Quantitative Information on Pore Size Distribution from the Tangents of

Oct 6, 2004 - It is known that the derivatives of the accumulating pore volume with respect to the pore size are the pore size distribution (PSD) of t...
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Langmuir 2004, 20, 10115-10122

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Quantitative Information on Pore Size Distribution from the Tangents of Comparison Plots Huai Yong Zhu,*,† Pegie Cool,‡ Etienne F. Vansant,‡ Bao Lian Su,§ and Xueping Gao| Australian Key Center for Microanalysis & Microscopy and School of Chemistry, The University of Sydney, NSW 2006, Australia, University of Antwerp (UIA), Laboratory of Adsorption and Catalysis, Universiteitsplein 1, B-2610 Wilrijk, Belgium, Laboratory of Inorganic Materials Chemistry, Faculties Universitaires Notre-Dame de la paix, 61 rue de Bruxelles, B-5000 Namur, Belgium, and Institute of New Energy Material Chemistry, Nankai University, Tianjin 300071, China Received April 16, 2004. In Final Form: August 16, 2004 The comparison plot obtained from the nitrogen adsorption data has a similar shape to that of the curve of accumulating pore volume of a solid. The intrinsic nature of this relation is discussed. It is known that the derivatives of the accumulating pore volume with respect to the pore size are the pore size distribution (PSD) of the solid. Thus, the tangent curve of the comparison plot can display, at least qualitatively, the PSD of a solid, over a wide range of pore sizes (from ∼1 to 50 nm) because the comparison plot is applicable to both micropores and mesopores. Quantitative pore structure information can be derived from the comparison plots by establishing a relationship between the t value and the pore size from the samples with uniform pore structure and known pore sizes, such as MCM-41 and alumina pillared clay samples. A calculation procedure to derive quantitative PSD from the comparison plots is suggested, giving reasonable results. This study proposes concise and reliable methods based on the comparison plots to derive information on pore structure in porous solids.

Introduction The measurement and analysis of N2 physical adsorption data is undoubtedly the most commonly employed method for the determination of pore size distribution (PSD) in solids.1,2 A number of theories, as well as methods, have been developed to derive information on pore structure from the isotherms of N2 adsorption by the solids.3-11 One class of methods is developed based on the interaction potential between adsorbate molecules and species in the adsorbent, which is generally expressed by * Corresponding author. Telephone: 61 2 9351 7549. Fax: 61 2 9351 7682. E-mail: [email protected]. † Australian Key center for Microanalysis & Microscopy and School of Chemistry, The University of Sydney. ‡ University of Antwerp (UIA), Laboratory of Adsorption and Catalysis. § Laboratory of Inorganic Materials Chemistry, Faculties Universitaires Nortre-Dame de la paix. | Institute of New Energy Material Chemistry, Nankai University. (1) Everett, D. H. In Characterisation of Porous Solids I; Unger, K. K., et al., Eds.; Elsevier Science: Amsterdam, 1988; p 1. (2) Sing, K. S. W. In Characterisation of Porous Solids II; Unger, K. K. et al., Eds.; Elsevier Science: Amsterdam, 1991; p 1. (3) Gregg, S. J.; Sing, K. S. W. Adsorption, Surface Area and Porosity, 2nd ed.; Academic Press: New York, 1982. (4) Rouquerol, F.; Rouquerol, J.; Sing, K. S. W. Adsorption by powders and porous solids: Principles, methodology and applications; Academic Press: San Diego, 1999. (5) Dubinin, M. M. In Progress in Surface and Membrane Science; Cadenhead, D. A., Danielli, J. F., Rosenberg, M. D., Eds.; Academic: London, 1975; p 1. (6) Everett, D. H.; Powl, J. C. J. Chem. Soc., Faraday Trans. 1 1976, 72, 619. (7) Horvath, G.; Kawazoe, K. J. Chem. Eng. Jpn. 1983, 16 (6), 470. (8) Seaton, N. A.; Walton, J. P R.; Quirke, N. Carbon 1989, 27, 853. (9) Kruk, M.; Jaroniec, M.; Sayari, A. Langmuir 1997, 13 (23), 6267. (10) Sonwane, C. G.; Bhatia, S. K. J. Phys. Chem. B 2000, 104 (39), 9099. (11) Neimark, A. V.; Ravikovitch, P. T. Microporous Mesoporous Mater. 2001, 44/45, 697.

