A stochastic theory for clustering of quenched-in vacancies—2. A solvable model

Pramin.a, Vol. 12, No. 5, May 1979, pp. 543-561, ¢g) printed in India

A stochastic theory for clustering of quenched-in vacancies-- 2. A solvable model

G A N A N T H A K R I S H N A

Reactor Research Centre, Kalpakkam 603 102 MS received 21 November 1978

Abstract. The model introduced for clustering of quenched-in vacancies in the first part of this series of papers is considered. Using a generating function, the rate equations are converted into a first order partial differential equation for the generat- ing function coupled to a differential equation for the rate of change of the concen- tration of single vacancy units. A decoupling scheme is effected which gives an exponentially decaying solution with a very short time constant for the concentration of single vacancy units. The differential equation for the generating function is solved for times larger than the time required for the concentration of single vacancy units to reach its asymptotic value. The distribution for the size of the clusters is obtained by inverting the solution thus obtained. Several results that follow are shown to be in reasonably good agreement with the experimental results.

Keywords. Vacancy units; vacancy loops; stacking fault tetrahedra; generating function; Fokker-Planck equation.

1. Introduction

In the previous p a p e r ( A n a n t h a k r i s h n a 1979a, referred to as p a p e r I hereafter), we introduced a m o d e l which can be solved in closed f o r m . This m o d e l specialises to a situation where the t o t a l n u m b e r o f vacancies in the cluster is q u a d r a t i c in the linear dimension o f the cluster. This corresponds to the f o r m a t i o n o f stacking fault tetrahedra a n d v a c a n c y loops. The a p p r o a c h o f this m o d e l consists in writing d o w n a system o f n-coupled rate equations (the differential-difference equations) f o r t h e growth o f clusters a s s u m i n g that only single vacancy units are mobile. The general rate e q u a t i o n given by equations (2), (3) and (4) of p a p e r I c a n n o t be solved even o n a c o m p u t e r when the n u m b e r o f such equations exceed a b o u t 103. I t is f o r this reason t h a t Kiritani (1973) resorts to a grouping m e t h o d (which in essence is coarse graining). T h e resulting equations are then solved on a c o m p u t e r . Thus the evolu- tion o f clusters c a n n o t be followed in a precise w a y when the n u m b e r o f equations is large. T h e g r o u p i n g operation itself has been criticised b y K i o w a (1974) b a s e d on c o m p a r i s o n o f results o f a solvable model ( M o n t r o l l 1967) with the numerical results o b t a i n e d o n a c o m p u t e r using the grouping operation. H a y n s (1976) h a s studied the effect o f the grouping operation and finds t h a t it yields consistent results when handled carefully. H e also observes that the g r o u p i n g m e t h o d is the only m e t h o d t h a t offers h o p e o f obtaining solutions to p r o b l e m s where such large n u m b e r o f coupled e q u a t i o n are involved in the absence o f a n y analytic method. T h u s it is desirable to evolve a ~olvable method which preserves the essential features o f the physical system a n d which can be solved in closed form, T h e model that we h a v e 543

544 G Ananthakrishna

introduced in paper I is a first step in this direction and will form the basis for further work presented in papers I11 and IV (Ananthakrishna 1979b and 1979c). Although the model gives a rather peaked distribution possibly due to the choice of the number of absorption and emission sites, and the assumption of constant binding energy, some comparison is possible with experiments. We shall discuss the advantages, disadvantages, possible reasons for the drawbacks and how these inadequacies will be overcome in subsequent papers. In the present paper we deal with the solution of this model in detail as this model will serve as a starting point for a continuum model (paper III, Ananthakrishna 1979c, Ananthakrishna 1977). We shall first recall the major idealizations in this model. We shall try to give some justification for these assumptions.

The rate equations have been written down by assuming the growth o f clusters proceeds via the absorption and emission o f single mobile units. (In the case of gold and aluminium, divacancies are considered as the mobile units.) This assumption is justified in most cases, particularly for the cases considered namely for aluminium and gold. For instance, the mobility of the monovacancies are two to three orders o f magnitude smaller than that of divacancies at temperatures of interest and the other clusters are practically immobile (Kiritani 1973). Further the results o f computer calculations of Kiritani (1973) with one mobile unit compare well with experiments thus offering a justification for this assumption. It is expected that the next mobile species may become important only at late stages o f clustering. We have also considered the system to have no sinks. This assumption is not serious since as far as the phenomenon of clustering is concerned, only supersaturation of vacancies is required. So ignoring sinks would effectively increase the supersaturation and would thus aid clustering. This in turn may lead to increased concentration of clusters. The support for this claim can again be had from the computer calculations o f Kiritani (1973). His work demonstrates that a system without sinks is sufficient to describe the phenomenon of clustering. Now if we wish to solve these equations in closed form further assumptions have to be made. This involves assuming that the binding energy o f a vacancy unit in a cluster is independent o f the size o f the cluster as far as the growth of clusters is concerned. The justification for this assumption is three-fold. First, the binding energy for vacancy loops and stacking fault tetrahedra saturates very rapidly, reaching a near asymptotic value for n =n~-~

250, whereas the number of vacancies in a cluster o f average size is of the order o f 105 , which suggests that the assumption is justified. In view of this, the emission rate depends on ( n ) as we will show later, where ( ) is taken over the distribution N~. In table l, we have given the variation o f b, as a function of n for aluminium.

The binding energy changes rapidly only for n ~ 50. Thus the variation of bn as a function o f n, can be seen to be small for the region o f interest. Second, by taking the same binding energy for all clusters, we will be essentially overestimating the density o f small clusters. This can be seen from the binding energy curve (Kiritani 1973 and table 1) which suggests that the probability o f emission from smaller clusters is larger than that from larger clusters. Third, and probably the best in our view, with this assumption the equations are mathematically tractable in closed form.

