Langmuir 2005, 21, 8297-8301
8297
Monte Carlo Simulation and Pore-Size Distribution Analysis of the Isosteric Heat of Adsorption of Methane in Activated Carbon Yufeng He† and Nigel A. Seaton* Institute for Materials and Processes, School of Engineering and Electronics, University of Edinburgh, Kenneth Denbigh Building, King’s Buildings, Mayfield Road, Edinburgh, EH9 3JL, United Kingdom Received March 15, 2005. In Final Form: June 14, 2005 The isosteric heat of adsorption of methane in an activated carbon adsorbent has been modeled by Monte Carlo simulation, using a pore-size distribution (PSD) to relate simulation results for pores of different sizes to the experimental adsorbent. Excellent fits were obtained between experimental and simulated isosteric heats of adsorption of methane in BPL activated carbon. The PSD was then used to predict the adsorption of methane and ethane in the same carbon adsorbent, with good results. The PSD derived from isosteric heat data was shown to be richer in information than PSDs obtained by the more conventional method of fitting to isotherm data.
1. Introduction The pore-size distribution (PSD) model is widely used to characterize the internal structure of activated carbons based on the adsorption integral equation
n(Pi) )
∫0∞ F(w,Pi)f(w) dw
i ) 1, ..., n
(1)
where n(Pi) is the experimentally determined amount adsorbed at pressure Pi, F(w,Pi) is the adsorbate density in a model pore of width w at pressure Pi, f(w) is the PSD, and n is the total number of experimental data points. The PSD is conventionally obtained by fitting to a suitably chosen adsorption isotherm and may then be used to predict pure-gas adsorption isotherms of the same or different adsorptive at the same or different temperatures or to predict multicomponent adsorption. The ability of a PSD, combined with a suitable model for adsorption in individual pores, to accurately predict adsorption across a wide range of conditions is a measure of its usefulness in technological applications.1,2 This PSD-based approach to the prediction of adsorption has been applied to adsorption in porous carbons by Davies et al.,1 McEnaney et al.,3 Davies and Seaton,2,4 Gusev et al.,5 Gusev and O’Brien,6 Saimos et al.,7 Lo´pez-Ramo´n et al.,8 and Sweatman and Quirke,9,10 using Grand Canonical Monte Carlo (GCMC) simulation to generate the single-pore isotherms, F(w,Pi). * To whom correspondence should be addressed. Telephone: 44-131-650-4867. Fax: 44-131-650-6551. E-mail:
[email protected]. † Current address: Department of Chemical Engineering, University of Bath, Bath BA2 7AY, United Kingdom. (1) Davies, G. M.; Seaton, N. A.; Vassiliadis, V. S. Langmuir, 1999, 15, 8235. (2) Davies, G. M.; Seaton, N. A. Langmuir 1999, 15, 6263. (3) McEnaney, B.; Mays, T. J.; Chen, X. Fuel 1998, 77, 557. (4) Davies, G. M.; Seaton, N. A. Carbon 1998, 36, 1473. (5) Gusev, V. Y.; O’Brien, J. A.; Seaton, N. A. Langmuir 1997, 13, 2815. (6) Gusev, V. Y.; O’Brien, J. A. Langmuir 1997, 13, 2822. (7) Samios, S.; Stubos, A. K.; Kanellopoulos, N. K.; Cracknell, R. F.; Papadopoulos, G. K.; Nicholson, D. Langmuir 1997, 13, 2795. (8) Lo´pez-Ramo´n, M. V.; Jagiello, J.; Bandosz, T. J.; Seaton, N. A. Langmuir 1997, 13, 4435. (9) Sweatman, M. B.; Quirke, N. Langmuir 2000, 17, 5011. (10) Sweatman, M. B.; Quirke, N. J. Phys. Chem. B 2001, 105, 1403.
