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Microscopic Mechanism of Adsorption in Cylindrical Nanopores with Heterogenous Wall Structure Bogdan Kuchta,*,† Lucyna Firlej,‡ Marcin Marzec,† and Pascal Boulet† Laboratoire des Mate´ riaux DiVises, ReVeˆ tement, Electroce´ ramiques (MADIREL), UniVersite´ de ProVence, Centre de Saint-Je´ roˆ me, 13397 Marseille, France, and Laboratoire des Colloı¨des, Verres et Nanomate´ riaux (LCVN), UniVersite´ Montpellier II, 34095 Montpellier, France ReceiVed December 23, 2007. In Final Form: January 31, 2008 We study the microscopic mechanism of adsorption in nanometric cylindrical pores with strongly heterogeneous walls using grand canonical Monte Carlo simulations. The pore surface structure is modeled by a new lattice-site approach. Each site is characterized by two amplitudessstructural and energeticsthat locally modify the structural and energetic properties of the surface. The amplitudes are randomly distributed over the pore wall. We have shown that different structural and energetic distribution functions lead to different mechanism of adsorption. The energetic site distribution plays the most crucial role in the submonolayer region. The structural site distribution modifies the multilayer adsorption. A method to analyze the stability of the adsorbed system using static susceptibility is proposed. Potential applications in multiscale modeling are discussed.
Introduction Computer simulations based on grand canonical Monte Carlo (GCMC) or density functional theory (DFT) methodology have proven their validity in investigating adsorption on various types of surfaces, including those of porous materials. However, in these latter cases, there are at least two important factors that make modeling of the adsorption mechanism difficult: the confined geometry and the heterogeneity of the adsorbing surface. In fact, the adsorption mechanism in porous materials is very sensitive to intricate local and global pore geometries, which include the possible heterogeneity of the adsorbing surface, pore size distributions, heterogeneous modulation of the pore size along the pore axis, and/or pores interconnectivity. Usually, experimental adsorption isotherms are used to characterize the adsorbing (macroscopic) materials and to deduce information about the microscopic parameters, for example, the accessible adsorbing surface, pore size distribution, and enthalpy of adsorption. This interpretation is not always unique because the observed isotherms, of apparently similar forms, may result from very different particular microscopic adsorption mechanisms. In such cases, only appropriate theoretical models can enhance our understanding of those mechanisms. Realistic atomic models of the adsorbing surfaces described in the literature are usually based on initially crystalline structures that are modified by the disordering of their surfaces. Stallons et al.1 gives an overview of such models developed for plane surfaces of amorphous silica. Some proposed techniques of surface construction are as follows: (1) an ordered surface is created by cutting a known silica polymorph, (2) an unrelaxed amorphous surface is obtained by cutting bulk amorphous silica, (3) a relaxed amorphous surface is created by relaxing the amorphous surface, and (4) a random surface is created by Monte Carlo simulations. These approaches are, in principle, general and may be applied to any type of surface. However, each model leads to a slightly different description of the adsorption mechanism. Porous materials with their curved walls introduce a new level * Corresponding author. E-mail:
[email protected]. † Universite ´ de Provence. ‡ Universite ´ Montpellier II. (1) Stallons, J. M.; Iglesia, E. Chem. Eng. Sci. 2001, 56, 4205.
