Effects of Adsorbent Deformation on the Adsorption of Gases in Slitlike

J. Phys. Chem. C 2008, 112, 14075–14089

14075

Effects of Adsorbent Deformation on the Adsorption of Gases in Slitlike Graphitic Pores: A Computer Simulation Study D. D. Do,*,† D. Nicholson,‡ and H. D. Do† School of Engineering, UniVersity of Queensland, St. Lucia, Qld 4072 Australia, and Theory and Simulation Group, Department of Chemistry, Imperial College, London, SW7 2AY ReceiVed: April 14, 2008; ReVised Manuscript ReceiVed: May 26, 2008

We present GCMC simulations of adsorption of argon in slitlike pores that can swell or shrink with adsorbate loading due to movement of the graphene layers. It is found that the shrinking or swelling depends on loading, pore size, and temperature. The complex functional dependence is illustrated by adsorption in slit pores of various sizes at a number of temperatures. For pores where the adsorbate layers are not commensurate with pore width, for example when the pore width is 8A, shrinkage is observed under subcritical conditions, while shrinkage followed by swelling occurs under supercritical conditions under extremely high external pressure. However for commensurate pores that can accommodate an integer number of layers under normal conditions, for example when the pore width is 10A, we observe only modest swelling at high pressures. For both suband supercritical conditions the dependence on loading is quite complex. When these results are compared with the equivalent rigid pores, we find that the difference in adsorption capacity, molar enthalpy of the adsorbed phase, and solvation pressure is significant only at high pressures under supercritical conditions. Under normal adsorption conditions, a rigid pore model should be a good approximation to the deformable model investigated here. 1. Introduction Deformation of adsorbent under the influence of an adsorbate has been studied experimentally as far back as 1927 (see the review by Fomkin1 for early references). A number of experimental investigations were made in the 1930s by Bangham and co-workers and by Flood and co-workers in the 1950s. Several Russian groups have reported studies of deformational changes in adsorbents, during adsorption; notably Fomkin et al.2 used a direct dilatometer method to measure dimensional changes and found that, when benzene is adsorbed onto activated carbon at 293 K, the adsorbent shrinks then swells as loading is increased. Similar behavior was observed for other adsorbates, including N2 for which it was found, however, that at temperatures above 293 K only swelling occurred.3 The swelling of carbon beads under the influence of benzene adsorption at 393 K has been noted.4 Shrinking and swelling of microporous carbon fibers during water and nitrogen adsorption was studied using X-ray diffraction by Kaneko and coauthors5-7 and work on ACFs has been reviewed by Mowla et al.8 Structural changes in coals during adsorption are well attested experimentally and are thought to play a crucial role in CO2 sequestration where extreme pressures and temperatures may be involved. The swelling or shrinking of coal under the influence of adsorption has been modeled using a thermodynamic framework with the inclusion of elastic moduli.9 At the molecular level, modeling of deformational changes in adsorbents has been restricted to density functional theory (DFT) theory studies of MCM silicas10 and zeolites11 and in carbon slit pores under subcritical conditions.12 Adsorption in porous solids has been studied extensively using molecular simulation techniques in the past three decades. * To whom correspondence should be addressed. E-mail: [email protected]. † University of Queensland. ‡ Imperial College.

Although these investigations are numerous they are restricted to rigid pores in which the solid properties remain constant during adsorption (in other words the solid is inert). The only influence of the solid on the adsorption process is through the external solid-fluid potential. It is clear, however, that the adsorbates can impart an interaction upon the solid by virtue of the law of action and reaction, causing a change in pore structure. It is also important to note that even though swelling or shrinkage might not be observed macroscopically they can occur internally in such a manner that swelling in some parts of the solid might cancel out the shrinkage of the other parts. The objective of this paper is to develop a simple model to investigate swelling/shrinkage and to apply Monte Carlo molecular simulation to explore how various parameters, such as pore size, temperature, and the extent of pore relaxation, can affect adsorption of simple gases in graphitic slit pores. The extent of pore relaxation is modeled as the number of lattice layers that are affected by the swelling/shrinkage. We have chosen very simple model systems with the intention of gaining insight into the basic underlying mechanisms that are likely to be operative in more realistic and complex materials. We will describe the model in details in section 2 and present our results with detailed discussion in section 3. 2. Theory For the purpose of illustrating the effects of swelling and shrinkage, we use argon as the model adsorbate and a graphitic slit-shaped pore as a model solid adsorbent. The schematic diagram of this pore model is shown in Figure 1. The pore has two fixed solid walls (shaded regions) and M movable layers on each wall. The internal width H varies with the extent of adsorption. The simulation box is shown as a dashed line, and the volume of this box is unchanged during the simulation so that we can

10.1021/jp8032269 CCC: $40.75  2008 American Chemical Society Published on Web 08/14/2008

14076 J. Phys. Chem. C, Vol. 112, No. 36, 2008

Do et al.

[ ( ) ( )] 2 σβ,β 5 z

f β,β (z) ) 2πFsεβ,β(σβ,β)2 (

)

10

-

σβ,β z

4

(2b)

where β represents the species of solid atom. 2.1.3. Interaction between Adsorbate and MoWable Layers. The interaction energy between N particles and NL movable layers is the usual sum of 10-4 potentials N

NL

∑ ∑ f R,β (zi,j) (

)

(3)

i)1 j)1

Figure 1. The schematic diagram of a slit-shaped pore with movable layers. There are M1 movable layers on the bottom wall and M2 movable layers on the top wall.

apply the grand canonical ensemble to this constant volume. The solids below the bottom dashed line and above the top dashed line are fixed in space, i.e., they are firmly mounted in space and do not move during the course of adsorption or desorption. On top of the two fixed solids are lattice layers whose location is affected by adsorption. Let us assume that there are M1 movable layers in the bottom and M2 movable layers at the top. We choose the z coordinate to be perpendicular to the walls, and z ) 0 is the position of the fixed bottom solid and z ) H′ is that of the fixed top solid (Figure 1). The positions of the jth movable layer is denoted zL,j, for j ) 1, 2,..., M1 + M2 ) NL. 2.1. Interaction Energies. Having defined the system as shown in Figure 1, the molecular simulation of adsorption with N particles in the box is simply the problem of minimizing the grand potential of the system comprising N particles and NL ) M1 + M2 movable layers. These interact with each other and also with the two fixed solid walls. The total potential energy of the system is the sum of all pairwise interaction energies. We list these interactions below: 2.1.1. Interaction between N Particles. The interaction energy among N particles is the sum of the pairwise interactions between two particles. It is calculated as follows, with the usual application of the periodic boundary conditions along the x and y directions and the minimum image convention in the calculation of energy N-1 N

