1456
Langmuir 1999, 15, 1456-1462
Adsorption of Nitrogen on Rutile (110): Monte Carlo Computer Simulations F. Rittner and B. Boddenberg Lehrstuhl fu¨ r Physikalische Chemie II, Universitat Dortmund, Otto-Hahn-Strasse 6, D-44227 Dortmund, Germany
M. J. Bojan and W. A. Steele* The Pennsylvania State University, 152 Davey Laboratory, University Park, Pennsylvania 16802 Received September 8, 1998. In Final Form: November 9, 1998 Grand canonical and canonical ensemble Monte Carlo computer simulations of the adsorption of N2 on the (110) face of rutile at 77 K are reported. A novel ab initio adsorbate-adsorbent interaction potential is employed in conjunction with the X1 nitrogen-nitrogen potential to investigate the adsorption mechanism. It is demonstrated that at low pressures (1 Torr and below) the Ti adsorption sites within the depressed rows of oxides on the rutile (110) face (denoted by A) are completely occupied by nitrogen molecules in end-on orientations with slight alternating tilts perpendicular to the row axis that are produced by repulsive lateral interactions. At higher pressures, adsorption on rows of exposed oxides (denoted by B) commences, typically with a side-on orientation of the N2 molecules. The calculated isotherm of adsorption exhibits type II behavior according to the Brunauer-Deming-Deming-Teller classification, in agreement with experimental findings. Although the experimental isotherms are often evaluated using the BrunauerEmmett-Teller adsorption model, our simulations indicate that the assumptions of this model are not fulfilled. The implications of these discrepancies and their influence on surface area determinations are discussed.
1. Introduction In the third part of this series of papers on the adsorption of nitrogen on rutile (110) we return to the question which initiated the present study: Which mechanism underlies the adsorption process? To throw more light on this problem, canonical and grand canonical ensemble Monte Carlo computer simulations were performed using the adsorbate-adsorbent interaction potential of the companion paper1 and the X1 nitrogen-nitrogen potential.2 The organization of this paper is as follows. In section 2 we describe the computational setup of our calculations. The results of the simulations will be reported in the third section where we present evaluations of the structural and energetic properties of the adsorbate as well as the adsorption isotherm for this system. Special emphasis will be put on the simulated adsorption isotherms and their dependence on technical parameters such as the choice of a starting configuration, the simulation time, and the dimensions of the simulation box. We finally compare the adsorption mechanism with the assumptions of the Brunauer-Emmett-Teller (BET) theory3 and discuss the accuracy of surface area determinations based on this approach. 2. Computational Details Canonical and grand canonical ensemble Monte Carlo (CEMC/GCEMC) computer simulations were performed using a modified version of the program GRAND, originally (1) Rittner, F.; Boddenberg, B.; Fink, R.; Staemmler, V. Langmuir 1999, 15, 1449. (2) Murthy, C. S.; Singer, K.; Klein, M. L. I. R. McDonald, Mol. Phys 1980, 41, 1387. (3) Brunauer, S.; Emmett, P. H.; Teller, E. J. Am. Chem. Soc. 1938, 60, 309.
