Evaluation of Microporosity, Pore Tortuosity, and Connectivity of

Langmuir 2003, 19, 8777-8786

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Evaluation of Microporosity, Pore Tortuosity, and Connectivity of Montmorillonite Solids Pillared with LaNiOx Binary Oxide. A Combined Application of the CPSM Model, the rs-Plot Method and a Pore Percolation-Connectivity Model C. E. Salmas,† A. K. Ladavos,‡ S. P. Skaribas,‡ P. J. Pomonis,‡ and G. P. Androutsopoulos*,† School of Chemical Engineering, Chemical Process Engineering Laboratory, National Technical University of Athens, GR 157 80, Athens, Greece, and Department of Chemistry, Laboratory of Industrial Chemistry, University of Ioannina, GR 451 00 Ioannina, Greece Received May 27, 2003. In Final Form: August 26, 2003 The present work deals with the comparative application of the CPSM (corrugated pore structure model), the Rs plot and the pore connectivity methods for pore structure analysis and especially the detection of microporosity of a category of montmorillonite clays pillared with LaNiOx binary oxide. A quite good agreement between the corresponding results deduced from the three methods was confirmed. The raising of the LaNiOx content caused a change of pore volume, surface area, and microporosity whereas the relevant curves pass through a maximum. The CPSM model, through the simulation of gas sorption hysteresis, enables the evaluation of unified mesomicropore size distributions, pore surface areas, tortuosity factors, τCPSM, and relative microporosities. It is also concluded that pore tortuosity, (τCPSM), is inversely proportional to the pore connectivity factor (c), and the nominal pore length (Ns) (i.e., a CPSM parameter) is proportional to the number of pore lengths (L) (i.e., a parameter of the pore connectivity model).

1. Introduction The development of separation and reaction processes utilizing porous membranes and catalysts of high activity and selectivity, requires well characterized porous materials exhibiting pore structures made up of micro-, meso-, and perhaps macropore sizes. The presence of microporosity constitutes the most crucial factor for the characterization of porous materials due to the adsorbentadsorbate pore-molecule size interactions, which control the adsorption mechanism. A quite detailed description of a three-stage micropore filling mechanism based on a study of ACF (activated carbon fibers was reported1). According to this mechanism primary pore filling occurs (i.e., formation of a bilayer) at very low P/P0 (i.e. < 0.005) in ultramicropores of 0.7 nm wide (i.e., two nitrogen molecular diameters in width). A second micropore filling process concerns monolayer formation in supermicropores of size 1.1 and 1.4 nm at P/P0 approximately in the range 0.005-0.04. In the third stage of micropore filling, termed cooperative process, nitrogen molecules fill the gaps between the micropore walls that are coated by N2 atoms. Cooperative adsorption takes place in supermicropores and wide micropores, at P/P0 > 0.04.2 Type I isotherms showing a sharp adsorption knee at very low relative pressure, being followed by a horizontal plateau up to P/P0 ) 1, without hysteresis, are characteristic of purely microporous structures typical of, e.g., large crystals of molecular sieve zeolites, ACF, microporous * To whom correspondence should be addressed. Telephone: (+3010) 772 3225. Fax: (+3010) 772 3155. E-mail: androuts@ chemeng.ntua.gr. † National Technical University of Athens. ‡ University of Ioannina. (1) Kakei, K.; Ozeki, S.; Suzuki, T.; Kaneko, K. J. Chem. Soc., Faraday Trans. 1990, 86, 371. (2) Sing, K. S. W. Carbon 1989, 27, 5.

silicas. Composite isotherms, characteristic of micromesoporous solids, exhibit a sloped straight-line segment instead of a horizontal plateau, following the initial steep gas uptake at very low pressures. Composite isotherms are indicative of the presence of wide micropores and/or mesopores while a hysteresis loop may also appear. Such isotherms are, for example, typical of a family of pillared clay materials. Conventionally, the characterization of composite isotherms requires the application of a combination of methods to deduce the PSD extending over the micro- and mesopore range. Thus, the BJH (BarrettJoyner-Halenda3) method to evaluate the PSD in the mesopore range and one of the DR (Dubinin-Radushkevitch4), DA (Dubinin-Astakhov5), DS (Dubinin-Stoeckli6), and HK (Horva´th-Kawazoe7) as well as the BP (Mikhail et al.8) methods are routinely used for the evaluation of the micropore capacity and the pore size distribution (PSD) over a relatively narrow range of micropores, (i.e., the method of PSD evaluation in the micropore region). However, unified micro-mesopore distributions are deduced via the application of various density functional theory (DFT) models.9,10 DFT models, are useful tools for pore size distribution determinations, but they are based on single pore systems of simplified (3) Barrett, L. P.; Joyner, L. G.; Halenda, P. P. J. Am. Chem. Soc. 1951, 73, 373. (4) Dubinin, M. M.; Radushkevich, L. V. Proc. Acad. Sci. USSR 1947, 55, 327. (5) Dubinin, M. M.; Astakhov, V. A. Adv. Chem. Ser. 1970, 102, 69. (6) Dubinin, M. M.; Stoeckli, H. F. J. Colloid Interface Sci. 1980, 75, 34. (7) Horva´th, G.; Kawazoe, K. J. Chem. Eng. of Japan 1983, 16, 470. (8) Mikhail, R. Sh.; Brunauer, S.; Bodor, E. E. J. Colloid Interface Sci. 1968, 26, 45. (9) Ravikovitch, P. I.; Haller, G. L.; Neimark, A. V. Adv. Colloid Interface Sci. 1998, 76-77, 203. (10) Olivier, J. P.; Occelli, M. L. Microporous Mesoporous Mater. 2003, 57, 291.

