Monomer Adsorption on Equilateral Triangular Lattices with Attractive

124

Langmuir 2008, 24, 124-134

Monomer Adsorption on Equilateral Triangular Lattices with Attractive First-neighbor Interactions Alain J. Phares,*,† David W. Grumbine, Jr.,‡ and Francis J. Wunderlich† Department of Physics, VillanoVa UniVersity, Mendel Hall, VillanoVa, PA 19085-1699, and Department of Physics, Saint Vincent College, Latrobe, PennsylVania 15650-4580 ReceiVed July 16, 2007. In Final Form: September 28, 2007 We have recently studied a model of monomer adsorption on infinitely long equilateral triangular lattices with terraces of finite width M and nonperiodic boundaries. This study was restricted to the case of repulsive adsorbateadsorbate first-neighbor interactions but included attractive, repulsive, and negligible second-neighbor interactions. The present work extends this study to the case of attractive first-neighbors, and the phases are determined, as before, with a confidence exceeding 10 significant figures. Phase diagrams are included for terrace widths M e11. Most of the occupational characteristics of the phases fit exact analytic expressions in M. The infinite-M limit of these expressions, combined with other analyses, provide the complete phase diagram for the infinite two-dimensional lattice. In addition to the empty and full coverage phases, there are three phases exhibiting stripe and cluster features that were not observed in the case of repulsive first-neighbors.

1. Introduction A complete discussion of the differences between our lattice model of surface adsorption and others is found in three recent publications1-3 and references cited therein. For completeness, we give a brief summary of these differences and a short presentation of the mathematical formulation and numerical procedure. Most lattice gas models that are applied to surface adsorption, ferromagnetism, or antiferromagnetism assume periodic boundaries and finite sizes. Here we consider an equilateral substrate lattice of finite length L and of finite width M, thus forming a terrace, but without a periodic boundary, thus forming a terrace. The lattice orientation is of the zigzag type, as in ref 1 and shown in Figure 1a. Lattice gas models have been often studied using transfer matrices whose eigenvalues provide the partition function of the system from which all pertinent thermodynamic quantities are derived. The infinite two-dimensional case is obtained by extrapolation of the numerical results for lattices of increasing size. Monte Carlo simulations have also been used, sometimes in conjunction with transfer matrices.4-6 Typical Monte Carlo simulations use an L × L net, which corresponds to a rhomboid for an equilateral triangular lattice.7,8 Extrapolation of the results with increasing values of L, for example from L ) 8 to 64, provides a good approximation of what to expect on the infinite 2-D lattice.7,8 Renormalization group methods have * To whom correspondence should be addressed. Phone: +1 610 519 4889. E-mail: [email protected]. † Villanova University. ‡ Saint Vincent College. (1) Phares, A. J.; Grumbine, D. W. Jr.; Wunderlich, F. J. Langmuir 2007, 23, 1928. (2) Phares, A. J.; Grumbine, D. W. Jr.; Wunderlich, F. J. Langmuir 2007, 23, 558. (3) Phares, A. J.; Grumbine, D. W. Jr.; Wunderlich, F. J. Langmuir 2006, 22, 7646. (4) Metcalf, B. D. Phys. Lett. 1973, 45A, 1. (5) Chin, K. K.; Landau, D. P. Phys. ReV. B 1987, 36, 275. (6) Pasinetti, P. M.; Roma´, S.; Riccardo, J. L.; Ramirez-Pastor, A. J. Phys. ReV. B 2006, 74, 155418. (7) Mihura, B.; Landau, D. P. Phys. ReV. Lett. 1977, 38, 977. (8) Gu, L.; Chakraborty, B.; Garrido, P. L.; Phani, M.; Lebowitz, J. L. Phys. ReV. B 1996, 53, 11985.

