Model Formulas for Facilitating Determination of Concentration-Dependent Diffusion Coefficients

摘要

Several formulas for determining the concentration dependence of diffusion coefficients are introduced for a one-dimensional semi-infinite diffusion problem, applying the Boltzmann-Matano method to "S-shaped" concentration profiles approximated by model functions. The functions are expressed in terms of the Gauss error function, hyperbolic tangent, exponential, and inverse tangent. For all model profiles the corresponding formulas for the diffusion coefficient are calculated. Rapid estimates of the diffusion coefficient are also provided as simple expressions obtained by evaluating the formulas at the center of the concentration profile. The results for the individual profiles are compared, and it is demonstrated that even very similar profiles can lead to rather different diffusion coefficients, especially at low concentrations. Using two examples of different diffusion processes, it is demonstrated that the results can be employed to rapidly calculate diffusion coefficients. In addition, it is shown that a finite diffusion coefficient at low concentrations only occurs if the corresponding concentration profile decays at a Gaussian rate or faster.