developed by Bakaev and Steele.4 For all simulations the temperature was set at the boiling point of nitrogen (77.4 K), which usually is employed in experimental adsorption studies. The analytic potential function used to describe the adsorbate-adsorbent interaction was tailored to reproduce the results of ab initio quantum chemical cluster calculations5 and is described in detail in the companion paper.1 The lateral interaction of the nitrogen molecules was modeled by the well established X1 effective pair potential.2,6 Here, two interaction sites are located at the N atoms, 1.098 Å apart, so that the Lennard-Jones (LJ) interaction of two N2 molecules is described by four LJ 12-6 contributions with parameters NN/k ) 36.4 K and σNN ) 3.318 Å. The quadrupolar electrostatic interaction is approximated by placing a positive charge of 0.810e at the center of symmetry of the molecule and two negative charges of half that value at the nitrogen atoms. Although several alternative nitrogen-nitrogen potentials are available in the literature, e.g., effective7 as well as pure8 pair potentials, these are computationally more involved and, for our purpose, are not likely to be superior to the X1 model, which has a long history in computer simulation studies of nitrogen adsorption.9-11 Nevertheless, the model has several well-known flaws, the most important of which is that it overestimates librational frequencies when used (4) Bakaev, V. A.; Steele, W. A. Langmuir 1992, 8, 148. (5) Rittner, F.; Boddenberg, B.; Fink, R.; Staemmler, V. Phys. Rev. B 1998, 57, 4160. (6) Murthy, C. S.; O’Shea, S. F. I. R. McDonald, Mol. Phys 1983, 50, 531. (7) Raich, J. C.; Gillis, N. S. J. Chem. Phys. 1977, 66, 846. (8) Bo¨hm, H.; Ahlrichs, R. Mol. Phys. 1985, 55, 1159. (9) Talbot J.; Tildesley, D. J.; Steele, W. A. Faraday Discuss. Chem. Soc. 1985, 80, 91. (10) Vernov, A. V.; Steele, W. A. Langmuir 1986, 2, 219. (11) Bottani, E. J.; Bakaev, V. A. Langmuir 1994, 10, 1550.
10.1021/la9812018 CCC: $18.00 © 1999 American Chemical Society Published on Web 01/12/1999
Adsorption of Nitrogen on Rutile (110)
in quasiharmonic lattice dynamics. This defect can be remedied in part by using a more detailed representation of the charge distribution, e.g., by increasing the number of point charges from three (X1 model) to five (X* model).6 However, one should recall that we are not primarily interested in the behavior of nitrogen molecules within a bulk phase but at a solid-gas interface. Since the lateral interaction between the nitrogen molecules will be affected by their proximity to the surface, the use of two-body adsorbate-adsorbate potentials is a more serious simplification than the details of the two-body potential used. However, in the case of the present system an analysis of the electronic properties of adsorbed nitrogen on the rutile (110) face disclosed no substantial differences in comparison to a nitrogen molecule within a diluted gas phase.1,5 Since many-body contributions to the molecular interactions are omitted throughout this study, the twobody functions used must be viewed as effective two-body potentials. Simulation boxes were constructed from rectangular arrangements of surface unit cells (a ) 6.496 Å, b ) 2.959 Å) each containing one coordinately unsaturated (cus) Ti atom.12 The base area of the boxes ranged from 45.4 × 44.4 Å (105 cus Ti atoms) to the most frequently used 51.9 Å × 47.4 Å (128 cus Ti atoms), and their height was fixed at 30 Å in conjunction with a hard wall boundary, which proved sufficient for all pressures studied. Periodic and minimum image boundary conditions in the plane of the surface were always applied. In general, the influence of the box dimensions on the simulation results turned out to be relatively small. However, some nontrivial aspects of the box geometry will be discussed in the following section. Computer simulations of nitrogen molecules inside the defined adsorption space were carried out on the basis of standard algorithms, using the Metropolis formula for creation, destruction, and translational/orientational moves.13-15 Technical details concerning the application to adsorbed nonspherical molecules can be found in ref 11. For GCEMC runs the probabilities of all types of moves were set equal to 1/3, which is generally believed to give the most rapid convergence of the Markov chain.14 By setting the probability of a translational/orientational move equal to 1, i.e., by keeping the number of particles fixed, a GCEMC run could be converted into a CEMC run. 3. Results and Discussion 3.1. Structural Aspects. In the course of a grand canonical ensemble Monte Carlo computer simulation the number of particles is a fluctuating property. Control charts reflect these fluctuations and, in conjunction with similar charts for the energy, are valuable indicators to assess whether a GCEMC simulation is well-behaved or not. In Figure 1 the control charts of some representative runs are presented. Here, each point is the arithmetic mean of 200 current values of the number of N2 molecules in the adsorption space, N, each of these values being taken after the completion of 200 trials. All charts exhibit a steep initial increase of N. After a sufficient number of steps the curves level off and fluctuate around a virtually constant value of the particle number which corresponds (12) Rittner, F.; Paschek, D.; Boddenberg, B. Langmuir 1995, 11, 3097. (13) Nicholson, D.; Parsonage, N. G. Computer Simulation and the Statistical Mechanics of Adsorption; Academic Press: London, 1982. (14) Allen, M. P.; Tildesley, D. J Computer Simulation of Liquids; Clarendon Press: Oxford, 1987. (15) Frenkel, D.; Smit, B. Understanding Molecular Simulation; Academic Press: San Diego, CA, 1996.
