Adsorption of Lennard-Jones Fluids in Carbon Slit Pores of a Finite Length. AComputer Simulation Study

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1 1 Invited Contribution Adsorption of Lennard-Jones Fluids in Carbon Slit Pores of a Finite Length. AComputer Simulation Study A. Wongkoblap 1, S. Junpirom 2 and D.D. Do 1 * (1) Department of Chemical Engineering, University of Queensland, St. Lucia, Queensland 472, Australia. (2) School of Chemical Engineering, Suranaree University of Technology, Nakhon Ratchasima 3, Thailand. (Received 13 September 24; accepted 27 September 24) ABSTRACT: The adsorption of simple Lennard-Jones fluids in a carbon slit pore of finite length was studied with Canonical Ensemble (NVT) and Gibbs Ensemble Monte Carlo Simulations (GEMC). The Canonical Ensemble was a collection of cubic simulation boxes in which a finite pore resides, while the Gibbs Ensemble was that of the pore space of the finite pore. Argon was used as a model for Lennard-Jones fluids, while the adsorbent was modelled as a finite carbon slit pore whose two walls were composed of three graphene layers with carbon atoms arranged in a hexagonal pattern. The Lennard-Jones (LJ) 12 6 potential model was used to compute the interaction energy between two fluid particles, and also between a fluid particle and a carbon atom. Argon adsorption isotherms were obtained at 87.3 K for pore widths of 1., 1.5 and 2. nm using both Canonical and Gibbs Ensembles. These results were compared with isotherms obtained with corresponding infinite pores using Grand Canonical Ensembles. The effects of the number of cycles necessary to reach equilibrium, the initial allocation of particles, the displacement step and the simulation box size were particularly investigated in the Monte Carlo simulation with Canonical Ensembles. Of these parameters, the displacement step had the most significant effect on the performance of the Monte Carlo simulation. The simulation box size was also important, especially at low pressures at which the size must be sufficiently large to have a statistically acceptable number of particles in the bulk phase. Finally, it was found that the Canonical Ensemble and the Gibbs Ensemble both yielded the same isotherm (within statistical error); however, the computation time for GEMC was shorter than that for canonical ensemble simulation. However, the latter method described the proper interface between the reservoir and the adsorbed phase (and hence the meniscus). 1. INTRODUCTION The separation of a mixture of substances into different chemical products can be effected by a number of methods, e.g. distillation, absorption, liquid liquid extraction and adsorption. Among these methods, the adsorption process has become one of the standard unit operations in the chemical industry because of its low energy consumption compared to other competing processes. In adsorption processes, most adsorbents are solid porous materials and adsorption occurs mainly in the micropore volume space because of the high adsorption potential of this type of pore. *Author to whom all correspondence should be addressed. duongd@cheque.uq.edu.au.

2 2 A. Wongkoblap et al./adsorption Science & Technology Vol. 23 No Separation is effected because differences in molecular weight, shape or polarity result in some molecules being held more strongly than the others or because the pores are too small to admit the larger molecules (McCabe et al. 21). A good design of an adsorber for either gas or liquid purification/separation requires information about the adsorption equilibrium and kinetics. Although equilibrium data can be readily obtained from experimental work or the literature, mathematical models to minimize the experimentation effort or to predict adsorption isotherms at temperatures other than those used in the experiments are also important. For this reason, the development of methods for describing adsorption isotherms is necessary. To this end, two molecular-based methods, Density Functional Theory (DFT) and Grand Canonical Monte Carlo (GCMC), are being increasingly applied to solve numerous adsorption problems, and refined methods based on classical approaches have also been developed. To characterize porous solids, argon and nitrogen are often used as adsorbates at their respective boiling points; however, argon is preferred because of its spherical particle shape and its zero dipole and quadrupole. Hence its adsorption on surfaces is non-specific compared to nitrogen. As a result, the adsorption of argon in porous solids has been studied by many workers using GCMC simulations (Do and Do 23; Maddox et al. 1995; Ravikovitch et al. 2; Miyahara et al. 2), DFT (Ravikovitch et al. 1998, 2; Ravikovitch and Neimark 21; Olivier 1998) and the enhanced potential method proposed by Do and co-workers (Do and Do 23; Kowalcyzyk et al. 23; Gauden et al. 24). To model porous solids, they are usually assumed to consist of pores of different widths, and for the purpose of modelling, they are assumed to be semi-infinite slit pores (Donohue and Aranovich 1998; Stroud et al. 21), infinite carbon slit pores (Do and Do 23; Kowalcyzyk et al. 23; Nguyen and Do 1999; Gauden et al. 24), finite-length slit pores (Donohue and Aranovich 1998; Stroud et al. 21) or infinite carbon cylindrical pores (Maddox et al. 1995). In most simulations, the pore consists of two smooth walls without atomic structure, while a facecentred cubic lattice structure has been used to construct the wall surface in some simulations. To model a real carbon pore of finite length (Franklin 1951) and carbon surfaces as graphene layers comprising of carbon atoms arranged in a hexagonal pattern, it is necessary to understand the difference between the finite length pore and the commonly assumed infinite pore. In this paper, argon adsorption in a carbon slit pore of finite length is considered. The adsorption isotherms of argon at 87.3 K were obtained for pore widths of 1., 1.5 and 2. nm by means of Monte Carlo simulation, these widths representing those commonly observed in activated carbons. Here, we use the Monte Carlo simulation method with a Canonical Ensemble (NVT) and a Gibbs Ensemble (GEMC). Detailed descriptions of these two ensembles are presented below. In particular, the effects of some parameters on the performance of the Canonical Ensemble simulation have been studied, i.e. the number of cycles necessary to attain equilibrium, the initial allocation of particles, the displacement step length and the simulation box size. The results obtained from this work on finite length pores have been compared with those obtained for infinite slit pores using a Grand Canonical Ensemble and periodic boundary conditions to reveal the effects of pore length on the behaviour of the adsorption isotherm. 2. THEORY 2.1. The fluid model Argon was used as a model for Lennard-Jones (LJ) fluids in the present work, together with the LJ parameters, σ ff =.345 nm and ε ff /k = K (Do and Do 23; Miyahara et al. 2;

