Adsorption in Slit-Like Pores of Activated Carbons: Improvement of the

Langmuir 2002, 18, 4637-4647

4637

Adsorption in Slit-Like Pores of Activated Carbons: Improvement of the Horvath and Kawazoe Method E. A. Ustinov† and D. D. Do* Department of Chemical Engineering, University of Queensland, St. Lucia, Queensland 4072, Australia Received April 10, 2001. In Final Form: April 1, 2002 A new thermodynamic approach has been developed in this paper to analyze adsorption in slitlike pores. The equilibrium is described by two thermodynamic conditions: the Helmholtz free energy must be minimal, and the grand potential functional at that minimum must be negative. This approach has led to local isotherms that describe adsorption in the form of a single layer or two layers near the pore walls. In narrow pores local isotherms have one step that could be either very sharp but continuous or discontinuous benchlike for a definite range of pore width. The latter reflects a so-called 0 f 1 monolayer transition. In relatively wide pores, local isotherms have two steps, of which the first step corresponds to the appearance of two layers near the pore walls, while the second step corresponds to the filling of the space between these layers. All features of local isotherms are in agreement with the results obtained from the density functional theory and Monte Carlo simulations. The approach is used for determining pore size distributions of carbon materials. We illustrate this with the benzene adsorption data on activated carbon at 20, 50, and 80 °C, argon adsorption on activated carbon Norit ROX at 87.3 K, and nitrogen adsorption on activated carbon Norit R1 at 77.3 K.

1. Introduction The mechanism of pore filling has attracted the attention of investigators for decades. Among the continuum macroscopic approaches, there were those that are based on the Kelvin equation of capillary equilibrium. This equation provides the explanation of phenomena such as hysteresis and allows us to calculate the pore volume distribution function for mesopores.1-3 It is known, however, that such approaches using the Kelvin equation cannot properly describe adsorption in pores having dimension less than 10 nm. The situation was substantially improved in a thermodynamic approach of Broekhoff and de Boer by introducing the potential field near the pore wall.4 Nevertheless, extending this approach to pores of molecular dimension faces great difficulties. Obviously, more reliable results can be achieved only by direct numerical calculations using the molecular dynamics (MD) technique5,6 and grand canonical Monte Carlo (GCMC) simulations.7-12 They have been used for modeling of local isotherms in micropores and mesopores of different geometries for supercritical and subcritical adsorption. * To whom correspondence should be addressed. † On leave from Saint Petersburg State Technological Institute (Technical University), 26 Moskovsky Prospect, St. Petersburg 198013, Russia. (1) Gregg, S. J.; Sing, K. S. W. Adsorption Surface Area and Porosity; Academic: New York, 1982. (2) Ruthven, D. M. Principles of Adsorption and Adsorption Processes; Willey: New York, 1984. (3) Yang, R. T. Gas Separation by Adsorption Processes; Butterworths: New York, 1987. (4) Broekhoff, J. C. P.; de Boer, J. H. J. Catal. 1967, 9, 8. (5) Yoshioka, T.; Miyahara, M.; Okazaki, M. J. Chem. Eng. Jpn. 1997, 30, 274. (6) Miyahara, M.; Kanda, H.; Yoshioka, T.; Okazaki, M. Langmuir 2000, 16, 4293. (7) Peterson, B. K.; Gubbins, K. E. Mol. Phys. 1987, 62, 215. (8) Panagiotopoulos, A. Z. Mol. Phys. 1987, 62, 701. (9) Maddox, M. W.; Olivier, J. P.; Gubbins, K. E. Langmuir 1997, 13, 1737. (10) Samios, S.; Stubos, A. K.; Kanellopoulos, N. K.; Cracknell, R. F.; Papadopoulos, G. K.; Nicholson, D. Langmuir 1997, 13, 2795. (11) Gusev, V. Yu.; O’Brien, J. A.; Seaton, N. A. Langmuir 1997, 13, 2815. (12) Gusev, V. Yu.; O’Brien, J. A. Langmuir 1997, 13, 2822.

These methods are rather time-consuming, which has stimulated the development of the density functional theory (DFT) based on a mean-field approximation.12-21 The DFT is successfully used for determination of pore size distributions (PSDs) of different materials from the analysis of low temperature argon and nitrogen adsorption and sometimes from adsorption isotherms of other gases, such as methane and carbon dioxide. There are, however, some difficulties. For example, very often a small gap of about 1 nm has been observed in PSDs obtained by these methods, and such a gap is obviously a model-induced artifact. The reason for this abnormality was discussed in refs 18 and 20, and it is not quite clear how to overcome this problem. Another problem associated with the MD, GCMC, and DFT methods is the extension of them to complex molecules. For the CO2 molecule, the authors of ref 20 used a 3-center model. Then, for the benzene molecule, it would be logical to apply a 12-center model. It is not only the question of the ability of modern computers but rather the possibility that the twoparametric Lennard-Jones potential may not adequately describe the behavior of real substances (surface tension, liquid and gas compressibility, critical temperature, triple point, saturation pressure as a function of temperature, heat capacity, and so onssimultaneously). This means that such approaches are not sophisticated enough to reproduce adequately the behavior of an adsorbed phase and, certainly, should be further developed. Such a situation leaves room for developing semiclassical ap(13) Evans, R.; Marconi, U. M. B.; Tarazona, P. J. Chem. Soc., Faraday Trans. 2 1986, 82, 1763. (14) Seaton, N. A.; Walton, J. P. R. B.; Quirke, N. Carbon 1989, 27, 853. (15) Lastoskie, C.; Gubbins, K. E.; Quirke, N. J. Phys. Chem. 1993, 97, 4786. (16) Lastoskie, C.; Gubbins, K. E.; Quirke, N. Langmuir 1993, 9, 2693. (17) Olivier, J. P. J. Porous Mater. 1995, 2, 9. (18) Olivier, J. P. Carbon 1998, 36, 1469. (19) Neimark, A. V.; Ravikovitch, P. I. Langmuir 1997, 13, 5148. (20) Ravikovitch, P. I.; Vishnyakov, A.; Russo, R.; Neimark, A. V. Langmuir 2000, 16, 2311. (21) Dombrovski, R. J.; Hyduke, D. R.; Lastoskie, C. Langmuir 2000, 16, 5041.

