Effect of pore morphology and topology on capillary condensation in nanopores: a theoretical and molecular simulation study

Studies in Surface Science and Catalysis 160 P.L. Llewellyn, F. Rodriquez-Reinoso, J. Rouqerol and N. Seaton (Editors)

9 2007 ElsevierB.V. All rights reserved

Effect of pore morphology and topology on capillary condensation in nanopores: a theoretical and molecular simulation study R. J.-M. Pellenq a, B. Coasne b, R. O. Denoyel c, J. Puibasset d CNRS, Campus de Luminy, 13288 Marseille cedex 09, France. [email protected], fr a CRMC-N,

b LCPMC, CNRS-Universit6 de Montpellier II, Place Eugene Bataillon, 34095 Montpellier cedex 5, France. c MADIREL, CNRS-Universit6 de Provence, Centre de Saint-J&6me,13397, Marseille cedex 20, France. a CRMD, CNRS-Universit6 d'Orl6ans, 45071 Orl6ans cedex 02, France. I. ABSTRACT We report a theoretical and simulation study of the temperature dependence of adsorption hysteresis for porous matrices having different morphologies and topologies. We used offlattice Grand Canonical Monte Carlo (GCMC) simulations and two Density Functional Theories (DFT): we used the standard DFT in the non local approximation for cylindrical pores and the coarse-grained lattice DFT developed by Kierlik et al. [7] for disordered porous materials. We aim at gaining some insights on the concept of critical hysteresis temperature defined as the temperature at which the adsorption/desorption isotherm becomes reversible. 2. INTRODUCTION Capillary condensation occurs when a fluid is confined within nanopores. It is analoguous to the usual gas-liquid transition but displaced towards lower pressure because of confinement. In fact, capillary condensation concerns pores large enough so that the transition can occur due to cooperative interactions between adsorbed molecules. In contrast, such a phenomenon is not expected in micropores of a few angstroms. The status of capillary condensation in nanoporous adsorbents as being or not a first order transition is still the subject of intense research. Theoretical works for slit pores have demonstrated the existence of a true first order transition with a so-called capillary critical point characterized by a critical temperature of the confined fluid T that is lower that of the bulk T 3 9 . Ip. a pioneering work on the criticality of cc

c

fluids between plates [ 1,2], Fisher and Nakanishi showed that the critical behavior is affected by the finite thickness of the adsorbed film and its interaction with the wall. This scaling theory predicts that pore condensation and the related hysteresis loop disappear at a temperature, Tc, that is below the bulk critical temperature, T~~ The shift in critical temperature ATc=T)~ c is dictated by the ratio of the pore width D and the correlation length (o of the density fluctuations in the bulk fluid:

2

R.J.-M. Pellenq et al.

In the 3D-Ising model, v = 0.63 and (o is of the order of magnitude of the molecular diameter or. C is a constant that takes the universal value of 1.658 for zero fluid/wall interactions or 3.432 for strong fluid/wall interactions. In the classical mean field van der Waals fluid theory, v = 0.5 and (o is again of the order of magnitude of or. C takes the universal constant value of 1.645 for zero fluid/wall interactions, which is close to the value found in the Ising approach, or takes the value of 4.284 for strong fluid/wall interactions. Critical-point shifts in a slit-like geometry can be rationalized by the facts that the fluid is between a 3D and a 2D states and that T2~is much smaller than T~~ . Evans et al. [3] used the Density Functional Theory (DFT) and derived another expression for the shift in the critical temperature for a fluid confined in small pores: 1 ~:~ ~ XD (2) where 1 / 2 is the range of the fluid-fluid interaction, which is equal to the adsorbate size in reference [4]. Equation (2) is compatible with the 2D-Ising fluid theory since v equals 1. An adsorption/desorption isotherm below Tee, exhibits a hysteresis loop. In between condensation and evaporation metastable pressures, there is an equilibrium pressure for which both the gas and liquid phases coexist. In the case of cylindrical pores of any dimensions, there are theoretical arguments to indicate that no first order transition can exist because at the critical point, the correlation length can diverge only in the direction of the pore axis; a cylindrical pore can be considered as a one-dimensional system whatever its diameter. However, DFT [4], simulations [5] and experiments suggest that fluids in both confining geometries (slit and cylinders with size of several nm) behave similarly as far as condensation and evaporation are concerned; the hysteresis loop shrinks as temperature increases and eventually disappears. We note that in a van der Waals picture of gas-liquid transitions in such simple systems, the temperature of hysteresis disappearance is the capillary critical temperature, i.e., Tr162 Over the last decade, significant theoretical advances have been achieved regarding the understanding of fluids confined in a disordered porous materials; it is now clear that fluids in a network of pores having topological and morphological disorders (such as Vycor or CPG) strongly affects capillary condensation as compared to that in independent pores of simple geometries [6]. The first effect is a flattening of the adsorption isotherm branch, which therefore does not exhibit any discontinuity as in a slit or a cylindrical pore. The desorption branch remains vertical and defines a hysteresis loop that shrinks and disappears with increasing temperature. In a random matrix, Kierlik et al. [7] used a coarse-grained lattice gas theory (see below) and showed that, for large values of the ratio of the surface-fluid to fluidfluid energies (parameter y), capillary condensation cannot be a first order transition when averaging over matrix disorder. The disorder generates a complex free energy landscape, with a large number of local minima (i.e., metastable states), in which the system remains trapped without finding the true equilibrium state; capillary condensation is then described as an outof-equilibrium phenomenon. Detcheverry et al. [8,9] used the same coarse-grained theory for fractal porous materials and found evidence for a first order capillary condensation that depends on the porosity; it is first order in aerogels with 98% porosity but no longer in aerogels with 87 % porosity. Woo and Monson [10] used again this on-lattice mean field theory for Vycor glasses (mean pore diameter 4-5 nm) and found evidence of a true first-order transition with a critical point at a temperature lower than Tcr (the shift of the real critical temperature compared to Tee increases with the coupling parameter y). However, the mapping