3 Adsorption of Lennard-Jones Fluids in Carbon Slit Pores of a Finite Length 3 Stroud et al. 21), and a cut-off radius for the interaction of two argon particles of five-times the collision diameter (5σ ff ). The potential energy of interaction between the two particles may be calculated using the following Lennard-Jones 12 6 equation (Do and Do 23; Do 1998; Toth 22): where r is the separation distance between the two particles The solid model ϕ ff () r = 4ε ff σ r In the present work, studies have been made of a carbon-based adsorbent with pores exhibiting a typical slit-shaped geometric structure. The pore width H is defined as the distance between a plane passing through all the carbon atom centres of the outmost layer of one wall and the corresponding plane of the other wall. The interlayer spacing between the two adjacent layers is assumed to be constant and is denoted as. A simple slit pore of finite length has been used in the study. It is assumed that the two walls of the pore consist of graphite layers which are perpendicular to the z-axis. Each wall consists of three graphite layers and these layers are stacked on top of each other with an interlayer spacing of.3354 nm. The configuration of the carbon atoms in each layer consists of condensed aromatic rings with each ring possessing six carbon atoms. The adjacent carbon carbon distance is.142 nm (Nguyen and Do 1999). Schematic diagrams of a carbon pore which is finite in length in both the x- and y-directions are depicted in Figures 1 and 2. It has been reported in the literature (Franklin 1951) that the graphite layer is between 2 Å and 7 Å in length. Hence, in this study it is assumed that all graphite layers are square, equal in size and have a linear dimension of 6 Å (i.e times the collision diameter of argon). Values of 1., 1.5 and 2. nm have been chosen for the pore width as such pore sizes are typical in most commercial activated carbons. The LJ parameters for carbon, σ ss and ε ss /k, are.34 nm and 28 K, respectively (Do and Do 23; Miyahara et al. 2; Stroud et al. 2). The interaction energy between an argon particle and a carbon atom may be calculated via the same equation [equation (1)] with σ ff and ε ff being replaced by σ sf and ε sf, respectively. These cross-molecular parameters are ff 12 6 ff σ r (1) Figure 1. Top view of the carbon configuration of one-quarter of one graphene layer.

4 4 A. Wongkoblap et al./adsorption Science & Technology Vol. 23 No nm Figure 2. Side view of one pore wall comprised of three graphene layers (only one-quarter of one layer is shown in each case). calculated from the usual Lorentz Berthelot rule: ε = ( ε ε ) 1/ 2 sf ss ff ( ) 2 σ = σ + σ sf ss ff (2a) (2b) where the subscripts f and s stand for fluid and solid, respectively. Assuming that pairwise additivity holds, the total energy can then be calculated by summing the pair interactions between argon particles, and between individual carbon atoms and argon particles, as follows: ( ) + U = ϕ r r ϕ r r ij i j i, j ik, ( ) ik i k (3) where r i and r j are the positions of argon particles i and j, respectively, r k is that of a carbon atom, ϕ ij is the pair interaction potential between argon particles and ϕ ik is that between an argon particle i and a carbon atom k The simulation box A Canonical Ensemble (NVT) and a Gibbs Ensemble are used to study the adsorption of Lennard- Jones fluids in a finite-length carbon slit pore. The simulation boxes for these ensembles are described below Canonical Ensemble In the Canonical Ensemble, the simulation box is cubic in shape and has the same length in the x-, y- and z-directions. A slit pore of finite length is positioned in the centre of this box and the box

