Journal qf Crvstal Growth 21 (I 914) 29-39 G North-Holland Publishing Co.

THE GROWTH OF CRYSTALS OF LOW SUPERSATURATION

I. THEORY

B. LEWIS

Allen Clark Research Centre, The Plessey Company Limited, Caswell, Towrester, Northants., England

Received 9 July 1971; revised manuscript received 9 September 1973

Growth of low energy planes of perfect crystals at low supersaturation requires two-dimensional nucleation. The formation energy of nuclei of i atoms may be written G’( = -ikT In cc+G,(i), where G( is the saturation ratio. An atomistic evaluation of G,, for a simple model, is in good agreement with the classical form G,(i) =

iiD’, with a constant edge energy coefficient b’, even for very small i. /I’ includes entropy terms and is signi- ficantly lower than the unmodified edge energy coefficient b when /l/kT < 5. The evaluation /3 1: 1-2~1 where % is the latent heat, or enthalpy of solution, and c is the surface energy (with c z A/6 and /? ‘v 21./3 on low energy planes) is a useful first approximation for any crystal. j? is the primary material parameter for two- dimensional nucleation (2DN) and also for screw disclocation growth (SDG) of imperfect crystals. When B/kT > 10, which is typical of growth from the vapour on low energy planes, 2DN is slower than SDG for c( 2 lOand is negligible for CC y 2. When /l/kT < 8, which is typical of melt growth, 2DN is faster than SDG at moderate supersaturation and lower at low supersaturation. When B/kT Q 2, B’ - 0 and growth by either mechanism is unimpeded at all supersaturations. For 2DN growth, simple crystal forms with only low energy planes are expected when growth is impeded; the occurrence of other crystal faces indicates SDG.

1. Introduction

Crystals grown at low supersaturation have forms

which are developed kinetically by the non-equilibrium

processes of growth. The mechanism is well-understood

in principle. On any plane, growth occurs at kink sites

at which the attachment energy is equal to the mean

binding energy per atom of bulk. On high energy planes

kink sites are plentiful and the growth rate is directly

proportional to the supersaturation. At high supersat-

uration kink sites are also freely nucleated on low

energy planes so that growth is again unimpeded. At

lower supersaturation the growth rate of low energy

planes becomes limited by the nucleation rate of kink

sites. High energy planes then grow out and a crystal

habit develops which is determined by the impeded

growth rates of the low energy planes.

The theory of the formation of growth sites on per-

fect crystal planes was established by Stranski’), Vol-

mer and Weber’), Becker and D6ring3) and others but

it was recognised by Volmer and Schultze4) that the

nucleation rate was too low to account for the ob-

served growth rate at low supersaturation in certain

cases. Frank’) then showed that steps associated with

dislocations in an imperfect crystal generated growth

spirals which did not grow out. The subsequent com-

prehensive treatment by Burton, Cabrera and Frank6)

applied surface diffusion concepts to screw dislocation

growth, and is the most frequently cited paper in the

field of crystal growth. Perhaps due to the emphasis

on the growth of imperfect crystals, there has been no

comparable treatment of growth by two-dimensional

nucleation.

The present paper considers the growth rate on low

energy planes of perfect crystals by two-dimensional

nucleation, and the dependence on material and growth

parameters. Comparison is also made with the growth

rate on crystal planes containing screw dislocations.

2. Condensation theory

2.1. THE DENSITY AND FORMATION ENERGY OF TWO-

DIMENSIONAL NUCLEI

The density ni of nuclei of i atoms on a substrate in

equilibrium with incident monomer, in the units nuclei

per site, is given by Lothe and Pound’) as

ni = exp (-G,/kT), (I)

Gi = G,(i) + G,(i). (2)

Gi is the cluster formation energy. The volume energy

G,(i) is -kTln CL, where CI = R/R, is the saturation

ratio, and R, R, are the actual and equilibrium inci-

29

30 R. LEWIS

dence fluxes. The “surface energy” term G,(i) repre-

sents the energy deficiency with respect to bulk. For the

crystal growth case, the nucleus is one atom thick and

the surface energy io of its major surface is balanced

by the eliminated surface energy -ia of the crystal

plane on which it lies. Hence G,(i) can be associated

with the periphery of the nucleus, whose length is pro-

portional to ii. We now consider the lowest energy or

ground state yi of a nucleus of i atoms and write g,(i)

= /Ii: so that eq. (2) becomes

Qi = -ikTlll sc+i'p. (321)

It is implicit in this relation that [j is independent of

i, and without this assumption we can make no pro-

gress. We therefore seek justification by considering a

particular example.

For this model calculation we consider a simple

cubic “Kossel” crystal, with nearest neighbour bond

strength 4 = i/3, where i is the binding energy per

atom. We first evaluate the ground state formation

energy, which we designate yi. Following the treatment

of Burton, Cabrera and Frank’), hereafter BCF, G,(i)

is 44 for each unsatisfied bond at the cluster edge, and

nuclei of size i are all treated as squares of side ii.

