(613d) Diffusion Bounds for the Interfacial Synthesis of Membrane Selective Layers

Over the last ten years, there has been a rapid increase in the synthesis and characterization of new membrane chemistries to tackle critical challenges across water treatment and brine valorization. From developing ion-selective nanofiltration membranes for critical metals separations to increasing the chlorine resistance of reverse osmosis membranes, researchers are studying the use of new monomers, solvents, and synthesis conditions. The thin selective layer at the heart of reverse osmosis and nanofiltration membranes is typically formed by condensation polymerization at the interface of aqueous- and organic-liquid phases. The aqueous-phase monomer (bifunctional amine in the case of polyamide reverse osmosis and nanofiltration membranes) partitions into and then diffuses through the organic phase where it reacts with the organic-phase monomer (trifunctional acid chloride in polyamide membranes), which is in stoichiometric excess, with approximately first-order kinetics in each reactant. Modeling membrane fabrication by interfacial polymerization is challenging given the complex interplay between reaction kinetics, acid formation, oligomer formation, and polymer precipitation. Nonetheless, a tractable understanding of how the choice of monomer, solvents, and synthesis conditions impacts the rate of selective layer formation is critical for the rational design of new membrane chemistries with precise control over membrane structure and morphology.

Here, we formulate an analytical model to bound the rate of reaction during the interfacial synthesis of thin membrane selective layers by considering the diffusion of the aqueous-phase monomer through the aqueous liquid, its partitioning through the liquid-liquid interface, and finally its diffusion and reaction in the organic phase. We begin by formulating the partial differential equations governing diffusion in phase 1 (aqueous phase) and reaction and diffusion in phase 2 (organic phase) along with boundary conditions for reactant partitioning. By nondimensionalizing the governing equations, we show that reaction dynamics are controlled by the lumped first-order rate constant (k′) and the partition-diffusion coefficient factor (κ = K (D₂/D₁)^½). Initially, we show that the reaction rate increases in proportion to η^½ (where η = t k′ is normalized reaction time), limited by the formation of a Danckwerts reaction-diffusion film in phase 2. As η approaches 1, the reaction rate plateaus with reactant consumption balanced by diffusion-limited reactant supply from phase 1. Eventually, as η increases beyond κ^–2, diffusion in phase 1 begins to limit reactant supply, leading to a decrease in reaction rate in proportion to η^–½.

By deriving analytic expressions for the reaction rate, interfacial flux, interfacial concentration, and cumulative rate of product formed; and exploring their asymptotic approximations at small, intermediate, and large times; we demonstrate how the transition in diffusion control from phase 2 to 1 is strongly affected by the reactant partition-diffusion coefficient factor. We finish by using the analytic expressions derived to explore implications for membrane development and interfacial synthesis. In particular, we highlight how the maximum rate of reaction scales with key parameters including the reaction rate constant, initial reactant concentrations, partition coefficient, and diffusion coefficient of reactant A (bifunctional amine) in phase 2 (organic phase). 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.