found to be

necessary to heat the crystal boule to about 90°C in order to keep the cutting

time reasonable. An infrared heat lamp with an aperture was used to heat

the boule so that heating could be concentrated in the cutting areas to avoid

softening of the glue which was used to hold the boule in a desired orientation.

This new highly volatile cutting solution and the addition of heat made

it necessary to establish a new way of mounting the oriented boules without

straining them. A silicone-rubber compound was found to withstand the conditions and was easily removed afterwards. It rapidly became obvious that any

solution which would cut nickel effectively would also attack stainless steel,

plastic and most other materials. This necessitated several modifications

to the acid saw which was described in Report III. The motor was removed to

a remote location and a drive train was installed. The stainless steel

rollers in the acid trays were replaced with precision-made quartz rollers.

A high grade tungsten wire was found to perform adequately for reasonable

amounts of time in carrying acid solutions. Since our acid saw was made of

stainless steel, a sprinkler system was installed to continuously wash the

cutting solution and reaction products off the saw to prevent the saw from

corroding.

11
After the boule was sliced, samples should be chemically polished to

obtain smooth surfaces without introducing dislocations in the samples.

After various solutions had been examined under a variety of conditions, an

optimum polishing solution and procedure were developed. The efficient

strain-free polishing can be best achieved when the samples were polished

with different solutions at several stages. The basic recipe consists of

nitric, hydrochloric, acetic acids and ethyl alcohol. The compositions

should be changed at each stage; ethyl alcohol again plays an active role

in making the solutions volatile. The solutions should be prepared shortly

before use. Because of the volatile nature of the solutions, the polisher

previously described in Reports I and II had to be modified. The present

polisher has several replacement parts made from teflon to enable it to

withstand such vigorous polishing solutions.

Besides the new developments described above, the other procedures are

almost the same as used for the previous copper work (Reports I and II)

The boules are mounted on a goniometer stage to be oriented in desired crystallographic directions and are transferred to the acid saw which can

accommodate the goniometer stage in precisely the same alignment. The slices

are handled very carefully and mounted on bakelite cylinders with the same

glue as developed for copper crystals for acid polishing. The sample discs

are later removed from the cylinders by dissolving the glue and mounted on

stainless steel rings which can attach to a goniometer head for x-ray topographic examination.

III. Dynamical Diffraction Topography

For the non-destructive assessment of the perfection of sample crystals,

two different x-ray optical arrangements were employed as in the previous

copper work, namely an asymmetric (double) crystal topographic (ACT) camera

12
and a high-resolution double-crystal scanning diffractometer (SCAD) . The

details of these systems have been described in Reports I, II and III and

5 6 also in previous papers '

. An x-ray image intensifying tube has been used

as described in Report III.

At the onset of this work, a rocking device was added to the ACT camera

to change the x-ray glancing angle continuously during photographic exposure.

Until one finds an optimum growth condition and chemical preparation methods,

the perfection of grown crystals are usually anticipated to vary over a large

range, possibly including a poly crystalline or multi-grained state. This

rocking device enables us to obtain topographs from such imperfect crystals.

From a view of x-ray diffraction optics, this device allows us to lower the

optical quality of our ACT system essentially down to the Berg-Barrett

method.

The thicknesses of the nickel discs were determined by the anomalous

transmission effect in the same fashion as described in Reports I, II and III

5 6 and previous papers ' for copper crystals. The SCAD system was used to

obtain the profiles and widths of rocking curves at various locations in the

crystal. CuKa radiation was used throughout the present work. The product

of the ordinary linear absorption coefficient, y = 43.9 mm-1

, and thickness,

L, ranged from 10 to 45 in the present work. This product plays an important

role in dynamical diffraction to indicate the degree of crystal perfection

in the interior of crystals. The higher the value of uL for which anomalous

transmission is observed, the better the crystal perfection.

X-ray diffraction topographs were taken both in the surface reflection

and in the transmission geometry with those two systems. The sample discs

were prepared in the (110) orientation to supply a maximum number of different lower index symmetric reflections; 220 surface reflection, 111, 111,

13
002 and 220 transmission topographs can be obtained from the same nickel

discs.

As mentioned before, the main efforts were concentrated on the development of the techniques. These are indispensable steps before the relationship between the crystal growth conditions and the resultant crystal perfection

is sought systematically with reasonable reproducibility. We undertook the

present work as a preliminary step to the establishment of a systematic procedure to achieve our final goal in the future. In the present work, we

therefore arranged various factors carefully to perform 24 nickel crystal

growth experiments. All the stages of the nickel growth and preparations

have taken much longer time than the previous copper procedures. For example,

the pre-baking of crucibles and extensive preparation of the charge have

created extra work which was not necessary for the copper work. The cutting

and polishing are much slower with nickel than copper. This fact, combined

with the highly volatile and corrosive nature of the solutions and their

effect on equipment, has severely limited the quantity of samples for a

given period.

Experimental Results

Large nickel single crystals were grown by the Czochralski technique

from either a polycrystalline or a [100] oriented single crystal seed. The

crystal boules grown with a polycrystalline seed are specified by the number

0 in the third digit of their sample number, for example, 170200, and the

boules grown from the [100] oriented seed by the digit 4. The melt and seed

were rotated in the same direction with approximate speeds of 10 r.p.m.; the

relative speed of rotation ranged from 0.2 to 0.6 r.p.m. This growth con14
dition is denoted by condition 2, which appears in the fourth digit of the

sample number.

The growth directions of the crystal boules grown with a polycrystalline

seed have been determined by the Laue method as shown in Fig. 1. The growth

direction of the boules grown with the oriented single crystal seed has

been confirmed to be the [100] direction, which is the crystallographic

orientation of the seed. As described previously, the bottlenecking was

not always achieved. An example of the boule shape with a successful bottleneck is shown in Fig. 2. The boules whose sample numbers are less than

120200 were exhausted during the process of collecting the necessary information for the development of nickel crystal growth method and chemical

processing methods described in a previous section.

Fig. 3 shows a 220 diffraction topograph of nickel sample #120223

taken with the ACT in the surface reflection geometry. This sample was

sliced out of boule #120200 which was grown without a successful bottleneck. During the growth, the diameter along the length of the boule was

not controlled satisfactorily. This topograph indicates that the sample

consists of many small grains, whose orientations spread out over more than

several minutes of arc. This sample should not be considered a single

crystal. Some sample discs show a distribution of much larger subgrains

in their surface reflection topographs as shown in Fig. 4, when they are

sliced out of boules grown under a similar poor diameter control condition

but with a reasonable bottleneck . No anomalous transmission was observed

in sample crystals whose surface reflection topographs were as poor as

those in Figs. 3 and 4. This observation was made without exception, in

particular, for the boules of the 0-series grown from a polycrystalline seed.

This is not true for sample crystals of the 4 series, as shown later.

15
In some occasions crystal boules were grown with a narrow bottleneck

and reasonable diameter control was achieved over the entire length. Surface reflection topographs of sample discs from these boules display a

drastic improvement in their crystal perfection. The topographic images

cover the entire exposed area of the sample discs, as shown in Figs. 5 and

6. It is seen in Fig. 5 that sample disc #140231 has many very low angle

subgrain boundaries, while sample #170231 has very few of them as shown in

Fig. 6. In the ACT system, the misorientations of these subgrains are

judged by the image coverage to be much smaller than 30 seconds of arc.

Horizontal line images in Fig. 6 correspond to scratches on the surface,

which were created in handling. Crystals grown in conditions similar to those

used for these two crystals produced surface topographs almost as good as (or

even better than) those shown in Fig. 5 and 6. All of these crystals demonstrate

anomalous transmission from their entire volume, as shown, for example,

in Fig. 7.

Fig. 7 shows 111 transmission topographs taken with the ACT in the

as-grown state of a sample crystal 0.9 mm thick, #140231. A topograph of

the 0-diffracted (transmitted) beam is shown in Fig. 7a and that of the

H-diffracted (Bragg-diff racted) beam is shown in Fig. 7b. A lighter (white)

area indicates that more x-ray photons have reached this area, while a dark

(black) area implies that photons have been disrupted by the crystal. Most

of the details in both topographs a and b are identical; in particular, the

images caused by the presence of crystal imperfections such as dislocations,

dislocation clusters and very small angle subgrain boundaries match perfectly

with each other. It should be noted that the topographs have revealed the

entire shape of the crystal disc, indicating that the prominent anomalous

transmission effect has taken place all over the crystal. Fig. 8 a and b

16
show 111 transmission topographs of the j^-dif f racted and the Jg-diff racted

beams taken with the ACT system in the as-grown state of sample crystal

#170231. This crystal is more perfect than #140231, as there are fewer

images caused by crystal imperfections in these topographs than in Fig. 7.

In addition to the crystal- imperfection images mentioned above, there

are extremely straight line images throughout the crystal. Also visible

are images of rectangular shape. Most of these straight line and rectangular images are nearly identical except for minor details in both topographs

a and b. However, as shown in the enclosed area A in Fig. 7, some of the

images are different in contrast in the two topographs. When the crystal

is magnetized by the application of a magnetic field of roughly 2.39 x 10 5

A/m in one of the easy directions of magnetization, [111], the straight

line images and rectangular image arrangements disappear completely, except

for one area at the right bottom corner, as shown in Fig. 9 a and b. In

such magnetic fields, nickel single crystals are magnetized to a single

magnetic domain.

The formation of more complicated patterns is shown in Fig. 10 a and

b, when a magnetic field of 1.43 * 10^ A/m is applied in one of the hard

magnetization directions, [001]. It should be noted that the disruption

images due to crystal imperfections have remained unchanged in the application of a magnetic field. After the magnetic field is removed, the

straight line and rectangular images recovered as shown in Fig. 11 a and

b. Almost all of these lines, including line segments, are parallel to

either the [001] or [110] direction. All the topographs give evidence

that those straight line and short line-segment images are indeed related

to the magnetic domains in the interior of crystals and that these images

very likely correspond to magnetic domain walls. A more detailed study

17
of the magnetic domain images is presented elsewhere. An example of

diffraction topographs taken from the O-diffracted and the H-diffracted

beams with the SCAD system is shown in Fig. 12 a and b.

Next, we describe the topographs obtained from the crystals of the

4-series which were grown in the (100) direction. So far we have not

succeeded in narrowing the boules initially to a reasonable bottleneck

diameter. Diameter control was achieved satisfactorily. An example of

ACT 220 surface reflection topographs is shown in Fig. 13. This topograph

indicates that the crystal consists of several large subgrains. Unlike the

crystals of the 0-series, anomalous transmission was obtained from the

crystals of the 4-series, even though subgrains exist. An example of an

ACT 002 transmission topograph is shown both for the 0-diffracted and for

the H-diffracted beam in Fig. 14 a and b. Images caused by magnetic domains

are visible, along with many disrupted images caused by crystal imperfections.

Usually spectroscopic data - namely, rocking curve profiles and

widths - alone would supply adequate information regarding the degree of

crystal perfection in sample crystals, as demonstrated previously. However, nickel crystals grown so far are not perfect enough to

be judged by their spectroscopic data. Here we only show those data

obtained from the sample crystals above a certain level of perfection.

The rocking curves were obtained from these sample discs at various locations with the SCAD for the 220 diffracted beam in the surface reflection

geometry and the 111 in the transmission geometry. The full width at

half maximum (FWHM) of the 220 rocking curve were 34 seconds of arc and

30 seconds of arc in average for #140231 and #170231, respectively, except

for an extremely distorted area. The perfection is not uniform throughout

18
the crystal. The local variations of rocking curve widths are quite

significant. The severely distorted area showed a width up to 64 seconds

of arc. An example of the 220 rocking curve profiles is shown in Fig. 15.

For the 111 rocking curves in the transmission geometry, the FWHM values

were in the neighborhood of 12 seconds of arc. An example of the profiles

is shown in Fig. 16a. The local variation of the profiles is dramatically

demonstrated in the transmission geometry as shown in Fig. 16b.

Conclusion and Discussion

Major emphasis of this work was placed on the development of chemical

processing procedures as well as the workable modification of the crystal

grower for nickel crystals. In this work we have only begun the first

step of the nickel crystal growth program. The entire procedures have

not become routine, nor become perfected . In this type of research, it is

extremely important to find the reproducibility of results under a given

set of conditions. Admittedly, we have not started to collect our data

along this desired line. We have even set a pulling speed at an unusually

high value so that we can explore all the other possibilities most effectively

to approach a final desired condition as quickly as possible. For this

reason, this report is merely a preliminary study of the perfection of

nickel crystals grown from the melt. The relationship between the growth

conditions and the resultant perfection cannot be discussed sensibly in

this report. We do not even know a reasonable set of controllable process

parameters for the growth at this stage. It is extremely important to

continue this effort to obtain optimum growth conditions for highly perfect

nickel single crystals before we undertake any growth experiments of nickel

crystals in space.

19
Even though the present work is preliminary to more systematic

investigations, we have obtained some facts which are comDatible with those

obtained in the previous work of copper. For example, diameter control

over the entire length of grown crystals is quite desirable to obtain the

crystal perfection above a certain level. Failure of diameter control

usually results from poor temperature control which increases the chance of

getting more dislocations into crystals through suddenly exposed new

crystal surfaces. Another factor which is consistent with the previous

results on copper is that bottlenecking is indispensable in obtaining

crystals of good quality. The present results indicate that the crystals

grown with a narrow neck have demonstrated the anomalous transmission

effect. It was concluded in the previous copper work that some crystallographic effects influence the resultant perfection of copper single

crystals. We have seen a similar effect on the crystal perfection of

nickel in the present work, although the present data may not be

enough to prove it. The crystals grown in a crystallographic direction, say, [100], have demonstrated the anomalous transmission

effect even though the crystals consist of many subgrains whose misorientations are very small.

A remarkable achievement of the present work is the observation of

magnetic domain arrangements in the interior of nickel single crystals.

No one has succeeded previously in observing such magnetic domain arrangements spreading over the entire crystal volume in nickel crystals. The x-ray

image formation of magnetic domains in crystals is purely dynamical. The

crystals are, therefore, required to be above a certain level of perfection.

Fortunately, we have achieved this level of perfection in the present work.

Since magnetic domain arrangements are closely related to the crystal

perfection in the bulk, it is desirable to find a correlation between the

20 (continued after figures)
Fig. 1. Stereographic plot of the growth directions

crystal boules grown from polycrystalline s by the Czochralski technique.

Fig. 2. Photograph of a Ni single crystal grown

by the Czochralski technique with

bottleneck diameter roughly 1 mm.

Fig. 3. ACT 220 surface reflection topograph

of a very poor quality as-grown Ni

crystal (#120223) grown without a

bottleneck and with poor diameter

control. Note the very small subgrains .

Fig. 4. ACT 220 surface reflection topograph

of as-grown Ni crystal #130221 grown

with a bottleneck and with poor

diameter control. Note the large

sub-grains

.

Fig. 5. ACT 220 surface reflection topograph

of as-grown Ni crystal #140231 grown

with a bottleneck and good diameter

control. Note that diffraction from

the entire area of the crystal indicates that misorientation of sub-grains

is less than 30 sec. of arc.

Fig. 6. ACT 220 surface reflection topograph of

a good quality as-grown Ni crystal

(#170231)

.

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.

macroscopic arrangements of magnetic domains and the crystal perfection in

a more systematic way. If x-ray intensities can be made stronger, dynamic

behaviors of magnetic domain walls and their distribution can be studied

in-situ by x-ray diffraction topography. It is obviously necessary to have

nickel single crystals of the present level of perfection or above in such

a dynamic investigation. Powerful x-ray sources, such as rotational targets

or synchrotron radiations, are available at present. It might not be so far

in the future that one could study cyclic magnetization processes by dynamical

diffraction topography, if one could systematically produce nickel single

crystals better in perfection than those reported here.

In view of these promising important applications, it is extremely

desirable to continue the present work in an effort to find whether an

optimum choice of the crystal growth conditions improves the resultant

crystal perfection, and if so, to evaluate the role of space in producing

nickel single crystals of better perfection.

Before concluding this report, we would like to express our appreciation to Roberta M. Eaton and Deborah A. Lavin for their technical assistance.

References

1. Chikaura, Y. , Fukumori, H. , and Nagakura, S., Japan J. Appl. Phys . 11 , 1582-1583 (1972)

.

2. Alex, V., Tikhonov, L. V., and Brummer, 0., Kristall u. Technik 9^, 643-645 (1974).

3. Boah, J. K. , Metall, Trans. 4, 1432 (1973).

4. Guterman, M. B. , and Trekalo, A. S., Zavodskaya Laboratoriya _34, 823 (1968).

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

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

6. Kuriyama, M. , Early, J. G. , and Burdette, H. E. , J. Appl. Cryst. 7_, 535-540 (1974).

7. Kuriyama, M. , Boettinger, W. J., and Burdette, H. E. , J. Appl. Cryst. (1976).

21
Task 2

Evaporative Preparation of Ultra-High Purity Materials

R. C. Paule

Inorganic Materials Division

W. J. Boettinger and F. S. Biancaniello

Metallurgy Division

Institute for Materials Research

Summary

This Task involves the development of techniques for the ultrapurification of refractory ceramics and metals by high temperature evaporation of volatile contaminants. The near zero-gravity environment of

space will be used to allow container less melting and evaporative purification of relatively large samples, and to provide high vacuum. These

ultra-high purity materials may provide future reference materials and

materials with unique physical properties. The work reported here is a

continuation of NASA contract W-13, 475 #3, Task 2.

Work on evaporative purification is divided into two separate but

closely related research areas. One area deals with the mass spectrometric investigation of the detailed mechanisms of the purification

process. Problems of measurement associated with purification experiments are also investigated. The second area of research consists of

studying the extent of purification that can be obtained in the laboratory using vacuum RF-levitation and heating of small refractory metal

samples. In many respects this latter work gives an overall check on

the total evaporative purification process. The reporting of work under

22
Task 2 has been broken into Parts A and B corresponding to the two general areas of research.

In anticipation of possible evaporative purification experiments with

Al^O^ in space, we have checked the thermodynamically predicted relative

abundances of the A^O^ vapor species with a mass spectrometer. With

the exception of the minor vapor species, AlC^, the predictions were

verified

.

In order to more accurately predict and control Al^O^ purification

conditions, the effect of non- equilibrium radiative heating on Langmuir

(open) vaporization of A^O^ has been investigated. This has lead to

the development of a radiative heating model to describe the purification

experiments both in the laboratory and in space. The model allows us

to anticipate moderate temperature measurement problems for A^O^ space

experiments. An alternate, mass spectrometric , method for the measurement of temperature is suggested. Future work will center on extending

the mass spectrometric measurements and the radiative heating model to

other chemical systems which show promise for evaporative purification.

The goal of the R.F. levitation experiments is to determine the

rate of purification of selected metallic systems under conditions which

can be conveniently compared to theory. In many evaporative purification

experiments reported previously, the rate of purification has been

limited by transport of impurities to the evaporation surface rather than

by the evaporation process itself. This transport is often impossible

to describe quantitatively and leads to experimental data which are

difficult to interpret. It is often assumed that the stirring induced

23
by R.F. fields is sufficient to prevent transport limitations. However,

for highly volatile impurities, this assumption has not been previously

examined. In this report the purification rate of Fe from Mo samples has

been measured. The Fe impurity was rapidly reduced from 50 ppm to 50 ppb

and insight into the degree of mixing present in R.F. levitated melts

has been obtained. Because of this insight, future experiments on less

volatile impurities can be interpreted on the basis of evaporation processes and not on liquid transport processes.

24
Part A: Mass Spectrometric Studies of Vaporization Processes

Introduction

It is anticipated that future vehicles such as the Space Shuttle

1 2 will allow processing and manufacturing of unique materials in space.

'

Greater isolation for processes such as evaporative purification of refractory materials will be possible. In vacuum and in near gravityfree space we should be able to conveniently levitate and purify materials such as molten A^O^. This technique will eliminate two common

sources of contamination, i.e., the ambient gas atmosphere and the container. Such isolation is highly advantageous, but it requires that

one know the non-equilibrium radiative and evaporative properties of

the material; one can no longer use conventional backscattering processes to produce effective unit emittance and unit evaporation coefficients. We have studied these non-equilibrium properties in order

to better understand the control parameters needed for possible future

space processing.

The evaporation of solid or molten A^O^ under equilibrium condi3-9

tions has been the subject of numerous investigations. For nonequilibrium conditions there have been fewer studies'^ and the reported Al^O^ evaporation coefficients have varied from < 0.005 to 1.

Upon melting the non-equilibrium vaporization of Al^O^ has been reported

8 9 - to increase discontinuously by a factor of about three. ' This increase in vaporization was attributed to an increase in the evaporation

25
coefficient; the value was believed to discontinuously increase from

1/3 to 1. In our investigation of evaporative purification of molten

A^O^ we have looked at this discontinuous increase in vaporization and

have studied the interaction of the radiative and the evaporative properties. Our investigation indicates, for the case of radiant heating

of an Al^O^ sample, that the observed jump in vaporization is due

primarily to the radiative rather than the evaporative properties of

AlyO^' We find that the evaporation coefficient does not change during

the melting of Al^O^.

Experimental

Our kl^O^ studies were made using a new modulated beam quadrupole

mass spectrometric facility. The mass spectrometer and associated

furnace are contained in a single, non-partitioned vacuum chamber. The

stainless steel chamber is pumped by two liquid-nitrogen-trapped, 4"

oil diffusion pumps (see Fig. 1). A tungsten coil resistance furnace

is contained in the lower portion of the vacuum system and is surrounded

by a water-cooled inner enclosure. The maintenance of cold walls for

this inner enclosure has largely eliminated the common problem of

furnace outgassing. A large annular opening is included in the bottom

of the furnace enclosure so that the furnace can be pumped. Background

gases can readily equilibrate between the furnace and the mass spectrometer regions, and ion gauges in the two regions show essentially identi

cal readings. This equilibration minimizes the modulated beam detection

of background gases. A 6 mm diameter opening in the top of the furnace

enclosure allows direct effusion of vapors from the hot sample to the

mass spectrometer. The effusing sample vapors can be shuttered from the

26
mass spectrometer ion source by use of a small movable paddle. The

sample vapors, after passing the region of the movable paddle, are

chopped by a rotating 10-tooth, 50% transmission disk. A synchronous

motor containing commercial dry M0S2 bearings drives the disk. The dry

M0S2 bearings were found to be very suitable for vacuum service.

The neutral molecular beam of vapor products, after being chopped,

passes on to the mass spectrometer where the mass-selected AC signal

is measured by a synchronized lock- in amplifier. The usual massselected DC signals from the background gases are also simultaneously

measured. Background pressures during the experiments were normally

about 1.3 x 10 ^ Pa (1 x 10 ^ torr) . The mass spectrometer was usually

operated using electron beam energies of 20-22 volts.

To measure the temperature, the sample is viewed by an optical

pyrometer through a window in the top vacuum flange. This window looks

directly through the ion source of the mass spectrometer and onto the

hot sample. Sample brightness temperature measurements were made with

the optical pyrometer and simultaneous A^O^ vaporization measurements

were made with the modulated- beam mass spectrometer.

Two types of solid and liquid sample containment systems were used

in the furnace. In the first type, a solid 20 mm long x 3 mm diameter

polycrystalline A^O^ rod was positioned such that only the top of the

rod was heated and melted, and the liquid Al^O^ was then contained as a

sessile drop. The mass spectrometer could see the fixed area on the

top surface of the sessile drop. For the second type of sample containment, a 12 mm long x 6 mm diameter Re rod with a 1 x 1 x 3 mm long

27
slot machined into the top surface was used as the sample holder. This

slot, which was tightly and fully packed with A^O^ powder also presented

a fixed area for vaporization. Both types of containment systems allowed

for uninhibited (Langmuir) vaporization. In regard to radiation, both

sample containment systems were far from equilibrium or black-body

conditions

Results

Several interesting radiation and evaporation phenomena occur when

the top of an A^O^ rod is melted and resolidified. The sample upon

melting shows a large drop in its brightness temperature and upon resolidification a large increase in brightness temperature. The corresponding mass spectrometric measurements show a large increase in A^O^

vaporization upon melting and an even larger decrease in vaporization

upon resolidif ication (see Fig. 2). This behavior was observed for all

vapor species (Al, A10, A^O, kl^O^, and 0). The general qualitative

features of these phenomena were reproduced upon repeated meltings and

resolidif icat ions

These phenomena have previously been observed, and the factor of

three change in vaporization was attributed to a discontinuous change

11 12 in the evaporation coefficient in going through the melting point. '

It has been claimed that, upon melting, the coefficient changes in

value from 1/3 to 1. There is no a priori reason to question the possibility of such a discontinuous change in the evaporation coefficient.