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.

also been used to successfully derive results for the infinite lattice.9,10 We use the same notation as the one found in ref 1. Here first-neighbor adsorbate-adsorbate interaction energy V is (9) Schick, M.; Walker, J. S.; Wortis, M. Phys. ReV. B 1977, 16, 2205. (10) Berker, A. N.; Ostlund, S.; Putnam, F. A. Phys. ReV. B 1978, 17, 3650.

10.1021/la702128f CCC: $40.75 © 2008 American Chemical Society Published on Web 11/30/2007

Monomer Adsorption on Triangular Lattices

Langmuir, Vol. 24, No. 1, 2008 125

positive, which corresponds to an attractive interaction. The second-neighbor adsorbate-adsorbate interaction energy W is allowed to be positive, negative, or negligible. No distinction is made between Ve and Vb, the adsorbate-substrate interactions at edge and bulk sites, so that Ve ) Vb ) V0. The chemical potential energy per particle in the gas phase µ′ is varied by changing the gas pressure. The system is at thermodynamic equilibrium and at absolute temperature T. The relevant quantities are the shifted chemical potential µ ) µ′ + V0, and the absolute activities, x ) exp(µ/kBT), y ) exp(V/kBT), and z ) exp(W/kBT), where kB is Boltzmann’s constant. As shown in ref 2 and applied in refs 1 and 3, the transfer matrix associated with adsorption on zigzag equilateral triangular terraces is obtained from 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. Index N refers to their rank 2N. These matrices satisfy the initial conditions

A0(R,β,γ;R′,β′) ) 1, B0(β,γ,δ;β′) ) 1

Table 1. Occupational Characteristics for Lattice Widths M ) 3-5 M

name

3 4

B31 A41 A42 B41 A51 A52 B51 C51

5

(

M

name

6

A61 B61 B62 C61 A71 A72 A73 B71 B72 C71 B73 B74 F71 C72 C73

(1)

BN-1(β,γ,δ;γ′) ) AN-2(γ,δ,0;γ′,0)

(

xyγ+δzβAN-2(γ,δ,1;γ′,0) γ+2δ 2β+1 z AN-2(γ,δ,1;γ,1) (γ,δ,0;γ′,1) xy

δ βAN-2

yz

) )

7

(2)

(3)

If the Mth site on the zigzag sequence shown in Figure 1a is at the top, the transfer matrix is BM(0,0,0;0); if it is at the bottom, the transfer matrix is AM(0,0,0;0,0). In both cases, the parity of M determines whether the right edge site (the first site in the sequence) is at the top or the bottom of the zigzag. Both formulations yield the same result. In the thermodynamic limit as the length L of the lattice becomes infinite while keeping the width M fixed, the partition function Z of the system simplifies to Z ) R1/M, where R is the eigenvalue of largest modulus of the transfer matrix. In this case R is real and positive as follows from the Frobenius-Perron theorem for matrices whose elements are real and non-negative. At thermodynamic equilibrium, the average coverage of the lattice θ0, and the numbers of first- and secondneighbors per site θ and β, follow as

θ0 )

x ∂R , MR ∂x

θ)

y ∂R , MR ∂y

β)

z ∂R MR ∂z

(4)

The average total energy per site and the entropy per site divided by Boltzmann’s constant, referred to as entropy S, are

) θ0 µ + θV + βW, S ) (1/M)lnR - /kBT

(5)

For a given set of values of V, W, µ, and T, the largest eigenvalue R and its first-order derivatives with respect to the three activities, x, y, and z, are numerically computed to at least 10 figures. In turn, these values provide the entropy S and the occupational characteristics listed as the set {θ0, θ, β}. A phase corresponds to the situation for which these characteristics remain unchanged over a wide range of the shifted chemical potential µ. A phase may be perfectly or partially ordered depending on whether or not the entropy S is zero. A threshold temperature T0 is determined below which no new phases are observed. Above T0, some of the phases begin to disappear. At high enough temperature, there is a smooth evolution from the empty lattice, E ) {0, 0, 0}, to full coverage, F ) {1, (3M - 4)/M, (3M - 6)/M}. In order to

