Monomer Adsorption on Terraces and Nanotubes - Langmuir (ACS

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Langmuir 2007, 23, 558-573

Monomer Adsorption on Terraces and Nanotubes Alain J. Phares,*,† David W. Grumbine, Jr.,‡ and Francis J. Wunderlich† Department of Physics, Mendel Hall, VillanoVa UniVersity, VillanoVa, PennsylVania 19085-1699, and Department of Physics, St. Vincent College, Latrobe, PennsylVania 15650-4580 ReceiVed July 13, 2006. In Final Form: September 22, 2006 We construct a nonsparse transfer matrix (T-matrix) for a lattice gas model of monomers adsorbed on planar and nanotube surfaces of arbitrary geometry. The model can accommodate any number of higher-order pairwise adsorbateadsorbate interactions. The technique is sufficiently general for application to nonequivalent adsorption sites and coadsorption of two or more monomer species. The T-matrices for monomer adsorption on a finite width terrace and for monomer adsorption on a nanotube, both of the same lattice geometry, share a basic G-matrix. First, the G-matrix is diagrammatically and recursively constructed. Then, its elements are modified to provide the T-matrix elements for either the terrace or the nanotube. The T-matrices for several particular lattice geometries previously studied as special cases are easily recovered with the generalized technique presented here. This generalization also provides a vectorized algorithm for efficient use on multi-parallel processors and supercomputers.

1. Introduction Lattice models have been successfully used in the study of many physical, chemical, and biological systems. Early models include the Langmuir adsorption theory1-8 and the Ising model applied to magnetism.9-20 The lattice gas model has been shown to be mathematically equivalent to the Ising model in an external field.18 The lattice model, as applied to surface adsorption, consists of gas particles (either atomic or molecular) and a crystalline substrate surface. At a particular pressure and temperature, a certain number of the gas particles are adsorbed on the substrate. The gas pressure determines the chemical potential energy per particle, µ′. If V0 is the interaction energy between an adsorbed particle and the substrate, then the shifted chemical potential µ ) µ′ + V0, which corresponds to the external field in the Ising model, characterizes the system. In general, the adsorbate coverage of the surface increases as the external pressure is increased. Experiments reveal a series of discrete adsorption patterns, or phases, as the external pressure increases rather than a smooth, continuous change in coverage. A particular type of particle adsorbed on different types of metallic * To whom correspondence should be addressed. Phone: +1 610 519 4889. E-mail: [email protected]. † Villanova University. ‡ St. Vincent College. (1) Langmuir, I. J. Am. Chem. Soc. 1912, 34, 1310. (2) Langmuir, I. J. Am. Chem. Soc. 1915, 37, 417. (3) Langmuir, I.; Kingdom, K. H. Phys. ReV. 1919, 34, 129. (4) Langmuir, I.; Kingdom, K. H. Proc. R. Soc. London A 1925, 107, 61. (5) Langmuir, I. Gen. Electron. ReV. 1926, 29, 143. (6) Langmuir, I.; Villars, D. S. J. Am. Chem. Soc. 1931, 53, 486. (7) Langmuir, I.; J. Am. Chem. Soc. 1932, 54, 2798. (8) Langmuir, I.; Taylor, J. B. Phys. ReV. 1933, 44, 423. (9) Izing, E. Z. Phys. 1925, 31, 253. (10) Kramers, H. A.; Wannier, G. H. Phys. ReV. 1941, 60, 252. (11) Kramers, H. A.; Wannier, G. H. Phys. ReV. 1941, 60, 263. (12) Montroll, E. J. Chem. Phys. 1941, 9, 706. (13) Onsager, L. Phys. ReV. 1944, 65, 117. (14) Kaufman, B. Phys. ReV. 1949, 76, 1232. (15) Kaufman, B.; Onsager, L. Phys. ReV. 1949, 76, 1244. (16) Yang, C. N. Phys. ReV. 1952, 85, 808. (17) Yang, C. N.; Lee, T. D. Phys. ReV. 1952, 87, 404. (18) Lee, T. D.; Yang, C. N. Phys. ReV. 1952, 87, 410. (19) McCoy, M.; Wu, T. T. The Two-dimensional Ising Model; Harvard University: Cambridge, MA, 1973. (20) Baxter, R. J. Exactly SolVed Models in Statistical Mechanics; Academic Press: New York, 1982.

surfaces (which have the same geometry) exhibits different adsorption phases. On the other hand, different adsorption phases occur for different types of particles adsorbed on the same metallic surface. Reference 21 contains an extensive review of experimental data on this subject. It follows that, for a given substrate lattice geometry, there is a direct relationship between the sequence of observed adsorption phases and the adsorbateadsorbate and adsorbate-substrate interaction energies. This is the paramount issue that our approach addresses. A realistic model of surface adsorption considers short-range, pairwise adsorbate-adsorbate interactions (or coupling constants). Nearest neighbors are first neighbors with interaction energy V1; next nearest neighbors are second neighbors with interaction energy V2; and i th order neighbors have interaction energies Vi. Ising models usually consider interactions to second neighbors on a lattice with periodic boundary conditions. For example, an L × M square lattice that has L vertical and M horizontal sites and periodic boundaries in either or both directions leads to a cylindrical or a toroidal surface, respectively. Other conditions have also been considered, such as a helical boundary.10 We do not make such boundary condition assumptions in the study of terraces. Our approach addresses an aspect of lattice models different from the point of view taken in other studies, such as finite-size scaling,22-26 group renormalization methods,27-30 and Monte Carlo sampling and simulation.31-34 It is impossible within the (21) Somorjai, G. A. Introduction to Surface Chemistry; John Wiley and Sons: New York, 1994. (22) Fowler, R. H.; Rushbrooke, G. S. Trans. Farday Soc. 1937, 33, 1272. (23) Fisher, M. E.; Barber, M. N. Phys. ReV. Lett. 1972, 28, 1516. (24) Nightingale, M. P. Physica A 1976, 83, 561. (25) Hamer, C. J.; Barber, M. N. J. Phys. A: Math. Gen. 1981, 14, 2009. (26) Nightingale, M. P.; Blote, H. W. J. J. Phys. A: Math. Gen. 1982, 15, L33. (27) Wilson, K. G.; Kogut, J. Phys. Rep. 1974, 120, 75. (28) Fisher, M. E. ReV. Mod. Phys. 1974, 46, 597. (29) Nightingale, M. P. Proc. Kon. Ned. Akad. Wet. B 1979, 82, 235. (30) Nightingale, M. P. J. Appl. Phys. 1982, 53, 7927. (31) Mon, K. K.; Jasnow, D. Phys. ReV. A 1984, 30, 670. (32) Mon, K. K.; Nightingale, M. P. Phys. ReV. B 1985, 31, 6137. (33) Binder, K.; Heermann, D. W. In Springer Series in Solid-State Sciences, Vol. 80: Monte Carlo Simulation in Statistical Physics: An Introduction, 3rd ed.; Fulde, P., Ed.; Springer: New York, 1997. (34) Nightingale, M. P.; Umrigar, C. J. In AdV. Chem. Phys., Vol. 105: Monte Carlo Methods in Chemistry; Ferguson, D. M., Siepman, J. I., Truhlar, D. G., series Eds; Prigogine, I., Rice, S. A., Eds.; Wiley: New York, 1998; Chapter 4.

10.1021/la0620354 CCC: $37.00 © 2007 American Chemical Society Published on Web 11/22/2006

Monomer Adsorption on Terraces and Nanotubes

scope of this article to give a proper account of all the advances made on the basis of these techniques. In many cases, the interest has been in determining, among other things, the behavior of the correlation length and verifying universality, or lack thereof, in calculating critical exponents and scaling functions. In most of these calculations, the eigenvalue of a Hamiltonian or transfer matrix based on a sequence of lattices, such as L × L square lattices of increasing size, is computed to estimate the bulk limit for an infinite lattice. Such simulations are quite remarkable and provide tremendous insight into problems that are otherwise intractable. In all the cases studied using our formulation of the adsorption problem, the numerical results fit exact analytic expressions in the increasing size of the lattice width, without relying on such simulations or approximations. This is the case for the occupational characteristics of the phases and for the conditions prevailing at the transitions between phases, which in many instances are found in terms of the golden ratio. Our numerical calculations are carried out using long double precision arithmetic in order to ensure agreement with the analytic expressions to at least 10 significant figures. In this sense, our approach is complementary to others, and it is the purpose of this article to extend its applicability to more complex adsorption systems. Density functional theory (DFT)35 has been successfully applied to surface adsorption but requires specifying the chemical system. For example, DFT has been used to describe the adsorption of CO on Pt(111).36 The embedded-atom method (EAM)37 is semiempirical and has been used to calculate approximate structure and energetics, from which many interesting properties of metals can be obtained. Again, our approach is complementary as it adds a different perspective to the study of surface adsorption. Unlike DFT and EAM, it makes no a priori assumptions as to whether the forces are attractive or repulsive or as to their relative strength. Our only inputs are the geometry of the periodic adsorbate lattice, the number of surface sites each particle may occupy (monomers, dimers, or polymers in general), and the number of pairwise interaction energies. Bounds on the possible interaction energies between adsorbates have been obtained from the observed adsorption phases and the conditions prevailing at the transitions between phases. Our goal is to model adsorption on surfaces consisting of infinitely long, identical stepped terraces. The terraces may have an arbitrary lattice geometry for which edge effects are important but not usually taken into account by standard models. There are extensive experimental data on the step effect in the case of CO adsorption on Pt(112) and on Pt(335).38-41 These Pt surfaces correspond to equilateral triangular terraces that are three and four atomic sites in width. Our theoretical work is applicable to such adsorption studies while also providing a unified approach for studying terraces and nanotubes. The starting point of our approach is to establish the formulation of a transfer matrix for a general lattice of finite width and finite length. There may be any number of pairwise, adsorbateadsorbate interaction energies, Vi, whether attractive or repulsive. If the lattice is a terrace, edge effects can be included. As is (35) Jones, R. O.; Gunnarsson, O. ReV. Mod. Phys. 1989, 61 689. (36) Gill, A.; Clotet, A.; Riccart, J. M.; Kreese, G.; Garcia-Hernandez, M.; Rosch, N.; Sautet, P. Surf. Sci. 2003, 530 71. (37) Daw, M. S.; Foiles, S. M.; Baskes, M. I. Materials Sci. Rep. 1993, 9, 251. (38) Henderson, M. A.; Szabo, A.; Yates, Jr., J. T. J. Chem. Phys. 1989, 91, 7245. (39) Henderson, M. A.; Szabo, A.; Yates, Jr., J. T. J. Chem. Phys. 1989, 91, 7255. (40) Henderson, M. A.; Szabo, A.; Yates, J. T., Jr. Chem. Phys. Lett. 1990, 168, 51. (41) Xu, J.; Yates, J. T., Jr. Surf. Sci. 1995, 327, 193.

