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11146

Langmuir 2005, 21, 11146-11160

Rigid Sphere Molecular Model Enables an Assessment of the Pore Curvature Effect upon Realistic Evaluations of Surface Areas of Mesoporous and Microporous Materials Constantinos E. Salmas and George P. Androutsopoulos* School of Chemical Engineering, Chemical Process Engineering Laboratory, National Technical University of Athens, 9 Heroon Polytechniou Street, GR 15 780 Athens, Greece Received April 1, 2005. In Final Form: August 12, 2005 A gas adsorption rigid spheres model (RSM) was incorporated into the CPSM model (corrugated pore structure model) to correlate the pore surface areas obtained from the BET and CPSM methods. The latter is a method simulating the gas sorption hysteresis loop and enables the evaluation of surface areas SCPSM through the integration of the pertinent pore size distributions. Thus, SCPSM values are inherently influenced by pore curvature. The new CPSM-RSM version estimates surface areas SCPSMfs that are independent of pore curvature and can be compared with the pertinent SBET values. The RSM exploits the fact that a curved pore surface accommodates fewer molecules, assumed to behave as rigid spheres, than an equal flat one. Thus, the RSM accounts for a higher molecular surface coverage Ac (nm2/molec.) in pores with marked curvature than that (i.e., Af) on a flat surface. The ratio Ac/Af for nitrogen adsorbed on single pore sizes varies in the range Ac/Af ) 1.44-1.03 for pore sizes D ) 1.5-15 nm, respectively. Also for D ) 1.5-5.0 nm the SCPSMfs and SBET values are lower by ∼10-45% than the SCPSM estimates. From the application of the CPSM-RSM model on several porous materials exhibiting all known types of sorption hysteresis loops, it was confirmed that SBET ≈ SCPSMfs ((5%) and (SCPSM - SBET)/SBET ) 3-68% for the materials examined. In conclusion, the BET method may produce quite conservative surface area estimates for materials exhibiting pore structures with appreciable pore curvature, whereas the CPSM-RSM model can reliably predict both SCPSM and SCPSMfs ) SBET values.

1. Introduction An early attempt to compute surface areas from pore size distribution (psd) data and check for concordance with the pertinent BET surface areas was reported by Wheeler in 1946.1-3 He proposed eq 1 to compute theoretical desorption isotherms by assuming a pore structure composed of cylindrical pores of distributed radius, thus

Vt - V a )

∫R∞ π(R - t)2 dR c

(1)

Vt is total specific pore volume, Va is volume occupied by adsorbed gas and condensed liquid, Rc is the pore radius currently considered, L(R) is the pore length distribution with respect to pore radius, and t is multilayer thickness of adsorbed gas. The latter is computed from a Halsey type correlation (eq 2)

t (Å) ) 4.3 (5/ln(P0/P))1/3

(2)

Equation 1 was applied for L(R) being of Gaussian or Maxwellian form, and the computed isotherms were compared with experimental nitrogen desorption isotherms. From the satisfactory matching of experimental with theoretical data, results were obtained for evaluating the actual shape of the pore volume and surface area distributions of the test materials. The use of the foregoing type of psd required the knowledge of the “average” pore radius rj ) 2Vg/Sg being connected to the most probable * Corresponding author. Tel. (+30) 210 772 3225. Fax: (+30) 210 772 3155. E-mail address: [email protected]. (1) Schull, C. G. J. Am. Chem. Soc. 1948, 70, 1405-1410. (2) Barrett, L. P.; Joyner, L. G.; Halenda, P. P. J. Am. Chem. Soc. 1951, 73, 373-380. (3) Wheeler, A. Reaction Rates and Selectivity in Catalyst Pores. Catalysis; Reinohold: New York, 1955; Vol. II.

radius R0. Therefore, the evaluation of the psd is based on the knowledge of the BET surface area. For internal consistency, this value should be comparable with the one deduced via the integration of the respective psd. The widespread method proposed by Barrett-JoynerHalenda2 performs a numerical stepwise integration of eq 1 starting from the high-pressure end of the isotherm and contains a fair number of approximations. As quoted in the original publication of the BJH method: “the procedure was devised not for the purpose of computing surface areas, but for the determination of the distributions of the areas and the volumes among pores of varying radii”. Any difference between surface area SBJH calculated from the psd and SBET was attributed to the fact that the SBET is computed from the low-pressure part of the isotherm, whereas estimation of SBJH involves primarily the highpressure portion. It was also suggested that the enforcing of internal consistency, i.e., SBJH ) SBET, requires adjustment of the parameter c ) (rjp - tr)/rjp where rjp and tr represent the Kelvin radius and the multilayer thickness at the mean P/P0 of each pressure decrement included in the BJH calculation procedure. Moreover, it should be noted that the Wheeler and BJH methods do not consider pore structure networking effects that contribute considerably to the appearance of gas sorption hysteresis. These and other similar methods of psd evaluation from gas sorption data employ either desorption or the adsorption data independently. As a result, the psds deduced by using gas adsorption data are usually different from those obtained when desorption data are used instead. This difference may be quite marked especially in cases of wide hysteresis loops. From the foregoing discussion, it becomes apparent that meaningful comparisons of surface areas deduced via the psd integration and the BET method need to be based on a systematic simulation of the entire gas sorption hys-