Further, the emission probability, as will be shown later, will be related to experi- mentally measurable quantities and therefore can be r e g a r d e d a s a parameter to be fixed by comparison with experiments. Thus xz which was originally a function o f n has been regarded as a function of (n). (For this reason, this factor takes into

Stochastic theory--2

Table 1, Binding energy bn as a function of n for alurninium.

n bn (eV) exp [--bn/kT]; T=230°K 10 0"303 1"14 X 10 -6 20 0"357 3"75 x 10 -~

30 0"382 1"33 x 10 -~

40 0"398 6-85 x 10 -7

50 0-407 4"72 x 10 -8

100 0"437 1"36 x 10 -~

150 0-450 7"94 x 10 -°

200 0"458 5"70 X 10 -°

300 0"470 3"46 x 10 -*

400 0'481 2"19 x 1 0 - °

545

account a n d corrects f o r the various idealisations in the theory. See § 4.) Once this assumption is m a d e , these equations can be solved using a generating function.

The p r o b l e m we wish to solve is: given that initially we h a v e a supersaturated system o f vacancy units with a concentration N 1, could we predict the distribution o f N , ( t ) at any instant o f time t ? O u r main aim would only be the statistical aspects o f clus- tering. G i v e n the transition probabilities for the association a n d dissociation, we would only be interested in obtaining the distribution function N,(t). T h e details a n d the precise m e c h a n i s m s b y which a particular type o f defect aggregate is f o r m e d will not be relevant. ( F o r example the growth o f t e t r a h e d r a m a y be due to the m o v e m e n t o f v a c a n c y ledges). However, the energies associated with these m e c h a - nisms o f f o r m a t i o n o f the defect aggregates would enter into the transition p r o b a b i - lities. M a k i n g use o f the m e t h o d of generating function we obtain a partial differential e q u a t i o n f o r the generating function coupled to a differential equation f o r rate o f change o f the concentration o f single vacancy units. A l t h o u g h we are n o t interested in short t i m e aspects o f clustering, we do obtain s o m e i n f o r m a t i o n a b o u t the nucleation times via the solution o f the concentration o f single vacancy units which has already been outlined in paper I and will not be repeated here. T h e basic idea involved is to d e c o u p l e the differential equation governing the rate of decay o f the concentration o f single vacancy units f r o m that o f the differential equation f o r the generating function in an a p p r o p r i a t e way. This is possible because the total n u m b e r o f vacancies in the system is conserved. As we have seen this yields (paper I) an exponentially decaying solution for the concentration o f single vacancy units with a small time constant, which means that the concentration o f single vacancy units attains an equilibrium with the configuration o f small clusters in a very short time.

T h e n using the a s y m p t o t i c value o f the concentration o f single vacancy units, the solution o f the e q u a t i o n for the generating function is obtained. By inverting this solution we obtain the distribution function for the cluster sizes. The results o f the average quantities t h a t follows are in reasonably g o o d a g r e e m e n t with the experi- mental results on the f o r m a t i o n of stacking-fault t e t r a h e d r a in quenched gold (Jain and Siegel 1972a), a n d the f o r m a t i o n of faulted vacancy loops in quenched a l u m i n i u m (Kiritani 1973; Y o s h i d a et al 1963).

546 G A n a n t h a k r i ~ h n a

2. Solution of the model

We shall briefly summarise a few results on the formation of stacking fault tetra- hedra and faulted vacancy loops in quenched gold and aluminium respectively.

Stacking fault tetrahedra are formed in quenched gold with their four stacking-fault sides on the four non-parallel {111} planes arranged in a tetrahedral shape. The edges of the tetrahedran are parallel to the six (110> directions. In the case of aluminium, the loops formed are hexagonal and are formed on the ~.111} planes bounded by (110> directions. As much as 9 5 ~ of the vacancy loops formed are faulted under certain conditions (Kiritani e t a l 1964). Other types of extended defects which are formed will not be considered in this work. (However, the theory holds for all planar configurations and stacking-fault tetrahedra.) For simplicity we shall assume that the tetrahedra are regular and the loops circular. We shall use the total number of vacancies contained in the cluster to be L ~ / a ~ or 4~- r 2 / a Z a / 3 , where a is the lattice constant, L the edge length of the tetrahedra and r the radius of the loop.

The energy of formation of such tetrahedra and loops as a function of the number of vacancies contained in the cluster is given by (Cotterill 1965).

en = d l n q - B l n 1/2 In (ct n); i = L , T, (1)

with d r = 7' a 2 x / 3 , A L = Y a S ' V ' ] , C r = 8, C L = ~, B r = G a S / [ 2 4 r r ( 1 - - v ' ) ] and

B L =- G a a / [ 4 w 61/~ (1--v')]

The subscripts refer to tetrahedra and loops respectively. (In the above expression, G is the shear modulus, v' the poissons ratio, and 7' the stacking-fault energy). The binding energy is

b,. = e f t - [ , . - ".-d, which for large n takes the form

b, "" E f t - - A t - - B , [ 2 + l n c l + l n ( n - - 1 ) I / 2 n m. (2) This formula has been obtained by using

( n - - 1) 1/z ~ n m - - n - m ~ 2 and In [1 -- (l/n)] ~ 1 / n .

Therefore (2) approximates the exact one for n ~, 1. (Indeed, it agrees reasonably well for n~100). This above equation tells us that b, is a slowly varying function of n for large n. An exact calculation indicates that b~ saturates very rapidly reaching the asymptotic value for n,~,200. (See for example Kiritani 1973.)

The concentration of single vacancies between the usual quenching and the ageing temperatures differs by ten orders of magnitude. This excess of vacancy concentra- tion leads to the formation of dusters of finite size via the collision of single mobile

!mits. We shall also assume that the quenching is instantaneous and nearly 50~/~

Stochastic theory---2 547 o f the single vacancies a r e in the f o r m o f divacancies before the s a m p l e t e m p e r a t u r e reaches t h a t o f the t e m p e r a t u r e o f the bath. In cases where there is s o m e u n c e r t a i n t y in the c o n c e n t r a t i o n o f vacancies stored in the clusters, we have estimated it b y u s i n g the a v e r a g e size a n d density (as we have done in the case o f the d a t a given by K i r i t a n i

1973, for a l u m i n i u m ) .

