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The influence of step motion and two-dimensional nucleation on the rate of crystal growth: Some computer simulated experiments
SURFACE
SCIENCE 15 (1969) 286-302 0 North-Holland
THE INFLUENCE
OF STEP MOTION
TWO-DIMENSIONAL OF CRYSTAL
Publishing Co., Amsterdam
NUCLEATION GROWTH:
SIMULATED
AND
ON THE RATE
SOME
COMPUTER
EXPERIMENTS
*
U. BERTOCCI Solid State Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37830, U.S.A. Received 22 November
1968
In order to investigate the influence of two-dimensional nucleation and step motion on the rate of crystal growth, the growth of a surface has been simulated with a digital computer. Two different models were employed. In the first, only a section of the surface was simulated and in the second a two-dimensional surface was simulated. Surfaces having fourfold and threefold symmetry were examined, both close-packed and vicinal. The results allow an estimate of the relative importance of random nucleation and step density on the growth rate, as a function of the rate of nucleation and step velocity. The effect of nucleation superimposed to growth from spirals arising from screw dislocations is also discussed. It is concluded that growth rate can be affected significantly by nucleation only for small misorientations and relatively large values of the free energy available for the crystallization process.
1. Introduction In the process
of crystal growth,
the growth
rate can be described
by the
velocity of the monatomic steps moving on the surface and by their density. Such steps might be present because of misorientation with respect to a close-packed plane, by the existence of dislocations having a screw component terminating at the surface, as well as two-dimensional nucleation occurring at random. The purpose of the present work is to estimate the relative importance of nucleation and step motion on the rate of crystal growth, both for close-packed and for stepped surfaces. Whereas earlier treatments assumed that nucleation was the most important process in determining the rate of growth’), since the realization that dislocations with a screw component intersecting the crystal surface have a step associated which can wind up on a long spiralsps), two-dimensional nucleation as an important rate-determining process in crystallization * Research sponsored
by the U.S. Atomic Union Carbide Corporation. 286
Energy Commission
under contract
with
INFLUENCE
OF STEP MOTION ANJJ NUCLEATION
287
has been generally disregarded. More recently, however, experimental observations on electrolytic deposition of silver on silver single crystalsd) have demonstrated that two-dimensional nucleation occurs with considerably large probability at relatively low overtensions. In some other studies of electrocrystallization processes5), difficulties were encountered in interpreting the results according to the most widely accepted models based exclusively on step motion. In the field of crystal growth of ionic salts from supersaturated solutions, devations from a “spiral growth” kinetics have been attributed to the effect of concurrent nucleatione). The treatment of the interaction of step motion and nucleation on the rate of crystal growth has received little attention. Some general relationships have been quoted by Chernov and Lyubov’), and Hilligs), and for electrochemical systems current-potential curves determined by nucleation have been briefly discussed by Damjanovic and Bockriss). The most detailed treatment of the influence of random nucleation on growth rate can be found in a book by Nielsenlo). This author has given an approximate solution for the case of a close-packed surface. As the theoretical analysis is exceedingly complex, computer simulations can be of interest; they have been proved to be a powerful tool for the study of randomly occurring processes on a crystal surface, and have been used for instance by Chernov to investigate radom walk processes and kinetic order-disorder transitions in crystal growthll), and by Moore to study surface diffusionlz). It was therefore thought that approximate solutions to the problem of finding the relative importance of step motion and nucleation on the crystallization rate could be found by simulating the events in a digital computer. The problem has been approached in two steps. The first model considered was that of a section of a crystal surface on which nucleation rate and step velocity are constant. Such a model is susceptible of analytical solution, and the computer calculations were used to verify the theoretical predictions. Such a model is obviously extremely primitive, and its relevance to real systems is open to serious doubts. Therefore, a second model was investigated in which a two-dimensional surface was simulated. Considerations about computation times and memory storage capability forced the simulation of a small surface and the use of drastically simplified laws for step motion and nucleation probability distribution. However, since the results obtained were not too sensitive to the details of the model employed, the formulae deduced are believed to be applicable to real crystal surfaces. 2. One-dimensional model As a first approximation
in studying
the effect of nucleation
and mis-
288
U. BERTOCCI
orientation on crystal growth, a section of a single crystal, so that its surface will be one-dimensional, is considered. On it a certain number of steps are present, all of them travelling at constant speed V and having the same height h. An act of nucleation consists in creating two steps at the same point on the surface, which will travel in opposite directions. The steps moving in a certain direction are called (+) steps and those moving in the opposite direction (-) steps. By convention the density of the (+) steps p +, defined as their average number per unit length (this being always measured in a close-packed direction) is chosen as being never less than p_. The average slope of the surface is tancc=hc (h being the height of a step) and i=p+
-P-
(1)
is a constant unaffected by an act of nucleation nor by the opposite process, the annihilation of two steps moving in opposite directions and meeting each other. The total step density p is defined by P=P+
(2)
+P-.
It is further assumed that the nucleation probability R,,expressed in nuclei per unit length and time, is a constant all over the surface. Let us divide the surface into segments I, the distance between two consecutive (+) steps. If n(Z) is the number of segments of length I, subject to the normalization condition a’
s s
n(Z)ZdZ=
1,
0
then
Co
n(Z) dZ = p+
(3)
0
The average lifetime of length I is
of a (-)
step generated
by nucleation
7(Z) = z/41/.
on a segment (4)
The density p_ is given by the product of the average lifetime of a (-) step and the rate of production of (-) steps on segments of length I, R,n(l) 1, integrated over the whole range of lengths m
p_
cc
R,?(Z) n(I) ZdZ = +
=
s
0
n(Z) l2 dl. s
0
(5)
As no correlation function
INFLUENCE
OF STEP
between
individual
events
is assumed,
289
the distribution
is: n(f) = n, exp(-
Therefore,
MOTION AND NUCLEATION
al) = p: exp(-
p+1).
from (1) (2), and (5) one obtains p2-c2=4p+p_
and therefore
the growth
=2R,/V,
(6)
rate G is
G = pV = VJ(2R,/V
+ c2).
(7)
This result was checked by simulating the experiment on a computer. The model of the “one-dimensional surface” was a 2000 place array, each place representing the unit of length of the surface; the integer contained in each place represented the close-packed layer exposed to the surface. At the beginning of the run, a number of steps at constant distances was established (the number of steps could also be zero, corresponding to zero misorientation). Periodic boundary conditions maintained the average misorientation of the surface. At each cycle, representing the unit of time, a random number generator determined whether and where new nuclei are formed. The program was written so that the formation of a nucleus did not cause a net growth of the crystal. The whole array was then scanned, and each step, represented by a difference in the integers of two consecutive places, was moved by one place (that is V= 1) in the appropriate direction. At the beginning of the run there is a transient, because the step distribution has yet to reach the steady state condition: accordingly, the data of the first several hundred cycles were discarded. Successively, the average number of steps as well as the distribution of distances between (+) steps were printed out at regular intervals. In the case of the computer simulation, a number of conditions differ from the continuous model employed in the preceding treatment. Sums instead of integrals have to be used, which have to be truncated to the longest step which can exist on the surface. The effect of the truncation, however, is negligible in the range of values investigated. The distribution function to be used is n(i) = n, exp( - ai), where i is an integer representing the number of places between two consecutive (+) steps. Subject to the conditions: i$0 n(i)*i
= 1
and
then cr=ln(l+p+)
(8)
and 110= ,6/(1
+ P+>.