θ

β

1/2 3/8 3/8 5/8 3/10 1/3 7/10 13/15

0 0 0 1/8 0 0 1/5 1/3

name C31 C41 F41 F42 C52 F51 F52 F53

θ0

θ

β

4/9 1/2 3/4 5/6 3/5 7/10 11/15 4/5

5/9 3/4 5/4 3/2 9/10 13/10 19/15 8/5

1/9 1/4 5/8 5/6 2/5 7/10 11/15 1

Table 2. Occupational Characteristics for Lattice Widths M ) 6-7

and are recursively related according to

AN(R,β,γ;R′,β′) ) BN-1(β,γ,0;γ′) xyβ+γ+γ′zR+R′BN-1(β,γ,0;γ′) BN-1(β,γ,0;γ′) xyβ+γ+γ′zR+R′+1BN-1(β,γ,0;γ′)

θ0 1/2 3/8 5/12 1/2 3/10 11/30 1/2 8/15

θ0

θ

1/3 5/12 1/2 1/2 2/7 1/3 1/3 3/7 3/7 10/21 1/2 4/7 4/7 13/21 31/49

1/3 17/24 3/4 5/6 2/7 2/7 8/21 1/2 3/4 17/21 11/14 9/14 8/7 1 61/49

β 0 7/24 1/4 1/3 0 0 1/21 1/7 9/28 1/3 2/7 3/7 9/14 4/7 37/49

name C62 F61 F62 F63 C74 C75 C76 C77 C78 F72 F73 F74 F75 F76 F77

θ0

θ

β

2/3 2/3 13/18 5/6 23/35 2/3 2/3 2/3 24/35 5/7 5/7 3/4 6/7 6/7 13/14

10/9 4/3 25/18 11/6 41/35 19/21 23/21 9/7 39/35 7/5 11/7 9/7 13/7 2 2

2/9 7/9 8/9 4/3 5/7 2/3 5/7 17/21 27/35 33/35 23/21 1 31/21 11/7 12/7

ensure that all phases are observed, the temperature is kept below T0, but well above 0 K, with V restricted to be positive. An energy region corresponds to the range of values of the ratio u ) W/V for which the phases observed with increasing pressure, or V ) µ/V, remain unchanged. As explained in refs 1-3 and a number of previous publications cited therein, at this relatively low temperature, the boundary between two phases is determined when the state of occupation corresponds to a local maximum of the entropy. Let {p1} ) {θ01 θ1 β1} and {p2} ) {θ02 θ2 β2} be two consecutive phases as µ is increased. The changes in the corresponding characteristics between the two phases are indicated as ∆θ0, ∆θ, and∆β. One defines µt to be the value of µ in the transition region linearly related to V and W according to

0 ) ∆θ0 µt + ∆θV + ∆βW

(6)

For a second-order transition, 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, and that the entropy reaches a maximum at µ ) µt. An infinite specific heat capacity indicates a first-order phase transition and corresponds to a discontinuity in the coverage as a function of µ. An excellent reference on the theory of firstorder phase transitions is given in a review article by Binder.11 This article also addresses a number of computational techniques to determine whether or not a transition is first-order. We have observed a number of cases, in our numerical results here, as well as in ref 2, in which there is a “discontinuous” jump between two phases, as µ was varied, at precisely the value µ ) µt defined in eq 6. The manner in which we have numerically handled the (11) Binder, K. Rep. Prog. Phys. 1987, 50. 783.