N-1 N

∑ ∑ φi,j(ri,j) ) ∑ ∑ 4ε R,R (

i)1 j>i

[( ) ( ) ] σ R,R ri,j (

)

i)1 j>i

)

12

σ R,R ri,j (

-

)

6

(1)

where R represents the species of the adsorbate, σ(R,R) and ε(R,R) are its collision diameter and well depth of the interaction energy, respectively, and the variable ri,j is the separation distance between particles i and j, ri,j ) |ri - rj|. 2.1.2. Interaction between MoWable Layers. For the interaction between movable layers, we sum the interactions between all adsorbent atoms within these layers

where zi,j is the z distance between the particle i and the jth layer, zi,j ) |zi - zL,j|, and the functional form of f is given in eq 2b with the molecular parameters σ (β,β) and ε(β,β) replaced by σ (R,β) and ε(R,β). The cross molecular parameters are calculated from the usual Lorentz-Berthelot rule. 2.1.4. Interaction between Particles and Two Fixed Solid Walls. We assume that the solid wall consists of semi-infinite lattice layers and apply the 10-4-3 potential energy equation of Steele.13 The interaction energy of all particles with the two fixed solid walls is N

∑

g R,β (zi) + (

)

i)1

N

∑ g R,β (H ’ -zi) (

)

(4a)

i)1

where g is the interaction energy between one particle and the semi-infinite lattices of constant surface density Fs, and it is given by

[( ) ( ) (

g R,β (z) ) 2πFsεR,β(σR,β)2 (

)

2 σR,β 5 z

10

-

(σR,β)4 σR,β 4 z 3∆(z + 0.61∆)3

)]

(4b)

2.1.5. Interaction between MoWable Layers and the Two Fixed Solid Walls. Finally the interaction energy of NL movable layers with the two fixed solid walls is NL

NL

∑ Nsg β,β (zL,i) + ∑ Nsg β,β (H ′ -zL,i) (

)

(

i)1

)

(5)

i)1

where g is given as in eq 4b with the molecular parameters σ (R,β) and ε(R,β) replaced by σ (β,β) and ε(β,β). Thus the total potential energy is simply the sum of eqs 1-5), that is N-1 N

U)

∑∑

N

φi,j(ri,j) +

i)1 j>i

N

∑ i)1

NL

∑∑

f R,β (zi,j) + (

)

i)1 j)1

N

∑ g R,β (zi) + (

i)1

NL-1 NL

g R,β (H ′ -zi) + (

)

∑∑ i)1

)

NL

Nsf β,β (zi,j) + (

)

j>i

∑ Nsg β,β (zL,i) + (

)

i)1

NL

∑ Nsg β,β (H ′ -zL,i) (

)

(6)

i)1

NL-1 NL

∑ ∑ Nsf β,β (zi,j) (

)

(2a)

i)1 j>i

where NL is the total number of movable layers, Ns is the number of solid atoms in the movable layer, Ns ) (LxLy)Fs, where Fs is the carbon atom surface density of the movable layer, Lx and Ly are the lengths of the simulation box in the x and y directions, and zi,j is the separation distance between the movable layers i and j, zi,j ) |zL,i - zL,j|. Here f(β,β) is the interaction energy between a solid atom and one lattice layer, and it takes the usual form of the 10-4 potential13

Minimization of this energy will give the equilibrium of the system and is done with Monte Carlo simulation. In simulations of adsorption on a surface, the top boundary of the simulation box is treated as a hard wall, and the fourth and seventh terms of eq 6 are not included. Before the adsorption occurs (vacuum condition), the movable layers are in equilibrium with the fixed solids underneath. Therefore the positions of these movable layers have to be determined from the minimization of the potential energy derived from (eq 6) by taking the number of particles, N, to be zero

Slitlike Graphitic Pores NL-1 NL

U0 )

J. Phys. Chem. C, Vol. 112, No. 36, 2008 14077 NL

∑ ∑ Nsf

(zi,j) + ∑ Nsg

(β,β)

i)1

j>i

(zL,i) +

(β,β)

i)1

NL

∑ Nsg β,β (H ′ -zL,i) (

)

(7)

i)1

This can be done with the usual search for minimum of the above equation by using any standard routine of optimization as detailed in Appendix 1. When fluid particles are introduced into the simulation box, the total potential energy will change relative to the initial potential energy of the system under vacuum. The difference is the configurational energy due to adsorption, ∆U ) U - U0. From fluctuation theory the change in the configuration energy per unit change in the number of particles in the simulation box is calculated as

- N ∆U [ ∂ ∂N∆U ] ) kT - N 〈∆U N 〉- N N (

〈 (

)

V

)〉

2

〈 〉〈

〈 〉〈 〉

〉

(8)

The change in the configurational energy is contributed by: 1. The fluid-fluid interaction, UFF. 2. The fluid-movable layer interaction, UFL. 3. The fluid-fixed solid interaction, UFS. 4. The layer-layer interaction, ULL. 5. The layer-fixed solid interaction, ULS. 2.2. Monte Carlo Algorithm. In the presence of adsorbates, we have to minimize the potential energy of e 6, and this can be done using Monte Carlo simulation in the grand ensemble.14,15 In the generation of the Markov chain, we execute four moves. 2.2.1. Particle Displacement MoWe. A particle is selected at random (say particle i) and is displaced to a new position. The acceptance of this new position follows the usual Metropolis rule, and the probability of acceptance is calculated from Prob ) min{1,exp (-δU/kT)} where δU is the difference between the potential energy after the move and that before the displacement, δU ) U(new) - U(old). Since only the particle i is displaced, the difference between the total potential energy δU is equal to the potential energy of the particle i with all other entities after the displacement and that before the move (Appendix 2). 2.2.2. Insertion and Deletion MoWe. Insertion and deletion are chosen with equal probability. In the insertion step, a particle is inserted at a random position over the pore volume space, and its acceptance follows the standard rule. In the deletion step a particle is chosen at random and is removed from the box. The volume for the insertion of a new particle is the whole volume of the simulation box (dashed line in Figure 1), but in practice the interstitial space between movable layers never becomes large enough to admit adsorbate. 2.2.3. Displacement of the MoWable Layers. A layer is chosen at random among the movable layers (say layer k) and is displaced to a new position. Its acceptance also follows the usual Metropolis rule. Again the difference in the total potential energy is equal to the difference in the potential energy of the layer k with all other entities after the move and that before the move. 2.3. Variables Obtained from the Simulation. 2.3.1. Density. From the GCMC simulation, we obtain the number of particles for a given chemical potential and temperature as well as the configurational energy of the system. The pore density can be calculated as the number of particles per initial unit accessible volume. This initial volume is the volume accessible to particle centers under vacuum. Since this accessible volume is changing due to swelling and shrinkage, it is convenient to

