Adsorption on an Equilateral Triangular Terrace Three Atomic Sites in

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Adsorption on an Equilateral Triangular Terrace Three Atomic Sites in Width: Application to Chemisorption of CO on Pt(112) 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 April 12, 2006. In Final Form: June 19, 2006 A model of monomer adsorption on equilateral triangular terraces three atomic sites in width is presented where step sites are considered first neighbors. Adsorbate-substrate interactions at the terrace step are treated differently than at bulk sites. Adsorbate-adsorbate first neighbor interactions are considered to be repulsive while second neighbors are allowed to be either repulsive or attractive. All low temperature phases have been identified under these conditions. The effect of increasing the temperature has also been investigated. Application of the model to chemisorption of CO on Pt(112) suggests experiments that would allow the various interaction energies to be obtained from a knowledge of the relatively low temperature phases and the conditions prevailing at the transitions between phases. Currently available experimental data is very extensive on the manner in which step sites are filled. However, there is insufficient data on the sequence of low temperature phases which appear when the pressure is gradually increased that would show the manner in which bulk sites are filled until full coverage of the terrace is reached.

1. Introduction Lattice models have been a powerful tool for providing insights in the study of many physical, chemical, and biological systems. Langmuir adsorption theory1-8 and the Ising model applied to ferromagnetism9-20 are among the earliest of these models. Matrix methods have been used extensively in lattice adsorption studies such as those presented in ref 19 or used in ref 21. The homonuclear22-25 and heteronuclear26-30 dimer problems have been investigated using Monte Carlo simulation. Two-dimen* 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. 1951, 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. J. Am. Chem. Soc. 1932, 54, 2798. (7) Langmuir, I.; Taylor, J. B. Phys. ReV. 1933, 44, 423. (8) Langmuir, I.; Villars, D. S. J. Am. Chem. Soc. 1931, 53, 486. (9) Ising, E. J. Phys. 1925, 31, 253. (10) Kramers, H. A.; Wannier, G. H. Phys. ReV. 1941, 60, 252. (11) Kramers, H. A.; Wannierm, 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.; Lee, T. D. Phys. ReV. 1952, 87, 404. (17) Lee, T. D.; Yang, C. N. Phys. ReV. 1952, 87, 410. (18) McCoy, M.; Wu, T. T. The Two-dimensional Ising Model; Harvard University: Cambridge, MA, 1973. (19) Baxter, R. J. Exactly SolVed Models in Statistical Mechanics; Academic Press: New York, 1982. (20) Schick, M.; Walker, J. S.; Wortis, M. Phys. ReV. B 1977, 16, 2205. (21) Bartlet, N. C.; Einstein, T. L.; Roelofs, L. D. Phys. ReV. B 1986, 34, 1616. (22) Ramirez-Pastor, A. J.; Riccardo, J. L.; Pereyra, V. D. Surf. Sci. 1998, 411, 294. (23) Roma´, F.; Ramirez-Pastor, A. J.; Riccardo, J. L. Langmuir 2000, 16, 9406. (24) Roma´, F.; Ramirez-Pastor, A. J.; Riccardo, J. L. J. Chem. Phys. 2001, 114, 10932. (25) Roma´, F.; Ramirez-Pastor, A. J.; Riccardo, J. L. Phys. ReV. B 2003, 68, 205407. (26) Rzysko, W.; Borowko, M. J. Chem. Phys. 2002, 117, 151. (27) Rzysko, W.; Borowko, M. Surf. Sci. 2002, 520, 151. (28) Rzysko, W.; Borowko, M. Physica A 2003, 326, 1. (29) Rzysko, W.; Borowko, M. Thin Solid Films 2003, 425, 304. (30) Rzysko, W.; Borowko, M. Surf. Sci. 2006, 600, 890.

