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Langmuir 2008, 24, 3836-3840
Comparative Study of Methane Adsorption on Graphite Alberto G. Albesa, Jorge L. Llanos, and Jose´ L. Vicente* Instituto de InVestigaciones Fisicoquı´micas Teo´ ricas y Aplicadas (INIFTA), Departamento de Quı´mica, Facultad de Ciencias Exactas, UNLP, CC 16, Sucursal 4 (1900) La Plata, Argentina ReceiVed NoVember 8, 2007. In Final Form: January 30, 2008 To study methane adsorption on graphite in a wide range of coverages and temperatures, we compare experimental results with Monte Carlo simulations (MCSs) of the grand canonical ensemble (GCE) and mean-field approximation (MFA) of the lattice gas model (LGM). MCSs were performed by employing two models for the substrate description; we utilized Steele’s 10-4-3 analytical potential, and as a second approach, we represented the graphite surface as composed of several graphene layers (at the atomic level). We obtained adsorption isotherms and density profiles that confirm a layer-by-layer mechanism at low temperatures; the later results in the analytical model having a denser condensed phase than the atomistic one. LGM calculations show a close-packed lattice configuration and also allow us to describe the adsorption mechanism changes with temperature. The isosteric heat of adsorption that was found was approximately 13 kJ/mol. We can also conclude that, in spite of the greater computational cost, the atomistic model could be employed for surfaces that are not necessarily homogeneous and beyond the low-pressure range that are not covered by the simple, fast description given by the analytical model.
Introduction There is increasing interest in search for and developing new storage systems and transport alternative fuels to traditional liquid petroleum.1 Because of its advantages from an environmental point of view and its natural abundance, natural gas is becoming one of these alternatives. It is composed of methane, with a lesser amount of ethane, propane, and butane; it contains few or no contaminants and burns cleanly and efficiently. The low heat of combustion per unit volume of natural gas compared with that of conventional fuels requires compressing it at high pressure in order to use it, for example, in gas fueled vehicles, with the associated disadvantages related to the weight and volume of the containment vessel required, safety risks, and the costs of multistage compression cycles.2 Alternatively, the adsorption of natural gas at relatively low pressures on new carbonaceous materials1,3,4 developed from activated carbons, fullerenes, and nanotubes opens the possibility of new applications. Taking into account that the graphite surface can be used to mimic many carbonaceous surfaces5 at determined scales, there is renewed interest in experimental and theoretical studies of methane adsorption on carbonaceous materials. From a theoretical point of view, there are two main techniques used in adsorption studies: computer Monte Carlo simulations (MCS) with the grand canonical ensemble6 and calculations using the mean-field approximation (MFA) of the lattice gas model.7 Both methods have been independently applied to describe the isotherms of adsorption of rare gases on graphite and have been compared with experimental results at different temperatures. However, in many surfaces studies it is interesting to compare both results8,9 * Corresponding author. E-mail:
[email protected]. (1) Radovic, L. R. Chemistry and Physics of Carbon; Marcel Dekker: New York, 2001; p 27. (2) Parkyns, N. D.; Quinn, D. F. In Porosity in Carbons; Patrick, J. W., Ed.; Edward Arnold: London, 1995; p 302. (3) Cracknell, R. F.; Gordon, P.; Gubbins, K. E. J. Phys. Chem. 1993, 97, 494. (4) Muris, M.; Dufau, N.; Bienfait, M.; Dupont-Pavlovsky, N.; Grillet, Y.; Palmari, J. P. Langmuir 2000, 16, 7019. (5) Radovic, L. R. Chemistry and Physics of Carbon; Marcel Dekker: New York, 2003; p 28. (6) Bottani, E. J.; Bakaev, V. A. Langmuir 1994, 10, 1550. (7) De Oliveira, M. J.; Griffiths, R. B. Surf. Sci. 1978, 71, 687. (8) Rafti, M.; Vicente, J. L.; Uecker, H.; Imbihl, R. Chem. Phys. Lett. 2006, 421, 577.
