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Nitrogen Physisorption on a Columnar Carbonaceous Solid Eduardo J. Bottani* Instituto de Investigaciones Fisicoquı´micas Teo´ ricas y Aplicadas (INIFTA-UNLP-CIC-CONICET), Casilla de Correo 16, Sucursal 4 (1900), La Plata, Argentina Received September 25, 2000. In Final Form: January 5, 2001 Grand canonical ensemble Monte Carlo computer simulation is employed to obtain the adsorption isotherms of nitrogen, at 77.5 K, on a columnar carbonaceous material. The solid constructed is based on the Bernal model. At moderate equilibrium pressures, the intercolumnar space is filled with nitrogen in a liquidlike state. The distributions of molecules with respect to gas-solid and gas-gas interaction energies are analyzed. The role played by quadrupolar interactions is discussed. The supersite theory of adsorption applied to this system gives a good description of the simulation results.
Introduction Some time ago, systems showing columnar structures were prepared and studied, for example, gold deposits on glass substrates (see for example refs 1-4). The chemisorption of pyridine has been studied on gold electrodes with columnar structures.5 In principle, columnar structures result during material deposition when surface diffusion cannot balance other effects characteristic of the system.6,7 More recently, columnar structures have been observed during the preparation of carbon nanotubes. In fact, during the synthesis different structures that are rich in multiwalled nanotubes are observed as in the case of carbon deposits obtained by arc-discharge. These nanotubes grow, under appropriate conditions, forming a deposit with a highly ordered columnar structure.8,9 It has also been observed that the columns form a regular hexagonal array.10 In this paper, we present the results of a series of grand canonical ensemble Monte Carlo (GCMC) computer simulations of nitrogen physisorption on a carbon columnar solid at 77.5 K. The solid consists of a regular array of amorphous carbon columns. Several features of nitrogen adsorption at 77.5 K will be presented. The effect of adsorbate-adsorbate interactions on the adsorption process and the structure of the adsorbed phase will be discussed. Local adsorption isotherms obtained on the intercolumnar regions are analyzed with the aid of the supersite adsorption theory. The adsorption energy dis* E-mail:
[email protected]. Fax: 54-221-425-4642. (1) Go´mez-Rodrı´guez, J. H.; Baro´, A. M.; Salvarezza, R. C. J. Vac. Sci. Technol., B 1991, 9, 495. (2) Herrasti, P.; Oco´n, P.; Va´zquez, L.; Salvarezza, R. C.; Vara, J. M.; Arvia, A. J. Phys. Rev. A 1992, 45, 7440. (3) Salvarezza, R. C.; Va´zquez, L.; Herrasti, P.; Oco´n, P.; Vara, J. M.; Arvia, A. J. Europhys. Lett. 1992, 20, 727. (4) Tong, W. M.; Snyder, E. J.; Williams, R. S.; Yanase, A.; Segawa, Y.; Anderson, M. S. Surf. Sci. Lett. 1992, 277, L63. (5) Go´mez, M. M.; Vara, J. M.; Herna´ndez, J. C.; Salvarezza, R. C.; Arvia, A. J. J. Electroanal. Chem. 1999, 474, 74. (6) Baraba´si, A. L.; Stanley, H. E. In Fractal Concepts in Surface Growth; Cambridge University Press: New York, 1995; Chapter 19. (7) Buzio, R.; Gnecco, E.; Boragno, C.; Valbusa, U.; Piseri, P.; Barborini, E.; Milani, P. Surf. Sci. 2000, 444, L1. (8) Kiselev, N. A.; Moravsky, A. P.; Ormont, A. B.; Zakharov, D. N. Carbon 1999, 37, 1093. (9) Saito, Y.; Uemura, S. Carbon 2000, 38, 169. (10) Colbert, D. T.; Zhang, J.; McClure, S. M.; Nikolaev, P.; Chen, Z.; Hafner, J. H.; Owens, D. W.; Kotula, P. G.; Carter, C. B.; Weaver, J. H.; Rinzler, A. G.; Smalley, R. E. Science 1994, 266, 1218.
