An Experimental Study of Gas Adsorption on Fractal Surfaces

Langmuir 2005, 21, 2281-2292

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An Experimental Study of Gas Adsorption on Fractal Surfaces Matthew J. Watt-Smith,† Karen J. Edler,‡ and Sean P. Rigby*,† Departments of Chemical Engineering and Chemistry, University of Bath, Claverton Down, Bath BA2 7AY, United Kingdom Received July 20, 2004. In Final Form: January 4, 2005 The validity of the fractal versions of the FHH and BET theories for describing the adsorption of butane and nitrogen on a variety of partially dehydroxylated silica surfaces has been tested. The fractal dimensions obtained from adsorption data have been compared with those obtained completely independently using SAXS. It was found that the fractal dimensions obtained from butane adsorption isotherms, using both the fractal FHH and fractal BET theories, agreed well with the corresponding values obtained from SAXS over overlapping length scales. However, in general, a systematic deviation between the fractal dimension obtained from nitrogen adsorption and that obtained from SAXS was observed. The fractal dimensions obtained from nitrogen adsorption were consistently larger than those obtained from SAXS, which is the opposite of what has often been found in the literature. It has been suggested that the differences in the suitability of the adsorption theories tested to describe butane and nitrogen adsorption is due to the significant difference between the interaction strengths of these two different molecules with silica surfaces. A modified theory that can account for the discrepancy between the fractal dimensions obtained from nitrogen adsorption and SAXS has been proposed. The implications of the new theory for the accuracy of nitrogen adsorption BET surface areas for silicas are discussed.

Introduction The internal surfaces of porous media, such as mesoporous alumina or silica supports for heterogeneous catalysts, are highly complex. The pores in these materials are often very different from the idealized, smooth cylinders that they are often assumed to be in models of porous media. Theoretical1-4 and computer simulation4,5 studies have shown that the degree of surface roughness of pores strongly influences the rate of mass transfer down those pores, particularly by surface diffusion1-3 and Knudsen regime pore diffusion.4,5 Simulations have also shown that the degree of surface roughness of a heterogeneous catalyst can affect the selectivity between competing diffusion-limited reactions.6 Hence, it is important to be able to quantify the degree of surface structural heterogeneity in a heterogeneous catalyst. The concept of fractals may offer a method by which apparently irregular surfaces may be both characterized and mathematically modeled.7 Fractals are objects that manifest the special property of self-similarity, in that closer and closer inspection at increasing magnifications reveals morphologically similar structures over smaller scales. For a fractal structure, the value of a particular property varies with the size of the yardstick used to measure it. For example, the area (A) obtained for a fractal surface depends on the resolution of the measurement (r) according to a power law of the form

(Rr)

A ∼ r2

D

(1)

where the exponent D is the fractal dimension of the surface and R is the overall linear dimension of the surface. * Corresponding author. Phone: +44 (0)1225 384978. E-mail: [email protected]. † Department of Chemical Engineering. ‡ Department of Chemistry. (1) Rigby, S. P. Langmuir 2002, 18, 1613-1618. (2) Rigby, S. P. Langmuir 2003, 19, 364-376.

Real objects tend only to obey scaling relations, such as eq 1, over limited length scale ranges between upper and lower cutoffs. Gas adsorption is a method that is frequently used to determine the surface fractal dimension of porous media. Several different theories have been developed to analyze gas adsorption data to obtain the fractal dimension. Initially, Avnir and co-workers8,9 attempted to use eq 1 directly by varying the resolution of measurement used to obtain the surface area by using various different chemical species with different molecular sizes. These workers found the surface area perceived by each molecule using the standard Brunauer-Emmett-Teller (BET) theory.10 Using this method, Avnir and co-workers8,9 proposed that a whole range of materials had fractal surfaces. However, a number of shortcomings in their method have been highlighted.11,12 The range of length scales that could be probed was limited by the small variation in molecular size over the set of molecules used. In addition, the effective cross-sectional area of an adsorbed molecule is known13-15 to depend on the strength (3) Rigby, S. P. In Catalysis in Application; Jackson, S. D., Hargreaves, J. S. J., Lennon, D., Eds.; Royal Society of Chemistry: Cambridge, U.K., 2003; pp 170-177. (4) Malek, K.; Coppens, M.-O. J. Chem. Phys. 2003, 119, 2801-2811. (5) Malek, K.; Coppens, M.-O. Colloids Surf., A 2002, 206, 335-348. (6) Rigby, S. P.; Gladden, L. F. J. Catal. 1998, 180, 44-50. (7) Mandelbrot, B. B. The Fractal Geometry of Nature; Freeman: San Francisco, CA, 1982. (8) Pfeifer, P.; Avnir, D. J. Chem. Phys. 1983, 79, 3558-3565. (9) Avnir, D.; Farin, D.; Pfeifer, P. Nature 1984, 308, 261. (10) Brunauer, S.; Emmett, P. H.; Teller, E. J. Am. Chem. Soc. 1938, 60, 309-319. (11) De Gennes, P. G. In Physics and Chemistry of Disordered Materials; Alder, D., Fritsche, H., Ovshinsky, S. R., Eds.; Plenum Press: New York, 1985; p 227. (12) Drake, J. M.; Levitz, P.; Klafter, J. New J. Chem. 1990, 14, 77-81. (13) Gregg, S. J.; Sing, K. S. W. Adsorption, Surface Area and Porosity, 2nd ed.; Academic Press: London, 1982. (14) Lowell, S.; Shields, J. E. Powder Surface Area and Porosity, 2nd ed.; Chapman and Hall: London, 1984. (15) Karnaukhov, A. P. J. Colloid Interface Sci. 1985, 103, 311-320.

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of the interaction with a surface and the conformation adopted by the molecule on a surface. The use of the standard BET equation is also questionable, given that it was derived for planar surfaces with uniform heats of adsorption, and not for chemically or structurally heterogeneous surfaces, and neglects intermolecular interactions of the adsorbate. However, the BET equation has subsequently been adapted for fractal surfaces by various workers.16,17 Unfortunately, the fractal BET equations still frequently omit the potential effects of chemical heterogeneity and/or adsorbate-adsorbate interactions. Pfeifer et al.18,19 proposed an alternative theory, based on the Frankel-Halsey-Hill (FHH)20 approach, for obtaining the fractal dimension of surfaces from gas adsorption data that only requires one type of adsorptive because it uses the variable thickness of the adsorbed multilayer film as the probe of varying length scale. This method is free from many of the complications in the original molecular tiling method developed by Avnir and co-workers8,9 and in the fractal BET theories.16,17 Any chemical heterogeneity of the surface should be unimportant because, after completion of the first monolayer, the substrate is mostly shielded. This method is frequently referred to as the fractal FHH method. In the fractal FHH approach, the relationship between the FHH exponent and the fractal dimension depends on whether the substrate (van der Waals) potential or the vapor-liquid surface tension dominates the adsorption. The various theories described above, for adsorption on fractal surfaces, have been used to analyze the gas adsorption data for many systems. These studies have also often shown that the experimental adsorption data do fit the relevant equation(s) for the various theories used. However, studies which actually directly test the validity of the fractal adsorption theories by the adsorption of gases onto surfaces with independently known fractal dimension are relatively much rarer in the literature. In validation studies, X-ray and/or neutron scattering experiments are generally used to obtain independent measures of surface fractal dimension to compare with the value obtained from gas adsorption. Ma et al.21 suggest that in only one study (by Rojanski et al.22) is it claimed that gas adsorption data agree with those from scattering. Ma et al.21 further suggest that some authors23 have claimed that even this agreement turned out to be a misinterpretation of the scattering data, though Pfeifer and Schmidt24 have contested that it was a misinterpretation. Ross et al.25 compared fractal dimensions obtained from gas adsorption using the molecular tiling and fractal BET methods with those obtained from small-angle X-ray (SAXS) and small-angle neutron (SANS) scattering. The adsorbates used in this work included nitrogen, argon, n-pentane, and n-heptane. For fumed silica materials, the adsorption and scattering methods were found to generally disagree, and this was attributed to the neglect of lateral (16) Fripiat, J. J.; Gatineau, L.; Van Damme, H. Langmuir 1986, 2, 562-567. (17) Mahnke, M.; Mo¨gel, H. J. Colloids Surf., A 2003, 216, 215-228. (18) Pfeifer, P.; Wu, Y. J.; Cole, M. W.; Krim, J. Phys. Rev. Lett. 1989, 62, 1997-2000. (19) Pfeifer, P.; Liu, K.-Y. Stud. Surf. Sci. Catal. 1997, 104, 625677. (20) Halsey, G. J. Chem. Phys. 1948, 16, 931-937. (21) Ma, J.; Qi, H.; Qong, P. Phys. Rev. E 1999, 59, 2049-2059. (22) Rojanski, D.; Huppert, D.; Bale, H. D.; Dacai, X.; Schmidt, P. W.; Farin, D.; Seri-Levy, A.; Avnir, D. Phys. Rev. Lett. 1986, 56, 25052508. (23) Wong, P.; Bray, A. J. Phys. Rev. Lett. 1988, 60, 1344. (24) Pfeifer, P.; Schmidt, P. W. Phys. Rev. Lett. 1988, 60, 1345. (25) Ross, S. B.; Smith, D. M.; Hurd, A. J.; Schaefer, D. W. Langmuir 1988, 4, 977-982.