Effect of pore morphology and topology on capillary condensation 35 a)

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_

~

b)

~:~

)-)~,.~,~

~

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,

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Fig. 1a. Xe Adsorption isotherms in Fig. 1b. Temperature dependence of Ar adsorption Vycor at different temperatures, isotherms in MCM-41 (D = 4.4 nm), Tcc= 102.5 K. From Ref. [25]. adapted from Ref. [26]. of these on-lattice results on more realistic situations attainable with atomistic off-lattice techniques [ 11 ] remains difficult (in particular concerning the dependence of on-lattice results with the lattice spacing and the absolute value of y). We note that Monte Carlo simulations on a single (small) periodic Vycor also suggest the existence of a first order transition [12,13,14], when analyzing adsorption data for Ar and H20 over a wide range of temperatures. However, such atomistic simulations do not consider any averaging over the matrix disorder. The extent of the pore topological disorder therefore seems to have some influence on the nature of capillary condensation. More recently, Coasne and Pellenq [15] showed the important effect on condensation and evaporation of extended morphological defects within a single cylindrical pore such as a constriction. Upon adsorption, the dense phase nucleates in the constrictions and subsequently propagates in the larger cavities (with a meniscus) until a gas bubble is formed and eventually collapses. Upon desorption, evaporation can follow either the so-called pore blocking mechanism (i.e., the larger pore region remains filled until the constricted region empties) or a cavitation process (i.e., the liquid in the largest pore regions undergoes a liquid-gas transition while the constricted regions remain filled), depending on the constriction size and the ratio of the fluid-fluid to fluid-wall energies [ 16]. These findings confirmed those obtained for constricted slit [17,18,19] and cylindrical [20] pores. Independent but constricted cylindrical pores give rise to adsorption/desorption isotherms, whose shape closely resembles that observed for connected porous networks, in agreement with the recent experiments by van der Voort et al. on plugged MCM-41 pores with silica clusters [21 ]. A constriction can therefore be viewed as a connection between two pores and a constricted single pore can be considered as the simplest model for connected porous materials. Up to now, we have only described results for infinite (periodic) systems, i.e. with pores that are not directly connected to the gas phase through the external surface of the material. If a simple cylindrical pore is connected to the external gas reservoir, then it was shown that desorption occurs at equilibrium [18]. A pore with a close end have a reversible adsorption isotherm with a sudden increase upon condensation (decrease upon evaporation) in the adsorbed amount at a pressure equal to the equilibrium pressure of the equivalent infinite pore (which corresponds to the desorption pressure of the open ended pore of the same diameter or width). Therefore, open ended pores or infinite pores of the same size will have the same hysteresis critical temperature. In disordered porous systems, the status of Tcc as being or not relevant to a true critical point is not clear as it seems to depend on the type and degree of disorder (random or fractal porous networks or correlated structures such as Vycor).