10.1021/la010535l CCC: $22.00 © 2002 American Chemical Society Published on Web 05/10/2002

4638

Langmuir, Vol. 18, No. 12, 2002

Ustinov and Do

proaches, which are not so rigorous but do allow deeper insight into the phenomenon by relatively simple means. Horvath and Kawazoe (HK)22 proposed a very attractive model of instant filling of a slitlike micropore at a definite pressure in the bulk phase. This model was extended to cylindrical23 and spherical pores.24 This model suggests that adsorption was described as a sequential filling of micropores from the smallest accessible pores to larger ones as the pressure increases. Such a sequential filling allows each micropore to be either empty or completely filled. It is a simple way to characterize a pore size distribution (PSD) of activated carbon. To determine the pressure of instant filling, the authors implicitly used the principle of equality of the chemical potential in the bulk phase and that in the adsorbed phase. It was implied that the adsorbed phase has the same properties as those of the corresponding liquid, which is affected by the external potential exerted by the adsorbent. The principal difficulty with their approach is the variation of the potential over the pore volume. It is negative for most of the pore volume and becomes positive infinity near the micropore walls, so it is not quite clear what value of the potential is needed to calculate the pressure for the spontaneous filling of the micropore. To circumvent this, Horvath and Kawazoe used the mean value of potential energy. Of course, it is not thermodynamically correct because, being a partial value, the chemical potential must not include any integral parameters. From our viewpoint, this suggests that an improvement of the HK model on the basis of a more rigorous thermodynamic approach is necessary. Other attempts to improve the HK model were made recently.25,26 The authors have replaced the 10-4 solid-fluid potential in the original HK theory by the 10-4-3 potential with the same parameters that were used in the DFT approach. Besides, the mean potential energy they calculated from a density weighted average over the adsorbed phase accounted for the density dependence on the distance from the pore walls. Interestingly, such a combination of the DFT and HK methods has led to poorer predictions of the filling pressures.26 This shows that the main drawback of the HK theory is associated with the thermodynamic inconsistency, which does not allow the model to predict the pore wall wetting before the phase transition. The main aim of this paper is to reconsider the mechanism of adsorption in micro- and mesopores proposed by Horvath and Kawazoe. Here we account for the change of free energy of the adsorbate in narrow slit pores (which has not been done in the original HK model) and apply thermodynamics more correctly. This will allow us to improve the HK algorithm for determination of the PSD. 2. Model In the framework of the HK model, the adsorbed phase is assumed to be like a liquid in an external potential field but the surface tension is not taken into account. For such a case, the minimum free energy condition leads to the following:

RT ln(p/p0) ) u(z)

(1)

where u is the value of the potential at the interface separating the vapor phase and the adsorbed phase inside (22) Horvath, G.; Kawazoe, K. J. Chem. Eng. Jpn. 1983, 16, 470. (23) Saito, A.; Foley, H. C. AIChE J. 1991, 37, 429. (24) Cheng, L. S.; Yang, R. T. Chem. Eng. Sci. 1994, 49, 2599. (25) Lastoskie, C. M. Stud. Surf. Sci. Catal. 2000, 128, 475. (26) Dombrowski, R. J.; Lastoskie, C. M.; Hyduke, D. R. Colloid Surf., A 2001, 187-188, 23.

a slitlike pore. However, Horvath and Kawazoe have replaced this potential by the following mean value: (H-d)/2 u dz ∫-(H-d)/2

1 H-d

(2)

Here H is the width of the pore, d is the sum of the diameter of a carbon atom and that of the adsorbate molecule, and z is the distance from the center of the pore. According to eq 1, the process of filling of a given pore starts from the value of pressure corresponding to the minimum potential, which lies at the pore center for small pores or near the walls for larger pores. At this pressure and depending on pore width, one or two liquid layers will appear inside a pore. Further increase in pressure leads to an increase in the thickness of the layers and, as a consequence, an increase in the amount adsorbed. So, a local isotherm should increase in a continuous fashion over a definite range of pressure, meaning that an abrupt change in the isotherm is not allowed. Nonetheless, Horvath and Kawazoe have grasped the main feature of adsorption mechanismsmicropore filling occurs over a very narrow range of pressure, which is known to be confirmed by molecular simulation methods and the DFT. In fact, this is the obligatory condition for the isotherms measured at various temperatures to be congruent, which is usually observed in cases of adsorption in activated carbon (so-called temperature invariance of characteristic curves). This peculiarity is a consequence of the mechanism of sequential filling of adsorption space elements. In particular, in the Polanyi theory that leads to the temperature invariance, such elements are thin layers between neighboring equipotential surfaces. This phenomenon is rather similar to the mechanism of capillary condensation. For this reason, Nguyen and Do27 have applied a combination of the BET theory and the Kelvin equation. At low pressures, adsorption occurring at micropore walls obeys the BET equation, which is followed by a spontaneous filling of the micropore when the threshold pressure has been reached. A similar argument was also used by Dobruskin.28 The widely used slitlike model for micropores in activated carbon implies that there could be many molecules in each micropore because of the infinite extent of the micropore walls. This means that, despite the molecular dimension of the pore width, such micropores may be considered as statistically independent macroscopic subsystems, which form a grand canonical ensemble. This allows us to use the grand canonical distribution for a group of equal size micropores. For the average number of molecules in a pore, one can write the following equation (see, for example, ref 29,29 Chapter 2):

d ln ξ

〈j〉 ) λ

dλ

∑j jλjQj

) ξ-1

(3a)

where

ξ)

∑j λjQj

λ ) exp(µ/kBT) Qj ) exp(-Fj/kBT)

(3b)

(27) Nguyen, C.; Do, D. D. Langmuir 1999, 15, 3608. (28) Dobruskin, V. H. Langmuir 1998, 14, 3840. (29) Davis, H. T. Statistical Mechanics of Phases, Interfaces and Thin Films; Wiley-VCH: New York, 1996.

Adsorption in Slit-Like Pores of Activated Carbons

Langmuir, Vol. 18, No. 12, 2002 4639

Here j and 〈j〉 are the number of adsorbate molecules in a pore and its mean value, respectively, ξ is the grand partition function, related to a single pore, Qj is the canonical partition function, µ is the chemical potential, Fj is the Helmholtz free energy of j molecules of adsorbate, T is the temperature, and kB is the Boltzmann constant. In such form, eqs 3 are mainly applied to zeolites, as their regular structure suggests an ideal example of the grand canonical ensemble of open systems (cavities). This was done for the first time by Bakaev.30 Later this approach was developed to obtain simplified isotherms and thermodynamic properties of an adsorbed phase and to predict multicomponent adsorption.31-37 In the context of this paper, we turn to eqs 3 to derive a condition, which, certainly, has not been overlooked, but draws insufficient attention. This condition can be easily obtained by noting that the first term in the grand partition function is equal to 1 because the Helmholtz free energy of zero molecules is equal to 0. In the macroscopic limiting case when the number of molecules j in a pore is large, all other terms in the grand partition function ξ are either much greater than 1 or negligibly small compared to 1. Therefore, the following two situations can be distinguished. In the first case, the maximal term in the sequence {λiQi} is greater than 1. This means that the state of the adsorbed phase in a slit pore as a macroscopic subsystem exactly corresponds to the minimum of Fi - iµ. Consequently, we obtain the well-known condition that the grand potential Ω ) Fi - iµ must be minimal for the equilibrium state (at constant volume of the system, pressure, and temperature). Since i is very large for a macroscopic system, it can be considered as the continuous variable, and the minimum of the grand potential is ensured by

dΩ d(Fi - iµ) ) )0 di di

(4)

