cavity center. Denoting by WS(r) this interaction energy at the saddle point configurations, the radial defect flux that includes the drift can be written as

Next, the modulus interaction with the strain field of the cavity is60

3aG 8 m2

2g (R)

W2(R/r)

[153]

jr = — exp[—fiWS(r)]—{DC(r)exp[bWS(r)]} [150] dr

The defect current J through each concentric sphere around the cavity is a constant, that is, 4яг2/р = J Integration of [150] then leads to

DC (r)exp[b WS(r)] =DC(R)exp[bW S(R)]

The strain field around the cavity is caused by both a pressure p when gas resides in the cavity, and by the surface stress g(R). Note that this surface stress is not equal to the surface energy or a constant as usually assumed. As shown in Section 1.01.7.4.3, it changes with the cavity radius and the gas pressure p.

For the case of voids, when p = 0, the surface tension is found to be

exp[b W S(r)]

2 me

1 + (1 — 2v)(R/ b)

Matching this solution to the boundary conditions at r! 1, where C(r) ! C, and at r = R, where C(R) = C0 is the defect concentration in local equi­librium with the cavity, gives the defect current

= 4pR(DC — DC0)

J0R=(R+h) exp [b WS(R/r)] d(R/r)

= 4pRZ0(DC — DC0) [151]

The integration extends up to the distance h(R) from the cavity surface where the strain energy of the defect becomes zero, as discussed in Section 1.01.5.3 and shown in Figure 17.

The interaction energy WS(R/r) consists of two parts, the image interaction and a modulus interac­tion. The former has been discussed in Section 1.01.5.3, and it has been given by Moon and Pao32 in the form

where e* is the residual surface strain of a planar surface. It is shown in Section 1.01.3.1 , that this residual surface strain parameter can be related to the relaxation volume of the vacancy, and according to eqn [11]

2

e* = —(1 — v)VVel/O [155]

With eqns [154] and [155], the modulus interaction then takes the final form

2

[156]

1

[157]

Z0(R)

(1+v)2m(vrel )2

36p(1 — v)R3

Uim (r, R)

requires numerical integration. As an example, the bias factors for self-interstitials and vacancies are com­puted for Ni, at a temperature of 773 K, and using the defect parameters in their stable configurations as given in Tables 2, 7, and 11. The results are presented in Figure 27. The solid curves are obtained when both the image and the modulus interactions are included, while the dashed curves neglect the contribution of

n(n — 1)(2n — 1)(2n + 1) R 2n+2 П 2 n2 + (1 — 2v)n + 1 — v r

[152]

The series converges slowly as r approaches the cavity radius R. However, when R/r < 0.99, no more than about 1000 terms are required to obtain accurate results.