Hence [j = 29’1 and

gi(BCF) = -ikTIn cc+2i*g. (4)

For comparison, we make an atomistic evaluation for

possible configurations of integral atoms. The ground

state is that with the smallest number pi of unsatisfied

edge bonds, and

I 2 ‘0

gi(atom) = -ikTln ~+;/>~,i&. (5)

The two evaluations of yi are compared in fig. I for

kT In c( = 0. I 4. The BCF evaluation gives ;I smooth

increase up to yj’ = c$‘/kT In CI (= loci, in our ex-

ample) at the critical size i* ( = 100 atoms). The atom-

istic evaluation has a maximum at i = 3 followed bl,

a minimum at i = 4. Then yi(atom) rises by (/,-kTlnr

for the first atom of each new row, and falls by kT In Y

for each atom added at a kink site to complete the row.

The minima for square nuclei (i = 4. 9, 16,. .) coin-

cide with gi(BCF) and the absolute maximum i5 at the

same critical size i*.

We now consider the reduction of energy due to

multiple configurational states. Following Frank’) the

atomistic evaluation with it’s ground state configura-

tions and 11.~’ singly excited states with excess energy (/I

becomes

- kT In [H.;+ I\.~’ exp( -&A T)]. (0)

Since eq. (5) represents every nucleus as ;I square there

is only one ground state configuration for each size and

correction is only required for the contigurational en-

tropy of excited states. BCF showed that the formation

energy becomes

G,(BCF) = -kT In cc+i’[2&41\7‘exp (-C/I/Z/CT)]. (7)

Fig. I shows that eqs. (6) and (7), for 4/l\ 7‘ = 6 (i,i/\7’

= 18), are in close agreement for all values of i. The

maxima and minima of y;(atom) are smoothed into

inflexions, and the reduction of energy due to excited

Fig. I. Formation encrgicsg, (excluding configurational entropy) and G, (including conligurational enlropy) for a clusier ofi adorn\ on the (001) plane of B simple cubic crystal. Atomistic and classical evaluations are compared.

THE GROWTH OF CRYSTALS OF LOW SUPERSATURATION. I 31

states is similar, for large i. The configurational en- the dimensions of length and is conveniently equated

tropy term becomes very significant when d/kTis small. to a diffusion distance x, atomic units. If q = 4 and

For example, when 4 = 0.15 kT the edge energy coef- vd = V, then

ficient is reduced from 0.3 kT for the ground state to

0.1 I kT. x, = exp [(es-e,,)/2kT]. (11)

We conclude that for this model the formation Considering nearest and next-nearest neighbour

energy of both regular and irregular shaped nuclei is bonding on the (001) plane of a simple cubic crystal:

represented quite accurately by treating each size as the e, = 41 f4& esd = 2$2, I. = 34, + 6$2 and e, -esd =

same shape, with edge energy proportional to i*. It is 343. Hence X, = exp (1/6kT). Similarly a mean edge

also clear that entropy terms may significantly reduce diffusion distance x, atomic units along a step edge can

Gi and we therefore introduce a modified edge energy be defined as

parameter /I’ and write

Gi = -ikTIn a+i+jY. (3b)

For a specific material we do not expect the value of p’

With constant /I’ we may differentiate eq. (3b) to

to conform with the simple bond model, except as a

find the maximum G* and the corresponding critical

first approximation, since it should include second and

third neighbour bonding, relaxation terms and vibra-

tional entropy. However we do not expect the propor-

size as

tionality to i+,

G* = /Y2/4kT In CY,

i.e. the constancy of /3’, to be upset.

(8)

i* = @‘/2kTln CC)~, (9)

x, = exp [(e, - e,,)/2kT]. (12)

For a [IOO] step of a simple cubic crystal, e, = 24, +

6429 eed = 442 and ee--eed = 21,/3. Hence x, = exp

(i./3kT).

Ti = [R&i + 1) b,x; . (13)

In this relation, bi is a dimensionless capture factor

BCF have obtained solutions of the diffusion equa-

tion for capture by a single nucleus, and by an array of

which depends on the nucleus size, shape and environ-

parallel steps. The net capture rate Ti for i z+ if 1, i.e.

allowing for decay, is

ment, and bixt is the effective catchment area of inci-

dent atoms. For an isolated nucleus of radius z atomic

and from eq. (1) the equilibrium density of critical units the capture factor b, is

nuclei is

n,, = exp (-G*/kT). 6; = 2~/K,(z/~~J lo(z/x,), (14)

(10) where K,,, I,, are modified Bessel functions of order

2.2. DIFFUSION AND CAPTURERELATIONS zero. For 0.005 < z/x, < 1, eq. ( 14) gives 1 < b < IO.

For a straight step of length z > 2x, separated from

The process by which atoms arriving on the sub- neighbouring steps by a distance y atomic units

strate combine to form critical and supercritical nuclei

is surface migration. Adda and Philibert’) derive the b = (2z/x,) tanh (y/2x,)

surface diffusion coefficient D, as 2 2z/x,, when y > 2x,, ( 15)

D, = +qv, exp ( - e,,/kT) ,

where e5d is the activation energy for surface migration,

and vd is the vibrational frequency for jumps in each of q

possible jump directions. The adatom desorption rate

may similarly be written

l/z, = rs exp (-e,/kT),

where rs, e, are the vibrational frequency and activa-

tion energy for desorption. In the solution of the diffu-

sion equations the term (D,r,)* appears, which has

per edge length Z.

