pubs.acs.org/Langmuir © 2009 American Chemical Society
Low Temperature Energy Phase Diagram for Adsorption on Fcc(112) Stepped Surfaces with Attractive First Neighbor Interactions Alain J. Phares,*,† David W. Grumbine Jr.,‡ and Francis J. Wunderlich† †
Villanova University, Department of Physics, Mendel Science Center, Villanova, Pennsylvania 19085-1699, and ‡ St. Vincent College, Department of Physics, Latrobe, Pennsylvania 15650-4580 Received June 2, 2009. Revised Manuscript Received July 17, 2009
The transfer matrix method developed for the study of monomer adsorption on terraces and nanotubes is applied to on top adsorption on fcc(112) stepped surfaces. The effect due to the step is taken into account by considering adsorbate-substrate interaction on step sites to be different from that on the other bulk sites. We also consider first- and second- neighbor adsorbate-adsorbate interactions, with attractive first-neighbors, thus completing the work published three years ago on repulsive first-neighbors. In the three-dimensional, low temperature energy phase diagram, other than empty and full coverage, there are eight phases: three cluster formations consisting of 4, 5, and 6 adatoms, and five stripe patterns parallel to the steps. Of the thirty-seven phases reported in the repulsive case, none exhibits clusters, and only four of them have stripes that match the ones found in the attractive case. These and other selection rules allow one to predict whether first-neighbors are attractive or repulsive, while the energy phase diagram can be used as a guide to obtain additional information on the remaining interaction energies.
1. Introduction The basic understanding of catalytic reactions on surfaces is one of the fundamental goals of surface adsorption studies. The well-known pressure gap problem is that one cannot extrapolate the experimentally obtained results in high vacuum, low temperature, to high pressure, high temperature.1 Another problem is the structure gap arising between the experimentally obtained results on ordered surfaces and commercial catalysts.2 A good starting point in addressing the structure gap is the study of crystals cut with low Miller indices, producing low coordinated sites, the steps. Adsorption phases on stepped surfaces are affected by the orientation of the terrace step, width, lattice geometry, and the chemical composition of the particles and substrate. The text by Somorjai provides an extensive review of experimental data on the subject.3 It follows that there is a direct relationship between the observed adsorption phases and the adsorbateadsorbate and adsorbate-substrate interaction energies. “Heterogeneous catalysis relies on the opening of new, fast reaction channels involving the adsorption and conversion of reactants on the surface of the catalyst. Consequently, the key to the success of catalytic processes lies in the details of the interactions between the surface and the adsorbed molecules;the adsorbates”.4 The fundamental question that our phenomenological approach intends to answer is: what are the constraints that can be obtained on the interaction energies from the experimental observation of one or more phases? This distinct point of view adds crucial information to the understanding of heterogeneous catalysis, and is complementary to other approaches such as the density *To whom correspondence should be addressed. Telephone: +1 610 519 4889. E-mail:
[email protected]. (1) Stampfl, C.; Ganduglia-Pirovano, M. V.; Reuter, K.; Scheffler, M. Surf. Sci. 2002, 500, 368. (2) Vattuone, L.; Savio, L.; Rocca, M. Surf. Sci. Rep. 2008, 63, 101. (3) Somorjai, G. A. Introduction to Surface Chemistry and Catalysis; John Wiley & Sons, Inc.: New York, 1994. (4) Zaera, F. Surf. Sci. 2002, 500, 947–965.
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functional theory,5 the embedded atom model,6 and Monte Carlo simulations.7 Section 2 is a brief review of the transfer matrix model used in the paper. Section 3 shows how the various thermodynamic properties of the system are numerically computed. Section 4 provides the numerical results and the construction of the low temperature threedimensional energy phase diagram. A study of the transitions, configurational entropy, and adsorption isotherms between phases are presented in section 5. Section 6 is the discussion and conclusion.