define the pore density in terms of the initial volume, thus allowing us to compare with the case where swelling and shrinkage do not occur. The initial accessible pore volume is defined as the volume accessible to the center of a particle under vacuum condition and can be found separately using Monte Carlo integration. We insert a particle at a random position in the simulation box (the dashed line in Figure 1), calculate its potential energy with the movable layers and the two solid walls, and then remove the particle from the box. If the solid-fluid potential of that particle is either zero or negative, the insertion is counted as a success. By repetition of this insertion process many times, the fraction of successes is f and the accessible volume is simply f times the volume of the simulation box, Vacc ) f(LxLyH′). This is the accessible volume to the center of a particle. It has been argued in the literature that the volume that is accessible to the whole particle is the sum of this volume and the volume occupied by one layer of molecules. This is not correct because the correct volume for the calculation of density is the accessible volume for the center of a particle. Imagine that we have a pore of width H and that we divide its volume into two equal parts with an imaginary hard wall of zero thickness. The width of one part is H/2, and the fluid-imaginary wall interaction energy is zero for all particle centers up to the wall. What that means is that the pore width available to the particle centers in this part is H/2, and if we use the conventional argument then the pore width available for the whole particle is H/2 + σ/2. The same applies to the other part, and that means that the total pore width of the two parts available to the whole particle is H + σ, which is actually incorrect (the available pore width is only H). We therefore conclude that the accessible volume is simply that available to the center of any particle that has zero or negative solid-fluid potential. Thus, having known the initial accessible volume obtained from the MC integration, the definition of the average absolute density is simply

Fav )

〈N 〉 〈N 〉 ) Vacc f(LxLyH′)

(9)

Recall that the initial accessible volume is used in the calculation of density, and the instant accessible volume changes with loading because of swelling or shrinkage. 2.3.2. Differential Molar Enthalpy of Adsorbed Phase. Another important variable that can be derived from the GCMC simulation is the differential molar enthalpy of the adsorbed phase. It is calculated from the well-known fluctuation formula15 as given in eq 8. When there is no swelling/shrinkage, the configurational energy U is the sum of the interaction energy due to fluid-fluid and fluid-solid interactions, when swelling/ shrinkage occurs this energy must also include the interaction among movable layers and that between movable layers and fixed solid walls. 2.3.3. Fluid-Wall Pressure. Another useful quantity relevant to adsorption in confined spaces is the pressure exerted by the fluid particles onto the two walls. Given the force exerted by each particle on the wall as ∂φi,w(zi)/∂z, the total force is simply the sum over all particles (including that due to the movable layers). In Monte Carlo simulation we can obtain this quantity as the ensemble average over the course of the simulation. Therefore the pressure exerted by the fluid on the solid is


psolv )

〈

N ∂φi(zi) 1 A j)1 ∂z

∑

〉

Do et al.

(10a)

where A is the surface area of on wall and φi is the fluid-surface interaction energy with one wall composed of M movable layers and one fixed solid wall M

φi )

∑ f R,β (zi,k) + g R,β (zi) (

)

(

)

(10b)

k)1

with the molecular parameters in the solid-fluid potential functions f and g being σ(R,β) and ε(R,β), and zi,k is the separation distance between the particle i and the movable layer k. Equation 10a is the same as the solvation pressure used by Snook and van Megen16-20 in considering the contact between a liquid and a solid. The pressure is termed the solvation pressure to reflect the effects of solvent on the attraction between two opposing walls. The minus sign is introduced in eq 10a to be consistent with the way the pore changes. A negative value of the fluid-wall pressure means a contracting pore, while a positive value means expansion. The derivative of the fluid-solid interaction energies, f and g, with z is given by eqsA1.3 and A1.4 in Appendix 1. 3. Results and Discussions To illustrate the new model of solid deformation, we use argon as the model adsorbate and study its adsorption in deformed pores at a range of temperatures. We choose 298 and 87.3 K to represent supercritical and subcritical conditions, respectively, and first consider a system with just one movable layer. The effects of pore size will be shown in section 3.1, those of the number of movable layers in section 3.2, and those of temperature are presented in section 3.3. 3.1. Effects of Pore Size. Deformation is rarely studied in the literature, partly due to the difficulty in measuring it and partly also due to the difficulty in formulation of an appropriate model to study it. To have a better understanding on how deformation could affect pores of various sizes, we study in a comprehensive manner this effect with a range of pore size, from the smallest pore that is accessible to the argon particle to the largest pore size in the micropore range. As a very broad classification, the pore widths may be divided into two general groups: commensurate and incommensurate with respect to the number of Ar layers that can be accommodated across the pore width. The commensurate pores with widths 7, 10, 13, 15, and 18 Å can accommodate 1-5 layers,

respectively, and show little or no dimensional change even at very high pressure of supercritical adsorbate. The remaining pore widths studied from 6.5-20 Å are incommensurate to some extent. We now summarize some of the detailed observations for pores in these two categories. 3.1.1. Incommensurate Pores. Results for the adsorption isotherm at 298 K for a 6.5-Å pore are shown in Figure 2a for rigid and deformable pores. Since this pore is too tight to fit one layer, we expect no shrinkage but rather an expansion at sufficiently high pressure under supercritical conditions. Expansion results in a lower solid-fluid potential energy and compensates for the energy expended in pushing the two walls apart. The variation of pore width with adsorption as a function of loading is presented in Figure 2b. We observe a distinct difference between the deformable pore and the rigid pore with the former having a slightly greater capacity because of the expansion of the pore as seen in Figure 2b. At the highest pressure studied, the pore expands by 2.3%. Despite the change in pore size due to the action of the adsorbates, the differential molar configuration energy of the adsorbed phase is practically the same between the deformed pore and the rigid pore as shown in the top plots of Figure 3a. The breakdown of this molar energy into the solid-fluid and fluid-fluid contributions is shown in the middle and bottom plots of Figure 3a. Although we do not observe any significant difference in the total configuration energy, the solid-fluid contributions for the deformable pore are greater than for the rigid pore. This difference is compensated by the energy expended in the layer-layer interactions and the layer-fixed solid wall interactions. The greater solid-fluid interaction for the deformable pore is explained by the increase in potential well depth before and after deformation (Figure 3b) when the pore is expanded from 6.5 to 6.65 Å and the adsorbate is in a less repulsive field which favors the expansion. The pressure exerted by the fluid onto the solid (movable layers + fixed solid walls) as a function of density is illustrated in Figure 4. For the rigid pore, where we have essentially twodimensional adsorption, we see clearly that the fluid-wall pressure is directly proportional to the loading (unfilled symbols) because all particles are at the same distance from the pore surface and exert the same force on the wall. This will not be true for larger pores as we shall see because molecules reside at various distances from the wall. However for deformable 6.5 Å pore, we see a lower fluid-wall pressure because of the expansion of the pore.