The term $(i) in eq. (13) is defined as ~-cc(i) where

CY = R/R, is the saturation ratio with respect to R, for

bulk crystal, and cc(i) = R/R,(i) is the saturation ratio

with respect to a nucleus of i atoms. A nucleus which

is in equilibrium with a saturation ratio c(, i.e. with a(i)

= c[, is a critical nucleus. Thus $(i) and the growth rate

Ti_l to size i are negative for i < i* and positive for

i > i*. At a kink site, k, at the edge of a supercritical

cluster, or on a step, the attachment energy is equal to

the crystal binding energy and a(k) = 1. Hence $(k)

32 B. LEWIS

= (r- 1). An atom which arrives at a step edge will

have a high probability of reaching a kink if the edge

diffusion distance .Y, exceeds the spacing _I-~ of kinks.

BCF showed that the equilibrium spacing of positive

kinks on a [OOI] step of a simple cubic crystal is _yI, =

exp (4/2kT) sites, which may be generalised to sI =

exp (i.-r,)/2kT. Hence sh < _vc if erd < 2e,-i. which

is well satisfied for this case.

< is a retardation factor (BCF’s symbol is /I) which

represents the fraction of atoms arriving at kink sites

which are incorporated (either permanently or tempora-

rily) into the crystal. For single atoms we expect [ = I,

but multiple rotational or configurational states of ad-

sorbed polyatomic molecules, or desolvation processes

in solution growth, may impede incorporation so that

< < I in these cases. We assume i is independent of c(

and to avoid carrying it through subsequent equations

we now introduce R, = CR,: ctR, = rx[R, is the effec-

tive incidence flux and the saturation ratio remains

r~ = R/R,. With $(k) = (a- I), the net capture rate

P, at kink sites on a supercritical nucleus edge now be-

comes

Tr = R,(a-I).x,‘O;, (16)

where h= is given by eq. (14). Using (I 5) the growth rate

of new rows on a long step, or on each edge of an iso-

lated nucleus (~3 > 2s,) with ; > 2.~, is

f-, = 2(x- I)R,.r,. (17)

2.3. NUCLLATION AND GROWTH KINETICS

We follow steady-state treatments of nucleation rate

such as have been given by Becker and Dtiring3) and

by Russell lo) for three-dimensional nucleation and by

Hirth’ ‘), Halpern”) and Franka) for the two-dimen-

sional case. The flux Ji between sizes i and if I is

Ji = f+Ni-f-i;,Ni+l, (18)

in which Ni is the steady-state density, and r+, rj- are

the capture and decay rates, of nuclei size i. Eq. (IX)

gives the flux through the system and we must use it up

to a supercritical size i = /I > i* for which r,- (which

decreases with increasing i) either becomes negligible

compared with f+ and riL or approximately constant.

In condensed phase nucleation, Russell”) has establish-

ed that the rate controlling step for promotion of a clus-

ter is not the interface jump but migration through the

parent phase lattice of the replacement atom required

to maintain equilibrium. In the present case, r,? and F,:

are the positive and negative components of the surface

diffusion capture relation, eq. (13). Thus putting

$(i+ I) = m-a(;+ I)

/-+ = stRobis,; , (l9a)

I‘.- ,+, = r(i+ I)R,b,xf. (19b)

In the usual treatments, r,y and Ni are eliminated in

favour of the equilibrium n, by considering first the

equilibrium state with Ji = 0 (obtained by a barrier to

growth beyond size /I) and then the steady state in

which the fluxes Ji are all equal. The result, with Jj now

written J* is

(P-l = f (r+llJ’. (20a)

i= 1

Since the 11~ are strongly size-dependent for small i,

with a minimum at i”, and f+ varies weakly with i. we

may conveniently write

J* = ZT;rli*, 00b)

z- ' = i (P,t*/l,*/l“,li) i 2 (H,,/Hi). (2OC) i= I i .z 1

in which Z is the Zeldovich non-equilibrium facto1

which allows for the departure of Ni, from the equilib-

rium I?~*, and for the size-dependent decay of super-

critical nuclei.

Frank*) and Russell lo) estimate Z by approximating

H~*//I~ = I for the range i, to i, over which G,,-G,

< h-T and IIJ/I; = 0 for all other 11~. Then Z =

(i, -i2)m’, This demonstrates that only the G, near the

absolute maximum are relevant and that other maxima

and minima in an atomistic evaluation have no effect

on the nucleation rate.

Eq. (3b) for Gi gives

IIT 12 = i* t_ 2 (i*/ In cc); + 11 In a!, (Zla)

which, with i* = G*/kT In CI, gives

Z = (kT/l6 G*)’ In s(. (21 b)

Treating Gj classically us a continuous function of i.

and replacing the summation by an integral gives

Z 2 - [(d2Gi/di2)i,j27ckT]‘.

From eq. (3b) we then find

Z = (G*/4nkT)+,‘i”

THE GROWTH OF CRYSTALS OF LOW SUPERSATURATION. I 33

as given by Hirth’ I), or in alternative form

Z = (kT/4xG*)’ In LX. (2lc)

Thus these two analytical approximations for Z are

almost identical.

We now use (20b), (lo), (19a) and (21~) to find the

nucleation rate, and note that J* is proportional to

b* R,a In CI through Ti, and Z. The nucleus growth rate,

fk or f,, is proportional to R,(cc- 1) and it is conve-

nient to write J* as

J* = C(cc-- l)R,xf exp (-G*/kT), (22a)

(22b)

where b* is given by eq. (I 4) and C - 1 is an order of

magnitude approximation.

