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New insights into the adsorption isotherm interpretation by a coupled molecular simulation—experimental procedure
Applied Surface Science 252 (2005) 519–528 www.elsevier.com/locate/apsusc
New insights into the adsorption isotherm interpretation by a coupled molecular simulation—experimental procedure M.J. Sa´nchez-Montero a, C. Herdes b, F. Salvador a, L.F. Vega b,* a
b
Departamento de Quı´mica-Fı´sica, Universidad de Salamanca, 37008 Salamanca, Spain Institut de Cie`ncia de Materials de Barcelona (ICMAB-CSIC), Campus de la UA, 08193 Bellaterra, Barcelona, Spain Available online 14 March 2005
Abstract We present here Grand Canonical Monte Carlo (GCMC) simulation results of nitrogen adsorption at 77 K on a crude model of activated carbon. The material is modeled as slit-like pores of different widths, with smooth surfaces. The individual adsorption isotherms serve as the basis to check the success and limitations of the assumptions made when using the BET model to characterize adsorbent materials, in particular to calculate the monolayer capacity and the C parameter. As done in our previous work with several experimental adsorption isotherms, different linearizations of the BETequation are used. The aim of this work is to quantify, using statistical mechanics tools, the changes in the C factor with surface coverage, showing that C is an intrinsically energetic meaningful quantity. The amount of molecules adsorbed at each pressure is calculated in the first and subsequent layers. We also keep track of the adsorbent–adsorbate and adsorbate–adsorbate energy along the simulations. The C factor is obtained following two different routes: as directly derived from the BET equation, once the monolayer capacity is known, and from the heat of adsorption obtained directly from the simulations. Results from simulations confirm the changes in the C values with surface coverage. In addition, molecular simulations provide independent and consistent ways of calculating the monolayer capacity. # 2005 Elsevier B.V. All rights reserved. Keywords: Nitrogen adsorption isotherms; Activated carbons; BET theory; Molecular simulations; Energetic heterogeneity; Monolayer capacity
1. Introduction A routine task performed before any adsorbent or catalytic material can be put into use is, among others, its textural characterization. Although it is true that a long effort has been devoted towards the development * Corresponding author. Tel.: +34 93 580 1853; fax: +34 93 580 5729. E-mail address: [email protected] (L.F. Vega).
of reliable methods to interpret the adsorption isotherm as a standard procedure to characterize materials, the most widely used method is still the BET model, due to Brunauer et al. [1]. The theory is based on kinetic arguments and it assumes a reversible adsorption/desorption process with multilayer formation. The BET model owes its widespread use to the fact that it is straightforward to apply and a linear transformation of the original equation, as proposed by the authors, provides two of the most meaningful
0169-4332/$ – see front matter # 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.apsusc.2005.02.068
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textural quantities, the monolayer capacity and the C parameter, giving fairly acceptable results for several solids. In spite of its success, BET is also one of the most criticized theories since its inception [2]. The main criticism of the BET model comes from two of the basic (inaccurate) assumptions made when developing it: (1) all adsorbing sites are energetically homogeneous; and (2) only vertical interactions between adsorbed molecules and the adsorbing surface take place, hence ignoring any lateral interaction. There have been several attempts to correct one or the two assumptions in order to improve the BET model, however, there is not general consensus on an accurate method to substitute it for surface area calculations [3]. Searching for a better understanding of the approximations behind the model, Seri-Levy and Avnir [2] performed Monte Carlo simulations of the BET conditions, finding excellent agreement with the model predictions. This allowed them to include lateral interactions into the simulations, identifying and analyzing deviations from the BET model. These results proved that the model is accurate under the conditions at which it was developed; however, it is clear that the model is applied most of the time for cases for which the assumptions made are too crude and far away from the experimental conditions. Hence, it would be desirable to have an independent test to check the accuracy of the textural properties obtained by applying the BET model to different real materials. In a recent work by some of us [4], we reviewed the BET plot proposing other possible linear transformations of the original equation, applying them to several different adsorbent materials. The main conclusions from that work were that the linear transformations of the BET equation can be classified into two categories, those unable to provide a linear regression, in which the energetic heterogeneity in the adsorption data is revealed, showing a C parameter changing with surface coverage; and those, to which the original transformation proposed by the authors belongs to, presenting a linear regression as a consequence of the high values taken by C, which neutralize the energetic heterogeneity of the adsorption data. A striking consequence of this work was that the energetic meaning of C can be inferred just by using different linear transformations of the original equation, without undertaking other more refined methods (i.e.
calorimetric methods). However, these results should be confirmed by independent techniques. It is proved now that molecular simulations provide an excellent tool in which to perform systematic studies towards the understanding of surface science [5]. They allow isolating and quantifying the effect of the different variables into the global behavior of the system. In this work, we use Grand Canonical Monte Carlo (GCMC) simulation results of adsorption on model surfaces as the ‘‘experimental’’ isotherms at which the BET treatment can be applied. Previous work done by Heuchel and Jaroniec [6] used GCMC adsorption isotherms to explore the possibility of extracting information about surface and structural heterogeneity of microporous solids from experimental data. These authors focused on evaluating the adsorption energy distribution function as a function of the degree of pore filling and pore width, pointing out that additional simulation studies are needed to understand the relationship between microporosity and surface heterogeneity of porous solids. Our goal here is to use the GCMC simulation results to get some insight into the values obtained when the BET model, with different linear transformations, is applied to some experimental isotherms. Since the simulations give independent ways of calculating the two quantities provided by BET, nm and C, they also allow validating, or not, the values obtained from the direct application of the model. It should be clear that our objective is not to improve the BET model, but to use it, as it is, and to understand the implications of applying it from a molecular perspective. The rest of the paper is organized as follows: we first present a summary of the BET model and some possible linearizations. In Section 3, we present the simulation details and the molecular model used in the calculations. Results are presented and discussed in Section 4; finally, some concluding remarks are presented in Section 5.