Figure 2. (a) Adsorption isotherms at 298 K in deformed pore and rigid pore. (b) Variation of pore width with loading.

Slitlike Graphitic Pores

J. Phys. Chem. C, Vol. 112, No. 36, 2008 14079

Figure 3. (a) Plots of the differential molar enthalpy of the adsorbed phase (top plots), the solid-fluid contribution to the enthalpy (middle plot), and the fluid-fluid contribution (bottom plot) for Ar adsorption at 298 K. (b)The solid-fluid potential energy profile for a single particle for 6.5and 6.65-Å pores.

Figure 5. Schematic diagram of adsorption in 6.5-Å deformable pore.

Figure 4. The fluid-wall pressure for a 6.5-Å pore at 298 K.

We summarize the adsorption behavior for this very narrow 6.5-Å pore in Figure 5. Under vacuum conditions, the pore is too tight for one layer, but upon adsorption the pore is slightly expanded to snugly accommodate one layer.

The next incommensurate pore (8 Å) is too large to accommodate one layer and too small to accommodate two layers under normal conditions of moderate pressures. The adsorption isotherms at 298 K are shown in Figure 6a for the deformable and rigid pores. The adsorption isotherms of these two pores do not show any differences for pressures less than 1000 atm, above which we see a significant (30%) increase in the adsorption capacity as a direct consequence of expansion of the pore as seen in Figure 6b. This expansion results in a surge of particles entering the pore and an increase from one to two layers of particles, resulting in a sharp transition in density. Steep transitions in density have been observed in microporous adsorbents such as zeolites, and structural change has been offered as a possible reason for these.21 The plot of the variation of pore size with loading shows a contraction at moderate


Do et al.

Figure 6. (a) Adsorption isotherms of argon in an 8-Å pore at 298 K. (b) Pore width of 8 Å with loading.

Figure 7. (a) Plot of the solid-fluid potential profile for 8- and 7.95-Å pores. (b) Differential molar enthalpy (top panel) and corresponding fluid wall and fluid-fluid contributions (lower two panels) of the adsorbed phase vs loading at 298 K in the 8-Å pore.

pressures, decreasing the pore width from 8 to 7.95 Å followed by an expansion at high pressures. For pressures less than 10000 atm, the contraction is about 0.5% which is sufficient to enhance the solid-fluid potential (Figure 7a), and as such we expect a higher contribution toward the differential molar enthalpy from the solid-fluid interactions in the deformable pore compared to that in rigid pore as seen in Figure 7b. As pressure is increased from 10000 atm, we observe a sharp swelling due to the enormous pressure applied. The swelling increases the pore size from 8 to 8.5 Å (an increase of 6%), a significant change in terms of molecular interactions. This is clearly reflected in the 30% increase in the adsorption isotherm (Figure 6a) and the molar configurational energy (Figure 7b). We now consider the behavior of the partial molar energy in Figure 7. For fractional loadings of less than 50%, the solid-fluid energy is fairly constant and then decreases beyond

50%. This decrease is due to more and more particles occupying the region where the solid-fluid potential becomes repulsive. In the rigid pore, we see that the solid-fluid energy becomes negative while that for the deformable pore remains positive. To explain this we show the plots of local density distributions for these two pores at a number of pressures (see Figure 8.) Also shown in this figure is the solid-fluid potential of a single particle (dashed line). Both pores start with a single layer and end with two layers at extremely high pressures; however, for the rigid pore we particularly note that at very high pressures a fraction of particles resides outside the accessible volume region where the solid-fluid potential is positive! While at lower pressures, for example 107 Pa, all particles reside within the region of negative solid-fluid potential. When the pressure is increased to 109 Pa, we see a small fraction of particles is in



Figure 8. (a) Plots of local density versus distance for rigid pore. (b) Plots of local density vs distance for deformable pore (dashed lines are solid-fluid potentials) at 298 K.

Figure 9. Fluid-wall pressure for adsorption of argon in an 8-Å pore at 298 K.

the positive solid-fluid potential region, and this fraction is increases with still higher pressures. For deformable pores, the plots of local density vs distance are shown in Figure 8b. It can be clearly seen in this figure that all the particles reside in the region of negative solid-fluid potential, and the expansion of the pore allows that to happen, in contrast to what is observed in the rigid pore. This explains why the solid-fluid energy for deformable pore is always positive (middle plot of Figure 7b). The fluid-fluid (FF) contribution to the partial molar energy of the adsorbed phase (the bottom plot of Figure 7c) shows that for loadings of less than 50% the initial linear increase is due to the fluid-fluid interaction. As loadings are increased the FF energy decreases because fluid particles approach closer together, and when two layers are formed the rearrangement of particles is such that the FF energy once again increases. Finally when the adsorbed phase is very dense, it decreases because of repulsive overlap between particles. The fluid-wall pressure (Figure 9) mirrors the change of pore width. Contraction of the pore is associated with a negative fluid-wall pressure while a positive pressure means expansion. Like the previous pore sizes, the fluid-wall pressure of the deformable pore is less than that for the rigid pore. This is characteristic of surface relaxation. We summarize the adsorption in an 8-Å pore with the schematic diagram in Figure 10.