Langmuir, Vol. 15, No. 4, 1999 1457
Figure 1. Control charts for the number of molecules Nads on the rutile surface at 77.4 K. From bottom to top, the plots are for pressures of 1, 250, 500, and 750 Torr.
to the preset activity, i.e., the pressure. At the lowest pressure, p ) 1 Torr, the final average value is N ) 128, which is the number of cus Ti sites within the simulation box (8 rows A, each containing 16 cus Ti). Obviously, the amplitudes of the fluctuations are extremely small for a range of pressures once these sites are occupied. This trend is accompanied by decreasing values of the acceptance ratios which at 1 Torr is about 1% for the translational/ orientational moves and is even less for creation and destruction, irrespective of the setting of the maximum allowed displacement. If the pressure is further drastically reduced, say to 10-3 Torr, the same observations are made. That is, the equilibrium number N of adsorbed particles does not change and eventually reaches again a value of ca. 128. Similarly, energy distributions and density profiles, which will be discussed later, do not show any relevant dependence on the pressure once p e 1 Torr. CEMC simulations within this low-pressure regime give an explanation for these findings. Suppose a nitrogen molecule is placed on the rutile (110) face and is moved at some low temperature, e.g., at 77 K. Very soon, the molecule will fall into the potential minimum above a cus Ti site. Since the energy barrier between neighboring adsorption sites (ca. 20 kJ/mol, cf. ref 1) is much larger than the thermal energy, it will be trapped there. Likewise, the chance of this molecule to be removed in a destruction step of a GCEMC run is negligibly small. Even the repulsive lateral interaction with other nitrogen molecules does not enable the trapped molecule to escape but leads to characteristic arrangements of the adsorbate which will be discussed later. In the case of a GCEMC run all cus Ti sites will eventually be occupied by exactly one nitrogen molecule. Actually, a steep rise of the particle number is expected when N < 128 at extremely low pressure, which is comparable to the experimental measurements of the low pressure part of the adsorption isotherm of a strongly adsorbing system. To get an impression of the structural properties of the adsorbate on the rutile (110) surface, perspective views of typical configurations taken from GCEMC simulations at four different pressures are shown in Figure 2. At p )
1458 Langmuir, Vol. 15, No. 4, 1999
Rittner et al.
Figure 2. Snapshots taken from simulations at 77.4 K and nitrogen pressures of (a) 1, (b) 250, (c) 500, and (d) 750 Torr. Nitrogen molecules in row A are shown by dark blue dumbbells and in light blue otherwise; oxygen and titanium atoms are represented by red and yellow spheres, respectively. In panel a, two characteristic types of the orientational ordering of row A nitrogens are highlighted.