This condition is equivalent to the following equation:

µ)

dFi di

(5)

The right hand side of the above equation is the chemical potential of the adsorbed phase, because the derivative of the Helmholtz free energy is taken at constant volume and temperature. So, eq 5 states the equality of the chemical potentials in different parts of an equilibrium system, which is physically expected. In the second situation, the maximal term in the sequence {λiQi} is less than 1. In this case, one can easily see from eq 3 that the amount adsorbed will be close to zero. This means that adsorption will not occur if the minimum value of the grand potential Ω is positive. Consequently, one can write the following equations describing the mechanism of adsorption: (30) Bakaev, V. A. Dokl. Akad. Nauk SSSR 1966, 167, 369. (31) Ruthven, D. M.; Loughlin, K. F.; Holbarow, K. A. Chem. Eng. Sci. 1973, 28, 701. (32) Ruthven, D. M. AIChE J. 1976, 22, 753. (33) Shekhovtseva, L. G.; Fomkin, A. A.; Bakaev, V. A. Bull. Acad. Sci. USSR, Div. Chem. Sci. 1987, 36, 2176. (34) Shekhovtseva, L. G.; Fomkin, A. A. Bull. Acad. Sci. USSR, Div. Chem. Sci. 1990, 39, 867. (35) Ustinov, E. A.; Polyakov, N. S.; Stoekli, F. Bull. Acad. Sci. USSR, Div. Chem. Sci. 1998, 47, 1873. (36) Ustinov, E. A.; Klyuev, L. E. Adsorption 1999, 5, 333. (37) Ustinov, E. A.; Polyakov, N. S. Russ. Chem. Bull. 2000, 49, 1011.

µ)

dFi di

Fi - iµ ) Ω e 0

(6)

Hence, the condition for adsorption is that the grand potential functional must be not only minimal but also negative. Now let us apply this approach to the analysis of adsorption in slitlike pores. In this analysis we make the following assumptions. (1) The adsorbed phase may be represented as one or two liquidlike layers between two parallel walls of a pore. (2) The layer thickness is the distance between two parallel planes confining centers of molecules, with the density of the layer being constant over its thickness and equal to that of the corresponding liquid. (3) The amount of free adsorbate phase inside a pore is negligible compared to that of the adsorbed layer. (4) The adsorption potential obeys the equation of Steele.38 The assumption of constant density inside a thin layer is the most vulnerable, as it is known that the density varies over the layer thickness. We resort to this simplification with the aim to elucidate factors that are responsible for the main features of the adsorbed phase behavior. The first part of assumption 2 corresponds to the only correct way to determine density as the limit of the ratio of the number of molecules to the volume confining their centers. Such a definition is used in molecular simulations and the DFT and helps to overcome the problem that the layer thickness cannot be less than one molecular diameter. 2.1. Helmholtz Free Energy of the Adsorbed Phase in a Pore and the Grand Potential. For further derivations, it is necessary to determine the Helmholtz free energy of the adsorbed phase. We restrict our consideration to subcritical species and assume that the vapor phase behaves as an ideal gas. For this reason, we need not introduce fugacities. The Helmholtz free energy of an adsorbed layer can be represented as the sum of three terms. The first one is the Helmholtz free energy of the corresponding bulk liquid, which is in equilibrium with the vapor. The second term is associated with the difference in the molecular structure between the adsorbed layer and the liquid. This term can include the work that is necessary to form free surfaces, which are the interfaces between the liquid layer and the vapor phase. In the general case, this additional work arises as a result of a lack of bonds between molecules in the adsorbed phase compared to those in the corresponding bulk liquid. In the macroscopic limit of the thick layer, this term may be expressed as the product of the surface tension γ and the surface area S. Furthermore, this term includes a change in the entropy and the work of adsorbate compression, because the pressure in the adsorbed phase may achieve several hundred or even thousand atmospheres.39 The last term contributing to the Helmholtz free energy of the adsorbed phase is the adsorbate-adsorbent interaction energy. For further derivations, we will use molar values for thermodynamic functions and number of moles instead of number of molecules. Either approach will obviously lead to the same results. The molar Helmholtz free energy of the bulk liquid is given by (38) Steele, W. A. The Interaction of Gases with Solid Surfaces; Pergamon Press: Oxford, 1974. (39) Ustinov, E. A. Russ. J. Phys. Chem. 1988, 62, 1575.

4640


Ustinov and Do

F h L ) [µ°(T) + RT ln(p0)] - p0v* Here p0 is the saturation pressure and v* is the liquid molar volume. The square bracket is the chemical potential of the bulk liquid, with µ°(T) being the standard value of the chemical potential. For subcritical species, the product p0v* is usually very small compared to that for vapor (which is close to RT). It can be neglected without loss of accuracy. Thus, the first contribution to the Helmholtz free energy of the adsorbed layer is

FL ) q[µ°(T) + RT ln(p0)] where q is the number of moles in the pore. The molar amount of the Helmholtz free energy change, which is associated with the structural change of adsorbate compared to the bulk liquid, is denoted as ∆F h s. The last term contributing to the Helmholtz free energy of the adsorbed phase is the potential energy U of the adsorbed layersolid interaction, which can be calculated by integrating the potential energy u over the whole volume of the adsorbed layer. Consequently, the Helmholtz free energy of the adsorbed phase may be expressed as follows:

hs + U F ) q[µ°(T) + RT ln(p0)] + q∆F

(7)

The chemical potential of the bulk phase is given by

µ ) µ°(T) + RT ln(p)

(8)

The grand potential Ω ) F - qµ is obtained by combining eqs 7 and 8 to give

Ω ) q∆F h s + U - qRT ln(p/p0)

(9)

As followed from eq 6, the state of an adsorbed phase in a pore must correspond to the minimum of the grand potential at constant pressure and temperature, but the nonzero value of the amount adsorbed q is possible only when the grand potential Ω is negative at its minimum. 2.2. Fluid-Solid Potential Interaction. The fluidsolid potential interaction U is obtained from integration of the Steele 10-4-3 potential:38

5 u ) /sf[φ(H/2 + z) + φ(H/2 - z)] 3 φ(x) )

10 σ4sf σ4sf 2σsf 5x10 x4 3∆(0.61∆ + x)3

(10)

In this equation z is the distance from the pore center; H is the pore width; ∆ is a distance between adjacent graphite lattice layers; and

The expression for the potential U in eq 9 calculated by eq 11 will be different in two cases of pore filling, depending on whether there is the single layer in the center of the pore or there are two layers near the pore walls. This will be elaborated later. 2.3. Correction of Helmholtz Free Energy for a Thin Layer. It is common that the formation of a free surface between the vapor and liquid phases leads to an increase in the Helmholtz free energy because of the deficiency of intermolecular interactions close to the surface compared to in the bulk liquid. In the macroscopic limit when the liquid film is thick enough, the Helmholtz free energy change can be easily expressed by the surface tension γ and the film thickness h:

