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Grand Canonical Monte Carlo Simulation for Determination of Optimum Parameters for Adsorption of Supercritical Methane in Pillared Layered Pores
Journal of Colloid and Interface Science 254, 1–7 (2002) doi:10.1006/jcis.2002.8543
Grand Canonical Monte Carlo Simulation for Determination of Optimum Parameters for Adsorption of Supercritical Methane in Pillared Layered Pores Dapeng Cao,∗,1 Wenchuan Wang,∗ and Xue Duan† ∗ Beijing University of Chemical Technology, College of Chemical Engineering, Beijing 100029, China; and †Key Laboratory of Controllable Chemical Reaction Science and Technology of the Ministry of Education, Beijing University of Chemical Technology, Beijing 100029, China Received December 19, 2001; accepted June 14, 2002; published online September 16, 2002
1. INTRODUCTION A grand canonical Monte Carlo (GCMC) method is carried out to determine optimum adsorptive storage pressures of supercritical methane in pillared layered pores. In the simulation, the pillared layered pore is modeled by a uniform distribution of pillars between two solid walls. Methane is described as a spherical Lennard-Jones molecule, and Steele’s 10-4-3 potential is used for representing the interaction between the fluid and a layered wall. The site-site interaction is also used for calculating the interaction energy between methane molecules and pillars. An effective potential model that reflects the characteristics of a real pillared layered material is proposed here. In the model, a binary interaction parameter, kfw , is introduced into the combining rule for the cross-energy parameter for the interaction between the fluid and a layered wall. Based on the experimental results for the Zr-pillared material synthesized and characterized by Boksh, Kikkinides, and Yang, the binary interaction parameter, kfw , is determined by fitting the simulation results to the experimental adsorption data of nitrogen at 77 K. Then, by taking it as a model of pillared layered material, a series of GCMC simulations have been carried out. The excess adsorption isotherms of methane in a pillared layered pore with three different pore widths and porosities are obtained at three supercritical temperatures T = 207.3, 237.0, and 266.6 K. Based on the simulation results at different porosities, various pore widths and different supercritical temperatures, the pillared layered pore with porosity ψ = 0.94 and pore width hσp = 1.02 nm is recommended as adsorption storage material of supercritical methane. Moreover, the optimum adsorption pressure is determined at a given temperature and a fixed width of the pillared layered pore. For example, at temperature T = 207.3 K, the optimum adsorption pressures are 3.1, 3.7, and 4.5 M Pa at H = 1.02, 1.70, and 2.38 nm, respectively. In summary, the GCMC method is a useful tool for optimizing adsorption storage of supercritical methane in pillared layered material. C
With the development and progress of human society, the fossil fuels of coal, petroleum, and others are consumed drastically. Accordingly, finding new and effective alternative energy sources is urgently required (1). At present, investigators (2–8) believe that natural gas and hydrogen are suitable alternatives without pollution. Therefore, effectively storing natural gas and hydrogen becomes an important subject. Adsorption storage of natural gas using micropore materials is a promising technology. Compared with compressed natural gas (CNG) technology, adsorption of natural gas (ANG) technology is safer and more economical. As is pointed out by Matranga et al. (2), CNG is stored in heavy steel cylinders at high pressures of 20–30 MPa, while ANG only requires a relatively low pressure about 4 MPa to be stored in a lightweight cylinder with micropore adsorbents. Therefore, a series of studies (9–13) on methane adsorbed in micropore materials were published in past decades. Quinn and MacDonald (9) and Zhou et al. (10) studied natural gas storage at ambient temperature by experimentation. Nicholas et al. (11) studied the diffusion behavior of methane in zeolites by using molecular dynamics (MD) simulation. In general, adsorption storage of methane takes place at supercritical temperatures. Therefore, Du et al. (12) investigated the adsorption of supercritical methane in zeolites by using mean-field theory (MFT) and Monte Carlo (MC) simulations. Tan and Gubbins (13) investigated the adsorption of methane in slit pores at supercritical temperatures and pointed out that classic excess adsorption isotherms from molecular simulation exhibit a maximum adsorption at a specific pressure, as expected by experiment (10). Based on the lattice theory, Aranovich and Donohue (14) developed an equation of state to describe the excess adsorption isotherms and observed the same shaped excess adsorption isotherm as Tan and Gubbins (13). However, the optimum adsorptive pressure that corresponds to a maximum adsorption at a fixed temperature was not determined yet. Among simulation and theoretical studies on the adsorption of fluids in micropore materials, slit carbon pores are mainly
2002 Elsevier Science (USA)
Key Words: grand canonical Monte Carlo simulation; supercritical methane; adsorption, pillared layered pores.