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usually the case, the eigenvalue of largest modulus of the transfer matrix provides the partition function of the system in the thermodynamic limit when the length of the lattice approaches infinity. To highlight the advances made in this paper and for completeness, we first present a brief summary of our approach as first derived in the context of the homonuclear dimer problem.42,43 We first considered homonuclear dimers on a square L × M lattice without any periodic boundary conditions and included only nearest neighbor interactions.42,43 To first approximation, no distinction was made between step edge and bulk sites, thus neglecting edge effects. As an adsorbed dimer might be oriented either horizontally or vertically on the lattice, this was a finite width two-dimensional problem rather than an assembly of onedimensional problems.44 In subsequent investigations of adsorption on square, rectangular, and isosceles triangular lattices, we only studied those cases that could be treated using the approach that was applied to the dimer problem on a square lattice. We started by obtaining the occupational degeneracy of the adsorbed dimers from which, in principle, the partition function of the system could be obtained. The occupational degeneracy is the number of possible arrangements on the lattice of a given number of dimers with a fixed number of first neighbor pairs, second neighbor pairs, and so on. In the standard scheme developed by McQuistan et al.,45-49 additional occupational degeneracies were introduced, each corresponding to an arrangement of dimers on a “restricted,” or “truncated,” lattice obtained from the original L × M lattice by restricting the occupation of a suitably chosen number of sites. In the case of first neighbor dimer interactions, there were only four additional degeneracies required. Five recursion relations were derived relating all five degeneracies, the physical occupational degeneracy on the nontruncated L × M lattice, and the four additional degeneracies. These recursion relations led to a complete set of five linear equations among the generating functions of the five degeneracies. The problem of obtaining the occupational degeneracy of first neighbor interacting dimers on the L × M square lattice was therefore reduced to obtaining its associated generating function.50 Suitable manipulations of the linear equations among the generating functions led to the introduction of a T-matrix, TM, whose eigenvalues were shown to provide the partition function of the finite system.42,43 Its rank was D(M) ) 3D(M - 1) + D(M - 2) with D(0) ) 1 and D(1) ) 3. TM was divided into 16 block-matrices: nine square and seven rectangular. With a suitable ordering of the rows and columns of the T-matrix, six of these blocks were zero-matrices. Within appropriate overall factors, two of the nonzero matrices were identified as TM-1. The remaining block-matrices were called PM-1, PM-2, JM-1, KM-1, and LM-1. The PM-1 and PM-2 matrices were square and of rank D(M - 1) and D(M - 2). JM-1 and LM-1 were rectangular D(M - 2) × D(M - 1) matrices. KM-1 appeared in two locations and was a rectangular D(M - 1) × D(M - 2) matrix. Each of these matrices, subdivided into smaller blocks, was related to itself (42) Phares, A. J.; Wunderlich, F. J.; Grumbine, D. W., Jr.; Curley, J. D. Phys. Lett. A 1993, 173, 365. (43) Phares, A. J.; Wunderlich, F. J.; Curley, J. D.; Grumbine, D. W., Jr. J. Phys. A 1993, 26, 6847. (44) Aranovich, G. L.; Wetzel, T. E.; Donohue, M. D. J. Phys. Chem. B 2005, 109, 10189. (45) Lichtman, D.; McQuistan, R. B. J. Math. Phys. 1967, 8, 2441. (46) McQuistan, R. B. NuoVo Cim. 1968, B58, 86. (47) McQuistan, R. B.; Lichtman, S. B. J. Math. Phys. 1970, 11, 3095. (48) McQuistan, R. B.; Hock, J. L. Fibonacci Quart. 1983, 21, 196. (49) Hock, J. L.; McQuistan, R. B. J. Math. Phys. 1983, 24, 1859. (50) Phares, A. J. J. Math. Phys. 1984, 25, 2169.

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Figure 1. (a) Square terrace with first neighbor edge sites; (b) square terrace with second neighbor edge sites; (c) rectangular fcc(110) terrace; (d) isosceles triangular bcc(110) terrace.

and to the others at lower ranks. This established a set of nonlinear recursive relations among five matrices, the result of which was the transfer matrix. We found that it was sufficient to construct the T-matrix recursively without the need to compute the degeneracies nor their generating functions. In this homonuclear dimer problem with first neighbor interactions, a one-to-one correspondence was established between the T-matrix elements and the states of occupation of two consecutive sequences of lattice sites of (horizontal) width, M, in a manner similar to but different from other T-matrix methods. The construction followed simple diagrammatic rules that were easily programmable. A given T-matrix element was related to a given state of occupation of the lower sequence of sites that was associated with the row index, as well as a given state of occupation of the upper sequence of sites that was associated with the column index. This feature continues to hold in more complex adsorption problems; however, the appropriate sequence of sites is not, in general, a single row across the physical width of the lattice but is related to it, as will be explained below. Our matrix method is essentially different from other matrix methods used in similar lattice studies, such as those presented in ref 20 or used in ref 51. Some of the results obtained from this method as applied to the homonuclear dimer problem42,43 have been recovered by others using Monte Carlo simulation.52-55 It is easily adapted to heteronuclear dimers, also recently studied using Monte Carlo simulation.56-60 The same steps, described above for first neighbor interacting dimers on a square lattice, were reported and used in all our subsequent adsorption studies involving monomers and dimers. The semi-infinite terraces studied to date using our T-matrix method are shown in Figure 1 and are sc, bcc, or fcc(100) corresponding to square lattices,42,43,61,62 fcc(110) or rectangular lattices,63,64 and bcc(110) or isosceles triangular lattices.65,66 (51) Bartlet, N. C.; Einstein, T. L.; Roelofs, L. D. Phys. ReV. B 1986, 34, 1616. (52) Ramirez-Pastor, A. J.; Riccardo, J. L.; Pereyra, V. D. Surf. Sci. 1998, 411, 294. (53) Roma´, F.; Ramirez-Pastor, A. J.; Riccardo, J. L. Langmuir 2000, 16, 9406. (54) Roma´, F.; Ramirez-Pastor, A. J.; Riccardo, J. L. J. Chem. Phys. 2001, 114, 10932. (55) Roma´, F.; Ramirez-Pastor, A. J.; Riccardo, J. L. Phys. ReV. B 2003, 68, 205407. (56) Rzysko, W.; Borowko, M. J. Chem. Phys. 2002, 117, 151. (57) Rzysko, W.; Borowko, M. Surf. Sci. 2002, 520, 151. (58) Rzysko, W.; Borowko, M. Physica A 2003, 326, 1. (59) Rzysko, W.; Borowko, M. Thin Solid Films 2003, 425, 304. (60) Rzysko, W.; Borowko, M. Surf. Sci. 2006, 600, 890. (61) Phares, A. J.; Wunderlich, F. J. Phys. ReV. E 1995, 52, 2236. (62) Phares, A. J.; Wunderlich, F. J. Phys. ReV. E 1997, 55, 2403. (63) Phares, A. J.; Wunderlich, F. J. Surf. Sci. 2001, 479, 43. (64) Phares, A. J.; Wunderlich, F. J. Int. J. Mod. Phys. B 2001, 15, 3323. (65) Phares, A. J.; Wunderlich, F. J. Surf. Sci. 1999, 425, 112. (66) Phares, A. J.; Wunderlich, F. J. Surf. Sci. 2000, 452, 108.

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Adsorbates in these studies were either monomers61-66 or dimers.42,43 In addition, for square lattices, we allowed edge sites to be either first or second neighbors, corresponding to two different lattice orientations, with adsorbate-substrate interactions on edge sites different from that in the bulk to take into account the effects of the edges and to model preferential adsorption on the steps.67,68 The model was also applied to some adsorption cases which resulted in surface reconstruction.63 With only pairwise interactions up to third neighbors, our T-matrix method showed the coexistence of phases, the existence of partially ordered phases, island formation, and domain boundaries, all of which have been observed experimentally. We have also studied coadsorption of monomers and dimers on square terraces.69 Coadsorption of two different species of monomers was also studied.70 In the absence of adsorbateadsorbate interactions on a one-dimensional lattice, we found the exact analytic solution of the coadsorption of any number of nonoverlapping polymeric species, each of which may occupy any number of consecutive sites.71 For monomer adsorption, our T-matrix method also established that the numerical occupational characteristics of the phases fit closed form analytic expressions in the width, M, of the terrace, as mentioned above. The infinite-M limit provided the phase characteristics on the infinite twodimensional lattice, recovering some of the results obtained by finite-size scaling and group renormalization techniques. In the cases of monomer and dimer adsorption studies mentioned above, there were never more than eight recursively related matrices which could be easily programmed and handled numerically. Each case was treated independently, as determined by the repetitive pattern of the lattice, the orientation of the edges, and the number of pairwise interactions. However, with the more general development of our T-matrix presented here for monomers, we show that all these seemingly independent cases actually follow from one unified formulation. In a more complex situation, such as monomer adsorption on equilateral triangular terraces of a given orientation with first and second neighbor interactions, this paper shows that there are 96 recursively related matrices, the result of which is the T-matrix whose eigenvalues lead to the partition function. Thus, an efficient computational algorithm is required to handle this increasing complexity. One example of additional complexity considered in this paper is the study of adsorption on surfaces made of regular hexagons or equilateral triangles for which adsorption may take place on top, in the hollows, and on bridges. Experimental data show that the adsorption of NO on Pt(111) takes place first on hollow sites, then on top, and finally on bridge sites.72,73 However, the adsorption of NO on Pd(111) occurs in a different order and is subject to further experimental and theoretical investigation.74 Similar studies have been made of the decomposition of NO2 into NO and O coadsorbed on Ag(111), with the NO adsorbed first on 3-fold bridge sites and then on top sites.75 Topologically, these three problems are the same yet the order in which adsorption (67) Phares, A. J.; Wunderlich, F. J.; Kumar, A. M. S. Surf. Sci. 2001, 495/ 1-2, 140. (68) Phares, A. J.; Wunderlich, F. J. Appl. Surf. Sci. 2003, 219/1-2, 174. (69) Phares, A. J.; Wunderlich, F. J. Phys. Lett. A 1997, 226, 336. (70) Phares, A. J.; Wunderlich, F. J.; Martin, J. P.; Burns, P. M.; Duda, G. K. Phys. ReV. E 1997, 56, 2447. (71) Phares, A. J.; Wunderlich, F. J. Phys. Lett. A 1987, 121, 347. (72) Zhu, J. F.; Kinne, M.; Fuhrmann, T.; Denecke, R.; Steinruck, H. P. Surf. Sci. 2003, 529, 384. (73) Matsumoto, M.; Fukunani, K.; Okano, T.; Miyake, K.; Shigekawa, H.; Kato, H.; Okuyama, H.; Kawai, M. Surf. Sci. 2000, 454-456, 101. (74) Hansen, K. H.; Slijvancanin, Z.; Hammer, B.; Laegsgaard, E.; Besenbacher, F.; Stensgaard, I. Surf. Sci. 2002, 496, 1. (75) Huang, W. X.; White, J. M. Surf. Sci. 2003, 529, 455.


occurs at the different types of sites is not the same. The alternative approach of our T-matrix method should provide additional insight into the reason for these experimental findings. As this article lays the groundwork for the study of a variety of problems, its applications will be presented in a series of forthcoming papers. The paper is organizated as follows. Section 2 presents a novel approach to the general structure of our T-matrix associated with monomer adsorption on both finite-width terraces and nanotubes but without specifying the lattice geometry and including any number of pairwise adsorbate interactions. The two T-matrices associated with adsorption on a terrace and a nanotube having the same lattice structure are shown to share a common, recursively constructed G-matrix. Section 2 also provides the equations relating the occupational characteristics and entropy of the adsorption system to the T-matrix and the process by which numerical computations are conducted. Section 3 outlines the systematic ordering of the row and column indices that is crucial for the recursive construction of the G-matrix. This leads to an efficient and fast computational algorithm. Sections 4-9 show how this is achieved in progressively more complex situations. Section 4 considers a parallelogram lattice, whether a terrace or a nanotube, and provides a unified treatment of all the monomer adsorption studies mentioned above. Sections 5 and 6 consider monomer adsorption on equilateral triangular lattices with two different orientations and first and second neighbor interactions. Sections 7 and 8 consider monomer adsorption on armchair and zigzag hexagonal lattices. Section 9 deals with monomer adsorption on a heterogeneous lattice composed of two sub-lattices made of equilateral triangles and hexagons. Section 10 is the summary and conclusion.

