A Stable and Efficient Regression Approach for Determination of Coefficients in Linear Multicomponent Diffusion

Abstract

Existing methods for determining constant multicomponent diffusion coefficients are typically based on generalizations of the established Boltzmann-Matano method. They often require performing numerical integration and differentiation operations on individual experimental concentration profiles. Scatter in these data, present due to uncertainties and variations in the measured concentrations, often necessitates the use of smoothing procedures for noise removal. A regression approach is presented here to determine constant multicomponent diffusion coefficients that eliminates the need to perform smoothing operations on the concentration profile data. This approach simultaneously fits the data from multiple diffusion couples to a functional form of the mathematical expression for the concentration profile, and allows us to determine the diffusivity matrix directly from the fitted parameters. Reformulation of the equation for the analytical solution is done in order to reduce the size of the problem and increase the convergence rate. The objective function for the regression can incorporate point estimations for uncertainty in the concentration, improving the statistical confidence in the estimated diffusivity matrix. Case studies are presented to demonstrate the reliability and the stability of the method.