Figure 1. (111) terraces, three atomic sites in width, exhibited in a (112)-plane cut of an fcc(111) crystal such as platinum, make an angle of 19.471° with the (112)-plane and an angle of 54.736° with the one-atom square step surfaces. Relative to the size of the fundamental cube, a ) 1/x2 is the size of the equilateral triangles making up the terraces as well as the squares making up the steps. The width of the terrace is b ) (3/2)1/2 and the height of the step is h ) 1/x3.

sional equilateral triangular lattices have been studied extensively within the context of the Ising model (see for example refs 19, 20, 31, and 32). The lattice model presented here deals with the general problem of monomer adsorption on very long equilateral triangular terraces, three atomic sites in width, where the edge sites are first neighbors. The terraces have two edges. We treat the sites on one edge, called step sites, differently than bulk sites, the sites making up the remainder of the terrace. The terraces are exposed to a gas of atoms or molecules that has a chemical potential energy µ′ per particle, which varies with the gas pressure. We assume that first neighbor adsorbates are repulsive and neglect adsorbate-adsorbate interactions beyond second neighbors. In this paper, we apply this lattice model to the adsorption of CO on Pt(112), which consists of equilateral triangular terraces three atomic sites in width separated by steps. The atomic sites on either side of the steps are first neighbors, as shown in Figure 1. CO bonds to Pt sites at the carbon end, and it is preferentially adsorbed on step-up sites.33-36 Experiments also indicate that (31) Walker, J. S.; Schick, M. Phys. ReV. B 1979, 20, 2088. (32) Kinzel, W.; Schick, M. Phys. ReV. B 1981, 23, 3435.

10.1021/la060999p CCC: $33.50 © 2006 American Chemical Society Published on Web 07/29/2006

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neighboring CO-CO interactions on the step-up sites are repulsive33-36 but show no reason for treating adsorption on step-down sites as different from adsorption on bulk sites. This is why preferential adsorption on step-up sites is modeled by considering adsorbate-substrate interaction energy at step-up sites Vs to be different from the absorbate-substrate interaction Vb on the remaining step-down and bulk sites. We therefore call the step-up sites simply “step sites” to differentiate them from the remaining sites, which are treated equivalently and collectively called “bulk sites”. The adsorbate-adsorbate interactions between first and second neighbors correspond to energies V and W, respectively, with positive energies corresponding to attractive forces and negative energies to repulsive forces. The differential adsorbate-substrate interaction energy between step sites and bulk sites is U ) (Vs - Vb) and the shifted chemical potential energy is µ ) µ′ + Vb. The activities associated with µ, V, W, and U are denoted

x ) exp(µ/kT), y ) exp(V/kT), z ) exp(W/kT), u ) exp(U/kT) (1) where k is Boltzmann’s constant and T is the absolute temperature. It then follows that the partition function Z3 of the adsorption system on this equilateral triangular terrace three atomic sites in width and of finite length is given in terms of all of the eigenvalues of a matrix of rank 26 because both first and second neighbor interactions have been considered. If only first neighbor interactions were considered, the rank of this matrix would have been 23. We have called this the T-matrix and have constructed it using the method derived in 1993,37,38 which we have applied to a number of other adsorption studies.39-47 This matrix method has been recently generalized for application to both terraces and nanotubes.48 The first operation recursively constructs the matrix associated with the adsorption problem but neglects the distinction between step and bulk sites (U ) 0, u ) 1).48 Two sets of matrices are required: AN(a, b, c, d;c′, d′) and BN(e, f, g, h;h′) where N is an integer indicating the rank of these matrices, 22N in this case. The arguments are either 0 or 1. It follows that there are 64 matrices of the A-type and 32 matrices of the B-type for any given N. These matrices are nonlinearly related according to the recursion relations in Chart 1 below.48

The matrix associated with the adsorption problem (neglecting the distinction between step and bulk sites) is given by A3(0, 0, 0, 0;0, 0) with the initial conditions A0 ) 1 and B0 ) 1.48 The distinction between step and bulk sites is then obtained by adding a factor of u to all elements of A3(0, 0, 0, 0;0, 0) in columns 2, 6, 10, ‚‚‚, 58, 62 and columns 3, 7, 11, ‚‚‚, 59, 63; and a factor of u2 to all elements in columns 4, 8, 12, ‚‚‚, 60, 64. In the limit of infinitely long terraces, the only contribution to Z3 is the eigenvalue of largest modulus R of the T-matrix. R is real and positive, as follows from the Frobenius-Perron theorem for matrices whose elements are real and nonnegative. Z3 is given by

