# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1906.06201