Atomically Detailed Models of the Effect of Thermal Roughening on

Langmuir 2002, 18, 3737-3748

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Atomically Detailed Models of the Effect of Thermal Roughening on the Enantiospecificity of Naturally Chiral Platinum Surfaces Timothy D. Power, Aravind Asthagiri, and David S. Sholl* Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213 Received October 10, 2001. In Final Form: February 11, 2002 Many high Miller index metal surfaces are naturally chiral and exhibit enantiospecific properties when chiral molecules are adsorbed on them. The structure of real metal surfaces typically deviates from those of perfect Miller index surfaces due to thermally induced surface roughening. Here, the effect of thermal surface roughening on the enantiospecific adsorption of (R/S)-trans-1,2-dimethylcyclopropane and (R/ S)-trans-1,2-dimethylcyclohexane on a series of chiral Pt surfaces vicinal to (111) is studied using atomistic and lattice models. Adsorption is probed in the dilute coverage limit and at higher coverages. The effect of thermal roughening is to diminish the differences between different Miller index surfaces with respect to their enantiospecificity. This implies that a chiral adsorbate that is not enantiodiscriminated on one surface vicinal to (111) with wide terraces is not likely to be enantiodiscriminated on any wide terrace surface vicinal to (111). Thus, the range of surfaces that are available for enantiospecific adsorption is not as large as the set of ideal Miller index surfaces would indicate.

I. Introduction The highly stereoselective nature of biological systems has deep consequences for the bioactivity of opposite enantiomers of chiral species. Enantiomers of pharmaceuticals, for example, may exhibit different efficacy or totally different functions. Thalidomide is a particular example in which the R enantiomer acts as a therapeutic whereas the S enantiomer has caused birth defects.1,2 The growth in the market for enantiomerically pure pharmaceuticals has grown over 3-fold in value over the period 1994-1999 reaching U.S. $115 billion in 1999.3,4 Preparation of enantiopure materials is often difficult. A potentially attractive way to discriminate between enantiomers is through the interaction of chiral species with solid surfaces that are chiral. One approach that has been successfully applied in the past involves the irreversible adsorption of chiral species on a nonchiral substrate.5,6 This process renders the surface chiral. Gellman and co-workers were the first to show that it is possible to prepare pure metal surfaces that are intrinsically chiral.7 An example of such a surface is shown in Figure 1a. The kinked surface steps on this surface render the surface chiral. A detailed description of the structure of chiral metal surfaces is given in a recent review article.8 Initial efforts at detecting enantiospecificity under ultrahigh vacuum (UHV) conditions on chiral Ag(643) were unsuccessful.7 Following theoretical predictions that chiral Pt surface should be enantiospecific,9 Attard and co* Corresponding author. Fax: (412) 268-7139. E-mail: sholl@ andrew.cmu.edu. (1) Muller, G. W.; Konnecke, W. E.; Smith, A. M.; Khetani, V. D. Org. Process Res. Dev. 1999, 3, 139-140. (2) Takeuchi, Y.; Shiragami, T.; Kimura, K.; Suzuki, E.; Shibata, N. Org. Lett. 1999, 1, 1571-1573. (3) Stinson, S. C. Chem. Eng. News 1994, 72, 38. (4) Stinson, S. C. Chem. Eng. News 2000, 78, 55. (5) Blaser, H. U.; Jalett, H. P.; Lottenbach, W.; Studer, M. J. Am. Chem. Soc. 2000, 122, 12675-12682. (6) Lorenzo, M. O.; Baddeley, C. J.; Muryn, C.; Raval, R. Nature 2000, 404, 376-379. (7) McFadden, C. F.; Cremer, P. S.; Gellman, A. J. Langmuir 1996, 12, 2483-2487. (8) Sholl, D. S.; Asthagiri, A.; Power, T. D. J. Phys. Chem. B 2001, 105, 4771-4782.

workers used electrochemical techniques to experimentally observe enantiospecificity on naturally chiral surfaces for the first time.10,11 Gellman and co-workers have since detected enantiospecificity with chiral Cu surfaces in UHV.12,13 In theoretical 8,9,14 and experimental10-13,15,16 work, it has been found that not all chiral adsorbate/ surface interactions are enantiospecific. To date, it has not been possible to elucidate simple rules that predict a priori what surfaces will or will not discriminate between given enantiomers of an adsorbate. Previous work on adsorption on naturally chiral surfaces has focused on surfaces that are envisioned as perfectly periodic Miller index surfaces.9-13,16,17 Under almost all practical conditions, however, surfaces are subjected to temperatures at which metal atoms on the surface can spontaneously diffuse. This so-called thermal roughening can result in gross distortions in surface structure.18 The effect of roughening on surface structure is shown pictorially in Figure 1, which shows an initially periodic Pt surface before and after the surface is held at 500 K for 60 min. The details of the model used to generate these images are given in section II. Typical UHV experiments on chiral metal surfaces involve cleaning regimes that involve heating the surface to temperatures exceeding 500 K, followed by adsorption experiments performed at room temperature and below.7,12,13,15 It is therefore likely that the surfaces used in these experiments are more like the one pictured in Figure 1b than the perfectly periodic (9) Sholl, D. S. Langmuir 1998, 14, 862-867. (10) Attard, G. A.; Ahmadi, A.; Feliu, J.; Rodes, A.; Herrero, E.; Blais, S.; Jerkiewicz, G. J. Phys. Chem. B 1999, 103, 1381-1385. (11) Ahmadi, A.; Attard, G.; Feliu, J.; Rodes, A. Langmuir 1999, 15, 2420-2424. (12) Gellman, A. J.; Horvath, J. D.; Buelow, M. T. J. Mol. Catal., A 2001, 167, 3-11. (13) Horvath, J. D.; Gellman, A. J. J. Am. Chem. Soc. 2001, 123, 7953-7954. (14) Power, T. D.; Sholl, D. S. Top. Catal. 2002, 18, 201-208. (15) Horvath, J. D.; Gellman, A. J.; Sholl, D. S.; Power, T. D. In Physical Chemistry of Chirality; Hicks, J. D., Ed.; American Chemical Society: Washington, DC, 2002, p 269-282. (16) Attard, G. A. J. Phys. Chem. B 2001, 105, 3158-3167. (17) Power, T. D.; Sholl, D. S. J. Vac. Sci. Technol., A 1999, 17, 17001704. (18) Jeong, H. C.; Williams, E. D. Surf. Sci. Rep. 1999, 34, 175-294.

10.1021/la011535o CCC: $22.00 © 2002 American Chemical Society Published on Web 03/30/2002

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Power et al.

