Nucleation of small clusters in supersaturated solu­tions has been of significant interest to several genera­tions ofscientists. The kinetic model for cluster growth and the rate of formation of stable droplets in vapor and second phase precipitation in alloys during aging was studied extensively. The similarity to the con­densation process in supersaturated solutions allows the results obtained to be used in RDT to describe the formation of defect clusters under irradiation.

The initial motivation for work in this area was to derive the nucleation rate of liquid drops. Farkas63 was first to develop a quantitative theory for the so-called homogeneous cluster nucleation. Then, a great number of publications were devoted to the kinetic nucleation theory, of which the works by Becker and Doring,64 Zeldovich,65 and Frenkel66 are most important. Although these publications by no means improved the result of Farkas, their treat­ment is mathematically more elegant and provided a proper background for subsequent works in for­mulating ME and revealing properties of the clus­ter evolution. A quite comprehensive description of the nucleation phenomenon was published by Goodrich.67,68 Detailed discussions of cluster nucle — ation can also be found in several comprehensive reviews.69,70 Generalizations of homogeneous cluster nucleation for the case of irradiation were developed by Katz and Wiedersich71 and Russell.72 Here we only give a short introduction to the theory.

For small cluster sizes at high enough tempera­ture, when the thermal stability of clusters is rela­tively low, the diffusion of clusters in the size-space governs the cluster evolution, which is nucleation of stable clusters. In cases where only FPs are produced by irradiation, the first term on the right-hand side of eqn [18] is equal to zero and cluster nucleation, for example, voids, proceeds via interaction between mobile vacancies to form divacancies, then between vacancies and divacancies to form trivacancies, and so on. By summing eqn [18] from x = 2 to 1, one finds

^ = J (x)k=i = Jnucl [24]

1

where Nc = ^2 f (x) is the total number of clusters.

x=2

The nucleation rate in this case, Jcnucl, is equal to the rate of production of the smallest cluster (divacancies in the case considered); hence the flux J(x)|x=1 is the main concern.

When calculating Jcnucl, one can obtain two limit­ing SDFs that correspond to two different steady — state solutions of eqn [18]: (1) when the flux J(x, t) = 0, for which the corresponding SDF is »(x), and, (2) when it is a constant: J(x, t) = Jc, with the SDF denoted as g(x). Let us first find »(x). Using equation P(x)n(x) — Q(x + 1)»(x + 1, t) = 0 and the condition »(1) = C, one finds that

P(y)

Q (y + 1)

Using function »(x), the flux J(x, t) can be derived as follows

J(x, t) = P(x)n(x)

The SDF g(x) corresponding to the constant flux, J(x, t) = Jc, can be found from eqn [26]:

1

PG0«M

Using the boundary conditions g(1) = »(1) = C one finds that Jcnucl is fully defined by »(x):

nucl _ ______ 1_______

Jc 1

[P(x)n(x)]—1

x=1

Generally, »(x) has a pronounced minimum at some critical size, x = xcr, and the main contribution to the denominator of eqn [28] comes from the clusters with size around xcr. Expanding »(x) in the vicinity of xcr up to the second derivative and replacing the sum­mation by the integration, one finds an equation for Jcnucl, which is equivalent to that for nucleation of second phase precipitate particles.64,65 Note that eqn [28] describes the cluster nucleation rate quite accu­rately even in cases where the nucleation stage coexists with the growth which leads to a decrease of the concentration of mobile defects, C. This can be seen from Figure 2, in which the results of numerical integration of ME for void nucleation are compared with that given by eqn [28].73

In the case of low temperature irradiation, when all vacancy clusters are thermally stable (C = Cv in the case) and only FPs are produced by irradiation,

the void nucleation rate, eqn [21], can be calculated analytically. Indeed, in the case where the binding energy of a vacancy with voids of all sizes is infinite, Ev(x) = 1 (see eqn [75]), it follows from eqn [25] that the function n(x) is equal to

C1 DyCy x 1

x1/3 DiCi

Substituting eqn [29] in eqn [28], one can easily find that the nucleation rate, Jcnucl, takes the form

rnucl

where w = (48я2/02)1/3 is a geometrical factor of the order of 1020m-2 (see Section 1.13.5). The sum in the dominant eqn [30] is a simple geometrical progression and therefore it is equal to

DyCy

DvCv — Di Ci

Substituting eqn [31] to eqn [30], one can finally obtain the following equation

Jcnucl = wCv(DvCv — DiCi) [32]

Note that the function g(x) in this case takes a very simple form, g(x) = Cc/x1/3, and hence decreases with increasing cluster size. In contrast, in R-space,
g(R) (see eqn [16]) increases with increasing cluster size: g(R) = (36я/0)1/3 CcR (see also eqns [43] and [44] in Feder et a/.69).

The real time-dependent SDF builds up around the function g(x) with the steadily increasing size range (see, e. g., Figure 2 in Feder et a/.69). Also note that homogeneous nucleation is the only case where an analytic equation for the nucleation rate exists. In more realistic scenarios, the nucleation is affected by the presence ofimpurities and other crystal imper­fections, and numerical calculations are the only means of investigation. Such calculations are not trivial because for practical purposes it is necessary to consider clusters containing very large numbers of defects and, hence, a large number of equations. This can make the direct numerical solution of ME impractical. As a result several methods have been developed to obtain an approximate numerical solu­tion of ME (see Section 1.13.4.4 for details).

The equations formulated in this section govern the evolution of mobile and immobile defects in solids under irradiation or aging and provide a frame­work, which has been used for about 50 years. Appli­cation of this framework to the models developed to date is presented in Sections 1.13.5 and 1.13.6.