Watt-Smith et al.

interactions between adsorbed molecules by the adsorption theories. Drake et al.12 characterized the surfaces of three silica gels using the molecular size dependent adsorption (molecular tiling) of a series of alcohol, alkane, and aromatic molecules. While these workers found a general trend in which larger molecules perceived lower surface areas, they did not observe scaling according to eq 1. In addition, SAXS studies suggested that the silica surfaces studied were smooth (i.e., nonfractal) despite the trend seen in the adsorption data. Drake et al.12 attributed the complexity observed in the adsorption behavior to the details of the specific interactions between the adsorbate and the surface. Since it only requires one adsorption isotherm, the fractal FHH theory19 has often been used to determine the fractal dimension of surfaces from gas adsorption. Ismail and Pfeifer26 have found that using the van der Waals adsorption version of the fractal FHH theory to analyze adsorption data often leads to unphysical values of fractal dimension that are less than 2. However, Ismail and Pfeifer26 suggest that the apparently unphysical results arise because, for many systems, the crossover from substrate (van der Waals)-dominated adsorption to the regime dominated by surface tension effects occurs very early on at around the completion of the monolayer. Hence, the expression for the FHH exponent based on surface tension effects should be used to analyze data, and this does give rise to physical values of the fractal dimension. Sahouli et al.27 have compared the surface fractal dimensions for carbon blacks obtained from SAXS with those obtained using the fractal FHH theory to analyze nitrogen adsorption data for the same materials. While the gas adsorption fractal dimensions obtained were physically meaningful, they disagreed with the corresponding value obtained from SAXS in most cases. Weidler et al.28 have studied the development of surface roughness of two synthetic goethites. These workers measured the fractal dimension of these materials using SAXS and a fractal FHH analysis of nitrogen adsorption data. They found that the SAXS and adsorption data exhibited the same trends but disagreed about the absolute value of the fractal dimension. Ma et al.21 compared the surface fractal dimensions for three shale samples obtained from a fractal FHH analysis of nitrogen adsorption data with those obtained from SANS. They found that the fractal dimensions obtained from gas adsorption were significantly lower than those from SANS. In light of numerical simulations of gas adsorption on a fractal substrate, Ma and coworkers29,30 have attributed this failure of the fractal FHH theory for nitrogen adsorption to the effects, on the slope of the FHH plot in the fitted data range, arising from the crossover from van der Waals-dominated adsorption to the surface tension regime. In many cases, the fractal FHH analysis of the multilayer region of a gas adsorption isotherm results in fractal dimensions that are significantly larger than those obtained from scattering methods. Tang et al.31 have found that the value of the fractal dimension obtained from a fractal FHH analysis of nitrogen adsorption data depends on the range of the number of adsorbed layers (n) that are (26) Ismail, I. M. K.; Pfeifer, P. Langmuir 1994, 10, 1532-1538. (27) Sahouli, B.; Blacher, S.; Brouers, S.; Sobry, R.; Van den Bossche, G.; Diez, B.; Darmstadt, H.; Roy, C.; Kaliaguine, S. Carbon 1996, 34, 633-637. (28) Weidler, P. G.; Degovics, G.; Laggner, P. J. Colloid Interface Sci. 1998, 197, 1-8. (29) Qi, H.; Ma, J.; Wong, P. Phys. Rev. E 2001, 64, 041601-1-4. (30) Qi, H.; Ma, J.; Wong, P. Colloids Surf., A 2002, 206, 401-407. (31) Tang, P.; Chew, N. Y. K.; Chan, H.-K.; Raper, J. A. Langmuir 2003, 19, 2632-2638.

Gas Adsorption on Fractal Surfaces

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fitted to the fractal FHH equation. These workers found that, for bovine serum albumin particles, the fractal dimension obtained from the upper multilayer region (n > 2) of a nitrogen adsorption isotherm differed from that obtained from an analysis of the adsorption region between monolayer coverage (n ) 1) and the completion of the second statistical layer (n ) 2). Tang et al.31 found that the fractal dimension obtained from around the monolayer region (1 < n < 2) agreed well with the value from scattering and scanning electron microscopy (SEM) data. These workers suggested that the change in the apparent fractal dimension with the amount adsorbed occurred for reasons concerned with the structure of the substrate and not because of crossover effects in the mechanism of adsorption. In the system studied by Tang et al.,31 the individual solid particles on which the nitrogen initially adsorbed were in close contact with each other such that at higher coverages the gas molecules probed the envelope surface of the aggregate instead of the surfaces of single particles. The above discussion of previous work clearly suggests that the fractal theories of adsorption have not been properly tested. In particular, all of the tests of the fractal FHH theories known to the authors have used nitrogen as the adsorptive. It is entirely possible that, whereas the fractal FHH theory may not apply to the nitrogen-silica system, it may apply in the case of a different adsorptive. It is the purpose of the work described here to compare the surface fractal dimensions obtained for a range of silica materials from fractal BET and fractal FHH analyses of nitrogen and n-butane adsorption isotherms with those obtained independently using scattering methods. A theory is also presented to suggest why, while data from nitrogen adsorption isotherms fit the mathematical form of the fractal BET and fractal FHH equations, the surface fractal dimensions derived from the fitting parameters are incorrect. Theory Fractal Analysis of Gas Adsorption Isotherms. The fractal dimension of a surface can be measured independently using a fractal version of the Frenkel-HalseyHill (FHH) equation for the analysis of gas adsorption data.18,19 The theory behind the derivation of the surface fractal dimension by this method is described in great detail elsewhere18,19 and thus will only be given briefly here. The method is based on an expression for the surface fractal dimension from an analysis of multilayer adsorption to a fractal surface such that

ln

[ ( )]

( )

P0 V ) C + S ln ln Vm P

(2)

where V is the volume of adsorbed gas at equilibrium pressure (P), Vm is the volume of gas in a monolayer, and P0 is the saturation pressure. The constant C is a preexponential factor, and S is a power law exponent dependent on D, the surface fractal dimension, and the mechanism of adsorption. There are two limiting cases: At the lower end of the isotherm, representing the early stages of multilayer buildup, the film/gas interface is controlled by attractive van der Waals forces between the gas and the solid which tends to make the film/gas interface replicate the surface roughness. In this case, the value of the constant S is given by

S)

D-3 3

(3)

At higher coverages, however, the interface is controlled by the liquid/gas surface tension which makes the interface move further away from the surface so as to reduce the surface area. In this second case, S is given by

S)D-3

(4)

Under both circumstances, when there is a single fractal scaling regime, the ratio V/Vm is related to the number of adsorbed layers (n) by

n)

( ) V Vm

1/(3-D)

(5)

The actual thickness of coverage is obtained by multiplying by the diameter of the adsorbate molecule (e.g., 0.35 nm for nitrogen). The length scale cutoffs encompassed by the surface fractal dimension are the thickness of the adsorbed multilayer over which the fractal dimension applies. Where there is a crossover between two fractal scaling regimes, within each of which different surface fractal dimensions apply, the total number of adsorbed layers up to a point in the upper regime (n2) is given by

n2 )

( ) V2 Vm

1/(3-D2)

-

( ) V1 Vm

1/(3-D2)

+

( ) V1 Vm

1/(3-D1)

(6)

where D1 and D2 are the fractal dimensions of the lower and upper (in terms of amount adsorbed) fractal scaling regimes, respectively. V1 is the volume of gas adsorbed in the layers up to the crossover between the two fractal regimes, and V2 is the total volume of gas adsorbed. Pfeifer and Cole32 have suggested that, for many systems, capillary condensation would occur once the adsorbed film thickness exceeds ∼1 nm. Below this thickness, they argued that the substrate potential would dominate. However, the range of thicknesses between one monolayer and 1 nm is too small for scaling behavior to be observed. Ma et al.21 have suggested that, when analyzing the data in the thick film regime, an additional constant term, VB, should therefore be included in eq 2 to represent the amount adsorbed before capillary condensation begins, such that

[ ( )]

V ) V0 ln

P0 P

D-3

+ VB

(7)

where V0 is a composite of the original constants in eq 2. As mentioned above, an alternative analysis method to derive surface fractal dimensions from gas adsorption data is to use the fractal version of the BET theory. The derivation16,17 of a fractal BET equation is based upon the fact that, for a fractal surface, the area available for adsorption in the ith layer of adsorptive decreases by the factor fi, given by

fi )

Ai ) iR-1 A1

(8)

where R ) 3 - D, from that (A1) which is available for the adsorption of the initial monolayer. Mahnke and Mo¨gel17 suggested an alternative derivation of a fractal BET equation to that initially proposed by Fripiat et al.16 The alternate derivation avoids the inconsistent behavior, for the earlier equation, in the case of a surface fractal dimension of 3. The isotherm equation derived by Mahnke and Mo¨gel17 is given by (32) Pfeifer, P.; Cole, M. W. New J. Chem. 1990, 14, 221-232.

