Diffusion-controlled reactions in lamellar systems

We study analytically the evolution of the concentrations of reactants initially contained in a one-dimensional array of lamellae. The model corresponds to the reaction of two mechanically mixed liquids. The asymptotic behavior of the reactant concentrations c(t) depends on the initial distribution p(x) of striation thicknesses. If the first two moments of p(x) are finite, then in general c(t)∼${\mathit{t}}^{\mathrm{-}1/4}$; the exception is the case of a strictly periodical system, for which c(t) decays exponentially. If the second moment of p(x) is infinite, c(t) depends in a sensitive way on p(x). Thus for p(x)∼${\mathit{x}}^{\mathrm{-}1\mathrm{-}\mathrm{\alpha }}$, as x→∞, with 1c(t)∼${\mathit{t}}^{\left(1\mathrm{}\mathrm{-}\mathrm{\alpha }\right)/4\mathrm{}}$. This case corresponds to initially poorly mixed liquids.