The 9-Å pore behaves very much like the 8-Å pore, although the extent of contraction and swelling is not as great as the 8-Å pore. The next incommensurate pore size (11 Å) loosely fits two layers under normal conditions. However, under extremely high pressures the external force is great enough to squeeze in an additional layer. It is interesting to enquire whether the extent of contraction at moderate pressures and swelling at very high pressures in this pore is as much as in the smaller incommensurate 8-Å pore. Figure 11a show the variation of the pore width with loading. The pore width increases by 3.2%, compared to 6% at an adsorptive pressure of 10000 atm in the smaller incommensurate 8-Å pore. Despite its lower expansion, we see that the fluid-wall pressure is comparable in both pores of the order of 20000 atm. The 12-Å pore behaves very much like the 11-Å pore in that it can accommodate two layers under normal conditions; at extremely high pressure three layers can be squeezed into this pore. Under normal conditions of moderate pressures, the 14-Å pore can accommodate three layers, but in the presence of very high external pressure four layers can be packed inside the pore. This occurs in both deformed and rigid pores, but the evolution from three to four is different between these pores. This is shown in Figure 12 where we plot the local density versus the distance for a number of pressures. The extent of swelling in the deformed pore is shown in Figure 13, where we see that the pore size can increase up to 2% at the highest pressure. The incommensurate 16-Å rigid pore holds four layers under all conditions, while the deformable pore accommodates four layers at moderate pressures but five layers at extremely high pressure (Figure 14.) When five layers are formed there is a significant jump in the pore width (Figure 15.) Similar behavior is observed when the pore width is increased by 1 to 17 Å. At 19 Å, the rigid pore is large enough to maintain five layers, but for the deformed pore we see the evolution of a sixth layer although there is a significant overlap between the two interior peaks as seen in Figure 16b. Finally we consider the 20-Å pore, which is the upper limit of the micropore range. Unlike the 19-Å pore, there is a complete evolution from 5 layers at moderate conditions to 6 layers at extremely high pressure conditions (Figure 17.) 3.1.2. Commensurate Pores. Unlike the small 6.5-Å pore, the 7-Å pore can accommodate one integer layer under normal conditions. Because of this perfect integer layer


Do et al.

Figure 10. (a) Pore under vacuum and very low loading, (b) pore under moderate loadings (contraction), (c) pore under very high loading (swelling).

Figure 11. Pore width and fluid-wall pressure for adsorption of argon in an 11-Å pore at 298 K.

Figure 12. (a) Local density plot for a rigid 14-Å pore. (b) Local density plot for a deformable 14-Å pore.

Figure 13. Pore width and fluid-wall pressure for adsorption of argon in 14-Å pore at 298 K.

arrangement, we find isotherms to be very much the same between the rigid pore and the deformable pore and observe much less expansion than for the smaller pore. In Figure 18a we see that the 0.4% expansion shows that pore width as a function of loading at the highest pressure studied. At

moderate pressures there is a slight contraction in the pore width despite the high fluctuations. This is seen more clearly in the plot of fluid-wall pressure as a function of pore density as presented in Figure 18b, where there is a small negative fluid-wall pressure at moderate pressures. As pressure



Figure 14. (a) Local density plot for a rigid 16-Å pore. (b) Local density plot for a deformable 16-Å pore.

Figure 15. Pore width variation with loading in a 13-Å pore at 298 K.

becomes very high, expansion is observed with positive values of fluid-wall pressure. By comparison of the fluid-wall pressure in this pore with that of the 6.5-Å pore, we notice that the pressure is at most 800 atm, compared to 8000 atm in the smaller pore, as expected. Like the 6.5-Å pore, the fluid-wall pressure of the deformable pore is less than that of the rigid pore due to the structural flexibility in the deformable pore. Because of the insignificant variation in the structure of this commensurate 7-Å pore, the partial molar enthalpy of the deformable pore is practically the same as that of the rigid pore. The 10-Å pore can pack two integer layers across its width. The change of the pore size with loading is shown in Figure 19a. Like the commensurate 7-Å pore with a single layer, this pore exhibits swelling only at extremely high pressure and the swelling is very small. For example, at the highest pressure studied (10000 atm), the swelling is merely 1%. The fluid-wall pressure is shown in Figure 19b to further support the insignificant contraction and swelling of this pore. The 10-Å pore retains its size up to 25% fractional loading compared to the 11-Å pore, which does so up to 50% fractional loading. The commensurate 13-Å pore can accommodate three layers which remain at all pressures. Hence the swelling is again very small, with an expansion of only 1% at the highest pressure studied (Figure 20.) The 15- and 18-Å pores pack 4 and 5layers, respectively, under all conditions. 3.1.3. ComparatiWe Swelling BehaWior. Having studied the details of how the deformation of each pore behaves we present,

in Figure 21, the collected results on pore width as a function of loading rescaled with respect to the initial pore width. From this plot we can see that there are four distinct patterns of deformation. (i) The first pattern is that of a 6.5-Å pore: This pore only swells because that is the only way the walls can move because the initial pore size is a bit too tight for one layer. (ii) The second is from a 7-Å pore. This pore fits nicely one layer and therefore there is insignificant change in pore size. (iii) The third is from an 8-Å pore. This pore of initial size is loose for one layer and too small for two layers. The contraction at the beginning is to enhance the solid-fluid potential while the swelling at higher loadings is due to the squeezing of an additional layer. (iv) The fourth is exhibited by larger pores. These pores exhibit little change at low loadings and all swell at high loadings due to the high force needed to add more molecules into the pore. 3.2. Effects of Number of Movable Layers. Next we study the effects of the number of movable layers in the deformation. As more layers become mobile we expect the effect of deformation increase, and this is exactly what we see in the following figures for adsorption isotherm and the variation of pore width with loading for the 8-Å pore. The contraction of the pore is greater at moderate pressures and the swelling is also greater at very high pressures. The adsorption isotherms show a sharp change in density when four layers are allowed to relax on each wall. This is induced by the degree of flexibility of the pore rather than due to the direct effects between solid-fluid and fluid-fluid interactions (Figure 22). 3.3. Effects of Temperature. We have seen how the pore size and the number of movable layers can affect pore deformation in significant ways. To examine the effects of temperature, we first study the adsorption of argon at 87.3 K in the smallest pore 6.5 Å. As we know one layer just fits into this pore, and therefore we would expect it to swell with loading, as observed earlier for high pressure supercritical adsorption. This is indeed the case as shown in the plot of variation of pore width with loading depicted in Figure 23a. The loading for the high pressure adsorption at 298 K is higher because of the extremely high pressure applied, and the swelling is also greater than for subcritical adsorption. This is due to the greater fluid-wall pressure exerted onto the wall by the fluid. It should be noted that for a given loading the adsorptive pressure in the supercritical adsorption is far greater than that in subcritical adsorption. For example at a loading of 100 kmol/m3, the external pressure in supercritical adsorption at 298 K is 5 × 106 Pa,


Do et al.

Figure 16. (a) Local density plot for a rigid 19-Å pore. (b) Local density plot for a deformable 19-Å pore at 298 K.

Figure 17. (a) Local density plot for a rigid 20-Å pore. (b) Local density plot for a deformable 20-Å pore at 298 K.

