On stochastic resonance in a model of double diffusion

What is it about?

The double diffusion model was proposed in the late 1970s and extended in 1980s and 1990s by Aifantis and co-workers. It is a continuum model that assumes two local non-equilibrium concentration fields abiding separate mass–momentum equations mediated by a linear mass exchange term. The present work deals with stochastic double diffusion of two competing phase densities to study transport in nanocrystals. We show that the presence of surface inhomogeneities and temporal delay, represented by stochastic dynamics, could have substantial impact at the interface of the two phases. The effect is most pronounced at a critical point (stochastic resonance), where maximum energy transfer occurs, thereby quantifying the impact of surface imperfections in nanocrystal dynamics. The results have been favourably compared with experimental data

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Why is it important?

The original continuum double diffusivity model [1,2,5] is based on the assumption of two distinct local non-equilibrium concentration fields, each obeying its own mass and momentum balance equation, as well as a linear relation describing the mass exchange term between the two diffusive paths. It comprises of a coupled system of reaction–diffusion-type linear partial differential equations representing the concentrations of the diffusive species in the two paths. The model was used to interpret solute diffusion in metal polycrystals, fluid flow in fissured rocks, and heat conduction in fibre reinforced composites. Its main advantage is the lack of any assumptions made regarding the geometry (shape, size, orientation) of the high diffusivity paths. It can also be generalised to account for a continuous distribution of multiple diffusion paths, e.g. in multiphase composites where additional phase boundaries may exist. Quantitative comparisons of the deterministic double-diffusion model with available experimental data for polycrystals and nanocrystalline materials can be found in [9–11]. The influence on stochasticity is qualitatively discussed here for representative values of the two diffusivities involved in order to illustrate the role of randomness in the dynamics. In both the present and the original double diffusivity models, the concentration ρ of the diffusive species represents the sumof the concentrations of the diffusive species in the ‘high’ (ρ1) and the ‘low’ (ρ2) diffusivity paths, i.e. ρ = ρ1 + ρ2. Its spatial distribution for a ‘constant source experiment’ (constant concentration of the diffusive species at the boundary) strongly depends on the time of the diffusion experiment, thereby closely resembling the spatial distribution plots as shown in Figure 3. For small time differences (t → 0), the slope of the descending curve is very steep (since most of the diffusive species have just started to diffuse). With increasing diffusion time, the slopes become less and less steeper, leading to sigmoidal-like curves. Work is presently underway to analyse competitive growth of multiple diffusion paths in nanocrystals.

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