Adsorption equilibrium is characterized not only by the adsorption isotherms but also by the isosteric heat of adsorption. The isosteric heat is a direct indicator of the heterogeneity of the interaction between the adsorptive (either a pure gas or a mixture) and the adsorbent.11 This variable is also required for the calculation of energy balances in industrial adsorption processes for gas separation and gas storage. Nicholson12 carried out GCMC simulations of the isosteric heat of adsorption of methane and carbon dioxide in smooth-walled slit pores based on three arbitrarily chosen PSDs and found that the isosteric heat of adsorption was highly sensitive to the presence of very small pores in the adsorbent; the isosteric heat in a single slit-shaped pore was obtained by analyzing fluctuations in a GCMC simulation. Pan et al.13,14 calculated the isosteric heat of propane and butane in the Westvaco BAX activated carbon by numerically differentiating the energy of adsorption calculated using nonlocal density functional theory, using the PSD of the samples calculated by fitting the isotherms of N2 in the same carbon at 77 K. While these studies elucidated the connection between isosteric heat and pore structure, they did not attempt a quantitative account of the isosteric heat in a real activated carbon adsorbent by Monte Carlo simulation. In this paper, we combine Monte Carlo simulation of adsorption and a PSD model of the pore structure to quantitatively describe the heat of adsorption in an activated carbon and go on to use this PSD model, with parameters fitted to the isosteric heat, to predict the adsorption isotherms. This allows us to make a rigorous test of the realism of the PSD and to test the ability of a PSD fitted to the isosteric heat (rather than the conventional approach of fitting to an isotherm) to predict adsorption. The heterogeneity of an adsorbent can be broken down into two aspects:12 energetic (or surface) heterogeneity (11) Dunne, J. A.; Mariwala, R.; Rao, M.; Sircar, S.; Gorte, R. J.; Myers, A. L. Langmuir 1996, 12, 5888. (12) Nicholson, N. D. Langmuir 1999, 15, 2508. (13) Pan, H. H.; Ritter, J. A.; Balbuena, P. B. Langmuir 1999, 15, 4570. (14) Pan, H. H.; Ritter, J. A.; Balbuena, P. B. Ind. Eng. Chem. Res. 1998, 37, 1159.
10.1021/la050694v CCC: $30.25 © 2005 American Chemical Society Published on Web 08/02/2005
8298
Langmuir, Vol. 21, No. 18, 2005
He and Seaton
and structural heterogeneity. Structural heterogeneity is caused by the presence of pores of different sizes, shapes, and connectivities,15 while energetic heterogeneity results from surface irregularities as well as from the presence of functional groups and impurities. In this paper, the energetic heterogeneity effect of activated carbon, such as the existence of surface groups,16 is omitted. We deal here only with the structural heterogeneity, which is accounted for by the PSD. 2. Isosteric Heat of Adsorption and the PSD 2.1. Experimental Measurement of the Isosteric Heat. Adsorption isotherms of methane and ethane in BPL were measured with a high-pressure volumetric apparatus, which has been described elsewhere.17 The experimental isosteric heat was obtained from the variation of adsorption with temperature, by differentiating the pure-gas adsorption isotherms at constant loading and using the Clausius-Clapeyron equation
qst ) -R
(
∂ lnP ∂(1/T)
)
( ) ∂ lnP ∂T
) RT2
n
(2)
n
where P is the pressure, n is the number of moles adsorbed, R is the universal gas constant, and T is the temperature. This equation can also be used to obtain the isosteric heats of adsorption in a Monte Carlo simulation
qst•abs ) -R
(
)
∂ lnP ∂(1/T)
( )
) RT2
Nabs
∂ lnP ∂T
(3)
Nabs
where Nabs is the absolute number of molecules in the pore, obtained in a GCMC simulation (see section 2.3). 2.2. Isosteric Heat from the Monte Carlo Simulation. The isosteric heat of adsorption may be obtained from fluctuations in the energy and in the amount adsorbed during a GCMC simulation18
qst ) NA
〈U h N〉 - 〈U h 〉〈N〉 〈N2〉 - 〈N〉2
(4)
where U and N are the internal energy of the adsorbate and the number of molecules adsorbed and the angle brackets indicate mean values. Avogadro’s number, NA, converts from the heat per molecule to the heat per mole. An alternative approach, proposed by Vuong and Monson,19 is to numerically differentiate the internal energy of the adsorbate, usually in a Canonical Monte Carlo (CMC) simulation
qst ) RT - NA
∂U ) (∂N
T
(5)
Vuong and Monson found that, for a given expenditure of computer time, this method is more accurate than the fluctuation method of eq 3. A third approach to obtaining the isosteric heat from simulation data is to apply the Clausius-Clapeyron equation, as for experimental data, using eq 3. This approach has been found to yield results that are at least as accurate as the method of Vuong and Monson.19,20 However, because adsorption isotherms of at least three different temperatures are needed to get accurate isosteric heat of adsorption by the ClausiusClapeyron method, the method of Vuong and Monson19 is used in this work. 2.3. Absolute and Excess Adsorption. Monte Carlo simulations generate absolute adsorption data, i.e., the actual number of molecules present in the simulated pore space. In contrast, (15) Kruk, M.; Jaroniec, M.; Sayari, A. Langmuir 1999, 15, 5683. (16) Boehm, H. P. Carbon 1994, 32, 759. (17) He, Y.; Yun, J. H.; Seaton, N. A. Langmuir 2004, 20, 6668. (18) Nicholson, D.; Parsonage, N. G. Computer Simulation and the Statistical Mechanics of Adsorption; Academic Press: London, U.K., 1982. (19) Vuong, T.; Monson, P. A. Langmuir 1996, 12, 5425. (20) Bakaev, V. A.; Steele, W. A. Langmuir 1992, 8, 148.