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geometry and do not allow for pore blocking due to networking effects or other form of diffusion control (ref 11, p 234). Micropore volume and surface area can be estimated by using the empirical Rs-plot method. In an Rs plot the amount n (µmol) adsorbed, is plotted against the reduced standard adsorption Rs ) (n/nx)s. The latter is plotted in the reduced form (n/nx)s vs P/P0. The normalization factor nx is taken to be the amount of adsorbed gas at P/P0 ) 0.4, of a reference nonporous material. Normally, Rs plots are composed of two straight-line sections. The low pressure (high slope) section is distorted in the case of enhanced adsorbent-adsorbate interactions in micropores, corresponds to monolayer adsorption on the walls of supermicropores and if extrapolated goes through the axis origin, indicating the presence of supermicropores only.2 The extrapolation of the low slope straight-line segment of the Rs plot gives the specific micropore capacity np(mic) as the intercept on the n axis. Pillared clays are a category of materials exhibiting a mixed micro-mesopore structure appropriate for a comparative application of methods like the Rs and CPSM (corrugated pore structure model12,13). Such an application enables the determination of pore structure properties, e.g., PSD, pore tortuosity, and connectivity. It is the scope of the present work to investigate the potential of the CPSM model to simulate composite gas sorption isotherms exhibiting hysteresis and evaluate the PSD covering both the micro- and mesopore size range. Furthermore, the aim is to calculate micropore volume and surface areas through the integration of the pertinent PSDs over the micropore range and compare the results with those obtained from the relevant Rs plots. The tests will involve a family of montmorillonite clays pillared with NiLaO3 and exhibiting a wide sorption hysteresis loop whose size depends on the contained amount of pillaring agent. A similar comparative study on the evaluation of microporosity of silica gels and clays intercalated with Al1-xFexOy oxidic species was reported in ref 14. An additional important aim of the present work is to compare the performance of the CPSM model12,13 against that of a more realistic pore structure connectivity model involving pore structure percolation approximations.15-19 Hence, the relation between pore structure tortuosity (i.e., a CPSM parameter20) and pore connectivity (c),21 is investigated. 2. Theoretical Approach 2.1. Corrugated Pore Structure Model (CPSM). The CPSM model is a unified theory enabling the simulation of nitrogen sorption and mercury porosimetry hysteresis phenomena as well as the prediction of pore structure tortuosity factors. (11) Rouquerol, F.; Rouquerol, J.; Sing, K. Adsorption by powders and porous solids; Academic Press: London, 1999. (12) Androutsopoulos, G. P.; Salmas, C. E. Ind. Eng. Chem. Res. 2000, 39, 3747. (13) Androutsopoulos, G. P.; Salmas, C. E. Ind. Eng. Chem. Res. 2000, 39, 3764. (14) Salmas, C. E.; Stathopoulos, V. N.; Ladavos, A. K.; Pomonis, P. J.; Androutsopoulos, G. P. In Rodriguez-Reinoso, F., McEnaney, B., Rouquerol, J., Unger, K., Eds.; Studies in Surface Science and Catalysis 144 (COPS VI); Elsevier: Amsterdam, 2002; p 27. (15) Liu, H.; Zhang, L.; Seaton, N. A. Chem. Eng. Sci. 1992, 47, 4393. (16) Liu, H.; Zhang, L.; Seaton, N. A. J. Colloid Interface Sci. 1993, 156, 285. (17) Liu, H.; Zhang, L.; Seaton, N. A. Langmuir 1993, 9, 2576. (18) Liu, H.; Seaton, N. A. Chem. Eng. Sci. 1994, 49, 1869. (19) Mason, G. Proc. R. Soc. London 1988, A415, 453. (20) Salmas, C. E.; Androutsopoulos, G. P. Ind. Eng. Chem. Res. 2001, 40, 721. (21) Seaton, N. A. Chem. Eng. Sci. 1991, 46, 1895.

Salmas et al.

•CPSM)Nitrogen. This is a probabilistic model simulating the gas sorption hysteresis phenomena, evaluates a single PSD by curve fitting the entire sorption hysteresis loop and is based on a corrugated pore structure configuration.12,13 The latter is envisaged to be composed of a sequence of Ns cylindrical pore segments of constant length and distributed diameter. 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.15-19 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. (i) Physical multilayer adsorption takes place on the surface of the corrugated pore as described by the modified Halsey equation (eq 2), which correlates the statistical multilayer thickness with the relative pressure P/P0

t (nm) ) n

[

5 ln(P0/P)

]

1/3

(P/P0)m

(1)

(n is usually taken equal to n ) 0.35 and m varies in the range m ) 0-0.2. For m ) 0, eq 1 is reduced to the conventional Halsey correlation22). (ii) Capillary condensation takes place according to the Kelvin equation

DK (nm) )

4γVL cosθ 1.908 ) cosθ RT ln(P0/P) ln(P0/P)

(2)

where for nitrogen γ ) 8.85 × 10-7 J/cm2, VL ) 34.71 cm3/mol, and T ) 77.2 K (ref 23, p 164). For a hemispherical liquid nitrogen meniscus cosθ ) 1 (i.e., cosθ ≡ cosθh), while for a cylindrical one cosθ ) 0.5 (i.e., cosθ ≡ cosθc) provided that γ, VL, and T are constant. Nitrogen condensation is assumed to commence preferably on a cylindrical interface at the narrowest pore segment and then proceeds through a hemispherical one in the adjacent segments, whereas evaporation proceeds through hemispherical interface geometry. The transition from a cylindrical interface geometry to that of a hemispherical accounts for the presence of condensation thermodynamic metastability effects, which may contribute to gas sorption hysteresis. Following a probabilistic analysis of the envisaged physical steps taking place within the corrugated pore, the pertinent formulas were developed: Adsorption-Capillary Condensation Isotherm. The relative volume Vads of gas adsorbed as a function of the relative pressure can be computed via eq 3.