5 Adsorption of Lennard-Jones Fluids in Carbon Slit Pores of a Finite Length 5 Figure 3. Initial positions of 2 argon particles in a simulation box of 28σ ff in length. size chosen such that the distance between any points of this pore to any boundaries of the simulation box is at least greater than five-times the collision diameter. Periodic boundary conditions are applied in all directions of the simulation box and the minimum image convention is used to calculate the intermolecular interaction energy. The effect of the simulation box size has been studied by varying its linear dimension from Å to 8 Å (i.e. from 28- to 235-times the fluid collision diameter). Figure 3 shows the initial random allocation of 2 particles in the simulation box. Note the central position of the slit pore with three graphene layers forming one wall of the pore. This simulation box is used as a Canonical Ensemble in which the volume of the box, the number of particles and the temperature are specified. Monte Carlo methods with the usual Metropolis scheme are used to determine the equilibrium distribution of the particles both inside and outside the pore. Those particles outside the pore and positioned at least five-times the collision diameter from any carbon atoms of the pore are treated as particles in the bulk phase, with the density of these particles being in equilibrium with the density inside the pore. The relationship between the densities of these two phases is simply the adsorption equilibrium Gibbs Ensemble The Canonical Ensemble described above involves a very large simulation box, which is necessary to accommodate a finite pore. The advantage of this simulation is that it correctly describes the adsorption of fluid from a reservoir into a pore of finite size. The effects of pore opening are implicitly accounted for in the simulation. However, the disadvantage of this method is the computation time associated with the MC simulation with this ensemble, simply due to the large size of the simulation box. In this paper, another ensemble smaller in size (and hence involving a smaller number of particles) is also studied to allow a smaller computation time. Two simulation boxes are involved in the Gibbs Ensemble. One box is cubic in shape. Its linear dimension is chosen as 4 Å and it is used to simulate the bulk gaseous phase. The second box corresponds to the finite slit pore. The total number of particles in these boxes is constant and so

6 6 A. Wongkoblap et al./adsorption Science & Technology Vol. 23 No is the total volume. Periodic boundary conditions are applied to the first box, while the properties of the second box (i.e. the finite carbon pore) are similar to those discussed for the solid model in Section Monte Carlo simulation method In this study, a Monte Carlo simulation method is used to study the adsorption of argon in a slit pore of finite length. The Monte Carlo simulations involve the use of random numbers to sample the phase space. Basically, a large number of possible configurations of particles in proportion to the probability of their occurrence is generated. This series of configurations is called the Markov chain. This chain of configurations is then used to compute the various properties using ensemble averages. In this paper, the Metropolis algorithm is adopted in the simulations (Frenkel and Smit 22). All length-scale and energy-scale variables are made non-dimensional in the Monte Carlo simulation by scaling against the collision diameter and the well-depth of the interaction energy of argon, respectively. A summary comparison of the two ensembles used in this paper is given in Table 1. TABLE 1. A Comparison between the Canonical Ensemble and Gibbs Ensemble Used in this Study Description Canonical Ensemble (NVT) Gibbs Ensemble (GEMC) General description Fixed number of molecules, Simulation of two phases with a fixed total volume and temperature. number of particles, total volume and temperature. Simulation result Stable at sufficient cycles, Stable at sufficient cycles. extensively great cycles. MC move Simple, only displacement Three basic moves are involved: of a randomly selected (i) displacement of a randomly selected particle is required. particle; (ii) transfer of a randomly selected particle from one box to the other; (iii) change in volume of the two boxes while keeping the total volume constant. Fluid densities Gives the equilibrium density in Gives the equilibrium density of the bulk the bulk as well as in the pore. phase as well as the density inside the pore. Surface Gives the equilibrium surface No surface concentration in equilibrium with concentration concentration and bulk density. bulk density is shown. Interface between Gives an interface between No interface joining the two phases is two phases the gas phase and the indicated. adsorbed phase. Simulation time Expensive computation time. Depends on the system size; for large systems, a long equilibrium time is needed.

