Bond energy formulations of heterogeneous nucleation theory

BOND ENERGY FORMULATIONS OF HETEROGENEOUS NUCLEATION THEORY B. LEWIS

Allen Clark Research Centre, The Plessey Company Limited, Caswell, Towcester, Northants. (Gt. Britain) (Received December 4, 1966; revised January 21, 1967)

SUMMARY

Condensation of a vapour beam on a substrate depends on the formation and growth of stable nuclei. By expressing the volume and surface energies of the capillarity model and also the cluster energies of the atomistic model in terms of nearestneighbour bond energies, these two approaches to nucleation theory have been put into forms which may be directly compared. This has revealed that their differences are confined to: (i) continuous and discontinuous variation of critical sizes; (ii) a numerical difference in the value of supersaturation for small nuclei which has the effect in the capillarity model of predicting a larger size for the critical nucleus and a lower nucleation rate; (iii) a compensating numerical difference which is revealed in a comparison of cluster energies, using the bond energy formulation, in which the idealized shape s of the capillarity model give higher cluster energies, smaller critical nuclei and a higher nucleation rate. It is shown that under nucleation conditions the capture rate of single atoms is independent of the sizes of the growing nuclei. In practical cases the critical nucleus is seldom larger than two atoms, and the atomistic model is preferred. Calculated values of nucleation rate are given in terms of incidence rate, substrate temperature, condensate bond strength and adsorption energy. The analysis of experimental data to derive material constants is discussed.

1. INTRODUCTION For net condensation of vapour on a substrate to occur the incidence pressure of the vapour stream must exceed the vapour pressure of bulk condensate, i.e. supersaturation conditions must exist. For a given supersaturation the formation rate of nuclei exceeds the decay rate only if nuclei are above a certain critical size. This is because the stability of a nucleus increases, usually by many orders of magnitude, as the size increases from a singlet to a large cluster. Thin Solid Films - Elsevier Publishing Company, Amsterdam - Printed in the Netherlands

86

B. LEWIS

A typical value of condensate vapour pressure is 10 -2o torr, or less, which gives a supersaturation of 1015 or more if the incidence pressure is 10 -5 tort, about 1 monolayer s-2. With such high supersaturations the critical nucleus contains only a few atoms. Recent contributions to nucleation theory may be distinguished as belonging either to the capillarity model, based on the classical treatment of Volmer and Weber 1, and Becker and D~Sring2 or to the atomistic model of Frenkel 3, Semenoff 4, and Walton 5. A statistical mechanical treatment by Reiss 6 circumvents several weaknesses of the classical treatment but gives no assistance on the essential problem, the evaluation of cluster energies. According to the capillarity model, macroscopic thermodynamic properties are ascribed to the subcritical and critical nuclei in order to estimate their free energy of formation. This procedure is open to serious objection for small nucleus sizes, but as yet no one has succeeded in evaluating the potential energies and internal partition functions of small clusters, and therefore the method is used even in these cases for want of anything better. In the atomistic model, as developed at the present time, the internal partition functions (for example, associated with orientation of clusters) are neglected and the problem is presented in terms of the binding energies of nuclei. Qualitatively, the increment of binding energy for each additional atom increases with nucleus size as the number of bonds per atom increases, but actual values must be determined by experiment. On the whole, these two theories have been developed separately and the different terminology and choice of parameters have made the expressions for nucleation rate appear quite different. It seemed worthwhile to undertake a comparative study of the two models with the object of revealing the nature and extent of agreement and difference between them. The agreement has proved sufficient to give confidence to predictions of nucleation rate and critical nucleus size based on an approximate evaluation of cluster energies in terms of cohesive and adhesive bond energies.

2.

BOND ENERGY FORMULATION OF THE ATOMISTIC MODEL

If the density of growing nuclei and their rate of growth is low enough not to significantly deplete the population of single atoms by capture, then single atoms are lost only by desorption. Equating the desorption rate to the incidence rate R, we find N 1 = -R- exp -Ea Vo

(1)

kT

where Ea is the adsorption energy of a single atom on the substrate, and vo is the

BOND ENERGY FORMULATIONS OF NUCLEATION THEORY

87

vibrational frequency of an adsorbed atom on the substrate. T is the temperature of the adsorbed atom, and if the adatoms are thermally accommodated, which will normally be the case, is equal to the substrate temperature. Walton 5 has shown that the metastable equilibrium population N~ of clusters of i atoms, up to and including the critical size, is given by the statistical probability of such a group in a random distribution of Nt atoms over No sites, with a weighting factor introduced to take into account the mutual binding energy E~ of the i atoms of a cluster. Thus

N~

[NIV

Ei

- kNo] exp k-T and substituting for

N 1

(2a)

from eqn. (1)

N i ( R ) i Eiq-iEa ~oV ° e x P - - k T

5/0-

(2b)

This equation is only valid at low surface coverage, i.e. when XiN~ ~ No. Walton 5 also introduces the condition N 1 ,,, ZiN~ to eliminate a constant and obtain eqn. (2a), his equation (6). However, by putting i = 1 in his equation (4) we can obtain this relation without any restriction of the relative values of N1, N2 . . . . . N~. The binding energies E i of clusters of i atoms should be evaluated by experiment. In order to permit comparison with the capillarity model and to predict values of the size i* of the critical nucleus under various experimental conditions, we will here evaluate E~ as the sum of the nearest-neighbour bonds, each o f energy Eb, for the most favourable configuration of atoms. The bond energy E b can be evaluated for close-packed structures, in which each atom has 12 shared nearest-neighbour bonds, as a sixth of the dissociation energy per atom. Experimental values of dissociation energies of homonuclear diatomic molecules have been listed by Verhaegen, Stafford, Goldfinger and Ackerman 7 and range from half to four times E b, so evaluated. We will, nevertheless, assume that the bond energy Eb is a constant for a given condensate material and is independent of the number of bonds, recognising that this is only a rough approximation. For a single atom the cluster binding energy El is zero. A pair has one bond so E 2 = E b. A triplet with the close-packed triangular configuration has three bonds, so E 2 = 3 Eb. A fourth atom, still on the substrate can make two more bonds, so E 4 = 5 Eb. If the fourth atom jumps off the substrate on to the top of a triangular pyramid the cluster has lost E, for this atom and there are six interatomic bonds, so E 4 = 6 Eb-- E,. Which of these two has the higher energy depends on the relative values of Eb and Ea and to make predictions covering all circumstances we must consider a range of values o f E, in terms of E b. When i atom~ join to form a cluster the binding energy is equal to the product of the number and strength o f the interatomic bonds, less the substrate adsorption

Thin Solid Films, 1 (1967) 85-107

88

B. LEWIS

energy for any atoms which are not in the base layer of the cluster and are thus no longer on the substrate. Variation of the adsorption energy associated with misfit of the condensate and substrate lattices should also be taken into account; this provides the driving energy for preferential orientation and epitaxial growth.

