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Langmuir 2007, 23, 1928-1936
Monomer Adsorption on Equilateral Triangular Lattices with Repulsive First-Neighbor Interactions 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 October 11, 2006. In Final Form: NoVember 20, 2006 A model of monomer adsorption on infinitely long, finite-width M equilateral triangular lattices with nonperiodic boundaries is presented. The study includes adsorbate-adsorbate first- and second-neighbor interactions with results obtained for repulsive first neighbors. The matrix method and numerical algorithms presented here allow determination of the occupational characteristics of the adsorption crystallization phases, which fit exact analytic expressions in the width M of the lattice. The limit as M approaches infinity provides the complete energy phase diagram for the infinite two-dimensional surface and recovers the results obtained by different methods that were often applied only in restricted energy regions of the phase diagram. The ordered phases are (2 × 1), (2 × 2), (3 × 1), (x3 × x3) R30°, and the complementary phases of (2 × 2) and (x3 × x3) R30°. Comparison is made with other theoretical studies and with experimental observations on adsorption systems consistent with the limitations of the model. In some cases, comparison with experimental data yields bounds on the interaction energies between adsorbates. On the basis of the model, suggestions are made on the manner in which to conduct relatively low temperature experiments to allow determination of most, if not all, of the interaction energies from the knowledge of the sequences of phases and the conditions prevailing at the transitions between phases.
1. Introduction Lattice models have a long history which includes Langmuir adsorption theory1-8 and the Ising9 and Potts10 models of magnetism applied to a variety of lattices, including equilateral triangular lattices11-36 in particular. The Ising model in an external * 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, Ser. A 1925, 107, 61. (5) Langmuir, I. Gen. Electr. 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) Ising, E. Z. Phys. 1925, 31, 253. (10) Potts, R. B. Phys. ReV. 1952, 88, 352. (11) Houtappel, R. M. F. Physica 1950, 16, 425. (12) Temperley, H. N. V. Proc. R. Soc. London, Ser. A 1950, A202, 202. (13) Wannier, G. H. Phys. ReV. 1950, 17, 357. (14) Husimi, K.; Syozi, I. Prog. Theor. Phys. V 1950, 177. (15) Fisher, M. E. Phys. ReV. 1959, 113, 969. (16) Fisher, M. E. J. Math. Phys. 1963, 4, 278. (17) Stephenson, J. J. Math. Phys. 1964, 5, 1009. (18) Runnels, L. K.; Combs, L. L. J. Chem. Phys. 1966, 45, 2482. (19) Campbell, C. E.; Schick, M. Phys. ReV. A 1972, 5, 1919. (20) Mihura, B.; Landau, D. P. Phys. ReV. Lett. 1977, 38, 977. (21) Schick, M.; Walker, J. S.; Wortis, M. Phys. ReV. B 1977, 16, 2205. (22) Baxter, R. J.; Temperley, H. N. V.; Ashley, S. E. Proc. R. Soc. London, Ser. A 1978, 358, 2535. (23) Berker, A. N.; Ostlund, S.; Putnam, F. A. Phys. ReV. B 1978, 17, 3650. (24) Domany, E.; Schick, M.; Walker, J. S.; Griffiths, R. B. Phys. ReV. B 1978, 18, 2209. (25) Ostlund, S.; Berker, A. N. Phys. ReV. Lett. 1979, 42, 843. (26) Walker, J. S.; Schick, M. Phys. ReV. B 1979, 20, 2088. (27) Kinzel, W.; Schick, M. Phys. ReV. B 1981, 23, 3435. (28) Roelofs, L. D.; Kortan, A. R.; Einstein, T. L.; Park, R. L. Phys. ReV. Lett. 1981, 46, 1465. (29) Baxter, R. J. Exactly SolVed Models in Statistical Mechanics; Academic Press Limited: London, 1982. (30) Wu, F. Y.; Schick, M. ReV. Mod. Phys. 1982, 54, 235. (31) Bartelt, N. C.; Einstein, T. L. Phys. ReV. B 1984, 30, 5339. (32) Blote, H. W. J.; Nightingale, M. P.; Wu, X. N.; Hoogland, A. Phys. ReV. B 1991, 43, 8751.
field is mathematically equivalent to the lattice-gas model.37 However, as we are interested in surface adsorption, we adopt the terminology appropriate for a lattice-gas system. The lattice-gas model of surface adsorption consists of a gas at a given temperature and pressure whose particles are adsorbed on a crystal surface, the substrate lattice. Under certain conditions, adsorption takes place at sites which form a periodic adsorbate lattice. Here, we have chosen an equilateral triangular adsorbate lattice. The substrate and adsorbate lattices need not be identical. For example, an equilateral triangular adsorbate lattice is formed when adsorption takes place only at the centers of a honeycomb substrate lattice such as graphite. In general, the adsorbate lattice coverage increases with the external pressure, producing a series of adsorption crystallization patterns, or phases. These phases, and the sequence in which they appear, are also influenced by the presence of terraces and steps. For a given surface geometry, there should be a direct relationship between adsorption phases and the pairwise adsorbate-adsorbate and adsorbate-substrate interaction energies. This is substantiated by the wide variety of phases that have been observed (see, for example, the extensive experimental reviews by Somorjai38 and Over39). Studies on two-dimensional lattices have used transfer matrix methods, which were first introduced by Montroll,40 Kramers, and Wannier.41 Simplifications occur by assuming periodic boundaries in one or both directions across the lattice: width or length. In general, the partition function of the system follows from knowledge of all of the eigenvalues of the transfer matrix. (33) de Queiroz, S. L. A.; Domany, E. Phys. ReV. E 1995, 52, 4768. (34) Blote, H. W. J.; Nightingale, M. P. Phys. ReV. B 1993, 47, 15046. (35) Jager, I. Surf. Sci. 2000, 454-456, 647. (36) de Queiroz, S. L. A. Phys. ReV. B 2006, 73, 064410. (37) Lee, T. D.; Yang, C. N. Phys. ReV. 1952, 87, 410. (38) Somorjai, G. A. Introduction to Surface Chemistry; John Wiley & Sons, Inc.: New York, 1994. (39) Over, H. Prog. Surf. Sci. 1998, 58, 249-376. (40) Montroll, E. W. J. Chem. Phys. 1941, 9, 706. (41) Kramers, H. A.; Wannier, G. H. Phys. ReV. 1941, 60, 252.