7 Adsorption of Lennard-Jones Fluids in Carbon Slit Pores of a Finite Length 7 In the Canonical Ensemble, one cycle consists of N trial moves (where N is the number of particles), and the number of cycles is varied from 5 to 1 cycles in the equilibrium step and maintained at 5 cycles in the sampling step. A cut-off radius of five-times the collision diameter is used in the calculation of the interaction energy between the argon particles while the calculated energy is not shifted. The initial distribution of particles in the simulation box is investigated either by (i) randomly positioning all the particles in the pore or (ii) randomly positioning all the particles in the bulk phase or (iii) randomly placing them anywhere in the whole simulation box (i.e. some are placed randomly in the pore and the others are also positioned randomly in the bulk phase). The displacement step length has also been studied by varying its value from.5σ ff to one-half the box length. Similarly, the box length is varied from 28σ ff to 235σ ff. The Gibbs Ensemble Monte Carlo (GEMC) method is also used to obtain the adsorption isotherm. Each simulation cycle consists of the displacement of a randomly selected particle and the transfer of a randomly selected particle from one box to the other. In this study, the pore volume (box 1) and the bulk volume (box 2) are fixed; hence, the volume change usually carried out in the Gibbs Ensemble (Panagiotopoulos 1987) has not been performed in this work. One GEMC cycle consists of N displacement moves and N particle swap attempts. It has been found that 5 cycles were needed for the system to reach an equilibrium state, and for this reason 5 cycles have been used to obtain various ensemble averages. A cut-off radius of 5σ ff, an initial displacement step length of.5σ ff and a bulk phase box length of 117.5σ ff in three directions have been selected and used in the simulation. The choice of displacement step length in the Canonical Ensemble is very critical. Unlike the Gibbs Ensemble in which the displacement step length was chosen as.5-times the collision diameter, the choice of displacement step length in the Canonical Ensemble is not as straight forward. This will be discussed in detail below in Section RESULTS AND DISCUSSION 3.1. Adsorption isotherm behaviour of finite pores The results are discussed below by first presenting the adsorption isotherms obtained from the Canonical Ensemble and later comparing them with those obtained from the Gibbs Ensemble. The adsorption isotherm of the 1.5-nm slit pore is first presented since this is a typical micropore size found in activated carbon. In this study, the linear dimension of this pore in the x- and y-directions was chosen as 6 Å which is typical for a carbon crystallite (Franklin 1951). A simulation box length of 58.7σ ff, a displacement step of 5σ ff (and a maximum at 29.3σ ff if the acceptance ratio is greater than.5) and 5 cycles were used in each simulation. The simulated curves depicted in Figure 4 have been plotted in linear and semi-logarithmic scales, respectively. It should be noted that the scattering of the simulated data points is significant over the low-pressure region due to the low number of particles assumed to be present in the fluidphase region outside the pore. This defect can be overcome by using a larger simulation box, allowing a larger number of particles in the bulk phase and thereby achieving a good statistical average. The adsorption branch of the isotherm was obtained by sequentially adding particles randomly into the simulation box after each equilibrium point is achieved, and the desorption branch was similarly obtained by removing some randomly selected particles from the simulation box. It will be noted that for a pore width of 1.5 nm (which can accommodate up to four layers of argon particles), the hysteresis loop spans a fairly wide range of pressure. Let us first consider the

8 8 A. Wongkoblap et al./adsorption Science & Technology Vol. 23 No (a) H = 1.5 nm 5 (b) H = 1.5 nm Pore density (kmol/m 3 ) Pore density (kmol/m 3 ) B C Pressure (kpa) 8 A Pressure (Pa) Figure 4. Argon adsorption isotherms in a finite-length slit pore of 1.5 nm width at a temperature of 87.3 K: (a) linear scale and (b) semi-logarithmic scale. In both parts of the figure, the filled data points refer to adsorption while the open data points refer to desorption. adsorption branch. Upon adding particles into the simulation box, a gradual change occurs in the adsorption isotherm brought about simply by the initial formation of two contact layers and the subsequent filling of the two inner core layers. The gradual filling of the inner core may be attributed to the fact that the solid fluid potential in the pore mouth region is lower than that in the region distant from the pore mouth. When the pressure is ca. 8 Pa, the adsorption isotherm exhibits a small jump in density due to compression of the argon particles. This compression leads to a density of 4 kmol/m 3, a value very close to the density of solid argon (42 kmol/m 3 ). However, it should also be noted that the density of saturated liquid argon at 87.3 K is 34.9 kmol/m 3. Thus, the behaviour of the adsorbed phase is similar to that of a highly compressed liquid at pressures close to the vapour pressure. When a number of particles are removed from the simulation box, it is then possible to trace along the desorption branch. Since the state of argon in the adsorbed phase is that of a highly compressed liquid, the pressure has to be reduced significantly before any change in the density of the adsorbed phase is observed. This occurs at a pressure of ca. 3 Pa, compared to the value of 8 Pa necessary to cause a change from the liquid-like state to a highly compressed liquid-like state along the adsorption branch. The desorption branch exactly coincides with the adsorption branch at lower pressures once the adsorbed phase has converted into the liquid-like phase at 3 Pa. In particular, the slanting hysteresis loop should be noted. This is very typical for many hysteresis loops observed with activated carbon. We shall compare this loop behaviour with that for the case of an infinite pore later. The potential of the Monte Carlo simulation is not just to predict the isotherm but also to provide the microscopic configuration of the particles. Thus, the particle configurations for the three values of the pressure indicated as points A, B and C in Figure 4 are presented in Figure 5. In the latter figure, (a) corresponds to the pressure at which the adsorbed phase is at sub-monolayer coverage and is confined to the two contact layers adjacent to the two pore walls (point A in Figure 4). The snapshot shown as (b) in Figure 5 corresponds to the situation when the contact layers are filled and the inner core is partially filled (this is point B in Figure 4). Finally, (c) in Figure 5 shows the snapshot when the pore is completely filled with saturated vapour (point C in Figure 4).