2. The BET model The original equation proposed by Brunauer et al. is [1]: n Cx ¼ nm ð1 xÞð1 x þ CxÞ
(1)
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where n represents the moles of gas adsorbed at the equilibrium pressure p; nm is the number of moles of gas adsorbed when the surface is covered by a complete monomolecular layer (the monolayer capacity); x is the relative pressure ( p/ p0), p0 is the vapor pressure of the adsorbate. According to the model, C is defined as: a1 n2 E1 EL C¼ exp (2) a2 n1 RT where a1 and a2 are the condensation coefficients for the first and second layers; n1 and n2 are the frequencies of oscillation of the molecule in a direction normal to the surface; E1 is the heat of adsorption of the first layer, and EL is the liquefaction heat of the adsorbate. The term (E1 EL) has been defined in some confusing ways in the literature; the accepted name now is the ‘‘net molar energy of adsorption’’. The C factor can be also obtained from the original model (Eq. (1)) for each surface coverage, once the monolayer capacity is known: C¼
nð1 xÞ2 x½nm nð1 xÞ
(3)
To check the validity of their model, the authors proposed the transformation of Eq. (1) into the linear form: x 1 C1 ¼ þ x nð1 xÞ nm C nm C
(4)
This linearization provides a direct way of obtaining the monolayer capacity value nm and the C factor, from the slope and the intercept of the line represented by this equation. Brunauer et al. observed that the linearity of the BET representation was reduced to a small range ( p/ p0 = 0.05 0.3). Later studies have shown that for many systems that range is much more reduced, although there is still no criterion to select the range. Moreover, for Type III and V isotherms, the BET plot is never linear. One also finds the paradox that in general, linearity occurs at low coverage values, where the energetic heterogeneity, not taken into account into the model, is higher. Although the most generalized way of using the BET model is through the linear form proposed by the authors (Eq. (3)); there are other possible linear transformations [4,7] of the original Eq. (1) that can be
521
used to check the BET model: x nð1 xÞ
2
¼
1 1 x þ nm C nm 1 x
nð1 xÞ2 ¼ nm C Cnð1 xÞ x 1 1 1 1x ¼ þ nð1 xÞ nm nm C x
(5)
(6) (7)
In fact, as pointed out by some authors [4,7], some of these equations have clear advantages over the original transformation; for instance, Eq. (5) provides the value of nm directly from the slope of the line, with no error associated to the intercept. Some authors use Eq. (7) as an alternative to the original linearization, as discussed in reference [7].
3. Molecular model and simulations We use the Grand Canonical Monte Carlo technique to obtain adsorption isotherms in which the BET model can be applied and discussed. Nitrogen molecules are modeled as 12–6 Lennard–Jones (LJ) spheres, with parameter values sff = 0.3575 nm and eff/k = 94.45 K, being k the Boltzmann’s constant. Those fluid–fluid parameters were chosen to fit the bulk properties of the adsorbate, including liquid–gas surface tension and reference adsorption isotherms on nonporous substrates [8]. The adsorbent is modeled as a collection of independent slit-like pores, with periodic boundary conditions and minimum image convention in the xy plane. Both walls are taken to be the basal plane of a graphite surface, made up of LJ atoms of diameter sss = 0.340 nm hexagonally arranged in each plane [9]. H is defined as the perpendicular distance separating the graphite planes from the center to center of carbon atoms. The interaction of the nitrogen molecules with each wall is calculated by the structureless 10–4–3 potential due to Steele [9]. " 2 s sf 10 2 fwall ðzÞ ¼ 2prss esf ðs sf Þ D 5 z 4 s 4sf s sf (8) z 3Dðz þ 0:61DÞ3 where D = 0.35 nm is the distance between carbon planes, rss = 114 nm3 is the solid density,
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ess = 28.0 K is the LJ solid energy parameter, and the unlike ssf, and esf solid–fluid interaction parameters are calculated following the standard Lorentz–Berthelot combining rules. For a given slit-like pore of width H, the external potential exerted to any molecule inside the pore is given by: fsf ðzÞ ¼ fwall ðzÞ þ fwall ðH zÞ
(9)
We have performed GCMC simulations for four different pore sizes H = 0.8, 1.125, 2.5 and 5.0 nm, at several chemical potentials (i.e. at several pressures). In GCMC, the temperature T, the volumes of the pore V, and the chemical potential of the adsorbate m are kept fixed, while the number of molecules in the pore is allowed to vary. The GCMC simulations provide the number of molecules inside the pore, the location of them, and the different contributions (solid–fluid and fluid–fluid) to the total energy. For simulation convenience, the chemical potential is given in terms of activity j, which is directly related to the pressure. Almost all simulation runs required 2.5 108 configurations to reach the equilibrium. Additional 5 108 configurations were performed for average purposes. Due to difficulties inherent to calculating long-range corrections to inhomogeneous fluids by simulation [10], no long-range corrections were added here; however, care was taken to consider a sufficient large cut-off value to the potential (5 sff). Assuming ideal behavior for the gas during the experiments, the reduced pressure is related to the reduced activities by ( p/ p0 = j/j0). The chosen state point for reduction of activities to pressures is the saturation point of pure nitrogen at 77 K, found to occur at j0 = 0.152 nm3. This result was calculated through a suitable equation of state, the soft-SAFT equation [11], using the nitrogen LJ parameters described above at needed conditions.
Fig. 1, the adsorption/desorption isotherm of an activated carbon obtained at high resolution ( p/ p0 = 107 1) after a prolonged outgassing of the sample at high temperature (623 K) during 24 h, using an ASAP 2010 device. The application of the different linear transformations of the BET equation to this data is presented in Fig. 2. It is observed that Eqs. (4) and (5) provide a linear regression to the data, while the representation of Eqs. (6) and (7) are far from being linear. The non-linear behavior observed for Eqs. (6) and (7) is a clear consequence of the non-constant value of the C parameter as already discussed in our previous work. This can be observed if the C parameter is plotted as directly obtained from the original model (Eq. (3)). Fig. 3 shows the value of C as calculated from Eq. (3), with the monolayer capacity value obtained from Eq. (5). This corresponds to a general behavior previously observed by us when applying this treatment to different materials [4], as already discussed in Section 1. These results give an additional importance to the C parameter as obtained from BET plots, as it can be interpreted as directly related to the energetic heterogeneity of the surface, as a consequence of the adsorption process. We discuss now the results of applying the same procedure as we did to the experimental data to simulated adsorption isotherms. Fig. 4 shows the individual adsorption isotherms obtained by GCMC for four slit-like pores of width H = 0.8, 1.125, 2.5 and 5.0 nm. Following the IUPAC classification, the two narrower pores
4. Results and discussion Since our goal is to provide complementary information on what happens at the molecular level in the adsorption process, and the meaning of the surface parameters as obtained from the BET treatment, we first applied the methodology to some experimental systems. As an example, we represent in
Fig. 1. Experimental nitrogen adsorption/desorption isotherm at 77 K on an activated carbon. Lines are guides to the eyes.