1 To whom correspondence should be addressed. E-mail: caodp@grad. buct.edu.cn.
1
0021-9797/02 $35.00
C 2002 Elsevier Science (USA)
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used as the absorbent (2–6). Recently, with the development of nanotechnology, new micropore pillared layered materials have been used in adsorption separation and shape-selective catalysis, because they have quasi-2-dimensional space and regularly controlled layer distances as well as the distribution of pillars. Du et al. (15) and Wei et al. (16) performed many experiments on the preparation and synthesis of pillared layered material and obtained successfully pillared hydrotalcites or pillared zirconium phosphate. In addition, Pereira et al. (17) and Yang and co-workers (18–20) performed some experiments on the adsorption behavior of pillared layered material, especially about the adsorption separation and distribution of micropores. In addition, there also are some papers (21–27) on computer simulation of fluids confined in pillared layered material. In 1995, Yi et al. (21) reported the diffusion of a simplified model fluid confined in pillared layered material by the MD method. In a subsequent paper of (22), they also reported the adsorption of the model fluid confined in pillared layered material. For the model fluid, based on their results, different distributions of pillars had very little effect on the adsorption, while the porosity of the pillared layered systems caused a significant fluctuation on the amount of adsorption. In fact, there are significant differences between the simplified model fluid and the methane fluid here. Consequently, it is important to investigate the behavior of real fluids confined in pillared layered material. Most recently, Ghassemzadeh et al. (23) addressed separation of gas mixtures confined in pillared layered material and molecular sieve membranes at temperature T = 303 K, using the MD method. In addition, in our previous paper (27), the phase behavior of methane molecules confined in the pillared layered material at low temperatures was reported. Capillary condensation and the hysteresis loop of methane in the pillared layered material at low temperatures were also observed. In contrast, little experimental work was reported on the layered pillared materials. Baksh et al. (19) synthesized five layered materials pillared by oxides of Zr, Al, Cr, Fe, and Ti. The adsorption isotherms of four probe molecules N2 (77 K), benzene (298 K), perfluorotributylamine (298 K), and H2 O (298 K) were measured. In addition, detailed characterization of the pillared layered material was carried out using thermogravimetric analysis, X-ray diffraction (XRD), scanning electron microscopy (SEM), and inductively coupled argon plasma atomic emission spectroscopy (ICAP-AES). The results indicate that the pillared layered materials are promising adsorbents for gas separation. In this work, we use the GCMC method to simulate adsorption storage of supercritical methane in layered pillared pores. A real pillared layered material, which has been well characterized by experiment, is taken as the prototype of the adsorbent. An effective potential model is proposed for all the simulations by introducing a binary interaction parameter that is fitted to the experimental data. Then, extensive simulations are carried out for adsorption storage of methane for diverse widths and porosities of pores at three supercritical tempera-
tures. The optimum adsorption pressures at fixed temperatures and widths are located, which can be used for determining operation conditions for adsorption storage of methane at supercritical temperatures.
2. MODEL OF LAYERED PILLARED PORES
In our simulation, a model from Yi et al. (23) with uniform distribution of pillars between two layered walls was used to describe a layered pillared pore. A schematic diagram of the layered pillared pore is shown in Fig. 1. Layered walls are represented by the (100) face of a face-centered cubic solid with a specified surface number density. The pillars are represented by rigid chains consisting of a given number of Lennard-Jones spheres with the size parameter σ p . Defining the perpendicular direction of layered walls as z direction, the z coordination of the center of the end molecules in pillar chains is 0.5 × (σ p + σw ), where σw is the size of atoms distributed in the layered walls. The distance (i.e., the pore width) between layered walls is defined by the pillar height H = hσ p (see Fig. 1), where h is the number of the pillar atoms. The porosity of the system ψ is defined as the volume faction of the system not occupied by the pillars, given by
ψ =1−
N p π σ p2 6·S
,
[1]
where N p is the number of pillars and S is the area in x − y surface in a simulation box. Although the model is a highly simple one, Ghassemzadeh et al. (23) used it as a model of pillared layered pores and have qualitatively simulated separation of binary mixtures among methane and nitrogen as well as CO2 at ambient temperature. Moreover, the model presents a good agreement with the result from statistical mechanical perturbation theory (23). As a result, we use the model to represent the pillared layered pore in this paper.
FIG. 1.
Schematic diagram of pillared layered material.
3
ADSORPTION OF METHANE IN PILLARED LAYERED PORES
teractions: the potential energy between fluid molecules, φff (r ), the potential energy between fluid molecules and pillars, φfp , and the potential energy between fluid molecules and layered walls, φfw (z):
3. POTENTIAL MODELS
3.1. Potential Models for Fluid-Fluid, Fluid-Pillars, and Fluid-Wall Interactions The fluid-fluid interaction is described by the cut and shifted Lennard Jones (LJ) potential (2, 3, 6), φff (r ) =
φLJ (r ) − φLJ (rc ) r < rc
[2]
r ≥ rc ,
0
where r is the intermolecular distance, rc is the cutoff radius, rc = 5σff , φLJ is the full LJ potential, φLJ (r ) = 4εff [(σff /r )12 − (σff /r )6 ], where εff and σff are the energy and size parameters of the fluid, respectively. Assuming that the pillars are composed of rigid chains with atom size σ p , the site-to-site method is used to calculate the interaction between fluid molecules and pillars, σfp , φfp = 4εfp
Nf h N p i=1 j=1
σfp ri j
12
−
σfp ri j
6 ,
[3]
where ri j is the distance between a fluid molecule and an atom of the pillars, N f is the number of fluid molecules in the simulation box, εfp and σfp are the cross-energy and size parameters, which are obtained from the Lorentz-Berthelot (LB) combining rules. The interaction between a layered wall and a fluid molecule is represented by the well-known Steele’s 10-4-3 potential (13, 28) 10 4 σfw σfw 2 φfw (z) = 2πρw εfw σfw 0.4 − z z 4 σfw , − 3 (0.61 + z)3
[4]
where ρw is the number density of the solid wall (27), and the subscript w represents the layered wall, is the distance between lattice planes, z is the normal distance between a fluid molecule and one of the layered walls, εfw and σfw are the crossinteraction parameters between the fluid and the wall. The energy and size parameters of the fluid molecule, pillar atom and the layered wall are shown in Table 1. Accordingly, the total potential energy φT (r ) of the fluid molecules confined in layered pillared pores is a sum of three inTABLE 1 Parameters of Potential Models for Methane, Pillar, and Layered Walls CH4 (27)
Pillar (27)
Layered wall (27)
σff (nm)
εff /k (K)
σpp (nm)
εpp /k (K)
σww (nm)
εww /k (K)
0.381
148.1
0.34
28.0
0.34
28.0
φT (r ) = φff (r ) + φfw (z) + φfp .
