Characterization of Pore Size Distributions of Mesoporous Materials

J. Phys. Chem. B 2000, 104, 9099-9110

9099

Characterization of Pore Size Distributions of Mesoporous Materials from Adsorption Isotherms C. G. Sonwane and S. K. Bhatia* Department of Chemical Engineering, The UniVersity of Queensland, St. Lucia, Brisbane, QLD 4072, Australia ReceiVed: March 8, 2000; In Final Form: June 30, 2000

In this article, a new hybrid model for estimating the pore size distribution of micro- and mesoporous materials is developed, and tested with the adsorption data of nitrogen, oxygen, and argon on ordered mesoporous materials reported in the literature. For the micropore region, the model uses the Dubinin-Rudushkevich (DR) isotherm with the Chen-Yang modification. A recent isotherm model of the authors for nonporous materials, which uses a continuum-mechanical model for the multilayer region and the Unilan model for the submonolayer region, has been extended for adsorption in mesopores. The experimental data is inverted using regularization to obtain the pore size distribution. The present model was found to be successful in predicting the pore size distribution of pure as well as binary physical mixtures of MCM-41 synthesized with different templates, with results in agreement with those from the XRD method and nonlocal density functional theory. It was found that various other recent methods, as well as the classical Broekhoff and de Boer method, underpredict the pore diameter of MCM-41. The present model has been successfully applied to MCM-48, SBA’s, CMK, KIT, HMS, FSM, MTS, mesoporous fly ash, and a large number of other regular mesoporous materials.

1. Introduction In recent years there has been a considerable increase in effort to synthesize new mesoporous materials of well-defined geometry. Such developments have been largely catalyzed by the synthesis of the M41S family of mesoporous materials,1 of which MCM-41 is perhaps the most widely known member. These materials are characterized by their unique pore topology, offering parallel pores of size tunable in the range of 2-10 nm, and high surface areas of about 600-1300 m2/g, without the complications of a network. Because of their unusual and tunable pore size and topology, several new and novel applications of such materials have been proposed and are being investigated by various researchers. Among these is the possibility of using them as hosts for a variety of quantum confined electronic and photonic materials. Their use as catalysts and as catalyst supports has also been extensively investigated, along with their potential in separation and adsorption for pollution control. All of these applications involve the filling and transport of molecules in the mesopore channels, the understanding of which requires a reliable characterization of the mesopore structure. Among the important characteristics are the surface area, pore volume, and the pore size distribution, which can be determined by a variety of techniques. These include gas adsorption, small-angle X-ray or neutron scattering (SAXS or SANS), porosimetry, and electron microscopy (scanning and/or transmission) among a battery of available methods.2,3 While each method has a limited length scale of applicability for probing the structure, nitrogen adsorption at its boiling point of 77.4 K is the most popular because of its utility for both micropores and mesopores, and because of its convenience and low cost. According to the recommendations of the IUPAC,4 micropores comprise the range below 2 nm, while mesopores lie in the range 2-50 nm. Most * To whom correspondence should be addressed. E-mail: sureshb@ cheque.uq.edu.au. Fax: +61 7 3365 4199. Telephone: +61 7 3365 4263.

catalysts and conventional as well as the newer mesoporous materials lie in this latter range of pore sizes. The explosive development of new mesoporous materials has also highlighted the longstanding need for accurate adsorption models in the range of pore sizes corresponding to mesopores. While a vast amount of data is now available in the adsorption literature, the models developed are applicable to microporous materials or large pore materials.2 In the former case the SaitoFoley5 or Dubinin models are commonly used, while in the latter case the Kelvin or BJH models are utilized. Although simple and elegant, these approaches do involve approximations that are often unjustifiable. In particular, for the mesopores the Kelvin and BJH models are inaccurate in pores below about 10 nm in diameter, because of their neglect of fluid-solid interaction, and therefore expected to be unreliable for characterizing most of the newer materials. In addition, these methods are inaccurate at low pressures at which the monolayer formation in the mesopores is not complete, as the state of the adsorbed molecules in this region differs considerably from that of the bulk phase. Consequently, there is a need to have a model that is applicable over a range of pore sizes, and valid for the isotherms of a variety of adsorbates and adsorbents. While this need is met by the newer molecular models and density functional theory approaches,6 they are computationally demanding and impractical for routine use. With the aim of treating both the monolayer and multilayer regions in a single model, we have recently7 proposed a new hybrid isotherm for interpreting the adsorption of condensable vapors on nonporous solids. The new model incorporates the fluid-solid interaction within the framework of classical Kelvin and BJH approaches in the multilayer region, while utilizing existing models such as the Freundlich or Unilan isotherms for submonolayer coverage. The model was successfully tested using isotherm data for nitrogen adsorption on nonporous silica, carbon, and alumina, as well as benzene and hexane adsorption

10.1021/jp000907j CCC: $19.00 © 2000 American Chemical Society Published on Web 09/09/2000

9100 J. Phys. Chem. B, Vol. 104, No. 39, 2000

Sonwane and Bhatia

Figure 1. Schematic of adsorption regimes over the entire pore size range.

on nonporous carbon. Based on the data fits, out of several different alternative choices of model for the submonolayer region, the hybrid model involving the Freundlich and the Unilan models were found to be the most successful when combined with the multilayer model to predict the whole isotherm. The model can be easily extended for porous solids having pores of cylindrical shape or any other arbitrary geometry. While the model for surface adsorption and subsequent condensation is well established, it is also recognized that in pores below a certain critical size (called as micropores) these concepts do not hold and a different pore filling mechanism is operative. Although other models such as the Harvath-Kawazoe (slit geometry)8 or the Saito-Foley (cylindrical)5 exist which consider the micropore filling to be instantaneous, the Dubinin (Dubinin-Astakhav or Dubinin-Radushkevich) model is still more widely used. According to the latter model, at any pressure adsorption in pores below a critical size will proceed by volume filling and the mesopores will have a submonolayer region (for low pressures). At higher pressures, in the case of multilayer region, at a given pressure, pores with size smaller than a particular size will be filled by capillary condensate while the larger ones will have a multilayer thickness of the adsorbate. This has been shown in Figure 1. The isotherm models for micropores and mesopores can be used9 along with the wellknown generalized adsorption integral

Ca(P) )