3. BONDENERGYFORMULATIONOF THE CAPILLARITYMODEL The classical model of Volmer and Weber 1, and Becker and DSring 2 has been further refined by later workers. The most complete recent expositions are by Hirth and Pound s and by Hirth, Hruska and Pound 9. The concentration of single atoms is again given by eqn. (1). The concentration of nuclei, up to and including the critical nucleus, is then obtained in terms of the Gibbs free energy of formation AGi from the Van't Hoff reaction isotherm for equilibrium between concentrations N 1 of single atoms and Ni of clusters size i. N i = N lexp--

AGi

(4)

kT

In the derivation of this equation it is usually assumed that the population of single atoms is much greater than that of all other classes. Hirth, Hruska and Pound 9 have shown that if AG~, is evaluated in terms of N1, No and E i, and substituted in eqn. (4) the result is eqn. (2). In the classical model AG~ is evaluated as the sum of the volume energy AGI, the surface energy AG2, and the distribution energy AG3, of a nucleus size i. An essential first step is to postulate a shape for the nuclei. The formulation for cap-shaped nuclei, radius of curvature q and contact angle 0, as shown in Fig. 1 will now be derived. Fricke 1° discusses volume and surface energies for particular structures and surfaces in terms of both nearest and next-to-nearest neighbours. Considering nearest neighbours only, an atom of bulk 1

Bulk atom has binding energy "~ Zb Eb / ~/ e energy u¢

b~/~

Substrate surface

energy = 6"s

~

Surface atom has b~nding energy ½(Zb-Zc)Ei /

-- ~

Single atom has binding energy E a Q

Fig. 1. Cap-shaped nucleus of radius q and contact angle O. Z b and (Zb--Zc) are, respectively, the numbers of nearest neighbours of a bulk atom and a surface atom. Eb is the strengh of each nearest-neighbour bond. Relations between ~c, %, °j, O, ZcE b and Ea are given in the text.

BOND ENERGY F O R M U L A T I O N S OF N U C L E A T I O N T H E O R Y

89

material has binding energy ½ Z b E b if it shares bonds of strength E b with Z b nearest neighbours. I f an a t o m in the surface has ( Z b - Z¢) nearest neighbours and binding energy ½(Z b - Z c ) E b, then the unsatisfied bonding represents surface energy and can be identified with the condensate surface free energy ac. Thus, for a surface atom (5)

acs = ½ZcE b

where s is the projected effective area per atom in the surface. The surface free energy densities cr¢ at the condensate-vapour surface, cs at the substrate-vapour surface, and crj at the condensate-substrate junction can be related to E a by considering the energy change on placing a single atom on the substrate: (6)

E a = (a¢ + as - aj)s

The contact angle 0 is normally given by the Young equation (7)

as--aJ

COS 0 - -

(7c

Using eqns. (5) and (6) gives COS 0 --

2Ea - Z~Eb

(8)

ZeEb

When the nucleus contains only a few atoms the actual shape will be the closest possible approximation to the idealised cap shape. Initially it will be a planar group of atoms; a second layer will form with a small value of i when 0 is large and with a larger value of i when 0 is small. The size of the cluster at which a second layer forms depends on 0 and thus on the relative values of c~, or its analogue ½ Z c E b, and E a, just as for the atomistic model. When E~ > 2Eb the corresponding value of 0 is < 70 ° and the cap shape predicts a nucleus which, for small i, is spread too thinly over the substrate surface to be a good approximation to a possible real cluster. When E~ > 2Eb it is preferable to use a disc-shaped monolayer nucleus as considered by Hruska 11. Continuing now with the cap shape, the volume energy AG 1 in terms of the cap radius of curvature q and cos 0 = c, and in terms of the number of atoms in the cluster i and the atomic volume v is 7rq3 AG1 = - ~ - (1 - c)2(2 + c)AGv (9a) (9b)

= ivAGv

where AGv is the Gibbs free energy difference per unit volume between the vapour and condensed phases. Now AGv -

-kT

Thin Solid Films,

v

In

p

p~

-

1 (1967) 8 5 - 1 0 7

-kT

v

R

In - R~

(10)

90

B. L E W I S

where p and R are respectively the pressure and arrival rate of incident single atoms andpo and R e are the vapour pressure and departure rate of single atoms from bulk condensate. Re can be evaluated from vapour-pressure data as po (2 r~ m k T ) - ½and experimentally it is found that lnR e is linear against 1/kT. Hence we can write ZbEb Re = Ncvc exp - - - kT

(11)

where N¢v c can be found from the intercept and ½ZbEb from the gradient of the lnR e vs. 1/kT relation. Physically ½ZbEb is the binding energy per atom, Nc is the surface density of atoms, and vo is a rate-determining coefficient, characteristic of the evaporation process for bulk material. Finally, from eqs. (9b), (10), (11) and (1) NcVc

AG a = i k T In - - - - ½i(ZbEb--2Ea) Nlvo

(12)

The surface energy AG2 for a spherical cap is the sum of 2 n q 2 ( l - c ) ~¢ and rrq 2 ( 1 - c ) (1 + c ) ( % - ~ s ) , from which (crj-crs) can be eliminated by eqn. (8) to give AG 2 =

nq2(1

- c)2(2 + c)tr e

(13a)

The radius q can be eliminated in favour of i by using the relation iv = lnq3 ( l - c ) 2 (2+c). v can in turn be eliminated to give an expression in terms of s; for spherical atoms, radius a, v = ] r t a 3, s = 7~a2 SO s = ~(3V/4) g~. Hence AG 2 = i~4~(1 - c)~(2 + c)+sa¢

(13b)

or substituting for cos 0 from eqn. (8) and ~ s from eqn. (5) AG2 = i~2[(ZoEb-- Ea)E(ZcEb + 2Ea)] ~r

(13c)

The distribution energy AG3 is a term first considered by Lothe and Pound 12 and represents the entropy involved in the distribution of Nx atoms over the N O sites. Hirth and Pound s have shown that when Z iNi ~ No AG 3 = - k T In No N1

(14)

Adding AG1, AGE and AG3 gives -AG~ = k T In No - - + i k T l n Nlvo- + i (ZbEb-2E~) N1 Ncv¢ -