Table 1. Lennard-Jones Interaction Parameters17 methane (X ) f) graphite (X ) s)
XX/kB
σXX/Å
148 K 28 K
3.81 3.40
because the MCS offers the possibility of studying the phenomenon on the molecular (microscopic) scale but is limited by computer capabilities, and MFA gives a global (mesoscopic) description of both the surface and the gas but ignores what happens on the molecular level. However, in their respective ranges, both models give complementary information of the same phenomena. The purpose of this article is to make a comparison between different MCSs, MFA calculations, and experimental data of methane adsorption isotherms on the basal plane of graphite at different temperatures. We also study the enthalpy of adsorption in order to compare the theoretical and experimental results. This article is organized as follows. First, we give a brief introduction of the MCS and MFA algorithms, and then we present a description of the experimental system. We then give our main experimental and theoretical results. Finally, a critical comparison of all of them is given.
Grand Canonical Monte Carlo Ensemble Adsorbate-Adsorbate Interaction Potential. We adopted the Martin and Siepmann model10 to calculate the adsorbateadsorbate interactions because of its capability to reproduce experimental data over a wide range of pressure and temperature.11 The interaction between methane molecules was modeled via a Lennard-Jones 12-6 potential
φff(r) ) 4ff
[( ) ( ) ] σff r
12
-
σff r
6
(1)
that describes the potential energy between two methane molecules at a distance r. The parameters employed in our calculations are given in Table 1. (9) Rafti, M.; Vicente, J. L. Phys. ReV. E 2007, 75, 061121. (10) Martin, M.; Siepmann, J. I. J. Phys. Chem. B 1998, 102, 2569. (11) Do, D. D.; Do, H. D. J. Phys. Chem. B 2005, 109, 19288.
10.1021/la7034938 CCC: $40.75 © 2008 American Chemical Society Published on Web 03/14/2008
Study of Methane Adsorption on Graphite
Langmuir, Vol. 24, No. 8, 2008 3837
Gas-Solid Potential. To calculate the gas-solid interactions, we employed two models, one of which we call Steele’s analytic model because it employs Steele’s 10-4-3 analytic potential12
φsf(r) ) φw
[(
) ( )
sf 10
1σ 5 z
-
sf 4
1σ 2 z
-
(σsf)4 6∆(z + 0.61∆)3
]
Qst ) RT (2)
where z is the distance between a fluid particle and the substrate surface and ∆ is the separation between lattice planes. The energy parameter φw is given by
φw ) 4πFssf(σsf)2
(3)
where Fs ) 0.382 A-2 is the carbon atoms density on the graphite slab surface. Interaction parameters σsf and sf were calculated using Lorentz-Berthelot combination rules (Table 1). The accuracy and advantages of the adsorption potential given in eq 2 are well known.13 In the other model, which we call the atomistic model, the substrate was represented by four graphene slabs, with a distance of 3.35 Å between planes. This number of planes was adopted because of computational costs and, after performing calculations with increasing structures of five and six slabs, to check that no significant changes occurred. The distance between carbon atoms adopted was 1.42 Å. The potential was calculated at each point and stored in a 200 × 200 × 200 matrix. Technical Simulation Details. The parameters used in the calculations associated with the simulations consisted of (1) a simulation box with a side length of 10 σ ff; (2) a cut-off radius of 2.5σ ff; (3) simulation runs consisting of 5 × 107 Monte Carlo steps each that included the possibility of a creation, destruction, or displacement attempt for each molecule, with 5 × 104 steps taken for statistical averages; (4) long-range corrections made because of the cutoff;14 and (5) periodic conditions in the x and y directions and a reflected plane in the z axis. Another advantage of MC methods is that it is also possible to obtain information about the isosteric heat of adsorption. The isosteric heat of adsorption -∆H0 is the difference between the molar enthalpy of the sorbate in the vapor phase and the partial molar enthalpy of the adsorbed phase.15 Enthalpy is a function of the internal energy and the product pV. For the vapor phase, pV is assumed to be equal to RT, and the molecular volume of the adsorbed phase is neglected. Assuming that the molar kinetic energy is the same in the gas and in the adsorbed state, the heat of adsorption can be expressed as a function of the total molar potential energy in the vapor phase, Egt , and in the adsorbed phase, Est .