∆F h s ) 2γv*/h

The factor 2 is associated with the two surfaces bounding the adsorbed layer. The situation becomes much more complicated when the layer thickness is of molecular dimension. In many theoretical investigations of filling of cylindrical mesopores, thermodynamic approaches often rely on the dependence of the surface tension on the meniscus curvature40,41 (the Tolman equation). As for flat surfaces, the common approach has been that no corrections are needed. It seems obvious that in very thin films the surface tension should decrease with a decrease in the layer thickness, but we are not aware of any correlations for this dependency in the literature. Moreover, the idea of surface tension loses its physical meaning when the layer thickness is comparable to a molecular diameter. Therefore, the Helmholtz free energy change ∆F h s must be used instead of the application of the surface tension concept. To estimate the dependence of ∆F h s on layer thickness, we consider the following situation. Let us imagine a continuous layer consisting of two parts: A and B. The larger part A exerts a potential field on part B, which is positioned at a distance h0 from the former, corresponding to a mechanical equilibrium. The thin layer B has a thickness h (the distance between two parallel planes confining centers of molecules). The point of the mechanical equilibrium corresponds to the minimum of the potential energy of layer B. As a first approximation, the potential energy of a molecule at a distance z from layer A comprising molecules of the same type can be calculated by summing the LJ potential energies of interaction between the molecule and all molecules within layer A. Further integration of this potential with respect to z from h0 to h0 + h gives the potential energy of the thin layer B of thickness h. The absolute value of this potential may be considered as that numerically equal to the Helmholtz free energy change. Details of the derivation are given in the Appendix. The resulting expression for ∆F h s is given by the following formula:

6 /sf ) πFssfσ2sf∆ 5

h∆F h s ) 2γv*C[s21 - s22 - (s81 - s82)/30]

where Fs is the number density of carbon atoms per unit volume and sf is the potential well depth. The integration of u corresponding to eq 10 with respect to z can be easily obtained in analytical form:

ω(x) )

x

-

σ4sf 2

3x

-

σ4sf 6∆(0.61∆ + x)2

s31 - s32 - (2/15)(s91 - s92) ) 0 where

s1 ) σff/h0, s2 ) σff/(h + h0)

∫0zu dz ) 35/sf[ω(H/2 - z) - ω(H/2 + z)] 10 2 σsf 45 9

(12)

C ) (4/3)(2/15)1/3 (11)

(40) Tolman, R. C. J. Chem. Phys. 1949, 17, 333. (41) Merlose, J. C. Ind. Eng. Chem. 1968, 60, 53.

(13)


Langmuir, Vol. 18, No. 12, 2002 4641

This set of equations defines the conditions of existence of a thermodynamically stable layer under the influence of an external potential field. Solution of this set suggests that the layer is stable over the range p1* e p e p0. The lower limit p1* is the pressure satisfying eq 15a and eq 15b when the left hand side has been equated to the right hand side. Here the subscript 1 denotes the lower limit of pressure for the case of a single adsorbed layer between the two pore walls. For any pressures less than p1*, the inequality of eq 5b is violated and hence adsorption is not possible. Once the pressure p1* has been reached, further adsorption with pressure is described solely by eq 15a and the inequality of eq 15b is always ensured. Equations 15 are equivalent to the condition that an increase in the amount adsorbed leads to an increase in the change in the molar Helmholtz free energy: Figure 1. Dependence of the Helmholtz free energy change on the layer thickness calculated by eq 13 for benzene at 20 °C. The asymptotic region where the ordinate of this curve is constant corresponds to macroscopic behavior of the liquid film.

Such a scheme has never been considered in the literature, though the interaction (without repulsive forces) between two thin layers situated on the opposite walls of a pore was accounted for by the Hamaker constants.42,43 The diagram calculated by eq 13 for the case of benzene at 20 °C is presented in Figure 1. It is seen from the figure that the behavior of a thin layer becomes quite close to that of a macroscopic film when the layer thickness exceeds approximately five molecular layers. As will be seen below, the mechanism of pore filling is different depending on the pore width. In narrow pores the adsorption occurs without prewetting on the pore walls. In wider pores the formation of two thin layers on the walls is followed by the first-order phase transition in the inner volume of the pore. For this reason we consider the pore filling separately for small pores and relatively large pores. This distinction between these pores is the difference in the adsorption mechanism we just described. 2.4. Case of Narrow Pores. In a pore which is narrow enough (small micropore), the potential energy distribution with respect to distance may have one minimum located at the center of the pore or two symmetrical minima near the pore walls. In either case, only one liquid layer may exist in the pore, for reasons that we will discuss below. The amount adsorbed q in the pore is hSF, where F is the mean value of the molar density of the liquidlike adsorbed phase, h is the layer thickness of the adsorbed layer, and S is the surface area of the pore wall. In this case, the grand potential can be written as follows:

Ω ) SF{h∆F hs +

h/2 u dz - hRT ln(p/p0)} ∫-h/2

(14)

Adsorption is possible when the grand potential is minimum and its minimum is negative (eq 6). For the grand potential of eq 14, this gives rise to the following two equations:

∂h∆F hs + u(h/2) RT ln(p/p0) ) ∂h 2 h/2 RT ln(p/p0) g ∆F hs + u dz h 0

∫

∆F h ) ∆F hs +

∫0h/2u dz

2 h

(16)

In this context there is an interesting question concerning the filling pressure. Of course, eq 15a itself could determine the lower limit of pressure at which the pore is partially filled with adsorbate molecules. However, even if eq 15a is satisfied, the inequality of eq 15b might be violated. Such a violation leads to a decrease of ∆F (eq 16) upon the addition of adsorbate, and therefore adsorption under such conditions is not feasible. Once the pressure p ) p1* has been reached, the pore is filled spontaneously with adsorbate molecules. This result (so-called 0 f 1 monolayer transition) is confirmed by GCMC and DFT simulations. Simultaneously solving eqs 15 will yield a local isotherm in the case of a single layer between two parallel walls. The shape of the isotherm depends on the pore width and the variation of ∆F h s with layer thickness. Under certain conditions, this shape can be continuous. This occurs when the pressure corresponding to the minimum of the potential energy satisfies the inequality of eq 15b. This pressure is the pressure for the onset of adsorption, and the film thickness will increase continuously from zero as the pressure increases, giving rise to the continuous variation of the local adsorption isotherm. This situation would not occur if the surface tension was assumed constant because, in this case, h∆F h s would be constant, while the right hand side of the inequality (eq 15b) increases with h. Thus, if the pressure p is less than p1*, the pore is empty and adsorption is impossible. 2.5. Case of Wide Pores. In the case of pores which are wide enough (mesopores and partly micropores), there are two local minima close to the walls. In this case, two liquid layers may exist in the same pore. Once they occur, these layers increase in thickness with pressure and the distance between them decreases. The two layers will coalesce into one layer once a definite pressure has been reached. For each layer (both layers are symmetrical) the grand potential is given by

Ω ) SF{h∆F hs +

∫zz u dz - hRT ln(p/p0)} 2

1

(17)

(15a) (15b)

(42) Derjaguin, B. V.; Churaev, N. V. J. Colloid Interface Sci. 1976, 54, 157. (43) Curry, J. E.; Christenson, H. K. Langmuir 1996, 12, 5729.