Vads ) [1/

∫DD

C

min

∫DD

[∫

D2F(D) dD] 4t

max

min

D2F(D) dD + 4t

∫DD

h

C

Dmax

(D - t)F(D) dD +

Dh

∫DD D2F(D) dD h

C

(D - t)F(D) dD

(

1

(

1

j)Ns

∑ (Pj)

Ns j)1

j)Ns

) )]

∑ (1 - Pj)

Ns j)1

+

(3)

Desorption-Capillary Evaporation Isotherm. The relative amount of nitrogen evaporated and/or desorbed Vder,

Montmorillonite Solids Pillared with LaNiOx

Langmuir, Vol. 19, No. 21, 2003 8779

as a function of the relative pressure, can be evaluated via eq 4 while the fraction of the pore volume Vsr saturated with nitrogen is given by eq 5

Vder )

∫DD

max

h

(D - 2t)2F(D) dD

(

2 1 - q Ns - qNs-1 Ns 1 - q

∫D

Dmax min

)/

D2F(D) dD (4)

Vsr ) 1 - Vder

(5)

Proofs of eqs 3-5 are presented elsewhere.12 Various applications covering the entire spectrum of hysteresis loop types according to the IUPAC classification have been worked out (e.g., refs 13, 20, 24-28). It should be emphasized, however, that the CPSM-nitrogen formulation is a simulation 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 “figure 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, cosθ, and Ns. Typical values of CPSM parameters to reproduce the four types of gas sorption hysteresis loops under the IUPAC classification were reported in ref 12. Ns was taken to be an integer in the formulation of eq 3. However, real values of Ns can be used in various applications of the CPSM model and bear a physical significance as follows: Consider for instance Ns ) 5.5, then we may assume that 50% of the CPSM-pore population is characterized by Ns ) 5 and the rest 50% by Ns ) 6. Thus, the application of eq 3 becomes possible while Ns ) 5.5 represents an average nominal pore length in the range Ns ) 5-6. The computation code of eq 3 includes a mathematical transformation of the summation terms j)Ns j)Ns ∑j)1 Pj and ∑j)1 (1 - Pj) of eq 3 by applying the formula for the estimation of the sum of terms of the pertinent geometric progressions that appear in the definition of probability Pj (ref 12, Appendix III). The latter transformation enables the application of eq 3 for real values of N s. •CPSM)Tortuosity. This model consists of an empirical correlation that is based on CPSM-nitrogen predictions of intrinsic pore size distribution and nominal pore length (i.e., Ns) data, thus

τCPSM ) 1 + 0.69

(

)

Dmax,eff - Dmin,eff (Ns - 2)0.58 Dmean

(6)

Dmean is the mean pore size calculated from the pore volume distribution, FV(D). Dmax,eff and Dmin,eff are calculated from

∫DD

max

max,eff

D2F(b;D) dD )

∫DD

min,eff

min

D2F(b;D) dD ) 0.025 (7)

The truncation margin, ca. 0.025, on both ends of the PSD, (22) Halsey, G. D. J. Chem. Phys. 1948, 16, 931. (23) Gregg, S. J.; Sing, K. S. W. Adsorption, Surface Area and Porosity, 2nd ed.; Academic Press: London, 1982. (24) Salmas, C. E.; Tsetsekou, A. H.; Hatzilyberis, K. S.; Androutsopoulos, G. P. Drying Technology 2001, 19, 35. (25) Salmas, C. E.; Androutsopoulos, G. P. Appl. Catal., A: Gen. 2001, 210, 329. (26) Salmas, C. E.; Stathopoulos, V. N.; Pomonis, P. J.; Rahiala, H.; Rosenholm, J. B.; Androutsopoulos, G. P. Appl. Catal., A: Gen. 2001, 216, 23.

is considered sufficiently small given the empirical nature of eq 7. It allows the calculation of realistic PSD limits by cutting off any long tail present. The CPSM-tortuosity model enables realistic predictions of pore structure tortuosity factors in satisfactory agreement with relevant literature data.29 Details on the formulation and applications of the CPSM-tortuosity theory are reported in ref 20. •CPSM)Mercury. The corrugated pore configuration has been also used to formulate mathematical relationships (Appendix I) simulating mercury porosimetry (MP) hysteresis observations, (e.g. CPSM-mercury30). Similar correlations have been developed simulating MP hysteresis loop scanning data and enabling the definition of pore structure tomography concepts.31,32 • Intrinsic Pore Size Distribution. A family of bell shape distribution functions (BSD) was chosen as the intrinsic pore number distribution fr(br;D). The normalized F(D) in its general form (i.e., composed of n unimodal PSDs) reads r)n

F(D) )

∑

r)1

()

wr fr(br;D) Rr

Φr

/ ∑(

)

r)n

wr

r)1

Rr

(8)

r)n wr ) 1 where wr is the weight of a unimodal psd, ∑r)1

fr(br;D) ) (D - Dmin) (D - Dmax) exp(bD) Φr )

∫DD

max

min

(9)

(D - Dmin) (D - Dmax) exp(bD) dD (10)

where Φr is the normalization factor of fr(br;D) and Rr ) Dmax ∫D D2(fr/Φr) dD is the normalization factor of each one min of the pore volume distributions composing the overall PSD. The choice of the BSD distribution function facilitates the application of the CPSM model for the simulation of gas sorption and mercury porosimetry hysteresis data and the evaluation of tortuosity factors. The mathematical form of BSD enables the analytical integration of the integrals appearing in the CPSM-nitrogen and CPSMmercury models.12,20,30-32 2.2. Procedure for the Evaluation of Pore Connectivity c. A summary of the procedure being applied for the calculation of pore connectivity c of pillared clays under study, according to the theory proposed by Seaton et al.15-18,21 is as follows. The bond (i.e., the pore segment connecting two neighboring sites) occupation probability (f), eq 11, is obtained as a function of the percolation probability (F), eq 12. In the present work, F(D), the probability density function with respect to pore number, is evaluated directly through the curve fitting of the pertinent sorption hysteresis loops using the CPSM model. (27) Salmas, C. E.; Androutsopoulos, G. P. J. Colloid Interface Sci. 2001, 239, 178. (28) Salmas, C. E.; Stathopoulos, V. N.; Pomonis, P. J.; Androutsopoulos, G. P. Langmuir 2000, 18, 423. (29) Satterfield, C. N. Mass transfer in heterogeneous catalysis; M.I.T. Press: London, 1970; p 7, Figure 1.5. (30) Androutsopoulos, G. P.; Salmas, C. E. Chem. Eng. Commun. 1999, 176, 1. (31) Androutsopoulos, G. P.; Salmas, C. E. Chem. Eng. Commun. 2000, 181, 137. (32) Androutsopoulos, G. P.; Salmas, C. E. Chem. Eng. Commun. 2000, 181, 179.