The rate e q u a t i o n s h a v e been written d o w n in the earlier p a p e r (paper I). We shall refer the e q u a t i o n s in p a p e r I by (I.n.). In the following we shall show t h a t xz in this picture should be r e g a r d e d as a function o f ( n ) instead o f n. This is a consequence o f o u r a s s u m p t i o n a b o u t the constancy o f the binding energy. F r o m (2) it is clear that the t e r m n -1/~ In ( n - - 1) 1/~ is the slower term o f the two. Using this fact a n d ~b(n)

= n - - 1 and ~ ( n ) = n in the equation for N 1, we have

dt 2 2

This can be written as

- - d N i : x i N ~ + x l N l f (t ) - - x i x 2 ( ( n ) ) f (t).

dt

I f bn rapidly saturates as we have pointed out namely for n ~-~200. This is permissible under the condition t h a t x2(n) is a rapidly saturating function whereas the p e a k o f the distribution occurs in experimental situations at a very large values. (see table 2).

Strictly

nx 2 (n) N,, = (nx~ ( n ) ) ~[, 2

where N = ~ N,. N o w if we use 2

( x d n ) n ) ~ ( n ) ( x , ( n ) ) -- ( n ) x ~ ((n)),

we get the result indicated above. This is s o m e w h a t the m a t h e m a t i c a l equivalent o f the a s s u m p t i o n a b o u t constant binding energy. W e shall later show t h a t this is w h a t follows in the final analysis. In fact, here it would have been sufficient to have t a k e n x~ as a constant. This a p p r o x i m a t i o n does not affect N 1 a n d the a v e r a g e values much, b u t affects the distribution. We remedy this p r o b l e m in the next p a p e r ( p a p e r

Table 2. The values of the parameters used

(eV) (eV) (eV) in sec -1

Z1 Z~

Au 0"94 0"86 0"55 1"0 3 × 10 ~s 10 10 AI 0"76 0"56 0-46 2"4 3 x 10 la 10 10

548 G Ananthakr~hna

III). T h e validity o f these two statements can be seen b y the fact that b o t h dN1/dt a n d d f / d t depend o n

x2 (n) n N , 2

which we have a p p r o x i m a t e d by x2 ( ( n ) ) f ( t ) and thus the effect on N 1 should be less in c o n t r a s t to the effect on iV, which depends on x 2 (n) a p p r o x i m a t e d by x2 ((n>).

It is worthwhile c o m m e n t i n g on the choice o f ~b (n) a n d ~ (n). This choice has been m a d e since it is the only choice that allows the decoupling o f equations (1.3) a n d (I.4) f r o m (I.2) even with the a s s u m p t i o n t h a t x~ is independent o f n.

W e shall d e m o n s t r a t e this point shortly. Before t h a t s o m e observations can be m a d e a b o u t ~b a n d ~. These factors correspond to the n u m b e r o f absorption and emission sites respectively. In a real physical situation the n u m b e r of emission and a b s o r p t i o n sites are functions o f the size o f the cluster. F o r example, in the case of a v a c a n c y l o o p the n u m b e r o f absorption and emission sites which depend linearly on n for small n can be expected to reach an a s y m p t o t i c value o f n x/2 f o r large loops. Never- theless, f o r simplicity we shall use the above relation to hold in o u r model. This m a y be one o f the factors which is responsible for the unphysical n a t u r e o f the distribu- tion. We shall c o m m e n t m o r e on this later. Rewriting equations (I.2), (1.3) and (I.4) using (I.17) a n d (I.18) we have,

dN1 _

dt x~ N~ + Xl [x~ (<n>) -- N~] f ( t ) , (3)

aN2 - - ½ x 1 N1 ~ - - x , x 2 (<n>) N,, -1- 3 x 1 x~ (<n>) Ns - - 2 x 1 N 1 N~,

dt ⁽⁴⁾

a n d d N . ^_ ( n - - l ) X 1 N 1 Nn_ 1 Af_ ( n + l ) x 1 x 2 ((n>) N,,+I dt

- n x l N1 N , - - n x , x~ ( ( n > ) N . . (5)

Since f ( t ) ---- ~ n IV. = N o - - N 1, (6)

2

e q u a t i o n (3) is entirely a function o f N 1 and thus leads to a n exact solution o f N 1.

We c a n n o w show that we c a n n o t decouple the e q u a t i o n f o r N1 f r o m that for N, unless ~b and ~ are chosen to be ( n - - 1) and n respectively. It is clear that any devia- tion f r o m linearity like ~b (n),-~n 1/~ or any power o f n leads to a hierarchy o f equation in functions

f(t),f1(t)

= ~ ~b(n)

N2, fo(t )

= ~ Nn

~2(n),

etc.

Thus the decoup]ing of the equation is not possible. Even a slightly different choice

Stochastic theory--2 5 4 9 o f ~b and ~ (preserving the linearity in n) would also disallow such a decoupling.

Consider using

~(n) --~ ~b(n) -F- a, and ,/;(,)-~

~(n)

+ b.

Then - - d t - - ( a ~ - x 1) N1 ~ - x t N ~ f ( t ) ~ - a x 1 N a IV.

2 K-"

- - x l x 2 . f ( t ) - b x l x~ , , 2 N , .

2

This equation cannot be solved unless ~ r = l 2 N, is known. The f a c t o r N is a strong function o f time for short times. This can be seen by the following argument. F o r short times after quenching, the rate at which the single vacancies are depleted is the cause for the p r o d u c t i o n o f small clusters. Also from the analysis when a and b are zero we know that N: has a very short time constant. S i n c e f ( t ) is linear in N 1 and

= f ( t ) / ( n ) , N would nearly have the same dependence as N 1 because in this short time ( n ) does not change much.

