A Fractal Approach to Adsorption on Heterogeneous Solid Surfaces. 1. The Relationship between Geometric and Energetic Surface Heterogeneities Wladyslaw Rudzinski,* ,² Shyi-Long Lee, ‡ Ching-Cher Sanders Yan, ‡ and Tomasz Panczyk ² Department of Theoretical Chemistry, Faculty of Chemistry, Maria Curie-Sklodowska UniVersity, Place Marii Curie-Sklodowskiej 3, Lublin, 20-031 Poland, and Department of Chemistry, National Chung-Cheng UniVersity, 160, San Hising, Ming-Hsiung, Chia-yi, 621 Taiwan, Republic of China ReceiVed: April 2, 2001; In Final Form: July 3, 2001 Solid surfaces are never ideally regular, that is, geometrically and energetically homogeneous, nor are they fully irregular or fractal. Instead, real solid surfaces exhibit a limited degree of organization quantified by the fractal dimension D.We find that there is a functional relationship between the differential distribution of adsorption energies and the differential distribution of pore sizes on such “partially correlated” surfaces. We also show that the differential pore size distribution reduces to the classical fractal pore size distribution in the limit of very small pore sizes, or when the fractal dimension D approaches 3. To do this, analytical expressions are developed describing pore size correlations, and correlations between adsorption energies. These correlation functions are then used to develop a general form of the interaction term in the equations for adsorption isotherms. Finally, using our theoretical approach, existing equations describing adsorption on heterogeneous surfaces are reexamined. It is shown that some of these equations have to be revised whereas others can be generalized to take into account both energetic and geometric heterogeneity. Introduction The main goal of surface science is to find fundamental features which are common to many different adsorption systems. The existence of such fundamental features was suggested by early adsorption experiments carried out at the end of the 19th century and at the beginning of the 20th. As early as 1906 Freundlich showed 1 that all the then-known adsorption isotherms can be described by a simple formula, now called the “Freundlich isotherm”. However, the fundamental physical property lying behind the common applicability of Freundlich’s empirical equation remained unexplained for the next 30 years. In 1916 Langmuir 2 published the first theoretical isotherm. This explained the “plateau” found in experimental adsorption isotherms. The related physical model assumed adsorption on a flat surface with energetically equivalent adsorption sites, one molecule per adsorption site. During the next 2 decades this model was elaborated, mainly to take into account possible interactions between adsorbed molecules. A similar model of mobile adsorption on a flat, energetically homogeneous surface, was also considered, but this and similar models all led to similar predictions. 3-6 A general prediction was that one should observe first-order and other kinds of phase transitions in the adsorbed phase. One should observe, for instance, a discontinuity (a sudden jump) in experimental adsorption isotherms, corresponding to the first- order transition at a certain surface coverage. Heats of adsorption should either be increasing functions of surface coverage or be constant in a region of surface coverage preceding the first- order transition. One should also observe discontinuities in the heat capacities of the adsorbed phases. These are only three examples of the predicted phenomena which, however, were not observed in the behavior of real adsorption systems. Instead, experimental isotherms sometimes showed a rapid increase in some region of surface coverage but were always continuous. Moreover, contrary to theoretical predictions the measured heats of adsorption were, as a rule, decreasing functions of surface coverage. Finally, instead of discontinuities, rounded peaks were sometimes observed in heat capacities. 6 More worryingly, the observed kinetics of adsorption did not follow the behavior predicted by Langmuir’s kinetic equations. Instead, the empirical Elovich equation was found to be generally applicable to adsorption kinetics. 7 All this suggested that a fundamental feature of adsorption systems had not been taken into consideration. Finally, at the beginning of the 1940s, a series of papers were published showing that this missing fundamental property is the energetic heterogeneity of solid surfaces. Most commonly that energetic heterogeneity was formulated as a dispersion of adsorption energy values on the available adsorption sites. By adding the factor of surface energetic heterogeneity one might expect a large variety of the adsorption behaviors to be made possible. Surprisingly, it appears that almost all the reported experimental isotherms can be well correlated using three simple empirical isotherm equations. In the case of nonporous solids Freundlich’s empirical equation applies at low pressures (i.e., low surface coverages). At higher surface coverages its empirically generalized form, the Langmuir- Freundlich isotherm, 6 can be used. Finally, the empirical Dubinin-Radushkevich equation describes adsorption in porous materials very well. Of course, a number of other isotherms was proposed to describe adsorption in specific adsorption systems in order to account for their individual features. * To whom correspondence should be addressed. Address: Department of Theoretical Chemistry, Faculty of Chemistry Maria Curie-Sklodowska University, Pl. Marii Curie-Sklodowskiej 3, Lublin, 20-031 Poland. Phone: +48 81 5375633. Fax: +48 81 5375685. E-mail: rudzinsk@ hermes.umcs.lublin.pl. ² Department of Theoretical Chemistry. ‡ Department of Chemistry. 10847 J. Phys. Chem. B 2001, 105, 10847-10856 10.1021/jp011225e CCC: $20.00 © 2001 American Chemical Society Published on Web 10/13/2001