-

(15)

i~2[(ZcEb- Ea)2(ZeEb + 2Ea)] ~

and from eqn. (4) N~ --

NO

=

[Nlvo\ ~ ½i(ZbEb--2E~)--i~2[(ZcEb--E~)2(ZcEb + 2E~)] ~ 1 - - / exp \Ncvc ] kT

(16a)

BOND ENERGY FORMULATIONS OF NUCLEATION THEORY

91

Substituting for N1 from eqn. (1)

Ni No-

[ R \i

½iZbEb_i~2[(ZcEb_Ea)z(Z¢Eb+2Ea)]~ ~ N ~ ) exp kT

(16b)

The equation has been derived in this form to facilitate comparison with eqn. (2), but it is fundamentally the same as conventional formulations for the capillarity model. Numerical evaluation is identical, using values of R from incidence pressure, Ncv¢ and ½ZbE b from vapour-pressure data, ZcEb/S = O"e from surface free energy density and E~ from adsorption energy or (via eqn. (8)) from the contact angle. The relation between E a and 0 has not previously been recognised. The alternative of a disc-shaped nucleus will now be considered. A discshaped monolayer nucleus of radius f containing i atoms has an area 7tf2. The periphery is 2nf and contains atoms of diameter 2a where na 2 = s. Eliminating3' and a, the periphery has a length 2(ins) ÷ and contains i~n atoms. Thus the surface and periphery energy, AG2, is AG 2 = istrc + is(trj - trs) + 2(ircs)~e

= i(ZcE b-Ea) +½i½1zZpEb

(17a) (17b)

where e is the edge free energy per unit length and Zp is the corresponding co-ordination number deficiency of an edge atom, with ½ZDEb = 2ae. AG1 and AG2 have the values given by eqns. (12) and (14). Adding AG1, AGE and AG3 to obtain AGi and substituting in eqn. (4), Ni - (NlV°] i

NO

\Ncv¢ / exp

½i(ZbEb--2ZcEb)--½i½rtZpEb kT

(18a)

Substituting for N1 from eqn. (1)

Ui No -

( R )i N~cv% exp

½i(ZbEb_2Z~Eb+2E,)_½i-~nZpEb kT

(18b)

4. COMPARISON OF ATOMISTIC AND CAPILLARITY MODELS

If the capillarity and atomistic models are equivalent then both the temperature-independent pre-exponential terms and the temperature-dependent exponential terms in eqns. (2) and (16) or (18) should be the same. Comparing the pre-exponential terms we find that (Arcvc)-i in eqns. (I 6) and (18) has replaced (Novo) -i in eqn. (2). The experimental value of N¢v¢, obtained by fitting vapour-pressure data to eqn. (11), varies from 1029 t o 1031 c m - 2 s - 1 for different materials, whereas Arc ~ No ~ 1015 cm-2 and v0 is generally taken as 10t2s - t giving Novo ~ 1027 c m - 2 s - 1. As discussed by Knacke and Stranski 13, the simple model of evaporation on which eqn. (11) is based does not take into Thin Solid Films, 1 (1967) 85-107

92

a. LEWIS

a c c o u n t the possibility o f indirect e v a p o r a t i o n by successive transitions o f a t o m s f r o m strongly b o u n d to m o r e weakly b o u n d positions. The overall p r o b a b i l i t y of escape is the p r o d u c t o f the probabilities o f intermediate transitions a n d the exp o n e n t is still the binding energy per a t o m ½ZbE b, but the pre-exponential factor is increased. A small cluster differs f r o m b u l k material in this respect in a d d i t i o n to the effect o f surface energy. I n considering the volume energies o f clusters with i = 1, 2 . . . . . different values, (Ncv~)~, (Nova)2 . . . . . should be assigned to the pree x p o n e n t i a l factor for each size. O f these, (N~v~) 1 = (Nova)2 = N~vo because for these cases step-wise e v a p o r a t i o n is impossible. W h e n i is small the error in using (NoVo) -~, as in the atomistic model, is small, b u t there is a significant error in using (N~vc)-~, f r o m v a p o u r - p r e s s u r e data, o r in any equivalent e v a l u a t i o n o f AGv f r o m m a c r o s c o p i c data. This has the effect, in the capillarity model, o f predicting a larger size for the critical nucleus a n d a lower nucleation rate. F o r the atomistic m o d e l the cluster energy E~ which a p p e a r s in the e x p o n e n tial t e r m can be evaluated as

El = X~Eb-- hEa

(19)

where the n u m b e r X~, o f i n t e r a t o m i c b o n d s a n d the n u m b e r h, o f a t o m s off the substrate are d e t e r m i n e d b y the a t o m i c configuration which maximises the value o f E i. C o m p a r i n g the e x p o n e n t i a l terms in eqns. (16) a n d (18) with eqn. (2), we find for the c a p - s h a p e d nucleus

E~ = ½i(ZbE b - 2Ea) - i~2[(Z~Eb-- Ea)2(Z~Eb + 2E,)] ~

(20)

a n d for the disc-shaped nucleus

E i = ½i(ZbE b - 2 Z ¢ E b ) - ½i½~zZpEb

(21)

W e are n o w in a p o s i t i o n to c o m p a r e cluster energies for the atomistic m o d e l with those for the capillarity m o d e l with b o t h shapes o f nucleus. I f we take, as an example, the case o f a structure with c l o s e - p a c k e d planes p a r a l l e l to the substrate a n d for which Zb = 12, Z~ = 3 a n d Zp = 2 a n d if we also choose a value for Ea/E b we can evaluate E i for all three models. It is convenient to express b o t h E i a n d Ea in units o f E b as the r e d u c e d energies e a = E,/Eb a n d ei = Ei/Eb. W e will consider the case e, = 1.5 which gives a h e m i s p h e r i c a l c a p with 0 = 90 °. Values o f .V~, h a n d e~ for the atomistic m o d e l are given in T a b l e I. F o r the capillarity m o d e l s TABLE

