Ind. Eng. Chem. Res. 2000, 39, 3747-3763 3747 A New Model for Capillary Condensation-Evaporation Hysteresis Based on a Random Corrugated Pore Structure Concept: Prediction of Intrinsic Pore Size Distributions. 1. Model Formulation George P. Androutsopoulos* and Constantinos E. Salmas Department of Chemical Engineering, Section II, Chemical Process Engineering Laboratory, National Technical University of Athens, 9 Heroon Polytechniou Street, GR 15 780 Athens, Greece The present article deals with the development of a new statistical model (corrugated pore structure model: CPSM) simulating capillary condensation-evaporation hysteresis. The formulation of analytical expressions is based on probability principles, an adsorbed layer thickness correlation, Kelvin equation, and a random corrugated pore concept. When the model is fitted over experimental hysteresis loop data, the respective intrinsic pore size distribution (psd) and the corrugated pore parameter, NS (frequency of pore cross-sectional area variation), can be determined. The predictive potential of the CPSM was successfully tested in part 1 (this work) by generating hysteresis loops that reproduced those included in the IUPAC classification as well as those of novel nanoporous MCM-41 materials. CPSM evaluations of intrinsic pore size distributions have been more realistic and accurate than those deduced by applying the conventional methods (e.g., Roberts). The model has been further tested successfully in part 2 (Ind. Eng. Chem. Res. 2000, 39, 3764) by the fitting of experimental hysteresis data of various porous materials, e.g., HDS catalysts, anodic oxide films, lignite, montmorillonite, pillared clays, and MCM-41 mesopore molecular sieves. Introduction example where the knowledge of the exact pore volume and surface area is of decisive importance. The technological significance of many meso- and The investigation of catalytic phenomena involving microporous materials (e.g., catalysts, inorganic mem- macromolecules necessitates even using well-character- branes, porous adsorbents, etc.) is widely recognized. ized regular pore structure, e.g., MCM-41 mesopore The design of heterogeneous processes involving such molecular sieves8 and anodic oxide films, which enable materials, namely, catalysts, separation means, raw the direct observation of catalytic phenomena at a materials, or desired products, requires their reliable single-pore level. Such studies have appeared in the characterization with regard to chemical composition literature in the past few years.9 The most important and physical structure. and hence popular method to determine pore volume The accuracy in the determination of pore structure and surface area distributions of meso- and microporous characteristics may be proved to be of vital importance materials is the nitrogen capillary condensation- in the elucidation of fundamental heterogeneous cataly- evaporation method. Other condensing gases can also sis process steps. Catalyst reactivity, selectivity, and be used. The adsorption isotherm at relative pressures, 1 resistance to deactivation (e.g., Petersen and Bell ) are P/P0 < 0.3, provides the basic experimental data for the controlled by two main factors, namely, chemical com- evaluation of the overall specific pore surface area by position of the deposited catalytic species in their active applying either BET theory or the B-point technique. form and pore structure properties of the catalyst The capillary condensation-evaporation hysteresis loop substrate. constitutes the basis for the deduction of pore volume In a catalyst porous substrate of a given porosity there and surface area distributions. is interplay between pore size and extent of specific Conventional Methods for the Evaluation of surface area; i.e., wider pore sizes are associated with Pore Volume and Surface Area Distributions from reduced specific surface areas. The two counteracting the Capillary Condensation-Evaporation Hyster- properties are the extent against accessibility of surface esis Loop. Various methods have been proposed in the area. Obviously, accessibility is related to the size of the literature for the calculation of the pore size distribution species being transported through the pores. Consider, with respect to pore volume and/or surface area by for instance, the hydrodesulfurization (HDS) and hy- means of the Kelvin equation. The most important drodeasphalting (HDA) of oil residua. Catalysts showing methods have been proposed by Barrett et al.10 (BJH higher porosity and wider pore sizes favor the conver- method), Brunauer et al.11 (ML, modeless method), sion of asphaltenes while catalysts with relatively Cranston and Inkley12 (C-I method), Orr and Dalla narrow pores favor HDS. Elucidation of configurational Valle (O-DV method) as cited in ref 13, and Roberts14 diffusion of macromolecules (i.e., asphaltenes) through (Rb method). All methods consider a progressive empty- - pore channels of comparable size2 7 is a characteristic ing of pores as steps in decreasing the pressure are taken. The starting point coincides with the relative * To whom correspondence should be addressed. Tel.: (+301) pressure P/P0 ) 0.95 that corresponds to a pore diam- 7723225. Fax: (+301) 7723155. E-mail: androuts@orfeas. eter equal to ≈40 nm. The pores are considered cylin- chemeng.ntua.gr. drical and independent and are divided into a number 10.1021/ie0001624 CCC: $19.00 © 2000 American Chemical Society Published on Web 09/15/2000 3748 Ind. Eng. Chem. Res., Vol. 39, No. 10, 2000 Table 1. Specific Surface Area (m2/g) of a Dried Lignite area distributions derived by Roberts method. Total Sample (Drying at 200 °C, for 1 h, under Vacuum ≈0.5 specific surface area data calculated by the five conven- Torr), Deduced from Nitrogen Sorption Data, by Using tional methods (i.e., BJH, Rb, C-I, ML, and O-DV) for Various Calculation Methods a specified sample of dry lignite are presented in Table total specific pore surface area of dry lignite (m2/g) 1. It is evident from Table 1 that four of the calculation method adsorption condensation evaporation methods (viz. BJH, C-I, Rb, and ML) yield comparable BJH 3.3 7.8 predictions of total specific surface areas that are more C-I 3.8 8.0 or less consistent with those estimated by the BET and Rb 3.5 7.8 B-point methods. The O-DV method predicts system- ML 3.9 7.9 atically much higher values (almost twice as much O-DV 8.0 10.2 compared to the BET value). It is also clear from Table BET 5.3 1 that, irrespective of the calculation method, total B-point 5.5 specific surface areas evaluated from the desorption- evaporation isotherm is higher (by 50%) compared to of groups. Each group is characterized by a mean pore the value deduced from the adsorption data. Similar diameter and the first group that has been assigned the observations have been made for nitrogen sorption data greatest mean diameter loses its condensate during the of several samples of partially dried lignite. first step of pressure reduction while an adsorbed Roberts method, as applied to the adsorption data, is multilayer of thickness t1 remains on the surface of the pores. During the second pressure reduction step, the taken to be the best representative of the four widely second group of pores loses its capillary condensate adopted methods being considered as practically equiva- while the thickness of the multilayer adsorbed on the lent. Modern psd Evaluation Methods. Among the more pore walls of group one is reduced from t1 to t2. This procedure is repeated until all pore groups become recent methods for psd evaluation from capillary sorp- 15 empty. tion data, the Horva´th-Kawazoe (HK) method has 8 All methods require a value for the multilayer thick- been applied to determine mesopore sizes. Beck et al. ness as a function of the relative pressure P/P and a have used the (HK) method to measure pores up to 6 0 nm in MCM-41 materials despite the recommendation value for the Kelvin radius (rk ) rp - t). The dilemma usually encountered when these methods are employed made by the authors (HK) for use in micropore size (i.e., is to decide the branch of the hysteresis loop to be below 1 nm) determinations. In addition to that, the applicability of the HK method has been questioned,16 considered in the evaluation of the intrinsic pore size 17 distribution. Despite the fact that the foregoing methods even for micropore size determinations. are based on the progressive emptying of fully saturated Contemporary efforts to elucidate the fundamental pores, when examined from a mathematical viewpoint, theoretical adsorption and transport phenomena through they are equally well applicable to both branches of the regular pore structures involve molecular modeling hysteresis loop. However, the pore size distributions methods.16,18 MCM-41 materials provide the pore struc- (psd’s) obtained for each particular branch are usually ture framework to develop molecular simulation models. markedly different while great discrepancies are ob- Such approaches, though promising, have not developed served when a comparison is carried out between the to the stage allowing a close simulation of gas sorption BET specific surface area and the respective surface hysteresis observations that would enable the deduction area estimated from condensation-evaporation data. of intrinsic psd’s and surface areas that compare Comparison of Specific Surface Areas Obtained satisfactorily with the pertinent BET estimates. The from Nitrogen Sorption Hysteresis Data and Those classical methods, e.g., BJH, Rb, etc., have also been Deduced by the BET Method. (1) Surface Areas of used in pore structure studies of mesoporous molecular 17 19 Hydrodesulfurization (HDS) Catalysts. In part 2 of sieves. For example, Storck et al. and Wu et al. have this work (Ind. Eng. Chem. Res. 2000, 39, 3764), applied successfully the BJH method in psd and surface nitrogen sorption hysteresis data for several commercial area determinations of MCM-41 materials. The latter grades of HDS catalysts are presented. The data have methods detected bimodal psd with one peak in the been employed in the derivation of the respective pore mesopore and another in the macropore region.