9 Adsorption of Lennard-Jones Fluids in Carbon Slit Pores of a Finite Length 9 (a) (b) (c) Figure 5. Snapshots of argon adsorption at 87.3 K for a pore width of 1.5 nm with the total number of particles corresponding to (a) 2, (b) 12 and (c) 25, respectively. What is important in (b) and (c) of Figure 5 is the meniscus of the adsorbed phase in the finite pore and the gas phase in the bulk fluid surrounding the pore. The meniscus has a cylindrical shape at point B (pressure of ca. 6 Pa), but when the pressure approaches that of the saturated vapour it becomes flat, as would be expected from physical considerations. However, such meniscus behaviour is not apparent in the simulation of infinite pore, and this is an advantage of the Canonical Ensemble used in this paper. Not only does the Canonical Ensemble provide the isotherm for adsorption inside the pore and the corresponding microscopic configuration of particles, it also provides the magnitude of the adsorption on the outer surface. The surface adsorption is simply the amount adsorbed on the two outer surfaces of the finite pore. This is calculated as the surface excess as follows. Thus, a volume space corresponding to three collision diameters above the outer carbon surface is used in the determination of the surface excess, which is the difference between the number of particles observed in the simulation for that volume and the number of particles if that volume has a density equal to the bulk density per unit surface area. The plot of this surface excess versus the pressure typically shows a multi-layering pattern. If the Langmuir equation is used to fit the initial part of this curve (sub-monolayer coverage), we obtain a monolayer coverage concentration of 12 µmol/m 2. This is equivalent to saying that the molecular projection area of argon is Å 2. This value agrees well with the value of Å 2 reported by Gregg and Sing (22) for argon, thereby confirming that the correct values of the molecular parameters have been used in the simulation Effects of finite length compared to infinite length We have shown that a Monte Carlo simulation with a Canonical Ensemble can provide some rich information about the adsorption behaviour in a finite pore. To further show the effects of finite pore length on the adsorption equilibrium, Figure 6 presents the isotherm obtained for an infinite pore of width 1.5 nm using the Grand Canonical Ensemble (circle symbols; filled symbols for adsorption and unfilled symbols for desorption) and that obtained for a finite pore of the same width using a Canonical Monte Carlo simulation (diamond symbols). The differences between these two isotherms are discussed below. (i) The isotherm for a finite pore is always lower than that for an infinite pore, simply due to the lower solid fluid potential in the pore mouth region where an argon particle at a given location interacts with a lesser number of carbon atoms in the case of the finite pore. This effect is further amplified in the case of the smaller pore width, as seen in Figure 8 below for the 1-nm pore width case. The reason simply arises because the 1 nm pore can

10 1 A. Wongkoblap et al./adsorption Science & Technology Vol. 23 No H = 1.5 nm Pore density (kmol/m 3 ) Pressure (Pa) Figure 6. Comparison between the adsorption isotherm of argon at 87.3 K for an infinite pore of 1.5 nm width (filled circles correspond to adsorption while open circles correspond to desorption) and that for a finite pore of the same width (filled diamonds refer to adsorption while open diamonds refer to desorption). (ii) (iii) accommodate exactly two layers of argon particles. The fluid fluid interaction in the case of the infinite pore is very strong, not only due to the stronger solid fluid interaction compared to the finite pore but also due to the greater fluid fluid interaction between particles in the same layer relative to particles across the layers. The isotherm for the finite pore (1.5 nm) is shallower than that for the infinite pore, showing a near completion of the two contact layers at ca. 5 Pa beyond which the two inner-core layers are then filled with adsorbate molecules. Compression of the adsorbed phase also occurs as for the case of the infinite pore, but this happens at a much higher pressure ca. 8 Pa compared to 3 Pa for the case of the infinite pore. This is once again due to the greater solid fluid interaction in the infinite pore, as discussed in point (i) above. For the finite pore, the higher pressure necessary for compression leads to a smaller hysteresis loop and somewhat easier evaporation. This is attributed to the presence of a meniscus at which evaporation of argon particles from the adsorbed phase into the surrounding bulk fluid is readily facilitated. Side-views of snapshots of argon particles in finite pores of 1., 1.5 and 2. nm width are shown in Figure 7 for various pressure values ranging from low pressure to saturated vapour pressure. The pressure values used in the snapshots for the 1 nm pore were 2, 13, 5 and 1 Pa [(a) in Figure 7], those for the 1.5 nm pore were 2, 14, 65 and 1 Pa [(b) in Figure 7], while those for the 2 nm pore were 4, 7, 12 and 1 Pa [(c) in Figure 7]. It should be remembered that a study of the meniscus is not possible with the simulation of infinite pores (Maddox et al. 1995). In each section of the figure, the black spheres represent carbon atoms while the white spheres are argon particles. The numbers of layers in the 1., 1.5 and 2 nm pores are 2, 4 and 5, respectively. It will be noted that the behaviour in each pore is similar irrespective of its width, i.e. the adsorbed phase is started at low pressure by forming the two contact layers adjacent to the two walls. When the pressure increases, these contact layers are completed and the inner cores are then filled. With the exception of the 1 nm pore where we have only two layers, the other pores show the presence of a meniscus at intermediate pressure. The meniscus is cylindrical in shape and