Kinetic treatments by Chakraverty13) and Kash-

chiev14) show that the steady-state flux J* is ap-

proached exponentially with time with a rate constant

* l-it; d2Gi \! = -

(H kT di2 i* (234

With eq. (3b) this induction rate becomes

\I* = r: In (x/2?. G=)

If we substitute (a- 1)/u for In c(, which is accurate for

(c(-- I) 4 1 and within a factor 3 for CI < 20, we see

that the induction time l/v* is approximately the time

required to collect 2i* atoms at the growth rate rk. It is

not quite clear whether v* takes us to i* + I, as we will

assume here, or to i, but the distinction is not impor-

tant. It will later be convenient to consider the induc-

tion time, which we now designate z,, as extending to

the size i = 4x,2 at which lY,, rather than rk, applies for

growth. Then

z, = ,j/(cc- l)R,b,$, (23~)

wherej is 2i* or 4.~: + i* whichever is the larger, and bj

lies between b* and 10 which are the values of bi for

i = I ‘*. . 2 dnd I = 4x,.

We now use eqs. (22) and (23) to find the growth

rate F of new crystal planes. Following Brice15) and

Bertocci16) we distinguish two cases.

When the crystal size and supersaturation are suffi-

ciently small that the growth time d/2r, to cover a face

of edge length d is small compared with J*-l, then F

is nucleation limited and is equal to d2J*. Using sub-

script 1 for this case,

F, = d2C (x- l)R,xz exp (- G*/kT). (24)

Introducing F, = (a- l)R, as the maximum possible

growth rate at supersaturation CI, the validity condition

for this relation is Fl/F, < 4x,/d.

For larger crystals we consider a nucleus which

grows at rate re from an edge length z = 2x, to z =

(2x,+T,t). The effective nucleation area on top of this

growing nucleus is zero at t = 0, because of monomer

depletion due to its own growth, and approximately

4rtt2 at time t, which gives a nucleation rate 41’tt2J*.

We now integrate4rft2J*dtfrom t = 0 to t0 and equate

to unity to find z0 = (4rtJ*/3)-” as the nucleus forma-

tion time, to which we add T, as the growth time to an

edge length z = 2x,. 70 + z, is independent of the size

of the growing face. Except at high supersaturation, T,

is negligible compared with z0 and the growth rate F,

of new planes is thus zO- ’ = (4/3)‘r!J*“. The nucleus

spacing is such that each nucleus coalesces with its

neighbours at about the same time as a new nucleus

forms on top. Hillig17), Brice’ 5, and Bertoccil’j) have

given growth relations similar except for the trivial

numerical term which depends on nucleus shape and

which we now drop. Substituting for re and J* we find

F, = (4C)“(c(- l)R,x: exp (- G*/3kT). (25a)

Subscript 2 denotes the two-dimensional nucleation

growth process which applies when F/F, > 4x,/d.

Eq. (8) applies until a approaches a saturation ratio

c(~ given by F, = (LY- l)F, which is the highest pos-

sible growth rate. Substituting for G* we find

In t12 = (fi’/kT2)/4 In (4Cx$). Wb)

Actually, when LY - x2 several corrections become nec-

essary. First is the restoration of the term tanh (y/2x,)

dropped from eq. (17); the nucleus spacing is now

- 2x, and re is therefore reduced. Allowance for com-

petitive capture effects in nucleation is also required.

The first steps in such a calculation have been taken by

Surek, Hirth and Pound’ “) for the analogous case of

evaporation under conditions with nucleus spacing

N 2x,. These corrections reduce F2 at CY = u2 and

cause it to approach F, only gradually with increasing

saturation. Thus if we are concerned with the relative

growth rates of two planes of a crystal, that with higher

34 IS. I_lIWIS

cz2 grows more slowly even when CA + Ed. But for ab-

solute growth rates we may regard eq. (25) as valid up

to 2 = 8x2, which we treat 21s the condition for un-

impeded growth, with F, = F,.

2.4. SCREW DISLOCATION GROWTH

We now introduce growth relations for imperfect

crystals containing screw dislocations. BCF showed

that a screw dislocation generates a spiral array of

steps with a spacing 47rv* between turns, where r* =

(i*/rt)’ is the two-dimensional critical radius and i” is

given by eq. (9). The growth rate is

F3 = (r- l)R,(.u,j2w*) tanh (~TII.*/.Y,)

= (x- I)/?, (In z/ In xj) tanh (In ,zJln x),

In a(3 = n+/Y/kT.\-,. (26)

Subscript 3 denotes growth from a single dislocation.

In a more accurate calculation Cabrera and Levine’ “)

find the step spacing as 19r* instead of 4xr”. but know-

ledge of values of .Y, and P’/kT is so uncertain as to

make this correction insignificant. Cabrera and Cole-

man”) have allowed for the depletion of adatoms by

the first turn of the spiral on r* at its centre. This

“back-stress” effect causes F3 to approach F, more

gradually than is given by eq. (26), when x > a3. Thus,

as for two-dimensional nucleation growth, two planes

with differing c(~ always grow at different rates, though

both approach F,, when E 9 c[~. However, for absolute

growth rates eq. (26) is sufficiently accurate.

Since tanh .v -+ I when .V > 1, and In x + x- 1 when

a < 2, it follows that FJ 4 (c(- l)‘Ro/ In c(~ when

2 > x < CLS, and since tanh .Y + .Y when .Y < I it fol-

lows that F3 + (x- l)R, when a > x3. Thus the growth

rate varies quadratically with (a- I) below z3, and

linearly above xj.

