Effective rate constant for nanostructured heterogeneous catalysts

There is a great deal of interest in the use of nanostructured heterogeneous catalysts, particularly those based on expensive precious metals, in order to maximise the surface to volume ratio of the catalyst, potentially reducing the cost without sacrificing performance. When there is an abundance of reactants available, the effective reactivity will depend on the surface density of the catalytically active sites. However, under diffusion-limited conditions, catalytically active sites may compete for reactants, potentially leading to diminishing returns from the use of nanostructures. In this paper we apply a mathematical homogenization approach to investigate the effect of scale and patterning on the effective activity of catalytic sites on a heterogeneous catalyst operating under diffusion-limited conditions. We test these theoretical results numerically using Monte Carlo simulations, and show that in the continuum limit the theory works well. In particular, in the limit where the mean free path is much less than the scale of patterning of catalytically active sites, the effective rate constant is found to be equal to the area-weighted harmonic mean of the rate constants on the surface. However, as the length scale of the patterns becomes comparable to the mean free path length, the simulations show that the effective activity of the system can exceed the theoretical limit suggested by the continuum theory.