of difficulty into the modeling of adsorption. The recent review by Sonwane et al.2 discusses some of the approaches proposed in the literature for the surface construction of cylindrical MCM41-like silica. In such materials, although the heterogeneity results from the structure of an amorphous wall, the exact structural and energetic site distribution functions are unknown. For this reason, in the first computer simulations of adsorption in nanopores,2,3 an ideal (smooth) cylinder was used as a model of the pore surface. In this approximation, the interaction between the adsorbed particle and the pore surface depended only on the distance between them. Today, the most realistic models of MCM41 pores are based either on explicit atomic structures with different levels of randomness of atomic positions3-14 or on an assumption of the distribution of adsorption sites in the pore, without any direct reference to the underlying atomic positions.15-19 Both approaches present different advantages when used in computer simulations. The atomistic models represent more accurately the real structure of pore walls. They could easily include some micro- and mesoscopic details of the pore surface (e.g., wall indentations) to reproduce such experimentally measurable quantities better as the effective pore volume or surface area. The stochastic models are often much faster in (2) Sonwane, C. G.; Jones, C. W.; Ludovice, P. J. J. Phys. Chem. B 2005, 109, 23395. (3) Rudzinski, W.; Everett, D. H. Adsorption of Gases on Heterogeneous Surfaces; Academic Press: London, 1992. (4) Nicholson, D.; Silvester, R. G. J. Colloid Interface Sci. 1977, 62, 447. (5) O’Brien, J. A.; Myers, A. L. J. Chem. Soc., Faraday Trans. 1 1985, 81, 355. (6) He, Y.; Seaton, N. A. Langmuir 2003, 19, 10132. (7) Yun, J.-H.; Duren, T.; Keil, F. J.; Seaton, N. A. Langmuir 2002, 18, 2693. (8) Feutson, B. P.; Higgins, J. B. J. Phys. Chem. 1994, 98, 4456. (9) Coasne, B.; Pellenq, R. J.-M. J. Chem. Phys. 2004, 120, 2913. (10) Gelb, L. D.; Gubbins, K. E. Langmuir 1998, 14, 2097. (11) Maddox, M. W.; Olivier, J. P.; Gubbins, K. E. Langmuir 1997, 13, 1737. (12) Koh, C. A.; Nooney, R. I.; Boissel, V.; Tahir, S. F.; Tricarico, V. Mol. Phys. 2002, 100, 2087. (13) Koh, C. A.; Montanari, T.; Tahir, S. F.; Westacott, R. E. Langmuir 1999, 15, 6043. (14) Vuong, T.; Monson, P. A. Langmuir 1998, 14, 4880. (15) Ravikovitch, P. I.; Neimark, A. V. Langmuir 2006, 22, 11171. (16) Kuchta, B.; Llewellyn, P.; Denoyel, R.; Firlej, L. Colloids Surf., A 2004, 241, 137. (17) Ravikovitch, P. I.; Vishnyakov, A.; Neimark, A. V.; Ribeiro, Carrott, M. M. L.; Russo, P. A.; Carrot, P. J. Langmuir 2006, 22, 513. (18) Bojan, M. J.; Steele, W. A. Surf. Sci. Lett. 1988, 199, L395. (19) Puibasset, J. J. Phys. Chem. B 2005, 109, 480.
10.1021/la704017u CCC: $40.75 © 2008 American Chemical Society Published on Web 03/05/2008
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numerical applications and, consequently, allow one to simulate larger systems. These models are also easy to parametrize, and the influence of each parameter on the mechanism of adsorption can be analyzed separately.16,19 The general picture emerging from the analysis of the literature on numerical studies of adsorption in mesopores is the following: (i) although several models of heterogeneous walls have been proposed, a unique model that reproduces all features observed in experiments does not exist; (ii) the energetic site distribution is the primary factor determining the adsorption at low pressure; (iii) the structural roughness seems to be the most important factor responsible for multilayer adsorption; (iv) pore wall heterogeneity on both the molecular scale and nanoscale is probably necessary to reproduce a continuous increase in adsorption in the multilayer regime; and (v) it is practically impossible to construct a unique multiscale model of adsorption if we persist in modeling the pore walls in an atomistic way. The first numerical studies of adsorption on heterogeneous surfaces used simple lattice site models of the adsorbent.4,5 Some of them are reviewed in the book by Rudzinski and Everett.3 All papers point out that an observation of smooth or stepped isotherms is not uniquely associated with the degree of heterogeneity of the adsorbing surface and emphasized the importance of the adsorbate-adsorbate potential as well. As a consequence, the temperature must be also considered as the important factor modifying the adsorption mechanism. This aspect is not always thoroughly emphasized in interpretations of either experimental or numerical studies of adsorption. Therefore, the effects related to temperature are always superimposed on those that result from the structural and energetic heterogeneity of the pore wall.16,20-22 However, the influence of temperature is purely kinetic. At high enough temperature, the entropic effect may screen some features of the surface structure and make the effective adsorption energy surface shallower. Consequently, the structure of the adsorbed fluid becomes more disordered, and the stepwise form of the isotherm can be rolled out. This effect emphasizes the necessity of the most rigorous tests of any structural model of the adsorbing surface, especially at low temperature. We discuss this aspect in more detail at the end of the article. The goal of this work is twofold. First, we introduce and characterize a new parametric lattice-site model of heterogeneity. This model mimics the heterogeneous situation of cylindrical walls through a statistical distribution of adsorption sites using energetic and structural distribution functions. Methane (CH4) adsorption in cylindrical MCM-41-type pores of 4 nm diameter serves as an example. We addressed the following questions: (i) What is the origin of the low-pressure adsorption that seems to depend mostly on the energetic distribution of the adsorption sites and determines the first layer formation mechanism?16,19 (ii) Is the structural heterogeneity of the surface the unique factor that is responsible for the stepwise character of the multilayer adsorption?5,18 We discuss the influence of different parameters of the model on the mechanism of adsorption and the isotherm profile. To test the validity of our approach, the results obtained using the parametric model are compared to those simulated in pores with atomistic walls. Second, we explore the simplicity of the former approach that consists of a small number of parameters that allow us to control the adsorbing sites’ distribution and to explain easily the influence of each aspect of surface heterogeneity on the mechanism of adsorption. The model, initially conceived (20) Kuchta, B.; Llewellin, P.; Denoyel, R.; Firlej, L. Low Temp. Phys. 2003, 29, 880. (21) Kuchta, B.; Firlej, L.; Maurin, G. J. Chem. Phys. 2005, 123, 174711. (22) Kuchta, B.; Firlej, L. Stud. Surf. Sci. Catal. 2005, 156, 683.