There are, however, a number of a posteriori reasons to question the

28
evaporation coefficient explanation for the case of Al^O^.

The first evidence that the evaporation coefficient explanation is

not correct is found in the observation that the vaporization increase

upon melting is smaller than the decrease upon resolidification. The

evaporation coefficient explanation would require symmetrical processes.

Furthermore, the vaporization increase upon melting was found not to be

constant for repeated sample meltings, but always increased. During

repeated meltings it increased from an initial factor of about three

to a factor of about ten.

The above observations, including the associated large changes in

sample brightness temperatures, suggest that a rapidly changing sample

temperature may be the primary cause of the vaporization "discontinuities." A slotted Re rod was therefore used to contain the and

to act as a temperature moderator for the sample. Under these conditions

where the sample temperature could not change rapidly, the vaporization

of klyOy did not show a discontinuity (see Fig. 3). Identical results

were obtained in repeated experiments. Melting was visually observed

during the experiments. Free vaporization conditions were maintained

with the Re cell since the liquid Al^O^ l evel was maintained near the

top of the shallow slot.

The above measurements show that the evaporation coefficient of

Al^O^ must be constant, and they also suggest that any rapid increase in

vaporization is caused by a corresponding rapid increase in surface

temperature of the melt due to a discontinuous increase in the emissi29
vity. There is additional reported evidence to substantiate an increase

in the emissivity . ^ '"^ A greater emissivity of the melt would result

in the sessile drop being more efficient in absorbing radiation from

the hot furnace coil and this in turn can cause a large increase in the

temperature of the melt. A temperature rise is possible since the

sample is not in a black-body environment. It should be noted that

Langmuir vaporizations, by their very nature, should be done in a relatively open furnace and are therefore not normally made under black-body

conditions. The unusual observed sample brightness temperatures shown

in Figs. 2 and 3 are due to the lack of black-body conditions. Further

discussion of these temperatures, and of the probable value for the

emissivity of liquid A^O^j will be deferred until a later section.

Mass spectrometric measurements, using 70 volt electrons , were made of the

relative pressures of species vaporizing from an Al

2 °3 sessile drop held at about

2450 K. All vapor species signals were modulated and all showed a

shutter (paddle) effect indicating that they were vaporization products .

from the A^O^ sample. The observed pressures relative to the Al vapor

species given in Table I were determined using Mann's cross sections'*"^

and the usual inverse square root relationship for the multiplier correction. Relative pressures calculated"*"^ from the JANAF Tables"^ are also

listed. With the exception of AlO^, we obtained reasonable agreement between the observations and the literature. Regarding AIO2 , we saw no

evidence for its existence. To minimize the possibility of ion fragmentation, the M.S. electron energy was reduced and a search was made

using 1C, 12, 14, 16, 18, 20, and 22 volt electrons. A10

2

was not

30
observed. The species Al^ with a pressure l/50th that reported for

A10

2

was easily seen. With our mass spectrometer, it is believed that

we could have detected A10

2

at pressures l/500th that reported in the

4 6 7

literature. Earlier mass spectrometric workers ' ' have also not seen

A102> but have seen Al,^.

Table I

Relative Pressures for Vacuum Vaporization of A^O^ at 2450 K

p • / P ai species Al

Vapor Species

Al

0

A10

A1

2

0

A10

2

A1

2

0

2

Observed

1.0

1.4

0.17

0.036

0.00023

JANAF

1.0

1.3

0.36

0.094

0.012

0.00026

Thermodynamic values for the vapor species A10

2

have only been

estimated by JANAF, and A10

2

has been seen by only one laboratory.

This laboratory used a vacuum enclosure with a partition between the

furnace and the mass spectrometer regions and it appears that a mass

59 background gas species may have been misidentif ied as A10,,. If the

furnace (mass 59) background pressure were higher than that in the mass

spectrometer region, then this background gas would be chopped and detected as a modulated beam. Identification of Al^O^ vapor species is

31
especially hazardous since there is no significant isotopic distribution

for either of the elements, Al or 0.

The absence of the vapor species AlC^ will have a small effect upon

the total rate of Al^O^ vaporization since the reported AlC^ partial

pressure was quite small (see Fig. 4).

In the current work, appearance potential measurements were made for

the vapor species Al, C, A10, A^O, and A^C^. The appearance potential

values obtained were 6.0 ± 1, 14.2 ± 1, 10.3 ± 1, 8.1 ± 1, and 10.0 ± 1,

7 20 respectively, and are in agreement with the literature. '

Some evidence of evaporative purification was noted during the

initial heating of the Al^O^ samples. Modulated beam signals were

initially seen at 40, 44, and 56 amu and are attributed to CaO, Si02

and FeO impurities. Rough calculations indicate that the initial respective concentrations were about 200, 3, and 3 ppm. The samples

were heated at 1900 K for several days and the impurities were vaporized

prior to melting the A^O^.

Discussion of Results

Emissive and radiative heat transfer calculations have been made

to evaluate the sample and furnace environment. The Langmuir (open)

vaporization of the Al^O^ was not under black-body conditions and as a result the calculations should be thought of as only semi-quantitative in

nature. From the radiation viewpoint, the experimental furnace arrangement was complex and the effective emissive characteristics of the

sample and furnace were not known with great accuracy. Nevertheless,

several large and important thermal effects can clearly be seen to be

operating during the melting of Al^O^.

32
To help understand the cause of the thermal effects we have formulated a radiative heating model. For our system let us consider an

energy balance per unit time for the top end section of the sample

that is to be later melted. Energy to this volume is received only

in the form of radiation, R, from the hot tungsten furnace coil. Note

that R is that part of the total furnace radiation that hits the

sample volume under consideration. The energy losses from this volume

can be grouped into three categories: (1) reflected radiation, pR,

where p is the total reflectance of the sample, (2) emitted radiation, er, , , bb

where e is the total emittance and r, , is the associated black-body

bb

radiation of the sample, and (3) energy lost by thermal conduction down

the A^O^ rod. The energy loss by thermal conduction is a function of

the sample geometry and the sample temperature. For a relatively constant

geometry and temperature, the thermal conduction losses can be expressed

as a constant, k, times r, , . By combining the three sources of energy

bb

loss we obtain the following energy balance equation:

R = pR + er,, + kr„ . (1) bb bb

Since the sample is normally either polycrystalline Al^O^ or a melt, it

is essentially opaque and we can replace the total reflectivity by the

term 1-e.

R = (1-OR + er^ + kr^ . (2) bb bb

33
Rearranging the terms we obtain:

R = ' rbb (3)

The thermal conductivity for our solid A^O^ rod results in a k value of

21 about 0.11. The total emissivity of solid A^O^ just below the melting

22 15 16 point is about 0.35 while the value for molten kl^O^ ^ s aDOUt 0.90. '

Equation (3) for solid A^O^ just below the melting point becomes

(

.35 + .11 1 .

( - |,bt

- 1.31 rbb

. (4)

When an inf initesimally small increment of radiation from the furnace is

added, the A^O^ melts and we obtain:

/ .90 + .11 V .

\ .90 1 1 R + dR =( — Q „ |r' bb

= 1.12 r'

bb (5)

where r' , refers to the melt. (The heat of fusion required for melting the Al^Oy is a transient effect and is not considered in this development of the radiative heating model.) Equating the furnace input radiation from eqs . (4) and (5), and rearranging we obtain:

l!bb = JL31 =

r., 1.12 ±,1/ ' bb

23 By using the Stef an-Boltzman law, and the melting point of Al,^ as

a point of temperature calibration, we obtain:

r

1 -^=1.17 -

rbb

T

1 2327 K

34
Solving:

T' (melt) = 2420 K

and

AT = T ' - 2327 K = 93 K .

Thus, the melt can be expected to spontaneously increase in temperature

to about 90 K above the melting point. This increase is due to the

combined effect of the thermal conductivity losses and the significantly

higher total emissivity of the melt.

A 90 K temperature rise corresponds to a factor of three increase

in the A^O^ vapor pressure (see Fig. 4). This closely corresponds to

the observed vapor pressure increase upon melting. A 90 K temperature

rise should result in all A^O^ vapor species increasing by nearly the

same amount. Our observations agree with this conclusion.

At this point in the model experiment the molten A^O^ is considerably above the melting point and the furnace power can be reduced somewhat without refreezing the A^O^. One should be able to reduce the

furnace power and the melt temperature until the freezing point temperature (and vapor pressure) are obtained. At that point the Al^ should

refreeze and the temperature should drop considerably below the freezing

point since the refrozen Al^ should have a lower radiation efficiency

(lower total emissivity) . In practice the general features of this

description are followed (see Fig. 2). The commencement of freezing,

however, appears to be somewhat early relative to the model in that

it starts at too high a vapor pressure (temperature) . This point of

35
transition may involve a more sensitive energy balance and a more detailed picture is probably required. Furthermore, the actual refreezing process is not truly reversible in that a clear, transparent

sapphire bead forms rather than the original polycrystalline Al^O^.

The development of a more detailed energy balance to describe this more

involved situation was not pursued since it requires significantly more

emissivity information, much of which is not currently available.

Let us now examine some of the effects that the thermal conductivity constant, k, has upon the experiments. If there were no thermal

conductivity (k = 0) then Eq. (3) tells us that R = r,, = r

1 ,, and

bb bb

there would be no temperature jump upon melting. Conversely, if k

should become larger then the temperature jump should become larger.

During repeated remeltings (and vaporizations) the Al^O^ rods used in

our experiments became shorter and as a result the k values became larger.

In agreement with the model, the observed temperature (and vaporization)

jumps did become larger with repeated remeltings. It should be noted,

however, that the remeltings were of sapphire beads formed from the previous melts. While the partial transparency of sapphire complicates

the analysis, it is believed that the general processes, reasoning and

conclusions would be the same.

The current model, in addition to explaining the sample temperature

jump, provides a qualitative explanation of the observed brightness temperatures. The observed brightness of the sample is due to two radiation

sources, i.e., the emitted and the reflected radiation. The reflected

radiation is from the tungsten furnace coil which is about 500 K hotter

36
than the sample. The furnace coil must be much hotter than the sample

since the furnace is quite open and there is relatively low efficiency

in radiative heat transfer to the sample. One consequence of this higher

temperature is that the spectra from the furnace is considerably richer

o in 6500 A radiation than that emitted from the sample. Therefore for a

condition of equal total radiation, e.g., R = *" bb , an optical pyrometer

o operating at 6500 A will see considerably more radiation from the furnace

than from the emitting sample. Equations (4) and (5) show that the condition of equal total radiation (R = will be approximately true for

our experiments.

Let us now examine what happens when polycrystalline Al^O^ melts.

The radiation measured by the pyrometer should be proportional to

PR + f rbb

= (1-0 R + er

bb

where we now consider the p, 1-e, R, and r,, values to be evaluated at bb

o 6500 A. The spectral emissivity values for polycrystalline A^O^ and

20 13 14 ° the melt are about .45 and .90, * respectively. The 6500 A radiation to the pyrometer, except for the fixed window-prism correction and

the fixed geometry factor, is therefore

.55 R + .45 rbb (polycrystalline Al^) (6)

and

.10 R + .90 r» bb

(A1

2

0

3

melt) . (7)

o Since the R of the furnace at 6500 A is considerably larger than either

rbb r 'bb^

° f the samP le > relationships 6 and 7 show that the

37
measured optical pyrometer brightness temperature for the melt will be

less than that for the polycrystalline A^O . This is observed in all

of our temperature measurements (see Figs. 2 and 3). The exact calculation of the predicted brightness temperature, however, is subject

o to moderate uncertainties since we can only estimate R at 6500 A.

Equations (4) and (5) are of no direct aid to us in this regard since

the R in these equations refer to the integrated (total) radiation.

o The direct observation of R at 6500 A was not possible in our experio mental arrangement. Any reasonable estimate, however, for R at 6500 A

will lead to a lower brightness temperature for the melt than for the

polycrystalline Al^O^.

Conclusions (Part A)

It is anticipated that the discontinuous nature of the emittance

of Al^Oy will have several consequences in any space manufacturing

venture using this material. Three of the more obvious consequences are:

(1) For an Al^O^ sample levitated in space, thermal conduction will

be absent and there will not be a temperature jump. The obo served spectral radiation from the sample at 6500 A, however,

will still depend on both the spectral emittance of the sample

and on the spectra of the input (heating) radiation. This

will cause difficulties in accurately measuring the temperature

with an optical pyrometer.

(2) Mass spectrometric measurement of the Al^O^ vaPor pressure

should be a reliable alternate method for continuously monitoring the temperature of the sample since these measurements

38
are not affected by the furnace or sample radiation. Since

the evaporation coefficient of Al^O^ is constant, the M.S.

vapor pressure measurements can be calibrated at the melting

point

.

(3) For non-radiative heating of Al^O^ there will not be a tempera

ture jump. There will, however, still be emissivity effects.

For the non-radiative heating case we have the energy balance

relationship, energy in = energy out = £ r^» and we still need

to accurately know the spectral emissivity to measure the

temperature with an optical pyrometer. Again, M.S. vapor

pressure measurements should be a useful alternate method for

continuously monitoring the temperature of the sample.

Additional conclusions from this work are that the thermodynamic

literature is reasonably reliable for predicting the relative abundance

of the A^O^ vapor species. The literature for the vapor species Alf^,

however, is a minor exception to this conclusion.

Finally, our observations have shown that CaO, SiC^, and FeO impuri

ties can be vaporized from Al o 0 .

39
Part B; Evaporative Purification of Refractory Metals Using R.F. Levitation

Introduction

The purification of metals and alloys by evaporation of volatile

impurities is a well-known technique of materials processing and is employed extensively in industry and in the laboratory. The basic principles of evaporative purification and distillation have been known since

24 25

the nineteenth century. ' More recently these techniques have been

applied to the refining of refractory metals such as Nb, Mo, Ta, and W by

various methods of vacuum melting. These refinings are frequently performed

in configurations which are convenient but not simple in terms of basic

purification processes. Often, transport of the impurities to the evaporating surface (or transport of the vaporized impurity away from the

evaporating surface in the case of refining in a poor vacuum or an atmosphere) is the limiting process in the purification rather than the evaporation process itself. Much data has been accumulated for electronbeam zone-refined materials. These data have the additional complications

that impurities are removed by evaporation from a non-isothermal sample

as well as by zone refining. Even for induction stirred melts in vacuum

2 6 it has been reported that there may be insufficient mixing to prevent

depletion of the impurity in the liquid near the evaporating surface and

this will prevent optimum purification rates. Because of these experimental complexities, rates of purification reported in the literature are

difficult to interpret in terms of basic evaporation processes. Yet the

success of any evaporative purification scheme, regardless of the details,

40
is based on these evaporation processes. Only two extremes of impurity

transport in the liquid lead without difficulty to a comparison of evaporation theory to experiment—the case of no impurity transport limitation

(completely stirred) and the case of impurity transport by diffusion only.

The general approach of this research has been to measure the rate

of loss of selected metallic impurities from refractory metals processed

by R.F. levitation melting in vacuum. Advantages of levitation melting

for this purpose include good stirring, rapid heating to and cooling from

temperature, a nearly spherical evaporation surface and the absence of

contamination from containers. In addition, this technique is quite

similar to proposed methods of positioning and heating refractory materials in a low gravity environment. Purification rates determined

with this method should avoid some of the complexities inherent

in other evaporative purification configurations and provide an overall

check on the total evaporation process.

2 The primary effort of last year's work was to develop specific

laboratory procedures for the use of R.F. levitation melting for the

measurement of purification rates for Mo. In this report, experiments

are described in which Mo samples with about 50 ppm Fe were levitation melted in vacuum for various lengths of time and initial and final

chemical analysis were used to measure the rate of purification. The

impurity Fe was chosen because it is the most abundant impurity in

commercial purity Mo and because its extreme volatility with respect

41
to Mo should give insight into whether evaporation or transport of impurity in the liquid is the rate limiting process for the purification

of R.F. levitated melts. The determination of this fact is essential

before quantitative comparisons of theory and experiment can be performed,

Purification Rate

In a previous report"'", calculations for the purification of materials where complex interactions between species are present has been

performed. This computer program can be used (in a trivial fashion)

to predict the rate of evaporation into vacuum of a well-stirred isothermal ideal binary solution of two metals.* However, in this simple

case the rate can be obtained analytically. Analytical solutions are

advantageous because they immediately show the effect of parameters

on the solution and in this case provide a convenient means of planning

and interpreting simple purifications experiments.

Let n^(t) an<i n^(t) be the number of moles of A and B present in

the liquid at time t. Assuming Knudsen evaporation conditions apply

the relevant equations describing the purification process are:

n

A

dn

A

PA nA + n "

~k

A dt~ (8)

A B

n

B

dn

B

PB nA + « = _k

B dT (9)

A B

It should be noted that for Mo-Fe evaporation, the computer

solution converged to the analytical solution when the time intervals

were short enough. This provided an overall check of the iterative

techniques used in the computer program.

42
where k. =

1

(2TrRTm.)

l

1/2

and

P

i =

m. =

l

A =

R =

T =

vapor pressure of pure i at temperature T

molecular weight of component i

area of sample available for evaporation

universal gas constant

absolute temperature.

The solution when A is a constant is given by

t = Va

N

A

"

\ B /

- 1

(10)

(11)

where and N

g

are the initial number of moles of A and B respectively

1/2 and a = (PA

/PB

)^m

B

/m

A

) • F°r a given number of moles of B remaining, the

Equation (10) is valid even if A is not constant and even if there are

other components in the solution. When a and N /N << 1, eqs. (10) and (11) B A

are approximated by n = N and n = N exp{-p t/k N }.

43
number of moles of A remaining is given by Eq. (10) and the time, t, required

to achieve these levels is given by Eq. (11) . From this the concentration

can then be simply calculated for any time. As has been known since the

nineteenth century, component A can be purified of component B when a is

less than one. It must be re-emphasized that this type of solution even

extended to multicomponent systems is only valid when there is no chemical

reaction of species. For the case where reactions are present, the

calculation developed for complex evaporation must be used.

Experimental Procedures

2

It was the purpose of previous work to develop specific procedures

suitable for the measurement of purification rates of refractory metals

using R.F. levitation melting in vacuum and most of the details are presented in that report. A schematic of the equipment used is shown in

Fig. 5. A product of last year's work was the development of a levitation

coil which produces high gradients of the magnetic field in the vertical

direction. Use of this type of coil enables a modest degree of temperature control of levitated samples.

At the start of the present work, modifications were made to the

R.F. generator which increased the maximum power and permitted smoother

power control. The increase in power permits experiments to be run for

longer periods of time. At constant power setting, the temperature of a

levitated sample decreases as the sample gets smaller. Because previously samples were levitated and melted at the maximum power setting,

there was only a finite time before the sample would freeze due to this

44
temperature decrease. With the added power, we have doubled the length

of time that Mo-based alloys can be processed in the liquid state in

earth gravity.

A second important improvement has been the use of an automatic

optical pyrometer and a vacuum window which prevents the accumulation of

metallic vapor. Hence, the temperature of levitated samples is monitored

continuously during melting, purification and freezing. The vacuum window

consists simply of a transparent teflon tape which moves across a glass

vacuum window. The window is 15 mm in diameter and the tape moves at

15 mm/sec. The output of the pyrometer along with that of an ionization

gauge in the levitation chamber is recorded on a dual channel recorder.

The effects of sample emissivity and absorption of the window are calibrated

for each experiment using the melting point of Mo as a reference.

A third improvement which is attributed to the coil design and the

improved power control of the R.F. generator is the ability to melt and

refreeze samples while levitated in vacuum. Much work performed in the

past by others required the introduction of a gas into the system to

solidify the sample. Currently 4 g Mo samples can be stably levitated

between the temperatures of 2600K and 3100K (M.P. Mo ^ 2896K) . These

accomplishments are very sensitive to sample size. Although the increase in power permits the levitation and melting of Mo samples down to

3 g, the optimum conditions are achieved with samples initially 4 g. Larger

samples tend to drip or pour while molten. This limitation will not be

present in low gravity environments and much larger quantities of material

can be processed. It should be noted here that it was not the purpose of

45
this work to engineer the most versatile levitation system for use on

earth but rather to employ readily available technology to perform evaporative purification experiments.

All purification experiments reported here were performed on etched

4 g samples taken from a single rod (8.4 mm diameter) of commerical purity

Mo. Three samples were taken from the rod at different positions along

the length and were chemically analyzed to determine the variation of the

initial composition of the samples.

Samples for purification were placed in the levitation coil and

rested directly on the lower turns of the R.F. coil, and the system was

evacuated to less than 7 x 10 ^ Pa 5 x 10 ^ torr) . The coil was

energized rapidly causing the sample to levitate. After melting,

the temperature was adjusted manually to about 3000K and held for various

lengths of time. Power was reduced causing the sample to freeze and

cool to about 2600K. Then the power was cut off and the sample fell into

a copper dish. A typical temperature- time, vacuum-time record (test 3-24)

is shown in Fig. 6. After cooling to room temperature, samples were removed from the vacuum and prepared for chemical analysis.

In order to avoid the effects of micro and/or macrosegregation of

impurities caused by solidification, chemical analysis was performed on

one entire half of the purified samples. The solidified samples were

cut in half through the geometrical symmetry axis which these samples

generally possess. This axis was vertical in the levitated sample at

the instant the sample surface solidified. The sawing was performed using

46
^2^2 water~ coo ^ ec^ wheel to minimize Fe contamination. The chemical

analyses reported here were performed by the Analytical Chemistry Division

of the National Bureau of Standards. Unpurified samples and four of the

purified samples were analyzed for Fe using Extraction-Spectrophotometric

analysis. The details of the technique are presented in Appendix I. The

sensitivity of this technique is 2 ppm by mass (weight) . Because all of the

purified samples had Fe levels below 2 ppm, the other halves of these samples

as well as an additional nine purified samples were analyzed using Isotope

Dilution Mass Spectroscopy*. The details of this technique are given in

Appendix II. The sensitivity of this technique (twice the blank Fe

concentration) is 50 ppb by weight. This is currently the most accurate

technique available for analysis at these levels and it is especially

suitable for analysis of bulk samples.

Equipment failures and time required to perform chemical analysis

greatly reduced the number of samples processed below the desired

number. Three months of experimentation were lost due to failures of

the R.F. generator and failure of the optical pyrometer. Additionally,

one month was required for the careful chemical analysis reported here.

Experimental Results

Table II shows the Fe concentration determined in duplicate for

three samples from the starting material. The average iron value is 48.4

ppm by weight. Table II also summarizes the experiments performed and

*Four samples were analyzed using both techniques and the results are

consistent. (see Appendix I and II.)

47
TABLE II - SUMMARY OF EVAPORATIVE PURIFICATION EXPERIMENTS ON MOLYBDENUM.

INITIAL MATERIAL

Sample Number

Fe Content ppm

(by mass)

A

54.

54.2

B 46.9 47.5

C 43.8 43.9

PURIFIED MATERIAL

Sample NumberInitial Mass

Final Mass

Initial Vacuum Maximun

Temp. + SuperCooling

Time On Melting

Isotherm

Time Completely

Molten

Time On Freezing

Isotherm

Fe Fraction

g g PaxlO^ torrxlO 7 K K s s s ppb

(by nass)

2-3 4.0190 3.8425 8.0 6.0 2950 0 23 57 18 22.9*

2-7 4.0182 3.7839 7.3 5.5 2959 41 33 88 11 26.3* 2-13 3.9378 3.8043 6.4 4.8 2936 19 38 6 3 45.0*

3-21 4.1634 4.0684 4.0 3.0 2896 54 0 1340.2

3-24 4.2271 3.6780 8.0 6.0 3035 . 62 18 198 6 350.3 4-2 4.1542 4.0651 1.3 1.0 2989 7 12 8 18 31.8* 4-9 4.2500 4.0665 4.8 3.6 3016 114 14 44 18 32.9* 4-14 4.2575 4.1792 2.7 2.0 2896 21 0 483.4

4-16 4.0815 3.8979 2.7 2.0 3013 0 21 41 21 78.0

5-2 4.1073 3. 7487 4.8 4.0 2944 54 51 195 21 20.1*

5-12 4.0769 3.5617 7.3 5.5 2983 0 36 306 32 175.2

5-15 4.1294 3.5394 24.0 18.0 -2950 M5 23 309 15 314.5

7-1 4.2350 3.4388 7.3 5.5 3016 0 30 470 15 33.7*

Based on melting point of Mo of 2896 K.

The sensitivity of this chemical analysis (as determined by the blank Fe level) is 50 ppb. Hence, these values

should be considered as <50 ppb.