10.1021/la0508644 CCC: $30.25 © 2005 American Chemical Society Published on Web 10/22/2005

Rigid Sphere Molecular Model

teresis loop and the investigation of the pore curvature effect upon surface area evaluation since in the BET theory gas adsorption is considered to occur on a “flat surface”. To this end, in the present paper, the corrugated pore structure model (CPSM)4,5 combined with the gas adsorption rigid spheres molecular model will be employed for quantifying the effect of pore curvature on surface area evaluation by carrying out sensible comparisons of SCPSM with SBET surface areas. A brief review of the basic aspects of the CPSM model is presented below. 2. Reviewing Methods for Pore Size Distribution and Surface Area Evaluation 2.1. Corrugated Pore Structure Model (CPSM). The CPSM model is a unified theory enabling the simulation of nitrogen sorption, mercury porosimetry hysteresis phenomena, the evaluation of intrinsic pore size distributions (psd), specific pore surface areas (ssa), and the prediction of pore structure tortuosity factors. (1) CPSM-Nitrogen. This is a probabilistic model simulating the gas sorption hysteresis phenomena, because it evaluates a single psd by curve fitting the entire sorption hysteresis loop and is based on a statistical corrugated pore structure configuration.4,5 The latter is envisaged to be composed of a sequence of Ns cylindrical pore segments of distributed diameter and constant length. Ns, is defined as the statistical nominal pore length (or frequency of corrugated pore cross sectional area variation), which accounts for pore structure networking effects. Generally, pore networking is considered to be an important factor inducing hysteresis in gas sorption measurements.6-11 It is obvious that in the case of Ns ) 2, the CPSM configuration reduces to the conventional model of “bundle of cylindrical pores distributed in size” and indicates the absence of pore structure networking effects. The development of the CPSM-nitrogen model is based on a number of assumptions that together with the related correlations are presented in Appendix IV. Numerous applications covering the entire spectrum of hysteresis loop types according to the IUPAC classification have been reported.5,12-19 It should be emphasized however, that the CPSM-nitrogen formulation is a simulation (4) Androutsopoulos, G. P.; Salmas, C. E. Ind. Eng. Chem. Res. 2000, Part I, 39, 3747-3763. (5) Androutsopoulos, G. P.; Salmas, C. E. Ind. Eng. Chem. Res. 2000, Part II, 39, 3764-3777. (6) Seaton, N. A. Chem. Eng. Sci. 1991, 46, 1895-1909. (7) Liu, H.; Zhang, L.; Seaton, N. A. Chem. Eng. Sci. 1992, 47, 43934404. (8) Liu, H.; Zhang, L.; Seaton, N. A. J. Colloid Interface Sci. 1993, 156, 285-293. (9) Liu, H.; Zhang, L.; Seaton, N. A. Langmuir 1993, 9, 2576-2582. (10) Liu, H.; Seaton, N. A. Chem. Eng. Sci. 1994, 49, 1869-1878. (11) Mason, G. Proc. R. Soc.: London 1988, A415, 453-486. (12) Salmas, C. E.; Androutsopoulos, G. P. Ind. Eng. Chem. Res. 2001, 40, 721-730. (13) Salmas, C. E.; Androutsopoulos, G. P. Appl. Catal A: General 2001, 210, 329-338. (14) Salmas, C. E.; Stathopoulos, V. N.; Pomonis, P. J.; Rahiala, H.; Rosenholm, J. B.; Androutsopoulos, G. P. Appl. Catal. A: General 2001, 216, 23-39. (15) Salmas, C. E.; Tsetsekou, A. H.; Hatzilyberis, K. S.; Androutsopoulos, G. P. Drying Technol. 2001, 19(1) 35-64. (16) Salmas, C. E.; Stathopoulos, V. N.; Ladavos A. K.; Pomonis, P. J.; Androutsopoulos, G. P. Proceedings of the 6th Symposium on the Characterization of Porous Solids (COPS-VI) Alicante, Spain; Rondriguez-Reinoso, F., McEnaney, B., Rouquerol, J., Unger, K., Eds.; Elsevier: Amsterdam, The Netherlands, 2002; Vol. 144, pp 27-34. (17) Salmas, C. E.; Stathopoulos, V. N.; Pomonis, P. J.; Androutsopoulos, G. P. Langmuir 2002, 18, 423-432. (18) Salmas, C.; Ladavos, A.; Skaribas, S.; Pomonis, P.; Androutsopoulos, G. Langmuir 2003, 19, 8777-8786. (19) Armatas, G. S.; Salmas, C. E.; Louloudi, M.; Androutsopoulos, G. P.; Pomonis, P. J. Langmuir 2003, 19, 3128-3136.