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Fig. 2. Four linear regressions of the BET model to the experimental data presented in Fig. 1. From top to bottom, from left to right, the figures correspond to Eqs. (4)–(7). Lines are guides to the eyes.
Fig. 3. The C parameter of the activated carbon shown in Fig. 1 as calculated from Eq. (3). The figure on the right is a zoom-in over the abscise axis. Lines are guides to the eyes.
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Fig. 4. Individual adsorption isotherms obtained by GCMC for four slit-like pores of width H = 0.8, 1.125, 2.5 and 5.0 nm represented by diamonds, squares, circles and triangles, respectively. Dash lines are guides to the eyes.
selected belong to the microporous region, while the pores of width 2.5 and 5.0 nm, belong to the mesoporous range. The four curves in Fig. 4 show marked differences, indicating that the adsorption process takes place in diverse ways, with a strong dependency on the confinement effect (distance between walls). The points at which adsorption begins to occur in the two microporous are three orders of magnitude apart from each other. The narrowest pore studied here is already filled (around 80% of the total pore adsorption capacity) even when the relative pressure is as low as 107 (high resolution physicsorption equipments have this as lower set point pressure). The adsorption behavior of the two widest pores is very similar at low and intermediate pressures, with multilayer adsorption. At high pressures, a capillary condensation is observed for H = 2.5 nm, while the widest pore (5 nm) shows the behavior of a non-porous surface, indicating that the pore is too
wide for the molecules to see the influence of both walls at the same time. The results of applying the different linear forms of the BET equation (Eqs. (4)–(7)) for the smallest and the widest pore are shown in Figs. 5 and 6, respectively. The same range of reduced pressures was considered in all cases. As observed for the activated carbon presented in Fig. 2, Eqs. (4) and (5) are linear in both cases, while Eqs. (6) and (7) are not linear for the same range of reduced pressures. This is consistent with the previously observed behavior in experimental systems. A possible criticism to this approach is that the BET model is not applicable to micropores, strictly speaking. Although this is true, we have decided to perform the study since it is also true that real materials have several micropores and the BET protocol is equally applied to them. The linear regression obtained by Eqs. (4) and (5) for these data allows obtaining the monolayer capacity value and the C parameter, from the slope and the intercept of the represented line. The particular values obtained for the experimental activated carbon and the two simulated pores (H = 0.8 and 5.0 nm) are presented in Table 1. It should be noted that, in spite of the fact that the simulations are performed on a crude model of carbons, the monolayer values are comparable for the experimental system and the widest pore. This is not the case of the micropore, as expected, since for this narrow pore, the two walls exert a strong attraction to the molecules at the same time and it is not possible to separate the monolayer capacity value from the total amount adsorbed [6]. The values of C are of the same order of magnitude for the experimental system and the widest pore, while the micropore shows a too high value. Let us now take advantage of the fact that the isotherms are obtained by molecular simulations. Instead of calculating nm and C from the BET
Table 1 The C parameter and monolayer capacity as obtained from the linear transformations of the BET model, Eqs. (4) and (5), for three different cases: the experimental material (activated carbon) and two simulated pores, a micropore and a mesopore Experimental
C nm (cm3/g)* nm (molecules/nm2) *
Simulation H = 0.8 nm
Simulation H = 5.0 nm
Eq. (4)
Eq. (5)
Eq. (4)
Eq. (5)
Eq. (4)
Eq. (5)
576 248.02 6.09
593 247.99 6.08
2.34 107 132.82 3.26
2.34 107 132.82 3.26
1530 271.60 6.66
1475 271.84 6.67
Specific surface area (SBET) 1095 m2/g.
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Fig. 5. Four linear regressions to the individual adsorption isotherm shown in Fig. 4 for the slit-like pore of H = 0.8 nm. From top to bottom, from left to right, the figures correspond to Eqs. (4)–(7). Lines are guides to the eyes.
treatment, we will obtain both values directly from the simulations. Fig. 7a shows the total number of molecules inside the pore H = 5.0 nm (squares) and the number of molecules adsorbed in the first layer (circles), as a function of reduced pressure. At relative pressures lower than 0.01, all molecules that get inside the pore reach the surface, hence the total adsorption and the adsorption at the first layer coincide. When p/ p0 = 0.01, the curves begin to fork, the total adsorption inside the pore increases continuously with the pressure, while the number of molecules in the first layer tends to stabilize; there is still a slight increase in adsorption with pressure, as shown in Fig. 7b. These simulations permit us to count the number of molecules within the first layer, providing the value of the monolayer capacity, nm = 6.11 molecules/nm2, obtained at the relative pressure 0.01 in this case. Note that this value is
comparable to 6.66 and 6.67 molecules/nm2, the values obtained by Eqs. (4) and (5). Fig. 8 presents the total energy of the molecules as a function of the total number of molecules inside the pore (number of molecules per nm2), as obtained from GCMC simulations for the pore of 5 nm. Two remarks are in order here: there is a continuous increase in energy as the number of molecules reaching the surface increases, and there is a change in the slope of the curve once the first layer is filled. The point at which this change occurs can be identified as the value of the monolayer capacity, 6.471 0.034 molecules/ nm2. Again, this value agrees, within the error uncertainties, with the value obtained by counting the number of molecules, and from Eqs. (4) and (5). Note that we have provided three different, independent ways of calculating the monolayer capacity, all of which are in excellent agreement with each other.