the Leonard-Jones potential,6,12 such as the methods proposed by Dubinin and his co-worker5 and by Horvath and Kawazoe7 (H-K method). They are valid for microporous solids but not for mesoporous solids because capillary condensation, the most important phenomenon for adsorption in mesopores,3,4 is not taken into account in these methods. It is known from the thermodynamics of adsorption that the condensation is caused by interface curvature.3,4,12,13 The Kelvin equation can be applied to evaluate the radius of the interface between the vapor and the adsorbed film in mesopores, at which the film becomes unstable and capillary condensation will occur filling the core space of the pores. Methods based on the equation, for instance, the Barrett-Joyner-Halenda (BJH) method,14 have been widely used to derive the PSD of mesoporous solids.3,4,13,15 Nevertheless, we have to bear in mind that in principle the methods based on the Kelvin equation are not valid for the microporous solids because the interaction potential between adsorbate molecules and species in the adsorbent is not considered in these methods. Besides, Broekhoff and De Boer16,17 predicted three decades ago that the thickness of the adsorbed film increases substantially with the curvature of the gasadsorbed phase interface, even at moderate relative pressures. The theory of Broekhoff and De Boer was verified by our recent studies18,19 on the nitrogen adsorption by MCM-41 samples, a novel family of mesoporous materials which consist mostly of long-range ordered (12) Steele, W. A. The interaction of gases with solid surfaces; Pergamon Press: Oxford, 1974. (13) Cole, M. W.; Saam, W. F. Phys. Rev. Lett. 1974, 32 (18), 985. (14) Barrett, E. P.; Joyner, L. G.; Halenda, P. H. J. Am. Chem. Soc. 1951, 73, 373. (15) Dollimore, D.; Heal, G. R. J. Colloid Interface Sci. 1970, 33, 508. (16) Broekhoff, J. C. P.; De Boer, J. H. J. Catal. 1967, 9, 8-14. (17) Broekhoff, J. C. P.; De Boer, J. H. J. Catal. 1967, 9, 15-27. (18) Zhu, H. Y.; Lu, G. Q.; Zhao, X. S. J. Phys. Chem. B 1998, 102, 7371. (19) Zhu, H. Y.; Ni, L. A.; Lu, G. Q. Langmuir 1999, 15 (10), 3632.

10.1021/la049041p CCC: $27.50 © 2004 American Chemical Society Published on Web 10/06/2004