R.J.-M. Pellenq et al.

o~

o

pore size ~ 5 nm

04

~

0

(f) (d)

(C)

03

(e) (b)

o2

(a) Ol

oo

0 O0

0 05

0 10

0 15

0 20

0 25

0 30

o/R o

Fig. 2. Experimental hysteresis critical temperatures for various confined fluids. T3~is the bulk c

critical temperature and ATe=[T~o-T ]. Experimental data from [26,22,27,28]. (a) 3D-classical fluid c

cc

theory with zero wall-fluid interaction, (b) 3D-Ising fluid theory with zero wall-fluid interaction, (c) 3D-classical fluid theory with strong wall-fluid interactions, (d) 3D-Ising fluid theory with zero wallfluid interactions, (e) 2D-Ising fluid theory with 1/~.=o, (f) same as (e) with 1/~,=2cr. Atter these theoretical considerations, one should turn to experimental results. Materials made of unconnected pores are for instance MCM-41 [23] and porous silicon for instance [24]. A standard nanoporous disordered material (with connected pores, and a mean pore size -~ 5 nm) is Vycor glass. Experimental data of Tcc in these porous systems are rather scarce as it requires adsorption/desorption isotherms over a wide range of temperatures. This was done for Vycor by Burgess et al. in the case of Xe and CO2 [25]. Very recently, Morishige et al. were able to measure a complete set of adsorption isotherms for several simple adsorbates such Ar, N2, 02, etc. in MCM-41 materials of several pore diameters [26]. Figure 1 presents experimental data for Xe in Vycor and Ar in MCM-41 (D = 4.4 nm). Gathering all available data together, one can plot the reduced shitt of Tc~ (with respect to the bulk fluid critical temperature) with the reduced pore dimension (with respect to the adsorbate diameter) as originally proposed by Thommes et al. [27] and Sing et al. [28] using experimental data of simple fluids in CPG and MCM-41 samples. This is presented in Figure 2 for a collection of fluids including Ar, Xe, N2, 02, SF6, CO2, C2H2 in various MCM-41 and Vycor. Clearly, none of the 3D (classical or Ising) theories is able to describe the data for MCM-41 with pore diameter smaller than 5 nm. Data for larger MCM-41 pores and disordered materials are reasonably well described using the 3D-Ising theory with strong wall-fluid interactions. To our knowledge, there is no available T~c data for fluids confined in disordered porous materials with mean pore size smaller than 5 nm. Data for MCM-41 pores with diameter smaller than 5 nm are adequately described using the 2D-Ising fluid theory with 1/2=2tr. It seems that the mean pore size matters more than the degree of disorder as data for large MCM-41 mix with those for disordered systems. In this work, we aim at gaining insights at a molecular level on the hysteretic behavior of adsorption/desorption isotherms of several fluids confined in a rather large collection of pore models that includes single pores of various geometries (cylinders, ellipsoids, constricted pores and Vycor).

Effect of pore morphology and topology on capillary condensation

3. COMPUTATIONAL DETAILS We used the standard Grand Canonical Monte Carlo (GCMC) technique to simulate capillary condensation and evaporation of molecular fluids such as Ar and H20 in connected and independent silica nanopores over a large range of temperatures; we calculated for each system a set of adsorption/desorption isotherms to determine Tcc. Details about the procedure used to generate realistic models of the porous materials (cylinders of various diameters, ellipsoidal pores, contricted cylinders, and Vycor) and intermolecular potentials can be found in Refs. [ 12,15,29,30]. The reference to the pore dimension corresponds either to the diameter in the case of cylinders or to the mean size of the largest cavities as given by the pore length distribution [13] since these are the location of condensation/evaporation. We also tested different mean field approaches: (i) the non local DFT in the case of Ar and N2 confined to perfectly smooth silica cylindrical pores as reported by Ravikovitch and Neimark [31 ], (iO the coarse-grained lattice theory (CGLT) originally developed by Kierlik et al. [7]. In order to compare precisely our data to other simulation or theoretical works on smooth structureless pore models, one should subtract 2.2 A to the value of the pore dimension of atomistic systems as it corresponds to the OH bond length plus the hydrogen van der Wall radius. 4. RESULTS AND DISCUSSION We first discuss GCMC results for Ar and water on the effect of the pore morphology (cylindrical, ellipsoidal, and constricted pores) and topology (Vycor) on the temperature dependence of capillary condensation hysteresis in nanoporous materials. In Figure 3, we plot the relative decrease of the hysteresis critical temperature, Tcc, with respect to the bulk critical temperature, T3~ as a function of the pore size R0 (reduced to the size of the adsorbate c

molecule taken as the tabulated kinetic diameter or). We also report data taken from the literature for Xe, o'= 0.39 nm, in a Vycor sample having a largest cavity of 5 nm [32], and for water, tr= 0.28 nm, in cylindrical pores of a diameter 2.4 and 4 nm [33]. The first important result is that GCMC simulations (this work and others) do follow the 2D Ising behavior ATcc/T 30 = 2cr/Ro for small independent nanopores (Ro < 5.0 nm) in agreement with c

experiments (see Figure 2). adsorbate since both water shape (cylinders, ellipsoids, 41 materials with diameter