2. The Model The model of monomer adsorption on terraces and nanotubes of reference8 is the starting point of this study. It is phenomenological as it is based on the geometry of the adsorbate lattice and on the number of pairwise interaction energies, but does not require the knowledge of the chemical composition of the substrate surface or the adsorbate particles. The mathematical framework for the numerical computations has been developed for a number of lattice geometries with first- and second-neighbor interactions, including the fcc(112) stepped surfaces considered here. The sites occupied by the atoms of these metallic stepped surfaces form a substrate lattice of infinitely long, armchair equilateral triangular, 3-atom wide terraces, as shown in Figure 1. In standard notation,9 such a surface is denoted [3(111) (100)], meaning that it is composed of 3-atom wide (111) terraces separated by monatomic (100) step heights. Here the orientation of the (111) terraces, referred to as armchair, is such that step sites are first-neighbors. The surface is exposed to a medium which contains the particles to be adsorbed, and whose chemical potential is μ0 . The system is at thermodynamic equilibrium and at absolute temperature T. From a phenomenological point of (5) Jones, R. O.; Gunnarsson, O. Rev. Mod. Phys. 1989, 61, 689. (6) Daw, M. S.; Foiles, S. M.; Baskes, M. I. Mater. Sci. Rep. 1993, 9, 251. (7) Binder, K., Heermann, D. W. Monte Carlo Simulation in Statistical Physics: An Introduction, Springer Series in Solid-State Sciences, 1997; p 80. (8) Phares, A. J.; Grumbine, D. W., Jr.; Wunderlich, F. J. Langmuir 2007, 23, 558. (9) Lang, B.; Joyner, R. W.; Somorjai, G. A. Surf. Sci. 1972, 30, 440.
Published on Web 08/18/2009
DOI: 10.1021/la901972h
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Phares et al.
Figure 1. Stepped fcc(112) surface having terraces three-atom wide, forming an armchair equilateral triangular lattice. The atoms on the one-step down form a square lattice.
view, the binding energy of adatoms on a step site should depend on whether the site is at the top or bottom of the step. Since experimental data shows preferential adsorption on the sites at the top of a step,11-14 we treat the sites at the step bottom as bulk sites. An infinitely long terrace has two edges one consists of sites on the top of a step and the other of sites at the bottom of a step, which are treated as bulk sites. Thus, we indicate adsorbatesubstrate interaction energy at top step sites as Vs, and at all other bulk sites as Vb. The differential adsorbate-substrate interaction energy between step and bulk sites is denoted U = Vs - Vb. Adsorbate-adsorbate first- and second-neighbor interaction energies are V and W, respectively. The relevant activities are (k is Boltzmann’s constant) x ¼ exp½ðμ0 þ Vb Þ=kT, z ¼ exp½W=kT,
y ¼ exp½V=kT, u ¼ exp½U=kT
ð1Þ
The quantity μ0 + Vb is named μ and will be referred to as the chemical potential. Reference 8 provides the numerical algorithm necessary for the construction of the transfer matrix whose elements are products of the activities, each raised to a non-negative integer power. The eigenvalue of largest modulus, R(x,y,z,u), is therefore real and positive and provides the partition function of the system. The numerical results have been obtained for repulsive first-neighbors, or V < 0, and applied to the chemisorption of CO on Pt(112) as discussed in reference 10. In this article, we use the same transfer matrix, and the only difference is that first-neighbors are attractive, V > 0.