[5]
3.2. An Effective Potential Model for the Fluid-Wall Interaction It is noted that the model aforementioned cannot well represent the real interaction between the fluid and a layered wall. In order to accurately represent the interaction between the fluid and a layered wall, a binary interaction parameter, kfw , is introduced here to modify the cross-interaction energy parameter in the LB combining rules, εfw = kfw · (εff εww )0.5
σfw = 0.5(σww + σff ),
[6]
where εfw and σfw are the cross-interaction parameters. kfw in Eq. [6] can be determined by fitting the simulation results to the experiment data, which is discussed in Section 5.
4. GRAND CANONICAL MONTE CARLO SIMULATION
Using the grand canonical Monte Carlo (GCMC) method, where the temperature, the chemical potential, and the micropore volume are the independent variables and are specified in advance, adsorption storage of methane in layered pillared pores at supercritical temperatures was simulated. In our simulation, the periodic boundary conditions were imposed only in x and y directions, because there is a layered wall in the z direction (see Fig. 1). The surface area S on the layered wall was set to 180 σff2 . An initial configuration was generated randomly. In general, the initial density of 0.2 was used. In our simulation, all variables were reduced with respect to methane parameters, given by µ∗ = µ/εff , z ∗ = z/σff , T ∗ = kT /εff , H ∗ = hσ p /σff , ρ ∗ = ρσff3 , S ∗ = S/σff2 , where superscript ∗ denotes reduced, µ is the chemical potential, T is the temperature of the system, ρ is the number density of the fluid, and k is the Boltzmann constant. In addition, for every state, 2 × 107 configurations were generated. The former 1 × 107 configurations were discarded to guarantee equilibration, whereas the latter 1 × 107 configurations were used to average the desired thermodynamic properties. The uncertainty on the final results (ensemble averages of the number of adsorbate molecules in the box and the total potential energy) was estimated to be less than 2%. Other simulation details can be referred to our previous work (6, 26, 27).
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TABLE 2 Physical Characterization of Zr-Pillared Pore (19): Surface Area, Pore Volume, and Interlayer Spacing (H)a
Zr-pillared pore a b
H, nm
BET, m2 /g
N2 , cm3 /g
0.96
322
0.177b
H is defined by Fig. 1. Measured by equilibrium pore filling with N2 (77 K).
5. DETERMINATION OF THE BINARY INTERACTION PARAMETER, kfw , BY GCMC AND EXPERIMENTAL DATA
As noted in Section 3.2, a binary interaction parameter is introduced (see Eq. [6]) to reflect the real interaction between a fluid molecule and a layered wall by fitting to the experimental data. Here, we take the Zr-pillared pore synthesized and characterized by Baksh et al. (19) as the prototype of adsorbents for adsorption storage of supercritical methane. The properties of the Zr-pillared layered pore are shown in Table 2. In the simulation, the width of the pore is exactly the same as H in Table 2, and the other parameters are taken from Table 1. The GCMC simulation for the modeled pillared layered pore was carried out to obtain the adsorption isotherm of nitrogen at 77 K, which is shown in Fig. 2. Note that the simulation isotherm exhibits pronounced discrepancy from the experimental isotherm (19), where the binary interaction parameter is not taken into consideration; i.e., kfw = 1.0. In particular, the pore filling pressures represented by points B for simulation and A
for experiment are of significant difference. It implies that a binary interaction parameter, kfw in Eq. [6], is needed to make the potential effective in the description of the fluid-wall interaction in pores. Therefore, we attempted a series of GCMC simulations in terms of different values of kfw . Figure 3 gives a comparison
FIG. 2. A comparison of experimental data and simulation results, when the binary interaction parameter kfw = 1.0. A and B represent the pore filing point for experiment and simulation, respectively.
FIG. 3. A Comparison of experimental data and simulation results, when the binary interaction parameter kfw = 0.65. A and B represent the pore filing point for experiment and simulation, respectively.
of the experimental and simulated isotherms when kfw = 0.65. It is found that a good agreement between the two isotherms is attained in this case. Particularly, the pressures for pore filling, denoted by B for experiment and A for simulation, coincide
FIG. 4. Excess adsorption isotherms of methane in pillared layered pores with different porosities at T = 207.3 K. (a) H = 1.02 nm, (b) H = 1.70 nm, (c) H = 2.38 nm.
ADSORPTION OF METHANE IN PILLARED LAYERED PORES
5
well. It indicates that the model proposed here, consisting of Eqs. [1]–[4] and [6] along with kfw = 0.65, is an effective model to reproduce the properties for the Zr-pillared porous material prepared and characterized by Baksh et al. (19). As a result, the model pillared layered pore is used for extensive GCMC simulations and to investigate the effects of variables on adsorption storage of methane in pillared layered material. 6. RESULTS AND DISCUSSION
6.1. Excess Adsorption Isotherms The overall average reduced density ρT∗ in pillared layered pores is given by ρT∗
1 = ∗ H
H ∗
ρ ∗ (z ∗ ) dz ∗ ,
[7]
0
where ρ ∗ (z ∗ ) is the local density distribution function, which can be solved by ρ ∗ (z ∗ ) = N (z ∗ ) /(S ∗ · ψ · z ∗ ),
[8]
FIG. 6. Excess adsorption isotherms of methane in pillared layered pores with different porosities at T = 266.6 K. (a) H = 1.02 nm, (b) H = 1.70 nm, (c) H = 2.38 nm.
where N (z ∗ ) is the ensemble average of the number of fluid molecules in a cell of S ∗ · ψ · z ∗ . The excess adsorption isotherm is represented by = ρT∗ − ρb∗ ,
[9]
where ρb∗ is the reduced density of the bulk fluid. In GCMC simulation, when the system reaches the adsorption equilibrium, the chemical potential, pressure, and temperature of the fluid confined in the micropore are the same as those of the bulk fluid. The pressure of the fluid in pillared layered pores can be solved by the MBWR equation (29), whose parameters are taken from the literature (30). Consequently, the reduced density of bulk fluid can also be obtained by the MBWR equation. 6.2. Effect of the Porosity
FIG. 5. Excess adsorption isotherms of methane in pillared layered pores with different porosities at T = 237.0 K. (a) H = 1.02 nm, (b) H = 1.70 nm, (c) H = 2.38 nm.