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Table 3. Occupational Characteristics for Lattice Widths M ) 8-9 M

name

8

A81 A82 A83 A84 B81 B82 B83 C81 C82 C83 F81 A91 B91 B92 B93 F91 C91

9

θ0

θ

β

1/4 1/3 1/3 3/8 7/16 1/2 1/2 31/52 5/8 5/8 5/8 1/3 4/9 4/9 1/2 5/9 2/3

1/4 7/24 5/12 3/8 25/32 13/16 1 63/52 7/8 31/24 11/8 1/3 29/36 8/9 5/6 11/9 10/9

0 0 1/12 1/16 11/32 5/16 11/20 3/4 9/16 5/6 15/16 0 13/36 13/27 1/3 37/45 2/3

name C84 C85 C86 C87 F82 F83 F84 F85 F86 F87 F88 F92 F93 F94 F95 F96

θ0

θ

β

9/14 2/3 2/3 15/22 11/14 4/5 7/10 3/4 5/6 7/8 7/8 2/3 7/9 7/9 5/6 8/9

33/28 1 4/3 5/4 45/28 31/20 7/5 7/4 43/24 29/16 17/8 14/9 44/27 17/9 31/18 29/9

5/7 2/3 7/8 9/11 17/28 6/5 19/20 4/3 17/12 3/2 7/4 7/6 32/27 41/27 4/3 17/9

Figure 2. Phase diagram for M ) 3.

Table 4. Occupational Characteristics for Lattice Widths M ) 10-11 M name 10

11

A10 1 A10 2 A10 3 A10 4 B10 1 B10 2 B10 3 B10 4 B10 5 B10 6 C10 1 A11 1 A11 2 A11 3 A11 4 A11 5 B11 1 B11 2 B11 3 B11 4 B11 5

θ0 3/10 1/3 9/25 28/75 2/5 13/30 9/20 1/2 1/2 8/15 3/5 3/11 1/3 4/11 53/143 62/165 32/77 5/11 5/11 1/2 6/11

q 3/10 3/10 21/50 11/25 4/5 3/5 33/40 17/20 11/10 17/20 1 3/11 10/33 19/44 67/143 72/165 46/77 2/3 37/44 19/22 28/33

b 0 0 3/50 2/25 3/7 1/5 3/8 7/20 11/15 2/5 8/15 0 0 5/66 14/143 14/165 2/11 8/33 17/44 4/11 14/33

name F10 1 C10 2 C10 3 B10 7 F10 2 C10 4 F10 3 F10 4 F10 5 F10 6 F10 7 B11 6 C11 1 C11 2 C11 3 C11 4 C11 5 F11 1 F11 2 F11 3

θ0

θ

b

3/5 16/25 7/10 7/10 7/10 11/15 4/5 4/5 5/6 17/20 9/10 13/22 7/11 2/3 23/33 8/11 17/22 5/6 19/22 10/11

7/5 28/25 13/10 59/40 17/10 3/2 33/20 2 7/4 37/20 23/10 1 13/11 14/11 46/33 31/22 35/22 59/33 43/22 26/11

26/25 16/25 17/20 41/40 27/20 31/30 5/4 5/3 7/5 3/2 2 6/11 15/22 26/33 10/11 1 13/11 16/11 18/11 23/11

existence of such a “discontinuity” is as follows. If phase {p1} occurs at µ < µt - δ, and phase {p2} occurs at µ > µt + δ, then it is a first-order phase transition if δ is exactly zero. Since our numerical computations have been carried out to a great accuracy, well beyond the precision available in experiments, we have used the numerical criterion that if δ is at least 6 orders of magnitude smaller than µt, then we interpret the transition as first-order. This is very much similar to determining the instant beyond which one can “physically” claim that a charging or discharging picofarad capacitor has been fully charged or fully discharged. There is a certain arbitrariness in that choice. In our study, we were guided by what appeared to be “numerically and physically” reasonable. For T < T0 and well above 0 K, a dimensionless phase diagram is numerically generated for a given value of V, chosen here to be positive, by plotting V ) µ/V versus u ) W/V. In this plot, the boundary points between phases correspond to µ ) µt. As



follows from eq 6, the boundary between two phases that have a nonvanishing difference in coverage (∆θ0 ) 0) is a straight line of slope β/θ0 with a vertical intercept ∆θ/∆θ0. When the phases have the same coverage (∆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. In Section 2, the phase characteristics, the occupational configurations, and the energy phase diagrams are presented for finite-width terraces up to and including M ) 11. Section 3 analyzes the evolution of the phases with increasing M and leads