11 Adsorption of Lennard-Jones Fluids in Carbon Slit Pores of a Finite Length 11 (a) H = 1. nm (b) H = 1.5 nm (c) H = 2. nm Figure 7. Snapshots of argon particles in finite-length pores of width, H, equal to (a) 1. nm, (b) 1.5 nm and (c) 2. nm at a temperature of 87.3 K and different pressures (P = 2, 13, 5 and 1 Pa for the 1. nm pore; P = 2, 14, 65 and 1 Pa for the 1.5 nm pore; and P = 4, 7, 12 and 1 Pa for the 2. nm pore). become flat at nearly saturated vapour pressure. Although the layers are quite distinct in the smaller pore, the layers are somewhat disordered in the case of the larger pore (2. nm) due to the combined effects of finite length and the weaker solid fluid potential. It is interesting to note that adsorbed argon particles exist just slightly outside the pore mouth. This is simply due to the fluid fluid interaction and (to a smaller extent) the weak interaction with carbon atoms near the pore mouth Isotherms from the Canonical Ensemble and the Gibbs Ensemble The adsorption behaviour of a finite pore and the effects of finite length have been discussed in the last two sections above. We now address the comparison between the adsorption isotherms for the finite pore obtained from the Canonical Ensemble (in which a finite pore is inside a large simulation box) and those for the same pore using the Gibbs Ensemble. These isotherms are shown in Figure 8 for pore widths of 1., 1.5 and 2 nm, respectively (the solid lines with filled diamond symbols are those from Canonical Ensemble simulation while those with unfilled upright triangle symbols are from the Gibbs Ensemble simulation). The simulation box length in the Canonical

12 12 A. Wongkoblap et al./adsorption Science & Technology Vol. 23 No Pore density (kmol/m 3 ) (a) 5 H = 1. nm 4 Infinite pore Pore density (kmol/m 3 ) (b) 4 H = 1.5 nm Pressure (Pa) Pressure (Pa) Pore density (kmol/m 3 ) (c) H = 2. nm Pressure (Pa) Figure 8. Adsorption isotherms of argon in finite-length pores of width, H, equal to (a) 1. nm, (b) 1.5 nm and (c) 2. nm at a temperature of 87.3 K. The data points in all three parts of the figure refer to the following:, finite pore NVT;, finite pore GEMC. Ensemble simulation was 4 Å. It should be noted that the amount adsorbed using Canonical Ensemble simulation was always less than that using the Gibbs Ensemble simulation, with this difference being significant in smaller pores but becoming much less so in larger pores (2 nm pores). This is because the interface between the adsorbed phase and the gas phase (gas reservoir surrounding the pore) always occurs in the Canonical Ensemble simulations, and hence energy is required to sustain the interface resulting in a lower amount being adsorbed. In contrast, no such interface exists in the Gibbs Ensemble simulations. This lack of interface also occurs in the case of GCMC simulation with infinite pores Behaviour of the average interaction energy with loading It is interesting to study the behaviour of the average interaction energy (solid fluid and fluid fluid) versus loading as depicted in Figure 9 for 1., 1.5 and 2 nm pores using the Canonical Ensemble. In all the curves depicted in the figure, a common feature is that the interaction energy is nearly constant when the loading is in the monolayer region of the isotherm. This indicates that

