Current concepts in crystal growth from the melt

2 C U R R E N T CONCEPTS IN CRYSTAL GROWTH FROM THE MELT K. A. JACKSON Bell Telephone Laboratories Incorporated, Murray Hill, New Jersey 07971

ABSTRACT This review article presents our current tmderstanding of crystal growth from the melt. Great advances have recently been made in tmderstanding the morphology of ¢rytrml growth. Both theory and experiment show that interface structure, that is, the degree of roughness of the crystal surface on an atomic scale, is the important consideration for determining crystal growth morphology. It is not yet possible, however, to predict the growth rate of ¢ry~als, nor the optimum ¢onditiom for their growth. A new theory of crystal growth, based on a very simple crystal model is presented, which takes proper account Of the structure of the interface, unlike earlier theories of crystal growth. The new theory contains many features which have been observed experimentally. The role of surface free energy is discussed briefly, mainly tocmphasizc that in most cases, the surface frec c~nergy does not determine crystal morphology. Recent advances in our understanding of diffusion controlled growth are presented. The most important of these is the work on interface stability. A solution to the diffusion equations for crystal growth must be stable against fluctuations if it is to be a valid solution. These instabilitiesplay a dominant role in Cellular and dendritic growth.

INTRODUCTION

Crystal growth is still very much an art. The present-day crystal grower relies to a large extent on past practice, modifying methods that have worked in the past. Indeed, the variety of phenomena encountered in crystal growth bewilders the novice, and constantly delights the experienced. Crystal growth has fascinated observers for centuries. However, the science of crystal growth is young, in the sense that many iphenomena of crystal science are incompletely or even incorrectly recorded. The theory has not yet advanced to the state where the best condition's to grow a particular crystal can be 53

$4

K . A . JACZSON

predicted. This state can only be approached slowly, with recourse to experiment and theoretical treatment along the way. In this article, our present position will be presented, with concentration on achieving an over-all view of recent developments. The two major topics to be discussed arc the classification of crystal growth according to the entropy change, based on a simple theory of interface structure, and the great advance in the theoretical treatment of diffusion in crystal growth, particularly the introduction of stability theory. As will become evident below, the growth rate of a crystal depends primarily on the structure of the interface. A model for predicting the roughness of the interface is reviewed, then photographs are presented for comparison with the predictions of the model. Three classes of crystal growth are identified in terms of the entropy of fusion: one containing the metals, one containing most molecular materials, and a third containing the polymers. Existing theoretical models for crystal growth are reviewed. These do not take proper account of the structure of the interface. A new model is presented which attempts to treat the structure of the interface. The model gives remarkable qualitative agreement with experiment. The second major topic in this article, diffusion controlled growth, is particularly important in low entropy of fusion materials, such as the metals. In high entropy of fusion materials, the growth process, rather than diffusion, dominates. The role of instabilities in the development of cellular and dendritic growth morphologies is outlined.

INTERFACE KINETICS One of the most striking facts about crystal growth is the difference in behavior between metals and nonmetals. This difference, which has been known for many years, has not been properly appreciated until recently. The discussion of interface roughness ( ~' 2) has led to a better understanding of growth morphology and of the difference between metals, which have rough interfaces (on an atomic scale), and nonmetals, most of which have atomically smooth interfaces. The important parameter governing this distinction is the entropy of the phase change, which is small for rough surfaces and large for smooth surfaces, as shown by the following simple statistical mechanical model. The roughness of the solid-liquid interface can be determined by considering an atomically plane interface, and calculating whether the interfacial free energy is raised or lowered when extra atoms are added to it. (1'2) It will be assumed that the energy of an atom can be divided equally among the nearest neighbor bonds: if there are v nearest neighbors, each bond has an energy 2L/v associated with it, where L is the energy required to

Current concepts in crystal growth from the melt

55

take the atom from the crystal and put it in the phase into which the crystal is growing. The change in free energy of the surface when atoms are added to it can be split into four terms: A F s = - A E o - A E 1 + T A S o - ]'AS 1 (1) AEo ffi 2 L N ~ o/V is the energy gained by putting NA single atoms on the surface if each has r/o neighbors in the plane below.

i

I

I

]

I

I

I

a = I0.0 1.5

1.0 la.i

,.=, ,.=

0.5 ,.-l, iv.

0

!

a=5.0

a'3.0 a. 2.0 ~=1.5

-0.51

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 OCCUPIED FRACTION OF SURFACE SITES

1.0

FIo. 1. Free energy of an interface vs. occupied fraction of surface sites. AE1 = L~I/v(N2A/N) is the energy gained because some of the extra atoms on the surface will, by chance, be adjacent. There are ~/1 possible sites foe another single atom to be adjacent to any given atom on the surface. ASo -- NAL/TE is the entropy change on adding Na atoms to the solid. AS1 = k N In [ N / ( N - Na)] + k N a In [ ( N - N ~ / N a] is the entropy gained by the random arrangement of the NA atoms in the N available surface sites.

56

K.A.

JACKSON

For T-- TB, the melting-point,and defining x equation (I) becomes:

= NA/N

and

AF, = ctx(1 - x ) + x In x+(1 - x ) In (1 - x ) NkT8

= rhlv,

(2)

where

= ~

L

~

(3)

Equation (2) is plotted in Fig. 1 for various values of u. The minima in the curves are given by x = e -* and 1 - x -- e -~ for large values ofu. If 0c is less than 2, the surface will have minimum free energy for x -- ½. If a is greater than 2 the surface has a minimum free energy with a small fraction of the surface sites filled, and a small number of holes. For large u, the fraction of extra atoms is e -~, and the fraction of holes is e -=. depends on the structure of the solid. It is always less than unity, and is closest to unity for the most closely packed planes of the structure. It decreases progressively for less closely packed faces of the crystal. L/kT 8 is about unity for all metals in contact with their melts. It is somewhat higher, 2 or 3, for a few near-metals and semiconductors in contact with their melts. It is larger still for most molecular materials growing from their melts. L/kTE is the order of 10 for many materials (including metals) when they are in contact with their vapors. The analysis presented above is limited to one-component systems. Recent work has considered surface roughness for alloys(3'4) and for crystals growing from solution. (5)

GROWTH OF MATERIALS HAVING VARIOUS ENTROPIES OF FUSION In this section, photographs of crystals having various entropies of fusion are presented. These will illustrate the various types of growth that occur. Recently, several low entropy of fusion compounds that freeze as metals do (6) have been investigated which open new possibilities for the study of phase transformations. In this section, the growth of these and of other materials which have higher entropies of fusion is examined. Although only growth from the melt is illustrated, the results are more general, and the growth characteristics are similar for growth from solution or vapor, provided the degree of surface roughness (related to the entropy change) is similar. As we will see below, there are in fact three distinct classes of crystal growth, not just the two indicated by the original rough or smooth interface classification. The boundaries for the three classes are not distinct, of course,

FIG. 2. Carbon tetrabromide (LIIkTE ffi 0.8) growing from the melt between glass cover slide approximately 25 ~ apart. The photographs were taken using a temperature gradient microscope stage. ~64~ 150x. (a) Planar interface growth.

FIo. 2(b) Cellular growth with a small amount of impurity.