10.1021/la062994y CCC: $37.00 © 2007 American Chemical Society Published on Web 01/04/2007
Monomer Adsorption on Triangular Lattices
In the thermodynamic limit as the lattice length becomes infinite (with or without a periodic boundary in this direction), the eigenvalue of largest modulus is the only contribution. In addition, if a periodic boundary condition is imposed on the lattice width, then the problem is that of a lattice wrapped on a cylinder which models a nanotube. In the thermodynamic limit (infinite length), whether the semi-infinite lattice has edges or is a nanotube, extrapolation of the numerical results with an increasing number of sites in the lattice width provides a reasonable approximation for what to expect when the two-dimensional lattice becomes infinite in both directions. This procedure, introduced by Fowler and Rushbrooke,42 has been followed by others.18,27,34,43 Numerical calculations have been carried out mainly on the basis of transfer matrices, Monte Carlo simulations, and a combination of both.44-46 Typical Monte Carlo calculations use an L × L net, which corresponds to a rhomboid-shaped net20 for an equilateral triangular lattice. Extrapolation of the results with increasing values of L again provides a good approximation of what to expect on the infinite 2-D lattice. Renormalization group methods have also been used to successfully derive results for the infinite lattice.21,23 We consider a triangular lattice of finite length L and finite width M, without periodic boundaries, thus forming a terrace. In our model, the first-neighbor adsorbate-adsorbate interaction V is repulsive (V < 0), and the second-neighbor interaction W may be attractive (W > 0), repulsive (W < 0), or zero. The chemical potential per particle in the gas phase µ′ is varied by changing the gas pressure. The adsorbate-substrate interaction energy V0 is arbitrary and assumed to be the same on both edge and bulk sites. However, the coordination numbers of edge and bulk sites are different, which affects the adsorption patterns. Therefore, this model allows the study of edge effects with increasing M. For simplicity, we consider only the thermodynamic limit (as L becomes infinite). The orientation of the edges is chosen such that consecutive sites on the edges are second neighbors as shown in Figure 1a. We refer to this lattice as a zigzag equilateral triangular terrace.47 When only first- and second-neighbor interactions are considered, a general transfer matrix method applied to surface adsorption of monomers on terraces and nanotubes47 has shown that the rank of the matrix for the zigzag terrace of Figure 1a is 2M. Adsorption on the armchair equilateral triangular terrace of Figure 1b, for which consecutive edge sites are first neighbors, has been recently investigated for a width M ) 3 and applied to the adsorption of CO/Pt(112).48 Pairwise interactions were included to first and second neighbors; however, the rank of the transfer matrix is 22M. From a computational point of view, the zigzag equilateral triangular lattice is therefore more manageable for higher values of M with the same number of pairwise interactions. The advantage of our method is that the occupational characteristics of the phases obtained numerically are again shown to fit exact analytic expressions in M, as has been the case for other lattices that we have investigated in the past, namely, lattices made of squares,49-52 rectangles,53,54 and isosceles triangles.55,56 Extrapolation of the results to the infinite two-dimensional lattice (42) Fowler, R. H.; Rushbrooke, G. S. Trans. Faraday Soc. 1937, 33, 1272. (43) Nightingale, M. P.; Blote, H. W. J. J. Phys. A: Math. Gen. 1982, 15, L33. (44) Metcalf, B. D. Phys. Lett. 1973, 45A, 1. (45) Chin, K. K.; Landau, D. P. Phys. ReV. B 1987, 36, 275. (46) Pasinetti, P. M.; Roma, S.; Riccardo, J. L.; Ramirez-Pastor, A. J. Phys. ReV. B 2006, 74, 155418. (47) Phares, A. J.; Grumbine, D. W., Jr.; Wunderlich, F. J. Langmuir, to be published. (48) Phares, A. J.; Grumbine, D. W., Jr.; Wunderlich, F. J. Langmuir 2006, 18, 7646. (49) Phares, A. J.; Wunderlich, F. J. Phys. ReV. E 1995, 52, 2236-2246. (50) Phares, A. J.; Wunderlich, F. J. Phys. ReV. E 1997, 55, 2403-2408.
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Figure 1. Equilateral triangular lattices: The highlighted zigzag sequence of M sites defines the width of the lattice and the appropriate sequence for the study of adsorption limited to first- and secondneighbor interactions. L indicates the number of sequences of M sites in the length of the lattice. (a) Zigzag lattice: edge sites are second neighbors. (b) Armchair lattice: edge sites are first neighbors.
is therefore straightforward and provides information complementary to that already obtained by other methods. The mathematical formulation of the problem and the numerical procedure are presented in section 2. Section 3 provides a summary of the occupational characteristics of the phases exhibiting an exact numerical fit in the width M of the lattice, including phase diagrams. A comparison with other theoretical calculations is presented in section 4, and comparison with experimental data is presented in section 5. Section 6 is the summary and conclusion.