48
includes the initial and final mass, the maximum temperature while molten,

the amount of supercooling before solidification , the time on the

melting isotherm, time completely molten, time on freezing isotherm, and

the final Fe content. The change in mass of the samples has no real

meaning due to excessive spitting that occurs initially during melting.

Because of the rapid purification rate expected, two experiments (3-21

and 4-14) were performed in which the samples were not quite completely

molten (or possibly were just molten) and were promptly solidified.

The iron level of these two samples is 1.3 ppm and 483.4 ppb which are

the highest levels of any of the processed samples. Figure 7 shows the

Fe content plotted as a function of time completely molten. The results

are summarized as follows:

1. Very rapid evaporation of Fe was observed with levels being

reduced to < 50 ppb in less than 6 seconds completely molten.

2. Samples which were not completely melted showed Fe levels far

below the starting level but above the levels obtained for samples which

were completely melted.

3. The Fe content of most of the completely melted samples was

beneath the sensitivity of the chemical analysis and, hence, might be much

lower than 50 ppb. However, these samples showed some evidence of a small

but consistent level above the blank which was not correlated with time

processed. This may be indicative of contamination after purification.

The amount of supercooling is included primarily for information only.

49
4. Thr ee of eleven experiments completely melted have comparatively

high levels of Fe and it is assumed that contamination of these samples

occurred during cooling in the copper dish and handling after purification.

Conclusions and Discussion (Part B)

Although Mo with extremely low Fe content has been obtained, a major

goal of this year's work was to try to obtain reproducible purification

data for the impurity Fe from Mo. This goal has certainly not been

met as even a casual glance at Fig. 7 will show. Many precautions

have been taken to eliminate scatter in the data. Surface cleaning

and sawing was performed carefully. Chemical analysis was performed on

bulk samples rather than small samples. Vacuum and temperature during

experiments was continuously monitored. However, two facts seem to be

causing the difficulty. First, the extremely rapid vaporization of

Fe is such that time required for heating to and cooling from the

test temperature is larger than times required to purify the sample.

It was previously thought that surface depletion of impurity might

have greatly slowed the purification process. Second, the extremely low

level of Fe present in most samples makes the effects of handling much

more severe. Nevertheless, a few conclusions can be drawn from this data.

Metallographic examination of several samples indicate that freezing

of the levitated samples occurs from the outside surface to the center.

For example, the shrinkage cavity is always in the interior of the sample.

Hence, purification ceases for all practical purposes as soon as solidification starts. Metallographic examination of sample 4-14 shows that the

50
sample was completely molten. Since the temperature of this sample did not

rise above the melting isotherm even though the power input of the sample

was high enough to do so, it is assumed that this sample was just barely

completely molten when freezing was initiated. Hence, the Fe content of

this sample (483 ppb) will be taken to represent the amount of Fe left

in samples after the melting stage. Two specimens, 2-13 and 4-2, were

completely molten for 6 and 8 seconds, respectively, and their final Fe

levels were less than 50 ppb. So we conclude that the Fe level in completely molten samples is reduced from 483 ppb to at least 50 ppb in

6 seconds or less.

Figure 8 shows the calculated change in Fe level versus time using

Eq. (10) and (11) for data representing the experimental configuration.

This is very closely approximated by the exponential form of this solution given in the footnote to Eq. (10). Hence it is valid to use this

curve to find the time interval to change from 483 ppb to 50 ppb. The

calculated time for this is 0.54 seconds. So we conclude that at the

least, the experimental purification rate is 9% of the calculated rate.

This is considered reasonable agreement for this impurity considering the

experimental difficulty of measuring such rapid evaporation rates.

Conclusions about the detailed assumptions of the purification model

cannot be made at this point. The assumptions about ideal solution behavior, non-reacting species, and evaporation coefficients cannot be

examined based on this data. However, some insight can be gained into

the degree of mixing present in R.F. levitated samples. As was described

in the introduction, the rate of purification is often limited by transport

of impurities to the evaporation surface (i.e., lack of complete mixing)

51
rather than by the evaporation process itself. Under these circumstances

the surface of the evaporating sample becomes depleted in the impurity and

purification is retarded. Because we want to measure purification rates

limited by basic evaporation processes and not by transport of impurities

in the liquid phase, it is important to assess the degree of mixing in

R.F. levitated samples. This is also important for space processing because the case of R.F. stirring is probably the best mixing that can be

currently obtained.

The rate of mixing required in a sample to prevent surface depletion

of the impurity is a function of the volatility of the impurity. A

highly volatile impurity (a << 1) requires more mixing than does a less

volatile one ( a < 1) . It has been shown here for the impurity Fe

-4

in Mo (a a. 10 ), that the purification rate is very rapid. At the

slowest the iron content has been reduced by a factor of 10 in less

than six seconds. This indicates very rapid circulation times and the

probable absence of dead zones in the liquid. Hence, limitations of

impurity transport to the evaporation surface are not large. Certainly

for less volatile impurities like Zr (a *v 10 ) , it can be stated

that there will be no measurable limitation to the rate of purification

caused by impurity transport in the liquid using levitation melting under

the same conditions. Next year's work will measure purification rates

for less volatile impurities from Mo and conclusions about the evaporation

process itself can be made.

52
Acknowledgment

The authors would like to thank I. L. Barnes, R. L. Burke,

B. I. Diamondstone, L. P. Dunstan, E. L. Garner, and E. J. Malenthal

of the Analytical Chemistry Division of the National Bureau of

Standards for the chemical analysis reported here.

53
References

Paule, R. C, NBSIR 73-402, 33 (1973).

Paule, R. C, Boettinger, W. J., Biancaniello, F.S., NBSIR 74-611,

24 (1974).

Brewer, L. and Searcy, A. W. , J. Am. Chem, Soc. 73., 5308 (1951).

Porter, R. F. , Schissel, P., Inghram, M. G., J. Chem. Phys. 23 , 339 (1955).

Ackermann, R. J. and Thorn, R. J., J. Am. Chem. Soc. 78, 4169 (1956)

DeMaria, G., Drowart, J. and Inghram, M. G. , J. Chem. Phys. 30 , 318 (1959).

Drowart, J., DeMaria, G. , Burns, R. P., and Inghram, M. G., J. Chem.

Phys. 32, 1366 (1960).

Farber, M. , Srivastava, R. D. , and Uy, 0. M. , J. Chem. Phys. 55 , 4142 (1971).

Farber, M. , Srivastava, R. D. , and Uy, 0. M. , J. Chem. Soc,

Faraday Trans. 1, 68, 249 (1972).

] Sears, G. W. and Navias, L., J. Chem. Phys. 30, 1111 (1959).

] Burns, R. P., Jason, A. J., and Inghram, M. G., J. Chem. Phys. 40 , 2739 (1964).

] Burns, R. P., J. Chem. Phys. 44, 3307 (1966).

] Alcock, C. B. and Peleg, M. , Trans. Brit. Cer. Soc. 66_, 217 (1967).

] Peleg, M. and Alcock, C. B., High Temp. Sci. 6_, 52 (1974).

] Diamond, J. J. and Schneider, S. J., J. Am. Ceram. Soc. 43_, 1 (1960)

] Gryvnak, D. A. and Burch, D. E., J. Opt. Soc. Am. 55, 625 (1965).

] Mann, J. B. , J. Chem. Phys. 46_, 1646 (1967).

] Paule, R. C, High Temp. Sci. 6, 267 (1974).

] JANAF Thermochemical Tables, Dow Chemical Co., Midland, MI (1975).

54
[20] Franklin, J. L., et al, Ionization Potentials, Appearance Potentials , and Heat, of Formation of Gaseous Positive Ions , NSRDS-NBS 26 (1969)

.

[21] Goldsmith, A, Waterman, T. E., and Hirschhorn, H. J., Handbook of

Thermophysical Properties of Solid Materials , Vol. 3, pp 36-7,

Macmillan Company, New York (1961).

[22] Touloukian, Y. S. and DeWitt, D. P., Thermophysical Properties of

Matter, Vol. 8, Thermal Radiative Properties Nonmetallic Solids , IFI/Plenum Press, New York-Washington (1972).

[23] Schneider, S. J., Pure & Applied Chem. 21 [1], 115 (1970).

[24] Brown, F. E., Trans. Chem. Soc. 35_, 550 (1879).

[25] Lord Rayleigh, Phil. Mag. 8_, 534 (1904).

[26] Obradovic, N. D. and Bennett, G. H. J., J. Inst. Metals 97_, 186

(1969)

.

55



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Fig. 4 A1 2 0

3 Equilibrium Vaporization

Pressures into Vacuum

Fig. 5. Schematic of the levitation melting facility. 1) Sample and

levitation coil; 2) copper cup to catch solidified sample;

3) R.F. leads to load-coil transformer and generator; 4) to

vacuum system; 5) ionization gauge; 6) window to prevent accumulation of metallic vapor; 7) automatic optical pyrometer.

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00

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(qdd) ©j NOIlDVdd SSVW

1 1 1 1 1

1 1 1 1 \

TIME (s)

. 8. Plot of calculated rate of purification for a 4g Mo sample

containing 48.4 ppm Fe (by mass) at 3000K. Based on remaining number of moles calculated using Eqs.10 & llwith P = 1.002 x 10_1+ atm., p = 0.521 atm. , A = 3.0484 cm2

, (a°=

1.4674 x lO- 4 ).
Appendix I

Determination of Iron in Molybdenum

Method: Extraction-spectrophotometric determination of iron as the

Fe (II) -Bathophenanthroline complex

Procedure: The molybdenum disks were cut in half to obtain samples

weighing 0.7-1.1 g. These samples were etched with dilute hydrochloric acid to remove any surface iron contamination. Subsequently,

the samples were dissolved in 100-ml Teflon beakers using a nitric

acid/sulfuric acid mixture. The nitric acid was removed by addition

of hydrochloric acid followed by preconcentration . After addition

of appropriate masking agents and adjustment of pH, the samples were

diluted to volume. Duplicate aliquots were taken for spectrophotometry analysis, using Bathophenanthroline as the chromogenic

reagent and 1-hexanol as the extraction solvent.

NBS high purity, sub-boiling acids were used throughout this study

and all reagents, (i.e. ammonium tartrate, ascorbic acid, hydroxylamine hydrochloride, boric acid, and Bathophenanthroline) were

freed of trace iron contamination by repeated extractions with

Bathophenanthroline-hexanol . To minimize the environmental blank,

all sample dissolutions were performed in the clean room facilities

of the Analytical Chemistry Division.

Sample Weight, g Iron

,

ppm

A 0.7 54.0; 54. 2

B 1.0 46.9; 47.5

C 1.1 43.8; 43.9

2-3 0.7 <2

2-7 1.0 <2

2-13 0.8 <2

3-2] 1.1 <2
Appendix II

Iron Concentration of High Purity Molybdenum Metal

by

Isotope Dilution Mass Spectrometry

The concentration of iron in high purity molybdenum metal was

determined by isotope dilution mass spectrometry. Samples

ranging in weight from 0.8 g to 1.6 g were etched in HNO3-H2SCU

and then HC1 to remove surface contamination. The samples were

weighed, spiked with 57 Fe, dissolved in a HNO 3 -H 2 S0 4 -H 2 0 solution, and evaporated to dryness. The residue was taken into

solution using NHi<OH and the solution again evaporated to

dryness. The iron was then separated from its matrix using

anion exchange chromatography. Reagents purified by sub-boilin

distillation were used throughout the analysis.

A triple filament rhenium ribbon ion source was used for mass

spectrometric analysis. FeCl 3 in nitric acid solution was

evaporated to dryness on each sample filament and then partically reduced in a hydrogen atmosphere to a mixture of iron

oxides and metallic iron.

The results of the iron concentration determination and the

analytical blank determination are shown in Tables 1 and 2,

respectively. Column 4 of Table 1 shows the ratio of the

total iron (sample + blank) to the average blank. Since the

iron concentration of most of the samples is less than 100 ng

and yields total iron to blank ratios <3, it is recommended

that, an upper limit statement be used Tor these samples. For

example, it is recommended that samples 1-5, 7, and 15 be

assigned a value <50 ng. The remaining samples have total

iron to blank ratios which are sufficiently large to quote

the values reported in Table 1. For these samples (6, 8-12),

the estimated accuracy of ±30 ng is the sum of the error

terms due to isotopic ratio measurement, spike calibration,

variability of the analytical blank and 56 Fe background

contribution from the ionizing filament.
Table 1. Concentration of Iron in High Purity Molybdenum Metal.

Sample No.

(1) 2-3

(2) 4-9

(3) 4-2

(4) 2-7

(5) 5-2

(6) 5-12

(7) 2-13

(8) 5-15

(9) 4-14

(10) 4-16

(11) 3-21

(12) 3-24

(13) 7-1

ng/g of Fe'

22.

9

b

32. b

31. 8

b

26. 3

b

20.

b

175.2

45.

0

b

314.5

483.4

78.0

1340.2

350.3

33.

b

Total Fe (ymols)

0.001095

0.0009862

0.0009064

0.001158

0.0008607

0.002908

0.001433

0.006468

0.011767

0.002100

0.039728

0.009944

0.001394

Total Fe/Blank

2.4

2.2

2.0

2.5

1.9

6.4

3.1

14.2

25.8

4.6

87.2

21.8

3.1

a Corrected for blank contribution of 0.0004554 ymols (25.4 ng)

.

Upper limit value of <50 ng recommended.
2. Iron Analytical Blank Determinations.

Blank No. n g

1 19.96

2 22.07

3 24.15

4 24.56

5 22.21

6 39.53

Average = 2 5.4

ymols

0.0003574

0.0003952

0.0004342

0.0004399

0.0003978

0.0007080

0.0004554
Task 3

Vacuum Effects in the Preparation of

Composite Materials

Harvey Yakowitz

Metallurgy Division

Institute for Materials Research

Summary

A method for determining the description in-depth of X-rays within

a material has been deduced. This method is applicable to composite

materials both conventional and laminates. The method is described

in detail along with examples. Other analytical tools for composites

such as energy dispersive analysis are described. A brief description

of the computer program for carrying out these analyses is given.

Some specific results on SiC-Al matrix composites supplied by Marshall

Space Flight Center are included.

56
Introduction

In order to characterize composite materials by means of electron

beam produced X-rays, the distribution in depth of these X-rays must

be determined. This distribution can be obtained experimentally but

only with extreme difficulty; the experimental errors are potentially

very large. Attempts to fit the distribution by means of

Monte Carlo techniques can be carried out but this method is cumbersome,

expensive, and not particularly sensitive. This report will

describe for the first time a simple analytical expression describing

this distribution function. Furthermore, the way in which the function

can be used to analyze layered structures or film composites and composites of cylindrical or spherical particles embedded in a matrix will

be given in detail.

Since many such analyses must be carried out with energy dispersive

X-ray microanalysis, this technique will be outlined briefly. Special

emphasis will be placed on characterization of the continuum X-ray

background produced during energy dispersive X-ray microanalysis. A

computer data reduction scheme incorporating this continuum background

correction will be described briefly as well.

Some further studies on the SiC filaments in Al matrix composite

provided by Marshall Space Flight Center will be discussed. Specifically,

the number and distribution of phases will be considered.

Finally, some by-products of the derivation of the X-ray distribution

function will be reported. These include a new absorption effect correction and atomic number correction for conventional X-ray microanalysis.

These new correction formulae can be used riot only in the analysis of

composite materials but also are applicable to general X-ray microanalysis.

57
Quantitative evaluation of thin film specimens, either unsupported

or upon a substrate, poses difficulties for the X-ray microanalyst

The chief problem is that the electron beam is not completely stopped

within the thin film so that conventional quantitation schemes cannot be safely applied. The purpose of this report is to describe the

difficulties encountered in thin film analysis in detail and to propose

a simple method for obtaining quantitative information from thin film

specimens

Nature of the Problem

An electron striking a thin unsupported film has three possible

interactions: backscatter, transmission or total energy loss within the

film, i.e., absorption. If the film is supported on a substrate, the

situation is more complex. The possibilities for this case are shown

schematically in Fig. (1). Any method for evaluating thin films must

account for reductions in X-ray emission due to electron transmission and

interactive effects due to the substrate, if any. The analyst usually

measures intensity ratios, i.e., the intensity from the thin film constituents with respect to bulk standards. Therefore, what is needed is

a means to predict this intensity ratio as a function of sample thickness and composition — both of which are usually unknown. In addition,

the method should also be applicable for non-normal electron beam incidence since many scanning electron microscopes (SEM's) require the specimen to be tilted for X- v ay analysis.

There have been a number of approaches taken toward obtaining a

thin film analytical procedure. For the unsupported specimen, there is

58
little electron-solid interaction so that the X-rays produced escape the

specimen after suffering negligible absorption and fluorescence effects.

Then observed intensity and concentration are linearly related. A number

of investigators have proposed techniques for determining the form of

equations taking advantage of this property of thin-film, X-ray interactions

[l] - [6 ]• These equations range from empirical relations in which the

proportionality factor is determined from intermediate standards to

complex theoretical schemes requiring fairly extensive computation facilities. For thin films mounted on substrates, two distinct approaches have

been taken. The first consists of attempting to determine the number

of ionizations produced by each group of electrons which follow any of

the paths delineated in Fig. (1). These contributions are summed to

obtain the total number of ionizations in the film; the cctal number of

ionizations from a bulk standard can be obtained [ 7 ]. The ratio of these

ionizations will be the observed intensity ratio. This method requires

knowledge of the X-ray range, the backs catter correction factor, R, the

ionization cross-section, Q, and the electron stopping power, S. Each

of these parameters is used in conventional electron probe microanalysis,

and each has been reviewed in detail in terms of derivation and uncertainties [8]. A further assumption of this method is that the substrate

does not contain any of the constituents found in the film. The final

result is a set of integral equations which are solved by normal equation

theory using matrix methods [9 J, [10]. Hence, fairly extensive computation facilities are needed and the method is subject to all of the

uncertainties associated with conventional X-ray microanalysis. Nevertheless, this approach has proved valuable in practice and is in current

use [ 9 ], [10].

59
The other approach is to attempt to determine the number of X-rays

generated as a function of depth in the specimen and in the standard.

The distribution in depth of X-ray generation is commonly referred to as

<Kz) , with z being the mass thickness of the specimen, following the

concept introduced by Castaing [11]. A plot of <j>(z) against z is shown

in Fig. (2); normal electron beam incidence was used [12]. The shape

of this cf>(z) plot is typical of all elements bombarded with electrons

at normal incidence. In effect, <}>(z) represents the number of X-rays

generated at depth z with respect to the number of X-rays generated from

a "free surface" of the same material, i.e. , a layer of vanishingly small

thickness. The peak occurs because electrons penetrate some distance

before producing ionizations and because electrons scatter back toward

the surface. If the specimen is tilted, the 4>(z) curves change shape

since the electrons remain nearer the surface; the effect has been

studied by means of Monte Carlo simulations [13]. Figure (3) shows some

results for a copper target.

The great value of the 4>(z) curve is that the integral of the curve

yields the total number of X-ray photons generated per incident electron.

Knowing this, one need only correct for absorption and fluorescence

effects in order to obtain the number of emitted X-rays from the specimen.

For thin film analysis, the <J>(z) curve for the thin film is needed as

well as that for the standard. If a bulk pure element is used as the

standard, the only correction needed in principle is for absorption;

fluorescence effects are nonexistent in the standard and negligible in

the unknown.

60
Relatively few <f>(z) curves have been determined experimentally;

the experiment is tedious and difficult [11] - [15]. Attempts have

been made to derive expressions enabling <(>(z) to be calculated. Analytical expressions have been discussed by Criss and Birks [16] and by

Reuter [17]. Many calculated <f>(z) curves have been obtained by Monte

Carlo methods [13], [18], [19]. The <f>(z) approach has been applied to thin

film analysis both from the point of view of measuring the thickness

of pure element films [17], [20]-[22] and actual analysis [22]. In this

report, we describe a simple method which can be used to obtain cf>(z)

for bulk specimens with any electron beam incidence angle; a procedure

based on this calculated <|>(z) to obtain quantitative thin film analysis

will then be described.

Thin Film Analysis with 4>(z) Data

In principle, one needs only to obtain the c|>(z) curves for the

specimen and standard. For the thin film specimen, the $(z) curve is

affected by the nature of the substrate or lack thereof. Hence, the

<|>(z) curve for the thin film case will depend upon the electron scattering properties of the substrate; when these factors are taken into

account, an appropriate <f>(z) relation can be obtained for the thin film.

In order to obtain the X-ray intensity emitted by the standard, the

integral of <f>(z) for the standard over the range of z from 0 to the

X-ray range, z^, must be corrected for absorption in the bulk specimen

[8]. The ratio of the two integrals gives the expected intensity ratio,

k^, for a film of thickness z with respect to a pure bulk standard. Thus,

one can prepare a curve of k^_ versus film thickness for pure element films.

61
In practice then, one requires a means to formulate <f>(z) for the

bulk standard as a function of incident electron beam energy, incidence

angle, and atomic number as well as a means to obtain the integral of

this expression. Furthermore, in order to obtain the required information from the film, the effect of the substrate must be taken into account

including the case where the atomic number of the substrate, Z , is

zero.

Formulation of ft( z ) for Bulk Material

Theoretically, the LaPlace transform of the absorption correction

yields the <)>(z) function equation in electron probe microanalysis.

Recently, direct investigation of the absorption effect resulted in a

proposed relation for the absorption correction factor, f(x)> such that

^= (1 + 1.2 x 10~6

xY)

2

(1) I

where x = ~~ esc ^, -jj is the X-ray mass attenuation coefficient, i>

is the X-ray emergence angle from the specimen and y = eJ"^^ - E

where E is the incident electron beam energy and E is the critical

o

bJ

q

excitation potential for the electron shell of interest, viz., q = K, L,

or M [23] . The inverse LaPlace transformation of Eqn (1) should yield

a valid expression of <J>(z). However, this procedure does not provide an

entirely satisfactory result because the value of <Kz) when z is zero,

called <j> , is zero. All experimental determinations of d> indicate that

o o this quantity is, in fact, greater than unity [11] , [12], [14] , [15] . Furthermor

the effect of varying the electron beam incidence angle on $(z) derived

in this way is not at all clear.

62
In view of this state of affairs, we examined both experimental and

Monte Carlo generated Kz) curves in the hope that a rational and consistent analytical <f>(z) formulation could be inferred. Consider Fig. (A)

in which the major features of a typical normal-incidence <f>(z) curve are

noted. Our observations based on all such $(z) results we could obtain

are that the function increases parabolically from <j> through the peak

o

at (h,k) , continues to follow the same parabola to about a z value of

1.5h and then drops off exponentially to the X-ray range, z^_. In particular, for the region of 0 <_ z <_ 1.5h

* (z) z<1.5h * h

~ 2

< 2 -h> 2

(Vk) + k (2a)

while for the region 1.5h <^ z <_ z^

(2b)

<Kz) ^, cu = 0.25(cf> +3k){exp[3h sec6/(z -1. 5h) J }{exp-[2zsec9/(z -1.5h)]}

z^l. in o r r

where 6 is the electron beam incidence angle. The major problems were

to obtain <J> o

, h, k and z^ as a function of incident beam energy, electron

shell and atomic number.

Reuter has discussed $ in detail and following Duncumb and Melford

o

&

formulated an expression [17] as

<J> = 1 + 2.8n (l - 0.9W) (3)

where n is the backs cat tering coefficient for electrons which has been

measured over a wide range of conditions [24] - [26] and W is E /E . q o One can fit n as a function of atomic number, Z, with a polynomial.

The X-ray range, z^, has been investigated by a number of authors [11]

[27] - [30]. Figure (5) shows that results from these expressions may

63
vary considerably with respect to one another. Hence, apparently there

will be an uncertainty introduced depending upon which expression for

is chosen. When our results were compared with experimental findings,

Heinrich's expression for z^ seemed to compare best:

z r = 0.007(E

o

1,65

- E

q

1 * 65)(mg/cm

2

) (4)

J

However, we are by no means certain that Eqn (4) is completely accurate.

The next task was to determine the values h and k. Examination of

the available evidence for normal electron beam incidence indicated that

the ratio (h/z^) was a decreasing function of atomic number and virtually

independent of beam energy. Apparently, the peak in the <Kz) curve

moves deeper into the material as a function of the parameter (E^* ^-E ^) ,

E values in keV, while the backs cattering propensity of the material n

opposes this tendency. Thus, the ratio h/z^ could be expressed in terms

of either n or for normal beam incidence, atomic number Z, since n depends

only weakly on beam energy:

h/z

r = 0.487 - 1.60n + 2.44n

2 - 1.33n

3

(5a)

h/z = z1/2 + io -V (5b) I r

Equation (5a) may be used with any electron beam incidence angle while

Eqn (5b) may only be used in the normal incidence case. The value of n

increases as a function of electron beam incidence angle, approaching

unity at grazing incidence [26]. If n in Eqn (5a) approaches unity

then h tends to zero in accord with Fig. (3).

