Tuning Selectivity in Adsorption on Composite Chiral Surfaces - The

In particular, we use a Monte Carlo search algorithm to predict the spatial distribution of active sites leading to the highest enantioselectivity. Ad...
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J. Phys. Chem. C 2007, 111, 11936-11942

Tuning Selectivity in Adsorption on Composite Chiral Surfaces Paweł Szabelski*,† and David S. Sholl‡ Department of Theoretical Chemistry, Maria-Curie Skłodowska UniVersity, Pl. M. C. Skłodowskiej 3, 20-031 Lublin, Poland, and Department of Chemical Engineering, Carnegie-Mellon UniVersity, Pittsburgh, PennsylVania 15213 ReceiVed: March 5, 2007; In Final Form: June 12, 2007

Designing the structure of enantioselective adsorbents is a difficult task. In this work we propose a simple theoretical method of optimization of selectivity in adsorption on a model chiral surface composed of inert and active sites. Our approach is based on using the molecular footprints of the adsorbing enantiomers to find the optimal arrangement of active sites on the surface. In particular, we use a Monte Carlo search algorithm to predict the spatial distribution of active sites leading to the highest enantioselectivity. Additionally, we present the results of grand canonical Monte Carlo simulations performed for the equilibrium adsorption of enantiomers from a racemic mixture on the surfaces obtained using the MC approach. The optimal pattern of the active sites is dependent on the footprint of the chiral molecule as well as on the pressure under which the separation is performed.

I. Introduction Adsorption on nanostructured solid surfaces has emerged as a promising method for chromatographic enantioseparations and asymmetric synthesis of stereoisomers in heterogeneous catalysis. This results mostly from a rapid development of methods to fabricate chemically and structurally modified surfaces to exhibit global or local chirality.1,2 A classical example of enantioselective adsorbents are chiral stationary phases (CSPs) used in high-performance liquid chromatography (HPLC) for the separation of enantiomeric pairs. The CSPs usually consist of a solid support, for example silica, with attached chiral ligands being organic molecules of a different size.3,4 The enantioselectivity of those materials originates mainly from specific intermolecular interaction between the chiral selector and the complementary enantiomer. In this case, a one-to-one correspondence is usually preserved such that, in principle, a larger number of chiral ligands means higher selectivity for the CSP.5-7 A different mechanism of enantiodiscrimination can occur for solid surfaces templated with chiral organic molecules, particularly metallic crystal faces templated with modifiers such as cinchona alkaloid1 or 2-butanol.8,9 In those cases, a suitable spatial distribution of the template molecules leads to the formation of chiral pockets promoting enantioselective adsorption or reactions. The structure of the template overlayer, including the number of the chiral pockets, their shape and spatial distribution is the deciding factor controlling enantioselectivity of the surface.10 Formation of this structure is greatly affected by both template geometry and coverage. For example, it has been demonstrated that selective adsorption of enantiomers of propylene oxide on a 2-butoxide templated Pd(111) surface occurs only within a narrow interval of the template coverage.8 In this case, preferential adsorption of one enantiomer of propylene oxide has been found to reach a maximum at 30% of the saturation coverage of the template. A similar tendency * Corresponding author. E-mail: [email protected]. † Maria-Curie Skłodowska University. ‡ Carnegie-Mellon University.

has been also observed for Pd(111) surface templated by chiral 2-aminobutanoate species.11 To explain the origins of enantioslectivity in the Pd(111)/2butanol system, Roma and co-workers12,13 have proposed a lattice gas model of enantioselective adsorption in which chiral pockets on a square lattice were formed by assemblies of up to four template molecules surrounding a given adsorption site. Using their model the authors were able to reproduce main trends in enantioselectivity observed experimentally. Further experimental studies of the chirally templated Pd(111) surface11 have provided information about the minimum requirement for an effective template, suggesting that the template molecule should be rigidly bonded to the surface in order to prevent azimuthal rotation of the chiral center. Theoretical efforts to understand the relation between structure and selectivity of the enantiospecific adsorbents have been also made in the case of CSPs14-19 and naturally chiral surfaces.20-22 Those attempts are of importance especially for the chromatographic separation and purification of enantiomers because of their use in pharmaceutical, agrochemical and food industries.3 In this case, the decision of which CSP is the most suitable for a given pair of enantiomers is usually based on testing ligands by trial and error. Computer simulations of the enantioselective binding are thus an alternative route of finding those factors which govern the enantiodiscrimination process. They involve, for example molecular dynamics14,16-19 or Monte Carlo simulations14-16 of the adsorption on CSPs based on cyclodextrins, proteins or synthetic receptors. Recently, the MC technique has been also successfully used for the prediction of enantioselectivity in the adsorption of chiral hydrocarbons on chiral Pt surfaces.20-22 A general picture emerging from both experimental and theoretical studies is that optimization of the structure of a surface to exhibit maximal selectivity toward a selected enantiomer is a complex problem which in most cases turns out to be case-sensitive. In practice there is no general method which hints at how to correlate the structure of the surface with the geometry of the adsorbate. This situation arises because a large