Figure 1. Ball model of (a) nonroughened and (b) roughened chiral fcc(643) surface. The surface shown in (b) is produced from that in (a) by thermal roughening for 60 min at 500 K. The step-edges are highlighted.

structure shown in Figure 1a. The aim of this paper is to use atomically detailed models of molecular adsorption to assess how the thermal roughening typified by Figure 1 changes the enantiospecific adsorption properties of chiral metal surfaces relative to their idealized Miller index states. To do this, we apply a lattice model we developed recently19 to predict the structure of a series of thermally roughened Pt surfaces vicinal to Pt(111) with atomic-scale detail. We then examine the adsorption of two chiral hydrocarbons, (R/S)-trans-1,2-dimethylcyclopropane (R/ S-DMCPr) and (R/S)-trans-1,2-dimethylcyclohexane (R/ S-DMCH), on these surfaces using atomistic methods we have previously developed for studying ideal high Miller index surfaces.8 The former molecule is a small molecule with little flexibility; the latter has a larger ring and can adopt several stable conformations.8,14 We show that, for both species, thermal roughening diminishes the differences between chiral surfaces vicinal to the same lowindex plane, effectively reducing the choice of substrates available for enantiospecific adsorption. The rest of this paper is organized as follows. In section II, we describe the lattice gas calculations used to predict roughened surface structures. In section III, we study the adsorption of the chiral hydrocarbons on roughened surfaces in the dilute coverage limit, i.e., when only one molecule is adsorbed on the surface. Our results are discussed in terms of both the adsorption states observed at low temperatures and the desorption behavior accessible from temperature-programmed desorption (TPD) experiments. The extension of our model to the coverage dependence of TPD spectra is described in section IV. We conclude in section V with a summary of our results and a discussion of strategies relevant to the problem of creating surfaces that are highly enantiospecific with respect to chiral adsorbates. II. Thermal Roughening of Chiral Pt Surfaces The starting point for a modeling study of thermally roughened surfaces is a description of the roughened surfaces’ structure. Because thermal roughening is driven by the surface diffusion of metal atoms, it is necessary to use a model that incorporates an accurate description of these processes. We have recently developed a lattice gas (LG) model using adatom diffusion barriers calculated by density functional theory (DFT) to quantitatively predict the effect of thermal roughening on the step structure of chiral Pt surfaces vicinal to Pt(111). A detailed description of the model can be found in ref 19, but we will outline its important assumptions and features below. (19) Asthagiri, A.; Feibelman, P. J.; Sholl, D. S. Top. Catal. 2002, 18, 193-200.

Our goal is to be able to make quantitative predictions on an atomic level of the roughened step structure of chiral Pt surfaces. This is possible if one knows the mechanism and rate of all the relevant surface diffusion processes. To limit the number of processes that must be considered, we restrict our attention to single surface steps bounded on both sides by (111) planes. That is, we consider only Pt surfaces vicinal to (111) with large step-step spacings. We discuss the effect of step spacing on enantiospecific adsorption in section III. On fcc surfaces vicinal to (111), kinks occur when a step edge with local (100) orientation intersects a step with local (110) orientation.8 Such steps are often referred to as A and B steps, respectively.8 We refer to kinks where the A step (B step) is the longer of the two step edges as being A-type (B-type) kinks. Kinks with equal A and B step lengths such as fcc(531) also exist and are chiral. A detailed description of these issues is provided in ref 8. The energy barriers to many of the relevant diffusion processes for these surfaces have been calculated using DFT by Feibelman.20 Energy barriers calculated via DFT for Pt(111) and Pt(331) are in good agreement with experimental results, which serves as an example of the accuracy of this technique.20 For this work we use the DFT results computed using the local density approximation (LDA). Results from the generalized gradient approximation (GGA) are qualitatively similar but not as quantitatively accurate.20 Although many energy barriers for Pt adatom diffusion have been calculated using DFT,20 it is not feasible to examine every process available to atoms on a surface. To overcome this difficulty we have derived a LG model for Pt diffusion along step edges on the basis of the barriers known from DFT. There are four assumptions in this model. First, we consider only diffusion of single Pt atoms via direct hopping mechanisms. DFT calculations indicate that these are preferred over concerted substitution mechanisms.20 Second, we do not allow step edge atoms to detach from the step, restricting our model to so-called periphery diffusion.18 DFT calculations indicate that the energy barrier for a step edge atom to move from a step to a terrace on these surfaces is much larger than diffusion along the step.19 STM experiments also indicate that periphery diffusion is the dominant mechanism for Pt steps on a range of relevant time scales with temperatures as high as 800 K.21 Third, we do not allow atoms to diffuse from 7- or 8-fold-coordinated sites in the step. Finally, the rate of each diffusion process is given by k ) ν exp(-Eb/ (20) Feibelman, P. J. Phys. Rev. B 1999, 60, 4972-4981. (21) Giesen, M.; IckingKonert, G. S.; Stapel, D.; Ibach, H. Surf. Sci. 1996, 366, 229-238.

Atomically Detailed Models of Thermal Roughening

kBT), where the preexponential factor is fixed to be ν ) 1013 s-1 for all processes and Eb is the process-dependent energy barrier. We used an extended bond counting scheme fitted to the known barriers to predict the barrier to any diffusion move along the step.19 The scheme includes 13 parameters that are determined using a least-squares fit to the 18 independent barriers known from DFT. One useful parameter for assessing the accuracy of our LG model is the effective energy barrier associated with step roughening, Eeff.19,21,22 This barrier is a function of several single atom diffusion barriers. For our purposes, we note that Eeff can be determined directly from the subset of diffusion barriers known from LDA-DFT: EeffDFT ) 0.61 (0.52) eV for B (A) steps.19 In our earlier work we equally weighted all the known DFT barriers when fitting the parameters of our LG model.19 This nonweighted approach yields EeffLG ) 0.56 (0.57) eV for B (A) steps. Although qualitatively correct, these results fail to capture the energetic differences between roughening of A and B steps. To address this issue, we have reparametrized our LG model by weighting the known diffusion barrier from a 6-foldcoordinated kink site to a 5-fold-coordinated straight step site as five times more important than the other known barriers. This revised LG model gives ELG eff ) 0.61 (0.53) eV, in excellent agreement with the DFT results. The rms deviation between the barriers predicted by this LG model and the 18 known DFT barriers is 0.047 eV. For completeness, the parameters and definition of our LG model are given in Appendix A. Our LG model gives a self-contained description of Pt step roughening. The dynamics of this model can be simulated using kinetic Monte Carlo (KMC). We use the N-fold way algorithm19,23 to select events and increment time. This algorithm is ideal for systems with a wide range of time scales and allows us to follow roughening on experimentally relevant time scales of minutes to hours. We show a typical result from our simulations in Figure 1, which shows the initial and final step structure of a step that was allowed to roughen for 60 min at T ) 500 K. The initial step was chosen to have the same structure as the steps on a chiral Pt(643) surface.7,8 The isolated step in our simulation has been mapped into Figure 1 using periodic boundary conditions for illustrative purposes. The step shows considerable rearrangement after roughening. While there was only one type of kink in the nonroughened surface, after roughening there are a distribution of kinks of various structures. We have emphasized in our earlier work that despite the disruption of the step structure that occurs due to thermal roughening, roughened steps that began as chiral surfaces are still chiral.8,19 One way to see this important fact is that in Figure 1b the kinks always move “up” the page and never “down” the page. The appearance of just one “direction” of step kinks is a natural feature of ideal chiral metal surfaces, and this property is retained on roughened surfaces because these surfaces retain the same average step direction as the original ideal surface from which they evolve. The net chirality of roughened surfaces such as the one in Figure 1b will also become apparent when we discuss adsorption of chiral molecules on these steps in the next section. The distribution of step lengths can be examined by defining PA(L) [PB(L)] to be the probability that a randomly chosen A [B] step has a length L (in lattice units). This (22) Natori, A.; Godby, R. W. Phys. Rev. B 1993, 47, 15816-15822. (23) Bortz, A. B.; Kalos, M. H.; Lebowitz, J. L. J. Comput. Phys. 1975, 17, 10-18.