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Figure 1. Definition of the lattice sites on the surface of the cylindrical pores.
for cylindrical pores, can be applied to simulate the heterogeneity of any surface, independently on its topology. Additionally, because it does not depend explicitly on the underlying atomic structure of the adsorbent, the model can be used in simulations not only on the atomic scale but also in multiscale approaches.
Heterogeneity Models Lattice-Site Model. Our model of pore surface heterogeneity is based on the concept of the lattice-site model. To prepare the lattice model of the pore wall, we proceed as follows. The surface of the smooth cylinder with radius R and length Lz (Lz is the length of the MC box in our simulations) is divided into small 2D cells of rectangular shape. The size of each cell is defined by two parameters: an angle ∆φ and a length ∆z (Figure 1). The angle ∆φ gives the length of its curved side () R*∆φ) along the pore circumference, and the value ∆z gives its length along the pore axis. Consequently, we obtain lattice-site elements sij (i ) 1, (2π/∆φ) and j ) 1, (Lz/∆z)) with the total number of cells sij equal to (2π/∆φ)(Lz/∆z). Heterogeneity of the lattice cell properties is introduced via a modulation of the uniform potential given by a function Vsmooth(r). It represents the interaction of particles adsorbed in the pore with ideal smooth walls. The variable r is the radial coordinate of the fluid atom with respect to the pore center. The modulation of the effective fluid-wall potential is defined in the following way. Two parameters are related to each lattice cell: the structural amplitude ∆rij and energetic relative amplitude ∆fij. They define the properties of the sij site. The distribution of ∆rij values characterizes the structural heterogeneity of the total surface. The variation of ∆fij relates to the chemical composition of the surface. In this article, the values of ∆rij and ∆fij are totally uncorrelated. The negative values of ∆rij locally shift the wall surface toward the center of the pore, and the positive values in the opposite direction locally modify the pore diameter. The energetic heterogeneity parameter ∆fij is defined as the ratio of the heterogeneous amplitude with respect to the uniform Vsmooth(r) function, and the energy of a particle situated above the site sij is calculated from
(1 + ∆fij)Vsmooth(r + ∆rij) Finally, we have introduced a parameter λij to define the spatial extent of the variation of the lattice-site energy defined by ∆fij. This parameter determines the “energetic deepness” of each site. In the model, λij is an exponentially decaying function. The final equation giving the heterogeneous energy is the following:
(
[
Vheter(r)ij ) 1 + ∆fij exp -
])
Rmin - r Vsmooth(r + ∆rij) λ
This equation represents the energy of the adsorbate-adsorbent interaction of a fluid particle located above the lattice site sij at
Mechanism of Adsorption in Cylindrical Nanopores Table 1. Parameters Defining the Surface Heterogeneity Model N ∆φ , ∆z p1, p2, ..., pN ∆r1, ∆r2,... ∆rN ∆f1, ∆f2,... ∆fN λ1, λ2, ... λN
number of different types of lattice sites (N-state model)a dimensions of the surface lattice site (Figure 1)b probabilities corresponding to each type of lattice site structural amplitudes of each type of lattice sitec energetic amplitudes of each type of lattice site energetic depth of lattice sitesd
a In this article, N ) 3 unless otherwise stated. b ∆φ ) 6° and ∆z ) 2.5 Å unless otherwise stated. c ∆ri expressed in Å. d In this article, λi ) 1.
