The free energy of a void or bubble, according to eqn [127], depends now on three surface parameters instead of just one as in eqn [105]: it depends on the surface energy g0 of a planar surface, on the residual surface strain e for such a planar surface, and on the
biaxial surface stretch modulus 2(ms + Is). As men­tioned above, the latter has the dimension of N m— , and we may then relate it to the corresponding bulk modulus 2mM/(1—2vM) by multiplying the latter with a surface layer thickness parameter h. The surface energy g0 has been determined both experimentally and from ab initio calculations and can be considered as known. The surface layer has been determined by Hamilton and Wolfer10 from atomistic simulations on Cu thin films to be one monolayer thick; hence d — b. A value for the residual surface strain parameter e* has been chosen in Section 1.01.3.1 such that it reproduces the relaxation volume of a vacancy according to eqn [11].

What if one selects the same value for voids con­taining n vacancies? The relative relaxation volume, that is, the ratio НП^/(пО) — 3e(R(n)), can now be computed with eqn [123] and the results are shown in Figure 23 by the solid curve. As it must, for n — 1 it reproduces the vacancy relaxation volume of—0.25 O. In addition, it also agrees with the overall trend of the atomistic results of Shimomura.48 Of course, the atomistic results for small vacancy cluster vary in a discontinuous manner with the cluster size. The sur­face stress model gives not only a reasonable approx­imation to these atomistic results, but also a valid extrapolation to relaxation volumes of large voids.

The chemical potential of vacancies for voids can now be computed with eqn [127] as Fc(R(n + 1)) — Fc(R(n)). Figure 24 shows the results for Ni as the solid curve. The vacancy chemical potentials for

voids are significantly lower than the capillary approximation predicts with a fixed surface energy (dashed curve). The chemical potentials from atom­istic simulations of voids in Ni have been obtained by Adams and Wolfer49 using the Ni-EAM potential of Foiles et a/.50 These results converge to those pre­dicted with the surface stress model.