Figure 18. (a) Plot of pore width of a deformable 7-Å pore with loading. (b) Plot of fluid-wall pressure vs loading at 298 K.

compared to 3 × 10-3 Pa in the subcritical adsorption of 87.3 K. The swelling in this pore at either supercritical or subcritical condition is reflected also in the fluid-wall pressure vs loading shown in Figure 23b. We next turn to the 7- and 8-Å pores. Both these pores can accommodate one layer of adsorbate. Here we can see how the differences in the conditions of adsorption affect swelling and contraction. The variation of the pore width with loading for these is shown in Figure 24. For subcritical adsorption at 87.3 K, we see only shrinkage because the adsorption at this low temperature is highly localized around the potential minimum at the center of the pore, and as a consequence contraction occurs. On the other hand, under

supercritical conditions we observed mainly swelling because of the higher entropic contribution from the thermal motion of molecules at high temperatures. In the 8-Å pore we observe a modest contraction before swelling because a little more volume is available than in the 7-Å pore. When temperature is raised to 400 K, we see, in Figure 24b, that the contraction is reduced and swelling increased at high pressures. This again emphasizes the importance of thermal motion of the molecules in swelling. So far we have seen that (i) for ultra fine pores, such as 6.5 Å, swelling occurs at all conditions because these pores are too narrow to fit one layer, and therefore the only mode of deformation is swelling, and (ii) for the more loosely fit onelayer pores such as those of 7 and 8 Å, contraction occurs under



Figure 19. Pore width and fluid-wall pressure for adsorption of argon in 10-Å pore at 298 K. TABLE 1 H′ (Å)

H

zL,1

zL,2

13.30730 13.79239 14.77619 15.76847 16.76443 17.76215 18.76079 19.75994 20.75938 21.75901 22.75875 23.75857 24.75843 25.75834 26.75826 106.75798

6.5 7 8 9 10 11 12 13 14 15 16 17 18 19 20 100

3.40365 3.3962 3.3881 3.3842 3.3822 3.3811 3.3804 3.3800 3.3797 3.3795 3.3794 3.3793 3.3792 3.37917 3.3791 3.37898

9.90365 10.3962 11.3881 12.3842 13.3822 14.3811 15.3804 16.3800 17.3797 18.3795 19.3794 20.3793 21.3792 22.37917 23.3791 103.37899

Figure 20. Pore width variation with loading in a 13-Å pore at 298 K. TABLE 2

Figure 21. Plots of the reduced pore width vs loading.

subcritical conditions because of the highly ordered single layer at the center of the pore that attracts the two walls. On the other hand under supercritical conditions swelling occurs because of the high thermal motion of the molecules. Nevertheless, contraction does occur at moderate loadings for pores with loosely fit one layer, such as the 8-Å pore. We now turn to the next pore size, 9 Å, and this pore provides an interesting feature that was not seen in the smaller pores. The variation of the pore width with loading is shown in Figure 25a.

H′

H

zL,1

zL,2

zL,3

zL,4

20.08313 20.55251 21.51895 22.50275 23.49416 24.48058 25.48628 26.48439 27.48315 28.48231 29.48172 30.48129 31.48099 32.48076 33.48058 113.47985

6.5 7 8 9 10 11 12 13 14 15 16 17 18 19 20 100

3.38680 3.37908 3.37054 3.36637 3.36414 3.36284 3.36205 3.36155 3.36121 3.36098 3.360816 3.36070 3.36061 3.36055 3.36050 3.360287

6.79157 6.77625 6.75947 6.75138 6.74708 6.74462 6.74314 6.74220 6.74158 6.74116 6.74086 6.74065 6.74049 6.74038 6.74029 6.739927

13.29157 13.77625 14.75948 15.75138 16.74708 17.74462 18.74314 19.74220 20.74157 21.74116 22.74086 23.74064 24.74049 25.74038 26.76029 106.73992

16.69633 17.17344 18.14841 19.13637 20.13002 21.12640 22.12423 23.12284 24.12194 25.12133 26.12090 27.12059 28.12038 29.12021 30.12008 110.11956

Under subcritical conditions, we observe contraction at moderate loadings (less than 50% fractional loading) for the same reason as for the 7- and 8-Å pores, the ordering of the adsorbed phase at the center of the pore. However at high loadings this pore swells because an additional layer squeezes in, which is not possible in smaller pores because the energy required to push the walls of smaller pores apart is too great. The contraction and swelling under supercritical conditions were explained above. The effect of squeezing in an additional layer disappears for pores that can accommodate two layers, such as the 10-Å pore as observed in Figure 25b. For subcritical conditions when the pressure is only moderate we see insignificant change, although


Do et al.

Figure 22. Isotherms for Ar at 298 K showing the effects of number of movable layers on (a) adsorption and (b) pore width.

Figure 23. Pore width and fluid-wall pressure for adsorption of argon in a 6.5-Å pore at 298 and 87.3 K.

Figure 24. (a) Pore width for adsorption of argon in a 7-Å pore at 298 and 87.3 K. (b) Pore width for adsorption of argon in an 8-Å pore at 400, 298, and 87.3 K.

a very small contraction is observed. For high-pressure supercritical adsorption, swelling does occur because of the thermal motion and we particularly note that this swelling is very small: note the scale of pore width in Figure 25b. The behavior of contraction and swelling for pores ranging from those with just one layer to those with two layers (6.5-10 Å) is repeated in larger pores; we show an example in Figure 26 for 11- and 12-Å pores. 3.4. Hysteresis Effects. Finally we consider the possibility of hysteresis of the swelling/shrinkage by studying adsorption and desorption branches. In rigid pores hysteresis is not observed for pores in this size range, but it is found to occur in the deformed pore as shown in Figure 27 for the 8-Å pore. Hysteresis at high pressures has been reported in the literature, and the deformation could be a possible reason for it.

Figure 28a shows the variation of pore width with pressure and demonstrates how pore width variation closely mimics the adsorption isotherm; by contrast the plot of pore width versus density in Figure 28b exhibits no hysteresis, which indicates that the swelling/shrinkage is the sole source of hysteresis. 4. Conclusion In this paper we have used computer simulation to study adsorption into graphitic slit pores that can deform in the presence of adsorbate. Swelling or shrinking of the adsorbate is incorporated into the model by introducing a number of graphene layers at the internal pores surfaces that can move in a direction normal to the surface under the constraint of van der Waals forces. In earlier work these interlayer forces



Figure 25. (a) Pore width for adsorption of argon in the 9-Å pore at 298 and 87.3 K. (b) Pore width for adsorption of argon in the 10-Å pore at 298 and 87.3 K.