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Figure 5. Phase diagram forM ) 6.


to the energy phase diagram on the infinite 2-D lattice presented in Section 4. Comparison with other theoretical work is discussed in Section 5, and comparison with observed experimental data is given in Section 6. Section 7 is the summary and conclusion.

2. Phase Characteristics, Phase Diagrams, and Occupational Configurations By ignoring the difference between adsorption on edge sites and adsorption on bulk sites, we have isolated the finite width


effect. This effect is expected to gradually diminish as the width M is increased. However, a great deal of structure is still observed at M ) 11. Moreover, it becomes apparent that certain adsorption features, such as stripes and triangular clusters, appear for all terrace widths, an indication that these features will persist in the infinite two-dimensional limit. The problem is to find the number of these distinct features that survive in this limit by considering a sequence of overlapping analyses.

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The numerical results provide the occupational characteristics of the phases and energy regions as determined by the sequences in which these phases appear. A phase diagram for a given M is unique and consists of a diagrammatic representation of the energy regions in a plot of V ) µ/V versus u ) W/V. Termination of all boundaries at critical points ensures that the phase diagram is complete and that no phases have been missed. The occupational characteristics of the phases and their designations are listed in Tables 1-4 according to increasing M. The phase diagrams are shown in Figures 2-10, where the phases are indicated by their designated names as found in the tables. The equations of the boundary lines between phases are not reported on these figures since they may be obtained from the occupational characteristics of the phases on either side of the boundary (listed in the tables) with eq 6. For example, consider in the phase diagram associated with the boundary line between phases B28 ) { 1/2, 13/16, 5/16} and C18 ) {31/52, 63/52, 3/4} for M ) 8. In this case, ∆θ0 ) 31/52 - 1/2 ) 5/52, ∆θ ) 63/52 - 13/16 ) 83/208, and ∆β ) 3/4 - 5/16 ) 7/16. The slope of the boundary line follows as -(∆β/∆θ0) ) -91/20 ,and the vertical intercept -(∆θ/∆θ0) ) -83/20. The boundary between B28 ) {1/2, 13/16, 5/16} and B38 ) {1/2, 1, 11/20} is vertical, as both phases have the same coverage, corresponding to u ) -(∆θ/∆β) ) -15/19. On the other hand, the boundary line between phases A18 ) {1/4, 1/4, 0} and A28 ) {1/3, 7/24, 0}, which have the same number of second neighbors per site (here zero), is horizontal as follows from eq 6. It corresponds to V ) -(∆θ/∆θ0) ) -1/2. The boundary equations have been verified numerically with an accuracy of ten significant figures.


Knowledge of the occupational characteristics of the phases allows one to obtain the manner in which the adsorbates are distributed on the terrace or their occupational configurations. Figures 11-21 show the occupational configurations for phases with patterns that repeat over a relatively small number of horizontal rows of M sites, which is the case for most of the phases. These figures provide a visual representation of the evolution of the effects due to the edges of the terrace as its width M increases. Most importantly, we looked, following analyses similar to those used for repulsive first neighbors,1 for features of the adsorption patterns within the bulk of the terrace which


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Figure 11. Occupational configurations for M ) 3. Figure 13. Occupational configurations for M ) 5.

Figure 12. Occupational configurations for M ) 4.

remain almost unchanged with increasing width. However, for attractive first-neighbors, the edge effects appear to be more pronounced than for repulsive first-neighbors.