(9)
U. RERTOCCI
290
In calculating the average lifetimes of the (-) steps it must be taken into account that in each instance the lifetime has to be an integral number of cycles: if the distance between a (-) step and the oncoming (+) step is an even number of places 2i, the lifetime will be iJV, but if the distance is an odd number 2if 1, the lifetime will still be i/V.Therefore ?(2j
1(2i) = &,
Ji+‘>f
+ 1) =
V(2i
the density of (-) p_
+
1)’
steps p- becomes
=Rjit!j-- [ T +
i.?.exp(-
2ai) 4- exp(-
rx) 2 i(i + l)expf-
i=O
?ai)]
i=O
R, (1 + I’+> = v p+(2+P+J’
(11)
By taking (I 1) into account, the formula analogous to (6) is
(12) Introducing $(p + i) for p+, and using the value of p given by the computer runs, the deviation from the expected values can be seen most clearly by dividing the right-hand side of (12) by the left-hand side. The computed ratios are given in table 1, for various values of R, and misorientation, expressed as c. The deviations from the expected value 1 are due to statistical fluctuations. TABLE 1 Values
of the
ratio --
computed
for various
I%+ -12)(2t_p+i
(&/V)
x 104 _---
_
.._~
.-.. 0 .
~_~. ---.
10-z ~~~ -
0.9798
i 2 x 10-Z 1.0062 0.9691
4
values of Rti/ V and i
5 x IO-2 ._ ..-.-. ~
10-l --
0.9887
0.9812
0.9925
0.9776
1.0038 1.0234
1.0112 i.ot3.5
0.9838 1.0254
1.0276
2
1.0188 1 0.5
0.9807
1.0162
0.9917
I .OOl I 1.0227
1.0220
1.0192
1.0052
INFLUENCE
OF STEP MOTION
AND
291
NUCLEATION
(x I o-31
0 0
10
20 N, STEP
Fig. 1.
30
40
50
SPACING
Distribution of (+) steps as a function of step spacing. Average over 266 samples. Rn/ V = 2 x 10-4; c = 5 x 1OW.
Fig. 1 gives the distribution of distances between (+) steps, taken over 266 samples. For comparison, the distribution curve calculated by means of (8), (9) and (12) is also shown. 3. Two-dimensional model 3.1. GENERAL CONSIDERATIONS For a more realistic treatment of the interaction and influence on the rate of growth of random nucleation and misorientation a two-dimensional surface must be considered. In this case, one should take into account that step velocities are in general a function of the step direction 4. Throughout this paper, two cases are discussed: 1) A step velocity function having fourfold symmetry whose polar plot is a square. The ratio between maximum and minimum velocity is 42 and the angle between the two directions is 45”. This fourfold symmetry system is meant to approximate the behavior of (100) surfaces and orientations close to it. 2) A step velocity function whose polar plot is an equilateral triangle. Here the ratio between maximum and minimum velocity is 2, and the angle
292
U BERTOCCI
between them 60”. This threefold symmetry system is meant to approximate the behavior of (111) surfaces and orientations close to it. In the absence of nucleation, if 8 is the tangent of the misorientation angle and h isthe height of a step, the rate of growth G expressed by the number of close-packed layers formed in unit time is (13) where V(4) is the velocity of the steps, here considered all parallel and moving at the same rate, and [ is the step density due to misorientation. Since no way to determine the growth rate of a close-packed surface (O=O) as a function of nucleation rate could be devised, as an approximate solution a system is considered, in which a non-random pattern of nuclei is put on the surface at fixed time intervals. For a surface having fourfold symmetry for the step velocities, a nucleation pattern of squares can be chosen. To make the problem amenable to simple analytical solution, the time interval between nucleation events is chosen so that no more than three layers will be present at any time on the surface, and no more than two at the instant when nucleation occurs. At this moment a fraction A of the surface is covered by the lower layer, and a fraction 1 -A covered by the growing upper layer. If the nucleation pattern is kept constant but its position on the surface is not, all nuclei will fall either on the upper or the lower layer. The growth of the next complete layer can occur in two mutually exclusive ways: If the nuclei fall on the upper layer, after a time t (time interval between nucleation events) the initial two-layered pattern will be found. The growth rate is G, = l/r and the probability of occurrence per nucleation event is PI = (1 - A)/( 1 + A). If the nuclei fall on the lower layer, after a time r the surface will be smooth, and after a second time z the initial situation
will be reproduced.
The average rate of growth G =
Therefore
G, = l/27 and P2 = 2A/( 1 + A).
is
G,P,+ G,P,= l/(1 + A) z;
(14)
with such a scheme either the upper or the lower layers will consist of individual squares at the instant of nucleation, depending on the value of A. It is possible to find an expression of G as a function of nucleation rate R,, maximum step velocity V,, the fraction of the surface A covered by the lower layer at the moment of nucleation and the angle 4 between the rows of nuclei and the direction of V,. Two formulae have to be used: the first is valid when the upper layer is divided in separate squares, the second when the lower layer is divided in separate squares. Furthermore, the condition that not more than two layers be present at the time of nucleation
INFLUENCE
must be satisfied.
OF STEP MOTION
The last condition A<
The first formula
AND
293
NUCLEATION
is met when
1 -+cos2~.
(15)
is valid as long as A~l-~os~(45~-~),
(16)
otherwise the second formula applies. Relatively tions give for the first formula
which is independent
is
2
=
(17)
- A) > ’
(1 +A)3(t
of 4. The second formula
’
considera-
+
2R,V,’ ‘=
simple geometric
(1 + A)3 ,c::;-
3
&A)]’
’
Subject to the limitations imposed by (15) and (16), it is found maximum rate of growth occurs for A = 0 and $J=45”, for which G = (2R,V;)+.
(18) that the
(19)
The minimum value occurs for 4 = 0 and A = 0, for which G = (R, Vi)*. An analogous treatment can be applied to threefold symmetry surfaces. The nucleation pattern is chosen as equilateral triangles. It is found that the maximum growth rate is achieved when the rows of nuclei are aligned so that the line joining two adjacent nuclei is along the direction of maximum velocity for the step issuing from one nucleus and in the direction of minimum velocity for the step issuing from the other. The value of A giving the maximum growth rate is 0.0552 for which G=(0.9866 &Vi)*. The minimum value occurs for the same angular positioning of the nucleation pattern and A =0.5, for which G=(0.7698 &Vi)*. The method for obtaining the preceding results obviously does not guarantee that the value of the coefficients in the expression for G valid for random nucleation will be close to those calculated for a highly nonrandom system. Nevertheless such results give an indication of the structure of the formulae relating growth to nucleation rate and step velocity. For the case in which both nucleation and misorientation contribute to crystal growth, no adequate analytical treatment, similar to that used for the one-dimensional case, could be devised. The computer simulations have provided the only means to obtain empirical formulae. 3.2. COMPUTER The computer
SIMULATIONS
simulation
of crystal growth was carried out in the following
U.BERTOCCI
294
way: the surface was mapped onto a square array, the integers contained in each element of the array indicating the layer of the crystal exposed to the surface. Steps are present if adjacent places contain different numbers. Before initiating a run, straight steps at uniform distances could be put on the surface, and the step direction could also be chosen. Periodic boundary conditions, giving toroidal shape to the surface, maintained constant the average misorientation. The average nucleation rate was selected for each run. At the beginning of each cycle (representing the unit of time), a random number generator determined how many nuclei would be formed and their location. The nuclei were generated by increasing by one the integer contained in an element of the array. The whole surface was then scanned, comparing the integer at every place with that of adjacent places and selecting the largest number. By this procedure, steps moved and interacted. After an initial transient, the growth per cycle was computed, and the average growth over a number of cycles was printed out at regular intervals. Computation times were about lo-’ set per place and per cycle, using an IBM System/360, Model 75 computer. Each run consisted of 4 x 107-5 x IO7 cycles x places. Two surfaces, one having fourfold symmetry and the other threefold symmetry were investigated. For the former, each element of the array has to be thought of as a square of unit side, and comparison occurs between an element and the four elements having a side in common. The maximum step velocity V, is therefore 1 place/cycle and its direction is parallel to a side of the array. For the threefold symmetry system, the elements have to be thought of as close-packed hexagons, and comparisons occurs between an element and three of the six adjacent to it. The maximum velocity is again 1 place/cycle in three directions at 120” from each other. The parameters examined have been the size of the array, which can give information about the influence of the boundaries on the results, the average misorientation, the direction of the steps with respect to the direction of the maximum step velocity and the nucleation rate. 3.3.