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hexagonal arrays of uniform mesopores (cylindrical tubes).20,21 The thickness of the film inside mesopores, particularly in small mesopores, has been substantially underestimated by the conventional methods, as has the pore radius.18,22 Moreover, in principle the methods based on the Kelvin equation are not valid for the microporous solids because the interaction between adsorbate molecules and the adsorbent is not taken into account. However, the presence of both microporosity and mesoporosity in solids is a commonly encountered situation. There are an increasing number of new porous materials, such as SBA23 and porous clay heterostructure (PCH) solids,24 exhibiting a structure of bimodal PSD. They have both micropores and mesopores, and thus their PSDs cannot be correctly derived by most existing methods. In principle, methods based on the density functional theory (DFT) can be applicable to such structures.8-11,25,26 But long and complicated derivations of these methods appear to be difficult barriers for many researchers. Apart from the theoretical treatments, the comparison plots, the t-plot27 and its improved version, the Rs-plot,28 are often used to obtain information on the micropore structure of a solid. The comparison plot methods are based on comparing adsorption of the solid being tested with that of a nonporous reference.3,4,27,28 From the N2 isotherm on a nonporous reference solid, we can obtain a relationship between the statistical thickness of the adsorbed film, t27 (or other quantity related to the adsorption on unit surface area3,4,28), and the relative pressure P/P0. The adsorption by a sample at various relative pressures is then plotted against the thickness of the adsorbed film on the nonporous reference at the same relative pressure. If the sample under test is a nonporous solid with the same surface chemistry properties as the reference sample, a multilayer adsorption mechanism (layer by layer) is applicable over the entire range of pressure, and the comparison plot is a straight line passing through the origin because the thickness of the adsorbed film on the sample surface is identical to that on the reference surface at all relative pressures and the adsorption amount is proportional to the surface area of the solid.3,4,29 For a porous solid, pore filling with adsorbate molecules causes remarkable increases in adsorption at some relative pressures and thus results in deviations from the straight line passing through the origin on the comparison plot of the solid. Such deviations clearly indicate the existence and dimension of the pores in the porous solid and can be readily observed.3,4,29 For instance, the deviation due to strong adsorption at small t values indicates the presence of micropores in the sample, while deviation at large t values is due to mesopores. The comparison plot method is a very important achievement in characterizing porous structures using adsorption data since the Brunauer(20) Kresge, C. T.; Leonowicz, M. E.; Roth, W. J.; Vartuli, J. C.; Beck, J. S. Nature 1992, 359, 710. (21) Inagaki, S.; Fukushima, Y.; Kuroda, K. J. Chem. Soc., Chem. Commun. 1993, 680. (22) Branton, P.; Hall, J.; Sing, K. S. W.; Reichert, H.; Schu¨th, F.; Unger, K. K. J. Chem. Soc., Faraday Trans. 1994, 90, 2965. (23) Zhao, D.; Feng, J.; Huo, Q.; Melosh, N.; Frederickson, G. H.; Chmelka, B. F.; Stucky, G. D. Science 1998, 279, 548. (24) Ahenach, J.; Cool, P.; Vansant, E. F. Phys. Chem. Chem. Phys. 2000, 2 (24), 5750. (25) Lastoskie, C. M.; Quirke, N.; Gubbins, K. E. Stud. Surf. Sci. Catal. 1997, 104 (Equilibria and Dynamics of Gas Adsorption on Heterogeneous Solid Surfaces), 745. (26) Maddox, M. W.; Gubbins, K. E. Langmuir, 1995, 11 (10), 3988. (27) Lippens, B. C.; De Boer, J. H. J. Catal. 1965, 4, 319. (28) Payne, D. A.; Sing, K. S. W.; Turk, D. H. J. Colloid Interface Sci. 1973, 43, 287. (29) Mikhail, R. S. H.; Brunauer, S.; Bodor, E. E. J. Colloid Interface Sci. 1968, 26, 45.

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Emmett-Teller (BET) theory. The shape of the plots can provide pore structure information in a clear and reliable manner without involving any theories that their assumptions could be invalid under some circumstances. It is even possible to derive the PSD from the comparison plots, if the important feature of the comparison plots is adequately studied as shown below. In previous works, we have shown that the tangent curve of a comparison plot can qualitatively reflect the PSD of a mesoporous30,31 or microporous solid.32 Moreover, we found that the pore diameter of MCM-41 samples could be estimated with a satisfied accuracy from the comparison plot.30 The value of the pore size thus obtained is in good agreement with the pore dimension obtained from powder X-ray diffraction (XRD) measurements and transmission electronic microscopy (TEM) images. Thus, we may construct a calibration curve with a series of samples that have uniform pore structure but with different pore size to convert the t or Rs value to pore dimension. This will allow us to derive the quantitative PSD of a porous sample over micro- and mesopore ranges. This can be an advantage of the approach based on the comparison plots, because it is not based on any theory and thus the restrictions and difficulties due to the assumptions of the theories can be avoided. In the present study, we illustrate the important similarity between the comparison plot and the curve of accumulating pore volume of a porous solid, both of which can be obtained from the nitrogen adsorption isotherms. The derivatives of the accumulating pore volume with respect to the pore size give the PSD of the solid. Mathematically, the tangent curve of the comparison plot of the solid is a close analogue of the PSD curve and can thus display qualitative PSD information for a porous solid over a broad range of micropores and mesopores. Furthermore, the procedures to derive qualitative and quantitative pore size distribution of porous solids are proposed. Experimental Section 1. Samples. The same MCM-41 samples in refs 18, 19, and 30 were used in the present study, as were the same alumina pillared clay samples in ref 32 and alumina samples in ref 33. The details on the preparations and characterization by powder X-ray diffraction and nitrogen isotherms are given in the references. Nitrogen isotherms of activated carbons were also measured under the same conditions as for MCM-41, the pillared clays, and aluminas on a Quantachrome Autosorb-1 surface area and pore size analyzer (measured at liquid nitrogen temperature and degassed at 10-5 Torr and 573 K for 3 h prior to the analysis). 2. Nonporous References. It is known that the detailed course of a nitrogen isotherm on a solid depends on the nature of the solid surface.3,4 In this study, several nonporous references of different compositions were used to achieve high precision of the calculation. The reference samples include TK 8003,28 for silicate surfaces, the data of Payne and Sing for alumina surfaces,34 and the data of Kaneko et al.35 for carbon surfaces, (30) Zhu, H. Y.; Zhao, X. S.; Lu, G. Q.; Do, D. D. Langmuir 1996, 12 (26), 6513. (31) Zhu, H. Y.; Lu, G. Q. Characterization of porous solids V; Unger, K. K., Rouquerol, J., Rodriguez-Reinoso, F., Sing, K. S. W., Eds.; Studies in Surface Science and Catalysis, Vol. 128; Elsevier Science: Amsterdam, 2000; p 243. (32) Zhu, H. Y.; Lu, G. Q.; Maes, N.; Vansant, E. F. J. Chem. Soc., Faraday Trans. 1997, 93 (7), 1417. (33) Zhu, H. Y.; Cool, P.; Lu, G. Q.; Vansant, E. F. Zeolites and Mesoporous Materials at the Dawn of the 21st Century: Proceedings of the 13th International Zeolite Conference, Montpellier, France, 8-13 July 2001; Studies in Surface Science and Catalysis, Vol. 135; Elsevier: Amsterdam, 2001; p 253. (34) Payne, D. A.; Sing, K. S. W. Chem. Ind. 1969, 918. (35) Kaneko, K.; Ishii, C.; Ruike, M.; Kuwabara, H. Carbon 1992, 30, 1075.