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Fig. 6. Four linear regressions to the individual adsorption isotherms shown in Fig. 4 for the slit-like pore of H = 5.0 nm. Symbols as in Fig. 5.
Fig. 7. Number of molecules adsorbed as a function of the relative pressure for the pore of H = 5.0 nm. (a) Total number of molecules inside the pore (squares) and number of molecules absorbed in the surfaces or superficial adsorption (circles), forming the first layer. (b) The superficial adsorption, the solid circle stands for the monolayer completion. Bars represent the estimated error over the calculated value. Dash lines are guides to the eyes.
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Fig. 10. The same as in Fig. 9 but for a simulated pore of H = 5.0 nm. Fig. 8. The total potential energy vs. total number of molecules in a simulated pore of H = 5.0 nm. The solid circle marks the change in slope (i.e. the monolayer completion). Dash line is a guide to the eyes.
We have obtained the value of C from two different routes, as directly calculated from Eq. (3), using the value of nm from the simulations, which we represent by Ci, and from the derivative of the energy resulting from the simulations, Cii, which is related to the calorimetric method of the heat evolved by adsorption [12]. Following Rouquerol et al. [12], we use the following expression for Cii: " # l ð˙uexc ðE1 EL Þ T;G uT Þ Cii exp ¼ exp (10) RT RT exc =dnexc Þ where u˙ exc T;V;A is the differential T;G ¼ ðdU surface excess internal energy and ul77 K ¼ 3:4419 kJ=mol is the molar internal energy of bulk liquid nitrogen [13]. Figs. 9 and 10 represent the C
Fig. 9. Variation of the C parameter vs. surface coverage calculated from Eq. (3) (left axis) and obtained from Eq. (10) (right axis), in a simulated pore H = 0.8 nm. Lines are guide to the eyes.
values as obtained from these two procedures for two of the pores considered in this study, H = 0.8 and 5.0 nm. Fig. 9 presents the C value for H = 0.8 nm versus surface coverage. Since this is a very narrow pore, the pore is already filled at very low pressures. The Ci parameter, obtained from Eq. (3), is represented in the left axis; it first increases reaching a maximum at n/nm 0.3 and then decreases, a behavior similar to that observed for the experimental system (see Fig. 3) at very low surface coverage. The shape of the curve can be justified by a competition between enthalpic and entropic effects: the first part of the adsorption process is dominated by the energy gain inside the pore, but once the pore has some molecules inside, there is an entropy lost by getting more molecules inside the pore. The right axis of Fig. 9 shows the same parameter for the same pore, but calculated from Eq. (10). There is not maximum observed in this case, but a continuous increase of the C value. Note, however, that this is an approximate calculation, since we have assumed a1 n2 =a2 n1 ¼ 1. Finally, the C parameter as calculated from the two routes for the widest pore is shown in Fig. 10. This pore corresponds to a case where the influence of one wall is not felt by molecules near the other wall, just the opposite of what happens in the narrow pore discussed above. The C parameter calculated from Eq. (3) (shown in the left axis) presents two maxima, the first one located at n/nm 0.7 and the second one at the point where the monomolecular layer is completed. It should be noted that the particular shape and relative height of the peaks are very
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sensitive to the nm value adopted (not shown here). The curve is in qualitative agreement to that shown in Fig. 3. No quantitative agreement is expected since simulations are performed for independent pores of fixed width, while the experimental material is polydisperse. The value of Cii for H = 5.0 nm is shown in the right axis of Fig. 10, where only one peak is observed, near the point at which the first monolayer is filled.
provides independent and consistent ways of obtaining the monolayer capacity value and the C parameter, including the calculation of energetic effects. An additional advantage of using molecular simulations is that the results from individual adsorption isotherms can be used to construct a kernel to obtain a pore distribution of adsorbent materials based on energetic heterogeneity, a subject of future work.
5. Conclusions
Acknowledgements
The main objective of this work was to apply the BET model, with different linear transformations, to a set of simulated adsorption isotherms, in order to confirm and get some additional insight into previous experimental results. We have applied the GCMC technique to four slit-like pores of different widths, with smooth walls, representing a crude model of activated carbons. Several conclusions can be drawn from the results presented here:
We acknowledge financial support from the Spanish Government (projects PPQ2001-0671, PPQ2003-0471 and CTQ2004-05985-C02-01). C. Herdes acknowledges a postgraduate I3P grant from CSIC-MATGAS Air Products.
The non-linearity observed with some of the applied linearizations of the BET equation shows the nonconstancy of the C factor from the model itself, contrary to what is obtained with the original linearization of the equation as proposed by Brunnaer et al. C varies as a function of surface coverage, a fact known from calorimetric and isosteric measurements of the same parameter. The linear fits obtained in some other linearizations, included the original linear transformation, come from the fact that, even though C is not constant, if C takes a high value, the factor (C 1)/C 1 and 1/C ! 0. It is possible then to get some information of the energetic heterogeneity of the surface just by applying the BET model, but with different linear transformations. The application of the BET model and its linear transformations to GCMC simulated adsorption isotherms has shown similar results to those obtained on different materials. In addition, GCMC
References [1] S. Brunauer, P.H. Emmett, E. Teller, J. Am. Chem. Soc. 60 (1938) 309. [2] A. Seri-Levy, D. Avnir, Langmuir 9 (1993) 2523. [3] S.J. Gregg, K.S.W. Sing, Adsorption, Surface Area and Porosity, Academic Press, London, 1982. [4] F. Salvador, C. Sa´nchez-Jime´nez, M.J. Sa´nchez-Montero, A. Salvador, Stud. Surf. Sci. Catal. 144 (2002) 379. [5] D. Nicholson, N.G. Parsonage, Computer Simulation and Statistical Mechanics of Adsorption, Academic Press, London, 1982. [6] M. Heuchel, M. Jaroniec, Langmuir 11 (1995) 4532. [7] J.B. Parra, J.C. de Sousa, R.C. Bansal, J.J. Pis, J.A. Pajares, Adsorpt. Sci. Tech. 11 (1994) 51. [8] A.V. Neimark, P.I. Ravikovitch, M. Gru¨n, F. Schu¨th, K.K. Unger, J. Colloid Interface Sci. 207 (1998) 159. [9] W.A. Steele, Surf. Sci. 36 (1973) 317; W.A. Steele, The Interaction of Gases with Solid Surfaces, Pergamon, Oxford, 1974, p. 56. [10] D. Duque, L.F. Vega, J. Chem. Phys. 121 (2004) 8611. [11] J.C. Pa`mies, L.F. Vega, Ind. Eng. Chem. Res. 40 (2001) 2532. [12] F. Rouquerol, J. Rouquerol, K. Sing, Adsorption by Powders and Porous Solids, Academic Press, London, 1999. [13] NIST Chemistry WebBook, Saturation properties of nitrogen at 77 K, NIST Standard Reference Database Number 69, March 2003 Release, http://webbook.nist.gov/cgi.