The solution o f (3) has already been given. (see equation 0.19)). Here we shall discuss the result. It is clear that the asymptotic value o f N 1 is Nox2/(No+x2). This enpression, apart from giving the time dependence o f N 1, also provides an estimate o f the nucleation time o f small clusters. I f we regard the nucleation time as the time required to reduce the concentration o f single vacancy units to a p p r o x i m a t e l y a few percent o f the initial value, then the nucleation time for gold aged at 313°K and alu- minium aged at 283°K is o f the order of 10 -3 sec. (The value o f N O used is 3 × 10 -4 and 2 × 10 -4 respectively. These numbers have been obtained using the parameters listed in table 2. In the case o f the formation o f tetrahedra, the effective migration energy o f divacancies decreases by 0-07 eV due to the t e m p e r a t u r e dependent sink efficiency o f tetrahedra. See Jain and Siegel 1972b: Sahu et al 1976.) This result agrees with the c o m p u t e r calculations o f Kiritani (1973) for the case o f aluminium.

The time constant is determined by the product o f the mobility a n d the initial concen- tration o f single vacancy units (since x2~: N o as will be shown later). Thus the nuclea- tion time rapidly increases as the temperature is reduced. (The above definition of the nucleation time has the simple interpretation that it is the time required for the formation o f clusters o f small sizes having a few vacancy units on an average.) The fact that the concentration o f single vacancy units decreases rapidly with a small time constant coupled with (6) tells us that the concentration o f single vacancy units attains an equilibrium with the configuration o f small clusters in a very short time. Since N1 is monotonically decreasing, we can decouple (5) and (4) from (3) and solve (5) and (4). This decoupling scheme is permissible because the decay time o f N~ is very small in contrast to the time scale involved for the formation o f large clusters as will be shown later on. Here it would be sufficient to indicate it by the following physical argument. In the initial stages o f clustering only small clusters are formed. In course o f time some clusters grow at the expense o f other clusters via a redistribution o f single vacancy units a m o n g themselves.

550 G Ananthakr~hna

In o r d e r to solve (4) and (5) we use the generating function approach. We define

x ( Z , t ) = Z N,,Z-"; [Z ~ 1 .

2

(7)

Clearly b o t h

X(1, t) = N , and - - ~ I z I = 1 2

=.f(t)

exist, since we know that the latter represents the total n u m b e r o f vacancy units in clusters o f all sizes. Therefore X(Z, t) is an analytic function o f Z for all Z>~I, and all its derivatives exist in that region. N o w we wish to t r a n s f o r m (5) and (4) into a differential equation for

X(Z, t).

This is done by multiplying (4) by Z -~ and (5) by Z - " and summing over n. After suitable manipulations we get

DXol -- x1N12Z-22

-b- ~ z ( Z - - 1 )

(Nl--Zx~) xl + xg, x1N2

( Z - 2 - - 2 Z - I ) -

(8)

We can see that if we set Z = I , we get the equation for f ( t ) . Equation (8) is a first o r d e r partial differential equation. We have obtained the solution o f this differential equation for regions o f time after N 1 has reached its asymptotic value with the assump- tion t h a t ignoring the term Nz does not affect the solution very much. The method o f obtaining the solution and the justification for the above assumption a b o u t ignor- ing the N 2 term has been outlined in appendix 1. The solution that we have obtained with the initial condition that

X(Z,

0) = 0, which reflects the fact that there are no clusters o f finite size at t = 0, is

x ( Z , t ) = a ~ l + b l n [

Z ( l - - b ) ] 1 l n q ~ - - 1 c 6 3 L Z - - ~ - - 1)J 2r- ( b - - l ) b'-Z

q_ ~ (Z--1)+b--Z

(9)

b[b--Z +c~b(Z--1)]'

where b --

No/(Noq-X2); c = x 1 x~

and

a= x z cb2/2.

In o r d e r to obtain iV,, we have to invert the above function, which involves per- f o r m i n g the c o n t o u r integral

1

Z"-~ X (Z, t) d Z.

(10)

2~ri J

Instead, we expand the left hand side o f (9) in inverse power o f Z and collect the co- efficient o f

Z-".

This gives

N.(t) ab"-2(!~_~):~l+b+

~b(b--l) ~ ]

c(1-b~)" L n

(l---~-(i----b~)J" ^(lJ)

S t o c h a s t i c t h e o r y - - 2 551 As t ~ 0, clearly N , = 0, which means that there are no vacancy clusters o f finite size at t = 0. As t ~ 0% N , takes the form given by

N . -- a ( l + b ) b " _ a,l_b,~¢ ~ exp [n In b], (12)

b2cn b2nc

which is the stationary solution o f (5). The distribution given by (1 I) and (12) are peaked at the origin a n d therefore is not a realistic distribution. T h e reason for the non-evolution o f the peak and the peaked nature o f the distribution can be m a n y - fold, some o f which have been mentioned in §1. We have investigated this p r o b l e m in detail a n d the d e v e l o p m e n t o f a continuous model presented in the next p a p e r (paper III) is the result o f this investigation. Here we mention a few points in this connection.

The basis for the c o n t i n u u m model is this discrete constant binding energy model.

A Taylor series expansion f o r P,±t is used upto second order. The resulting partial differential equation is slightly modified to satisfy the conservation o f the total n u m - ber o f vacancies. This equation looks very similar to a F o k k e r - P l a n c k equation but should be only regarded as a differential equation for the distribution function for the sizes o f the clusters. The method used in developing the c o n t i n u u m model sheds some light on the p r o b l e m at hand.

It m a y be recalled that the choice of the number o f a b s o r p t i o n and emission sites have been chosen to be if(n)----n-- 1 and ~(n) = n respectively. In a realistic situation these can vary as the clusters grow for instance f r o m n for small loops to n 1/~ f o r large loops. Such a change in the choice o f if and ~ can give rise to very different partial differential equation for the distribution o f clusters. The solutions will be very different and the position o f the peak will be at non-zero value o f n. Even in the case o f the choice we have made for if(n) and ~(n), if we make a small change, it produces a different distribution as we have shown in the next paper. (see p a p e r III).

Indeed the differential equation which we have evolved self-consistently corresponds to a change by a f a c t o r o f ½ for b o t h if(n) and ~ n ) . A n o t h e r factor which m a y have contributed is the assumption about constant binding energy.