3. Evolution of Phases with Increasing Width M Triangular clusters are observed in phases designated as AiM where the upper index M indicates the width of the terrace and the lower index enumerates different configurations. Most of the A -type phases belong to two series valid for M g6, namely

{

1 , 3

3

}{

}

], 0 , [M3 ], [M3 ], 0 [M3 ] + [M - 3[M/3] 2 3M

M

M

(7)

where the square brackets refer to the integer part of the enclosed quantity. Both series merge to {1/3, 1/3, 0} in the infinite 2-D limit. The number of phases in which the triangular clusters are


affected by the edges and that do not belong to either series varies with M. The two series of eq 7, when evaluated at M )

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Figure 16. Further occupational configurations for M ) 7.

We have designated these phases as BM i using the same convention for the indices as above. Most of these phases belong to two series, namely

{21, 2M2M- 3, M2M- 3},

M g3

{M2M- 1, 4M4M- 7, 2M4M- 5}, Figure 15. Occupational configurations for M ) 7.

6 modulo 3, are identical. The first series, at M ) 7,8 modulo 3, has two possible occupational configurations: triangular clusters, and rows of sites at 60° with every third row of sites fully occupied. This indicates that two distinct phases with the same occupational characteristics can coexist. The remaining phases of the two series of eq 7 have only the triangular cluster pattern. Therefore, in the infinite 2-D limit, there are two possible occupational configurations with the same characteristics {1/3, 1/3, 0}, namely stripes and triangular clusters. This feature was not encountered for repulsive first-neighbors for which we found the same 2-D limit of {1/3, 1/3, 0}. There, however, the occupational characteristics at any finite width were different from those of eq 7 and realizable only when adsorbates occupied every third row of horizontal stripes with no possibility of forming triangular clusters. Consequently, according to this model, the experimental observation of triangular clusters on the infinite 2-D surface indicates that first-neighbors are attractive. However, the observation of stripes alone at one-third coverage would not be sufficient to determine whether first-neighbors are attractive or repulsive. Stripes consisting of two rows of occupied sites followed by two rows of vacant sites are also observed with some finite width effects. This was not the case when first-neighbors are repulsive.1

M g6

(8) (9)

The first is a series of horizontal stripes, and the second is a series of stripes at an angle of 60°. In the infinite 2-D limit, the orientation is irrelevant and both series merge to {1/2,1,1/2}, which is distinct from the one-half coverage observed for repulsive first-neighbors, namely {1/2,1/2,1/2}.1 The phase in the first series, evaluated at M ) 3, may also be viewed as a configuration of triangular clusters. However, since its characteristics fit the series given by eq 8, we have indicated it as belonging to the B-type series. Another distinctive pattern that is apparent in many phases is two rows of occupied sites at 60°, separated by one row of vacant sites, which are strongly affected by the finite width of the terrace. Such phases are indicated as CM i . The occupational characteristics do not belong to any particular series. The occupational characteristics of these patterns, undisturbed by the finite width effect, are calculated explicitly in terms of M as

{32, 4M3M- 5, MM- 2}

(10)

These characteristics are valid for only M ) 7 and correspond to the phase named C76. The phases with patterns closest to those given by eq 10 are designated C31, C41, C51, C61, and C71. Other phases also designated as C correspond to coverages close to 2/3, and also depend on their location in the phase diagram. Indeed, we have found it reasonable to also designate as C-type regions of the phase diagram surrounded by regions already definitely identified as C-type. Such a designation has no other justification than as an aid in following the evolution of the


Langmuir, Vol. 24, No. 1, 2008 131


Finally, a number of phases fit series which merge, in the infinite 2-D limit, to full coverage. These are designated FiM and are

{MM- 1, 3MM- 7, 3MM- 10}, M g5 {M m- 2, 3MM- 10, 9M3M- 40}, M ) 6,7,8,9,10 {MM- 3, 3MM- 13, 6M2M- 33}, M ) 7,8,10 {MM- 4, 3MM- 16, 15M5M- 98}, M ) 8,9,10 Figure 17. Occupational configurations for M ) 8.

phases with increasing M, which leads to the phase diagram in the infinite 2-D limit.