2. Mathematical Formulation and Numerical Procedure In ref 47, we have shown that the transfer matrices associated with adsorption on a terrace and its associated nanotube are derived from a single G-matrix. This matrix is constructed recursively and then modified to account for either the difference between adsorbate-substrate interactions at edge and bulk sites for the terrace or for the wrapping of the lattice to form a nanotube. This technique is different from other approaches and provides a vectorized algorithm for efficient use on multiparallel processors and supercomputers. In the present study, the transfer matrix is the G-matrix for a zigzag equilateral triangular lattice, which accounts for first- and second-neighbor interactions as derived in ref 47. (51) Phares, A. J.; Wunderlich, F. J.; Kumar, A. M. S. Surf. Sci. 2001, 495/ 1-2, 140-152. (52) Phares, A. J.; Wunderlich, F. J. Appl. Surf. Sci. 2003, 219/1-2, 174-190. (53) Phares, A. J.; Wunderlich, F. J. Surf. Sci. 2001, 479, 43-68. (54) Phares, A. J.; Wunderlich, F. J. Int. J. Mod. Phys. B 2001, 15, 33233330. (55) Phares, A. J.; Wunderlich, F. J. Surf. Sci. 1999, 425, 112-130. (56) Phares, A. J.; Wunderlich, F. J. Surf. Sci. 2000, 452, 108-116.
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Phares et al.
The relevant quantity is the shifted chemical potential per particle in the gas phase, namely, µ ) µ′ + V0. The absolute activities associated with µ, V, and W are
x ) exp(µ/kBT) y ) exp(V/kBT) z ) exp(W/kBT)
(1)
where kB is Boltzmann’s constant and T the absolute temperature. The G-matrix of rank 2M is given in terms of two sets of matrices, AN(R, β, γ; R′, β′) and BN(β, γ, δ; β′), whose arguments take on the values 0 or 1. The index N refers to their rank 2N. These matrices satisfy the initial conditions47
A0(R,β,γ;R′,β′) ) 1, B0(β,γ,δ;β′) ) 1
(2)
and are recursively related according to47
AN(R,β,γ;R′,β′) ) BN-1(β,γ,0;γ′) xyβ+γ+γ′zR+R′BN-1(β,γ,0;R′) BN-1(β,γ,0;γ′) xyβ+γ+γ′zR+R′+1BN-1(β,γ,0;γ′)
(
BN-1(β,γ,δ;γ′) ) AN-2(γ,δ,0;γ′,0)
(
xyγ+δzβAN-2(γ,δ,1;γ′,0) y z AN-2(γ,δ,0;γ′,1) xyγ + 2δz2β+1AN-2(γ,δ,1;γ′,1) δ β
)
(3)
)
(4)
If the Mth site on the zigzag sequence shown in Figure 1a is at the top, the G-matrix is BM(0, 0, 0; 0); if it is at the bottom, the G-matrix is AM(0, 0, 0; 0, 0). In both cases, the parity of M determines whether the right edge site (the first site) is at the top or the bottom of the zigzag. Both formulations yield the same result. Note that all of the matrix elements are real and nonnegative, and consequently, the eigenvalue of largest modulus R is real and positive. In the limit as the length L of the lattice becomes infinite while keeping its width M fixed, the partition function Z of the system simplifies to
Z ) R1/M
(5)
At thermodynamic equilibrium, the average coverage of the lattice θ0 and the numbers of first and second neighbors per site θ and β follow as
θ0 )
x ∂R MR ∂x
θ)
y ∂R MR ∂y
β)
z ∂R MR ∂z
Figure 2. Temperature phase diagram for a terrace width M ) 7 with W/|V| ) -1/4 exhibited as a plot of kBT/|V| vs µ/|V|. The threshold temperature T0 below which all the phases exist is kBT0/|V| ) 0.0288.
(6)
We have used a recently developed numerical algorithm for computing derivatives to arbitrary precision57 to obtain the partial derivatives of R with respect to the various activities. In addition, calculations have been conducted with long double precision arithmetic for repulsive first neighbors (V < 0) so that our quoted results are accurate to at least ten decimal places. As described in ref 48, a threshold temperature T0 is determined for a given set of values of V and W, below which no new phases are observed. Above T0, some of the phases begin to disappear. This threshold temperature may be as high as 800 K.48 Phase diagrams are generated below T0 but well above 0 K. The effect of temperatures above T0 will also be discussed. Keeping the temperature at a fixed value below T0 for a given set of values of µ, V, and W, eigenvalue R and its first derivatives with respect to the three activities provide the occupational characteristics, listed as {θ0 θ β}. The entropy S is then computed. By gradually increasing the external gas pressure, and therefore µ, a series of crystallization patterns, or phases, are observed which have a minimum entropy. When the minimum is zero, the phase is perfectly ordered; otherwise, it is partially ordered. At an entropy minimum, the occupational characteristics that are given by the set {θ0 θ β} completely identify the pattern of the phase. For consistency, whenever a partially ordered phase was found, the closed form expression of its nonzero entropy was analytically derived and agrees with the value obtained numerically. At low pressure, the lattice is empty, and the state of occupation is E ) {0 0 0}. At high pressure, the lattice is full and, for any width M, the state of occupation is F ){1, (3M 4)/M, (3M - 6)/M}. At these extreme points, the entropy is zero. In a plot of entropy S versus µ, a minimum resembles a plateau58,59 indicating that the ordering remains over a wide range of pressures. On the other hand, in a plot of entropy S versus the coverage θ0, a minimum of S corresponds to a cusp.58,59 This has been confirmed by others using Monte Carlo simulation.60,61 For given values of first- and second-neighbor interaction energies, we determine the threshold temperature T0 below which all possible phases are observed. A sample is presented in Figure 2 corresponding to a terrace of width M ) 7 with W/|V| ) -1/4. In this case, the threshold temperature is numerically obtained
The average total energy per site is
E ) θ0µ + θV + βW
(7)
and the entropy per site divided by Boltzmann’s constant, referred to as entropy S, is
S ) (1/M) ln R - E/kBT
(8)
(57) Phares, A. J.; Wunderlich, F. J. Int. J. Theor. Phys., Group Theory Nonlinear Opt. 2003, 10, 415. (58) Phares, A. J.; Wunderlich, F. J.; Grumbine, D. W., Jr.; Curley, J. D. Phys. Lett. A 1993, 173, 365-368. (59) Phares, A. J.; Wunderlich, F. J.; Curley, J. D.; Grumbine, D. W., Jr. J. Phys. A 1993, 26, 6847-6877. (60) Roma´, F.; Ramirez-Pastor, A. J.; Riccardo, J. L. Langmuir 2000, 16, 9406-9409. (61) Roma´, F.; Ramirez-Pastor, A. J.; Riccardo, J. L. J. Chem. Phys. 2001, 114, 10932-10937.