Figure 1. Schematic representation of the columnar structure generated. The shaded circles represent the columns and are in scale.
tribution function has been calculated from the simulated isotherms employing a constrained regularization method.11,12 Technical Details The model solid contains 5504 carbon atoms, and the apparent density is 2.2 g/mL. The solid is built in three steps. First, an amorphous carbon is constructed as described elsewhere.13 On a copy of this surface, 40 columns with a diameter of 6.2 Å are defined in a regular hexagonal array. Then, all atoms falling outside the columns are deleted down to a depth of ca. 20 Å. A schematic representation of the columnar array is shown in Figure 1. The circles representing the columns are in scale. Finally, the solid is constructed by adding the modified structure on top of the original one. The result is a columnar structure with an amorphous surface. The geometric area of the simulation box is 2778.3 Å2. Periodic boundary conditions have been established in the x and y directions. A reflection plane was kept at different heights from the top of the columns, to optimize the simulations. These heights ranged from 3σN-N up to 12σN-N for the highest equilibrium pressures, where σN-N is the LennardJones distance parameter for nitrogen-nitrogen interactions. The simulation algorithm has been described elsewhere.14 Each point of the simulation is the result of 1.5 × 109 movement and (11) Provencher, S. W. Comput. Phys. Commun. 1982, 27, 213. (12) Provencher, S. W. Comput. Phys. Commun. 1982, 27, 229. (13) Cascarini de Torre, L. E.; Bottani, E. J. Langmuir 1995, 11, 221. (14) Martı´nez-Alonso, A.; Tasco´n, J. M. D.; Bottani, E. J. Langmuir 2000, 16, 1343.
10.1021/la0013648 CCC: $20.00 © 2001 American Chemical Society Published on Web 03/27/2001
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The first characteristic observed in the simulated isotherms, shown in Figure 2, is that the isotherms are fully reversible. To study the effect of the interactions between adsorbed molecules, isotherms (adsorptiondesorption) have been calculated including dispersion and quadrupolar terms (referred to as full interaction) and neglecting quadrupole-quadrupole interaction, and finally one set has been calculated using a hard-sphere potential to calculate gas-gas interactions. The hardsphere potential just avoids the case in which two molecules approach closer than the “collision” diameter; in our case, it has been made equal to the molecular diameter. The isotherms calculated neglecting quadrupolar interactions are coincident with the ones that include them. This fact is quite unexpected, at least in the multilayer region where this interaction can represent up to 20% of the total lateral interaction. This behavior can be explained by taking into consideration the structure of the solid. In fact, the adsorbed molecules are restricted to the intercolumnar space, and thus there are fewer
molecules within the interaction sphere of each one than in other solids (porous or nonporous). The isotherms obtained with a hard-sphere potential and neglecting the quadrupolar interaction are also shown in Figure 2. The shape and relative position of the adsorption isotherms depicted in Figure 2 are in good agreement with the gas-gas interaction potential employed in each case. In fact, the isotherms calculated including the nitrogen quadrupole moment are coincident with the ones calculated without this contribution. The simulations where the gas-gas interactions have been suppressed (hard-sphere potential) produced larger amounts of adsorbed gas at each equilibrium pressure. As could be expected, this model cannot produce the adsorbate condensation; this is why a plateau is reached at sufficiently high pressures. Neither of these isotherms show hysteresis. The Brunauer-Emmett-Teller (BET) surface area obtained from the simulated isotherm, including full interactions, is 4610.2 Å2, that is, 1.7 times the geometric area of the simulation box. Figure 3 depicts the total potential, gas-gas, and gassolid energies obtained with and without quadrupolar interaction and with the hard-sphere potential. The points corresponding to the systems with and without quadrupolar interactions fall on the same curve as could be expected from the adsorption isotherms. The total potential energy profiles can be compared with the results obtained for nitrogen