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is a measure of the local step structure. We find that these distributions equilibrate in experimentally relevant time scales of less than 5 min at temperatures of 500 K or above for all the Pt surfaces we discuss in this paper8,19 and are only very weakly dependent on the roughening temperature. We have also determined that the distribution of step lengths that appear on roughened surfaces are uncorrelated. That is, the appearance of one type of step length does not influence the identity of the adjacent step lengths.8 We also observe that the time scale for significant rearrangement of the surface becomes extremely long (.1 day) at room temperature and below. Thus, to model real surfaces that are used in experiments at room temperature and below, we can generate a roughened structure using a simulation at 500 K or above to generate a structure that would arise during thermal annealing and then hold the surface structure fixed when we consider the surface’s low-temperature properties. The step length distribution, PA(L) and PB(L), for each step structure discussed in this paper are averaged over 20 simulations, each containing step lengths of 2000 atoms and roughened at T ) 500 K for t ) 1 h. Once we have measured PA(L) and PB(L) for a particular step structure, we can generate roughened steps of arbitrary length by alternately choosing step lengths at random from PA(L) and PB(L). This procedure is valid because, as noted above, our detailed simulations show that the identity of adjacent step lengths on roughened steps is uncorrelated. All of the roughened structures used in the remainder of this paper were generated using this method. It is interesting to note that experimental studies of naturally chiral surfaces using LEED have typically shown sharp diffraction patterns.7,10,15,16 Although it is tempting to conclude from these sharp patterns that real chiral surfaces exhibit very little disorder, there is considerable evidence from theoretical studies of LEED that very irregular arrays of steps can result in nearly ideal LEED patterns.24,25 Thus, the experimental data currently available on the structure of chiral metal surfaces are consistent with the surfaces having disordered structures such as the one shown in Figure 1b. III. Enantiospecific Adsorption in the Dilute Coverage Limit Having described our model for thermally roughened chiral Pt surfaces, we now move to describing molecular adsorption on these surfaces. We have modeled adsorption of two chiral hydrocarbons, (R/S)-trans-1,2-dimethylcyclopropane (DMCPr) and (R/S)-trans-1,2-dimethylcyclohexane (DMCH), on roughened surfaces generated from five chiral Miller index surfaces: Pt(531)S; Pt(643)S; Pt(653)S; Pt(854)S; Pt(874)S. These surfaces have been chosen since they contain members of the three fundamental kink structures present on surfaces vicinal to (111).8 For a detailed description of chiral surface structure and the R/S notation used to distinguish between enantiomorphic surface pairs, the reader is referred to our recent review article.8 In this paper we approximate the periodic terrace structure of these Miller index surfaces by surfaces with infinite terrace width, i.e., surfaces containing a single step-edge. The nonroughened surfaces have periodic kinks of fixed structure along the step-edge, whereas the roughened surfaces contain a distribution of kinks generated as described in section II. In this section we study adsorption in the dilute limit, i.e., where a single adsorbate molecule is adsorbed on the (24) Rhead, G. E. Surf. Sci. 1977, 68, 20. (25) Ertl, G.; Kuppers, J. Low Energy Electrons and Surface Chemistry, 2nd ed.; VCH Publishers: Deerfield Beach, FL, 1985.

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surface. Following our previous work, we have modeled the interaction of the adsorbate atoms with the surface Pt atoms using a potential devised by Weaver and Madix.8,26 Briefly, the adsorbate hydrocarbon is modeled as a set of united atoms that interact with each Pt atom via a Morse potential. The potential was parametrized to fit scattering data for neopentane on Pt(111).26 This potential accurately predicts the observed binding energies for isobutane, n-butane, and neopentane on Pt(111). Within each molecule, bond bending, bond stretching, torsion, and antiinversion potentials are used as described in our earlier work.8,14 In addition to thermal roughening, metal surface structures deviate from ideality by virtue of the local relaxation of surface atoms required to balance surface forces. The impact of this phenomenon on enantiospecific adsorption has been studied by us previously.14 In that work the structure of relaxed surfaces had been determined using DFT. It was found that chiral Pt surfaces exhibited greater relaxation than atomically flat surfaces such as Pt(111). The amount of relaxation varied along the kinked step edge with the greatest degree of relaxation being exhibited by the least coordinated atoms. Although relaxation does not result in a gross distortion of surface structure, adsorption enantiospecificity is altered when it is taken into account. In the majority of cases we studied, using a relaxed surface structure slightly reduced the magnitude of enantiospecificity but did not change the main qualitative features of these systems.14 It is impractical to apply DFT to nonperiodic systems such as those necessary to describe roughened surfaces. This means that quantitatively determining the fully relaxed geometry of surfaces such as those in Figure 1b is not currently possible. As such, all of the results in this paper, both for roughened and nonroughened surfaces, correspond to surfaces where all atoms are in the positions obtained from their geometry in the bulk solid. Nevertheless, the known changes in enantiospecificity due to relaxation on periodic surfaces14 strongly suggest that the conclusions we draw in this paper are also applicable to fully relaxed roughened surfaces. A useful measure of the enantiospecificity of adsorption is the difference in binding energies between enantiomers on a chiral surface. To this end, we define ∆U ) 〈UR〉 〈US〉, where 〈‚‚‚〉 indicates a canonical average at temperature T and UR and US are the potential energy of the R and S enantiomers, respectively, on the surface.8,9,14,17 Initially, we examine the enantiomeric energy difference at 0 K, denoted ∆U0. Since physisorbed species are weakly bound to the surface, they can adopt many adsorption geometries. Each local minimum of the potential energy surface can therefore be regarded as a distinct adsorption state. Our method for identifying adsorption states is described elsewhere.8,14 Briefly, we sample the full potential energy surface of the adsorbed molecule using Monte Carlo sampling and then minimize the energy of many configurations to locate local energy minima. On a chiral surface, the kink provides greatest coordination to an adsorbate. Therefore, the global minimum energy state of physisorbed species is typically located at a kink site. The global minimum energy state for an adsorbed species in the dilute limit will be the state that it occupies at T S ) 0 K. Therefore, ∆U0 ) UR min - Umin, where Umin is the energy of the global minimum energy state for the adsorbate being considered. In Table 1 we present ∆U0 for DMCPr and DMCH on all five (single-step) roughened and nonroughened surfaces. We note that energy differences of a fraction of a kcal/mol are known to be sufficient to allow effective