BCF also considered a distribution of dislocations

of both signs. Pairs of dislocations of opposite sign

closer together than 2rm are inactive. An array, length

L, of ,c dislocations of the same sign (or an excess S of

one sign for a mixed array) with maximum separation

/ < 27~~” between neighbours, has activity E times that

of a single dislocation. c = S when 2nr* B L, c =$/2

when 2rrr* = L, and e = 2w*SjL when L $ 2w*.

Hence, neglecting a small error when 2nr* - L,

S,S 2m” F,, = (x-I)Ro

2xr” tanh

x,s ’ 2rrr” > L, (27a)

s,s L F,, = (x- I )K, tan Ii

.Y,S ’ L> 2~‘” > I, (27b)

L

The growth rate of a face is determined by the group ol

dislocations with highest S/L. The activity lies between

i: = S when 2rr~* > L and c = 5 = I. for which F, =

FI, when the separation of all like dislocations exceeds

2rrr”. Thus eqs. (27) and subscript 4 represent the

general case of dislocation growth. A critical saturation

31~ can be defined for which F5 = F,, which occurs

when .v,.c = 2rrr* or L. whichever is the larger.

For an array of length L. separated by at least L

from other dislocations. FSh falls as (x - I ). This is the

“second linear law” discussed by Bennema”). which

holds until, with increasing r*. the validity condition

L > 2rrr* is violated, and FSh changes to F,,, falling x

(x- l)2. Thus ,with multiple screw dislocations, a lineut

dependence on (c(- I) does not necessarily imply un-

impeded growth, F,, and a change to the square I;IW

(x- I)’ can occur below the saturation ratio transition

x3 for a single screw.

When the step spacings j‘2 and J’~ are both large

compared with _v,, two-dimensional nucleation and

screw dislocations produce steps independently and are

additive, so that F = Fz + Fa. As for either process on

its own, the growth rate cannot exceed F, and the

approach to F, is gradual.

3. The dependence of growth rate on supersaturation

3. I. MATERIAL ANI) GROWTH t’At~~Mr-rt.Rs

The experimental and material parameters which

appear in the equations above are R,,, “A, 11”. T and .\-,.

We presume that R,. a and 7‘ are known and are in-

terested in the growth laws and growth rates by each

mechanism as dependent on the saturation ratio rl and

supersaturation (x- I ). We therefore need to select rep-

resentative pairs of values of/Y and .v, to illustrate the

range of growth behaviour, but unfortunately there are

no reliable independent measurements of these quan-

tities. For low energy planes of the simple cubic crystal

considered in section 2.1, allowing for second nearest

neighbour bonding,

/I = 241, +44, = 2il3, (xh)

/i’ = p-4kTexp (-P14kT). (28b)

X, = exp (4, +2q5,)/2kT = exp (i/hkT), (28c)

THE GROWTH OF CRYSTALS OF LOW SUPERSATURATION. 1 3s

where A is the binding energy difference per atom be-

tween the parent and condensed phases. For the general

case we may expect relations similar to eqs. (28) to

apply but with differing numerical factors for each

different crystal structure or growth plane.

In growth from the vapour the incidence flux R can

be obtained from the vapour pressure p as p(2rmzkT))~

and CI = R/R, = p/p,, wherep,, R, are the equilibrium

vapour pressure and flux at the growth temperature.

R, = CR, is the vacuum sublimation flux.

R, can be written in terms of the latent heat of vapor-

ization is, as

R, = v, exp ( -MkT),

where V, - IOr monolayers set- ’ for many ma-

terials22).

In growth from the melt,

In c( = &,(T,-T)/kT,T,

where is, is the latent heat of solidification, r, is the

melting point and T is the growth temperature. For

small supercoolings (z- 1) = In CY and is thus propor-

tional to( TM - T)/T,. Following Turnbull and Fisher23)

the supply rate of atoms is

R = (kT/h) exp (-gJkT),

where gd is the activation energy for the transition from

the liquid state to surface adsorption, and we assume

the solid and liquid phases have similar densities. Typi-

cal values are exp (-gd/kT) - 10m2 and R - IO”

monolayers set- ’ at the melting point of metals24).

Then R, = CR, = [R/x. Since CI + 1 and we expect

[ = 1 in most cases, we can generally put R, = R.

In growth from solution’5),

In a = i.,,(T,-T)/kT,T,

where i.,, is the enthalpy of solution, defined by the

equilibrium relation

C, = A exp ( -&,jkT,).

For small supercoolings, (a- I) K (T,--T)/T,. R is

given by a relation similar to Turnbull and Fisher’s

above; desolvation energy occurs in either gd or in c or

in both, and may depend on the orientation of the

growing face15). For complex molecules, which are

common in solution growth, [ may include other imped-

iments. Hence in solution growth R, is unknown and

may differ between planes.

p’ is only directly measurable by nucleation rate ex-

periments, but p may be estimated from the surface

energy c. Frank’) suggests that /I is lower on less closely

packed planes (higher o) and may be obtainable from

the orientation-dependence of surface free energy. We

also expect proportionality to A for crystal planes of

similar structure.

