(46a) Partition-Diffusion-Reaction Bounds for Thin-Film Membrane Formation Kinetics

Over the past decade, the synthesis and characterization of novel membrane chemistries have rapidly advanced to address critical challenges in water treatment and brine valorization. From designing ion-selective nanofiltration membranes for critical metal recovery to enhancing the chlorine resistance of reverse osmosis membranes, researchers have explored new monomers, solvents, support layers, and synthesis conditions. At the core of these membranes is a thin selective layer typically formed by interfacial condensation polymerization—most commonly between an aqueous-phase amine and an organic-phase acid chloride. Despite its industrial importance, modeling interfacial polymerization remains challenging due to the complex interplay of diffusion, reaction kinetics, acid generation, oligomer formation, and polymer precipitation. A tractable framework linking synthesis conditions to selective layer growth is essential for the rational design of next-generation membranes.

Here, we develop an analytical model that bounds the rate of reaction during interfacial polymerization by capturing key steps: diffusion of the aqueous-phase monomer through the support layer, its partitioning across the interface, and subsequent diffusion and reaction in the organic phase. Starting from the governing partial differential equations, we nondimensionalize the system to reveal the roles of key design parameters—monomer choice, solvent properties, and support layer thickness—on the normalized reaction rate and interfacial flux. We find that the reaction rate initially scales with the square root of normalized time as a Danckwerts-type reaction-diffusion zone forms in the organic phase. If the organic phase is sufficiently thick, the rate plateaus near a normalized value of 1, limited by diffusive supply from the aqueous phase. At longer times, the rate declines due to combined resistances in the aqueous phase and depletion of the finite reactant pool.

Using asymptotic analysis, we derive closed-form expressions for the reaction rate, interfacial flux, and cumulative product formation in both short- and long-time regimes. We demonstrate how the maximum reaction rate scales with key mass transfer parameters, emphasizing the dominant influence of the partition coefficient—strongly governed by monomer and solvent selection. To support this, we develop a computational workflow that predicts partitioning and diffusion coefficients across 624 combinations of common reactants—including aromatic and aliphatic diamines, polyamines, alcohols, and saccharide macrocycles—and solvents, including linear and branched alkanes, cycloalkanes, aromatics, and organochlorides. This workflow enables rapid estimation of mass transfer parameters and provides predictive bounds on reaction rates and interfacial fluxes across a broad chemical design space for new thin-film composite membranes. By developing a tractable model that provides rigorous diffusion bounds for the reaction rate during interfacial synthesis, we strive to guide the strategic development of new membrane chemistries and morphologies.