I

C L U S T E R E N E R G I E S e i - - E i / E b FOR T H E ATOMISTIC M O D E L FOR T H E CASE e a =

i Xi h ei

1 0 0 0

2 1 0 1

3 3 0 3

4 5 0 5

5 7 0 7

6 9 0 9

7 12 0 12

8 14 0 14

9 10 11 12 21 24 26 31 3 3 3 4 16.5 19.5 21.5 25

Ea/E b --

13 33 4 27

1.5

14 35 4 29

15 40 5 32.5

93

BOND ENERGY FORMULATIONS OF NUCLEATION THEORY

e~ = 4.5i-4.75i ~ for a cap-shaped nucleus and e~ = 2.65i-rci ½ for a disc-shaped nucleus. On adding iea we obtain the total binding energy e~+ ie,, which is the parameter appearing in eqns. (2b), (16b) and (18b) which give Nl in terms of R and T. In Fig. 2(a) the points were obtained for the atomistic model and the lines for the capillarity models. The points are below the lines because while i is small a real cluster cannot conform to the ideal shapes of the capillarity model. For large values of i the spherical cap and the atomistic models both give values of binding energy converging towards 6 bonds per atom, as for bulk material. For small values of i the monolayer disc model agrees extremely well with the atomistic model but beyond i = 15, when a second atomic layer has formed, it gives too low a value of binding energy. When ea -- 3 the disc model gives excellent agreement for all values of i, whereas the spherical cap model (0 = 0) predicts Ei + ie, = 6 for all values of i. (o)

6O

(b) Spherical

.#

ea = 1.5

+

eo = 6 - 3.17 [ "1/3

I

4o

L

de~ ~T

cop

8

Mono/oyee disc dd--~JT"+ e~ = 4 . 5 - 1 / 2 T( i "1/2

Monoloyer disc iea = 4.15{ - ~ IV2.

U 20 Atomistic

Ae~ +' + e= - "%el" + 1.5 "o

g~ 0

0

10 Cluster

size

[

15

5

10

Ctuster

15

SIZe[

Fig. 2. (a) T o t a l cluster b i n d i n g energy e i + ie a a n d (b) differential t ot a l cluster b i n d i n g energy A e i i + l + ea o r d e i / d i + e a for atomistic, spherical cap and m o n o l a y e r disc models. Energies are expressed in units o f the n e a r e s t - n e i g h b o u r b o n d strength Eb, i.e. e i = E i / E b a nd ea = E a / E b. The case illustrated is a c o n d e n s a t e with close-packed structure E a = 1.5 E b a n d c ont a c t angle 0 = 90 °.

The essential difference between the models is that one uses only discrete arrangements of atoms, whereas the others employ simple idealised geometrical shapes for the cluster. This difference is most striking when we plot the differential or incremental energy as in Fig. 2(b), and the capillarity models also gives higher values than the atomistic model. The effect is that the capillarity models predict a continuous variation of critical size and, for any given conditions, a smaller critical size and a higher nucleation rate than the atomistic model. This compensates or overcompensates for the differences arising from the higher value of Ncvc compared with Novo. The discontinuities of the atomistic model are more realistic than the smooth curves o f the capillarity models. It is also notable that beyond i = 6, whereas some atoms make 4.5 or 5 bonds others can only make 3.5 bonds, so that when the condition for stability is Thin S o l i d Films, 1 ( 1 9 6 7 ) 8 5 - 1 0 7

94

B. LEWIS

that all atoms have more than 3.5 bonds, the atomistic model predicts that the critical size, beyond which clusters are stable, is quite large. In fact, growth can only occur at a lattice defect at which there is sufficiently high binding energy for atoms to be stable and which never grows out to form a perfect crystal plane. Spiral growth at a screw dislocation is a well-known example. We note that when i = 1, E~ should be zero. The capillarity models with ea = 1.5 are very nearly correct. Substituting El = 0 and, as discussed above, Ncv~ = Novo, eqns. (16a) and (18a) correctly give N i = N1 when i = 1, which is a convincing justification for the inclusion of the distributional term A G 3 ; without it, when i = 1 and Ei = 0 Ni = No, which is wrong. We conclude that, although our evaluation of cluster energies for the atomistic model may not be correct, it is a reasonable approximation and is essentially similar to that for the capillarity models. We will now use the atomistic model, with its discontinuous variation of cluster energy with size, to predict nucleation rates.

5. NUCLEATION RATE

The value of i for which AG~ is a maximum and Ni is a minimum is written i = i*, designating the critical nucleus. The smallest stable cluster is one atom larger that the critical nucleus and has an equilibrium population, neglecting growth, higher than N~.. Its formation rate is the nucleation rate. The capture rate of single atoms by a nucleus of size i, W x~, is usually evaluated as the rate at which single atoms from the population N1, moving hop distances d with a hop frequency vI e x p - (Ed/kT), cross a boundary of length b i equal to the diameter 14 or the periphery s of the nucleus:

E~

WI~ = N1 dbivl exp - - kT

(22a)

or, substituting for N1 from eqn. (1) R Ea-E d Wli = - - d b y x exp - Vo kT

(22b)

However, this expression does not take into account that capture is localised at the capturing nucleus, and that an atom moving in random hops over the substrate has a high probability of visiting several sites in a group of adjacent sites, if it reaches any one of them. Since capture can only occur once, if the area A = (vl/Novo) exp [ ( E , - E d ) / k T ] traversed by a single atom in its lifetime is large compared with the area of the capturing nucleus, the capture probability and capture rate are nearly independent of the nucleus size. Thus in eqn. (22) we put b i = d, the value appropriate to capture at a single site, and d 2 = l / N o and obtain

BOND ENERGYFORMULATIONSOF NUCLEATIONTHEORY

WI -

Rvl

Ea-E d exp . - Nov o kT

95

(23)

dropping the subscript i, which is no longer relevant. Now, the right-hand side of eqn. (23) is R A , so the nucleus is capturing the number of atoms incident in a lifetime area A which surrounds it. Note that although we can assign a size to this capture area this does not imply a definite or fixed shape. The capture rate of atoms executing a random walk can be numerically evaluated, but the arrival points, paths and capture sites of particular atoms are not predictable. The limits of validity of eqn. (23) depend on the value of A and on the size and surface density of the capturing nuclei. When A is comparable with the nucleus size the effective capture area of a nucleus is larger than A. (When A is small compared with the nucleus size the capture area is nearly equal to the nucleus size, and the capture rate is the direct impingement rate.) When the surface density Ng of growing nuclei is such that Ngd = 1, covering the whole substrate area, all single atoms are absorbed by growth and this, therefore, represents the saturation density of nuclei, as pointed out by Walton, Rhodin and Rollins 15. In the case of nucleation, these validity limits correspond, respectively, to nucleation rates too small to detect, and to a limit of validity of nucleation theory itself, so we can write the formation rate of nuclei (neglecting growth) as W1 from eqn. (23) multiplied by the population N~, of the critical nuclei from eqn. (2). We should also include the Zeldovich factor, which corrects for the disturbance of the equilibrium numbers of critical nuclei by growth. For small nuclei the Zeldovich factor is approximately l/i .5 so we can allow for it by defining the nucleation rate J~, as the rate at which single atoms combine to form nuclei. Hence Ji* _ ( R / I'+1 Ei,+(i*+l)E,,-Ed Nov1 \Novo/ exp kT