In the discrete model, it is possible to make the peak o f the distribution evolve in time by choosing i f ( n ) = I n - - n o - 11 and ~ ( n ) = I n - - n o J, where n o is the position o f the peak at t-+ oo. T h e value o f n o has to be supplied by experiments. T h e m e t h o d o f attack is exactly the same and will not be given here as it does not give a n y better insight into the p r o b l e m except for the final result. T h e distribution obtained in this case is

f A No (n0-- 1) (b~--b ~) x2, for n <n0 (no--n) (l--b)

A No(no--1 ) (ba--b"o) b"-"o

( n _ n o ) x2, f o r n > n o ,

where A is normalisation constant. Although the distribution is not strictly symmetric about no, a plot o f N, for typical values o f No, x2 and n o shows that it is nearly symmetric.

55~ G Ananthakrishna

In view o f all the above statements we regard n in (I 1) and (12) to be measured f r o m the peak, or we can rewrite the distribution in the f o r m

N , = 2xz (2N°-q-xz) b~n-n°l

(No-}-xa) [ n--no 1" (13)

The position o f the peak n o should be supplied by experiments.

3. Comparison with experiments

Having o b t a i n e d the distribution function, we can c o m p a r e several results with experiments. These are: (a) the decay o f the c o n c e n t r a t i o n o f single vacancy units and the nucleation time o f small clusters, (b) the total density o f the clusters as a function o f time a n d temperature, (c) the average size o f the clusters as a function o f time and temperature, (d) the characteristic time for the growth o f large clusters. We shall c o m p a r e these quantities with the results o n quenched gold and aluminium.

(Kiritani 1973; Y o s h i d a et al 1963; Jain and Siegel 1972a).

The first main result that emerges out of o u r t h e o r y is that the concentration o f single vacancy units decreases exponentially with a very short time constant. This indirectly leads to information about the nucleation time. In a realistic situation, a cluster that becomes stable at some size may be regarded as a nucleus. In the case o f vacancy loops, it has been experimentally well established that there is no stable cluster and the binding energy increases gradually with size (Kiritani et al 1969). In contrast there is sufficient indication that the f o r m a t i o n o f the stacking-fault tetra- hedra has a stable size below a certain temperature. T h e exact mechanism leading to the f o r m a t i o n o f stable nucleus is not clear. In a n y case, o u r theory provides a useful i n f o r m a t i o n a b o u t the nucleation time. We have defined the nucleation time r', as the time required for the concentration o f single vacancy units to reduce to a few per cent o f its initial value. This definition has a simple interpretation that there are few vacancy units in each cluster on an average. This definition yields a value o f 10 -2 sec for ~-' at 283°K for aluminium which agrees with the value obtained by Kiri- tani (1973) o n computer. In his approach he defines two quantities namely the tran- sient time and the nucleation time. Our definition o f the nucleation time is the same as his transient time. The value o f the nucleation time (in his definition) is larger by an order o f magnitude than the transient time. W e do not have an equivalent o f Kiritani's nucleation time due to the nature o f the decoupling used. Although we have only indirectly obtained information about the nucleation time, it does give an idea o f the time scales involved. Experimentally the time scales involved can only be estimated indirectly and therefore no quantitative comparison can be made.

(Kiritani et al 1969). A plot o f the nucleation time for the f o r m a t i o n o f vacancy loops in quenched aluminium and the formation of stacking-fault tetrahedra in quenched gold is shown in figure 1. The nucleation time rapidly decreases with temperature.

This should be expected as the mobility o f the vacancy units increases. In all p r o b a - bility, the estimated nucleation time is only a lower b o u n d .

The Shape o f the distribution is not realistic. Experimental curves are nearly Gaussian often with some ~/symmetry. Howe~ver, as for the calculation o f averages,

Stochastic theory--2

553 I 0 °

1 0 -I

16 2

10 -3 -6 z i ~ 4

- 2 0

- Au

I I t I

-10 0 10 20 ^{3 0}

Ageing temperature ^(°C)

Figure 1. The nucleation time for faulted l o o p s a n d stacking-fault tetrahedra in aluminium a n d gold respectively. The curve A1 (K) corresponds to the work o f Kiritani (1973).

the a g r e e m e n t with the experiment will be shown to be quite good. Consider the calculation o f m e a n square n u m b e r o f vacancies in a cluster.

oo oo

Zn' /7

2 1 ~ - - ~ ° 1 / I . - ~ o l

T h e p r i m e on the s u m m a t i o n indicates n ~no in the s u m m a t i o n . T h e value o f no is generally large, at least at temperatures o f interest in this paper. I f we use this fact, we have

oo

( n 2 > =

~'

- - n o + 2

---2

~o b,o,

(m q-- no) z irnl /-n.L~+ 2 im I ,

co

1 1

N°(N° + x2)

(14)

So ( n ~ > - - no = _{X 2} 2 In [(N 0-[-x~''x~'')/J

Thus we see t h a t x 2 is directly related to <n 2) a n d no, and therefore x~ can be regarded as a p a r a m e t e r to be determined f r o m experiments. Given the value o f ( n 2 ) a n d n o at a n y t e m p e r a t u r e x2 can be determined. T h e total concentration o f the clusters,

N r = ~ iV, = 4 x~ (2N°~-x~)

In

( No+ x~ I .

2 ( N o + x D ~ x~ /

( i s )

554 G A n a n t h a k r i s h n a

It is clear f r o m (14) a n d (15) t h a t the t e m p e r a t u r e d e p e n d e n c e o f the total density N s a n d the m e a n s q u a r e d average o f the n u m b e r o f vacancies is opposite. ( I f (nZ) 1/2 increases with t e m p e r a t u r e , n o also increases. H o w e v e r , the t e m p e r a t u r e dependence o f these two quantities would be in general different.) T h u s if ( n 2) increases with temperature, N r decreases a n d vice versa.