This scaling law seems to be independent (i) of the nature of the data and Ar data for argon fall on this master curve, (ii) of the presence of a constriction) of the (independent) pore. For MCMlarger than 5 nm, the relative shift of Tcc with respect to Trio is

closer to the 3D Ising model, again with no sensitivity to the adsorbate as both data for Ar, Xe and water are reported. Data for independent pores can be rationalized as a 3D to 2D transition. The locus of this transition, 2cr/Ro .~ 0.27 (see Figure 3) corresponds to the point above which (small pores) the number of adsorbate particles under the influence of the porewall potential (i.e. at a distance of approximately two adsorbate diameters) after capillary condensation is larger than that of particles in the pore core; the system adopts a 2D behavior. Reciprocally, below this point (large pores), there is a larger number of molecules in the core of the pore rather than the number of particles that feel the wall potential. If we now consider data for Vycor materials, it is clear that systems with large pores behave as large MCM-41. Again, this behavior seems to be independent of the nature of the adsorbate. Surprisingly, Vycor with smaller pores do not follow the scaling behavior as found for MCM-41 of smaller pore dimension. The reasons for such a behavior in the case of Vycor samples with large cavities is not yet clear. This can be interpreted as a strong effect of the pore disorder in

R.J.-M. Pellenq et al. 1 : this work, Ar, 2 : this work, Ar, 3 : this work, Ar, 4 : this work, Ar, 5 :this work, Ar, 6 : this work, Ar, 7 : this work, Ar, 8 : this work, Ar, 9 : this work, Ar, 10 : Brovchenko 11 12 13 14 15 16 17 18

cyl-2.8 nm cyl-3.6 nm cyl-4.4nm cyl-5.0 nm e11~-5.0-2.5 nrn cyl-5.0 nrn + constriction cyl-5.6 nm cyl-6.1 nm cyl-7.2nm et al, H20, cyl-2.4 nm [33]

: Brovchenko et al, H20, cyl-4.0 nm [33] : Gelb et al, Xe, cyl-5.0 nm [33] : this work, H20, Vycor-5.0 nm : this work, Ar, Vycor-2.5 nm : this work, Ar, vycor-3.47 nm : Pellenq et al, Ar, Vycor-5.0 nm [13] : Gelb et al, Xe, Vycor-5.5 nm [32] : Gelb et al, Xe, CPG-5.5 nm [32]

Fig. 3. Hysteresis critical temperature as a function of the ratio 2cr/Ro : GCMC results, the number of the plot reads: n~ author, adsorbate, susbstrate-pore,dimension. networked materials. For such systems, we expect the correlation length on the condensed fluid to be able to grow to larger values, on average, so that the depression of the pore critical point would be less. In Figure 4, we report DFT results for argon and nitrogen in cylindrical pores of various diameters. As the pore wall are structureless (perfectly smooth pore surface), DFT in the non local density approximation produces a spurious multi-layering of the fluid close to the pore wall resulting in steps in the adsorption isotherm prior to condensation that are not observed experimentally. The adsorbate-fluid (Lennard-Jones) potential parameters are taken from the literature and reasonably mimic the experimental adsorption and condensation/evaporation of nitrogen and argon in MCM-41 materials [34]. It is striking that DFT follows rather closely a 3D-Ising behavior far away from the experimental data in the case of small MCM-41. This result may be attributed to the genuine mean-field character of DFT. As in the case of GCMC and experimental results, DFT calculations are nearly independent of the nature of the confined fluid. 06

--

ATc/TOD=

[ 2*a / Ro]159 (3D Ismg)

0 5 -"=" ATc/Tc3D= 2"~/R o (2D I s m g ) 9 OFT [N2, e=94 45 K, ~=3.575 A]

o~

~176

. J J J

/

[]

