The concentration of vacancies in equilibrium with a void of radius R, C[;q (R), which enters eqn [74], can be obtained by considering the free energy of a crystal with a void and a solution of vacancies. Let x be the number of vacancies taken from a solution of vacan­cies to make a spherical void of a radius R = (3 хО/4я)1/3. The associated free energy change is given by

4pR3 2 r n

DF = —— mv + 4P~R2 [75]

where mv = kBT ln(Cv/C*) is the chemical poten­tial of a vacancy (C* is the equilibrium concentration in a perfect crystal) and у is the void surface energy. By differentiating this equation with respect to radius and equating the result to zero, one can find the equilibrium vacancy concentration, which is given by

c?(r)=[76]

Absorption and emission of PDs change a void vol­ume on the basis of the flux of PDs dDV/dt = 4pR2(dR/dt) = (Jv — Ji — Jvem). With the aid of eqns [51], [52], [74] and keeping the leading term
proportional to R only and [76], the growth rate of a void due to absorption of vacancies and SIAs and vacancy emission can be written as ’ 2O~ ‘

RkBT

Neglecting the entropy factor for simplicity, one can find that CJh = exp(-Ef / kBT), where Ef is the vacancy formation energy. The last term in the square brackets on the right-hand side of eqn [66] can be then represented in the following form ’ 2O~ ‘

RkBT where [79] is a well-known equation for the binding energy of a vacancy with a void that is valid for large enough radius. For voids of small sizes, the value Eb has to be calculated by using ab initio or MD methods.

Equation [77] is used in calculations of void swelling. Note that the vacancy and SIA fluxes, the first and second terms, enter this equation sym­metrically and this is because of the neglect of the difference in the interactions of SIAs and vacancies with voids. Also, when the sum of the second and third terms in the right-hand side of this equation is larger than the first term, the voids shrink. Such a shrinking takes place during annealing of preirra­diated samples or, in some cases, during irradiation, if the irradiation conditions are changed. However, in the majority of cases, voids grow under irradiation because dislocations interact more strongly with SIAs than vacancies.