## Effective rate constants for nanostructured heterogeneous catalysts

#### Shaun HendyVictoria University of Wellington

With precious metal catalysts such as platinum, palladium and gold in high de-mand, the use of these materials in nanoparticle form can substantially reduce the cost of a catalyst through the exposure of more surface area for the same volume of material [1]. In addition, the density of active sites can also be higher on nanostructured materials [2,3]. When reactants are plentiful, the activity by mass of a nanoparticulate catalyst will in-crease as its surface area increases. Under diffusion-limited conditions, however, the reac-tant must diffuse to active sites on the catalyst, so a high surface area and a high density of active sites may bring diminishing returns if reactant is consumed faster than it arrives. Here we apply a mathematical homogenisation approach [4-6] to derive simple expres-sions for the effective reactivity of a nanostructured catalyst under diffusion limited condi-tions that relate the intrinsic rate constants of the surface sites presented by the catalyst to an effective rate constant. We show that distinct limiting cases emerge depending on the degree of overlap of the reactant depletion zone about each site. When highly active cata-lytic sites, such as step edges or other defects are present, we show that distinct limiting cases emerge depending on the degree of overlap of the reactant depletion zone about each site. In gases, the size of this depletion zone is approximately the mean free path of the gas molecules, so the effective reactivity will depend on the structure of the catalyst on that scale. We test these results using numerical simulations, firstly in the continuum limit by solving the diffusion equation, and then in the limit where the Knudsen number is large using kinetic Monte Carlo simulations.
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[4] A. Bensoussan, J. L. Lions, and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, North-Holland, 1978.
[5] S. C. Hendy and N. J. Lund “Effective slip boundary conditions for flows over nanoscale chemical heterogeneities”, Physical Review E 76, 066313, 2007.
[6] N. Lund, X. Y. Zhang, K. Mahelona and S. C. Hendy, “Calculation of effective slip on rough chemically heterogeneous surfaces using a homogenization approach”, Physical Review E, in press, 2012.

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