FIG. 2(c) Dendritic growth with several per cent impurity.

F1o. 3. t-Butyl alcohol

(LI/RTE = 2.6) growing from the melt as in Fig. 4. 150x (a) Faceted growth.

FIo. 3(Io)Pseudo-dendritic growth.

Flo. 4. Benzil

(Ll/kTz ffi 6) growing from the melt as in Fig. 4. 150x.

Fxo. 5. Beazil-azobenzene eutectic growing from the melt as in Fig. 4. 150x

Fie. 6. Salol (Ll/kT~ ----7) growing from the melt as in Fig. 4. Notice the "autonucleation" of crystals of a different orientation, which occurs only at largo undercoolings in saloi. 150 x.

FIc. 7. Tristearin

(LI/kTE -- 63) growing from the melt as in Fig. 4. 150x

Current conceptsin crystalgrowth from the melt

57

but in fact it is usual to find workers concentrating their efforts in one of these classes. The lowest entropy group contains all the metals and certain compounds; the intermediate entropy group contains most organic and inorganic crystals, growing from the melt or solution, as well as most vapor growth; the high entropy group includes the polymers. The lowest entropy of fusion range is typified by CBr4(Ls[kT ~ -- 0.8) as shown in Fig. 2. The growth is isotropic, or almost so. Cells and dendrites occur with the addition of impurity. These growth forms are discussed in more detail below, t-Butyl alcohol (LflkTe = 2.6) is just in the next range, Fig. 3. This material grows with facets when pure, and can grow in a faceted pseudo-dendritic manner when enough impurity is added. The tips of the pseudo-dendrites can be rounded. For crystals with entropies of fusion higher than 4 or so, the anisotropy of growth is sufficiently large that facets, at least in some growth directions, are always observed. Pseudo-dendrites are not observed even with a large amount of second component. Figure 4 shows bem:il (LflkT~ = 6) growing, and Fig. 5 shows a eutectic mixture of azobenzene and benzil. Both primary phases are growing as faceted crystals. The photograph of salol (Ls/kT E = 7) in Fig. 6 illustrates "autonueleation": the nucleation of a crystal of a different orientation on a crystal of same material. Tristearin is not a polymer, but is in the high entropy of fusion range (Lr/kTE = 63). It grows spherulitically, as shown in Fig. 7. The undercooling at the interface is sufficiently large so that nucleation can occur ahead of the interface. Spherulitic growth is characterized by frequent autonueleation and often by changes in crystallographic orientation during growth.(7) The structures illustrated are generally typical of the types of growth found for materials with the various entropies of fusion. Increase or decrease of the growth rate by a factor of 10 usually makes little difference to the growth morphology. There are some differences due to the anisotropy of the crystal structure: for example, long-chain molecules exhibit strong anisotropy of growth, being nonfaceted in one direction and strongly faeeted in another, and so belong in two groups because of their anisotropy. Nonfaceted growth and growth instabilities leading to dendritic growth occur in the lower entropy of fusion materials. Spherulitic growth occurs in the highest group. Most compounds are between these two extremes: some can be pushed to grow as pseudo-dendrites, some can be pushed to grow spherulitically. Usually they grow as faeeted crystals over a wide range of conditions. It is evident from these photographs that the anisotropy of growth rate is greater for materials with large entropies of fusion. The degree of anisotropy does not depend strongly on the growth rate or the undercooling, but is similar for a wide range of growth conditions. Growth-rate measurements have been made on several systems. A review of these data has recently been presented, tS~ Most of these data are for

58

K.A. JACKSON

systems in the second category, that is, for materials in which the kinetic undercooling is sufficiently large that it can be measured readily. The temperature dependence of the. growth rate indicates growth controlled either by screw dislocation or surface nucleation. The undercooling for a given growth rate increases with the viscosity of the liquid. Satisfactory measurements have not yet been made on low entropy of fusion materials, although strenuous efforts have been made. (9,1°) The difficulty is basically that the departure of the interface temperature from the melting-point during growth is too small. Measurements of dendrite growth rates have been made for several materials. In these experiments, the temperature of the bath into which the dendrites grow is recorded. This temperature is significantly different from the interface temperature. The dendrite measurements will be discussed in more detail below. The entropy of the transformation defines a class to which the growth process under consideration belongs. In the lowest entropy of fusion group, planar interfaces, cells or dendrites occur, depending on the impurity level and growth conditions. The occurrence of these phenomena depends on small kinetic undercooling at the interface. These phenomena have been analyzed in some detail, and these analyses are discussed below. In the intermediate entropy range, faceted growth occurs and the growth morphology depends only in a minor way on impurity content. For these growth processes, the kinetic undercooling is sufficiently large so that it dominates the diffusion processes. In the highest entropy of fusion group, the anisotropy of growth and the undercooling required for growth are still larger. Spike growth or needle growth due to the growth anisotropy are observed. Autonucleation and changes in orientation of crystals during growth which lead to spherulitic growth are facilitated by the large undercooling at the interface. INTERFACE KINETICS The analysis of surface roughness indicates the condition of the crystal surface. As illustrated by the photographs, the roughness of the interface plays an important role in determining the crystal growth rate. Before outlining a new theory of crystal growth which attempts to treat surface roughness properly, a review of previous theories which have been proposed for crystal growth will be presented. These theories are based on models of the crystal surface and make implicit assumptions about its roughness. Three theories will be considered in detail. The first is normal growth, where it is assumed that growth can take place at any surface site. This is an idealization of rough interface growth. The second is screw dislocation growth, where it is assumed that the only growth sites are those provided by spiral steps due to dislocations. This model assumes a highly idealized surface structure which might be attained in high entropy of fusion materials

Current concepts in crystal growth from the melt

59

(where the surface is smooth) which contain dislocations. The third growth model which will be reviewed is two-dimensional surface nucleation. The treatment here involves the use of equations which were developed for nucleation of droplets in a vapor. The surface in this model is smooth except for isolated atoms and clusters of atoms on the surface. This model applies only t ° high entropy of fusion materials which have no dislocation growth sites. The model does not apply to rough surfaces. In addition, the Cahn theory(1 :,12) of interface motion will be discussed briefly.