2. T-matrix, Phases, Numerical Computations, and Motivation As mentioned above, there is a one-to-one correspondence between T-matrix elements and the states of occupancy of two vertically consecutive sequences of sites across the physical width of the lattice. The highest order, I, of pairwise adsorbate interactions under consideration dictates which sequence to choose. Consequently, the sites making up the sequence must be chosen such that all I th neighbors of any site in a given sequence are in the same or an adjacent sequence, but not beyond. This in turn determines the number of sites in a given sequence, as shown in the following two examples where only first and second neighbor interactions are considered. The simplest case is a square terrace M atoms in physical width. Figure 2a shows such a terrace with three consecutive horizontal rows of M sites. The first and second neighbors of a site in the middle row are located in the same row, the row above it, or the row below it, but not beyond. Therefore, the appropriate sequence to use in this case is a single horizontal row of M sites. Since each site in this sequence can either be vacant or occupied by a monomer, there are 2M possible occupational states. Thus, when only first and second neighbor interactions are considered, the 2M × 2M states of two consecutive sequences determine the elements of the T-matrix of rank 2M. The situation is quite different when the adsorption takes place on an equilateral triangular terrace oriented as exemplified in Figure 2b where four rows of such a terrace M atomic sites in physical width are shown. The first neighbors of the site indicated by the filled circle in the second row from the top are located in the same row of M sites or in adjacent rows above and below it, but not beyond. On the other hand, its second neighbors are located in the same row, the row above, and in two rows below it, but not beyond. The sequence of sites relevant to the

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Figure 2. (a) Square terrace and (b) an equilateral triangular terrace. Both terraces are M atomic sites in physical width. Dashed segments connect sites belonging to the same sequence extending across the physical width of the lattice. A site in a given sequence is connected to first neighbor sites by thick line segments and to second neighbor sites by thin line segments. In (b), the highlighted sequence consists of two rows of M sites.

construction of the T-matrix thus requires two rows of M sites each, as highlighted in Figure 2b, yielding a matrix with rank 22M. The one-to-one correspondence between the states of these two physical rows and the elements of the T-matrix still holds, as long as it is not necessary to consider pairwise interactions of higher order. As is evident from the two examples considered above, the relevant quantity is not necessarily the physical width M of the terrace, but the number of sites, M′, in the sequence consistent with the highest order of pairwise interaction under consideration. This subsequently determines the number of sites which are on the right or left edge of any particular sequence. There are two sites on the right edge and two sites on the left edge in the Figure 2b, with M′ ) 2M. There are features common to both terraces and nanotubes. A nanotube may be viewed as a terrace wrapped around a cylinder resulting in a periodic boundary in one direction. Wrapping a terrace in this manner is possible for any value of M for the lattice shown in Figure 2a, but is only possible when M is even for the lattice in Figure 2b. Edge effects on a terrace are replaced by effects due to wrapping for a nanotube. While we only consider one monomer species throughout this paper, extension of this method to coadsorption of several monomer species is relatively straightforward. In this section, all bulk sites are treated equivalently. In Section 9, this constraint is removed for the discussion of adsorption on tops and in hollows in a heterogeneous lattice. We begin by considering terraces. The chemical potential energy per particle in the gas is µ′. The adsorbate-substrate interaction energy at bulk sites is Vb. The exposed surface of the cut crystal may consist of terraces aligned as stairs or as a combination of trenches and plateaus. To account for the effect of terrace edges, including the possibility of interaction between

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adjacent terraces, the adsorbate-substrate interaction on edge sites is different from bulk sites. The adsorbate-substrate interaction energies at sites on the right and left edges of the terrace are Vright and Vleft, respectively. These are different from Vb at bulk sites. The differential interaction energies U1 ) (Vright - Vb) and U2 ) (Vleft - Vb) are equal when the terraces are symmetrical trenches or plateaus. If the terrace is wrapped on a cylinder to form a nanotube, U1 ) U2 ) 0. Finally, the pairwise adsorbate-adsorbate interaction energy between i th neighbors is Vi, and the highest order interaction under consideration is I. Activity, x0, is associated with the shifted chemical potential, µ ) (µ′ + Vb). The activity associated with Vi is xi. The activities associated with U1 and U2 are u1 and u2, respectively. These are given by

x0 ) exp[µ/kBT] xi ) exp[Vi/kBT] uj ) exp[Uj/kB T]

(1)

where kB is Boltzmann’s constant and T is the absolute temperature. In the special case of wrapping to form a nanotube, u1 ) u2 ) 1. We now provide the one-to-one correspondence between a T-matrix element and a state of two consecutive sequences of sites as determined above. There are three classes of pairwise ith neighbor monomers in any particular state: pairs within the upper sequence, pairs within the lower sequence, and pairs with one monomer in the upper and one in the lower sequence. The quantities that are relevant to the one-to-one correspondence are: the number a0 of monomers in the upper sequence, including edge sites; the numbers b1 and b2 of occupied sites found on the right and left edges of the upper sequence; and the numbers ai of ith-neighbor monomers, excluding those within the lower sequence. It then follows that the T-matrix element, Tmn, of row m and column n associated with this state of occupation is the product of the activities, each raised to a nonnegative integer power, as61-70

Tmn ) Gmnu1

u2

b2(m,n)

Gmn )

∏ i)0

Z ) R1/M′

xai i(m,n)

(2)

We have adopted the one-to-one correspondence such that the row index m corresponds to the state of occupation of the lower sequence and the column index n corresponds to the state of occupation of the upper sequence. For a nanotube, not only does u1 ) u2 ) 1, but the wrapping introduces new neighbors as the right and left edges meet. Consequently, the exponents of the x activities are incremented to account for these new neighbors. Thus, the factor ub11(m,n) × I ub22(m,n) in eq 2 is replaced by a factor of the form ∏i)1 × xci i(m,n). th The exponent ci(m, n) is the number of additional i neighbor monomers created by the wrapping, excluding those found in the lower sequence. One of the main consequences of this analysis, overlooked in the past, is that the G-matrix may be recursively constructed in the same manner irrespective of whether the lattice is a terrace or a nanotube. Then, once the G-matrix is constructed, the appropriate factor is added to each element depending on the adsorption problem under consideration, ub11(m,n)ub22(m,n) for adI sorption on terraces or ∏i)1 xci i(m,n) for adsorption on nanotubes, to obtain the T-matrix. The next section addresses the problem of appropriate ordering of the states of occupation for any sequence of sites which then dictates the ordering of the rows and columns of the G-matrix and subsequently the T-matrix. At thermodynamic equilibrium, the average occupational characteristics of the adsorption system are the coverage, θ0, the number per site of ith neighbor pairs, θi, and when considering

(3)

The occupational characteristics are derived from the value, and first partial derivatives of R with respect to the various activities, namely,

θ0 )

x0 ∂R xi ∂R uj ∂R θi ) γj ) RM′ ∂x0 RM′ ∂xi RM′ ∂uj

(4)

The average energy per site, , follows as,

) µ θ0 +

∑i Vi θi + U1 γ1 + U2 γ2

(5)

where the nanotube case has fewer energy parameters since γ1 ) γ2 ) 0. The entropy per site divided by Boltzmann’s constant is

S)

I

b1(m,n)

terraces, the number per site of occupied sites on the right and left edges of the terrace, γ1 and γ2. When considering nanotubes, γ1 ) γ2 ) 0 because there are no edge sites. As all of the matrix elements are real and non-negative, the Frobenius-Perron theorem states that the eigenvalue of largest modulus, R, of the T-matrix is real and positive. All of the T-matrix eigenvalues contribute to the partition function of the adsorption system at equilibrium when the terrace or nanotube has a finite vertical length.50 The number, M′, of sites in a given sequence is not necessarily the same as the number, M, of sites in the physical width of the terrace. A major simplification occurs in the thermodynamic limit for which the number of sequences of M′ sites making up the vertical length of the lattice approaches infinity. In this limit, R is the only contribution to the partition function Z which, for monomers, is given by

1 ln R M′ kBT

(6)

The numerical computations are based on eqs 1-6. A series of crystallization patterns, or phases, occur at minima of the entropy as the external gas pressure (and therefore µ) is gradually increased while the system is kept at a relatively low temperature. A perfectly ordered phase occurs when the entropy minimum is zero; otherwise, the phase is partially ordered. The closed form expression of the nonzero entropy for each partially ordered phase is analytically derived to check the numerically computed value. At each entropy minimum, the phase is identified by its occupational characteristics listed as the set {θ0, θi, γj}. Entropy minima resemble plateaus42,43 when S is plotted against µ indicating that the phases are persistent over a wide range of pressure. On the other hand, the entropy minima correspond to cusps in a plot of S versus the coverage θ0,42,43 which has been confirmed by others using Monte Carlo simulation.52-55 At low pressure, the lattice is empty and the state of occupation is {0, ..., 0}. At high pressure, the lattice is fully covered and, for all values of M, the state of occupation is θ0 ) 1. The entropy is zero at these extreme points. The entropy maximum between two consecutive phases characterizes the phase transition. At relatively low temperature, it has been shown, as well as numerically verified, that the transition between two phases occurs at a value of µ which is linearly related to U1, U2, and the adsorbate-adsorbate interaction energies, Vi. This follows from the requirement that energy be conserved across the transition. If the changes in the occupational


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characteristics between two consecutive phases are ∆θ0, ∆θi, and ∆γj, eq 5 yields

0 ) µ(∆θ0) +

∑i Vi(∆θi) + U1(∆γ1) + U2(∆γ2)

(7)

It has also been numerically verified that, at relatively low temperature, each occupational characteristic varies linearly as the coverage changes.61-70 The numerical computations are conducted as follows. For a given M, values are selected for the energy parameters Vi and Uj. The value of µ is varied over the range necessary to produce coverages from empty (θ0 ) 0) to full (θ0 ) 1). The resulting sequence of phases and phase transitions are recorded. An “energy region” consists of all values of Vi and Uj for which the sequence of phases is the same. In all cases, these regions give rise to a set of linear inequalities among the energy parameters. For a given M, all possible energy regions are tabulated for which a given sequence of phases is observed. The phases observed experimentally and the tabulated list of phase sequences obtained numerically can thus be matched in order to predict the energy region to which the adsorption system belongs. Knowledge of the experimentally observed phases is sufficient to find bounds on the interaction energies. When additional information is available on all of the transitions between phases, the model can predict the values of the interaction energies from eq 7. This predictive capability is the real motivation for extending our T-matrix approach to more complex adsorption problems.