10.1021/jp0717948 CCC: $37.00 © 2007 American Chemical Society Published on Web 07/25/2007

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Figure 2. Influence of the number of the active sites in the (5 × 5) unit cell on the maximal selectivity at the zero-pressure limit (b). The inset shows the distributions of the active sites which maximize the selectivity for n ) 5 (A) and n ) 7 (B). The distribution (C) corresponds to the system with n ) 5 studied in refs 26 and 30; the selectivity for that system is marked by (O).

Figure 1. Schematic structures of the enantiomers of 1,2-dimethylcyclopropane (top part) and their simplified footprints on a square lattice (middle part). The second type of nonsuperimposable footprints possible for a chiral molecule occupying four adsorption sites (bottom part).

number of factors including chemical and structural properties of both adsorbate and surface which have to be considered while designing the optimal enantioselective adsorbent. This task is somewhat simpler when the chirality of the surface is imposed, for example, by purely geometrical factors like it is in the case of naturally chiral surfaces.1,20-25 In this case, tuning of the enantioselectivity involves change in the symmetry of the chiral surface so that terraces and steps of a different size are created while the chemical composition of the adsorbent remains the same. In this context, minimization of the number of factors which influence enantioselective properties of a surface is an important objective in designing effective chiral adsorbents and catalysts. Another helpful approach is to define chiral adsorption centers and determine their spatial distribution, leading to selective recognition of the enantiomers. In this contribution we propose a simple method for finding the optimal structure of a model chiral surface with active sites. The approach described in this study was meant to be quite general being based on a molecular footprint of an adsorbing enantiomer. In particular, our main objective was to determine the distribution of the active sites leading to the highest selectivity in the enantioseparation. Additionally, this study aims at understanding of the relation between energetic and structural factors which influence the adsorption of chiral molecules on a composite surface. II. Theory We have considered lattice models that describe chiral adsorption on two-dimensional surfaces in a simplified way, as in our earlier work.26 In these models, the surface is represented by a square lattice of binding sites with well-defined binding

energies. The chiral molecules used in this work were assumed to be rigid structures having footprints shown in the middle part in Figure 1. As seen in the figure, the footprints consist of four adsorption sites, each occupied by one segment of the adsorbed enantiomer. To give an example of a chiral molecule whose footprints are similar to those in our model, the top part of Figure 1 shows schematic structures of the enantiomers of 1,2dimethylcyclopropane. In this simplified description, only that part of the molecule which directly contacts the surface is considered. The remaining part of the molecule which is not involved in the adsorption is disregarded and assumed to be responsible only for preservation of chirality in the bulk phase. For example, in the case of 1,2-dimethylcyclopropane, this approach requires neglecting of the carbon atom in the C-C-C ring which is not connected with the methylene groups. Obviously, for more complex molecules one or more atoms or functional groups would be neglected. The adsorption of the enantiomers was assumed to be submonolayer with no lateral interactions in the adsorbed phase. Moreover, no surface diffusion of the adsorbed molecules was allowed. The simulations were carried out on a two-dimensional square lattice repeated periodically in both directions. Each unit cell consisted of inert and active sites whose interaction energy with a segment of a molecule was equal to i and a, respectively. All of the simulations described in this work were performed for i/kT ) 0.1 and a/kT ) 2. We studied square unit cells of a side l including (3 × 3), (4 × 4) and (5 × 5) cells with a different number of the active sites, n. For any unit cell with n g 4, regardless of the enantiomer type, there are five values of the adsorption energy accessible to a single molecule, depending on the number of active and inert sites the molecule occupies. This energy can be expressed as

j ) (4 - j)i + ja

j)0...4

(1)

where j denotes the number of active sites occupied by a molecule. The question of whether all of the energy modes or just some of them will be observed for a given n depends on how the sites are distributed within the unit cell. The number

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Figure 4. The maximal selectivity at the zero-pressure limit as a function of the number of the active sites in the (4 × 4) (b) and (3 × 3) (O) unit cell. The inset shows the distributions of the active sites which maximize the selectivity for n ) 4, for both distributions D and E.