1 Torr (Figure 2a) the cus Ti atoms within rows A of the rutile (110) surface are completely covered 1:1 with nitrogen molecules. The molecules are adsorbed end-on but slightly tilted and offset from the ideal row A lines. This minimizes the repulsive lateral interaction energy without significantly altering the N2-surface interactions. In this way almost perfect zigzag chains are generated. Closer inspection reveals domains with two different types of characteristic nitrogen arrangements depending on whether the parallel chains are in-phase or out-of-phase. In the former case narrow channels are formed whereas in the out-of-phase configuration, hexagonal arrays of endon N2 molecules are generated inside of which another nitrogen molecule can be stabilized in a side-on position above a row B oxide which lies between the A rows of oxides. The narrow channel of an in-phase configuration as well as the characteristic hexagonal arrangement of an out-of-phase configuration are highlighted in Figure 2a. At pressures of 250-750 Torr (Figure 2b-d), further nitrogen molecules will adsorb, preferentially in side-on orientation above the row B oxides. With increasing pressure, higher layers are populated without any clear orientational ordering. Sometimes, larger clusters are formed. Sporadically, isolated molecules are observed which are associated with the gas phase. Figure 3 shows the particle density F(z) ) ∆N/∆z as a function of the coordinate z, where z is the distance of the center of mass of N2 from the plane through the centers of the cus Ti atoms. At a pressure of 1 Torr, F(z) exhibits essentially only one maximum at z ) 2.8 Å, corresponding
to nitrogen molecules attached to the cus Ti atoms. At higher pressures, additional maxima of F(z) appear which can be associated with the increasing adsorption on regions of the surface which are atomically higher than the cus Ti atoms. The question of whether these adsorbed atoms should be assigned to the monolayer or not is nontrivial since the definition of a surface “layer” is somewhat arbitrary for atomically rough surfaces as has been pointed out earlier.16 For the present system, we can ascribe at least two of the peaks of the density distribution function to adsorption on different kinds of surface site. Hence, the first peak corresponds to completely filled rows A and the second peak (z ≈ 4.5 Å) can be attributed to nitrogen molecules which are adsorbed on row B oxygen atoms, mainly side-on in the centers of hexagonal arrangements such as that shown in Figure 2a. If perfect arrays of hexagonal sites are formed by the row A N2 molecules, one would expect 64 N2 molecules to occupy these sites. The actual numbers obtained by integrating under the second peaks in the density distributions of Figure 3 give 52, 63, and 71 for pressures of 250, 500, and 750 Torr. A close look at these peaks reveals that the peak for 71 molecules is broader than those for the lower occupancy numbers. This is most likely due to reorientation of sideon adsorbed molecules into more nearly end-on positions. This results in closer packing and thus allows adsorption in excess of the estimated number of 64 over the row B (16) Cascarini de Torre, L. E.; Bottani, E. J., Steele, W. A. Langmuir 1996, 12, 5399.
Adsorption of Nitrogen on Rutile (110)
Figure 3. Adsorbate density for N2 on the (110) face of rutile at 77 K is shown as a function of the distance z between the molecular center-of-mass and the surface. Pressures for these curves are as follows: 1 Torr, solid black line; 250 Torr, solid gray line; 500 Torr, dashed gray line; 750 Torr, dashed black line.
Ti atoms. The third peak that appears in Figure 3 at high pressures is less sharply defined and is due to “multilayer” adsorption on top of the row A and row B nitrogen molecules. As can be seen in the snapshots, the nitrogen molecules within the rows A are tilted along the x-axis out of the end-on orientation, which was found to be the optimum geometry for an isolated N2 molecule.1 On the average, this tilt amounts to 20-30° and reflects an attempt to minimize the repulsion between neighboring molecules. This repulsion could be anticipated since the distance between the cus Ti atoms (2.959 Å) is noticeably smaller than the van der Waals size parameter σNN ) 3.318 Å. An analysis shows that the repulsive interaction reaches its maximum in the collinear end-on arrangement due to Pauli repulsion and, to a smaller degree, to the unfavorable electrostatic interaction between the aligned N2 quadrupole moments. 3.2. Energetic Aspects. Of course, molecular potential energies in systems at a finite temperature are not at their minimum values but are distributed over a range determined by the temperature and by the nature of the interactions in the neighborhood of the minimum. Thus, distributions of the adsorbate-adsorbent and adsorbateadsorbate interaction energies U were calculated in analogy to the particle density distribution. The results, normalized to give areas equal to the total number of adsorbed molecules, are presented in Figures 4 and 5, respectively. Since the adsorption potential is closely related to the distance z between the molecules and the surface, one can expect a resemblance between the distribution of the nitrogen-solid interaction energies and the aforementioned adsorbate density profiles of Figure 3. As can be seen in Figure 4, this is the case. Maxima in the numbers of particles occur at typical values of the adsorption potential within row A (ca. -40.9 kJ/mol) and row B (ca. -5 kJ/mol). In the case of molecules adsorbed in more distant (higher) layers, the interaction with the
Langmuir, Vol. 15, No. 4, 1999 1459
Figure 4. Distributions of the adsorbate-adsorbent interaction energies for N2 on the (110) face of rutile at 77.4 K are shown. Adsorbate pressures are as follows: 1 Torr, solid black line; 250 Torr, solid gray line; 500 Torr, dashed gray line; 750 Torr, dashed black line.
