At the cross-over of double diffusivity–spinodal decomposition models

Document Type

Article

Publication Date

10-1-2026

Abstract

In a preceding paper of the journal we considered the effect of stochastic forcing on the deterministic Aifantis’ double diffusivity (DD) model. Herein we dispense with stochasticity altogether and consider a fully deterministic model that incorporates higher order diffusion (HOD) terms of the linearized Cahn-Hilliard (CH) type introduced in their seminal spinodal decomposition model. The HOD terms in the CH model are introduced in conjunction with a double-well homogeneous free energy potential with a gradient (interfacial) energy contribution. It is the latter that gives rise to the presence of a ∇4 in addition to the ∇2 term of Fick’s classical diffusion (with negative diffusivity within the spinodal region). The governing equations of the DD model are two Fick-type equations (with positive diffusivities) coupled through a linear source/mass exchange term to account for the diffusing species jumps between the two transport paths. The aforementioned ∇4 term also emerges in curvature-controlled studies for generic interfaces. The present study is the first generalization of the DD model with densities ρ1 and ρ2, interacting through a linearized CH HOD enhancement of classical diffusion, thus providing a novel framework at the crossover of the DD and CH models for describing related spinodal decomposition processes and curvature controlled interface growth. The discussion focuses on unbounded domain and localized initial data for a system of two coupled partial differential equations (PDEs), recast in dimensionless form, and solved exactly by a combined Laplace–Fourier technique. The exact solutions are validated against a direct finite-difference integration of the governing equations, and the conditions for partial and global uphill diffusion within the linear model are identified.

Publication Title

Meccanica

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