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pores Vp and the amount adsorbed on the external surface Vs:

Vt ) V p + V s

Figure 1. The t-plots and the curves of accumulating pore volume for a γ-alumina (a) and a commercial silica gel Kielselgel 40 sample (b). respectively. To facilitate calculation, we derived empirical equations which give the t values on the nonporous surfaces at various relative pressures by fitting the adsorption data of reference samples. The equation for silica gel surface is

t ) 4.47 - 0.17/log x - 2.75/[10000(log x)2] - 1.26 log x + 0.10(log x)2 + 1.08x + 4.54x2 (1) where x is the relative pressure of nitrogen. The equation for activated alumina is

t ) 3.51 - 0.20/log x + 1.49/[1000(log x)2] - 0.04 log x 0.32(log x)2 + 2.44x + 2.24x2 (2) and the one for activated carbons is

t )3.65 - 8.51/(1000 log x) + 0.8/[1000(log x)2] + 0.31 log x - 0.11(log x)2 + 4.14x + 2.68x2 (3) These empirical equations are similar, in format, to the Harkins and Jura equation,36 while the coefficients vary with the reference materials. 3. t-Plots. The t values at various relative pressures were calculated using the above equations. These values increase with the relative pressure. The adsorption by a sample at various relative pressures is plotted against the t values at the corresponding relative pressures. As examples, the t-plots of a γ-alumina with a narrow pore size distribution33 and a commercial silica gel (Kielselgel 40 from Aldrich) are illustrated in Figure 1.

Results and Discussion 1. Comparison Plots and the Curve of Accumulating Pore Volume. The amount adsorbed at a relative pressure P/P0, Vt, is the sum of the amount adsorbed in (36) Harkins, W. D.; Jura, G. J. Am. Chem. Soc. 1944, 66, 1366.

(4)