Here z1 and z2 are the coordinates of the internal and external boundaries of the adsorbed layer, respectively, with h ) z2 - z1 being its thickness. The grand potential must be negative and minimal simultaneously (otherwise the pore will be empty). Following the same procedure as that described in the previous section, the following equations for the case of wide pores are obtained:

4642


RT ln(p/p0) )

Ustinov and Do

|

∂h∆F hs ∂h

+ u(z2)

h)z2-z1

u(z1) ) u(z2) hs + RT ln(p/p0) g ∆F

(18a) (18b)

∫zz u dz

1 z2 - z1

2

1

(18c)

Inequality eq 18c requires that an increase of the amount adsorbed is accompanied by an increase in the molar Helmholtz free energy change; otherwise adsorption is not feasible. The filling pressure in this case is given by

hs + RT ln(p2*/p0) ) ∆F

∫zz u dz ) min

1 z2 - z1

2

1

(19)

Here the subscript 2 denotes the lower limit of pressure for the case of double layers between two pore walls. Once the pressure p2* has been reached, further adsorption is described by eq 18a. However, it is necessary to keep in mind that eqs 18 do not always have roots. There will be certain conditions when both eqs 15 for a single layer and eqs 18 for two layers have roots. The question is then raised about which solution is physically accepted? The choice is based on the value of the Helmholtz free energy. The solution corresponding to lower Helmholtz free energy will be accepted. The system of eqs 18 together with eqs 10 and 11 for adsorption potential and eq 13 for the structural part of the Helmholtz free energy change can be solved. The lower limit of pressure p2* that satisfies eqs 18a and 18b, and eq 18c being equal to zero is the filling pressure in the case of two layers for a relatively wide pore when the single layer cannot exist because such pressure does not satisfy eqs 15. However, further increase of pressure in the bulk phase may lead to the situation when both eqs 15 and eqs 18 have roots. So the question is whether there will be a transition from two layers to a single layer. To see whether such a transition is possible, we compare the Helmholtz free energies corresponding to these cases. If this transition is accompanied by a decrease in the free energy, it will be thermodynamically possible. However, this situation is not quite obvious. Two local minima in the free energy are divided by the local maxima, which is similar to a potential barrier that has to be overcome in order for the transition to occur. Undoubtedly, in the macroscopic limit, the case corresponding to the lower value of the Helmholtz free energy would be metastable. If the distance between the two layers is of molecular dimension, fluctuations and their mutual attraction will result in spontaneous pore filling. Of course, it is possible only if the Helmholtz free energy of one layer is less than that for two layers at the same pressure in the bulk phase. It is difficult to draw the border between the microscopic and macroscopic systems exactly, but we can rely on MC and DFT simulation results, which show that in slitlike pores such a transition takes place in all pores investigated. So, we assume in this paper that metastable states are not allowed. During decreasing pressure, the single liquid layer exists in the pore. But at the moment when the bulk pressure equals the lowest limit of stability of a single layer p1*, this single layer will split into two layers and capillary evaporation will occur. 2.5.1. Kelvin Equation. It is of interest to elucidate whether the lower limit of the single layer stability when the grand potential is equal to zero corresponds to the Kelvin equation. As is shown below, this capillary evaporation can be easily explained by considering the

following hypothetical situation. Let us consider a pore that is large enough so that the adsorption potential can be approximated by a δ-function being operative in an infinitely narrow range of z. Consequently, the liquidsolid interaction is implied to occur directly on their boundaries. This is a classical case of the application of the Kelvin equation. In the macroscopic case under consideration (dγ/dh ) 0), eq 15b can be rewritten as follows.

hRT ln(p/p0) g 2γv* + 2U

(20)

where U is the potential energy interaction of the liquid film with one of the walls (related to unit area):

U)

∫0h/2u dz

The layer boundary is affected by repulsive forces, which change very sharply with the distance from the wall. For this reason, the potential energy U in this case of a wide pore is nearly constant. What does inequality 20 mean? The sum of the two terms on the right hand side is negative and can be expressed by means of a contact angle φ:

γv* + U ) -γv* cos φ

(21)

Supposing that the layer thickness h is close to the pore width H (due to its large value), eqs 20 and 21 can be replaced by the following equation:

γv* cos φ rRT

ln(p/p0) g -

(22)

where r ) H/2. This is the Kelvin equation for a cylindrical meniscus of radius r. Despite the infinite size of the pore walls, which is assumed in the model, it is correct to consider the entrance to the slitlike pore. It is obvious that in the case of a single layer inside the pore the meniscus at the entrance has a cylindrical form. Hence, eq 22 expresses the condition of existence of a cylindrical meniscus. A decrease of pressure leads to a decrease in the meniscus radius. However, this radius cannot be less than the pore half width r, and therefore, when a threshold pressure has been reached, capillary evaporation will occur. Consequently, the approach is confirmed by the classical theory of capillary condensation, but of course, in practical cases, when the adsorption potential is distributed over the volume of a pore, it is more fundamentally sound to apply eqs 15 or 18 rather than to resort to some idealized simplification. 2.6. Pore Size Distribution. Knowing the local isotherms for pores of various sizes, the total isotherm can be obtained by summing the amounts adsorbed of all pores. As is known, this is an ill-posed problem, which can be solved by a method of regularization44 and is repeatedly discussed nearly in each paper devoted to the PSD analysis. A review of methods for determination of physical pore width may be found in ref 19. It is obvious that the space between two neighboring graphite planes of distance ∆ is not a pore. Otherwise, we should consider that the pore volume always equals the whole volume of a carbon adsorbent. If the distance between neighboring graphite planes exceeds ∆ (which is close to σss), the space between these two planes will constitute a pore. It should be noted, however, that the distance of the closest approach of a molecule to the graphite wall corresponds to a plane at (44) Tikhonov, A. N. Dokl. Akad. Nauk SSSR 1943, 39, 195.