Figure 5. Distributions of the adsorbate-adsorbate interaction energies for the systems of Figure 4.
surface is very small and gives rise to the sharp spike extending from U ) 0 to -0.8 kJ/mol, which dominates the spectrum at p ) 750 Torr. (Note that this energy range is comparable to the thermal energy RT ) 0.6 kJ/mol.) Despite the small adsorbate-adsorbent interaction of the 80 molecules that produce the spike for p ) 750 Torr, they are not all isolated but are the multilayer part of the adsorbed fluid. This characterization is supported by the rather diffuse peak at 7 Å in the adsorbate density profile that contains roughly 80 molecules for p ) 750 Torr and by the observation that the density at large distances is nearly zero.
1460 Langmuir, Vol. 15, No. 4, 1999
Distributions of the lateral interaction energies of the adsorbed molecules (Figure 5) show more complex shapes. At p ) 1 Torr the distribution essentially consists of a single peak centered around U ) +2.5 kJ/mol, which is due to the repulsive interaction of the neighboring molecules in the troughs. The width of the distribution originates from deviations from the perfect zigzag ordering, interactions between N2 molecules in different rows, and the thermal motion of the adsorbate. Also visible is a weak shoulder at ca. -4 kJ/mol which is due to the attractive interaction of a few row B molecules with their neighbors within the troughs. This shoulder grows as the pressure is increased and eventually results in bimodal distributions at p ) 250-750 Torr. As the number of attractive interactions between row B and row A molecules increases, the repulsive peak is affected as well and steadily moves to more negative energies. Interestingly, for p ) 500 Torr and p ) 750 Torr the right-hand parts of the distributions nearly coincide. A straightforward explanation seems to be that the limited number of row B positions are all occupied. However, inspection of the snapshots in Figure 2c and Figure 2d shows that the situation is more complex. In fact, there seems to be a reordering of the row B nitrogen molecules in an effort to accommodate more nitrogens with favorable lateral interactions, which is documented by a further shift to the left for the low energy part of the distribution. As mentioned above, the changing orientations of these molecules also affects the density versus distance distributions shown in Figure 3. Finally, we comment on the origin of the spike in the lateral interaction energy distribution at U ≈ 0. This feature cannot be explained simply by the cancellation of attractive and repulsive interactions since these should result in a smooth distribution. The additional spike is rather due to the presence of molecules in the gas phase which experience no lateral interactions with other molecules. The number of such isolated particles obtained from integrating the area under the sharp peak is rather small (ca. 7 for 750 Torr) and less than 10% of the number of particles which experience small interactions with the surface. (This estimate agrees fairly well with an ideal gas calculation of ca. 6 for the number of molecules in the computer box at 750 Torr.) Areas under the peaks in three histograms for p ) 750 Torr that are shown in Figures 3-5 lead to the conclusion that the approximate numbers of adsorbed atoms for this pressure are 129 over row A, 71 over row B, 70 in the multilayer, and 7 in the gas phase, giving a total of 277. (The uncertainty in each of these numbers is about (1.) The pros and cons of this assignment of the row B molecules to the monolayer will be discussed more extensively below. However, the molecule/surface and molecule/molecule energy distribution functions reveals that the lateral interaction between adsorbed molecules has a noticeable influence on the adsorption of molecules within the row B and higher layers, whereas the geometry of the molecules in rows A is almost entirely dictated by the interaction with the surface. This is in contrast to cases with energetically smooth homogeneous surfaces, e.g., the basal planes of graphite, where the well-known herringbone structures of many of the monolayers on this adsorbent are formed due to the lateral interactions (see ref 17 for a review). 3.3. Ordering and Adsorption Isotherms. As is indicated in Figure 2a, the 1:1 occupation of the cus Ti sites by nitrogen molecules at very low adsorbate pressures produces linear chains of end-on molecules that exhibit (17) Steele, W. A. Chem. Rev. 1993, 93, 2355.
