Langmuir 1998, 14, 4953-4954
4953
Analysis of the Spatial Variation of the Pore Network Coordination Number of Porous Solids Using Nitrogen Sorption Measurements K. L. Murray,† N. A. Seaton,*,‡ and M. A. Day§ Department of Chemical Engineering, University of Cambridge, Pembroke Street, Cambridge CB2 3RA, United Kingdom, Department of Chemical Engineering, University of Edinburgh, King’s Buildings, Mayfield Road, Edinburgh EH9 3JL, United Kingdom, and ICI Technology, Research and Technology Centre, P.O. Box 90, Wilton, Middlesborough, Cleveland TS90 8JE, United Kingdom Received January 29, 1998. In Final Form: April 16, 1998
1. Introduction Since the 1950s, mesoporous solids have been characterized in terms of their pore size distribution (PSD), obtained by analyzing nitrogen sorption isotherms using methods based on the Kelvin equation.1-3 The connectivity (the way the pores are connected together) is also an important aspect of the pore structure. Seaton and coworkers4-6 have developed an analysis that uses sorption hysteresis to characterize the connectivity of pore networks. In this method, the solid is modeled as an array of three-dimensional lattices (each lattice corresponding to the pore network of one of the microparticles that make up the solid), and the hysteresis between the adsorption and desorption isotherms is viewed as a percolation phenomenon. The sharp decrease in adsorption at the knee of the desorption isotherm corresponds to the formation of a percolating network of vapor-filled pores. The size and shape of the hysteresis loop are dictated by the mean coordination number of the pore network (the average number of pores that intersect at a pore junction), Z, and the size of a typical microparticle (expressed as a number of pore lengths), L. In the analysis of Seaton and co-workers, the fraction of pores below their condensation pressure (i.e., the fraction of pores from which nitrogen would have vaporized if all the pore had access to the vapor phase) and the fraction of pores from which nitrogen has actually vaporized are calculated at different points on the isotherms (using the PSD as an input), so as to transform the experimental adsorption and desorption isotherms into the variables of percolation theory. The transformed experimental data are then fitted to simulation data for percolation in a model pore network, to solve for Z and L. A good fit of the simulated with the experimental data indicates that the structural model used to analyze the data, with the fitted values of Z and L, and the calculated * To whom correspondence may be addressed: telephone, (+44) 131 650 4867; fax (+44) 131 650 6551; e-mail,
[email protected]. † University of Cambridge. ‡ University of Edinburgh. § ICI Technology. (1) Barrett, E. P.; Joyner, L. G.; Halenda, P. H. J. Am. Chem. Soc. 1951, 73, 373. (2) Cranston, R. W.; Inkley, F. A. Advances in catalysis; Academic Press: New York and London, 1957; Vol. 9, p 143. (3) Brunauer, S.; Mikhail, R. Sh.; Bodor, E. E. J. Colloid Interface Sci. 1967, 24, 451. (4) Seaton, N. A. Chem. Eng. Sci. 1991, 46, 1895. (5) Liu, H.; Zhang, L.; Seaton, N. A. Chem. Eng. Sci. 1992, 47, 4393. (6) Liu, H.; Seaton, N. A. Chem. Eng. Sci. 1994, 49, 1869.
Figure 1. Experimental adsorption and desorption isotherms and simulated desorption isotherm for sample A, using the original connectivity analysis. The symbols are the experimental data and the solid line is a curve fit to the simulated data.
Figure 2. Experimental adsorption and desorption isotherms and simulated desorption isotherms for sample B, using both original and new connectivity analyses. The symbols are the experimental data; the solid and dashed lines are curve fits to the simulated data using the original and new analyses, respectively.