The porosity, which is directly related to the number of pillars between two layered walls, has a significant effect on the adsorption. We simulated the excess adsorption isotherms for three different porosities (ψ = 0.88, 0.94, 0.98) and pore widths (H = 1.02, 1.70, 2.38 nm) at three different supercritical temperatures, T = 207.3, 237.0, 266.6 K. The excess adsorption isotherms are shown in Figs. 4–6, respectively. For a model
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CAO, WANG, AND DUAN
FIG. 9. Optimum adsorption storage pressure for supercritical methane changing with temperature at three different pore widths.
it is observed from Figs. 4–6 that at the three fixed pore widths (hσ p = 1.02, 1.70, and 2.38 nm), and three supercritical temperatures, the isotherms for the porosity of 0.94, rather than 0.98, give the largest adsorption amount. This observation indicates that there is an optimized porosity, ψ = 0.94, for methane storage at supercritical temperatures. It coincides with our previous results at ambient temperature (26). Consequently, we focused our studies on the porosity of 0.94 subsequently. 6.3. The Optimum Adsorption Pressure FIG. 7. Excess adsorption isotherms of methane in pillared layered pores with different pore widths at porosity of ψ = 0.94. (a) T = 207.3 K, (b) T = 237.0 K, (c) T = 266.6 K. A, B, and C represent the maximum adsorption at H = 1.02, 1.70, and 2.38 nm, respectively.
fluid, the results of Yi et al. (9) suggest that at low pressures (P < 2 MPa), the adsorption amount decreases with the increase of porosity, while the adsorption amount increases with porosity at high pressures (P > 2 MPa). In contrast, for methane here,
It can be seen from Figs. 4–6 that the excess adsorption isotherms are of a classical shape, where the excess adsorption amount increases to a maximum, then decreases with the increase of pressure. As a result, the optimum adsorption pressure, corresponding to the maximum adsorption amount, exists at each pore width and temperature at ψ = 0.94, as shown in Fig. 7. It can be seen from Fig. 7 that the optimum adsorption pressure increases with the increase of pore width at a fixed temperature. For example, the optimum adsorption pressures are 3.1, 3.7, and 4.5 M Pa, at H = 1.02, 1.70, and 2.38 nm and T = 207.3 K, respectively. For a fixed pore width, the fact that the optimum adsorption pressure increases with temperature can also be found from Fig. 7. As illustration, Fig. 8 gives the relationship between the optimum pressure and pore width at three fixed temperatures. In addition, Fig. 9 presents the relationship between the optimum pressure and the temperature at three fixed pore widths. 7. CONCLUSIONS
FIG. 8. Optimum adsorption storage pressure for supercritical methane changing with pore width at three different temperatures.
Adsorption storage of supercritical methane in layered pillared pores has been simulated, by using the grand canonical Monte Carlo method. Based on the experimental data for Zrpillared porous material synthesized and characterized by Baksh et al. (19), an effective potential model that reflects the characteristics of a real pillared layered material is proposed. In the model, the binary interaction parameter, kfw , for the cross-interaction
ADSORPTION OF METHANE IN PILLARED LAYERED PORES
energy between a methane molecule and a layered wall in Eq. [6] is introduced and is determined by fitting simulation results to the experimental adsorption data of nitrogen at 77 K (19). It is noted that because the binary interaction parameter kfw was obtained by fitting to the Zr-pillared porous material, it is expected that its value would vary with different materials and even the fluids of interest. Nevertheless, this approach provides a useful tool to model pillared porous materials. By taking this model pillared layered pore as a prototype of adsorbents, extensive simulations have been carried out. The excess adsorption isotherms of methane for three different pore widths (H = 1.02, 1.70, 2.38 nm) and porosities (ψ = 0.88, 0.94, 0.98) are obtained at three supercritical temperatures, T = 207.3, 237.0, and 266.6 K. As is shown in Figs. 4–6, ψ = 0.94 is the optimum porosity, which gives relatively larger uptakes for adsorption storage of supercritical methane in the model pillared layered pores. Figure 7 shows the excess adsorption (see definition in Eq. [9]) changing with pressure for different size pores at the porosity of 0.94. It is found that compared with the pores of H = 1.70 and 2.38 nm, the pore of H = 1.02 nm gives considerably larger excess adsorption because of the stronger fluid-wall interaction in the relatively narrower pore. Accordingly, combined with the above results, the pillared layered pore with the ψ = 0.94 and pore width hσ p = 1.02 nm is recommended as adsorption material of supercritical methane. Importantly, all the excess adsorption increases with pressure and reaches a maximum for each isotherm at a constant pore width. It implies that the optimum adsorption pressure that corresponds to the maximum excess adsorption in Fig. 7 can be located. We have plotted the optimum pressure against pore width and temperature in Figs. 8 and 9, respectively. It is found that the optimum adsorption pressure increases monotonically with temperature at a fixed pore width, and also increases monotonically with pore width at a fixed temperature in the ranges of variables studied. In summary, the GCMC method and the model proposed in this work are a useful tool for preparing of the layered pillared material and optimizing the operation for adsorption storage of methane in the porous materials. ACKNOWLEDGMENTS This work was supported by the Key Fundamental Research Plan (No. G2000048010), and the Key Laboratory of Controllable Chemical Re-
7
action Science and Technology of the Ministry of Education (China), and the National Natural Science Foundation of China under Grant No. 2977604, and the National High Performance Computing Foundation of China (No. 99118).