a distance r from the mean (smooth) wall surface. Rmin is the distance from the pore center where the Vsmooth(r) function takes a minimum value. Table 1 gives the complete list of parameters used in our model and the necessity of defining the heterogeneity of the surface. Atomistic Model. To compare our parametric approach with the atomistic model of the adsorbing surface, we have also prepared a model of the cylindrical pore with explicit positions of atoms in the pore wall. In this model, the pore has been generated using a stochastic distribution of silicon and oxygen atoms in a ratio of 2:1.23-25 The atoms are randomly distributed to form an amorphous silica wall with a density of ∼2.1 g/cm3 around a pore of 4 nm diameter. Two limiting situations have been analyzed. In the first one, the structural wall heterogeneity mimics the parametric model presented above. This is achieved by imposing a distribution of atom positions on the surface of the cylinder within a distance R ( ∆r from the pore center (∆r ) (1 Å). In the second one, the wall surface is regular (∆r ) 0 Å). In each case, the structural site distribution is programmed as a constraint in the wall preparation procedure. Isotherms of adsorption simulated using atomistic models of the pore have been compared with corresponding parametric model simulations (adsorption on heterogeneous and smooth surfaces). Numerical Techniques. The simulations have been carried out in a conventional grand canonical MC ensemble. The simulation box contained one pore of diameter 2R ) 4 nm. Periodic boundary conditions were applied along the pore axis. Most of the calculations were performed for a pore of length Lz ) 5 nm. In some cases, longer boxes (Lz ) 10 nm) were checked to verify the importance of the size effect. The adsorbed system was assumed to be in equilibrium with the bulk gas, which obeyed the ideal gas law. Trial moves included the translations of particles, insertion of new particles, and removal of existing ones. The system typically contained from 600 to 1300 adsorbed particles in the Monte Carlo box. Typical production runs consisted of at least 106 MC steps (per atom) and were started from the previously equilibrated runs. The interaction model used the classical 6-12 Lennard-Jones (LJ) model with parameters and σ taken from the literature. The numerical results presented in this article relate to methane CH4 adsorbed in MCM-41 pores. The methane molecule was considered to be a structureless superatom. In such a standard approximation, according to ref 26 the methanemethane LJ parameters are f-f ) 148.1 K and σf-f ) 0.381 nm. However, in this article we used slightly modified values (f-f ) 151 K, σf-f ) 0.375 nm) to match the pressure of capillary condensation better. The final Vsmooth(r) potential has been calculated as the average of the interaction energy of methane with all oxygen atoms in the wall (methane-oxygen interaction parameters f-wall ) 172.7 K and σf-wall ) 0.3215 nm). (23) He, Y.; Seaton, N. A. Langmuir 2006, 22, 1150. (24) Coasne, B.; Hung, F. R.; Pellenq, R. J. M.; Siperstein, F. R.; Gubbins, K. E. Langmuir 2006, 22, 194. (25) Coasne, B.; Galerneau, A.; Di Renzo, F.; Pellenq, R. J. M. Langmuir 2006, 22, 11097. (26) Radhakrishnan, R.; Gubbins, K.; Sliwinska-Bartkowiak, M. J. Chem. Phys. 2000, 112, 11048.
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The distributions of the values of structural (∆r) and energetic (∆f) amplitudes define the nature of the surface in the pore. In this article, we used a simple three-state distribution defined by three structural and three energetic amplitudes, ∆rij and ∆fij, respectively, and three probabilities pij. The amplitudes have been randomly distributed among the sij sites. We also preformed some tests calculations with different N-state distributions (3 < N < 12), but they have not revealed any new features with respect to the three-states assumption. The atomistic wall calculations were performed with the TOWHEE program package using the grand canonical ensemble routines. The interactions between the adsorbed particles and the atoms in the wall (O and Si) were defined directly by the LennardJones potential using the same LJ parameters as in the parametric model. These simulations were run for at least 106 cycles per methane molecule, and data were collected during the last 107 cycles.