Z3 ) R1/6

(2)

At thermodynamic equilibrium, the average coverage θ0, the average numbers per site of first and second neighbor adsorbates θ and β, and the average number per site γ of occupied step sites are given by

θ0 )

x ∂R y ∂R z ∂R u ∂R , θ) , β) , γ) 6R ∂x 6R ∂y 6R ∂z 6R ∂u

(3)

The average energy per site of the adsorption system is then

) µθ0 + Vθ + Wβ + Uγ

(4)

and the entropy per site divided by Boltzmann’s constant is given by

S ) (1/6) ln(R) - /(kT)

(5)

2. Numerical Results The ratio µ/V was varied, for a chosen set of energy parameters V, W, and U, such that the computed coverage ranged from empty to full at a particular temperature. For each value of µ/V, the four activities were calculated from eq 1. The activities were then substituted into the T-matrix. The largest eigenvalue R and its partial derivatives with respect to the four activities were then obtained which provided the coverage θ0, the numbers per site of first and second neighbors θ and β, and the number per site γ of occupied step sites, from eq 3. The energy per site and the entropy S follow from eqs 4 and 5.

Chart 1

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Figure 2. Plot of coverage θ0 vs -µ/V at 40 K with |V| ) 4 kcal/ mol, W/V ) 3, and -U/V ) 5.

Figure 4. Occupational characteristics and configurations of phases p1 through p14.

Figure 3. Plot of S vs θ0 at various temperatures with the same values of V, W, and U as Figure 2.

With |V|/k ) 2000 K per adsorbate, or about 4 kcal/mol, W/V ) 3, -U/V ) 5, and a temperature of T ) 40 K, the coverage θ0 exhibits a series of plateaus as - µ/V varies from -0.5 to 1.6. These plateaus occur at θ0 ) 0, 1/6, 1/3, 1/2, 2/3, 5/6, and 1, as shown in Figure 2, and correspond to perfectly ordered phases. Each of these phases corresponds to a cusp in the plot of S vs θ0 where S ) 0, evident in Figure 3. These phases have occupational characteristics given by the set {θ0,θ,β,γ}. The complete list of phases and occupational configurations, denoted p1 to p37, are shown in Figures 4-7, excluding the empty phase E and the full coverage phase F, {1, 7/3, 5/3, 1/3}. In these (33) Henderson, M. A.; Szabo, A.; Yates, J. T., Jr. J. Chem. Phys. 1989, 91, 7245. (34) Henderson, M. A.; Szabo, A.; Yates, J. T., Jr. J. Chem. Phys. 1989, 91, 7255. (35) Henderson, M. A.; Szabo, A.; Yates, J. T., Jr. Chem. Phys. Lett. 1990, 168, 51. (36) Xu, J.; Yates, J. T., Jr. Surf. Sci. 1995, 327, 193. (37) Phares, A. J.; Wunderlich, F. J.; Grumbine, D. W., Jr.; Curley, J. D. Phys. Lett. A 1993, 173, 365. (38) Phares, A. J.; Wunderlich, F. W.; Curley, J. D.; Grumbine, D. W., Jr. J. Phys. A 1993, 26, 6847. (39) Phares, A. J.; Wunderlich, F. J. Phys. ReV. E 1995, 52, 2236. (40) Phares, A. J.; Wunderlich, F. J. Phys. ReV. E 1997, 55, 2403. (41) Phares, A. J.; Wunderlich, F. J.; Martin, J. P.; Burns, P. M.; Gintaras, G. K. Phys. ReV. E 1997, 55, 2447. (42) Phares, A. J.; Wunderlich, F. J. Phys. Lett. A 1997, 226, 336. (43) Phares, A. J.; Wunderlich, F. J. Surf. Sci. 1999, 425, 112. (44) Phares, A. J.; Wunderlich, F. J. Surf. Sci. 2000, 452, 108. (45) Phares, A. J.; Wunderlich, F. J. Int. J. Mod. Phys. B 2001, 15, 3323. (46) Phares, A. J.; Wunderlich, F. J.; Kumar, A. Surf. Sci. 2001, 495, 140. (47) Phares, A. J.; Wunderlich, F. J. Appl. Surf. Sci. 2003, 219, 174. (48) Phares, A. J.; Grumbine, D. W., Jr.; Wunderlich, F. J. “Monomer adsorption on terraces and nanotubes”. To be submitted for publication.


