1.01.7.1 Introduction

Global thermodynamic equilibrium of a crystalline solid is nearly impossible to realize. Such a system would have to be a single crystal free of any defects except for a small concentration of vacancies. This concentration depends on the temperature. If the temperature is changed, these vacancies could not spontaneously appear and disappear within the per­fect crystal lattice. The thermal fluctuations are too low in energy to nucleate a Frenkel pair, that is, a vacancy and a self-interstitial. However, at surfaces of this perfect solid, vacancies can form without the creation of a self-interstitial. Of course, surfaces are in reality extended defects just as grain boundaries are in real crystals. The latter contain also dislocations, which are lattice imperfections formed during the solidification of the crystal from its melt, and when thermal stresses are relaxed by plastic deformation.

The conclusion one reaches is that the extended crystal defects, namely grain boundaries, dislocation cores, incoherent interfaces, and surfaces, are the places where vacancies are produced by thermal fluctuations and where they also disappear. It is at these places, called sinks (or sources), where a local equilibrium concentration of vacancies is established and maintained. The vacancy concentration in the remaining, perfect part of the crystal, adjusts by dif­fusion to the local equilibrium concentrations at the sinks. However, these local equilibrium concentra­tions are not necessarily all equal. In fact, they can differ depending on the type of sink, the local state of stress, and the local composition in the case of alloys.

Here, we shall derive local equilibrium vacancy concentrations for some of the important types of sinks that are typically found in irradiated metals and alloys. The procedure we will employ is the same for all sink types, namely we will consider the reaction of species indicated by square brackets

N [V] + [S] $ (n* + 1)[V] + [S’]

in which a sink of type [S] surrounded by Nv vacan­cies [ V] will emit one more vacancy and change to a different sink [S’]. The two sinks [S] and [S0] do not differ from each other in type, but may differ in their internal energies. The change in Gibbs free energy, AG(T), for this reaction will be equal to zero when local thermodynamic equilibrium exists around the sink. The following specific example will clarify the meaning of the procedure. The outcome of this procedure is a thermal vacancy concentration that is in local thermodynamic equilibrium with the particu­lar sink, and this concentration defines a boundary condition for the diffusion of vacancy into or out of the sink or source, respectively. Henceforth, sink and source are considered synonymous.