To obtain a relation between /?, I. and g we consider a

single layer of i atoms with the equilibrium shape of a

two-dimensional nucleus. Treating both the surface

energy 2ia and edge energy i*/3 as unsatisfied bonding

and 1. as the total bonding energy per atom,

il = Mi+2ia+if~. (29)

where Mi is the mutual bonding between the i atoms of

the layer. If we put i = co we find MJi = A-2a as the

binding energy per atom within the plane, and if we

put i = I, for which M, = 0, we find

p = A-20. (30)

In support of an evaluation which depends on con-

stancy of /I and a down to i = 1, we recall that this as-

sumption gave agreement with gi (atom) for i = I in

section 2. I. Furthermore, we are particularly interested

in critical sizes between I and 10 atoms, so that para-

meters valid for small i are relevant. If /I and p’ are i-

dependent, the differentiation which gave eq. (8) for G*

does not hold. If /I’ varies slowly with size near i* our

equations can be used for analysis of nucleation and

growth measurements to find an effective value of j?’

for the particular critical size which occurs. For predic-

tions, we have no other course than to assume that B

and a are size independent and to find b from eq. (30)

and known or estimated values of A and a.

Creep measurements of asv for solids show a strong

correlation between asv and A,v2’), and range from

A,,/8 for Sn to A,,/4 for Zn (except for Hg, which has

the particularly high value asv = 0.4 Asv). From eq.

(30) we thus expect /J to lie between 3114 and A/2. We

note that a = A/6 and /3 = 2i/3, which are about the

middle of the range above, were the values for the (001)

plane of a simple cubic crystal. Interfacial solid-liquid

surface energies asL have been obtained from three-

dimensional nucleation measurements and are around

0.45 is, for metalsz6) and 0.35 Lsr_ for non-metals2’).

Used in eq. (30) very low values would be obtained for p.

However, Zell and Mutaftschiev”) have found by ex-

36 B. LEWIS

amination of a ball model that the liquid structure in

contact with solid is disorganised. The energy associat-

ed with liquid disorganisation was not considered in

deriving eq. (30), and there is no reason to suppose that

the edge energy is particularly low. Hence for melt

growth, and also for solution growth, we will tenta-

tively assume that /I lies in the range 3?./4 to ;/2, as for

vapour growth.

For the entropy correction, to obtain /I’ from /I, the

general case differs in two ways from the simple model

previously considered. Firstly, [I is no longer just a

bond energy term since we now derive it from the lat-

tice and surface energies, and secondly, vibrational and

rotational as well as configurational terms should be

included. However, we tentatively adopt eq. (28b) as an

approximate evaluation and note that this predicts

/?‘/kT = 0, which gives no nucleation barrier, when

[I/kT = 2.3.

The low jI/kT condition for unimpeded growth is

closely related to the equilibrium structure of crystal

planes which has been considered by BCF, Jackson,

Temkin and others6.29P”3 ). Jackson introduced a ma-

terial parameter CI equal to IjkT times the interatomic

binding energy within the plane considered, which, as

discussed above can also be written as M,/i = ,! -2~,

so that Jackson’s CI is our /?/kT. A roughening transition

is predicted as /i’/kT decreases, occurring between

[j/kT = 56) and 1.23’), depending on the treatment.

Since growth kinetics is essentially concerned with ini-

tiation of a new layer of atoms on a completed layer,

the two-level treatments of Jackson and Mutaftschiev

are most closely relevant to the transition to unimpeded

growth. These both predict that the surface becomes

rough when B/kT -c 2.

Experimentally, in growth from the melt interfaces

for which A/kT ? 6 (p/kT y 4) exhibit facetted growth,

while for i/kT 2 3 (jI/kT ?: 2) the growth front is

smooth, as expected if the nucleation barrier is minimal

and the growth rate is dominated by heat flow con-

siderations. Thus the prediction of eq. (28a) that growth

is unimpeded when jI/kT < 2.3 is in general agreement

with experimental and theoretical findings of interface

structure.

The next parameter to consider is the diffusion dis-

tance s, given by eq. (1 I). Growth behaviour is depen-

dent on _I+~ to approximately the first power and a rough

evaluation is adequate. For low energy planes of a

TARLI I

Material parameters for representative growth systems

System Material T (eV) I./X T /CAT \-.

(K)

Vaponr Iodine 273 0.7 30 20 I50 growth Cadmium 573 I .2 23 I5 50

Ice 270 0.4 IX I2 20

Melt SZllOl 314 0.94 is 23 300

growth Germanium 1’10 0.33 3.2 2. I 1.7

Ice 260 0.06 7.7 1.x 1.5

Solution Alum 313 0.29 12 8 7

growth NaClO, 310 0.09 3.2 2.1 1.7

Sucrose 273 0.03 I.1 0.7 I.2

/I 2il3 and .\, cxp (1/6kT) are estimated values for IOU

energy planes.

simple cubic crystal we estimated c’,-P_, = ;./3 and

similar values can be expected for low energy planes of

other structures. For higher energy planes both P, and

P,~ increase. Volmer’) has calculated that for (100) and

(1 IO) planes of bee, fee and hcp crystals P, -L’_, lies be-

tween 0.4 i. and 0.5 i. Thus we expect c,-P,~ to lie be-

tween 3./3 and iL/2 and _I-~ to lie between exp (i,/bkT)

and exp (/1/4kT) in all cases.