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Langmuir, Vol. 23, No. 4, 2007 1931
as kBT0/|V| ) 0.0288. For a typical first-neighbor repulsion of 15 kcal/mol, T0 ) 218 K. Figure 2 exhibits a series of mounds and spikes enclosing the phases. For T < T0, the phases between empty (E) and full coverage (F) are, in order, A1 ) {2/7, 0, 0}, A2 ) {3/7, 1/7, 2/7}, C1 ) {4/7, 3/7, 4/7}, C2 ) {5/7, 6/7, 9/7}, D ) {6/7, 11/7, 10/7}, and F1 ) {13/14, 2, 12/7}. As T increases beyond T0, some of the phases cease to be observed, first F1, then A2, A1, C2, D, and finally C1. Beyond that point, there remains only a smooth transition from empty to full coverage. For given values of V and W with the temperature below T0, consider two consecutive phases {p1} ) {θ01 θ1 β1} and {p2} ) {θ02 θ2 β2} as µ is increased. With the changes in the corresponding characteristics between the two phases indicated as ∆θ0, ∆θ, and ∆β, we define µt as the value of µ in the transition region linearly related to V and W according to
0 ) ∆θ0µt + ∆θV + ∆βW
(9)
If the transition is second-order, it has been shown and numerically verified that θ and β are linearly related to the coverage θ0 as it increases from θ01 to θ02. Simultaneously, the entropy reaches a maximum at µ ) µt. In Figure 2, the values of µt/|V| at each transition are, in order, 0, 5/4, 10/4, 17/4, 21/4, 28/4, 30/2. An infinite specific heat capacity indicating a first-order phase transition corresponds to a discontinuity of the coverage as a function of µ. This has been observed in a number of cases between two phases at precisely the value µ ) µt as defined in eq 9. Numerically, if {p1} occurs at µ < µt - and {p2} occurs at µ > µt + for values of at least 6 orders of magnitude smaller than µt, we interpret the transition as first-order. In the example of Figure 2, the transition between C1 and C2 is first-order below a critical temperature kBTC/|V| ) 0.0328. The transition between C2 and D is first-order below kBTC/|V| ) 0.0146. Above these critical temperatures, the transitions are second-order. For |V| ) 15 kcal/mol, these critical temperatures are 248 and 110 K, respectively, while the corresponding threshold temperature T0 is 218 K. In this example, all other transitions are second-order at nonzero temperatures. For first-order transitions, the numerically computed entropy is zero at µ ) µt, and the occupational characteristics are the average of the characteristics of the phases {p1} and {p2} on either side of µt. For T < T0, and well above 0 K, as exemplified above, a dimensionless phase diagram is numerically generated for a given value of V, chosen here to be negative, by plotting V ) µ/|V| versus u ) W/|V|. The absolute value is used so that V and u have the same sign, as do µ and W, respectively. In this plot, the boundary points between phases correspond to µ ) µt. As follows from eq 9, the boundary between two phases that have different coverages (∆θ0 * 0) is a straight line of slope -(∆β/∆θ0) and vertical intercept (∆θ/∆θ0). When the phases have the same coverages (∆θ0 ) 0), they must have different values of β to be distinct, and the boundary is a vertical line corresponding to the value of u given by (∆θ/∆β).
3. Occupational Characteristics of Phases and Phase Diagrams Calculations have been carried out for lattice widths M ) 3 to 11, with corresponding matrix ranks from 23 to 211. Most of the phases are perfectly ordered. Their characteristics fit exact analytic expressions in terms of M. In the infinite M limit, the phases merge to only six in addition to empty (E) and full coverage (F). • A ) {1/4 0 0} corresponding to a (2 × 2). • B ) {1/3 1/3 0} corresponding to a (3 × 1). • C ) {1/2 1/2 1/2} corresponding to a (2 × 1).
