Nonlinear Multi-Component Maxwell-Stefan Diffusion Model In Deforming Rocks: Chemo-Mechanical Coupling

Widely accepted model of Fickian linear diffusion of inert or trace-like elements is restricted to ideal solution models of components with equal molar mass. Simultaneous diffusion of multiple concentrations without mechanical stresses is well-described by the classical Maxwell-Stefan model, which is limited to the use of concentration gradients. Quantitative predictions of concentrations evolution in real mixtures should be treated instead by modified Maxwell-Stefan closure relations, which result in a correct equilibrium limit due to the use of the chemical potential gradients instead of concentration gradients. There is no linearity and tensorial homogeneity assumptions on flux-force relationships of classical irreversible thermodynamics. Coupling the multicomponent diffusion to mechanics results in pressure gradients that contribute to the Gibbs-Duhem relationship. Note, it was demonstrated that current models used for describing chemical diffusion in presence of stress gradient don’t remain invariance with respect to the choice of units, such as mole and mass, and the thermodynamic admissibility is doubted [1].

We develop a new thermodynamically admissible model for multicomponent diffusion in viscously deformable rocks. Thermodynamical admissibility of this model ensures non-negative entropy production, while maintaining invariance with respect to the choice of units and reference frame. We demonstrate the correct Fickian limit and equilibrium limit with zero gradients of chemical potentials of individual components instead of concentration gradients in classical Maxwell-Stefan model. The model satisfies conventional Gibbs-Duhem and Maxwell relationships under pressure gradients and represents the natural coupling to the viscous multi-phase models featuring spontaneous flow localization.

For numerical purposes, we develop the optimal pseudo-transient scheme for diffusion fluxes coupled to viscoelastic bulk deformation. This new effective damping techniques are compared to analytical solutions. The developed model is applied for radial garnet growth with multicomponent diffusion under pressure gradient, hydration porosity waves and melt transport in the Earth’s crust.

1. Tajčmanová, L., Podladchikov, Y., Moulas, E., & Khakimova, L. (2021). The choice of a thermodynamic formulation dramatically affects modelled chemical zoning in minerals. Scientific reports, 11(1), 1-9.