Figure 26. Pore width as a function of density for adsorption of argon at 298 and 87.3 K (a) in the 11-Å pore and (b) in the 12-Å pore.

Figure 27. Hysteresis in the adsorption of argon at 298 K in a deformable 8-Å pore.

have been directly identified with graphite repulsive potential under compression22 and recent density functional calculations show excellent agreement with the experimental bulk modulus.23 We have examined slit pore widths ranging from 6.5 up to 20 Å in steps of 1 Å for and argon adsorbate at a subcritical temperature of 87.3 K and a supercritical temperature of 298 K. The pore sizes cover the whole of the micropore range and can accommodate from 1 to 6 layers of adsorbate in rigid (nondeformable) pores. Some pore widths accommodate an integer number of layers very closely, while others (e.g., 8, 11, 14, 17, and 20 Å) are strongly incommensurate with respect to an exact number of layers, and these show the

greatest deformation especially at high pressure and at the higher temperature, showing expansions of up to 6% increase in the original width for a high adsorbate density in the 8-Å pore. Smaller deformations were found for other pores, and near-commensurate adsorbates promote very little change in pore size. An interesting observation is that the 8-Å pore actually contracts at lower loading at the supercritical temperature. Significant variation between rigid pore and deformed pore isosteric heats were found for this pore size. When a larger number of movable layers are included in the calculation, deformation is greater and occurs at lower pressures. A steep transition in the isotherm observed at very high pressure also becomes more pronounced when the number of movable layers is increased. At subcritical adsorption temperatures, pore width typically shows an initial contraction followed by an increase in pore size for some cases as the pore density reaches saturation. Deformation is generally much less at subcritical temperatures, since adsorptive pressure is restricted to values of less than 1 atm and because there is a smaller kinetic energy contribution. The possibility of hysteresis was examined for 8A adsorption and was found to occur at the supercritical temperature, despite this pore being well below the micro- meso- pore boundary. We conclude that graphitic pore deformation can make some contribution to adsorption in carbon materials, especially at supercritical temperatures and high pressures. It is especially notable that these dimensional changes are highly dependent on the local pore width in the original material which may have interesting ramifications in materials with a distribution in pore sizes. Our simulations were performed for an argon adsorbate


Do et al.

Figure 28. (a) Plot of pore width vs pressure in the 8-Å pore at 298 K. (b) Plot of pore width vs loading. TABLE 3 H′ (Å)

H (Å)

zL,1

zL,2

zL,3

zL,4

zL,5

zL,6

28.256085

8

3.36797

6.73919

10.12804

18.12804

21.51690

24.88812

TABLE 4 H′ (Å)

H (Å)

zL,1

zL,2

zL,3

zL,4

zL,5

zL,6

zL,7

zL,8

34.99177

8

3.36734

6.73594

10.10706

13.49589

21.49588

24.88471

28.25583

31.62443

for which the fluid interactions are relatively small and a rigid pore model is acceptable under moderate conditions. It would be interesting to extend these studies to more aggressive adsorbates such as Xe or water. Acknowledgment. Support from the Australian Research Council is gratefully acknowledged. Appendix 1 The potential energy to minimize in order to determine the positions of all movable layers prior to adsorption is derived from eq 6 NL-1 NL

U)

∑∑ i)1

NL

Nsf(zi,j) +

j>i

∑

NL

Nsg(zL,i) +

i)1

∑ Nsg(H ′ -zL,i) i)1

(A1.1) Solution for the positions can be obtained from ∂U/∂zL,k ) 0, for k ) 1, 2,..., NL. Taking the partial derivatives of eq A1.1, we get NL df(z dg(zL,k) dg(H - zL,k) L,k - zL,j) 1 ∂U ) Σ + )0 )1 NS ∂zL,k j j*k dz dz dz

(A1.2) The functions f and g are given in eqs 2b and 4b, and their derivatives are

[

]

df 4 4 ( ) ( ) (A1.3) ) 2πFsε β,β (σ β,β )3 - * 11 + * 5 dz (z ) (z ) dg 4 4 1 ( ) ( ) ) 2πFsε β,β (σ β,β )3 - * 11 + * 5 + * * dz (z ) (z ) ∆ (z + 0.61∆*)4 (A1.4)

[

]

where the reduced variables z* and ∆* are z* ) z/σ(β,β) and ∆* ) ∆/σ(β,β). Therefore the minimization is reduced to solving NL equations (eq A1.2) for the positions of movable layers. We first consider the system with one movable layer on each wall,

Figure A1.1. Positions of movable layers in vacuum. Red symbols represent only one movable layer on each wall. Red and green symbols represent positions of two movable layers on each wall.

and Tables 1 and 2 shows the positions of the two movable layers as a function of internal pore width under vacuum. It is seen from this figure is that the separation distance of the outermost layers are greater than the separation distance in graphite. For three movable layers in each wall, the width H′ and the positions of the movable layers are given in Table 3 for the internal pore width of 8 Å. Similarly, Table 4 shows the positions for the case of four movable layers. Appendix 2 Particle Displacement Move The total potential energy is given in eq 6. When the particle i is displaced, the difference in the total potential energy is equal to the potential energy of the particle i with all other entities after the displacement and that before the move

U new - U old ) φi new - φi old (

)

(

)

(

)

(

)

(A2.1)

The potential energy of the particle i with others is given by



N

NL

j)1

k)1

φi ) Σ φi,j(ri,j) + Σ f(zi,k) + g(zi) + g(H - zi) (A2.2) j*i

where ri,j ) |ri - rj| and zi,k ) |zi - zL,k|. The first term in eq A2.2 is the interaction with all other particles, Ui,F; the second term is the interaction with all movable layers, Ui,L, and the last two terms are the interactions with two fixed solid walls, Ui,S. The potential energy of the particle i before the move, φ(old) i is calculated from eq A2.2 with r(old) being its position before i the move, while that after the move, φ(new) , is obtained from i the same equation with the new position being r(new) . The i acceptance of the displacement follows the usual Metropolis algorithm.14 If the move is accepted, only the energies of the fluid-fluid interaction, fluid-layer interaction, and fluid-solid interaction are updated as follows

UFFnew - UFFold ) Ui,Fnew - Ui,Fold (

)

(

(new)

UFL

)

(

)

(

( ) ( ) old - UFL ) Ui,Lnew - Ui,Lold

old ) Ui,Snew - Ui,Sold UFSnew - UFS (

)

)

(

)

(

)

(A2.3a) (A2.3b)