Finally, the value of k was estimated by fitting available data to the

relation

64
k = <f> (1 + 0.35 cos 91n U) (6)

where U = E /E and 6 is the electron beam incidence angle. While the

reasons for the exact form of the multiplier for <j> in Eqn (6) are not

readily apparent, the dependence on a slowly varying function of U is

not altogether surprising. As U increases, the depth at which the peak

occurs increases, i.e., h increases. But, the average electron is

steadily losing energy as it proceeds along its path. Apparently then,

the number of ionizations produced at depth h follows a roughly similar

relation as does the ionization cross-section [31] which also increases

with In U. Results from Eqn (2) are shown in Figs. (2) and (3) for comparison purposes; the agreement is typical of many such tests.

The integration of Eqn (2) is straightforward; the result for the

intensity generated per incident electron, I , is

The appropriate cf>(z) relation for the thin film specimen and its integral

are now needed.

The (ft(z) Relation for a Thin Film Specimen

The thin film specimen has an effective backscattering coefficient,

n£

. If there is no substrate, then n E

is the backscattering coefficient

for the film, r\ . If there is a substrate then

o q

z

r r I. = <Kz)d(z) = 0.125 (4> + 3k) (Q + 3h)

Lj O O

(7)

with Q = (z - 1.5h)(l - e"

2 or ) = 0.865(z - 1.5h)

r

- n BF

+ n SS n TF (1 - V (8)

65
where n is the backscattering coefficient for the bulk substrate having

atomic number Z , and r\ is the fraction of electrons transmitted through

SS ir

the thin film. We have been able to fit transmission and backscattering

data available as follows:

nBF * 4n

Z

Cz/RK-0) (9)

where is the bulk backscattering coefficient for atomic number Z, z

2

is the film thickness in mg/ cm and R is the maximum electron range, in

2 mg/cm , of Kanaya and Okayama [32J:

R^_

Q = 2.76 x

10~ 3

E

q

5/3 A/Z 8/9

(10)

A being the atomic weight of the target.

For Tirr-r, we obtained

lr

n TF

= exp - {(0.03X2X10"4

E

q

) Z (z/\_

Q ) (z/\_

Q

) 1 > (H)

These expressions differ in form to some extent from those suggested

by Reuter [17], but the results are similar.

To obtain 4>(z) for the thin film specimen, one needs to substitute

n into Eqns (3) and (5a). Then the values of <j> , h and k for the thin

c. o film specimen, called cf>^, h

+

and k

+

for the remainder of the discussion,

can be obtained. The form of the 4>(z)

+

curve is assumed to be as given

in Eqn (2). The integral relationships are:

For the region o <_ z <^ 1.5h

+

z<1.5h

+

_ z(* - k ) 2

f(z)d(z) = ^

Ej" " h z + h J + k 2 ^ 12 > o h

66
and for the region 1.5h

+

<_ z ^_ z^

z>1.5h

+

t

o

<f>+

(z)d(z) = (<J>+

+ 3k

+

)(3h

+

+ Q)0.125 (13)

where Q = ( z r ~ l-5h

+

) {[1 - exp - [ (2z - 3h

+

)/(z

r

- 1.5h

+

)J}

The required ratio of photons generated in the thin film specimen

to those generated in the standard can be obtained by dividing Eqn (12)

or Eqn (13) by Eqn (7) depending on whether z _> 1.5h

+

or not. Figure (4)

shows <f)(z) for bulk Au and for Au films on Al^O^ as calculated from Eqns

(2) and (12).

But the value of k measured is not l^/lr but rather

k = ^1 f+ (X> (14)

r I

G

f( X )

The value of f(x) can be obtained from Eqn (1). For many thin film

specimens f

+

(x) is virtually unity. We have assumed that f

+

(x) varies

linearly with specimen thickness, being unity at z = 0 and being f(x) at

z = z : r

f

+

(x) = 1 + [(-T-)(f(x) - D] (15)

r

The required computations for obtaining k^ as a function of specimen

thickness for a pure element can be carried out with the aid of a desk

calculator. We have also written a short (40 statements) computer program in the BASIC language to obtain this function.

Comparison of Results with Experimental Evidence

Several investigators have reported results of measurement of film

thickness as a function of X-ray intensity; parameters such as incident

67
beam energy and substrate material were varied as well [17], [20]-[22].

Tables I-III give results of the method proposed in this report as

compared with experimental results obtained by Reuter [17] , and by

Bishop and Poole [20]. Figure (6) shows a comparison between results

calculated by our method and Monte Carlo and experimental results reported by Kyser and Murata [22]. Additional results for gold films on

various substrates were reported by Bolon and Lifshin [20]. These

workers utilized the Au-Ma line in conjunction with an instrument having

an X-ray emergence angle of 18°. Therefore, the absorption correction

factor for the pure element standard was about 2.5. Uncertainties in

the absorption effect in this case make comparison of these experimental

values with results from our model ambiguous as indicated in Fig. (7)..

The thin film procedure we propose reproduces, within about ±10%

relative, experimental data taken under a wide variety of conditions of

incident beam energy, specimen thickness, and substrate material. But

these comparisons are only for materials of known composition. The

effect of introducing the complication of unknown composition as well

as unknown thickness still requires consideration.

Compositional Analysis of Thin Film Specimens

Presumably, the investigator measures the emitted intensity for

element i from the thin film specimen, i"!" , and the emitted intensity

from a bulk specimen. We will assume that the intensity from bulk pure

"i" is I since one can obtain this quantity even if multielement

standards [33] or calculational schemes for pure element intensities [34]

are used. If the intensity I. p emitted from the pure element having the

exact thickness of the thin film specimen could be obtained, assuming both

unknown

68
and standard to be measured under identical experimental conditions,

i.e., same substrate, specimen tilt, incident beam energy and current,

then the ratio

where is the true concentration of element "i" in a film of thickness

z. Equation (16) is valid since absorption, electron scattering, and

retardation as well as fluorescence effects are negligible in thin films.

— z The value of I. „ can be calculated as a function of z as has been desi,E

cribed. The value of i"!" -/I. „ can be measured. But the film thickness

i,E i,E

z is unknown.

However, if the ratio iVl^ can be obtained for all components of

E

the thin film specimen, a good approximation to z can be made. This

possibility exists because emitted X-ray intensity from a thin film is

a nearly linear function of thickness as shown by experimental results

[21], [22]. So we presume data for all components to be measured;

we can calculate for all components straightforwardly. We mav then

z +

obtain the value of z for which is equal to I calling this

& h E

value (z)^ for the "±^" element, we may similarly obtain (z)^. and so on

to ( z ) n

for an "n" component sample. Finally,

z = (z) + (z). + . . . + (z) (17) 1 3 n

where z is the mass thickness of the unknown thin film specimen. Thus

the intensities from a homogeneous mixture of thickness z are equivalent

to the intensities from a hypothetical layered structure of thickness

(z) + (z) . + ... + (z) = z. i J n

69
z , Having thus obtained the value of z, we may now obtain 1^ •g'^-^ £•

Then

I

+

I

2

I

+

f Jul = Jul . c+ III (18)

2

^i.E i,E

xi,E

These assertions can be tested using the experimental data of Kyser and

Murata for Mn-Bi and Co-Pt films on A1

2

0 substrates [22]. Figure (8)

shows I /I for all of these elements as a function of z. Table IV

lists the measured values of lT _ reported for several specimens.

X y

Ed 1) L

Table IV also gives the z values found from using Fig. ( 8 ) and Eqn (17)

and the values subsequently obtained by using Eqn (18) as compared

with those reported by Kyser and Murata [22],

Extension to Spherical and Cylindrical Particles

The major difference between the particle and thin film case is

that the number of electrons which escape the particle of interest

depends on the particle size; Fig. (9) shows the effect of bombarding

spheres of Al with electron beams of varying incident energy. Calling

the fraction of electrons escaping from the particle n T

> experimental

evidence as well as Monte Carlo calculations led to the formulation of

an empirical expression for nT

Li

n T = 1-d IU-n)/R] (19) L p

where d^ is the particle diameter and n and R are as defined previously.

If n T is substituted into Eqns (3) and (5a) , then values of <p , h and k

Li O

for particles can be obtained. Then the ratio of generated intensities

from the particle to the bulk can be obtained for the particle in an

70
analogous fashion to that discussed previously (Eqns 12, 13). However,

this ratio is not sufficient by itself since the particle shape must

be taken into account as well. By using Monte Carlo calculations for

comparison, the best approximation for spheres and cylinders appears

to be:

kj = (^)

3 {i + (f(x)-i)J K10

^°0

3Z) (20)

G r

where 1^ is the generated intensity for the particle, d^ is the particle

diameter and all other terms are as defined previously. Equation (20)

is only valid up to d = z . Then k^ becomes unity. However, some of

p r r

J

the generated intensity will be lost due to the shape of the particle as

shown schematically in Fig. (10) . For spheres and cylinders the volume

ratio of the segment containing the X-^ray emission with respect to that

of the bulk is given by

V 1.5d - z _P_ = — —.

L (2i) V 2z ^

J z r

r

when d > z . Equation (21) predicts that this ratio is unity when

p r

J

d = 2z . Hence, for the region z - d < 2z we have

p r - x p — r

k? - V /V (22)

i P z r

The need for the cubing in Eqn (20) is apparently due to the 3

dimensional X-ray distribution in depth; we are investigating this distribution in more detail. Figures (11)&(12) show results from Eqns (20)

and (22) as compared with Monte Cerlo calculations for particles.

Agreement is reasonably good.

71
The Absorption Correction Factor in X-ray Microanalysis

The LaPlace transform of the function <j>(z) is given by:

00 F(0) = <j)(z)dz (23)

*" o

If there were no absorption of electron beam produced X-rays by the

sample, then the intensity of emitted X-rays would be F(0). But as

these X-rays travel out of the specimen in a direction toward the de-

— YZ

tector, they suffer absorption according to Beer's Law viz, <})(z)e

where X is (u/p) esc tJj , being the average angle of emergence of the

X-rays from the specimen and (y/p) being the mass attenuation coefficient

for the X-ray line of interest. In this situation,

F(X ) -

f

e"

XZ

(j>(z) dz (24)

' o

The intensity of emitted X-rays is given by:

h-hrlfc-h*TM (25)

This absorption correction factor, f(x) has been the subject of much

study [8 ], [14], [23], [27]. The value of f( x ), which ranges from 0 to 1, is

probably the single most important parameter in quantitative X-ray microanalysis. Accurate f(x) values are especially crucial for soft X-ray

analysis, i.e., when x is large. In fact, the accuracy of X-ray microanalysis deteriorates rapidly if f(x) < 0.7 [23], [35]. One reason

that f(x) is not well known is that few determinations of this parameter

have been made experimentally and also that few $(z) curves have been

available. With the relation given in Eqn (2) , one can derive an expression for f(x):

72
f(x) = {x"

3

[0.25X

2

[44» -e

_1 * 5hx

(cf> +3k) J+xfh'^k-* ) (e"

1 * 51^) +

2h"

2

(<j) -k)(l-e"1,5hX

)] -

o

-1.5Xh

0.25(<J)o

+3k)e A (z r-1.5h)

g

[ 1.5hX - xz

r"2 ]>

(<t» o+3k)(z

r+1.5h) (26)

This rather cumbersome expression — compare with Eqn (1) — is easily

programmed for computer reduction. Its chief value should lie in the

region 0.5>_f(x). As an example, the value of f(X ) from Eqn (26) is

plotted against x for Cu-La @ E

q = 25keV in Fig. (13). The results

compare well with the Monte Carlo derived curve.

The Atomic Number Effect Correction in X-ray Microanalysis

The so-called atomic number correction arises because the specimen

is usually different in average atomic number than the standard. This

difference is reflected in differences in electron backscatter and retardation between sample and standard. This correction is often called

k [8]. In fact,

z

'

. F(0) STD k

* (27)

But since F(0) = 0. 125(<fr +3k) (z +1.5h) based on Eqn (2), then

k

z - w

Y*-Y*W (28)

1 2

W

where Y, = (1+2. 8n) (1+1. 5h ), h being h/z

i o o r

Y. = n(2.52+3.78h ) L O

73
Y*, Y* » Y

1

and Y

2

for the sample

h - 10"5 Z* + Z" 1/2

o

n* = ic± n.

Z* = C, Z^, being the true mass fraction of the i^ element

in the specimen. In practice, Y* and Y* must be obtained from the iteration loop of the correction program. Equation (28) has been tested for

Ag-Au and Au-Cu alloys (SRM 481, 482) comprising four compositions and

a range of incident beam energies. The results are good as shown by the

comparison in Tables V and VI.

Energy Dispersive X-ray Spectrometry

An energy dispersive X-ray spectrometer (EDS) was first used for

electron microanalysis in 1968 [36]. Since then the number and quality

of EDS systems has steadily and rapidly increased. Today a large

majority of scanning electron microscopes (SEM) and many transmission

electron microscopes (TEM) are equipped with EDS facilities. Thus,

the electron micros copist can perform qualitative and quantitative

analyses and can also map elemental distribution in the area scanned by

the electron beam.

EDS Systems

The heart of the EDS detector is a p-i-n (p-type, intrinsic,

n-type) lithium-drifted silicon crystal. X-rays generated from the

specimen are absorbed in the intrinsic region creating electron-hole

pairs. These pairs are separated and collected by bias applied to the

Si(Li) crystal to form a charge pulse. A charge sensitive f ield-ef f ect74
transistor preamplifier converts this pulse to a voltage pulse. After

further amplification and pulse shaping, this signal is sent to a

multichannel analyzer (MCA). The function of the MCA is to collect,

classify, and store each pulse. Pulse voltage remains proportional

to initial photon energy. Pulse amplitude is converted to time so that

an input signal can be linearly converted to some number say between

0 and 1024. This number is proportional to the incident X-ray photon

energy from the sample and also serves as an address of storage location in the MCA memory. Thus, if a 5 volt pulse corresponded to address

(channel) 1024, a 1 volt pulse arriving at the MCA would produce an

addition of one unit at channel 205. The energy distribution spectrum

is built up in this way over a period of time; the spectrum can be displayed or sent to a computer for further processing.

Detector resolution is typically quoted at 150 to 170 eV at a

5.9 keV reference energy [37]. The resolution figure usually meant

is the full width at half-maximum (FWHM) of the peak. There are two

contributions to the value, (1) the inherent statistical aspect of the

energy transfer from the photon in the Si (Li) crystal and (2) electronic

noise arising from the preamplifier. If the electronic noise were

reduced to zero, then the theoretical resolution (FWHM) at 5.9 keV would

be about 100 eV [37]. In practice, noise reduction is accomplished by

cooling the Si (Li) crystal as well as the first-stage preamplifier field

effect transistor to about 77°K. (This cooling also prevents the lithium

from diffusing out of the intrinsic region.) Noise reduction can also

be obtained by using pulsed optical feedback systems. These systems

activate a light-emitting diode when the preamplifier output reaches a

75
preset voltage level. The diode causes a leakage current to flow in

the field-effect-transistor which restores it to the initial operating

point [38]. This scheme eliminates noise usually associated with resistive feedback in conventional preamplifiers.

As stated, present-day EDS systems are supplied with resolution

figures quoted at about 150 eV; in 1968, the figure was 600 eV. Most of

the reduction was due to improved noise minimization procedures. However,

this means of resolution improvement is nearly exhausted [39]. Thus,

in the near future, dramatic resolution improvement is unlikely. Complete details concerning the principles and operation of energy dispersive

X-ray spectrometer systems can be found in the literature [37]-[4l].

Interpretation of Spectra

The MCA is usually calibrated to correspond to some convenient

energy increment, say 20 or 50 eV, per channel. For example, a 1024

channel MCA with a 20 eV per channel calibration will collect all

peaks in the energy range 0 to 20.5 keV. Moseley's Law relates the

characteristic energy of an X-ray peak and the atomic number, Z, of the

element responsible for the peak. In the context of an EDS spectrum,

Moseley's Law can be expressed as:

Z = exptK^ln^ + K^] (29)

where C is the peak channel number and K.. and K„ are numerical coefn 12

ficients depending on the MCA calibration. Thus, appropriate atomic

numbers can be assigned for the elements present in the electron irradiated region. Collection of the spectrum often takes only one or two

minutes; hence, qualitative analysis is in principle very rapid and

spectral interpretation is simple.

76
However, there are some major potential sources of difficulty

in using EDS techniques. These include (1) pulse pile-up effects;

(2) difficulties in accurate determination of the spectral background

caused by the X-ray continuum; (3) overlaps or interferences occurring

when spectral peaks are closer in energy than the resolution of the

EDS system; (4) problems associated with positive location and identification of weak peaks in the spectrum, i.e., detectability limits;

and (5) difficulties associated with analysis of elements with

4 <_ Z <_ 11, i.e. , the light elements

(a) Pulse Pile-up

The main amplifier of the EDS system can only process incoming

signal pulses at some finite rate. When the time interval between

pulses approaches this processing time, pulse pile-up results. Thus,

high input count rates to the EDS system can cause pulse pile-up

which appears as an energy distortion of one or both of the two closely

spaced pulses. The net effect is to cause such pulses to be added to

channels representing higher energy than the true pulse energy. Thus,

extraneous peaks may appear in the final spectrum causing confusion

and potential error in its interpretation. Particularly insidious is

the appearance of sum-peaks representing the effective coincidence in

time and summation of two strong monoenergetic pulses, e.g.,

Ka + Ka, Ka + KB, KB + KB, within the timing resolution of the detection and pulse processing system [37], [39].

Therefore, most modern EDS systems provide for pulse pile-up

rejection by inserting a circuit to inspect the rate at which pulses

arrive in the main amplifier. This circuit will switch the MCA input

77
off should a second pulse arrive before the main amplifier processes

its predecessor. Pulse shaping time in the amplifier interacts as well

shorter shaping times permit higher input count rates at the expense of

energy resolution. At present, input rates of 15 to 20,000 counts per

second can be achieved without output count rate loss or serious resolu

tion degradation in conventional systems [37], [39]. Special systems,

such as switched beam apparatus, appear capable of accepting even highe

input count rates [42].

All of the pulse processing methods cause the MCA to be inactive

for finite time periods after arrival of a pulse. Hence, most MCA's

have a so-called live-time option. If this option is selected, then

counting time is adjusted to compensate for these inactive periods.

Any comparative measurements ought to be performed with the live-time

option [39].

(b) Background Correction

The development given here is based on a method developed by

Fiori et al . [43] following Ware and Reed [44] and Lifshin et al . [39].

For background correction with an EDS system, an estimate of the

continuum background intensity below a peak having photon energy, E,

is required. Therefore, absorption of generated continuum photons

within the specimen must be taken into account. Furthermore, the

efficiency for photon absorption in the detector window and p-type

silicon dead layer in the Si (Li) crystal as well as for- transmission

of photons through the Si (Li) crystal must be considered. Thus, the

number of detected continuum photons N within the small energy range

E to E + AE may be written as:

78
N

E

- IAE f

E

P

£

(30)

where IAE is the probability of photon production (due to one electron)

in the energy range from E to E + AE. The form of the absorption

effect designated f_ has been expressed as [43]

l/f

E

= (1 + 1.2 x

10"6

yx)

2

(3D

y = E

1M - E

1 - 65

' o

E

Q

being the incident electron beam energy and

X = cscif; [M(target, E) ] with

i|> being the X-ray emergence angle and

M(target, E) being the X-ray mass attenuation coefficient of the

target for photons of energy E.

The value of M(target, E) is £C^M^ where i refers to the i*"*

1 element in the target. Hence, the background correction is a function of

the target composition, since is the true mass fraction of element i.

The detector efficiency, P , can be expressed as

P_ = exp-{M(Be,E)t„ + M(SiD

,E) t° }{l-exp-[M(Si,E) t_ . ] } (32)

£• fie bi oi

where M(Be,E), M(SiD

,E), M(Si,E), t

fi , t^ and t_. are the respective

X-ray mass attenuation coefficients and mass thicknesses of the Be window,

Si dead zone and Si active zone. There is also a gold conductive layer

about 20 nm thick flashed onto the Si (Li) crystal which does not contribute significantly to P_ [43], [44]

.

Lifshin et al . found that IAE in Eqn. (30) could be expressed as

[39]:

79
IAE = Zla(E -E)/E + b (E -E)

2

/E]

o o

(33)

where Z is the average atomic number of the target equivalent to EC^Z^

and "a" and "b" are empirical fitting factors.

In practice, N is measured at two widely separated points in the

spectrum, both free of all peak interferences and "a" and "b" determined

by solving the resulting two equations in two unknowns. Then the entire

curve of N as a function of energy can be calculated. Figure (14)

shows an as-collected spectrum of the mineral Kakanui Hornblende while

Fig. (15) shows the background corrected spectrum. This method of

background correction has been tested by comparison with measured spectra

and by application in quantitative X-ray analyses [43].

(c) Peak Interference

Many X-ray peaks lie within 100 eV of one another, e.g., Pb-H and

S-K [45] • Hence, peak overlap or interference may occur in the EDS

spectrum. The usual means of attacking this problem is by means of

spectrum stripping. However, to carry out spectrum stripping properly

usually requires computer processing of the spectrum.

Gaussian parameters of each peak detected in the spectrum may be

written as follows:

I =1 exp - [(E-E ) s P

* p

2 2

] (34)

where I is the stripped intensity at energy E

I is the peak intensity

E is the peak energy and

is the detector resolution (FWHM) at energy E . P

80
For Eqn (34) to be valid, the background due to the continuum must be

accurately subtracted from the raw spectrum.

In practice, spectrum stripping is carried out by stripping major

peaks from the spectrum in regions where overlap is suspected or known

to be present. Consider the spectrum in Fig. (15) in the region around

the CaKa peak. The K-Ka peak lies on the low energy shoulder of the

CaKa peak; the K-K3 peak is totally obscured. For quantitative analysis

of calcium and for proof positive of the existence of the K-KB peak in

the spectrum, one would have to first estimate the Ka/KB ratio in

potassium in the spectrum and then strip the KB potassium peak from the

calcium Ka peak. Results are shown in Fig. (16).

At this point, the need for computer interfacing with the EDS

system should be clear. Background correction and spectrum stripping

both require such a facility. The most advantageous arrangement would

be for real-time interaction of the operator, EDS system and computer.

For example , background correction could be carried out and the result

displayed on the MCA. Spectrum stripping could then be performed

interactively by the operator in control of the computer. The stripped

spectrum would be displayed for immediate analysis [45]. Furthermore,

computer data reduction is required for quantitative analysis; small

on-line programs exist for this purpose [47].

(d) Detection of Weak Peaks

Lifshin and co-workers [39], [48] have pointed out that the limiting

count-rate in an EDS system, which was suggested to be 15-20,000 counts

per second, is the total system count-rate for all energies. This fact

can lead to serious difficulties if one is attempting to optimize

81
sensitivity for a weak peak since the accompanying stronger peaks from

other elements contained in the specimen will account for most of the

observed total count-rate [483 • In addition, even with pure elements,

peak-to-background ratios with the EDS system are usually less than

50 to 1 and often less than 30 to 1 [49]. Thus, unambiguous detection

of weak peaks in the spectrum may be difficult.

Such considerations underscore the need for accurate background

corrections and adequate spectrum stripping methods via the interactive

computer mode. Nevertheless, after these procedures have been carried

out, there is perhaps no better instrument than the human eye-brain

combination for deciding whether some feature in the spectrum is worth

further investigation. Then a long collection time can be used to

ascertain whether a real peak is present. Clearly, this procedure is

subject to compromises dictated by available time, potential specimen

damage by the primary beam and instrument stability.

The detectability limit of EDS facilities has been compared to

that attainable with wavelength dispersive spectrometers [39], [49].

For example, elemental concentrations of silicon or molybdenum on the

order of 0.1% were easily detected in steel standards [49] by EDS.

Usually, EDS and wavelength dispersive methods give comparable detection

limits [48].

(e) Light Element Analysis

Because of X-ray absorption in the beryllium window of the detector

(typically 8-12 ym Be is used) and in the silicon dead-layer, elements

with atomic number less than eleven are usually not analyzed by means

of EDS methods. For example, the detector efficiency, P„, at the energy

E

82
of oxygen Ka (0.52 keV) is about 0.2% if an 8 m Be window intervenes

[37] . But if the window is removed and Eqn (32) solved for P

£

with

t^. of 0.2 um and a mass attenuation coefficient of Si for oxygen Ka

of 8790 [50], then P is 66%. Therefore, windowless detectors offer

£

a means for analysis of the light elements. However, the characteristic

energies of adjacent elements in the region 11 > Z are 150 eV or less.