Langmuir, Vol. 18, No. 12, 2002 4643

Figure 2. Filling pressure as a function of pore width. Benzene-slitlike pore at 20 °C. (Solid line) Dependence corresponding to the minimum of the Helmholtz free energy of a single or double layer. (Dashed line) Pressure of the two layerone layer transition.

which the potential energy is zero. This distance is greater than σss/2, which was emphasized in ref 45. We completely agree with the authors of that paper, but the problem is that this distance depends on the collision diameter of the adsorbate molecule. This means that the pore volume will be overestimated if the physical width is taken as H - ∆. Thus, the total amount adsorbed can be calculated by following equation:

a)

W0 v*

∫∆/2∞ f(r) θ(r, p/p0, T) dr

(23)

Here the local isotherm is given by

θ(r, p/p0, T) )

{

p1* e p h/(H - ∆), 2(z2 - z1)/(H - ∆), p2* e p < p1* (24) 3. Results

To analyze the main features of a single pore filling, we consider the case of benzene adsorption in activated carbon PAU46 at 20, 50, and 80 °C. The values of the collision diameter σff and the interaction energy ff/kB are taken as 0.5349 nm and 412.3 K. The values of ∆, Fs, σss, and ss/kB for the carbon graphite lattice are well-known38 and are taken as 0.335 nm, 114 nm-3, 0.34 nm, and 28 K, respectively. For adsorbate-adsorbent interaction, we accept that /sf equals 32 kJ/mol and σsf ) (σs + σff) ) 0.4374 nm. For the surface tension γ (as the asymptotical value in the macroscopic case) and the molar volume v* at 20 °C we use the values 0.02888 N m-1 and 8.68 × 10-5 m3 mol-1, respectively. The variation of the surface tension with temperature is described well with a linear function, with the rate of change of the surface tension with T given by dγ/dT ) -1.285 × 10-4 N m-1 K-1. The molar volume at different temperatures is considered to be the same as that for the bulk liquid. Figure 2 presents the dependence of filling pressure on the pore width at the instant when one or two liquid layers appear inside a pore. One can see from the figure that adsorption does not occur in pores having width less than 0.747 nm, though this pore is not so narrow because the (45) Kaneko, K.; Cracknell, R. F.; Nicholson, D. Langmuir 1994, 10, 4606. (46) Astakhov, V. A. Theoretical and Experimental Investigations of Adsorption Processes. Thesis, Leningrad, LTI, 1972.

Figure 3. Local isotherms for the system benzene-slitlike pore at 20 °C. (Solid lines) from left to right: H (nm) ) 0.88, 0.92, 0.96, 1.0, 1.08, 1.16, 1.24, 1.32, and 1.4. For pore widths below 0.866 nm, the filling pressure decreases with increasing pore width. (Dashed lines) from right to left: H (nm) ) 0.8 and 0.84.

minimum of the potential energy at the center of the pore is -13.62 kJ/mol. So attractive forces are dominating. Adsorption cannot occur if p/p0 < 2.25 × 10-12. Once this pressure is achieved, adsorption starts from micropores of 0.866 nm width. A further increase in pressure will initiate adsorption in pores either larger or smaller than 0.866 nm. The initiation of adsorption in smaller pores at higher pressure is attributed to the strong repulsive forces in those pores. When the pore width is greater than 1.06 nm, two situations are possible. In the first situation, the adsorption is initiated with the formation of one layer while, in the second situation, the adsorption is started with the instant formation of two layers. However, for pores having width greater than 1.06 nm but less than 1.16 nm, the grand potential corresponding to the single layer at the start of adsorption is less than that for the double layers. This means that the adsorption is initiated with a single layer and remains when the bulk pressure increases. When the pore width is greater than 1.16 nm, the two layers in a pore are initiated because the grand potential for this situation is lower. The dashed line in Figure 2 corresponds to the lower limit of stability of a single layer. Once the bulk pressure reaches the pressure given by the dashed line, the empty space between the two layers will be immediately filled and a single layer will form because it leads to a lower value of the Helmholtz free energy. Calculated local isotherms for relatively small pores (pore width ranging from 0.8 to 1.4 nm) are presented in Figure 3. The isotherm is described as the layer thickness scaled against the pore width. Dashed lines are plotted for pores of width 0.8 and 0.84 nm, with the isotherm shifting to lower pressure when the pore width is increased from 0.8 to 0.84 nm. This arises because of the lower potential energy in 0.84 nm compared to 0.8 nm pores, resulting from the greater repulsive force in 0.8 nm pores. For pores having width greater than 0.866 nm, the pore filling pressure increases with pore width and the local isotherm is shifted to the right, as expected. It is interesting to note that for pores having width falling in the narrow range from 0.96 to 1.16 nm, the thickness of the layer at the instant when adsorption starts is nonzero. For example, for the pore of 1.1 nm width, the layer thickness abruptly increases from 0 to 0.292 nm at the pressure p/p0 ) 8.38 × 10-8. This is because a thinner layer than 0.146 nm would have too large a free surface per unit mass and,

4644


Figure 4. Local isotherms for the system benzene-slitlike pore at 20 °C. Reading from left to right: H (nm) ) 1.2, 1.4, 1.6, 1.8, 2, 3, 4, and 6.

consequently, would require too high an amount of work to produce the change in molar Helmholtz free energy, which could not be compensated by a decrease in the potential energy. For pores having widths up to 1.16 nm, only a single layer can exist in a pore, even though, for pores having width greater than 0.99 nm, there are two minima in the potential energy distribution. However, in wider pores, two layers appear at the first instant of adsorption because at this point the existence of the single layer is not possible (i.e. no solution of eqs 15). For wider pores in the region of width from 1.2 to 6 nm, various local isotherms are plotted in Figure 4. In each curve there is a threshold pressure at which capillary condensation occurs. For bulk pressures greater than this pressure, both a single layer and double layers could exist, but the former, corresponding to a smaller value of the Helmholtz free energy, provides the stable equilibrium solution. As is easily seen, the section of the isotherms associated with the two layers is continuous and tends to an asymptote with an increase of pore width. This asymptote is simply the isotherm of nonporous carbon black. This behavior of local isotherms is in good agreement with results obtained by sophisticated methods such as MC simulations and DFT. The distribution of potential energy with respect to distance normal to the pore wall for the case H ) 1.1 nm is shown in Figure 5. The thickness of the adsorbed layer at the moment of its inception is equal to the distance between vertical dashed lines. The possible physical range of ln(p/p0) occurs in the region bounded by the two horizontal lines. It should be noted that the potential energy at the liquid-vapor boundary is not equal to zero when the pressure in the bulk phase approaches the saturation pressure p0. This is because of the nonconstant value of the product h∆F h s in accordance with eq 13. An analogous potential energy distribution is presented in Figure 6 for a wider pore of 2 nm width. In this case, two symmetrical layers close to the pore walls will appear at a definite pressure in the bulk phase because the single layer can exist only at higher pressures. A further increase of pressure will lead to an increase in the thickness of these layers up to the moment when the existence of a single layer is allowed for by eqs 15, which corresponds to the dotted line in the figure. When the pressure decreases, the single layer inside a pore retains its entity to the moment when p is equal to the pressure corresponding to the lower limit of stability of a single layer. When this point is reached, this layer is spontaneously divided into two layers.