∫0∞F(r,P)f(r) dr

(1)

to give a hybrid model for estimating the pore size distribution of the porous materials. Here Ca(P) is the total amount adsorbed, and F(r,P) is the local effective density of the adsorbate in a pore of size r at pressure P. A key unknown in such a model is the critical micropore size below which the volume filling mechanism exists. While the IUPAC has recommended a limit of 2 nm (diameter) as standard (for both capillary condensation and the absence of hysteresis),2 recent studies with adsorption in regular mesoporous materials (MCM-41 type) suggest that the actual critical pore sizes vary and can be significantly different.10-13 The adsorption of condensable gases and vapors on mesoporous materials is generally characterized by multilayer adsorption followed by a distinct vertical step (capillary condensation) in the isotherm accompanied by a hysteresis loop.2 However, studies of adsorption on newer mesoporous materials with wellorganized structures, such as MCM-41, have also demonstrated the absence of hysteresis for materials having pore size below a critical value. While this has been previously reported2 for

silica gel and chromium oxide containing some mesopores, no consistent explanation has hitherto been made available. Such conventional porous materials, having interconnected pores with a broader size distribution, are generally known to display a hysteresis loop with a point of closure which is characteristic of the adsorptive. Recent studies11,12 now suggest that the pore size for which capillary condensation occurs at this point of closure corresponds to the critical size below which this mechanism cannot occur. This critical size is smaller than that below which the condensation is reversible, as is evident from the results for MCM-41,11 and both can be interpreted by application of classical tensile stress hypothesis.14,15 In the former case the hypothesis is applied to the cylindrical meniscus during adsorption, while in the latter case it is applied to the hemispherical case during desorption. As indicated earlier, the family of these recently invented mesoporous materials, MCM-41, has attracted significant attention from a fundamental as well as applied perspective. They are considered as the most suitable model adsorbents currently available due to their array of uniform size pore channels (hexagonal/cylindrical pores) with negligible pore-networking or pore-blocking effects. Aside from the tunability, their pore diameter, and high surface area, these materials provide high thermal, hydrothermal, and mechanical stability, as well as ease of modification of the surface properties by incorporating heteroatoms such as Al, B, Ti, V, and Mo or anchoring organic ligands. This has also led to interest in their use as host materials for the construction of nano-structured materials by host-guest technology.16,17 In this paper we present a new model for determining the pore size distribution of microporous and mesoporous materials. The model has been tested using the adsorption isotherms on pure as well as mixtures of MCM-41 materials. These materials have an independent method of estimating the pore size from XRD and TEM, which allows comparison with theoretical results. Consequently, we have chosen these materials as reference materials to test the proposed model. The experimental data of adsorption of nitrogen at 77.4 K has been inverted using the regularization technique of Bhatia.9 The results for the PSD by the present model are compared with the pore size obtained by other models such as the nonlocal density functional theory (NLDFT)6 and the model of Broekhoff and de Boer (BdB)18 as well as methods proposed by Zhu et al.,19 Kruk et al.,20 and Lukens et al.21 In addition, comparison is made with the pore diameter obtained from X-ray diffraction using geometrical considerations. The comparison of the PSD’s of MCM-41 obtained by the present model using adsorption data of nitrogen, argon, and oxygen is also provided. The model is also applied to estimate pore size distribution of numerous novel mesoporous materials such as MCM-48, SBA’s, HMS, FSM, KIT, CMK, and several others, providing improvements over earlier published results based on Kelvin or BJH treatments of nitrogen adsorption data. The comparison of the pore size distributions of MCM-41 based on nitrogen, argon, and oxygen is particularly relevant in view of their reported unusual behavior in some cases. In particular, induced polarization has been reported to distort the adsorptive force fields for the first layer for nitrogen and argon,22 while cluster formation of oxygen in nanospaces has been reported in other instances.23 All of these effects, if prevalent, should lead to differences in estimated PSD’s, indicating the need for improved potential models. 2. Theory 2.1. Micropores. The fractional pore filling of the micropores of radius r at a given pressure P is given by the Dubinin-

Pore Size Distributions of Mesoporous Materials

J. Phys. Chem. B, Vol. 104, No. 39, 2000 9101

Radushkevich (DR) isotherm

φ˜ (t,r) +

θ(r,P) ) exp[-{RT ln(P0/P)/βE0(r)}2]

KN0Φ ) E0β

(3)

where N0 is the Avogadro’s number and K is a constant. Here this approach can be used for cylindrical micropores using the Saito-Foley potential. The mean potential, Φ, for a cylindrical micropore system is given by5

Φ)

3πN0 (NAAA-A + NEAE-E) 4

d40

∞

[ ( ) { ( ) ( ) }] 1

∑ k+1

k)0

1-

d0

21 d0 Rk 32 r

2k

r

10

- βk

d0 r

Γ(-4.5) Γ(-4.5 - k)Γ(k + 1)

(5)

βk1/2 )

Γ(-1.5) Γ(-1.5 - k)Γ(k + 1)

(6)

Thus using eqs 3-6, the values of the characteristic energy E(r) can be obtained in terms of pore diameter. 2.2. Mesopores. The adsorbate molecule in a mesopore is present in entirely different states when it is in the submonolayer and in the multilayer regions. Therefore, most prior models have attempted to describe these two parts of the isotherm using two different models independently. In the present study, two different models are conjugated to form a single hybrid isotherm to describe the entire pressure range. A smooth transition between the two models is achieved by equating the slopes and the amount adsorbed obtained by the two models at the monolayer completion. For the submonolayer region we employ the widely used Unilan model

[

]

FL [(2r - t(P/P0) - tpm)(t(P/P0) - tpm - λ)] 2r

Im )

(9)

where tpm is the location of potential minimum. A smooth transition from submonolayer to multilayer region of the isotherm is possible if the parameters of the Unilan model are obtained by matching the amounts adsorbed and the slopes obtained by the two models. Accordingly, the two eqs

(4)

Rk1/2 )

Cm 1 + bPes ln 2s 1 + bPe-s

where φ˜ (t,r) is the position-dependent incremental local potential due to the solid and λ is the interlayer spacing in the saturated adsorbate which is estimated by assuming the hexagonal close packed (hcp) arrangement of adsorbate molecules. The integral (second term) was obtained using the Benedict-Web-Rubin (BWR) equation of state. The amount adsorbed at a given pressure in a cylindrical pore is given by

4

where NA and NE are number of oxygen atoms on the surface of zeolite and molecules/atoms in adsorbent respectively, d0 is the arithmetic mean of the diameter of the adsorbent atoms and adsorbate atoms, and Rk and βk are given by

Ism(P) )

(8)

0

(2)

where R is the ideal gas constant, T is temperature, E0(r) is characteristic energy, and β is similarity coefficient. Chen and Yang24 have shown that the characteristic energy of adsorption is related to the mean potential, Φ, inside the pores by

γ Vs(r - t)

∫PPVg dP ) (r -∞ tl - λ/2)2

(7)

where Cm is the monolayer capacity and b and s are constants. There are several other models in the literature, such as the Freundlich, the Langmuir, the Temkin, the Sips, or the FowlerGuggenheim isotherms, which could be used to describe the submonolayer region.25 However, the Unilan model is chosen because, as seen in our previous article,7 this model along with the molecular continuum model in the form of a hybrid isotherm was found to be very successful in describing the adsorption isotherm of various adsorbates on nonporous materials over the entire pressure range.7 The adsorption in the multilayer region has been described by our recent molecular continuum model. Accordingly, in a mesopore of radius r, the equilibrium thickness t of the adsorbed layer at pressure P is given by26-28