RESULTS OF THESIMULATIONS
3.3.1. Fourfold symmetry system The results obtained on surfaces with no misorientation are given in fig. 2, for arrays having 30 x 30, 72 x 72, and 108 x 108 places. The rate of growth G = 1IN, where N is the number of cycles necessary to grow a layer is given as a function of R,, the nucleation rate per cycle and per unit area, the unit area being a place on the array. It must be remembered that the
INFLUENCE
y
OF STEP MOTlON
295
AND NUCLEATION
0.08
i?? 0.06
0.02
0.01
to-6
2
5
10-5 R,,
2
5
to-4
2
5
fO-3
NUCLEATION RATE
Fig. 2. Growth rate versus nucleation rate for surfaces having zero misorientation. Fourfold symmetry. [ = 0.
maximum step velocity V,, parallei to the side of the array, is kept constant as one place/cycle. The results show clearly an influence of the size of the array. This is understandable, since for very low nucleation rates G must be proportional to R,. If N>L (L=side of the array), that is if the layer spreading from a single nucleus can cover the whole surface before another nucleus is formed, it is R,= l/NL', and therefore G=R,L? (20)
Lines corresponding to eq. (20) are included in fig. 2. When the nucleation rate is such that layers spreading from different nuclei will interfere, formula (20) is no longer valid. If the size of the array could be made infinite, the true relationship between G and R, would be similar to that found for high nucleation rates on surfaces of limited size. A good approximation for it is
The results of computer runs on surfaces having non-zero misorientation and for steps moving in the direction of maximum velocity are given in fig. 3. There the effect of the size of the array seems to be negligible, probably because the misorientation steps moving in and out the surface unaffected by the boundary, erase most of its artificial effect on the results.
296
U.BERTOCCI
0 20
0.03 i 10-6
2
5
10-5 R,,
Fig. 3.
2
5
NUCLEATION
10-4
2
5
40-3
RATE
Growth rate versus nucleation rate for various misorientations. Misorientation steps moving in direction of maximum step velocity. Fourfold symmetry.
The rate of growth
obtained
can be described
G=V() A systematic being higher is very likely fact that the estimate the non-random
(
by the formula
3
2;+y3 . 0
1
(24
deviation has been observed however, the computed values than the values predicted by (22) at high R,. Such an effect due to the discontinuous nature of the computer model. The coefficient 2 is found both in (22) and (19) suggested a way to effect of the discontinuous model: in the calculation of G for nucleation, analogous to the development of (19), it is found
that the area covered by a spreading nucleus after Ncycles is(2N2 + 2N+ 1) U2, where U is the length of the side of the square representing the unit of the array, and N is the number of cycles. The time necessary for the formation of one layer is expressed by the product of N and the time U/V, corresponding to a cycle: z=NlJ/I/,. Therefore the rate of growth is
G =
!z = V.
2GU G2U2 2 f-v+-_r 0
0
)I
R, I/
3 .
(23)
0
Remembering that for the computer calculation U= V, = 1, comparison of (23) with (19) shows that deviations from the behavior of the continuous model should occur at high R,, when G is not negligible with respect to 2. If a similar correction is applied to (22), the agreement with the computed results becomes excellent, the maximum difference found being less than
INFLUENCE
OF STEP MOTION
AND
NUCLEATION
29-i
3%. This in turn suggests that the systematic deviations observed are indeed caused by the discontinuous nature of the computer model. In fig. 3, the solid lines are calculated using the correction suggested by (23). In fig, 2, lines corresponding to both (23) and (21) are drawn. Formula (22), however, does not represent very well the results, if the misorientation steps are moving in the direction of minimum velocity. In this case one would expect that instead of [, the value c/,/2 should be used, as the minimum step vetocity is V,/,/‘Z. Fig. 4 gives the results for two mis-
72x72
408x4
R,
Fig. 4.
Growth
,
NUCLEATION
RATE
rate versus nucleation rate. Misorientation steps moving in direction of minimum step velocity. Fourfold symmetry.
orientations, with caiculated curves from (22) both uncorrected and corrected by the factor J2. If the corrected equation obviously holds when growth due to nucleation is negligible, the uncorrected formula seems to be more accurate at high nucleation rates, with an intermediate behavior when both nucleation and misorientation influence the growth rate in a comparabIe way. 3.3.2. Threefold symmetry
system
For surfaces whose polar plot of the step velocity is an equilateral triangle, an expression similar to (22) describes with good approximation the results, when misorientation steps move in the direction of maximum velocity. The
298
formula is (24) where the numerical coefficient .f varies from 1.5 to 2, as shown in fig. 5. Two different array sizes were used, 72 x 72 and 144 x 144 places. It is possible that f is a function of [ and perhaps of R,, but since G depends on the cube root off, the accuracy is insufficient to determine whether the variations are spurious, due to the limited size of the model surface and to the discontinuous nature of the computer model. An attempt to evaluate the latter effect in a way analogous to that used for the fourfold symmetry system has given uncertain results, only suggesting that a more probable average value for f is about 1.6 rather than the uncorrected value of 1.7. 2.6 2.4 2.2 2.0
f 1.6 1.6 1.4 1.2 1.0
J
lo-=
2
I?,, , NUCLEATION
Fig. 5.
5
16”
2
5
10-3
RATE
Values of coefficient fas a function of nucleation rate for various values of misorientation and array size. Threefold symmetry.
For steps moving in the direction of minimum velocity, it was found, as in the fourfold symmetry case, that the corrected value $5 is better at low R,, whereas no correction is necessary at high R,. 3.4.
EFFECT
OF NON-UNIFORM
DISTRIBUTION
OF
NUCLEI
On crystal surfaces it is generally thought that the probability of nucleation is reduced in the vicinity of steps. It was therefore interesting to know if a non-random distribution of nuclei would cause significant deviations from the behavior found when nucleation probability was independent of the presence of steps.