N o w if we wish to c o m p a r e with experiments, a little care would be needed in identifying the theoretical distribution in n-space with the experimental distribution in the r-space. We have shown in appendix B t h a t the total densities in n- a n d r- space are the same. With this identification, we shall p r o c e e d to c o m p a r e the theo- retical a n d the experimental densities. (The total density in theory is denoted by N ~ = Z N if, where Z is the total n u m b e r o f a t o m s p e r cm-3.) T h e p r o b l e m in using (14) is that ( n 2 ) is n o t generally supplied by experiments, i n cases where the distri- bution is given, these can be easily calculated. T h e case o f gold (Jain and Siegel 1972a) a n d a l u m i n i u m (Kiritani 1973, Jain and Siegel 1972b) where the distributions are given will be first considered. The concentration o f vacancies contained in the clusters is calculated f r o m the experimental d a t a using the expression N o = a a ( r ~ N ~ / 4 where Ns e is the total density of the d u s t e r s e x p e r i m e n t a l l y measured, ( r 2) is the m e a n squared radius o f the clusters a n d ~ is equal to unity f o r tetrahedra, a n d 4rr/~¢/3 f o r vacancy loops. (We wish to r e m a r k that the value o f N O t h a t we will use for c o m p u t i n g N s r will be different f r o m that used f o r calculating the nucleation time for vacancy loops earlier. While computing the nucleation time, we used a value o f 2 x 10 -4 to facilitate c o m p a r i s o n with the value o b t a i n e d b y Kiritani (1973). In the present case, the value o f N O h a s to be consistent with the total n u m b e r o f vacancies in the clusters.) T h e value o f ( n 2 ) t n , n o a n d N O h a v e been listed in table 3. (Table 3 also contains d a t a as given in Kiritani 1973, Jain a n d Siegel 1972a, Yoshida et al 1963. Incidentally, there is an error in scale o f the y-axis o f figure 16 o f Kiritani 1973. T h e y-axis should be multiplied b y a f a c t o r 10 xl f o r experimental curve a n d a factor o f 10 t° f o r the theoretical one (Kiritani 1977).) T h e calculated values o f the total density a l o n g with the corresponding e x p e r i m e n t a l values have also been listed in table 3. T h e a g r e e m e n t in the case o f g o l d is seen to be quite good. I n the case o f aluminium, the theoretical values are nearly 25 times larger t h a n the corres- p o n d i n g experimental values. This is due to the fact t h a t the theoretical distribution is highly p e a k e d in c o n t r a s t to the experimental ones. I n a d d i t i o n the experimental distributions are nearly symmetric which generally lead t o small values o f variance.

(The value o f ( n ) is close to n o in these cases. F o r example, in the case o f alumi- n i u m quoted in c o l u m n one, row three, ( r ) = 2 7 6 A with r0=275,~. ) Thus these values

Table 3. Total density of dusters

Total density per cm 8

( j~ ) no No xz Experimental Theory

z N ~ r Au a 410 85259 62977 3 x 10-' 1"51 x 10-* 2-00 x 10 Is 4"34 x 1016 A1 b 642 1 9 3 3 4 7 172897 6 x 10 -6 1"96 × 10 -l° 2-00 x 10 ~s 5"96 × 10 a' A1 c 276 36204 33716 5"4 x 10 -6 1"25 x 10 -1° 1-00 x 10 ~" 3"26 × 10 a6 a .lain and Siegel 1972a; b Yosh/da el al 1963; c Kiritani 1973.

i I

Stochast~ theory--2 555 o f {n~) 1/2 a n d n o lead to larger theoretical densities. I n the case o f gold, due to the a s y m m e t r y the difference between {n 2) and n o is larger t h a n the c o r r e s p o n d i n g symmetric case, i.e, a distribution with the same value o f ( n 2) a n d n 0. T h e r e f o r e the agreement is better.

N o w we c a n consider the t e m p e r a t u r e dependence o f N r. I n order to o b t a i n this, we need the t e m p e r a t u r e dependence o f ( n ~ ) and no. (This is necessary since xz is n o w a p a r a m e t e r . ) G e n e r a l l y (n ~) and n o are not given. These c a n be o b t a i n e d in the cases where the distribution is given. The two distributions given f o r v a c a n c y loops (Kiritani 1973, Y o s h i d a et al 1963) indicate t h a t there is very little a s y m m e t r y in these distributions. W e take this feature and other features o f these two cases to be representative at o t h e r temperatures as well. F o r these t w o cases, we find t h a t the values o f (n2)l/4/(rZ/aZ) 1/z and no~/Z/(r~/aZ} 1/2 to be very close. T h e values o f these quantities t h a t we h a v e used for calculating n o a n d ( n z} at other t e m p e r a t u r e s are 1.047 a n d 0.996 respectively. The data used (first f o u r c o l u m n o f table 4) h a v e been o b t a i n e d f r o m figure 18 o f Kiritani (1973). W e n o t e t h a t the c o n c e n t r a t i o n o f vacancy units in these cases (calculated f r o m the d a t a using the f o r m u l a N O = a a

~r 2) N~/4) is n o t constant. The values o f the theoretical densities Z N r at v a r i o u s t e m p e r a t u r e s a l o n g with the corresponding experimental values h a v e been t a b u l a t e d in table 4. A p l o t o f the t e m p e r a t u r e dependence o f the total densities is s h o w n in figure 2. I t c a n be seen t h a t the theoretical and e x p e r i m e n t a l curves are n e a r l y parallel.

T h e t i m e evolution o f the distribution has two contributions, one o f which vanishes identically as t - + oo. T h e t i m e required for the f o r m a t i o n o f large clusters is given b y

• ~ , ~ (No+xz)/xlx2L T h e value o f ~r" that we obtain f o r gold a n d a l u m i n i u m a r e larger t h a n the c o r r e s p o n d i n g experimental values b y t w o orders o f magnitude. This is due to the fact t h a t ~," is sensitive to the value o f x 2. T h e additional r e a s o n m a y be in the very n a t u r e o f the distribution.