∂Im ∂Ism | | ) ∂P C)Cm ∂P C)Cm

(10)

Ism|C)Cm ) Im|C)Cm

(11)

need to be solved. Substitution of eqs 7-9 in eqs 10 and 11 and rearrangement leads to the two simultaneous equations

[

]

Cmb (es - e-s) ) 2s (1 + bPe-s)(1 + bPes)

[

{

}

]

∫

P0 ∂ ( V dPg) FL(2r - 2t - λ) ∂Pg Pg (12) s 2r ∂φ˜ (r,t) Vl γ∞(r - t - λ/2) ∂t (r - t - λ/2)3

{ [ {[

b)

1 - exp

[

FL(2r - t - tpm)(t - tpm + λ) -1 rCm

Pg exp s

]

FLs(2r - t - tpm)(t - tpm + λ) rCm

]}

}

]

- exp[s]

(13)

Therefore, for a given mesopore, the amount adsorbed in the pore can be obtained by solving the above equations depending upon the pressures. Consistent with eq 2, with different adsorption regimes as shown in Figure 1, the integral can be split as

Ca(P) )

∫rr

m

min

Fm(r,P)f(r,P) +

∫rr (P)Fc(r,P)f(r,P) + ∫rr(P)Fs(r,P)f(r,P) P

m

max

(14)

P

permitting different forms of F(r,P) in each integral. Here Fc(r,P) represents the effective density for pores in which capillary condensation occurs, and Fs(r,P) that for pores in which multilayer surface coverage occurs. In the present case for the multilayer region

Fc(r,P) )

(2r - t - tpm)(t + x2/3σ - tpm)

Fl (15)

[(r - tpm)2 + x2/3{σ(r - tpm) + σ2/4}]


Sonwane and Bhatia

and for low pressure

Fs(r,P) )

2rIsm [(r - tpm) + x2/3{σ(r - tpm) + σ2/4}] 2

(16)

with the effective density in the micropores given by

Fm(r,P) ) Fl exp[-{RT ln(P0/P)/βE0(r)}2]

(17)

A key unknown in such a model is the critical micropore size below which the adsorption occurs by the volume filling mechanism. Conventionally, this critical size has been taken according to the IUPAC recommendation which is 2 nm (diameter);4 however, recent studies with adsorption in regular mesoporous materials (MCM-41 type) suggest that the actual critical pore sizes depends on numerous factors such as the adsorbate-adsorbent system and temperature. Therefore, in the present calculations, this limit has been estimated as described in our recent article.12 Based on the mechanical stability criteria for the cylindrical meniscus (during adsorption), the critical size is obtained by solving the tensile strength hypothesis

P g + τ0 g

γ∞ (r - t - λ/2)

(18)

Figure 2. Pore size distribution of MCM-41 samples by regularization and XRD using (a) nitrogen (b) argon and (c) oxygen adsorption data28 at 77.4 K. Present model, s; XRD: - - -.

TABLE 1: Lennard-Jones Parameters and Other Properties

along with the stability criterion for the cylindrical meniscus s ∂φ˜ γ∞Vl (r - t + λ/2) ) ∂t (r - t - λ/2)3

(19)

3. Results and Discussion 3.1. Pore Size Dependence of Characteristic Energy. The use of the Lennard-Jones potential for defining the characteristic energy has been successfully implemented for nitrogen-activated carbon.24 The mean potential energy for slit pores based on the 6-12 Lennard-Jones potential was found to give moderately good fit for the Dubinin-Stoeckli equation, E(r) ) X/r for the 1-2 nm range of pore size. However, as is well-known, the Dubinin-Stoeckli equation is an approximate expression. In the present work, along with this approach, we have also studied the applicability of the McEnaney expression given by29

E(r) ) arb

(20)

where a and b are constants evaluated by fitting eq 20 to the mean potential energy given by eq 4. The values of rE(r) for the cylindrical pores for the nitrogen-silica system were found to be fairly linear with an average of 2.47 nm kJ/mol and a mean error of 4.9%. These values are significantly higher than those obtained for the activated carbon-nitrogen slit pore system (average ) 1.24 nm kJ/mol, mean error ) 7.3%). For cylindrical pores in silica using eqs 3-6 and eq 20, the regression yielded a ) 1.129, b ) -1.218 with r2 ) 0.999 for nitrogen, and a ) 1.172, b ) -1.203 with r2 ) 0.999 for argon, indicating an excellent fit. The estimates of the constants are in the same range of the constants that have been reported for slit pores in the literature.29 These results suggest that the empirical approach (eq 20) proposed by McEnaney for the slit pore system (where the constants were originally estimated from SAXS and molecular probe data) is valid also for a cylindrical system, when the constants are estimated theoretically from the mean potential energy obtained by the Lennard-Jones potential. 3.2. Determination of PSD of MCM-41 Based on Nitrogen Adsorption. The model was applied to our nitrogen isotherm

σFF (Å) FF (K) σFS (Å) FS (K) P0 (mmHg) Vls (m3/K mol) γ∞ (N/m)

O2 (77.35 K)

Ar (77.35 K)

Ar (87.46 K)

N2 (77.35 K)

3.467 106.7 3.71 51.86 160 0.026531 0.015535

3.418 124 3.55 60.35 200 0.02757 0.01407

3.418 124 3.89 56.35 760 0.028728 0.011772

3.681 91.5 3.65 71.52 760 0.03468 0.00888

data for C8-C18 MCM-41 samples to extract pore size distributions (PSD’s). Inversion of the adsorption integral was accomplished using the regularization package of Bhatia.9 Figure 2a, b, and c represents the PSD results for pure C10-C18 MCM-41 samples, along with the XRD estimates of the pore size, obtained using nitrogen, oxygen, and argon adsorption data at 77.4 K.27,28 The details of the calculation of pore size from the XRD and gas adsorption data have been given elsewhere.20,28 In the present section, we discuss the nitrogen results, while those for oxygen and argon are discussed in the next. The LJ parameters and other adsorbate properties used in the present calculations for the PSD are given in Table 1. The fluidsolid interaction parameters obtained in the present work are also shown in Table 1. The fluid was represented by the BWR equation of state. The properties of the adsorbate fluid (nitrogen) were taken as the bulk saturated liquid properties. For each isotherm, the values of the small regularization parameter were varied and the standard deviation, δ, was estimated. For low values of the regularization parameter, the solution for the PSD is unstable, leading to significant variation of standard deviation. The final value was chosen in such a way that the standard deviation was constant and a stable nonnegative PSD was obtained as described earlier.9 In the present model, the lower mesopore limit for nitrogen adsorption on silica at 77.4 K was taken as 2.4 nm which was estimated from our model describing stability of the adsorbate meniscus in mesopores (by solving eqs 18 and 19). Such a limit for other adsorbates, argon and oxygen, was obtained as 2.5 nm each. The details of the calculations for this criticality have been given in our previous article. For nitrogen adsorption, the DR isotherm with characteristic energy as obtained above was applied for all the pores