Table 1. Phases for M ) 9 M)9 name
θ0
θ
β
name
θ0
θ
β
A1 G1 G2 A2 A3 A4 G3 A5 C1 B C2
5/18 1/3 1/3 1/3 7/18 7/18 4/9 1/2 5/9 5/9 11/18
0 0 0 5/27 1/9 2/9 1/9 4/9 4/9 16/27 7/9
0 1/3 7/9 0 2/9 1/9 8/9 1/3 5/9 4/9 5/9
C3 H1 C4 C5 C6 C7 H2 C8 D1 C9 D2
2/3 2/4 2/3 2/3 13/18 13/18 7/9 7/9 5/6 5/6 8/9
7/9 7/9 26/27 10/9 1 11/9 11/9 4/3 14/9 5/3 17/9
10/9 14/9 19/27 2/3 4/3 8/9 5/3 11/9 14/9 4/3 5/3
Table 2. Phases for M ) 11 M ) 11W name
θ0
θ
β
name
θ0
θ
β
A1 B1 G2 G1 A3 A2 B2 B3 G3 G4 G5 B4 C1 C2 C3 C4
3/11 1/3 4/11 4/11 4/11 4/11 4/11 13/33 9/22 5/11 5/11 5/11 38/77 1/2 17/33 6/11
0 3/11 0 0 1/11 2/11 3/11 10/33 1/11 2/11 2/11 14/33 36/77 5/11 20/33 5/11
0 0 10/11 8/11 2/11 1/11 2/33 4/33 8/11 9/11 1 8/33 36/77 4/11 4/11 6/11
C5 C6 C7 C8 C9 C10 C11 C12 C13 C14 H C15 C16 D1 D2 D3
6/11 25/44 13/22 20/33 34/55 7/11 7/11 7/11 2/3 23/33 8/11 17/22 17/22 9/11 5/6 19/22
25/33 8/11 19/22 9/11 54/55 8/11 9/11 1 35/33 37/33 1 15/11 16/11 17/11 58/33 20/11
14/33 1/2 6/11 20/33 34/55 1 8/11 15/22 26/33 10/11 20/11 14/11 13/11 17/11 16/11 18/11
• D ) {3/4 3/2 3/2} with the adsorption sites being the vacancies of the (2 × 2) or its complement. • G ) {1/3 0 1} corresponding to a (x3 × x3) R30°. • H ) {2/3 1 2} forming a honeycomb, with the adsorption sites being the vacancies of the (x3 × x3) R30° or its complement. It is therefore convenient to name the phases encountered in the finite width lattices according to the adsorption features approaching those of the A, B, C, D, E, F, G, or H phases. The occupational characteristics of the phases for M ) 9 and 11 are provided as a sample. For convenience, short names are assigned to these phases. They begin with a letter ranging from A to H, to highlight their family features as previously mentioned, followed by an order number when there are several phases belonging to the same family, as exhibited in Tables 1 and 2. The same naming convention has been applied in the example of Figure 2 for M ) 7. The corresponding energy phase diagrams are given in Figures 3 and 4, where the boundary of each family of phases has been highlighted because the members of each family merge to the same phase in the infinite width limit. Figure 5 shows the evolution of two series of phases with increasing values of M whose occupational characteristics fit the analytical formulas, for any M g 3
+ 1)]/2 0 0 98 1 0 0 ) A ) (2 × 2) {[(M 2M } {4 } Mf∞
(10)
+ 1 0 3[(M - 1)/3] + 1 98 1 0 1 ) {[(M - 1)/3] } {3 } M M Mf∞
G ) (x3 × x3) R30°(1>) >)
(11)
The square brackets refer to the integer part of the enclosed quantity. Figure 6 shows the evolution of two sequences of phases,
1932 Langmuir, Vol. 23, No. 4, 2007
Figure 3. Energy phase diagram for adsorption on a lattice of width M ) 9. The equation of the straight-line boundary between any two phases is obtained from the knowledge of the occupational characteristics of each phase as explained in the text and numerically verified.
Figure 4. Energy phase diagram for adsorption on a lattice of width M ) 11. The equation of the straight-line boundary between any two phases is obtained from the knowledge of the occupational characteristics of each phase as explained in the text and numerically verified.
M ) 3p and M ) 3p + 2, where p is an integer. These sequences merge to the G-phase. This shows that there are several phase sequences on finite width lattices that merge to the same in the infinite width limit. We have observed that some sequences apply for even or odd widths in addition to those of widths M ) 3, M ) 4, and M ) 5 modulo 3. It is in this manner that we arrived at the merging, in the infinite limit, to the phases named A to
Phares et al.
Figure 5. Occupational characteristics and configurations of two series of phases with increasing lattice width M valid for all values of M g 3, which merge in the infinite width limit: one to the A phase, {1/4 0 0}, and the other to the G phase {1/3 0 1}. The occupied sites are the black circles.
Figure 6. Occupational characteristics and configurations of two series of phases, one valid for width M ) 3 modulo 3 and the other for width M ) 5 modulo 3, which merge in the infinite width limit to the G phase. The occupied sites are the black circles.
H. The occupational configurations of phases A and G are exhibited in Figures 5 and 6. The occupational configurations of phases B, C, D, and H are provided in Figure 7. While the G and H phases are complementary, as are A and D phases, the complement of the B phase does not occur. Therefore, there is no particle-vacancy symmetry with the pairwise interactions considered in this study. The energy phase diagram for the infinite two-dimensional lattice follows and is shown in Figure 8. The case M ) 7 with W/|V| ) -1/4 is used again as an example providing the temperature and chemical potential dependence of
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Langmuir, Vol. 23, No. 4, 2007 1933
Figure 9. Three-dimensional plot of the coverage vs kBT/|V| and µ/|V| complementing the temperature phase diagram of Figure 2.
Figure 7. Occupational characteristics and configurations of phases B, C, D, and H. The occupied sites are the dark circles. The B and C phases are (1 × 3) and (1 × 2). The D phase is the complement of the (2 × 2), and the H phase is the complement of the (x3 × x3) R30°.
Figure 10. Three-dimensional plot of entropy S vs coverage θ0 and kBT/|V| complementing the temperature phase diagram of Figure 2.
from 9/14 to 5/7 (phase C2), from 5/7 to 11/14, and from 11/14 to 6/7 (phase D). Since the transitions from phases C1 to C2 to D are not gradual, the transitions between these phases are first-order as noted in section 2, which explains these discontinuities. The cusps in the entropy correspond to the remaining phases between empty and full coverage, namely, A1 (θ0 ) 2/7), A2 (θ0 ) 3/7), and F1 (θ0 ) 13/14). Disappearance of the cusps with increasing temperature indicates the disappearance of the phases. This does not occur simultaneously to all phases as discussed in section 2 and in connection with Figure 2. The disappearance of a first-order transition beyond a certain critical temperature TC has also been analyzed by looking at the isotherms in the plot of coverage θ0 versus µ/|V|. A sample is presented for M ) 7 with W/|V| ) -1/4. Figure 11 exhibits the transition from C1 to C2 which is first-order for kBT/|V| < 0.0328, and Figure 12 exhibits the transition from C2 to D which is first-order for kBT/|V| < 0.0146. Figure 8. Energy phase diagram on the infinite two-dimensional equilateral triangular lattice with repulsive first-neighbor adsorbateadsorbate interactions (V < 0).