3. Ordering of the Indices of the T-matrix Elements The following provides a systematic enumeration of all the states of a given sequence of sites associated with a given element of the G-matrix, which exhibits the features common to both a terrace and the associated nanotube. The M′ sites in the sequence (determined by the highest order adsorbate interaction) are ordered from the right edge of the lattice to the left edge. The state of occupancy of the jth site in this sequence is Rj, with Rj ) 0 if the site is vacant or Rj ) 1 if the site is occupied. Thus, the state of a sequence of M′ sites may be written (RM′, RM′-1, ..., R1). The total number of possible states is 2M′. Because each site is either vacant or occupied, numbers in base 2 are convenient to enumerate the occupational states. The decimal number corresponding to the state represented by the set (RM′, RM′-1, ..., R1) is

RM'(2M'-1) + RM'-1(2M'-2) + ‚‚‚ + R1(20) + 1

(8)

The row and column indices of any particular element of the G-matrix are determined by the states of two vertically consecutive sequences of M′ sites. Therefore, the row and column indices of the G-matrix may also be ordered in the manner described above, yielding a G-matrix of rank 2M′. The columns of the matrix are ordered so that the indices increase from left to right. The rows are ordered such that the indices increase from top to bottom. The (decimal) row and column indices of the G-matrix element, Gmn, whose row m corresponds to the state (RM′, RM′-1, ..., R1) and whose column n corresponds to the state (βM′, βM′-1, ..., β1) are given by

m ) RM′(2M'-1) + RM'-1(2M'-2) + ‚‚‚ + R1(20) + 1 (9) n ) βM'(2M'-1) + βM'-1(2M'-2) + ‚‚‚ + β1(20) + 1 (10) As exemplified in the forthcoming sections, this ordering is crucial to account for edge effects on terraces or wrapping effects on

Figure 3. Two consecutive sequences of M sites on a lattice made of parallelograms. The order of the column and row indices of the G-matrix follows the order of the states of the sites in the upper and lower sequences, respectively. To develop a recursive construction of the G-matrix, the sequences are divided into two sections by placing a partition between sites N + 1 and N. The first section contains the sites from M to N + 1, and the second contains sites from N to 1. The sites in the first section are in a given occupational state while sites in the second section are allowed to be in all possible states.

nanotubes. In addition, it leads to the development of a fast and efficient recursive algorithm to construct the G-matrix, and subsequently the T-matrix, whether the lattice is a terrace or a nanotube. Element Gmn, with indices m and n given by eqs 9 and 10, is of the form given by eq 2. Exponent a0 is the number of adsorbates in the nth occupational configuration and is given by the number of β-coefficients equal to 1 in eq 10. Exponent ai is the number of ith neighbors between adsorbates in the nth configuration and the adsorbates in both the nth configuration and those in the adjacent sequence corresponding to the mth configuration. The rank of the T- and G-matrices for monomer adsorption on a lattice with M′ ) 10 is 210 ) 1024. The number of matrix elements is over a million. Including the shifted chemical potential, µ, each element involves J + 1 energy parameters, where J is the total number of pairwise adsorbate interactions under consideration (of highest order I). Computing numerical values for all of the matrix elements requires the development of an algorithm that is both efficient and easily programmed. The examples presented in the following sections show how this is accomplished on the basis of this general formulation.

4. General Method Applied to an Adsorbate Lattice Made of Parallelograms Parallelogram lattices provide a unified treatment of all the monomer adsorption systems studied in the past, including lattice edges of differing orientations. They also provide a simple example of how the G-matrix is recursively constructed, based on the ordering presented in Section 3. The T-matrix for the associated terrace or nanotube is then derived from the G-matrix, as explained in Section 2. Two consecutive rows of M sites along the horizontal width of the parallelogram lattice are shown in Figure 3. The orientation and length of the sides of the parallelograms are arbitrary. By symmetry, it is sufficient to consider the angle in the lower left corner of the parallelograms as less than or equal to 90°. Here we only consider interactions between first neighbor adsorbates in the same horizontal row and between adjacent neighbors of first through fourth order in consecutive horizontal rows. Under these conditions, the number of sites in the sequence that is consistent with these pairwise interactions is M′ ) M. In the previous sections, the ith order activity is associated with the ith order pairwise interaction, which, in this case, depends

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on the geometry of the parallelogram (size and orientation). As this geometry has not been specified, the adsorbate-substrate activity previously called x0 is renamed simply x. The activity between first neighbor adsorbates on the same sequence of M sites is named y. For example, y is the activity of adsorbates occupying the sites β and γ in Figure 3. The activities between adjacent neighbor adsorbates in consecutive sequences are likewise named q, r, s, and t. In Figure 3, activity q is associated with adsorbates located at sites γ and γ′, activity r between sites at β and γ′, activity s between sites γ and β′, and activity t between sites R and γ′. Matrix CN is composed of G-matrix elements with column and row indices corresponding to a selected number of occupational states of two consecutive sequences, as indicated in Figure 3. These sequences are partitioned into two sections. The partition is placed between locations N + 1 and N. The first section, consisting of locations M to N + 1, has a given state. The second section, consisting of locations N to 1, has all possible states. Thus, the rank of CN is 2N where N indicates its rank. In the first section of the upper sequence there are a adsorbates and b adjacent adsorbate neighbor pairs. Similarly, c, d, e, and f are the numbers of pairs of adjacent adsorbate neighbors in the first section which have activities of types q, r, s, and t, respectively. Hence, all CN elements have a common factor of the form xa yb qc rd se tf due to the given state of the first section. After factoring out xa yb qc rd se tf, the remaining matrix, denoted AN and of rank 2N, is associated with the second section having all possible states. The rows and columns of AN and GN are ordered in the same manner. Matrix CN may therefore be rewritten as

CN ) xaybqcrdsetfAN

(11)

In Figure 3, the sites at locations N + 2, N + 1, and N are designated R, β, and γ on the upper sequence, and R′, β′, and γ′ on the lower sequence. These designations are also used to indicate the state of occupation (0 or 1) of the corresponding sites. When the elements of matrix AN are constructed, the states R and β contribute an activity y when β ) γ ) 1, an activity r when β ) γ′ ) 1, and an activity t when R ) γ′ ) 1. Similarly, β′ contributes an activity s when β′ ) γ ) 1. On the other hand, the state R′ does not affect any of the elements of AN. Therefore, AN is denoted AN(R,β;β′). For a given N, there are eight distinct AN matrices because R, β, and β′ can each be either 0 or 1. There are two extreme partitioning cases. If the entire set of sites in the two consecutive sequences are in the first section, then there are no sites in the second section and N ) 0. In this case, the C0 has one element of the form xa yb qc rd se tf. Thus, A0(R,β;β′) is 1 for all possible values of (R, β; β′) making the initial conditions for the eight AN matrices

A0(R,β;β') ) 1

(12)

If the entire set of sites in the two consecutive sequences are in the second section, leaving no sites in the first section, the adsorption problem is unaffected if two vacant sites are added to each sequence at locations M + 1 and M + 2. In this case, the partition is between locations M and M + 1. The states of the four “new” sites on the two consecutive sequences (at M + 1 and M + 2) are all 0. Matrix CM is the G-matrix and is given by

CM ) G ) AM(0, 0;0)

(13)

All eight matrices AN may now be recursively constructed, for any N, which ultimately provide the solution for the G-matrix from eq 13.

A shift from sites at the corners of the shaded region of Figure 3, (R,β;R′,β′), to (β,γ;β′,γ′) maintains the same parallelogram topology. Consequently, when the partition is moved from between locations N + 1 and N to between locations N and N - 1, the matrix associated with the second section is of the same type as AN but of lower rank, 2N-1, and depends on the states β, γ, and γ′, namely, AN-1(β,γ;γ′). As a result, AN(R,β;β′) can be divided, with the appropriate activity factors (as explained above), into four block matrices AN-1(β,γ;γ′) of lower rank, which depend on the states γ and γ′ of the sites at location N. Therefore, the column indices of these block matrices are ordered according to the values γ ) 0 or γ ) 1. Similarly, the row indices are ordered according to the values γ′ ) 0 or γ′ ) 1. The matrix recursion is then

AN(R,β;β′)

γ)0

γ)1

γ′)0

AN-1(β, 0;0)

x yβ sβ' AN-1(β, 1;0)

γ′)1

rβ tR AN-1(β, 0;1) x yβ q rβ sβ' tR AN-1(β, 1;1) (14)

This ordering ensures that the numbering of rows and columns of AN(R,β;β′) follows the binary system described earlier when the matrices are recursively constructed. The computational program follows the algorithm set by eqs 12-14 and is especially suited for multi-parallel processing. Modifications to the G-matrix, given by AM(0, 0;0), to provide a T-matrix which accounts for the edge effects on terraces and for wrapping of the lattice to form a nanotube are now presented. 4.1. Edge Effects on Terraces Made of Parallelograms. Edge effects are modeled by considering that the adsorption on edge sites is different from that on bulk sites. This requires additional differential activities u1 and u2, as explained in Section 2. A factor of ub11ub22 is added to G-matrix elements in columns with indices for which the states β1 (first site on the right edge) and βM (the Mth site at the left edge) are either (β1 ) 0, βM ) 1), (β1 ) 1, βM ) 0), or (β1 ) 1, βM ) 1). According to eq 10, this corresponds to column numbers 2n, 2M-1 + n - 1, and 2M-1 + 2n, where n ) 1, ..., 2M-2, respectively. Thus, a factor of u1 (b1 ) 1) is added to all elements in even-numbered columns of the G-matrix and another factor of u2 (b2 ) 1) is added to the elements in columns of the second half of the G-matrix. The special case u1 ) u2 corresponds to both edge sites being symmetrically either step-up or step-down, as noted earlier. All the T-matrices for monomer adsorption problems constructed in the past as separate cases are recovered from this compact generalized formulation. These special cases of the parallelogram lattice are shown in Figure 1a-d. The square lattice of Figure 1a corresponds to activities q ) y (first neighbor activity) and r ) s (second neighbor activity). Activity t is equal to 1 since this higher order interaction was neglected.61,62,67 In ref 61, we neglected edge effects and considered only first neighbor interactions, which led to recursive relations involving two matrices. In ref 62, edge effects were neglected but second neighbor interactions were included, which led to recursive relations among four matrices. In ref 67, edge effects were included with u1 ) u2, as well as first and second neighbor interactions. Again, this resulted in recursive relations among four matrices. Figure 1b is a square lattice with a linear array of edge sites at 45° which corresponds to the parallelogram structure shown highlighted in the figure. This was studied in ref 68 including edge effects and first and second neighbor interactions. In this case, y ) r corresponded to the first neighbor activity, q ) t to


Figure 4. Wrapping a parallelogram lattice on a cylinder to form a nanotube. Sites at locations numbered 1 and M on the same sequence of M sites become first neighbors. We limit this analysis to interactions between first neighbor adsorbates on the upper sequence and between first, second, third, and fourth neighbor adsorbates belonging to the upper and lower sequences. Solid lines connect pairs of sites, not previously considered, which could account for additional activity factors when both sites in a given pair are occupied, such as y, r, s, or t.

