- Email: [email protected]
Characterization of nanoporous carbons by combining CO2 and H2 sorption data with the Monte Carlo simulations
Applied Surface Science 253 (2007) 5715–5720 www.elsevier.com/locate/apsusc
Characterization of nanoporous carbons by combining CO2 and H2 sorption data with the Monte Carlo simulations M. Konstantakou a,b, Th.A. Steriotis a,*, G.K. Papadopoulos c, M. Kainourgiakis a, E.S. Kikkinides b, A.K. Stubos a b
a National Center for Scientific Research Demokritos, 15310 Ag. Paraskevi, Athens, Greece Department of Engineering and Management of Energy Resources, University of Western Macedonia, 50100 Kozani, Greece c School of Chemical Engineering, National Technical University of Athens, 15780 Zografou Campus, Athens, Greece
Available online 27 December 2006
Abstract The Monte Carlo method in its grand ensemble variant (GCMC) is used in combination with experimental data in order to characterize microporous carbons and obtain the optimal pore size distribution (PSD). In particular, the method is applied in the case of AX-21 carbon. Adsorption isotherms of CO2 (253 and 298 K) and H2 (77 K) up to 20 bar have been measured, while the computed isotherms resulted from the GCMC simulations for several pore widths up to 3.0 nm. For the case of H2 at 77 K quantum corrections were introduced with the application of the Feynman–Hibbs (FH) effective potential. The adsorption isotherms were used either individually or in a combined manner in order to deduce PSDs and their reliability was examined by the ability to predict the experimental adsorption isotherms. The combined approach was found to be capable of reproducing more accurately all the available experimental isotherms. # 2007 Elsevier B.V. All rights reserved. PACS : 81.05.Rm; 82.20.Wt; 61.43.Gt; 68.43.h Keywords: Activated carbons; Pore size distribution; GCMC simulations
1. Introduction The reliable characterization of activated carbons (ACs) in terms of pore size distribution (PSD) is an important issue for their efficient utilization in industrial applications but also for the development of novel improved carbon adsorbents. Gas sorption is routinely used to characterize porous materials with mesopores (2–50 nm) as well as with micropores (<2 nm). Contrary to the mesopore case the mechanism(s) of adsorption in micropores is still under active debate and in this respect the limitations of conventional micropore characterization methods have been repeatedly discussed in literature [1]. On the other hand, the development of improved molecular level approaches has led to a better understanding of the adsorption processes in micropores. For instance, the density functional theory (DFT) [2,3] and the Grand Canonical Monte Carlo (GCMC) technique
* Corresponding author. E-mail address: [email protected] (T.A. Steriotis). 0169-4332/$ – see front matter # 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.apsusc.2006.12.053
have been established recently as an efficient alternative approach for the simulation of adsorption processes in carbons [4,5] and the determination of PSDs [6,7]. Regardless the theoretical approach, the determination of PSDs of ACs is commonly based on N2 adsorption isotherms at 77 K. However, in many microporous networks very slow adsorption processes are observed and such diffusional limitations can lead to significant under estimation of the adsorption isotherm [8] especially for the ultra-microporous (<0.7 nm) samples. Adsorption measurements at higher temperatures represent a more convenient alternative in terms of both experimental time and resolution [9]. For instance CO2 has been extensively used at room temperatures because it can easily access micropores which would present diffusion resistance for N2 at 77 K. Similarly, H2 is considered an excellent probe for very fine pores due to its small size, while at 77 K it is far above its supercritical temperature ensuring fast equilibration kinetics. Pertinent adsorption measurements have been used lately for the determination of PSDs [10,11].
5716
M. Konstantakou et al. / Applied Surface Science 253 (2007) 5715–5720
In this work we report the experimental adsorption isotherms of CO2 at 253 and 298 K as well as H2 at 77 K on a large surface area AC and the respective GCMC calculations for individual slit-shaped graphitic pores of varying width. The experimental and theoretical data were initially used in order to derive individual PSDs (one for each experimental isotherm) of AC. The data were also fitted simultaneously and new PSDs that respect combinations of the experimental isotherms were deduced. Finally, the ability of the calculated PSDs to predict the experimental isotherms was used as a measure of their reliability. It should be mentioned that the present study is an extension of our previous work [11] with the updated GCMC databases, namely the use of alternative CO2 potential parameters as well as the introduction of quantum corrections for H2 adsorption at 77 K. 2. Experimental The AC used in this work is the large surface area KOH activated carbon AX-21, made by Amoco Co. and kindly provided by S.R. Tennisson, MAST Carbon. The adsorption isotherms of CO2 at 253 and 298 K as well as H2 at 77 K were measured in a pressure range 0–20 bar on the Intelligent Gravimetric Analyzer (IGA, Hiden Analytical Ltd.).
