On the Calculation of Diffusion Coefficients from Sorption Data

The general equation of diffusion in which the diffusion coefficient D(c) is given by D₀(1+αc+βc²+···), where α, β··· are small, can be solved by the method of successive approximation: The solution has the form $${\mathrm{c(\xi )=c}}_0{\text{(1−Φ(ξ))+c}}_0^2{\mathrm{\alpha \psi (\xi )+c}}_0^3{\mathrm{[\alpha }}^2{\chi }_1{\mathrm{(\xi )+\beta \chi }}_2\mathrm{(\xi )]+···,}$$ where Φ(ξ) is the error function and ψ(ξ), χ₁(ξ)··· may be computed by numerical integrations. On the other hand, one knows from the sorption experiments the quantities defined by Kc₀=2D₀^½∫₀^∞ c(ξ)dξ as a function of c₀. Thus, by the above solution one may obtain a set of linear equations, from which the constants D₀, α, β··· can be found with a reasonable amount of labor. For practical purposes it would be necessary to tabulate the functions ψ(ξ), χ₁(ξ), ···, because they are needed for all the problems of this kind.