Power et al. Table 1. ∆U0 for DMCH and DMCPr on Five Roughened and Nonroughened Surfacesa surf type

Pt(531)S Pt(643)S Pt(653)S Pt(854)S Pt(874)S

nonroughened roughened

-0.66 -0.76

nonroughened roughened

0.02 0.03

a

DMCPr -0.45 -0.85 -0.76 -0.76 DMCH 0.03 0.03

0.38 0.03

-0.23 -0.76

-0.72 -0.76

0.02 0.03

0.36 0.03

Energy is shown in kcal/mol.

separation of enantiomers in chromatographic applications.8,27 For each adsorbate, ∆U0 varies between the different nonroughened surfaces, but ∆U0 is identical on every roughened surface. That is, with respect to ∆U0, roughening renders the surfaces indistinguishable. This observation can be understood as follows. Kinks are structures that are present where two step edges meet. On all nonroughened Miller index surfaces of fcc metals, at least one step-edge forming a kink is one atom in length.8,19 Thermal roughening allows rearrangement of atoms on a surface, leading to structures that are not found on nonroughened surfaces. Specifically, roughened surfaces contain kinks where both step edges can be more than one atom in length. We call these structures nonMiller index kinks (NM kinks).8,19 An example of a NM kink is shown in Figure 2. NM kinks offer physisorbed molecules a means to increase their coordination to surface atoms relative to standard Miller index kinks. As an example, the global minimum energy state for R-DMCPr is shown on an NM kink and on the nonroughened infinite terrace analogue of Pt(643)S in Figure 2. In this example, the adsorbate is more firmly bound to the NM structure by 0.91 kcal/mol. Examples of NM kinks can also be seen in Figure 1b. The lowest energy state of an adsorbate within an NM kink is typically positioned close to where the step edges meet. Thus, once the steps defining a NM kink are longer than a critical size, they make no further contribution to an adsorbate’s binding energy. All NM kinks that are greater in size than the critical size will therefore yield the same minimum energy state at the kink center. Furthermore, these NM kinks provide the greatest possible coordination and therefore the tightest possible binding for phyisorbed species. Consequently, the global minimum energy state for a physisorbed molecule on a roughened surface is typically located at a NM kink that is larger than the critical size. is independent of the initial The fact that ∆Urough 0 surface structure can now be rationalized by noting that NM kinks are structurally independent of the roughened surface’s initial structure. For example, the NM kink shown in Figure 2 is an S-NM kink and was found on a roughened Pt(643)S surface. However it can also be found on roughened Pt(854)S or any S-oriented roughened surface vicinal to (111). A complete description of which Miller index surfaces fall into this category can be found in ref 8. Within this class of surfaces, the ideal (i.e. nonroughened) surface structure merely dictates the relative abundance of different kink structures on the roughened surface. Thus, the global minimum energy state for a physisorbed molecule will be identical on any S-oriented roughened surface vicinal to (111). Since the same set of kink structures (with opposite handedness) (26) Weaver, J. F.; Madix, R. J. J. Chem. Phys. 1999, 110, 1058510598. (27) Lipkowitz, K. B.; Coner, R.; Peterson, M. A.; Morreale, A.; Shackelford, J. J. Org. Chem. 1998, 63, 732-745.


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Figure 2. R-DMCPr adsorbed in (a) a Miller index (ideal) kink and in (b) a non-Miller index kink. The molecule is more firmly bound to the kink in (b) than in (a) by 0.91 kcal/mol. The step-edges are highlighted with the A step (B step) shown as a dashed (solid) line.

will appear on R-oriented roughened surfaces, ∆Urough 0 will be the same for all surfaces in this class and opposite in sign to the S-oriented roughened surfaces. Our discussion to this point has focused on enantiospecific adsorption in the dilute limit at T ) 0 K. We now consider higher temperatures, where the global minimum energy state is no longer the only relevant adsorption state. To do so, we apply a harmonic approximation to the potential energy surface.28 This approximation expresses the potential energy surface as a set of harmonic oscillators centered at each local minimum of the full potential energy surface. The harmonic approximation is most accurate at low T. With this approximation, Nstates

〈U〉 )

∑i niEi exp(-Ei /kT)

Nstates

∑i

(1)

ni exp(-Ei /kT)

where Ei (ni) is the binding energy (relative degeneracy) of state i and there are Nstates distinct adsorption states. This expression assumes that the product of normal-mode frequencies for each state is equal. We have performed calculations in which actual normal-mode frequencies are included and repeated them when their product is equal; the value of 〈U〉 calculated for each case is indistinguishable. Once the adsorption energies and their relative degeneracies {Ei,ni} are identified, eq 1 gives 〈U〉 as a function of T. Our goal, therefore, for modeling physisorption on roughened surfaces is to find {Ei,ni} for each system of interest. In previous work,8,14 we have performed extensive Monte Carlo (MC) simulations of the adsorption of DMCH and DMCPr on a series of Miller index surfaces, specifically, Pt(531)S, Pt(643)S, Pt(854)S, and Pt(874)S. In addition, we have performed analogous simulations of Pt(653)S and achiral Pt(322) and Pt(332). The latter two surfaces are vicinal to Pt(111) with straight A-type and B-type step edges, respectively. This set of seven surfaces spans all fundamental structures present on ideal surfaces vicinal to (111).8 From these MC studies, we have isolated the individual adsorption states on each surface for each (28) Fowles, G. R. Analytical mechanics, 4th ed.; Saunders College Pub.: Philadelphia, PA, 1986.

adsorbate we have simulated.8 The adsorption states located on this set of (ideal) Miller index surfaces forms the basis of our calculations to determine {Ei,ni} on roughened surfaces. For each adsorbate on each roughened surface we considered we used the following scheme: (1) We construct a roughened surface step of fixed length as described in section II. Every atom along the resulting step edge is considered to be an independent site. (2) The three lowest energy states from each nonroughened Miller index surface listed above (a total of 21 states) are mapped into the coordinate system of the roughened surface. (3) The first of the 21 mapped states is placed at a site at one end of the roughened step edge, and its energy is minimized. If the minimized structure is not located too far away (0.75 of a lattice unit or more) from the starting position, the energy of the minimized structure is recorded and associated with that site; otherwise the minimum is rejected and no record is made. (4) The unminimized mapped state is moved to the next site along the roughened step edge, and the same minimization procedure is carried out. (5) Step 4 is repeated along the step-edge to the end. Steps 3-5 are repeated with the rest of the 21 mapped states. For every state at each site, provided that its minimum state is acceptable, the minimum energy is compared with the existing states associated with that site. If the energy is new, it is recorded as a new state associated with that site. For all the results below, we performed simulations on five realizations of each roughened surface with 2000 atoms along the step edge. Each adsorption site j along the (total) roughened step-edge is associated with a number of accessible energy states, qjtot, with corresponding energies {Ejq} (q ) 1, 2, ..., qjtot). This information proves useful for studying high-coverage effects. By compiling these data, we identify all distinct adsorption energies and their relative degeneracies, {Ei,ni}, along each step-edge. This procedure does not include any adsorption on the terrace. Adsorption on the terrace is considerably less energetically favorable than adsorption against the step-edge on account of the terrace’s lower coordination. Before returning to the effects of surface roughening, we can use the technique described above to examine the differences between ideal Miller index chiral surfaces and