C–O distances of the model are 0.2298 and 0.1149 nm, respectively. The intermolecular potential uij is assumed to be a sum of the interatomic potentials between the atoms of molecules i and j, plus the electrostatic interactions due to CO2 quadrupole moment (point partial charges qO = 0.3256e, qC = +0.6512e). 3.2. Solid–fluid interactions Each wall of the graphite surface consists of stacked planes of carbon atoms separated by a distance D = 0.335 nm and having a number density rw = 114 atoms/nm3. The interaction of CO2 and H2 with the carbon walls is calculated by the 10-4-3 potential of Steele [20]. Additionally for hydrogen, the quantum term [21]: 12 2 bh2 X 1 s ab uqu ðr z Þ ¼ 176prs eab 24mm j¼0 2 r z þ jD 6 1 s ab (2) 4:4 r z þ jD is added to the Steele potential so that the total solid–fluid potential, uH–w, is given by uHw ðr z Þ ¼ u10-4-3 ðr z Þ þ uqu ðr z Þ
(3)
3. Simulation model The Grand Canonical Monte Carlo method [12,13] was employed for the simulation of H2 and CO2 adsorption in idealized graphite slit-like pores. Details can be found in Ref. [11]. 3.1. Fluid–fluid interactions Despite earlier claims [14], recent results suggest that at 77 K quantum effects are expected to contribute significantly to the adsorption process of H2, especially inside nanopores [15,16]. Thus, in our simulations H2 is represented by a Gaussian wave packet of width £/(12 mkbT)1/2. The Feynman–Hibbs (FH) effective potential is obtained by averaging the classical Lennard–Jones (LJ) potential over the Gaussian wave packet. The second order FH effective potential is given by [17]: uFH ðrÞ ¼ uLJ ðrÞ þ
b h2 24mm
d2 2 d u u ðrÞ þ ðrÞ LJ LJ r dr dr 2
(1)
where b = (kbT)1, £ = h/2p, mm is the reduced mass of a pair of interacting hydrogen molecules (mm = m/2) and uLJ is the energy of the pairwise interaction between LJ sites. H2 was treated as a two center LJ molecule with eHH/ kB = 12.5 K and sHH = 0.259 nm. The H–H distance in the model was taken to be the actual bond length (0.074 nm). The parameters for the two-site model were adopted from Ref. [18]. CO2 was modeled as a three charged center LJ molecule with the parameters eOO/kB = 80.507 K, sOO = 0.3033 nm, eCC/kB = 28.129 K and sCC = 0.2757 nm [19]. The O–O and
where rz is the distance between the L–J site on the adsorbent and the plane of carbon atoms, mm is the reduced mass of a hydrogen molecule with a carbon atom (we use mm = m by assuming that the carbon atoms are linked together rigidly) and rs = 38.19 nm2 the surface density of a single graphite layer. The potential parameters of the solid surface are ess/k = 28.0 K and sss = 0.34 nm, while all the cross interaction potential parameters between different sites (a 6¼ b) were calculated according to the Lorentz–Berthelot rules. The overall potential energy Uw due to the walls inside a slitlike pore is calculated by the sum of the interactions between the adsorbate and both pore walls: U w ¼ uw ðr z Þ þ uw ðH r z Þ;
H 0 ¼ H 2z0 þ s g
(3)
where H is the distance between the carbon centers across the slit pore model (physical width). In principle the corrected width H0 (chemical width) should be used (H0 is the ‘‘available’’ pore width), where sg is the root of the adsorbate–adsorbent LJ function, and z0 the root of its first derivative. In the present H2 or CO2–graphite systems, it is found that about 0.24 nm should be subtracted from H to define H0 . 4. GCMC simulation results Indicative GCMC calculated isotherms for the slit pores of different sizes (H = 0.6, 1.2, 1.8, 2.4 and 3.0 nm) are presented in Fig. 1. The shape of the CO2 isotherms at 253 K gradually changes from a clear microporous-type I (0.6 and 1.2 nm) to a mesoporous-type IV (2.4 and 3.0 nm) as the pore width is increased. On the other hand, at 298 K the microporous character transforms to an almost linear-Henry
M. Konstantakou et al. / Applied Surface Science 253 (2007) 5715–5720
5717
5. Determination of micropore size distribution The procedure for the determination of the optimal PSD involves the numerical solution of a minimization problem under certain constraints. In practice, the problem consists of inverting the adsorption integral equation: Nð pÞ ¼
Z
H max
f ðHÞnðH; pÞ dH
(4)
H min
where N( p) is the experimentally measured amount of adsorbate, n(H, p) the average density of adsorbate at pressure p in a pore of width H, and f(H) is the PSD sought. The solution of Eq. (4) is an ill-posed problem. Depending on the form of the kernel n(H, p) and the isotherm N( p), there can be from zero to an infinity of solutions for f(H). Detailed discussions on the methods for the solution of Eq. (4) and the application of suitable constraints to force physically sound or appealing solutions including constraints on the smoothness of f(H) and the range of H, can be found in Refs. [22,23]. Our work aims at finding useful solutions in the sense that the gas adsorption properties of microporous carbons can be reliably predicted. The process of defining the optimal PSD begins by assigning an assumed initial volume, Vj to every pore group Hj and constructing thus a computed isotherm. Consequently, this constructed isotherm is compared to its experimental counterpart and by iterative variation of the Vj matrix elements (which are subject to the constraint of non-negativity), the ‘‘optimum’’ micropore size distribution, namely the one providing the best fit, is selected. For this reason, the routine E04NCF of NAG library has been implemented. It is a routine solving linearly constrained linear least-squares problems based on a two-phase (primal) quadratic programming method with features to exploit the convexity of the objective function [24,25].