Figure 3. Theoretical and experimental adsorption28 isotherms of nitrogen adsorption at 77.4 K on MCM-41 samples: (a) C18, (b) C16, (c) C14, (d) C12, (e) C10.

below 2.4 nm. Although the present calculations show that the samples do not contain micropores, the applicability of the DR equation up to 2.4 nm pore size needs to be further studied. Further, artifacts may arise at the transition between the different isotherms for the micropores and mesopores, but for the predominantly mesoporous MCM-41 this effect is not of significance. The nitrogen adsorption results indicated that the average pore size of C18 was 4.25 nm as compared to 4.36 nm obtained by XRD which shows a deviation of 0.11 nm for nitrogen adsorption studies. The pore size of other samples C16, C14, C12, and C10 obtained by the present model as 3.75, 3.36, 3.0, and 2.80 nm were very close to the XRD diameters 3.78, 3.38, 2.96, and 2.73 nm, respectively. The predicted isotherm by the present model was found to be in excellent agreement with the experimental adsorption isotherm in the case of all the samples, as shown in Figure 3. It should be noted that the good agreement of the nitrogen PSD’s with the XRD characterization in Figure 2a for the MCM-41’s obtained using C10-C18 chain length templates is due to the ordered nature of these samples. Indeed, the results for the C8 sample were much less satisfactory. In the latter case, a relatively broader pore size distribution was obtained, with a peak at about 2.55 nm, deviating from the XRD determined pore diameter of about 2.31 nm. This deviation is most likely due to the disordered nature of the C8 template based MCM-41, demonstrated in our recent articles,30,31 with inherent structural differences requiring potential parameters different from those used for the C10-C18 materials. This disordered nature was confirmed from TEM observations31 and from SAXS results which gave surface areas matching with BET values only for C8 sample and not for C10-C18 samples.30 The latter was explained on the grounds that the Debye equation used in

J. Phys. Chem. B, Vol. 104, No. 39, 2000 9103 estimating areas by analysis of the SAXS data is developed for random media. Its success only for the C8 sample is therefore an indicator of its disordered nature and the more organized nature of the C10-C18 samples. Thus, the XRD estimate of the C8-MCM-41 based on the hexagonal lattice assumption, and the same fluid-solid LJ parameters as the C10-C18 samples, may be expected to be erroneous. In view of this difference, the application to the C8 material is not discussed here. 3.3. PSD of MCM-41 Using Argon and Oxygen Adsorption. The pore size distribution of MCM-41 samples obtained using argon and oxygen adsorption data at 77.4 K28 are shown in Figure 2, b and c, respectively. The pore sizes are compared with the XRD pore sizes obtained using pore volume of the respective gases. An excellent agreement can be seen between the modal pore size obtained by present model and the XRD results in the case of argon adsorption. The pore sizes of samples C18, C16, C14, C12, and C10 were obtained as 4.26, 3.67, 3.31, 2.61, and 2.38 nm, respectively. Only in the case of sample C10, a small deviation (0.06 nm) was observed between the model predictions and the XRD diameter. In contrast to the argon results, the pore sizes obtained from the current model for oxygen adsorption had small deviations from the XRD diameters. The pore sizes of C18, C16, C14, C12, and C10 samples by the present model were 4.16, 3.56, 3.18, 2.88, 2.49 nm as compared to 4.28, 3.52, 3.11, 2.75, and 2.42 nm, respectively. Although the difference appears significant in Figure 2c, in reality the deviations for each sample are only 0.12, 0.04, 0.07, 0.13, and 0.07 nm, respectively. The predicted isotherms by the present model were found to be in good agreement with the experimental adsorption isotherms (for argon as well as oxygen), as depicted in Figure 3 for nitrogen. Although the peaks of the argon-based pore size distribution match the XRD-determined pore sizes exceptionally well, some differences in the distribution with those from nitrogen are apparent. Similar differences have been reported earlier by other workers,32 based on NLDFT calculations, indicating that the discrepancy is not related to isotherm choice. In this regard, the observations of Olivier et al.,22 for carbons, that induced polarization in the first layer of argon and nitrogen affects the isotherms, may be relevant to this case as well. While it is not obvious whether this effect is important for adsorption on MCM41 as well, it is clear that the presence of the discrepancy with the NLDFT isotherms suggests the deviation to be related to differences in the fluid-solid interaction. This may arise either due to the above effect or due to differences in crystallinity among samples produced in different batches. Thus, further studies directed along these lines are needed to resolve this issue. For oxygen, significant deviations from the XRD-based diameter is seen from the PSD’s, and in this case may be due to its unusual behavior in nanospaces, reported recently by Kaneko et al.,23 or due to sample differences discussed above. These authors measured the magnetic susceptibility of oxygen in the nanospaces of the activated carbon fibers and deduced the presence of oxygen cluster-dominated states in oblition to the gaseous and liquidlike condensed states. Such states precede capillary condensation and may be due to orientational ordering on the pore walls seen in GCMC simulations.33 If they occur in wider pores of MCM-41 as well then deviations of the LJ potential based PSD from the true XRD-based diameter are to be expected. Further studies to elucidate the behavior of oxygen in these pores are therefore needed to clarify this issue. 3.4. PSD of MCM-41 Mixtures Using Nitrogen Adsorption. Although the PSD of MCM-41 shows a distribution (with


Sonwane and Bhatia

Figure 4. Pore size distribution of mixtures of MCM-41: (a) C12 + C18, (b) C10 + C14, (c) C12 + C16. Results obtained from adsorption on mixture, s; average of parents samples, -‚-‚-; XRD, - - -. Isotherm data used from Sonwane and Bhatia.12

Figure 5. Theoretical and experimental12 adsorption isotherms for nitrogen adsorption at 77.4 K on mixtures of MCM-41: (a) C12 + C18, (b) C10 + C14, (c) C12 + C16.