3. Determination of the Thermodynamic Properties of the System For completeness, we provide the basic equations found in ref 10 still valid in this case. The partition function, Z3, for the adsorption on armchair, infinitely long, 3-atom wide terraces is10 Z3 ¼ ½Rðx, y, z, uÞ1=6
ð2Þ
The statistical average of the coverage, θ0, the numbers per site of first- and second-neighbor adsorbates, θ and β, and the number per site of adsorbates on step sites, γ, are θ0 ¼
x DR , 6R Dx
θ ¼
y DR , 6R Dy
β ¼
z DR , 6R Dz
γ ¼
u DR ð3Þ 6R Du
The average energy per site ε, and the entropy per site divided by Boltzmann’s constant S follow as: ε ¼ μθ0 þVθþWβþγU, (10) 7646. (11) (12) (13) (14)
S ¼
1 ε lnðRÞ 6 kT
ð4Þ
Phares, A. J.; Grumbine, D. W., Jr.; Wunderlich, F. J. Langmuir 2006, 22, Henderson, M. A.; Szabo, A.; Yates, J. T., Jr. J. Chem. Phys. 1989, 91, 7245. Henderson, M. A.; Szabo, A.; Yates, J. T., Jr. J. Chem. Phys. 1989, 91, 7255. Henderson, M. A.; Szabo, A.; Yates, J. T., Jr. J. Chem. Phys. 1990, 168, 51. Xu, J.; Yates, J. T., Jr. Surf. Sci. 1995, 327, 193.
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Figure 2. Low temperature energy phase diagram in the first w-region, w < -2. There are eight triple critical points. In this figure and in Figures 3-7, the equations of the boundary lines between phases are obtained by using eq 5 and the occupational characteristics of phases Pn found in Figure 8. The diagram is generated with w = -2.4.
A state of the system is determined by its occupational characteristics {θ0, θ, β, γ}. A phase is found when the occupational characteristics remain unchanged over a wide range of values of the chemical potential μ. A phase is perfectly ordered and its entropy is zero when its occupational configuration corresponds to a unique crystallization pattern. Otherwise, the phase is partially ordered and its nonzero entropy is a local minimum. The temperature-energy phase diagram is four-dimensional with scaled axes: u = U/V, v = μ/V, w = W/V, and t = kT/V. A numerical scanning at a given t generates a three-dimensional phase diagram. At high temperatures, the system smoothly evolves from empty (E = {0, 0, 0, 0}) to full coverage (F = {1, 7/3, 5/3, 1/3}), and in between there are no crystallization patterns or phases. As the temperature is decreased, intermediate phases begin to appear, until, below a certain temperature T0, corresponding to t0 on the order of 1/100, no new phase is observed. For example, with V=20 kcal/mol, T0 =100 K. The low-temperature energy phase diagram is generated at values of T below T0. When the transition between two phases is secondorder, the entropy has a local maximum. As follows from eq 4, this maximum occurs when υ ¼ υt ¼ -
Δγ Δβ Δθ uwΔθ0 Δθ0 θ0
ð5Þ
Here Δθ0, Δθ, Δβ, and Δγ are the changes between the occupational characteristics of the two phases on either side of the boundary. Equation 5 is used to define the boundary plane between phases. There are cases for which a transition between two phases becomes first-order below some critical temperature Tc. We have numerically verified that the discontinuity occurs on the boundary plane given by eq 5. We may view the low temperature 3-D phase diagram (u, v, w) as right-handed with the w-axis as vertical. A horizontal plane corresponds to a given value of w, and a numerical scanning in both u and v generates a 2-D phase diagram, where the boundaries between phases become straight-lines. A w-region between two wboundary planes corresponds to 2-D phase diagrams having the same general structure. For repulsive first-neighbors, reference 10 identified 31 w-regions and 37 phases. The information provided there focused on the application to CO/Pt(112). Langmuir 2009, 25(23), 13467–13471
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Figure 3. Low temperature energy phase diagram in the second wregion, -2 < w < -7/4. There are 10 triple critical points and the equations of the boundary lines are obtained as explained in Figure 2. The diagram is generated with w = -1.8.