INFLUENCE
For this purpose,
OF STEP MOTION
a modified
of the place of nucleation,
program
AND
299
NUCLEATION
was used in which, upon selection
the immediate
vicinity
of the place was scanned
to find if a step was present. Two of such modified programs were used, valid for a fourfold symmetry surface, the first checking only nearest neighbors, the second also second nearest neighbors, If a step was found in the area surrounding the place selected for nucleation, the probability of nucleus formation was reduced. The results are given in table 2, where TABLE
Type of program Nearest Nearest Nearest Nearest Nearest Nearest Nearest Nearest Nearest Nearest Nearest Second Second Second Second Second Second Second
2
i
neighbors neighbors neighbors neighbors neighbors neighbors neighbors neighbors neighbors neighbors neighbors nearest neighbors nearest neighbors nearest neighbors nearest neighbors nearest neighbors nearest neighbors nearest neighbors
0.041667 0.041667 0.041667 0.041667 0.041667 0.041667 0.041667 0.055556 0.055556 0.055556 0.055556 0.041667 0.041667 0.041667 0.055556 0.055556 0.055556 0.055556
1.0457 5.2250 3.5435 1.8458 9.3439 3.7126 I.8116 5.3399 1.8185 1.1815 1.9561 9.3861 1.5608 1.8040 6.9553 1.5449 6.4199 1.3003
x x x x x x x x x x x x x x x x x x
10-3
lo+ 10-4 10-d 10-s 1O-5 10-S 10-a lo-* 10m5 10m5 1O-4 1O-4 10m5 1O-4 10-4
1O-5 10-s
Error % on G
‘A Reduction on Rn
+ 1.24 zk 0.50 - 0.92 & 0.83 - 0.42 i 0.65 + 0.33 + 1.03 + 0.98 i 0.97 + 0.47 zt 0.88 +0.14zkO.63 + 1.39 i 0.57 + 1.05 i 0.89 + 1.02 & 0.63 + 1.55 + 0.47 - 2.88 + 0.62 - 1.11 kO.78 f0.70 i 0.78 - 0.78 * 0.41 + 1.09 i 0.62 + 0.97 f 0.56 + 1.03 kO.44
9.6 9.7 8.2 4.3 3.1 3.7 6.1 7.7 5.7 4.8 1.4 27.5 19.1 6.5 27.9 19.9 16.8 15.7
the deviations from the values predicted by (22) as well as the decrease in nucleation rate due to non-random selection of the nucleation places are reported. No significant effect was detected, and formula (22) holds if the average value of the nucleation rate over the whole surface is used instead of the maximum value at large distances from steps. It is appreciated, however, that the modified computer program is a crude approximation of the reduction
in nucleation
probability
caused
by the presence
of steps.
4. Discussion The formula describing the effect of random nucleation and surface misorientation valid for the one-dimensional and two-dimensional surface present considerable formal analogies: nevertheless, the large differences between them in the estimate of the importance of nucleation, show the
300
Xl. BERTOCCI
inadvisability of simplifying problems related to surface processes by considering only sections of the surface. For the two-dimensional model the results of the computer runs indicate that the effect of random nucleation on growth rate is very close to that calculated for nucleation in a regular pattern. The results are also rather insensitive to surface symmetry, and lend support to the impression that in practice this parameter can be safely disregarded. About this point, it is interesting that Nielsen’s approximate solution for a close-packed surface and for circle-shaped spreading nucleig) is very close to the computer results. In most of the computer simulation the nucleation probability was constant over the surface. However, the results obtained when the nucleation rate was decreased in the vicinity of steps indicate that if the average nucleation rate, whose value would depend on step density, can be calculated, the formulae deduced from the simulations can be applied with some degree of confidence. As far as misoriented surfaces are concerned, it is perhaps regrettable that an exact formula could be found only for misorientation steps moving in the direction of maximum velocity, which is a less likely occurrence during crystal growth. However, without resorting to complicated corrections, formula (22) should give fairly reliable results in all cases. The results obtained here, since the model employed presupposes that the step velocity is independent of step density, can be applied to the case of the parabolic law for the growth of a crystal due to screw dislocations6s13). The rate of growth can be written as the product of step density 5, due to the winding up of the spirals, times a step velocity Y’ which includes the effect of random nucleation. From (22) one obtains
rF G = v’[ = [ 2!!$!? + v,” . > C
(25)
The step density [ is inversely proportional to the radius of the critical nucleus pC (ref. 14) which, in turn, is inversely proportional to the free energy available dp (here taken as a positive quantity):
where y is the surface energy and V, is the molar volume. If Ap is smaller than RT, the nucleation rate can be expressed by14) (27)
INFLUENCE
whereas,
if the parabolic
OF STEP MOTION
law is valid,
301
AND NUCLEATION
the step velocity
is proportional
to
A,u : V, = K’A,u. Therefore G = K
A,u K13 Ap3 + K” Ay -*(I
+g)exp(-$)-/,
(28)
where K” contains all terms independent of Ap. The constant B, for a discshaped nucleus, is equal to nhy* V,/kT and its value is at least of the order of RT. It can be shown, therefore, that the contribution of random nucleation on growth rate increases with increasing Ap in the range where the approximations are valid, which should correspond to the range of Ap’s where a parabolic law is likely to hold. It is probable that (22) can be used to describe the influence of random nucleation also when the step velocity is dependent on step density, if an average step velocity value is taken, based on an appropriate statistical distribution of step spacings: however, it is obvious that when a linear law for spiral growth holds, random nucleation cannot influence the rate of growth, as a linear law is valid when the crystallization process is no longer rate determining. Formally, this is expressed by the fact that step velocity is inversely proportional to step density: formula (22) can be considered as the product of step velocity and of a step density which depends on the slope of the surface (due to the winding of the spirals or otherwise) as well as on random nucleation. It is then clear that if the step velocity decreases proportionally to the increase of step density, no effect on G can be caused by random nucleation. In order to estimate the influence of nucleation on the growth rate by means of (23), expressions for the nucleation rate and step velocity as a function of the free energy available Ap have to be constructed. Since such expressions would be strongly dependent on the models chosen, and quantitative evaluations would entail the use of parameters whose values are very poorly known, it is perhaps convenient to examine formula (22) in a very simple way, in order to draw conclusions of more general validity. It is evident that random nucleation would increase the rate of growth if the term 2RJV is not negligible compared with c3. Since 5 represents the step density over the surface, only in exceptional cases can one expect it to be less than 10’ cm- I. Therefore, the ratio R,/V must be of the order of 1014 cmm3 if it has to play a significant role in the crystal growth rate. The step velocity in general will range between 10e3 and lO-‘j cm/set, and accordingly R, must be at least between 10” and lo* nuclei/cm2.sec. Therefore, the conditions in which random nucleation is a significant factor must be necessarily limited to small misorientations and relatively large values of A,u.