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Figure 3. ∆U is shown for DMCH (circles) and DMCPr (triangles) adsorbed on a Miller index (unfilled symbols) and single step-edge (solid symbols): (a) Pt(531)S and (b) Pt(874)S kink structure surfaces.

surfaces with just one step surrounded by extremely wide terraces. In Figure 3, ∆U is shown for DMCH and DMCPr adsorbed on Miller index surfaces and on surfaces with the same step structure but infinitely wide terraces. In each case, ∆U was calculated using eq 1 after finding {Ei,ni}. When the adsorption energy on Miller index surfaces is dominated by interactions that do not span more than one terrace, it is reasonable to expect that the finite and infinite terrace structures yield similar results. From this figure, it is clear that, for Pt(531)S, the infinite terrace approximation to the true finite terrace structure is poor (see Figure 3a). This is not surprising since the fcc(531) surface has practically no terrace.8,16 However, the comparison between the finite and infinite terrace data for the Pt(874)S structures is good (see Figure 3b). The terraces in the Miller index Pt(874)S surface are sufficiently wide for interterrace adsorption interactions to be mild. We have observed similar results with the other chiral surfaces for the adsorbates we have examined; with the exception of Pt(531)S, the representation of surface structures as single steps surrounded by extremely large (111) terraces captures the adsorption properties of the Miller index surfaces well. For this reason, we only present results from steps with infinitely wide terraces below. For convenience we will refer to these steps using their Miller index analogue in the remainder of this paper. Although this notation is convenient, it is more accurate to consider each surface used below as representing a family of stepped surfaces with identical step structures and variable terrace widths with the caveat that the terrace widths are larger than the adsorbates we consider.

Power et al.

In Table 1, the effect of roughening on ∆U at 0 K was shown. In Figure 4, ∆U is shown over a temperature range for DMCH and DMCPr on the series of roughened and nonroughened surfaces. Our main observation from 0 K holds true at higher temperatures also; while adsorption behavior on nonroughened surfaces varies from surface to surface, the differences between surfaces are diminished by roughening. For both DMCPr and DMCH the value of ∆U ) UR - US on roughened surfaces is identical at low T, but the ∆U curves for different surfaces separate as T increases. Once the curves separate, the data can be ordered by surface type. That is, the steps with longer B-type steps29 (namely Pt(874)S and Pt(653)S) give similar results and the same is true for A-type stepped surfaces (Pt(854)S and Pt(643)S). Pt(531)S, which has equal lengths of A and B steps, lies between the other two groups of surfaces. This behavior can be understood by consideration of the distinct adsorption states on each surface. For roughened surfaces, the low-temperature behavior is dominated by the global minimum energy state of each enantiomer, which is the same for a given enantiomer of an adsorbate on all wide terrace surfaces vicinal to (111) (see above). At higher temperatures other states become important. Although many of the same states appear on each surface, the relative degeneracies of each state vary from surface to surface. This leads to the variation in ∆U with surface structure at higher temperatures. ∆U ) 〈UR〉 - 〈US〉 is a useful way to characterize adsorption enantiospecificity in our calculations, but it is not precisely the quantity measured in experimental studies of chiral surfaces. Attard and co-workers study differences in electrochemical responses to adsorption and have related these to adsorption energies.10,11,16 In the UHV experiments by Gellman et al., temperatureprogrammed desorption (TPD) is used to infer a barrier to desorption for an adsorbed layer.7,12,15 In a TPD experiment the temperature of the peak desorption rate can be used to determine a desorption barrier using the Redhead analysis if first-order desorption kinetics are assumed.30 The experimental observable is the desorption rate as a function of time, dn/dt, where n is the number of adsorbed molecules. For first-order kinetics, the desorption rate can be written in terms of an effective rate constant r as

dn ) nr dt

(2)

The desorption of molecules located at sites with energy Ei is described by ri ) υ exp(Ei/kT)Pi, where Pi is the probability of observing a molecule in that state. For a dilute system of adsorption states with energies/relative degeneracies given as {Ei,ni},

Pi )

ni exp(-Ei /kT) Nstates

ni exp(-Ei /kT) ∑ i)1

Combining these expressions and summing over all states yields r as shown in Appendix B. The binding energy of state i, Ei, is negative, and ν is the desorption frequency (which we assumed to be the typically accepted 1013 s -1). We can integrate eq 2 using r to predict the TPD result that would be observed for a dilute adlayer using {Ei,ni} (29) Horvath, J. D.; Gellman, A. J. J. Am. Chem. Soc. 2002, 124, 2384-2392. (30) Redhead, P. A. Vacuum 1962, 12, 203.


Langmuir, Vol. 18, No. 9, 2002 3743

Figure 4. ∆U as a function of temperature for DMCH on (a) nonroughened surfaces and (b) roughened surfaces and for DMCPr on (c) nonroughened surfaces and (d) roughened surfaces.

for our model system. Figure 5 shows TPD spectra calculatedin this way for R/S-DMCH and DMCPr on roughened and nonroughened Pt(531)S and Pt(874)S surfaces. All TPD spectra described in this paper were generated with the heating rate β ) 1 K/s. The temperature of the peak desorption rate, Tp, can be found either by curve-fitting the data in Figure 5 or by using the analytic method described in Appendix B. Both methods give equivalent results. It is apparent from Figure 5 that the TPD spectrum for both adsorbates on the Pt(874)S surface is virtually unaffected by roughening (see Figure 5b,d), whereas roughening has a significant impact on adsorption onto Pt(531)S (Figure 5a,c). This occurs because roughening results in an increase of the average local coordination of step-edge atoms, but the relative change compared to the initial structure is highest for the step-edge with the highest kink density, namely Pt(531)S. After determining Tp from our TPD data, we follow the experimental procedure of using the Redhead analysis to assign a desorption barrier to the TPD data.7,12,15,30 As in experimental studies,7,12,13,15 we assume a constant preexponential factor of ν ) 1013 s-1 in our Redhead analysis. The results are presented in Tables 2 and 3. The enantiospecificity measured as the differences in desorpS tion barriers, ∆Udes ) (UR des - Udes), is also shown in Tables 2 and 3. This is precisely the property measured experimentally by Gellman et al.7,12,15 It is interesting to note that, for these examples, roughening changes the TPD curves for Pt(531)S significantly, but ∆Udes is not greatly changed. The values of ∆Udes measured using TPD are not the same as those discussed above for lowtemperature adsorption, ∆U0. Comparing Tables 2 and 3 with Table 1 shows that, in each of the 8 examples