4 . S u m m a r y a n d d i s c u s s i o n

T o s u m m a r i s e the r a t e equations for clustering o f vacancies were written d o w n a s s u m - ing only single v a c a n c y units to be mobile. By defining a generating function, the coupled differential-difference equations were converted into a first order p a r t i a l differential e q u a t i o n f o r the generating function. U s i n g the fact t h a t the t o t a l

Table 4. Temperature dependence of the total density A~.eing

temim- ( r D x/2 ratures (~,)

(°C)

(n~) l/j no No/10 -e xs

Total density/cm a Experimental Theory

--20 66

--10 97

0 150

10 276 20 438 30 790

i

P . - - 9

2173 1973 2-75 1"I x 10 -8 4591 4131 4"11 6-89 x 10 -°

10976 9887 4"36 3-40x 10 -0 36204 33716 5"4 1.25 x 10 -1°

93082 84296 9"4 6"2 x 10 -1°

302812 274238 134 2"95 X 10 -1°

- , - i

8"6 X 10 ^TM 2"1 X 10 ^TM 6"2 x 1014 1"46 X 10 TM

3"0 X 1014 7"9 x 1015 1"0 X 101. 3"2 x 1015 6"8 × 10 rs 1"6 X 101°

3"2 x 10 x8 9"2 x 10 t4

, , , , J - - - i

556 G A n a n t h a k r ~ h n a

1016~

I E

u

0

~ 10 ~5

>,

"~ 101,4

lo 3:

Figure 2.

Q

I I , . i I 1. I

- 2 0 - 1 0 0 10 2 0 3 0 Ageing temperoture (°C)

Temperature dependence of the total density of clusters.

number o f the vacancies is conserved, an exact solution for the decay of the concen- tration of single vacancy units was obtained. The form of the solution is exponential with a short time constant. This allows us to decouple it from the differential equation for the generating function. The solution of N z apart from giving the nucleation time for small clusters, also suggests that the collapse time for cascades produced during irradiation would be (Ncxz) -1 where Arc is the local concentration of the vacancies produced during irradiation. This time is expected to be of the order of 10 -~ to 10 -4 see at 300°C in aluminium if we assume N c ~ 10 -4 to 10 -6. Then, the solution of the differential equation for the generating function was obtained. The distribution function which was obtained by inverting the generating function predicts the total density as a function of temperature and time given ( n 2) and n o as a function o f temperature. The results agree well with the experiments. The theory also predicts the characteristic time required for the formation of large clusters.

We shall recall the idealisations and approximations made in obtaining the solution o f the problem and comment on them. The number of absorption (and emission) sites has been taken to be proportional to n-- 1 (and n) in order to facilitate decoupling as has been mentioned earlier. In the case of vacancy loops the number of absorption sites and emission sites are functions of the size of the loop and for small size it is linear in n, changing over to n 1/~ for large sizes. From the connection that we have investigated between the solution of differential-difference equation and the associ- ated continuum equation, it appears that this may be one of the factors contributing to the non-evolution of the peak. The major idealisation in modelling the physical situation is in the assumption about the binding energy of a vacancy unit being inde- pendent of the size of the cluster. Some justification has been offered in §1. The idealisation is possibly another factor responsible for the unphysical features of this distribution. In a separate paper (paper No. III) we shall see how the unphysical pature of the distribution can be overcome. The quantity x~ is dependent on (n~

Stochastic theory--2

557 and has to be regarded as a parameter, and therfore many factors which have not been taken into consideration while modelling such as the effect of impurities, the presence of various kinds of sinks has been taken care of (at least partially), since these have a direct effect on the sizes of the clusters and the total densities which are in turn a function of x2.

Two approximations have been made in obtaining the solution of the differential equation for the generating function. The first one is to ignore the term containing N~. The justification for this has been given in appendix A. The other approxi- mation is the decoupling approximation, wherein we use only the asymptotic value N 1 to solve for

X(Z, t).

This approximation assumes that there is a smooth evolution of

N,(t)

for shorter times which matches with the solution obtained at the start of the growth process (i.e., after N i has reached its asymptotic value). This statement appears to be justified since the solution

N,(t)

obtained is consistent with the initial condition that we have used in obtaining the solution

X(z, t).

This is to be expected for two reasons namely, first, the decay time for N1 to reach its asymptotic value is very small and second, after decoupling, the vacancy units redistribute themselves among the several clusters. Thus the evolution clusters proceeds in a smooth way.

This is supported by experiments done by Jain and Siegel (1972a). For shorter times the theory is incapable of predicting the distribution. However, there may be short transients superimposed on the present solution obtained. This aspect can be seen from the fact that

N,(t)

goes to zero as t is allowed to go to zero.

Acknowledgements

The author is very thankful to Prof. S K Rangarajan for many helpful discussions and valuable comments, to Dr G Venkataraman for his interest in this work and to Drs S Dattagupta, D Sahoo and V Balakrishnan for their suggestions and help.

Appendix A

In this appendix we shall outline the method of obtaining the solution of the differen- tial equation for the generating function. At the outset we wish to state that we have been able to obtain the solution only if the term containing N~ can be disre- garded and over the interval of time after N1 has reached its asymptotic value. Consi- der the equation without N2 term. Later we shall provide some justification for this assumption that ignoring the term containing N~ does not affect the solution of

X(Z, t)

seriously. First, we observe that even with this assumption, this equation cannot be solved in dosed form if the full time dependence of N i is used. Fortu- nately Nx has a very short time constant and N 1 attains its asymptotic value in less than a second. So we could use the asymptotic value of N i in (8) and solve the diffe- rential equation assuming that there is some kind of a smooth evolution of the dus- ters for time interval shorter than this. This.would mean that we have to extend the time domine upto zero. Using the asymptotic value of N 1 in (8) we get

07 _ aZ_~+c(Z_ 1) (O--Z) ~ ,

(A. 1)

558

G Ananthakrishna

where the constants are

b~-No/(Noq-x2) , c:-xlx 2

and a =

x2cb2/2.

