Strict equivalence between Maxwell-Stefan and fast-mode theory for multicomponent polymer mixtures

TL;DR

This paper proves the strict equivalence between Maxwell-Stefan and fast-mode theories for multicomponent polymer mixtures, simplifying the selection of modeling approaches based on computational efficiency.

Contribution

It demonstrates the exact equivalence of Maxwell-Stefan and fast-mode theories for complex multicomponent systems, extending previous two-component results.

Findings

Fast-mode and Maxwell-Stefan theories are strictly equivalent for multicomponent mixtures.

The equivalence holds even with different molecular sizes and complex diffusivity dependencies.

This reduces the modeling choice to computational efficiency considerations.

Abstract

The applicability of theories describing the kinetic evolution of fluid mixtures depends on the underlying physical assumptions. The Maxwell-Stefan equations, widely used for miscible fluids, express forces depending on coupled fluxes. They need to be inverted to recover a Fickian form which is generally impossible analytically. Moreover, the concentration dependence of the diffusivities has to be modelled, e.g. by the multicomponent Darken equation. Cahn-Hilliard type equations are preferred for immiscible mixtures, whereby different assumptions on the coupling of fluxes lead to the slow-mode and fast-mode theories. For two components, these were derived from the Maxwell-Stefan theory in the past. Here, we prove that the fast-mode theory and the generalized Maxwell-Stefan theory together with the multicomponent Darken equation are strictly equivalent even for multicomponent systems with very different molecular sizes. Our findings allow to reduce the choice of a suitable theory to the most efficient algorithm for solving the underlying equations.