Rittner et al.
Figure 6. Control charts for the number of adsorbed molecules Nads obtained from restarted simulations: 1 Torr, hexagonal start configuration, lower black line; 1 Torr, parallel zigzag chains, lower gray line; 250 Torr, hexagonal start configuration, upper black line; 250 Torr, parallel zigzag chains, upper gray line.
predominantly zigzag ordering due to alternating tilts of the nitrogen molecular axes. A zigzag chain can be either in- or out-of-phase with a neighboring chain and the calculations show that these arrangements have nearly identical potential energy. The figure also shows a few breaks in the zigzag ordering produced either by thermal disorder or by incomplete equilibration in the simulation. Since the orientational configurations of nitrogen molecules adsorbed over row B atoms depend on the configurations of the nearby nitrogen molecules over row A sites, the possibility of an incomplete equilibration of the A site nitrogen molecules is of considerable significance to the adsorption isotherm. Figure 2a shows that an outof-phase arrangement of neighboring row A nitrogen chains produces a chain of hexagonal nitrogen sites that are are energetically more favorable for side-on adsorption of row B nitrogen molecule than for adsorption between a pair of the in-phase zigzag chains that are too close together to favor the side-on orientation of nitrogens adsorbed in the gap. The attainment of equilibrium in the simulations was studied by generating configurations for the adsorbed nitrogen molecules with p ) 1 and 250 Torr starting from three different initial configurations: the usual empty surface (in Figure 1) and surfaces with 128 N2 molecules in perfect zigzag chains that are either in- or out-of- phase. The control charts for the last four configurations are given in Figure 6 and show that the limiting number of molecules is essentially independent of the starting configuration when p ) 1 Torr. In contrast, the limiting numbers of molecules at p ) 250 Torr depends significantly on the starting configurations, at least for the numbers of Monte Carlo trials used in these calculations. Figure 6 shows that the limiting number of adsorbed particles is smaller for the in-phase start than for either the empty or the out-of-phase initial configurations, which yield essentially equal limiting numbers of adsorbed molecules. Evidently, the limiting results can be affected by the choice of initial
Adsorption of Nitrogen on Rutile (110)
Figure 7. Simulated adsorption isotherms for N2 on the (110) face of rutile at 77.4 K are shown. 〈Nads〉 is the average number of molecules in the adsorption volume and the various isotherm points show the following: the “long” (regular) simulation, solid diamonds; simulations restarted from an initial configuration of parallel zigzag chains, circles; simulations restarted from an initial configuration giving hexagonal N2 arrays, triangles.