NORMAL GROWTH

The theory of normal growth was developed by Wilson(13) and by Frenkel.(l 4) The treatment given here is from Jackson and Chalmers. (15) Assuming that each atom at the interface moves independently, the individual atomic processes of melting and freezing are simple activated processes. The net rate of growth is given by:

= RA--RL = R~

exp(--QF[kT)--R[ exp(--QM/kT)

(4)

where R~" and Rr are the rates at which atoms join and leave the crystal The atom movements at the interface are similar to those elsewhere in the liquid, so that Qr should be approximately the same as the activation energy for liquid diffusion. Also we have QM-

,(5)

Q~ = L

Where L is the latent heat of fusion, For equilibrium

~=~ or

R,~/Rt = exp

(03

So that equation (4) can be written = R~ exp --

1 - exp

fr)l

(7)

Writing

AT= TE-T we have

_o / QALAr LAr ,~ = ~ . exp ~ - ~ k~k'F~T for krEr "~ 1

(s)

60

K.A. JACmSON

For a numerical estimate we can use

the interatomic distance times the atomic vibration frequency. We have, then, the approximate equation: =av~exp

-

exp

-L

The net rate of growth depends on the jump distance, the vibrational frequency, the mobility of the atoms and on the thermodynamic driving force for the transformation. Equation (9) predicts quite large growth rates for fairly small values of AT, about 100 era/see for I degree undercooling in copper and the same magnitude for many metals. Although there are no reliable experimental data with which to compare this number, such a large number indicates that measurement of AT should be difficult, as is indeed the case. ¢9'1o) The interface of a metal and other low entropy of fusion materials is very close to the melting-point for all conditions of growth,

GROWTH ON SCREW DISLOCATIONS The screw dislocation mechanism was first proposed by F. C. Frank (re) to provide a continuous repeatable step for crystal growth. Hillig and Turnbull(1 ~) applied this mechanism to growth from liquids. If the growth step advances everywhere at a constant rate, the growth step will form an Archimedes spiral. The minimum curvature of the spiral at the core is the radius of the critical nucleus on the interface. In fact, a step with the critical curvature is one which has zero growth velocity, and the growth rate does depend on curvature, becoming faster as the radius of curvature increases. An Archimedes spiral will thus give an upper limit to the number of growth steps. The free energy of a disc on the surface of a crystal is given by A F .--- - o q n A F o +uza,~/n

(10)

where AFo is the volume free energy change per atom associated with the transformation, o', is the surface free energy per atom, a is the interatomic spacing and n is the number of atoms in the disc. ul and u2 are shape factors which relate the number of atoms in the disc to its area and peripheral length. The critical size cluster is given by a,T:2 ~/n* = LAT2al

(11)

Current concepts in crystal growth from the melt The distance between steps on the surface is the order of steps advance laterally at a rate v,. given by eq. (9).

61

av'n*, and the

The growth velocity normal to the interface is =

LAT2~ILAT 1' Qf~ (-L) = avk--~rT ~ exp ~,-~-~) exp

(12)

For most materials, nucleation experiments (1 s-21) show that ½L~. so that

L

v ~ av~

( Q-k~ (-L'~~ (AT'~2 exp \kTE,]o~2 \ T~]

exp -

(13)

This equation predicts a growth rate that is proportional to A T 2. The undercooling for a given growth rate is greaterthan predicted by eq. (9).There are several systems where the growth rate is proportional to the square of the undercooling,¢8) however, the growth rate is differentfrom that predicted by eq. (13).

SURFACE NUCLEATION In this model, the growth occurs when the discs of the next atomic layer reach critical size. Each new atomic layer is formed by nucleation. The critical size disc is given by eq. (11). The number of these discs on the surface is given by: N,. -- N exp

(

4~2t~T

/

(14)

The average distance between these discs is approximately l/tiN.. so that the rate of advance of the intxrface is: v

-- a v L 4 N . .

• 2 LAT

4a~ LkTAT]

(I5)

The exponential term with AT in the denominator of the exponent is the dominant part of this equation. For all AT, very small compared to TB,

62

K.A. JACKSON

the growth rate is extremely slow. At a critical AT* (when the exponent is about I) the growth rate increases very rapidly. There are systems which exhibit growth rates with this functional dependence on AT (s) However, this equation does not correctly predict either the pre-exponential factor nor the coefficient in the exponent. The analyses presented above do not take the structure of the interface into account properly. The normal growth theory assumes that growth can occur at any surface site. This may or may not be precisely true for a rough surface, although it should be approximately correct. The screw dislocation theory and the surface nucleation theory consider separately the origin and motion of steps. They may give a reasonable model for smooth surfaces, but are unrealistic for rough surfaces or for surfaces which are between rough and smooth, which are typical of the systems for which measurements have been made. It is encouraging that the predicted functional forms of the growth rate have been observed. It is hoped that a general theory of crystal growth, taking surface structure into account, will have the screw dislocation theory and surface nucleation theory as appropriate limiting cases, and that the experimental systems which have been studied are near, but not at, those limits. The theory of interface motion developed by Calm et a[.(I i, 12) was an attempt to develop a general theory of crystal growth. This theory has recently been discredited, (s) although it was a brilliant attempt in the right direction. Only a few brief comments will be made about it here, and readers are referred to the original papers for details. The theory assumed that interfaces could have various degrees of roughness, and that the growth rate depended on this roughness. The basic difficulty with the theory is that the structure of the interface is averaged out, and replaced with a diffuse region, the motion of which is considered. The growth rate of a crystal, however, depends on many discrete atomic processes. It is the average rate of these processes which gives the growth rate of the crystal. These processes depend on the detailed atomic configuration of the interface, which are averaged out of this theory too early. In addition to this and other theoretical difficulties with the theory, none of the observed crystal growth data are of the form that can be fitted by the theory.(s) There is thus not available at present a theory of crystal growth which adequately describes experimental results. Furthermore, none of these theories takes surface roughness into account properly. It is evident, however, that there is a good correlation between surface roughness as predicted by the entropy of fusion and the morphology of crystal growth. In the next section an analysis of the growth of a very simple crystal will be presented. The model is sufficiently simple that it can be analyzed in detail, and proper account taken of surface roughness. The simplicity of the model means that it has inherent limitations which will become evident.

Current concepts in crystal growth from the melt

63

THE GROWTH OF A TWO-DIMENSIONAL SQUARE CRYSTAL The growth of a two-dimensional square crystal at its edge has been considered in some detail, <22) but an expression for the growth rate was not obtained in closed form. The growth of such a crystal with an edge three atom distances in length has been solved in closed form; <22) this analysis is presented below. Two assumptions are made which permit the model to be analyzed. The first is that steps on the surface may be of single height only. This assumption is not too severe, since the number of double steps will usually be small corn-

__TL -q_y-

-71

!

IL_ 2

I

_J

1 3

FIO. 8. Possible configurations for the edge of a two-dimensional square crysta: three atom distances in length.

pared to the number of single steps, and they will therefore not contribute importantly to the growth rate. The second assumption, that the left end of the crystal is joined (mathematically) to the right end, is necessary to obtain closure of the distribution. That is, one of the nearest neighbors of each end site is the other end site. With these assumptions, only three distinct configurations of the surface are possible. These are illustrated in Fig. 8. Since the left end is connected to the right end, the three configurations labelled 1 are equivalent, as are those labelled 2. The energy scheme for the crystal surface is shown in Fig. 9. The difference in energy between the liquid and the repeatable step (a two-bonded site) is the latent heat of fusion. QF is the activation energy associated with the transition of a molecule from the liquid to the crystal. QF is approximately the same as the activation energy for viscosity or diffusion in the liquid. The energy of one-bonded surface sites is L/2 above the repeatable step, the energy of the three-bonded site is L/2 below it.

64

K . A . JACXSON

The number of atoms per siteper second arrivingon the surfaceisgiven by

[cf.eq. (4)] R,l = R,~ exp (-Qe/klO

(16)

The rate a t which atoms leave the surface depends on n, the number of nearest neighbors of the atom on the surface

Q~

1 -BONDED

L

2

2-BONDED 3-BONDED

LiQUiD

SOLID

FIo. 9. Energy scheme for liquid and various solid sites.