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ca. 2 nm) or wider micropores (i.e., 1.0-2 nm), is given by

Thus:

f)

∫DD

max

∫DD

F(D) dD/

max

min

F(D) dD

Vmax - Vdes f ) F Vmax - Vads

(11) (12)

where Vmax ≈ Vflat max and Vdes, Vads the corresponding values in the desorption and adsorption CPSM-fitted isotherms for specified (P/P0) values. The best connectivity c (i.e., the average number of bonds envisaged converging in a single site) and L (i.e., the characteristic size of particles that express the number of pore length) are obtained by fitting the experimental scaling data (F, f), deduced via eqs 11 and 12, to the generalized scaling relation (eq 13),15-18,21 thus

Lβ/νcF ) G[(cf - 3/2)]L1/ν

(13)

In the present analysis the generalized scaling relation was constructed using the critical exponents β ) 0.41 and ν ) 0.88 given by Kirkpatrick.33 Typical fitting results are shown in Figure 7. The variation of c with the percentage of LaNiOx content is displayed in Figure 8. The sigmoid scaling function considered in this work reads

F ) [exp(-exp(-((c - 2) f - 1.5)(L1/ν)))/ ((c - 2)(Lβ/ν))]/Q (14) where Q is the normalization factor i.e., for f ) 1, w F ) 1. Equation 14 provides for a minimum connectivity equal to c ) 2. 2.3. Surface Area Evaluation Methods. •BET Surface Area Variants. The Brunauer-Emmett-Teller (BET) method, proposed in the pioneer work of Brunauer et al.,34 has been, over several decades, the dominant method23 for the determination of the specific surface area (SSA), despite the simplified assumptions that involves and the ambiguity accruing from the choice of the linear part of the BET plot. Two BET variants were proposed in ref 34. The formula for multilayer adsorption with an envisaged infinite number of adsorbed gas monolayers, reads

C-1 P 1 P + ) nwC P0 na(P0 - P) nwC

(15)

where na is the amount (µmol/g) of gas adsorbed at relative pressure (P/P0), nw is the monolayer capacity (i.e., µmol/ g), and C is the BET constant. A second BET formula was also proposed34 for multiplayer adsorption assuming a finite number of adsorbed monolayers n, thus

na(1 - P/P0) nwC(P/P0)

)

1 - (n + 1)(P/P0)n + n(P/P0)n+1 1 + (C - 1)(P/P0) - C(P/P0)n+1

(16)

The BET formula for monolayer, restricted adsorption (Langmuir model), applicable in cases of narrow mesopores (i.e., approaching the upper limit of the micropore region (33) Kirkpatrick, S. 1979. Models of Disordered Materials. In Illcondensed Matter; Balian, R., Mayward, R., Toulouse, G., Eds.; NorthHolland: Amsterdam, 1979. (34) Brunauer, S.; Emmett, P. H.; Teller, E. J. Am. Chem. Soc. 1938, 60, 309.

(P/P0) 1 P 1 + ) na nwC nw P0

(17)

It should be noted that for pure physical adsorption the acceptable range of the C constant variation is ca. 20 < C < 200, since C is related to the enthalpy of adsorption and provides information for the magnitude of adsorbentadsorbate energy of interaction.23,34 •Surface Area Evaluation from rs-Plots. The low pressure (high slope) section of an Rs plot goes through the axis origin, as long as the multilayer of adsorbate is formed unhindered, indicating the presence of wide micropores.2,11 Under these circumstances the slope of the line through the axis origin is a measure of SSA according to the equation Sa ) 2.87(Vads/Rs).35 •CPSM Surface Area. The SSA deduced from the CPSM analysis of the N2 sorption data represents the cumulative surface area calculated from the integration of the relevant surface area differential PSD, over the detected pore size range.12 3. Results and Discussion 3.1. Materials. Nitrogen sorption literature data36 of a family of montmorillonite pillared clays are being subjected to an analysis using the CPSM model. The sorption data of the pillared clays samples exhibit a lowpressure steep adsorption step and at higher relative pressure an H4-type hysteresis loop (IUPAC classification) that is transformed into an H2-type loop as the cationic complex loading is being increased. The materials under consideration are high surface area solids containing both micro- and mesopores and were synthesized by intercalation of the heterobinuclear cation of the complex NiLa(fsaen)NO3 (ca. H4fsaen ) N,N′-3-hydroxysalicylideneethylenediamine) between the layers of Na montmorillonite clay and calcined at 500 °C. Calcination of the complex and removal of the organic matter causes the formation of perovskite binary oxide pillars of the form LaNiO3.36 The dried samples, originally identified as NiLa-Mont0.2, are designated here as NiLa-M-0.2, NiLa-M-0.4, NiLaM-0.8, NiLa-M-2.0, NiLa-M-4.0 and NiLa-M-8.0, whereas the digits indicate the cationic complex loading (CCL) mmol/g of clay. The results of solid characterization tests e.g. gas adsorption, XRD, IR, and ζ potential measurements were reported in ref 36. In the latter citation, the tracing of microporosity present in these materials was accomplished by the construction of the pertinent Rs plots. The isotherm of the Na-montmorillonite (ca. Na-M), Figure 1, was chosen as the standard isotherm by assuming that it does not possess microporosity. However, this material possess a surface area S ) 40-45 m2/g, as reported in ref 36 and in Table 2, this work, as well as limited microporosity as evidenced from the PSD deduced from the CPSM model, Figure 2. As the construction of the Rs plots requires the use of a standard isotherm of a nonporous reference material, in the present work a low surface area (ca. 20 m2/g) Na-rich montmorillonite from the literature (ref 23, p 132, Figure 327a) was chosen as the reference material. 3.2. Simulation of N2-Sorption Data. The corrugated pore structure model (CPSM-nitrogen),12,13 was used to simulate the sorption data of the materials under consideration and the results are presented in Figure 1 (35) Lecloux, A.; Pirard, J. P. J. Colloid Interface Sci. 1979, 70, 265. (36) Skaribas, S. K.; Pomonis, P. J.; Grange, P.; Delmon, B. J. Chem. Soc., Faraday Trans. 1992, 88, 3217.


