To evaluate the diffusion tensor and the drift force, we consider here the diffusion of self-interstitials and vacancies in fcc crystals, and begin first with the ideal case of a crystal free of any stresses other than those produced by the migrating defect itself. The jump vectors for both self-interstitials and vacancies coin­cide with the n = 12 nearest neighbor locations of the defect in its stable configuration. Table 12 lists the components Xi of these jump vectors, and they are divided into three groups according to the orienta­tions of the dumbbell axes before and after the jump. These are indicated in the first row of Table 12 by the indices 1, 2, or 3 when the dumbbell axis is aligned with the xb x2, or x3 crystal coordinate direc­tion, respectively.

In the absence of stress, all saddle point energies are equal, say ES, and ES — E = Em is just the defect migration energy. It is then a simple matter to

 

-Kv (r + — R)

 

[741

 

This latter function may be viewed as an analytic function of r, since the saddle point energies vary with strains that are obtained from continuum elas­ticity theory, and they are by definition analytic func­tions of r. Expanding the first term in eqn [73] into a Taylor series up to second order, and then reverting back to the real defect concentration C(r, t), one arrives at the diffusion equation @C 82 8 87 = @785D(r)C(r — t)-a7iF(r)C(r-t) [75] with the diffusion tensor defined as

 

Table 12 Components of the jump vectors R in units of d0/V2. 1 $ 2 2 $ 3 3 $ 1 X! 11 —1 —1 0000 11 —1 —1 X2 1 —11 —1 11 —1 —1 0000 X3 0000 1 —11 —1 1 —11 —1 d0 is the nearest neighbor distance between atoms.

 

V (r)

 

[76]

 

-R

perform the summations in the definitions of the diffusion tensor and the drift force to show that

Dij = lAod02 exp(-b£m)dy = D0(T)% [79]

and Fi = 0.

Next, let us consider the case of an applied spatially uniform stress field. According to Section 1.01.5, eqn [44], the interaction energy to linear order in the applied stress will change the energy ofthe defect in its stable configurations to

Efm = Ef + Wf = Ef — OeiSj [80]

and in its saddle point configurations to eS + WSn

Here, ef is the transformation strain tensor of the defect in its stable configuration with orientation f. We had specified this tensor for self-interstitials in Section

1.01.4 for an orientation in the [001] direction, that is, for m = 3, although the nonlinear contributions were not included. Below, it will be given with these con­tributions for self-interstitials in Cu and for the same orientation. The corresponding transformation strain tensors for the other two orientations, for m = 1, 2, can be obtained from ej by appropriate coordinate rotations.

Similarly, efn is the transformation strain tensor of the defect under consideration in its saddle point con­figuration while changing its orientation from m to n. Specifying it for one particular jump is sufficient to obtain the transformation strain tensors for all other jumps by appropriate coordinate rotations. The inspec­tion of the jump directions for a particular pair mn reveals that there are equal but opposite jump directions with the same saddle point interaction energy; the only difference for such a pair of jump directions is that the components of R and — R have equal and opposite signs. As a result, the drift force F vanishes again. However, for the diffusion tensor, the two opposite jump directions make positive and equal contribution. Suppose, the applied stress is uniaxial with the only nonvanishing component a033 = a. When the crystal is oriented such that this uniaxial stress is perpendicular to a (001) plane, then Chan et a/.35 have shown that the diffusion within the (001) plane and perpendicular to it are given by

D(001) ^ D(001) ^ D0 3 exp [/i°ef3 a] + exp(eSi + e2S2 )a/2] 11 22 2 2exp[bOef1 a] + exp[bOef3 a]

D(001 ) = D0 3exp[b°(e1S1 + e2S2 )a/2]

33 2exp[bOef1 a] + exp[bOef3 a]

respectively.

When the uniaxial stress is perpendicular to a (111) crystal plane, then the diffusion tensor in the reference frame of the stress tensor has the

components35

D(111)- D(111)

D11 = D22

3exp[bO(2e|2 + eS3)a/3] +exp[bO(2e1S1 + e3S3)a/3]
4exp[bO(e[1 + f2 + ef3)a/3]

exp[b°(2e1S1+4,)а/з] [83]

exp[b°(ef1 + e|2 + e33)a/3]

In order to obtain the transformation strains for the saddle point configurations, Chan, Averback, and Ashkenazy35 carried out molecular dynamics simula­tions of diffusion as a function of the applied stress. Fitting their results to the above equations enabled them to determine the tensors ej and ej. Their princi­pal values are listed in Table 13 for self-interstitials and vacancies in Cu.

The ratio of the diffusion coefficients in the plane perpendicular to the uniaxial stress, namely D11/D0, is shown in Figures 19 and 20 as solid curves, while

Uniaxial stress (GPa)

diffusion parallel to the stress, D33/D0, is shown as dashed curves. The enhancement of diffusion by tensile (positive) stresses in the (001) planes is much larger for vacancies than for self-interstitials. However, when tensile stress is applied to (111) crystal planes, diffusion in these planes is reduced for self-interstitials but remains almost unaltered for vacancies.