a half peak width of 0.2-0.3 nm), ideally, they may be expected to have an almost uniform pore size. The observed distribution obtained by gas adsorption could be either due to limitations of the technique, heterogeneity of the walls, or less crystallinity of the sample. Here, in an attempt to create an artificial distribution of the pores over a broader range of pore size, different samples of MCM-41 were mixed and characterized using nitrogen adsorption. The mixtures of MCM-41 were prepared by physical mixing of C10 and C14 samples, and C12 and C16 as well as C12 and C18 samples12 in approximately equal proportion,12 and it is expected that they will show peaks close to the parent sample. This was indeed confirmed by our results shown in Figure 4, which provides the mixture PSD’s in each case computed using the present model based on the nitrogen adsorption data published earlier.12 It can be seen from Figure 4a that the computed pore size distribution of the mixture shows two peaks for C12 + C18 at 3.04 and 4.20 nm which are very close to the values of 2.96 and 4.36 nm obtained by the XRD method for the parent samples. The expected mixture pore size distribution obtained by averaging the individual pure component PSD’s of the C12 and C18 samples is also shown in Figure 4a, and it can be seen that the peaks are at 3.0 and 4.25 nm. These are very close to the pore sizes computed by present model and also by the XRD method. Similar results were obtained for the samples C10 + C14 and C12 + C16 and the results are shown in Figure 4b,c. For C10 + C14, the peaks obtained by the current method were at 2.7 and 3.36 nm respectively as compared to 2.73 and 3.38 nm obtained by XRD. The pore size distribution obtained by averaging the individual pure component PSD’s C10 and C14 is also shown in Figure 4b, and it can be seen that the peaks are at 2.71 and 3.36 nm, respectively, very close to the pore size by the present model and also by XRD. However, the expected pore size distribution near the first peak (at about 2.71 nm) was broader than that actually obtained, possibly reflecting greater crystallinity in the C10 sample used, compared to that synthesized in the earlier work28 in which the pure component isotherms were measured. Alternately, the difference could also be due to the presence of fewer data points in the appropriate region. For the sample C12 + C16, the peaks computed by the

present model were found to be at 3.04 and 3.72 nm, very close to 2.96 and 3.78 nm obtained by XRD. Also, the pore sizes obtained by averaging of the pure component PSD’s were at 3.04 and 3.75 nm, close to the XRD results as shown in Figure 4c. Interestingly, the predicted isotherm by the present model was found to be in excellent agreement with the experimental adsorption isotherm in case of all the samples as shown in Figure 5. Therefore, it can be observed that present model provides a consistent way of analyzing the real pore structure of the system. 3.5. Comparison with Other Models. As discussed earlier, the development of MCM-41 materials has catalyzed much activity in the area of characterization of mesoporous solids, and several newer modified approaches have been proposed. Here we have studied several such models and compared their results for MCM-41. The models examined are those of Broekhoff and de Boer,18 Zhu et al.,19 Kruk et al.,20 and Lukens et al.21 as well as the NLDFT.6 For comparison of the results on a common basis, the sample AM-1 of Neimark et al.6 has been chosen, for which the nitrogen adsorption data and the NLDFT results for the PSD were provided by them. It should be noted here that the NLDFT method uses the desorption branch of the isotherm while our method uses the adsorption branch. Figure 6 depicts the sample PSD as determined from the various methods. These are discussed below. The drawbacks of one of the popular models, the BroekhoffdeBoer model, have already been discussed in our previous articles.26,27 The model fails to incorporate the pore size dependent incremental potential as well as the effect of pore size on surface tension. It was also found that the model does not predict the criticalities for capillary condensation and for hysteresis present in the isotherms.12 However, many such classical methods are popular because they are embodied in the software supplied along with commercial adsorption equipment. As can be seen from Figure 6, the BdB method underpredicts the pore diameter of MCM-41 by 7.7 Å. Recently, the BdB approach has been revisited and modified by Lukens et al.21 They proposed a simplified BdB method which uses Hill’s approximation for the thickness of the adsorbed gas layer2 in the classical BdB equation, and found that it simplifies the model to



ln(P/P0) ) A -

B t2

(25)

where A ) c1/RT and B ) c3/RT. The constants in eq 25 were obtained by fitting the equation to the experimental data of adsorption of nitrogen on MCM-41. Although eq 24 appears simple and easy to use, it has not been tested for the thermodynamic stability criterion. The stability criterion for the interface is given by

d2(∆G) dN2

g0

(26)

Using the equality sign in the above equation along with

N ) πL[r2 - (r - t)2]

(27)

provides for the capillary condensation pore radius Figure 6. Pore size distribution of MCM-41 sample AM-16 by various models.

fγVsl d(∆G) R ) ln(P0/P) - 3 dN RT(r - t) t

(21)

where f is 1 for a cylindrical meniscus and 2 for a hemispherical meniscus. In estimating the pore size using the adsorption branch of the isotherm, eq 21 is solved along with the stability criterion

d2(∆G) dN2

fγVsl 3R ) 4 )0 t RT(r - t)2

(22)

using f ) 1 for the cylindrical meniscus. For desorption, f ) 2, and a different criterion is obtained, but is not used in the present study where only the adsorption branch is used. The above approach is a simplified form of BroekhoffdeBoer18 equation, given by

ln(P0/P) - F(t) )

γVsl RT(r - t)

(23)

in which F(t) was given a complex empirical form. It was proposed that the method was useful for materials without welldefined pore structure. It was successfully applied for estimation of the PSD of MCM-41 and SBA-15 materials.21 However, the method uses an approximated form of the BdB equation and suffers from the same deficiencies discussed above. As seen from Figure 6, this approach has been found to significantly underpredict the pore diameter for MCM-41 sample AM-1 (by 8.9 Å). Recently, a new method has been proposed by Zhu et al.19 and applied for estimation of the PSD of MCM-41 materials. According to the method, the thickness of the adsorbate layer in a cylindrical mesopore at a given pressure Pg is given by

c2 c3 d(∆G) γVl ) RT ln(P0/P) + c1 + n - 2 ) 0 (24) dN r - tp r t p

where R is the universal gas constant, c1, c2, c3, and n are constants, while r and tp are pore radius and thickness, respectively. This approach considers the effect of curvature on the thickness of the layer and it reduces to the established Harkins and Jura equation for thickness of adsorbed layer on nonporous materials when R f ∞, given by

r ) tP + 0.27tP3/2

(28)

in which reported values of c1, c2, and c319 have been used. Equations 25 and 28 were then solved simultaneously, and the results obtained are plotted in Figure 7. It can be seen that the relative pressure at the criticality for a given pore radius is larger than unity when the pore radius increases beyond 7.2 nm, which is unrealistic. Although it is possible that this may not be related to the reasons for underprediction of pore diameter shown in Figure 6, it is likely that the values of the constants c1-c3 need modification, and that approach needs further refinement. The modal pore size predicted by the method of Zhu et al.19 was found to underpredict the actual XRD pore size of MCM-41 by about 10.9 Å as seen from Figure 6. The approach proposed by Kruk et al.20 for the estimation of pore size uses a corrected form of Kelvin equation given by

r)

2γVsl RT ln[(P0/Pg)]

