22.08.2015   Comprehensive nuclear materials

Dislocation bias with size and modulus interactions

The modulus interaction has been discussed in Section 1.01.5.3. Treating both the material as well as the defect inclusion as elastically isotropic, the modulus interaction depends on two diaelastic polar­izabilities, aK and aG, for which values are provided in Table 11.

For this isotropic case, the modulus interaction for edge dislocations is58

W2 (r;’) = — (Ao + A2cos2′)(b/r)2 [145]

with

 

aK(1 — 2n)2 + |am(1 — n + n2)

[4p(1 — n)]2

 

[146]

 

Ao

 

Zedge = 1 + [ГС/(2ro)] — [142]

ln(R/ro) + m[rc/(2ro)]2

with m = 3 fits the exact results for 0 < rCJb < 6 as seen in Figure 26.

Very accurate results for the bias factors of edge dislocations can therefore be produced with eqn [142] for small capture radii and with eqn [141] for large capture radii. The two approximations transition extremely well at rc/b = 6. These approximations sug­gest a way to proceed when the interaction energy assumes a more complicated form than in eqn [133].

It is therefore tempting to see if an angular aver­age of the size interaction energy, eqn [134], could be used to evaluate at least approximately the bias factors. Obviously, the angular average of sinj is zero. The diffusion flux will wind around the dislocation as it approaches the core in order to avoid regions where the interaction energy W1 becomes strongly repulsive. Therefore, an average should only be taken over the angular range where W1 is attractive, that is, negative. So, ifwe replace W1(r, ‘) in eqn [134] with W1(r, p/2)/2 in the case of interstitials and with W1(r, 3я/2)/2 in the case of vacancies and evaluate the equation

Zd « — ‘n(R/rd) [143]

rd exp[b W1 (r)]d’n(r)

we obtain58

 

and

 

(aK — |am)(1 — 2n)2 + 4amn(1 — n)
[4p(1 — n)]2

 

[147]

 

A2

 

The perturbation theory of Wolfer and Ashkin58 with the sum of the size interaction [130] and the modulus interaction [145] gives the result

 

2Ao

kT

  image150

Zedge

 

[148]

 

This suggests that an effective capture radius can be defined as

 

and replacing rc with it in eqn [144] yields bias factors that include both the effects of size and mod­ulus interactions. Using the values given in Table 11 for the diaelastic polarizabilities, effective capture radii and new bias factors are obtained and presented in Table 15. Comparing these results with the corresponding ones in Table 14 shows that the mod­ulus interaction contributes to the net bias about 25% for fcc metals, and about 1o—15% for bcc metals.

 

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