Langmuir, Vol. 19, No. 21, 2003 8781

Figure 2. Surface area variation of montmorillonite clays pillared with LaNiOx, as a function of pillaring agent load, CCL.

of the BSD (bell shape distribution) indicates and Figure 3 illustrates. The method of combining two or more BSD distributions was described in the Theoretical Approach section of this work. In all simulation situations the equality cosθc ) cosθh ) cosθ holds and indicates the absence of liquid/vapor interface metastability effects appearing between capillary condensation and evaporation processes. Plausibly therefore, the observed hysteresis phenomena should be attributed to the pore structure networking effects. CPSM is a one-dimensional model of constant connectivity that equals 2 (c ) 2) (i.e., one segment is connected to two neighboring segments) but networking in terms of the CPSM theory depends also on the value of the tortuosity factor.20 The latter combines the effect of Ns and the characteristics of the intrinsic pore volume distribution as being calculated from the CPSM theory. The CPSM model does not allow for dynamic effects other than those caused by pore networking as relates to pore tortuosity (e.g., τCPSM > 1) and capillary condensation metastability effects. An indication for the presence of microporosity is that cosθ < 0.22 for most of the materials studied in this work. In only two pillared clay samples i.e. those two with minimum and maximum pillaring agent load, microporos-

Figure 1. CPSM simulation of nitrogen-sorption hysteresis (literature data36) at 77.5 K for montmorillonite samples pillared with LaNiOx.

(continuous lines through the points). It is obvious from the plots of Figure 1 that in all cases an excellent fit was achieved for the set of CPSM parameters reported in Table 1. The intrinsic pore volume distributions, PSDs, are of bimodal type as the number of the required (bi) parameters

Table 1. CPSM-Nitrogen Parameters for the Fitting of N2-Sorption Hysteresis of Pillared Montmorillonite Materials material designation montmorillonite standard material Na-M NiLa-M-02 NiLa-M-04 NiLa-M-08 NiLa-M-20 NiLa-M-40 NiLa-M-80

F(D) Distribution Characterization Parametersb b3 w1 w2 w3 Pce/P0

Vgmax (cm3 N2 (STP)/g)

Nsa

b1

b2

48

2.6

-5000

-150

-35

2

10

58 45 47 104 180 165 186

3.7 7.0 12.5 5.5 9.0 4.9 4.8

-5000 -6000 -6000 -2000 -2000 -3000 -1500

-250 -140 -115 -100 -150 -180 -250

-70 -50 -50

1 1 1 1 1 1 1

5 8 7 3 5.5 10 7

(P/P0)m

Kelvin parameters cos θC cos θh

t (nm)c m n

25

0.45

0.998

0.46

0.46

0.00

0.35

8

0.425 0.45 0.47 0.43 0.41 0.40 0.33

0.999 0.999 0.999 0.999 0.999 0.999 0.999

0.22 0.13 0.082 0.10 0.15 0.18 0.37

0.22 0.13 0.082 0.10 0.15 0.18 0.37

0.05 0.00 0.05 0.07 0.00 0.03 0.16

0.35 0.35 0.35 0.35 0.35 0.35 0.35

3 1

a Nominal pore length (frequency of corrugated pore cross-sectional area variation.12 b BSD: bell shape distribution function (see the “Theoretical Approach” section of the present work). c Parameters of the modified Halsey equation (eq 1).

Table 2. Specific Surface Areas of Montmorillonite Samples Pillared with LaNiOx, Deduced from BET, Langmuir, CPSM, and rs Plot Methods material designation

SBET(mu) (m2/g)

montmorillonite standard material Na-M NiLa-M-02 NiLa-M-04 NiLa-M-08 NiLa-M-20 NiLa-M-40 NiLa-M-80

20.5 42.8 55.4 64.8 143.5 224.0 153.0 126.0

CBET(mu)

P/P0 linearity limits

SBET(mo) (Langmuir) (m2/g)

443

0.01-0.12

23.7

191 738 208 179 898 259 50

0.02-0.15 0.02-0.15 0.05-0.15 0.02-0.15 0.00-0.15 0.02-0.15 0.05-0.20

49.2 64.2 71.6 167.3 248.5 161.3 143.2

CBET(mo)

P/P0 linearity limits

SCPSM (m2/g)