302
U. BERTOCCI
Acknowledgement The author wishes to thank Robinson for the many helpful
W. E. Atkinson, D. K. Holmes discussions and suggestions.
and
M. T.
References 1) M. Volmer and A. Weber, Z. Physik. Chem. 119 (1926) 277; H. Brandes, Z. Physik. Chem. 126 (1927) 196; T. Erdey-Gruz and M. Volmer, Z. Physik. Chem. 157 (1931) 165; M. Volmer, Kinetik der Phasenbildung (Steinkopf, Dresden, 1939). 2) F. C. Frank, Disc. Faraday Sot. 5 (1949) 48. 3) W. K. Burton, N. Cabrera and F. C. Frank, Phil. Trans. Roy. Sot. London A 243 (1950) 299. 4) E. Budevski, W. Bostanoff, T. Vitanoff, Z. Stoinoff, A. Kotzewa and R. Kaischev, Phys. Status Solidi 13 (1966) 577; Electrochim. Acta 11 (1966) 1697. 5) U. Bertocci, J. Electrochem. Sot. 113 (1966) 604; U. Bertocci, L. D. Hulett, Jr., L. H. Jenkins and F. W. Young, Jr., in: Proceedings of the 6th Znternational Symposium on the Reactivity of Solids, 1968, in press. 6) P. Bennema, R. Kern and B. Simon, Phys. Status Solidi 19 (1967) 211. 7) A. A. Chernov and B. Ya. Lyubov, in: Growth of Crystals, Vol. 5A (Consultants Bureau, New York, 1968) p. 7. 8) W. B. Hillig, in: Growth and Perfection of Crystals, eds. R. H. Doremus, B. W. Roberts and D. Turnbull (Wiley, New York 1958) p. 350. W. B. Hillig, Kinetics of High-Temperature Processes, ed. W. D. Kingery (Wiley, New York, 1959) p. 127; J. W. Cahn, W. B. Hillig and G. W. Sears, Acta Met. 12 (1964) 1421. 9) A. Damjanovic and J. O’M. Bockris, J. Electrochem. Sot. 110 (1963) 1035. 10) A. E. Nielsen, Kinetics of Precipitation (Pergamon, New York, 1964) p. 46. 11) A. A. Chemov and J. Lewis, J. Phys. Chem. Solids 28 (1967) 2185. 12) A. J. W. Moore, J. Australian Inst. Met. 11 (1966) 220. 13) N. Cabrera and W. K. Burton, Disc. Faraday Sot. 5 (1949) 40. 14) N. Cabrera and M. M. Levine, Phil. Mag. 1 (1956) 450. 15) G. M. Pound, M. T. Simnad and Ling Yang, J. Chem. Phys. 22 (1954) 1215; J. P. Hirth, Acta Met. 7 (1959) 755; 3. Lothe and G. M. Pound, J. Chem. Phys. 36 (1962) 2080; J. P. Hirth and G. M. Pound, in: Progress in Materials Science, Vol. 11 (Pergamon, New York, 1963).
SCIENCE 15 (1969) 286-302 0 North-Holland
THE INFLUENCE
OF STEP MOTION
TWO-DIMENSIONAL OF CRYSTAL
Publishing Co., Amsterdam
NUCLEATION GROWTH:
SIMULATED
AND
ON THE RATE
SOME
COMPUTER
EXPERIMENTS
*
U. BERTOCCI Solid State Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37830, U.S.A. Received 22 November
1968
In order to investigate the influence of two-dimensional nucleation and step motion on the rate of crystal growth, the growth of a surface has been simulated with a digital computer. Two different models were employed. In the first, only a section of the surface was simulated and in the second a two-dimensional surface was simulated. Surfaces having fourfold and threefold symmetry were examined, both close-packed and vicinal. The results allow an estimate of the relative importance of random nucleation and step density on the growth rate, as a function of the rate of nucleation and step velocity. The effect of nucleation superimposed to growth from spirals arising from screw dislocations is also discussed. It is concluded that growth rate can be affected significantly by nucleation only for small misorientations and relatively large values of the free energy available for the crystallization process.
1. Introduction In the process
of crystal growth,
the growth
rate can be described
by the
velocity of the monatomic steps moving on the surface and by their density. Such steps might be present because of misorientation with respect to a close-packed plane, by the existence of dislocations having a screw component terminating at the surface, as well as two-dimensional nucleation occurring at random. The purpose of the present work is to estimate the relative importance of nucleation and step motion on the rate of crystal growth, both for close-packed and for stepped surfaces. Whereas earlier treatments assumed that nucleation was the most important process in determining the rate of growth’), since the realization that dislocations with a screw component intersecting the crystal surface have a step associated which can wind up on a long spiralsps), two-dimensional nucleation as an important rate-determining process in crystallization * Research sponsored
by the U.S. Atomic Union Carbide Corporation. 286
Energy Commission
under contract
with
INFLUENCE
OF STEP MOTION ANJJ NUCLEATION
287
has been generally disregarded. More recently, however, experimental observations on electrolytic deposition of silver on silver single crystalsd) have demonstrated that two-dimensional nucleation occurs with considerably large probability at relatively low overtensions. In some other studies of electrocrystallization processes5), difficulties were encountered in interpreting the results according to the most widely accepted models based exclusively on step motion. In the field of crystal growth of ionic salts from supersaturated solutions, devations from a “spiral growth” kinetics have been attributed to the effect of concurrent nucleatione). The treatment of the interaction of step motion and nucleation on the rate of crystal growth has received little attention. Some general relationships have been quoted by Chernov and Lyubov’), and Hilligs), and for electrochemical systems current-potential curves determined by nucleation have been briefly discussed by Damjanovic and Bockriss). The most detailed treatment of the influence of random nucleation on growth rate can be found in a book by Nielsenlo). This author has given an approximate solution for the case of a close-packed surface. As the theoretical analysis is exceedingly complex, computer simulations can be of interest; they have been proved to be a powerful tool for the study of randomly occurring processes on a crystal surface, and have been used for instance by Chernov to investigate radom walk processes and kinetic order-disorder transitions in crystal growthll), and by Moore to study surface diffusionlz). It was therefore thought that approximate solutions to the problem of finding the relative importance of step motion and nucleation on the crystallization rate could be found by simulating the events in a digital computer. The problem has been approached in two steps. The first model considered was that of a section of a crystal surface on which nucleation rate and step velocity are constant. Such a model is susceptible of analytical solution, and the computer calculations were used to verify the theoretical predictions. Such a model is obviously extremely primitive, and its relevance to real systems is open to serious doubts. Therefore, a second model was investigated in which a two-dimensional surface was simulated. Considerations about computation times and memory storage capability forced the simulation of a small surface and the use of drastically simplified laws for step motion and nucleation probability distribution. However, since the results obtained were not too sensitive to the details of the model employed, the formulae deduced are believed to be applicable to real crystal surfaces. 2. One-dimensional model As a first approximation
in studying
the effect of nucleation
and mis-
288
U. BERTOCCI
orientation on crystal growth, a section of a single crystal, so that its surface will be one-dimensional, is considered. On it a certain number of steps are present, all of them travelling at constant speed V and having the same height h. An act of nucleation consists in creating two steps at the same point on the surface, which will travel in opposite directions. The steps moving in a certain direction are called (+) steps and those moving in the opposite direction (-) steps. By convention the density of the (+) steps p +, defined as their average number per unit length (this being always measured in a close-packed direction) is chosen as being never less than p_. The average slope of the surface is tancc=hc (h being the height of a step) and i=p+
-P-
(1)
is a constant unaffected by an act of nucleation nor by the opposite process, the annihilation of two steps moving in opposite directions and meeting each other. The total step density p is defined by P=P+
(2)
+P-.
It is further assumed that the nucleation probability R,,expressed in nuclei per unit length and time, is a constant all over the surface. Let us divide the surface into segments I, the distance between two consecutive (+) steps. If n(Z) is the number of segments of length I, subject to the normalization condition a’
s s
n(Z)ZdZ=
1,
0
then
Co
n(Z) dZ = p+
(3)
0
The average lifetime of length I is
of a (-)
step generated
by nucleation
7(Z) = z/41/.
on a segment (4)
The density p_ is given by the product of the average lifetime of a (-) step and the rate of production of (-) steps on segments of length I, R,n(l) 1, integrated over the whole range of lengths m
p_
cc
R,?(Z) n(I) ZdZ = +
=
s
0
n(Z) l2 dl. s
0
(5)
As no correlation function
INFLUENCE
OF STEP
between
individual
events
is assumed,
289
the distribution
is: n(f) = n, exp(-
Therefore,
MOTION AND NUCLEATION
al) = p: exp(-
p+1).
from (1) (2), and (5) one obtains p2-c2=4p+p_
and therefore
the growth
=2R,/V,
(6)
rate G is
G = pV = VJ(2R,/V
+ c2).