(24a)

We are interested in J as a function of R and T with bond strength E b and adsorption energy E a as material parameters. To plot generalised curves it is convenient to use reduced units, Ji* Ji* = l o g - Nov1

r =log---

R

ei,Eb = Ei, eaEb = Ea

NOV0

1

t

log e

--

kT

edE b = E d

and rewrite eqn. (24a) as Ji* = (i* + 1)r + [el, + (i* + 1)ea - ed] E~ t Thin Solid Films, 1 (1967) 85-107

(24b)

96

B. LEWIS

The equation Novo

--

= exp

R

AEi~+ 1 + Ea kT

(atomistic model)

(25a)

or its reduced equivalent -

r = (aei i + 1 + e,) Eb

(25b)

t

represents the transition relation for i* = i ~ i + 1. Physically, at a transition the rate of formation of a cluster with (i+ 1) atoms is equal to its decay rate, so that the populations Ni and Ni+l are equal. We must also consider the case when condensation conditions (i.e., R, T, E a and Ed) are sufficiently favourable for the population of single atoms to exceed one per capture area A so that capture is more probable than evaporation. This is the case of a time-dependent adatom concentration considered by Hirth and Pound 8 : Single atoms accumulate on the substrate and if i* is defined as the size having the smallest quasi-equilibrium population, eqns. (2a), (16a) or (18a) still apply. However, eqns. (1), (2b), (16b), (18b) and (24) dot not apply because the population N l is determined by capture rather than by evaporation. We will define a "half condensation" transition occurring when, at the earliest stage of condensation, half of the incident atoms are lost by mutual capture and half by evaporation. Very soon, the growing nuclei will capture all the incident atoms, so this case corresponds to virtually complete condensation. For nucleation, when i* = 1 Rvl

½R = dl = N1W1 = N1 Novo exp

Ea-Ed kT

(26)

and for evaporation Ea

½R = N l v o exp - kT

(27)

The transition relation is thus Novo R

2Ea- E a

- e x p -

kT

(28a)

or

-- I" = (2e a - ed) Eb

(28b)

t

when vx = Vo. And, neglecting the trivial factor ½, Jl = r

(29)

An alternative view of this situation is that the transition relation, eqn. (28), is the limit of validity of eqn. (24), and is obtained from eqn. (24) by puttingj x = r

BOND ENERGY FORMULATIONS OF NUCLEATION THEORY

97

and i* = 1. W h e n e a - e a > 1 eqn. (24) can give j2 = r w h e n i* = 2 and i* = 1 does not occur, i.e., pairs are never stable; the half-condensation transition relation for this case is

(30)

- r = ½(e2 + 3 e . - ea) Eb

Similarly, when e a - e d > 2 e 3 - 3 e 2 , i* = 2 does not occur and triplets are never stable; this situation requires the binding energy o f a condensate a t o m to the substrate to be greater than to a plane surface o f condensate atoms. In order to reduce the number o f independent parameters w e will assume a constant ratio between the values o f the related quantities Ea, surface adsorption energy, and E d, surface diffusion activation energy. E d is always less than Ea and we will assume a value Ea/3. Scole of ~ 0

2

for r : - 1 0 , 4

6

R=200 ~ s-~ B

10

12017

1.2

1015

1.4

1013

"7 ~

1 1.6

1011 o4

,

log ~,

O O

g,

ul 1.8

E 2

IO 7

2.2

IO 5

II c

o

d i

~j

2.4

o 3 ,"

2.6

"o 2.8

o_ ~ ~,E u

3

O-a "-i .*~

3.2

O-5 -8

3AI

0-7

u

-0.2r

-0.4r

-0.6r

-O.8r

(Reduced t e r n p e r o t u r e ) -I

Eb

t-=

-r

-1.2r

log e Eb

RT

Fig. 3. N u c l e a t i o n rate as a f u n c t i o n o f reciprocal t e m p e r a t u r e for a range o f v a l u e s o f adsorption e n e r g y and for a fixed incidence rate r = l o g R/Novo. Scales are also given for the particular incidence rate 200 A s -1. T h e b r o k e n line is Ji* = 105 c m - : s -1 and b e l o w this line n u c l e a t i o n rate is t o o l o w to detect.

Thin Solid Films, 1 (1967) 8 5 - 1 0 7

98

B. LEWIS

Reduced nucleation rate j~, evaluated from eqn. (24b) for the atomistic m o d e l is plotted against reduced b o n d strength and temperature Eb/t as variable with a fixed value o f r as scale factor in Fig. 3, and against r as variable with a fixed value o f Eb/t as scale factor in Fig. 4. The value o f R for an incidence rate o f 200A s - 1 is about 1017 c m - 2 s - ~. If Novo = 1027, the corresponding value o f r is - 10. The range o f practical values of

1 0 -3

Incidence 10 2 10 7

rate R (cm-2s -I) for ~9=5 1012 1017 1022 10 27 10 27 1022 cfl

:L:;-2

o17£ 1012

tl t ~z

J 10 7

c

_,

102

o

c

u~

"0

K

-~

I0 -3

0 -8 z

-:

0-~3

10-18 -1(

[0_23 t--6-~

-5-~-

-4-~-

-3-~-

-2~

--E---b-bt

0

Reduced incidence rote r'= to 9 e ~-oVo Fig. 4. Nucleation rate as a function of incidence rate for a range of values of adsorption energy and for fixed values of bond strength and temperature. Scales are also given for Eb/t = 5, e.g. aluminium at r o o m temperature. The broken line parallelogram represents experimentally attainable limits of R and Ji* for Eb/t = 5. The diagonal broken line is the detection limit for variable Eb/t, and values of Eb/t are marked along this line.

r is between about - 9 and - 15. Scales ofEb/t andj~, for r = - 10 are included in Fig. 3. Practical values o f j,. range from an upper limit o f r, (assuming v 1 = Vo) to a lower experimental limit for detection by electron m i c r o s c o p y o f about 1 nucleus p - 2 (1000s)- 1. At 200 A s - t the total a m o u n t o f material incident in this period is equivalent to a 20 # deposit. This gives Jm;n = 10 s c m - 2 S - 1 andjml" = _ 22. The