Table 1. Summary of Lennard-Jones Parameters for the Adsorptive Species adsorptive
σ (Å)
/kB (K)
bond length (Å)
methane22 ethane23
3.81 3.512
148.2 139.8
2.353
the experimental results, whether obtained by volumetric or gravimetric methods, are Gibbs excess properties.21 Therefore, the simulation results must be converted to their excess counterparts before they can be used to analyze experimental data.22 The excess amount adsorbed, Nex, is given
Nex ) Nabs - Nb ) Nabs - FbV
(6)
where Nabs is the simulated absolute amount adsorbed, Nb is the number of moles of gas that would be present in the pore without adsorption at the bulk density, V is the accessible volume in the model pore, and Fb is the density of the bulk gas at the same temperature and pressure. The value of V is obtained by simulating the adsorption of helium in the model pore, because this is consistent with the use of helium (a very weakly adsorbing gas) to determine the experimental dead volume.22 The absolute and excess isosteric heats are related by the result of Myers et al.23 For an ideal bulk phase (a very good approximation under the experimental conditions), the difference between absolute and excess isosteric heats is
qst•ex - qst•abs )
Nb(qst•abs - RT) ∂Nex P ∂P T
( )
(7)
where qst•ex and qst•abs are the excess and absolute isosteric heats, respectively. 2.4. Calculation of the Isosteric Heat of Adsorption using the PSD. The isosteric heat of adsorption in the adsorbent as a whole is related to the isosteric heat in individual pores and the PSD by an adsorption integral equation analogous to eq 1
qst(Pi) )
∫
∞
0
qst(w,Pi)F(w,Pi)f(w) dw
∫
∞
0
(8)
F(w,Pi)f(w) dw
where qst(Pi) is the experimental isosteric heat of adsorption at pressure Pi and qst(w,Pi) is the simulated isosteric heat of adsorption in an individual pore of width w at pressure Pi. The PSD, f(w), is obtained by solving eq 8 using isosteric heat data as the experimental input, rather than the isotherm data as in the case of eq 1. Equation 8 was solved by converting the integrals to summations and summing over 18 pore sizes. The PSD obtained is then used to predict the adsorption isotherms or the isosteric heats of adsorption of the same or different adsorbates, at the same or different temperatures and over a range of pressures.
3. Molecular Models for the Adsorptive and the Adsorbent The interaction between adsorbate molecules is modeled using the Lennard-Jones potential. The methane molecule is represented by a single Lennard-Jones site, while ethane consists of two sites. The Lennard-Jones parameters for the adsorptive species are given in Table 1. The pores in the activated carbon adsorbent are modeled as slits. Each pore wall consists of an infinite number of structureless graphitic layers composed of Lennard-Jones sites. The interaction between a site on an adsorptive molecule and a single semi(21) Sircar, S. Ind. Eng. Chem. Res. 1999, 38, 3670. (22) Talu, O.; Myers, A. L. AIChE J. 2001, 47, 1160. (23) Myers, A. L.; Calles, J. A.; Calleja, G. Adsorption 1997, 3, 107.