(7)
This result was checked by simulating the experiment on a computer. The model of the “one-dimensional surface” was a 2000 place array, each place representing the unit of length of the surface; the integer contained in each place represented the close-packed layer exposed to the surface. At the beginning of the run, a number of steps at constant distances was established (the number of steps could also be zero, corresponding to zero misorientation). Periodic boundary conditions maintained the average misorientation of the surface. At each cycle, representing the unit of time, a random number generator determined whether and where new nuclei are formed. The program was written so that the formation of a nucleus did not cause a net growth of the crystal. The whole array was then scanned, and each step, represented by a difference in the integers of two consecutive places, was moved by one place (that is V= 1) in the appropriate direction. At the beginning of the run there is a transient, because the step distribution has yet to reach the steady state condition: accordingly, the data of the first several hundred cycles were discarded. Successively, the average number of steps as well as the distribution of distances between (+) steps were printed out at regular intervals. In the case of the computer simulation, a number of conditions differ from the continuous model employed in the preceding treatment. Sums instead of integrals have to be used, which have to be truncated to the longest step which can exist on the surface. The effect of the truncation, however, is negligible in the range of values investigated. The distribution function to be used is n(i) = n, exp( - ai), where i is an integer representing the number of places between two consecutive (+) steps. Subject to the conditions: i$0 n(i)*i
= 1
and
then cr=ln(l+p+)
(8)
and 110= ,6/(1
+ P+>.
(9)
U. RERTOCCI
290
In calculating the average lifetimes of the (-) steps it must be taken into account that in each instance the lifetime has to be an integral number of cycles: if the distance between a (-) step and the oncoming (+) step is an even number of places 2i, the lifetime will be iJV, but if the distance is an odd number 2if 1, the lifetime will still be i/V.Therefore ?(2j
1(2i) = &,
Ji+‘>f
+ 1) =
V(2i
the density of (-) p_
+
1)’
steps p- becomes
=Rjit!j-- [ T +
i.?.exp(-
2ai) 4- exp(-
rx) 2 i(i + l)expf-
i=O
?ai)]
i=O
R, (1 + I’+> = v p+(2+P+J’
(11)
By taking (I 1) into account, the formula analogous to (6) is
(12) Introducing $(p + i) for p+, and using the value of p given by the computer runs, the deviation from the expected values can be seen most clearly by dividing the right-hand side of (12) by the left-hand side. The computed ratios are given in table 1, for various values of R, and misorientation, expressed as c. The deviations from the expected value 1 are due to statistical fluctuations. TABLE 1 Values
of the
ratio --
computed
for various
I%+ -12)(2t_p+i
(&/V)
x 104 _---
_
.._~
.-.. 0 .
~_~. ---.
10-z ~~~ -
0.9798
i 2 x 10-Z 1.0062 0.9691
4
values of Rti/ V and i
5 x IO-2 ._ ..-.-. ~
10-l --
0.9887
0.9812
0.9925
0.9776
1.0038 1.0234
1.0112 i.ot3.5
0.9838 1.0254
1.0276
2
1.0188 1 0.5
0.9807
1.0162
0.9917
I .OOl I 1.0227
1.0220
1.0192
1.0052
INFLUENCE
OF STEP MOTION
AND
291
NUCLEATION
(x I o-31
0 0
10
20 N, STEP
Fig. 1.
30
40
50
SPACING
Distribution of (+) steps as a function of step spacing. Average over 266 samples. Rn/ V = 2 x 10-4; c = 5 x 1OW.
Fig. 1 gives the distribution of distances between (+) steps, taken over 266 samples. For comparison, the distribution curve calculated by means of (8), (9) and (12) is also shown. 3. Two-dimensional model 3.1. GENERAL CONSIDERATIONS For a more realistic treatment of the interaction and influence on the rate of growth of random nucleation and misorientation a two-dimensional surface must be considered. In this case, one should take into account that step velocities are in general a function of the step direction 4. Throughout this paper, two cases are discussed: 1) A step velocity function having fourfold symmetry whose polar plot is a square. The ratio between maximum and minimum velocity is 42 and the angle between the two directions is 45”. This fourfold symmetry system is meant to approximate the behavior of (100) surfaces and orientations close to it. 2) A step velocity function whose polar plot is an equilateral triangle. Here the ratio between maximum and minimum velocity is 2, and the angle
292
U BERTOCCI
between them 60”. This threefold symmetry system is meant to approximate the behavior of (111) surfaces and orientations close to it. In the absence of nucleation, if 8 is the tangent of the misorientation angle and h isthe height of a step, the rate of growth G expressed by the number of close-packed layers formed in unit time is (13) where V(4) is the velocity of the steps, here considered all parallel and moving at the same rate, and [ is the step density due to misorientation. Since no way to determine the growth rate of a close-packed surface (O=O) as a function of nucleation rate could be devised, as an approximate solution a system is considered, in which a non-random pattern of nuclei is put on the surface at fixed time intervals. For a surface having fourfold symmetry for the step velocities, a nucleation pattern of squares can be chosen. To make the problem amenable to simple analytical solution, the time interval between nucleation events is chosen so that no more than three layers will be present at any time on the surface, and no more than two at the instant when nucleation occurs. At this moment a fraction A of the surface is covered by the lower layer, and a fraction 1 -A covered by the growing upper layer. If the nucleation pattern is kept constant but its position on the surface is not, all nuclei will fall either on the upper or the lower layer. The growth of the next complete layer can occur in two mutually exclusive ways: If the nuclei fall on the upper layer, after a time t (time interval between nucleation events) the initial two-layered pattern will be found. The growth rate is G, = l/r and the probability of occurrence per nucleation event is PI = (1 - A)/( 1 + A). If the nuclei fall on the lower layer, after a time r the surface will be smooth, and after a second time z the initial situation
will be reproduced.
The average rate of growth G =
Therefore
G, = l/27 and P2 = 2A/( 1 + A).
is
G,P,+ G,P,= l/(1 + A) z;
(14)
with such a scheme either the upper or the lower layers will consist of individual squares at the instant of nucleation, depending on the value of A. It is possible to find an expression of G as a function of nucleation rate R,, maximum step velocity V,, the fraction of the surface A covered by the lower layer at the moment of nucleation and the angle 4 between the rows of nuclei and the direction of V,. Two formulae have to be used: the first is valid when the upper layer is divided in separate squares, the second when the lower layer is divided in separate squares. Furthermore, the condition that not more than two layers be present at the time of nucleation
INFLUENCE
must be satisfied.
OF STEP MOTION
The last condition A<
The first formula
AND
293
NUCLEATION
is met when
1 -+cos2~.
(15)
is valid as long as A~l-~os~(45~-~),
(16)
otherwise the second formula applies. Relatively tions give for the first formula
which is independent
is
2
=
(17)
- A) > ’
(1 +A)3(t
of 4. The second formula
’
considera-
+
2R,V,’ ‘=
simple geometric
(1 + A)3 ,c::;-
3
&A)]’
’
Subject to the limitations imposed by (15) and (16), it is found maximum rate of growth occurs for A = 0 and $J=45”, for which G = (2R,V;)+.
