Design formulae for reactive barrier membranes

Polymers films and membranes with immobile and irreversible reactive sites can provide significant barrier properties for packaging materials. The reactive term that consumes the mobile species in the governing transport equations for such materials is a function of both the mobile species and the immobilized reactive sites, leading to non-linear partial differential equations that typically have to be solved numerically. Here we present analytic design formulae to estimate the time varying flux, kill time and time lag through homogenous reactive membranes, obviating the need to numerically solve the model's non-linear equations.Modeling reveals three regimes for the time varying flux. For early times most reactive sites are still present, and an initial flux plateau is observed. For intermediate times, a moving reaction front is found to travel across the film coupled with an increasing flux of the mobile species. Finally, for long times when most reactive sites are consumed, the transient flux approaches its steady value. Analytic design equations characterizing the flux in these three regimes are developed based on perturbation theory and matched asymptotic expansions. They agree closely with the exact numerical solutions. Algebraic formulae for the kill time, time lag and speed of the reaction front also correspond to the full numerical solutions.