Anomalous Grain Boundary Diffusion: Fractional Calculus Approach

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Grain boundary (GB) diffusion in engineering materials at elevated temperatures often determines the evolution of microstructure, phase transformations, and certain regimes of plastic deformation and fracture. Interpreting experimental data with the use of the classical Fisher model sometimes encounters contradictions that can be related to violation of Fick’s law. Here, we generalize the Fisher model to the case of non-Fickian (anomalous) diffusion ubiquitous in disordered materials. The process is formulated in terms of the subdiffusion equations with time-fractional derivatives of order α∈(0,1] and β∈(0,1] for grain volume and GB, respectively. It is shown that propagation along GB for the case of a localized instantaneous source and weak localization in GB (β>α/2) is approximately described by distributed-order subdiffusion with exponents α/2 and β. The mean square displacement is calculated with the use of the alternating renewal process model. The tail of the impurity concentration profiles along the z axis is approximately described by the dependence ∝exp(-Az6/5) for all 0<α≤1, as in the case of normal GB diffusion, so the 6/5-law itself can serve as an identifier of a more general phenomenon, namely, anomalous GB diffusion.