Fig. 1. GCMC adsorption isotherms of CO2 at 253 K (a) and 298 K (b) and H2 at 77 K (c) for 0.6 nm (diamonds), 1.2 nm (squares), 1.8 nm (triangles), 2.4 nm (circles) and 3.0 nm (star) carbon slits.
type behavior, beyond 1.8 nm. This observation is expected since the isotherms have been measured in the same pressure range and thus the 298 K data refer to much lower relative pressures. In this respect the data pertain rather to surface coverage than pore filling. It is noteworthy that in contrast to the 253 K case where each curve is more or less a fingerprint of the pore size, the set of Henry-type isotherms at 298 K does not allow clear distinction of different pore sizes. The adsorption isotherms of H2 at 77 K reveal that adsorption is enhanced at very narrow pores. The Langmuir-type shape smoothens out gradually as pore size is increased and for pores wider than 1.2 nm the majority of data points (e.g. above 3 bar) fall into a set of almost parallel straight lines. As in the case of CO2 at 298 K this behavior implies that H2 (at 77 K) is not sensitive enough for the distinction of large micropores and/or mesopores.
6. Results and discussion n the first stage, the technique is applied for determination of PSD of microporous carbon AX-21 based on individual experimental isotherms. In this respect three different PSDs based on CO2 at 253 K, CO2 at 298 K and H2 at 77 K adsorption data were deduced (Fig. 2). The PSD obtained from the H2 data differs considerably from the CO2 PSDs as it practically reveals 3 classes of pores centered at around H = 0.7, 1.6 and 2.7 nm. On the contrary, the PSDs of CO2 at both temperatures reveal a much broader distribution of sizes covering the entire range studied. In order to examine the reliability of the results, each PSD was used in a reverse, manner, i.e. for predicting the adsorption isotherms of CO2 (253 and 298 K) and H2 (77 K). The results are shown in Fig. 3. It can be easily seen that PSDs obtained from the individual isotherms have in general, a limited predictive potential. Of course all the PSDs can quite accurately predict their experimental counterparts, nevertheless experimental data on different molecules and/or temperatures cannot be accurately reproduced. CO2 PSDs produce CO2 isotherms that more or less resemble those experimentally
5718
M. Konstantakou et al. / Applied Surface Science 253 (2007) 5715–5720
Fig. 4. PSDs based on combinations of adsorption isotherms (Vj: volume of each class of pores and Vt: total pore volume). Fig. 2. PSDs based on individual adsorption isotherms (Vj: volume of each class of pores and Vt: total pore volume).
obtained, however their predictions for H2 at 77 K are not even qualitatively similar to the measured isotherm. The H2 PSD provides the worst predictions. For instance, this PSD produces a step isotherm for the full relative pressure range isotherm (CO2, 253 K), presumably pertaining to filling of the two larger classes of pores (1.6 and 2.7 nm). Such poor predictions are compatible with the fact that both H2 at 77 K and CO2 at 298 K isotherms contain insufficient information about wide pores (which are significant for the CO2 isotherm at 253 K). As mentioned above H2 calculated isotherms for different slit widths differ considerably for H < 1.2 nm (and H < 1.8 nm for CO2 at 298 K), while for larger pore sizes they are similar. In this respect, the minimization process for the deduction of the PSD fails to distinguish the contribution of different pore sizes as there are almost infinite combinations of different pores having H > 1.2 nm for H2 (H < 1.8 nm for CO2 at 298 K) that can reproduce the experimentally determined
amount adsorbed. On the other hand, CO2 at 253 K offers a fullscale description of the pore system but since ultramicropores are filled at very low pressures it fails to capture their contribution (which mainly determines, e.g. the H2 isotherm) accurately. A more reliable PSD should contain complete information of a full relative pressure range isotherm and an accurate description of ultramicropores. This can be implemented by simultaneously inverting different adsorption integral equations (different experimental and respective GCMC data sets but the same f(H) function), seeking thus a common PSD that respects more than one experimental isotherm. The above approach has been followed for combinations of our data and the PSDs obtained are shown in Fig. 4. The pertinent predictions of the experimental isotherms are presented in Fig. 5. The similarity of the ‘‘combined’’ PSDs coupled with their ability to better reproduce the experimental isotherms than the individual PSDs, imply that such combinations depict more accurately the actual porous system of AX-21. As expected, the optimal PSD is
Fig. 3. Experimental isotherms (open circles) of CO2 at 253 K (a), CO2 at 298 K (b) and H2 at 77 K (c) on AX-21 and the predictions (lines with symbols) based on the individual PSDs of Fig. 2.