adsorption on amorphous carbons13 and porous amorphous carbons.16 As could be expected, columnar and porous solids show the same total energy values that are larger than the ones obtained for nonporous amorphous carbons. When gas-gas interactions are switched off (hardsphere potential), gas-solid energies are larger (see Figure 3). This fact is a direct consequence of the gas-gas potential adopted. In fact, the hard-sphere potential allows the adsorbed molecules to be almost in touch with their nearest neighbors, adopting configurations that are normally energetically unfavorable. The density profiles depicted in Figure 4 that have been calculated with full interactions, no quadrupole interaction, and hard-sphere potential clearly show this effect. The number of adsorbed molecules is almost the same for the three curves shown. Again, the lack of a major effect of quadrupolar interactions is observed because the corresponding profile is almost coincident with the one calculated with full interactions. The profile obtained with the hard-sphere potential indicates that almost all the adsorbed molecules are in the intercolumnar spaces. The calculated average density for molecules in intercolumnar space is 0.015 molecule/Å3 when full gas-gas interactions are included in the simulation. This value agrees with liquid nitrogen experimental density (0.017 molecule/Å3). On the other hand, the density obtained with the hardsphere potential is 0.03 molecule/Å3, which represents a phase denser than the real solid nitrogen. The tops of the columns are located on average at z/σ ≈ 6, where σ is the Lennard-Jones gas-solid potential’s distance parameter. When gas-gas interactions are included in the simulation, the density profiles show details of the surface that are not observed when the hardsphere potential is employed. The peak located at z/σ < 4 corresponds to the adsorption on the bottom of the intercolumnar spaces. Peaks between z/σ > 4 and z/σ ) 6 are due to the adsorption on the walls of the columns
(15) Cascarini de Torre, L. E.; Flores, E. S.; Llanos, J. L.; Bottani, E. J. Langmuir 1995, 11, 4742.
(16) Cascarini de Torre, L. E.; Bottani, E. J. Langmuir 1997, 13, 2499.
Figure 2. Simulated nitrogen adsorption isotherms at 77.5 K. Open symbols correspond to adsorption and filled symbols to desorption: full interactions (O), hard-sphere potential (0), and without quadrupolar interaction (4). creation/destruction attempts, except for the first point where 4.5 × 109 attempts were employed. The maximum displacement and rotation have been defined to have 40-60% of successful movements (displacement/rotation) and to have 1-3% of successful creation/destruction attempts. Gas-solid interactions were modeled via a 12-6 LennardJones potential as in previous papers,13 employing the same values for the interaction parameters: N-C ) 34.65 K and σN-C ) 3.36 Å. These parameters have been calculated using the standard Lorentz-Bethelot combining rules. Nitrogen-carbon well depth has been increased by 10% to reproduce the experimental adsorption isotherms.13 Later, these values were confirmed through a virial analysis of the adsorption isotherms.15 The nitrogen molecule is simulated as a set of two LennardJones spherical interaction sites with N-N ) 36.4 K and σN-N ) 3.32 Å.13 The quadrupole moment of the molecule has been simulated by placing a positive charge (12.98 × 10-20 C) at the symmetry center of the molecule and one negative charge (6.49 × 10-20 C) on each nitrogen atom. The obtained quadrupole moment reproduces the experimental one. The calculated interaction energy between adsorbed molecules (gas-gas or lateral interactions) is the result of adding two terms, one that represents the dispersion energy and an electrostatic one that models the quadrupolar interaction.
Results and Discussion
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Figure 3. Total potential energy with full interactions (O) and without quadrupolar interaction ()), gas-gas energy with full interactions (0) and without quadrupolar interaction ("), and gas-solid energies with full interactions (4), with hard-sphere potential (3), and without quadrupolar interaction (*).
Figure 4. Density profiles obtained at approximately the same equilibrium pressure with full interactions (thick continuous line), without quadrupolar interaction (thin continuous line), and with hard-sphere potential (broken line).
