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We present a detailed theory for the total reaction rate constant of acomposite core-shell nanoreactor, consisting of a central solid core surroundedby a hydrogel layer of variable thickness, where a given number of smallcatalytic nanoparticles are embedded at prescribed positions and are endowedwith a prescribed surface reaction rate constant. Besides the precise geometryof the assembly, our theory accounts explicitly for the diffusion coefficientsof the reactants in the hydrogel and in the bulk as well as for their transferfree energy jump upon entering the hydrogel shell. Moreover, we work out anapproximate analytical formula for the overall rate constant, which is valid inthe physically relevant range of geometrical and chemical parameters. Wediscuss in depth how the diffusion-controlled part of the rate depends on theessential variables, including the size of the central core. In particular, wederive some simple rules for estimating the number of nanocatalysts pernanoreactor for an efficient catalytic performance in the case of small tointermediate core sizes. Our theoretical treatment promises to provide a veryuseful and flexible tool for the design of superior performing nanoreactorgeometries and with optimized nanoparticle load.

Reaction rate of a composite core-shell nanoreactor with multiple, spatially distributed embedded nano-catalysts