New insights into the adsorption isotherm interpretation by a coupled molecular simulation—experimental procedure M.J. Sa´nchez-Montero a, C. Herdes b, F. Salvador a, L.F. Vega b,* a
b
Departamento de Quı´mica-Fı´sica, Universidad de Salamanca, 37008 Salamanca, Spain Institut de Cie`ncia de Materials de Barcelona (ICMAB-CSIC), Campus de la UA, 08193 Bellaterra, Barcelona, Spain Available online 14 March 2005
Abstract We present here Grand Canonical Monte Carlo (GCMC) simulation results of nitrogen adsorption at 77 K on a crude model of activated carbon. The material is modeled as slit-like pores of different widths, with smooth surfaces. The individual adsorption isotherms serve as the basis to check the success and limitations of the assumptions made when using the BET model to characterize adsorbent materials, in particular to calculate the monolayer capacity and the C parameter. As done in our previous work with several experimental adsorption isotherms, different linearizations of the BETequation are used. The aim of this work is to quantify, using statistical mechanics tools, the changes in the C factor with surface coverage, showing that C is an intrinsically energetic meaningful quantity. The amount of molecules adsorbed at each pressure is calculated in the first and subsequent layers. We also keep track of the adsorbent–adsorbate and adsorbate–adsorbate energy along the simulations. The C factor is obtained following two different routes: as directly derived from the BET equation, once the monolayer capacity is known, and from the heat of adsorption obtained directly from the simulations. Results from simulations confirm the changes in the C values with surface coverage. In addition, molecular simulations provide independent and consistent ways of calculating the monolayer capacity. # 2005 Elsevier B.V. All rights reserved. Keywords: Nitrogen adsorption isotherms; Activated carbons; BET theory; Molecular simulations; Energetic heterogeneity; Monolayer capacity
1. Introduction A routine task performed before any adsorbent or catalytic material can be put into use is, among others, its textural characterization. Although it is true that a long effort has been devoted towards the development * Corresponding author. Tel.: +34 93 580 1853; fax: +34 93 580 5729. E-mail address: [email protected] (L.F. Vega).
of reliable methods to interpret the adsorption isotherm as a standard procedure to characterize materials, the most widely used method is still the BET model, due to Brunauer et al. [1]. The theory is based on kinetic arguments and it assumes a reversible adsorption/desorption process with multilayer formation. The BET model owes its widespread use to the fact that it is straightforward to apply and a linear transformation of the original equation, as proposed by the authors, provides two of the most meaningful
0169-4332/$ – see front matter # 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.apsusc.2005.02.068
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textural quantities, the monolayer capacity and the C parameter, giving fairly acceptable results for several solids. In spite of its success, BET is also one of the most criticized theories since its inception [2]. The main criticism of the BET model comes from two of the basic (inaccurate) assumptions made when developing it: (1) all adsorbing sites are energetically homogeneous; and (2) only vertical interactions between adsorbed molecules and the adsorbing surface take place, hence ignoring any lateral interaction. There have been several attempts to correct one or the two assumptions in order to improve the BET model, however, there is not general consensus on an accurate method to substitute it for surface area calculations [3]. Searching for a better understanding of the approximations behind the model, Seri-Levy and Avnir [2] performed Monte Carlo simulations of the BET conditions, finding excellent agreement with the model predictions. This allowed them to include lateral interactions into the simulations, identifying and analyzing deviations from the BET model. These results proved that the model is accurate under the conditions at which it was developed; however, it is clear that the model is applied most of the time for cases for which the assumptions made are too crude and far away from the experimental conditions. Hence, it would be desirable to have an independent test to check the accuracy of the textural properties obtained by applying the BET model to different real materials. In a recent work by some of us [4], we reviewed the BET plot proposing other possible linear transformations of the original equation, applying them to several different adsorbent materials. The main conclusions from that work were that the linear transformations of the BET equation can be classified into two categories, those unable to provide a linear regression, in which the energetic heterogeneity in the adsorption data is revealed, showing a C parameter changing with surface coverage; and those, to which the original transformation proposed by the authors belongs to, presenting a linear regression as a consequence of the high values taken by C, which neutralize the energetic heterogeneity of the adsorption data. A striking consequence of this work was that the energetic meaning of C can be inferred just by using different linear transformations of the original equation, without undertaking other more refined methods (i.e.
calorimetric methods). However, these results should be confirmed by independent techniques. It is proved now that molecular simulations provide an excellent tool in which to perform systematic studies towards the understanding of surface science [5]. They allow isolating and quantifying the effect of the different variables into the global behavior of the system. In this work, we use Grand Canonical Monte Carlo (GCMC) simulation results of adsorption on model surfaces as the ‘‘experimental’’ isotherms at which the BET treatment can be applied. Previous work done by Heuchel and Jaroniec [6] used GCMC adsorption isotherms to explore the possibility of extracting information about surface and structural heterogeneity of microporous solids from experimental data. These authors focused on evaluating the adsorption energy distribution function as a function of the degree of pore filling and pore width, pointing out that additional simulation studies are needed to understand the relationship between microporosity and surface heterogeneity of porous solids. Our goal here is to use the GCMC simulation results to get some insight into the values obtained when the BET model, with different linear transformations, is applied to some experimental isotherms. Since the simulations give independent ways of calculating the two quantities provided by BET, nm and C, they also allow validating, or not, the values obtained from the direct application of the model. It should be clear that our objective is not to improve the BET model, but to use it, as it is, and to understand the implications of applying it from a molecular perspective. The rest of the paper is organized as follows: we first present a summary of the BET model and some possible linearizations. In Section 3, we present the simulation details and the molecular model used in the calculations. Results are presented and discussed in Section 4; finally, some concluding remarks are presented in Section 5.