BOND

ENERGY

FORMULATIONS

OF NUCLEATION

99

THEORY

even lower figure o f Jmi, = 1 c m - 2s- 2, which has sometimes been used in discussing critical condensation experiments, corresponds to Jmi, = - 2 7 . N o t e t h a t j and r are always negative. The b o n d strength Eb and reduced temperature t occur together as the dimensionless quantity Eb/t. Some values o f E b evaluated as !6 the dissociation energy, and corresponding values o f Eb/t for a range o f temperatures, are given in Table II. Practical values lie between 1 and 100. F o r a single material, variation of temperature from - 196 to 500 °C changes the value by a factor 10, e.#., from 30 to 3 for aluminium. A much wider range o f experimental conditions can be covered by varying substrate temperature than by varying incidence rate. TABLE II & VALUES OF THE REDUCED BOND STRENGTH AND TEMPERATURE TERM - t

Atom

Eb (eV)

--196 77

--148 125

K Na Pb AI Ti Mo

0.15 0.2 0.3 0.5 0.8 1.2

9 13 20 32 52 78

5 8 12 20 32 48

--73 200 3 5 8 12 20 30

e

~

log ~

Eb

27 300

227 500

527 °C 800 °K

2 3 5 8 13 20

1.5 2 3 5 8 12

1 1.5 2 3 5 8

F r o m eqn. (24) the gradient ofji, against r is (i* + 1), as can clearly be seen in Fig. 4. W h e n j~. = r the gradient becomes unity and we have virtually complete condensation. As the incidence rate is decreased, i* takes successively larger values. W h e n i* > 2 the next possible value is five or more, depending on the value o f ea, and the nucleation rate rapidly becomes very small. W h e n the independent variable is Eb/t, as in Fig. 3, eqn. (24) shows that the gradient is e~. + ( i * + 1 ) e a - e a. The gradient is zero when Ji* = r, and increases very rapidly with the value o f i*, so the nucleation rate plot shows long regions with j~. = r, shorter regions with i* --- 1 or 2, and then plunges steeply with i* = 5 or more. Figures 3 and 4 ale for the atomistic model. F o r the capillarity model, using eqn. (20) or (21) for E~ the predicted nucleation rate for any chosen condition is somewhat higher and there is a continuous variation o f the slope o f the curves and the size o f the critical nucleus. To obtain values o f nucleation rate from Figs. 3 and 4 the appropriate values o f E J t and r must be inserted. As an example, in Fig. 3 scales are included for R -- 200A s - 2, r = - 10, and the lower limit o f detectable nucleation r a t e j = - 22 is marked as a broken line. Values o f Eb/t are shown in Table 1I, and for materials with b o n d energies below 0.3 eV the whole area o f interest in Fig. 3 can be explored by varying temperature over the range - 196 ° to 500 °C. F o r materials with Eb > 0.5 Thin Solid Films, 1 (1967) 85-107

100

B. LEWIS

eV, and high values o f E~, a substrate temperature over 500 °C is required to make Ji* < r.

Figure 4 includes scales o f r and j for Eb/t = 5, which is the case o f aluminium at r o o m temperature. The experimental limits o f attainable values o f r and j for Eb/t = 5 have been marked on Fig. 4 as a broken-line parallelogram. For other values o f Eb/t a broken line has been drawn which represents the lower limit of detectible nucleation rate, j = - 22 for r = - 10. A n important feature o f e q n . (24) is the dependence o f nucleation rate on E~. In order to show this dependence, the variables r and t can be combined into a single parameter

kT

rt

-

e

log

R Nov o

log

-

kT -

log e

log

Novo ---

R

A high temperature and a low incidence rate are external factors which oppose condensation. W h e n condensation occurs, the parameter - r t is thus a measure o f the driving force of the substrate and condensate system towards condensation and will be called the condensation energy o f the system. In Fig. 5(a) nucleation rate jl./r is plotted against e~ w i t h - r t as parameter. Zinsmeister ]4, has pointed out that, when E, is low, pair evaporation must be t r =1

i o

*~

:-Tz

r

2 2.5

3

3.5

/ / / / / / G / ]:' / / / f

II

lie"

I/

,"

#

s

L

g

•

-~ a

.-"

o

,'

', /

,", •

tr

-~=4

,,

,'

•

,",

,"•

,'1 7 L

,, -110 o

-"

,~_.

,"

1°-~-

I

i

I

I

1

1.5

2

2.5

(a)

eO

i -2Eb .~

~

-3Eb

'

,

I

energy

10 2 e~ ~

s

0.5

Adsorption

i

,,

4

4.s

-

~-

a

•

-

,1

#

,,

• "

,,

,'

......

e

••

,

s

,'

/

,o

,r

tw / / _ - _--__,___~

(IJ

~

4

10 -s

3

"~

"6 -~ ~

.co _4Eb ~

-5E b O

0.5 I Adsorption

1,5 2 energy

2.5 eo

(b)

Fig. 5. Relations between nucleation rate, adsorption energy a n d c o n d e n s a t i o n energy. C o n d e n s a tion energy tr = (kT/log e) log (R/No ~o) is expressed in units o f the nearest-neighbour b o n d energy El0, a n d a d s o r p t i o n energy e a = Ea/E b.

considered. Evaporation o f a pair requires energy 2E,, whereas dissociation and migration one hop away requires Eb+Ed. Hence when e, = E,/Eb < 0.5 pair evaporation becomes more likely than dissociation. Due to neglect o f this process eqn. (24) is not valid when e, < 0.5, so predictions for e~ < 0.5 have been excluded from Fig. 5(a). The lines are shown dotted for i* > 2 because the information in Figs. 3 and 4 from which Fig. 5(a) was derived is insufficient for complete accuracy.

3

BOND ENERGY FORMULATIONS OF NUCLEATION THEORY

101

A scale for r = - 1 0 and the lower detectible limit j = - 2 2 are shown. Except when condensation is complete, a change of ea of 0.5 changes nucleation rate by about five orders of magnitude. It follows that for the appropriate value o f - rt nucleation can discriminate strongly between different values of ea either at different sites or for differently oriented nuclei on a single-crystal substrate. This provides a physical basis for decoration and epitaxy respectively. Figure 5(b) was derived from Fig. 5(a) by plotting corresponding values of condensation energy - rt and adsorption energy ea for complete condensation (full line) and for just detectable nucleation (broken line). Figures 3, 4 and 5(a) are all drawn for e a = ea/3 but complete condensation boundary lines for e d = eJ 2 and e d = 3e J 4 are included in Fig. 5(b).

6.