Finally, we consider the relation P,~ < 2~, -i which

is required for validity of eq. (I 7). For square nuclei the

edge energy per atom ‘/ is /I/4. By consideration of the

energy change on placing an atom on the surface, c, =

2a, and on placing an atom in an edge adsorbed posi-

tion, ce 2 20+2y. Then with /j = L -2~ we tind

2e, - i =I= c,. Hence our condition reduces to ecd 2 r,,

which is probably satisfied in most cases. It is least

likely to be satisfied for a structure with a smooth

growth plane and a rough step edge.

In the expressions above i is the binding energy

difference per atom between the parent and condensed

phases and is j.sv and i,,, in growth from the vapour

and melt, respectively. In growth from solution, i,, is

analogous to jLsL. A selection of values of i and A/kT

are given in table I, assuming a growth temperature

near the melting point for vapour or melt growth and

near room temperature for solution growth. P/kT for

p = 2i/3 and s, = exp (A/6kT) are also shown and are

seen to cover wide ranges of values.

3.2. THE DEPENDENCE OF GROWTH RATL ON SUPER-

SATURATION FOR A SIMPLL CUBIC C‘RYSTAL

The full lines in fig. 2 show the growth rate by two-

dimensional nucleation plotted as F,/R, against super-

THE GROWTH OF CRYSTALS OF LOW SUPERSATURATION. I 37

Fig. 2. The variation with supersaturation (G(- 1) of the reduced

growth rate F/R0 of the (001) plane of a simple cubic crystal, for the edge energy parameter B = 21/3 (edge energy y = 1/6 per edge atom for square nuclei) and diffusion distance x, = exp (,%/~/CT) atomic sites, for ,I/kr = 18 and 4.5. The full lines give F2, and the broken lines FL for a crystal of edge length d = I Mm, for two-dimensional nucleation; the numbers against the curves are the critical nucleus size i*. The screw lines give F3 for screw dislocation growth.

saturation (a- 1) on logarithmic scales for a simple

cubic crystal with j? and X, given by eqs. (28). The bro-

ken lines show F,/R,, for a crystal of edge length 1 pm.

Results are given for AjkT = 18 which is representative

of vapour growth, with an F scale for R, = lOI exp

(- 18) monolayers see- ’ . Results are also shown for

13/kT = 4.5, which is typical of melt growth, with a

supercooling scale and an F scale for R, = IO’ ’ mono-

layers set- ‘. For these two values of /Z/kT the ratio

?./kTx, is the same so that a3 = 3 in both cases. The

screw line in fig. 2 thus shows F3 for both values of

AlkT. When CI > tlZ, two-dimensional nucleation is un-

impeded by nucleation, and F, varies linearly with

(a- 1). When CI < c(~, the growth rate is impeded and

falls rapidly with decreasing cI. From eqs. (8) and (9), i* varies as (l/ In a)’ and G* as l/In cx. Hence i* increases

with decreasing a, and F2, which is proportional to .*

exp (- G*/3kT), varies as CI’ j3. The values of i* are

shown against the curves. The critical size for unimpe-

ded growth at CI = t12 is i * = 1 when AjkT = 4.5 and

i* = 3 when A/kT = 18, for which x, is much larger.

Fl varies as exp (-G*/kT) and therefore as cli* which

is steeper than F,. Fl only applies for small crystals and

low growth rates.

When AjkT = 18, x3 < CQ so growth with a single

screw dislocation is easier than two-dimensional nu-

cleation. When i,/kT = 4.5 the equations give c(~ < cl3

and two-dimensional nucleation is the easier process

down to a = 1.02. The reason for this result is that x,

= 2 and eq. (27) demands r* = xc-’ and i* = nr*2 = 71 -+ for unimpeded growth, whereas eqs. (25b) and (9)

give unimpeded nucleation growth when i* = 1. It is

perhaps more realistic to accept that when i* = 1 both

processes give unimpeded growth. Below CI = 1.02 a

single screw gives faster growth than two-dimensional

nucleation.

Gilmer and Bennema34) have examined simple-cubic

crystal growth by computer simulation, for P/kT (their

y) = 2.5to4andx, = Oto3,i.e.fori.lkT - 4.5.Growth

by two-dimensional nucleation fitted an equation of

similar form to eq. (25a), and “experimental” values of

j?‘/kT (their 2n’o/kT) were obtained. For example,

jl/kT = 3.5 gave /Y/kT = 1.1, in substantial agreement

with eq. (28b) which gives b’/kT = 1.1 when P/kT = 3. This agreement is expected since theory and simulation

are based on the same simplified model of crystal

structure.

3.3. THE DEPENDENCE OF GROWTH RATE ON SUPER-

SATURATION FOR THE GENERAL CASE

The growth rate relations and the values of j”/kT and

x, in terms of /l/kT for the general case are similar to

those for simple cubic crystals. Hence fig. 2 also repre-

sents the general case. Additional data, plotted in fig. 3,

include A/kT = 30 and 12, so that the experimental

range of values of 3,/kT is well represented. For each

value of IlkTin fig. 3, F2 is plotted for x, = exp (/1/6kT)

and j’ = /1/2 and 3 3.14, which roughly covers the range

of uncertainty of the edge energy on low energy planes,

and also indicates the difference of growth rates on

different planes of one crystal, e.g. between (111) and

(100) planes of a close-packed cubic crystal. Values of

a3 for screw dislocation growth for these values of i./kT and x, are also shown in fig. 3. c(~ is low when i+/kT =

30 because x, is then high and is also low when AlkT =

4.5 and /I = L/2 because j?lkT is then very low. F3 al-

ways has the same shape as in fig. 2 and is not plotted

in fig. 3. F4, for multiple dislocations, lies between F3

and FE.