2. The BET model The original equation proposed by Brunauer et al. is [1]: n Cx ¼ nm ð1 xÞð1 x þ CxÞ
(1)
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where n represents the moles of gas adsorbed at the equilibrium pressure p; nm is the number of moles of gas adsorbed when the surface is covered by a complete monomolecular layer (the monolayer capacity); x is the relative pressure ( p/ p0), p0 is the vapor pressure of the adsorbate. According to the model, C is defined as: a1 n2 E1 EL C¼ exp (2) a2 n1 RT where a1 and a2 are the condensation coefficients for the first and second layers; n1 and n2 are the frequencies of oscillation of the molecule in a direction normal to the surface; E1 is the heat of adsorption of the first layer, and EL is the liquefaction heat of the adsorbate. The term (E1 EL) has been defined in some confusing ways in the literature; the accepted name now is the ‘‘net molar energy of adsorption’’. The C factor can be also obtained from the original model (Eq. (1)) for each surface coverage, once the monolayer capacity is known: C¼
nð1 xÞ2 x½nm nð1 xÞ
(3)
To check the validity of their model, the authors proposed the transformation of Eq. (1) into the linear form: x 1 C1 ¼ þ x nð1 xÞ nm C nm C
(4)
This linearization provides a direct way of obtaining the monolayer capacity value nm and the C factor, from the slope and the intercept of the line represented by this equation. Brunauer et al. observed that the linearity of the BET representation was reduced to a small range ( p/ p0 = 0.05 0.3). Later studies have shown that for many systems that range is much more reduced, although there is still no criterion to select the range. Moreover, for Type III and V isotherms, the BET plot is never linear. One also finds the paradox that in general, linearity occurs at low coverage values, where the energetic heterogeneity, not taken into account into the model, is higher. Although the most generalized way of using the BET model is through the linear form proposed by the authors (Eq. (3)); there are other possible linear transformations [4,7] of the original Eq. (1) that can be
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used to check the BET model: x nð1 xÞ
2
¼
1 1 x þ nm C nm 1 x
nð1 xÞ2 ¼ nm C Cnð1 xÞ x 1 1 1 1x ¼ þ nð1 xÞ nm nm C x
(5)
(6) (7)
In fact, as pointed out by some authors [4,7], some of these equations have clear advantages over the original transformation; for instance, Eq. (5) provides the value of nm directly from the slope of the line, with no error associated to the intercept. Some authors use Eq. (7) as an alternative to the original linearization, as discussed in reference [7].
3. Molecular model and simulations We use the Grand Canonical Monte Carlo technique to obtain adsorption isotherms in which the BET model can be applied and discussed. Nitrogen molecules are modeled as 12–6 Lennard–Jones (LJ) spheres, with parameter values sff = 0.3575 nm and eff/k = 94.45 K, being k the Boltzmann’s constant. Those fluid–fluid parameters were chosen to fit the bulk properties of the adsorbate, including liquid–gas surface tension and reference adsorption isotherms on nonporous substrates [8]. The adsorbent is modeled as a collection of independent slit-like pores, with periodic boundary conditions and minimum image convention in the xy plane. Both walls are taken to be the basal plane of a graphite surface, made up of LJ atoms of diameter sss = 0.340 nm hexagonally arranged in each plane [9]. H is defined as the perpendicular distance separating the graphite planes from the center to center of carbon atoms. The interaction of the nitrogen molecules with each wall is calculated by the structureless 10–4–3 potential due to Steele [9]. " 2 s sf 10 2 fwall ðzÞ ¼ 2prss esf ðs sf Þ D 5 z 4 s 4sf s sf (8) z 3Dðz þ 0:61DÞ3 where D = 0.35 nm is the distance between carbon planes, rss = 114 nm3 is the solid density,
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ess = 28.0 K is the LJ solid energy parameter, and the unlike ssf, and esf solid–fluid interaction parameters are calculated following the standard Lorentz–Berthelot combining rules. For a given slit-like pore of width H, the external potential exerted to any molecule inside the pore is given by: fsf ðzÞ ¼ fwall ðzÞ þ fwall ðH zÞ
(9)
We have performed GCMC simulations for four different pore sizes H = 0.8, 1.125, 2.5 and 5.0 nm, at several chemical potentials (i.e. at several pressures). In GCMC, the temperature T, the volumes of the pore V, and the chemical potential of the adsorbate m are kept fixed, while the number of molecules in the pore is allowed to vary. The GCMC simulations provide the number of molecules inside the pore, the location of them, and the different contributions (solid–fluid and fluid–fluid) to the total energy. For simulation convenience, the chemical potential is given in terms of activity j, which is directly related to the pressure. Almost all simulation runs required 2.5 108 configurations to reach the equilibrium. Additional 5 108 configurations were performed for average purposes. Due to difficulties inherent to calculating long-range corrections to inhomogeneous fluids by simulation [10], no long-range corrections were added here; however, care was taken to consider a sufficient large cut-off value to the potential (5 sff). Assuming ideal behavior for the gas during the experiments, the reduced pressure is related to the reduced activities by ( p/ p0 = j/j0). The chosen state point for reduction of activities to pressures is the saturation point of pure nitrogen at 77 K, found to occur at j0 = 0.152 nm3. This result was calculated through a suitable equation of state, the soft-SAFT equation [11], using the nitrogen LJ parameters described above at needed conditions.