the second neighbor activity, and s was equal to 1 since it corresponded to an activity of higher order which was neglected. The T-matrix was the result of recursion relations between eight matrices. Adsorption on fcc(110) as shown in Figure 1c, with first, second, and third neighbor interactions, was studied in refs 63 and 64. The parallelogram becomes a rectangle for which y is the first neighbor activity, q is the second neighbor activity, r ) s is the third neighbor activity, and t ) 1 corresponds to neglected fourth neighbor interactions. Adsorption on bcc(110), studied in refs 65 and 66, is another special case of the parallelogram lattice shown in Figure 1d. The lower left corner of the highlighted parallelogram has an angle of 54.74°. In this geometry, y ) r is the first neighbor activity, and q, t, and s are the second, third, and fourth neighbor activities, respectively. However, only first and second neighbor interactions were included in ref 65, which resulted in recursion relations involving three matrices. In ref 66, third neighbor interactions were added, which resulted in eight recursively related matrices. The general formulation with the parallelogram lattice has therefore provided a unified treatment of all of the previous monomer adsorption studies. The T-matrices associated with all of these problems are recovered as special cases. This same formulation is now used to discuss adsorption on nanotube surfaces. 4.2. Nanotube Surfaces Made of Parallelograms. Wrapping a parallelogram lattice on a cylinder causes the first site in a given sequence to be adjacent to the Mth site in the same sequence, as shown in Figure 4. In this figure, the states of occupation of the sites on the upper sequence, at locations 2, 1, M, and M 1, are R, β, γ, and δ, respectively. Similarly, the states of the sites at the same locations on the lower sequence are R′, β′, γ′, and δ′, respectively. This “untwisted” nanotube may also be viewed as twisted. This becomes apparent when considering sequences along the helical lines connecting sites such as R′, β, etc., or β′, γ, etc. To obtain the T-matrix due to this wrapping, as indicated in Figure 4, a factor y is added to the elements of the G-matrix corresponding to β ) γ ) 1. Similarly, a factor r is added to elements corresponding to β ) γ′ ) 1. A factor s is added to elements corresponding to γ ) β′ ) 1. Finally, a factor t is added to elements corresponding to β ) δ′ ) 1 or R ) γ′ ) 1. Using the enumeration of rows and columns defined by eqs 9 and 10, the following conclusions are drawn. • β ) γ ) 1 corresponds to the elements in column numbers 2M-1 + 2n, with n ranging from 1 to 2M-2. Therefore, a factor

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Figure 5. Hexagonal lattices with different orientations: (a) zigzag and (b) armchair. When adsorption sites are located at the center of the hexagonal ring, the adsorbate lattice is composed of equilateral triangles. In each case, two consecutive rows of sites across the width of the lattice are connected by a solid line.

of y is added to the elements in even columns of the second half of the G-matrix. • β ) γ′ ) 1 corresponds to the elements in column numbers 2n, with n ranging from 1 to 2M-1, and in row numbers 2M-1 + m, with m also ranging from 1 to 2M-1. Therefore, a factor r is added to the elements of columns in the second half of the G-matrix which are also in even numbered rows. • γ ) β′ ) 1 corresponds to the elements in row numbers 2M-1 + 2m, with m ranging from 1 to 2M-2 and in column numbers 2n, with n ranging from 1 to 2MM-1. Therefore, a factor of s is added to the elements of rows in the second half of the G-matrix which are also in even columns. • β ) δ′ ) 1 corresponds to the elements in even columns (β ) 1) of rows in the second and fourth quarters (δ′ ) 1) of the G-matrix. Also, R ) γ′ ) 1 corresponds to the elements in columns 4n - 1 and 4n, with n ranging from 1 to 2M-3 (R ) 1) and the second half of the rows of the G-matrix (γ′ ) 1). In each case, a factor t is added to the corresponding matrix elements. This completes the construction of the T-matrix for nanotubes made of parallelograms.

5. Adsorbate Lattice Made of “Zigzag” Equilateral Triangles An equilateral triangular lattice is composed of either a bcc or fcc(111) surface. An adsorbate lattice with the same geometry is also obtained when adsorption on a hexagonal surface takes place preferentially at the center of the hexagons, as may be the case on single-walled carbon nanotubes.76 There are two particular orientations we consider in this paper: “zigzag” and “armchair,” as shown in Figure 5a,b. These also correspond to zigzag and armchair carbon nanotubes, respectively. For terraces, edge sites are first or second neighbors depending on whether the equilateral triangular lattice is an armchair or zigzag. In this section, we consider the zigzag case of Figure 5a with first and second neighbor interactions corresponding to activities y and z, respectively. Thus, the appropriate sequence of sites relevant to the construction of the G-matrix is the same as the zigzag of the sites across the physical width of the lattice. Unlike Section 4, there are two types of matrices here, CN and DN-1, whose elements are found within the G-matrix with column and row indices specified by the states of the two consecutive zigzag sequences highlighted in Figure 6a,b. These sequences are again partitioned into two sections where the first has a given state and the second has all possible states. However, depending on where the partition is placed, the sites in the two zigzag sequences on the left of the patition, and closest to it, could be at the top or the bottom of the zigzag. In Figure 6a, the first (76) Dai, H. Surf. Sci. 2002, 500, 218.

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In constructing the elements of matrix BN-1, the states β, γ, δ, and γ′ account for an activity y whenever γ ) ) 1, δ ) ) 1, δ ) φ ) 1, or δ ) ′ ) 1 and for an activity z whenever β ) ) 1, β ) ′ ) 1, γ ) φ ) 1, γ′ ) φ ) 1, δ ) ν ) 1, or δ ) ν′ ) 1. However, the states β′ and δ′ have no effect as long as higher order interactions are not considered. Thus, matrix BN-1 is denoted BN-1(β,γ,δ;γ′) and depends on the states of occupation of four of the six sites making up the corners of the shaded polygon of Figure 6b. For a given N, there are 16 BN matrices. As for the parallelogram lattice, we consider the limiting case when the first section of the upper and lower sequences contains all 2M sites in a given state. If the first sites are at the top of the zigzag, A0(R,β,γ;R′,γ′) ) 1; if they are at the bottom, B0(β,γ,δ;γ′) ) 1, valid for all possible values of their arguments. This yields the initial conditions

A0(R,β,γ;R′,γ′) ) 1 or B0(β,γ,δ;γ′) ) 1

Figure 6. A lattice made of equilateral triangles of the type shown in Figure 5a is partitioned into two sections as for the parallelogram lattice of Figure 3. Two consecutive sequences of M sites, connected by a heavy dashed line, are shown. The state of these sequences is related to the G-matrix elements associated with the adsorption of monomers with first and second neighbor interactions.

section is from M to N + 1 with the two sites at N + 1 in the top position. This leads to a matrix CN of rank 2N. In Figure 6b, the first section is from M to N with sites at N in the bottom position, leading to a matrix DN-1 of rank 2N-1. In constructing the elements of CN or DN-1, we designate the number of adsorbates to be a in the first section of the upper sequence of sites, whether from M to N + 1 or from M to N. Similarly, the numbers of first and second neighbor adsorbate pairs in the first section of the combined upper and lower sequences are denoted b and c, respectively, excluding those found within the lower sequence. It then follows that all of the elements of CN or DN-1 have a common factor of the form xa yb zc. Factoring out this term, we obtain

CN ) xaybzcAN

(15)

DN-1 ) xaybzcBN-1

(16)

In Figure 6a,b, the states of occupation of the sites at locations N + 3, N + 2, N + 1, N, N - 1, N - 2, and N - 3 in the upper sequence are designated R, β, γ, δ, , φ, and ν, respectively, and in the lower sequence R′, β′, γ′, δ′, ′, φ′, and ν′, respectively. In the process of constructing the elements of matrix AN, the states R, β, γ, R′, and γ′ account for an activity y whenever β ) δ ) 1, γ ) ) 1, γ ) δ ) 1, or γ′ ) δ ) 1; and for an activity z whenever R ) δ ) 1, β ) ) 1, β ) ′ ) 1, γ ) φ ) 1, R′ ) δ ) 1, or γ′ ) φ ) 1. However, the state β′ has no effect as long as higher order interactions are not considered. Thus, matrix AN is denoted AN(R,β,γ;R′,γ′) and depends on the state of occupation of five of the six sites making up the corners of the shaded polygon of Figure 6a. For a given N, there are 32 AN matrices.

(17)

A shift from sites (R,β,γ;R′,β′,γ′), which are at the corners of the shaded polygon of Figure 6a, to the sites (β,γ,δ;β′,γ′,δ′), exhibits a different topology represented by the shaded polygon of Figure 6b. Referring to Figure 6a, AN(R,β,γ;R′,γ′) is divided into four blocks which depend on the states δ and δ′ of the sites at location N. The column indices of these blocks are ordered according to δ ) 0 or 1 and the row indices according to δ′ ) 0 or 1. These blocks are obtained in terms of B-matrices of lower rank, 2N-1, as follows

AN(R,β,γ;R′,β′)

δ)0

δ)1

δ′)0

BN-1(β,γ, 0;γ′)

x yβ+γ+γ′ zR+R′ × BN-1(β,γ, 1;γ′)

δ′)1

β+γ+γ′ R+R′+1 z × BN-1(β,γ, 0;γ′) x y BN-1(β,γ, 1;γ′)

(18)

Exchanging R and R′ does not change this recursion relation, consequently

AN(R,β,γ;R′,γ′) ) AN(R′,β,γ;R,γ′)

(19)

Since this is true for any set of values of (β, γ, γ′) but is meaningful only when R * R′, there are eight relationships between the 32 AN matrices and therefore only 24 are linearly independent. A shift from sites (β, γ, δ; β′, γ′, δ′), which are at the corners of the shaded polygon in Figure 6b, to sites (γ, δ, ; γ′, δ′, ′), returns to the same topology as that of the shaded polygon in Figure 6a. This allows the recursive construction of matrices BN-1(β,γ,δ;γ′) in terms of matrices of the A-type of lower rank. This is apparent when the BN-1 matrix is divided into four block matrices with column indices ordered according to state ) 0 or 1, and row indices ordered according to state ′ ) 0 or 1. With this ordering, the B-matrices are related to the A-matrices as

BN-1(β,γ,δ;γ′)

)0

)1

AN- 2(γ,δ, 0;γ′, 0)

z × xy AN-2(γ,δ, 1;γ′, 0) γ+δ β

′)0 )1

γ+2δ 2β+1 z × yδ zβ AN-2(γ,δ, 0;γ′, 1) xAy (γ,δ, 1;γ′, 1) N-2 (20)

All 16 B-matrices are linearly independent. The chosen ordering ensures recursively that the numbering of rows and columns of the A- and B-matrices follows the binary system described earlier.


Langmuir, Vol. 23, No. 2, 2007 567

may require an additional factor of y, y2, y3, or y4, as expected from the number of solid line segments connecting first neighbor sites in Figure 7a. In Figure 7b, the six solid line segments crossing the boundary between the first and Mth sites are connections between second neighbor sites created by the wrapping. Thus, certain G-matrix elements require an additional factor of z, z2, z3, z4, z5, or z.6 These depend on the states of sites , R, β, γ, δ, and ν in the upper sequence associated with the column indices of the A-matrix, and the states of occupation of sites ′, β′, and δ′ in the lower sequence associated with the row indices of A. As before, this determines the column and row indices of the G-matrix elements which require additional factors of z, z2, z3, z4, z5, or z6.

6. Adsorbate Lattice Made of “Armchair” Equilateral Triangles

Figure 7. The analogue of Figure 4 adapted to wrapping a zigzag equilateral triangular lattice of width M around a cylinder, possible only when M is even. The relevant section of the two consecutive sequences of sites shows that the first and Mth sites on each sequence are now first neighbors. The contributing first neighbor sites are shown connected by solid segments in (a). The contributing second neighbor sites are shown connected by solid segments in (b). Note that these segments cross the boundary between the first and Mth sites.