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model, for gas physisorption, based on the adoption of a simplified idealized pore structure model that allows for pore networking and capillary condensation metastability effects. However, the pore structure “finger prints” of a specified material are traced through the CPSM fit of the corresponding gas sorption hysteresis data and the evaluation of the CPSM parameters, e.g., m (Halsey eq IV-1), cos θ (Kelvin eq IV-2), Ns. Typical values of CPSM parameters to reproduce the four types of gas sorption hysteresis loops under the IUPAC classification were reported.4 (2) CPSM-Tortuosity. The CPSM-tortuosity model enables realistic predictions of pore structure tortuosity factors in satisfactory agreement with relevant literature data20 and consists of an empirical correlation that is based on CPSM-nitrogen predictions of intrinsic pore size distribution and the nominal pore length Ns (Appendix IV). Details on the formulation and applications of the CPSM-tortuosity theory are reported.12 (3) CPSM-Microporosity. The CPSM model can be effectively employed for determining the micropore volume via the integration of the intrinsic pore volume distribution in the range D ) Dmin - 2 nm. CPSM microporosity detection results together with comparisons with relevant microporosity data from Rs plots have been reported in various citations.16,18 (4) CPSM-Surface Area. The ssa deduced from the CPSM analysis of nitrogen sorption data represents the cumulative surface area calculated from the integration of the relevant surface area differential (psad) over the detected pore size range.4 Similarly micropore surface areas can be computed by integrating the CPSM surface area distribution over the pore size range Dmin - D()2 nm).18 (5) CPSM-Mercury. The corrugated pore configuration has been also used to formulate mathematical relationships simulating mercury porosimetry (MP) hysteresis observations.21 Similar correlations have been developed simulating MP hysteresis loop scanning data and enabling the definition of pore structure tomography concepts.22 CPSM-mercury correlations are provided in Appendix V. (6) CPSM-Contact Angle Hysteresis. This model represents a combined application of the CPSM-nitrogen and CPSM-mercury models for simulating experimental hysteresis data obtained from both experimental techniques applied on samples of the same material. MP hysteresis is attributed to both pore structure networking and contact angle hystetresis between the mercury penetration and retraction branches of the pertinent hysteresis loop. Overall psds extending over the macromeso-micropore range are derived. This model enables the testing of the CPSM model as an integrated theory through the comparison of the psds yielded by both CPSM methods for a specified porous material and over a common pore size range. Applications of the CPSM-Contact Angle Hysteresis model are reported.23 (7) Intrinsic Pore Size Distribution. A family of bell shape distribution functions (BSD) was chosen as the intrinsic pore number distribution fr (br, D). The analytical form of the normalized F(D) in its general form (i.e., composed of (20) Satterfield, C. N. Mass Transfer in Heterogeneous Catalysis; M. I. T. Press, London, 1970. (21) Androutsopoulos, G. P.; Salmas, C. E. Chem. Eng. Commun. 1999, 176, 1-42. (22) Androutsopoulos, G. P.; Salmas, C. E. Chem. Eng. Commun. 2000, Part I, 181 137-177; Part II, 181, 179-202. (23) Salmas, C. E.; Androutsopoulos, G. P. J. Colloid Interface Sci. 2001, 239, 178-189.

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n unimodal psds) is provided in Appendix VI. The choice of the BSD distribution function facilitates the application of the CPSM model for simulating gas sorption and mercury porosimetry hysteresis since the mathematical form of BSD enables the analytical integration of the integrals involved in both CPSM models.4-5,21-22 2.2. Surface Area Evaluation Methods. 2.2.1. Methods Based on the Flat Surface Assumption. (1) BETSurface Area Variants. The Brunauer-Emmett-Teller (BET) method, proposed in the pioneer work of Brunauer et al.,24 has been, over several decades, the dominant method25 for ssa determination, despite the simplified assumptions that it involves and the ambiguity related with the choice of the linear part of the BET plot. Two BET variants were proposed in Brunauer et al.24 A finite number (n) of adsorbed gas monolayers was assumed in the first variant, whereas its linearized counterpart for n f ∞ is the well-known model that finds widespread use over several decades. The Langmuir model for restricted adsorption (BET monolayer model24) is valid for narrow mesopores (i.e., approaching the limit of micropores ca. 2 nm) or wide micropores (i.e., 1.0-2.0 nm). The above-mentioned models correlate the amount (na, µmol/g) of adsorbed gas with the relative pressure P/P0 and enable the evaluation of the monolayer capacity (nw, µmol/g) as well as the characteristic constant C. The surface area is computed from eq 3