Fig. 1, the adsorption/desorption isotherm of an activated carbon obtained at high resolution ( p/ p0 = 107 1) after a prolonged outgassing of the sample at high temperature (623 K) during 24 h, using an ASAP 2010 device. The application of the different linear transformations of the BET equation to this data is presented in Fig. 2. It is observed that Eqs. (4) and (5) provide a linear regression to the data, while the representation of Eqs. (6) and (7) are far from being linear. The non-linear behavior observed for Eqs. (6) and (7) is a clear consequence of the non-constant value of the C parameter as already discussed in our previous work. This can be observed if the C parameter is plotted as directly obtained from the original model (Eq. (3)). Fig. 3 shows the value of C as calculated from Eq. (3), with the monolayer capacity value obtained from Eq. (5). This corresponds to a general behavior previously observed by us when applying this treatment to different materials [4], as already discussed in Section 1. These results give an additional importance to the C parameter as obtained from BET plots, as it can be interpreted as directly related to the energetic heterogeneity of the surface, as a consequence of the adsorption process. We discuss now the results of applying the same procedure as we did to the experimental data to simulated adsorption isotherms. Fig. 4 shows the individual adsorption isotherms obtained by GCMC for four slit-like pores of width H = 0.8, 1.125, 2.5 and 5.0 nm. Following the IUPAC classification, the two narrower pores
4. Results and discussion Since our goal is to provide complementary information on what happens at the molecular level in the adsorption process, and the meaning of the surface parameters as obtained from the BET treatment, we first applied the methodology to some experimental systems. As an example, we represent in
Fig. 1. Experimental nitrogen adsorption/desorption isotherm at 77 K on an activated carbon. Lines are guides to the eyes.
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Fig. 2. Four linear regressions of the BET model to the experimental data presented in Fig. 1. From top to bottom, from left to right, the figures correspond to Eqs. (4)–(7). Lines are guides to the eyes.
Fig. 3. The C parameter of the activated carbon shown in Fig. 1 as calculated from Eq. (3). The figure on the right is a zoom-in over the abscise axis. Lines are guides to the eyes.
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Fig. 4. Individual adsorption isotherms obtained by GCMC for four slit-like pores of width H = 0.8, 1.125, 2.5 and 5.0 nm represented by diamonds, squares, circles and triangles, respectively. Dash lines are guides to the eyes.
selected belong to the microporous region, while the pores of width 2.5 and 5.0 nm, belong to the mesoporous range. The four curves in Fig. 4 show marked differences, indicating that the adsorption process takes place in diverse ways, with a strong dependency on the confinement effect (distance between walls). The points at which adsorption begins to occur in the two microporous are three orders of magnitude apart from each other. The narrowest pore studied here is already filled (around 80% of the total pore adsorption capacity) even when the relative pressure is as low as 107 (high resolution physicsorption equipments have this as lower set point pressure). The adsorption behavior of the two widest pores is very similar at low and intermediate pressures, with multilayer adsorption. At high pressures, a capillary condensation is observed for H = 2.5 nm, while the widest pore (5 nm) shows the behavior of a non-porous surface, indicating that the pore is too
wide for the molecules to see the influence of both walls at the same time. The results of applying the different linear forms of the BET equation (Eqs. (4)–(7)) for the smallest and the widest pore are shown in Figs. 5 and 6, respectively. The same range of reduced pressures was considered in all cases. As observed for the activated carbon presented in Fig. 2, Eqs. (4) and (5) are linear in both cases, while Eqs. (6) and (7) are not linear for the same range of reduced pressures. This is consistent with the previously observed behavior in experimental systems. A possible criticism to this approach is that the BET model is not applicable to micropores, strictly speaking. Although this is true, we have decided to perform the study since it is also true that real materials have several micropores and the BET protocol is equally applied to them. The linear regression obtained by Eqs. (4) and (5) for these data allows obtaining the monolayer capacity value and the C parameter, from the slope and the intercept of the represented line. The particular values obtained for the experimental activated carbon and the two simulated pores (H = 0.8 and 5.0 nm) are presented in Table 1. It should be noted that, in spite of the fact that the simulations are performed on a crude model of carbons, the monolayer values are comparable for the experimental system and the widest pore. This is not the case of the micropore, as expected, since for this narrow pore, the two walls exert a strong attraction to the molecules at the same time and it is not possible to separate the monolayer capacity value from the total amount adsorbed [6]. The values of C are of the same order of magnitude for the experimental system and the widest pore, while the micropore shows a too high value. Let us now take advantage of the fact that the isotherms are obtained by molecular simulations. Instead of calculating nm and C from the BET
Table 1 The C parameter and monolayer capacity as obtained from the linear transformations of the BET model, Eqs. (4) and (5), for three different cases: the experimental material (activated carbon) and two simulated pores, a micropore and a mesopore Experimental
C nm (cm3/g)* nm (molecules/nm2) *
Simulation H = 0.8 nm
Simulation H = 5.0 nm
Eq. (4)
Eq. (5)
Eq. (4)
Eq. (5)
Eq. (4)
Eq. (5)
576 248.02 6.09
593 247.99 6.08
2.34 107 132.82 3.26
2.34 107 132.82 3.26
1530 271.60 6.66
1475 271.84 6.67
Specific surface area (SBET) 1095 m2/g.