Monte Carlo Simulation and Pore-Size Distribution Analysis
Figure 1. Simulated excess isosteric heats of adsorption of methane at T ) 301.4 K. infinite slab of graphite is given by Steele’s 10-4-3 potential26
u/sf(zi) ) 2πsfFsσ2sf∆
[( ) ( ) 2 σsf 5 zi
10
-
σsf zi
4
-
σ4sf
]
3∆(zi + 0.61∆)3
(9)
where Fs is the number of carbon atoms per unit volume in the graphitic layer (0.114 Å-3 ), ∆ is the separation distance between layers of graphitic carbon (3.35 Å), and zi is the distance between the site and the surface. The parameters σ and /kB are 3.40 Å and 28.0 K, respectively. The solid-fluid Lennard-Jones parameters are calculated by applying the Lorentz-Berthelot combining rules. The overall interaction between an adsorbate site with the two pore walls is given by
usf ) u/sf(zi) + u/sf(w - zi)
(10)
GCMC simulations, in which the temperature, the volume of the simulation cell, and the chemical potential of the adsorbate are kept constant, were carried out for the adsorption isotherms of pure methane and ethane in single slit-shaped pores. The absolute configurational energy of the adsorbates was obtained by a CMC simulation, in which the number of molecules in the pore, the temperature, and the volume of the simulation cell are kept constant. More details of the GCMC and CMC simulations can be found in refs 27 and 28. Both types of simulation were carried out in a rectangular simulation cell, which is bounded in the z direction by the pore walls and replicated in the x and y directions. The cell length in the x and y directions is 57.15 Å, and periodic boundary conditions are applied in these directions. The cutoff distance, beyond which the potential is neglected, is set to be 15.24 Å. The system was equilibrated with 6 × 106 steps, and then the variables of interest were averaged over another 1 × 107 steps.
4. Results and Discussion Figure 1 shows the simulated excess isosteric heats of adsorption of methane in a series of slit-shaped model pores with different pore sizes at 301.4 K, using eq 7 converted between absolute and excess heats. The isosteric (24) Fischer, J.; Heinbuch, U.; Wendland, M. Mol. Phys. 1987, 61, 953. (25) Hirschfelder, J. O.; Curtiss, C. F.; Bird, R. B. Molecular Theory of Gases and Liquids; John Wiley and Sons Press: New York, 1954. (26) Steele, W. A. The Interaction of Gases with Solid Surfaces; Pergamon Press: London, U.K., 1974. (27) Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids; Clarendon: Oxford, U.K., 1987. (28) Frenkel, D.; Smit, B. Understanding Molecular Simulations; Academic Press: San Diego, CA, 1996.
Langmuir, Vol. 21, No. 18, 2005 8299
Figure 2. Methane-pore potential as a function of the distance to the center of the pore, with pores size of 6.50, 7.62, and 15.46 Å.
heats in all of the pores are a weakly increasing function of pressure, indicating the adsorption of methane in slitshaped model pores is a highly homogeneous adsorption system. This reflects the nature of the slit-shaped model pores: structure-less and with no defects or chemical heterogeneity. The isosteric heats of adsorption have two physically distinct contributions: the interaction between the adsorbate molecules and the adsorbent and the interaction between the adsorbate molecules themselves. The adsorbate-adsorbent interaction remains essentially constant as the loading increases, at least as long as the adsorbate molecules are close to the surface, while the adsorbate-adsorbate contribution increases as the adsorbed phase becomes more dense. The combination of these two contributions gives the gently increasing function shown in Figure 1. The isosteric heat increases with the pore size from 6.29 Å and reaches a maximum at a pore size of 7.62 Å, and then the isosteric heats start to decrease and are essentially independent of the pore size for pores larger than 30 Å (where adsorption proceeds essentially independently on the two pore walls). The maximum at 7.62 Å corresponds to a maximum in the depth of the adsorption potential, as shown in Figure 2. When the pore size is smaller than 7.62 Å, the repulsive interactions become more important and adsorption is possible only in a narrow region near the center of the pore. The absolute adsorption isotherms and the isosteric heats of methane in pores of width 7.62 and 26.7 Å are shown in Figure 3, along with their excess counterparts, obtained by eqs 6 and 7, respectively. In small pores, because the accessible pore volume is small and the adsorbate density is high, the difference between absolute and excess adsorption isotherms is small at low pressures (for example, less than 2% for adsorption in the 7.62 Å pore at pressures less than 1 MPa), but it becomes substantial at high pressures (about 10% at 2 MPa in the 7.62 Å pore). The difference between absolute and excess adsorption isotherms is much larger in bigger pores (greater than 14% for the 26.7 Å pore even at pressures less than 1 kPa) even at low pressures. For all of the pores, the excess isosteric heats of adsorption are larger than the absolute values, as found for a similar adsorption system by Myers et al.22 This can be understood by considering the relationship between the excess and
8300
Langmuir, Vol. 21, No. 18, 2005
He and Seaton
Figure 3. Simulated absolute and excess adsorption of methane at 301.4 K. (a) Isotherms and (b) isosteric heats.