figures, the open circles correspond to unoccupied lattice sites, filled circles are occupied sites, and the sites surrounded by a rectangle are the step sites. The numerical computations were carried out with long double-precision arithmetic in order to


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Figure 8. Coverage as a function of -µ/V for |V| ) 4 kcal/mol at several different temperatures. The curves have been shifted on the vertical axis for clarity.



reliably obtain rational expressions for the occupational characteristics of these perfectly ordered phases. In this specific example, the phases encountered with increasing values of -µ/V are

E f p2 f p10 f p17 f p27 f p36 f F

(6)

In the transition region between any two phases, the point at which the entropy reaches a maximum occurs, at sufficiently low temperature, when

0 ) µ∆θ0 + V∆θ + W∆β + U∆γ

(7)

where ∆θ0, ∆θ, ∆β, and ∆γ are the changes in the occupational characteristics between phases on either side of the local entropy maximum. This has been verified here and in all previous studies.37-47 Lowering the temperature below 40 K while keeping the interaction energies V, W, and U constant does not alter the sequence of phases observed. However, as shown in the plot of θ0 versus -µ/V, the range of the ratio -µ/V that corresponds to the passage from one plateau to the next becomes narrower as the temperature is decreased. This width reduces to zero at zero temperature. Since numerical calculations cannot be carried out at T ) 0 K, this has been verified numerically by extrapolation. On the other hand, the transition regions become wider as the plateaus become narrow when the temperature is increased while the interaction energies V, W, and U are held constant. The cusps in the entropy curves gradually disappear, as shown in Figure 2, and the local minima are no longer zero. The temperature dependence of the transition width is shown for |V| ) 4 kcal/mol in Figure 8. The choice of 4 kcal/mol is too small for the case of CO/ Pt(112). However, as temperature is only a scaling factor, increasing the chosen value of |V| by a factor of f leads to the same result when temperature T is increased by the same factor. For example, with f ) 4 (or |V| ) 16 kcal/mol), the isotherms of Figure 8 would correspond to temperatures scaled by the same factor. Consequently, the phase sequence of eq 6 is observable well beyond room temperature. Thus, the sufficiently low temperature required to observe all of the phases predicted by the model depends on the chemisorption system. Quantification of the width of the transition region requires the choice of a criterion from which to determine whether a certain numerical coverage is within a particular phase. Experimentalists have used coverages to within two or three significant figures33-36 as the criterion. Here, we consider a particular phase to be present as long as the computed coverage agrees with the exact, rational value to three decimal places. Therefore, as the temperature is increased and the transitions broaden, the extent over which each phase is present diminishes until the phase completely disappears, as shown in Figure 9. This criterion, when applied to a particular adsorption system, yields a “critical temperature” below which all of the low-temperature phases predicted by the model are present. In the numerical example considered, |V| ) 4 kcal/mol, W/V ) 3 and -U/V ) 5, the critical temperature is 154 K, as follows from Figure 9. With |V| ) 16 kcal/mol, the critical temperature is 716 K which is well above room temperature.