13 Adsorption of Lennard-Jones Fluids in Carbon Slit Pores of a Finite Length 13 2 Interaction energy (kj/mol) Pore density (kmol/m 3 ) 3 35 Figure 9. The relationship between interaction energy and pore density at pore widths, H, of 1. nm ( ), 1.5 nm ( ) and 2. nm ( ), respectively. the argon particles reside on the same planes parallel to the pore walls. Another observation is that adsorption is preferable in smaller pores. Particles occupy spaces away from the two contact layers as the loading is increased, resulting in an increase of the potential energy due to the weaker solid fluid interaction. The results for the interaction energy behaviour are similar to the results for isosteric heat predicted by Balbuena and Gubbins (1993) for semi-infinite slit pores Effect of the parameters employed in the Canonical Monte Carlo simulation Having discussed the effects of pore length on the adsorption isotherm using both the Canonical Ensemble and Gibbs Ensemble, we now turn to the effects of the parameters employed in the Monte Carlo simulation with the Canonical Ensemble. The critical parameters are the number of cycles to reach equilibrium, the initial allocation of particles, the displacement step length and the simulation box size. A serious loss in accuracy may occur if the number of cycles used in the Canonical Ensemble is not sufficient. This is discussed below in Section Similarly, the effect of the initial particle allocation is discussed in Section However, the choice of displacement step length in the Canonical Ensemble simulation is not as straightforward. We shall discuss this effect in Section The simulation box size is also important as its size should be sufficiently large to have a statistically acceptable number of particles in the bulk phase. We shall present this effect in Section Effect of the number of cycles necessary to attain equilibrium on the Canonical simulation The time-consuming step in MC simulation with a Canonical Ensemble lies in the displacement movement of the particles and the calculation of the new energies associated with it. Generally, many such moves are required to obtain a reasonable accuracy in the calculation of the ensemble average. In Figures 1 and 11, the numbers of particles in the pore, on the carbon surface, at the

14 14 A. Wongkoblap et al./adsorption Science & Technology Vol. 23 No Number of particles Number of cycles ( 1 3 ) 4 5 Figure 1. Number of particles in the pore ( ), on the surface ( ), in the bulk ( ) and at the edge ( ) at every 1 cycles for 2 particles and a displacement step length of.5σ ff. edges of the carbon layer and in the bulk phase are plotted against the number of Monte Carlo cycles. We start the simulation after placing 2 argon particles randomly in the cubic simulation box. Because of this random placement, particles can be found in all parts of the simulation box; in the pore, on the carbon surface, at the edges and in the bulk phase. We carried out simulations for two cases. In one case, the maximum displacement step length is kept at one-half of the collision diameter, while it is maintained at five-times the collision diameter for the other case. In each case, we monitor the variation in the number of particles in each part of the simulation box every 1 cycles. In the first case where the displacement step length is.5σ ff, equilibrium is 2 Number of particles Number of cycles ( 1 3 ) 4 5 Figure 11. Number of particles in the pore ( ), on the surface ( ), in the bulk ( ) and at the edge ( ) at every 1 cycles for 2 particles and a displacement step length of 5σ ff.

15 Adsorption of Lennard-Jones Fluids in Carbon Slit Pores of a Finite Length 15 attained at ca. 1 cycles, while in the second case with a larger displacement step length at least 35 cycles are required. At first, it would seem that the small displacement step length is ideal because of the low number of cycles required, but on closer inspection we note that the number of particles on the outer surface is greater than that inside the pore. This is physically impossible because the solid fluid interaction on the outer surface is lower than that inside the pore. The reason for this is because the displacement step length is too short and it is impossible to move the particles away from the outer surface once they are trapped in the local minimum of the potential energy. With a displacement step length of.5σ ff, any attempt to move a particle from the surface is faced with an increase in the potential energy. We shall discuss the effects of the displacement step length in greater detail in Section 3.5.3, but in general the displacement step length has to be sufficiently large. For this reason, the number of cycles used in our subsequent simulation is at least Effect of the initial allocation of particles on the Canonical simulation Next, we present the effects of the initial allocation of particles on the performance of the Canonical simulation. The initial distribution of particles in the box is investigated by either placing particles randomly over the whole volume of the simulation box (but, of course, excluding the volume space occupied by the graphite layers), or placing all the particles in the bulk phase randomly, or placing all the particles in the pore randomly. The displacement step length of 5σ ff is used in the simulation when the acceptance ratio (defined as the ratio of the number of successful moves to the number of attempts) is less than.5. In contrast, the maximum length of 14σ ff is chosen when the acceptance ratio is greater than.5. The number of Monte Carlo cycles necessary to attain equilibrium of 4 is used in each simulation. We will study three cases where the corresponding numbers of particles are 1, 3 and 5, respectively. In each of these cases, we carry out three simulations with three different ways of allocating these particles. Table 2 shows the number of particles in the pore at equilibrium for each of these cases. It is seen that irrespective of the way the particles are allocated there is no difference in the results; thus, as a general rule, we use the random allocation of particles over the whole volume space of the simulation boxes in our subsequent simulations Effect of the displacement step length on the Canonical simulation The most important parameter in the Canonical Monte Carlo simulation is the displacement step length, as this dictates how far a particle can be displaced during a move. To study the effects of TABLE 2. Number of Particles in a Finite-length Carbon Slit Pore of 1. nm Width for Three Different Initial Particle Allocations Total number of particles Resulting number of particles in the pore at equilibrium Anywhere in the whole box All in the bulk phase All inside the pore