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Grand Canonical Monte Carlo Simulation for Determination of Optimum Parameters for Adsorption of Supercritical Methane in Pillared Layered Pores Dapeng Cao,∗,1 Wenchuan Wang,∗ and Xue Duan† ∗ Beijing University of Chemical Technology, College of Chemical Engineering, Beijing 100029, China; and †Key Laboratory of Controllable Chemical Reaction Science and Technology of the Ministry of Education, Beijing University of Chemical Technology, Beijing 100029, China Received December 19, 2001; accepted June 14, 2002; published online September 16, 2002
1. INTRODUCTION A grand canonical Monte Carlo (GCMC) method is carried out to determine optimum adsorptive storage pressures of supercritical methane in pillared layered pores. In the simulation, the pillared layered pore is modeled by a uniform distribution of pillars between two solid walls. Methane is described as a spherical Lennard-Jones molecule, and Steele’s 10-4-3 potential is used for representing the interaction between the fluid and a layered wall. The site-site interaction is also used for calculating the interaction energy between methane molecules and pillars. An effective potential model that reflects the characteristics of a real pillared layered material is proposed here. In the model, a binary interaction parameter, kfw , is introduced into the combining rule for the cross-energy parameter for the interaction between the fluid and a layered wall. Based on the experimental results for the Zr-pillared material synthesized and characterized by Boksh, Kikkinides, and Yang, the binary interaction parameter, kfw , is determined by fitting the simulation results to the experimental adsorption data of nitrogen at 77 K. Then, by taking it as a model of pillared layered material, a series of GCMC simulations have been carried out. The excess adsorption isotherms of methane in a pillared layered pore with three different pore widths and porosities are obtained at three supercritical temperatures T = 207.3, 237.0, and 266.6 K. Based on the simulation results at different porosities, various pore widths and different supercritical temperatures, the pillared layered pore with porosity ψ = 0.94 and pore width hσp = 1.02 nm is recommended as adsorption storage material of supercritical methane. Moreover, the optimum adsorption pressure is determined at a given temperature and a fixed width of the pillared layered pore. For example, at temperature T = 207.3 K, the optimum adsorption pressures are 3.1, 3.7, and 4.5 M Pa at H = 1.02, 1.70, and 2.38 nm, respectively. In summary, the GCMC method is a useful tool for optimizing adsorption storage of supercritical methane in pillared layered material. C
With the development and progress of human society, the fossil fuels of coal, petroleum, and others are consumed drastically. Accordingly, finding new and effective alternative energy sources is urgently required (1). At present, investigators (2–8) believe that natural gas and hydrogen are suitable alternatives without pollution. Therefore, effectively storing natural gas and hydrogen becomes an important subject. Adsorption storage of natural gas using micropore materials is a promising technology. Compared with compressed natural gas (CNG) technology, adsorption of natural gas (ANG) technology is safer and more economical. As is pointed out by Matranga et al. (2), CNG is stored in heavy steel cylinders at high pressures of 20–30 MPa, while ANG only requires a relatively low pressure about 4 MPa to be stored in a lightweight cylinder with micropore adsorbents. Therefore, a series of studies (9–13) on methane adsorbed in micropore materials were published in past decades. Quinn and MacDonald (9) and Zhou et al. (10) studied natural gas storage at ambient temperature by experimentation. Nicholas et al. (11) studied the diffusion behavior of methane in zeolites by using molecular dynamics (MD) simulation. In general, adsorption storage of methane takes place at supercritical temperatures. Therefore, Du et al. (12) investigated the adsorption of supercritical methane in zeolites by using mean-field theory (MFT) and Monte Carlo (MC) simulations. Tan and Gubbins (13) investigated the adsorption of methane in slit pores at supercritical temperatures and pointed out that classic excess adsorption isotherms from molecular simulation exhibit a maximum adsorption at a specific pressure, as expected by experiment (10). Based on the lattice theory, Aranovich and Donohue (14) developed an equation of state to describe the excess adsorption isotherms and observed the same shaped excess adsorption isotherm as Tan and Gubbins (13). However, the optimum adsorptive pressure that corresponds to a maximum adsorption at a fixed temperature was not determined yet. Among simulation and theoretical studies on the adsorption of fluids in micropore materials, slit carbon pores are mainly
2002 Elsevier Science (USA)
Key Words: grand canonical Monte Carlo simulation; supercritical methane; adsorption, pillared layered pores.