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Table 1. U Subregions and Phase Sequences for W < V when Nearest Neighbor Interactions Are Repulsive (V < 0) W 2 the sequence of phases is always the sequence given in eq 6. As -U/V is decreased from 2 to -11/2, several different phase sequences occur, each corresponding to a given range of -U/V. Finally, for -U/V < -11/2, the sequence is always

E f p1 f p4 f p9 f p12 f p20 f p32 f p37 f F (8) This scan is continued for different values of W relative to V, keeping V negative, thus delineating all of the energy regions. The regions are grouped according to W relative to V (W regions) and are subdivided according to the values of U relative to V and W (U subregions) to highlight the step effects. There are 31 W regions of which only the first is shown in Table 1. This W region

Figure 9. Plot of T vs -µ/V providing the order-disorder phase diagram at |V| ) 4 kcal/mol, W/V ) 3, and -U/V ) 5.

E f p1 f p4 f p9 f p12 f p20 f p32 f p37 f F E f p1 f p4 f p9 f p12 f p20 f p26 f p32 f p33 f p37 f F E f p1 f p4 f p9 f p12 f p16 f p26 f p27 f p33 f p36 f F E f p1 f p4 f p9 f p12 f p16 f p27 f p36 f F E f p1 f p4 f p9 f p16 f p27 f p36 f F E f p1 f p4 f p5 f p16 f p27 f p36 f F E f p1 f p5 f p16 f p27 f p36 f F E f p2 f p5 f p17 f p27 f p36 f F E f p2 f p10 f p17 f p27 f p36 f F

has 9 U subregions. Certain W regions have up to 24 U subregions. The 30 boundaries of the 31 W regions are

V, V/2, V/3, 0, -V/13, -V/12, -V/11, -V/10, -V/9, -V/8, -V/7, -V/6, -5V/27, -V/5, -2V/9, -5V/22, -V/4, -5V/18, -V/3, -5V/14, -7V/18, -2V/5, -5V/12, -4V/9, -V/2, -5V/9, -2V/3, -5V/6, -8V/9, -V There is a dramatic change at the crossing from W < 0 (repulsive second neighbors) to W > 0 (attractive second neighbors). The phases occurring only for W < 0 are p1, p4, p5, p9, p12, p16, p20, p22, p26, and p33. In addition, p9 appears only in the first W region and p22 only in the fourth W region. Another major change occurs at the crossing of the boundary W ) -V/3, after which the following phases no longer occur: p6, p8, and p27. Thus, it is possible to determine the energy relationship between U, V, and W for a physical system from an experimental knowledge of the phases occurring at sufficiently low temperature and their sequence. If it is also possible to obtain the conditions prevailing at the transition between these phases, the precise values of U, V, and W may be determined from eq 7, which is valid for every transition. To ensure that no phases have been missed during the numerical search, the low-temperature phase diagram of -µ/V versus -U/V is constructed for every W region. The phase diagram associated with the W region W/V > 1 for V < 0 is shown in Figure 10. It provides a visualization of the step effect as U varies across all of the U subregions. The phase boundaries are straight lines corresponding to the points between two adjacent phases where the entropy is a local maximum. As mentioned above, the equations of these straight lines are obtained using eq 7. The low-temperature phase diagrams for the remaining 30 W regions may be generated in the same manner. Finally, the effect of temperature on the order-disorder transition may be studied in every energy region in the same manner as the example above.

3. Application to Chemisorption of CO on Pt(112) The experimental evidence, based on electron simulated desorption-ion angular distribution, low-energy electron diffraction, and temperature programmed desorption, shows that CO preferentially adsorbs on the steps of Pt(112).33-36 The observation is that there is “one-dimensional CO- -CO repulsions (which) result in tilting along the steps”.35 The available experimental data focuses on the angular orientation of CO relative to the step and gives details only about the filling of step sites: step sites are filled first. We have assumed that the bulk sites are covered next. Under these conditions, the phases which have only step sites occupied are p2 and p10. Therefore, according to the results of our model, the only phase sequences having this property are

E f p2 f p10 f p17 f p27 f p36 f F Figure 10. Low-temperature phase diagram for energy region W/V > 1.