M. Konstantakou et al. / Applied Surface Science 253 (2007) 5715–5720
5719
Fig. 5. Experimental isotherms (open circles) of CO2 at 253 K (a), CO2 at 298 K (b) and H2 at 77 K (c) on AX-21 and the predictions (lines with symbols) based on the combined PSDs of Fig. 4.
obtained from the combination of all the data. In the GCMC PSD approach, the adsorption integral inversion mathematically degenerates to a minimization problem, while the experimental data probe different pore sizes with varying accuracy (from fine to gross in the order H2 77 K > CO2 298 K > CO2 253 K). We can thus conclude that in order to deduce a PSD that can adequately describe the pore system, a high temperature (in the sense that diffusion limitations are negligible and equilibrium is easily attained) and a full relative pressure range isotherm (in order to capture the contribution of all the pore sizes) should be the basis. This set should then be improved by using data on finer pores simultaneously. For instance, the incorporation of H2 data in the calculation finetunes the analysis by introducing small pores sizes, not detectable by CO2. 7. Conclusions Adsorption isotherms of CO2 at 253 and 298 K as well as H2 at 77 K have been performed on the AX-21 activated carbon, while GCMC-based isotherms for the same cases were calculated for model carbon slits having widths varying from 0.6 to 3.0 nm. For the latter the Harris and Yung [19] potential parameters were used for CO2, while quantum corrections were included in H2 calculations. Each experimental isotherm was used individually, with its corresponding GCMC kernel, for the calculation of PSDs. The three calculated PSDs differ significantly (especially that of H2) and exhibit limited use for the general prediction of adsorption behavior. Subsequently, the individual isotherms were combined and new PSDs that respect combinations of the experimental isotherms were deduced. These combined PSDs can reproduce much more accurately the experimental data. From our results it can be concluded that the use of different probes is essential for a reliable pore size analysis of polydisperse
porous materials. Good combinations will be a high temperature full relative pressure scale adsorption isotherm (such as CO2 at 253 K, 0–20 bar) that can capture the contribution of all the sample pores with isotherms of small and very mobile molecules (preferably close or above their critical temperature) such as H2 (77 K) and CO2 (298 K), that give a detailed picture of very fine pores. References [1] F. Stoeckli, A. Guillot, D. Hugi-Cleary, A.M. Slasli, Carbon 38 (2000) 938. [2] P.I. Ravikovitch, A. Vishnyakov, A.V. Neimark, M.M.L. Ribeiro Carrott, P.A. Russo, P.J. Carrott, Langmuir 22 (2006) 513. [3] T.X. Nguyen, S.K. Bhatia, Langmuir 20 (2004) 3532. [4] D.D. Do, H.D. Do, Colloids Surf. A: Physicochem. Eng. Aspects 252 (2005) 7. [5] T.X. Nguyen, S.K. Bhatia, D. Nicholson, Langmuir 21 (2005) 3187. [6] X. Shao, W. Wang, R. Xue, Z. Shen, J. Phys. Chem. B 108 (2004) 2970. [7] D.D. Do, H.D. Do, Adsorption Sci. Technol. 21 (2003) 389. [8] F. Rodriguez-Reinoso, A. Linares-Solano, in: P.A. Thrower (Ed.), Chemistry and Physics of Carbon, vol. 21, Marcel Dekker, New York, 1988. [9] J. Garrido, A. Linares-Solano, J.M. Martin-Martinez, M. Molina-Sabio, F. Rodriguez-Reinoso, R. Torregosa, Langmuir 3 (1987) 76. [10] J. Jagiello, M. Thommes, Carbon 42 (2004) 1227. [11] M. Konstantakou, S. Samios, Th.A. Steriotis, M. Kainourgiakis, G.K. Papadopoulos, E.S. Kikkinides, A.K. Stubos, Stud. Surf. Sci. Catal. 160 (2006) 543. [12] M. Allen, D.J. Tildesley, Computer Simulation of Liquids, Clarendon, Oxford, 1987. [13] D. Nicholson, N.G. Parsonage, Computer Simulation and the Statistical Mechanics of Adsorption, Academic Press, London, 1982. [14] M. Rzepka, P. Lamp, M.A. de la Casa-Lillo, J. Phys. Chem. B 110 (1999) 4020. [15] Q. Wang, J.K. Johnson, J. Phys. Chem. B 103 (1999) 277. [16] H. Tanaka, J. Fan, H. Kanoh, M. Yudasaka, S. Iijima, K. Kaneko, Mol. Simul. 31 (2005) 465. [17] L.M. Sese, Mol. Phys. 85 (1995) 931.
5720
M. Konstantakou et al. / Applied Surface Science 253 (2007) 5715–5720
[18] R.F. Cracknell, Phys. Chem. Chem. Phys. 3 (2001) 2091. [19] J.G. Harris, K.H. Yung, J. Phys. Chem. 99 (1995) 12021. [20] W.A. Steele, The Interaction of Gases with Solid Surfaces, Pergamon, Oxford, 1974. [21] H. Tanaka, H. Kanoh, M. Yudasaka, S. Iijima, K. Kaneko, J. Am. Chem. Soc. 127 (2005) 7511.
[22] P.A. Gauden, P. Kowalczyk, A.P. Terzyk, Langmuir 19 (2003) 4253. [23] C. Herdes, M.A. Santos, F. Medina, L.F. Vega, Langmuir 21 (2005) 8733. [24] P.E. Gill, W. Murray, M.A. Saunders, M.H. Wright, ACM Trans. Math. Software 10 (1984) 282. [25] J. Stoer, SIAM J. Numer. Anal. 8 (1971) 382.