As in the example of the parallelogram lattice, any number of additional vacant sites may always be added to the left of every sequence of M sites without changing the adsorption problem. In this case, it is sufficient to add three vacant sites to each sequence. Since the Mth site on a sequence may be at the top or bottom of the zigzag, there are two possibilities. If it is at the top, the G-matrix is given by BM(0, 0, 0;0), and if it is at the bottom, the G-matrix is given by AM(0, 0, 0;0, 0). In both cases, the parity of M determines whether the right edge site (the first site) is in the top or bottom position of the zigzag. The T-matrix for terraces made of equilateral triangles, with the orientation chosen in Figure 6, is obtained from the same analysis as that in Section 4. For a nanotube, wrapping the zigzag triangular lattice on a cylinder is possible if, and only if, M is an even number. If the Mth site on a sequence is at the bottom of the zigzag, the first site of that sequence is at the top, as indicated in Figure 7. Figure 7a shows the first neighbor sites left unaccounted when the G-matrix was constructed. They are shown connected by solid line segments. An additional factor of y is required whenever two first neighbor sites are occupied. Similarly, Figure 7b demonstrates when additional factors of z are required due to the new second neighbor sites. Since in this example the Mth site on the sequence is at the bottom of the zigzag, the G-matrix is given by AM(0, 0, 0;0, 0). Following Figure 7a, an additional factor of y is is required in all the elements of the G-matrix in columns corresponding to the states in which β ) γ ) 1, β ) δ ) 1, and R ) γ ) 1. These correspond to columns numbered 2M-1 + 2n where n ) 1, ..., 2M-1; 2M-2 + 2n and 2M-1 + 2M-2 + 2n where n ) 1, ..., 2M-3; and 2M-1 - 1 + 4n and 2M-1 + 4n where n ) 1, ..., 2M-3. Yet another factor of y is required in matrix elements in columns corresponding to the state of occupation γ ) 1 and in rows corresponding to β′ ) 1. The column numbers are 2M-1 + n where n ) 1,..., 2M-2 and the row numbers are 2m where m ) 1,..., 2M-1. The G-matrix elements affected by this analysis

In the previous cases, we have shown that recursive construction of the G-matrix is linked to topologically distinct polygonal structures. The generalization of this construction to more complex situations is straightforward and leads to a simple, diagrammatic recursive construction of the G-matrix which avoids unnecessary repetitions. We have identified two vertically consecutive sequences of sites across the horizontal width of the lattice whose occupational states are linked to the elements of the G-matrix. The state of the upper sequence is associated with the column index of the matrix element while the state of the lower sequence is associated with the row index. These sequences are partitioned into two sections. The sites in the first section have a given state and the sites in the second section have all possible states. In the process, polygons are identified with corners that consist of sites from the two consecutive sequences at two or more consecutive locations. These polygons correspond to the shaded region of Figure 3 for the parallelogram lattice and the two shaded regions of Figure 6 for the zigzag equilateral triangular lattice. Matrices of a given type are associated with every polygon, and their rank depends on the location of the polygon within the two consecutive sequences. This in turn provides the initial conditions of the matrices. Matrices of a given type depend on the states of a number of sites found at the corners of the associated polygon. Their elements are selected from within the G-matrix as specified by the states of the sites of the two partitions mentioned above. Recursion relations are then established among all matrices of all types. The result of the recursions is the single G-matrix. In this section we consider adsorption in the hollows of an armchair hexagonal substrate, as shown in Figure 5b, which forms an adsorbate lattice of armchair equilateral triangles with an orientation of 90° to the one treated in Section 5. Once again, we include only first and second neighbor interactions. The correct sequence consists of two consecutive armchair rows as explained in Section 2 and exhibited in Figure 8a. The sites in the lower armchair row are assigned odd numbers and to those in the upper armchair row, even numbers. The sites are numbered sequentially from right to left. In a given sequence, the two sites at location N are numbered 2N - 1 and 2N, as shown in Figure 8a. We now consider a subset of the states of the two vertically consecutive sequences according to the partitioning technique described above. Figure 8b shows a section from location N + 2 to N - 2. The topologically distinct polygonal structures identified in this case are two parallelograms of different orientation. These are indicated by the shaded and unshaded regions of Figure 8b. When the partition is placed between locations N + 1 and N, the matrix identified as AN depends on the states of sites a, b,

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Figure 9. Diagrammatic representation of the four sets of blockmatrices A, B, C, and D, making up the G-matrix associated with adsorption on a finite width lattice made of armchair hexagons.

of the pair (h, g) label the columns and the states of the pair (h′, g′) label the rows, following the same binary order. This gives rise to blocks depending on matrices of the A-type and the recursive relation is Figure 8. (a) Numbering of the sites in the sequence made of two consecutive rows of sites across the width of an armchair equilateral triangular lattice. (b) A diagrammatic representation of the two sets of block-matrices, A and B, making up the G-matrix associated with adsorption on a finite width lattice made of armchair equilateral triangles as highlighted in Figure 5b.

c, d, c′, and d′, but not on those of a′ and b′ on the perimeter of the shaded parallelogram. Its rank is 22N and there are 2N sites in the second partition from location N to 1. When the partition is placed between locations N and N - 1, the matrix identified as BN-1, of rank 22N-2, depends on the states of sites c, d, e, f, and f ′, but not on those of c′, d′, and e′ on the perimeter of the unshaded parallelogram. If the first sites are at the top of the armchair, A0(a, b, c, d;c′, d′) ) 1, and if they are at the bottom, B0(c, d, e, f;f ′) ) 1, valid for all possible values of their arguments. This yields the initial conditions

A0(a, b, c, d;c′, d′) ) 1 or B0(c, d, e, f;f') ) 1

(21)

Matrix AN(a, b, c, d;c′, d′) may itself be divided into 16 blocks depending on the states of the sites at location N. The states of the pair (f, e) label the columns and the states of (f ′, e′) label the rows, following the binary order (0, 0), (0, 1), (1, 0), and (1, 1). This ordering results in blocks depending on matrices of the B-type with the recursive relation

[

AN(a, b, c, d;c′, d′) )

BN-1(c,d,0,0;0)

xyc+d′ za+d+c′ × BN-1(c, d, 1, 0; 0)

xyc+d zb+d′ × BN-1(c, d, 0, 1; 0)

x2y1+2c+d+d′ za+b+d+c′+d′ × BN-1(c, d, 1, 1; 0)

BN-1(c,d,0,0;0)

xyc+d′ za+d+c′ × BN-1(c, d, 1, 0; 0)

xyc+d zb+d′ × BN-1(c, d, 0, 1; 0)


xyc+d′+1 za+c+d+c′ × BN-1(c, d, 1, 0; 1)

xyc+d zb+c+d′ × BN-1(c, d, 0, 1; 1)


xyc+d′+1 za+c+d+c′ × BN-1(c, d, 1, 0; 1)

xyc+d zb+c+d′ × BN-1(c, d, 0, 1; 1)


z × c

BN-1(c,d,0,0;1) zc × BN-1(c,d,0,0;1)

]

(22)

Similarly, BN-1(c, d, e, f;f ′) is divided into 16 blocks which depend on the states of the sites at location N - 1. The states

[

BN-1(c, d, e, f;f') )

AN-2(e,f,0,0;0)

xye+f zc+f′ × AN-2(e, f, 1, 0; 0, 0)

xyf zd+e × AN-2(e, f, 0, 1; 0, 0)

x2y1+e+2f zc+d+e+f′ × AN-2(e, f, 1, 1; 0, 0)

ze × AN-2(e,f,0,0;1,0)

xye+f zc+e+f′ × AN-2(e, f, 1, 0; 0)

xyf zd+2e × AN-2(e, f, 0, 1; 1,0)

x2y1+e+2f zc+d+2e+f′ × AN-2(e, f, 1, 1; 1, 0)

yezf × AN-2(e,f,0,0;0,1)

xy1+2e+f zc+f+f′ × AN-2(e, f, 1, 0; 1)

xye+f zd+e+f × AN-2(e, f, 0, 1; 0,1)

x2y2+2e+2f zc+d+e+f+f′ × AN-2(e, f, 1, 1; 0, 1)

yeze+f × AN-1(e,f,0,0;0,1)

xy1+2e+f zc+e+f+f′ × AN-2(e, f, 1, 0; 1)

xye+f zd+2e+f × AN-2(e, f, 0, 1; 1, 1)

x2y2+2e+2f zc+d+2e+d+f′ × AN-2(e, f, 1, 1; 1, 1)

]

(23) th

If the M location on the armchair row is at the top, the G-matrix is given by AM(0, 0, 0, 0;0, 0); if at the bottom, it is given by BM(0, 0, 0, 0;0). The T-matrix follows by adding the appropriate factors to the elements of the G-matrix to account for edge effects on terraces, or the wrapping to form a nanotube (possible only if M is even). The armchair equilateral triangular terrace three atomic sites in width has been investigated and applied to the chemisorption of CO on Pt(112).78 Our model suggests experiments that would allow the various interaction energies to be obtained from the knowledge of the relatively low-temperature phases and the conditions prevailing at the transitions between the phases. The case of the nanotube is of interest, as it corresponds to an armchair carbon nanotube for which adsorption takes place at the centers of the hexagonal carbon rings.

7. Adsorbate Lattice Made of Armchair Hexagons In this and the following section, the adsorbate lattice is hexagonal. The model is applicable to preferential adsorption in the hollows of fcc or bcc(111) surfaces. It is also applicable to on-top adsorption on graphite or carbon nanotube surfaces. From consideration of only first and second neighbor adsorbateadsorbate interactions, it follows that the appropriate sequence of sites across the width of the lattice is the armchair row of M sites shown in Figure 9a. The diagrammatic short-cut presented in Section 6 again leads to the recursive construction of the G-matrix.


Langmuir, Vol. 23, No. 2, 2007 569

Figure 9b shows a section of two consecutive sequences of M sites at locations ranging from N + 2 to N - 3, with states of occupation a, b, c, d, e, and f on the upper sequence and a′, b′, c′, d′, e′, and f ′ on the lower sequence, respectively. The polygonal structures consist of two topologically distinct parallelograms (shaded regions) and two topologically distinct rectangles (unshaded regions). The two types of parallelograms are distinguished by an angle of either 150° or 30° at the lower left corner. The two types of rectangles are distinguished by their relative vertical shift. Consequently, there are four pairs of locations between which the partition can be placed: N + 1 and N, N and N - 1, N - 1 and N - 2, or N - 2 and N - 3. Different matrices are introduced with each partition, AN, BN-1, CN-2, and DN-3, respectively. As mentioned before, the subscript indicates the rank of the matrix which is the total number of states that the sites, in the section of the sequence to the right of the partition, may have. It is straightforward to show their dependence on the following states, AN ) AN(a, b;a′), BN-1 ) BN-1(b, c), CN-2 ) CN-2(c, d;d′), and DN-3 ) DN-3(d, e; d′, e′). There is a total of 36 matrices, for a given N, satisfying the initial conditions

A0(a, b; b′) ) 1 B0(b, c) ) 1 C0(c, d; d′) ) 1 D0(e, d; d′, e′) ) 1 (24) It is also straightforward to show that these matrices are recursively related according to

AN(a, b;c)

c)0

c)1

c′)0

BN-1(b, 0)

x ybza+a′ BN-1(b, 1)

c′)1

BN-1(b, 0)

x ybza+a′+1 BN-1(b, 1)