conditions, for the numbers of steps that made up these Monte Carlo chains. However, snapshots of the later configurations reveal that a small amount of disordering in the chains has occurred for all three initial configurations. The reason is simply that the second layer nitrogen molecules are much easily removed in the case of the inphase zigzag chains, as pointed out before. Of course, in the limit of an infinite simulation time, the starting conditions should not matter, as the first layer molecules should eventually attain statistical ordering. However, transitions between the in-phase and out-of phase arrangements are less likely to occur at higher pressures/ adsorbate densities, since random displacements of first layer nitrogen molecules will mostly result in strongly repulsive interactions with the surrounding higher layer molecules resulting in a poor sampling of phase space. For practical reasons, it is therefore necessary to choose reasonable starting conditions. In this case, this means a start either from an empty box or an out-of-phase arrangement. Adsorption isotherms at 77.4 K are shown in Figure 7. In addition to the points which were obtained from regular simulations, results from restarted simulations are shown as well. In the low and intermediate pressure region, the regular adsorption isotherm agrees very well with that obtained from simulations restarted from an out-of-phase start configuration but differs markedly from the isotherm obtained from an in-phase start. At higher pressures, where the interaction between nitrogen molecules in the second and higher layers gains importance, the simple picture of in-phase and out-of-phase zigzag chains obviously breaks down (compare Figure 2d). Interestingly, at 750 Torr the number of adsorbed molecules is highest in the case of a restart from an in-phase start configuration.
Langmuir, Vol. 15, No. 4, 1999 1461
However, since the differences between the different runs were not substantial, we did not pursue this point any further. Variations of the simulation time generally resulted in only minor differences and are not documented in Figure 7. Likewise, different box geometries (7 × 15 instead of 8 × 16 surface unit cells) did not have a significant impact on the number of adsorbed particles once these values were rescaled according to the surface area. However, it should be noted that simulation boxes consisting of an odd number of surface unit cells, e.g., 7 × 15, do not allow perfect hexagonal or parallel zigzag geometries. As a consequence, a larger number of defects is inevitably produced and leads to a visibly lower degree of ordering. Although it is possible to convert the number of adsorbed molecules into a macroscopic amount and thus compare the simulated isotherm directly with their experimental counterparts, little additional information would be gained by this procedure since the experimental isotherms strongly depend on the characteristics of the sample used. To account for sample specific properties, such as the contribution of the (110) faces to the overall powder surface, different scaling factors must be introduced which are a priori unknown. Although these factors will not change the shape of the isotherm and hence its classification, one can achieve arbitrarily good or poor agreement with the simulation by varying the parameters. We therefore do not compare our calculated isotherm with a specific measurement but rather concentrate on their shape. Clearly, the simulated adsorption isotherm can be classified according to Brunauer-Deming-DemingTeller (BDDT) as type II.18 It hence matches the typical course of the experimental isotherms (see, e.g.,19,20 which are usually evaluated using the Brunauer-EmmettTeller (BET) isotherm equation.3 It is instructive to apply the BET theory to the calculated isotherm of adsorption where the surface area is precisely defined by the dimensions of the simulation box. Unfortunately, a direct evaluation of the data points using the linearized version of the BET equation strongly depends on the value of the condensation pressure p0. In the case of the computer simulations p0 is noticeably higher than the ideal value of 760 Torr, as can be seen in Figure 7. The accurate determination of the liquid-gas phase transition in the presence of a surface is not trivial and requires additional expensive calculations. We therefore approximated the monolayer capacity nm by using the point B approach,18 which is known to usually agree very well with the results obtained from linearization. In the present case one obtains nm ) 170 ( 10. Using a ) 16.1 Å2 the BET surface area is 2700 ( 200 Å2, which is ca. 10% larger than the exact value of 2460 Å2. Although this agreement is encouraging, the analysis of the isotherms given here shows that the monolayer capacity is not at all close to 170 molecules, but is 128 if only the row A molecules are included and 200 if the row A plus row B molecules are defined as the monolayer. Of course this adsorbent is certainly not an array of identical adsorption sites. Although rutile (110) is, strictly speaking, a homogeneous surface (the adsorption potential exhibits only one kind of minima, namely the cus Ti atoms) secondary “sites” (on rows B) are formed at increased pressure. These positions are minima of the total potential energy of an adsorbed (18) Gregg, S. J.; Sing, K. S. W. Adsorption, Surface Area and Porosity; Academic Press: London, 1982. (19) Furlong, D. N.; Rouquerol, F.; Rouquerol, J.; Sing, K. S. W J. Chem. Soc., Faraday Trans. 1 1980, 76, 774. (20) Boddenberg, B.; Horstmann, W. Ber. Bunsen-Ges. Phys. Chem. 1988, 92, 519.