-∆H0 ) RT - Est + Egt
(4)
In the GCMC simulation, it is equivalent to calculating -∆H0 using partial derivatives of the average total energy with respect to the average number 〈N〉 of adsorbed molecules.
-∆H0 ) RT -
∂〈Est 〉 ∂〈N〉
+
∂〈Egt 〉 ∂〈N〉
The isosteric heat of adsorption Qst can thus be calculated through the fluctuations method
(5)
(12) Steele, W. A. Surf. Sci. 1973, 36, 317. (13) Bottani, E. J. Langmuir 1999, 15, 5574. (14) Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids; Clarendon Press: Oxford, England, 1987. (15) Pascual, P.; Ungerer, P.; Taviatian, B.; Pernot, P.; Boutin, A. Phys. Chem. Chem. Phys. 2003, 5, 3684.
{
}
[〈EtN〉 - 〈Et〉〈N〉] [〈N2〉 - 〈N〉2]
(6a)
where Et is the sum of two terms: the potential energy between adsorbed molecules and the potential energy between adsorbed molecules and the solid substrate surface.11 The potential energy of the interaction can be broken down into contributions of fluidfluid and fluid-solid interactions.
[
Qst ) RT -
]
〈Egt N〉 - 〈Egt 〉〈N〉 〈N2〉 - 〈N〉2
-
〈Est N〉 - 〈Est 〉〈N〉 〈N2〉 - 〈N〉2
(6b)
The square-bracketed term in eq 6b is the contribution of the fluid-fluid interaction to the isosteric heat of adsorption, whereas the last term is the contribution from the fluid-solid interaction.
Lattice Gas Model in the Mean-Field Approximation In this section, we follow the De Oliveira and Griffiths proposal,7 assuming as in the usual lattice gas model16 that the region above the substrate that is accessible to methane molecules is divided into a set of cells whose centers form a regular lattice. No more than one molecule is permitted in a cell, and each pair of gas molecules in adjacent cells contributes an amount -∈ to the potential energy. All molecules in the jth layer experience an additional potential energy -Vj due to the substrate, with
[
Vj ) ∈ δj1D +
]
(1 - δj1)C j3
(7)
If njk is the occupation number of the kth cell in the jth layer (0 empty and 1 occupied), then δjk is 1 when k ) j and 0 otherwise. C and D are proportional to the minimum energies for the interaction of a methane molecule with a semi-infinite continuous slab (9-3 potential)17 and to the summation of the potential over different surface sites, respectively. Then Gibbs-Boltzmann probability for a configuration {njk}has to be proportional to exp(-βH), with β ) (kT)-1 where
H ) -∈
∑
(j,k),(j′,k′)
njknj′k′ -
∑j (µ + Vj) ∑k njk
(8)
Here, µ is the chemical potential of the gas, apart from a temperature-dependent constant, where ( ) denotes a nearestneighbor pair of cells. Let Fj be the average value of njk in the jth layer. In the meanfield approximation,18 the grand potential is obtained by minimizing Ω, where
ΩL-2 ) kT
∑{Fj ln Fj + (1 - Fj) ln(1 - Fj)} 1 ∑(µ + Vj)Fj - ∈[2 a∑Fj2 + b∑FjFj+1]
(9)
as a function of F1, F2, . . . Here, L2 is the number of cells in a single layer, and each cell has a nearest neighbors in the same layer and b nearest neighbors just above it. At a minimum of Ω, the coupled set of equations (16) Stanley, H. E. Phase Transitions and Critical Phenomena; Oxford University Press: London, 1971; p 260. (17) Steele, W. A. The Interaction of Gases with Solid Surfaces; Pergamon: Oxford, England, 1974; Vol, 57, p 160. (18) Burley, D. M. In Phase Transitions and Critical Phenomena; Domb C., Green, M. S., Eds.; Academic Press: London, 1972; Vol. 2, p 329.
