Fickian Insights Using Probability Theory as Logic

TL;DR

This paper develops a probabilistic framework based on Bayesian probability theory to derive and improve constitutive equations for diffusion in multicomponent systems, addressing limitations of existing models.

Contribution

It introduces a new set of diffusion equations incorporating physical constraints and inter-species correlations, extending Jaynes' probabilistic approach to multicomponent diffusion.

Findings

New constitutive equations include Darken's form as a limit.

Explicit limits on non-ideal diffusion behavior are provided.

Applications to published data demonstrate model effectiveness.

Abstract

In Clearing Up Mysteries -- The Original Goal (Maximum Entropy and Bayesian Methods: Cambridge, England, 1988 Springer, pp. 1-27) Jaynes derived Fick's Law for a dilute binary solution from Bayes' Theorem by considering, probabilistically, the motion of dilute solute molecules. Modifying Jaynes' prior, changing the frame of reference, and allowing for multicomponent systems, one can follow Jaynes' logic to arrive at several expressions for the diffusion coefficient that are widely used in application to solvent-polymer systems. These results, however, do not generally satisfy required conditions over the full concentration range. This limitation is resolved by considering the joint motion of all components in the solution with the inclusion of known physical constraints. Doing so, one arrives at a new set of constitutive equations for binary and multicomponent diffusion that include Darken's form as a limit and provide means to characterize inter-species correlations, removing the need for the ad hoc modifications of the thermodynamic term or substitution of bulk compositions with local compositions that have been proposed in the literature. In binary systems, the Bayesian expression for the mutual diffusion coefficient provides explicit limits on non-ideal mutual diffusion behavior, given component self-diffusion coefficients, that can be attributed to inter-species correlation versus stronger intra- and inter-species clustering. Applications of the model to published data for a set of binary systems are presented, along with pointers to further directions for research.