ssa(m2/g) )

nw N A × 1014 MW A f

(3)

where MW (N2) ) 28, NA) 6.06 × 1023 (Avogadro number) and Af ) 0.162 nm2 (N2 molecular occupation area on a flat pore surface). For pure physical adsorption, the acceptable range of variation of the C constant is ca. 20 < C d, Figure 1. 3.2. Gas Monolayer Adsorption on Pore Surfaces of Varying Curvature. The calculation of the number of molecules (Nc) forming a close hexagonal monolayer packing along the circumference of a cylindrical pore cross section proceeds as follows. A single molecule occupies a section of the cylindrical surface, and (CE) is the corresponding increment on the circumference (Figure 1). The number Nc of the molecules of the adsorptive required to form a monolayer that fully occupies the circumference of the pore cross section, can be computed from eq 6

Nc ) π/sin-1(d/(D - d))

(6)

where sin(d/(D - d)) is expressed in rad. The derivation of eq 6 is based on the geometric features of Figure 1, as follows:

sin Rˆ ) (AB)/(OA) w sin Rˆ ) (d/2)/(D/2 - d/2)

(7)

-1

w Rˆ ) sin (d/(D - d)) and ω ˆ ) 2Rˆ w ω ˆ ) 2 sin-1(d/(D - d)) (8) ˆ w Nc ) w Nc ) 2π/ω π/sin-1(d/(D - d)) (9) By taking account of a dense hexagonal arrangement of the adsorbed gas molecules, the area Ac on the concave side of the cylindrical pore being occupied by a single gas molecule is equal to the sum of the areas Aell of an ellipse (lightly shaded area of Figure 2) with axis a ) d κRι b ) D sin-1(d/(D - d)), (i.e., from Figure 1, b ) (CE) and (CE) ) πD/(π/sin-1(d/(D - d)), proof in Appendix I) and the interstitial area Aintr enclosed between the ellipse under consideration and the three neighboring ellipses (Figure 2, dark gray area)). Thus, Ac ) Aell + Aintr or according to eq I-5 of Appendix I, we obtain eq 10

Ac ) 0.87ab

molecules will be: Sc ) NcAc or from eqs 6 and 11

w Sc ) 0.87πdD

(12)

3.3. Molecular Diameter of Adsorptive Gas. The formulation of the rigid spheres molecular model requires the knowledge of a gas molecule size, i.e., a nominal molecule diameter and a molecular surface area coverage or area of occupation (Af) on an assumed flat and smooth solid surface. Af is usually obtained from eq 13, assuming a hexagonal close packing of spherical molecules [ref 32, p 170, eq 6.11, and ref 24]

Af ) f(M/FL)2/3

(13)

where M is the molar mass of the adsorptive, L is the Avogadro constant, F is the absolute density of the bulk liquid adsorptive at the operational temperature, and f is a packing factor taken to be f ) 1.091 for hexagonal closepacking. Thus, for nitrogen M ) 28, F ) 0.808 cm3/g, L) 6.023 × 1023, and f ) 1.091, we calculate Af ) 0.162 nm2. Supposing that the surface area (M/FL)2/3 ) 0.149 nm2 is circular, then πd2/4 ) 0.149 nm2, and hence, the nitrogen molecular diameter is d ) 0.436 nm. Other literature citations,3,34,35 suggest the value d ) 0.43 nm, for a hexagonal close-packing of the adsorbed molecules. This value according to Lippens et al.34 is related with an adsorbed gas monolayer thickness equal to n ) 0.35 nm and is included in both the conventional Halsey36 and the modified Halsey correlation,4,5 eq 14

t(nm) ) n[5/ln(P0/P)]1/3(P/P0)m

(14)

where n ) 0.35 nm and m is an adjustable parameter. The value n ) 0.35 nm has been verified experimentally.26 Based on plain geometry considerations, an assumed hexagonal close packing of N2 molecules adsorbed on a hypothetical flat solid surface and taking d ) 0.43 nm, the values Af ) 0.1602 nm2 and n ) 0.37 nm were calculated (Appendix I). The value of Af ) 0.1602 nm2 is practically identical with the pertinent values reported by Emmett and Brunauer,37 ca. Af ) 0.162 nm2 and Rhodin,38 ca. Af ) 0.16 nm2. The geometric monolayer

(10)

Following substitution for a and b, we derive eq 11

Ac ) 0.87dD sin-1(d/(D - d))

Figure 2. View of a dense hexagonal arrangement of equal size ellipses, with axis (a) and (b). The molecular area coverage on a cylindrical surface Ac is represented by the sum of the two shaded areas, i.e., Aell (light gray) and Aintr (dark gray).