the phases. Figure 9 is a three-dimensional plot of coverage θ0 versus kBT/|V| and µ/|V|. At temperatures less than kBT0/|V| ) 0.0288, the surface has plateaus corresponding to the coverage of the phases identified in Figure 2. As follows from the discussion provided in the previous section in connection with Figure 2, as temperature increases, the plateaus gradually merge leading to a smooth increase in the coverage with the chemical potential. A plot of the entropy S versus coverage θ0 and kBT/|V| is shown in Figure 10. At temperatures below T0, the entropy curves consist of a series of humps and a flat region (at S ) 0) connected by cusps at S ) 0. In the region that appears to be flat below T0, the coverage jumps discontinuously from 4/7 (phase C1) to 9/14,
Again for M ) 7 with W/|V| ) -1/4, Figures 13 and 14 provide a sample of the relationship between θ and θ0 and between β and θ0 at various temperatures. In particular, at kBT/|V| ) 1/60, which is well below the threshold of 0.0288, both plots consist of straight segments connected at points with coordinates that are the characteristics of the phases encountered in this energy region from empty to full coverage. After the expected discontinuity for the two first-order transitions occurring below the critical temperatures mentioned above is excluded, the remaining segments pertain to second-order transitions. The latter segments verify the property that, in these transition regions, the number of first and second neighbors per site, θ and β, are linearly related to the coverage θ0. Linearity and discontinuity are eventually replaced by a smooth rise as exhibited by the curves at kBT/|V| ) 5/6 and beyond.
1934 Langmuir, Vol. 23, No. 4, 2007
Figure 11. Adsorption isotherms between phases C1 and C2 of Figure 2. Below the critical value of kBTC/|V| ) 0.0328, the transition is first-order. The isotherms are calculated for T ) TC, 2TC, 3TC, and 4TC.
Figure 12. Adsorption isotherms between phases C2 and D of Figure 2. Below the critical value of kBTC/|V| ) 0.0146, the transition is first-order. The isotherms are calculated for T ) TC, 2TC, 3TC, and 4TC.
Phares et al.
Figure 14. This plot is the analog of Figure 13 with θ replaced by β, the number of second neighbors per site.
(-3, +3) while changing the temperature. We recover their results as noted in the section of the energy phase diagram of Figure 8 that corresponds to the segment u ) 1 and -3 e V e +3. In the range -3 < V < 0, their 1/3 coverage phase is our G phase. In the range 0 < V < +3, their 2/3 phase is the honeycomb H phase complementary to the G phase. Their gas phase corresponds to our E phase, and their liquid phase corresponds to our F phase. At low temperature, they observe that the transitions from E to G and from H to F occur at V ) -3 and V ) +3, respectively, by assuming particle-vacancy symmetry. We agree with these results. The particle-vacancy symmetry that they have noted holds within the region they have considered. However, since the B phase of the infinite 2-D lattice does not have a complement, particle-vacancy symmetry only holds for u g 0, or attractive second-neighbors, as exhibited in Figure 8. For all lattice widths that we have considered with u g 1, we have found that there are only G- and H-type phases between empty and full coverage, namely
{GM} )
{
}
[(M - 1)/3] + 1 3[(M - 1)/3] + 1 0 M M
(12)
and
{HM} ) M - [M/3] 2M - 2 - 3[M/3] 3M - 4 - 3[M/3] M M M
{
Figure 13. This plot of the number of first neighbors per site θ as a function of the coverage θ0 at different temperatures complements the information provided in Figures 2, 9, 10, 11, and 12. At temperatures below the threshold T0, the curve consists of a series of connected straight segments, as is the case shown for kBT/|V| ) 1/ , which is less than k T /|V| ) 0.0288. 60 B 0
4. Comparison With Other Theoretical Studies Mihura and Landau20 studied adsorption on an L × L rhomboidshaped triangular lattice with periodic boundary conditions using Monte Carlo simulation with 12 e L e 60. In our notation, this corresponds to W/|V| ) u ) 1 with µ/|V| ) V varying in the range
} (13)
For all values of M, the transitions E f {GM}, {GM} f {HM}, and {HM} f F are all first-order below critical temperatures (TC) that are higher than those in a different energy region for M ) 7, as presented in section 3. Therefore, in the infinite 2-D limit, the corresponding transitions are also expected to be first-order. This agrees with the results obtained by Mihura and Landau valid for u ) 1. Walker and Schick26 have studied the adsorption problem where first- and second-neighbor interactions are both repulsive. Using our notation, their study covered the region -1 < V < 0 and 0 < u < 3. The only phases that appear in this region of Figure 8 are A ) (2 × 2), C ) (2 × 1), and G ) (x3 × x3) R30°, in agreement with their results. As follows from their analysis, we also agree on the equations of the boundary lines, from eq 9, between the A and C phases, between the A and G phases, and between the G and C phases. These are V ) 2(1 - u), V ) -12u, and V ) 3(1 + u), respectively. A recent work by Jager35 obtains the section of our phase diagram which exhibits the E, A, C, and G phases, but just misses the B and H phases.
Monomer Adsorption on Triangular Lattices
Figure 15. An fcc (111) or hcp (0001) surface highlighting the on-top, fcc-hollow, hcp-hollow, and bridge sites.