BN-1(b, c)

d)0

d′)1

d)1

CN-2(c, 0;0)

d′)0

(25)

c b

x y z CN-2(c, 1;0)

yc zbCN-2(c, 0;1) x y2c z2b+1 CN-2(c, 1;1) (26)

CN-2(c, d;d′)

e)0

e)1

DN-3(d, 0;d′, 0)

xy z × DN-3(d, 1;d′, 0) d c

e′)0 e′)1

d 2c+1 × zc × DN-3(d, 0;d′, 1) xDy z (d, 1;d′, 1) (27) N-3

DN-3(d, e;d′, e′)

f)0

f)1

f ′)0

AN-4(e, 0;e′)

f ′)1

AN-4(e, 0;e′) x ye+e′ zd+d′+1 AN-4(e, 1;e′) (28)

xy

e+e′ d+d′

z

AN-4(e, 1;e′)

These relations exhibit simple symmetries, namely AN(a, b; a′) ) AN(a′, b; a) and DN(d, e; d′, e′) ) DN(d′, e;d, e′). Based on these properties, we also find that DN(a, b; a′, b′) ) DN(a′, b′; a, b). Thus, of the 36 matrices of a given rank, only 27 are linearly independent. The Mth site on a sequence could be in one of two top locations, such as d or e, or one of two bottom locations, such as b or c. In these cases, the G-matrix is given as CM(0, 0;0) or DM(0, 0;0, 0), or AM(0, 0;0) or BM(0, 0), respectively. For M ) 1, all of these matrices are identical. This is not surprising, as they all provide the G-matrix for the one-dimensional problem. The changes to the G-matrix required to obtain the T-matrix which describes adsorption on terraces and on armchair carbon

Figure 10. (a) Analogue of Figure 8a. It exhibits the numbering of the sites in the sequence appropriate to adsorption on an armchair hexagonal lattice with first and second neighbor interactions. (b) The analogue of Figure 8b.

nanotubes (possible now only if M is a multiple of four) are similar to those described in previous cases.

8. Adsorbate Lattice Made of Zigzag Hexagons The substrate is a zigzag hexagonal lattice with on-top adsorption, and the corresponding model is applicable to on-top adsorption on graphite or zigzag single-walled carbon nanotubes. The finite width lattice consists of a zigzag row of M sites. We consider again only first and second neighbor adsorbate-adsorbate interactions. The two consecutive sequences of sites, whose states provide the elements of the G-matrix, are those in the two shaded regions of Figure 10a, where each sequence has 2M sites. There are two vertical sites belonging to a given sequence at any location. Successive pairs of sites in a sequence can be in either of two vertical positions. The ordering of the sites is such that, at location N indicated in Figure 10a, the lower site in a pair is numbered 2N - 1 and the upper site in that pair is numbered 2N. Figure 10b shows a section of two consecutive sequences at locations ranging from N + 2 to N - 2, with states a, b, c, d, e, f, g, h, i, and j on the upper sequence and a′, b′, c′, d′, e′, f ′, g′, h′, i′, and j′ on the lower sequence, respectively. The pair of sites at location N in either sequence is in the upper vertical position. When the partition is placed between locations N + 1 and N, matrix AN, associated with all possible states of the sites to the right of the partition, depends on the states of sites a, b, c, and d but not on a′, b′, c′, or d′. Sites a and d′ must be added to sites b, c, d, a′, b′, and c′ on the perimeter of the shaded parallelogram in order to complete the topological figure associated with A-type matrices. When the partition is placed between locations N and N - 1, matrix BN-1, associated with all possible states to the right of the partition, depends on the states of sites c, d, e, f, e′, and f ′, but not on c′ or d′. Sites e and

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d′ must be added to sites c, d, f, c′, f ′, and e′ on the perimeter of the nonshaded parallelogram to complete the topological figure associated with B-type matrices. As before, subscript N indicates the rank 22N of the corresponding matrix, and we have

AN ) AN(a, b, c, d ) BN-1 ) BN-1(c, d, e, f; e′, f′)

(29)

If the first pair of sites in a sequence is in the lower vertical position, then A0(a, b, c, d) ) 1 for all possible values of its arguments. If the first pair of sites in a sequence is in upper vertical position, then B0(c, d, e, f; e′, f ′) ) 1 for all possible values of its arguments. Consequently, the initial conditions are

A0(a, b, c, d ) ) 1 or B0(c, d, e, f; e′, f') ) 1

(30)

Matrix AN(a, b, c, d) is divided into 16 blocks consisting of B-type matrices of lower rank with an appropriate activity factor. The columns of these blocks are labeled by the states of the pair of sites (e, f) at location N on the upper sequence. The rows are labeled by the states of the pair of sites (e′, f ′) at location N on the lower sequence. The ordering of these pairs follows the binary system (0, 0), (0, 1), (1, 0), and (1, 1). The recursion relation is then

[ [

AN(a, b, c, d ) )

BN-1(c,d,0, 0; 0, 0)

xyc zb+d × BN-1(c, d, 0, 1, 0; 0)

x zc × BN-1(c, d, 1, 0, 0; 0)

x2yc+1 zb+c+d × BN-1(c, d, 1, 1; 0, 0)

zd × BN-1(c,d,0, 0; 0, 1)

xyc zb+2d × BN-1(c, d, 0, 1, 0; 1)

x zc+d × BN-1(c, d, 1, 0, 0; 1)

x2yc+1 zb+c+2d × BN-1(c, d, 1, 1; 0, 1)

zdzc × BN-1(c,d,0, 0;1, 0)

xyc+d zb+c+d × BN-1(c, d, 0, 1, 1; 0)

xyd z2c × BN-1(c, d, 1, 0, 1; 0)

x2yc+d+1 zb+2c+d × BN-1(c, d, 1, 1; 1, 0)

yczc+d × BN-1(c,d,0, 0;1,1)

xyc+d zb+c+2d × BN-1(c, d, 0, 1, 1; 1)

xyd z2c+d × BN-1(c, d, 1, 0, 1; 1)

x2yc+d+1 zb+2c+2d × BN-1(c, d, 1, 1; 1, 1)

] ]

(31)

Similarly, the BN-1-matrices are recursively related to the AN-2 matrices as BN-1(c, d, e, f; e', f') )

xy z × xyf zc+e+e′ × x2yf+e′+1 zc+d+e+f+e′+f′ × AN-2(e, f, 0, 0) AN-2(e, f, 1, 0; 0, 0) AN-2(e, f, 0, 1; 0, 0) AN-2(e, f, 1, 1) e′ d+f+f′

AN-2(e, f, 0, 0)

xye′ zd+f+f′ × AN-2(e, f, 0, 1)

xyf zc+e+e′ × AN-2(e, f, 1, 0)

x2yf+e′+1 zc+d+e+f+e′+f′ × AN-2(e, f, 1, 1)

AN-2(e, f, 0, 0)

xye′ zd+f+f′ × AN-2(e, f, 0, 1)

xyf zc+e+e′ × AN-2(e, f, 1, 0)


AN-2(e, f, 0, 0)

xye′ zd+f+f′ × ×AN-2(e, f, 0, 1)

xyf zc+e+e′ × AN-2(e, f, 1, 0)


(32)

Not all B-matrices are linearly independent, since BN-1(c, d, e, f;e′, f ′) ) BN-1(c, f ′, e, f;e′, d). The G-matrix is given by BM(0, 0, 0, 0;0, 0) when the Mth pair of sites in a sequence is in the lower vertical position. If this pair is in the upper vertical position, the G-matrix is given by AM(0, 0, 0, 0). The one-dimensional lattice consists of pairs of sites separated by a vacancy, and the corresponding G-matrix is obtained by setting M ) 1, which could be either B1(0, 0, 0, 0;0, 0) or A1(0, 0, 0, 0). The fact that these two matrices are identical provides a consistency check of our general formalism. Addition of appropriate factors to account for the effects due to the edges of a terrace or the wrapping of the lattice on a cylinder to describe adsorption on a zigzag carbon nanotube

Figure 11. Section of a heterogeneous surface consisting of the combination of two lattices, one made of equilateral triangles (large circle sites) and the other made of hexagons (small circle sites). On-site adsorption in one lattice corresponds to hollow adsorption on the other.

proceed as in the previous cases. As mentioned earlier, M must be even for zigzag carbon nanotubes.

9. Adsorption on Heterogeneous Lattices This section addresses the modeling of monomer adsorption on surfaces consisting of either equilateral triangles or hexagons. We consider armchair orientation in both cases. For simplicity, bridge-site adsorption between two adjacent sites of either the triangular or hexagonal surface has not been included in this analysis. Rather, we consider adsorption on tops and in hollows, which are the centers of either the equilateral triangles or the hexagons. This lattice geometry has numerous applications, as in the case of hollow and on-top adsorption of nitric oxide on silver (111).75,77 In this case, the adsorbate lattice consists of two sub-lattices exhibited in Figure 11. The large circles are the sites of the equilateral triangular lattice, and the small circles are the sites of the hexagonal lattice. If the substrate consists of an equilateral triangular lattice, on-top adsorption sites are the large circles and hollow adsorption sites are the small circles. If the substrate consists of a hexagonal lattice, on-top adsorption sites are the small circles and hollow adsorption sites are the large circles. Therefore, the adsorbate lattice consists of zigzag equilateral triangles and its sites cannot be treated equivalently. This is the reason for referring to this adsorbate lattice as heterogeneous. We consider only first and second neighbor interaction on this heterogeneous adsorbate lattice. Therefore, the sequence of sites appropriate for construction of the G-matrix is, as in Section 5, the zigzag row of M sites in the physical width shown in Figure 11. Label “1” is used to indicate the hexagonal sub-lattice, and label “2” is used to indicate the triangular sub-lattice. The notation of Section 5 must be extended to reflect the heterogeneous aspect of the lattice as follows. • The chemical potential of a particle in the gas phase is µ′. • The adsorbate-substrate interaction energy on the hexagonal sub-lattice is V01. • The adsorbate-substrate interaction energy on the triangular sub-lattice is V02. • The interaction energy between first neighbors on the hexagonal sub-lattice is V11; the interaction energy between firstneighbors on the two sub-lattices is V12; and the interaction energy between first-neighbors on the triangular sub-lattice is V22. • The interaction energy between second-neighbors on the hexagonal sub-lattice is W11. (77) Itoyama, T.; Wilde, M.; Matsumoto, M.; Okano, T.; Fukutani, K. Surf. Sci. 2001, 493, 84.