To find the pattern which maximizes the selectivity for a given 2 l and n, we generated Z ) N l random spatial distributions of n the active sites using N varying from 100 to 10 000. For each of the distributions, Eo was calculated. The pattern with the largest Eo was considered to be the optimal pattern (an assumption that is rigorously correct in the limit N f ∞). The results from the calculations defined above compare the adsorption of a pure R-enantiomer to the adsorption of a pure S-enantiomer in the low-pressure limit. Understanding this situation is not necessarily sufficient to accurately characterize adsorption of mixtures of enantiomers.26 To compare the enantioselectivity derived from the results obtained for pure enantiomers using eqs 2 and 3 with analogous data corresponding the mixed adsorption we carried out additional MC simulations. The simulations examined equilibrium adsorption of the racemate on the S-selective surface using a standard Grand Canonical MC technique for localized adsorption of polyatomics on a square lattice.26-29 An S-selective surface is a surface whose unit cell was characterized by Eo > 1. Our computer simulations were performed on a square lattice of a side length L ) zl for integer z for each (l × l) unit cell. Specifically, in the case of the (3 × 3), (4 × 4), and (5 × 5) unit cell we used an array of 17 × 17, 13 × 13, and 10 × 10 cells, respectively. A fixed molar composition of the gas phase was set to 1:1 such that partial pressure of each enantiomer was equal to 0.5p with p being the total pressure of the racemate. The fractional coverage of each enantiomer at a fixed p was an average over 100 independent runs. For each value of p, up to 2 × 104 Monte Carlo steps, where one Monte Carlo step is defined as L2 attempts of changing the system state, were performed to reach equilibrium.

()

Figure 3. Periodic distribution of the active sites (black squares) leading to the maximal selectivity at the zero-pressure limit. The fragment of the lattice shown in the top part consists of nine (5 × 5) unit cells B from Figure 2. The fragments of the surfaces shown in the bottom part consists of nine (4 × 4) (left panel) and (3 × 3) (right panel) unit cells from Figure 4. The letters describing the surfaces are consistent with the notation used in Figures 2 and 4.

of the modes can be different for the two enantiomers of the adsorbing molecule. For this reason one can expect that for a certain spatial distribution of the active sites the ratio of the adsorbed amounts of the enantiomers will be maximized. This distribution corresponds to the maximal enantioselectivity in the adsorption model proposed here. Our primary goal is to determine both the number and spatial distribution of the active sites resulting in the highest enantioselectivity for a selected (l × l) unit cell. To find the optimal distribution of the active sites, we used the Monte Carlo search algorithm. The screening method proposed here involves calculation of the adsorption energy distribution functions for the enantiomers adsorbing at zeropressure limit. Specifically, for each pattern of the active sites, calculation of the AEDs requires considering of all possible 2l2 configurations of a single molecule within the unit cell and enumerating those configurations which have the same j. This procedure gives the normalized energy distributions ΓS and ΓR, which are used for the calculation of the associated Henry’s constants:

Kx )

∑j Γx(j) exp(j/kT)

x ) R,S

(2)

Having determined the Henry’s constants, the selectivity is defined using the conventional expression

Eo ) KS /KR

(3)

III. Results and Discussion A. The Optimal Pattern of Active Sites. Figure 2 shows the influence of the number of the active sites in the (5 × 5) unit cell on the maximal selectivity calculated at the zero pressure limit. Each point on the curve corresponds to the distribution of n active sites which gives the largest value of Eo. The selectivity is a nonmonotonic function of the number of the active sites with maximum at n ) 7. At this maximal

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Figure 7. Influence of the number of the active sites on the Henry’s constants (i ) R, S) corresponding to the maximal selectivity obtained for the unit cell of a different size indicated in the figure.