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[

log[V(P/P0)] ) log(Vm) + log

C(P/P0)

Watt-Smith et al.

]

1 - (P/P0) + C(P/P0) R log[1 - (P/P0)] (9)

where C is the BET constant. To obtain an estimate for R, the following procedure suggested by Mahnke and Mo¨gel17 is followed: (i) Estimate C and Vm by the standard BET10 formalism. (ii) Evaluate the plot of

log

[

]

V 1 - (P/P0) + C(P/P0) ()log(Z)) vs -log[1 Vm C(P/P0) (P/P0)]

It should be linear, and the slope is an estimate for R ) 3 - D. A New Variant of the BET Equation. The fractal BET equation is only one of a number of proposed modifications to the original BET theory.10 These modifications to the basic BET theory have been proposed largely due to the generally limited range of relative pressures (typically 0.05-0.35) over which the basic BET theory has been found to be applicable to data. Pickett33 proposed a modified BET theory where it was assumed that the number of layers on a part of the surface is limited to n and that more than n layers can be accommodated on the rest of the surface. The fraction of the surface accommodating a given number of layers was related to the number exponentially such that

dA ∝ e-Ri di

(10)

where dA is the portion of the surface on which from i to i + di layers can be built and R is a constant. Anderson34 proposed a variant of the original BET model where the volume of adsorbate required to completely cover the ith layer of adsorbate is less than that required to complete the monolayer. It was assumed that the area available to each layer of adsorbate is a constant fraction, β, of the area available to the previous layer such that

fi )

Ai ) βi-1 A1

(11)

Figure 1. Example of a two-component BET model (eq 12) composite isotherm (thick line) and its two constituent isotherms (thin lines).

dehydroxylated by increasingly intensive thermal pretreatment, it has been observed that the BET constant also steadily decreases.13,38 This correlation is probably due to the loss of higher energy sites for nitrogen adsorption by the removal of hydroxyls. NMR studies37,39,40 have shown that the distribution of individual hydroxyl groups on a partially dehydroxylated silica surface is not homogeneous. Hence, it is proposed that a realistic model for the surface of a partially dehydroxylated silica, as perceived by nitrogen, consists of a patchwork of regions of high heat of adsorption (corresponding to patches of the surface still retaining many hydroxyls) and regions of low heat of adsorption (corresponding to virtually dehydroxylated zones). Hence, the surface is thought to consist of a random patchwork where the values of the BET constant may take one of two very different values, one high and one low. As is common practice with homotattic patch models of surfaces, it is assumed that the patches are large, such that edge effects are small, and the adsorption behavior in each patch is independent of that in surrounding patches. For a homotattic patch variant35,36 on the original BET model, the overall isotherm is thus a composite of two BET-type equations, one for each of the two types of surface patches, that are each weighted by the fraction of the total surface area they occupy:

[

This equation is obviously different from eq 8. Several sets of workers35,36 have modified the BET equation to account for variations in the heat of adsorption across the surface of the substrate. These models generally consist of postulating separate patches on the surface that possess different values of the BET constant. In this work, a new variant on the basic BET model is proposed. In the new model, both a variation in the heat of adsorption across the surface of the substrate and a decrease in the area available for adsorption in each successive layer of adsorbate is postulated. Silica surfaces are known to possess hydroxyl groups.13,37 Nitrogen possesses a quadrupole moment which is believed13 to interact strongly with hydroxyl groups leading to some specific adsorption. As silica surfaces are progressively

C1 and C2 are the BET constants for surface patches of types 1 and 2, respectively, and θ is the fraction of the surface occupied by patches of type 1. An example of an overall isotherm given by eq 12 is shown in Figure 1. Also shown in Figure 1 are the two constituent isotherms, for the two types of surface patches, making up the composite isotherm. From Figure 1, it can be seen that, when one type of patch has a very low BET constant while the other

(33) Pickett, G. J. Am. Chem. Soc. 1945, 67, 1958-1962. (34) Anderson, R. B. J. Am. Chem. Soc. 1946, 68, 686-691. (35) McMillan, W. G. J. Chem. Phys. 1947, 15, 390-397. (36) Walker, W. C.; Zettlemoyer, A. C. J. Phys. Chem. 1948, 52, 4764. (37) Liu, C. C.; Maciel, G. E. J. Am. Chem. Soc. 1996, 118, 51035119.

(38) Rouquerol, F.; Rouquerol, J.; Sing, K. Adsorption by Powders and Porous Solids: Principles, Methodology and Applications; Academic Press: London, 1999. (39) Chuang, I.-S.; Maciel, G. E. J. Phys. Chem. B 1997, 101, 30523064. (40) Hwang, S.-J.; Uner, D. O.; King, T. S.; Pruski, M.; Gerstein, B. C. J. Phys. Chem. 1995, 99, 3697-3703.

θC1 (P/P0) V ) + Vm 1 - (P/P0) 1 + (C1 - 1)(P/P0) (1 - θ)C2

]

1 + (C2 - 1)(P/P0)

(12)


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Figure 2. Schematic diagram of a lower layer (open molecule symbol) and upper layer (shaded molecule symbol) of closepacked structures that form the basis of (a) triangular pyramidal and (b) square pyramidal structures.

is much larger, at lower relative pressures the amount adsorbed on the high BET constant patches is substantial, while there is very little adsorbed on the low BET constant patches. In the new model, layers of adsorbed molecules are treated like close-packings of spheres. As seen above, in the earlier stages of adsorption on a surface consisting of a random patchwork, where patches have either a high or a low heat of adsorption, the molecules are almost completely concentrated on only those patches of the surface with a high heat of adsorption. This causes the first close-packed layer of adsorbed molecules to consist only of generally isolated patches of relatively small spatial extent. In finite packings of spheres, supported at their base by a planar surface, the sites for the spheres in each layer above the first are located above the gaps between the spheres in the layer below. If the first layer of closepacked molecules is only of a limited spatial extent, then the number of gaps between molecules is less than the number of surrounding molecules in the patch, and hence, the number of sites available for adsorption decreases in each subsequent layer. Hence, the stack of molecules is, overall, pyramidal in form. There are a variety of alternative close-packed geometries for the first layer, such as the square or hexagonal close-packings shown in Figure 2. The different possibilities for the geometric arrangement, and the finite spatial extent, of the first layer lead to different potential overall geometries for the subsequent close-packed stack above the first layer, such as square pyramidal or triangular pyramidal for square and hexagonal close-packed arrangements, respectively. These pyramidal stacks are in contrast with the arrangement of adsorbed molecules envisaged in the original BET model,10 where the individual molecules were adsorbed onto individual vertical stacks directly above each adsorption site on the surface. In the vertical stacks of the BET model, the number of sites available for adsorption did not decrease in subsequent layers above the first. However, if the adsorption of molecules in each subsequent layer occurs above the gaps between molecules in the layer below, but the first layer also completely covers the surface (rather than being patchy), then each layer would have the same number of adsorption sites as all the others and the model would be indistinguishable from the original BET model. In the original BET model,10 individual molecules only interacted with molecules above and below them in the stack in which they were situated. Hence, all sites above the first layer were equivalent to each other. In the new model, it is also assumed that molecules only interact with other molecules above and below them in the stack and that all sites are equivalent above the first layer. However, the locations of molecules in each layer are dictated by the potential wells formed by the gaps in the close-packed molecules in the layer below, rather than lateral interactions with their neighboring molecules in the same layer. It is also implicitly assumed that the

Figure 3. Variation of the ratio (f) of the number of adsorption sites available in level i to that in the base level of triangular pyramidal stacks with base side lengths of 50 (+) and 100 (×). Also shown (solid lines) are fits of the data to eq 13 (or 8) for 2 < i < 11 and the apparent “fractal dimensions” that would be obtained from eq 8.