Figure 5. Distributions of molecules with respect to gassolid energy with full lateral interactions (thick line) and with the hard-sphere potential (thin line).
themselves. At this point of the isotherm, the intercolumnar space is filled with adsorbate and the adsorption continues as in the case of an ordinary amorphous surface. Integration of the density profile between z/σ ) 6.8 and z/σ ) 7.9 gives the number of adsorbed molecules in a layer directly on top of the columns which in turn is equivalent to an area of 2487 Å2. This value is very close to the geometric area of the cell. Figure 5 shows the distributions of adsorbed molecules with respect to their gas-solid energy obtained from simulations with and without gas-gas interactions. The number of adsorbed molecules is approximately the same for both curves. The range of gas-solid energy covered by both distributions is the same; the main difference is in the peak located at zero energy. In fact, a large number
of molecules are adsorbed on higher layers because the interactions between adsorbed molecules are limiting the density of the adsorbed phase in the intercolumnar space, which is not the case when gas-gas interactions are described with the hard-sphere potential. Figure 6 shows a map of the adsorption energy of a nitrogen molecule in which the regions with the largest energies have been blackened. The energy in those areas is between -30 and -12 kJ/mol, and all correspond to intercolumnar regions. The blackened regions are very different in shape, and no regularities are observed as could be expected for a random heterogeneous surface. This point will be retaken later on. Figure 7 shows the distribution of molecules with respect to the gas-gas energy obtained at low, moderate, and
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Figure 8. Distribution of molecules adsorbed on two of the blackened areas of Figure 6 marked as 1 (thin line) and 7 (thick line) with respect to their gas-gas energy.
Figure 6. Map of the gas-solid energy. Blackened regions have energies between -30 and -12 kJ/mol. All correspond to intercolumnar spaces. The supersites marked as 1 and 7 are discussed in the text.
Figure 9. Local adsorption isotherms corresponding to three representative intercolumnar spaces with areas of ca. 335 (O), 235 (3), and 146 Å2 (0). The lines are guides to the eye.
Figure 7. Distributions of molecules with respect to gas-gas energy at three different surface coverages corresponding to pressures of 30 (broken line), 390 (thick line), and 660 Torr (thin line).
high surface coverages. These distributions are very similar to the ones obtained on porous solids.16 From the equilibrated configurations, it is possible to calculate the distribution of molecules with respect to their gas-gas interaction energy that are experiencing a certain interaction with the solid and are located on a given part of the surface. Figure 8 shows the corresponding distribution for molecules whose gas-solid energy is between -30 and -20 kJ/mol and, at the same time, are adsorbed on two of the blackened areas of Figure 6 (marked as 1 and 7). Both distributions have common features such as a large peak at zero energy that corresponds to molecules located outside the interaction sphere. The peaks close to -1 kJ/ mol are very similar and correspond to pairs of molecules oriented parallel to each other. The gas-gas potential gives a minimum (-1.2 kJ/mol) for two molecules parallel to each other separated at a distance of ca. 3.4 Å. The existence of peaks corresponding to repulsive interactions is indicating that gas-solid interaction is determining the structure of the adsorbed phase. For intermolecular
distances of ca. 3.4 Å, a minimum departure from a parallel relative orientation converts lateral interactions from attraction to repulsion. This could be the origin of the peaks at positive energies shown by the distributions of Figure 8. Given that the surface is an amorphous one and that molecules will tend to be adsorbed parallel to “local” planes defined by the closest solid atoms, the change in orientation previously mentioned seems to be very possible. These distributions are very similar for all the blackened areas of Figure 6. The observed differences are restricted to the adsorbate population of each region that is a function of the pressure, average adsorption energy, and the area of the blackened regions. From the configurations generated during the simulation, it is possible to calculate the local adsorption isotherms on arbitrarily chosen parts of the surface. Selecting the intercolumnar spaces seems to be the obvious choice to calculate the local isotherms. Figure 9 shows the obtained results for three of these regions. These isotherms are representative of all the local isotherms obtained on the intercolumnar regions. The upper isotherm represents the adsorption on surface regions with an average area of 345 ( 7 Å2, the middle isotherm corresponds to regions with an average area of 250 ( 25 Å2, and the lower one corresponds to an area of 151 ( 15 Å2. These are the BET areas obtained from the local isotherms. All the isotherms fall over the same curve, within the statistical error, if