Figure 5. Adsorption energy distributions (x ) R, S) corresponding to the S-selective surface composed of the patterns A-D shown in Figures 2 and 4. The bars plotted in black are the AEDs for the S-enantiomer, while those plotted in gray are the AEDs for the R-enantiomer.

Figure 8. The maximal selectivity at the zero-pressure limit as a function of the number of the active sites in the (5 × 5) (b), (4 × 4) (O), and (3 × 3) (9) unit cell, obtained for the S-enantiomer from the bottom part of Figure 1. The inset shows the distributions of the active sites which maximize the selectivity for n ) 4, for all of the distributions shown in the figure.

Figure 6. Influence on the amount fraction of the active sites in the (l × l) unit cell, γ ) n/l2, on the average energy of adsorption on the S-selective surface. The results shown in the figure correspond to both enantiomers.

point, the selectivity is equal to 2.64 and it decreases slightly when n changes from 7 to 10. When n is smaller than 7, the selectivity drops rapidly. The results shown in Figure 2 indicate clearly that there is an optimal distribution of seven active sites arranged in the pattern B for which Eo reaches a maximum. To give a clearer picture of the corresponding surface, Figure 3B

shows a region of the surface composed of nine unit cells of pattern B. It can be seen from Figure 3B that the active sites form assemblies composed of two overlapped footprints of the S-enantiomer. The footprint of the S-enantiomer appears also on the surface with five active sites that is shown in the inset of Figure 2 as unit cell A, for which Eo ) 2.47. In our previous work, we had studied in detail the unit cell shown as unit cell C in the inset of Figure 2.26,30 The structure of this unit cell was motivated by the structure of intrinsically chiral surfaces that can be created using high Miller index metal surfaces.1,20-25 Unit cell C does not allow formation of the superstructure containing assemblies of the active sites matching the footprint of the S-enantiomer. Consequently, the selectivity obtained for that unit cell, shown with an open symbol in Figure 2, is markedly lower than for the unit cell A. This observation suggests that the presence of the assemblies of the active sites which are of the shape of the molecular footprint are important for achieving enhanced selectivity. The optimal pattern simul-

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Figure 9. Equilibrium adsorption isotherms of the enantiomers calculated for the S-selective surface composed of the patterns A, B, D, and E. Pressure is given in arbitrary unis.

taneously provides suitable spacing between the assemblies and allows their proper relative orientation. To examine the effect of l on the selectivity, Figure 4 shows the optimal Eo(n) for the (4 × 4) and (3 × 3) unit cells. For both cells the number of the active sites that maximizes Eo is 4. However, the maximal selectivity corresponding to the (4 × 4) cell D is nearly 1.5 times larger than the maximal selectivity obtained for the (3 × 3) cell E. This observation suggests that the reduction of the unit cell size results in a lowered maximal selectivity (cf. Figure 2). On the other hand, the maximal Eo obtained for the (4 × 4) unit cell is larger than the maximal selectivity calculated for the pattern C, the surface pattern we studied in our previous work.26,30 The bottom part of Figure 3 shows fragments of surfaces composed of the patterns D and E. We now consider the energetic properties of the surfaces discussed above. Figure 5 shows the AED functions calculated for the patterns A - E, for both enantiomers. The distributions for these unit cells differ considerably in shape. An extreme example is the pattern C, for which ΓS and ΓR consist only of three and two energy modes, respectively. In this case, the highenergy modes 4 and 3 are not accessible to either adsorbed enantiomer. This effect is a direct consequence of the structure of the pattern C, which does not provide suitable assemblies of four (or even three) active sites. In the case of the remaining distributions mode 4 appears exclusively for the S-enantiomer. On the other hand, the second highest energy mode, 3, is present either for both enantiomers (A, B) or only for the R-enantiomer (D, E). The results presented so far indicate that the selectivity in our model is dictated by the difference between properties of

