13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- Reckoning up the hops of particles from A to B sites through control surface S at position x, using Taylor series expansions of involved variables about x with respect to Δx and truncating them at the second term, the net partial transfer by forth and back A→B jumps can be obtained ( ) { } d Y C C Y Y U B AB B A A B A B − ∇ − ∇ − ∇ = β \ J , (5) where the diffusivity 2 AB AB AB l d Ω = when Δx = lAB is taken. The total diffuser flux vector J is the sum of the net partial ones JA\B over all pairs of site kinds A and B, which reads: ( ) ∑ ∑ ∇ −= = A B A B U AB B A A B A B d Y C e C Y , , \ / ln β J J . (6) This holds for arbitrary number m of site types, e.g., the L-sites and (m – 1) kinds of traps Ti (i=1,…,m – 1). Presented flux equations advance those derived elsewhere [18] in that arbitrary occupation degrees θi (i = 1,…,m) towards saturation are here admitted for all kinds of sites, and that alteration of lattice potential relief by some superposed field U(x) is taken into account. Description of diffusion in terms of specified partial fluxes can be supplemented with mass-balance relation being now the usual continuity equation: ( ) [ ] ∇ − ∇ ∇⋅ = −∇⋅ = ∂ ∑ − A B B U AB B A A d Y C C e Y t C t , ln ( , ) β J x . (7) 3.2. Mass Balance In contrast to the one-level system (single kind of sites), Eq. (7) does not accomplish description of diffusion in terms of concentrations Ci (i = 1,…,m) for the m-level case (m > 1) where a system of m balance equations must be built up. This requires to combine the same diffusion steps as shown in Fig. 2d to gather all forth and back jumps across S that fill/vacate the sites of the sort A in a region Δx around a point x by surmounting saddle points at x ± Δx/2. Desired equations are derived here following the random walk theory and its continuum implementation [19,20]. Considering two-level system and addressing the net species supply into A-sites in a domain Δx∍x by overcoming saddle point at x – Δx/2 from all outer A- and B-sites, involved diffusion steps include the net income flux JA\A, and the resulting BA-exchange flux, which is as follows: [ ] [ ] [ ] [ ] { } A AB AB x B x x x U B AB BA x x A x x U B A Y Ω Y e C l Ω J e C l x x Δ Δβ Δ Δβ − ∇ − − ∇ − = 2 1 2 1 . (8) The net exit from A-sites in a domain Δx towards all sites beyond x + Δx/2 is defined similarly. Then, balancing transitions at both extremes of Δx, using power series expansions and disregarding higher order terms with Δx, mass balance for the species dwelling in A-sites can be derived. However, calculations in general terms are overly long and tedious, so that now we content ourselves with particular case when the sites characteristics are isotropic and uniform, i.e., jump frequencies Ω… and lengths l… are constant, for which the following is deduced:
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