+t+c

(29)

where c is a constant which has a value of 0.3 nm. The value of constant c has been proposed empirically based on the observation that pore diameter obtained by the original Kelvin equation (with the statistical thickness, t) was found to underpredict the pore diameter of MCM-41 by 0.3 nm. The method was found to be quite successful for predicting the pore diameter of MCM-41 materials in the range of 2-6.5 nm. When this method was applied for sample AM-1 in the present study, it was found that it predicts the pore diameter with an accuracy of about 2 Å as shown in Figure 6. One of the important developments in this area is the application of NLDFT for determination of PSD of MCM-41.6 It was reported that NLDFT gives consistent results with respect to PSD and wall thickness. The comparison of the PSD of AM-1 obtained6 by NLDFT with the XRD pore diameter, as shown in Figure 6, clearly indicates that it is in excellent agreement. Also shown in the figure are the results based on the present model, indicating excellent agreement with the XRD pore size and the NLDFT-based PSD. In our interpretation of literature data, and of our own data discussed earlier, small differences among the samples and therefore of adsorption results under similar conditions become apparent. Therefore, additional comparisons were made with the NLDFT results for other reported nitrogen and argon adsorption data32 at 77.4 K on MCM-41, as depicted in Figure 8. The inset shows the comparison of the PSD by the present


Sonwane and Bhatia

Figure 7. Variation of capillary condensation pressure with pore radius for the model of Zhu et al.19 Figure 9. Theoretical and experimental adsorption isotherm and PSD obtained using argon adsorption data34 at 87 K on MCM-41 samples (a) sample 2.8, (b) sample 3.1, (c) sample 3.6, (d) sample 3.9. The inset shows the comparison of PSD by present model (solid line), modified BJH20 (dashed line), and XRD (vertical line).

Figure 8. Theoretical and experimental adsorption isotherm and PSD obtained using data32 for argon adsorption at 77.4 K on MCM-41 samples: (a) C16, (b) C14, as well as data32 for nitrogen adsorption at 77.4 K on (c) C16 and (d) C14. The inset shows the comparison of PSD by present model (solid line), NLDFT (dashed line) and XRD (vertical line).

model and from the NLDFT as reported by Ravikovitch et al.,32 as well as XRD pore diameters. It can be seen that for nitrogen as well as argon adsorption at 77.4 K, the predictions of amounts adsorbed are very close to the experimental data. For nitrogen adsorption, the modal pore sizes by the present model were found to be 3.41 and 3.15 nm, as compared to 3.35 and 3.05 nm obtained by XRD for MCM-41 samples C16 and C14, respectively, indicating a close resemblance. However, the NLDFT indicated pore sizes of 3.62 and 3.4 nm respectively, which show slightly larger deviations from the XRD diameter. Similarly, for argon adsorption at 77.4 K results, the pore size for C16 and C14 samples by the present model was 3.08 and 2.85 nm as compared to XRD pore diameter 3.31 and 2.9 nm, respectively. It can be seen that there is a deviation of 0.23 nm for C16 but the pore sizes of C14 by XRD and the present model are very close (deviation ) 0.05 nm). The corresponding results by NLDFT show a pore size of 3.6 and 3.26 nm which show a deviation of 0.29 and 0.36 nm, only slightly larger than those

obtained with the present model. Therefore, it can be seen that the present model is found to perform comparably to the NLDFT. Comparison of the results for AM1 and C16 as well as C14 samples indicates differences of NLDFT in predicting the pore diameter due to changes in potential parameters used.6 The predicted pore diameters by NLDFT were almost matching with the XRD diameter for AM1 sample but some deviations were found for C16 and C14 samples. Accurate data for argon adsorption at 87 K is also available in the literature,34 and was used here for further evaluation of the approach. However, it is to be noted that the parameter values for the fluid-solid interaction in our model were slightly adjusted for argon on their sample at 87 K, as seen in Table 1, possibly because of sample-to-sample differences. Alternately, it is also possible that the induced polarization in the first layer, referred to earlier, is affected by temperature. Figure 9 depicts the fitted isotherms and PSD’s determined. The solid curves depict our PSD results, while the dashed curves depict those of Kruk and Jaroniec34 using their modified BJH approach,20 and the vertical line the XRD diameter. In general, good agreement with the XRD size is seen for both methods, indicating their suitability. For their modified BJH method, given in eq 29, the constant c was chosen34 as 0.438 for argon, indicating that some degree of parameter estimation is needed. This parameter value, once chosen, may be used for argon adsorption at 87 K on other MCM-41 samples as well, but its suitability at other temperatures is uncertain. Nevertheless, the method is simple and readily usable. Based on the current evaluations, it would therefore appear that the present model and the NLDFT have the advantage of a thermodynamic basis, with the NLDFT being the most rigorous, while the approach of Kruk et al.16,34 has the feature of simplicity and accuracy close to that of these two. 3.6. PSD of Various Mesoporous Materials. After the development of M41S type materials, a new era in the field of mesoporous materials was initiated. Several new mesoporous materials have since been synthesized which have well-defined pore structure. Some of these materials include MCM-41, MCM48, SBA-n, FSM, HMS, CMK, and MSU, among a host of others. Nevertheless, MCM-41 has been the only material that



Figure 10. Pore size distributions of various mesoporous materials. Curves with solid lines represents results of present calculations, while those having dashed lines the reported PSD based on BJH or HK analysis. The solid vertical line (when reported) represents the XRD diameter.

has attracted attention from researchers from the theoretical standpoint, as a result of its nonintersecting pore structure with a uniform array of cylindrical pores. This renders MCM-41 an ideal material for the testing of fundamentals of physicochemical processes such as phase transitions and gaseous diffusion, and of the structure of molecules in the adsorbed state. The mesoporous materials also have potential application in polymerization for the synthesis of nanowires and immobilizing enzymes, ligands, complexes, and chemically active colloids and nanoparticles. As a result of the ensuing surge in the research interest in these materials, it becomes necessary to get a clear picture of the structure of these materials. Here, an attempt has been made to estimate the PSD of various mesoporous materials synthesized in the past few years using the present model. In most of these cases the results obtained by the present theory are compared with those reported by the

authors in their original papers. Our results for the different materials are depicted in Figure 10 and are based on the nitrogen isotherms provided by the authors referenced. Table 2 lists the reported pore size and structure, and our estimated modal pore size. Generally, the originally reported pore size is based on the use of the Kelvin equation or the BJH method in conjunction with nitrogen adsorption on a commercial apparatus, and may be expected to be inaccurate. FSM-16 are folded sheet materials synthesized by intercalation of surfactant cations into the bilayers of kanemite via an ion exchange process.35 As the ion exchange proceeds, the interlayer cross-linking occurs by condensation of silanol groups. The alkyl chain length of the surfactant used is 16. The pore diameter of these materials was found to be 3.85 nm as shown in Figure 10 as compared to the reported pore diameter of 2.2 nm by the BJH method, and 3.75 nm by XRD.


