Langmuir 1995,11, 1178-1183
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Correlated Site-Bond Percolation Model: Application to Catalytic Deactivation and Desorption from Porous Solids A. M. Vidales,* R. J. Faccio,? and G. Zgrablich***,* Departamento de Fisica y CONICET, Universidad Nacional de San Luis, Chacabuco y Pedernera, 5700 San Luis, Argentina, and Centro Regional de Estudios Avanzados (CREA), Gobierno de la Provincia de San Luis, C. C. 256, 5700 San Luis, Argentina Received June 28,1994. In Final Form: January 6, 1995@ A model for percolative correlated networks of sites and bonds is used in the calculation of catalytic deactivation and the adsorption-desorption hysteresis in porous materials. This model takes into account the important characteristics of porous materials such as correlations between voids and throats (sites and bonds). Experimental data on catalytic deactivation by coke formation and nitrogen sorption on alumina sustrates are analyzed and compared with model predictions.
Introduction High surface area porous solids are widely used in many diverse fields of science and engineering, ranging from agriculture to powder metallurgy. The behavior of these materials is strongly dependent on the structure and morphology of the void space (pores) and this fact makes it important to try to obtain better and more detailed models to describe the porous space and the interconnectivity between pores. It has proven to be useful to consider the porous network as interconnected constrictions (throats) and openings (voids), which we will call more generally by bonds and sites, respectively. See Figure 1. The structure and morphology of porous solids used as catalyst supports and adsorbents control their transport and reaction properties. Many efforts have been made to characterize these media and to explain such phenomena like catalytic deactivation by coke f ~ r m a t i o n ~and -~ nitrogen sorption h y ~ t e r e s i s . ~Percolation *~ concepts have been applied as the main bases to understand the behavior of the system under study. The models used, however, did not take into account explicitly correlations in sizes between pore voids and throats. In this work, we briefly review a site-bond model for porous media7-9 which is capable of describing different kinds of structures through a unique parameter Q and we apply it for modeling catalytic deactivation by coke depositionloand nitrogen sorption hysteresis" contrasting
* To whom correspondenceshould be sent.
Universidad Nacional de San Luis. * Gobierno de la Provincia de San Luis. Abstract published in Advance ACS Abstracts, March 1,1995.
@
(1)Sahimi, M. Rev. Mod. Phys. 1993,65,1393; Chem.Eng.Sei. 1990, 45, 1443. (2) Marin, G. B.; Beeckman, J. W.; Froment, G. F;J.Catal. 1986,97, 416. (3) Beeckman, J. W.;Froment, G. F. Znd. Eng.Fundam. 1979,18, 245. (4) Sahimi, M.; Tsotsis, T. T. J. Catal. 1985,96, 552. ( 5 ) Seaton, N. A. Chem. Eng. Sei. 1991,46,1895. ( 6 ) Hailing Liu; Lin Zhang; Seaton, N. A. Chen. Eng. Sei. 1992,47 (No. 17/18), 4393. (7) Mayagoitia, V.; Kornhauser, I. In Principles and Applications of
Pore Structural Characterization;Haynes, J. M., Rossi-Doria, P., Eds.; Arrowsmith: Bristol, 1985; p 15. (8) Mayagoitia, V.; Rojas, F.; Kornhauser, I. J.Chem.Soc.,Faraday Trans. 1 1985,81,2931; 1988,84,785. (9) Mayagoitia, V.; Gilot, B.; Rojas, F.; Kornhauser, I. J.Chem.Soc., Faraday Trans. 1 1988,84,801. (10) Faccio, R. J.; Vidales, A. M.; Zgrablich, G.; Zhdanov, V. P. Langmuir 1993,9, 2499. (11) Zgrablich, G.; Mendioroz, S.; Daza, L.; Pajares, J.; Mayagoitia, V.; Rojas, F.; Conner, W. C. Langmuir 1991, 7, 779.