(11)

In cylindrical pores of size D, the area covered by Nc gas

(34) Lippens, B. C.; Linsen, B. G.; De Boer, J. H. J. Catal. 1964, 3, 32-37. (35) Schull, C. G.; Elkin P. B.; Roess, L. C. J. Am. Chem. Soc. 1948, 70, 1410-1414. (36) Halsey, G. D. J. Chem. Phys. 1948, 16 (10), 931-937. (37) Emmett, P. H.; Brunauer, S. J. Am. Chem. Soc. 1937, 59, 15531564. (38) Rhodin, T. N. J. Am. Chem. Soc. 1950, 72, 5691-5695.

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Table 1. Pore Surface Molecular Occupation Area Ac vs Pore Sizea

pore size range wide micropores

narrow mesopores medium size mesopores wide mesopores

a

pore diameter, D (nm)

Ac/Afb

Ac (nm)

(Ac - Af)/Af (%)

1.0 1.2 1.5 2.0 2.2 2.5 3.0 5.0 10.0 15.0 30.0 50.0 100.0

1.988 1.654 1.443 1.290 1.256 1.217 1.173 1.096 1.045 1.030 1.015 1.009 1.004

0.318 0.265 0.231 0.207 0.201 0.195 0.188 0.176 0.167 0.165 0.163 0.162 0.161

98.8 65.4 44.3 29.0 25.6 21.7 17.3 9.6 4.5 3.0 1.5 0.9 0.4

Given Af ) 0.1602 nm2 and d ) 0.43 nm for N2. b Ac/Af ) D - d))/d, from eq 18.

sin-1(d/(D

thickness n ) 0.37 nm approaches the experimentally observed value n ) 0.35 nm and deviates substantially from the molecular diameter d)0.43 nm. In general, the rigid spheres model can be applied for any set of Af and d values one might consider appropriate. However, in the present study, the CPSM-RSM was applied for Af ) 0.1602 nm2 and d ) 0.43 nm. 3.4. Relating Molecular Occupation Areas on Flat and Curved Surfaces. The ratio Ac/Af of the pore surface areas occupied by a single adsorbed molecule is given in eqs 15 and 16

Ac/Af ) [0.87dD sin-1(d/(D - d))]/0.87d2

(15)

Ac/Af ) D sin-1(d/(D - d))/d

(16)

From the application of eq 16 for single pore sizes, the results listed in Table 1 are deduced. It is readily seen from Table 1 that the surface area of wide micropores may exceed that obtained by methods relying on the assumption of flat surface area by ∼30-65%. Appreciable deviations of 20-30% may be also observed for narrow mesopores (i.e., D ) 2.2-3.0 nm), whereas for medium (i.e., D ) 5-15 nm) and wide (i.e., 40-100 nm) mesopores, sizes the deviations of surface areas due to pore wall curvature fall below 10%. A more representative picture of the relative deviation, (Ac - Af)/Af %, of surface area estimates due to pore wall curvature, is presented in Figure 3. It should be emphasized that the application of eq 16 should be made with care when D < 2 nm and not for D/d < 3 since for finer micropores abnormally low molecular packing densities may occur.39 3.5. Pore Surface Area Evaluation from Nitrogen Sorption Data using the CPSM-Rigid Spheres (CPSM-RSM) and the BET Methods. Pore surface area distributions (psads) can be computed by means of the CPSM model through the curve fitting of gas sorption hysteresis loop data. Total surface areas, SCPSM, can be obtained through the integration of the pertinent psads over the range Dmin - Dmax. This evaluation includes inherently the pore curvature effect. It would be very interesting to calculate surface areas SCPSMfs that are independent of the pore curvature effect. The latter requirement can be satisfied in the present study through the application of the rigid spheres molecular model (39) Carrott, P. J. M.; Roberts, R. A.; Sing, K. S. W. Chem. Ind. 1987, 855-856.

Figure 3. Percent deviation of curved pore surface areas compared with the corresponding flat surface area estimates.

(CPSM-RSM) for gas adsorption and the comparison of the SCPSMfs results with the respective SCPSM and the SBET values. Computation of the SCPSMfs over the pore sizes Dmin-D can be carried out using eqs 17 and 18

∫DD

SCPSMfs(D) ) C1

min

Af DF(D) dD Ac

(17)

or by substitution for Af/AC according to eq 16, we obtain

∫DD

SCPSMfs(D) ) C1 d

min

DF(D) dD D sin-1(d/(D - d))

