Symmetries and modelling functions for diffusion processes

Abstract

Short title: Symmetries and modelling functionsPACS numbers: 68.43.Jk -Diffusion of adsorbates; 66.30.Dn -Theory of diffusion and ionic conduction in solids. AbstractA constructive approach to theory of diffusion processes is proposed, which is based on application of both the symmetry analysis and method of modelling functions. An algorithm for construction of the modelling functions is suggested. This algorithm is based on the error functions expansion (ERFEX) of experimental concentration profiles. The high-accuracy analytical description of the profiles provided by ERFEX approximation allows a convenient extraction of the concentration dependence of diffusivity from experimental data and prediction of the diffusion process. Our analysis is exemplified by its employment to experimental results obtained for surface diffusion of lithium on the molybdenum (112) surface pre-covered with dysprosium. The ERFEX approximation can be directly extended to many other diffusion systems.

Model formulas for facilitating determination of concentration-dependent diffusion coefficients

“…(2). For profiles 1 and 2, the formulas for D(C) were already derived in [31] and [32], respectively. Note that since the inverse to profile 7 cannot be obtained in a closed form, we just use I Y to denote it.…”

“…All these model profiles are specified by two parameters, a > 0 and η 0 > 0, which represent the profile's largest slope and its position, respectively. The remaining three profiles-numbers 2 (already considered in [32]), 4, and 5-have the shape of a spike positioned at η = 0, and only their half on 0  η < ∞ is used. Thus, η 0 is absent from these profiles.…”

“…Nevertheless, model concentration profiles expressed via analytical formulas have been also proposed to fit the data, making it possible to find the derivative C' more reliably and thus to calculate D(C) more accurately and rapidly (as in example 2 below). The error function (denoted as erf) is a very popular model profile [31][32][33], because it is a solution to the one-dimensional diffusion equation with a constant D [29,34]. Other model profiles use an integral of the error function [14,29], the hyperbolic tangent (i.e., the sigmoid function) [24,[35][36][37], or linear rational functions [38].…”

“…Such an analytical approach goes back to the work of Hall [31] and has also been used lately [35,32,38]. An advantage of this approach is, undoubtedly, that the formulas show how the details of a concentration profile-for instance, its slope-affect the resulting diffusion coefficient (cf.…”

Coupled Water and Salt Transport in Porous Materials: Rapid Determination of a Varying Diffusion Coefficient from Experimental Data

“…On the other hand, one can avoid all of these drawbacks and make the determination mathematically reliable and rapid (within seconds) by employing an analytical approach. Then the experimental profiles are approximated by analytical curves from which formulas for the parameter dependences are derived (Hall 1953;Kailasam et al 1998;Nikitin et al 2009;Vach and Svojtka 2012).…”