Results This section is organized as follows. In the first part, we analyze the influence of the structural and energetic roughness of the adsorbing surface on the mechanism of adsorption. The second part is devoted to the analysis of the properties of the lattice-site model itself. In particular, we focus on the detailed analysis of the role of selected model parameters (Table 1) in the shape of simulated adsorption isotherms. Finally, in the last subsection we compare the lattice-site model with that of the atomistic walls. Energetic versus Structural Roughness. In the following text, we discuss the principal features of adsorption on the heterogeneous surface with respect to the reference situation, that is, the adsorption on the wall that is smooth. Our reference system is a cylindrical pore with smooth wall. At 77 K, the simulated isotherm of CH4 (the smooth model, Figure 2) has a stepwise form. At low pressure, adsorption is negligible, and the first layer formation and the capillary condensation occur at well-defined pressures. A similar form of the isotherm is experimentally observed when adsorption is occurring on smooth or structurally regular surfaces such as a graphite surface or carbon nanotube walls. However, the stepwise isotherm was never observed for adsorption in the pores of MCM-41-type materials where the walls are known to be heterogeneous. To illustrate this feature, Figure 2 compares the experimental isotherms of methane adsorbed in the MCM-41 sample with cylindrical pores of diameter d ) 4 nm and the results of simulations in the same pores assuming that the pore surface is smooth. The main characteristics of the experimental curve27 that differ from the simulated one are (i) strong adsorption and rapid first layer formation at low pressure, (ii) a continuous increase in the adsorbed amount above the first-layer level, and (iii) progressive capillary condensation indicating narrow but non-negligible distribution of the pore sizes. These differences are crucial and should be addressed to understand any evolution of the adsorption mechanism as a function of pore wall heterogeneity. It is also worth remembering that the abovementioned differences are more pronounced at low temperatures. In fact, high temperature introduces a smoothing of the effective potential exercised by a wall on the adsorbed particle that can be interpreted as a screening of the local wall structure. We will discuss this aspect latter in this section. For comparison, Figure 2a also presents adsorption isotherms calculated for two limiting models of a heterogeneous wall: (27) Coulomb, J. P.; Grillet, Y.; Llewellyn, P.; Martin, C.; Andre G. In Proceedings of the 6th International Conference of Fundamentals of Adsorption; Meunier, F., ed.; Elsevier: Amsterdam, 1998; p 147.
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Figure 2. Isotherms of methane absorbed in MCM-41 pores, calculated assuming two models of heterogeneity: purely structural (S) and purely energetic (E). The model parameters are defined in the following way. (a) Model S: (∆r1, ∆r2, ∆r3) ) (0.5, 0.0, -0.5); (∆f1, ∆f2, ∆f3) ) (0.0, 0.0, 0.0); (p1, p2, p3) ) (0.4, 0.2, 0.4). Model E: (∆r1, ∆r2, ∆r3) ) (0.0, 0.0, 0.0); (∆f1, ∆f2, ∆f3) ) (0.3, 0.0, -0.2); (p1, p2, p3) ) (0.4, 0.2, 0.4). For comparison, the isotherm calculated using the smooth wall model and the experimental data are also presented. (b) Purely structural (S) models of heterogeneity [(∆f1, ∆f2, ∆f3) ) (0.0, 0.0, 0.0)]. Model S1: (∆r1, ∆r2, ∆r3) ) (1.0, 0.0; -1.0); model S2: (∆r1, ∆r2, ∆r3) ) (1.0, 0.0, -1.5); model S3 ) model S1 for 2 times larger lattice site dimensions: ∆φ ) 12° and ∆z ) 5 Å.
Figure 3. Influence of the site distributions (three examples, denoted as d1, d2, and d3) on the mechanism of adsorption: (a) adsorption isotherms and (b) radial density of particles in the pore. The vertical dashed line indicates the approximate limit of the first layer. For all distributions, the parameters of the structural and energetic roughness of the wall are the same: (∆r1, ∆r2, ∆r3) ) (1.0, 0.0, -1.0) and (∆f1, ∆f2, ∆f3) ) (0.8, 0.0, 0.0).
assuming pure structural (S) corrugation or purely energetic (E) corrugation. When the origin of heterogeneity is purely structural (model S), a continuous increase in the amount of the adsorbed mass is observed. The main factor that is responsible for such a form of the isotherm is the disorder introduced by structural heterogeneity. This structural roughness of the wall leads to a disordered structure of the first adsorbed layer that, in turn, becomes a rough substrate for particles adsorbed in the second layer. In such a way, the initial roughness of the pore wall is transmitted from layer to layer: each adsorbed layer produces a new distribution of adsorption sites for the next layer. As a consequence, the adsorption mechanism shows a continuous multilayer character before capillary condensation. It is interesting to see that the character of the isotherm does not change significantly when the structural amplitudes increase (Figure 2b, models S1 and S2). Actually, other structural parameters: the size of the adsorption sites (Figure 2b, model S3) and the site distribution (see below) are more important. However, at low pressure (before the first layer formation, P/P0 < 0.025), the adsorption is negligible as for the smooth pore situation. This behavior proves that at the low-density limit the adsorption progresses not because of the structural corrugation of the surface but because of the existence of adsorption sites that differ in strength. In fact, in the second limit, when the pore wall is structurally smooth and the wall heterogeneity is purely energetic, the low-pressure (P/P0 < 0.025) adsorption is much stronger and qualitatively similar to that observed experimentally. It is the consequence of energetic heterogeneity of the surface and the
