Abstract
The paper is devoted to the existence and rigorous homogenisation of the generalised Poisson–Nernst–Planck problem describing the transport of charged species in a two-phase domain. By this, inhomogeneous conditions are supposed at the interface between the pore and solid phases. The solution of the doubly non-linear cross-diffusion model is discontinuous and allows a jump across the phase interface. To prove an averaged problem, the two-scale convergence method over periodic cells is applied and formulated simultaneously in the two phases and at the interface. In the limit, we obtain a non-linear system of equations with averaged matrices of the coefficients, which are based on cell problems due to diffusivity, permittivity and interface electric flux. The first-order corrector due to the inhomogeneous interface condition is derived as the solution to a non-local problem.
Highlights
We study a generalised Poisson–Nernst–Planck (PNP) problem formulated in a twophase domain composed of periodic cells
Non-linear interface conditions are strongly motivated by electro-chemical interfacial reactions, which are of primary importance for electrokinetic applications modelling, for example, electrolyte, Li-Ion batteries and fuel cells
Depending on available a priori estimates, we provide a general principle of the two-scale convergence for a family of interfacediscontinuous functions, its gradient, boundary traces and interface jumps, which have different asymptotic orders
Summary
We study a generalised Poisson–Nernst–Planck (PNP) problem formulated in a twophase domain composed of periodic cells. Homogenisation of the PNP system in a one-phase perforated domain with the homogeneous Neumann boundary condition and with a jump of the electrical flux was studied in [19]. Coupled multi-component reaction-diffusion systems were treated with respect to non-linear reaction terms over the domain in [10] and examined for degenerate asymptotic behaviour in [32] using the two-scale convergence. The suggested inhomogeneous interface conditions are non-linear They do not satisfy any periodicity assumptions since depending on the state variables distributed over the domain. Based on the two-scale convergence and using the scale transformation between twophase domains, we rigorously prove a new homogenisation result for the doubly non-linear drift-diffusion system of PNP focusing on the inhomogeneous flux interface conditions.
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