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Figure 6. Low temperature energy phase diagram in the fifth wregion, -1 < w < -1/2. The diagram is generated with w = -0.7. Phases P1, P3, and P4 have disappeared. There are two triple critical points at u = 0, with v = -3w - 3 and v = -w - 2, and a third at (u = 2w þ 2, v = -2w - 3). As w increases and approaches -1/2, the first two points eventually merge to form a quadruple critical point between E, F, P7, and P8. Beyond w = -1/2, one reaches the sixth w-region shown in Figure 7.
Figure 4. Low temperature energy phase diagram in the third wregion, -7/4 < w < -5/3. The diagram is generated with w = -1.7, and the phases are the same as those found in Figure 3. However, the critical points at - w/3 - 1/3 and at -w - 5/3 appear in reverse order. One also observes the shrinking of the region in which P5 is found, which is expected to disappear when reaching the next w-region shown in Figure 5. Figure 7. Low temperature energy phase diagram in the sixth w-region, -1/2 < w < 0. The diagram is generated with w=-0.2. Phases E, F, P2, P7, and P8 show up as in Figure 6. However the three triple critical points occur at (u=-w - 2, v=-2w - 1), at (u= -3w - 3, v=4w þ 2) and at (u = 2w þ 2, v = - 2w - 3). Here there is a direct transition from empty to full coverage which occurs in the u-subregion, -2w - 1 < u < 4w þ 2. The seventh phase diagram is the same, but with P8 removed.
Figure 5. Low temperature energy phase diagram in the fourth wregion, -5/3 < w < -1. The diagram is generated with w = -1.4. Phase P5 has disappeared, and there is a quadruple critical point at the merging of P3, P4, P7, and P8. As the boundary line v = -2w - 3 approaches v = -1, then w approaches - 1, and the regions occupied by P1, P2, P3, and P4 gradually shrink, and should disappear in the next w-region of Figure 6.
4. Numerical Results: Phases and Energy Phase Diagram In the present case of attractive first-neighbors, the low temperature 3-D energy phase diagram is less complex. Our numerical computations show seven w-regions. Figures 2-7 provide the six 2-D phase diagrams associated with w-regions, w < -2, -2 < w < -7/4, -7/4 < w 0, is not shown as it is similar to that of region six without phase P8. In the sixth and seventh regions there is a direct transition from E to F, which is not the case in the fifth region. 13470 DOI: 10.1021/la901972h
Figure 9. Configurational entropy curve of S versus v = μ/V at three temperatures, T, for the P1 f P3 transition.
5. Phase Transitions, Configurational Entropy, and Adsorption Isotherms The six w-regions of the 3-D low temperature energy phase diagram exemplified by Figures 2-7 show that there are 22 phase transitions, 16 of which are first-order and 6 of which are secondorder. These transitions are listed in Table 1 along with the occupational characteristics obtained at v = vt given by eq 5, and numerically determined to better than 9 decimal places. When a transition is first-order, there is a discontinuity in the values of the occupational characteristics at v = vt. At this transition point, the numerical computations give a zero-entropy and the occupational characteristic is the average of the corresponding characteristics of the phases on either side of the transition. That it is the average indicates the presence of an equal mixture of the two ordered phases on either side of the transition, and is therefore consistent with the zero value numerically obtained for the entropy. Of the 6 second-order transitions, the transition P1 f P3 evaluated at v=vt is the only one that has closed form occupational characteristics, namely, {7/18, 4/9, 1/18, 1/18}. The corresponding occupational configuration is obtained by removing, in the occupational configuration of phase P3 (see Figure 8), half of Langmuir 2009, 25(23), 13467–13471
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temperature predictions. This effect is also exemplified in the plot of S versus the coverage θ0, again limited to the transition region, as shown in Figure 10. Figure 11 exhibits the adsorption isotherms of θ0 versus v. As expected, one observes widening of the transition with increasing temperature. As shown in Table 1, there exists some symmetry among the remaining 5 second-order transitions. The maximum entropy in the transition E f P1, and that in the transition P1 f P7, are identical. Furthermore, adding the coverage at one maximum to that at the other yields exactly 2/3. Similarly, the maximum entropy in the transition P3 f P4 and that in the transition P4 f P5 are also identical. Adding the coverage at one maximum to that at the other in this case yields exactly 1. Finally, the transitions P3 f P4 and P3 f P7 differ only in the number of adsorbates on step sites, γ. The configurational entropy and adsorption isotherms for these transitions are very similar to those of Figures 9 and 11. Figure 10. Configurational entropy S versus coverage θ0 at three temperatures, T, for the P1 f P3 transition.
