Dissociation of vacancies from voids and other defects is an important process, which significantly affects their evolution under irradiation and during aging. Similar to the absorption rate eqn [54], it has been shown that the dissociation rate is proportional to the void radius. Such a result can readily be obtained by using the so-called detailed balance con­dition. However, as the evaporation takes place from the void surface, the frequency of emission events is proportional to the radius squared. In the following lines, we clarify why the dissociation rate is propor­tional to the void radius and elucidate how diffusion operates in this case.

Consider a void of radius R, which emits ndiss = t-d. vacancies per second per surface site in a spherical coordinate system. Vacancies migrate 3D with the diffusion coefficient Dv = a2/6t, where a is the vacancy jump distance and t is the mean time delay before a jump. The diffusion equation for the vacancy concentration Cv is

r2Cv = 0 [69]

To calculate the number of vacancies emitted from the void and reach some distance R1 from the void surface, we use absorbing boundary conditions at this distance

CV(R1) = 0 [70]

An additional boundary condition must specify the vacancy-void interaction. Assuming that vacancies are absorbed by the void, which is a realistic scenario, the vacancy concentration at one jump distance a from the surface can be written as

Cv(R V a) Cv(R V 2a)

ndiss V

t 2t

The left-hand side of the equation describes the fre­quency with which vacancies leave the site. The first term on the right-hand side accounts for the produc­tion of vacancies due to evaporation from the void. The last term on the right-hand side accounts for vacancies coming to this site from sites further way from the void surface. After representing the latter term using a Taylor series, in the limit of R ^ a, the boundary condition, eqn [71], assumes the follow­ing form

CV(R)=2tndiss V arCY(R) [72]

Using this condition and eqns [69] and [70], one finds the vacancy concentration, Cv(r), is equal to

1 — (R1) 1 R-1 — (R1)-1

It can readily be estimated using the last two equations that the gradient of concentration in eqn [72] is smal­ler than the other terms by a factor of a/r0 and does not contribute to eqn [73]. This means that most vacancies emitted from the void return to it. As a result, the equilibrium condition for the concentration near the void surface is defined by the equality ofthe frequency of evaporation and the frequency of jumps back to the surface and is not affected by the flux of vacancies away from the surface. The vacancy equilib­rium concentration at the void surface is readily obtained from eqn [73] as Cvq(R) = Cv(R) = 2tVjiss.

The total number of vacancies passing through a spherical surface of radius R and area S = 4nR2 per unit time, that is, the rate of vacancy emission from the void, is equal to

SD

Jvem = — VVCv(rV=R

D Cvq 4PR — A Cvq 1pR

O 1 — R/R1 ~ O

There are three points to be made. First, eqn [73] becomes independent of the distance r from the surface, when r ^ R. Thus, vacancies reaching this distance are effectively independent of their origin and can be counted as dissociated from the void. Second, despite the fact that the total vacancy

emission frequency is proportional to the void sur­face area, the total vacancy flux far away from the surface is proportional to the void radius. This is a well-known result of the reaction-diffusion theory40 considering the void capture efficiency. Third, as can be seen from eqn [74], significant deviation from the proportionality to the void radius occurs at distances of the order of the void radius.

As discussed above, most emitted vacancies return to the void. The fraction of vacancies which do not return is equal to the ratio of the frequency defined by eqn [63] and the total frequency of vacancy emis­sion ~ 4nR2Vnss/a2. It is thus equal to a/R. The same result can be demonstrated considering another, although unrealistic, scenario in which vacancies are reflected by the voids.102 We also note that the first nonvanishing correction to the proportionality of the vacancy flux to the void radius is positive and pro­portional to the void radius squared, see eqn [74], where R(1 — R/R1)-1 « R + R2/R1. The same result was obtained previously by Gosele40 when con­sidering void capture efficiency. Thus, with increasing volume fraction more and more vacancies become absorbed at other voids and the proportionality to the void radius squared would be restored. The first cor­rection term just shows the right tendency.