The subsidiary system of equations take the form

dt dZ d x

1 c ( Z - - 1)(Z--b) d Z -~

The solution for these equations are

a [ _ ) + l + b Z 1

x - - - In +

L

c b 2

Z - - b 1--b

Z - - l ] ,

In

Z - - b = K ,

^(A.2) and Z - - 1 exp

[--c(l--b)t] Z--1

^K".

z---b = Z - - b ( t ) = (A.3)

Eliminating k' and k" from the above two equations we get a F l + b l n l n Z - - 1

x(Z, t) = ~ I. b 3 zZ~--b +

_{i - - b}

1

Z----b-- ~ z ] + F ( Z - - b 4 (t)), (A.4)

Z--1

where ~(y) is an arbitrary function of its argument to be determined by using the initial conditions. At t = 0 , all

N,

are zero except n = 1, which implies that

x(Z,

t ) = 0 . At this point it may be appropriate to point out the apparent inconsistency of using the asymptotic value of N 1 and using the initial condition

x(Z, t)=O.

After having used asymptotic value of N1, we do not expect the solution of

x(Z, t)

to be valid for times shorter than the time required for N 1 to reach its asymptotic value. (Note that this is less than a second.) Therefore there will be inconsistency for shorter times, in particular at t = 0 . ( t = 0 is a point in this region.) If the full time dependence of N 1 was used, then it is clear that more and more single vacancy units would flow back f r o m

N ,

to N1. Thus by disregarding the time dependence of N I, we have prevented the back flow and therefore cannot account for some vacancy units in this region, in .particular

No--N 1

(c~) vacancy units at t = 0 . (An alternate initial condition that suggests itself as a possibility is to use

x(Z, O)=N~/(No+x2)

which means that at t = 0 itself so many clusters are formed. Apart from the fact that this would be unphysical, it also raises a question about the distribution for N, to be used for computing

x(Z,

0) which is required for the initial condition. Thus we will use

x(Z,

0 ) = 0 as the initial condition. As we see later, this condition gives

iV.

which is consistent with the initial condition i.e., as

t->O, N,->O.)

With these remarks in mind, we shall use the condition

X(z,

0)=0. Then the solution that we obtain is

x(Z,

t) = a ~ld-b In~ Z(1--b) ] + 1 1

-~ (--~- LZ--b--~b(Z--1)J

1--b In ~b--b--Z

d?(Z--1)+b--Z

+ _(A.5)

Stochastic theory--2 559

Now we shall consider the justification that can be offered in support of the assumption that throwing away the term containing N~ should aot affect the solution of x(Z, t) seriously. First, recall that we solve for x(Z, t) only after N 1 reaches its asymptotic value. By this time several clusters would have formed and N2 happens to be just one of them. Thus ignoring the term

xlxzN~(Z-2--2 Z -1)

may change only the magni- tude of the coefficient N, in the expansion of x(Z, t), since this term is equivalent to a source term. However, the total number of vacancy units contained is proportional to 2Ne, whereas, X(Z, t) contains several At,. Thus ignoring this term should have very little effect on x(Z, t). Further there are other terms which contain N2 in the equation for x(Z, t). These terms arise from

_ ax xxx~ ( z - l ) ( z - b ) ,

and give a contribution of

+2z-~ x~x~N2 ( z - o (Z--b).

The term with the coefficient Z -t cancels the corresponding term in the source term.

The other terms are, however, larger by a factor of two. Thus the number of vacancy units that we have ignored (contaicmd in the term

NO

is small, tn addition the value of N~ at the beginning of the process of growth of the dusters is small. This can be seen by the following argument. We shall first argue that N~ would have reached equilibrium with N 1 by the time N 1 reaches its asymptotic value. Since N 1 is mono- tonically decreasing and since we have used the asymptotic value of N 1 (due to its short time constant), it means that N 1 has attained equilibrium with the remaining clusters, particularly with N 2. 0nly a redistribution among the various clusters can occur and any vacancy emitted from N~ or any other clusters is to be interpreted as going into another cluster. This is due to the fact N x has already reached its asymp- totic value. (It may be noted that a similar argument would not hold for shorter times.) Thus N 1 is in equilibrium with N~ and therefore, it is most likely to be of the order N~ ~ exp

b2/kT,

where b~ is the binding energy of the div'acancy unit. Thus the number of vacancy units contained in this term is small right at the beginning of the process of clustering. (Note that there is possibly a time dependence of N~. This may give raise to a short transient.) Also we wish to stress that there may be some effect of this approximation on the value of N2 at any instant of time, but we would not be interested in small clusters, since (n) is of the order of 105.

Appendix B

in this appendix we will show that the total density of the dusters measured in n-space is equal to the total density of clusters in r-space. This follows from fine fact that the total number of vacancy units ia both the spaces should be the same. Con- sider the concentration of vacancies in the clust~s in n-space.

5 6 0 G Ananthakrishna

w h e r e N(n) is t h e c o n c e n t r a t i o n o f clusters in this s p a c e . U s i n g the relation n =ar2/a) (a---1 f o r t e t r a h e d r a a n d a = # r r / ~ / 3 f o r loops), we h a v e

a ~ _ (ar~/a~) 2rdr. (B.1)

C ~ = _{a ~.} N _{a S}

T h e e x p e r i m e n t a l distributions are m e a s u r e d in r- o r in L - s p a c e a n d t h e c o n c e n t r a - t i o n o f v a c a n c i e s s t o r e d in the clusters is given b y

Co = a ( r z > iV, aI4, (B.2)

where Ns is the number density or the total density per cm -3 in r-space.

C v = a a[4 f r ~ ~a (r)dr : aa f rZ/a 2 ~ c (r)dr/a. (B.3) T h e d i s t r i b u t i o n o f clusters

N a (r) = 4/a a N c (r)

is in r- space (per cm -a per Angstrom unit), and NC(r) is the corresponding concentra- tion. Comparing (B.1) and (B.3), we have

2 a r/a~N(r2/a~) = N ~ (r).

The total density of vacancy clusters in n-space is

Z f N(n)dn = f N*(n)dn = 2a f rdr[a ~ zN(r~la 2)

-~ f 2a rdr/a 2 N * (r2/a ~) = f ~ d (r) dr : N --a. (B.4) T h u s t h e t o t a l density in n a n d r-spaces are equal.

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Stochastic theory--2

561

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