5. Experimental Applications When considering closed-packed fcc(111) and hcp(0001) surfaces, possible adsorption sites include the on-top, fcc-hollow, hcp-hollow, and bridge sites shown in Figure 15. Each of the on-top, fcc-hollow, and hcp-hollow sites forms an equilateral triangular lattice of the same size. The lattice formed by their combination is also an equilateral triangular but with triangles of a smaller size. The on-top, fcc-hollow, and hcp-hollow sites on this combined lattice are each independently arranged in a (x3 × x3) R30°, or G configuration. If, on this combined lattice, the G phase is followed by the H phase, and then full coverage, the adsorption occurs first on one type of site, followed by adsorption on two different types of sites, and then on all three. However, if there is a preferential order in which sites are covered, then the adsorbate-substrate energy is different for different types of sites, and the current model must be modified. The adsorption of CO on Pd(111) was studied with scanning tunneling microscopy (STM). It was “shown that at low coverage, CO molecules adsorb with almost similar probability on two different sites of the Pd(111) surface. From the STM images alone, possible pairs of sites include top-hollow (fcc or hcp) and hcp-fcc hollow sites... Total energy calculations indicate that the most favorable sites are the two threefold hollow sites, with only small energy difference between them”.62 In this case, the adsorbate lattice is honeycomb, consisting of the hcp- and fcchollow sites, and not equilateral triangular. However, a highresolution core-level photoemission and electron energy loss spectroscopy study of CO/Pd(111) has also shown, under certain conditions, bridge-site adsorption, as well as patches of (x3 × x3) R30°, with domain boundaries, (2 × 2)-3CO, and c(4 × 2).63 Here, the accepted terminology for coverage is not relative to the adsorbate lattice but relative to the substrate lattice. Therefore, the appropriate model of adsorption on the equilateral triangular substrate lattice for this study must include bridge, hcp-hollow, and fcc-hollow sites, with adsorbate-substrate interaction on bridge sites different from those at the other sites. Clearly, our current model does not apply to similar adsorption problems on equilateral triangular surfaces, since the adsorbate lattice is not an equilateral triangular. We therefore only consider experiments that fall within the limitations of our model. (62) Sautet, P.; Rose, M. K.; Dunphy, J. C.; Behler, S.; Salmeron, M. Surf. Sci. 2000, 453, 25-31. (63) Surnev, S.; Sock, M.; Ramsey, M. G.; Netzer, F. P.; Wiklund, M.; Borg, M.; Andersen, J. N. Surf. Sci. 2000, 470, 171-185.
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In our study of adsorbate lattices which are square (fcc or bcc(100)), isosceles triangular (bcc(110)), and rectangular (fcc(110)), we found that the phases with repulsive first neighbors are distinct from those with attractive first neighbors.50,53,55 This is expected for all lattice geometries. Therefore, if any of the crystallization patterns obtained in this study are observed experimentally, whether or not any other information is available, the model automatically predicts repulsive first neighbors. In the study of surface diffusion of Xe on Pt(111), the G phase has been observed at 85 K.64 It was found that the diffusion coefficient decreases with increasing coverage, suggesting attractive interactions between the adsorbed molecules. Occurrence of the G phase, according to the model, requires firstneighbor repulsion and second-neighbor attraction. From the energy phase diagram of Figure 8, a second-neighbor attraction that overcomes the first-neighbor repulsion, corresponding to u ) (W/|V|) > 1, might explain the decrease in the coefficient of surface diffusion. In general, the G phase is a well-known structure for a metal adsorbate on Pt(111) surface, for example, Ag/Pt(111),65 K/Pt(111),66 and Ni/Pt(111).67 Another example is the adsorption of Sb/Ag(111).68 Therefore, the existence of the G phase should automatically prescribe first-neighbor repulsion. In addition, if no other information is available, the phase diagram shows that the G phase occurs only when u ) (W/|V|) > -1/5. The (x3 × x3) R30°, or G phase, and the (2 × 2), or A phase, of Sn/Pt(111) have been experimentally prepared for further adsorption and reaction studies of gaseous H(D) atoms with D(H) adatoms.69 The very fact that these two phases occur on Pt(111) with Sn as the adsorbate shows that u ) (W/|V|) must be greater than -1/5 and less than 0, as follows from Figure 8. Therefore, both first- and second-neighbor Sn-Sn interactions must be repulsive, and their ratio W/V must be in the range (0, 1/ ). 5 There are many cases in which alkali metals that are adsorbed on fcc surfaces go from the (2 × 2) to the (x3 × x3) R30°.39 Again, from the phase diagram in that energy range, it should be possible, in principle, to prepare a 2 × 1, or C phase, and then the honeycomb H phase, followed by the phase complementary to the 2 × 2, or D phase, before reaching full coverage. Similar situations occur with oxygen gas adsorbed on Ag(111), Cr(111), Ni(111), Pd(111), Pt(111), and Rh(111).38,70 Reference 39 also cites numerous examples which can be analyzed using our model, such as the adsorption of H2S, H2O, S2, and H2Se. Another example is the adsorption of H/Ni(111) observed on hcp- and fcc-hollow sites thus forming an H phase.71 As mentioned earlier, the model is applicable to the adsorption on graphite for which adsorption takes place at the centers of the hexagonal structures, forming an equilateral triangular adsorbate lattice. This is the case of xenon72 as well as argon and krypton73 adsorbed on graphite. (64) Meixner, D. L.; George, S. M. J. Chem. Phys. 1993, 98, 9115-9125. (65) Hubbard, A. T.; Stickney, J. L.; Rosasco, S. D.; Soriaga, M. P.; Song, D. J. Electroanal. Chem. Interfacial Electrochem. 1983, 150, 165. (66) Pirug, G.; Bonzel, H. P.; Brode´n, G. Surf. Sci. 1982, 122, 1. (67) Su, C. W.; Ho, H. Y.; Shern, C. S.; Chen, R. H. Surf. Sci. 2002, 499, 103-108. (68) Soares, E.; Bittencourt, C.; Nascimento, V.; de Carvalho, V.; de Castilho, C.; McConville, C.; de Carvalho, A.; Woodruff, D. Phys. ReV. B 2000, 61, 13983. (69) Busse, H.; Voss, M. R.; Jerdev, D.; Koel, B. E.; Paffet, M. T. Surf. Sci. 2001, 490, 133-143. (70) Bolotin, I. L.; Kutuna, A.; Makarenko, B.; Rabalais, J. W. Surf. Sci. 2001, 472, 205-222. (71) Ito, T.; Umezawa, K.; Nakanishi, S. Appl. Surf. Sci. 1999, 147, 146-152. (72) Pussi, K.; Smerdon, J.; Ferralis, N.; Lindroos, M.; McGrath, R.; Diehl, R. D. Surf. Sci. 2004, 548, 157-162. (73) Chen, X-R.; Oshiyama, A.; Okada, S. Chem. Phys. Lett. 2003, 371, 528.
