Development and Comparison of Solution Methods for Modeling Reaction and Diffusion in “Molecular-Square” Membranes

“Molecular squares” are novel self-assembling structures, typically containing metal-ion corners and porphyrin sides and centers, that have exciting promise as stereoselective catalysts that mimic enzymes. Recently, films of molecular squares have been fashioned on membrane supports, and their efficacy for oxidation of styrene has been demonstrated. To aid in the design of these materials, continuum models were developed that allow the effects of the film thickness, pore size, and reactant ratios to be explored. The coupled set of nonlinear partial differential equations describing transient reaction and diffusion was solved numerically using three different classes of methods: a split-step method using finite difference (FD); domain decomposition in two different forms, one involving three overlapping subdomains and the other involving a gap−tooth scheme; and the multiple-time-step method. The first two classes of methods were implemented based on literature methods, and some modifications, particularly to the gap−tooth scheme, were made to handle the problem at hand. The multiple-time-step method was developed in this work and was shown to be about 250 times faster than the split-step method and a factor of 60 faster than domain decomposition using the gap−tooth scheme.