The rate of arrival and leaving of atoms at repeatable step sites must be zero at the melting point so that R~ exp (-QF/k.TB) =

R~ exp [-(QF+ L)/kT~]

(18)

or

Rt/-R~ =

exp (z/kr~)

Combining eqs. (17) and (18) we obtain

(-QF/kT)P,

(19)

P, -- exp (L/kTE-nL/2kT)

(20)

= R,~ exp where

Taking into account that molecules may only arrive at or leave from sites such that no double steps are created, the average growth rate of the threesite crystal is given by

R= R°aexP(--~T)I3N3+2NI+N2-3N3P3-NIPI-2N2P2 + hr2 + N3)

(21)

Where N1, N2 and Ns are the fractional numbers of type 1, 2 and 3 type configurations respectively. These fractions may be considered to be either a

Current concepts in crystalgrowth from the melt

65

time average for one crystal or the instantaneous average for a large number of separate three-site crystals. N1, N2 and N3 are determined by the condition that these numbers remain constant in time. This condition is imposed by considering all the possible transitions between the three types of site and the probability of these transitions occurring. For example: the type 3 site can gain a molecule converting it to a type I site; or a type 3 sitecan lose a molecule, becoming a type 2 site. The probab~lity of a site gaining a molecule may be taken as I. The probability that a site loses a molecule depends on n, its number of nearest neighbors, and is P,, as given by eq. (20). Consideration of these transitions results in the equations: N~PI + N 2 - 3 ( I + Ps)N s = 0

- (2 + PI)N~ + 2N2P2 + 3N3 = 0

(22)

2N1 - (1 + 2P2)N2 + 3N3Ps -- 0 or

N1(2 + P1 + 2PxP2) - 3N3(1 + 2P2 + 2PzP3) N2(2+P1 +2P1P2) ffi 3N3(2+2P3 +P1P3)

(23)

The implication of these equations can be seen by noting that for the crystal under consideration [see eq. (3)] L

2kT~ For • large, eq. (23) reduces to [using eq. (20)] N 1 ~ 3Nse-"

N2 ,~,3N3e-" and for • ~ ~, I,

(24) N2 ~ 3N3,

NI ~ 3N3

A one-configuration has an extra atom, and a two-configuration has a hole. For large values of u, the number of holes and extra atoms is small, and is the same as that given by the earlier analysis [eq. (2)].(The number of unoccupied sites is 3N3.) The number of holes and extra atoms becomes large for small values of ~, indicating a rough surface, also in agreement with the earlier analysis. The growth rate of the crystal is obtained by combining eqs. (23) and (21), giving: -

11+Pl+6Pz+6~+6P2V3+3p,p3

J (25)

66

K.A.J.cxsoN

Using eq. (20), defining AT -exp

kTBT/

TE--T, and expanding

1 kTRT

R = R~ exp(- Q~T)

18(LAT/kTET)

(26)

11 + Pl + 6P2 + 6P3 + 2P~P2 + 6P2P3 + 3P1P3 This equation predicts a growth rate linear in AT. For e ~ ~ 1, eq. (26) becomes R -- R~ exp -

6k-~zTe

For e" ~ 1, eq. (26) becomes

(27)

R ~ R~ exp(-.Q--_r~ 18 LAT \kTJ 35 kTET Although these results have been obtained for a specific case, they can be generalized as follows: 22) For small values of u, the growth rate is g/ven approximately by the classical Wilson-Frenkel ~13,14) law. For large values of u, the growth rate is reduced by a factor of about 12e-'. The u-factor depends on direction as well as the entropy of fusion [eq. (3)].For low entropy of fusion mater/als, th/s analysis predicts almost isotropic growth that is relatively independent of u and therefore of orientation. For large entropy of fusion, the growth rate will be strongly an/sotropic, very slow on the closest packed faces and approaching the classical value for the high index faces. Moreover, the degree of anisotropy (ratio of growth rates in various directions) does not depend strongly on AT. In all cases the growth rate is proportional to a factor exp (-QF/kT) which is a measure of the mobility in the liqu/d. For low :,-factor faces the density of step sites is large, so defects will be unimportant. For high u-factor faces, the density of steps is small, so that extra steps, introduced by defects, will be important. These conclusions from the model are all in agreement with the observations on crystal growth reported above. The model has one outstanding defect, relating to its geometry. Surface nucleation is not possible on a two-dimensional crystal. The nuclei would be one-dimensional, and th/s does not present a nucleation barr/er. Surface nucleation thus cannot be incorporated into th/s model. O n a three-dimensional crystal,surface nucleationwill be important for highu-factor faces leading to even slower growth on these faces than predicted here. For a three-dimensional crystal, results rather similar to those obtained here are expected for a low u-factor face. For a high u-factor face, the dependence of growth rate on AT, on the entropy of fusion and on crystal geometry will be more complex than in the present model. Quantitative agreement with

Current concepts in crystal growth from the melt

67

experiment is therefore not anticipated with the present model. It is anticipated that a three-dimensional model of a crystal based on the same assumptions used here will make satisfactory quantitative predictions, and will retain the correct qualitative aspects of the present model.

SURFACE ENERGY The surface energy of a crystal is not sufficient to drive transport processes such as diffusion over large distances. As F. C. Frank (2s) has pointed out, the surface energy of a crystal is important only when the dimensions of the crystal are small, in the micron range. In the usual case, inhomogeneities of temperature and composition in the system will be much larger than the effects of surface energy. The "equilibrium shape" is never observed on macroscopic crystals (anything larger than a fraction of a millimeter): their shapes are determined by diffusion processes, which are by nature isotropic in liquids; or by the kinetics of the growth process,, which are usually anisotropic. Surface energy is important when the crystals are microscopic or submicroscopic in size, for example, during nucleation, during the growth of eutectics, during dendritic growth, or in determining the structure of the interface on a small scale, as in cellular growth. One exception to this generalization is the case in which the surface energy can influence mass motion by fluid flow. This occurs when the last layer of liquid freezes at the free surface of a sample, whether the rest of the liquid was decanted or whether the liquid is freezing without decanting. Fluid flow occurs under forces very small compared with those normally present in crystal growth: for e~ample, compare the effect of a hydrostatic head on the melting-point of a material, as against the fluid flow which the same hydrostatic head will produce. During the freezing of a liquid layer, the liquid does, in some cases, flow on the surface of the solid to produce a faceted external surface on the solid as shown in Fig. 10. The facets which develop in this way have been called "terraces", and have been studied by several investigators. ~24"2e) They have mistakenly been considered to play a role in the growth of the crystal. They are in fact irrelevant to the growth process, as has been shown by several investigators. ¢27-29) The layer of liquid which exists on a decanted solid makes it ditficult to determine the detailed shape of the solid-liquidinterface prior to decanting. Not only can false detail be added, but also the shapes of ceils and dendrites will be obscured as the liquid layer freezes. This problem does not arise, of course, during growth from solution or growth from the vapor, where the crystal can usually be separated more readily from the medium in which it is growing. The general conclusion then is that surface energy can influence the macroscopic shape of a crystal only if it" can do so by fluid flow. The

68

K.A. J A c x ~

morphology of crystals will usually not depend directly on the surface energy. Adsorption strongly influences crystal growth. Adsorption can be considered to occur because of the change in surface free energy that it produces. However, the important effect of adsorption is due not to the change in surface free energy per se, but to the effect of the adsorbed atoms on the growth rate. That is, the adsorbed atoms change the rates at which molecules can join or leave the crystal. In lamellar eutectic growth the surface energy is very important. (a°-a2) A large surface area is built into the structure in the form of boundaries between alternate layers of the two phases which are usually a micron or so thick. On this scale the surface energy is large enough to drive transport by diffusion. The combined effects of surface energy and diffusion account for the principal phenomena of lamellar eutectic growth. In Fig. 11 the calculated shape of a lamellar interface is compared with the observed shape in a eutectic where both primary phases have low entropies of fusion.(an) Regular lame[lax and rod eutectic structures are only observed in the cases where both primary phases have low entropies of fusion,(a4) otherwise the anisotropy of growth becomes dominant.