1 To whom correspondence should be addressed. E-mail: caodp@grad. buct.edu.cn.
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0021-9797/02 $35.00
C 2002 Elsevier Science (USA)
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CAO, WANG, AND DUAN
used as the absorbent (2–6). Recently, with the development of nanotechnology, new micropore pillared layered materials have been used in adsorption separation and shape-selective catalysis, because they have quasi-2-dimensional space and regularly controlled layer distances as well as the distribution of pillars. Du et al. (15) and Wei et al. (16) performed many experiments on the preparation and synthesis of pillared layered material and obtained successfully pillared hydrotalcites or pillared zirconium phosphate. In addition, Pereira et al. (17) and Yang and co-workers (18–20) performed some experiments on the adsorption behavior of pillared layered material, especially about the adsorption separation and distribution of micropores. In addition, there also are some papers (21–27) on computer simulation of fluids confined in pillared layered material. In 1995, Yi et al. (21) reported the diffusion of a simplified model fluid confined in pillared layered material by the MD method. In a subsequent paper of (22), they also reported the adsorption of the model fluid confined in pillared layered material. For the model fluid, based on their results, different distributions of pillars had very little effect on the adsorption, while the porosity of the pillared layered systems caused a significant fluctuation on the amount of adsorption. In fact, there are significant differences between the simplified model fluid and the methane fluid here. Consequently, it is important to investigate the behavior of real fluids confined in pillared layered material. Most recently, Ghassemzadeh et al. (23) addressed separation of gas mixtures confined in pillared layered material and molecular sieve membranes at temperature T = 303 K, using the MD method. In addition, in our previous paper (27), the phase behavior of methane molecules confined in the pillared layered material at low temperatures was reported. Capillary condensation and the hysteresis loop of methane in the pillared layered material at low temperatures were also observed. In contrast, little experimental work was reported on the layered pillared materials. Baksh et al. (19) synthesized five layered materials pillared by oxides of Zr, Al, Cr, Fe, and Ti. The adsorption isotherms of four probe molecules N2 (77 K), benzene (298 K), perfluorotributylamine (298 K), and H2 O (298 K) were measured. In addition, detailed characterization of the pillared layered material was carried out using thermogravimetric analysis, X-ray diffraction (XRD), scanning electron microscopy (SEM), and inductively coupled argon plasma atomic emission spectroscopy (ICAP-AES). The results indicate that the pillared layered materials are promising adsorbents for gas separation. In this work, we use the GCMC method to simulate adsorption storage of supercritical methane in layered pillared pores. A real pillared layered material, which has been well characterized by experiment, is taken as the prototype of the adsorbent. An effective potential model is proposed for all the simulations by introducing a binary interaction parameter that is fitted to the experimental data. Then, extensive simulations are carried out for adsorption storage of methane for diverse widths and porosities of pores at three supercritical tempera-
tures. The optimum adsorption pressures at fixed temperatures and widths are located, which can be used for determining operation conditions for adsorption storage of methane at supercritical temperatures.
2. MODEL OF LAYERED PILLARED PORES
In our simulation, a model from Yi et al. (23) with uniform distribution of pillars between two layered walls was used to describe a layered pillared pore. A schematic diagram of the layered pillared pore is shown in Fig. 1. Layered walls are represented by the (100) face of a face-centered cubic solid with a specified surface number density. The pillars are represented by rigid chains consisting of a given number of Lennard-Jones spheres with the size parameter σ p . Defining the perpendicular direction of layered walls as z direction, the z coordination of the center of the end molecules in pillar chains is 0.5 × (σ p + σw ), where σw is the size of atoms distributed in the layered walls. The distance (i.e., the pore width) between layered walls is defined by the pillar height H = hσ p (see Fig. 1), where h is the number of the pillar atoms. The porosity of the system ψ is defined as the volume faction of the system not occupied by the pillars, given by
ψ =1−
N p π σ p2 6·S
,
[1]
where N p is the number of pillars and S is the area in x − y surface in a simulation box. Although the model is a highly simple one, Ghassemzadeh et al. (23) used it as a model of pillared layered pores and have qualitatively simulated separation of binary mixtures among methane and nitrogen as well as CO2 at ambient temperature. Moreover, the model presents a good agreement with the result from statistical mechanical perturbation theory (23). As a result, we use the model to represent the pillared layered pore in this paper.
FIG. 1.
Schematic diagram of pillared layered material.
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ADSORPTION OF METHANE IN PILLARED LAYERED PORES
teractions: the potential energy between fluid molecules, φff (r ), the potential energy between fluid molecules and pillars, φfp , and the potential energy between fluid molecules and layered walls, φfw (z):
3. POTENTIAL MODELS
3.1. Potential Models for Fluid-Fluid, Fluid-Pillars, and Fluid-Wall Interactions The fluid-fluid interaction is described by the cut and shifted Lennard Jones (LJ) potential (2, 3, 6), φff (r ) =
φLJ (r ) − φLJ (rc ) r < rc
[2]
r ≥ rc ,
0
where r is the intermolecular distance, rc is the cutoff radius, rc = 5σff , φLJ is the full LJ potential, φLJ (r ) = 4εff [(σff /r )12 − (σff /r )6 ], where εff and σff are the energy and size parameters of the fluid, respectively. Assuming that the pillars are composed of rigid chains with atom size σ p , the site-to-site method is used to calculate the interaction between fluid molecules and pillars, σfp , φfp = 4εfp
Nf h N p i=1 j=1
σfp ri j
12
−
σfp ri j
6 ,
[3]
where ri j is the distance between a fluid molecule and an atom of the pillars, N f is the number of fluid molecules in the simulation box, εfp and σfp are the cross-energy and size parameters, which are obtained from the Lorentz-Berthelot (LB) combining rules. The interaction between a layered wall and a fluid molecule is represented by the well-known Steele’s 10-4-3 potential (13, 28) 10 4 σfw σfw 2 φfw (z) = 2πρw εfw σfw 0.4 − z z 4 σfw , − 3 (0.61 + z)3
[4]
where ρw is the number density of the solid wall (27), and the subscript w represents the layered wall, is the distance between lattice planes, z is the normal distance between a fluid molecule and one of the layered walls, εfw and σfw are the crossinteraction parameters between the fluid and the wall. The energy and size parameters of the fluid molecule, pillar atom and the layered wall are shown in Table 1. Accordingly, the total potential energy φT (r ) of the fluid molecules confined in layered pillared pores is a sum of three inTABLE 1 Parameters of Potential Models for Methane, Pillar, and Layered Walls CH4 (27)
Pillar (27)
Layered wall (27)
σff (nm)
εff /k (K)
σpp (nm)
εpp /k (K)
σww (nm)
εww /k (K)
0.381
148.1
0.34
28.0
0.34
28.0
φT (r ) = φff (r ) + φfw (z) + φfp .