Hence, adjacent peaks will probably not be clearly resolved with EDS

facilities

.

Windowless detector systems have been developed [51], [52l and are

available. With such systems, elements down to carbon have been observed

and a detectability limit of about 0.4% for oxygen in aluminum found [5ll«

Oxide thicknesses of SiO^ on silicon substrates have been successfully

measured [52]- Flow proportional counters having an ultrathin window

can be used for the detection of light elements as well.

Quantitative Analysis

All schemes for carrying out quantitative analysis use a standard

of known composition. In many cases, especially metals, pure elements

are suitable. In the case of mineral or petrological samples, homogenous,

compound standards close in average atomic number to the unknown are

usually chosen. What the analyst measures is the relative X-ray intensity ratio from the elements of interest in the specimen to the same

elements in the standard. Both specimen and standard are examined under

identical experimental conditions. The measured relative intensity ratio

(commonly called k) must be accurately determined or any quantitative

analysis scheme will result in the same inaccuracy. Peak intensities

83
can usually be measured more easily and with better accuracy than integrated

Intensities. Using peak intensity values gives results as accurate as

integrated intensities [53]; hence, in all that follows, peak intensity

is meant.

A useful exercise is to insert a pure element and treat one side as

"specimen" and the other as "standard." Then have several operators

measure "relative intensity" values (one side to the other) using different operating conditions. The results will indicate the maximum

accuracy with which a relative intensity ratio can be obtained; values

of 1.00 for the ratio in this test will not be obtained except by good

fortune.

Once the k values have been obtained, they must be corrected for

several effects including (1) effects resulting from the nature of the

differences in electron scattering and retardation between the specimen

and the standard, i.e., the so-called atomic number effect, k^, (2)

absorption of X-rays within the specimen, k , (3) fluorescence effects,

kj,, and (4) continuum fluorescence, k^. A common form of the correction

equation is

C = k • 1^ • k

A

• k

F

• k

c (35)

where C is the weight fraction of- the element of interest. This method

is often referred to as the ZAF method. Each of the effects listed has

been studied in detail [54] and reviews have been written [8 ] , [55] , [56]

Empirical methods for data reduction have also been developed [57], [58].

A procedure using a mini-computer for on-line correction of X-ray

data from electron microprobe analysis has been developed [47]. This

84

!

[
program, called FRAME, allows mass fractions to be computed on-line by

the ZAF approach. Besides X-ray intensities, the only input data required

are (1) the atomic numbers of the elements present, (2) the analytical

line being used, and (3) the operating voltage. Other required parameters

such as atomic weights and X-ray mass attenuation coefficients are stored

or calculated by the program. FRAME need not be used on-line. If desired,

the program can be run as a batch or time-shared program for all analyses.

The program is flexible, short, simple to use, and accurate. Furthermore,

the absorption term f in the expression for background subtraction in

E

EDS spectra (Eqns 2, 3) depends upon the specimen concentration. Such

a program is needed then in order to obtain f properly and in turn to

Ci

carry out the background correction. Clearly, incorrect background subtraction will lead to errors in quantitation.

The correction procedure FRAME B has been developed for electron

probe microanalysis with a lithium-drifted silicon detector. This program,

which computes on-line mass fractions of the elements contained in the

specimen, is a modification of the correction procedure FRAME [33]. The

input data required are: X-ray intensities from the peaks of interest

and the continuum intensities at two energies, the atomic numbers of

all the elements present, the analytical lines used and the operating

voltage. Other required parameters, such as atomic weights, the X-ray

emergence angle and X-ray mass attenuation coefficients, are stored or

calculated by the program. FRAME B is in FORTRAN IV; it requires less

than 5,000 words of computer core, and will handle up to 13 elements

The continuous spectrum for a specific set of conditions is obtained

as follows. From the value of N(E) measured at two energy levels where

85
there are no characteristic lines or detector artifacts (incomplete

charge collection, pulse pile-up, double energy peaks and escape peaks)

,

the parameters a and b in Eqn (33) are calculated. They in turn define,

through Eqn (30), the value of N(E) at an energy E.

As pointed out by Ware and Reed [44] , the absorption term f

£

is

dependent upon the composition of the target. Therefore, when the composition of the target is unknown, it is necessary to include Eqn (31

)

in an iteration loop as shown in Fig. (17). This flow diagram is a

simplified description of the program.

Elemental Mapping and SiC in Al Composite

The technique of X-ray area scanning provides the investigator

with what amounts to a scanning X-ray microscope. The amplified signal

from the detector system — energy-dispersive or crystal spectrometer —

is made to modulate the brightness of a cathode ray tube (CRT) scanned

in synchronism with the electron probe. Thus, on the CRT, a picture is

obtained using the spatial variauxon of X-ray emission from the surface. The same

magnification controls, sweep system and amplifier as in the scanning

electron microscope, are used. An X-ray area scan can show tones ranging

from black to white, depending on experimental conditions. In places of

high concentration of the element in the scanned area, the picture will

be nearly photographically white; it will be gray where the element's

concentration is lower and black where the element is absent.

When an energy-dispersive system is used to prepare the area scan

photograph, care must be taken to ensure that no other peak interferes

with the signal of the desired element. The desired peak should be

86
carefully Isolated by means of a single-channel analyzer; this analyzer

is often built into multichannel analyzers. The amplified output of the

single-channel analyzer is displayed on a CRT and so provides the elemental

distribution desired. Magnification can be varied in the usual way.

Separate micrographs of all elements of interest in the scanned area can

be built up to give the complete elemental distribution. A detailed

discussion of X-ray area scanning has been given by Heinrich [59].

In the case of an energy-dispersive detector, the time required to

build up a satisfactory X-ray area scan may be very long. This hindrance

results from the requirement imposed by Poisson statistics to provide

at least 20,000 photons per micrograph; frequently 200,000 are needed

and occasionally as many as 500,000 photons must be used. For the SEM,

often run with beam currents on the order of 10 or 10 A, the total

4

number of photons produced is fairly low (^10 per second) . Since one

peak obtained from the total spectrum is of interest, the time to obtain

a satisfactory micrograph is correspondingly long. Times in excess of

15 min are not uncommon in the preparation of X-ray area scans.

If the specimen will not be harmed, higher beam currents are

useful — not only for elemental mapping but for all phases of X-ray

analysis. The loss of resolution resulting from increasing beam current matters little since the X-ray emission volume is largely determined by electron diffusion and not by probe size except for thin

specimens. Since it is difficult to decrease the effective volume of

X-ray emission below 1 um, X-ray scanning pictures are limited to

magnifications of about 3000 diameters.

87
Some further studies using elemental mapping were made on the SiC

in aluminum composite provided by Marshall Space Flight Center. The

manufacturing method has led to the formation of several phases: 1) SiC

unreacted, 2) Al-Si, 3) Al-Si-C, 4) Al-C and silicon is present in the

grain boundaries. The tungsten phases produced by aluminum reaction

with the tungsten core of the SiC filament are Al^W and Al^W. Less

than 25% of the diameter of a SiC filament remains as SiC; the remainder

has reacted. The results are shown in Fig. (18).

Discussion and Conclusions

A simple analytical method for the analysis of thin film laminate

composite specimens has been devised. This method can be used to calculate

film thickness and/or film composition. If a substrate is present, its

composition is presumed known. For compositional determination, the

method suffers the drawback that all elements present must have their

intensities measured. Results from this method agree with results reported in the literature to within about ±10% relative.

For pure element films, the thin-film technique predicts the intensity

ratio of film-to-bulk for thicknesses up to the X-ray range, i.e.,

effectively bulk material. In the analysis of multielement films, the

thickness range over which the technique can be applied is more limited.

This is a result of the assumption that a thin, multi component film can

be approximated as a layered structure of the pure element but having

the same total mass thickness. As long as there is negligible elastic

scattering within the film, this approximation appears to be reasonable.

As the thickness is increased, elastic scattering, which is composition

88
dependent, will cause this simple approximation to fail. The available

data for binary films permitted us to test the thin film technique up

2

to a mass thickness of 100 pg/cm in Co-Pt and Mn-Bi samples in which the

intensity from the thin film is about 20% of the bulk value. Further

testing with data from thicker films is necessary to determine the range

of thickness where the technique is applicable. The method can be used

with tilted specimens in principle. However, no experimental data comparisons have as yet been carried out for these cases. The computational

procedure is not overly cumbersome; results can be obtained with a desk

calculator in a few minutes. In addition, the distribution in-depth of

electron beam produced X-rays in a solid can be deduced with this method;

agreement with published data is good.

The method can be extended to composites consisting of particulate

reenforcers in a metal matrix provided the geometry of the particles is

specified or known. The extension has been carried out for spherical

and cylindrical particles.

89
References

1. T. A. Hall in Quantitative Electron Probe Analysis (K. F. J. Heinrich,

ed.) NBS Spec. Publ. 298 , 269 (1968).

2. T. A. Hall in Physical Techniques in Biological Research (G. Oster,

ed.) Vol. IA, p. 157 (1971).

3. T. A. Hall and P. D. Peters in Microprobe Analysis as Applied to

Cells and Tissues (T. Hall, P. Echlin, and R. Kaufmann, eds.)

Academic (New York and London) p. 229 (1974).

4. R. Tixier and J. Philibert, J. de Microscopie 1, 588 (1968).

5. M. H. Jacobs and J. Baborovska, Proc. 5th Eur. Conf. Elect. Micros.

(Inst. Phys. London) p. 136 (1972).

6. G. Cliff and G. W. Lorimer, Proc. 5th Eur. Conf. Elect. Micros.

(Inst. Phys. London) p. 140 (1972).

7. J. I. Goldstein and J. W. Colby in Practical Scanning Electron

Microscopy (J. I. Goldstein and H. Yakowitz, eds.) Plenum (New York),

p. 470 (1975).

8. H. Yakowitz in Practical Scanning Electron Microscopy (J. I. Goldstein

and H. Yakowitz, eds.) Plenum (New York), p. 327 (1975)-.

9. J. W. Colby, Adv. X-ray Anal. 11, 287 (1968).

10. Y. Oda and K. Nakajima, J. Jap. Inst. Met. 3_7, 673 (1973).

11. R. Castaing, Thesis to Univ. of Paris (1951).

12. R. Castaing and J. Henoc, Proc. 4th Int'l Congr. X-ray Optics and

Microanalysis (R. Castaing, P. Deschamps and J. Philibert, eds.)

Hermann (Paris) p. 120 (1966).

13. P. Duncumb in Electron Microscopy and Analysis , Conf. Ser. #10

(Inst. Phys. Lond) , p. 132 (1971).

14. R. Castaing and J. Descamps, J. Phys. Radium 1_6, 304 (1955).

15. J. D. Brown and L. Parobek, Proc. 6th Int'l Congr. X-ray Optics

and Microanalysis (G. Shinoda, K. Kohra and T. Ichinokawa, eds.)

Univ. Tokyo Press, p. 163 (1972).

16. J. W. Criss and L. S. Birks in The Electron Microprobe , (T. D. McKinley,

K. F. J. Heinrich, and D. B. Wit try, eds.) Wiley (New York) p. 217

(1966).

17. W. Reuter, Proc. 6th Int'l. Congr. X-ray Optics and Microanalysis

(G. Shinoda, K. Kohra, and T. Ichinokawa, eds.) Univ. Tokyo Press,

p. 121 (1972).

90
18. G. Shinoda, K. Murata, and R. Shimizu in Quantitative Electron

Probe Microanalysis (K. F. J. Heinrich, ed.) NBS Spec. Publ. 298 , 155 (1968).

19. Proceedings of Workshop on Monte Carlo Calculations in Electron

Probe Microanalysis (K. F. J. Heinrich, H. Yakowitz, and D. E. Newbury,

eds.) NBS Spec. Publ. (1976) - in press.

20. H. E. Bishop and D. M. Poole, J. Phys . D: Appl. Phys. 6^, 1142

(1973).

21. R. B. Bolon and E. Lifshin, SEM/73 (0. Johari, ed.) IITRI (Chicago),

p. 285 (1973).

22. D. F. Kyser and K. Murata, IBM J. Res. Devel. 18 (4), 352 (1974).

23. K. F. J. Heinrich and H. Yakowitz, Analytical Chemistry — in press.

24. K. F. J. Heinrich, Proc. 4th Int'l. Congr. X-ray Optics and

Microanalysis (R. Castaing, P. Deschamps and J. Philibert, eds.)

Herman (Paris), p. 159 (1966).

25. H. E. Bishop, Proc. 4th Int'l Congr. X-ray Optics and Microanalysis

(R. Castaing, P. Deschamps and J. Philibert, eds.) Hermann (Paris)

p. 153 (1966).

26. D. J. Fathers, J. P. Jakubovics, D. C. Joy, D. E. Newbury and

H. Yakowitz, phys. stat. sol. (a) 22, 609 (1974).

27. C. A. Andersen and M. F. Hasler, Proc. 4th Int'l. Congr. X-ray

Optics and Microanalysis (R. Castaing, P. Deschamps, and J. Philibert,

eds.) Hermann (Paris), p. 310 (1966).

28. S. J. B. Reed, Proc. 4th Int'l. Congr. X-ray Optics and Microanalysis

(R. Castaing, P. Deschamps, and J. Philibert, eds.) Hermann (Paris),

p. 339 (1966).

29. P. Duncumb, Proc. 2nd Int'l. Congr. X-ray Optics and Microanalysis

(A. Engstrom, V. E. Cosslett, and H. Pattee, eds.) Elsevier

(Amsterdam), p. 365 (1960).

30. K. F. J. Heinrich, private communication (1975).

31. M. Green, Thesis to Cambridge University (1962).

32. K. Kanaya and S. Okayama, J. Phys. D: Appl. Phys. 5_, 43 (1972).

33. C. Fiori, R. L. Myklebust, K. F. J. Heinrich and H. Yakowitz, Proc.

Microbeam Analysis Soc. (10th Nat'l. Conf.) Las Vegas, Nev. (1975)

Paper #20.

34. J. C. Russ, Proc. Microbeam Analysis Soc. (9th Nat'l. Conf.) Ottawa,

Can. (1974) Paper #22.

91
54. Quantitative Electron Probe Microanalysis , (K. F. J. Heinrich,

ed.) NBS Spec. Publ. 298 (1968).

55. S. J. B. Reed in Microprobe Analysis (C. A. Andersen, ed.) Wiley,

New York (1973) p. 53.

56. S. J. B. Reed, Electron Microprobe Analysis , University Fress,

Cambridge (1975).

57. T. 0. Ziebold and R. E. Ogilvie, Anal. Chem. 36, 322 (1964).

58. A. E. Bence and A. Albee, J. Geol. 76.» 382 (1968).

59. K. F. J. Heinrich, NBS Tech. Note 278 (1967).

60. M. Bayard in Microprobe Analysis (C. A. Andersen, ed.) Wiley,

New York (1973) p. 323.

92
Table I: Comparison of Thin Film Data

of Reuter [17] with NBS Results

Film Subs trate E

o

z 2

( u e / cm )

k k

NBS

z„„ _ NBS

z REUTER

Al Be 30 73 0.046 0.038 84 75

Al Fe 30 73 0.073 0.065 86 83

Al Au 30 73 0.093 0.092 86 85

Al B 12 14.5 0.0324 0.0339 13.9 14.2

Al B 12 7.3 0.0166 0.0163 7.3 7.6

Al B 12 4.4 0.0098 0.0096 4.4 4.6

Al B 12 3.4 0.0075 0 0074 3.4 3.6

Al B 4 14.5 0.407 0.351 16.5 14.6

Al B 4 7.3 0.182 0.164 8.0 7.0

Al B 4 4.4 0.109 0.095 5.0 4.3

Al B 4 3.4 0.084 0.072 3.9 3.3

Cu-Ka Z = 10 30 83 0.063 0.062 84 72

Cu-Ka Au 30 83 0.104 0.092 94 85

Cu-La Z = 10 10 83 0.433 0.452 80 71

Cu-La Au 10 83 0.590 0.662 73 82

Ni-La Z = 10 6 48 0.612 0.596 49.5 45

Ni-La Z = 10 12 48 0.232 0.214 51 43

Ni-La Z = 10 20 48 0.131 0.122 51 40

Ni-La Z = 10 30 48 0.117 0.101 54 47

Ni-Ka Z = 10 15 26 0.094 0.087 28 23

Ni-Ka Z = 10 18 26 0.053 0.053 26 23

Ni-Ka Z = 10 22 26 0.035 0.033 27 24

Ni-Ka Z - 10 30 26 0.017 0.017 26 23

Au-La Pt 30 77 0.122 0.088 105 94

Au-La Pt 25 77 0.178 0.136 98 91

Au-La Pt 30 193 0.276 0.243 215 198

Au-La Pt 25 193 0.366 0.368 192 183

93
Table II

Bishop and Poole: Thickness Measurement of Copper

on Nickel Substrate [20] Compared with NBS Calculations

(Nominal Cu thickness was 45 yg/cm^; 9=0°, ^=75°)

E (keV)

o Cu

Z

B-P

Z

NBS

10.75 0.476 29 26

12.88 0.263 31 35

14.76 0.176 35 37

17.86 0.105 37 38

20.9 0.075 36 40

26.0 0.045 35 39

31.1 0.033 37 41

94
Table III

Bishop and Poole: Thickness Measurement of MgF^ on Si

Substrate [20] Compared with NBS Calculations

2 (Nominal MgF thickness was 30 yg/cm ; Mg-Ka monitored, 9=0°, $=75°)

ti ^Kev;

O

Z

B-P

Z

NBS

6.6 0.478 36 44

8.7 0.263 35 40

10.75 0.172 34 39

13.34 0.109 31 36

15.83 0.087 31 35

20.9 0.061 27 34

26.0 0.052 25 34

31.1 0.044 20 32

95
Table IV

Kyser-Murata: Analysis of Binary Films [22]

Co-Pt on Si02

Mn-Bi on SiC>2

Compared with NBS Calculations

z-values in yg/cm^

9 = 0°

i|) = 52.5°

^t rK-M rNBS

^Co ZK-M

Z

NBS

L

Co

L

Co

0.0177 0.0581 53 51 0.185 0.199

0.0305 0.114 89 99 0.163 0.197

0.0148 0.0675 56 60 0.140 0.154

0.0067 0.0465 40 41 0.101 0.103

hi Z K-M

Z

NBS

,NBS

'Mn

0.0226

0.0293

0.0329

0.0610

0.0766

0.0515

0.0220

0.00668

54

53

38

29

42

60

59

41

30

41

0.255

0.510

0.800

1.0

0 0

0.263

0.518

0.796

1.0

96
Table V

Analysis of Au-Ag Alloys

E

o

(keV) AHALYTE CCERT _ W (Eqn

±U

48.5

Ag-La

Ag-La

u. lyyo

0.1996

0. 191

0.0795

A TOT 0. lo

7

0.0826

i n48.5

Au—Ma

Au-Ma

U

.

OUUj

0.8005

n 7"7i

0.755

r\ 7 q /. U. /o4

0.753

48.5

Ag-La

Ag-La 0.3992 0.181

n n/i U . J m

0.184

i n48.5

Au—Ma

Au-Ma

U . DUUD

0.6005 0.537

u . joy

0.532

48.5

Ag-La

Ag-La

n ^qq

0.5993

U . DOH

0.339

n *;7 s 0.341

10

48.5

Au-Ma

Au-Ma

0.4003

0.4003

0.362

0.337

0.369

0.334

10

48.5

Ag-La

Ag-La

0.7758

0.7758

0.764

0.542

0.758

0.550

10

48.5

Au-Ma

Au-Ma

0.2243

0.2243

0.201

0.181

0.202

0.177

97
Table VI

Analysis of Au-Cu Alloys

E

o (k£V) ANALYTE ^ALC. 28)

10 Cu-La 0.1983 0.166 0.153

40 Cu-La 0.1983 0.067 0.074

10 Au-Ma 0.8015 0.749 0.756

40 Au-Ma 0.8015 0.675 0.676

10 Cu-La 0.3964 0.354 0.327

40 Cu-La 0.3964 0.166 0.179

10 Au-Ma 0.6036 0.531 0.538

40 Au-Ma 0.6036 0.428 0.436

10 Cu-La 0.5992 0.557 0.529

40 Cu-La 0.5992 0.313 0.336

10 Au-Ma 0.4010 0.332 0.336

40 Au-Ma 0.4010 0.253 0.246

10 Cu-La 0.7985 0.764 0.752

40 Cu-La 0.7985 0.550 0.576

10 Au-Ma 0.2012 0.154 0.157

40 Au-Ma 0.2012 0.110 0.104

98
Incident Beam

Film

Substrate

Fig. (1) Schematic representation of possible electron trajectories

in a thin film-substrate target.

,0

9

8

7

6

3

-

Al, 20keV

Castaing-Henoc Experimental

Monte Carlo Calculation

0.2 0.4 0.6 0.8 1.0

Mass Thickness pz (mg/cm 2

)

1.2

Fig. (2) Function <|>(z) plotted against z for aluminum target bombarded

normally by 20 keV electrons. After Castaing and Henoc [12].

\

N

CO> >O

CMO

Id

ri «> 8 * ° z *

to

£

C

c _i 6

10

0>

a> rii

OO

(7)

O Ct> 2 * r/ ob

CMEE

ii / m

CM6

00 •

ri en u <—• *i

a)OC

<D

Q

T3U

U

cd a) S <4-l o <J

J3

(D CO oo a

Q)

ouuo

a.o >o a) o o

<4-l <N N >%

Xi

CO CO > QJ rH

oo

-ea) C o o c

•rl 0) U "O O -H

(3

O 3 C

oO

•i CD

oo

•H

JL. O

CM

M

Fig. (4) Function $(z) vs. z for Au-Ma radiation from solid gold and

from 300 A thick gold on AI2O3 substrate bombarded normally

by 20 keV electrons. The parameters <f> Q, h, k, and (see

text) are indicated.

liJ

us>

o>co

QC

>»o

q:i x >

IO

UJ

MoII _ C E co 0 T3 CO

C - o c a> 3 a> u < a: o x G O 0 S> ©

IIc >

cr UJ

cfl

O

•h com 4-1 4J i—i a r-i i <U 3 -—

4J CO v© O CD CN

(X pti GO • •H > H 4J 11 H « AS —

4J •h

m co

O

H

W

X <u o c U (3

cu a)

"4-1 o

cso

•H

4Jo

CD *H

00

CO

a)

n

cou

4-1

0) CD

CO

CO

4-1 0) o

a, "tij •H M 4J N

J-l cd 4J

uo

CD

(3o

•H

CO

CO

cuM

scu0oM<4-l

CO 00 T3 h fi O

rj

c

&.

S tH 3 cfl CD CO 4-1 H CO O

do •H ( zwo/6uu) a6uoy

0.25

005

0025

00025

0.01 -

0 005 -

0001

250 500 1000 2500

Film Thickness (A)

5000 10,000

Fig. (6) Comparison of analytic model with experimental results taken

from Kyser and Murata [22].

(6a) Results for Si films on Al^ substrate.

1.0

0.9

0.8

0.7

0.6

kcu 0 5 ~

0.4 -

0.3 -

0.2 -

1 1 1 r Cu-Ka on Al 2 03

-r Thin

x Cu

rBulk A

Cu * = 52.5°

0=0°

Solid Lines : <f> (z) Calc.

<8> Kyser - Murata Expt'l Data

!5keV.

i r

500 1000 1500 2000 2500

Thickness (X)

3000 3500

(6b) Results for Cu films on A^O^ substrate.

I

0.01

Au-Mct on Al 2 03

T Thin

Ao / yBglh

*«52.5°; B'Oe

Solid Lines : 4> (z) Cole.

® : Kyser-Muroto Expt'l Data

100

X

250 500

Film

1000

Thickness

2500 5000 I0.O

(6c) Results for Au films on A^O^ substrate.

0.34

Au-Ma

10 20 30 40 50 60 70 80

Z- Substrate

Fig. (7) The intensity ratio of Au-Ma X-rays from a 500 A thick gold

film to Au-Ma from a solid block of gold as a function of

substrate atomic number. Experimental data from Bolon and

Lifshin [21]. Data calculated using two absorption correction

expressions, Eqn (1) and the Philibert-Duncumb-Heinrich

relation (see Ref [8]).

k

z (/ig/cm2

)

Fig. (8) Calculated intensity ratios as a function of thickness for

films of fo, Pt, Mn, and Bi against the pure solid element.