DIFFUSION Diffusion processes are generally well understood in principle. There are, however, many details to be worked out. The equations become complex when the boundary conditions are to be applied on a surface whose shape, temperature and composition depend on the growth process. Often the calculations can be simplified by considering cases in which either heat or matter diffusion dominates or where some of the complicating factors in applying the boundary conditions can be ignored. Heat Flow The latent heat of fusion for most materials is about ~ to ½ CT B, where C is the specific heat and Te is the melting temperature in degrees absolute. This is a large-amount of heat, which must be carried away from the region of the interface before solidification can continue. The thermal diffusivity of most metals is high: 0.1 to 1 cm2/sec, so that the gradients set up by the heat flow are usually macroscopic. For poor thermal conductors the thermal gradients can be microscopic. Chemical Diffusion In all systerrls other than "pure" materials, solutes are present. These, in general, have a different solubility in the solid and the liquid. The measure

FIG. 10. Terraces on the surface of a lead crystal. 1 5 0 x .

FIG. 11. Comparison of calculated and observed eutectic interface shapes. Carbon tetrabromide-hexachioroethane eutectic growing as in Fig. 4 (phase contrast). 2000 x .

Current concepts in crystalgrowthfrom the melt

69

of this difference in solubility is the distribution coefficient, k. The rejected solute must also be removed from the interface before solidification can continue. The diffusion coefficient D, for liquid diffusion, is usually about 5 × 10- s cm2/sec. This is so slow that the solute usually forms a boundary layer whose width is about the diffusion distance, D/v, where v is the growth rate, as shown in Fig. 12. A typical value of D/v is (5 x 10-5)/0.1 cm or about a few microns. This boundary layer can show up during crystal growth as "banding", the formation of impurity layers in the solid due to growth rate

SOLID

[ LIQUID

V u

DISTANCE

>

FIG. 12. The diffusionboundarylayerahead of a plane interfacehas a widththe order of the diffusiondistance,D/v. fluctuations.(S 5) The diffusion boundary layer at the crystal surface becomes partially trapped during rapid growth, and has time to diffuse away during slow growth. The effect of convection is to cause growth rate fluctuations and thereby to produce bands, parallel to the growth front which are depleted or enriched in solutes.(s6,sT)

Low Entropy of Fusion Growth In low entropy of fusion growth, the rate controlling factor is usually the rate of removal of heat from the system. The material solidifies, as rapidly as the latent heat can be evacuated. The kinetics conform to the imposed growth rate. The interface becomes established at the required temperature for the imposed growth rate, as shown in Fig. 2(a).

Small Amounts of Impurities The growth rate is still imposed by the heat flow. The rejected constituents pile up in a boundary layer ahead of the interface. The temperature at the

70

IL A. JAc~oN

/nterface becomes that necessary to freeze the boundary layer liquid at the /reposed growth rate. The distribution of impurity in the solid after solidification depends primarily on convection. The main effect is due to the enrichment of the liquid far from the interface by convection. This is described by an effective distribution coefficient, k, = Cs/C~ which is usually different from the distribution coefficient, k = Cs/CL obtained from the phase diagram, k, depends on k, the growth rate, the crucible shape, stirring, etc. Diffusion usually causes only a narrow depleted zone at one end of the crystal and a narrow zone of high solute content at the other. These are due to the build-up of, and ultimate freezing of, the boundary layer. During most of the solidification, the boundary layer moves along with the interface. In some cases, particularly for small k, the build-up of the boundary layer takes most of the length of the crystal. (3 5)

I

A

I

co

_

t

Y

i

q/k COMPOSITION OF B

FIo. 13. Part of a phase diagram. The temperature differeace between the solidus and liquidus line for composition Co is mCo(1-k)/k.

CELLULAR GROWTH

The addition of more solute, or a higher growth rate can result in the boundary layer depressing the interface temperature so that liquid not adjacent to the interface is below its freezing temperature. This is known as constitutional supercooling. The condition for constitutional supercooling has been developed formally, (aS) but it can be derived from dimensional arguments. At composition Co (Fig. 13) the difference in temperature between the solidus and liquidus line is mCo(1 - k ) / k , where m is the slope of

C t ~ t concepts in crystalgrowthfrom the melt

71

the fiquidus and k is the distribution coefficient. This temperature, divided by the diffusion distance D/v, gives a temperature gradient. If this gradient is greater than the imposed temperature gradient G,

a< D \ k / V then there is constitutional supercooling. Under these conditions a plane interface becomes unstable, so that if part of the interface advances slightly, it finds liquid that can freeze more readily. This results in the cellular growth pattern shown in Fig. 2(b). These normally form an hexagonal pattern on the interface as shown in Fig. 14. The boundary layer over most of the interface is reduced by lateral diffusion. The onset of instability associated with constitutional supercooling has been verified by several experimenters.(s*-*°) The cell boundaries have a much higher concentration of solute than the rest of the crystal. The cells are typically 50/z across. Mullins and Sekerka¢.1'4~) have analyzed the conditions for the stability of solutions to the diffusion equations for crystal growth and have stimulated considerable work on interface stability. The original papers have been followed by several extensions¢43-s°) of this work. The basis of this work is that a solution to the diffusion equations which satisfies all the boundary conditions may be unstable under conditions where the liquid (or gas) phase is supercooled. To examine the stability, a solution to the diffusion equation is first obtained, assuming some interface geometry (plane, spherical, etc.). A small periodic perturbation is introduced (mathematically) into the interface shape. The eft'vet of this perturbation on the diffusion is then calculated to determine whether the perturbation will grow or shrink as growth proceeds. Recent work has been to relax successively the assumptions of the simplest theory and to consider different geometries. Later stages in the development of the perturbation are also being considered34s) The effect of kinetic undereooling has recently been considered by several authors. ¢4~s°) The effects of surface tension anisotropy have also been considered35°) We will present the analysis of Mullins and Sekerka<'2) for a plane interface, adding the effvet of interface kinetics. Consider a coordinate system moving with the interface at a uniform velocity, v. The mean interface is the plane Z = 0, and Z is positive in the liquid. It is assumed that there is no convection and that steady state growth conditions consist of a temperature gradient G in the liquid and G 1 in the solid. There is a diffusion boundary layer in the liquid with a composition gradient Gc at the interface. Consider a sinusoidal ripple on the interface of infinitesimal amplitude: Z ffi 6(0 sin w x