Ustinov and Do

Figure 5. Distribution of the potential energy over the normal to the pore walls for benzene at 20 °C. The pore width H equals 1.1 nm. The vertical dashed lines show the coordinates of the layer borders at the moment of its appearance. The dashdotted line corresponds to the saturate pressure in the bulk phase. The region of existence of the liquid phase in the pore is between the dotted and dash-dotted lines.

Figure 6. Distribution of the potential energy over the normal to the pore walls for benzene at 20 °C. The pore width H equals 2.0 nm. The dashed lines show the coordinates of the layer borders at the moment of the two layer-one layer transition when u/RT ) -6.995 (dotted line). Other notations are the same as those in Figure 4.

The isotherm equation (eq 23) is tested against the experimental data of benzene on activated carbon at three different temperatures: 20, 50, and 80 °C. In the PSD analysis we define the accessible pore volume as the one bounded by two planes at which the potential energy of the solid-fluid interaction is zero. Such a definition of the the accessible volume of a pore is numerically close to that proposed in the original HK theory. Figure 7 shows the pore size distributions obtained separately for each isotherm. Despite the relatively wide range of temperature, the PSD curves are very close to each other, with the main feature of two distinct peaks being observed for each curve. What is interesting in the figure is the narrow gap between these peaks, occurring at approximately 1 nm pore width. The same result is nearly always obtained by DFT. The reason for this feature is associated with the change of the mechanism of pore filling. In pores having width H - ∆ greater than 1 nm, the onset of adsorption corresponds to the formation of two layers close to the pore walls followed by a 3-D phase transition to a single layer at higher pressure. In such pores the effect of overlapping of potentials exerted by the opposite walls is not so pronounced as it is in the case of smaller pores and,


Figure 7. Pore size distribution for benzene adsorption on activated carbon PAU46 obtained for different temperatures. Temperature (°C): (solid line with points) 20; (dashed line) 50; (dash-dotted line) 80.

Figure 8. Benzene adsorption isotherm on activated carbon PAU:46 (points) experiment; (solid line) approximation by the theory. Temperature (°C): (b) 20; (O) 50; (0) 80.

consequently, the pore filling pressure varies weakly with pore width. What this means is that when a certain pressure p* has been achieved and with a small increase in pressure many pores will be filled with adsorbate molecules because of the weak variation of the pore filling pressure with the pore width. If this happens, there should exist a large increase in the adsorption isotherm around the pressure p*. But such an increase is not observed in experimental adsorption isotherms. Therefore, a gradual change in the adsorption isotherm requires the absence of certain pores in the neighborhood of 1 nm width. The question is whether the absence of pores having widths about 1 nm reflects a real feature of carbon materials or is an artifact of model approximations and assumptions. Most likely, this is an artifact resulting from the model of slit pores formed by semi-infinite planes of homogeneous graphite layers, which was emphasized by Olivier.18 Figure 8 shows the excellent fit of the theory for isotherms plotted at different temperatures. The determined value of the total pore volume W0 is 0.39 cm3/g at 20 °C. Some part of the volume corresponding to pores having width less than 0.747 nm is inaccessible to benzene molecules. The comparison of our approach with the HK method is presented in Figure 9 for the case of nitrogen adsorption in activated carbon Norit R1 at 77.35 K. The experimental data were kindly given to us by P. Harting (Institute of Non-Classical Chemistry, Leipzig, Germany). The LJ

Langmuir, Vol. 18, No. 12, 2002 4645

Figure 9. Pore size distribution in the case of nitrogen adsorption in activated carbon Norit R1 at 77.35 K (P. Harting): (solid line) the new approach; (dashed line) the original Horvath and Kawazoe theory;22 (dashed-dotted line) the HK theory for the 10-4-3 Steele potential.

Figure 10. Nitrogen isotherm in activated carbon Norit R1 at 77.35 K. The solid line is the correlation by the theory developed in this paper. Points are experimental data.

parameters of fluid-fluid interaction were taken from the paper of Neimark et al.47 (ff/kB ) 94.45 K, σff ) 0.3575 nm). For the solid-fluid interactions, we used the Lorentz-Berthelot rule. The solid line is plotted by our theory. The dashed line corresponds to the original theory of Horvath and Kawazoe. As might be expected, this curve indicates the presence of smaller pores, as the original HK is known to overestimate the filling pressure. However, the overestimation of the filling pressure arises mainly from the use of the 10-4 potential instead of the 10-4-3 potential and some arithmetic mistakes.25,26 When corrected values and the 10-4-3 potential are used in the HK model, we derive the new PSD presented the as dashed-dotted line in Figure 9. In this case the PSD curve is shifted to the right in the region of small pores. The main difference between our approach and the HK method is associated with the form of the PSD. Our theory shows the distinct second peak in the PSD in the region between 1 and 2 nm, while the HK method shows smoother peaks. The excellent agreement between experimental data and the correlated nitrogen isotherm is shown in Figure 10. The last system that we consider in this paper is the system of argon-activated carbon Norit ROX 0.8 at 87.29 K.48 Tabulated data and the DFT analysis were kindly (47) Neimark, A. V.; Ravikovitch, P. I.; Gru¨n, M.; Schu¨th, F.; Unger, K. K. J. Colloid Interface Sci. 1998, 207, 159.

4646


Figure 11. PSD for the system argon-activated carbon Norit ROX 0.8 at 87.29 K:48 (solid line) the present theory; (dashed line) the DFT.

Ustinov and Do

requires two conditions simultaneously: the Helmholtz free energy must be minimal and the grand potential functional must be less than zero. In the general case, a local isotherm may exhibit one or two jumps. In narrow pores, adsorption may proceed with formation of a single layer between the pore walls at a definite pressure. This threshold pressure may not correspond to the minimum of the potential energy distribution. In relatively wide pores, the onset of adsorption occurs with the formation of two layers close to the pore walls. A further increase of pressure in the bulk phase leads to an increase in their thicknesses followed by the second jump of adsorption, which is associated with the formation of the single layer. The dependence of the surface tension on layer thickness may lead to the appearance of a sharp but continuous local isotherm instead of a benchlike one. This is the case for the benzene-activated carbon system investigated in this paper. In particular, the formation of two symmetrical layers in larger pores always occurs continuously for this system. In narrow pores, local isotherms may be both abrupt and continuous. These results are in good agreement with those obtained by MC simulations and the DFT, and they allow us to determine the pore size distribution of activated carbons with significantly less effort. The model was successfully illustrated with the benzene adsorption data on activated carbon at 20, 50, and 80 °C, argon adsorption data on activated carbon Norit ROX at 87.3 K, and nitrogen adsorption data on activated carbon Norit R1 at 77.3 K. Acknowledgment. Support from the Australian Research Council is gratefully acknowledged. Appendix