Similarly when the movable layer k is displaced to a new position, z(new) L,k , the difference in the total potential energy is equal to the difference of the potential energy of the layer k with all other entities after the displacement and that before the move. (

)

(

)

(

)

(

)

(A2.7a)

old new ) UN+1,L UFLnew - UFL )

(A2.7b)

old new ) UN+1,S UFSnew - UFS

(A2.7c)

(

)

(

)

(

)

(

)

Deletion of a Particle. In the deletion step, a particle is chosen at random and an attempt made to delete it from the box. Let this particle be the ith particle. Its interaction energy with other entities is N

NL

j)1

k)1

φi ) Σ φ(ri,j) + Σ f(zj,k) + g(zi) + g(H ′ -zi) (A2.8) j*l

where the first term is the interaction of the ith particle with (old) the remaining N-1 particles, Ui,F , the second term is its interaction with all movable layers, U(old) i,L , and the last two terms are the interaction with two fixed solid walls, U(old) i,S . The variables in eq A2.8 are ri,j ) |ri - rj| and zi,k ) |zi - zL,k|. If the deletion is successful, the fluid-fluid, fluid-layer, and fluid-solid energies are updated according to

(A2.4a)

old UFFnew - UFF ) Ui,Fold (

)

)

(A2.9a)

(new)

( ) old - UFL ) Ui,Lold

(A2.9b)

(new)

( ) old - UFS ) Ui,Sold

(A2.9c)

UFL

NL

φk ) Σ f(zi,k) + Ns Σ f(zk,l) + Nsg(zL,k) + Nsg(H ′ -zL,k)

UFS

k-1

i)1

)

(

The potential energy of the layer k with others is given by N

old new UFFnew - UFF ) UN+1,F (

(A2.3c)

Displacement of Movable Layer

U new - U old ) φknew - φkold

(new) interaction with all movable layers, UN+1,L , and the last two terms are the interaction with two fixed solid walls, U(new) N+1,S. The variables in eq A2.6 are rN+1,i ) |rN+1 - ri| and zN+1,k ) |zN+1 - zL,k|. If the insertion is successful, the fluid-fluid, fluid-layer, and fluid-solid energies are updated according to

(

k*l

(A2.4b) where zi,k ) |zi - zL,k| and zk,l ) |zL,k - zL,l|. The first term is the interaction of the kth movable layer with all other particles, Uk,F; the second term is with the other movable layers, Uk,L, and the last two are the interactions with two fixed solid walls, Uk,S. The potential energy of the kth movable layer before the move, φ(old) is calculated from eq A2.4a with z(old) k L,k being its position before the move, while that after the move, φ(new) , is obtained k from the same equation with the new position being z(new) L,k . The acceptance of the displacement follows the usual Metropolis algorithm. If the move is accepted, only the energies of the fluid-layer interaction, layer-layer interaction, and layer-solid interaction are updated as follows old new old UFLnew - UFL ) Uk,F - Uk,F (

)

)

(A2.5a)

(new) (old) old - ULL ) Uk,L - Uk,L

(A2.5b)

(new) (old) (old) - ULS ) Uk,S - Uk,S

(A2.5c)

(new)

ULL

(new)

ULS

(

)

(

Insertion of a Fluid Particle. When a fluid particle is inserted into the box, its interaction energy with other entities in the simulation box is calculated from:

φN+1 )

N

NL

i)1

k)1

∑ φ(rN+1,i) + ∑ f(zN+1,k) + g(zN+1) + g(H ′ -zN+1) (A2.6)

where the first term is the interaction of the new particle with (new) the already existing N particles, UN+1,F , the second term is its

References and Notes (1) Fomkin, A. A. Adsorption 2005, 11, 425. (2) Fomkin, A. A.; Regent, N. I.; Sinitsyn, V. A. Russ. Chem. Bull. 2000, 49, 1012. (3) Yakovlev, V. Y.; Fomkin, A. A.; Tvardovski, A. V.; Sinitsyn, V. A.; Pulin, A. L. Russ. Chem. Bull. 2003, 52, 354. (4) Kamegawa, K.; Yoshida, H. J. Mater. Sci. 1999, 34, 3105. (5) (a) Kaneko, K.; Fujiwara, Y.; Nishikawa, K. J. Colloid Interface Sci. 1989, 127, 298. (b) Suzuki, T.; Kaneko, K. Carbon 1988, 26, 743. (6) Kaneko, K.; Suzuki, T.; Fujiwara, Y.; Nishikawa, K. Dynamic micropore structures of micrographitic carbons during adsorption. In Characterization of Porous Solids II; Rodriguez-Reinoso, F., Rouquerol, J., Sing, K. S. W., Unger, K. K. Elsevier: Amsterdam, 1991. (7) Kaneko, K.; Ishii, C.; Ruike, M.; Kuwabara, H. Carbon 1992, 30, 1075. (8) Mowla, D.; Do, D. D.; Kaneko, K. Chem. Phys. Carbon 2003, 28, 229. (9) Pan, Z. J.; Connell, L. D. Int. J. Coal Geol. 2007, 69, 243. (10) Ravikovitch, P. I.; Neimark, A. V. Langmuir 2006, 22, 11171. (11) Ravikovitch, P. I.; Neimark, A. V. Langmuir 2006, 22, 10864. (12) Ustinov, E. A.; Do, D. D. Carbon 2006, 44, 2652. (13) Steele, W. A. Surf. Sci. 1973, 36, 317. (14) Frenkel D.; Smit D. Understanding Molecular Simulation; Academic Press: Sydney, 1991. (15) Nicholson, D.; Parsonage, G. Computer Simulation and the Statistical Mechanics of Adsorption; Academic Press: London, 1982. (16) van Megen, W.; Snook, I. Mol. Phys. 1982, 45, 629. (17) van Megen, W.; Snook, I. Mol. Phys. 1982, 47, 1417. (18) van Megen, W.; Snook, I. Mol. Phys. 1985, 54, 741. (19) Snook, I.; van Megen, W. J. Chem. Phys. 1979, 70, 3099. (20) Snook, I.; van Megen, W. J. Chem. Phys. 1980, 72, 2907. (21) Pellenq, R. J.-M.; Nicholson, D. Langmuir 1995, 11, 1626. (22) Crowell, A. D. Surf. Sci. 1985, 161, L597. (23) Rydberg, H.; Jacobson, N.; Hyldgaard, P.; Simak, S. I.; Lundqvist, B. I.; Langreth, D. C. Surf. Sci. 2003, 606, 532.

JP8032269

Effects of Adsorbent Deformation on the Adsorption of Gases in Slitlike

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