ΓS and ΓR that are inherent to a given surface. However, this conclusion refers exclusively to the shapes (or variances) of the distribution functions, not to their mean values 〈〉x, x ) R, S. We would like to emphasize that the mean values 〈〉x, referred to here as the average adsorption energies, are statistical properties of the distributions ΓS and ΓR that are distinct from the Boltzmann-weighted adsorption energies that would be used, for example, to define the isosteric heat of adsorption. For a given n and l in all of the cases studied here, we observed that 〈〉R ) 〈〉S ) 〈〉, regardless of the pattern formed by the active sites. To illustrate this observation, Figure 6 shows 〈〉 as a function of the fraction of the active sites in the unit cell: γ ) n/l2. In Figure 6 all of the datapoints lie on a straight line defined by 〈〉 ) 7.6γ + 0.4, independent of which enantiomer is adsorbed on the surface. This result shows that the average energy of adsorption of either enantiomer is only a function of n and l; that is, it is independent of the way in which the active sites are distributed in the unit cell. In consequence, the enantioselectivity that can arise in these systems is a matter of how the enantiomers share the energy modes offered by the surface. Figure 7 presents the effect of the number of active sites on the Henry’s constants Ki (i ) R, S) calculated for the unit cell of a different size. The data plotted in the figure for a given n and l correspond to the same set of the unit cells for which Eo was plotted in Figures 2 and 4. From Figure 7 it follows that both KR and KS increase markedly with increasing n and that the increase is, in general, more intense for smaller unit cells. This originates from the fact that for a smaller unit cell a lower absolute number of the active sites is needed to saturate the cell such that 〈〉 ) 4. It can also be seen that the gap between

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the solid and the dashed line calculated for a given l increases markedly when l changes from 3 to 5. This confirms the previous results indicating that the maximal Eo becomes larger for a bigger unit cell. The results presented in the preceding paragraphs refer to a chiral molecule whose footprints are nonsuperimposable mirror images composed of four adsorption sites (see the middle part of Figure 1). The footprint of this molecule is one of two possible chiral footprints that can be constructed using four adsorption sites. The bottom part of Figure 1 presents another chiral molecular footprint consisting of four sites. Figure 8 shows the influence of n on the maximal selectivity calculated for these footprints using the screening method described before. For these molecules, the maximal selectivity occurs for n ) 4 for all three of the unit cell sizes we examined. This result differs from the results for the chiral molecule shown in the middle part of Figure 1. Similar to the earlier molecular footprint, however, in Figure 8 we can also observe that the maximal Eo increases when the unit cell size becomes larger. The maximal values of Eo obtained for the patterns F, G and H shown in Figure 2 are 3.11, 2.47, and 2.01, respectively. These values are larger than the maximal selectivities obtained for the footprints from the middle part of Figure 1 for the corresponding cell sizes (cf. Figure 2). As can be seen in the inset of Figure 8, each of the patterns F, G and H contain an assembly of the active sites forming the exact footprint of the S-enantiomer. Thus, the differences in the selectivity observed for those patterns originate mainly from different spacing between the assemblies forming the associated superstructure. The discussion above related to the average adsorption energies also applies to the molecular footprint shown in the bottom part of Figure 1. That is, the average adsorption energy of the S-enantiomer, 〈〉S, is equal to that of the R-enantiomer, 〈〉R, for any specific pattern of active sites. This energy is a function of γ only, being independent of the actual spatial distribution of the active sites. Moreover, the functional relationship between 〈〉 ) 〈〉R ) 〈〉S and γ for the molecules shown in the bottom part of Figure 1 is identical with the result plotted in Figure 6. That is, the average adsorption energy in our model is not dependent on the footprint shape. In consequence, there should exist 〈〉-γ master-curve for any triplet n, l, and m, where m is the number of adsorption sites forming the footprint. B. Equilibrium Adsorption of the Racemate. We now consider the adsorption of enantiomers from a racemic mixture onto the surfaces discussed above. Figure 9 shows the equilibrium adsorption isotherms of the enantiomers calculated for the surfaces composed of the unit cells A, B, D, and E. The results shown in the figure correspond to equimolar mixture of the enantiomers adsorbing on the S-selective surface. As shown in the figure, in all the cases examined the adsorption of the S-enantiomer prevails, especially for low pressures (see panels D and E). When the pressure increases, the distance between the isotherm shrinks. This can be most clearly seen in the case of the patterns D and E. For the latter pattern, we can also observe that the surface coverage of the S-enantiomer at high pressures reaches the largest value among analogous results shown in the figure. Furthermore, the initial parts of the isotherms shown in panel E are much steeper than those of the isotherms from panels A-D. This effect is consistent with the Henry’s constants shown in Figure 7. The adsorption isotherms shown in Figure 9 can be used to define the pressure-dependent selectivity using