149

0.02-0.10

19.2

103 163 153 95 287 282 43

0.02-0.10 0.02-0.10 0.01-0.08 0.02-0.11 0.01-0.08 0.00-0.07 0.04-0.09

44.8 56.0 71.5 168.3 227.1 155.8 143.4

SR ) 2.87*Vads/Rs

(SBET SCPSM)/SCPSM (%) +1.8

41.3 51.7 63.0 135.2 214.5 144.0

-4.5 -1.1 -9.4 -15.0 -1.4 -1.8 -12.0

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Halsey correlation parameters, the simulation of a specified hysteresis loop is achieved for a unique set of PSD and Kelvin parameters. 3.3. Pore Volume and Surface Area Evolution. The variation of pore volume and surface area as a function of the pillaring agent is depicted in Figure 2. For CCL up to 20 mmol/g, there is a steady increase of pore volume and surface area up to a maximum value. For CCL > 20 the pore volume is stabilized to about its maximum value (i.e., 0.25-0.28 cm3/g) while the surface area undergoes a drop from about 225 to 150 m2/g. Surface area evaluations were accomplished using the four different methods outlined in the theory section of this work and the results are given in Table 2 and in graphical form in Figure 2. The BET and Langmuir,34 methods were applied for the P/P0 linearity region indicated in Table 2 and by assuming a close-packed monolayer with am ) 0.162 nm2 at 77 K. It is interesting to observe that the four methods evaluate comparable results despite the deficiencies involved in each one of them, (Figure 3). It is also worth noting that the C values for both the BET and Langmuir methods are in their majority within acceptable limits, Table 2. 3.4. Pore Size Distributions. The application of the CPSM model, by curve fitting the sorption hysteresis, enables the evaluation of the intrinsic pore volume PSDs that cover both the mesopore and micropore size range. The PSDs deduced for the indicated pillared clay materials being examined in this work are illustrated in Figure 3. It is evident from the later figure that the parent material Na-M exhibits a bimodal PSD with two completely distinct peaks, one sharp peak between 1 and 2 nm in the micropore range and a wide one stretched over the mesopore range 2-50 nm. We mention that the microporosity is a typical feature of natural montmorillonites as shown in Figures 1 and 3 for a standard sample reproduced from ref 23 (i.e., p 132, Figure 327a). The montmorillonite used had a cation exchange capacity (CEC) equal to ∼1.20 mequiv/g.36 Therefore, intercalation of the cations NiLa(fsaen)+ of the binuclear complex NiLa(fsaen)+NO- could theoretically proceed up to this level by exchange with Na+ and any further insertion of complex cation between the layers is expected to proceed in a disorganized and random way. Indeed we observe that initially loading of cationic complex up to 0.2 and 0.4 mmol/g, i.e., ∼17% and ∼35% of CEC, increases the surface area to a rather small extend, from ∼4.3 to 55 and further to 65 m2 g-1 while the pore volume also increases. At the same time the microporosity is affected positively but mildly. Then loading up to 0.8 mol/g (70% of CEC) results in further increase of surface area to ∼143 m2 g-1 and the pore volume also increases (see Figure 2). The microporosity is now affected in an appreciable degree (Table 3). The maximum of Sp (∼215

Figure 3. Pore volume intrinsic PSDs evaluated from the CPSM method for the indicated materials.

ity detection and hysteresis simulation were achieved for cosθ ) 0.46 and cosθ ) 0.37, respectively. It should be emphasized, however, that the CPSM simulations presented in this work, were implemented for A ) (4VL γ cosθ/RT), the Kelvin parameter, varied over the range A ) 0.16-0.88. These values are well below those either for a hemispherical interface geometry (i.e., A ) 1.908) or a cylindrical one (i.e., A ) 0.954) and may be attributed to the change of either VL and/or γ cosθ due to the presence of microporosity. In all simulations parameter (n) of the modified Halsey correlation was kept equal to n ) 0.35 while parameter (m) was varied over the range m ) 0.0-0.16 (Table 1). This minor deviation from the conventional Halsey correlation, in the low relative pressure region, may be induced by the pore size, surface roughness and/or chemical composition, which affect the mode of molecular packing and its deviation from the assumed normal hexagonal type. The nominal pore length varies in the range Ns ) 2.612.5. Obviously, this variation is reflected in the values of the respective tortuosity factors. Given the modified

Table 3. Evaluation of Microporosity Ratios Obtained from the CPSM and the rs-Plot Methods for the Montmorillonite Materials Pillared with LaNiOx pore vol (cm3/g) material designation montmorillonite standard material Na-M NiLa-M-02 NiLa-M-04 NiLa-M-08 NiLa-M-20 NiLa-M-40 NiLa-M-80

total Vp

micro- Vmi

surface area (m2/g)

Vmi/Vp (%) CPSM as

0.074 0.090 0.070 0.073 0.160 0.278 0.255 0.288

SCPSM

Smi

Smi/S (%) CPSM as

19.2 0.0052 0.0078 0.0086 0.0390 0.0310 0.0154

5.8 11.2 11.8 24.0 11.1 6.0

4.3 12.6 15.1 19.8 13.7 6.0

44.8 56.0 71.5 168.3 227.1 155.8 143.4

12.3 18.6 24.8 97.0 73.2 34.7

27.5 33.0 34.7 57.6 32.3 22.2

16.6 30.6 32.4 42.2 33.1 19.9


Langmuir, Vol. 19, No. 21, 2003 8783

Figure 5. Comparison of relative microporosities deduced from the Rs plots and the corresponding CPSM simulations of N2 sorption hysteresis data for the pillared clays under consideration.

Figure 6. Variation of relative micropore volume (a) and surface area (b) (predicted by the Rs plot and the CPSM method) with respect to pillaring agent loading.

Figure 4. Typical Rs plots of montmorillonite samples pillared with LaNiOx. N2 Sorption data for the standard nonporous montmorillonite material were from ref 23, p 132, Figure 317a.

m2/g) and Vp (0.28 cm3/g) as well as the microporosity is observed when the loading reaches 2 mmol/g, which should be compared with CEC of 1.2 mequiv/g. At this point, it seems that the maximum number of nanopillars of LaNiOx have been inserted into the layered structure which reserves its ordering, as testified also by XRD (ref 36, Figure 3). It is exactly this delaminated structure, at this composition, which is the reason for the increase of mesoporosity created by the spaces between the disordered aluminosilicate layers and the LaNiOx relative large particles. Thereafter further loading up to 4 mmol/g and 8 mmol/g results in a drop of both Sp and Vp but also microporosity. This effect is attributed to blocking of micropores by relative large LaNiOx particles and the development of further extended delaminated structure as testified by low angle XRD data (ref 36, Figure 3). Again this extended delaminated structure is the reason for the increase of mesoporosity created by the spaces between the disordered aluminosilicate layers and the LaNiOx relatively large particles. 3.5. Microporosity. The relative micropore volume (Vmi/Vt) and surface area (Smi/St), for all materials studied here, were calculated by the Rs plots (Figure 4) and the CPSM methods and numerical results are provided in Table 3. In the case of the CPSM model the evaluation of microporosity involves the integration of the respective intrinsic PSDs of Figure 3 over the micropore range i.e., from Dmin to D ) 2 nm. Comparisons of microporosities deduced by the Rs and CPSM methods for both pore volume

