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

Dieter Bothe, Pierre‐Étienne Druet|arXiv (Cornell University)|Aug 12, 2020

High Temperature Alloys and Creep4 citations

TL;DR

This paper proposes a novel closure scheme for multicomponent diffusion fluxes in continuum thermodynamics that avoids the computationally expensive matrix inversion required by the Maxwell-Stefan approach while preserving non-negativity of partial densities. It demonstrates equivalence among the Fick-Onsager, Maxwell-Stefan, and the new core-diagonal closure schemes under positivity constraints, with the latter providing a rigorous foundation for the multicomponent Darken equation and revealing that cross-diffusion effects arise primarily from mass conservation and composition-dependent diffusivities rather than true cross-coupling.

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.

Motivation & Objective

• To address the computational inefficiency of the Maxwell-Stefan approach, which requires solving implicit equations via matrix inversion.
• To develop a more direct and computationally efficient closure for multicomponent diffusion fluxes within irreversible thermodynamics.
• To clarify the structural relationship between the Fick-Onsager, Maxwell-Stefan, and novel closure schemes under thermodynamic consistency constraints.
• To provide a rigorous mathematical foundation for the multicomponent extension of the Darken equation.
• To investigate the physical origin of cross-diffusion effects, distinguishing between constraint-induced coupling and true cross-diffusion.

Proposed method

• Proposes a core-diagonal closure scheme where diffusion fluxes depend only on diagonal diffusivities and a projection operator that enforces mass conservation.
• Derives the new closure from the entropy production functional, ensuring thermodynamic consistency and non-negativity of partial densities.
• Establishes equivalence between the novel closure, Fick-Onsager, and Maxwell-Stefan formulations under the condition that all concentrations remain non-negative.
• Uses a projection matrix P to enforce the continuity equation, which introduces effective cross-coupling without requiring off-diagonal coefficients in the diffusivity matrix.
• Demonstrates that in ternary systems (N=3), the system is inherently core-diagonal, with the reciprocal diagonal diffusivities expressible in terms of Maxwell-Stefan diffusivities and mole fractions.
• Analyzes the sign of diffusivity tensors, showing that the second law requires positivity of the entire tensor or operator, not just scalar diffusivities.

Experimental results

Research questions

• RQ1 Can a direct, non-implicit closure for multicomponent diffusion fluxes be constructed that avoids matrix inversion while preserving thermodynamic consistency?
• RQ2 What is the structural relationship between the Fick-Onsager, Maxwell-Stefan, and the proposed core-diagonal closure schemes?
• RQ3 Why do molecular dynamics simulations and experiments often suggest that cross-diffusion effects are dominated by composition-dependent diffusivities rather than true off-diagonal coupling?
• RQ4 In what cases is the system effectively core-diagonal, and what does this imply for modeling and simulation efficiency?
• RQ5 What is the physical origin of apparent cross-diffusion in multicomponent systems—mass conservation or true inter-species coupling?

Key findings

• The proposed core-diagonal closure scheme is mathematically equivalent to the Maxwell-Stefan and Fick-Onsager approaches under the condition of non-negative concentrations, proving that all three formulations describe the same physical fluxes.
• The core-diagonal scheme avoids matrix inversion, offering a computationally more efficient alternative to the implicit Maxwell-Stefan formulation while maintaining thermodynamic consistency.
• For ternary mixtures (N=3), the system is inherently core-diagonal, with the reciprocal diagonal diffusivities given explicitly in terms of Maxwell-Stefan diffusivities and mole fractions.
• The multicomponent Darken equation is rigorously justified as a special case of the proposed core-diagonal closure, with cross-effects arising from the projection operator and composition-dependent diffusivities rather than true cross-coupling.
• The second law of thermodynamics requires positivity of the full diffusivity tensor or operator, not just scalar diffusivities, resolving long-standing ambiguities about the sign of Fickian diffusivities.
• The analysis reveals that apparent cross-diffusion in multicomponent systems is primarily due to the continuity constraint and compositional dependence of diffusivities, suggesting that true cross-diffusion may be rare in complex mixtures.