In general, after micropores and mesopores in a sample are filled with adsorbate, the adsorption proceeds only on the external surface. A linear portion on the comparison plot of the sample can be observed because the adsorption on the external surface is multilayer adsorption that is the same as the adsorption on the nonporous reference.27,29 If we extrapolated the linear portion to the adsorption axis, the obtained straight line reflects the adsorption course on the external surface.3,4,22,27-30 The external surface area, Sext, can be obtained from the slope of the line, and the intercept on the axis indicates the volume of the pores which have been filled. This method has been widely used to evaluate micropore volume of a solid.3,4 The adsorption on the external surface at a certain relative pressure P/P0 is the difference in the adsorption between the intercept (corresponding to P/P0 ) 0) and the t value that corresponds to the P/P0. We can readily calculate the adsorption from the comparison plots and deduct it from the overall adsorption to get the adsorption by the pores. Generally, the plots of the adsorption by the pores are similar to the t-plots for the overall adsorption because the adsorption by the external surface is quite small, compared with that by pores of the solids. Therefore, the t-plots can represent the course in which the pores in the samples are filled with nitrogen molecules. It is known that the size of the pore filled by nitrogen increases with increasing relative pressures P/P0, and the t value increases with relative pressure, too. When the accumulating pore volume obtained from adsorption data is plotted versus the increasing pore diameter, the plot reflects the analogous course that is shown by t-plots. In Figure 1, the plots of accumulating pore volume (the right vertical axis) against the pore diameter (the horizontal axis on the top), calculated using the method of Dollimore and Heal,15 are also given for comparison. The two curves in each panel of Figure 1 are very similar. Evidently, for the solid with a small external surface area the t-plots are intrinsically associated with accumulating pore volume curves. 2. Tangent Curve and Qualitative PSD. The PSD is the derivative plot of the accumulating pore volume curve. Naturally, we anticipate that a curve of the tangents of the comparison plot has a similar shape because the contribution from the adsorption by the external surface will be a small constant (if the external surface area is small) in the tangent curve, which has no substantial influence on the shape of the curve. Consequently, such a tangent curve should reflect a qualitative pore size distribution of the sample over t that increases with the pore diameter. The tangent plots, dVt/dt, versus the corresponding t obtained from the comparison plots in Figure 1 are depicted in Figure 2. Figure 3a-c illustrates the nitrogen sorption isotherms, t-plots, and tangent plots of activated carbons which were obtained by heating a commercial activated carbon at 500 °C in air for various time periods. A shift of the PSD of the carbon samples to large pore size can be seen as the burnoff increases. Figure 4a-c shows the situation for a group of MCM-41 samples with different pore diameters. The mean pore size of the samples was calculated from the ratio of pore volume and surface area of the pore walls; both were derived from the t-plot.30 It is in most cases in good agreement with the results of XRD measurements.

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Figure 2. The tangent plots, dVt/dt, versus the corresponding t, obtained from the comparison plots in Figure 1.

The sharp peak of the tangent plots for MCM-41 samples shifts to the right as the pore diameter increases. We mechanically mixed two MCM-41 samples with different pore diameters in equal parts and measured the nitrogen sorption isotherm of the mixture. The tangent plot of the mixed MCM41 sample is shown in Figure 5. As shown in these figures, the shape of the peak of the tangent plots is certainly related to the pore structure. For the samples having a uniform pore size (MCM-41 samples, according to transmission electron microscopy18,20), a sharp peak is observed on their tangent plots (Figure 4c). For the mixed MCM41 sample, two peaks are observed (Figure 5). However, the peak on the tangent plot of silica gel (Kielselgel 40) is broad, because the sample has a broader PSD than MCM-41 samples. Obviously the tangent plots do reflect information on pore size distribution. In the adsorption by a porous solid, the pore fillings are always accompanied by a steep increase in adsorption, and the comparison plot of the sample will swing upward significantly, so does the differentials curve. After the filling, the surface of the pore walls is not available to further adsorption. The adsorption proceeds on much less surface area, giving substantially low differential values. A peak is thus observed on the tangent curve, and it identifies the existence of pores. The relationship between the pore dimension of the sample and the peak position on the t-axis is also understandable: as the pore size is larger, the peak which corresponds to the filling of a group of pores occurs at a larger t value. For the sample of uniform pore structure, the pore filling occurs in a very narrow range of t values (and P/P0), resulting in a sharp peak on the tangent plot. As the area under the PSD curves indicates the pore volume, the area under the tangent curve is also proportional to the pore volume. With the features shown above, the tangent curves from comparison plots can be used for the purpose of comparing the pore structures of a series of samples or monitoring the

Figure 3. The nitrogen sorption isotherms (a), t-plots (b), and tangent plots (c) of three activated carbons with burnoff of 0 (trace A), 13 wt % (trace B), and 30 wt % (trace C), respectively.

evolution of the pore structures of a series of samples. These curves can be readily derived from the adsorption data and provide visual pictures of the pore structures of the samples. Nevertheless, there are two important differences between the comparison plots and the curves of accumulating pore volume. First, the t value cannot be used directly for indicating pore dimension although the t value varies synchronously with the pore size.28 As can be seen from the tangent plots, the value at the peak position is far below the radius of the pores. Second, in Figure 1a,b, the t-plot is always above the corresponding accumulating

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Figure 5. The tangent plot of the sample obtained by mixing two MCM-41 samples in Figure 4 in equal masses.