and surface area are shown in Figure 5. The micropore volume comparison plot for the full series of materials confirms a satisfactory agreement between the results obtained by the two methods, as the data points approach closely the diagonal line. However, slightly higher relative microporosities are calculated from the CPSM method. These differences become more pronounced in the micropore surface area comparison graph of Figure 5b and may be attributed to the errors involved in both methods, e.g., the quality of fit of the CPSM model and the inaccuracies of graphical determination of the slope and intercept on the Rs plots. The effect of CCL loading on relative microporosity is illustrated in the relevant graphs of Figure 6. The addition of CCL pillaring agent up to 20 mmol/g causes a rise of relative microporosity, which declines for CCL > 20 mmol/ g, Figure 6. 3.6. Pore Structure Tortuosity and Connectivity. The pillared clays under examination, exhibit a tortuosity factor τCPSM ) 2.6-12.5 (Table 4). Such values are typical of many porous catalytic supports.29 The variation of τCPSM with respect to CCL loading is illustrated in Figure 8. A general monotonic increase of τCPSM is observed, when the τCPSM value for NiLa-M-04 is excluded. A maximum value τCPSM ) 12.5 is reached for CCL ) 8 mmol/g. The high CCL apart from creating microporosity, it probably intensifies the irregularities of the corrugated pore structural elements. This can be also readily seen from the characteristic gradual widening of the hysteresis loops, e.g., from (Na-M) to (NiLa-M-20), that are indicative of increasing tortuosity, in view of the absence of capillarity metastability effects. In addition to pillared clays pore structure tortuosity investigations an interesting comparison would be that of pore structure connectivity analysis that involves percolation concepts, (e.g., refs 15-19, 21). A brief outline

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Salmas et al.

Table 4. Tortuosity Factors (τCPSM) and Pore Connectivity (c) Data for Montmorillonite Pillared with LaNiOx Materials, Obtained by Jointly Applying the CPSM12,13 and Seaton21 Methods material designation montmorillonite standard material Na-M NiLa-M-02 NiLa-M-04 NiLa-M-08 NiLa-M-20 NiLa-M-40 NiLa-M-80

Ns

Dmin,eff (nm)

Dmax,eff (nm)

Dmean (nm)

τCPSM

L

c

2.6

2.56

102.20

37.64

2.35

0.838

5.05

3.7 7.0 12.5 5.5 9.0 4.9 4.8

1.87 1.65 1.51 1.50 1.74 1.79 2.32

48.86 16.52 13.05 17.31 17.44 45.25 75.01

17.09 7.36 5.85 6.87 7.54 12.29 16.01

3.58 4.54 6.32 4.28 5.44 5.52 6.69

0.911 1.078 1.316 1.014 1.156 0.984 0.981

4.48 3.55 2.65 3.91 3.20 4.05 4.09

of the procedure being applied here is presented in the “Theoretical Approach” section of this work. It is of interest to note that the required intrinsic pore number PSD to determine the occupation probability f (i.e., eq 11), is calculated from the CPSM model. Similarly, the CPSM isotherms fitted over the pertinent experimental nitrogen sorption hysteresis loop provide for the determination of Vflat max, Vads, and Vdes data. Numerical results of pore connectivity (c) and number of pore lengths (L) for the pillared clay samples are presented in Table 4. The best connectivity (c) and pore lengths (L) are obtained by fitting the generalized scaling relation (eq 13) over the experimental scaling data (F, f), deduced via eqs 11 and 12. The fitting results are quite satisfactory and they are illustrated in Figure 7. The variation of connectivity (c) vs the CCL loading is depicted in Figure 8. It is readily seen from the latter figure that the increase of CCL loading causes an almost linear drop of c that is reversed compared to that of τCPSM. Indeed, this is expected because in the absence of liquid/vapor interface metastability effects, wide gas sorption hysteresis loops are indicative of a tortuous pore structure of restricted connectivity. The data in Figure 8 illustrate that the tortuosity factor variation is more sensitive than of the connectivity factor.

There is a linear correlation between the c and τCPSM data of Figure 4 of the form

c ) 6.50 - 0.62(τCPSM)

(18)

An additional observation is that a similar mode of variation of Ns (Table 1) and L (Table 4) parameters with the change of the CCL content occurs (Figure 9). Both Ns and L, though numerically different, are characteristic pore length parameters defined in their respective models.

Figure 8. Variation of tortuosity factors and pore connectivity with respect to LaNiOx content.

Figure 7. Curve fitting result of eq 13 over the relevant data valid for the indicated materials and the corresponding c and L values.

Figure 9. Comparison of Ns, nominal pore length, a CPSM model parameter,12 and L, the characteristic number of pore lengths, a connectivity model parameter.21


Langmuir, Vol. 19, No. 21, 2003 8785

A ) k for Ps ) 1

A linear correlation of the form

L ) 0.74 + 0.05Ns

(19)

was deduced. It is a striking result the fact that two entirely different theoretical approaches being used in the simulation of gas sorption hysteresis phenomena yield quite consistent results. 4. Conclusions The characterization of the family of montmorillonite clays pillared with a NiLaO3 binary oxide using the CPSM model confirmed that this method can be effectively applied for pore structure analysis of materials exhibiting mixed meso-micropore structures and yields unified pore size distributions extended over the full pore size rang detected by gas sorption. As regards the assessment of microporosity, the CPSM method is most appropriate for the tracing of micropores in the range Dp ) 1.5-2.0 nm, as it is based on the assumption of adsorbed gas monolayer formation. The CPSM model, through the simulation of gas sorption hysteresis, apart from the evaluation of meso-micropore size distribution, enables also the determination of pore surface areas in good agreement with those deduced from other methods i.e., BET, Langmuir and Rs, the evaluation of tortuosity factors τCPSM and the detection of relative microporosity in satisfactory agreement with relevant data obtained by the Rs method. It is further concluded that, for the materials examined in this work, pore connectivity factors are inversely proportional to pore tortuosity whereas the nominal pore length Ns (i.e., a CPSM parameter) is proportional to the number of pore length L (a parameter of the pore percolationconnectivity model). These results are indicative of how well two entirely different in conception theoretical approaches being used in the simulation of gas sorption hysteresis phenomena generate quite meaningful and consistent results. Acknowledgment. The authors are grateful to the State Scholarships Foundation of Greece for the financial support provided to C.E.S. Appendix I CPSM)Mercury. Following a probabilistic approach and assuming a corrugated pore structure configuration, mathematical relationships were formulated, simulating mercury porosimetry (MP) hysteresis observations.30 The Washburn equation is used to relate pore size with the applied pressure, thus