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they are plotted as adsorbed quantity per unit surface area. The local isotherms can be analyzed with the supersite theory of adsorption.17,18 A supersite is a region of the surface capable of accommodating several molecules on it and is energetically heterogeneous with variable size and shape. In this theory, the adsorption isotherm on each supersite is given by a truncated virial equation:
( )
ln
Ns ) ln Khs - 2BsNs p
(1)
where Ns is the number of adsorbed molecules on supersite s at pressure p and Khs and Bs are the Henry’s Law constant and the second virial coefficient for supersite s. Because the supersites are independent of each other, it is possible to write
1 ns
ns
∑ s)1
Ns(pi)
)
Nms
Nt(pi) Nm
i ) 1, ..., ns
∫V{
[ ( ) ]}
s exp -1 kT
dx dy dz
(3)
where x and y are restricted to the supersite under consideration; s that strictly speaking is a function of x, y, and z is the gas-solid energy for a molecule adsorbed on supersite s. The second virial coefficient, Bs, is given by
Bs ) -
2qs(2) - q2s (1) 2q2s (1)
1 2 3 4 5 6 7 8
-7.68 -9.74 -9.64 -8.17 -9.14 -11.79 -11.99 -9.95
6.35 6.85 6.49 6.81 6.64 7.88 7.64 6.40
1.54 1.53 1.93 2.03 1.93 1.81 2.01 1.62
9 10 11 12 13 14 15 16
-9.53 -12.02 -12.20 -9.91 -7.15 -10.16 -9.95 -8.89
6.95 6.61 7.39 6.55 4.40 6.57 7.30 6.74
1.74 1.90 1.95 1.72 1.36 1.96 1.88 1.89
Figure 10. Calculated (line) and experimental (O) nitrogen adsorption isotherms at 77.5 K.
Once the local isotherms have been calculated, it is possible to calculate the monolayer capacity corresponding to each supersite. From the adsorption energy distribution function, the range of adsorption energies is known. The system represented by eq 2 is an ill-posed one as in the case of the general adsorption equation. Thus, an alternate procedure based on the validity of eq 5 is tested.
Nmref s Nms ) Ns(pi) ref Ns (pi)
(5)
where the superscript ref indicates the reference surface. This equation simply shows that the local coverages on the reference solid and the columnar one are coincident provided that the corresponding supersites have the same average gas-solid energy. The monolayer capacity of each supersite is calculated, and the obtained values are averaged over all the simulated points. Then, the total monolayer capacity is obtained from ns
(4)
where qs(1) and qs(2) are the partition functions for a single molecule on supersite s and for two particles adsorbed on the same supersite, respectively. Complete expressions for the partition functions have been published in a previous paper.17 Table 1 shows the obtained values for Khs and Bs for the reference surface STR5 including the average energy of each supersite. (17) Bottani, E. J.; Steele, W. A. Adsorption 1999, 5, 81. (18) Steele, W. A. Langmuir 1999, 15, 6083.
super〈s〉 Khs super〈s〉 Khs site [kJ/mol] [Torr-1] 2Bs site [kJ/mol] [Torr-1] 2Bs
(2)
where Ns(pi) is the number of adsorbed molecules on supersite s when the equilibrium pressure is pi, Nms is the monolayer capacity of supersite s, Nm is the monolayer capacity of the full adsorption isotherm, Nt(pi) is the number of adsorbed molecules on the whole surface, and ns is the number of supersites. Equation 2 represents a set of linear equations from which the Nms could be determined. The local isotherms can be calculated from eq 1. We have employed Newton-Raphson’s numerical method to calculate the local isotherms at the same equilibrium pressures that we have for the total isotherm. To determine Khs and Bs, it is necessary either to have a set of adsorption isotherms (see for example ref 15) or to know the gas-solid and gas-gas interaction potentials. Implicit is that a detailed knowledge of the solid surface structure is required to perform such calculation, which is not the case for real heterogeneous surfaces. The local isotherms shown in Figure 9 have the same shape as the ones obtained with other carbonaceous solids.17,18 This fact suggests that it is possible to take a surface as reference for which Bs and Khs are known for different supersites. Using the results previously published17 more precisely for a solid denoted STR5, it is possible to calculate Bs and Khs using the expressions
