We describe a general theory for surface-catalyzed bimolecular reactions in responsive nanoreactors, catalytically active nanoparticles coated by a stimuli-responsive "gating" shell, whose permeability controls the activity of the process. We address two archetypal scenarios encountered in this system: the first, where two species diffusing from a bulk solution react at the catalyst's surface, and the second, where only one of the reactants diffuses from the bulk while the other is produced at the nanoparticle surface, e.g., by light conversion. We find that in both scenarios the total catalytic rate has the same mathematical structure, once diffusion rates are properly redefined. Moreover, the diffusional fluxes of the different reactants are strongly coupled, providing a behavior richer than that arising in unimolecular reactions. We also show that, in stark contrast to bulk reactions, the identification of a limiting reactant is not simply determined by the relative bulk concentrations but is controlled by the nanoreactor shell permeability. Finally, we describe an application of our theory by analyzing experimental data on the reaction between hexacyanoferrate(III) and borohydride ions in responsive hydrogel-based core-shell nanoreactors.
Keywords: bimolecular reactions; catalysis; diffusion-reaction theory; nanoreactors; permeability; smart polymers.