APPLICATION OF NUCLEATION THEORY

Nucleation experiments The smallest clusters that can be detected by electron microscopy are ,~ 10A diameter and contain about 6 atoms. Field-emission microscopy is somewhat less sensitive. In the early stages of growth, single atoms combine to form nuclei at the rate J~, appropriate to the conditions of deposition, and a range of sizes of nuclei develops. The number of nuclei increases asymptotically towards a saturation density (Novo/vl) exp ( - ( E ~ - E o ) / k T ) . Either before or after this stage agglomeration occurs and fresh nucleation may be seen in denuded areas of substrate between large islands. The circumstances in which the nucleation rate J~. can be determined are limited to the early stages when nucleation density is still increasing linearly with time. Under these conditions i* can be obtained from the rate-dependence of J, and E a and E~ from the temperature-dependence of J and one or more transition relations. Walton, Rhodin and Rollins ~5 have measured nucleation rates for silver on rocksalt and found a transition from i* = 1 to i* = 3. The values Novo = 1027, E 3 = 2.1 eV, E~ = 0.4 eV and E d = < 0.2 eV can be obtained from these measurements. The absence of i* --- 2 is ascribed to lack of fit with the substrate for this configuration which reduces the value of E 3 + 3Ea. These are the only nucleation rate measurements known to the author from which a value of i* > 1 can be deduced. Complete condensation The condition for complete condensation is the most straightforward prediction to apply to experimental conditions. Normally, this depends on the product rt and on 2Ea - Ea, by eqn. (28). When Eb < Ea -- Ed, pairs are never stable and eqn. (30) applies. The dependence on EJEb is shown in Fig. 5(b). -

Thin Solid Films, 1 (1967) 85-107

102

B. LEWIS

Critical condensation In critical condensation experiments, starting from a condition when condensation does not occur, incidence rate is raised at constant substrate temperature, or substrate temperature is lowered at constant incidence rate until condensation is observed. The relation between critical rate and reciprocal temperature is usually linear and the gradient yields a critical condensation energy. The problem is to interpret this relation in terms of the material constants. In his classic paper on this subject, Frenkel 3 introduced the concept of adsorbed single atoms and pairs, with equilibrium populations determined by incidence rate and lifetime. He interpreted a relation which defined a validity limit for his equations as the condition for critical condensation. Using the symbols and terminology of the present paper, his relation gives a condensation energy of Ea + Eb, which is the same as that for the transition i* = 1 ~ 2. Frenkel's treatment is discussed in Appendix II and shown to have several defects, but since his relation has often been used with critical condensation measurements it is shown in Fig. 5(b) as a chain line. Our acceptance of a range of critical sizes and nucleation rates dependent on condensation energy makes any critical size transition inadequate as a universal condensation condition. Indeed, as pointed out by Fuchs 16, in a paper which is a reasonable summary of the present position, both nucleation and growth are concerned in critical condensation and a particular detection limit is involved in any experiment. Yang, Birchenall, Pound and Simnad 1v have suggested Ji* = 1 nucleus c m - 2 s -1 as the critical condensation condition, but this ignores growth and can only be considered as a rough approximation. Eqn. (24) gives the nucleation rate and if eqn. (23) were valid for growth a critical condensation equation could readily be established. However, eqn. (23) written as W 1 = RA shows that after a time • --- No/R the number of atoms captured is equal to ANo and hence the size of the nucleus becomes comparable with the capture area A. In critical condensation experiments R is typically between 10 ~2 a n d 1016 c m 2 S -1, N O is 1015 c m - 2 , and the experiment time is between 10 3 and 104 s. Hence z is exceeded and eqn. (23) becomes invalid. Zinsmeister 14 has appreciated that critical condensation is essentially a growth problem and has shown how the condensation coefficient may remain low for a certain period and then increase sharply towards unity; and that the required waiting period varies steeply with incidence rate and with temperature. However, he restricted his attention to cases in which pairs do not dissociate, i.e. i* = 1, and the form of his results does not readily permit the evaluation of adsorption energy from critical condensation measurements. The capture factor he uses is also ot doubtful validity. Thus nucleation theory has not yet been properly applied to critical condensation.

BOND ENERGY FORMULATIONS OF NUCLEATION THEORY

103

7. CONCLUSIONS It has been shown that the agreement between the capillarity and atomistic models of nucleation theory is wider than the usual formulations suggest. The fundamental concepts upon which they are based are, in fact, identical. The synthesis presented here, in which macroscopic data is applied to the atomistic model, combines the merits of both approaches, but does not, of course, eliminate their limitations. The principles involved in the derivation of a critical condensation equation have been outlined and this problem is now being examined. Growth equations for the conditions which exist when the condensation coefficient approaches unity have also been studied and will be published elsewhere.

ACKNOWLEDGEMENTS

The author wishes to thank Dr. V. Halpern and Dr. G. Zinsmeister for helpful discussion and his colleagues Mr. D. S. Campbell, Mr. D. J. Stirland and Mr. B. N. Chapman for their active interest and criticism. He also thanks the Plessey Company Limited for permission to publish. This work was supported by the Ministry of Aviation.

APPENDIX I

List of symbols A

substrate area traversed by a single atom in its lifetime atomic radius b~ linear size parameter for a cluster of i atoms C cos 0 d hop distance of a migrating atom adsorption energy of a single atom e~ reduced adsorption energy of a single atom Ea/Eb strength of a nearest-neighbour interatomic bond Ed activation energy for surface diffusion reduced surface diffusion energy Ea/Eb ed binding energy for a cluster of i atoms Ei reduced cluster binding energy EJEb ei AE~ +1 increment of cluster binding energy for the (i+ 1)th atom Ael +1 reduced incremental binding energy AE~+ 1/Eb radius of disc-shaped monolayer nucleus f a