(11) (12) (13) (14)

These series of phases may not necessarily continue for values of M > 11, as the number of F-phases depends on width M, and its parity. Phases which are very close to full coverage but do not fit any of the F-series are F14, F24, F15, F25, F26, F27, F37, F57, F77, F38, F68, F78, F29, F39, F59, F310, F510, F610, and F211. The remaining few phases were designated as F based solely on their location in the phase diagrams, as mentioned above for the C-phases.

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Figure 20. Further occupational configurations for M ) 10.

) {1/3, 1/3, 0}, B ) {1/2, 1, 1/2}, and C ) {2/3, 4/3, 1}. However, as the phase diagram is unique, the final validity test of such an extrapolation is whether it leads to a phase diagram in which the boundaries between these three phases perfectly meet at critical points. This is indeed verified, as shown in Figure 22, which exhibits three triple critical points located at (-1, -1), (-1,1), and (-1/2, -3/2). All possible phases have therefore been identified under the assumption that interactions beyond second-neighbor adsorbates are negligible. A closer look at the phase diagram shows that when u < -1, all three phases are observed. In the region -1 < u < -1/2, only the B-phase is observed, and for u > -1/2, none of these phases are observed, with only one transition from empty directly to full coverage. Thus, for clusters or stripes to occur, the second-neighbor interaction must be repulsive with u ) W/V < -1/2.

5. Comparison with Other Theoretical Work


4. Phases and Phase Diagram on the Infinite 2-D Surface On the basis of the above analyses, the only phases, in addition to empty and full coverage, that appear to persist on the infinite 2-D surface are those whose occupational characteristics are A

A number of theoretical studies have been conducted on equilateral triangular lattices with attractive first-neighbors, most of which used ab initio calculations with DFT and the EAM. For example, Stasevich, Einstein, and Stolbov12 studied Cu on Cu(111) and Cu(001). The input of their model was a particular occupational configuration of various clusters and stripes on large supercells that contained up to (14 × 4 × 2) atoms for Cu(111) and (14 × 3 × 2) atoms for Cu(001). On the basis of this input and making a distinction between the two-atom steps on either side of the three- and four-atom-wide terraces, they provide energy calculations using the DFT up to third-neighbor pair interactions on a strip 14 atoms long. Our calculations consider infinitely long terraces and are carried out with a width up to and including 11 atoms. However, we did not make any distinction between step-up and step-down of the edges, nor did we consider third-neighbor interactions or specify the adsorbate or substrate. Fichthorn and Scheffler13 used combined kinetic (12) Stasevich, T. J.; Einstein, T. L.; Stolbov, S. Phys. ReV. B 2006, 73, 115426.


Langmuir, Vol. 24, No. 1, 2008 133

Chang, Wei, and Chen14 used ab initio DFT supplemented by EAM to study self-diffusion of small clusters on the (111) surface. Even though this is unrelated to what we have done, it is interesting to note that they found the compact (equilateral) triangular Al trimers more stable than both the linear trimers and the noncompact (isosceles) triangular trimers. Additional theoretical work has been directed toward the study of the energetics of clusters. For example, the energetics of small Pt clusters on Pt(111) were calculated using EAM.15,16 Our approach is different since we consider thermodynamic equilibrium and generate low-temperature energy phase diagrams without the need to specify the chemical composition of adsorbates or substrates.