Figure 12. Argon isotherm in Norit ROX 0.8 at 87.29 K: (points) experiment; (solid line) correlation by the present approach.

given to us by S. Bhatia. The fluid-fluid LJ parameters were also taken from the paper of Neimark et al.47 (ff/kB ) 118.05 K, σff ) 0.3305 nm). Figure 11 shows the PSD determined by our approach (solid line). The DFT pore size distribution is plotted by the dashed line. These curves are shown to be very similar. The main difference between them is associated not with the form of the PSD but rather with the areas under these curves. The total pore volume determined by our approach and by the DFT is 0.673 and 0.488 cm3 g-1, respectively. The argon density averaged over the pore volume obtained by the DFT is 48 mmol cm-3, which seems to be overestimated (the liquidphase density for argon at the same temperature is about 35 mmol cm-3). The argon isotherm is presented in Figure 12. The degree of correlation by our model is again very high. 4. Conclusion The Horvath and Kawazoe approach is reconsidered in this paper on the basis of a more rigorous thermodynamic analysis. Consideration of the mechanism of slitlike pore filling has shown that there is a pressure region where adsorption does not occur, although the minimum of the Helmholtz free energy corresponds to the nonzero layer thickness. It was shown that condensation in a pore (48) Ismadji, S.; Bhatia, S. K. Langmuir 2000, 16, 9303.

Here we present a simplified approach to determine (at least qualitatively) the dependence of the Helmholtz free energy change on layer thickness h. One can represent a semi-infinite liquid layer as consisting of two parts: a larger part A and a thin new part B close to the boundary of the layer. Part B is affected by the potential field exerted by the larger part A, and its state corresponds to a mechanical equilibrium for which the potential energy U is minimal. If we separate part B from part A so that these parts do not interact, this separation requires a work which is equal to -U. This work may be considered as the change in the Helmholtz free energy. At equilibrium, the thinner part B is positioned at a distance h0 from the larger part A. As a first approximation, the potential energy of a molecule at a distance z from layer A can be calculated by integrating the LJ potential energies of interaction between the molecule and all molecules within layer A. The result is the following 9-3 potential:

u)

[ ( ) ( )]

2 NA 2 σff πffσ3ff 3 v* 15 z

9

-

σff z

3

(A1)

To evaluate the potential of layer B of unit surface area, the potential u of eq A1 must be further integrated with respect to z from h0 to h + h0, where h is the thickness of layer B. Accounting for the molecular density NA/v*, one can obtain

U)

( ) [() () ( ) ( )]

1 NA 2 1 σff πffσ4ff 3 v* 30 h0

8

-

σff h0

2

-

σff 1 30 h0 + h σff h0 + h

8

+

2

(A2)


Langmuir, Vol. 18, No. 12, 2002 4647

A balance of forces between layers A and B requires U to be minimal, so the derivative of U with respect to h0 must be zero, which leads to the following equation:

() () (

2 σff 15 h0

9

-

σff h0

3

)

σff 2 15 h0 + h

) ( 9

-

)

σff h0 + h

3

(A3)

The work -U to be done to remove a unit area of layer B of thickness h away from layer A to infinite distance may be considered as the work that is needed to create two free surfaces of layer B, having a thickness of h. Assuming that the entropy does not change in this process, we may h s/v*. In the macroscopic case, replace -U by ∆Fs ) h∆F this allows us to express h∆F h s/v* as 2γv*. As the thickness h approaches infinity, the equilibrium distance between the two layers A and B is

lim h0 ) (2/15)1/6σff hf∞

In this limit of a thick layer, the surface tension can be expressed in terms of molecular parameters as follows:

γ ) (NA/v*)2(π/8)ffσ4ff(15/2)1/3

(A4)

In the case of benzene at 20 °C the liquid molar volume v* is equal to 8.769 × 10-5 m3 mol-1. The collision diameter σff and the reduced interaction energy ff/kB are taken as 0.5349 nm and 412.3 K, respectively. The calculated value of γ equals 0.017 24 N m-1, compared to the experimental value 0.028 88 N m-1. This may be regarded as a reasonable agreement in light of the very approximate approach and the great sensitivity of the result to collision diameter σff. In the general case, we can replace the right hand side of equality A4 by the known value of γ. Thus, one can write the following final equation:

h∆F h s ) 2γv*C[s21 - s22 - (s81 - s82)/30] s31 - s32 - (2/15)(s91 - s92) ) 0

(A5)

where

s1 ) σff/h0, s2 ) σff/(h + h0) C ) (4/3)(2/15)1/3 When h f 0, h∆F h s f 0, which is quite realistic. However, the first derivative of h∆F h s with respect to h is not equal

to zero at h f 0, which is very important because this shifts the pore filling pressure to a much higher value. Nomenclature a ) amount adsorbed of component i from the mixture (mol kg-1) f(r) ) pore size distribution function Fi ) Helmholtz free energy of i molecules in a pore (J) F h ) molar Helmholtz free energy (J mol-1) F h L ) molar Helmholtz free energy of liquid (J mol-1) ∆F h s ) structural change of molar Helmholtz free energy due to adsorption (J mol-1) h ) liquidlike layer thickness inside a pore (nm) i ) number of molecules in a pore kB ) Boltzman’s constant NA ) Avogadro’s number p ) pressure in the bulk phase (Pa) p0 ) saturation vapor pressure of adsorbate (Pa) p1*, p2* ) filling pressure for a single layer and double layers in a pore (Pa) H ) half width of a slitlike pore (nm) R ) 8.314 41 J mol-1 K-1 S ) surface area (m2) T ) absolute temperature (K) U ) potential of the fluid-solid interaction (J mol-1 m-2) v* ) molar volume of adsorbate (m3 mol-1) W0 ) pore volume of adsorbent (cm3 g-1) z ) distance of adsorbate molecule from the center of the slitlike pore Greek Letters ∆ ) graphite layer spacing (nm) ) Lennard-Jones pairwise interaction potential (J) γ ) surface tension for the liquid layer (N m-1) φ ) contact angle µ ) chemical potential of the adsorbate (J mol-1) µ°(T) ) standard chemical potential of the adsorbate (J mol-1) θ ) relative amount adsorbed in a pore F ) adsorbed phase density (mol m-3) Fs ) graphite molecular density (114 nm-3) σ ) collision diameter (nm) ξ ) grand partition function related to a single pore Θ ) grand potential functional Superscripts 1 ) nearest border of the liquid layer to the center of a slitlike pore 2 ) nearest border of the liquid layer to a wall of a slitlike pore ss ) solid-solid interaction sf ) solid-fluid interaction ff ) fluid-fluid interaction LA010535L

Adsorption in Slit-Like Pores of Activated Carbons: Improvement of the

Recommend Documents