(18)

max

D where C1 ) 4Vsp/∫D D2F(D) dD, F(D) is the intrinsic min CPSM pore number distribution function. The derivation of eqs 17 and 18 is presented in Appendix II. The distribution F(D) is a probability density function and can assume the analytical form of any statistical distribution truncated in the range Dmin - Dmax. In various applications of the CPSM model4,5,18 F(D) was taken to be a mono-parametric distribution of BSD-type as described in Appendix VI. The differential form of the pore surface area distribution for the CPSM-flat surface version, ca. psad ≡ FS-CPSMfs can be deduced from eq 19

FS-CPSMfs(D) ≡

C1dDF(D)dD dSCPSMfs(D) ) (19) dD D sin-1(d/(D - d))

4. Applications of the Rigid Spheres Model for Characterizing Various Porous Materials (Results and Discussion) 4.1. Materials. The CPSM-RSM molecular model was applied for characterizing several porous materials exhibiting different pore structure properties. Emphasis was placed in relating the pore size distribution with the pore wall curvature effect on surface area determination. The selected materials are listed in Table 2 and have gas sorption hysteresis loops covering the entire series of loop type included in the IUPAC classification as well as gas sorption features typical of the novel mesoporous materials MCM.40 4.2. Nitrogen Sorption Hysteresis Loop Data. The nitrogen sorption (77.4 K) hysteresis loops for the materials under investigation are presented in Figure 4. The continuous lines through the data points represent the result of fitting the CPSM model. It is apparent from

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Table 2. Materials Designation and the Corresponding Type of the Gas Sorption Hysteresis Loop sample no.a

material

hysteresis loop type

sample no.

material

hysteresis loop type

1 2 3 4 5 6 7 8 9

anodic alumina vycor glass Comox 451b Harshaw 400Eb ICI-41-6b ICI61-1b G-51-15b G-51-30b lignite

H-1 H-2 H-2 H-(2-3) H-(2-3) H-(2-3) H-(2-3) H-(2-3) H-3

10 11 12 13 14 15 16 17 18

montmorillonite pillared clay-A pillared clay-B MCM-41 (Rav.) MCM-39 MCM-123 MCM-128 MCM-133 MCM-134

H-3 H-4 H-4 c c c c c c

a Literature data as follows: sample nos. 1, 3-12,5 sample no. 2,12 sample no. 13,41 and sample nos. 14-18.14 b Hydrodesulfurisation catalysts (Co, Mo oxides on γ-Al2O3 substrate). c Gas sorption characteristic of novel mesoporous materials of regular structure (MCM).

Figure 4 that the CPSM fit of the sorption data is quite successful, all types of hysteresis loops according to the IUPAC classification are represented and gas sorption data typical of the novel mesoporous materials MCM are included. The CPSM fitting parameters being used in the simulation of the experimental nitrogen sorption hysteresis loop data of Figure 4 are listed in Table 3. The simulation of the gas sorption data in most of the cases required the combination of three unimodal psds of BSDtype. A single Kelvin parameter, i.e., cos θc ) cos θh, is employed for the majority of materials being examined. Thermodynamic hysteresis, i.e., cos θc * cos θh, is considered in the simulation of hysteresis exhibiting by the most of the MCM materials. The Halsey parameter (m) is being varied in the range m ) 0.07-0.17 (except materials nos. 11 and 16). The nominal pore length parameter Ns influences the width of the hysteresis loop when Ns > 2, whereas Ns ) 2 indicates the absence of hysteresis. Generally Ns varies in the range Ns ) 2-10.4,5,14 However. the simulation of H-type hysteresis loops require higher values, e.g., Ns ) 60 for Vycor Glass (Table 3). Big Ns values are associated with high tortuosity factors τCPSM (eq IV-6). 4.3. Pore Volume Distributions. The pore volume distributions (psvd) of the materials examined here (Figure 5) were deduced from the CPSM simulation of the hysteresis loops shown in Figure 4. Some of the materials exhibit bimodal psd (e.g., material nos. 3, 5, 9, 10, 12, 16, and 18 in Table 2), and a few of them (e.g., material nos. 11, 12, 14, 16, and 18 in Table 2) possess a small percentage of microporosity being determined via the integration of the CPSM surface area psd over the micropore size range. Pore surface area psds (not shown here) were reproduced from the respective pore volume distributions for cylindrical pore geometry, which is one of the main assumptions of the CPSM model. Specific surface areas, SCPSM, were calculated by integrating the pertinent pore surface area psads over the maximum detected pore size range and the results are provided in Table 4. 4.4. Surface Area Evaluations. The specific surface areas of Table 4 were determined by five different methods using the nitrogen sorption data of Figure 4. BET values were obtained from the conventional BET linear plots over the range: P/P0 ) 0.05-0.20. The I-point method employs an alternative procedure of determining the monolayer capacity and hence the corresponding ssa, SI-point. Surface area SCPSMfs data calculated from the CPSM-RSM model demonstrate how the surface area estimates are reduced (40) Beck, J. S.; Vartuli, J. C.; Roth, W. J.; Leonowitcz, M. E.; Kresge, C. T.; Scmitt, K. D.; Chu, C. T-W.; Olson, D. H.; Sheppard, E. W.; McCullen, S. B.; Higgins, J. B.; Schlenker, J. L. J. Am. Chem. Soc. 1992, 114, 10834-10843. (41) Ravikovitch, P. I.; Haller, G. L.; Neimark, A. V. Adv. Colloid Interface Sci. 1998, 76-77, 203-226.