pyramidal stacks of molecules are large compared to the size of a single molecule and hence have low surface-areato-volume ratios, and thus, there are relatively few molecules in the sites on the edge of the packing where the sites are not completely equivalent to those in the interior of the stack. This is consistent with the assumptions of the homotattic patch model. In contrast to the original BET model, in a homotattic patch model with pyramidal molecular stacks, as described above, the number of adsorption sites available in each layer in the multilayer stacks decreases in higher layers. The decrease in the numbers of sites with height above the base layer for two differently sized triangular pyramidal stacks is shown in Figure 3. Fits to isotherm equations, such as the fractal BET and FHH equations, are typically made for amounts adsorbed that would only correspond to the first few adsorbed layers above the base of the pyramidal stack (e.g., layers 1, or 2, up to 10). As shown in Figure 3, over this particular range, the number of available adsorption sites in a pyramidal stack is a good fit to a power law of the form

fi )

Ai ) iγ A1

(13)

Equation 13 is of the same mathematical form as eq 8 for fractal surfaces. It can also be seen from Figure 3 that the exponent of the power law in eq 13 is a function of the size of the base of the stack. An equation of the form of eq 13 can be inserted into the derivations of the “fractal” BET equation at the same point as eq 8 and give rise to variants of the original BET equation of the same mathematical form as the fractal BET equation itself. However, whereas, with the fractal BET equation, in addition to the BET constant and the monolayer coverage, the fitted parameters include the surface fractal dimension, with the new model variant of the BET equation, the third fitting parameter is ultimately the overall linear dimension of the patch forming the base of the pyramidal stack. It is also conceivable that a surface may both possess a fractal geometry and be energetically heterogeneous and thus have surface patches with differing strengths of interaction with the adsorbate. For an adsorbed molecule, within a pyramidal stack on a fractal surface, the number of potential adsorption sites available at each subsequent level will decrease relative to the levels below because of

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the overall geometry of the stack but also be further reduced because of the effect of concavities in a rough fractal surface. Hence, since the right-hand side of eq 8 is independent of the overall size of the fractal surface, it is presumed that the decrease in the number of adsorption sites with increasing numbers of adsorbed layers in a pyramidal stack will be enhanced by the fractal geometry, such that

fi )

Ai ) iR-1iγ ) i(R+γ-1) ) iω A1

(14)

The multiplicatory form of eq 14 is such that it has the correct limiting behavior, where when D ) 2 and the surface is flat, then eq 14 reduces to eq 13 as required. Equation 14 also has the same overall mathematical form as eq 8 and can thus also be substituted into the derivation of the fractal BET equation in the same way. Analysis of SAXS Data. It is well-known21 that the intensity of radiation scattered by a fractal surface is often proportional to a negative power of the q vector, such that

I(q) ∝ q-η

(15)

where q ) 4πλ-1 sin(φ/2), λ is the wavelength of the radiation, and φ is the scattering angle. Usually this dependence is observed only when q satisfies the inequality qξ . 1, where ξ is the characteristic length of the structure producing the scattering. From the value of η, one can determine the fractal nature of the system under investigation. The exponent 1 < η < 3 describes the mass fractal of dimension (Dm)SAXS ) η, and 3 < η < 4 describes surface fractals of dimension (DS)SAXS ) 6 - η. For η ) 4, eq 15 leads to Porod’s law, where (DS)SAXS ) 2 and the surface is smooth. The surface fractal dimension obtained from SAXS can also be used as the basis to obtain, independently of gas adsorption, an estimate of the specific surface area, as perceived by a given adsorbate, of a porous solid with a fractally rough internal surface. If the porous solid consists of a close-packing of spherical particles of radius R which each possess a fractally rough surface of dimension D, then the specific surface area of the solid (As) as perceived by a particular adsorbate species is given by

(Rr)

3r2 As )

D

R3Fs

(16)

where r2 is the cross-sectional area of a close-packed adsorbate molecule and Fs is the skeletal mass density of the solid. For porous solids consisting of a packing of particles, the characteristic particle size (R) can be obtained using electron microscopy,41 mercury porosimetry,42,43 or SAXS.44,45 Experimental Section The commercially available materials studied in this work were the three sol-gel silica materials G1, G2, and S2 and the fumed silica pellet A1. The pore structures of batches G1, G2, and S2 have been characterized extensively elsewhere43,46 using (41) Reyes, S. C.; Iglesia, E. J. Catal. 1991, 129, 457-472. (42) Mayer, R. P.; Stowe, R. A. J. Colloid Sci. 1965, 20, 893-911. (43) Rigby, S. P.; Edler, K. J. J. Colloid Interface Sci. 2002, 250, 175-190. (44) Guinier, A.; Fournet, G.; Walker, C. B.; Yudowitch, K. L. Smallangle scattering of x-rays; Wiley: New York, 1955; p 25. (45) Debye, P.; Anderson, H. R.; Brumberger, H. J. Appl. Phys. 1957, 28, 679.

nitrogen adsorption and mercury porosimetry, and hence, only a summary of the relevant details is given here. These materials have been found to possess a solely mesoporous void space with negligible microporosity. More detail concerning batch A1 is given below. Nitrogen Adsorption. Samples for nitrogen adsorption consisted of one (for batch A1) or more (typically three to four for batches G1, G2, and S2) pellets taken from the same batch. Nitrogen adsorption experiments were performed at 77 K by the use of a Micromeritics Accelerated Surface Area and Porosimetry (ASAP) 2010 apparatus. The sample was placed into a preweighed round-bottomed glass tube, and a glass spacer was placed gently into the sample tube, so as to reduce the free-space volume. A sealing frit was then fitted to the opening of the tube, so as to seal the tube after degassing prior to analysis. The sample tube and its contents were then loaded into the degassing port of the apparatus. A heating jacket was applied to the tube, and the sample was heated under vacuum, at a fixed temperature for 24 h. Temperatures in the range 200-400 °C were used. The purpose of the thermal pretreatment for each particular sample was to drive off any physisorbed water on the sample but to leave the morphology of the sample itself unchanged. Once the preparation was completed, the heating jacket was removed and the sample allowed to cool to room temperature. The sample tube and its contents were then reweighed to obtain the dry weight of the sample, which was entered into the necessary section of the software program, prior to transfer of the sample tube to the analysis port for the automated analysis procedure. The sample tube was immersed in liquid nitrogen at 77 K before the adsorption measurements were taken in the relative pressure region 0.05-1.00 P/P0. The software controlling the automated apparatus performed a leak-checking procedure and an equilibration time of 45 s for each adsorption point taken. Once the experiment was completed, the data values of the adsorption/ desorption isotherms, BET surface area, and microporosity of the sample were available for analysis. The cross-sectional area (σ) of nitrogen was taken to be 0.162 nm2. Butane Adsorption. Butane adsorption experiments were performed on a Hiden intelligent gravimetric analyzer (IGA). The sample was loaded into the IGA and the reactor chamber sealed tightly. The sample (similar in size to that for nitrogen) was then evacuated to vacuum and heated to 250 °C for 24 h. Once completed, the reactor chamber was allowed to cool to room temperature. A water jacket was placed around the chamber, and the water bath was set to 0 °C. Once the temperature was set to 0 °C, the isothermal analysis was performed in the relative pressure region 0.004-1.00 P/P0. The experiment was all software-controlled, and once completed, the data values of the adsorption, BET surface area, and kinetic data of adsorption were available for analysis. The exact cross-sectional area (σ) of a butane molecule that is appropriate for adsorption at 0 °C is uncertain. Values of 0.321 and 0.469 nm2 have both been suggested in the literature.13,14 Hence, when calculating the appropriate length scale cutoffs for the fractal dimensions obtained from butane adsorption, both values have been used and the resulting range of uncertainty has been quoted. SAXS. Small-angle X-ray scattering (SAXS) patterns were run on the materials listed above using both the DUBBLE SAXS beamline on BM26B at the ESRF47 and on the SAXS apparatus in the laboratory of Professor Robert Richardson in the Physics Department at Bristol University.48 On BM26B, a single pellet was simply placed in the beam, while, at Bristol, a single pellet was ground into a powder and loaded into a 1 mm X-ray capillary. Several different pellets from the same batch of pellets were run to identify variations in the scattering due to this sampling procedure. In both cases, the two-dimensional patterns were corrected for detector efficiency and calibrated against silver behenate and pixels at the same radius were averaged to produce one-dimensional patterns. Scattering from an empty capillary was subtracted from the samples run at Bristol to remove background scattering. (46) Rigby, S. P. Catal. Today 1999, 53, 207-223. (47) Bras, W. J. Macromol. Sci., Phys. 1998, B37, 557-565. (48) Bateman, J. E. Nucl. Instrum. Methods Phys. Res. 1987, A259, 506-520.