6. Comparison with Experimental Data Triangular clusters have been observed in a wide variety of adsorption systems, such as I/Cu(111),17 Pt/Pt(111),18 Cr/Pt(111),19 and Pd/W(111).20 The most stable clusters contain three or six adsorbates in a triangular form. In ref 20 the piling of layers of adsorbates to form triangular base pyramids of various sizes is observed. Our model is restricted to single monolayer adsorption at thermodynamic equilibrium, but reproduces the fundamental three-adsorbate triangular cluster from which larger structures may develop. Stripes have also been observed for example in the adsorption of Ag/Cu(111),21 of Cu/Ru(0001),22 and I/Cu(111).23 The model has shown only three types of stripes, one stripe of adsorbates separated by two stripes of vacancies (A-phase), and two consecutive stripes of adsorbates separated by either one (Cphase) or two (B-phase) consecutive stripes of vacancies.

7. Summary and Conclusion


Figure 22. Phase diagram for the infinite width lattice.

Monte Carlo simulation and first-principles DFT to study the growth of Ag on a monolayer of Ag on Pt(111). Their main objective was to study island formation in thin-film epitaxy on both strained and unstrained Ag(111). They found that the pair interaction is strongly attractive at the nearest-neighbor distance but repulsive at longer distances, in agreement with our work. (13) Fichthorn, K. A.; Scheffler, M. Phys. ReV. Lett. 2000, 84, 5371.

We have presented a model of surface adsorption on infinitely long, finite width, equilateral triangular terraces, with attractive first-neighbors and attractive, repulsive, or negligible secondneighbors. There are no a priori approximations. The numerical computations were carried to better than 10 significant figures. This paper completes the study of adsorption on equilateral triangular terraces for arbitrary first- and second-neighbor interactions. The generalization to adsorption on the infinite twodimensional lattice in this case was not as straightforward as it was for repulsive first-neighbors. Additional analyses were required to establish the manner in which the phase diagram on the infinite 2-D lattice was obtained. We have not discussed first- and second-order phase transitions and temperature effects in this paper, since they lead to results similar to those presented for the repulsive case. As anticipated, the phases observed with attractive first-neighbors are not the same as those with repulsive first-neighbors. In the infinite 2-D limit, the phase with characteristics of {1/3,1/3,0} observed in the attractive case shows (14) Chang, C. M.; Wei, C. M.; Chen, S. P. Phys. ReV. Lett. 2000, 85, 1044. (15) Fallis, M. C.; Daw, M. S.; Fong, C. Y. Phys. ReV. B 1995, 51, 7817. (16) Feibelman, P. J.; Nelson, J. S.; Kellogg, G. L. Phys. ReV. B 1994, 49, 10548. (17) Andryusheshkin, B. V.; Baranovsky, R. E.; Eltsov, K. N.; Yurov, Yu, V. Surf. Sci. 2001, 488, L541. (18) Michely, T.; Hohage, M.; Bott, M.; Comsa, G. Phys. ReV. Lett. 1993, 70, 3943. (19) Zhang, L. P.; van Ek, J.; Diebold, U. Phys. ReV. B 1999, 59, 5837. (20) Nien, C.-H.; Madey, T. E.; Tai, Y. W.; Leung, T. C.; Che, J. G.; Chan, C. T. Phys. ReV. B 1999, 59, 10335. (21) Bachmann, A. R.; Mugarza, A.; Ortega, J. E.; Speller, S. Phys. ReV. B 2001, 64, 153409. (22) Gu¨nther, C.; Vrijmoeth, J.; Hwang, R. Q.; Behm, R. J. Phys. ReV. Lett. 1995, 74, 754. (23) Andryusheshkin, B. V.; Eltsov, K. N.; Shevlyuga, V. M. J. Surf. Sci. Nanotech. 2004, 2, 234.

134 Langmuir, Vol. 24, No. 1, 2008

the coexistence of two configurations, triangular clusters and stripes. The phase with the same characteristics observed in the repulsive case has only one possible configuration: stripes. Excluding empty and full coverage, the remaining two phases in the attractive case have characteristics and configurations not found in the repulsive case.

Phares et al.

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. LA702128F

Monomer Adsorption on Equilateral Triangular Lattices with Attractive

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