Figure 4. The nitrogen sorption isotherms (a), t-plots (b), and tangent plots (c) of three MCM-41 samples of pure silica with different pore diameters (2.4, 3.3, and 3.8 nm). They were prepared with surfactants as described in ref 30.

pore volume curve at small t values (and pore sizes). The reason is that the adsorption at small t values (i.e., low relative pressures) includes the adsorption on the pore walls of the large pores so that the adsorption in this region is above the accumulating pore volume. As a consequence, the tangent values on the left-hand side of the peak (pore filling) are always substantially higher than those on the right-hand side of the peak for all samples. This cannot be attributed to the existence of smaller pores but is due to the multilayer adsorption on the wall of the pores which have not been filled. Before the pore filling, there is an adsorbate film on the pore walls.3,4 The area under the peak is proportional to the pore volume but not the real pore volume.

To derive quantitative pore structure information, a relationship between the t value and the pore size is needed, and such an empirical relationship can be established from the samples with pore sizes known from other techniques, such as XRD or TEM. Besides, a correction on the adsorption in the tangent curve is necessary to give correct pore volumes over various pore sizes. 3. Calibration Curve. MCM-41 samples have uniform cylindrical mesopores,18,20 and we found that the mean hydraulic diameter37,38 of the mesopores in these samples is in good agreement with the pore diameter obtained from powder XRD measurements.30 Therefore, we can use the position of the peak on the t-axis of the tangent curve and the mean hydraulic diameter of these samples to derive the relation between the pore size and t value. In micropore range, the slit-pore widths of alumina pillared clays obtained from powder XRD measurements and from nitrogen adsorption are in good agreement.30 The width of slit-pores is equivalent to the hydraulic radius of the pores. A clear peak can be seen in the tangent plot (Figure 6a). Zeolites have definite crystal structures, and their pore size can be calculated precisely.39 However, it is very difficult to observe a sharp peak on their tangent plots (Figure 6b). The pore filling begins at very low relative pressures because of the strong adsorption force in small cages or channels of zeolites, which usually have a diameter below 1 nm. In these small pores, there is no stage prior to the pore filling, in which adsorption occurs only on pore walls. Therefore, it is difficult to use zeolites of small micropores to establish the empirical relation, and the proposed approach in this study could not be applicable to these very fine micropores (so-called ultramicropores, for instance, cylindrical pores with a diameter below 1 nm). In general, to achieve a precise and reliable calibration curve, the reference samples should possess monomodal pores of a uniform size, and the tangent curve of such samples always shows a sharp peak. The mean size of the framework pores, rather than interparticle voids, can be calculated from their t-plot. In addition, the reference samples should exhibit a relative sharp diffraction peak in their XRD pattern, which is associated with pore size. (37) The linear portion of the plot after the pore condensation was extrapolated to the adsorption axis of the plot; this intercept is used to obtain the volume of the mesopores, and the slope of this linear portion is used to obtain the external surface area, Sext. Subtracting this external surface area from the BET surface area, one can obtain the surface area from the framework pores. The mean hydraulic diameter can be derived from the ratio of the pore volume and the surface area of the pore walls. (38) Brunauer, S.; Mikhail, R. S. H.; Bodor, E. E. J. Colloid Interface Sci. 1967, 24, 451. (39) Barrer, R. M., Zeolites and Clay Minerals as Sorbents and Molecular Sieves; Academic Press: London, 1975; p 7.

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Figure 8. PSDs of a γ-alumina (a) and a silica gel Kielselgel 40 (b), calculated with the procedures listed in Table 1.

Figure 6. The tangent plots of alumina pillared clays (a), prepared as described in ref 32, and a commercial zeolite Y (b). The mean pore width of the framework pore in the pillared clay is 0.73 nm. A clear peak can be seen for the pillared clay but not for the zeolite because the pore filling in the small cage or channels of the zeolite begins at very low relative pressures.

Figure 7. The relation between the pore sizes of several MCM41 samples and an alumina pillared clay (Al-PILC used in ref 32) and the peak positions on the tangent curves (the solid circles). The curve in the figure is from the equation which best fitted the points.