(29)

72

K.A. JACKSON

where ,t = 2~t/co is the wavelength of the ripple. The steady state diffusion equations in the moving coordinate system in the liquid are: v 0C V2C+~-~ = 0 V2T+

(30a)

or

OZ - 0

(30b)

and in the solid v aT ~

V2Ti4 . . . . Ds dZ

0

(30C)

where v is the growth rate, and D, Dr. and Ds are the solute, liquid thermal and solid thermal diffusivities, respectively. The temperature of the interface is given by

T® -- Te-mC® +(T~o/L)6co 2 sin cox-fly

(31)

Te is the melting-point of pure material with a plane interface, mC® is the depression of the local equilibrium temperature due to solute, m being the slope of the liquidus, cr is the specific surface free energy, L is the heat of fusion. The third term takes into account surface tension effects due to the curvature of the surface. The last term is due to growth kinetics (assumed linear), fl being the proportionality constant between the growth rate and actual interface undercooling [cf. eq. (9)]. Both o and fl are assumed to be isotropic. The conservation of heat and solute at the interface require:

1FK/OTI'

[OT'-]

D

(OC)

Ks and Ez. are the solid and liquid thermal conductivities, and k is the distribution coefficient. In addition, the equations must reduce to the steady state diffusion field for the unperturbed system far from the interface. We look for a serfconsistent solution of the form: To = To + a6 sin cox

(33a)

C® = Co+b5 sin cox

(33b)

The solutions which satisfy eq. 00), give the unperturbed fields far from the interface, and reduce to eq. (33) at the interface are: (34a)

= ,.o+OO,r, ~ k

.,. \/)L/.J

o_..+

(34b)

Current conceptsin crystalgrowthfrom the malt

73

and in the solid: T~(x, Z ) =

T0+-~[1-exp(-~s)+/J(a-G~)sin

cox e -'~"z

(34c)

Where: v

F / v \2

°"

-1~

+°U .

co,~ = ~ - Z + collliI

=

V

+co Fi t/

V

~l i

,

(35)

2"] 'i

J

The gradients at the interface are given by: a sin cox+ G~

=

(36a)

O

0(~ = - m ( a - G ) ~ sin c o x + G \~,z,/ @

(oT" ~ ' ~ /® ffi + c o ( a - G i ) ~

(36b)

(36c)

sin cox + G i

The last two equations have been simplified using

COthDL ~ CO~Ds < 1

Substituting eqs. (34) into eq. (31) and (32), using eq. (36) and equating nke Fourier components, we obtain: T ~-tr 6

~

fl I

co* - v / D

( f f 1 - ~#')/2-- tomGc[(co* - v(1 - k ) / D ) + flcov

Where ~ ffi (KL/K)G, @1 = (Ks/FOG1and R = (Ks + KD/2. The denominator is always positive. The perturbation shrinks if the numerator is negative and grows if it is positive. The first two terms in the bracket in the numerator are negative (provided the temperature increases into the liquid). The last term is positive since Gc is negative (i.e. the composition decreases into the liquid away from the interface). The term T~.~co2/L is usually less than the gradient

74

K . A . JAcr.mN

term (if' +if)/2. The stability of the interface is therefore usually determined by the condition

> -rnGc

2 for small k, and instability occurs for

- T - - < -m o

(38)

The unperturbed composition gradient at the interface is given by

G,

1 -k

v

Co B .

:

So that the instability occurs for

ffl +ff 2

mCo/1-k~ <-~--~----~-)v

(39)

as Compared with eq. (28) <

mCo/1-k'

(28)

for the onset of constitutional undercooling (for G -- G l, (fCl +~)/2 ffi O). The conditions for the onset of instability are thus similar to the condition for constitutional supercooling over a wide range of growth conditions. The difference between the two conditions is small compared to the experimental error involved in detecting the onset of instability, or to the uncertainty in some of the material parameters involved, such as the liquid diffusion coet~cient, D. The experimental work which has been done to test the constitutional supercooling equation °s-4°) may also be regarded as providing direct verification of the instability theory. The surface energy term [the first term in the brackets in eg. (37)] tends to stabilize the interface. When the temperature gradients are small, this term will represent the dominant stabilizing influence, and the stability condition will be different from the constitutional supercooling condition. The interface should remain stable even though there is constitutional supercooling. This effect has not yet been detected experimentally. The interface kinetics term does not appear explicitly in the numerator of the stability equation. This means that interface kinetics do not directly affect the stability condition, but only the rate of growth of an instability. If a crystal is grown under conditions where the growth rate and temperature gradient are controlled (Bridgman or Czochralski methods, for example), the interface kinetics determine the temperature of the interface, but not the onset of instability. On the other hand, for growth under

FIG. 14. Cellular growth on a lead crystal interface as revealed by decanting. 150x.

Fxo. 15. Crystal of quartz grown hydrothermally. Notice the instabilities which are present on the fast-growing faces, but not on the slow-growing faces.

Current concepts in crystal growth from the melt

75

conditions where the temperature or supersaturation are specified, and the growth rate is not specified, as in flux growth, a material with a small kinetic coefficient will grow more slowly than one with a large kinetic coefficient. The composition gradients will therefore also be smaller and the interface will tend towards stability. For the case where the temperature or supersaturation is specified, the crystal will be stable if the interface temperature is close to the bath temperature (i.e. if the kinetic term is large) and unstable if the interface temperature is close to the equilibrium temperature (i.e. if the kinetic term is small). (5°) A crystal may have some faces which are stable and some which are unstable depending on their kinetic coefficients. An interesting example of this is the hydrothermal growth of quartz. The seed crystal is cut so that the fastest-growing face is normal to its thin dimensions. (51) The slow-growing faces are smooth, whereas the fast-growing faces have an irregular cellular structure, Fig. 15. Adsorption of a second component on a crystallographic face will tend to decrease the growth rate, and thus stabilize the face on which the adsorption is occurring. Impurity in the parts-per-million range are sufficient to completely "poison" the growth and change the growth morphology. The effects of adsorption on the growth rate of a face has been discussed using the kinematic wave theory. Cs2) The growth of the perturbation in time has been studied for the early stages of development,(4s) but the complete development from the onset of instability to stable cellular growth has not yet been analyzed.