Sonwane and Bhatia

TABLE 2: Pore Sizes of Various Mesoporous Materials mean pore size (nm) material CMK-1

structure

3-D regular mesopores, disordered carbon framework DMS disordered, narrow and bimodal PSD FSM hexagonal, highly ordered, wall made from double SiO4 layers HMS hexagonal, disordered, uniform pore size, cross-linked framework KIT fully disordered 3-D network wormlike channels, uniform pore size MAMCM highly ordered mesoporous hybrid material MCM-48 cubic, 3-D, highly ordered, unit cell parameter: 8.3 nm meso-fly ash highly ordered hexagonal, contains inpurities MSF optically transparent, highly ordered, 2-D, uniform size MTS space between monodispersed particles gives mesopores SBA-1 highly ordered 3-D cubic, two type of cages SBA-11 highly ordered 3-D cubic, unit cell par, a ) 10.6 nm SBA-2 3-D symmetric, hexagonal SBA-8 2-D highly ordered hexagonal, centered rectangular symmetry UOFMN hexagonal, less ordered, highly uniform and narrow pore size

ref reported

by present model

47

3.01

3.84

39

2.56

3.84

35

2.20

3.85

45

2.80

3.50

41

3.52

3.89

48

3.17

2.95

46

3.49

3.50

37

2.74

3.51

42

1.60

2.81

44

3.42

3.64

46

3.05

3.05

38

2.41

3.60

36 43

2.22 1.87

3.41 2.98

40

2.40

3.40

SBA-2 is a novel regular caged mesoporous silicate material and has a 3D symmetric structure.36 It is synthesized by divalent quaternary ammonium surfactants. The average pore size (whose basis is not mentioned, though it is expected that it has been obtained by BJH or HK analysis) was reported as 2.22 nm which is much lower than 3.41 nm obtained by the present model. The latter estimate of the diameter should be used for future use. Mesoporous hexagonal highly ordered aluminosilicate material is obtained from coal combustion byproduct fly ash.37 However, the material contains abundant impurities in the form of nonporous particles and fibrils. The pore size obtained by the present model was 3.51 nm as against 2.74 nm (BJH) reported. As the lattice spacing is of the order of 4.1 nm, the wall thickness is about 0.6 nm. SBA-11 is a highly ordered 3-D cubic mesoporous silica structure material38 which was found to have a pore diameter of 3.6 nm as against 2.41 nm (BJH) reported as shown in Figure 10. The pore size distribution of other mesoporous materials is shown in Figure 10 and Table 2. It can be seen that in most of the cases the pore diameter by the present model is higher than that reported in the literature. The reported distribution is obtained by one of the several classical methods such as BJH, HK, and BdB or their modifications. Therefore in all of the cases the distribution obtained by the present model is expected to be more accurate. Figure 11 depicts the isotherm data and model fits for the different materials examined. Double mesopore silica (DMS) is a disordered structure with a very narrow bimodal PSD synthesized rapidly at room

temperature using aqueous ammonia as a catalyst.39 Its XRD pattern is similar to that of HMS and MSU. The sample was reported to have first peak of pore size in the range of 2.6-3.0 nm, with a second at about 19 nm based on an unspecified classical method. The pore size of the first peak obtained by the present model was 3.84 nm. Unified hybrid inorganic/organic frameworks (UOFMN) are hexagonal mesoporous materials consisting of organosilicate framework with covalently bonded silica and ethane/ethylene groups.40 These materials have thicker walls and higher thermal stability and are less ordered as compared to MCM-41. The pore size was found to be 3.4 nm by the present model, as compared to the reported 2.4 nm based on the BJH method. KIT41 are disordered mesoporous silica materials prepared using polymerization of silicate anions surrounding surfactant micelle in the presence of organic salts. These materials have 3-D disordered channels branching similar to a true fractal and they have high thermal stability. The pore diameter obtained by the present model was 3.89 nm as compared to the reported estimate of 3.52 nm based on HK method. Mesoporous silica films (MSF) are oriented optically transparent silica films which are crack-free up to centimeters in size and 0.5 mm in thickness synthesized by a sol-gel process.42 They have a uniform and ordered mesopore structure. The pore diameter obtained by the present model was found to be 2.81 nm as compared to the reported value 1.6 nm based on BJH method. SBA-8 are silicate mesoporous materials prepared by boloform surfactant.43 They have a 2-D pore structure with an ordered mesopore structure. The pore diameter obtained by the present model was 2.98 nm as compared to the reported value 1.87 nm. Micelle-templated structures (MTS) are a family of mesoporous materials synthesized using nonionic ethoxylated sorbitan esters.44 They possess a wormhole-like framework with intraparticle pore size in the range of 3-4 nm and have monodispersed particles with size in the range of 30-60 nm. These are supposed to be useful for diffusional limited reactions because of their small particle size as compared to M41S materials. The pore diameter obtained by the present model was 3.64 nm as compared to the reported value of 3.42 nm obtained by the BdB model. Hexagonal mesoporous materials (HMS) are hexagonal, 2-D structures with narrow pore size distribution and thicker walls.45 They have small crystallite domains and are less ordered as compared to MCM-41. They are prepared by a neutral templating method using primary amines as templates. The pore size of HMS was found to be 3.5 nm by the present model as compared to the reported value of 2.8 nm by the HK method. SBA-1 are cubic 3-D ordered mesoporous materials having two types of globular cages forming a continuous structure. The pore size was found to be 3.05 nm by the present model, in precise agreement with reported value of 3.05 nm46 obtained using the model of Kruk et al.20 MCM-48 are highly ordered 3-D cubic mesoporous materials. The pore size was found to be 3.5 nm as compared to the reported value of 3.49 nm46 obtained using the model of Kruk et al.20 CMK are 3-D ordered carbon mesoporous molecular sieves obtained using MCM-48 as template.47 They are obtained by carbonizing glucose in pores of MCM-48 and then by removing the silica framework by dissolving. The pore size by the present model was found to be 3.81 nm as compared to the reported



Figure 11. Theoretical and experimental adsorption isotherms of nitrogen at 77.4 K on various mesoporous materials.

3.01 nm obtained using the BJH method. In this case as well as for MA-MCM discussed below, we used the carbon-nitrogen interaction paramters reported earlier.7 MA-MCM are highly ordered functionalized mesoporous sieves with methacrylate groups in the host channel system.48 The pore size was found to 2.95 nm as compared to the reported value of 3.17 nm using an unspecified method. 4. Conclusion A new model for determining the pore size distribution of micro- and mesoporous materials from the gas adsorption isotherm has been proposed and successfully tested against the mesoporous MCM-41 reference materials. The model was found to be successful in predicting the pore size distribution of pure as well as binary physical mixtures of MCM-41, and the results were found to be in agreement with the pore size obtained from XRD using geometrical considerations. The model was also utilized for determining pore size distribution of MCM-41 samples using argon and oxygen adsorption at 77.4 K, with deviations from the nitrogen PSD suggestive of sample differ-

ences, or of the need for more refined interaction models in these cases. The fitted isotherms were in excellent agreement with the experimental adsorption isotherms. It was found that some other recently proposed methods, as well as the classical BdB method,18 underpredicted the pore diameter of MCM-41 materials. The method of prediction currently proposed, NLDFT, and the empirical correction20,34 of the BJH method were found to be closest to the experimental value of the pore size obtained by XRD. Acknowledgment. The authors acknowledge Mr. Russell Williams for his help in the computations. The award of University of Queensland Graduate Research Scholarship to C.G.S. is gratefully acknowledged. This research has been supported by a grant from the Australian Research Council. References and Notes (1) Kresge, C. T.; Leonowicz, M. E.; Roth, W. J.; Vartuli, J. C.; Beck, J. S. Nature 1992, 359, 710. (2) Gregg, S. J.; Sing, K. S. Adsorption, Surface Area and Porosity, 2nd ed.; Academic Press: New York, 1982. (3) Kaneko, K. J. Membr. Sci. 1994, 96, 59.