Characterization of nanoporous carbons by combining CO2 and H2 sorption data with the Monte Carlo simulations M. Konstantakou a,b, Th.A. Steriotis a,*, G.K. Papadopoulos c, M. Kainourgiakis a, E.S. Kikkinides b, A.K. Stubos a b
a National Center for Scientific Research Demokritos, 15310 Ag. Paraskevi, Athens, Greece Department of Engineering and Management of Energy Resources, University of Western Macedonia, 50100 Kozani, Greece c School of Chemical Engineering, National Technical University of Athens, 15780 Zografou Campus, Athens, Greece
Available online 27 December 2006
Abstract The Monte Carlo method in its grand ensemble variant (GCMC) is used in combination with experimental data in order to characterize microporous carbons and obtain the optimal pore size distribution (PSD). In particular, the method is applied in the case of AX-21 carbon. Adsorption isotherms of CO2 (253 and 298 K) and H2 (77 K) up to 20 bar have been measured, while the computed isotherms resulted from the GCMC simulations for several pore widths up to 3.0 nm. For the case of H2 at 77 K quantum corrections were introduced with the application of the Feynman–Hibbs (FH) effective potential. The adsorption isotherms were used either individually or in a combined manner in order to deduce PSDs and their reliability was examined by the ability to predict the experimental adsorption isotherms. The combined approach was found to be capable of reproducing more accurately all the available experimental isotherms. # 2007 Elsevier B.V. All rights reserved. PACS : 81.05.Rm; 82.20.Wt; 61.43.Gt; 68.43.h Keywords: Activated carbons; Pore size distribution; GCMC simulations
1. Introduction The reliable characterization of activated carbons (ACs) in terms of pore size distribution (PSD) is an important issue for their efficient utilization in industrial applications but also for the development of novel improved carbon adsorbents. Gas sorption is routinely used to characterize porous materials with mesopores (2–50 nm) as well as with micropores (<2 nm). Contrary to the mesopore case the mechanism(s) of adsorption in micropores is still under active debate and in this respect the limitations of conventional micropore characterization methods have been repeatedly discussed in literature [1]. On the other hand, the development of improved molecular level approaches has led to a better understanding of the adsorption processes in micropores. For instance, the density functional theory (DFT) [2,3] and the Grand Canonical Monte Carlo (GCMC) technique
* Corresponding author. E-mail address: [email protected] (T.A. Steriotis). 0169-4332/$ – see front matter # 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.apsusc.2006.12.053
have been established recently as an efficient alternative approach for the simulation of adsorption processes in carbons [4,5] and the determination of PSDs [6,7]. Regardless the theoretical approach, the determination of PSDs of ACs is commonly based on N2 adsorption isotherms at 77 K. However, in many microporous networks very slow adsorption processes are observed and such diffusional limitations can lead to significant under estimation of the adsorption isotherm [8] especially for the ultra-microporous (<0.7 nm) samples. Adsorption measurements at higher temperatures represent a more convenient alternative in terms of both experimental time and resolution [9]. For instance CO2 has been extensively used at room temperatures because it can easily access micropores which would present diffusion resistance for N2 at 77 K. Similarly, H2 is considered an excellent probe for very fine pores due to its small size, while at 77 K it is far above its supercritical temperature ensuring fast equilibration kinetics. Pertinent adsorption measurements have been used lately for the determination of PSDs [10,11].
5716
M. Konstantakou et al. / Applied Surface Science 253 (2007) 5715–5720
In this work we report the experimental adsorption isotherms of CO2 at 253 and 298 K as well as H2 at 77 K on a large surface area AC and the respective GCMC calculations for individual slit-shaped graphitic pores of varying width. The experimental and theoretical data were initially used in order to derive individual PSDs (one for each experimental isotherm) of AC. The data were also fitted simultaneously and new PSDs that respect combinations of the experimental isotherms were deduced. Finally, the ability of the calculated PSDs to predict the experimental isotherms was used as a measure of their reliability. It should be mentioned that the present study is an extension of our previous work [11] with the updated GCMC databases, namely the use of alternative CO2 potential parameters as well as the introduction of quantum corrections for H2 adsorption at 77 K. 2. Experimental The AC used in this work is the large surface area KOH activated carbon AX-21, made by Amoco Co. and kindly provided by S.R. Tennisson, MAST Carbon. The adsorption isotherms of CO2 at 253 and 298 K as well as H2 at 77 K were measured in a pressure range 0–20 bar on the Intelligent Gravimetric Analyzer (IGA, Hiden Analytical Ltd.).