P ) -4γmcosθm/D

(I-1)

where γm and θm are surface tension and contact angle of mercury. •Mercury Penetration. The CPSM prediction of the intruded mercury volume as a function of the applied pressure reads

∫DD

A VP(P(D)) ) ( k

max

D2F(D) dD/

∫DD

max

min

D2F(D) dD) (I-2)

where

A ) (1 - Ps2k)/(1 - Ps) - k Ps2k-1 for Ps < 1 or

and

Ps )

∫DD

max

F(D) dD

Parameter k is related to the corrugated pore nominal length, i.e., k ) Ns/2. •Mercury Retraction. The CPSM prediction of the volume of mercury withdrawn (i.e., VW), under decreasing pressure conditions, can be computed from the following equations:

Case i: for Dmin < D < (Dmin/λ) VW(P(D)) )

(

B / 2k2

∫DD

max

min

)∫

D2F(D) dD

Case ii: for (Dmin/λ < D < Dmax) VW(P(D)) ) [

∫D

(

∫DD

B / 2k2

D min/λ

max

min

D

Dmin

D2F(D) dD (I-3)

)

D2F(D) dD ×

(D2F(D)

∫D

Dmax

F(D) dD) dD +

∫DD

min/λ

min

D2F(D) dD] (I-4)

where

B ) (2k/(1 - Pc)) - ((1 - Pck) (Pc + Pck))/ (1 - Pc2) for Pc < 1 or

B ) 2k2 for Pc ) 1 and

Pc ) 1 - P s The pore constriction parameter λ is the ratio Di+1/Di, with i increasing in the direction of mercury motion. A mercury snap-off mechanism is attributed to corrugated pore crosssectional variation, and λ is a key parameter to account for the pore geometry irregularity. The CPSM predicted mercury retraction line is deduced from the following equation:

VR ) (1 - VW)

(I-5)

Proofs of eqs I-2-I-5 are presented elsewhere.30 Similar correlations have been developed for the simulation of MP hysteresis loop scanning, that enable the definition of pore structure tomography concepts.31,32 Nomenclature bi ) parameter of unimodal PSD of BSD type (eq 9), (i ) 1, 2, 3, ...) c ) connectivity factor (i.e., the average number of bonds envisaged converging in a single site as defined in the pore connectivity model by Seaton15-19,21) C ) BET constant D ) pore diameter (nm) DC, Dh ) diameter of pore filled up with condensate through a cylindrical or a hemispherical l/v interface shape (nm) DK ) Kelvin core diameter (nm) (eq 2) Dmin, Dmax ) minimum and maximum pore diameter

8786


Dmin,eff, Dmax,eff ) effective minimum and maximum pore diameters as defined by eq 7 Dmean ) mean pore diameter as predicted by the intrinsic pore size distribution f ) occupation probability (defined in the pore connectivity model of Seaton15-19,21) F ) percolation probability from the adsorption isotherms.15-19,21 fr(br;D) ) component pore number distribution function of BSD type used in the synthesis of F(D), eq 9 F(D) ) intrinsic pore number distribution density function, as defined by eq 8 FV(D) ) intrinsic pore volume distribution, F(D) ) D2F(D) k ) number of cylindrical segments of equal length, halfway through a corrugated pore, i.e., k ) Ns/2, eqs I-2 and I-3 L ) number of pore length (as defined in the connectivity model15-19,21) m ) exponent of the adjustment factor of the modified Halsey correlation, eq 1 n ) monolayer thickness of Halsey correlation, eq 1 na ) amount (µmol/g) of gas adsorbed at relative pressure P/P0, eqs 15-17 nw ) monolayer capacity (i.e., µmol/g), eqs 15-17 Ns ) number of pore segments forming a corrugated pore (also nominal pore length or frequency of corrugated pore cross-sectional area variation) P(D) ) pressure applied on mercury bulk, sufficient to force it through a cylindrical pore of size D, according to the Washburn equation, eq I-1 P/P0 ) relative gas pressure Pce/P0 ) minimum relative pressure for the initiation of capillary condensation according to the CPSM model (P/P0)max ) maximum relative pressure for the termination of the gas sorption process according to the CPSM model Pj ) probability for at least one l/v hemispherical interface to be adjacent to the general jth segment of a corrugated pore12 q ) probability for a pore segment of size Dj to be within the range Dh - Dmax12

Salmas et al. t ) thickness of physically adsorbed nitrogen multilayer (nm), eq 1 T ) temperature (K) Vads ) relative saturation during capillary condensation Vder ) relative pore volume evacuated from desorbedevaporated nitrogen, eq 4 Vg ) volume of gas adsorbed at STP (cm3/g) Vp ) specific pore volume (cm3/g) Vsr ) relative saturation during capillary evaporation (1 - Vder) VP(P(D)) ) relative volume of mercury penetrated the pore structure at P(D), eq I-2 VR(P(D)) ) relative volume of mercury retained in the pores at pressure P(D), eq I-5 VW(P(D)) ) relative volume of mercury withdrawn from the pore structure at a pressure P(D), eqs I-3 and I-5 wi ) relative weight of the ith unimodal PSD participating in the synthesis of a multimodal PSD, eq 8 Acronyms BSD ) bell shape distribution function: (D - Dmin)(D Dmax) exp(bD) CPSM ) corrugated pore structure model DFT ) density functional theory PSD ) pore size distribution SSA ) specific surface area Greek Letters γ ) liquid nitrogen surface tension (γ ) 8.85 × 10-7 J/cm2) γm ) surface tension of mercury (γm ) 4.80 × 10-3 J/cm2) θ ) mercury contact angle, 140° θC, θh ) liquid nitrogen contact angle for cylindrical and hemispherical liquid nitrogen meniscuses, respectively λ ) critical pore constriction ratio, λ ) Di+1/Di (i increases in the direction of mercury flow) τCPSM ) tortuosity factor as defined by eq 6 LA034913T

Evaluation of Microporosity, Pore Tortuosity, and Connectivity of

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