Specimens bombarded normally by 20 keV electrons, \\> = 52.5°.

This figure can be used to calculate the results given in

Table IV.

>»O

CO

<uIo > a >

rH

CO

<4-l O •

COCD i M 42

CU CO 43

(X CO

CO

43

4-> •H

u > CUa

C3 CU O

•rl

u

4J

c

o CU

ctj •H

(1) CJ U C3 c *

eso CU

S-i 4-1 +Jo

CU i

•H

w T3

(30

J*4

Beam

Fig. (10) Schematic representation of "missing" X-ray emission due to

particle geometry.

10 I 1 1 r" 1 1 1 1 1 1 r

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5

Spherical Particle Diameter (/im)

Fig. (11) Results of calculation for Fe-Si spheres.

100

80

^60

c

0>

£40

20

®

Aluminum Spheres: E0

= 17 keV

M. Bayard Data

Zr ® Polygon Approximation

o 3 4 5

Diameter,

Fig. (12) Results of calculation for Al spheres as compared with experimental data of Bayard [60].

0 1000 2000 3000 4000 5000 6000 7000

X (cm2/g)

(13) Parameter lit (x) vs. x for Cu-La; Eo - 25 keV.

\



180 1

1 1 1 1 1 1 1 1

1 r

Energy (keV)

Fig. (16) Spectrum analysis of selected energy region from Fig. (15).

CALCULATE BACKGROUND

ON STANDARDS

i

ESTIMATE BACKGROUND

ON SPECIMEN

i

CALCULATE RELATIVE

INTENSITIES, k|j

CALCULATE ANALYTICAL

FUNCTION, Fj

ki-Mj * F

j

C

ij

i

HYPERBOLIC

APPROXIMATION

Ci+l,j"f(Cjj,kij f k| + , tj)

CALCULATE BACKGROUND

ON SPECIMEN

Fig. (17) Flow diagram of program FRAME B.

Fig. (18) Phase locations in SiC reenforced-Al matrix composite.

Task 4

Melt Shape in Weightless Crystal Growth

S. R. Coriell, S. C. Hardy

Metallurgy Division

Institute for Materials Research

and

M. R. Cordes

Applied Mathematics Division

Institute for Basic Standards

Summary

Experimental and theoretical investigations of the floating zone

process of crystal growth and purification have been carried out with

particular emphasis on microgravity conditions. The shape and stability

of a vertical rotating liquid zone subject to a vertical gravitational

field have been calculated numerically. For a cylindrical volume and a

weak gravitational field, analytic results have been obtained for the

shape and stability of a zone for arbitrary direction of the gravitational

field. Comparison of the numerical results with experiments on vertical

water zones between two metal rods indicate that the liquid zone is

slightly less stable than predicted theoretically.

99
Introduction

The floating zone process is widely used to purify materials and to

grow single crystals"'". For example, in recent years several hundred tons

2

per year of silicon are prepared by this process . One of its advantages

is that the melt is not in contact with a crucible and contamination problems are minimized. In the earth's gravitational field there is apparently

a maximum zone length (approximately 1.5 cm for silicon) above which the

1 3

zone is unstable ' (i.e., the liquid zone fails). In microgravity zones

of greater length can be stable. The shape of the liquid zone, in particular, the angle the liquid-vapor interface makes with the solid-vapor

interface at the freezing interface, determines the shape of the resulting

3— 6 solid . Thus, the shape of the zone is important for growing crystals pf

constant diameter.

We have carried out theoretical investigations of the shape and

stability of liquid zones, including both numerical and analytical calculations. In addition, we have experimentally measured liquid zone shape

and stability to compare with the predictions of theory.

Theory

As illustrated in Fig. 1, we consider a liquid zone of length L between

two solids which are circular cylinders and are coaxial. The radii of the

upper and lower solids are and R, respectively. Defining the usual

cylindrical coordinate system (r, 0, z) with origin midway between the two

solids and coaxial with them, we study the effects of a constant gravitational acceleration g along the negative z-axis and of rotation with constant

angular velocity ft about the z-axis on the shape and stability of the liquid

zone. For example, in the experiments to be described later, water zones

100
between steel end plates were studied. In the studies of floating liquid

7 8

zones in simulated zero gravity by Carruthers and Grasso ' , the solids

were glass and the liquid was a water-ethanol solution which was surrounded

by nujol (a high molecular weight mineral oil). In a typical floating zone

process the solids might be silicon and the liquid molten silicon. However,

the present theory assumes that the liquid is either static or uniformly

rotating. Thus, in applying the results of the calculations to the floating zone process, the effect of fluid flow in the melt is neglected. The

error due to this approximation is undetermined at present and requires

additional theoretical and experimental work.

We assume that the liquid-vapor interface is axisymmetric and denote

the shape by the function r(z). This assumption is consistent with the

direction of the gravitational field and the axis of rotation. (In the

Appendix we obtain some approximate analytic results for the case where the

gravitational field is at an arbitrary angle to the liquid zone, and the

liquid shape is not axisymmetric.) Since the liquid is static (or uniformly

rotating), the energy E of the liquid depends only on the shape r(z) of the

liquid-vapor interface. The energy is composed of gravitational, rotational,

and surface contributions. In addition, there is the constraint that the

liquid volume remain constant (we assume constant liquid density). Appli9 cation of the calculus of variations to this problem yields differential

equations which govern the shape and stability of the liquid. Since this

procedure has been previously described"^ and a similar calculation is given

in an Appendix, we will simply state the basic equations. The requirement

that the first variation SE vanish gives the Euler equation for r(z), i.e.,

y{r~ 1 (l+r 2 )~h - r (1+r 2 )" 3/2

} = -pgz + (pfi 2 r

2 /2) + p , (1)

101
where y is liquid-vapor surface energy, the subscript z denotes differentiation, i.e., = (dr/dz), p is density difference between liquid and vapor

(although we will use the word vapor for the fluid surrounding the liquid

zone, it should be recognized that the calculation is valid for the vapor

replaced by any fluid which is immiscible with the liquid zone), and p is

a Lagrange multiplier (independent of z) which results from the volume constraint, i.e. , L/2

V = J7rr 2 dz = CONSTANT (2)

-L/2

The left-hand side of Eq. (1) is the product of surface tension and the

mean curvature of the interface. Thus, the equation is the familar Laplace

equation which gives the pressure difference across a curved interface. The

equation can be written in dimensionless form by introducing the variables

H = z/a, \p = r/a, e = pga 2 /y, u = pn 2 a

3 /2y, and P = ap/y, where a is an

arbitrary length (to correspond with our previous calculations we take

a = R/[l+e ] 2 where e = pfi 2 R

3 /y). The differential equation for i>(n) is

R R

then

ijr 1 (l-HJ> 2 )~^ - $ (l-hp 2)"3/2 = _ en + ^2 + P ( 3 ) n nn n

where $ = (dip/dri). In this form it is apparent that the shape of the

liquid zone is determined by the two dimensionless parameters e and to.

If the shape r(z) of the zone satisfies the Euler or Laplace equation,

then the energy is stationary with respect to small variations in shape.

In order to determine whether the energy is a minimum and the liquid zone

stable with respect to small variations in the shape one must consider the

second variation. If the second variation 6 2 E is positive then the energy

102
9 is a minimum. From the calculus of variations in order for 6 Z E to be

positive, the Legendre condition is necessary and the Jacobi differential

equation must possess certain properties. For our problem the Legendre

condition is always satisfied"^. The Jacobi differential equation is

[d/dz] [ Y rW /(1+r )

'

] + [pfi 2 r

2+Y/r (1+r 2

)

2 ]W = -ur/2

7 7 7 (4)

where W = 6r and u is a Lagrange multiplier used to satisfy the volume

constraint on W, viz.,

L/2

r W dz = 0 (5) /

-L/2

Consider a non-trivial W which satisfies Eqs . (4) and (5) and the boundary

condition W(-L/2) = 0. Since r is positive, it follows from Eq. (5) that

W must vanish once in the interval (-L/2) < z < (L/2). The Jacobi condition which ensures stability (6 2 E>0) requires that W should not vanish again

in the interval. Since W satisfies a linear differential equation, the

stability criterion may be written as

z* > (L/2) (6)

where z* (called the conjugate point) satisfies the equation

z* z*

u(z*) f v r dz - v(z*) / u r dz = 0 (7)

J -L/2 J -L/2

and is the first root of this equation greater than (-L/2) . The functions

u(z) and v(z) are non-trivial solutions of the homogeneous (u = 0) and

inhomogeneous (u ^ 0) Jacobi equation which vanish at z = (-L/2) . Since

z* is independent of the choice of y, u (-L/2) and v (-L/2), any convenient

z z value of these quantities may be used. Thus, the shape and stability of a

103
liquid zone are determined by the solutions of the Euler and Jacobi equations, respectively.

We discuss the appropriate boundary conditions for r(z), the zone shape.

If the trij unction (intersection of solid, liquid, and vapor phases) does not

occur at a corner of the solid, as illustrated in Fig. 2, then the balance

of forces in the direction tangential to the solid requires that

Y SL

+ YLV

COS 9

L

= Y SV (8)

where YgL

J Y^y and Ygy are the solid-liquid, liquid-vapor, and solid-vapor

surface energies, respectively, and 6* is the angle between the solid-liquid

Li and liquid-vapor interface measured in the liquid. Since for given substances

and conditions the surface tensions are fixed, this equation specifies 6* at

the trij unction. For a liquid between two solids without corners, the slope

r

z of the liquid shape is then specified at the trijunctions . If the trij unction occurs at a corner in the solid as illustrated in Fig. 2, the

following inequalities must be satisfied''""'"

either - Ygv > yLV

cos 9

y

or 0

y

> Tr

and either Y

gV

- Y SL 1 YLV

cos 6

L

or 8

L

> it

where 6 and 9 are defined in Fig. 2. The above inequalities are equivalent

V Li

to

e* < e_ < tt - e_ + e* (9)

Li Li O Li

where 6

S = 2tt - (9L

+ 9

y

) and cos 9* = (Y gv

- Y sl )/ylv

« Note that for

9 = tt, the inequalities demand that 9 = 9*. For 9 < it, there is a range

O Li Li O

of possible 9 values. If one considers a rounded corner from a macroscopic

Li

viewpoint, an identical range of angles is possible. The appropriate boundary

condition is that the liquid be at the corner. If as a result of a calculation

of r(z) one finds that 9 does not satisfy the appropriate inequalities, then

Li obviously such a liquid zone is unstable (i.e., the liquid will pull away

IDA
from the corner) and the calculation should be repeated with different boundary conditions. Experimentally the range of allowed 9 values can be increased

Li by using composite solids, i.e., each edge is a different material. For

7—8

example, Carruthers and Grasso coated the vertical surface of their glass

with grease to prevent capillary run-away. In the floating zone process, the

temperature gradients presumably prevent the liquid from moving along the

solid-vapor interface. Thus, for the situation with corners shown in Fig.

1, we will use the boundary conditions r(-L/2) = R and r(L/2) = R^. In

other circumstances one might wish to specify r at one or both of the endz points (z = ± L/2) rather than r.

Since there is an unknown Lagrange multiplier p in the Euler equation,

an additional condition on the liquid zone is required. In an experiment

where given volume of liquid is placed between two inert solids, the additional

condition is to specify the volume of liquid in the zone. For the floating

zone process, one can also specify the volume in the zone. However, in order

to grow a solid of uniform diameter, recent evidence^ ^ indicates that the

slope at the freezing interface must have a fixed value (e.g., arctan

(r ) 11 degrees for silicon ). Thus, rather than specifying the volume,

z

one could specify r' at the freezing interface.

Numerical Calculations

An algorithm and computer program has been developed for solving the

Euler equation (with boundary conditions r(-L/2) = R, r(L/2) = R^, and

the volume specified) to find the liquid shape r(z) and the Jacobi equation

to determine the conjugate point and thus the stability of the zone. Since

the boundary conditions on the Jacobi equation specify the function and its

derivative at z = (-L/2), i.e., an initial value problem, the solution

of this equation presents no problems. The Euler equation with r(-L/2) = R

105
and r(L/2) = R^ is a two point boundary-value problem with the additional

complication that the Lagrange multiplier p is unknown and must be chosen to

00 yield the correct zone volume which we denote by V . We will solve the

Euler equation as an initial value problem by guessing values of r (-L/2)

z

- s and p. In general, the values of r(L/2) and the volume obtained from

our initial guess will not be the desired ones. An iteration procedure

CO which allows us to find s and p such that r(L/2) = R^ and V = V will now

be described.

We denote the values of s and p at the n*"*

1 iteration by s11 and p

n

,

respectively. We assume that r(L/2, s

11

, p

n

) - E R° - and V(s

n

, p

n

)

00 - V are sufficiently small that we can expand in a Taylor series about

(s

11

, p

n

) , retain only linear terms in the expansion, and equate the result

to zero (Newton's method). This yields

(s

n+1

-s

n

) (3R^/3s) + (P

n+1

-p

n

) (3R^/3 P ) = R

u

- r(|, s

11

, P

n

) (10)

(s

n+1

-s

n

) (3V/3s) + (p

n+1

-p

n

) (3V/3p) = V°° - V(sR

, p

n

) . (11)

Here (3R*/3s) = 3r(L/2 , s

11

, p

n

)/3s, (3R*/3p) = 3r(L/2, S

n

, p

n

)/3p,

(3V/3s) = 3V(s

n

, p

n

)/3s, and (3V/3p) = 3V(s

n

, p

n

)/3p. These equations

can be solved to find p

n+^ and sn+\ The required partial derivatives are

calculated numerically by the following procedure. Since the operators

(3/3s) and (3/3p) commute with (d/dz), the functions r = 3r(z,s,p)/3s and

s

r^ = 3r(z,s,p)/3p satisfy the differential equations

r - r (3f/3r ) - r (3f/3r) = 0 (12)

szz sz z s

r - r (3f/3r ) - r (3f/3r) = (3f/3p). (13)

pzz pz z p

106
Here the subscripts indicate differentiation and f(r,r ,p) = [(1+r 2 )/r]

z z

o In + [1+r 2

] [(pgz/y) - (pfi 2 r

2 /2y) - (p/y)l = r • The appropriate boundary

z z z conditions at z = (-L/2) are r = 0, r =l,r =0, and r =0. These

s sz p pz

are initial value problems which are solved numerically to obtain r

g

and

r . The (3V/3s) and (3V/3p) are obtained by numerical evaluation of the

P

integrals

L/2

/

-L/2

d 3s) -

|; 2irrr

g

dz (14)

and

L/2

/

S ^ = ; 2irrr

p

dz (15)

-L/2

Equations (1), (2), and (12-15) can be written as a system of nine

first order differential equations. The method used to solve the initial

value problem is a variable order, variable step formulation of the

12 classic Adams methods as implemented by Gear . As previously described,

Newton's method is used to find new values of s and p and the system

of differential equations is solved again. This iteration is continued

until the relative change in p and s is less than a certain value (usually

10~ 3 ). An absolute change criterion is used if s and p are close to zero.

The iteration procedure can fail for a number of reasons. Frequently

r becomes very large and the calculation fails. If r remained finite,

one could presumably solve this problem by allowing the step size to become

13

very small. However, it is known that solutions of the Euler equation

can be multi-valued and can become infinite. Presumably to solve such

problems it would be necessary to use a different coordinate system than

107
the (r,z) system; we have not attempted this. Computations with r —

z 37 have been successfully performed.

It is difficult to make any general statements about the initial

guesses for s and p. For certain cases, the initial s and p can be quite

far from the final values and the iteration method will work. In other

cases, it is necessary that the initial choice be close to the final values.

In this case one can find reasonable initial choices by slowly incrementing

some parameter from a known solution, i.e., a primitive continuation procedure. For example, if a solution is known for zero gravity and one wants

a solution for a large gravitational field, one can slowly increase the

gravitational field using the values of s and p at the smaller gravitational

field as the initial choice of s and p for the larger gravitational field.

Numerical Results

Before presenting the results of a number of calculations, we will

describe in some detail a particular stability calculation. We take

R = = 1 cm, L = 6.2 cm, Q = 0, V = ttL cm3

, p = 1 gm/cm3

, and y = 100

mJ/m2 (although we use these values, it should be realized that only

certain dimensionless combinations of these quantities are important). We

wish to find the maximum value of the gravitational acceleration for which

the zone is stable (we denote this value by g* and the corresponding e

value by e* = pg*R2 /y). It is evident that for g = 0, a solution of Eq. (1)

108
is r = R with p = y/R. This provides initial values for p and s, viz.,

p = 100 and s = 0. Since we know that for L = 2tt, stability occurs at

zero gravitational field we expect g* to be small. Upon carrying out

calculations we find that for g < g*, the iteration procedure works very

well. For example, for g = 0.05 cm/s 2

, only four iterations are necessary

to obtain a solution with s = 0.041378 and p = 100.08. However, for

g > g*, the iteration procedure fails. For example, for g = 0.071 cm/s 2

and initial values of s = 0.075677 and p = 100.27 (which are the final

values of s and p for g = .070 cm/s 2

) after 10 iterations the relative

change in s for the next iteration is 0.25. It is also apparent from the

calculations that s is not converging toward a definite value but is oscillating; thus, it is unlikely that a greater number of iterations would lead

to a solution.

Occasionally, large changes in s and p will occur and a solution of

the Euler equation will be found which is unstable. For example, for

g = 0.074 cm/s 2 and initially s = 0.075677 and p = 100.27, a solution was

found with s = -0.15319 and p = 101.21. The conjugate point occurred at

2.848, which is less than L/2, which indicates that the solution was

unstable. Physically, it was obvious that the solution was unstable

since it bulged at the top and was constricted at the bottom. Using s =

-0.15319 and p = 101.21 as initial values and changing g one can find a

109
family of unstable solutions, e.g., s = -0.13340 and p = 100.89 yields a

similar unstable solution for g = 0. In general, the Euler equation with

the boundary conditions r(-L/2) = R, r(L/2) = and fixed volume has a

number of solutions. We have found solutions where r =0 for more than

z

two values of z. Two solutions of the Euler equation are illustrated in

Fig. 3. Both solutions satisfy the boundary conditions and have the same

volume. The solid curve is the stable solution while the dashed curve is

unstable. Physically, it is reasonable to assume that there is never more

than one stable solution.

By approaching g* very slowly from below, it is possible to show that

the failure of the iteration procedure corresponds to the solution of the

Euler equation becoming unstable. For example for g = 0.07014, 0.07017

and 0.07020 cm/s 2

, the conjugate point occurs at 3.102, slightly greater

than 3.100 and slightly less than 3.100, respectively ( a conjugate point

less than 3.100 indicates instability). For g = 0.07020 the magnitude

of the relative change in s and p for the last iteration was 0.019 and

0.00011, respectively. If the usual criterion (that the relative change

be less than 0.001) had been used, the iteration process would have failed

for g = 0.07017 and 0.07020 and succeeded for g = 0.07014. However, the

solution for g = 0.07020 satisfies the boundary conditions r(L/2) = 1.0000

110
and the volume constraint (e.g., r (-L/2) = 0.078917 and r (L/2) =

z z

0.078915; these derivatives should be equal if V = ttL) .

In general, we will identify the failure of the iteration procedure

as g (or some other parameter) is slowly incremented with the onset of

instability. This may slightly underestimate g*; in the preceding example

(requiring that the relative change in s and p be less than 0.001) we would

take g* = 0.07014 whereas the solution is actually stable for g = 0.07017.

If r

z is large, only a lower bound for g* can be obtained since it is

difficult to tell whether the failure of the iteration process is due to

instability or to large values of r . z With a gravitational field present we have not been able to find

in the unstable region a solution of the Euler equation which appears

to be a continuation from the stable region of the stable solution

(although solutions exist in the unstable region, they are qualitatively

different from the stable solution) . The basic question of whether the

solution does not exist or whether the present numerical method is

failing to find it is unresolved at present and requires further investigation. In zero gravity it is well known that such unstable solutions

exist, e.g., r = R for L > 2itR (the present numerical program also calculates this rather trivial example)

.

A number of tests of the numerical calculations have been performed.

We will describe a few of these. For V = ttR2L, g = 0, R^ = R, and finite

rotation, the computed conjugate point agrees with analytic results"^.

For g=0, = 0, R=R^, and L/R = 2, we find that the smallest volume

for which there is a stable zone is V = . 387irR2L (the iteration procedure

fails below this volume) ; this appears to be in agreement with the

111
analytic results of Gillette and Dyson (to the accuracy obtainable

from their Fig. 6). For R = R^, V = ttR2L, Q = 0 and e small, we can

compare the numerical solution for r(z) with an approximate analytic

solution (see reference 10 or the appendix). In Table 1, we compare

the numerical values of s and p with the analytic values. For sufficiently small e there is agreement between the numerical and analytic

results. How small an e is required is clearly a strong function of

L/R. In fact, as e approaches e* (where instability occurs; for (L/R)

= 6.2, e* = 7.02 (10_1+

) and for (L/R) = 5, e* = 0.073), the analytic

and numerical results do not agree. This indicates that the analytic

results are incorrect at e* (even though e* can be very small). Thus,

the analytic prediction of the stability-instability demarcation will

not be correct; this will be discussed subsequently.

The liquid zone shape depends on the five dimensionless parameters

R /R, V/ttR2 L, L/R, e and e_. It is clear that with this many variables

u R

one cannot study all possible combinations. Our initial efforts have

been to study the case R /R = 1, V/ttR2 L =1, e„ = 0 with L/R and e u R

variable. This allows comparison of our numerical results with our

previous analytic studies for small e (see ref. 10 or the Appendix) and

with the present experimental results and those of Carruthers and Grasso^

We have also carried out a small number of calculations with other values

of the parameters.

For R = R, V = ttR2 L, and e„ = 0, we have plotted in Figure 4 the

u R

maximum stable value of L/2ttR as a function of log e. For a given e 'the

zone is stable if L/2ttR lies to the left of the curve and unstable if to

112
(L/R) = 6.2

(L/R) =5.0

Table 1

Comparison of Numerical and Analytic Values

of s and p [V = irR 2L, R = R and ft = 0]

u

NUMERICAL ANALYTIC

10 3 s P 10 3 s P

1 1 .DD ID ±UU. UU -7 C/,QQ 1UU . UU

2.5 19.208 100.02 18.872 100.00

5 41.378 100.08 34.745 100.00

6 53.015 100.13 45.293 100.00

7 75.677 100.27 52.843 100.00

7.02 78.917 100.30 52.994 100.00

10 1* e 10 3 s p 10 3 s p

10 xuu . uu • DH DO _LUU . uu

20 8 6928 100 00 8 6932 100 00

50 21.745 100.01 21.733 100.00

100 43.570 100.03 43.466 100.00

200 87.811 100.11 86.932 100.00

500 234.11 100.83 217.33 100.01

700 376.62 102.47 304.26 100.01

113
the right. For large values of s, r becomes large (e.g., for e = 2.89,

z

log e = 0.461, r (z = -L/2) = 23.6) and as previously indicated we may

z underestimate the stability of the zone. This is indicated by the dashed

curve in Fig. 4; to the left of this curve the zone is certainly stable.

As indicated previously, the analytic results for small e do not agree

with the numerical results; a comparison and tabulation of the numerical

results are given in Table 2. In the table the largest discrepancy

between numerical and analytic results actually occurs for the smallest

non-zero value of e; at L/R = 6.24 the ratio of the analytic e and the

numerical e is 1.47. This disagreement is not understood at present and

the development of a valid analytic approximation requires further research.

No values of the analytic £ are given for e > 0.025 since there is no

reason to expect agreement with the numerical results

We briefly give the results of a few other calculations to illustrate

the types of problems which can be investigated. For R = R^, (V/ttR2L) =

0.5, L/R = 2 and e = 0, the maximum value of e for which the zone is

stable is 1.537. For R = R

u

/2, (V/irR2L) = 7/3 (a conical volume), L/R =

5, and e = 0, the maximum value of e for which the zone is stable is

0.134. The zone shape for this case is illustrated in Fig. 1; the solid

and dashed lines are for e = 0 and e = 0.123, respectively. For R = K K

R

u

, L/R = 10, and (V/ttR2L) = 16.667 (a spherical volume, V = ttL 3 /6) , and

e = 0, the maximum value of e_ for which the zone is stable is e = 0.0361.

R K

For this geometry the initial slopes are 3.89, 29.1, and 0.016 for the

cases (e = 0, e = 0) , (e = 0.054, £t3 = 0) and (e = 0, r = 0.0361),

K K K

respectively.