1462 Langmuir, Vol. 15, No. 4, 1999
molecule, a phenomenon that is well-known in the case of the present substrate.21,12 Likewise, the sites are not independent, but lateral interactions between the adsorbed molecules exist and have a characteristic influence on the adsorbate structure. Furthermore, the area occupied by 128 row A N2 on a surface of 2460 Å is 19.2 Å2 whereas that occupied by 200 row A plus row B molecules is 12.3 Å2sneither value is even close to the usual one taken in the BET surface area calculation and one concludes that the initial agreement between the BET surface area and the actual value is obtained as the result of two compensating errors. 4. Conclusion In the previous papers in this series, the interaction of a nitrogen molecule with the TiO2(110) surface has been shown to be extremely strong (about -40 kJ/mol) for such an inert molecule. The consequences of this for the adsorption isotherm are as expected: even at very small pressure, the regions of large interaction that correspond to molecules in the depressed rows on this surface (denoted by A) are covered by adsorbed species that are almost completely localized in space and, to a somewhat lesser extent, in orientation. Indeed, the localization is sufficient to jam the molecules into configurations where the N2N2 energies are repulsive rather than the usual case of attractive lateral interactions. Once this region of the surface is completely occupied by molecules (128, for the surface studied here), adsorption commences on other regions where the N2-surface interaction is weaker (-10 kJ/mol). This second type of adsorption (on row B areas) is characterized by attractive (negative) N2-N2 interactions both with the neighboring row A and with the neighboring row B molecules. In contrast to the row A adsorbed species, the row B occupancy at “completion” (21) Bakaev, V. A.; Steele, W. A. Langmuir 1992, 8, 1372. (22) Lide, D. R., Frederikse, H. P. R., Eds. Handbook of Chemistry and Physics, 76th ed.; CRC Press: Boca Raton, FL, 1995.
Rittner et al.
can vary from 64 molecules up to 71, with these changes facilitated by reorientation of the molecules from side-on to partly end-on as the adsorption pressure increases. Despite the well-known unrealistic simplifications of the BET model (lattice gas adsorption on a homogeneous array of sites, combined with a very crude treatment of lateral interactions), its many applications to adsorption on strongly interacting surfaces have shown that it can be used to give moderately realistic surface areas when one combines the monolayer capacities obtained from BET with estimates of the area per molecule in the layer. Although the simulated isotherm data for the present system give a point B estimate for monolayer capacity of 170 molecules, this does not correspond to either the row A number of 128 or the alternative estimate of about 199 in rows A plus B. Furthermore, the area per molecule in the fully covered surface is quite different for the row A and the row B molecules. Each of the two kinds of row amounts to half of the total surface area of 2460 Å2; thus the area per row A molecule is 9.6 Å2 and for row B, 17.3 Å2. Neither value is equal to the usual 16.2 Å2. It is evident that the characteristics of this adsorption system are too different from the BET model to allow its application to the isotherm data. This is a particularly disturbing result in view of the fact that N2 on rutile at the nitrogen boiling point is hardly an exotic system. The only feature that might qualify it as unusual is its singlecrystal nature, but single-crystal adsorbents are much more often encountered in modern studies than 30 years ago in the hay-day of the BET theory. Acknowledgment. Financial support of this work by Deutsche Forschungsgemeinschaft and Fonds der Chemischen Industrie is gratefully acknowledged. Professor V. Staemmler and Dr. R. Fink (both from the University of Bochum) are thanked for valuable discussions. LA9812018
