13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- ( ) [ ] ( ) [ ] ( ) ( ) ( ) [ ] , ln / ln 2 AB A B B BA B A A B B AB A B U B BA A A A U A A AA A k C N C k C N C d YC eCYdYC eCdCY YU t C − − − + + − ∇ − ∇ ⋅∇ ∇⋅ ∇ + ∇⋅ ∇ = ∂ ∂ β β β (9) where ) /( / 2 1 AB AB AB AB fNl f N d k = = − Ω . Similarly, balance for a diffuser dwelling in B-sites can be derived rendering the result that differs from Eq. (9) by permutation of the site labels A and B. This description is extensible for arbitrary number m of site types by taking in the right-hand part of Eq. (9) the sum over all site kinds from a set {B;B≠A}. The last term in brackets in Eq. (9) represents transitions between the nearest neighbour sites of different types in a volume d3x around a point x. This way, the system of nonlinear differential equations (9) is derived for partial concentrations Ci (i=A,B,… or 1, …,m). Balance in terms of total concentration C = ΣCi can be obtained summing up the equations of the system (9) over site kinds totality {A}. After all, the result coincides with Eq. (7). One may notice here the similarity with equations built up by Leblond-Dubois [10] following distinct approach based on construction of Boltzmann type transport equations. Present derivation advances the previous one [19] with respect to the sites saturability, their concentrations nonuniformity, and the contribution of a field U. 3.3. Equilibrium The chemical potential µA of hydrogen residing in metal in sites of whichever type A is [13,15] ( )) ln (1 ( ) A A A A A G RT θ θ θ µ − = + , (10) where AG is the free energy at the site with account for imposed potential U, G G U A A = + (Fig, 1c). At equilibrium, µΑ must be the same throughout a solid and in equilibrium with the input fugacity of hydrogen imposed by an environment, e.g., H2 gas at pressure p that has chemical potential of hydrogen µp = const. Then, for all sites at equilibrium µA = µp, which yields ( ) ( ) [ ] G U G A eq A p A p A e e + − − = = − µ β µ β θ θ 1 or ( ) p e Y C NS N C S C A eq A A A A A eq A βµ = = − 1 , (11) where ( )A A G S β − =exp is the solubility factor. From the kinetics point of view, the numbers of forward and backward jumps between the nearest neighbour sites of distinct types in equilibrium are equal to each other in a volume d3x around a point x. This is expressed by detailed balance relation [20] being nothing else than equilibrium partition Eq. (3), which gets now more forms: A AB B B BA A C Y C Y Ω Ω = or A AB B B BA A C d Y C d Y = or ( ) ( ) ( ) [ ] A B B A A B G G− − = − β θ θ θ θ 1 exp 1 . (12) One can verify that fluxes JI\I (I = A,B,…) are nil at equilibrium by virtue of Eq. (11), and that the sum JA\B + JB\A, A ≠ B, does the same with the aid of detailed balance (12). So, the total flux (6) is nil at equilibrium. As well, Eq. (9) at equilibrium yields ∂CA/∂t = 0 for all A-s by virtue of Eqs. (11) and (12), being the last term in brackets in Eq. (9) a paraphrase of the detailed balance (12).
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