Thin SoRd Films, 1 (1967) 85-107

104 AG~ AGv AG 1 AG 2 AG 3 h i i* Ji j~ k m Arc N~ Ng No p Pe q R Re r s T t o W~ ~

Xi Zb Zc Zp e 0 Vo vl vc ~ ~j crs

B. LEWIS Gibbs free energy of formation of a cluster of i atoms volume free energy difference between vapour and condensed phases volume free energy of a cluster surface free energy of a cluster distributional energy of a cluster number of atoms in a cluster which are not on the substrate number of atoms in a cluster number of atoms in a critical cluster rate of formation of single atoms into clusters of i atoms reduced nucleation rate log (J/No Vo) Boltzmann's constant atomic mass surface density of atoms of bulk condensate surface density of clusters of i atoms surface density of growing nuclei surface density of substrate-adsorption sites vapour pressure of incident beam of atoms vapour pressure of bulk condensate radius of curvature of cap-shaped nucleus incidence rate of single atoms departure rate of single atoms from bulk condensate reduced incidence rate log (R/Novo) projected effective area per atom absolute temperature of substrate reduced temperature log e/kT atomic volume capture rate of single atoms by a cluster of i atoms number of interatomic bonds of a cluster of i atoms co-ordination number of an atom of bulk condensate co-ordination deficiency of a surface atom cf. Z b co-ordination deficiency of an edge atom cf. Zb edge free energy density of condensate contact angle between condensate and substrate rate constant associated with desorption rate constant associated with surface diffusion rate constant associated with evaporation of bulk condensate surface free energy density of condensate-vapour surface surface free energy density of condensate-substrate interface surface free energy density of substrate-vapour surface period in which 1 monolayer of atoms is incident on the substrate

BOND ENERGY FORMULATIONS OF NUCLEATION THEORY

105

APPENDIX I1

Frenkel's theory o f adsorption and condensation Frenkel 3 laid the foundation o f nucleation theory, and his results have been widely quoted. In this discussion we will use his symbols. He introduced the concepts o f lifetimes x --- "co exp (uo/kT) for a single a t o m with absorption energy Uo, and "ci exp [(Uo+ iAui)/kT] for an a t o m with i neighbours and b o n d strength Aui, which provide the basis for the present paper. Following Frenkel, let us consider the numbers o f single atoms n o and paired atoms ni a m o n g a total o f n atoms on a substrate o f area s. I f tro is the area occupied by an atom, and there is no cohesive energy (Le. Au I = 0), then

no = n [ 1 - ( n - 1 )

?]

= n-n I

nl = n ( n - 1) ao _.._ n2 tro s

(la) (2a)

s

These equations are only valid when the total coverage is low, i.e. nCo/S ~ 1, so we can write them in alternative f o r m as /1o = n

(lb)

nl = n 2 tr° --

(2b)

s

Here, and below, only the equations numbered (-a) are in Frenkel's paper. Introducing finite cohesive energy Au~, the probability o f occurrence o f the associated state is enhanced by a factor exp (Aul/kT). Frenkel expresses this condition by replacing ~o by an effective a t o m area AUl tr = cro exp - - -

(3a)

kT

and, from eqn. (2), obtains nl .

n2 tr

.

.

s

. n2 tro exp A- -u 1

s

kT

(4a)

If, instead, we use eqn. (2b) we find ao Aul nz = n°2_tr = n 2 - e x p -

s

s

(4b)

kT

Frenkel then uses eqn. (4a) in conjunction with the equilibrium relation dn -

-

dt

= 0 = v-otno-~'n 1

Thin Solid Films, 1 (1967) 85-107

(5a)

106

B. LEWIS

where v is the incidence rate o f single atoms, ~ is the loss rate o f single atoms by desorption and 0( is the loss rate o f paired atoms by dissociation and desorption. He finds a singularity when 0~2S

v, - 4 a ( ~ - ~ ' )

(6a)

and takes this as the condition for condensation. N o w ='/= ,~ 1 so eqn. (6a) can be rewritten v~ =

(~+~')s

(6b)

4a

and substitution into eqns. (4a) and (5a) then gives n 1 = no = ½n and ng/s = ½. The singularity which gives eqn. (6a) is actually a m a x i m u m value o f no which occurs because the expanded area c is used, rather than %, for all the atoms on the substrate. It is preferable to consider the factor exp ( A u l / k T ) as increasing n a by increasing the associated lifetime o f pairs but leaving no unchanged, and it is now clear that eqn. (4b) should be used instead o f eqn. (4a). Confirmation is provided by comparing with the treatment o f W a l t o n 5. Substituting i = 2, N o = s / % , and E2 = Aul in eqn. (2a) of the present paper gives eqn. (4b) above. Equation (4b) in conjunction with eqn. (5a) has no singularities but gives n 1 = n o (which defines the transition i* = 1 ~ 2) when s s Aul vK = (ct + ~') - = (~ + ~') exp a ao kT

(6c)

Neglecting ~' and converting the rest o f eqn. (6c) into the symbols o f the present paper (vK = R~, o~ = Vo e x p - ( E a / k T ) , S/~o = No, Aul = Eb) gives Eb + E a

R~ = Novo exp . . . . .

kT

(6d)

which is eqn. (25) o f this paper for the transition i* = 1 -~ 2.

REFERENCES 1 2 3 4 5 6 7 8

M. VOLMER AND A. WEaER, Z. Phys. Chem., 26B (1925) 277. R. BECKER AND W. D()RING, Ann. Phys., [5] 24 (1935) 719. J. FRENKEL, Z. Phys., 26 (1924) 117. N. SEMENOFF,Z. Phys. Chem., B7 (1930) 741. D. WALTON,J. Chem. Phys., 37 (1962) 2182. H. REISS, J. Chem. Phys., 20 (1952) 1216. G. VERHAEGEN,F. E. STAFFORD, P. GOLDFINGER AND A. ACKERMAN,Trans. Faraday Soc. 58 (478) (1962) 1926. J . P . HIRTH AND G. M. POUND, Condensation and Evaporation, P e r g a m o n , Oxford, 1963.

BOND ENERGY FORMULATIONS OF NUCLEATION THEORY 9 10 II 12 13 14 15 16 17

107

J . P . HIRTH, So J. HRUSKA AND G. M. POUND, in M. H. FRANCOMBE AND H. SATO (eds.), Single Crystal Films, P e r g a m o n , L o n d o n , 1964, p. 9. R. FRICKE, Kolloid Z., 36 (1941) 211. S . J . HRUSKA, Trans. A.LM.E., 227 (1963) 248. J. LOTHE AND G. M. POUND, J. Chem. Phys., 36 (1962) 2080. O. KNACKE AND I. N. STRANSKI,Progr. MetalPhys., 6 (1956) 181. G. ZINSMEISTER, in R. NIEDERMEYER AND H. MAYER (eds.), Basic Problems in Thin Film Physics, V a n d e n h o e c k a n d Ruprecht, G6ttingen, 1966, p. 33. n . WALTON, T. N. RHODIN AND R. ROLLINS, J. Chem. Phys., 38 (1963) 2695. N. FUCHS, Phys. Z. Sowjetun., 4 (1933) 481. L. YANG, C. E. BIRCHENALL,G. M. POUND AND M. T. SIMNAD,Acta. Met., 2 (1954) 462.

Thin Solid Films, 1 (1967) 85-107