The Double Diffusivity Model under Stochastic Forcing

The “double diffusivity” model was proposed in the late seventies, and reworked in the early eighties, as a continuum counterpart to existing discrete models of diffusion corresponding to high diffusivity paths, such as grain boundaries and dislocation lines. It was later rejuvenated in the nineties to interpret experimental results on diffusion in polycrystalline and nanocrystalline specimens where grain boundaries and triple grain boundary junctions act has high diffusivity paths. Technically, the model pans out as a system of coupled Fick type diffusion equations to represent “regular” and “high” diffusivity paths with “source terms” accounting for the mass exchange between the two paths. The model remit was extended by analogy to describe flow in porous media with double porosity, as well as to model heat conduction in media with two non-equilibrium local temperature baths e.g. ion and electron baths. Uncoupling of the two partial differential equations leads to a higher-ordered diffusion equation, solutions of which could be obtained in terms of clasical diffusion equation solutions. Similar equations could also be derived within an “internal length” gradient (ILG) mechanics formulation applied to diffusion problems, i. e. by introducing nonlocal effects, together with inertia and viscosity, in a mechanics based formulation of diffusion theory. While being remarkably successful in studies related to various aspects of transport in inhomogeneous media with deterministic micro/nano structures, its implications in the presence of stochasticity have not yet been considered. This issue becomes particularly important in the case of diffusion in nanopolycrystals whose deterministic ILG based theoretical calculations predict a relaxation time that is only about one-tenth of the actual experimentally verified timescale. This article provides the “missing link” in this estimation by adding a vital element in the ILG structure, that of stochasticity, that takes into account all boundary layer fluctuations. Our stochastic-ILG diffusion calculation confirms rapprochement between theory and experiment, thereby benchmarking a new generation of gradient-based continuum models that conform closer to real life fluctuating environments.