114
TABLE 2

The Stability-Instability Demarcation:

Bond Number £ versus Zone Length for

Ru

= R, V = ttR 2 L and = 0. Comparison of Numerical and Analytic Values.

L/R

£

NUMERICAL ANALYTIC

2tt 0. 0 0.0

6. 24 0.000258 0.\-/ • 00C380 vy vy vy ^y vy vy

6. 20 0.000702 0.00102

6.10 0.00239 0.00332 vy • vy ~—' ^y

5.92 0.00723 0.00927

5.85 0.0100 0.0121

5.70 0.0162 \y • u -X- Vy 0. 0189

5. 63 0.\y • 0200 \y i— v vy 0.0224

5.20 0. 0518

4.81 0.100

4.31 0.200

3.99 0.300

3.56 0.500

2.94 1.00

2.00 >2.89

1.00 >12.8

0.50 >52

U5
Also of interest is a computation with R = R , L/R =7, e = 0, e„=0,

u R

and various values of (V/ttR2L). It was possible to approach the instabilitystability demarcation from both the stable and unstable regime [r(z) = R is

a solution in the unstable region for (V/frR2L) = 1]. For (V/irR2L) = 1.270

and (V/ttR2L) = 1.255 the conjugate points occur at 3.518 and 3.485, respectively (i.e., the liquid zones are stable and unstable, respectively). Both

zones are symmetric about z = 0; however, the unstable zone has a negative

slope at z = (-L/2) while the stable zone has a positive slope at z = (-L/2).

Thus, the unstable zone has r =0 for three values of z (viz., z = 0 and

z

z near ±L/2) whereas for the stable zone r =0 only at z = 0. If we require

z

that the relative change in s be less than 10" 3

, the iteration procedure

fails for (V/ttR2L) equal to 1.260 and 1.265; it would work if we slightly

relaxed the 10" 3 requirement . The (V/ttR 2L) obtained for the stabilityinstability demarcation appears to agree with the results of Gillette and

Dyson

Experimental Observations

We have made some observations of water zones between non-rotating flat

faces of coaxial circular cylinders of equal diameter in order to compare the shape

and stability with the predictions of the calculations. By using very

small radii, one can simulate microgravity , e.g., for R = 2.7 x 10~ 2 cm,

e = 10~2. The solid surfaces used have generally been steel, but some

measurements have also been made with glass surfaces. The top cylinder is

rigidly mounted and the bottom cylinder is clamped to a micromanipulator

that permits precise variation of the vertical position. To minimize

volume changes during measurement due to evaporation, the steel rods and

water zone are enclosed in a glass tube sealed at the top and containing

116
a reservoir of water at the bottom.

The theoretical calculations previously discussed have found the

maximum stable length L of a zone whose volume is ttR 2L, where R is the

radius of the end surfaces. The technique used experimentally to reproduce the conditions assumed in the theory is to find the maximum stable

length as a function of zone volume. The stability of the zone of volume

ttR 2L can then be determined from this function.

The experiments are performed by injecting a small drop of known

volume into the space between the two coaxial rods using a micrometer

syringe. The liquid zone is then extended by slowly lowering the bottom

rod. When the zone fails, the length is recorded, and the process repeated

with a different volume. The zone is observed through the microscope during

the extension to verify that failure is due to necking of the zone rather

than detachment from the upper surface. In all cases the zone wetted the

upper face up to the edge.

The result of a typical measurement using this process is presented in

Figure 5 which shows the variation of maximum length with zone volume for

an end face of radius 0.1205 cm. The line through the origin gives the

volume of a cylindrical zone of length L and radius 0.1205 cm. Thus, the

intersection of this line with the experimental curve determines the length

at which a cylindrical zone of volume = irR 2L will fail. For Figure 5, this

intersection is at V = 2.45 x 10-2

cm 3 and L = 0.515 cm.

A series of measurements of maximum stable length L for a zone of volume

ttR 2L have been made using rods of different R. The results are shown in

Figure 6 where (L/R) is plotted as a function of the Bond number,

exp

2. e = . Figure 6 also shows the theoretical prediction of the numerical

calculations summarized in Table 2 and the dependence suggested by Carruthers

117
and Grasso, i.e., L/R = 27t/(l+e) (the range of e in their experiments

was approximately 0.4 to 14). Our experimentally measured (L/R) are within

about 2% of the theoretical prediction for all values of e. An observed

zone is compared with a calculated profile in Figure 7. The agreement is

excellent.

There is a large difference between our numerical and experimental

results and those of Carruthers and Grasso, e.g., for e = 1.35, they find

an L/R that is about 55% larger than our theoretical prediction and our

experimentally observed value. The reasons for such large differences are

unclear. Perhaps the boundary conditions used in our experimental and

theoretical work, viz., r(-L/2) = r(L/2) = R and V = ttR2 L, do not model

their experimental situation. We have done a few calculations with volumes

slightly different than TR L; these have only a slight effect on the maximum

stable length. For example, for £ = 0.5, the maximum stable L/R is 3.51,

3.56 and 3.60 for (V/ttR2 L) equal to 0.95, 1.0 and 1.05, respectively. We

plan further investigations of the effect of the boundary conditions on

the stability criterion. In addition, it is desirable to study the effect

of various containing fluids on the zone stability, e.g., are there differences

between a vapor and a liquid as containing fluid?

In addition to the measurements of water zones, observations of molten

zones in thin tin cylinders have also been made. This work was initiated

to simulate an experiment proposed for the sounding rocket program.

We thank H. W. McLaughlin and R. F. Sekerka for many stimulating discussions.

118

)
References

1. W. G. Pfann, Zone Melting (John Wiley & Sons, Inc., New York, 1966).

2. W. Keller, Third American Conference on Crystal Growth, Stanford

University, 1975, p. 67.

3. W. Heywang, Z. Naturforschg. 11a, 238 (1956).

4. P. I. Antonov, in: Growth of Crystals, Vol. 6a, Ed. N. N. Sheftal

(Consultants Bureau, New York, 1968) p. 145.

5. W. Bardsley, F. C. Frank, G. W. Green, and D. T. J. Hurle, J. Crystal

Growth 23, 341 (1974).

6. T. Surek and B. Chalmers, J. Crystal Growth 29, 1 (1975).

7. J. R. Carruthers and M. Grasso, J. Crystal Growth 13/14 , 611 (1972).

8. J. R. Carruthers and M. Grasso, J. Appl. Phys. 43, 436 (1972).

9. 0. Bolza, Lectures on the Calculus of Variations (Dover Publications,

Inc. , New York, 1961).

10. S. C. Hardy and S. R. Coriell, NBS Space Processing Research (NBSIR

74-611) November 1974, p. 60.

11. J. W. Gibbs, The Scientific Papers of J. Willard Gibbs, Vol. I (Dover

Publications, New York, 1961) p. 326.

12. C. W. Gear, Numerical Initial Value Problems in Ordinary Differential

Equations (Prentice-Hall, Inc., Englewood Cliffs, N.J., 1971).

13. J. F. Padday, Phil. Trans. Roy. Soc. London A269, 265 (1971).

14. R. D. Gillette and D. C. Dyson, Chemical Engineering _2, 44 (1971).

119

Fig. 1 Stable liquid zones between solids of radii Ru and R. For these

zones (L/R) = 5, 2R = Ru , the volume V = (7/3)ttr2l, and the gravitational tield is zero. The solid curve corresponds to zero rotation while the dashed curve corresponds to finite rotation (eR

= 0.123); a 20% increase in thxs rotation rate would make the zone unstable.

Fig. 2 Trijunctions where solid, liquid and vapor phases meet. In the

top figure, the solid surface is flat; in the bottom figure the trijunction occurs at a corner of the solid.

Fig. 3 Two liquid zones satisfying the same boundary conditions and having

the same volume. For these zones, (L/R) = 5.5, R = Ru , V = frR 2L, er = 0,

and e = 0.01. The zone indicated by the solid curve is stable (the con- jugate point z* = 3.512 R) ; the zone indicated by the dashed curve is

unstable (z* = 2.020 R) . Both zone shapes satisfy the Euler equation

(with different values of p).

D.1 0.3 0.5 0.7 0.9

277" R

Fig. 4 Numerical results for the stability- instability demarcation; log ln e as a function of (L/2*R) for R = Ru , V = ttR^L, and no rotation. The liquid

zones are stable to the left of the curve. The Bond number e

= pgR2/Y

Fig. 5 The maximum stable zone length as a function of zone volume. The

straight line indicates the volume of a cylindrical zone.

Fig. 6 A comparison of numerical and experimental results for the maximum

stable zone length as a function of e. The dashed line is the results of

Carruthers and Grasso.

Fig. 7 A comparison of observed and calculated zone shape (V = .020 cm 3

, R = 0.1205 cm, L = 0.497 cm, y = 72 mJm-2

, no rotation).
Appendix

In this appendix we formulate the relevant differential equations for

the shape and stability of a liquid zone when the gravitational field is

in an arbitrary direction. For the special case of R = R^, the volume V =

ttR 2L, no rotation, and a small gravitational field, we obtain approximate

analytic results for the zone shape and stability; in particular, we find

the maximum stable zone length as a function of gravitational field. We

also find that a horizontal liquid zone is more stable than a vertical liquid

zone.

9

We apply the calculus of variations to the stability of a static

liquid zone between two solid circular surfaces (plates) which are parallel

and coaxial. We assume that the liquid remains at the edges of the plates

and that the gravitational field vector makes an angle A with the line joining the center of the circular plates. The distance between the plates is

L and the radii of the upper and lower plates are R^ and R, respectively

(subsequently, we take R^ = R) . We define a Cartesian coordinate system

(x, y, z) with origin at the center of the plates and midway between them.

The z-axis coincides with the line joining the center of the plates and is

directed toward the upper plate. The x-axis is chosen such that the gravitational field vector lies in the (x-z) plane and makes an angle A with the

(-z) axis, i.e., A = 0 corresponds to a gravitational field in the (-z)

direction. It is more convenient to use a cylindrical coordinate system

(r, 6, z) which is related to the (x, y, z) system by the usual transformation.

The shape of the liquid zone may then be written as r = r(0,z) with the

boundary conditions r(6, -L/2) = R and r(0, L/2) = R^. We also assume that

the volume in the zone is fixed, i.e.,
V =

L/2

-L/2

2tt

dz d6 {r 2 (6,z)/2} = CONSTANT (Al)

0

For a static liquid, the energy is completely determined by the shape

of the liquid-vapor interface r(9, z). The calculus of variations allows

us to determine r(9, z) such that the energy is stationary and in addition

to determine whether the energy is a minimum and the shape stable. The

energy can be written as a sum of surface and gravitational energies, i.e.,

L/2

E' = dz

-L/2

2tt

d9 F(r,r

Q

,r

z ,e,z) (A2)

0

with

F(r,r ,r ,9,z) = y [r ^r^+r^2 ]^ + pg[(r 3 /3)cos 8 sin A + (r 2 z/2) cos A]

\D Z \J Z

(A3)

Here, y is the liquid-vapor surface tension, p is the density difference

between liquid and vapor, g is the magnitude of the gravitational acceleration, and the subscripts 6 and z denote partial differentiation, e.g.,

r = (3r/39) and r = (3r/3z). Since the volume is fixed we introduce the

y z Lagrange multiplier p and define H(r,r ,r ,9,z) = F(r,r ,r ,9,z) - p r

2 /2

\j 2 \j Z

f L/2

and E = dz

-L/2

2tt

d9 {H(r,r

Q

,r

z ,9,z) . (A4)

0

If E' is a minimum, then E is also a minimum since the integral of r

2 is

a constant. In order for E to be stationary, it is necessary that

(3H/3r) - (3/39)(3H/3r ) - (3/3z)(3H/3r ) = 0.

o z (A5)
Upon evaluation of this expression, we obtain the Euler equation

P¥-e[¥2 cos 9 sin A+¥ r\ cos A] = {^2+^2+^2vj,2}- 3 / 2

6 ri

{y3-\<i2y +2^<i/2+y3^2_y2y2^ +2¥ 2,F ¥ ¥ -¥**¥ -y 2 y2y } (A6)

69 6 n ri 66 8 n 6ri nn 6 r\r\

where we have introduced the dimensionless variables, P = R p/y 5 n = z/R,

¥ = r/R, e = p g R

2 /y and the subscripts indicate partial derivatives. The

expression on the right hand side of the Euler equation is equal to ¥K,

where K is the mean curvature of the surface ¥=¥(6,n)« The boundary conditions on ¥ are ¥(8,-3) = 1 and ¥(6,3) = R

u

/R with 3 = L/2R. The dimensionless

Lagrange multiplier P is chosen to give the desired volume in the zone.

Using standard techniques from the calculus of variations one calculates

the second variation of E. Considering small variations in ¥, i.e.,

¥-*¥+6¥ = ¥+W(6,ri), one obtains the Jacobi equation

(3J/3W)-(3/38)(3J/3Wj-(3/3ri)(3J/3W ) = 0 (A7)

6 T) with

+ 2H ll^t 2H W

e + M * W (A8)

where the subscripts denote partial derivatives and we have introduced the

Lagrange multiplier M in order that the integral

f 3 f 2Tr

dn d6 ¥ W = 0 , (A9)

-3 Jo

which ensures that the volume in the liquid zone is constant. The Jacobi

equation may be written as

W{2e¥cos6sinA+encosA-P+B¥ 2 (l+¥ 2 )}

8 n - W A {B[¥¥ +¥¥„¥2 ]}-W {B[¥ 3 ¥ (l+¥ 2 )+2¥¥ ¥

2 ]}

8 8 8n n n n n 8
+ {8/3e}{B[W(1ni' rt +^rt Y

2 )+W CV 2x¥ V n )-\Jn (V 2+V2V

2 ootiti )]}

rj 0 9 Ti - 0/3n}{B[W('F 31' +^ 3^ 3+2^^ ¥

2 )+W (V k+y 2V

2 )-Wn (Y 2 V Y )]}

ti n n 6 n ee 8n

= - MW ,

where B = {V 2+V 2+V 2 V

2 }~ 31' 2

6 n

(A10)

The stability of the liquid zone is determined by the properties of the

Jacobi equation. Let u(6,n) and v(6,r|) be solutions of the homogeneous (M=0)

and inhomogeneous (M^O) Jacobi equation, respectively, such that u(6,-3) =

v(6,-3) = 0. Further, let

m =

n =

c2tt

de

0

de

n(6)

dn{^ u}

-3

n(6)

(All)

J 0

J -3

D[n(9),-3] = m v - n u

(A12)

(A13)

The liquid is unstable if in the region 0 < 6 <_ 2ir and -3 < n £ 3 > there

exists a curve (or boundary) n*=n*(9) such that D[n*(6), -3] - 0. If no

such n* exists, the liquid zone is stable.

The preceding paragraphs give the basic equations for the shape and

stability of a liquid zone. In principle these equations can be solved

numerically for arbitrary volumes, gravitational fields, and upper plate

radius R^. We proceed however to obtain an analytic solution for the special

case where V = ttR 2L, the gravitational field is small (e«l) and = R. We

expand all quantities in a power series in e, and retain terms of the order

of e

2

, e.g.,

¥ = ¥ + e + e z y 2 o

1

P = P + e Pi + e

z

P.

(A14)

(A15)
W = W + e Wi + e z Wo , etc.

o

1 z (A16)

In view of the quantitative disagreement between our numerical and analytic

results for A = 0 or it (see Tables 1 and 2), this expansion procedure is

apparently not rigorous even if e«l; the results obtained should be regarded

as approximations.

For ¥(e,Ti) , we obtain

(A17)

(A18)

¥ = 1

o

A = -Pi + cos 9 sin A + n cos A

A y 2 = -P 2 + ^l cos 9 sin A + 2Vj ¥j + ff + hv\

Q ~ ^fn (A19)

with P

q = 1 and the operator A = 1 + (9 2 /3 9 2

) + (3 2 /8n 2) - The boundary

conditions on and V 2 are that (6 , ±3) = I^C 8 *^) = 0. The constants

Pi and P 2 are chosen so that the integrals

3

dn

f 2tt

de{ (f

1 }

J -3 J o

6

-3

dn

2tt

d9{^i 2 +2 y 2 > = 0 . (A20)

We find that

XV

1 = cos A {n-[B sin n/sin 3]} + sin A cos 6 (n

2~3 2 )/2 with Pj = 0 and

cos 2 A{n 2 -(3 2 /2)-(3 2 /2) (sin2 n/sin2 3)

+ [3/sin 3][(n 2 /2) cos n-(3 2 /2) cos n + 2 cos n - 2 cos 3]}

+ sin A cos A cos 0{3/sin 3}{3sin n - 3(n/3) sin 3 - n cos n + n cos 3}

+ sin2 A{-(n 2 -3 2

) 2 /16 + (3n 2 /4) - (33 2 /4)

+ (3 3 /2) [cos 3-cos n]/[3 cos 3-sin3]

+ cos 26[(n 2-3 2 )/16 + (n

2 /4) + (1/6)

- (cosh {3h n>/cosh {3h

3))({3 2 /4} + {1/6})]} (A21)

with

*2 = cos 2 A {(3 2 /2)-(5/2) + (23 cos 3/sin g) + [3 2 /(2 sin2 3)]}
+ sin2 A{-(3/2) - (3 2 /4) + [3 cos 3 + sin3]/[3 cos 3 - sin3]> .

(A22)

Expanding the Jacobi equation in powers of e, we obtain

AW = M (A23)

0 o

A Wi = Mi + M fi + W [2 sin A cos 6 + n cos A - Pi + ¥ loo 1 1 - ] o

1 o

1 i 99 1 nn

- W fi-W (A24)

onn

1 on n

A W2 = M2 + + M ? 2 + W

Q

[2V

2ee

+ ^2 + v l cos 9 sin A

- (3/2)^

Q - 4*i V 1qq - yf + (¥? /2)]

+ W1 [2vf lee+ *! + cos 6 sin A]

+ W [2* lo V la + *! * loo + *i - ¥ 2 ]

+ W

onn

[ '*2 + ( %/2) + t^2 ^!^

+ WlQ0y 1 + Wle

VF le - Wi

n*i

n - Wi^f ! (A25)

In obtaining the equations for Wj and W2 , we have used the result that W

q

is independent of 9. In fact, the solutions u

q

and v

q

of the homogeneous

and inhomogeneous Jacobi equations, respectively, are

u = sin(n+3) (A26)

o

and

v = M [l-cos(n+3)] (A27)

o o

To zeroth order in e a simple calculation gives n*(9) = 2ir— B and the

stability criterion is n* > 3> i.e., the well known result 3 < tt(L < 2ttR) .

It is convenient to evaluate quantities at n* = 2tt— 3 rather than at n* =

nQ

* + erii* + e

2 n 2 *. Thus, we expand in a Taylor series about n o

*« For

example, given a function f(9,n) = f + efj+e 2 f 2 , we evaluate this function

at n=n*(6) by

f(9,n*) - f

D

+ e[ni*f

on

+ fi] + e

2 [(f

orin /2)(nf)

2 + n 2 *f

on

+ f 2 + ni * fj ], (A28)
where f and its derivatives on the right hand side of the equation are

evaluated at n *• Applying the above formula to the quantities appearing

in D[n(6),-B], we obtain the following expressions for m* and ri2* :

Hi* = -Ui(n *) (A29)

n 2 * = -uo(ti *) - ni* ui (n *)

•2tt r n o + {vi(n *)/(4tt 2

x M )}

o o de

0

dn{u + Ul } (A30)

-6

Since vj(n

o*) is proportional to M

q

, the right hand side of the equation

for r\2* i s independent of M^.

In general m* and ri2* could depend on 9; however, for the particular

problem being considered, we will indicate why m* and n

2 * are independent

of 6. Specifically we consider ni* = -ui (n *); now ui (n *) may be written

o o

as

u l(n Q

*) = -{cos A}{2tt(tt-B)}

00 + I 1o{B sin (n6)+ B cos(n0)} (A31>

ii cn

where the coefficients B and B are arbitrary constants. The ui (n) which

sn cn 1

gives rise to the above form of ui(n *) satisfies the appropriate differential

equation and the boundary condition uj(-3) = 0. It is clear that the average

value of uj(ti *) over the interval 0 < 6 <_ 2tt is the 6 independent term

{-[cos A] [2it(tt-$)] }. Further unless B = B =0 for all n, uifn *) will

sn cn x o

be both larger than and smaller than this average value. This will impose

a less stringent test of stability than that obtained by setting B^ = B^

= 0 for all n. Thus, we require that m* and * be independent of 9.

Evaluation of the expressions for m* and ri2* yields

ni* = cos A {2tt(tt-B)} (A32)
n 2 * = if cos 2 A {-9-67tB+4tt 2+43 2- 143(cos g/sinP)- [9B 2

/ (2 sin2 3)}}

+ tt sin2 A {-[3$ 2 /2] + [3 3 cos 3/(6 cos 3-sin 3)]} (A33)

The stability criterion 3 < n* can be written

0 < (tt-3)(1+ e ttcos A) + (e 2 /2) n 2 * (A34

Assuming 0 < Tr-3=y<<l and expanding the above inequality in powers of y,

we obtain

e

2 {cos 2 A[(97r 3 /4y 3

) - (23tt 2 /2u 2

) + (55Tr/4y) - (tt 3 /4u)]

+ sin2 A[(u 3 /4y)]} < 1 + eir (cos A) (A35)

For A = 0 or tt, the stability criterion is

3 < - (3e/2) 2/3

] (A36)

while for A = ±(tt/2), the stability criterion is

3 < tt[1 - (e 27r 2 /4)] (A37)

Thus a gravitational field along the z-axis (which connects the centers of

the two end plates) is more destabilizing than a gravitational field perpendicular to the z-axis.
Task 5

VAPOR TRANSPORT SYNTHESIS AND CRYSTAL GROWTH

H. S. Parker

Inorganic Materials Division

Institute for Materials Research

Summary

The major objective of this study is the investigation of advanced

techniques for vapor transport synthesis and crystal growth and definition

of the limitations imposed by convective effects in a temperature gradient,

for comparison with results of similar experiments in near zero-gravity

conditions. The growth of mercurous chloride by an evaporation-condensation

mechanism has been chosen not only for its adaptability to this method

but also for the potential technological importance as an infra-red

polarizing material. During the previous contract period, transport rates

were shown to be high at temperatures in the 400°C range and temperature

differences (AT) of 60-80°C.

Seeded growth experiments were undertaken during this contract period

and improvements in the transport tube geometry and technique are described.

Growth at temperatures in the 350°-390°C range at lower AT has resulted in

good quality growth of 0.6 cm diameter boules over 1 cm in length.

The existence of a tetragonal-monoclinic transition in mercurous

chloride was found to occur at about 168 ±5 K as determined by low

temperature, x-ray powder diffraction. This transition should be investigated further to determine if the monoclinic phase is non-centrosymmetric in view of the potential as an infra-red laser modulator.

120
Introduction

The major objective of this study is the investigation of advanced

techniques for vapor transport synthesis and crystal growth and definition

of the limitations imposed by convective effects in a temperature gradient,

for comparison with results of similar experiments in near zero-gravity

conditions. 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 the technique have been described by Schafer

[1], Laudise [2], and more recently by Kaldis [3]. In growth by chemical

transport, the reversible reaction occurring at source and growth regions

between the material and transporting agent must be considered, and

separation of the effect from other growth parameters is often difficult.

In those cases where the vapor pressure of the material is sufficiently

high, transport and growth can occur via a simpler evaporation-condensation

process. However, 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. In both cases, however,

convective and diffusive effects are closely 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

.

121
The work described in this report is a continuation of the study of

the growth of mercurous chloride by an evaporation-condensation mechanism

as described in the previous report [4]. Although the growth of mercurous

chloride by the vapor Bridgman technique has been demonstrated by previous

work at NBS and elsewhere [5,6,7], the present study is primarily concerned

with attempts to successfully achieve single crystal growth in a static

system. That is, by evaporation-condensation in a temperature gradient

with no relative motion of transport tube and gradient.

In addition, the behavior of mercurous chloride at low temperatures

was examined by x-ray diffraction to determine if a phase transition

existed. Such a transition, if to a phase which lacked a center of symmetry,

could make mercurous chloride (cooled below the transition temperature) an

important material for infra-red laser modulator or phase matching applications. A brief survey of compounds in the Tl-Pb-Cl-Br systems was made

to establish if compounds of technological importance as infra-red materials

might exist in these systems and be adaptable to vapor growth techniques.

Experimental Procedures

The details of the crystal growth procedure, including a description

of the multiple-tapped gradient furnace and ampoule loading procedure was

reported previously [4] and only modifications will be described here.