Langmuir, Vol. 21, No. 6, 2005 2287

Table 1. Parameters Obtained from Fits of the Standard BET10 Model to Typical Examples of Nitrogen and Butane Adsorption Isotherms for Each of the Batches Studied in This Work butane batch

monolayer coverage (mmol/g)

BET constant

G1 G2 S2 A1

0.894 ( 0.006 0.321 ( 0.005 0.223 ( 0.001 0.572 ( 0.004

4.596 2.934 9.032 4.822

nitrogen fitted P/P0 range

BET fit R2

monolayer coverage (mmol/g)

BET constant

0.1-0.3 0.1-0.25 0.1- 0.3 0.1-0.25

0.9995 0.9979 0.9997 0.9987

3.098 ( 0.005 1.013 ( 0.004 0.818 ( 0.004 1.60 ( 0.01

75.33 119.2 90.3 247.1

fitted P/P0 range

BET fit R2

0.05-0.2 0.05-0.2 0.05-0.2 0.1-0.25

0.9999 0.9999 0.9999 0.9998

Table 2. Parameters Obtained from Fits of the Fractal FHH19 and Fractal BET17 Models to Butane Adsorption Isotherms for Each of the Batches Studied in This Worka FHH19 sample G1#1 G1#2 G2#1 G2#2 G2#3 S2#1 S2#2 A1#1a A1#1c A1#1d A1#1e

BET17

D

n

length scale (nm)

R2 for fit

D

n

length scale (nm)

R2 for fit

2.23 ( 0.01 2.27 ( 0.01 2.18 ( 0.01 2.21 ( 0.01 2.41 ( 0.02 2.553 ( 0.004 2.52 ( 0.01 2.41 ( 0.02 2.558 ( 0.004 2.52 ( 0.01 2.39 ( 0.02 2.531 ( 0.004 2.48 ( 0.01 2.33 ( 0.02 2.58 ( 0.01 2.50 ( 0.01 2.35 ( 0.01 2.557 ( 0.004 2.50 ( 0.01 2.17 ( 0.01 2.26 ( 0.01 2.28 ( 0.01 2.382 ( 0.007 2.27 ( 0.01 2.300 ( 0.002 2.26 ( 0.01 2.307 ( 0.003

1.15-1.84 1.84-8.69 1.11-1.90 1.29-5.55 1.0-2.0 2.0-8.63 1.0-7.22 1.0-2.14 2.14-8.64 1.0-7.44 1.0-2.14 2.14-5.35 1.0-4.56 1.0-2.12 2.12-14.5 1.0-10.34 1.0-2.25 2.25-18.36 1.0-14.16 1.0-6.92 1.0-2.21 1.0-2.19 1.1-9.1 1.0-2.3 1.0-18.9 0.98-2.12 0.98-12.33

0.65-0.89 to 1.04-1.26 1.04-1.26 to 4.92-5.95 0.629-0.760 to 1.08-1.30 0.731-0.883 to 3.14-3.80 0.567-0.685 to 1.13-1.37 1.13-1.37 to 4.89-5.91 0.567-0.685 to 4.09-4.94 0.567-0.685 to 1.21-1.47 1.21-1.47 to 4.90-5.92 0.567-0.685 to 4.22-5.09 0.567-0.685 to 1.21-1.47 1.21-1.47 to 3.03-3.66 0.567-0.685 to 2.58-3.12 0.567-0.685 to 1.20-1.45 1.20-1.45 to 8.22-9.93 0.567-0.685 to 5.86-7.08 0.567-0.685 to 1.27-1.54 1.27-1.54 to 10.4-12.6 0.567-0.685 to 8.02-9.70 0.567-0.685 to 3.92-4.74 0.567-0.685 to 1.25-1.51 0.567-0.685 to 1.24-1.50 0.623-0.753 to 5.16-6.23 0.567-0.685 to 1.30-1.58 0.567-0.685 to 10.7-12.9 0.555-0.671 to 1.20-1.45 0.555-0.671 to 6.99-8.44

0.9987 0.9983 0.9990 0.9995 0.9947 0.9993 0.9948 0.9930 0.9994 0.9937 0.9932 0.9995 0.9928 0.9955 0.9981 0.9856 0.9957 0.9990 0.9911 0.9966 0.9997 0.9974 0.9971 0.9977 0.9997 0.9982 0.9995

2.25 ( 0.01

0.33-10.63

0.187-0.226 to 6.02-7.28

0.9976

2.14 ( 0.01

0.41-6.86

0.232-0.281 to 3.89-4.70

0.9982

2.661 ( 0.001

1.91-6.35

1.08-1.31 to 3.60-4.35

0.9999

2.643 ( 0.001

2.0-6.24

1.13-1.37 to 3.54-4.27

0.9999

2.632 ( 0.003

2.04-6.35

1.16-1.40 to 3.60-4.35

0.9993

2.571 ( 0.005

1.83-4.29

1.04-1.25 to 2.43-2.94

0.9976

2.551 ( 0.003

4.14-12.02

2.35-2.84 to 6.81-8.23

0.9997

2.17 ( 0.01

1.51-3.88

0.856-1.03 to 2.20-2.66

0.9982

2.426 ( 0.003

1.2-10.8

0.680-0.822 to 6.12-7.40

0.9994

2.288 ( 0.004

1.26-13.43

0.714-0.863 to 7.61-9.20

0.9995

2.325 ( 0.003

1.2-13.2

0.680-0.822 to 7.48-9.04

0.9997

a The errors shown are one standard error. The ranges in the upper and lower length scale cutoffs result from the uncertainty in the size of a butane molecule, as described in the text.

Results Fractal FHH and BET Analysis of the Gas Adsorption Isotherms. Butane and nitrogen adsorption isotherms were obtained for all of the materials studied, as described above. An example of a set of raw adsorption isotherms for batch G1 is shown in Figure 4. For all materials studied, the butane sorption isotherms were typically of transitional type IV/V.38 The nitrogen sorption isotherms were always clearly type IV. The butane and nitrogen adsorption isotherms were initially analyzed using the original BET equation to obtain the BET monolayer coverage. Typical results of this analysis are shown in Table 1. Subsequently, the butane and nitrogen isotherms were analyzed using the fractal BET and fractal FHH methods. The raw adsorption data were fitted to the fractal BET theory of Mahnke and Mo¨gel,17 using the approach described above. Examples of the fits to data for the different batches studied are shown in Figures 5 and 6. The ranges of the various fits, the length scales to which they correspond, and the fitted fractal dimensions are shown in Tables 2 and 3. Since the fractal FHH equation is strictly valid only above monolayer coverage, the BET monolayer coverage was chosen as the minimum lower cutoff for all of the fractal FHH fits. In the initial fractal FHH fits, eqs 2 and 4 were fitted to the largest range of amount adsorbed (above monolayer coverage) for which a straight line fit was statistically reasonable. To give rise to reliable fractal dimensions, this range typically

Figure 4. Examples of typical adsorption isotherms for (a) nitrogen and (b) butane on a sample from batch G1.

covered of the order of a decade in the numbers of adsorbed layers and always satisfied the minimum criterion for an

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Table 3. Parameters Obtained from Fits of the Fractal FHH19 and Fractal BET17 Models to Nitrogen Adsorption Isotherms for Each of the Batches Studied in This Worka FHH19 sample

D

n

G1#1

2.533 ( 0.005 2.541 ( 0.006 2.505 ( 0.007 2.525 ( 0.007 2.525 ( 0.005 2.538 ( 0.005 2.640 ( 0.004 2.59 ( 0.01 2.62 ( 0.01 2.585 ( 0.007 2.658 ( 0.005 2.58 ( 0.01 2.686 ( 0.003 2.54 ( 0.01 2.580 ( 0.008 2.6984 ( 0.0004 2.596 ( 0.003 2.571 ( 0.007

0.76-8.23 1.08-2.25 0.75-11.1 0.96-2.19 0.79-7.95 1.06-2.16 0.96-11.22 0.96-2.08 1.65-56.5 1.0-2.1 0.85-33.71 1.0-1.8 3.6-10.0 1.0-2.2 1.1-2.0 3.52-21.87 0.94-4.33 0.94-1.93

G1#2 G1#3 G2#1 G2#2 G2#3 S2#1 S2#2 A1#1b a

length scale (nm)

BET17 R2

0.266-2.881 0.378-0.788 0.263-3.873 0.336-0.767 0.277-2.781 0.371-0.756 0.335-3.926 0.335-0.728 0.577-19.79 0.35-0.735 0.299-11.80 0.35-0.63 1.26-3.5 0.35-0.77 0.385-0.70 1.23-7.65 0.329-1.52 0.329-0.676

for fit

0.9983 0.9991 0.9961 0.9986 0.9994 0.9994 0.9978 0.9975 0.9847 0.9981 0.9963 0.9963 0.9969 0.9981 0.9987 0.9999 0.9988 0.9981

n

2.34 ( 0.01

1.76-5.79

0.616-2.03

0.9973

2.347 ( 0.007

1.89-4.43

0.662-1.55

0.9992

2.0-6.9

0.70-2.42

0.9982

2.593 ( 0.004

2.98-11.4

1.04-3.99

0.9989

2.560 ( 0.005

3.96-12.7

1.39-4.44

0.9987

2.638 ( 0.008

3.4-20.6

1.19-7.21

0.9987

2.658 ( 0.005

6.0-8.3

2.1-2.91

0.9961

2.656 ( 0.001

5.5-16.3

1.93-5.71

0.9995

2.350 ( 0.004

1.5-4.6

0.525-1.61

0.9994

2.31 ( 0.01

length scale (nm)

R2 for fit

D

The errors shown are one standard error.