9110 J. Phys. Chem. B, Vol. 104, No. 39, 2000 (4) Rouquerol, J.; Avnir, D.; Fairbridge, C. W.; Evertt, D. H.; Haynes, J. H.; Pernicone, N.; Ramsay, J. D.; Sing, K. S. W.; Unger, K. K. Pure Appl. Chem. 1994, 66, 1739. (5) Saito, A.; Foley, H. C. AIChE J. 1991, 37, 429. (6) Neimark, A. V.; Ravikovitch, P. I.; Unger, K. K. J. Colloid Interface Sci. 1998, 207, 159. (7) Sonwane, C. G.; Bhatia, S. K. Sep. Purif. Tech. 2000, 20, 25. (8) Horvath, G.; Kawazoe, K. J. Chem. Eng. Jpn. 1983, 16, 470. (9) Bhatia, S. K. Chem. Eng. Sci. 1998, 53, 3239. (10) Inoue, S.; Hanzawa, Y.; Kaneko, K. Langmuir 1998, 14, 3079. (11) Morishige, K.; Shikimi, M. J. Chem. Phys. 1998, 108, 7821. (12) Sonwane, C. G.; Bhatia, S. K. Langmuir 1999, 15, 5347. (13) Maddox, M. W.; Olivier, J. P.; Gubbins, K. E. Langmuir 1997, 13, 1737. (14) Kadlec, O.; Dubinin, M. M. J. Colloid Interface Sci. 1969, 31, 479. (15) Burgess, C. G. V.; Everett, D. H. J. Colloid Interface Sci. 1970, 33, 611. (16) Kruk, M.; Jaroniec, M.; Sayari, A. J. Phys. Chem. B 1997, 101, 583. (17) Sayari, A.; Yang, Y.; Kruk, M.; Jaroniec, M. J. Phys. Chem. B 1999, 103, 3651. (18) Broekhoff, J. C. P.; De Boer, J. H. J. Catal. 1967, 9, 8. (19) Zhu, H. Y.; Lu, G. Q.; Zhao, X. S. J. Phys. Chem. B. 1998, 102, 7371. (20) Kruk, M.; Jaroniec, M.; Sayari, A. Langmuir 1997, 13, 6267. (21) Lukens, W. W., Jr.; Schmidt-Winkel, P.; Zhao, D.; Feng, J.; Stucky, G. D. Langmuir 1999, 15, 5403. (22) Olivier, J. P.; Conklin, W. B.; Szombathely, M. V. In Characterization of Porous Solids III; Rouquerol, J., Rodriguez-Reinoso, F., Sing, K. S. W., Unger, K. K., F., Eds.; Elsevier: Amsterdam, 1994. (23) Kanoh, H.; Zamma, A.; Setoyama, N.; Hanzawa, Y.; Kaneko, K. Langmuir 1997, 13, 1047. (24) Chen, S. G.; Yang, R. T. Langmuir 1994, 10, 4244. (25) Do, D. D. Adsorption Analysis: Equilibria and Kinetics; Series on Chemical Engineering, Vol. 2; Imperial College Press: London, 1998. (26) Bhatia, S. K.; Sonwane, C. G. Langmuir 1998, 14, 1521.

Sonwane and Bhatia (27) Sonwane, C. G.; Bhatia, S. K. Chem. Eng. Sci. 1998, 53, 3143. (28) Sonwane, C. G.; Bhatia, S. K.; Calos, N. Ind. Eng. Chem. Res. 1998, 37, 2271. (29) McEnaney, B. Carbon 1987, 25, 69. (30) Sonwane, C. G.; Bhatia, S. K. Langmuir 1999, 15, 2809. (31) Sonwane, C. G.; Bhatia, S. K.; Calos, N. Langmuir 1999, 15, 4603. (32) Ravikovitch, P. I.; Wei, D.; Chueh, W. T.; Haller, G. L.; Neimark, A. V. J. Phys. Chem. B. 1997, 101, 2671. (33) Sokolowski, S. Mol. Phys. 1992, 75, 999. (34) Kruk, M.; Jaroniec, M. Chem. Mater. 2000, 12, 222. (35) Inagaki, S.; Koiwai, A.; Suzuki, N.; Fukushima, Y.; Kuroda, K. Bull. Chem. Soc. Jpn. 1996, 69, 1449. (36) Huo, Q.; Margolese, D. I.; Stucky, G. D. Chem. Mater. 1996, 8, 1147. (37) Chang, H.-L.; Chun, C.-M.; Aksay, I. A.; Shih, W.-H. Ind. Eng. Chem. Res. 1999, 38, 973. (38) Zhao, D.; Huo, Q.; Feng, J.; Chmelka, B. F.; Stucky, G. D. J. Am. Chem. Soc. 1998, 120, 6024. (39) Wang, X.; Dou, T.; Xiao, Y. Chem. Commun. 1998, 1035. (40) Melde, B. J.; Holland, B. T.; Blanford, C. F.; Stein, A. Chem. Mater. 1999, 11, 3302. (41) Ryoo, R.; Kim, J. M.; Ko, C. H.; Shin, C. H. J. Phys. Chem. 1996, 100, 17718. (42) Ryoo, R.; Ko, C. H.; Cho, S. J.; Kim, J. M. J. Phys. Chem. B 1997, 101, 10610. (43) Zhao, D.; Huo, Q.; Feng, J.; Kim, J.; Han, Y.; Stucky, G. D. Chem. Mater. 1999, 11, 2668. (44) Prouzet, E.; Cot, F.; Nabias, G.; Larbot, A.; Kooyman, P.; Pinnavaia, T. J. Chem. Mater. 1999, 11, 1498. (45) Zhang, W.; Pauly, T. R.; Pinnavaia, T. J. Chem. Mater. 1997, 9, 2491. (46) Kruk, M.; Jaroniec, M.; Ryoo, R.; Kim, J. M. Chem. Mater. 1999, 11, 2568. (47) Ryoo, R.; Joo, S. H.; Jun, S. J. Phys. Chem. B 1999, 103, 7743. (48) Moller, K.; Bein, T.; Fischer, R. X. Chem. Mater. 1999, 11, 665.

Characterization of Pore Size Distributions of Mesoporous Materials

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