Langmuir, Vol. 23, No. 2, 2007 571

At thermodynamic equilibrium, the occupational characteristics per lattice site of the adsorption system are computed on the basis of the total number of sites in the adsorbate lattice. The average number of adsorbates per lattice site on the hexagonal sub-lattice is θ01, and the average number of adsorbates per lattice site on the triangular sub-lattice is θ02. Then, θ0 ) θ01 + θ02 is the coverage of the heterogeneous lattice. Similarly, θ1 is the number per lattice site of first neighbor adsorbates on the hexagonal sub-lattice, and θ12 is the number per lattice site of first neighbor adsorbates with one adsorbate on the hexagonal sub-lattice and the other on the triangular sub-lattice. The total number per site of nearest neighbor adsorbates on the heterogeneous lattice is θ ) θ1 + θ12. Finally, β1 is the number per lattice site of second neighbor adsorbates on the hexagonal sublattice, and β2 is the number per lattice site of first neighbor adsorbates on the triangular sub-lattice. Thus, β ) β1 + β2 is the total number per lattice site of adsorbates which are next nearest neighbors on the heterogeneous lattice. The energy per lattice site on the heterogeneous lattice follows as

) θ01(µ′ + V01) + θ02(µ′ + V02) + θ1V11 + θ12V12 + β1W11 + β2V22 (33) We will not further discuss treatment of edge effects for terraces or the effect of wrapping the lattice on a cylinder to form a nanotube, as these are treated in a manner similar to all previous sections. We redefine U ) (V02 - V01) to be the difference between the adsorbate-substrate interaction energies on the triangular sub-lattice and on the hexagonal sub-lattice. Similarly, V ) (V12 - V11) is the differential nearest neighbor interaction energy, and W ) (V22 - W11) is the differential next nearest neighbor interaction energy. In this manner, a first order approximation is readily obtained by setting V and W equal to zero. In general, with µ ) (µ′ + V01), the total energy per site on the adsorbate lattice is equivalently written as

) θ0µ + θV11 + βW11 + θ02V + β2W

(34)

There are six activities, x, y, z, u, V, and w, associated with the energies µ, V11, W11, U, V, and W, respectively. The occupational characteristics of the adsorption system {θ0,θ,β,θ02,θ12,β2} are computed in the same manner as before

x ∂R y ∂R z ∂R θ0 ) θ) β) MR ∂x MR ∂y MR ∂z θ02 )

u ∂R V ∂R w ∂R θ12 ) β2 ) MR ∂u MR ∂V MR ∂w

(35) (36)

where R is the eigenvalue of the T-matrix of largest modulus and M the total number of sites in the zigzag sequence of the adsorbate lattice, as shown in Figure 11. Recursive construction of the G-matrix is based on six types of matrices. Following the general procedure, we consider two consecutive sequences of M sites whose states are related to the row and column indices of the G-matrix. The diagrams of Figure 12 consist of six sets of the same section of two consecutive sequences taken from locations N + 3 to N - 5, with sites a, b, c, d, e, f, g, h, and i on the upper sequence, and sites a′, b′, c′, d′, e′, f ′, g′, h′, and i′ on the lower sequence, respectively. The topmost pair of sequences show the partition placed locations N + 1 and N. The sites to the left of the partition have a given state, and those to the right have all possible states, which are associated with matrices of rank 2N of the AN-type that

Figure 12. Diagrammatic representation of the six block matrices of type A, B, C, D, E, and F which make up the G-matrix associated with adsorption on a heterogeneous surface comprised of equilateral triangles and hexagons. The large circles are the sites of the triangular lattice, and the small ones are the sites of the hexagonal lattice. The letters refer to the state of occupation of the sites.

depend on the states of sites a, b, c, a′, and c′ at the corners of the shaded polygon. In the next pair of sequences, the partition is placed between locations N and N - 1, and the matrices of rank 2N-1 associated with the states of the sites to the right of the partition are of the BN-1-type. These matrices depend on the states of sites b, c, d, and c′ at the corners of the second shaded polygon. The procedure of moving the partition to the right is repeated. Each new location generates a different type of matrix: CN-2(c, d, e;c′, e′), DN-3(d, e, f;e′), EN-4(e, f, g;e′, f ′), and FN-5(f, g, h;g′). The shaded polygons associated with matrices C and E have the same shape as the polygon associated with A-matrices; however, their corners are sites which belong to different sub-lattices; consequently, they are topologically distinct. Similarly, the shaded polygons associated with matrices D and F have the same shape as the polygon associated with B-matrices, but they are topologically distinct since their corner sites again belong to different sub-lattices. Matrices of the A-type of rank 2N-6 reappear when the partition is placed between locations N - 5 and N 6. When the section to the left of the partition contains all the sites in both sequences, then N ) 0 and the initial conditions

572 Langmuir, Vol. 23, No. 2, 2007

depend on the six sites in the upper and lower sequence at locations 1, 2, and 3. These sites are at the corners of a polygon which has one of the six topologies indicated above. Since each topology is associated with a matrix of a given type, the corresponding matrix equals one. The initial conditions follow as A0(a, b, c;a′, c′) ) 1, B0(b, c, d;c′) ) 1, C0(c, d, e;c′, e′) ) 1, D0(d, e, f;e′) ) 1, E0(e, f, g;e′, f ′) ) 1, or F0(f, g, h;g′) ) 1. When the section to the right of the partition contains all sites in both sequences, the adsorption problem is not affected by adding three vacant sites to each sequence at locations M + 1, M + 2, and M + 3, as mentioned in previous sections, with the partition placed between locations M + 1 and M. However, the extended lattice must retain the same heterogeneous property as that of the original lattice. Consequently, the topology of the polygon consisting of the six new sites determines the G-matrix as either AM(0, 0, 0;0, 0), BM(0, 0, 0;0), CM(0, 0, 0;0, 0), DM(0, 0, 0;0), EM(0, 0, 0;0, 0), or FM(0, 0, 0;0). When the recursive relations among the six types of matrices are constructed, the necessary changes due to the heterogeneous lattice follow from eq 34. An adsorbate at a site on the upper sequence that belongs to the hexagonal lattice (small circles) accounts for a factor x, and when it belongs to the triangular lattice (large circles) it accounts for a factor xu. A factor y is added for any nearest neighbor adsorbates, whether on the upper sequence or between upper and lower sequences; there is also an additional factor V if one of these neighbors is on a site belonging to the triangular lattice. Finally, a factor z is added for any next nearest neighbor adsorbates, whether on the upper sequence or between the upper and lower sequences; and there is an additional factor w when these neighbors are both on the triangular lattice. Following the same procedure explained earlier, but as applied to Figure 12, the recursive relations are

[

[

AN(a, b, c;a', c') ) x(yV)b yc+c′ za+a′ × BN-1(b, c, 1; c′) ×BN-1(b, c, 0; c′) ×BN-1(b, c, 0; c′)

x(yV)b yc+c′ za+a′+1 × BN-1(b, c, 1; c′)

BN-1(b, c, d;c') ) (xu)(yV)c+d zwb × CN-2(c, d, 1; c′, 0) ×CN-2(c, d, 0; c′, 0) (yV)d (zw)b × CN-2(c, d, 0; c′, 1)

[ [

(xu)(yV)(c+2d) (zw) × CN-2(c, d, 1; c′, 1)

CN - 2(c, d, e;c′, e′) ) x(yV)e+e′ yd zc+c′ × ×DN-3(b, c, 1; c′) DN-3(b, c, 1; c′) ×DN-3(b, c, 1; c′)

] ]

]

(38)

(39)

x(yV)e+e′ yd zc+c′+1 × DN-3(b, c, 1; c′)

DN-3(d, e, f;e') ) xyf(yV)e zd × ×EN-4(e, f, 0; e′, 0) EN-4(e, f, 1; e′, 0) yfzd × EN-4(e, f, 0; e′, 1)

]

(37)

xy (yV) z × EN-4(e, f, 1; e′, 1) 2f

e 2d+1

(40)

[

Phares et al.

EN-4(e, f, g;e', g') ) (xu)(yV)f+g+g′zwe+e′ × FN-5(f, g, 1; g′) ×FN-5(f, g, 0; g′) ×FN-5(f, g, 0; g′)

[

(xu)(yV)f+g+g′(zw) × FN-5(f, g, 1; g′)

FN-5(f, g, h;g′) ) xyg(yV)hzf × ×AN-6(b, c, 1; c′) AN-6(b, c, 1; c′) (yV) z × AN-6(b, c, 1; c′) h f

xy (yV) z × AN-6(b, c, 1; c′) g

2h 2f+1

]

]

(41)

(42)

Again, edge effects for terraces and wrapping effects for nanotubes are handled as in previous cases.

10. Summary and Conclusion A lattice gas model is used to study surface adsorption. The partition function of gas particles adsorbed on a surface of finite width and length with a relatively simple lattice geometry was shown in 1993 to be related to the eigenvalues of a recursively constructed transfer matrix (T-matrix). The eigenvalue of largest modulus is the only contribution to the partition function in the thermodynamic limit when the length of the surface becomes infinite. This work has now been generalized to the systematic construction of T-matrices associated with monomer adsorption on terraces and nanotubes of arbitrary lattice geometry, accommodating any number of pairwise attractive or repulsive adsorbate-adsorbate interactions of arbitrary strength. A terrace of a given lattice geometry wrapped on a cylinder forms a nanotube (i.e., a periodic boundary in the width of the terrace). It follows that the two T-matrices associated with monomer adsorption on a terrace and on its companion nanotube have in common a G-matrix with the same rank. The G-matrix is first constructed and then modified to obtain the T-matrix. These modifications account for the effects due to the edges of the terrace or due to the wrapping of the terrace to form a nanotube. We have presented a systematic recursive construction of the G-matrix based on the following considerations: • Each element of the G-matrix is the product of the pairwise activities under consideration with each activity raised to a nonnegative integer power. • There is a one-to-one correspondence between the exponents of the activities of a given G-matrix element and a given state of occupation of two vertically consecutive sequences of sites across the physical width of the lattice. • The sequence of sites for a given lattice geometry is determined by the highest order adsorbate-adsorbate pairwise interaction considered in the model. • The rank of the G-matrix is the total number of states of the sites in a sequence, which are ordered according to the binary system. The order of the column and row indices of the G-matrix follows the order of the states of the sites in the upper and lower sequences, respectively. • Recursive construction of the G-matrix follows from partitioning the two consecutive sequences into two sections. The sites in the first section have a given state, and the sites in the second section have all possible states. Matrices of different types are then formed with their elements being the G-matrix elements corresponding to these selected states. • The topologies exhibited when the location of the partition is changed determine the types of matrices generated. The rank


of the matrices of a given type is the total number of states of one sequence of sites in the second section of the partition. Matrices of a given type and rank are then related to matrices of lower rank, thus establishing recursive relations. • Initial conditions on these matrices are obtained when the first section contains all the sites in both sequences and the second section contains no sites. When the first section contains no sites and the second contains all the sites, the corresponding matrix is the G-matrix, which is then determined to be one of the matrices of a particular type. The general technique has been applied to parallelogram adsorbate lattices. It has provided a unified formulation for all T-matrices derived in the past as separate cases of monomer adsorption on sc, bcc or fcc (100), and on fcc and bcc (110), with and without edge effects, and with various edge orientations. The technique has also been applied to adsorption on terraces and nanotubes with adsorption sites forming equilateral triangular lattices, hexagonal lattices, and a combination of both lattice types. These cases have been selected because of their experimental relevance. The model has already been applied to the

Langmuir, Vol. 23, No. 2, 2007 573

adsorption of CO on Pt(112).78 The CO molecules play the role of monomers, and the Pt surface consists of equilateral triangular terraces three atomic sites in width. As problems become more complex, so does the coding of computational programs, and the length of computing time becomes a major factor. The new mathematical algorithm developed in this paper increases computing efficiency while allowing the study of models reflecting the complexity of the experimental conditions. The recursion relations of a number of problems of immediate interest presented here provide the necessary vectorization for efficient multi-parallel processing code. Acknowledgment. This research is supported by an allocation of advanced computing resources supported by the National Science Foundation and the Pittsburgh Supercomputing Center. LA0620354 (78) Phares, A. J.; Grumbine, D. W., Jr.; Wunderlich, F. J. Langmuir 2006, 22, 7646.

Monomer Adsorption on Terraces and Nanotubes - Langmuir (ACS

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