PSD, give a good description of the aggregate structure of the sample. Figures 1 and 2 show comparisons between simulation and experiment for samples A and B, both γ-aluminas, in the original experimental coordinates rather than those of percolation theory. Figure 1 (sample A) shows a good fit between simulation and experimental data, resulting in the values Z ) 5.2 and L ) 45. (In this figure, p is the experimental pressure and p0 is the saturation vapor pressure, so that p/p0 is a relative pressure running from 0 to 1.) This good fit is characteristic of samples with hysteresis loops that close relatively soon after the percolation threshold. In contrast, for instances where the hysteresis loop has a long tail extending to low pressure, the fit of experimental to simulated data is often poor, as in the case of sample B, shown in Figure 2. (The simulated desorption isotherm for this analysis is given by the solid line, rather than the dashed one, in this figure.) The fitted parameters are Z ) 9.2 and L ) 45. For this sample, the simulated desorption isotherm matches the experimental one very poorly except in the vicinity of the percolation threshold
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Figure 3. Schematic illustration of the pore network model for the new connectivity analysis.
and gives a much earlier hysteresis loop closure than in the experiment. This result demonstrates that the model used in the original connectivity analysis of Seaton and co-workers is sometimes inadequate. We investigate the possibility that the long tail on some hysteresis loops (as in the case of sample B) is due to spatial variation of Z among the microparticles, which might arise from irregularities during the manufacture of the solid. In this case we might expect the location of the apparent percolation threshold to be governed by the dominant value of Z, while regions of the sample with smaller Z values could inhibit desorption well beyond the predominant percolation threshold, giving the hysteresis loop a long tail. The variation in Z would make the experimentally observed percolation transition more diffuse, being effectively the aggregate of a set of individual transitions. 2. New Analysis In a real amorphous (or only locally crystalline) solid, there is likely to be a continuous distribution of Z values among the microparticles. Given the limited number of experimental data points in the hysteresis loop region, it is not possible to reasonably attempt to fit a continuous distribution of Z. Thus to obtain a measure of the spatial variation of coordination number, we chose a model with two values of Z. While the original connectivity analysis has two fitted parameters (Z and L), our modified version has four: Z1, coordination number of microparticles of type 1; Z2, coordination number of microparticles of type 2; F, fraction of microparticles of type 1; L, characteristic dimension of the microparticles (common to both types). Our new model is illustrated schematically in Figure 3, for F ) 0.8. As with the original method, the new analysis fits the simulation data to the experimental desorption isotherm. Figure 2 shows the application of the new
Notes
connectivity analysis to sample B; the new simulated desorption isotherm (the dashed line in the figure) gives a very good fit to the experimental data. (It is nevertheless noticeable that the simulated isotherm has three points of inflection in the region of the hysteresis loop, whereas the experimental isotherm has only one; this discrepancy is a consequence of fitting two discrete values of Z rather than a continuous distribution.) The fitted parameters are Z1 ) 9.2, Z2 ) 4.0, F ) 0.15, and L ) 50. As the second percolation threshold in the model is well beyond the first, the value of Z corresponding to the dominant percolation threshold is unchanged (to two significant figures) from the original analysis. The good agreement between simulation and experiment suggests that our model is a plausible one. However, we note that the spatial variation of Z among the microparticles is not the only possible explanation for a long tail in the hysteresis loop: a variation in the PSD among the microparticles or, indeed, the variation of either Z or the PSD within individual microparticles would also cause the percolation transition to become more diffuse. The experimental data do not allow us to discriminate between these possibilities, and our choice of model is in that respect arbitrary. However, even if a particular sample were not to conform to our structural model, the modified analysis nevertheless provides a simple quantification of the variation in the percolation threshold within the sample, which is closely related to the transport properties of the solid.5 Last, we point out that there exists a purely thermodynamic (i.e., nonstructural) contribution to hysteresis, which is ignored in our new analysis (as well as in the original analysis of Seaton and co-workers). In a nitrogen sorption experiment on an amorphous solid with a substantial hysteresis loop, this contribution cannot be separated from the effect of connectivity.7 3. Summary Good agreement between simulated and experimental desorption isotherms for sample B, also found in other samples having a hysteresis loop with a long tail, suggests that spatial variation of coordination number among the microparticles is a plausible interpretation of the long hysteresis loop tails that are sometimes observed experimentally. Our new analysis allows us to estimate the variation in coordination number among the microparticles by assuming that two values of Z are present in the solid. Acknowledgment. The authors thank ICI for their support for this research and for permission to publish. K.L.M. thanks Merck and Co. for supporting her on their doctoral study program. LA980113K (7) Liu, H.; Zhang, L.; Seaton, N. A. Langmuir 1993, 9, 2576.