considered here, the sign of ∆U0 and ∆Udes are the same. There are several examples where the magnitude of the energy difference measured using TPD is reduced relative to the low-temperature value. This is not a universal observation however; the enantiospecificity of DMCH on both roughened and nonroughened Pt(531)S is negligible, but there is considerable enantiospecificity for these two cases when desorption is measured using TPD. We have performed similar dilute limit TPD calculations for Pt(854)S, Pt(653)S, and Pt(643)S. Like Pt(874)S, nonroughened Pt(854)S has relatively long steps in its structure; therefore, it is not surprising that TPD curves from this surface are not greatly affected by roughening. Both nonroughened Pt(643)S and Pt(653)S have step lengths that are intermediate between the long Pt(854)S/ Pt(874)S structures and the short Pt(531)S structures. TPD curves from these surfaces exhibit some changes on roughening but are not so dramatic as those from Pt(531)S. The ∆Udes data for DMCH and DMCPr on the entire series of roughened and nonroughened surfaces are given in Table 4. An interesting observation can be made from viewing the data for a given enantiomer of an adsorbate on the full series of roughened and nonroughened surfaces. In Figure 6 such data are shown for R-DMCPr. It is clear from Figure 6 that as expected, different nonroughened surfaces yield quite different TPD curves. Surfaces with long A-type step edges (namely Pt(643)S and Pt(854)S) give similar results. The same can be said for surfaces with long B-type step edges (Pt(653)S and Pt(874)S). As described above, Pt(531)S does not fall into either of these categories, and the curve resulting from this surface in Figure 6a is quite different from the other four curves. For our purposes, the most important observation from Figure 6 is that the

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Figure 5. Dilute limit TPD spectra for (a) DMCH on Pt(531)S, (b) DMCH on Pt(874)S, (c) DMCPr on Pt(531)S, and (d) DMCPr on Pt(874)S. Curves for nonroughened (roughened) surfaces are shown with filled (unfilled) symbols, and circles (triangles) represent the R (S) enantiomer. Table 2. Peak Temperatures and Desorption Barriers for R/S-DMCH and R/S-DMCPr on Roughened and Nonroughened Pt(874)S in the Dilute Limita DMCPr

DMCH

Tp (K) Udes (kcal/mol) Tp (K) Udes (kcal/mol) R nonroughened R roughened S nonroughened S roughened ∆Udesnonroughened ∆Udesroughened

299.93 301.18 294.09 294.92 -0.38 -0.41

-19.16 -19.24 -18.78 -18.83

446.36 444.66 444.39 446.22 -0.13 0.11

-28.86 -28.74 -28.73 -28.85

a Energies are negative for consistency with energy datum as gas-phase minimum energy. Enantiospecificity is shown also.

Table 3. Peak Temperatures and Desorption Barriers for R/S-DMCH and R/S-DMCPr on Roughened and Nonroughened Pt(531)S in the Dilute Limita DMCPr

DMCH

Tp (K) Udes (kcal/mol) Tp (K) Udes (kcal/mol) R nonroughened R roughened S nonroughened S roughened ∆Udesnonroughened ∆Udesroughened

294.94 303.12 287.97 296.44 -0.46 -0.44

-18.83 -19.37 -18.37 -18.93

437.52 448.45 440.14 451.11 -0.17 0.18

-28.27 -29.00 -28.44 -29.17

a Energies are negative for consistency with energy datum as gas-phase minimum energy. Enantiospecificity is shown also.

differences between the curves are greatly diminished for roughened surfaces. This observation is true for both enantiomers of each adsorbate that we have examined. Tables 1 and 4 summarize our dilute limit data. They show the dilute coverage limit enantiospecificity as both

Table 4. Dilute Limit Enantiospecificity, ∆U, for DMCPr and DMCH on the Series of Roughened and Nonroughened Surfaces As Determined from the Simulated TPD Spectrum ∆U

Pt(531)S Pt(643)S Pt(653)S Pt(854)S Pt(874)S

roughened nonroughened

-0.44 -0.46

roughened nonroughened

0.18 0.17

DMCPr -0.43 -0.43 -0.50 -0.47 DMCH 0.18 0.23

0.15 0.01

-0.39 -0.45

-0.41 -0.38

0.18 0.23

0.11 -0.13

the 0 K difference in binding energy, ∆U0, and the difference in desorption barriers assigned from TPD, ∆Udes, for DMCH and DMCPr on the series of roughened and nonroughened surfaces. The same data are shown pictorially in Figure 7. In the dilute limit, whether enantiospecificity is measured as a difference in binding energies or as a difference in TPD desorption barriers, roughening effectively diminishes the differences between the surfaces vicinal to (111). This implies that the range of surfaces that appear to be available for enantiodiscrimination is greatly reduced. Thus, if a given wide terrace surface vicinal to (111) is not found to discriminate between enantiomers of a particular adsorbate, then it is not likely that any wide terrace surface vicinal to (111) will do so either. IV. Enantiospecific Adsorption at High Coverage The discussion above has addressed adsorption in the limit of dilute coverage only. In this section we examine enantiospecific TPD at higher coverages with the assumption that the adsorbates are noninteracting and


Langmuir, Vol. 18, No. 9, 2002 3745

Figure 6. Dilute TPD curves for R-DMCPr on series of (a) nonroughened and (b) roughened surfaces. Differences between surfaces are greatly diminished once roughening is considered.

Figure 7. Summary of dilute enantiospecificity for (a) DMCPr and (b) DMCH. Roughening diminishes the differences between the surfaces.

adsorption occurs only along step-edges. Since the adsorbates are weakly interacting hydrocarbons,31 this assumption is reasonable. In addition, we ensure that the coverages are sufficiently low that interactions are unlikely (see below). A method for modeling TPD in lattice systems has been developed by Meng and Weinberg.32 We can consider our system to be a one-dimensional lattice of adsorption sites. We define 100% coverage to be the (unphysical) case in which there is an adsorbate present at every step edge site. For DMCH, molecules will begin to sterically overlap at approximately 25% coverage, and DMCPr molecules do so at 33% coverage. We consider these to be saturation coverages, and we chose 25% as the maximum coverage we studied. To incorporate steric repulsions in our model we apply an extended hard-wall region about each adsorbate. We have observed that when adsorbed along a step-edge, DMCH (DMCPr) lies against three (two) atoms on the step edge. Consequently, for DMCH (DMCPr), we allow no adsorbate to be present in any site within three (two) sites on either side of an occupied site. We refer later to the number of excluded sites about an occupied site as nexclude. It is important to point out that a set of TPD curves (simulated or real) will show no coverage dependence for a system of noninteracting adsorbates on an energetically homogeneous surface. That is, if the binding energy of every adsorbate is identical, then the peak temperature

observed in TPD is not a function of coverage. This will not necessarily be the case on chiral surfaces (roughened or ideal), because of the energetically heterogeneous nature of the adsorption sites. To give one simple example, if one molecule adsorbs at a kink site on a chiral surface, it then sterically blocks a number of nearby binding sites, as described above. This type of effect could lead to systematic variations in the observed peak temperatures with coverage for a model of our type. Since the aim of our work is to attempt to make relatively general statements about the impact of surface roughening on adsorption enantiospecificity, it is useful to be able to quantify how strong these coverage dependent effects might be. The calculations described in section III provide qtot j , the number of available energy states at each site j, and their energies Eqj (where q ) 1, 2, ..., qtot j ) for individual adsorbates. This facilitates implementation of the Meng and Weinberg Monte Carlo (MWMC)32 method for modeling TPD as follows: (1) Populate the L site lattice with N0 adsorbates at the initial coverage θ0 (N0 ) θ0L) while excluding particle overlaps. At each occupied site j the state assigned is chosen at random from state 1 to state qtot j . Each adsorbate is assigned a label i (i ) 1, 2, ..., N0). Periodic boundary conditions are applied at the ends of the lattice. (2) To thermally equilibrate the system, perform a MC move as follows. An adsorbate i at site j is chosen at random. A diffusion or change of adsorption-state move is chosen, each with 50% probability. Move acceptance/ rejection is decided using the standard Metropolis condition.32 In change of state moves, adsorbate i remains located at site j and a new adsorption state is chosen at