Figure 5. Examples of fits (solid lines) of the Mahnke and Mo¨gel17 model (eq 9) to the butane adsorption isotherms for typical samples taken from batches A1 (#1d, 0), G1 (#1, O), G2 (#1, +), and S2 (#2, ×). The fitted ranges in the number of adsorbed layers were the following: G1, 0.33-10.63; G2, 1.916.35; S2, 4.14-12.02; A1, 1.26-13.43.

acceptable fractal dimension given by Pfeifer.49 In addition, fits of the data to the version of the fractal FHH equation that also included VB as an additional fitting parameter (eq 7) were also carried out. However, in all cases, the best-fit value of VB obtained was 0. This finding suggests that the van der Waals regime is entirely suppressed for butane, as has also been found previously46 for nitrogen adsorption on the same materials as those studied here and by past workers26 for other materials. This result may arise because the molecular size of butane is such (∼0.7 nm) that it is unlikely that any additional adsorption occurs beyond a monolayer before the commencement of capillary condensation. The iterative procedure described by Tang et al.31 was also used to obtain fits of the fractal FHH equation for the region of the adsorption isotherms corresponding to ∼1 < n < ∼2. In cases of interest, the region of the isotherm for n > 2 was also separately fitted to eq 2. Typical examples of these fits for each batch of pellets are shown in Figures 7 and 8. The ranges of the various fits, the length scales to which they correspond, and the fitted fractal dimensions are shown in Tables 2 and 3. From Tables 2 and 3, it can be seen that the results (49) Pfeifer, P. Appl. Surf. Sci. 1984, 18, 146-164.

Figure 6. Examples of fits (solid lines) of the Mahnke and Mo¨gel17 model (eq 9) to the nitrogen adsorption isotherms for typical samples taken from batches A1 (#1b, 0), G1 (#1, O), G2 (#1, +), and S2 (#2, ×). The fitted ranges in the number of adsorbed layers were the following: G1, 1.76-5.79; G2, 2.9811.4; S2, 5.5-16.3; A1, 1.5-4.6.

from fitting the gas adsorption data are highly repeatable between samples from the same batch, within experimental error. From a comparison of the respective results using each theory, for butane isotherms for each batch of pellets shown in Tables 2 and 3, it can be seen that the fractal dimensions obtained using the fractal BET theory are identical, or very similar, to those obtained using the fractal FHH theory, over an overlapping range in length scale. For the FHH analysis of the butane adsorption isotherms for batches A1 and G1, there is generally very little, or no, significant difference between the fractal dimensions obtained from fits of eq 2 to the region where ∼1 < n < ∼2 and for fits over much larger ranges of n above the completion of the monolayer. For batch S2, there is little difference between the fractal dimensions obtained for fits where n > ∼1 and where n > ∼2. However, for the butane adsorption isotherms for the other batch of pellets (G2), the fractal dimensions obtained for the region where ∼1 < n < ∼2 are always less than the fractal dimensions fitted to the extended region where n > ∼1, and also the region where n > ∼2.


Figure 7. Examples of fits (solid lines) of the fractal FHH19 model (eq 2) to the butane adsorption isotherms for typical samples taken from batches A1 (#1d, 0), G1 (#1, O), G2 (#1, +), and S2 (#2, ×). The fitted ranges in the number of adsorbed layers were the folllowing: G1, 1.84-8.69; G2, 1.0-2.0; S2, 1.0-14.16; A1, 1.0-18.9.

Figure 8. Examples of fits (solid lines) of the fractal FHH19 model (eq 2) to the nitrogen adsorption isotherms for typical samples taken from batches A1 (#1b, 0), G1 (#1, O), G2 (#1, +), and S2 (#2, ×). The fitted ranges in the number of adsorbed layers were the following: G1, 0.76-8.23; G2, 0.96-11.22; S2, 3.52-21.87; A1, 0.94-4.33.

A particular series of experiments was conducted on the single pellet constituting sample 1 from batch A1. This single sample underwent a series of adsorption isotherms interspersed by repeated thermal treatments (as described in the paragraph titled Butane Adsorption in the Experimental Section above) at 250 °C. Butane adsorption isotherms were measured after the sample had undergone one, three, four, and five separate thermal treatments (samples 1a, 1c, 1d, and 1e, respectively). A nitrogen adsorption isotherm was conducted after the sample had received two thermal treatments to 250 °C (sample 1b). After each isotherm, the sample was evacuated to vacuum. From Table 2, it can be seen that for sample 1 from batch A1 the surface fractal dimension obtained from butane adsorption remains relatively constant following repeated thermal pretreatments. As described above, the nitrogen adsorption isotherms for various samples taken from each batch were also analyzed using both the fractal BET and FHH theories. From Table 3, it can be seen that the fractal dimensions obtained from nitrogen adsorption are generally significantly higher than the corresponding values obtained, using the same theory, from butane isotherms (see Table

Langmuir, Vol. 21, No. 6, 2005 2289

Figure 9. Examples of fits (solid lines) of eq 15 to the smallangle X-ray scattering data for typical samples taken from batches A1 (0), G1 (O), G2 (+), and S2 (×). For clarity, the fitted lines have been displaced slightly from the corresponding experimental data. Also shown is the fractal dimension obtained from each fit.

2), over overlapping length scales, for G1 and S2 using both theories and A1 and G2 using fractal FHH theory. However, the fractal dimensions obtained using fractal BET theory from nitrogen adsorption isotherms for A1 and G2 are generally relatively similar to the corresponding values obtained, using the same theory, from butane adsorption. Fractal Analysis of SAXS Data. To validate the fractal BET and FHH theories of gas adsorption on fractal surfaces, the surface fractal dimensions for the materials under study were also obtained independently using SAXS. Examples of the experimental data obtained for each batch of pellets and the fits of these data to eq 15 are shown in Figure 9. The results of the fits of eq 15 to all of the SAXS data obtained are shown in Table 4. For each batch, one or more fits to eq 15 ranged over the order of a decade in q values, and all fractal dimensions obtained satisfied the minimum criterion for a well-defined fractal dimension specified by Pfeifer.49 From Table 4, it can be seen that, with SAXS, there is much more scatter among the fractal dimensions obtained for different samples taken from the same batch than that which was obtained with gas adsorption. This is probably because, as described above, the samples for gas adsorption typically consisted of three to four individual pellets, whereas the samples for SAXS consisted of only a single pellet, or less (when powdered), from a given batch. The larger sample size used for gas adsorption damped out much of the intrabatch variability observed with the SAXS data. However, it can also be seen from Table 4 that there is no significant difference between the fractal dimensions obtained, for samples taken from the same batch of pellets, at the ESRF or in Bristol, and thus, equipment type has no effect on the results obtained. From a comparison of the fractal dimensions obtained from butane adsorption and SAXS, it can be seen that, generally, for batches G1, S2, and A1 (both for individual samples and overall batch averages), the surface fractal dimensions obtained from SAXS match well with those obtained from butane adsorption above monolayer coverage over the same, or largely overlapping, length scale range. As mentioned above, in the literature,13,14 there is some variation in the cross-sectional area quoted for adsorbed butane molecules. However, as for example with batch A1, there is often a wide overlap between the length scale ranges covered by butane adsorption and SAXS data, and thus, this uncertainty in the exact length scale represented by a single adsorbed butane molecule does

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Table 4. Parameters Obtained from Fits of eq 15 (Fractal Porod Law) to SAXS Data for Samples of Each of the Batches Studied in This Worka batch G1

G2

S2

A1

a

sample fractal dimensions

q (nm-1)

length scale cutoff (nm)