Dendritic Growth Increasing solute content in low entropy of fusion materials results in the cell boundaries becoming progressively deeper until branching and dendritic growth occur. The cells grow normal to the interface, but as they approach dendritic form, they grow more and more toward the dendrite direction. For several per cent of a second component, well-developed dendrites with branches are observed in Fig. 2(c). Dendritic growth only occurs when the growth process is diffusion limited. The rnajn features of dendrite growth can be understood from a consideration of heat or solute diffusion. It occurs in an unstable system when the growth rate of the crystal growth is fast compared to diffusion. The growing crystal subdivides the liquid on a scale the order of the diffusion distance (D/v) and the instability is removed from the remaining liquid by a change in temperature or composition due to diffusion. MEASUREMENT

OF GROWTH VELOCITY

The growth rate of dendrites for various amounts of undercooling has been measured in pure water, c53) tint5 s) and nickel. (5o Measurements have

K.A. JACXSON

76

also been made in aqueous solutions ~53) and dilute tin and lead alloys. ~55~ The experimental data usually conform to a curve of the form v -~ A A T "

(40)

where v is the growth velocity, AT is the undercooling and A and n are constants for a given material, n for all the measurements lies between 1.5 and 3.0. Alloy elements generally retard the growth. The measurements are made by undercooling a quantity of the liquid, nucleating the solid, and measuring the growth rate. The temperature measured is the initial undercooling before growth starts. The growth rate has been determined by a variety of methods: it has been observed visually in water, <53~ detected by thermocouples in lead and tin, CsSJ and detected optically by the recalescence in nickel. ~5~) Observation of growth on the surface of the liquid ~5.) can give incorrect results, because of non-uniform temperatures near the surface. In most systems which have been investigated, the preferred dendrite growth direction depends on the configuration of the closest-packed planes in the crystal. The dendrite growth directions are in the directions of the corners of the solid figure made up of the closest packed planes. For example, in hexagonal materials, these are the (1120) directions, and in fcc and bec the (I00) directions. There are, however, cases in which dendrites do not grow in crystallographic directions. In ice, the dendrite directions are the (l12X) for growth in aqueous solutions. X depends on concentration and growth rate, and has been measured between 0 and 0.35 for various growth conditions/5v) Ice dendrites also grow noncrystallographically across solid surfaces/53~ The dendrite growth in tin has recently been reported to be 12° from the (110) direction. ~Ss~ The tip of a dendrite always appears roughly parabolic when examined under a microscope, Fig. 2(c). Despite the strong crystallographic features of dendrite growth, it appears that the crystallography enters in only to steer the dendrite, and that small differences in growth rate are magnified by the instability of the growth conditions, to result in these strong crystallographic features. The diffusion around a dendrite has been treated by several authors. ~59-61) The models used have looked for an isothermal body which preserves its shape as it grows. The diffusion field around the body must carry away the heat generated by the freezing process. The heat generated depends on the local growth rate of the interface, which is less than the axial velocity of the dendrite except at the dendrite tip. It was suggested some time ago (59) that the shape which satisfied these conditions was a paraboloid of revolution. Ivantsov ~6°) has shown by an elegant mathematical method that this is indeed the correct solution, and that the dendrite obeys the equation -- V p / 2 D L

exp ( v p / 2 D L ) E i ( - vp/2Dz) = A T C / L

(41)

Current concepts in crystal growth from the melt

77

where v is the growth rate, p is the dendrite tip radius, DL and C are the diffusivity and specific heat of the melt, L is the heat of fusion, AT is the difference between the interface temperature and the bath temperature, Ei is the integral error function. It has subsequently been shown ~61) that paraboloids of elliptical cross section are also valid solutions. Unfortunately, the diffusion solution does not completely specify the conditions of growth. A diffusion solution can be obtained for a given undercooling provided the product vp is constant, as is evident from eq. (41). Thus for a given bath undercooling, thin dendrites growing rapidly or thick dendrites growing slowly can both satisfy the diffusion conditions. For a given bath temperature, the dendrites are observed to grow at one rate, with one tip radius. How does one of the many possible dendrite sizes come to dominate? This difficulty is similar to growth of a spherical particle or of a plane interface with constitutional supercooling. Many solutions are possible, but they are not all stable against fluctuations. A stability analysis has not yet been done for the dendrite. A partial solution has been obtained t62,6s) by taking into account the effect of surface energy and interface kinetics on the growth. The dendrite was still considered to be isothermal, but the equilibrium condition was imposed only at the dendrite tip. An equation similar to eq. (41) was obtained which contained additional terms due to surface free energy and interface kinetics:

vp exp

2DL

Ei -

=

L

~L 1 "["L1/flP-k L2a/vp2

(42)

Here fl is the interface kinetic coefficient as defined above in eq. (31), ~ is the specific surface free energy and 7L and ~s are the densities of the liquid and solid. L, and L2 are defined by:

L1 = (1.33-t-0.60KL/Ks)~-~- -

a n d L 2 = (3.86+2.08KJKs) DLCTr

7LL 2

(43)

where Ks and KL are the thermal conductivitics of the solid and liquid. It was then assumed that the dendrite tip radius was such that the growth rate was a maximum. This maximum growth rate is probably close to the growth rate which is stable against fluctuations. The maximization used should, however, be regarded as a method of obtaining a solution when the stability analysis is too complex. For (vp/2DL) ~ 1 eq. (42) is approximately:

vp

[.

L TC/L

T

2-D'L -- ~1 L_~s[1 +L,fflp+L2tr/vp2]] (44) where =1 = 0.457 and n = 1.21. Differentiating this equation with respect to p, and setting dv/dp = 0 gives:

(2+~)"=

'~/-2~1Dl"

[(2n-I)ATC-~V

78

K.A. JACr~ON

and

( 2 n - l)L2o"

v.

(45b)

pip-(n- 1)Ld~]

Fitting dendrite growth rate measurements on tin (5.) gives p ffi 32 cm/sec °C [c/'. eq. (9)]. There are no reliable data on the kinetic undercooling of tin available for direct comparison. (8) The growth direction of dendrites can be understood readily in terms of these same factors. The shape of the dendrite is such that the radius of curvature at the tip is smallest, and increases away from the tip. This produces the largest undercooling due to surface energy at the tip. In order to maintain an isothermal shape-preserving solution, this can be compensated if a fastgrowing direction which requires smaller undcrcooling is located at the tip, while slower-growing directions are located nearby where the curvature is less. Thus in fcc materials, the faster-growing (100) direction is along the dendrite axis, while the slower-growing (111) directions are located symmetrically off-axis. This gives a region near the tip where the surface can remain isothermal. Further back from the tip, the curvature decreases further, and the direction moves away from the slowest growing direction. The resulting instabilities develop into branches. SUMMARY

T h r ~ classes of crystal growth have been identified. The lowest entropy of fusion group contains the metals. The intermediate group contains most organic and inorganic compounds. The highest entropy of fusion group includes polymers. A simple model of crystal growth has been outlined, which predicts many of the important features of cTystal growth morphology. In the low entropy of fusion materials, diffusion is the dominant factor in crystal growth. Instabilities play an important role in producing c~Uular and dendritic growth morphologies in these materials.

REFERENCES 1. K. A. JACIL~ON,Liquid Metals and Solidification, p. 174, ASM, Cleveland (1958). 2. K. A. JACKSON, Growth and Perfection of Crystals, ed. by R. H. DOR~S, B. W. ROASTS and D. TU~SULL, p. 319, Wiley, New York (1958). 3. B. MUT.~O'SO~F,Colloqu¢~ Intemationaux du CNRS, No. 152, Adsorption et Croissance Crystalline, p. 283 (1965). 4. H. W. KERR and W. C. WIN~ARD, Crystal Growth, ed. by H. S. P~Ea, Suppl. to J. Phys. Chem. Solids 179 (1967). 5. V. V. VORONKOVand A. A. ~ o v , Crystal Growth, ed. by H. S. l~Is~, Suppl. to J. Phys. Chem. Solids 593 (1967). 6. K. A. JACKSONand J. D. HUNT, Acta. Met. 13, 1212 (1965). 7. H. D. I~rrH and F. J. PAVD~, J. Appi. Phys. 35, 1270 (1964) 35, 1286 (1964).