existence of strongly attractive adsorption sites that induce the local formation of clusters and the rapid formation of the first layer when the pressure increases. At the same time, as the surface is structurally smooth, the layer is ordered, and adsorption progresses at higher pressures with stepwise character typical of structureless substrates. This simultaneous analysis of limiting situations shows that an appropriate modeling of both types of heterogeneity is needed to explain adsorption isotherms quantitatively in MCM-type materials. Actually, there are more parameters that affect the mechanism of adsorption, sometimes in a more subtle way. We discuss this aspect in the following sections. Properties of the Lattice-Site Model of Roughness. Following the previous conclusions, one can see that the parameters that affect the mechanism of adsorption and, in consequence, the shape of the adsorption isotherm most dramatically are (i) the site-type probability distribution (pi) and (ii) the amplitudes of structural heterogeneity (∆ri). Figure 3a presents the evolution of the shape of the isotherm when varying the distribution of the adsorption sites. For all cases presented here, structural and energetic amplitudes are the same. The main information coming out of the Figure is the following: the ratio between more and less attractive sites dramatically modifies the form of isotherm. It is interesting to see that both asymmetric distributions (d1 and d3, Figure 3) make the adsorption isotherm more stepwise. However, the mechanism of adsorption is different in each case. The large number of strongly attractive sites (d1) stabilizes a relatively
Mechanism of Adsorption in Cylindrical Nanopores
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Figure 4. Influence of the values of structural amplitudes (∆r1, ∆r2, ∆r3) on the mechanism of adsorption: (a) adsorption isotherms and (b) radial density of particles in the pore. The vertical dashed line indicates the approximate limit of the first layer. For all structural modulations, the parameters of energetic roughness and site distributions are the same: (p1, p2, p3) ) (0.5, 0.2, 0.3) and (∆f1, ∆f2, ∆f3) ) (0.8, 0.0, 0.0).
ordered first layer and provokes the rapid formation of the second layer. In the other limit (d3), a small number of attractive sites stabilize the first layer over a larger range of pressure. Obviously, to reproduce a real experimental situation a kind of balance in the distribution of the adsorption sites must be conserved. If the ratio between the sites of different strength and deepness is correctly balanced (distribution d2), then the shape of simulated adsorption isotherm reproduces the experimental one. At the same time, the adsorbed structure is disordered, as indicated by the radial density distribution. When the number of strongly attractive sites is large (d1), the less attractive sites are very weakly populated. Therefore, the first layer is not (structurally) disordered enough to affect adsorption of higher layers. The resulting structure of the adsorbate is only weakly disordered, as for smooth or weakly heterogeneous substrates. Obviously, in such a case the adsorption isotherm preserves the stepwise form. It is worth noticing that different probability distributions may also modify the effective local radius of the pore. In the case presented in the Figure 3, larger p1 probability also indicates a larger effective pore radius. As a result, the number of molecules adsorbed in the pore (the height of the first layer level and the plateau of capillary condensation) varies as a function of the distribution model. The effect of the variation of structural parameters of the model on the adsorption mechanism is illustrated in Figure 4. As in the case of site probability distributions, if one wants to modify the stepwise character of adsorption, structural amplitudes should be properly balanced. The shape of isotherms evolves when the value of the structural amplitude increases. The stepwise character of the adsorption progressively disappears when the amplitudes approach (1.0 A. Again, this behavior is a consequence of the disorder introduced by corrugated surface but amplified by the energetic site distribution. However, it is interesting that when the amplitudes continue to increase, at (1.5 A the stepwise shape of the isotherm reappears (Figure 4b): two (sub)layers are formed at low pressure, and the third layer appears just before the capillary condensation. At the same time, the structure of the adsorbed layers differs from that formed on a slightly corrugated surface. In fact, because the amplitude of the substrate structure is large (∼3 A), when the more attractive (deeper) sites are populated, the effective surface for the adsorption of subsequent molecules becomes smoother. Effectively, there is no real completion of the first layer: the second layer builds up simultaneously with the first one, starting from the less attractive (shallower) sites of the substrate. As a result, starting from the second layer the
Figure 5. Influence of the fluid-fluid interaction parameter f-f on the capillary condensation pressure. The parameters of structural and energetic roughness of the wall are the same: (∆r1, ∆r2, ∆r3) ) (1.0, 0.0, -1.0) and (∆f1, ∆f2, ∆f3) ) (0.8, 0.0, 0.0).