Figure 11. Adsorption isotherms of θ0 versus v = μ/V at three temperatures, T, for the P1 f P3 transition.
the adsorbates found on the step of the terrace. As a consistency check, we prove that entropy S of this occupational configuration is (1/9) ln 2, as listed in Table 1. Consider a 3-atom wide terrace of length N. For simplicity, assume N to be a multiple of 3 or 3n, with n approaching infinity. Then the number of sites in the terrace is 9n and the number of adsorbates on step sites in the P3-phase is n. The number of ways to remove half of these adsorbates is the combinatorial factor n!/[(n/2)!]2 which, with increasing values of n, approaches 2n/Γ(1/2). Thus S, the entropy per site divided by Boltzmann’s constant, approaches the quantity, (1/9n) ln(2n/Γ(1/2)). In the infinite-n limit, one obtains (1/9) ln 2. To complete the study of this transition, we provide in Figure 9 the configurational entropy curve of S versus v = μ/V with increasing temperature T, limited to the transition region. For definiteness, in this plot and the plots shown in Figures 10 and 11, we choose a typical value of first-neighbor interaction energy V= 15 kcal/mol, to compute T given by (V/k)t. As one should expect, Figure 9 shows a widening of the entropy curve, as its maximum maintains the same value of (1/9) ln 2 over a wide range of the temperature. It is only when the temperature exceeds about 300 K that one begins to observe a slight deviation from the low
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6. Discussion and Conclusion Consider the case for which adsorbate-adsorbate interaction energies are independent of the coverage. Then, the differential energy parameter u and the interaction energy parameter w remain unchanged. Varying v corresponds to a change in the chemical potential of the particles in the surrounding medium. If the medium is a gas, this is achieved by changing the pressure, if a liquid by changing the concentration. As the system must be in one of the six w-regions, its evolution by changing v is represented in the energy phase diagram by a displacement parallel to the v-axis. As follows from the low temperature energy phase diagrams, when it is feasible to experimentally observe the phases encountered with increasing chemical potential, and to measure the chemical potentials at which the transitions between phases occur, the model predicts the values of the interaction energies. If the adsorbate-adsorbate interaction energies of the system depend on the coverage, the model can still be applied. However, the evolution of the system is no longer represented by a simple displacement parallel to the v axis. This low temperature model also provides a number of selection rules. For example, the direct transition from empty to full coverage without any intermediate phase does not occur in the repulsive case,10 but only when first-neighbors are attractive and in the sixth w-region, -1/2 < w < 0, with u in the range (-2w - 1, 4w þ 2). The fact that CO is preferentially adsorbed on the steps of Pt(112), with repulsive CO-CO first-neighbors,11-14 is what justified our study presented in reference 10. In that case, the model showed that the filling of step sites proceeds in stages before the filling of the bulk sites, and this is precisely what is experimentally observed. The filling of step sites first is also predicted when first-neighbors are attractive, this is phase P2, which appears for u > 0 and w < -1, as shown in Figures 2-5. However, the complete filling of the step sites occurs with no intermediate phase between the E phase and P2. This shows the self-consistency of the model in complete agreement with experimental observation. Acknowledgment. This research was supported in part by Grant No. CHE050015N from the Pittsburgh Supercomputing Center, supported by several federal agencies, the Commonwealth of Pennsylvania, and private industry.
DOI: 10.1021/la901972h
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