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Fig. 5. Four linear regressions to the individual adsorption isotherm shown in Fig. 4 for the slit-like pore of H = 0.8 nm. From top to bottom, from left to right, the figures correspond to Eqs. (4)–(7). Lines are guides to the eyes.
treatment, we will obtain both values directly from the simulations. Fig. 7a shows the total number of molecules inside the pore H = 5.0 nm (squares) and the number of molecules adsorbed in the first layer (circles), as a function of reduced pressure. At relative pressures lower than 0.01, all molecules that get inside the pore reach the surface, hence the total adsorption and the adsorption at the first layer coincide. When p/ p0 = 0.01, the curves begin to fork, the total adsorption inside the pore increases continuously with the pressure, while the number of molecules in the first layer tends to stabilize; there is still a slight increase in adsorption with pressure, as shown in Fig. 7b. These simulations permit us to count the number of molecules within the first layer, providing the value of the monolayer capacity, nm = 6.11 molecules/nm2, obtained at the relative pressure 0.01 in this case. Note that this value is
comparable to 6.66 and 6.67 molecules/nm2, the values obtained by Eqs. (4) and (5). Fig. 8 presents the total energy of the molecules as a function of the total number of molecules inside the pore (number of molecules per nm2), as obtained from GCMC simulations for the pore of 5 nm. Two remarks are in order here: there is a continuous increase in energy as the number of molecules reaching the surface increases, and there is a change in the slope of the curve once the first layer is filled. The point at which this change occurs can be identified as the value of the monolayer capacity, 6.471 0.034 molecules/ nm2. Again, this value agrees, within the error uncertainties, with the value obtained by counting the number of molecules, and from Eqs. (4) and (5). Note that we have provided three different, independent ways of calculating the monolayer capacity, all of which are in excellent agreement with each other.
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Fig. 6. Four linear regressions to the individual adsorption isotherms shown in Fig. 4 for the slit-like pore of H = 5.0 nm. Symbols as in Fig. 5.
Fig. 7. Number of molecules adsorbed as a function of the relative pressure for the pore of H = 5.0 nm. (a) Total number of molecules inside the pore (squares) and number of molecules absorbed in the surfaces or superficial adsorption (circles), forming the first layer. (b) The superficial adsorption, the solid circle stands for the monolayer completion. Bars represent the estimated error over the calculated value. Dash lines are guides to the eyes.
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Fig. 10. The same as in Fig. 9 but for a simulated pore of H = 5.0 nm. Fig. 8. The total potential energy vs. total number of molecules in a simulated pore of H = 5.0 nm. The solid circle marks the change in slope (i.e. the monolayer completion). Dash line is a guide to the eyes.
We have obtained the value of C from two different routes, as directly calculated from Eq. (3), using the value of nm from the simulations, which we represent by Ci, and from the derivative of the energy resulting from the simulations, Cii, which is related to the calorimetric method of the heat evolved by adsorption [12]. Following Rouquerol et al. [12], we use the following expression for Cii: " # l ð˙uexc ðE1 EL Þ T;G uT Þ Cii exp ¼ exp (10) RT RT exc =dnexc Þ where u˙ exc T;V;A is the differential T;G ¼ ðdU surface excess internal energy and ul77 K ¼ 3:4419 kJ=mol is the molar internal energy of bulk liquid nitrogen [13]. Figs. 9 and 10 represent the C
Fig. 9. Variation of the C parameter vs. surface coverage calculated from Eq. (3) (left axis) and obtained from Eq. (10) (right axis), in a simulated pore H = 0.8 nm. Lines are guide to the eyes.
values as obtained from these two procedures for two of the pores considered in this study, H = 0.8 and 5.0 nm. Fig. 9 presents the C value for H = 0.8 nm versus surface coverage. Since this is a very narrow pore, the pore is already filled at very low pressures. The Ci parameter, obtained from Eq. (3), is represented in the left axis; it first increases reaching a maximum at n/nm 0.3 and then decreases, a behavior similar to that observed for the experimental system (see Fig. 3) at very low surface coverage. The shape of the curve can be justified by a competition between enthalpic and entropic effects: the first part of the adsorption process is dominated by the energy gain inside the pore, but once the pore has some molecules inside, there is an entropy lost by getting more molecules inside the pore. The right axis of Fig. 9 shows the same parameter for the same pore, but calculated from Eq. (10). There is not maximum observed in this case, but a continuous increase of the C value. Note, however, that this is an approximate calculation, since we have assumed a1 n2 =a2 n1 ¼ 1. Finally, the C parameter as calculated from the two routes for the widest pore is shown in Fig. 10. This pore corresponds to a case where the influence of one wall is not felt by molecules near the other wall, just the opposite of what happens in the narrow pore discussed above. The C parameter calculated from Eq. (3) (shown in the left axis) presents two maxima, the first one located at n/nm 0.7 and the second one at the point where the monomolecular layer is completed. It should be noted that the particular shape and relative height of the peaks are very
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sensitive to the nm value adopted (not shown here). The curve is in qualitative agreement to that shown in Fig. 3. No quantitative agreement is expected since simulations are performed for independent pores of fixed width, while the experimental material is polydisperse. The value of Cii for H = 5.0 nm is shown in the right axis of Fig. 10, where only one peak is observed, near the point at which the first monolayer is filled.
provides independent and consistent ways of obtaining the monolayer capacity value and the C parameter, including the calculation of energetic effects. An additional advantage of using molecular simulations is that the results from individual adsorption isotherms can be used to construct a kernel to obtain a pore distribution of adsorbent materials based on energetic heterogeneity, a subject of future work.
5. Conclusions
Acknowledgements
The main objective of this work was to apply the BET model, with different linear transformations, to a set of simulated adsorption isotherms, in order to confirm and get some additional insight into previous experimental results. We have applied the GCMC technique to four slit-like pores of different widths, with smooth walls, representing a crude model of activated carbons. Several conclusions can be drawn from the results presented here:
We acknowledge financial support from the Spanish Government (projects PPQ2001-0671, PPQ2003-0471 and CTQ2004-05985-C02-01). C. Herdes acknowledges a postgraduate I3P grant from CSIC-MATGAS Air Products.
The non-linearity observed with some of the applied linearizations of the BET equation shows the nonconstancy of the C factor from the model itself, contrary to what is obtained with the original linearization of the equation as proposed by Brunnaer et al. C varies as a function of surface coverage, a fact known from calorimetric and isosteric measurements of the same parameter. The linear fits obtained in some other linearizations, included the original linear transformation, come from the fact that, even though C is not constant, if C takes a high value, the factor (C 1)/C 1 and 1/C ! 0. It is possible then to get some information of the energetic heterogeneity of the surface just by applying the BET model, but with different linear transformations. The application of the BET model and its linear transformations to GCMC simulated adsorption isotherms has shown similar results to those obtained on different materials. In addition, GCMC
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