3838 Langmuir, Vol. 24, No. 8, 2008
mj ) tanh
Albesa et al.
{21 β[∆µ + V + 21 ∈(am + bm j
j
j-1
]} (10)
+ bmj+1)
is satisfied for j ) 1, 2, . . . Here, mj ) 2Fj - 1 (m0 ) -1) and ∆µ ) µ + ∈ (a/2 + b) is the chemical potential minus the resulting value for an adsorbed layer of infinite thickness. To compute adsorption isotherms, we truncated eq 10 at j ) 20 (m21 ) m∞), where m∞ is the negative solution of
m∞ ) tanh
{21 β[∆µ + 21 ∈(a + 2b)m ]} ) 2F ∞
∞
-1
(11)
The equation is numerically solved for different values of a, b that are appropriate for each lattice configurations. For each value of β and ∆µ, we used the solutions that minimize eq 8.
Figure 1. Isotherms at 80.2 K. Circles are obtained from experimental results, and simulations are represented by dashed black lines for the analytic model and solid gray lines for the atomistic model.
Experimental Section The adsorption isotherms at 80.2, 93, 103, 113, and 123 K were volumetrically determined by employing conventional Pyrex equipment. The corresponding values for 80.2, 103, and 113 K are shown in Figures 1-3, respectively. Pressures were determined using absolute capacitance manometers MKS and Baratron 122 AA-00010AB with a 1.0 × 10-3 Torr maximum error. Working temperatures were thermostatically controlled. Temperature was measured with a digital thermometer, Altronix, with a Pt-100 (DIN) sensor head previously calibrated against an oxygen vapor pressure thermometer with 0.1 K precision.13 The gas was employed without any previous treatment, and it was of high purity, greater than 99%, provided by Matheson Gas Products. The sample was a carbon black Sterling MT-FF(D) graphitized at 3100 °C, which was previously characterized as energetically homogeneous with a BET specific surface area of 7.7 m2 g-1. From two isotherms determined at similar but different temperatures, T1 and T2, it is possible to calculate the isosteric heat of adsorption qst of the gas used via eq 12 qst ) (RT) ln
() p2 p1
Figure 2. Isotherms at 103 K. Circles are obtained from experimental results, and simulations are represented by dashed black lines for the analytic model and solid gray lines for the atomistic model.
(12)
where p1 and p2 are the equilibrium pressures at temperatures T1 and T2, respectively, when the adsorbed volume is constant and T is the corresponding mean temperature.
Results and Discussions Isotherms and Density Profiles. Figures 1-3 show the experimental isotherms at 80.2, 103, and 113 K, respectively. The lower-temperature isotherms measured (from 80.2 to 103 K) give two clear horizontal steps located at relative pressures of 0.325 and 0.725, respectively. These steps are absent in highertemperatures isotherms (from 113 K, the boiling point, to 123 K). These steps can be attributed to the completion of the first and second layers, confirming a layer-by-layer adsorption mechanism.19 As the temperature increases, this kind of ordered adsorption is less significant, and above 103 K, there is multilayer adsorption before monolayer completion. All isotherms above and below the triple point of methane (90.7 K) showed that the film thickness increases asymptotically as the saturated vapor pressure is approached, indicating complete surface wetting.20 (19) Hamilton, J. J.; Goodstein, D. L. Phys ReV. B 28 1983, 3838. (20) Hess, G. B. In Phase Transitions in Surface Films 2; Taub, H., Torzo, G., Lauter, H. J., Fain, S. C., Jr., Eds.; NATO ASI Series B; Plenum Press: New York, 1991; Vol. 267.