Current concepts in crystal growth from the melt

79

8. K. A. JACKSON, D. R. U m . ~ x ~ and J. D. Him'r, J. Crystal Growth I, ! (1967). 9. J. J. ~ and W. A. TILLL~V,,J. Chem. Phys. 42, 257 (1965). I0. D. R l o ~ Y and J. BLAX.~Y, Acta Met. 14, 1375 (1966). II. J. W. CAt-I~,Acta Met. 8, 554 (1960). 12. J. W. CAm~, W. B. Hn.~G and G. W. SEARS, Acta Met. 12, 1421 (1964). 13. H. A. WILSON, Phil. Mag. 50, 238 (1900). 14. J. Fmu~cm., Physik Z. der Sowjet Union 1,498 (1932). 15. K. A. Jxcg.~aN and B. ~ , Can. J. Phys. 34, 473 (1956). 16. F. C. FRANK, Disc. Faraday Sac. 5, 48 (1949). 17. W. B. I-IIX~G and D. TURNaUt.L, J. Chem. Phys. 24, 914 (1956). 18. D. Tu'm~ut2. and R. E. C-~'H, J. Appl. Phys. 21, 804 (1950). 19. D. G. T~OMAS and L. A. K. STAVm.eY, J. Chem. Sac. 4569 (1952). 20. H. J. DENor, DWAX.L and H. J. STAWL~'V, J. Chem. Sac. 224 (1954). 21. E. R. Buctt~ and A. R. UBe~LOI-IDE, Proc. Roy. Sac. A 259, 325 (1961). 22. IC A. JACKSON, tO be published. 23. F. C. FRANK, "Growth and Perfection of Crystals, ed. by R. H. DOREMUS, B. W. Rom~'rs and D. Tum~u'x~ p. 3, WHey, N e w York (1958). 24. C. E I ~ u M and B. ~ , Can. J. Phys. 33, 196 (1955). 25. A. Rosle3,~a~o and W. A. TwT.v~t,Acts Met. 5, 565 (1957). 26. W. A. Tn.T-~.~Journal of Metals 9, 847 (1957). 27. H. A. ATWXTm~, A. R. LANO and B. ~ , Can. 3. Phys. 33, 352 (1955). 28. G. A. CHADWICK, Acts Met. 10, l (1962). 29. J. L. SAMPSON and K. A. McCAR~,iY, J. AppL Phys. 34, 142 (1963). 30. C. Zla,~, A I M E Trans. 167, 550 (1946). 31. W. A. Tn.~.~, Liquid Metals and Solidification,A S M , Cleveland (1958), p. 276. 32. M. H~T~ ~a.T,Jernkintorets Ann. 144, 520 (1960). 33. K. A. JACKSON and J. D. HUNT, Trans. Met. Sac. AIME236, 1129 (1966). 34. J. D. H U N T and K. hi. JACKSON, Trans. Met. Sac. A I M E 236, 843 (1966). 35. W. A. "I'u.t.~,,K. A. JACKSON, J. W. Rurr~x and B. C ~ I ~ , Acta Met. I, 428 (1953). 36. A. MuLx.~ and M. Wn.x-m~, Z, Naturtarsch 199, 254 (1963). 37. H. P. UT~I-x and M. C. Ft.~IrNGS, Crystal Growth, ed. by H. S. PEIsr~, Suppl. to J. Phys. Chem. Solids 651 (1967). 38. D. WALTON, W. A. Tn~l.~, J. W. Ru'rr~ and W. C. Wn~r~A~, Trans. A I M E 203~ 1023 (1955). 39. W. A. ~ and J. W. Rurrr.g, Can. J. Phys. 34, 96 (1956). 40. E. L. HoD~nes, J. W. RurreR and W. C. Wr~naaARD, Can. J. Phys. 35, 1223 (1957). 41. W. W. MULLINS and R. F. ~ , J. Appl. Phys. 34, 323 (1963). 42. W. W. MtJX2.n~ and R. F. S~.mcA, J. Appl. Phys. 35, 444 (1964). 43. R. F. ~ , J. AppL Phys. 36, 264 (1965). 44. S. R. CarroLL and R. L. P~duc~ J. AppL Phys. 36, 632 (1965). 45. R. F. S~g.em~, Crystal Growth, ed. by H. S. PEISER, Suppl. to J. Phys. Chem. Solids 691 (1967). 46. S. R. Co~n~J~ and R. L. P A m a ~ Crystal Growth, ed. by H. S. P~SER, Suppl. to J. Phys. Chem. Solids 703 (1967). 47. L. A. TAm, ms and W. A. T~.~v~, Crystal Growth, ed. by H. S. PE~SER, Suppl. to J. Phys. Chem. Solids 709 (1967). 48. G. R. K o ~ and W. A. TILLER, Crystal Growth, ed. by H. S. PE~SER, Suppl. tO Y. Phys. Chem. Solids 721 (1967). 49. R. G. S~D~NS'nCKea., Crystal Growth, ed. by H. S. PE~S~, Suppl. to J. Phys. Chem. Solids 733 (1967). 50. J.W. CAH~, Crystal Growth, ed. by H. S. PEIS~R, Suppl. to Y. Phys. Chem. Solids 681 (1967). 51. R. A. LAUDIS~, J. Am. Chem. Sac. 81, 562 (1959). 52. N. C ~ m ~ and D. V ~ X L V r ~ Growth and Perfection of Crystals, ed. by R. H. DOR~MUS, B. W. ROBERTS and D. TURNBULL, p. 393, Wiley, N e w York (1958). 53. C. S. L ~ D ~ N ~ , Ph.D. Thesis, Harvard Univ. (1960).

80

K . A . JAC'K~N

54. A. R O ~ E a O and W. C. W i g a n , Acta Met. 2, 242 (1954)~ 55. G. T. Or~OK, Ph.D. Thesis, Harvard Univ. (1960). ..56. J. WM.x~, unpublished. 57. C. S. l.~wasYr~a and B. ~ , J. Chem. Phys. 45, 2804 and 2807 (I966). 58, S. O ' H a ~ and W. A. TILLER, to be published. 59. A. P~,c~'rRou, Z. Krist. 92, 89 (1936). 60. (3. P. Iv~rrsov, Doklady Akad. Nauk. SSSR 58, 567 (1947), transl.: Growth o f Crystals, Vol. 1, Cor~ultants Bureau Inc., N.Y. (1958). 61. G. HORVAYand J. W. CA~n~,Acta Met. 9, 695 (1961). 62. D. E. TSS~IN, Soy. Phys., Doklady 5, 609 (1960). 63. G. F. BOLLn~Oand W. A. ~ , J. AppL Phys. 32, 2587 (1961). 64. J. D. HUNT, K. A. JACKSONand H. BROWN,Rev. Sci. Instr. 37, 805 (1966).