when the pore curvature effect is neglected. Percent relative deviation data are listed in Table 4 for SCPSM and SCPSMfs vs SBET, SCPSM vs SCPSMfs, SI-point vs SBET, SRo(co) vs SBET, and SRo(ev) vs SBET. Comparing SCPSM and SCPSMfs vs SBET. The ssa comparison plots of Figure 6 were constructed by using data extracted from Table 4 and depict the difference between SCPSM and SBET which is quite big for the MCM and pillared clay materials and moderate for the HDS catalysts and the rest (miscellaneous) materials examined in the present study. Figure 6a illustrates the potential of the CPSMRSM for evaluating the pertinent SCPSMfs in quite satisfactory agreement with the corresponding SBET data (i.e., all open circle points fall on the diagonal of Figure 6a). Magnified comparison plots for materials with medium (i.e., ssa > 400 m2/g) or low (i.e., ssa > 70 m2/g) ssas are those shown in Figure 6b,c. It is evident from the latter plots that SCPSMfs deviate from the SBET values within the (5% margin. However, the best performance of the CPSM-RSM model concerns the MCM materials (i.e., Figure 6a). This result may be attributed to the fact that MCM materials possess a regular pore structure composed of cylindrical independent pores with more or less smooth pore walls. It should be reminded that such pore structure characteristics have been incorporated in the formulation of the CPSM model. It seems that the absolute deviation %|(SCPSMfs - SBET)/SBET| tend to increase from 0 to 1.5% (e.g., MCM materials) to 4-5% (e.g., some HDS catalysts) when the tortuosity factor τCPSM increases from 2 to 4 (e.g., MCM materials) to 9-10 (e.g., some HDS catalysts; Table 4). Comparing SI-point vs SBET. Furthermore, a quite close agreement is evidenced between the SI-point and SBET values valid for the HDS catalysts, the pillared clay samples, and the rest of the low ssa materials (Figure 7a). Nevertheless, SI-point surface areas for the novel mesoporous materials MCM are substantially higher than the corresponding SBET data (i.e., relative deviation outside the (5% limit: % (SI-point - SBET)/SBET) (-7.4)-(+23). A similar observation is also quoted in the original work by Pomonis et al.27 At the present time, it is not clear how and why the I-point method predicts substantially higher ssa than the BET method for materials exhibiting narrow mesopores and/or micropores. However, the present work provides additional information supporting the fact that the I-point method can be reliably used for determining ssa of mesoporous materials having mean pore size D > 5 nm. Comparing SRo vs SBET. The application of the Roberts method, being a reliable representative of the approximate methods for computing psd from gas sorption data, produced ssas with considerable deviation from the relevant SBET values (Figure 7b). It is readily seen in Figure 7b that both SRo(co) and SRo(ev) for the MCM materials, the MCM-41 Rav being an exception, are lower than the SBET

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Salmas and Androutsopoulos

Figure 4. Nitrogen sorption hysteresis loops (77.4 K). (b) Adsorption, (O) desorption data. Continuous line CPSM fitting result.

data and clearly outside the 5% limit. Moreover, it is worth noting from Figure 7b that for the HDS catalyst and the pillared clay materials the following relation holds: SRo(ev)>SBET > SRo(co). Percent deviations for the latter category of materials are as follows: %(SRo(co) - SBET)/SBET ) (+2.2)-(-90.5) and %(SRo(ev) - SBET)/SBET ) (+28.8)-

(-65.4). These results are examples supporting the view that the conventional methods of psd calculation from gas sorption data are not adequate for accurate and reliable ssa evaluations. 4.5. Correlating Pore Surface Curvature with Mean Pore Size. The effect of pore surface curvature on