Porod fit R2

(i) 1.977 ( 0.004b (ii) 2.246 ( 0.004b (iii) 2.34 ( 0.04c (iv) 2.51 ( 0.05c (i) 2.411 ( 0.004b (ii) 2.23 ( 0.05c (iii) 2.19 ( 0.06c (iv) 2.36 ( 0.03c (i) 2.51 ( 0.02c (ii) 2.46 ( 0.02c (iii) 2.46 ( 0.01b 2.50 ( 0.02b (iv) 2.50 ( 0.01b 2.53 ( 0.02b (i) 2.245 ( 0.007b (ii) 2.266 ( 0.008b (iii) 2.250 ( 0.007b

0.626-3.23 1.0-3.54 0.566-0.997 0.503-0.998 0.40-3.50 0.442-0.998 0.395-0.998 0.442-1.12 0.411-1.24 0.411-1.24 1.004-2.72 0.132-0.363 1.004-2.72 0.132-0.363 0.607-2.72 0.607-2.72 0.607-2.72

0.310-1.60 0.282-1.0 1.0-1.768 1.0-1.988 0.286-2.50 1.0-2.27 1.0-2.53 0.892-2.27 0.806-2.44 0.806-2.44 0.368-0.996 2.75-7.57 0.368-0.996 2.75-7.57 0.368-1.65 0.368-1.65 0.368-1.65

0.9996 0.9996 0.9967 0.9945 0.9995 0.9933 0.9920 0.9965 0.9974 0.9990 0.9962 0.9989 0.9977 0.9984 0.9979 0.9985 0.9988

batch average fractal dimension 2.27 ( 0.11

2.30 ( 0.05

2.49 ( 0.01

2.254 ( 0.005

The errors shown are one standard error. b Obtained at the ESRF. c Obtained at the University of Bristol. Table 5. Comparison of Geometric and BET Specific Surface Areas

sample

throat radius (nm)

G1 G2 S2 A1

3.53 14.6 17.6 6.29

porosity

particle radius (nm)

geometric surface area for nitrogen (m2‚g-1)

geometric surface area for butane (m2‚g-1)

BET surface area for nitrogen (m2‚g-1)

BET surface area for butane (m2‚g-1)

0.66 0.69 0.69 0.65

4.82 20.0 24.0 8.59

543 ( 150 219 ( 43 414 ( 17 340 ( 5

495 ( 118 199 ( 35 350 ( 13 311 ( 4

302.2 ( 0.5 98.8 ( 0.4 79.8 ( 0.4 156 ( 2

399 ( 3 143 ( 2 80.6 ( 0.4 255 ( 2

not affect the above finding. However, for batch G2, there is good agreement between the surface fractal dimension obtained from the butane isotherms for adsorption in the range ∼1 < n < ∼2 and that obtained from the same overlapping length scale range using SAXS. This result suggests that the adsorption of butane on the surface of G2 may follow a similar scenario to that described by Tang et al.31 Over length scales larger than the second adsorbed layer, the structure of the butane multilayer may be affected by the morphology of the packing of elementary spheres making up the solid of G2, rather than just the surface roughness of an individual sphere. Comparison of Gas Adsorption Data with Mercury Porosimetry and SAXS Data. It has been found above that, where the fractal dimension obtained from nitrogen adsorption differs from that obtained by SAXS, the deviation is always such that the fractal dimension from adsorption is higher than that obtained from SAXS. As described above in the Theory section, both the specific adsorption on patches of an energetically heterogeneous surface and the fractality of the surface can cause a decrease in the number of adsorption sites available in progressively higher levels within the adsorbed multilayer. Therefore, it is suggested that, for those materials where there was a significant difference observed between the fractal dimensions obtained from nitrogen adsorption and SAXS, then both of the above effects may be giving rise to a decrease in the number of adsorption sites and thus the fractal dimension would be overestimated. If this is the correct explanation of the discrepancy between the fractal dimensions obtained by nitrogen adsorption and SAXS, then nitrogen adsorption data should show some indication of more specific adsorption on the high energy patches. If nitrogen molecules are undergoing adsorption of a more specific character than what is occurring with butane, then the surface area as perceived by nitrogen should potentially be less than that perceived by butane despite the fact that a nitrogen molecule is smaller than a butane molecule.

The specific surface areas of the materials studied here have been obtained independently of gas adsorption using the method utilizing SAXS data described above. Electron microscopy studies38,41 have shown that sol-gel silica and fumed silica materials, such as those studied here, consist of packings of elemental spheres. Using a method described in earlier work,43 the radii of the elemental spheres making up these materials can be obtained using the point of inflection in mercury porosimetry intrusion curves. Mercury porosimetry was chosen, since it should give rise to estimates of sphere sizes that are more statistically representative of a macroscopic sample than what is possible with electron microscopy. Mercury porosimetry also only probes the pore structure accessible to molecules invading from the exterior, whereas SAXS probes the entire structure. The porosity (voidage fraction) of the pellets has been determined using helium pycnometry.50 The radii of the connecting throats between the cavities formed between packed spheres determined from mercury porosimetry43 are given in Table 5. The porosities determined using helium pycnometry are also shown in Table 5. It can be seen that the porosities of the pellets are generally very close to the porosity of 0.66 that is expected for a tetrahedral packing of spherical particles. Hence, a tetrahedral packing will be used as the model for the materials studied here. From the known properties of tetrahedral packings38 the radii of the packed particles are a factor of 1/0.732 larger than the throat radii. Hence, the particle radii for the materials studied here can be estimated, and these values are also shown in Table 5. In previous work,43 it has been found that the value of the modal radius for the elementary packing particles in G1 obtained by the above method is identical, within experimental error, to the estimates of the same parameter obtained from SAXS using either of the methods of Guinier44 or Debye.45 The butane adsorption studies (50) Rigby, S. P.; Gladden, L. F. Chem. Eng. Sci. 1999, 54, 35033512.


described above suggest that, for example, for batches G1 and A1, the same fractal dimension applies from molecular length scales up to the calculated particle size. The total geometric specific surface areas that would be perceived by nitrogen and butane can then be calculated using eq 16. Since the maximum geometric area accessible to a given molecule is required, then in this calculation the dimensions of the molecules corresponding to the most dense packing in the volume (coordination number 12) and on the surface (coordination number 6) are utilized. The effective cross-sectional areas (r2) used are thus 0.162 nm2 for nitrogen14,38 and 0.321 nm2 for butane.14 The calculated specific geometric surface areas are also shown in Table 5. The quoted errors in the surface areas arise predominantly from the errors in the fractal dimensions obtained by SAXS. The estimates of the total geometric surface area accessible to a particular species, obtained using a combination of SAXS and mercury porosimetry, can be compared with the values obtained using the standard BET10 method for butane and nitrogen. However, the BET method requires an appropriate value for the effective cross-sectional area of a molecule to calculate the specific surface area from the monolayer coverage. It is wellknown13,14,38 that the effective cross-sectional area of molecules varies strongly with the strength of interaction with the surface. Both nitrogen15 and butane51 have been found to exhibit a hyperbolic dependence of their effective cross-sectional areas on the BET constant, which is a measure of the relative strength of the adsorbate-surface interaction. When an adsorbate is strongly tied to a surface, it is constrained to specific adsorption sites, and the adsorbate cross-sectional area is more determined by the adsorbent lattice structure than by the adsorbate dimensions. However, if the value of the BET constant is sufficiently small, the adsorbate lateral mobility on the surface will tend to disrupt any tendency for an organized structure to develop and the adsorbate layer might appear more like a two-dimensional gas. Therefore, the effective surface area occupied per molecule increases with decresing strength of interaction with the surface because the typical spacing of molecules on the surface increases, and not because the geometric surface accessible to the molecule changes. The correct effective cross-sectional area for a given BET constant value can be obtained from adsorption studies on standard materials with specific surface areas already known by other means. For nitrogen, the values of the BET constant obtained for the materials studied here are in the range ∼75-250 (see Table 1). Studies involving model adsorbents by Karnaukhov15 suggest that the standard effective cross-sectional area 0.16 nm2 generally assumed for nitrogen is valid for values of the BET constant in this range. However, the values of the BET constant for butane on the materials studied here are in the range ∼3-9 (see Table 1). Previous work by Lowell et al.51 suggests that, for values of the BET constant in this range, the effective cross-sectional area of butane significantly exceeds the value 0.321 nm2 obtained assuming the most dense packing of molecules both in the volume and on the surface. Lowell et al.51 found that, for example, for values of the BET constant of 7 and 10 for butane, the correct effective cross-sectional areas are ∼0.741 and ∼0.5 nm2, respectively. In their work, Lowell et al.51 also found that, for values of the BET constant less than ∼40, the effective cross-sectional area of butane increases sharply as the BET constant is (51) Lowell, S.; Shields, J.; Charalambous, G.; Manzione, J. J. Colloid Interface Sci. 1982, 86, 191-195.