Figure 3. Isotherms at 113 K. Circles are obtained from experimental results, and simulations are represented by dashed black lines for the analytic model and solid gray lines for the atomistic model.
Atomistically simulated adsorption isotherms at 80.2, 103, and 113 K and those obtained by the analytic model are also compared with the experimental ones in Figure 1-3. The agreement between experiments and simulation is very good in all cases for the atomistic model, and for the analytic one, there is a very good fit in the monolayer region, giving a greater adsorption at higher pressures and finally condensing after the third monolayer is formed. The observed behavior can be rationalized by taking into account the density profiles F* shown in Figures 4-6 as a function of the distance to the surface and the degree of coverage θ (θ ) Vads/Vmono where Vads is the adsorbed volume and Vmono the volume of the monolayer). At lower temperatures, for the same coverage θ, the analytic model density profiles show a condensed phase that is denser (approximately 30%) than that of the atomistic
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Langmuir, Vol. 24, No. 8, 2008 3839
Figure 4. Density profiles F* as a function of the distance to the surface in angstroms and the degree of coverage θ at 80.2 K when the isotherm is completed. Solid lines represent the results from the analytic model, and broken lines represent the atomistic model at θ ) 3.0.
Figure 7. Experimental and lattice gas model isotherms at reduced temperatures τ ) 0.5383 (T ) 103 K) and τ ) 0.9 (T ) 171 K), where (τ ) 4[β∈(a + 2b)]-1). Figure 5. Density profiles F* as a function of the distance to the surface in angstroms and the degree of coverage θ at 103 K when the isotherm is completed. Solid lines represent the results from the analytic model, and broken lines represent the atomistic model at θ ) 2.0.
Figure 6. Density profiles F* as a function of the distance to the surface in angstroms and the degree of coverage θ at 113 K when the isotherm is completed. Solid lines represent the results from the analytic model, and broken lines represent the atomistic model at θ ) 2.5.
model (Figure 5). The difference in density profiles in both models at these temperatures is due to the fact that the surface is completely flat and smooth in the analytic case and the methane molecules packing is more effective than in the atomistic one, which means that corrugation surface effects become important. The above discussion can be summarized by saying that, in spite of its simplicity, the atomistic model gives an excellent description of the adsorbed phase over the whole range of pressures as a result of better substrate representation. The presented model also allows for a description of different substrate geometries; the increase in computational resources needed in this case would be compensated for by the valuable results
obtained for the high-pressure-regime investigations (e.g., research on methane storage technologies). However, the analytic model gives a simple, fast description of homogeneous surfaces, even with a low degree of corrugation21 when results at low-pressure values have to be explored. We also use the lattice gas model, from the mean-field approach, to analyze the effect of the adsorbed phase density. To reproduce experimental isotherms, we take C ) 12 and D ) 20, values that are greater than those obtained following the same arguments given by De Oliveira et al.7 for the case of methane. Nevertheless, these changes do not affect significantly the results obtained. Increasing the packing parameters a and b (a ) 9 and b ) 4) gives a density of the condensed phase that is greater than the experimental value. Figure 7 shows the results for reduced temperatures (τ ) 4 [β∈(a + 2 b) ]-1) of 0.5383 and 0.9, T = 103 and 171 K, respectively. The last value was chosen because it is lower than the critical temperature and high enough to show temperature effects. Following this model, we see that at low temperatures the isotherms show a greater dependence on the density of the condensed phase (a and b values), but when the temperature increases and approaches the critical one, this dependence disappears. Isosteric Heat of Adsorption. Figures 8-10 show isosteric heats of adsorption from simulations, calculated by employing eq 6. Experimental results, where qst as a function of the degree of coverage θ was obtained from eq 12, are also shown in Figures 9-10. We cannot measure values at 80.2 K, but results given by other authors for similar substrates and temperatures22 confirm the following conclusions. At lower temperatures, qst shows two clear peaks that can be attributed to the first and the second layer completion. The heat of adsorption obtained from the atomistic model is 12.6 kJ/mol, and the corresponding value for the analytic model (21) Kim, H. Y.; Steele, W. A. Phys. ReV. B 1992, 45, 6226. (22) Piper J.; Morrison, J. A. Phys. ReV. B 1984, 30, 3486.
