13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- At any rate, hydrogen delivery to fracture sites proceeds by thermally activated hopping of H atoms between available sites in metal, i.e., by diffusion that turns out to be affected by trapping [2,4,6,7,10,13]. In this context, diffusion modelling is crucial for HAF analysis, prediction and control, as far as, combining with HAF experiments able to reveal fracture initiation time tcr and location xcr, this allows to specify the critical concentration Ccr(σ,εp), i.e., the fracture criterion (1), and to employ this criterion for assessment of fracture time of structures [12,16]. 3. Modelling of Hydrogen Diffusion with Trapping: Backgrounds Revisited Both atomistic and thermodynamic arguments have been used to derive diffusion equations [1,411,13,15]. These approaches are not contradictory, but complementary [11] and capable to converge into the same field equations with certain insights about specific factors. Concerning traps, they were incorporated into resulting field equations in some cases via plausible postulating [4,6,7,17], but not from background principles (excepting few attempts with limited offspring [10,18,19] for HAF). In this section, diffusion equations grounding upon diffuser jumps probabilities is revisited. 3.1. Flux Equations Isothermal diffusion by particles hopping among sites of kinds A and B is considered adopting the techniques used elsewhere [11,13,15,18-20]. Concerning the flux of the species through unit surface S normal to x-axis and situated there at position x, eight possible elementary steps can be grouped in pairs as shown in Fig. 1b, where lIJ (I,J = A or B) are jump distances between specified sites, so that each couple renders the net flux JA\B by forth and back hops between transboundary A and B sites, which are here diffuser releasers and receptors, respectively. The transition frequency ΓAB from Asites located at x′ to B-ones at x″ per unit time depends on attempt frequency ΩAB controlled by particle vibration frequency at given site ω0A and by potential barrier ΔEAB = EAB – GA, where EAB is the free energy at saddle point of lattice potential G(x) between A and B (Fig. 1a). The frequency of successful hops depends on combined probability YB, which merges the probabilities γB that encountered receptor sites are the B-type ones and ΘB that they are empty, that is [ ] [ ] AB x B x AB Y ′ ′ = Ω Γ at ( ) AB x A AB EΔβ ω Ω − = exp 0 , B B BΘ Y γ = , γB = NB/N , ΘB = 1 – θB, (4) where x A A x A f 0 0 ω ω = to reckon up the fraction fA x of hops that contribute to the flux through S having directions towards it, and N = ΣNi is the volume concentration of all available sites. (Note, that process parameters fA, lAB and ΔEAB, according to the crystal symmetry, can depend on the x-axis orientation with respect to the lattice, causing this way diffusion anisotropy. They are isotropic, e.g., for interstitial sites in cubic lattice, and can be for “spherical” point-wise defects there, yielding fA = f = 1/6 irrespectively of orientation.) Jump probability to any site is assumed to be not conditioned by that to another. When some imposed potential field U(x) distorts the lattice relief G(x), Fig. 1c, ΔEAB depends on jump sense, which biases the hopping probabilities. This implies modification of hopping frequency (4) with the factor exp(±½βΔx∇xU), where the lower/upper sign corresponds to jumps pro-/counterwise the x-direction, Δx = x′ – x″, and ∇x is the x-component of a gradient.
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