Consider 3D diffusion of mobile defects near a spherical cavity of radius R, which is embedded in a lossy-medium of the sink strength k2:
G — k2 D(C — Ceq) — VJ = 0 [46]
where Ceq is the thermal-equilibrium concentration of mobile defects and the defect flux is
J = — d(vC + kCrVtf) [47]
Here, D is the diffusion coefficient, U is the interaction energy of the defect with the void, kB is the Boltzmann constant, and T the absolute temperature. The boundary conditions for the defect concentration, C, at the void surface and at infinity are
C(R) = Ceq [48]
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G
C1 = Ceq + _ [49]
k2D
Equation [49] follows from eqn [46] and the requirement that the gradients vanish at large distances. Here, all other sinks in the system, voids, dislocations, etc. are considered in the MFA and contribute to the total sink strength k2. This procedure is selfconsistent.
The interaction energy of a defect with the void in eqn [47] is small and usually neglected. The solution of eqn [46] for a void located at the origin of the coordinate system, r = 0, is then
C(r) = Ceq + (c1 — Ceq) 1 — Rexp[-k(r — r)] [50]
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In order to predict the evolution of mobile PDs and their impact on immobile defects, one needs to know the sink strength of different defects for vacancies and SIAs and the rate of their mutual recombination. The reaction kinetics of 3D migrating defects is considered to be of the second order because the rate equations contain terms with defect concentrations to the second power.40 An important property of such kinetics is that the leading term in the sink strength of any individual defect depends on the characteristics of this defect only. Thus,
N
ka = k2j [45]
j =1
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The total defect flux, I, through the void surface S = 4nR2 is given by
I = — SJ(R) = kC(R)D(C1 — Ceq) [51]
where the void sink strength is
kC (R) = 4pR(1 + kR) [52]
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The sink strength of all voids in the system is obtained by integrating over the SDF, f (R):
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where a = v, i and N is the total number of sinks per unit volume. For example, the total sink strength of an ensemble of voids of the same radius, R, is equal to ka = Nka(R). The individual sink strength such as a void or a dislocation loop may be obtained from a solution to the PD diffusion equation. In the
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where NC = dRf(R) is the void number density, (R) is the void mean radius and (R2) is the mean radius squared. Typically, k2 « 1014m-2, that is,
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k—1 « 100 nm, while the void radii are much smaller, so that one can omit the term proportional to the radius squared:
kC = 4k(R)Nc [54]
Equation [52] is derived by neglecting the interaction between the void and mobile defect. There is a difference between the interaction of SIAs and vacancies with voids due to differences in the corresponding dilatation volumes. As a result, the void capture radius for an SIA is slightly larger than that for a vacancy (see, e. g., Golubov and Minashin94). However, this difference is usually negligible compared to that for an edge dislocation, which is described below.
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where p, is the dislocation density and Zd the capture efficiency. The capture efficiencies for vacancies and SIAs, Za and Za, are different because of the difference in their dilatation volumes (see eqn [56])
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mb l + n eg
3p 1 — n 4kB T
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The dilatation volume of SIAs is larger than that of vacancies, hence RD > RD and the absorption rate of dislocations is higher for SIAs: Zd > Z,. This is the reason for void swelling, which is shown below in Section 1.13.5.2.1. A more detailed analysis of the sink strengths of dislocations and voids for 3D diffusing PDs can be found in a recent paper by Wolfer.96
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