When the mean pore size obtained is in good agreement with the results of XRD measurements, the mean hydraulic diameter is reliable and can be used for the calibration curve. This is the case for MCM-41 and AlPILC samples. In Figure 7, the pore sizes of several MCM-41 samples and an alumina pillared clay (Al-PILC used in ref 32) are plotted against the corresponding peak positions on the tangent curves (the solid circles). The equation that best fitted the points is

D ) 0.653 - 0.572t + 10.764t2

(5)

and is expressed by a dashed line in the figure. We can convert the term dVt/dt to dV/dD and t to D using eq 5, and the obtained curves exhibit proper pore dimension but retain the shape of the tangent plots in Figures 2, 3c, 4c, and 5. Such curves are capable of providing important PSD information, and one can use

Figure 9. (a) PSDs of three activated carbons with burnoff of 0 (trace A), 13 wt % (trace B), and 30 wt % (trace C), respectively. (b) PSDs of three MCM-41 samples with different pore sizes (the same samples shown in Figure 4). The single and sharp peaks of the PSDs are at 2.43, 3.27, and 3.76 nm, respectively.

them for comparing the structures of a series of samples with similar surface properties or examining the structure evolution. However, such curves are not completely quantitative PSD. We have to take the adsorbed film on the pore walls into consideration to get the correct PSD of the samples as illustrated in Figures 8-11. In this study, we put forward a calculation procedure to derive a quantitative PSD. The arithmetical steps of

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Table 1. Calculation of Pore Size Distribution of a Silica Sample (1) P/P0 1 2 3 4 ... n-2 n-1 n

(2) (3) (4) V t Vl (cm3/g) (nm) (cm3/g)

(5) R (nm)

(6) Rm (nm)

(7) tm (nm)

(8) ∆t (nm)

(9) ∆Vl (cm3/g)

(10) ∆Rm (nm)

8.12 6.76

1.27 1.15

0.14 0.09

0.0060 1.71 0.0056 1.32

(11) Sn (m2/g)

(12) (13) (14) (15) ∆Vfp ∆S ∆Vrp dV/dR(∆Vrp/∆Rm) (cm3/nm/g) (cm3/g) (m2/g) (cm3/g)

0.91 0.88 0.86

749.5 745.6 742.0

1.31 1.18 1.12

1.160 1.154 1.148

8.98 7.27 6.26

7.8 9.5 0.0047 11.5 0.0043

0.075 0.051

276.3 252.5

0.34 0.32

0.428 0.391

0.870 0.913 0.356 0.024 0.0327 0.087 1364.6 0.0005 0.778 0.824 0.330 0.026 0.0369 0.092 1370.2 0.0008

1.7 2.0

0.0068 0.0064

0.005 0.007

5.6

0.0013 0.0023

0.017 0.026

Table 2. Comparison of the Accumulation Surface Area with the BET Surface Area and of the Accumulating Pore Volume with the Overall Pore Volume for Some Samples sample

Sn (m2/g)

SBET (m2/g)

Kielselgel 40 γ-alumina

631.5 326.6

632.8 337.9

∑∆Vrp Vt Sn/SBET (cm3/g) (cm3/g) ∑∆Vrp/Vt 1.00 0.97

0.593 0.232

0.619 0.241

0.96 0.96

C12 C16 C12 + C16 C18

1328.8 1206.6 1244.0 1141.0

MCM-41 Samples 1320.9 1.01 0.722 1242.1 0.97 0.964 1285.8 1.03 0.792 1125.9 1.01 0.980

0.790 1.041 0.879 1.074

0.91 0.90 0.90 0.91

original 13% burnoff 30% burnoff 42% burnoff

Activated Carbons 1076.3 1045.5 1.03 0.460 1476.5 1346.6 1.10 0.628 1298.6 1283.0 1.01 0.731 1020.8 987.1 1.03 0.520

0.571 0.710 0.805 0.578

0.81 0.89 0.91 0.90

If the pore geometry is known, we can calculate the volume of the pores filled in this pressure interval, ∆Vrp (column 13). For instance, for cylindrical pores, Figure 10. (a) PSDs of the mixture of two MCM-41 samples with different pore diameters in equal masses. They are the same samples shown in Figure 5, and the pore diameters are 2.43 and 3.27 nm, respectively, as shown in the figure. (b) PSD of a pore clay heterostructure sample (described in ref 24). The sample has both micropores (peaked at 1.02 nm) and mesopores (peaked at 2.85 nm).