E ) θS/θR

(4)

Figure 10. The selectivity as a function of pressure, simulated for the surfaces composed of different patterns of active sites indicated in the figure. Pressure is given in arbitrary units.

where θS and θR are the surface coverages of the enantiomers taken at a fixed pressure. This definition must give limpf0 E ) Eo. Figure 10 shows E plotted as a function of pressure defined inside two intervals. The top part of the figure presents the results obtained for p changing from 0 to 0.5. The bottom part shows a magnified fragment of the curves from the top part, that is the results corresponding to p less than 0.05. Let us first focus on the values of E simulated at the zero pressure limit, that is, at p ) 10-5. The limiting values of E calculated for the patterns A-E are equal to 2.45, 2.65, 2.12, 2.38, and 1.69, respectively. These values are nearly identical with the corresponding values of Eo obtained by means of eq 3, that is, 2.47, 2.64, 2.15, 2.37, and 1.69. As the pressure increases, the behavior of E becomes more complex. First, as we can observe for the pattern E, the selectivity is a nonmonotonic function of pressure. Specifically, the selectivity reaches a maximum at p ≈ 0.007 and it drops gradually as the pressure grows. This behavior is different from the monotonic decrease in the selectivity observed for the remaining systems from Figure 10 as well as for other models of enantioselective adsorption.5,6 Another important observation is that the pattern which maximizes the selectivity at the zeropressure limit does not have to be the optimal pattern at elevated pressures. This can be easily seen for the (5 × 5) unit cell, that is, for the curves A and B. At pressures close to zero E(B) > E(A), but this trend reverses when the pressure exceeds ∼0.02. For this unit cell we can also observe large differences between the selectivities obtained for the patterns with the same number of the active sites, that is, for patterns A and C with n ) 5. At low pressures we have E(A) > E(C), but when the pressure increases, E(C) becomes the largest of any of the selectivities shown in Figure 10. This observation demonstrates that the

11942 J. Phys. Chem. C, Vol. 111, No. 32, 2007 relation between the optimal pattern and the pressure at which adsorption occurs is nontrivial. A similar situation arises for the (3 × 3) unit cell (curve E) for which the selectivity at p ) 10-5 is the smallest among the systems studied. At sufficiently high pressures, the selectivity for pattern E becomes larger than those obtained for the (5 × 5) unit cell using patterns A or B. Systematically searching for the pattern of active sites that optimizes the adsorption selectivity at high pressures would be much more difficult than for the low-pressure limit. In the latter limit, the methods we introduced above were able to find the optimal pattern by essentially considering all possible patterns because the calculations required for each pattern are very simple. At high pressures, however, an independent GCMC simulation is required for every pattern that is considered. As a result, we have not attempted to systematically optimize the patterns that can maximize the high-pressure adsorption selectivities. IV. Conclusions The results of this study show that the MC search technique can be helpful in predicting of the pattern of the active sites leading to the highest enantioselectivity in a model adsorption system. The approach developed here has several practical limitations associated with both adsorbing molecules and chiral surface. For example, our model describes molecules that are rigid and that consist of chemically homogeneous building blocks. Moreover, the presence of intermolecular interactions was neglected. These simplifications can have considerable impact on the effectiveness of adsorption-based separation, especially at high pressures/coverages. We also note that the structure of the adsorbent we used was idealized compared to real materials which often have surface heterogeneity due to defects of chemical disorder. Despite these simplifications, the insights possible from our study should be useful in describing the behavior of more realistic and complex materials. It would not be difficult to extend the lattice model we have used to include heterogeneity in binding site energies or in the chemical functionality of the adsorbing molecules. We have shown that our methods allow for tuning the selectivity by suitable modification of the pattern of the active sites distributed on the adsorbing surface. This general approach requires only knowledge about the configuration of an adsorbed chiral molecule, in particular about the footprint the selected enantiomer leaves on the surface with well-defined adsorption centers. In the zero pressure limit, we demonstrated a screening procedure that rigorously converges to the pattern of active sites maximizing the enantioselectivity of adsorption. Our subsequent GCMC simulations of mixed adsorption indicate that the selectivity calculated at elevated pressures for the optimal pattern does not have to remain maximal among the corresponding

