On the structure of continuum thermodynamical diffusion fluxes -- A novel closure scheme and its relation to the Maxwell-Stefan and the Fick-Onsager approach

Abstract

This paper revisits the modeling of multicomponent diffusion within the framework of thermodynamics of irreversible processes. We briefly review the two well-known main approaches, leading to the generalized Fick-Onsager multicomponent diffusion fluxes or to the generalized Maxwell-Stefan equations. The latter approach has the advantage that the resulting fluxes are consistent with non-negativity of the partial mass densities for non-singular and non-degenerate Maxwell-Stefan diffusivities. On the other hand, this approach requires computationally expensive matrix inversions since the fluxes are only implicitly given. We propose and discuss a novel and more direct closure which avoids the inversion of the Maxwell-Stefan equations. It is shown that all three closures are actually equivalent under the natural requirement of positivity for the concentrations, thus revealing the general structure of continuum thermodynamical diffusion fluxes. As a special case, the new closure also gives rise to a core-diagonal diffusion model in which only those cross-effects are present that are necessary to guarantee consistency with total mass conservation, plus a compositional dependence of the diffusivity. This core-diagonal closure turns out to provide a rigorous fundament for recent extensions of the Darken equation from binary mixtures to the general multicomponent case. As an outcome of our investigation, we also address different questions related to the sign of multicomponent thermodynamic or Fickian diffusion coefficients. We show rigorously that in general the second law requires positivity properties for tensors and operators rather than for scalar diffusivities.

Weak-strong uniqueness for Maxwell-Stefan systems

“…Second, the introduction of the Bott-Duffin inverse provides an efficient way to reduce the Maxwell-Stefan system to a (degenerate) parabolic system formulated in the square roots √ c i . (Related formulations using the chemical potentials δH/δc i can be found in [6].) Third, we show that our technique can be extended to more general Maxwell-Stefan systems which may have degeneracy at zero.…”

“…If L = ran(A) and A is symmetric, the Bott-Duffin inverse is the same as the group inverse, which was investigated in the context of Maxwell-Stefan systems in [6].…”

“…The last integral is well defined, since the gradient bound for √ c i implies that √ c i u i ∈ L 2 loc (0, ∞; L 2 (Ω)) (see Lemma 7 below) and thus, because of the property 0 ≤ c i ≤ 1, c i u i ∈ L 2 loc (0, ∞; L 2 (Ω)). Finally, a weak solution is required to satisfy the entropy inequality (6) H(c(t)) + 1 2…”

“…Compared to [6], we work here with the square roots √ c i instead of the chemical potentials log c i (see [6, (4.25)]). The relative entropy inequality (8) is rewritten in the form (see Lemma 9)…”

A kinetic model of a polyelectrolyte gel undergoing phase separation

“…By accounting for cross-diffusion, we avoid anomalous diffusivities which arise when the underlying theory assumes that the diffusive flux of a species is solely driven by the gradient of its own chemical potential [31]. The Stefan-Maxwell approach [32,33] correctly captures the hydrodynamic drag between different components of the mixture by balancing the friction forces between the different species [34]. While having been previously ignored for polyelectrolyte gels [35], the recent work by Zhang et al [36] has highlighted the role of cross-diffusion in modelling effects such as a temporary excess of salt entering the hydrogel during swelling, which is subsequently rejected as the gel approaches its new equilibrium.…”