16 16 A. Wongkoblap et al./adsorption Science & Technology Vol. 23 No TABLE 3. Number of Particles at Equilibrium in the Pore and on the Surface with Different Displacement Step Lengths for a Finite Pore Width of 1.5 nm and 2 Argon Particles Displacement step Number of particles length, σ ff in the pore on the surface and the displacement step length, we chose a simulation box length of 28-times the collision diameter and a set of 2 particles initially randomly placed throughout the volume space of the box. A range of displacement step lengths between.5σ ff and 14σ ff has been investigated. The upper limit of this range corresponds to one-half the length of the simulation box. The results of the simulations are listed in Table 3, in which the number of particles in the pore and on the surface are presented as a function of the displacement step length. As seen from the table, the number of particles in the pore increases when the displacement step length increases from.5σ ff to 7σ ff. Further increase in the step length does not change the particle distribution. As mentioned earlier in Section 3.5.1, particles can be artificially trapped in the local minimum of the potential energy located on the outer surface when the displacement step length is small. In that case, it does not matter in which direction we might wish to displace a particle from the surface with a very small step length (.5σ ff ) since the new configuration (after the displacement move) has much higher potential energy than that of the old configuration. Hence, the probability of rejection is very high, resulting in the system being trapped in a rather sharp local minimum of the outer surface. This problem no longer exists with a larger step length. Thus, in our subsequent study we choose a displacement step length of five-times the collision diameter if the acceptance ratio is less than.5, and use a value of one-half the box length when this ratio is greater than Effect of the simulation box size on the Canonical simulation Finally, we study the effects of the size of the simulation box by carrying out the Canonical simulation with a finite pore having a width of 1.5 nm. Box sizes ranging from 28σ ff to 176.2σ ff are studied. Following the recommendation made in our investigation of the displacement step length in Section 3.5.3, we choose a displacement step length of 5σ ff and the maximum at onehalf the box length when the acceptance ratio is greater than.5. The results of these simulations indicate that the box size does not have much effect when the pressure is high. However, when the pressure is low, there is a consistent deviation. This may be attributed to the number of particles in the bulk fluid being not sufficiently high to give a good statistical description of the fluid phase

17 Adsorption of Lennard-Jones Fluids in Carbon Slit Pores of a Finite Length 17 surrounding the finite pore. Thus, it is recommended that a sufficiently large box size be used to achieve a good statistical average of the particle number in the bulk phase. 4. CONCLUSIONS In this paper, we have presented the adsorption of argon in a carbon slit pore of finite length using Canonical Ensemble (NVT) and Gibbs Ensemble Monte Carlo simulations. Argon adsorption isotherms for pore widths of 1., 1.5 and 2. nm obtained from these two methods have been compared with isotherms obtained with the corresponding infinite slit pores using a Grand Canonical Monte Carlo (GCMC) simulation and periodic boundary conditions. It is found that adsorption in a finite pore is always less than adsorption in the corresponding infinite pore of the same pore width, with the magnitude of derivation decreasing as the pore size increases. In the case of a finite pore, adsorption hysteresis is smaller, round and shifted to high pressure when compared with that in an infinite pore of the same width. The microscopic configuration of particles inside the finitelength pore using the Canonical Ensemble shows not only an adsorption mechanism involving layering and pore filling but also the generation of a meniscus. Such a meniscus is not possible in the simulation of the infinite pore. In comparing the isotherms obtained from the Canonical Ensemble and the Gibbs Ensemble for finite pores, we find that the isotherm of the Gibbs Ensemble is slightly greater than that of the Canonical Ensemble due to the existence of the meniscus in the Canonical Ensemble. In the Canonical Ensemble simulation, the effects of the number of Monte Carlo cycles necessary to reach equilibrium, the initial allocation of particles, the displacement step length and the system size on the simulation performance have been particularly studied. Among these parameters, the displacement step length has the most significant effect on the performance of the simulation. The system size is also important, especially at low pressures when the size must be sufficiently large to enable the bulk phase to contain a statistically acceptable number of particles. ACKNOWLEDGEMENTS This project was supported by the Australian Research Council. We also acknowledge financial support from the Royal Thai Government in the form of scholarships to AW and SJ. REFERENCES Balbuena, P.B. and Gubbins, K.E. (1993) Langmuir 9, 181. Do, D.D. (1998) Adsorption Analysis: Equilibria and Kinetics, Imperial College Press, NJ, USA. Do, D.D. and Do, H.D. (23) Langmuir 19, 832. Donohue, M.D. and Aranovich, G.L. (1998) J. Colloid Interface Sci. 25, 121. Franklin, R.E. (1951) Proc. R. Soc. London, Ser. A 29, 196. Frenkel, D. and Smit, B. (22) Understanding Molecular Simulation, Academic Press, New York. Gauden, P.A., Terzyk, A.P., Rychlicki, G., Kowalczyk, P., Cwiertnia, M.S. and Garbacz, J.K. (24) J. Colloid Interface Sci. 273, 39. Gregg, S.J. and Sing, K.S.W. (22) Adsorption, Surface Area and Porosity, Academic Press, San Diego, CA, USA.

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