C–O distances of the model are 0.2298 and 0.1149 nm, respectively. The intermolecular potential uij is assumed to be a sum of the interatomic potentials between the atoms of molecules i and j, plus the electrostatic interactions due to CO2 quadrupole moment (point partial charges qO = 0.3256e, qC = +0.6512e). 3.2. Solid–fluid interactions Each wall of the graphite surface consists of stacked planes of carbon atoms separated by a distance D = 0.335 nm and having a number density rw = 114 atoms/nm3. The interaction of CO2 and H2 with the carbon walls is calculated by the 10-4-3 potential of Steele [20]. Additionally for hydrogen, the quantum term [21]: 12 2 bh2 X 1 s ab uqu ðr z Þ ¼ 176prs eab 24mm j¼0 2 r z þ jD 6 1 s ab (2) 4:4 r z þ jD is added to the Steele potential so that the total solid–fluid potential, uH–w, is given by uHw ðr z Þ ¼ u10-4-3 ðr z Þ þ uqu ðr z Þ
(3)
3. Simulation model The Grand Canonical Monte Carlo method [12,13] was employed for the simulation of H2 and CO2 adsorption in idealized graphite slit-like pores. Details can be found in Ref. [11]. 3.1. Fluid–fluid interactions Despite earlier claims [14], recent results suggest that at 77 K quantum effects are expected to contribute significantly to the adsorption process of H2, especially inside nanopores [15,16]. Thus, in our simulations H2 is represented by a Gaussian wave packet of width £/(12 mkbT)1/2. The Feynman–Hibbs (FH) effective potential is obtained by averaging the classical Lennard–Jones (LJ) potential over the Gaussian wave packet. The second order FH effective potential is given by [17]: uFH ðrÞ ¼ uLJ ðrÞ þ
b h2 24mm
d2 2 d u u ðrÞ þ ðrÞ LJ LJ r dr dr 2
(1)
where b = (kbT)1, £ = h/2p, mm is the reduced mass of a pair of interacting hydrogen molecules (mm = m/2) and uLJ is the energy of the pairwise interaction between LJ sites. H2 was treated as a two center LJ molecule with eHH/ kB = 12.5 K and sHH = 0.259 nm. The H–H distance in the model was taken to be the actual bond length (0.074 nm). The parameters for the two-site model were adopted from Ref. [18]. CO2 was modeled as a three charged center LJ molecule with the parameters eOO/kB = 80.507 K, sOO = 0.3033 nm, eCC/kB = 28.129 K and sCC = 0.2757 nm [19]. The O–O and
where rz is the distance between the L–J site on the adsorbent and the plane of carbon atoms, mm is the reduced mass of a hydrogen molecule with a carbon atom (we use mm = m by assuming that the carbon atoms are linked together rigidly) and rs = 38.19 nm2 the surface density of a single graphite layer. The potential parameters of the solid surface are ess/k = 28.0 K and sss = 0.34 nm, while all the cross interaction potential parameters between different sites (a 6¼ b) were calculated according to the Lorentz–Berthelot rules. The overall potential energy Uw due to the walls inside a slitlike pore is calculated by the sum of the interactions between the adsorbate and both pore walls: U w ¼ uw ðr z Þ þ uw ðH r z Þ;
H 0 ¼ H 2z0 þ s g
(3)
where H is the distance between the carbon centers across the slit pore model (physical width). In principle the corrected width H0 (chemical width) should be used (H0 is the ‘‘available’’ pore width), where sg is the root of the adsorbate–adsorbent LJ function, and z0 the root of its first derivative. In the present H2 or CO2–graphite systems, it is found that about 0.24 nm should be subtracted from H to define H0 . 4. GCMC simulation results Indicative GCMC calculated isotherms for the slit pores of different sizes (H = 0.6, 1.2, 1.8, 2.4 and 3.0 nm) are presented in Fig. 1. The shape of the CO2 isotherms at 253 K gradually changes from a clear microporous-type I (0.6 and 1.2 nm) to a mesoporous-type IV (2.4 and 3.0 nm) as the pore width is increased. On the other hand, at 298 K the microporous character transforms to an almost linear-Henry
M. Konstantakou et al. / Applied Surface Science 253 (2007) 5715–5720
5717
5. Determination of micropore size distribution The procedure for the determination of the optimal PSD involves the numerical solution of a minimization problem under certain constraints. In practice, the problem consists of inverting the adsorption integral equation: Nð pÞ ¼
Z
H max
f ðHÞnðH; pÞ dH
(4)
H min
where N( p) is the experimentally measured amount of adsorbate, n(H, p) the average density of adsorbate at pressure p in a pore of width H, and f(H) is the PSD sought. The solution of Eq. (4) is an ill-posed problem. Depending on the form of the kernel n(H, p) and the isotherm N( p), there can be from zero to an infinity of solutions for f(H). Detailed discussions on the methods for the solution of Eq. (4) and the application of suitable constraints to force physically sound or appealing solutions including constraints on the smoothness of f(H) and the range of H, can be found in Refs. [22,23]. Our work aims at finding useful solutions in the sense that the gas adsorption properties of microporous carbons can be reliably predicted. The process of defining the optimal PSD begins by assigning an assumed initial volume, Vj to every pore group Hj and constructing thus a computed isotherm. Consequently, this constructed isotherm is compared to its experimental counterpart and by iterative variation of the Vj matrix elements (which are subject to the constraint of non-negativity), the ‘‘optimum’’ micropore size distribution, namely the one providing the best fit, is selected. For this reason, the routine E04NCF of NAG library has been implemented. It is a routine solving linearly constrained linear least-squares problems based on a two-phase (primal) quadratic programming method with features to exploit the convexity of the objective function [24,25].