During the period of this report, efforts were devoted primarily to seeded

growth experiments. All experiments conducted in the tapped gradient

furnace were made utilizing a geometry which minimized the convective

effect, that is, the transport ampoule was vertical, with the hotter

(source) end at the top. A "transparent" furnace with a gold-coated

122
quartz tube as thermal insulation was used for some growth runs, permitting continuous observation of charge and seed. This furnace does

not have a tapped winding and was operated horizontally to provide a

suitable thermal environment.

Seeds of suitable dimensions (approximately 3x3x6 mm) were prepared by cleaving from a vapor Bridgman grown boule of good optical

quality. Prior to insertion of the ampoule into the transport furnace,

a brief thermal etch was given the seed in a separate furnace.

In the initial seeded experiments, the transport tube was inserted

into the gradient furnace at temperature. In subsequent experiments,

an auxiliary heater was added to the seed end of the transport tube to

maintain a small reverse gradient until the tube equilibrated with the

furnace temperature. At that time, power to the auxiliary heater was

manually reduced slowly in an attempt to provide a gradual increase in

the transport rate.

Because of continuing problems with nucleation on several sites at

the seed end of the tube, two major modifications were adopted. Experimental runs were started with transport tube and furnace at room temperature

and raised to the desired growth temperature over a six hour period,

maintaining the desired final gradient during the heating period. The

second modification was to the transport tube. Two concentric quartz

tubes are utilized at the seed end of the ampoule, with a 2-3 mm annular

space. The inner tube is pointed and sealed at one end and attached to

the closed end of the outer £ube by a quartz rod of 2-4 mm diameter. The

annular space is then evacuated and sealed off prior to charging the tube.

123
In this fashion, the heat flow is primarily axial in the growth region,

with the tip the coldest part. After an initial period of heating in a

reverse temperature gradient, the normal gradient conditions are established

and transport proceeds.

In order to obtain x-ray diffraction patterns of mercurous chloride

at low temperatures, use was made of a commercial powder diffraction unit

which uses a Geiger-Muller counter in conjunction with a high angle goniometer having a horizontal axis. A copper casing was designed to surround

the conventional glass slide normally used with this equipment. Nitrogen

gas, chilled by passing through about 14 meters of copper tubing passing

through a liquid nitrogen bath, was led to the copper casing through

insulated tubing and directed over the slide and specimen. A rotating

seal, coaxial with the goniometer, permitted operation of the goniometer

in the usual fashion. Temperatures were monitored by a copper-constantan

thermocouple attached to the slide at the edge of the specimen. Temperatures

as low as about 95 K were easily reached. To prevent formation of ice on

the specimen surface, the x-ray shield of the goniometer was covered with a

thin plastic film which did not seriously attenuate the diffracted beam.

Phase relations in the systems Tl-Pb-Cl-Br were examined by conventional phase equilibrium techniques. Mixtures were prepared from

end-member compounds, reacted at various temperatures in evacuated sealed

quartz tubes, quenched and examined by petrographic microscope and x-ray

diffraction methods.

124
Experimental Results

Transport experiments reported previously [4] have shown that at

source temperatures in the 405-410°C range and a AT of 60-80°C, corresponding to a Ap of about 1.5 atmospheres, transport rates in excess of

0.4 grams per hour are obtained. Similar experiments conducted in the

same minimum convection geometry but at much lower AT have shown similar

rates. With the source at 411°C and a AT at 15° (corresponding to a

Ap of 0.8 atm), a rate of 0.21 grams per hour over an average transport

path length of 8 cm was observed. With the same source temperature and

a AT of 27° (corresponding to a Ap of 1.0 atm), a rate of 0.76 grams per

hour was observed. The reason for the inconsistency in the relationship

of rate to AT is not presently known. A possible explanation may be that

the actual AT within the transport tube differs considerably from that

shown by the monitoring thermocouples.

All seeded experiments in which the transport tube was inserted with

the furnace at temperature were unsuccessful. The initial deposit was

polycrystalline in all cases. Use of a small auxiliary heater to provide

a small reverse gradient in the initial heating to temperature of the

transport tube did not aid in preventing formation of a polycrystalline

deposit. The use of the modified transport tube described above yielded

much more satisfactory results. After an initial period of heating in a

reverse gradient, normal gradient conditions are established and growth

proceeds. At a source temperature of 390°C and a AT of 20°C (Ap = 0.6

atm) boules 6 mm diameter and 6-8 mm in length were obtained in eight

hour growth runs. It was found that if the vacuum- insula ted portion of

125
the tube is too long, it is impossible to maintain a temperature gradient

over the entire region available for growth. As a result, growth continues only until the source and growth interfaces are essentially isothermal with the free volume in the tube. The optimum length of the

insulated region in the present furnace appears to be about 2 cm.

The most success in seeded growth was obtained by placing the modified

transport tube in the furnace at room temperature and raising the furnace

temperature to the desired growth temperature over a six hour period.

The final AT was maintained during this heating period. At a source

temperature of 355°C and AT of 15° (Ap = 0.1 atm) over the last six

centimeters of transport distance, transport times of 36 hours were

used to provide specimens of reasonable size for evaluation. The total

transport distance was 12 centimeters. Optical quality of the resulting

boules was much better than previously obtained, and single crystal areas

in excess of 1 cm in length were obtained. See Figure 1.

The use of the horizontal "transparent" furnace for visual monitoring

of transport rates and progress of growth is still undergoing development.

Attempts at seeded growth runs were unsuccessful, resulting in polycrystalline deposits, apparently due to too large a AT between source and

seed for this furnace geometry. Several growth runs were made with the

source at 390°C and a AT as low as 10°C (Ap = 0.2 atm). In future

experiments, it is planned to use a small negative gradient (seed hotter)

during the heating up cycle. In this case, it will be necessary to use

a seed of sufficient size to prevent complete evaporation before the

gradient is reversed.

126
As previously indicated, if mercurous chloride exhibits a phase

transition to a non-centrosymmetric form, it would have the potential

for phase-matching and modulator applications over a broad infra-red

range. For this reason, it seemed desirable to investigate the possibility

of such a transition. An x-ray powder diffraction pattern of mercurous

chloride was obtained at 98 K. The pattern was indexed as monoclinic

using a least squares calculation yielding unti cell dimensions of

a = c = 4.4382 ± 0.0005A*, b = 10.905 ± 0.001A and 6 = 90.40°. The

o room temperature is tetragonal with a = 4.4801 ± 0.0005A and

c = 10.907 ± O.OOlA. In order to determine the transition temperature,

the split of the tetragonal 220 reflection into the monoclinic 202 and 202

reflections were observed at a series of temperatures between 98 K and

298 K. These data are shown in Figure 2 and indicate the transition to

occur at 168 ± 5 K. No discontinuous effects were noted for the change

of the tetragonal 008 reflection to the monoclinic 080 reflection

indicative of no change in the tetragonal c-axis at the transition. A

manuscript describing this work in greater detail is in preparation.

It is hoped that in the future a definitive study can be made of the lowtemperature phase to determine presence or absence of a center of symmetry.

The compounds of major interest in the Tl-Pb-Cl-Br systems were

found to be Tl^PbCl,- and the corresponding bromide, Tl^PbBr,.. Based on

x-ray precession photographs of Tl^PbCl,., the compound appears to be of

tetragonal, non-centrosymmetric symmetry. The bromide was found to be

orthorhombic at room temperature, with a transformation to tetragonal

127
symmetry at about 280°C as determined by elevated temperature x-ray

powder diffraction. Considerable solid solubility was found to exist

between these compounds. A lowering of the transition temperature of

the bromide could be followed with increasing additions of the chloride,

up to 20 mole percent chloride and a transition temperature of about 155°C.

Conclusion and Discussion

Vapor transport experiments for the growth of mercurous chloride

during the course of the current and previous work of this project have

shown that high transport rates can be achieved at temperatures below

400° C. One of the major problems in the growth of mercurous chloride

by an evaporation-condensation in a temperature gradient which is fixed

with respect to the growth tube as been nucleation at several sites near

the colder end, even with seeds present. In the usual vapor Bridgman

technique where the growth tube is slowly moved through the gradient,

the problem is less common, although extremely slow growth rates, of

the order of 2 mm per day, are required for high quality growth. To

provide specimen crystals in shorter growth times in order to be compatible with expected durations of low gravity experiments in space,

much effort has been devoted to utilizing faster transport rates which

means greater supersaturation at the growing interface. On the basis

of results obtained thus far, it appears that growth periods of at

least 24 - 36 hours will be required for good quality growth under

optimum conditions. The use of lower temperature gradients to reduce

transport rates and improve crystal quality will also impose a greater

128
burden on the temperature control of the furnace. The use of an axial

heat sink and vacuum insulation at the growth end of the transport tube

shows considerable promise.

On the basis of the results of this study and the technological

potential of mercurous chloride as an infra-red material, a proposal was

submitted to NASA in response to NASA-AAFE request AO No. OA75-3 suggesting the vapor growth of mercurous chloride by evaporation-condensation

as a part of the Advanced Applications Flight Experiment Program.

Further investigation of the low temperature, monoclinic form of

mercurous chloride should be pursued to determine the presence or absence

of a center of symmetry in view of the further potential application of a

non-centrosymmetric form as a modulator or phase-matching material for

infra-red laser applications. As a further extension of growth by

evaporation-condensation, other potentially useful materials such as the

thallium-lead halides should be examined for adaptability to this technique.

129
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. Kaldis, in "Crystal Growth, Theory and Techniques, Volume I",

C.H.L. Goodman, Ed., 49-191, Plenum Press, Mew York, 1974.

[4] E. Passaglia and R. L. Parker, "NBS Space Processing Research",

NBSIR 74-611, 69-78, November 1974.

[5] R. A. Forman, W. S. Brower and H. S. Parker, "Mercurous Chloride

Polarizer Material", AFML-TR-74-56 , April 1974.

[6] R. A. Forman, W. S. Brower and H. S. Parker, "U.S. Patent 3,902,782,

Mercurous Chloride Prism Polarizers", Assigned to United State of

America, September 2, 1975.

[7] C. Barta, Kristall und Technik 5, 541 (1970).

130
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lo

COLO 1 I* 1 , x o o o o o LO LO00 r>. LO

LO LO LO LO • • V 'P

<o cu

sO -(-> •i- (O

iQJ Q_ 4^ §

ai +-> sCJ) o c

CM •!- CM i

i— <e to 4- cO II cr> re -—

s_ • -M - cu

+-> 4— CU O i-3

CD+-> C ro

•i- s- o cu

ro

C

Q. f=

i/l CU

+-> CUO 4— •- o +->

+->

c

(OOO• c

CNJ3 r— 4-

<_3OJrO CD :r «/i ra c

•I- </) c c o o •<- I- +->

+->

o

•i- CO

to I

C 4-

ta cu

s- S-

+->I CM ZJ

(_>

O -t-> •i— CM fO c s_

•r- -O CU r— C Q. O E o 93 c cm +-> 0 oE CM CD 1 c

i—

o

rOC

C

O •<- o^tra U

SO 4-> £=

—

CU

OX

CU

II

CM

CU

Task 6

Surface Traction and Other Surface Phenomena

Alan L. Dragoo

Inorganic Materials Division

Institute for Materials Research

Summary

Ground-based experiments and apparatus were developed for studies

of convection in drops. A cell consisting of a spherical glass bulb

with a heater below and a heat exchanger above to establish a vertical

temperature gradient was assembled for studies of convection in a

spherical volume of fluid. Preliminary work with this apparatus

indicated some of the photographic procedures required in a similar

study of thermocapillary convection at low-gravity and provided an

estimate of the temperature gradient required for such an experiment

using a drop of light fluorocarbon oil. A neutral buoyancy cell was

assembled for studies of the coalescence of drops and of possible

non-thermal driving-forces for convection.

131
Introduction

Although thermocapillary flow in drops may require low-gravity

so that a drop can be levitated for a long period, ground -based

studies can be used to develop experimental techniques, to provide

a buoyancy analogue upon which thermocapillary flow can be modeled

and to estimate conditions for thermocapillary flow. This report is

devoted largely to two such ground-based experiments which were

initiated in the last year.

The first experiment consists of observations and measurements

of flow in a liquid contained in a spherical bulb which is heated

from below and cooled from above. The objectives of this experiment

are to develop a buoyancy analogue to first-order themocapillary flow

in a drop, to develop experimental procedures and to estimate

conditions for thermocapillary flow. In conjunction with this

experiment, some theoretical work was performed on the problem of

buoyancy convection in a fluid sphere with rigid walls. Preliminary experimental results and observations about experimental

techniques which are relevant to a study of thermocapillarity will

be presented in the report.

The second experiment is intended to study the behavior of

silicone oil drops in a neutral buoyancy cell. This cell will be

used for studies of the coalescence of drops and to investigate

possible non-thermal driving-forces for convection—for example,

oscillating electric and acoustic fields. In this report, the

neutral buoyancy cell will be described.

132
Although buoyancy-driven convection in a bulb of liquid is not

exactly comparable to thermocapillary convection in a drop with a

free surface, there is sufficient similarity between the two phenomena

that buoyancy flow can serve as a model for thermocapillary flow. The

two types of flow are approximately similar in two different liquids

if

Buoyancy force in liquid #1 ^ Surface traction of liquid #2

Viscosity of liquid #1 Viscosity of liquid #2.

For buoyancy convection, this ratio is the Grashof number, N . For

Gr

thermocapillary convection, this ratio is given by the ratio, ^

a/^pr

»

of the Marangoni number to the Prandtl number. These numbers are

expressed in terms of the fluid properties in the table of nomenclature at the end of this chapter of the report. As shown in

Table 1, by choosing an oil with a kinematic viscosity on the order

of 100 cs* for the ground-based, buoyancy study, a condition can be

achieved which approximates the thermocapillary convection in a drop

of Krytox oil**' with a kinematic viscosity of about 35 cs.

Using a model similar to the one developed by Chandrasekar [1]

for the problem of thermal instability in fluid spheres, the hydrodynamic equations were solved in the linearized Boussinesq approximation. The present model differs from that of Chandrasekar in

assuming an applied temperature field with a cos6-dependence in contrast to an applied field with only a radial dependence

—2 2 —6 2

*The unit "cs" stands for centistokes, 1 cs • 10 cm /s = 10 m /s.

**The use of a commercial product name is intended only for the purpose

of identification and does not imply approval or acceptance by National

Bureau of Standards nor does it imply that the product is necessarily

the best material for the purpose described in this report.

133
Table 1. Comparison of Liquids for Ground-Based Study (G-B) of a Liquid

in a Spherical Bulb and for Low-Gravity Study (L-G) of a Levitated or Tethered Drop

Liquid

Krytox 143 AA fluorocarbon oil

Glycerol

DC 200 silicone oil

Olive Oil

Study

L-G

G-B

ii

ii

ii

ii

in" 2 2 / v, 10 cm /s

36

757

100

200

500

1000

91.5

Pr

0.411

N

Gr

0.00659

0.490

0.118

0.0188

0.00470

0.399

These comparisons are made for a drop or a bulb with a radius of 1 cm arid

a constant temperature gradient in vertical direction of l°C/cm.

as in his work. The complete temperature field is specified by

T(r,6) = T

q - T^cose + 0(r,6),

where 0(r,6) is the perturbation associated with the fluid flow. The

present model also was solved only for steady flow since only a comparison with previous work [2] on steady thermocapillary flow was

desired. By making a further assumption about the solution for 0,

single mode solutions can be obtained for the velocity-field problem —

that is,

(1)

w = ru

r = W(r)P

£ (6), (2)

1

134
where is the radial component of the velocity and

W(r) = aiA a

3

f

£

(Xr) (3)

for the mode of order I. The function f^(Xr) is the spherical Bessel

function j^(Xr) if X is real and the modified spherical Bessel function

i^(Xr) if X is imaginary . The set {X} of wavenumbers is defined by the

condition f^

+^ (^*0 = 0« The theta-component of the velocity is

Considering the £ = 1 mode,

u

r = + a

3

f x (Xr)/r] cos 9 (5a)

u

0 = - [ &1 + a

3

X(2f

Q

(Xr) - f

2

(Xr) ) ] sine. (5b)

Eqns. (5a) and (5b) can be compared to the similar results for steady

thermocapillary flow—eqns. (31a) and 31b) in [2]. Moreover, for

small r, eqns. (5a) and (5b) become approximately

»

3 2

u^_ « [a^ + a^X - a^X r /6]cos0 (6a)

3 2

Uq « ~t a

i

+ ^ a3^ ~ a3^ r /3]sin6. (6b)

It is this region of small r which is most amenable for study by photographic techniques.

Photographic techniques for observing circulation in drops rising

or descending in liquid columns were reviewed by Kintner, et_. al. [3].

In these techniques, flow patterns were traced out by fine particles

entrained in the drops. These authors discussed a variety of stilland motion-picture techniques, methods of illumination, methods for

time-marking and a procedure for data analysis.

135
Particles moving near the center of the drop yield photographs

with the least amount of distortion of their image traces due to the

spherical lens effect of the drop. The motions of particles in the

neighborhood of the drop's center are also the easiest to interpret

in terms of the first-order convective mode. For convection driven

by a vertical temperature gradient, the flow in the neighborhood of

the equatorial plane will be predominately upward or downward across

the plane. Such flow is characteristic of all odd-order modes;

even-order modes give flows which are parallel to the equatorial plane.

By the proper design of temperature conditions, the first-order mode

can be made to predominate. This mode is the only mode with flow

through the center of the drop. In practice, the convection may be

composed of several modes. However, if axisymmetric conditions can

be realized, the rate of flow is expected to tend toward an extremum

at the axis of the drop. If even-order modes are negligible, an

extremum also is to be expected at the center of the drop. A significant even-order component is evidenced if the trajectories of

particles across the equatorial plane are not perpendicular to the

plane.

Experimental Apparatus

1. Buoyancy Convection Apparatus

Fig. 1 shows a sample bulb, heater and heat exchanger used

in the convection experiment and indicates the positions of the

heater and of the heat exchanger. The spherical portion of the

3 bulb has a volume of 62.7 cm . The lower stem of the bulb admits

a thermocouple to the lower pole of the bulb. A plastic tube connects this stem to a reservoir of the liquid for the study. The

136
upper stem admits a thermocouple to the upper pole of the bulb.

Three other thermocouples (not shown) are located about the

outside of the bulb: one below, one above and one that is movable.

The heater consists of a coil of Kanthal wire wound in a

second helix inside a short length of alumina tube. The wire is

held in place by a castable alumina cement . The heater is powered

by a filament power supply and, at the maximum output voltage (8 V),

draws about 2.1 A of current.

The inner wall of the heat exchanger is a copper tube which fits

around the upper glass stem of the sample bulb. The outer wall is a

plastic tube. The space between these two tubes is loosely packed

with copper ribbon to improve the heat transfer to the circulating

water.

Fig. 2 shows the apparatus assembled for bright-field illumination:

the light beam from the microscope lamp is passing through the box

containing the sample bulb and is incident upon the camera. Darkfield illumination dan be obtained by moving the camera to a port

on the right-hand side of the box so that the camera views the bulb

perpendicular to the incident beam of light. The port for dark-field

illumination is not shown in the picture.

The box shields the sample bulb from stray light and reduces

the temperature fluctuations near the bulb. The inside of the box

is painted with matte black paint to remove the light scattered

inside the box. The box consists of two compartments: the upper,

and larger, compartment houses the sample bulb and the heat

exchanger; the lower compartment houses the heater. The two

137
compartments are separated by a floor of corrugated paper backed by

a layer of asbestos felt and a thin sheet of molybdenum. A small

hole at the center of the floor allows the lower stem of the sample

bulb to pass into the lower chamber. A coil of copper tubing circulates water around the perimeter of the heater compartment

.

A single-lens reflex camera with a lens focusable down to 0.24 m

(9 1/2 inches) and panatomic-X film are used to record the flow

patterns. To improve detection of the particles in the liquid, the

camera can be equipped with a bellows.

Glycerol was used in the preliminary study because it was readily

available and its properties were sufficiently well known. Aluminum

dust was used as the tracer.

2 . Neutral Buoyancy Cell

Fig. 3 shows the neutral buoyancy cell. It is fabricated

from 9.5 mm thick acrylic plastic sheet and is 25 cm high. One pair

of opposite walls have ports for f eedthroughs . One feedthrough is

shown in the picture. This feedthrough contains a teflon core with

a hole which matches the outer diameter of the syringe needle and a

rubber septum. The teflon core and the rubber septum can be

replaced with cores with holes up to 9.5 mm diameter so that larger tub

can be inserted into the neutral buoyancy cell.

Experimental Results

At a power setting of 9.7 W (6.0 V output; 1.62 A through the

heater) glycerol convects slowly. At this power setting, a temperature

difference of 101. 8°C (21.6°, lower pole; 123.4°, upper pole) was

measured. For glycerol this temperature difference across the bulb

yields a Rayleigh number, Ra^, of 1.03xl0"\

138
Fig. 4 is a photograph of particle trajectories in glycerol with

a temperature difference of 168° across the bulb. The particles were

photographed using the bellows to give a reproduction ratio of 1:1

and with dark-field illumination. The exposure time was 20s.

Conclusions and Discussion

These preliminary results with glycerol show first that buoyancy

convection can be established in this viscous liquid and second that

the observation of the convection is not obscured by the spherical

lens effect. If the Grashof number for the observed buoyancy convection in glycerol is equated to the ratio N^

a

/Np

r for thermocapillary

convection in Krytox 143 AA oil, the temperature gradient of 168° used

for the glycerol implies a temperature difference of 34° for a

drop of Krytox oil with the same radius (2.46 cm).

The best photographs were obtained using dark-field illumination

and the bellows. When bright-field illumination was used, the

illumination could be improved somewhat if the incident light were

oblique to the optic' axis of the bulb and camera rather than along

the axis so that the condensing effect of the bulb was reduced.

In the present work, the exposures lasted on the order of 5 to

30 s so that they could be timed with a stopwatch. However, if motionpictures are taken, such as with a DAC (data acquisition camera) , then

timing marks must be recorded on the film or a clock must be included

in the objective field of the camera.

139
Nomenclature

g

= magnitude of the gravitational acceleration

N

Gr

3 2 2 = R p g 3(T-T )/y , Grashof number

o N

Ma

3 = a VT R /ya, Marangoni number

1 1 K

Pr

= u/pa , Prandtl number

R = radius of bulb or of drop

Ra

2

3 = R pg3(T-T

o )/ya, Rayleigh number

T = local temperature

T

o = average temperature

a = thermal diffusivity

3 = bulk thermal expansion coefficient

y = viscosity

V = y/p , kinematic viscosity

p = density

a = surface tension

= (da/dT)

T

o

Acknowledgments: Thanks are due to Mr. Arnold Driver for his

assistance with photographing the convection in glycerol.

140
References

1. Chandrasekar , S., Hydrodynamlc and Hydro-magnetic Stability ,

Oxford University, Clarendon Press, London (1961).

2. Dragoo, A. L., "Steady Thermocapillary Convection Cells in

Liquid Drops," to be published: cf . NBS Space Processing

Research NBSIR 74-611, E. Passaglia and R. L. Parker, Eds.

(1974)

.

3. Kintner, R. C, Horton, T. J., Grauman, R. E., and Ambeckar, S.,

Can. J. Chem. Eng., 39, 235 (1961).

141



Fig. 2 — Buoyancy convection apparatus arranged for bright-field illumination.

Fig. 3 — Neutral buoyancy cell.



Task 7

Consulting Support

NBS personnel continue to consult extensively with NASA Headquarters

and NASA field offices as required. We have visited Marshall Space Flight

Center, Alabama on numerous occasions and have provided advice on many

aspects of the Space Processing Program, including Shuttle Payloads. We

continue our Space Processing Seminar at NBS.

142

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NBS-114A (REV. 7-73)

U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET

1. PUBLIC ATION OR REPORT NO.

NBSIR 76-980

2, Gov't Accession

No.

3. Recipient's Accession No.

4. TITLE AND SUBTITLE

NBS Space Processing Research

5. Publication Date January 1976

6. Performing Organization Code

7. AUTHOR(S)

E. Passaglia, R. L. Parker

8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Project/Task/Work Unit No.

i 1 1 r\ n t r <x r~ t /Ctr^rxt \I /-\

W-13,475, Mod. 4

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 —11/74-12/75

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

Annual Report

16. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here.)

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

Program covering the period November 1, 1974 to December 31, 1975. The objectives

of the NBS program are to perform ground-based studies (and, where appropriate,

space-based studies) of those aspects of space that cculd 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.

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

)

Convection; crystal growth; crystal perfection; microgravity ; purification; space

processing.

18. AVAILABILITY QT Unlimited

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|X I Order From National Technical Information Service (NTIS)

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