(31) Raut, J. S.; Sholl, D. S.; Fichthorn, K. A. Surf. Sci. 1997, 389, 88-102. (32) Meng, B.; Weinberg, W. H. J. Chem. Phys. 1994, 100, 52805289.

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Power et al.

Figure 8. TPD spectra for R-DMCH on roughened Pt(531)S for a range of coverages.

random from 1 to qtot j . In diffusion moves, one of the sites on either side of the adsorption site j is chosen with 50% probability as a target site for diffusion. If it is possible to place an adsorbate in the target state without overlapping an existing adsorbate or the sites blocked by steric repulsions, and then the MC trial state becomes adsorbate i located at the target site with the adsorption state assigned at random from 1 to qtot k . If the diffusion attempt is infeasible due to steric overlaps, the move is abandoned and the move is restarted with another randomly chosen adsorbate. (3) Repeat step 2 until thermodynamic equilibrium is achieved. We have found that 100 000 single molecule MC moves are sufficient for this task. (4) The desorption rate for each adsorbate is calculated using a desorption frequency of 1013 Hz, and the molecule with the highest desorption rate, rmax, is identified. An adsorbate i is identified at random and allowed to desorb with probability Pi ) ri/rmax. If no desorption occurs, this procedure is repeated until desorption is successful. After N ri)-1, where each desorption, time is updated by δt ) (Σi)1 N is the number of adsorbate molecules, and temperature is increased by δT ) βTHδt. The observed desorption rate is recorded as ∆N/∆t, where ∆N is the number of molecules desorbed during the time interval ∆t. We have performed simulations on five realizations of every roughened surface, each containing 2000 sites. Therefore, L in our lattice is 10 000. To achieve better statistical accuracy, we perform multiple MWMC simulations for every adsorbate/surface pair and average the resulting TPD spectra. Lower coverages require more simulations to achieve good statistical coverage. The number of independent simulations was varied from 100 at 5% coverage to 20 at 25% coverage. In Figure 8, TPD curves for R-DMCH on roughened Pt(531)S for various coverages are shown. By fitting a cubic polynomial to the TPD curve using data from 10 K either side of Tp, the peak temperature can be isolated. Applying the Redhead equation converts the temperature to the apparent desorption barrier. Our approach to analyzing these data is exactly the same as that used by Horvath and Gellman for their experimental data.33 The peak temperatures and associated desorption barriers are reported in Table 5. Since a set of individual simulations were used to calculate the average curves shown in Figure 8, differences between those curves were used to estimate the error in peak temperature and desorption energy. The enantiospecificities for DMCH on roughened Pt(531)S and (33) Horvath, J. D. Private communication.

Figure 9. Enantiospecificity for DMCH on roughened (a) Pt(874)S and (b) Pt(531)S for a range of coverages. Table 5. TPD Peak Temperatures and Associated Desorption Barriers for R-DMCH on Roughened Pt(531)S coverage (%)

Tp (K)

Udes (kcal/mol)

0 (dilute) 5 10 15 20 25

448.45 447.43 ( 0.97 447.48 ( 1.13 447.78 ( 1.40 445.43 ( 1.10 444.83 ( 1.34

29.00 28.92 ( 0.06 28.92 ( 0.07 28.94 ( 0.09 28.79 ( 0.07 28.75 ( 0.09

Pt(874)S are shown in Figure 9. Although the uncertainty in the measured desorption energies is considerable, clearly the enantiospecificity is not a strong function of coverage. Extending the results shown in Figure 9, we have generated TPD curves for DMCPr and DMCH adsorption on the full series of roughened and nonroughened surfaces we considered in section III. In every case the results are analogous to those in Figure 9 in the sense that there is at most a weak dependence of the observed enantiospecificity on the adsorbate coverage. Our dilute coverage calculations showed that the main effect of roughening is to greatly diminish the variation in enantiospecificity between various vicinal surfaces. The same conclusion applies to our MC simulations of TPD at higher coverages. To show one example, the TPD curves for R-DMCH on a series of roughened and nonroughened surfaces at 20% coverage are shown in Figure 10. This example shows the least dramatic roughening effect of all the examples we have studied in the sense that there is the most variation between the TPD curves on the roughened surfaces. In all of the other examples we have examined, the similarities between the TPD curves on the series of roughened surfaces are greater than in Figure 10b. Our results apply to the coverage dependence of TPD curves on chiral surfaces in cases where adsorbateadsorbate interactions are limited to steric repulsions and where adsorption only occurs along step edges. In these cases, the coverage dependence of the desorption energies observed via TPD is weak. As a consequence, the conclu-


Figure 10. TPD spectra for R-DMCH on a series of (a) nonroughened and (b) roughened chiral surfaces at 20% coverage.

sions we drew in the limit of dilute adsorption also apply over this range of coverages. Most importantly, thermal roughening of chiral surfaces greatly diminishes the differences in adsorption enantiospecificity on surfaces vicinal to (111) relative to their ideal Miller Index counterparts. We note that a more complex range of possibilities can be envisioned in systems where adsorbate-adsorbate interactions are significant. For example, the existence of an orientationally ordered set of adsorbates along a chiral step edge could induce chiral order in nearby adsorbates on surface terraces. V. Discussion and Conclusion Many high Miller index surfaces of simple metals are chiral. Theoretical studies of ideal Miller Index surfaces8,9,14,17 andexperimentalstudiesofrealsurfaces10-13,15,16 have repeatedly shown that chiral molecules can adsorb on these surfaces in an enantiospecific manner. A natural question arising from these studies is how a surface may be chosen that maximizes the enantiospecificity of adsorption for a particular molecule. Theoretical studies of ideal Miller index surfaces surfaces8,9,14,17 show that each surface (Pt(643)S, Pt(653)S, Pt(854)S, etc.) exhibits a different degree of enantiospecificity. This observation suggests that many different surfaces must be screened if the surface maximizing an enantiospecific effect is to be identified. The main conclusion from the results presented here is that the number of chiral surfaces that will yield significantly different enantiospecific responses is considerably smaller than suggested by the discussion above. Real stepped metal surfaces deviate almost unavoidably from their ideal Miller index structure due to thermally induced surface roughening. Although the structural disorder caused by roughening does not remove the net chirality of the surface, it does, not unexpectedly, quantitatively change the enantiospecific adsorption properties of the surface. The detailed simulations of Pt surfaces reported here indicate that the differences in adsorption enan-

