Effect of heterogeneity on the lag time in one-dimensional diffusion

A new general equation has been developed for the lag time for a dilute diffusant permeating through a one-dimensional membrane in which both diffusivity D and partition coefficient K are dependent on position. The lag time is given by t_L=∫^h₀K(x)[∫^x₀ R(y)dy][∫^h_xR(y)dy] ×dx/ ∫^h₀R(x)dx, where h is the membrane thickness and R(x)=1/[D(x)K(x)]. Formulas derived from this equation include those for: lag time in the presence of an electrical potential gradient; lag time in a multilaminar periodic membrane; and an upper bound for lag time given by t_L <(h²/ 4D_eff), where D_eff is the effective diffusivity as deduced from steady-state partitioning and permeability measurements. The lag time is shown to be independent of the direction of diffusion. A computational technique is presented in which the equation for t_L is rearranged to a system of two differential equations, which are readily solvable by numerical integration, even when the system has discontinuities in D and K which would increase the difficulty of evaluating the double integral directly.