[5]
3.2. An Effective Potential Model for the Fluid-Wall Interaction It is noted that the model aforementioned cannot well represent the real interaction between the fluid and a layered wall. In order to accurately represent the interaction between the fluid and a layered wall, a binary interaction parameter, kfw , is introduced here to modify the cross-interaction energy parameter in the LB combining rules, εfw = kfw · (εff εww )0.5
σfw = 0.5(σww + σff ),
[6]
where εfw and σfw are the cross-interaction parameters. kfw in Eq. [6] can be determined by fitting the simulation results to the experiment data, which is discussed in Section 5.
4. GRAND CANONICAL MONTE CARLO SIMULATION
Using the grand canonical Monte Carlo (GCMC) method, where the temperature, the chemical potential, and the micropore volume are the independent variables and are specified in advance, adsorption storage of methane in layered pillared pores at supercritical temperatures was simulated. In our simulation, the periodic boundary conditions were imposed only in x and y directions, because there is a layered wall in the z direction (see Fig. 1). The surface area S on the layered wall was set to 180 σff2 . An initial configuration was generated randomly. In general, the initial density of 0.2 was used. In our simulation, all variables were reduced with respect to methane parameters, given by µ∗ = µ/εff , z ∗ = z/σff , T ∗ = kT /εff , H ∗ = hσ p /σff , ρ ∗ = ρσff3 , S ∗ = S/σff2 , where superscript ∗ denotes reduced, µ is the chemical potential, T is the temperature of the system, ρ is the number density of the fluid, and k is the Boltzmann constant. In addition, for every state, 2 × 107 configurations were generated. The former 1 × 107 configurations were discarded to guarantee equilibration, whereas the latter 1 × 107 configurations were used to average the desired thermodynamic properties. The uncertainty on the final results (ensemble averages of the number of adsorbate molecules in the box and the total potential energy) was estimated to be less than 2%. Other simulation details can be referred to our previous work (6, 26, 27).
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CAO, WANG, AND DUAN
TABLE 2 Physical Characterization of Zr-Pillared Pore (19): Surface Area, Pore Volume, and Interlayer Spacing (H)a
Zr-pillared pore a b
H, nm
BET, m2 /g
N2 , cm3 /g
0.96
322
0.177b
H is defined by Fig. 1. Measured by equilibrium pore filling with N2 (77 K).
5. DETERMINATION OF THE BINARY INTERACTION PARAMETER, kfw , BY GCMC AND EXPERIMENTAL DATA
As noted in Section 3.2, a binary interaction parameter is introduced (see Eq. [6]) to reflect the real interaction between a fluid molecule and a layered wall by fitting to the experimental data. Here, we take the Zr-pillared pore synthesized and characterized by Baksh et al. (19) as the prototype of adsorbents for adsorption storage of supercritical methane. The properties of the Zr-pillared layered pore are shown in Table 2. In the simulation, the width of the pore is exactly the same as H in Table 2, and the other parameters are taken from Table 1. The GCMC simulation for the modeled pillared layered pore was carried out to obtain the adsorption isotherm of nitrogen at 77 K, which is shown in Fig. 2. Note that the simulation isotherm exhibits pronounced discrepancy from the experimental isotherm (19), where the binary interaction parameter is not taken into consideration; i.e., kfw = 1.0. In particular, the pore filling pressures represented by points B for simulation and A
for experiment are of significant difference. It implies that a binary interaction parameter, kfw in Eq. [6], is needed to make the potential effective in the description of the fluid-wall interaction in pores. Therefore, we attempted a series of GCMC simulations in terms of different values of kfw . Figure 3 gives a comparison
FIG. 2. A comparison of experimental data and simulation results, when the binary interaction parameter kfw = 1.0. A and B represent the pore filing point for experiment and simulation, respectively.
FIG. 3. A Comparison of experimental data and simulation results, when the binary interaction parameter kfw = 0.65. A and B represent the pore filing point for experiment and simulation, respectively.
of the experimental and simulated isotherms when kfw = 0.65. It is found that a good agreement between the two isotherms is attained in this case. Particularly, the pressures for pore filling, denoted by B for experiment and A for simulation, coincide
FIG. 4. Excess adsorption isotherms of methane in pillared layered pores with different porosities at T = 207.3 K. (a) H = 1.02 nm, (b) H = 1.70 nm, (c) H = 2.38 nm.
ADSORPTION OF METHANE IN PILLARED LAYERED PORES
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well. It indicates that the model proposed here, consisting of Eqs. [1]–[4] and [6] along with kfw = 0.65, is an effective model to reproduce the properties for the Zr-pillared porous material prepared and characterized by Baksh et al. (19). As a result, the model pillared layered pore is used for extensive GCMC simulations and to investigate the effects of variables on adsorption storage of methane in pillared layered material. 6. RESULTS AND DISCUSSION
6.1. Excess Adsorption Isotherms The overall average reduced density ρT∗ in pillared layered pores is given by ρT∗
1 = ∗ H
H ∗
ρ ∗ (z ∗ ) dz ∗ ,
[7]
0
where ρ ∗ (z ∗ ) is the local density distribution function, which can be solved by ρ ∗ (z ∗ ) = N (z ∗ ) /(S ∗ · ψ · z ∗ ),
[8]
FIG. 6. Excess adsorption isotherms of methane in pillared layered pores with different porosities at T = 266.6 K. (a) H = 1.02 nm, (b) H = 1.70 nm, (c) H = 2.38 nm.
where N (z ∗ ) is the ensemble average of the number of fluid molecules in a cell of S ∗ · ψ · z ∗ . The excess adsorption isotherm is represented by = ρT∗ − ρb∗ ,
[9]
where ρb∗ is the reduced density of the bulk fluid. In GCMC simulation, when the system reaches the adsorption equilibrium, the chemical potential, pressure, and temperature of the fluid confined in the micropore are the same as those of the bulk fluid. The pressure of the fluid in pillared layered pores can be solved by the MBWR equation (29), whose parameters are taken from the literature (30). Consequently, the reduced density of bulk fluid can also be obtained by the MBWR equation. 6.2. Effect of the Porosity
FIG. 5. Excess adsorption isotherms of methane in pillared layered pores with different porosities at T = 237.0 K. (a) H = 1.02 nm, (b) H = 1.70 nm, (c) H = 2.38 nm.