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The adsorption system O/Ru(0001) has been experimentally investigated, and comparison was made with lattice-gas model calculations.74 The oxygen adsorbs on hcp-hollow sites with the observed phases (2 × 2)-O and (2 × 1)-O but not the (x3 × x3) R30°. We agree with the comment made in ref 74: “Repulsive interactions between first- and second-neighbors have been shown previously to lead to p(2 × 2) and p(2 × 1) phases near coverages of 1/4 and 1/2, respectively. Furthermore, to suppress the formation of a (x3 × x3) R30° ordered phase at coverages near 1/3, the second-neighbor interaction must be sufficiently strong. Ground-state arguments require that E2/E1 (sic W/V) > 0.2 to ensure this.” Indeed, according to the energy phase diagram of Figure 8, this can only occur if W/V > 1/5. In this study, however, there is no mention of another phase, the (3 × 1), or B phase, which is present between the (2 × 1) and (2 × 2) phases, for W/V > 1. As this phase has not been observed in this study, it is probable that 1/5 < W/V < 1. We obtained these results without having to add higher-order interactions to the theory, as has been suggested and achieved with appropriately chosen parameters to fit the experimental data.74 In our finite width calculations, we have not observed first-order phase transitions in the energy region mentioned above. Analytical extrapolation to infinite width maintains the absence of firstorder transitions, which is in agreement with the experimental data. On the other hand, in the electro-oxidation of CO at the Ru(0001) single-crystal surface, oxygen preadsorbed on the crystal has been observed to form a (3 × 1)-O phase in addition to the (2 × 2)-O and (2 × 1)-O.75 Furthermore, the occupation of the hcp-hollow sites (2 × 2)-3O, or our D phase, has been observed,76 as well as the full-coverage (1 × 1)-O.77 In this case, the A, B, C, D, and F phases are possible only if W/V > 1, as follows from Figure 8. It may very well be that adsorption taking place from solution affects the adsorbate interaction energies, since it is now in a different energy region of the phase diagram.
6. Summary and Conclusion We have presented a model of monomer adsorption on a semiinfinite zigzag equilateral triangular terrace without periodic boundaries which includes first- and second-neighbor adsorbateadsorbate interactions. First-neighbor interactions were restricted to be repulsive. The relatively low temperature (on the order of 150 K) phase diagrams for terraces of width M, in the range 3-11, were obtained and extrapolated to generate the complete phase diagram for the infinite two-dimensional lattice. The numerical computations of the occupational characteristics for (74) Piercy, P.; De’Bell, K.; Pfnu¨r, H. Phys. ReV. B 1992, 45, 1869. (75) Lin, W. F.; Christensen, P. A.; Hamnett, A.; Zei, M. S.; Ertl, G. J. Phys. Chem. B 2000, 104, 6642. (76) Kim, Y. D.; Wendt, S.; Schwegmann, S.; Over, H.; Scheffler M.; Ertl, G. Phys. ReV. Lett. 1996, 77, 3371. (77) Stampfl, C.; Schwegmann, S.; Over, H.; Ertl, G. Surf. Sci. 1998, 418, 267.
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a given M were determined with a precision exceeding 10 decimal places. As in previous studies, the occupational characteristics of the phases were found to fit exact analytic expressions in M to within the stated precision. The infinite M limit of these exact results provided all of the phases as well as the energy phase diagram on the infinite two-dimensional lattice. We have recovered results obtained by other theoretical methods and found the existence of additional phases, the (3 × 1), or B phase, which has no complement, and the complement of the (2 × 2), or D phase. These phases occur in interaction energy regions not previously investigated. For first- (repulsive) and second-neighbor interactions, the particle-vacancy symmetry holds only in a limited region but not throughout the entire phase diagram. First- and second-order phase transitions have also been found over a wide temperature range in several energy regions. The model has been applied to a variety of experiments. Information on interaction energies between adsorbates, beyond what has been reported in section 5, could be obtained whenever it is feasible to conduct adsorption experiments at relatively low temperature in the following manner. The relatively low temperature depends on the adsorption system under study. For substrates such as Pt, Pd, graphite, and adsorbates such as O, CO, and NO, which have been extensively studied with many different techniques, a relatively low temperature is on the order of 150 K. This choice will ensure that all possible phases are detected as the pressure of the gas is gradually increased, and will determine whether certain phase transitions are first-order. At each incremental step in the pressure, corresponding to a given chemical potential per particle µ′, the system is allowed to reach thermodynamic equilibrium. The sequence of crystallization patterns are recorded as the adsorption system evolves from empty to full coverage. The values of µ′ at the transitions between the phases are also determined, whether the transitions are first- or second-order. Then, according to the model, knowledge of the ordered patterns of the phases, the sequence in which the phases occur, and the values of µ′ at the transitions between phases, will make it possible to determine all of the interaction energies. This particular feature of the model has already been mentioned and discussed in earlier theoretical adsorption studies of one monomer species,47,48,50-56 the coadsorption of two monomer species,78 and the coadsorption of one species of monomer and another of dimer.79 Acknowledgment. This research was supported by an allocation of advanced computing resources supported by the National Science Foundation. The computations were performed in part on the TG PSC TCS1 at the Pittsburgh Supercomputing Center. LA062994Y (78) Phares, A. J.; Wunderlich, F. J.; Martin, J. P.; Duda, G. K. Phys. ReV. E 1997, 56, 2447. (79) Phares, A. J.; Wunderlich, F. J. Phys. Lett. A 1997, 226, 336.
