Effective transient behaviour of heterogeneous media in diffusion problems with a large contrast in the phase diffusivities

Abstract

This paper presents a homogenisation-based constitutive model to describe the effective transient diffusion behaviour in heterogeneous media in which there is a large contrast between the phase diffusivities. In this case mobile species can diffuse over long distances through the fast phase in the time scale of diffusion in the slow phase. At macroscopic scale, contrasted phase diffusivities lead to a memory effect that cannot be properly described by classical Fick's second law. Here we obtain effective governing equations through a two-scale approach for composite materials consisting of a fast matrix and slow inclusions. The micro-macro transition is similar to first-order computational homogenisation, and involves the solution of a transient diffusion boundary-value problem in a Representative Volume Element of the microstructure. Different from computational homogenisation, we propose a semi-analytical mean-field estimate of the composite response based on the exact solution for a single inclusion developed in our previous work [Brassart, L., Stainier, L., 2018. Effective transient behaviour of inclusions in diffusion problems. Z. Angew Math. Mech. 98, 981-998]. A key outcome of the model is that the macroscopic concentration is not one-to-one related to the macroscopic chemical potential, but obeys a local kinetic equation associated with diffusion in the slow phase. The history-dependent macroscopic response admits a representation based on internal variables, enabling efficient time integration. We show that the local chemical kinetics can result in non-Fickian behaviour in macroscale boundary-value problems.

Enriched continuum for multi-scale transient diffusion coupled to mechanics

“…In diffusion-mechanics, the coupling is governed by the partial molar volume parameter γ in Eq. (28). The higher the value of γ , the higher the coupling will be.…”

“…The effective behavior is computed from a representative microscopic element (RVE) [27] and transferred to the macroscale. The computational homogenization of transient phenomena, as associated with lithium diffusion in batteries, has been the focus of research recently [24,28]. Effective responses have to be computed at each macroscopic material point at each time step, making homogenization of transient phenomena computationally demanding.…”

“…The mechanical response relies on the assumption of full scale separation since the characteristic time of the elastic deformation for the considered problem is very small compared to the characteristic diffusion time [2,9]. Moreover, the characteristic diffusion time in the active material particles is several orders of magnitude larger, than the one of electrolyte (considered here as a matrix in which active particles are embedded) [28]. Therefore, the lithium ions travel instantly in the electrolyte as compared to their diffusion in the active material.…”

Data-driven reduced homogenization for transient diffusion problems with emergent history effects

“…Assuming that the micro-scale material properties and microstructural topology are known, the non-Fickian behavior at the macro-scale can be captured through a multi-scale approach such as transient computational homogenization [4,5,7]. For diffusion problems, the macroscopic behavior, in terms of the macroscopic chemical potentialμ as the primary unknown field, is obtained by solving a macroscopic transient mass balance equation…”

“…Computational homogenization represents the heterogeneous domain by a homogeneous macro-scale and a heterogeneous microscale and solves the transient diffusion phenomena in a coupled two-scale setting. Despite the 1 fact that the individual micro-scale constituents might reveal instantaneously linear behavior, the homogenization of transient mass diffusion phenomena in heterogeneous materials provides an emergent non-Fickian diffusion behavior [6,7]. This lagging and history-dependent diffusion behavior obtained at the macro-scale is due to the transient nature of the mass diffusion occurring at the micro-scale.…”