Anomalous Grain Boundary Diffusion: Fractional Calculus Approach
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 encoun...
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Wiley
2019-01-01
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| Series: | Advances in Mathematical Physics |
| Online Access: | http://dx.doi.org/10.1155/2019/8017363 |
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| author | Renat T. Sibatov |
| author_facet | Renat T. Sibatov |
| author_sort | Renat T. Sibatov |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-4c609dfc41aa4ec4b336d868112476ea |
| institution | Kabale University |
| issn | 1687-9120 1687-9139 |
| language | English |
| publishDate | 2019-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Advances in Mathematical Physics |
| spelling | doaj-art-4c609dfc41aa4ec4b336d868112476ea2025-02-03T06:06:41ZengWileyAdvances in Mathematical Physics1687-91201687-91392019-01-01201910.1155/2019/80173638017363Anomalous Grain Boundary Diffusion: Fractional Calculus ApproachRenat T. Sibatov0Ulyanovsk State University, 432017, 42 Leo Tolstoy Street, Ulyanovsk, RussiaGrain 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.http://dx.doi.org/10.1155/2019/8017363 |
| spellingShingle | Renat T. Sibatov Anomalous Grain Boundary Diffusion: Fractional Calculus Approach Advances in Mathematical Physics |
| title | Anomalous Grain Boundary Diffusion: Fractional Calculus Approach |
| title_full | Anomalous Grain Boundary Diffusion: Fractional Calculus Approach |
| title_fullStr | Anomalous Grain Boundary Diffusion: Fractional Calculus Approach |
| title_full_unstemmed | Anomalous Grain Boundary Diffusion: Fractional Calculus Approach |
| title_short | Anomalous Grain Boundary Diffusion: Fractional Calculus Approach |
| title_sort | anomalous grain boundary diffusion fractional calculus approach |
| url | http://dx.doi.org/10.1155/2019/8017363 |
| work_keys_str_mv | AT renattsibatov anomalousgrainboundarydiffusionfractionalcalculusapproach |