(18) that the
(19)
The minimum value occurs for 4 = 0 and A = 0, for which G = (R, Vi)*. An analogous treatment can be applied to threefold symmetry surfaces. The nucleation pattern is chosen as equilateral triangles. It is found that the maximum growth rate is achieved when the rows of nuclei are aligned so that the line joining two adjacent nuclei is along the direction of maximum velocity for the step issuing from one nucleus and in the direction of minimum velocity for the step issuing from the other. The value of A giving the maximum growth rate is 0.0552 for which G=(0.9866 &Vi)*. The minimum value occurs for the same angular positioning of the nucleation pattern and A =0.5, for which G=(0.7698 &Vi)*. The method for obtaining the preceding results obviously does not guarantee that the value of the coefficients in the expression for G valid for random nucleation will be close to those calculated for a highly nonrandom system. Nevertheless such results give an indication of the structure of the formulae relating growth to nucleation rate and step velocity. For the case in which both nucleation and misorientation contribute to crystal growth, no adequate analytical treatment, similar to that used for the one-dimensional case, could be devised. The computer simulations have provided the only means to obtain empirical formulae. 3.2. COMPUTER The computer
SIMULATIONS
simulation
of crystal growth was carried out in the following
U.BERTOCCI
294
way: the surface was mapped onto a square array, the integers contained in each element of the array indicating the layer of the crystal exposed to the surface. Steps are present if adjacent places contain different numbers. Before initiating a run, straight steps at uniform distances could be put on the surface, and the step direction could also be chosen. Periodic boundary conditions, giving toroidal shape to the surface, maintained constant the average misorientation. The average nucleation rate was selected for each run. At the beginning of each cycle (representing the unit of time), a random number generator determined how many nuclei would be formed and their location. The nuclei were generated by increasing by one the integer contained in an element of the array. The whole surface was then scanned, comparing the integer at every place with that of adjacent places and selecting the largest number. By this procedure, steps moved and interacted. After an initial transient, the growth per cycle was computed, and the average growth over a number of cycles was printed out at regular intervals. Computation times were about lo-’ set per place and per cycle, using an IBM System/360, Model 75 computer. Each run consisted of 4 x 107-5 x IO7 cycles x places. Two surfaces, one having fourfold symmetry and the other threefold symmetry were investigated. For the former, each element of the array has to be thought of as a square of unit side, and comparison occurs between an element and the four elements having a side in common. The maximum step velocity V, is therefore 1 place/cycle and its direction is parallel to a side of the array. For the threefold symmetry system, the elements have to be thought of as close-packed hexagons, and comparisons occurs between an element and three of the six adjacent to it. The maximum velocity is again 1 place/cycle in three directions at 120” from each other. The parameters examined have been the size of the array, which can give information about the influence of the boundaries on the results, the average misorientation, the direction of the steps with respect to the direction of the maximum step velocity and the nucleation rate. 3.3.
RESULTS OF THESIMULATIONS
3.3.1. Fourfold symmetry system The results obtained on surfaces with no misorientation are given in fig. 2, for arrays having 30 x 30, 72 x 72, and 108 x 108 places. The rate of growth G = 1IN, where N is the number of cycles necessary to grow a layer is given as a function of R,, the nucleation rate per cycle and per unit area, the unit area being a place on the array. It must be remembered that the
INFLUENCE
y
OF STEP MOTlON
295
AND NUCLEATION
0.08
i?? 0.06
0.02
0.01
to-6
2
5
10-5 R,,
2
5
to-4
2
5
fO-3
NUCLEATION RATE
Fig. 2. Growth rate versus nucleation rate for surfaces having zero misorientation. Fourfold symmetry. [ = 0.
maximum step velocity V,, parallei to the side of the array, is kept constant as one place/cycle. The results show clearly an influence of the size of the array. This is understandable, since for very low nucleation rates G must be proportional to R,. If N>L (L=side of the array), that is if the layer spreading from a single nucleus can cover the whole surface before another nucleus is formed, it is R,= l/NL', and therefore G=R,L? (20)
Lines corresponding to eq. (20) are included in fig. 2. When the nucleation rate is such that layers spreading from different nuclei will interfere, formula (20) is no longer valid. If the size of the array could be made infinite, the true relationship between G and R, would be similar to that found for high nucleation rates on surfaces of limited size. A good approximation for it is
The results of computer runs on surfaces having non-zero misorientation and for steps moving in the direction of maximum velocity are given in fig. 3. There the effect of the size of the array seems to be negligible, probably because the misorientation steps moving in and out the surface unaffected by the boundary, erase most of its artificial effect on the results.
296
U.BERTOCCI
0 20
0.03 i 10-6
2
5
10-5 R,,
Fig. 3.
2
5
NUCLEATION
10-4
2
5
40-3
RATE
Growth rate versus nucleation rate for various misorientations. Misorientation steps moving in direction of maximum step velocity. Fourfold symmetry.
The rate of growth
obtained
can be described
G=V() A systematic being higher is very likely fact that the estimate the non-random
(
by the formula
3
2;+y3 . 0
1
(24
deviation has been observed however, the computed values than the values predicted by (22) at high R,. Such an effect due to the discontinuous nature of the computer model. The coefficient 2 is found both in (22) and (19) suggested a way to effect of the discontinuous model: in the calculation of G for nucleation, analogous to the development of (19), it is found
that the area covered by a spreading nucleus after Ncycles is(2N2 + 2N+ 1) U2, where U is the length of the side of the square representing the unit of the array, and N is the number of cycles. The time necessary for the formation of one layer is expressed by the product of N and the time U/V, corresponding to a cycle: z=NlJ/I/,. Therefore the rate of growth is
G =
!z = V.
2GU G2U2 2 f-v+-_r 0
0
)I
R, I/
3 .
(23)
0
Remembering that for the computer calculation U= V, = 1, comparison of (23) with (19) shows that deviations from the behavior of the continuous model should occur at high R,, when G is not negligible with respect to 2. If a similar correction is applied to (22), the agreement with the computed results becomes excellent, the maximum difference found being less than
INFLUENCE
OF STEP MOTION
AND
NUCLEATION
29-i
3%. This in turn suggests that the systematic deviations observed are indeed caused by the discontinuous nature of the computer model. In fig. 3, the solid lines are calculated using the correction suggested by (23). In fig, 2, lines corresponding to both (23) and (21) are drawn. Formula (22), however, does not represent very well the results, if the misorientation steps are moving in the direction of minimum velocity. In this case one would expect that instead of [, the value c/,/2 should be used, as the minimum step vetocity is V,/,/‘Z. Fig. 4 gives the results for two mis-
72x72
408x4
R,
Fig. 4.
Growth
,
NUCLEATION
RATE
rate versus nucleation rate. Misorientation steps moving in direction of minimum step velocity. Fourfold symmetry.
orientations, with caiculated curves from (22) both uncorrected and corrected by the factor J2. If the corrected equation obviously holds when growth due to nucleation is negligible, the uncorrected formula seems to be more accurate at high nucleation rates, with an intermediate behavior when both nucleation and misorientation influence the growth rate in a comparabIe way. 3.3.2. Threefold symmetry
system
For surfaces whose polar plot of the step velocity is an equilateral triangle, an expression similar to (22) describes with good approximation the results, when misorientation steps move in the direction of maximum velocity. The
298
formula is (24) where the numerical coefficient .f varies from 1.5 to 2, as shown in fig. 5. Two different array sizes were used, 72 x 72 and 144 x 144 places. It is possible that f is a function of [ and perhaps of R,, but since G depends on the cube root off, the accuracy is insufficient to determine whether the variations are spurious, due to the limited size of the model surface and to the discontinuous nature of the computer model. An attempt to evaluate the latter effect in a way analogous to that used for the fourfold symmetry system has given uncertain results, only suggesting that a more probable average value for f is about 1.6 rather than the uncorrected value of 1.7. 2.6 2.4 2.2 2.0
f 1.6 1.6 1.4 1.2 1.0
J
lo-=
2
I?,, , NUCLEATION
Fig. 5.
5
16”
2
5
10-3
RATE
Values of coefficient fas a function of nucleation rate for various values of misorientation and array size. Threefold symmetry.