the procedure are summarized in a calculation table (Table 1) in which a silica sample is used as an example. The relative pressures (P/P0) and the nitrogen uptakes at these pressures are listed in columns 1 and 2, respectively. The t values at these relative pressures are listed in column 3, which can be calculated using eq 1. The nitrogen uptake is converted into a liquid volume (column 4). The radius of the pore filled at the relative pressure is obtained from the empirical equation and listed in column 5. We can follow the conventional manner to conduct the calculation from the high relative pressure to low relative pressure.3,4,13-15 The highest relative pressure for the calculation is chosen a little arbitrarily. For instance, we can start from a P/P0 of about 0.91 and calculate the surface area available for further adsorption, S1, from the t-plot. This is similar to the calculation of nonmicroporous surface area for which the comparison methods are generally used. For the sample shown in Table 1, S1 is 7.8 m2/g (the first row in column 11). The difference in N2 adsorption between the starting pressure and the next relative pressure is ∆Vl (column 4), including contributions from pore filling, ∆Vfp (column 12), and the increase in the adsorbed film on the wall of the unfilled pore or external surface, ∆t S1.

∆Vfp ) ∆Vl - ∆t S1

(6)

∆Vrp ) ∆Vfp[Rm/(Rm - tm)]2

(7)

where Rm (column 6) and tm (column 7) are the mean radius of these pores, respectively. The ratio ∆Vrp/∆Rm can be readily obtained and used as an approximation of dV/dR (column 14). Meanwhile, the surface area of these pores is also derived from ∆Vrp,

∆S2 ) 2πRL ) 2∆Vrp/R ) 2∆VfpR/(R - t2)2

(8)

Subsequently, we can calculate the surface area of the walls from unfilled pores and the external surface at the pressure next to the starting pressure (0.88 in Table 1) by S2 ) S1 + ∆S2. One can obtain the dV/dR data by repeating the arithmetic step by step to lower relative pressures and then plot dV/dR against Rm. Such a plot is the pore size distribution of the porous sample. The repeating calculation can be conducted using an Excel sheet. Figures 8-10 are the PSD curves of the samples in Figures 1 and 3-5. Figure 10b is the PSD for a porous clay heterostructure sample24 which has both micropores and mesopores. In Table 2, the surface area Sn is compared with the BET specific surface area (SBET), and the accumulating pore volume, ∑∆Vrp, is compared with the overall pore volume (the nitrogen uptake in liquid volume at the highest relative pressure P/P0 ) 0.90), Vt, for the samples. The differences between the surface areas derived from the two methods are below 10%, and the differences between the two pore volumes are below 8%. These results obtained by the proposed procedures are reasonably good. Nevertheless, it is not suitable to extend this approach to fine micropores (for instance, ultramicropores far below 1 nm) as we mentioned above. Also the BET equation

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could not be applied to calculate the specific surface area from these fine pores. We have no reliable criteria for the significance of the obtained data for fine micropores. Conclusions The comparison plot obtained directly from the nitrogen sorption isotherms is intrinsically related to the curve of accumulating pore volume of the solid. Therefore, the tangent curve of the comparison plots can display qualitative distributions of pore volume over a wide range of pore sizes (from ∼1 to 50 nm) in porous solids. This is a convenient and reliable means of providing visual pictures of the pore structure of the samples. To derive quantitative pore structure information, a relationship between the t value and the pore size is established from the samples with uniform pore structure and known pore sizes, such as MCM-41 and alumina pillared clay samples. A calculation procedure to derive quantitative PSD from the

Zhu et al.

comparison plots is suggested. The surface area and pore volume from the calculation are close to those calculated from the BET equation and overall adsorption, respectively. This suggests that we can derive reliable information on pore structure, qualitatively and quantitatively, from the comparison plots. The tangent curve and PSD calculation can be applied to a wide range of pore sizes, because the comparison plots are not based on a particular hypothesis which may incur unreliable results beyond a certain pore size range. Acknowledgment. Financial support from the Australian Research Council (ARC) to this project is gratefully acknowledged. H.Y.Z. is indebted to ARC for the QE II fellowship. P.C. acknowledges the FWO-Flanders (Fund for Scientific Research Flanders-Belgium) for financial support. LA049041P