Fig. 1. GCMC adsorption isotherms of CO2 at 253 K (a) and 298 K (b) and H2 at 77 K (c) for 0.6 nm (diamonds), 1.2 nm (squares), 1.8 nm (triangles), 2.4 nm (circles) and 3.0 nm (star) carbon slits.
type behavior, beyond 1.8 nm. This observation is expected since the isotherms have been measured in the same pressure range and thus the 298 K data refer to much lower relative pressures. In this respect the data pertain rather to surface coverage than pore filling. It is noteworthy that in contrast to the 253 K case where each curve is more or less a fingerprint of the pore size, the set of Henry-type isotherms at 298 K does not allow clear distinction of different pore sizes. The adsorption isotherms of H2 at 77 K reveal that adsorption is enhanced at very narrow pores. The Langmuir-type shape smoothens out gradually as pore size is increased and for pores wider than 1.2 nm the majority of data points (e.g. above 3 bar) fall into a set of almost parallel straight lines. As in the case of CO2 at 298 K this behavior implies that H2 (at 77 K) is not sensitive enough for the distinction of large micropores and/or mesopores.
6. Results and discussion n the first stage, the technique is applied for determination of PSD of microporous carbon AX-21 based on individual experimental isotherms. In this respect three different PSDs based on CO2 at 253 K, CO2 at 298 K and H2 at 77 K adsorption data were deduced (Fig. 2). The PSD obtained from the H2 data differs considerably from the CO2 PSDs as it practically reveals 3 classes of pores centered at around H = 0.7, 1.6 and 2.7 nm. On the contrary, the PSDs of CO2 at both temperatures reveal a much broader distribution of sizes covering the entire range studied. In order to examine the reliability of the results, each PSD was used in a reverse, manner, i.e. for predicting the adsorption isotherms of CO2 (253 and 298 K) and H2 (77 K). The results are shown in Fig. 3. It can be easily seen that PSDs obtained from the individual isotherms have in general, a limited predictive potential. Of course all the PSDs can quite accurately predict their experimental counterparts, nevertheless experimental data on different molecules and/or temperatures cannot be accurately reproduced. CO2 PSDs produce CO2 isotherms that more or less resemble those experimentally
5718
M. Konstantakou et al. / Applied Surface Science 253 (2007) 5715–5720
Fig. 4. PSDs based on combinations of adsorption isotherms (Vj: volume of each class of pores and Vt: total pore volume). Fig. 2. PSDs based on individual adsorption isotherms (Vj: volume of each class of pores and Vt: total pore volume).
obtained, however their predictions for H2 at 77 K are not even qualitatively similar to the measured isotherm. The H2 PSD provides the worst predictions. For instance, this PSD produces a step isotherm for the full relative pressure range isotherm (CO2, 253 K), presumably pertaining to filling of the two larger classes of pores (1.6 and 2.7 nm). Such poor predictions are compatible with the fact that both H2 at 77 K and CO2 at 298 K isotherms contain insufficient information about wide pores (which are significant for the CO2 isotherm at 253 K). As mentioned above H2 calculated isotherms for different slit widths differ considerably for H < 1.2 nm (and H < 1.8 nm for CO2 at 298 K), while for larger pore sizes they are similar. In this respect, the minimization process for the deduction of the PSD fails to distinguish the contribution of different pore sizes as there are almost infinite combinations of different pores having H > 1.2 nm for H2 (H < 1.8 nm for CO2 at 298 K) that can reproduce the experimentally determined
amount adsorbed. On the other hand, CO2 at 253 K offers a fullscale description of the pore system but since ultramicropores are filled at very low pressures it fails to capture their contribution (which mainly determines, e.g. the H2 isotherm) accurately. A more reliable PSD should contain complete information of a full relative pressure range isotherm and an accurate description of ultramicropores. This can be implemented by simultaneously inverting different adsorption integral equations (different experimental and respective GCMC data sets but the same f(H) function), seeking thus a common PSD that respects more than one experimental isotherm. The above approach has been followed for combinations of our data and the PSDs obtained are shown in Fig. 4. The pertinent predictions of the experimental isotherms are presented in Fig. 5. The similarity of the ‘‘combined’’ PSDs coupled with their ability to better reproduce the experimental isotherms than the individual PSDs, imply that such combinations depict more accurately the actual porous system of AX-21. As expected, the optimal PSD is
Fig. 3. Experimental isotherms (open circles) of CO2 at 253 K (a), CO2 at 298 K (b) and H2 at 77 K (c) on AX-21 and the predictions (lines with symbols) based on the individual PSDs of Fig. 2.