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Albesa et al.
analytic model is 3 kJ/mol greater than the atomistic model value. In this region, there is better agreement with the experimental results for the analytic model. The difference between both models is due to the fluid-solid contribution because the fluid-fluid contribution is the same in both models
Conclusions
Figure 8. Isosteric heat of adsorption as a function of the degree of coverage θ at 80.2 K. Simulations are represented by dashed black lines for the analytic model and solid gray lines for the atomistic model.
Figure 9. Isosteric heat of adsorption as a function of the degree of coverage θ at 103 K. Circles are obtained from experimental results, and simulations are represented by dashed black lines for the analytic model and solid gray lines for the atomistic model.
Figure 10. Isosteric heat of adsorption as a function of the degree of coverage θ at 113 K. Circles are obtained from experimental results, and simulations are represented by dashed black lines for the analytic model and solid gray lines for the atomistic model.
is 13.5 kJ/mol, in good agreement with others values reported in the literature. Do and Do11 obtained an isosteric heat at zero coverage of 12.6 kJ/mol, and Piper and Morrison22 obtained values at fairly similar temperatures of 13.4 and 13.7 kJ/mol. We see agreement between the analytic and atomistic models over the whole range of coverage except near the first monolayer completion, where the isosteric heat of adsorption given by the
- At low temperature, the experimental results confirm the completion of the first and second methane layers at approximately 0.325 and 0.725 relative pressure, respectively. Around the triple point (90.7 K), liquid and solid methane completely wet the graphite surface. Above 113 K (the boiling point), there is no layer-by-layer filling mechanism, and all layers are available to be filled. - At low temperature, MCS gives methane adsorption as a layer-by-layer mechanism in both models, but the completion of each layer fits the experimental results better in the atomistic case, whereas the analytic model is shifted to lower pressures. For the same degree of coverage, the analytic model density profiles show a denser condensed phase than does the atomistic model because methane molecules can be arranged better on the plane surface. Both models give the same isosteric heat of adsorption over all ranges, except near the completion of the first monolayer, where the analytic model gives a greater value as a result of the fluid-solid contribution. The heat of adsorption obtained from the atomistic model is 12.6 kJ/mol, in good agreement with others values reported in the literature,11,22 and the corresponding value for the analytic model is 13.5 kJ/mol. - The lattice gas model in the mean-field approximation confirms the close-packed lattice configuration and how the dependence of the layer-by-layer mechanism with the density of the condensed phase disappears as the temperature increases. - The analytic model gives a simple, fast description of homogeneous surfaces, especially when results at low-pressure values have to be explored. - The atomistic model allows a very simple description of all kinds of substrate geometries over the whole pressures range, including high values, which are of great interest in methane storage studies. - The better performance of the atomistic model is based on a more accurate description of the molecule-substrate interaction taking into account the positions of the carbon atoms in the graphite surface. Calculations performed using DFT23 have shown that energy differences occurring at different orientations of the methane molecule with respect to the substrate are not significant. Acknowledgment. We gratefully acknowledge financial support from the UNLP (Universidad Nacional de La Plata), CICPBA (Comisio´n de Investigaciones Cientı´ficas de la Provincia de Buenos Aires), and CONICET (Consejo de Investigaciones Cientı´ficas y Tecnolo´gicas). LA7034938 (23) Albesa, A. G.; Vicente, J. L. J. Argent. Chem. Soc., in press, 2008 (http:// www.aqa.org.ar/analesi.htm).
