Algebraic, differential, and integral relations for membranes in series and other multilaminar media: Permeabilities, solute consumption, lag times, and mean first passage times

The problem of transport with possible solute binding and consumption is considered for membranes in series, as well as other multilaminar media, in which transport and reaction processes are linear and time invariant. By analogy with electrical networks, the properties of a membrane are summarized by an admittance (or alternatively, impedance) matrix that relates fluxes to concentrations on either side of the membrane. The elements of the admittance matrix are shown to be related to the permeability, solute consumption, and certain lead and lag time parameters of the membrane. The admittance matrix can then be transformed into a transmission matrix. The product of transmission matrices for two membranes in series is equal to the transmission matrix for the series combination. From these facts, simple algebraic relations are derived that relate parameters for membranes in series to the same parameters for the individual membranes, where the latter parameters can be estimated theoretically or regarded as phenomenological quantities to be determined by experiment. These relations have differential counterparts when the properties of the medium vary continuously in space. In the absence of solute consumption, integral relations are also derived for continuously varying media. Results for continuous media can be used to extend the analysis to cases where convection and/or an externally applied force field is present. The relationship of the lead and lag times to mean first passage times is noted. The algebraic, differential, and integral procedures yield results identical with those obtained previously for special cases but with considerably less effort. Also, some new results are presented as examples.