absolute isosteric heats given by eq 7. The derivative in the denominator of this equation is the slope of the excess adsorption isotherm, which is greater at low pressure than at high pressure and is greater for smaller pores than for larger pores (see Figure 3). From eq 7, then, one would expect the difference between absolute and excess isosteric heats to be greatest at high pressure and to generally increase with pore width. Both of these effects are seen in Figure 3. As discussed in the Introduction, PSDs are conventionally obtained by using eq 1 to match the model isotherm to its experimental counterpart. Figure 4 shows two different PSDs (“PSD-1” in panel 1 of Figure 4 and “PSD2” in panel 2 of Figure 4), obtained by this approach, using different initial values for the PSD. These two PSDs are similar in the small pore-size range, but there is a second peak at 30 Å in PSD-1. These two PSDs both fit the experimental isotherm very accurately throughout the pressure range, as shown in Figure 5. The PSDs were then used to predict the isosteric heats of methane in BPL using eq 8; the results are shown in Figure 6. PSD-1 greatly underestimates the isosteric heat across the pressure range. PSD-2 underestimates the isosteric heats at low pressures but greatly overestimates the isosteric heats when the pressure is higher than 500 kPa. The poor representation of the isosteric heat at low pressure by PSD-1 and PSD-2 reflects the fact that these fits understate the level of heterogeneity of the adsorbent; this is
Figure 4. Pore-size distributions for BPL obtained by different methods. PSDs 1 and 2 are obtained using eq 1, and PSD 3 is obtained using eq 19.
reflected in the smaller volume of the very smallest pores in these PSDs, in comparison with PSD-3. We conclude, first, that a good fit of the adsorption isotherm does not guarantee that a PSD can accurately predict the isosteric heat and, second, that different PSDs consistent with the adsorption isotherm may give quite different predictions of the isosteric heat. Having established that fitting the PSD to isotherm data using eq 1 does not necessarily give a good account of the isosteric heat, we now fit the experimental isosteric heat of methane using eq 8 to get another PSD. The resulting PSD (“PSD-3”) is shown in panel 3 of Figure 4, and the very good fit between the experimental and simulated isosteric heats of adsorption of methane in BPL activated is shown in Figure 7. The simulated isosteric heat of adsorption, as well as the experimental one, decreases with pressure and becomes nearly constant in the high-pressure range. The PSD obtained by fitting to the isosteric heat is quite different from the two previously
Monte Carlo Simulation and Pore-Size Distribution Analysis
Langmuir, Vol. 21, No. 18, 2005 8301
Figure 7. Fit of the PSD model to the isosteric heat data. Figure 5. Fits of the PSD model to the experimental adsorption isotherm of methane at 301.4 K.
Figure 6. Predictions of the isosteric heats of methane at 301.4 K.
obtained by fitting to the isotherms. In particular, its apparent “information content” is greater in the small pore-size range, giving a much greater variation in the PSD with pore width. PSD-3 was then used to predict the adsorption of methane at 301.4 and 264.6 K and also the adsorption isotherm of ethane at 301.4 K. The results are shown in Figure 8. Clearly, this PSD gives accurate predictions of the isotherms of methane in BPL at different temperatures with a relatively small error (mostly less than 3%) and also gives accurate predictions for ethane at the same temperature. We thus conclude that a PSD obtained by fitting to the isosteric heat is richer in information than a PSD fitted to the isotherm and has a greater predictive capability.
Figure 8. Prediction of adsorption isotherms based on the PSD obtained by fitting the isosteric heat.
5. Conclusions We used a combined Monte Carlo simulation and PSD analysis to investigate adsorption in an activated carbon, using the isosteric heat, rather than the isotherm, as the experimental input. We found that the PSD obtained from the isosteric heat can be used to predict the isotherms of the two components that we studied. The reverse process, fitting to an isotherm and predicting the isosteric heat, is in contrast much less accurate. We conclude that the isosteric heat is a more sensitive measure of the structure of activated carbon adsorbents. Acknowledgment. The financial support of the U.K. Engineering and Physical Sciences Research Council and an ORS reward from the Universities U.K. are gratefully acknowledged. LA050694V
