In order to obtain the sink strength, one first solves the steady state diffusion equation
V-j = V2(DC) =0
without a drift term. For the case of a straight dislocation, the solution depends only on the radial direction in a cylindrical coordinate system with the dislocation as the axis. The total current of defects is then given by
2p
J0 = над [°C — ОСЧ1 I130′
per unit length of the dislocation. Here, R is an outer cut-off radius taken as half the average distance between dislocations, and rd is the dislocation core radius. It is assumed that at this radius the defect concentration becomes equal to the local, thermal equilibrium concentration.
The total defect current to all dislocations is then proportional to the
Dislocation sink strength = -—P [131]
£n(R/rA)
where p is the dislocation density.
When the drift term is now included, the defect current changes to
J = ZJo [132]
Z is called the bias factor, and there are as many such factors as there are diffusing defects and different types of sinks. The complexity of the interaction of a migrating defect with the strain field of the sink makes it difficult to find an analytical solution to the diffusion equation with drift. However, there exist a few important solutions.
1.01.8.2.1 The solution of Ham
One is for the diffusion to an edge dislocation when the interaction energy is given by eqn [45] and the stress field is that in an isotropic material. In this case
M 1 + V Vrel sin’
3% 1 — V r expressed in polar coordinates (r, ‘).
The solution of the diffusion equation with drift determined by the size interaction energy [133] can be obtained in terms of products of modified Bessel functions with cosine functions, Kn cos(n’) and In cos(n’),56 and the edge dislocation bias factor is then obtained 57 in the form
Kn(rC/rd ) fn(rc/rd )
1
[134]
where the capture radius is defined as
_ (1 + n)b mvrel|
c 6я(1 — v) kT
The series converges very rapidly for R ^ rc, and it is sufficient, as the numerical evaluation shows, to retain only the zeroth order term. As before, 2R is the average distance between dislocations. If this distance becomes small as in a dense dislocation cell wall with narrow dislocation multipoles, the long — range stress fields of individual edge dislocations cancel each other, and the net interaction energy of eqn [133] is no longer valid.
For example, consider an edge dislocation dipole as shown in Figure 25. The interaction energy with the migrating defects is now given by
4b 1 + v V rel 3p 1 — v
y + h y — h
![Подпись: [136]](/img/1185/image145_0.png "image145")
(x + h)2 + (y + h)2 (x — h)2 + (y — h)2
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Figure 25 Edge dislocation dipole.
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At distances r energy becomes
Wl(r; «)«
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: Jx2 + y2 >> h, this interaction
ub 1 + n = sin2a
—A Vrel23/2h —- [1371
3я 1 — n r2
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Table 14 lists values for both the interstitial and vacancy capture radii evaluated at half the melting points for various metals, and the bias factors for rd = 2b and R/b = 2250, corresponding to a dislocation density of 1012m—2 The vacancy capture radii are small for all metals, and about equal to what one would expect for the sum of the dislocation core radius plus the point defect radius, namely rd ‘ 1—2b. In contrast, the interstitial capture radii are significantly larger, in particular for fcc metals when compared with bcc metals.
The evaluation of eqn [134] gives the solid curve displayed in Figure 26. As already mentioned above, terms in the sum with n > 1 contribute less than 0.00025 to the dislocation bias factors. In addition, for large values of R/rC, the term that depends on rC/r0 can be neglected, and the series expansions can be used for the modified Bessel functions K0 and I0. As a result, one then obtains the asymptotic approximation59
Zedge _ ‘n(R/ro)
£n(2R/rc) — g
where g = 0.577216 is the Euler constant. This approximation is also shown in Figure 26, and it is seen that it coincides with the exact results for rCJb> 6. However, for rc/b < 2, eqn [140] gives incorrect bias factors less than one.
For small values of rc/b, Wolfer and Ashkin58 have obtained from perturbation theory the following expression:
Zedge * 1 + mR/^2 — °([rc/(2ro)]4) [141]
As indicated, extension of this perturbation theory to higher orders shows that an alternating series is obtained with poor convergence. This then suggests to seek a Pade approximation that may extend the usefulness of eqn [141]. For example,58
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when expressed in the polar coordinates indicated in Figure 25. Comparing this with the interaction energy for a singular edge dislocation, eqn [133], we see that interaction with an edge dislocation dipole falls off as r-2, and it has an angular periodicity of twice that for the single edge dislocation.
The solution of the diffusion equation can again be constructed in terms of products of cosine functions, cos(2na), and the modified Bessel functions, but now of a different argument, namely 2rcrD/r2, where rD = 21/2h is the radius of the dipole. If we take this radius to be the sink radius, then the bias factor for the dipole is
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Again, just as for the bias of a single edge dislocation, eqn [134], we find when evaluating eqn [138] numerically that the first term, as shown, in this series already provides accurate results.
The important material parameter that determines the bias factor of edge dislocation is the capture radius defined in eqn [135]. At this radius, and at polar angles perpendicular to the direction of the Burgers vector, the interaction energy is of the magnitude
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We may then attach the following physical meaning to this capture radius: when a defect reaches the dislocation at this distance from its core, and the interaction is attractive, meaning negative, it is inevitably pulled into the core, and when it is repulsive, that is, positive, the defect is definitively repelled.
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where Ei is the exponential integral function. Evaluation of eqn [144] gives the curve labeled as ‘Average’ in Figure 26. The angular average approximation [144] compares well with the exact result for large values of rc/b, that is, for interstitial capture radii, but slightly over-predicts vacancy bias factors.
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Figure 26 Edge dislocation bias factors based on Ham’s solution and various approximations to it.
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