Internal stress fields from dislocations and other defects such as precipitates are spatially varying, that is, they are functions of the location r of the migrat­ing point defect. However, the stresses o;y(r) may be viewed as continuous functions except at material interfaces such as grain boundaries and free surfaces, and their differences between adjacent lattice sites may be approximated by і R where a summation over the repeated index k is implied. The saddle point energy as given in eqn [81] is now replaced by

= ES — °ер1 spq(r)-0

When this is inserted into eqn [76], a new diffusion tensor is given obtained that, to linear order in the stress gradients, is given by

Xjk

R, m,v

exp [-bELM + b/(r)] [86]

Here, the first term is just the diffusion tensor obtained for a uniform stress field, but now with the stress replaced by the local stress field at r. The second term is a correction linear in the stress gradient. How­ever, this term vanishes for the following reasons. We have seen that for fcc crystals there are pairs of oppo­site jump directions that have identical transformation strain values ep^, but they differ only in the signs of the jump vector components Xk. Hence, these opposite pairs of jump directions cancel each other’s contribu­tion to the sum in eqn [86]. As a result, there is no correction to the diffusion tensor that is linear in stress gradients.

Spatially varying stress field, however, produce a drift force. Inserting eqn [84] into the formula [77] leads to

-bEL(r) + bEf (r)

The factor with the sum has a remarkable resem­blance with the expression for the diffusion tensor. Indeed, it is straightforward to show that

d @

F‘(r) = — dXkD, k(r) — Dik(r)bOfdXkspq(r) [88]

Since the average energy of the point defect in its stable configuration in the presence of a stress field is given by

EF (r) = Ef — Of spq (r) [89]

we may also write eqn [88] as

F (r) = — exp [b£f (r)] dL {exp [-b£f (r)] D, k(r)} [90]

In this last expression, the product of the two func­tions in the curly brackets depends now only on the saddle point energies, and this shows that it is the spatial dependence of the saddle point energy only that gives rise to a drift force, while the variation of the defect formation energy is not contributing to the drift force.

The general formulae [88] or [90] reveal that stress — induced diffusion anisotropy affects the direction of the drift force such that it is in general not collinear with the stress-induced interaction force.