Diffusion and reaction in percolating pore networks J. S. Andrade, Jr., 1,2 D. A. Street, 3 Y. Shibusa, 3 S. Havlin, 2,4 and H. E. Stanley 2 1 Departamento de Fı ´sica, Universidade Federal do Ceara ´, 60451-970 Fortaleza, Ceara ´, Brazil 2 Center for Polymer Studies and Department of Physics, Boston University, Boston, Massachusetts 02215 3 Department of Production Technology, Showa Denko K. K., 13-9 Shiba Daimon, 1-Chome, Minato-ku, Tokyo 105, Japan 4 Department of Physics, Bar-Ilan University, Ramat-Gan, Israel Received 12 April 1996 We address the problem of diffusion and reaction in porous catalysts subjected to percolation disorder. The results with an idealized pore network indicate that the fractal characteristics of the void space can have a remarkable influence on the transport and reactive properties of the system. Within a specific range of length scales, we observe scaling behavior relating the catalytic effectiveness of the network and the diffusion- reaction ratio J ¯ N  ( D/ K ) 1/d R . In addition, the exponent d R is consistently in the range d w d R d w ' , where d w is the two-dimensional random walk exponent on the incipient infinite cluster and d w ' is the corresponding diffusion exponent which includes all clusters of the system at the percolation threshold. Moreover, in contrast with diffusion under ‘‘inert’’ conditions, where the ‘‘dangling’’ bonds in the percolating cluster do not play any role in transport, these elements become active zones due to the reaction mechanism. We also outline some specific guidelines to demonstrate the relevance of these results in the context of design and characterization problems in heterogeneous catalysis. S1063-651X9710801-7 PACS numbers: 47.55,Mh, 05.40.+j I. INTRODUCTION The development of modeling techniques for the descrip- tion of transport phenomena through the interstitial void space of disordered porous catalysts represents a genuine challenge, due to inherent limitations of traditional models which cannot explicitly account for topological and morpho- logical characteristics of real porous media 1. The classical approach to model diffusion and reaction in porous catalysts is to consider the catalyst particle as a homogeneous system where reagents and products can freely diffuse and react ac- cording to a given effective transport coefficient and an in- trinsic reaction mechanism. Under steady-state conditions, this situation can be mathematically formulated as D eff  2 C +R =0, 1 where C is the concentration of the reacting species within the catalyst, D eff its effective diffusivity, and R represents the intrinsic kinetics, expressing the local rate of creation or annihilation per unit volume of the species one desires to trace in the system. Recently, it became clear that the classical diffusion for- malism, which is valid for Euclidean geometries, cannot be used to provide a macroscopic description of transport phe- nomena in many disordered materials. In the case of porous media, the breakdown of this conventional transport theory can be clearly understood as a consequence of the intrinsic structural heterogeneity of the complex void space geometry, causing significant modifications in the diffusional character- istics of the system. Generally speaking, the departure from the classical behavior usually occurs in the form of a subdif- fusive regime which has been extensively investigated 1–4. The mathematical concept of fractals and the use of percola- tion models as an idealized description for disordered media turned out to be fundamental ingredients to analyze and pre- dict theoretical properties of anomalously diffusive systems of transport 3,5–10. There are a number of experimental works showing strong evidence that, within some limited range of length scales, many porous catalysts can be consid- ered as realizations of fractal morphologies 11. Much less effort has been dedicated to the study of diffusion and reac- tion in fractal geometries, and its consequences on the reac- tive properties of porous catalysts 12–16. However, it is not easy to transpose and systematically apply fractal con- cepts in order to solve problems in catalysis. An important issue in the design of most catalytic reactors is the choice of the size of the catalyst pellet. Diffusion is normally considered to be a deleterious mechanism because it might restrict the transport of reagent into the deepest re- gions of the pellet, reducing the overall reactivity of the available active surface area. Under these circumstances, the smallest pellet would be the preferred material. On the other hand, it is well known that small particles produce ‘‘tight’’ packings, which require a large consumption of energy to pump the reacting species through the extraparticle void space in a fixed bed reactor. Thus, there is an important trade-off between catalyst efficiency and energy consump- tion. The problem could be better analyzed if we had a more realistic model for the structure and phenomenology of the diffusion-reaction system, but few attempts have been made to develop a coherent framework where this problem could be properly examined. Sahimi applied the network of pores model to simulate the effectiveness of an idealized catalyst under different diffusion-reaction conditions 17. The re- sults with a disordered and fully occupied lattice show a marked contrast when compared with the classical descrip- tion, but no reference is made relating the structural features of the pore space and its transport properties. Just above the critical point, the incipient infinite percola- tion cluster is an example of a random fractal that can be used as a conceptual model for real pore catalysts Fig. 1. PHYSICAL REVIEW E JANUARY 1997 VOLUME 55, NUMBER 1 55 1063-651X/97/551/7726/$10.00 772 © 1997 The American Physical Society