adsorbed phase is more ordered and promotes the stepwise form of the isotherm, in a similar way as in the case of small structural amplitudes (Figure 4b). Obviously, structural heterogeneity can be interpreted as a local modification of the effective pore radius. This perception of heterogeneity can explain the apparent variations of the average density of the adsorbed matter in pores of nominally the same radius. (See Figure 4a above the capillary condensation pressure.) However, we have observed that the pressure of capillary condensation is not very sensitive to small variations in pore radius. At the same time, the capillary condensation pressure depends critically on the strength of the fluid-fluid interaction. Figure 5 shows the adsorption isotherms calculated using the same model of surface heterogeneity and when varying the f-f parameter of the methane-methane interaction only. The calculations show that relatively small changes in f-f lead to important modifications of the system behavior. This observation is very important because it suggests that small changes in the (semiempirical) potential parameters could be necessary to obtain perfect quantitative agreement between simulated and measured values. In our calculations, we have increased the standard value of f-f () 148.10 K) by a factor as small as 2.13%; this has modified the pressure of the capillary condensation by ∼25%. Atomistic Model of Roughness. The calculations performed using an atomistic model of the pore wall led essentially to the same conclusions as those obtained using the lattice-site approach.
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βT )
2 V 〈(δN) 〉 N2 kBT
In adsorbed systems, the value of 〈N〉 can be considered to be a macroscopic parameter characterizing the formation of layers during adsorption. Assuming that the adsorbed system is in equilibrium with an external ideal gas, one can easily calculate the following relation:21
χN ) Figure 6. Radial density distribution in the atomistic pore wall model: (a) regular cylindrical wall and (b) random distribution of structural corrugation.
When the surface is atomically regular, the layers of adsorbate grow in a stepwise manner. On the contrary, when surface heterogeneities are accounted for in the model, the adsorption isotherm smoothly increases up to capillary condensation. The radial density distribution of fluid molecules in the pore (Figure 6) exhibits the most important characteristics of the adsorption mechanism: in the regular pore, the density shows well-defined layer structure that is absent when the pore surface is heterogeneous. The atomistic calculations confirm our previous conclusion that surface disorder, on a scale larger than the atomic size, is indispensable to the introduction of energetic heterogeneities in the pore, and, consequently, the multilayer adsorption of fluids. Similar conclusions have been presented in recent papers.6,23-25 It is interesting to compare the radial density distribution in atomistic and in lattice-site calculations (Figures 4 and 6). The most important difference appears in the first-layer profile. The three-peaks structure observed for the lattice-site model results from our choice of a three-state representation of the surface corrugation and progressively disappears when the number of states increases. For the atomistic model, n f ∞ and the layer structure tends to smear out. However, it is important to emphasize that the n-state representation does not seem to affect the mechanism of adsorption in the multilayer region.
( ) ∂〈N〉 ∂Pext
V,T
)
〈(δN)2〉 Pext
The susceptibility χN(Pext) is sensitive to any mechanical instabilities in the system that are induced by a variation of the external pressure: the smaller the susceptibility, the more stable the corresponding system. Figure 7 shows the fluctuations 〈(δN)2〉 as a function of the relative external gas pressure (p/p0) calculated for three models of wall heterogeneity. In all cases, when the first layer is formed (p/p0 < 0.05), the fluctuations are comparable and relatively small. At higher pressures, the evolution of 〈(δN)2〉 depends on the heterogeneity model. In the case of pure energetic corrugation (the wall is structurally smooth), the first-layer structure is stable (very low fluctuations) up to the capillary condensation. In the pores that are structurally corrugated, the fluctuations of 〈(δN)2〉 depend on the structure of the first layer. If the number of strongly attractive sites is large (model M, 40% of attractive sites), then the second layer is formed in a way that resembles the stepwise mechanism. In this case, the system is destabilized (above p/p0 = 0.09, see Figure 7) after the first layer is formed; consequently, the formation of the second layer is accelerated. However, when the number of attractive sites is reduced (model L, 20% of attractive sites), the adsorption is continuous, and the system stability decreases more steadily. The formation of the second layer is delayed and shifted to higher pressure (p/p0 = 0.24). When the third layer starts to grow, the multilayer structure is totally destabilized, and capillary condensation occurs. Above the pressure of capillary condensation, the amplitude of fluctuation always drops because the filled pore is always stable.
Conclusions Discussion The formation of subsequent layers during fluid adsorption in a pore can be interpreted as a sequence of transformations in the system. Quantitatively, these transformations can be discussed using the notion of static susceptibility. It allows one to analyze the evolution of the adsorption isotherm, between a stepwise and continuous shape, from the point of view of adsorbate internal stability. In the grand canonical ensemble, the response of the system to an external perturbation (defined by a variation of the chemical potential ∂µ) is monitored through the evolution of the mean number of particles 〈N〉:
( ) ∂〈N〉 ∂µ
VT
)
〈N2〉 - 〈N〉2 〈(δN)2〉 ) kBT kBT
(1)
In bulk materials, this quantity is directly related to system isothermal compressibility28 and can be easily calculated from the fluctuations of the number of adsorbed particles: (28) Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids; Clarendon Press: Oxford, England, 1987.