~

;~

o~-0-~-

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T~

0o

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Fig. 4. Hysteresis critical temperature as a function of the ratio 2 cr/Ro" DFT results Finally, we employed the CGLT of Kierlik et al. [7], which is a simplified DFT (nearest neighbor), to model adsorption in our 5 nm Vycor sample. We used the adsorbate-substrate energy grid that was used in the GCMC simulations [13] with variable lattice spacing; all

Effect of pore morphology and topology on capillary condensation negative values (in-pore locations) of this 3D grid were set to zero, while positive values were set to unity at the beginning of each CGLT calculations. We first tested our CGLT code for a bulk fluid and verified that the critical temperature is simply k8 T9 / w f f = c/4, where wry is the fluid-fluid energy parameter and c the number of neighbors for a given lattice symmetry. Of course, this result is independent of the lattice spacing as expected for a bulk system. For a porous system, one adds a characteristic length that corresponds to the pore size. In a nearest neighbor approximation, a lattice theory will be equivalent to off lattice methods when the lattice spacing is chosen so that there is one molecule per site. In their work, Kierlik et al. chose a coarse-grained lattice in which a site is no longer one molecule but several. In other words, site occupancy is no longer zero or unity but any value in-between allowing to consider very large disordered porous systems. This implies the use of rather large lattice spacings (15 ./k for instance) [10]; for instance, a pore width in a lattice model of Vycor is described by three lattice sites only. In Figure 5, we show that for a given set of fluid-fluid, fluid-wall energy parameters, and a given lattice symmetry, the value of T~c strongly depends on the lattice spacing. Increasing the lattice spacing is equivalent to artificially increase the fluid-wall interaction hence decrease the fluid-fluid interaction rending the CGLT approach difficult to be mapped out onto off-lattice GCMC results or experiment, unless restraining the theory to a classical version allowing one molecule per lattice site (Figure 5a and b).

0.8 0.5

iii i GCMC(ref 13) ~

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'

o~176176 Oo

lattice spacing : --e-- 1.069 nm + 0.430 nm ........... 0.891 am

E

s

/tv

"-- 0.4 I-

J

~oo ....

L

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4

6

8

10 12 14 lattice spacing (A)

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16

18

20

0.0 -25

/~ / / ~

~

~0 0

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nm

/v'~ ~' ,#~A, ~

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00

, -20

, -15

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, -10

-5,

0,

Fig. 5a. Hysteresis critical temperature as a Fig. 5b. Adsorption/desorption isotherms for Ar function of the lattice spacing in CGLT. Lattice in Vycor-5 nm at T= 77 K, sff=120 K, Fv~=330K, spacing values are the one given in Fig. 5b. simple cubic lattice for various lattice spacings. Inset : comparison with off-lattice GCMC results. 5. CONCLUSION

We report a theoretical and simulation study of the temperature dependence of adsorption/desorption hysteresis for porous matrices having different morphologies and topologies. We used the off-lattice Grand Canonical Monte Carlo (GCMC) simulation technique combined with two Density Functional Theory approaches: the first in the non local approximation and the second being a coarse-grained lattice theory. We gained some insights on the concept of critical hysteresis temperature, Tcc, defined as the temperature at which the adsorption/desorption isotherm becomes reversible. GCMC-simulated Tcc for unconnected cylindrical, ellipsoidal, and constricted pores follow the experimental scaling law ~/O-Tc~T/O=2a/R established experimentally for MCM-41 silica materials with pore radius Ro

8

R.J.-M. Pellenq et al.

lower that 5 nm (or is the adsorbate diameter). In contrast, G C M C - o b t a i n e d Tcc for larger cylindrical pores and V y c o r samples with various mean pore sizes obeys a different relationship, in agreement with experiments. The determination o f Tcc (for a given the adsorbate in a given porous matrix) thus appears to be a reliable tool to characterize the m e a n pore size. W e infer that it can replace all Kelvin-equation based procedures that rely on m a n y u n k n o w n properties o f the confined fluids such as surface, tension, density, etc... W e also found that the genuine mean-field nature o f DFT gives a rather poor description o f the Tcc for systems such as Ar and N2 in cylindrical silica pores. W e show that providing that the lattice spacings is chosen close to the adsorbate size, the coarse-grained D F T lattice theory gives acceptable results compared to G C M C simulations o f critical hysteresis temperature. Since it is computationally less expensive that off lattice methods, it can be used as a quantitative predicting tool to study large systems: recently in the condition o f small lattice spacing, Salazar and Gelb [35] proposed an improved version o f C G L T that allows taking into account pair correlation effects rending C G L T even more effective.

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