1 Khs ) kT
Table 1. Supersite Average Energies, Henry’s Law Constants, and Virial Coefficients for Nitrogen Adsorbed on Reference Solid STR5
Nm0 )
Nmi ∑ i)1
(6)
where Nm0 is the first approximation to the best solution, and it should be equal to Nm in eq 2. Our reference surface has been arbitrarily divided into 16 supersites; thus, Nm0 is based on 16 supersites too. If Nm0 is smaller than the Nm value corresponding to the simulated isotherm, it means that a larger number of supersites is needed to reproduce the simulated Nm. In the opposite situation, fewer supersites are needed. In either case, the process is repeated, changing the number of supersites, until the monolayer capacity is as close as possible to the experi-
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Table 2. Supersite Areas Obtained for the Columnar Surfacea supersite
〈s〉 [kJ/mol]
As [Å2]
1 2 3 4 5 6
-7.68 -9.74 -9.64 -8.17 -9.14 -9.95
382.2 352.7 352.7 361.3 353.2 352.5
supersite
〈s〉 [kJ/mol]
As [Å2]
7 8 9 10 11
-9.53 -9.91 -7.15 -9.95 -8.89
352.8 352.7 419.7 352.5 353.8
a The area has been calculated taking 16.2 Å2 as the nitrogen cross-sectional area.
mental value. The final step is to calculate the adsorption isotherm and to compare it with the simulated one. In Figure 10, the isotherm obtained from the simulation is compared to the supersite isotherm, and the area of each supersite is quoted in Table 2. The agreement with the simulation results is quite good, as can be seen in Figure 10. The supersite theory reproduces the simulated isotherm up to a surface coverage close to two monolayers. This fact can be explained based on the reduced effect of gas-gas interaction energy due to the structure of the solid. As could be expected at a higher surface coverage, the supersite theory is no longer adequate, at least in its present form, to describe the simulated isotherm. The total monolayer capacity is 246.1 molecules, which is 13% lower than the BET value. The surface area obtained with the supersite model is 4287 Å2, which differs by 7% from the BET value (4610 Å2). All the supersites have almost the same area, which is not an unexpected result given that the columnar structure is a regular array of columns on the surface. The average energy corresponding to the whole surface is ca. -9.1 kJ/ mol, which is in agreement with the distribution function obtained with a constrained regularization method. This distribution, not included in the paper, shows two main peaks, one centered at ca. -5 kJ/mol and the second one, much less intense, centered at -25 kJ/mol. Thus, the weighted energy average obtained from this distribution suggests that the value obtained from the supersite theory is quite reasonable.
Finally, another interesting feature of the supersite model is that it would be possible to recover the full adsorption isotherm from one data point provided that the monolayer capacity of the whole surface is known from other sources. In the present work, the monolayer capacity employed is determined from the simulated isotherm averaging over all the isotherm points. The adsorption on columnar surfaces of molecules with different sizes, shapes, and dipole moments is currently being studied. Conclusions The columnar structure studied here presents, with respect to nitrogen physisorption, several common characteristics with porous materials. Hysteresis is not observed in desorption isotherms, probably because of the fact that the intercolumnar spaces are all interconnected. Quadrupolar interactions between adsorbed molecules do not influence the adsorption as in the case of open surfaces because of the topography of the columnar surface. When lateral interactions are described in terms of a hard-sphere potential, the intercolumnar space is filled with molecules in a denser, unrealistic, state than the normal solid nitrogen; otherwise, nitrogen is in a liquidlike state. The supersite theory of adsorption gives a reasonable description of the surface energetic structure. The theory also reproduces the adsorption isotherm, up to two monolayers, and the monolayer capacity. Preliminary calculations performed on real experimental isotherms show very good results. Acknowledgment. E.J.B. is Associate Professor of the National University of “El Litoral” (UNL) and Researcher to the Comisio´n de Investigaciones Cientı´ficas de la Provincia de Buenos Aires (CIC). The research project is financed by Consejo Nacional de Investigaciones Cientı´ficas y Te´cnicas (CONICET) (PID: 0488)-CIC and UNLP (Project 11-X223). LA0013648