We can understand mechanism of adsorption as a competition between the energetic requirements that generally prefer more ordered structure and the structural disorder induced by the wall structure and temperature. From such a perspective, it is very helpful to analyze the system stability in terms of a susceptibility related to microscopic fluctuations of the number of adsorbed particles. This information can be easily obtained according to the model of the adsorbing surface or the simulation technique and can be used when describing the adsorption mechanism. The proposed lattice-site model of pore wall heterogeneity presents at least two advantages with respect to the previous models: (i) it is defined by a relatively small number of parameters that (ii) allow one to characterize energetic and structural heterogeneity separately. We have shown that the evolution between stepwise and continuous isotherms of adsorption could be easily modeled when structural heterogeneity in the form of shallow micropores of irregular shapes is introduced onto the surface. The role of this structural roughness is most important at intermediate pressures in the multilayer adsorption region. Once the first layer is formed, the structural roughness is transmitted to the second (and possibly higher) layer of adsorbed particles, producing a distribution of adsorption energies for the higher layers. On the contrary, the energetic heterogeneity is
Mechanism of Adsorption in Cylindrical Nanopores
Figure 7. Fluctuation of the number of molecules in the pore for three heterogeneity models: purely energetic corrugation [(∆r1, ∆r2, ∆r3) ) (0.0, 0.0, 0.0), (∆f1, ∆f2, ∆f3) ) (0.3, 0.0, -0.2), and (p1, p2, p3) ) (0.4, 0.2, 0.4)] and two models with the same structural and energetic heterogeneity values [(∆r1, ∆r2, ∆r3) ) (1.0, 0.0, -1.0) and (∆f1, ∆f2, ∆f3) ) (0.6, 0.0, 0.0)] that differ only in the proportion of strongly attractive sites [model M (more attractive): (p1, p2, p3) ) (0.4, 0.2, 0.4) and model L (less attractive): (p1, p2, p3) ) (0.2, 0.4, 0.4)].
most important at low pressure and determines the mechanism of formation of the first layer. We have shown that a certain minimum number of energetically strong adsorption sites are necessary to observe the rapid formation of the first layer. Actually, there are two factors that are important here. First, the energetic strength and distribution of adsorption sites. Second, the competition between intermolecular and particle-wall interactions. Depending on these, the first-layer formation could be more uniform (delocalized) or could go through the initial adsorption of well-localized clusters. However, the details of the surface heterogeneity become screened when the temperature increases. Both effects (thermal fluctuations and surface heterogeneity) induce disorder in the adsorbed system and are always superimposed, which means that any valuable test of the surface
Langmuir, Vol. 24, No. 8, 2008 4019
model should be performed at low temperature, especially if the exact physical meaning of the model’s parameters should be established. The presented results show that the lattice-site model of surface heterogeneity is able to reproduce the experimental features of adsorption isotherms. It was not our intent to give an exact quantitative fit to any experimental data. We aimed to identify the main variables affecting the microscopic mechanism of adsorption: the effective pore radius, the structural amplitude of heterogeneity, and the probability distribution of lattice (surface) sites with different attraction strengths. There is another parameter, not explicitly analyzed in this article but indirectly related to the previous ones, that can also influence the adsorption process: the form of the lattice sites. We have tested the model using the simplest cells of rectangular shape. More exotic, irregular cells may be necessary to fit the experimental observations more closely. In any case, the main advantage of the parametric latticesite model is the possibility of separate analysis of structural and energetic aspects of material heterogeneity, which are correlated in real situations and in any atomistic models.20,28 Our calculations show that the form of the isotherms in the multilayer region depends on the multiscale structural heterogeneity of the wall, from atomic to nanometric length. This conclusion has been supported by both our lattice-site model and atomic walls simulations. It is also consistent with recent studies24 where different atomistic models have been tested, as well as with the classical calculations of Steele et al. performed for flat surfaces.29,30 All numerical results, those presented in this article and those referred to in the literature, emphasized that the influence of heterogeneity is a multiscale problem. As a consequence, the modeling of the adsorption mechanism will require a multiscale simulation approach as proposed recently by Puibasset.31 In this context, our parametric model appears to be an excellent tool that is easy to use and possible to implement on any scale. LA704017U (29) Bakaev, V. A.; Steele, W. A. Langmuir 1992, 8, 1379. (30) Bakaev, V. A.; Steele, W. A. Langmuir 1992, 8, 148. (31) Puibasset, J. J. Chem. Phys. 2007, 127, 154701.