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reduced. Hence, it may be reasonably assumed that, in cases where the values of the BET constant are less than 7 (such as G1, G2, and A1), the effective cross-sectional area of butane may well significantly exceed 0.741 nm2. In light of this previous work, the specific surface areas for the materials studied here have also been estimated from the values for monolayer coverage obtained by the standard BET method given in Table 1. The effective crosssectional area for nitrogen was taken as 0.162 nm2 in all cases, while the effective cross-sectional area for butane was taken as 0.741 nm2 for G1, G2, and A1 and 0.6 nm2 for S2 (obtained from linear interpolation of the values for BET constants of 7 and 10 for butane of Lowell et al.51). The calculated values of the BET surface areas are shown in Table 5. From Table 5, it can be seen that, for all the materials studied, the BET areas for nitrogen are significantly less than the corresponding estimates (including quoted errors) for the total accessible geometric surface area. However, it can also be seen that the values of the BET surface area obtained using butane are generally larger than the corresponding values obtained using nitrogen and much closer to the relevant estimates of the geometric surface area, often being within the quoted errors for this parameter. In light of the uncertainty in the correct effective cross-sectional area of butane to be used at the very low values of the BET constant encountered here, the assumptions made in the estimation of the geometric surface area, and the errors in the fractal dimension obtained using SAXS, it is probably reasonable to conclude that the BET areas obtained for butane are likely to be similar to the geometric surface area of the materials. It is also noted that, for butane, the BET constants for A1 and G1 are both equal to ∼5, and thus, the effective crosssectional area for butane on both surfaces should also be equal and probably larger than the value of 0.741 nm2 found for a BET constant of 7 by Lowell et al.51 If the butane BET surface area for both batches A1 and G1 is calculated using an effective cross-sectional area for butane of ∼0.91 nm2, then that area is found to be equal to the corresponding best estimate for the geometric surface area in both cases. Hence, these results for butane are consistent with each other, as would be expected if accurate values of the geometric surface area have been obtained from combined mercury porosimetry and SAXS in each case. Hence, it is proposed that, once the variation of effective cross-sectional area with BET constant has been taken into account, then the above results suggest that nitrogen undergoes adsorption with some specific character. It is likely that nitrogen only significantly adsorbs on part of the total geometric surface that is, in principle, accessible to molecules of its size, on the materials studied here. Hence, the adsorption of nitrogen on the silica materials studied here probably does follow a pattern similar to that given in Figure 1. Since SAXS gives rise to an independent estimate of the surface fractal dimension of the materials, it is possible to deconvolve the two effects in eq 14 giving rise to a decrease in the numbers of potential adsorption sites in higher levels of the adsorbed multilayer. Hence, it is possible to use gas adsorption and SAXS in combination to estimate the size of the surface patches. For example, the apparent value of R obtained for nitrogen on A1 is 0.650 ( 0.004. As shown in Table 4, the value of the surface fractal dimension (D) obtained independently using SAXS is 2.254 ( 0.005. Hence, from eq 14, the value of γ for nitrogen on A1 is -0.096 ( 0.009. From the data shown in Figure 3, this value corresponds to a characteristic patch size ∼100 times the size of a nitrogen molecule. This value

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is relatively large, and thus, the assumption that the boundary effects at the edge of adsorbate stacks can be ignored is reasonable. The area of the base of the stack is ∼88% of the surface area of one elemental sphere in A1 (or also, in a tetrahedral packing, to the surface area of the spheres inscribed in the cavities between the elemental solid spheres). Hence, one might speculate that the high energy patches correspond to particular cavities within which hydroxyl groups have been retained. Discussion The close similarities between the fractal dimensions obtained completely independently from butane adsorption and from SAXS suggest that the adsorption of butane on the fractal surfaces of the various different silica materials investigated here is consistently well described by both the fractal FHH and fractal BET theories. This finding strongly contrasts with the corresponding results observed for nitrogen, which are, relatively, more mixed. For some materials, there is good agreement between the fractal dimensions obtained from nitrogen adsorption and SAXS, whereas, for others, there is not. This result is similar to the mixed findings from the ensemble of results from the range of previous nitrogen adsorption studies described in the Introduction. As mentioned above, it is well-known13,38 that nitrogen may show some specific adsorption on the surface of hydroxylated silicas, whereas butane is unlikely to show any specific adsorption.13,38 The above comparison of the geometric surface area accessible to molecules of a particular size with the corresponding BET surface areas suggests that nitrogen only adsorbs on a fraction of the silica surface area potentially accessible to it, whereas butane probably adsorbs across the whole surface. This finding is in accordance with what would be expected if nitrogen tends to adsorb upon specific surface patches on partially dehydroxylated silicas. It is suggested that these patches may be associated with regions with higher concentrations of hydroxyl groups on the surface. The heterogeneous distribution of hydroxyl groups on a partially dehydroxylated silica surface could potentially be shown using multiple quantum NMR,40 thereby confirming this suggestion. In addition, it is noted that the deviation, between the fractal dimension obtained from scattering data and the corresponding value obtained from nitrogen adsorption, observed here for silicas occurred in the opposite direction to that found for the deviation between scattering and nitrogen adsorption data for shales by Ma et al.21 As a result of some simulation studies, Ma and co-workers29,30 have suggested that a deviation between the scattering and adsorption data, such that the fractal dimension from adsorption is lower than that from scattering, can be explained by crossover effects from the van der Waals (substrate potential)-dominated to the capillary-condensation-dominated domains. However, since the observed deviation for nitrogen adsorption occurs in the opposite direction here, then a different explanation, such as that based on surface hetrogeneity effects described above, is required to account for the contrasting results obtained. However, in light of the work of Ma and co-workers,21 the good agreement between the fractal dimensions obtained from SAXS and from the application of the fractal FHH

Watt-Smith et al.

equation to butane adsorption isotherms, suggests that the surface tension of butane may be, relatively, much stronger than the substrate potential. This is consistent with the lower strength of interaction (and hence lower values of the BET constant) for butane, compared to nitrogen, on partially dehydroxylated silicas, as shown by the shapes of their respective adsorption isotherms. The discrepancies between the fractal dimensions obtained from SAXS and nitrogen adsorption for batches A1 and G1 also cannot be explained by the theory of Tang et al.31 As mentioned above, Tang et al.31 suggested that a discrepancy between scattering and adsorption data may arise because adsorbed molecules may actually probe the envelope surface of the aggregates of elementary particles rather than the surface of the particles themselves as probed by scattering. However, in the cases of A1 and G1, there is only disagreement between the fractal dimensions from SAXS and gas adsorption for nitrogen but not for butane. This is despite the fact that the adsorbed layers of both these species probed overlapping ranges of length scale. If the influence of the shape of the aggregate envelope surface was responsible for the observed discrepancy, then the effect ought to be seen for both gases, but this was not found. Hence, in summary, the alternative explanations, for the difference between the fractal dimensions obtained by scattering and nitrogen adsorption methods, presented by Tang et al.31 and Ma and co-workers21,29,30 have both been found to be inconsistent with the above data for silicas. However, the data reported here do suggest that nitrogen undergoes patchy adsorption on silica surfaces. Hence, the data obtained are consistent with the explanation proposed above for the discrepancy between nitrogen adsorption and SAXS. Conclusions The surface fractal dimensions obtained from butane adsorption are very similar, within experimental error, to those obtained from SAXS for silica materials. Hence, the adsorption of butane on the rough surface of partially dehydroxylated silicas is consistently well described by the fractal FHH and fractal BET theories. However, in contrast, the differences observed between the surface fractal dimensions obtained from SAXS and those obtained from nitrogen adsorption suggest that these same theories are less generally applicable to the adsorption of nitrogen on the same silica surfaces. The deviation between the fractal adsorption theories and experiment for nitrogen contrasts in direction from that found in previous work and may arise because of an additional effect due to specific adsorption not contained within the original theories. Acknowledgment. We acknowledge the ESRF for provision of synchrotron radiation facilities. We are grateful for the kind assistance of Dr. Igor Dolbyna and the other staff on the DUBBLE beamline at the ESRF for help in collecting the SAXS data. The authors also wish to thank Prof. Robert Richardson of the University of Bristol for the use of his SAXS equipment. S.P.R. and M.J.W.-S. thank the EPSRC for financial support (under Grant No. GR/R61680/01). K.J.E. thanks the Royal Society for a Dorothy Hodgkin Research Fellowship. LA048186T

An Experimental Study of Gas Adsorption on Fractal Surfaces

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