Szabelski and Sholl values obtained for other possible patterns of the active sites. One interesting challenge for future work in this area is to develop methods that allow the optimization of surface patterns for mixture adsorption as a function of pressure. Acknowledgment. This work was supported by the Polish Ministry of Science and Higher Education Grant Number 1 T09A 103 30. PS is grateful to the Polish-U.S. Fulbright Commission for the award of an Advanced Research Scholarship. References and Notes (1) Rampulla, D.; Gellman, A. J. In Dekker Encyclopedia of Nanoscience and Nanotechnology; Schwarz, J. A., Contescu, C. I., Putyera, K., Eds.; Marcel Dekker, Inc.: New York, 2004; pp 1113-1123. (2) Humbolt, V.; Barlow, S. M.; Raval, R. Prog. Surf. Sci. 2004, 76, 1. (3) Maier, N. M.; Franco, P.; Lindner, W. J. J. Chromatogr., A 2001, 906, 3. (4) Ahuja, S. Chiral Separations by Chromatography; Oxford University Press: Washington, DC, 2000. (5) Szabelski, P.; Talbot, J. J. Comput. Chem. 2004, 25, 1779. (6) Szabelski, P. Appl. Surf. Sci. 2004, 227, 94. (7) Szabelski, P.; Kaczmarski, K. J. Chromatogr., A 2006, 1113, 74. (8) Stacchiola, D.; Burkholder, L.; Tysoe, W. T. J. Am. Chem. Soc. 2002, 124, 8984. (9) Lee, I.; Zaera, F. J. Phys. Chem. B 2005, 109, 12920. (10) Ortega-Lorenzo, M.; Haq, S.; Bertrams, T.; Murray, P.; Raval, R.; Baddeley, C. J. J. Phys. Chem. B 1999, 103, 10661 (11) Stacchiola, D.; Burkholder, L.; Zheng, T.; Weinert, M.; Tysoe, W. T. J. Phys. Chem. B 2005, 109, 851. (12) Roma, F.; Zgrablich, G.; Stacchiola, D.; Tysoe, W. T. J. Chem. Phys. 2003, 118, 6030. (13) Roma, F.; Stacchiola, D.; Tysoe, W. T.; Zgrablich, G. Physica A 2004, 338, 493. (14) Lipkowitz, K. B. Acc. Chem. Res. 2000, 33, 555. (15) Choi, Y.-H.; Yang, C.-H.; Kim, H.-W.; Jung, S. Carbohydrate Res. 2000, 328, 393. (16) Lipkowitz, K. B. J. Chromatogr., A 2001, 906, 417. (17) Schefzick, S.; Lindner, W.; Lipkowitz, K. B.; Jalaie, M. Chirality 2000, 12, 7. (18) Lipkowitz, K. B.; Coner, R.; Peterson, M. A. J. Am. Chem. Soc. 1997, 119, 11269. (19) Alvira, E.; Garcı´a, J. I.; Mayoral, J. A. Chem. Phys. 1999, 240, 101. (20) Sholl, D. S.; Asthagiri, A.; Power, T. D. J. Phys. Chem. B 2001, 105, 4771. (21) Sholl, D. S. Langmuir 1998, 14, 862. (22) Power, T. D.; Sholl, D. S. J. Vac. Sci. Technol., A 1999, 17, 1700. (23) Rankin, R. B.; Sholl, D. S. J. Chem. Phys. 2006, 124, 074703. (24) Bhatia, B.; Sholl, D. S. Angew. Chem., Int. Ed. 2005, 44, 7761. (25) Hanzen, R. M.; Sholl, D. S. Nat. Mater. 2003, 2, 367. (26) Szabelski, P.; Sholl, D. S. J. Chem. Phys. 2007, 126, 144709. (27) Ramirez-Pastor, A. J.; Nazarro, M. S.; Riccardo, J. L.; Zgrablich, G. Surf. Sci. 1995, 341, 249. (28) Ramirez-Pastor, A. J.; Riccardo, J. L.; Pereyra, V. Langmuir 2006, 16, 682. (29) Ramirez-Pastor, A. J.; Pereyra, V.; Riccardo, J. L. Langmuir 1999, 15, 5705. (30) Szabelski, P. Appl. Surf. Sci. 2007, 253, 5387.