E f p2 f p10 f p17 f p23 f p29 f p35 f p36 f F

(9) (10)



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(11)

F. Applying eq 7 at each of these six transitions respectively leads to

E f p2 f p10 f p14 f p17 f p23 f p29 f p35 f p36 f F (12)

µ′1 + Vb ) -U; µ′2 + Vb ) -2V - U; µ′3 + Vb ) -W; µ′4 + Vb ) -2V - W; µ′5 + Vb ) -4V - 4W; µ′6 + Vb ) -6V - 4W


(13)

E f p2 f p10 f p14 f p23 f p29 f p35 f F

(14)

E f p2 f p10 f p23 f p29 f p35 f F

(15)

Sequence (9) occurs in either of the following energy regions:

W < 0 and -2V < U; 0 < W < - V/3 and -2V + W < U Sequence (10) occurs in the regions

-V/3 < W < -(5/9)V and -(3/2)V + (5/2)W < U Sequence (11) occurs in the regions

-(5/9)V < W < -(2/3)V and -(3/2)V + (5/2)W < U < V + 7W

The unknown interaction energies are the adsorbate-substrate interaction energy at bulk sites Vb, the adsorbate-substrate interaction energy at step sites Vs, and adsorbate-adsorbate first and second neighbor interaction energies V and W. Here U ) Vs - Vb. Therefore, it would be sufficient to measure the chemical potential at any four of these transitions to determine all the energies. It is interesting to note that the numerical occupational characteristics and the entropy at the transitions between phases p2 and p10, and between p10 and p17, fit exact analytic expressions in terms of the golden ratio φ ) (1 + x5)/2. At the transition between p2 and p10, we have

θ0 ) γ )

Sequence (12) occurs in the regions

-(5/9)V < W < -(2/3)V and V + 7W < U < -V + 4W Sequence (13) occurs in the regions

-(5/9)V < W < -(2/3)V and -V + 4W < U Sequence (14) occurs in the regions

-(2/3)V < W and - (3/2)V + (5/2)W < U < -2V + (5/2)W Finally, sequence (15) occurs in regions

-(2/3)V < W and -2V + (5/2)W < U Therefore, knowledge of the order in which bulk sites are filled as the chemical potential increases is necessary to further narrow the possible energy regions to which the adsorption system CO/ Pt(112) belongs. For instance, take the sequence of phases given in (9). The aim is to obtain the interaction energies from the knowledge of the sequence of phases encountered at relatively low temperature, ranging from empty to full coverage, and from the conditions at the transition between phases determined by the point where the entropy is a local maximum. Suppose that it is experimentally possible to determine the values of the pressure and consequently of the chemical potential energy per CO molecule at every transition. Then let the corresponding chemical potentials be µ′1 at the transition between E and p2, µ′2 between p2 and p10, µ′3 between p10 and p17, µ′4 between p17 and p27, µ′5 between p27 and p36, and finally µ′6 between p36 and

φ+1 φ 1 , θ) , β ) 0, S ) ln φ 3 3(φ + 2) 3(φ + 2) (16)

At the transition between p10 and p17, we have

θ0 )

φ+3 1 1 1 , θ)γ) , β) , S ) ln φ 3 3 3(φ + 2) 3(φ + 2) (17) 4. Summary and Conclusion

The proposed model is phenomenological in the sense that the only assumptions made are that (a) the geometry of the terrace is equilateral triangular with edge sites being first neighbors, (b) the adsorbates are monomers, (c) first neighbor adsorbates are repulsive, and (d) adsorbate interactions are negligible beyond second neighbors. As interaction energies are allowed to take on any set of values, keeping the temperature relatively low (as defined above), we identified all possible energy regions and the corresponding phases and phase sequences. A sample phase diagram and the effect of temperature is presented. Based on the available experimental data on CO/Pt(112), it is possible to make some predictions on the interaction between first and second CO-CO neighbors, CO-Pt interaction on bulk sites, and COPt interaction on step sites. Further experimental work along the lines suggested above is required for the model to provide definite values for these interaction energies. 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. LA060999P

Adsorption on an Equilateral Triangular Terrace Three Atomic Sites in

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