38 13. Ll WIS

Fig. 3. Growth rntc I’L/Ro plotted against supersaturation (g I ) for the general USC. for ).:A 7. 20. II and

of l//,7’ gro\cth c‘urvcs are gibcn for p i,i2 and 3i.i4, corresponding to moderate and very low cncrgy planes.

which scrcu dislocarion growth is unimpeded. are also shov.n for each value of i.,‘XTand /I 1.:2 and 3P4.

When P//CT 5 IO, which includes all cases of growth

from the vapour, perfect crystals cannot grow at c( 2 2.

When /j//CT ?_ IO, which includes most cases of growth

from the melt or from solution, %2 < xJ and perfect

crystals can grow by two-dimensional nucleation even

at low supersaturation. However, screw dislocation

growth becomes more favourable at very low super-

saturation, because the F, and F3 growth curves cross,

as shown in fig. 2 for i/kT = 4.5.

When, due to variation of [j between planes, x 3 a,.

x3 or xq on some planes and x < x2, x3 and a4 on others,

growth is anisotropic. The strong dependence of Fz on

/I’ causes two-dimensional nucleation growth to be

strongly anisotropic, so that cubic crystals have only

the slowest growing low energy planes, and lower sym-

metry crystals form needles and plates. Screw disloca-

tions on one or two faces may generate whiskers or

platelets whose symmetry does not correspond to that

of the crystal. With screw dislocations on all faces, the

inverse relations between bi and /I’, and between /j and

0 for each plane, cause the growth rates to vary in order

of surface energy, but consistent crystal habits are not

expected if growth is impeded. i.e. for x < x5 on any

plane.

When r > c(~, CI~ or 8x4 on all planes, flat faces may

still develop because of the slow approach to F, as-

sociated with competitive capture effects. The growth

rates of different planes in monolayers set-’ are ex-

pected to be equal, and in cm set- ’ to be proportional

to the planar lattice parameters.

4. Summary and conclusions

The classical evaluation of cluster formation energies

is based on idealised nucleus shapes and on surface ot

edge energy parameters which are assumed to be inde-

pendent of cluster size. For small nuclei an atomistic

evaluation is more realistic. However. comparison for

a simple model shows that the evaluation

G, = -iliTln r+i’ [[I-4kTexp (-/{:‘4k’/‘)]

with constant [I’ agrees closely with an atomistic eval-

uation even for very small nuclei. The growth rate of

perfect planes of it small crystal is equal to the two-

dimensional nucleation rate. The growth rate of macro-

scopic crystals is proportional to the < power of the

advance rate of steps and to the _i power of the two-

dimensional nucleation rate.

The material parameters which appear in the growth

rate expressions are the edge energy coefficient /i and

the diffusion distance I,. /j is approximately related to

THE GROWTH OF CRYSTALS

the binding energy difference per atom between solid

and fluid phases iti and the surface energy per atom of

the growth plane o by /I = /1-2~. For low energy

planes Q - 46 and p - 2i/3. x, is approximately exp

(46kT) atomic sites for low energy planes.

The saturation ratio c(~ is defined, above which two-

dimensional nucleation (2DN) does not impede growth.

c(~ is strongly dependent on /?/kT, and below zz the

two-dimensional nucleation growth rate varies as

c?*j3 (or as ui* for small crystals). As c( decreases, i*

increases, and the growth rate falls steeply. For screw

dislocation growth (SDG) saturation ratios x3 for

single screws (SSDG) and x4 for multiple screws

(MSDG) can similarly be defined. For SSDG below

r3 the growth rate varies approximately as (a- 1)2. For

MSDG below c(~ the supersaturation dependence of

growth rate generally lies between (CI- 1)2 and (a- 1).

Above z2, c(~ and cz4 the growth rate is almost unim-

peded and varies as (c( - 1).

When /?/kT > 10, which is typical of growth from

the vapour on low energy planes, c(~ < x2, i.e. 2DN is

more difficult than SSDG; for IX 2 2, 2DN is negli-

gibly small. When /l/kT < 8, which is typical of melt

and solution growth, 2DN may give faster growth

than SSDG; however, at some value of CI below a2 the

two growth curves cross and SSDG becomes the faster.

When fi/kT 7 2, x1 = cc3 = 1 and growth is unim-

peded at all supersaturations.

In many practical cases only qualitative comparison

is possible with the predictions above because of lack of

data. Some examples which permit quantitative com-

parison between theory and experiment are considered

in Part I13”).

Acknowledgements

In addition to the specific references cited, acknow-

ledgement is made to many reviews, discussions and

other sources. The comprehensive theoretical and ex-

perimental accounts by Strickland-Constable36) and by

Hirth and Pound37), have been particularly valuable.

I also thank Professor F. C. Frank for discussions

which clarified initial misconceptions in my atomistic

treatment of nucleation, Professors J. P. Hirth, J.

Lothe and K. C. Russel for correspondence regarding

the nucleation capture factor, and the Plessey Company

for support and for permission to publish.

OF LOW SUPERSATURATION. I 39

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