Rigid Sphere Molecular Model

Langmuir, Vol. 21, No. 24, 2005 11153

Table 3. CPSM Parameters for the Fitting of Nitrogen Sorption Hysteresis Data F(D) distribution characterization parameters (BSD- type)a sample no. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Vg,max

c

Ns

b1

b2

b3

b4

0.237 4.3 -18 29.64 60.0 -200 304.20 5.0 -70 -15 -1 308.10 4.3 -100 -1 394.00 5.7 -85 -10 -2 344.00 8.7 -90 -25 -10 404.00 3.4 -150 -45 -2 328.40 5.0 -350 -130 12.876 3.3 -5400 -600 -100 48.00 2.45 -1470 -30 -1 107.4 4.7 -1100 -200 120.00 6.60 -11000 -300 949.40 2.05 -1700 -200 -5 472.64 2.3 -1200 -100 -15 525.26 2.5 -1400 -150 -25 346.88 2.3 -140 -10 -2 462.64 2.2 -2100 -150 -15 430.20 2.15 -11000 -1000 -170 -30

w1

w2

w3

1 1 20 10.0 10.0 20 1.0 12 1.5 1.0 13 1.0 1.0 15 10.0 6.0 13 10.0 35 650 350.0 1.5 1.0 25.0 1.0 1.0 1.2 2.27 6.5 1.0 4.0 40 4.0 1.0 20 1.0 1.0 8 1.0 0.8 9 1.0 1.0 9 3.0 5.0

w4 Pce/P0

2

0.500 0.460 0.400 0.320 0.440 0.550 0.400 0.390 0.250 0.480 0.44 0.450 0.235 0.235 0.290 0.250 0.310 0.290

Kelvin parameters.

t (nm)b

Pma/P0

cos θc

cos θh

m

0.97656 0.990 0.965 0.990 0.990 0.990 0.9912 0.995 0.9996 0.9800 0.999475 0.99946 0.994 0.991 0.990 0.962 0.990 0.999

1.00 0.33 0.65 0.84 0.70 0.70 0.60 0.80 0.45 0.38 0.085 0.08 0.85 0.68 0.75 0.17 0.69 0.62

1.00 0.33 0.65 0.84 0.70 0.70 0.60 0.80 0.45 0.38 0.085 0.08 0.85 0.70 0.80 0.17 0.72 0.65

0.07 0.09 0.09 0.08 0.17 0.18 0.13 0.11 0.10 0.10 0.00 0.12 0.14 0.14 0.14 0.25 0.15 0.13

BSD: bell shape distribution function (Appendix VI). Parameters of the modified Halsey equation (eq IV-1). n ) 0.35 nm, λ1 ) 1/3, (excepting sample no. 1, λ1 ) 1/5). c Nominal pore length (frequency of corrugated pore cross sectional area variation4). a

b

surface area evaluation depends on pore size. This dependence on a single pore size and a single molecule coverage scale has already been examined and the results are presented in Table 1 and graphically illustrated in Figure 3. It is more important though to correlate the pore curvature effect on the evaluation of ssa as a function of pore size distribution of real porous materials. The pore curvature effect upon ssa estimation being expressed as the percent deviation of SCPSM from SCPSMfs (i.e., %(SCPSM - SCPSMfs)/SCPSMfs) is plotted versus the mean pore size DmN (nm) (Figure 8, data in Table 4). The mean value DmN is calculated from the pore population psd. Definitions and relations for evaluating mean pore sizes are presented in Appendix III. The function of surface area deviations due to pore curvature versus the mean pore size, plotted in Figure 8, is identical to that of Figure 3. It is apparent from Figure 8 that the pore wall curvature has an appreciable impact on ssa estimates for pore structures with mean pore sizes DmN < 10 nm. Obviously, this impact increases exponentially when the decreasing DmN values approach the micropore size range. 4.6. Comparing Pore Size Distributions Deduced from the CPSM and CPSMfs Models. The CPSMRSM model apart from evaluating overall surface areas that involve or not the pore curvature effect, i.e., SCPSM and SCPSMfs ) SBET, enables the calculation of FS-BET(D) distributions through its equivalent distribution FS-CPSMfs(D) deduced from eq 21. The graphical comparison of FS-CPSM(D) ) dSCPSM/dD and FS-CPSMfs(D) ) dSCPSMfs/dD psds for six representative materials are portrayed in Figure 9. The area difference of psd pairs in each plot of Figure 9 represents the SCPSM - SCPSMfs deviation. Relative differences %(SCPSM - SCPSMfs)/SCPSMfs), corresponding to the data of Figure 9, range between 4.7% (e.g., anodic alumina) and 66.5% (e.g., MCM-128; Table 4).

investigation of the pore wall curvature effect on surface area determinations. The simulation of the gas sorption hysteresis loop by means of the corrugated pore structure model (CPSM) is an appropriate method for evaluating intrinsic pore surface area distributions and hence the overall specific surface area that inherently includes the pore wall curvature effect. The CPSM-RSM was derived from the CPSM model by incorporating a gas adsorption rigid spheres molecular model and enables ssa evaluations through CPSM simulation of the hysteresis loop by either considering or neglecting pore curvature. In the latter case, the CPSMRSM predicts FS-CPSMfs (D) (psds) and SCPSMfs (ssa) in quite satisfactory agreement with the corresponding SBET ssas ((5%). This deviation margin is considerably lower (i.e.,