Langmuir, Vol. 18, No. 9, 2002 3747

tiospecificity between a broad range of chiral Pt surfaces with wide (111) terraces are greatly diminished by surface roughening. This suggests that if a particular molecule does not show strong enantiospecific properties on one chiral surface with wide (111) terraces, then it is likely to show the same result on other chiral surfaces with wide (111) terraces. There are several experimental avenues suggested by our findings. First, it would be useful to test our conclusion that chiral surfaces with the same low index terrace yield similar enantiospecific adsorption properties by measuring adsorption of chiral species on several surfaces with the same terrace but differing step structure. To date, the only experiments performed on multiple chiral surfaces have been the electrochemistry studies of Pt(643), Pt(321), Pt(431), and Pt(531) performed by Attard et al.10,11,16 Attard reported that the enantiospecificity in these systems scales qualitatively with kink density,16 although the quantitative values of enantiospecificity were not given explicitly. Taken in the simplest form, our hypothesis would suggest that the measured enantiospecificity in these experiments would be insensitive to the surface structure. Unfortunately, our conclusions cannot be directly compared to these experiments because in every case the adsorbate used experimentally (glucose) is large compared to the average terrace width of the surface. Our studies only strictly apply to examples where the adsorbate is narrow compared to the average terrace width. Indeed, our simulations of the Miller index Pt(531) surface and the related surface with the same step structure but very wide terraces show that surfaces with narrow terraces can exhibit substantially different properties than the related surfaces with wide terraces (see Figure 3a and associated discussion). This discussion indicates that a more accurate statement of our main conclusion is that families of surfaces vicinal to the same low index plane with wide terraces will exhibit similar enantiospecificities but surfaces vicinal to the same plane with narrow terraces may show variation in their adsorption properties. A second direction for future experimental work would be to seek to maximize the enantiospecificity of adsorption of a molecule by sampling a range of chiral surfaces that are expected to show significantly different properties. Since our arguments above are based on classifying all surfaces with wide (111) terraces as similar, a minimal set of surfaces spanning the range of possible chiral surfaces would include a surface with a wide (111) terrace, a surface with a wide (110) terrace, and a surface with a wide (100) terrace. Chiral surfaces with each of these properties exist and simple rules are known to generate Miller indices corresponding to each case.8 In addition to surfaces from each of these three classes, it would be sensible to also examine several surfaces with narrow terraces such as the (531)S face. We do not claim that all surfaces with wide terraces defined by the same low-index plane will give identical results, just that the results will be qualitatively similar. Thus, if it is found for a species of interest that the surface from the list above with the (110) terrace gives much greater enantiospecificity than the other surfaces tested, it may well be fruitful to examine a range of chiral surfaces with (110) terraces to more finely optimize the observed enantiospecificity. Although the discussion above provides a quite general framework for considering the enantiospecificity of molecular adsorption on chiral metal surfaces, several caveats should be kept in mind. Our simulations describe physisorbed molecules that do not interact strongly with one another. Moreover, our description of step roughening has been restricted to steps that remain well separated, that

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Power et al.

Table 6. Parameters for Eqs A.1 and A.2 Parameterized Using a Weighted Least-Squares Fit of Barriers Computed Using LDA-DFTa atom type (i)

transn site (TS) EiTS

initial site (S) EiS

A step B step 120° corner 240° corner 300° A corner 300° B corner

-0.105 0.016 -0.037 -0.039 0.165 -0.030

-0.512 -0.458 -0.450 -0.509 -0.335 -0.470

a

possible states. This is calculated by Nstates

∑i ri

r)

(B.2)

For each state i,

ri ) υ exp(Ei /kT)Pi

(B.3)

The value of γ is 0.50 and energies are in eV.

is, surfaces with relatively wide terraces. Scenarios can be envisioned in which the specific structure of a chiral surface may be especially important. For example, if enantiospecific effects occur because chiral order in molecules adsorbed on terraces is induced by molecules adsorbed at step edges, this behavior will only occur on surfaces with terraces wide enough to allow adsorption of multiple molecules and these effects will vary as a function of the terrace width. Nevertheless, the observation that thermal roughening tends to diminish the differences between chiral metal surfaces with the same low index terrace appears to be quite general. It is hoped that this idea and the detailed results reported here will spur continued experimental study of chiral metal surfaces and provide a useful means for classifying experimental data from the wide range of possible surfaces.

where υ is the desorption frequency, Ei is the binding energy, which is negative, and Pi is the probability of observing a molecule in that state. Pi is given in the dilute limit by

Pi )

Nstates

(B.4)


Combining (B.4) and (B.3) gives

ri )

υni

(B.5)

Nstates

∑ i)1

Appendix A Our LG model for periphery diffusion of Pt atoms along Pt surface steps has been described in detail elsewhere.19 Briefly, the activation energy to move between two sites is given by Eb ) ETS - ES, where ETS and ES are the moving atom’s energy at the transition state (TS) and the initial state (S). These energies are defined in terms of 6 types of nearest neighbor atoms: A-step atom, B-step atom, 120° corner atom, 240° corner atom, 300° A corner atom, and 300° B corner atom.19 We define

ni exp(-Ei /kT)

exp(-Ei /kT)

Substitution of (B.5) into (B.2) yields r as Nstates

υ r)

ni ∑ i)1

(B.6)

Nstates


6

NiEiS ∑ i)1

(A.1)

(Ni + γMi)EiS ∑ i)1

(A.2)

ES ) and 6

ETS )

where i ) 1, .., 6 labels the atom types, EiS and EiTS are energies of interactions, and Ni (Mi) are the number of “fully” (“partially”) bonded nearest neighbors.19 The parameters of the LG model used throughout this paper are summarized in Table 6. Appendix B The desorption rate of n molecules from the surface is given as

dn ) rn dt

(B.1)

where r is the aggregate desorption rate constant from all

The temperature corresponding to the peak desorption rate, Tp, can be determined by solving an implicit equation that arises from generalizing the usual Redhead analysis30 to a system with multiple adsorption sates. By determination of the conditions where d2n/dt2 ) 0, the peak temperature is found to satisfy Nstates

kTp2

∑i

β

Nstates

ni )-

∑i

niEi exp(-Ei /kTp) υ

(B.7)

Acknowledgment. This work was supported by the NSF under Grant No. CTS-9813937. Computations were performed at the Pittsburgh Supercomputer Center and on a computer cluster supported by the NSF (Grant No. CTS-0094407) and Intel. T.D.P. received a graduate fellowship from Air Products and Chemicals Inc., and D.S.S. is an Alfred P. Sloan research fellow. LA011535O

Atomically Detailed Models of the Effect of Thermal Roughening on

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