M. Konstantakou et al. / Applied Surface Science 253 (2007) 5715–5720
5719
Fig. 5. Experimental isotherms (open circles) of CO2 at 253 K (a), CO2 at 298 K (b) and H2 at 77 K (c) on AX-21 and the predictions (lines with symbols) based on the combined PSDs of Fig. 4.
obtained from the combination of all the data. In the GCMC PSD approach, the adsorption integral inversion mathematically degenerates to a minimization problem, while the experimental data probe different pore sizes with varying accuracy (from fine to gross in the order H2 77 K > CO2 298 K > CO2 253 K). We can thus conclude that in order to deduce a PSD that can adequately describe the pore system, a high temperature (in the sense that diffusion limitations are negligible and equilibrium is easily attained) and a full relative pressure range isotherm (in order to capture the contribution of all the pore sizes) should be the basis. This set should then be improved by using data on finer pores simultaneously. For instance, the incorporation of H2 data in the calculation finetunes the analysis by introducing small pores sizes, not detectable by CO2. 7. Conclusions Adsorption isotherms of CO2 at 253 and 298 K as well as H2 at 77 K have been performed on the AX-21 activated carbon, while GCMC-based isotherms for the same cases were calculated for model carbon slits having widths varying from 0.6 to 3.0 nm. For the latter the Harris and Yung [19] potential parameters were used for CO2, while quantum corrections were included in H2 calculations. Each experimental isotherm was used individually, with its corresponding GCMC kernel, for the calculation of PSDs. The three calculated PSDs differ significantly (especially that of H2) and exhibit limited use for the general prediction of adsorption behavior. Subsequently, the individual isotherms were combined and new PSDs that respect combinations of the experimental isotherms were deduced. These combined PSDs can reproduce much more accurately the experimental data. From our results it can be concluded that the use of different probes is essential for a reliable pore size analysis of polydisperse
porous materials. Good combinations will be a high temperature full relative pressure scale adsorption isotherm (such as CO2 at 253 K, 0–20 bar) that can capture the contribution of all the sample pores with isotherms of small and very mobile molecules (preferably close or above their critical temperature) such as H2 (77 K) and CO2 (298 K), that give a detailed picture of very fine pores. References [1] F. Stoeckli, A. Guillot, D. Hugi-Cleary, A.M. Slasli, Carbon 38 (2000) 938. [2] P.I. Ravikovitch, A. Vishnyakov, A.V. Neimark, M.M.L. Ribeiro Carrott, P.A. Russo, P.J. Carrott, Langmuir 22 (2006) 513. [3] T.X. Nguyen, S.K. Bhatia, Langmuir 20 (2004) 3532. [4] D.D. Do, H.D. Do, Colloids Surf. A: Physicochem. Eng. Aspects 252 (2005) 7. [5] T.X. Nguyen, S.K. Bhatia, D. Nicholson, Langmuir 21 (2005) 3187. [6] X. Shao, W. Wang, R. Xue, Z. Shen, J. Phys. Chem. B 108 (2004) 2970. [7] D.D. Do, H.D. Do, Adsorption Sci. Technol. 21 (2003) 389. [8] F. Rodriguez-Reinoso, A. Linares-Solano, in: P.A. Thrower (Ed.), Chemistry and Physics of Carbon, vol. 21, Marcel Dekker, New York, 1988. [9] J. Garrido, A. Linares-Solano, J.M. Martin-Martinez, M. Molina-Sabio, F. Rodriguez-Reinoso, R. Torregosa, Langmuir 3 (1987) 76. [10] J. Jagiello, M. Thommes, Carbon 42 (2004) 1227. [11] M. Konstantakou, S. Samios, Th.A. Steriotis, M. Kainourgiakis, G.K. Papadopoulos, E.S. Kikkinides, A.K. Stubos, Stud. Surf. Sci. Catal. 160 (2006) 543. [12] M. Allen, D.J. Tildesley, Computer Simulation of Liquids, Clarendon, Oxford, 1987. [13] D. Nicholson, N.G. Parsonage, Computer Simulation and the Statistical Mechanics of Adsorption, Academic Press, London, 1982. [14] M. Rzepka, P. Lamp, M.A. de la Casa-Lillo, J. Phys. Chem. B 110 (1999) 4020. [15] Q. Wang, J.K. Johnson, J. Phys. Chem. B 103 (1999) 277. [16] H. Tanaka, J. Fan, H. Kanoh, M. Yudasaka, S. Iijima, K. Kaneko, Mol. Simul. 31 (2005) 465. [17] L.M. Sese, Mol. Phys. 85 (1995) 931.
5720
M. Konstantakou et al. / Applied Surface Science 253 (2007) 5715–5720
[18] R.F. Cracknell, Phys. Chem. Chem. Phys. 3 (2001) 2091. [19] J.G. Harris, K.H. Yung, J. Phys. Chem. 99 (1995) 12021. [20] W.A. Steele, The Interaction of Gases with Solid Surfaces, Pergamon, Oxford, 1974. [21] H. Tanaka, H. Kanoh, M. Yudasaka, S. Iijima, K. Kaneko, J. Am. Chem. Soc. 127 (2005) 7511.
[22] P.A. Gauden, P. Kowalczyk, A.P. Terzyk, Langmuir 19 (2003) 4253. [23] C. Herdes, M.A. Santos, F. Medina, L.F. Vega, Langmuir 21 (2005) 8733. [24] P.E. Gill, W. Murray, M.A. Saunders, M.H. Wright, ACM Trans. Math. Software 10 (1984) 282. [25] J. Stoer, SIAM J. Numer. Anal. 8 (1971) 382.