Figure 1. Schematic representation of porous space of voids and throats.
our predictions with experimental data due to Froment et a1.2 and Seaton et aL6for alumina sustrates. It should be stressed that, even though the porous solids used in the two independent experiments are not the same, they are similar kinds of alumina and then they are expected to present similar porous structures. The fact that our analysis of these two sets of experimental data, correspondingtotwo different kinds of processes, leads to almost the same value for the structure parameter 51 is encouraging and indicates that the proposed model could be a consistent starting point for the description of porous media.
Dual Site-Bond Model for Porous Media The porous solid is modeled by a three-dimensional network of N (N -) spherical sites (or voids) connected by cylindrical bonds (or throats), with probability density functions for sites and bonds Fs(R)andFB(R), respectively. The connectivity of the sites, z , is assumed to be the same for all of them. The site and bond distribution functions, S(R)and B(R), are given by
-
and represent the probability of finding a site or a bond, respectively, with a radius smaller or equal to R. In order that the construction principle be (no site can be smaller than its connecting bonds) it is necessary that
B(R) IS(R)for every R
(2)
To construct the network, sites and bonds cannot be chosen independently. Thus, the joint probability of finding a site and a connected bond with radius in the
0743-7463/95/2411-1178$09.00/0 1995 American Chemical Society
Correlated Site-Bond Percolation Model ranges (RsPs
+ dRs) and (R&B + ~ R B )respectively, ,
is determined by a correlation function ~ R S which ~ B is ) equal to 1if FS and FB do not overlap. The function 4, carries the structural information characterizing the porous solid. Its simplest form, which is used here, is the one that allows the maximum degree of randomness compatible with the construction principle and is given b9z9
for R, < R, In the simple case of uniform probability density functions for sites and bonds, the calculation of 4 is straightforward and a close expressionlocan be obtained in which a unique structure parameter is involved: Q, the overlap between the distributions Fs(R)and Fs(R).If Q = 0, one has an uncorrelated structure where sites and bonds are very well differentiated entities randomly assigned to each other. Q = 1is the opposite case, namely, a very correlated structure where sites are connected with bonds with almost the same radii. This kind of topology may be viewed as a collection of macroscopic domains of uniform pore sizes. Intermediate values of !2 indicate a quite intricate structure with a certain short range order. This case may be representative of the great majority of porous media. It is expected that these different porous structures will present different percolationbehavior and, in consequence, different characteristics in catalytic deactivation phenomena and nitrogen sorption hystersis loops.
Percolation in Random and Correlated Site-Bond Lattices Percolation theory was developed for the description of disordered systems and has been the subject of intensive research due both to its theoretical importance and to its many applications for describing a diversity of phenomena.’ In the classical percolation theory two fundamental problems exist: the random site percolation problem and the random bond percolation problem. In random site (bond) percolation the sites (bonds)of and infinite network are either randomly occupied with probability ps (PB)or vacant with probability 1-ps (1 -PB). Two sites (bonds) are connected if there exists at least one path of occupied sites (bonds) between them. A set of connected sites (bonds) surrounded by vacant ones is called a cluster. If p s (PB)is small, the size of any cluster is probably small, but at some defined value of p s (PB) an infinite cluster entirely connecting the network will appear. This defined value of p s (PB) is called the site (bond) percolation threshold,pcs (Peg). Exact values forpCsandpeghave been found only for a few two-dimensional lattices and for the Bethe lattices (which are infinitely ramified lattices without closed loops). In the case of a Bethe lattice with coordination number z , both ps and PB can be calculated and are given by the same equation12
(12)Stauffer, D. Introduction to Percolation Theory; Taylor and Francis: London, 1985.
Langmuir, Vol. 11, No. 4,1995 1179 The site (bond)percolation probability P (PB) is defined as the probability that an arbitrary chosen site (bond), supposed to be open, belongs to the infinite cluster. For the Bethe lattice, these probabilities are given, respectively, by
Note that for p C s ,