The porosity, which is directly related to the number of pillars between two layered walls, has a significant effect on the adsorption. We simulated the excess adsorption isotherms for three different porosities (ψ = 0.88, 0.94, 0.98) and pore widths (H = 1.02, 1.70, 2.38 nm) at three different supercritical temperatures, T = 207.3, 237.0, 266.6 K. The excess adsorption isotherms are shown in Figs. 4–6, respectively. For a model
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CAO, WANG, AND DUAN
FIG. 9. Optimum adsorption storage pressure for supercritical methane changing with temperature at three different pore widths.
it is observed from Figs. 4–6 that at the three fixed pore widths (hσ p = 1.02, 1.70, and 2.38 nm), and three supercritical temperatures, the isotherms for the porosity of 0.94, rather than 0.98, give the largest adsorption amount. This observation indicates that there is an optimized porosity, ψ = 0.94, for methane storage at supercritical temperatures. It coincides with our previous results at ambient temperature (26). Consequently, we focused our studies on the porosity of 0.94 subsequently. 6.3. The Optimum Adsorption Pressure FIG. 7. Excess adsorption isotherms of methane in pillared layered pores with different pore widths at porosity of ψ = 0.94. (a) T = 207.3 K, (b) T = 237.0 K, (c) T = 266.6 K. A, B, and C represent the maximum adsorption at H = 1.02, 1.70, and 2.38 nm, respectively.
fluid, the results of Yi et al. (9) suggest that at low pressures (P < 2 MPa), the adsorption amount decreases with the increase of porosity, while the adsorption amount increases with porosity at high pressures (P > 2 MPa). In contrast, for methane here,
It can be seen from Figs. 4–6 that the excess adsorption isotherms are of a classical shape, where the excess adsorption amount increases to a maximum, then decreases with the increase of pressure. As a result, the optimum adsorption pressure, corresponding to the maximum adsorption amount, exists at each pore width and temperature at ψ = 0.94, as shown in Fig. 7. It can be seen from Fig. 7 that the optimum adsorption pressure increases with the increase of pore width at a fixed temperature. For example, the optimum adsorption pressures are 3.1, 3.7, and 4.5 M Pa, at H = 1.02, 1.70, and 2.38 nm and T = 207.3 K, respectively. For a fixed pore width, the fact that the optimum adsorption pressure increases with temperature can also be found from Fig. 7. As illustration, Fig. 8 gives the relationship between the optimum pressure and pore width at three fixed temperatures. In addition, Fig. 9 presents the relationship between the optimum pressure and the temperature at three fixed pore widths. 7. CONCLUSIONS
FIG. 8. Optimum adsorption storage pressure for supercritical methane changing with pore width at three different temperatures.
Adsorption storage of supercritical methane in layered pillared pores has been simulated, by using the grand canonical Monte Carlo method. Based on the experimental data for Zrpillared porous material synthesized and characterized by Baksh et al. (19), an effective potential model that reflects the characteristics of a real pillared layered material is proposed. In the model, the binary interaction parameter, kfw , for the cross-interaction
ADSORPTION OF METHANE IN PILLARED LAYERED PORES
energy between a methane molecule and a layered wall in Eq. [6] is introduced and is determined by fitting simulation results to the experimental adsorption data of nitrogen at 77 K (19). It is noted that because the binary interaction parameter kfw was obtained by fitting to the Zr-pillared porous material, it is expected that its value would vary with different materials and even the fluids of interest. Nevertheless, this approach provides a useful tool to model pillared porous materials. By taking this model pillared layered pore as a prototype of adsorbents, extensive simulations have been carried out. The excess adsorption isotherms of methane for three different pore widths (H = 1.02, 1.70, 2.38 nm) and porosities (ψ = 0.88, 0.94, 0.98) are obtained at three supercritical temperatures, T = 207.3, 237.0, and 266.6 K. As is shown in Figs. 4–6, ψ = 0.94 is the optimum porosity, which gives relatively larger uptakes for adsorption storage of supercritical methane in the model pillared layered pores. Figure 7 shows the excess adsorption (see definition in Eq. [9]) changing with pressure for different size pores at the porosity of 0.94. It is found that compared with the pores of H = 1.70 and 2.38 nm, the pore of H = 1.02 nm gives considerably larger excess adsorption because of the stronger fluid-wall interaction in the relatively narrower pore. Accordingly, combined with the above results, the pillared layered pore with the ψ = 0.94 and pore width hσ p = 1.02 nm is recommended as adsorption material of supercritical methane. Importantly, all the excess adsorption increases with pressure and reaches a maximum for each isotherm at a constant pore width. It implies that the optimum adsorption pressure that corresponds to the maximum excess adsorption in Fig. 7 can be located. We have plotted the optimum pressure against pore width and temperature in Figs. 8 and 9, respectively. It is found that the optimum adsorption pressure increases monotonically with temperature at a fixed pore width, and also increases monotonically with pore width at a fixed temperature in the ranges of variables studied. In summary, the GCMC method and the model proposed in this work are a useful tool for preparing of the layered pillared material and optimizing the operation for adsorption storage of methane in the porous materials. ACKNOWLEDGMENTS This work was supported by the Key Fundamental Research Plan (No. G2000048010), and the Key Laboratory of Controllable Chemical Re-
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action Science and Technology of the Ministry of Education (China), and the National Natural Science Foundation of China under Grant No. 2977604, and the National High Performance Computing Foundation of China (No. 99118).
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