For steps moving in the direction of minimum velocity, it was found, as in the fourfold symmetry case, that the corrected value $5 is better at low R,, whereas no correction is necessary at high R,. 3.4.
EFFECT
OF NON-UNIFORM
DISTRIBUTION
OF
NUCLEI
On crystal surfaces it is generally thought that the probability of nucleation is reduced in the vicinity of steps. It was therefore interesting to know if a non-random distribution of nuclei would cause significant deviations from the behavior found when nucleation probability was independent of the presence of steps.
INFLUENCE
For this purpose,
OF STEP MOTION
a modified
of the place of nucleation,
program
AND
299
NUCLEATION
was used in which, upon selection
the immediate
vicinity
of the place was scanned
to find if a step was present. Two of such modified programs were used, valid for a fourfold symmetry surface, the first checking only nearest neighbors, the second also second nearest neighbors, If a step was found in the area surrounding the place selected for nucleation, the probability of nucleus formation was reduced. The results are given in table 2, where TABLE
Type of program Nearest Nearest Nearest Nearest Nearest Nearest Nearest Nearest Nearest Nearest Nearest Second Second Second Second Second Second Second
2
i
neighbors neighbors neighbors neighbors neighbors neighbors neighbors neighbors neighbors neighbors neighbors nearest neighbors nearest neighbors nearest neighbors nearest neighbors nearest neighbors nearest neighbors nearest neighbors
0.041667 0.041667 0.041667 0.041667 0.041667 0.041667 0.041667 0.055556 0.055556 0.055556 0.055556 0.041667 0.041667 0.041667 0.055556 0.055556 0.055556 0.055556
1.0457 5.2250 3.5435 1.8458 9.3439 3.7126 I.8116 5.3399 1.8185 1.1815 1.9561 9.3861 1.5608 1.8040 6.9553 1.5449 6.4199 1.3003
x x x x x x x x x x x x x x x x x x
10-3
lo+ 10-4 10-d 10-s 1O-5 10-S 10-a lo-* 10m5 10m5 1O-4 1O-4 10m5 1O-4 10-4
1O-5 10-s
Error % on G
‘A Reduction on Rn
+ 1.24 zk 0.50 - 0.92 & 0.83 - 0.42 i 0.65 + 0.33 + 1.03 + 0.98 i 0.97 + 0.47 zt 0.88 +0.14zkO.63 + 1.39 i 0.57 + 1.05 i 0.89 + 1.02 & 0.63 + 1.55 + 0.47 - 2.88 + 0.62 - 1.11 kO.78 f0.70 i 0.78 - 0.78 * 0.41 + 1.09 i 0.62 + 0.97 f 0.56 + 1.03 kO.44
9.6 9.7 8.2 4.3 3.1 3.7 6.1 7.7 5.7 4.8 1.4 27.5 19.1 6.5 27.9 19.9 16.8 15.7
the deviations from the values predicted by (22) as well as the decrease in nucleation rate due to non-random selection of the nucleation places are reported. No significant effect was detected, and formula (22) holds if the average value of the nucleation rate over the whole surface is used instead of the maximum value at large distances from steps. It is appreciated, however, that the modified computer program is a crude approximation of the reduction
in nucleation
probability
caused
by the presence
of steps.
4. Discussion The formula describing the effect of random nucleation and surface misorientation valid for the one-dimensional and two-dimensional surface present considerable formal analogies: nevertheless, the large differences between them in the estimate of the importance of nucleation, show the
300
Xl. BERTOCCI
inadvisability of simplifying problems related to surface processes by considering only sections of the surface. For the two-dimensional model the results of the computer runs indicate that the effect of random nucleation on growth rate is very close to that calculated for nucleation in a regular pattern. The results are also rather insensitive to surface symmetry, and lend support to the impression that in practice this parameter can be safely disregarded. About this point, it is interesting that Nielsen’s approximate solution for a close-packed surface and for circle-shaped spreading nucleig) is very close to the computer results. In most of the computer simulation the nucleation probability was constant over the surface. However, the results obtained when the nucleation rate was decreased in the vicinity of steps indicate that if the average nucleation rate, whose value would depend on step density, can be calculated, the formulae deduced from the simulations can be applied with some degree of confidence. As far as misoriented surfaces are concerned, it is perhaps regrettable that an exact formula could be found only for misorientation steps moving in the direction of maximum velocity, which is a less likely occurrence during crystal growth. However, without resorting to complicated corrections, formula (22) should give fairly reliable results in all cases. The results obtained here, since the model employed presupposes that the step velocity is independent of step density, can be applied to the case of the parabolic law for the growth of a crystal due to screw dislocations6s13). The rate of growth can be written as the product of step density 5, due to the winding up of the spirals, times a step velocity Y’ which includes the effect of random nucleation. From (22) one obtains
rF G = v’[ = [ 2!!$!? + v,” . > C
(25)
The step density [ is inversely proportional to the radius of the critical nucleus pC (ref. 14) which, in turn, is inversely proportional to the free energy available dp (here taken as a positive quantity):
where y is the surface energy and V, is the molar volume. If Ap is smaller than RT, the nucleation rate can be expressed by14) (27)
INFLUENCE
whereas,
if the parabolic
OF STEP MOTION
law is valid,
301
AND NUCLEATION
the step velocity
is proportional
to
A,u : V, = K’A,u. Therefore G = K
A,u K13 Ap3 + K” Ay -*(I
+g)exp(-$)-/,
(28)
where K” contains all terms independent of Ap. The constant B, for a discshaped nucleus, is equal to nhy* V,/kT and its value is at least of the order of RT. It can be shown, therefore, that the contribution of random nucleation on growth rate increases with increasing Ap in the range where the approximations are valid, which should correspond to the range of Ap’s where a parabolic law is likely to hold. It is probable that (22) can be used to describe the influence of random nucleation also when the step velocity is dependent on step density, if an average step velocity value is taken, based on an appropriate statistical distribution of step spacings: however, it is obvious that when a linear law for spiral growth holds, random nucleation cannot influence the rate of growth, as a linear law is valid when the crystallization process is no longer rate determining. Formally, this is expressed by the fact that step velocity is inversely proportional to step density: formula (22) can be considered as the product of step velocity and of a step density which depends on the slope of the surface (due to the winding of the spirals or otherwise) as well as on random nucleation. It is then clear that if the step velocity decreases proportionally to the increase of step density, no effect on G can be caused by random nucleation. In order to estimate the influence of nucleation on the growth rate by means of (23), expressions for the nucleation rate and step velocity as a function of the free energy available Ap have to be constructed. Since such expressions would be strongly dependent on the models chosen, and quantitative evaluations would entail the use of parameters whose values are very poorly known, it is perhaps convenient to examine formula (22) in a very simple way, in order to draw conclusions of more general validity. It is evident that random nucleation would increase the rate of growth if the term 2RJV is not negligible compared with c3. Since 5 represents the step density over the surface, only in exceptional cases can one expect it to be less than 10’ cm- I. Therefore, the ratio R,/V must be of the order of 1014 cmm3 if it has to play a significant role in the crystal growth rate. The step velocity in general will range between 10e3 and lO-‘j cm/set, and accordingly R, must be at least between 10” and lo* nuclei/cm2.sec. Therefore, the conditions in which random nucleation is a significant factor must be necessarily limited to small misorientations and relatively large values of A,u.
302
U. BERTOCCI
Acknowledgement The author wishes to thank Robinson for the many helpful
W. E. Atkinson, D. K. Holmes discussions and suggestions.
and
M. T.
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