Phenomenological Study of Monomer Adsorption on fcc - American

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Phenomenological Study of Monomer Adsorption on fcc (335) Surfaces With Application to CO, O, and N2 Adsorption on Pt(335) Alain J. Phares,*,† David W. Grumbine, Jr.,‡ and Francis J. Wunderlich† Department of Physics, Mendel Science Center, VillanoVa UniVersity, VillanoVa, PennsylVania 19085-1699, and Department of Physics, St. Vincent College, Latrobe, PennsylVania 15650-4580 ReceiVed August 27, 2008. ReVised Manuscript ReceiVed October 10, 2008 We extend our recent study of adsorption on fcc (112) to fcc (335) surfaces, still considering only first- and secondneighbor interactions with repulsive first-neighbors. We consider the adsorbate-substrate interaction on the step sites of one of the two edges of the infinitely long terraces to be different from that on the remaining sites. The adsorption features on fcc (335) surfaces are richer than those on fcc (112), which can be attributed to the fact that the equilateral triangular terraces are now four-atoms wide rather than three. Our approach is independent of the chemical composition of the substrate and adsorbates and consequently may be applied to a variety of adsorption systems on fcc (335) surfaces which satisfy the limitations of our model. The basic 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 question is answered in the cases of CO, O, and N2 adsorbed on Pt(335).

1. Introduction Experimental observations of preferential adsorption on fcc (335) step sites have been extensively reported. In the case of adsorption on Pt (335), step adsorption occurs with adsorbates such as carbon monoxide,1-6 oxygen,7-10 hydrogen,11,12 nitrogen,13-15 nitric oxide,16-18 gold,19 copper, and cadmium.20 This is the motivation for choosing the adsorbate-substrate interaction energy at step sites to be different than at other sites. We intend to show that our phenomenological approach, which does not specify the chemical composition of substrate or adsorbates, enables one to obtain a number of restrictions on the interaction energies between adsorbates and between adsorbates and substrate from the knowledge of the experimentally observed phases. * To whom correspondence should be addressed. Phone: +1 610 519 4889. E-mail: [email protected]. † Villanova University. ‡ St. Vincent College.

(1) Xu, J.; Henriksen, P.; Yates, J. T., Jr. J. Chem. Phys. 1992, 97, 5250. (2) Jøansch, H. J.; Xu, J.; Yates, J. T., Jr. J. Chem. Phys. 1993, 99, 721. (3) Xu, J.; Yates, J. T., Jr. J. Chem. Phys. 1993, 99, 725. (4) Xu, J.; Yates, J. T., Jr. Surf. Sci. 1995, 327, 193. (5) Wang, H.; Tobin, R. G.; Lambert, D. K.; Fisher, G. B.; DiMaggio, C. L. J. Chem. Phys. 1995, 103, 2711. (6) Skelton, D. C.; Tobin, R. G.; Lambert, D. K.; DiMaggio, C. L.; Fisher, G. B. J. Phys. Chem. B 1999, 103, 964. (7) Heyd, D. V.; Scharff, R. J.; Yates, J. T., Jr. J. Chem. Phys. 1999, 110, 6939. (8) Tripa, C. E.; Yates, J. T., Jr. J. Chem. Phys. 2000, 112, 2463. (9) Gee, A. T.; Hayden, B. E. J. Chem. Phys. 2000, 113, 10333. (10) Tripa, C. E.; Yates, J. T., Jr. J. Chem. Phys. 2001, 115, 8552. (11) Gee, A. T.; Hayden, B. E.; Mormiche, C.; Nunney, T. S. J. Chem. Phys. 2000, 112, 7660. (12) Gee, A. T.; Hayden, B. E.; Mormiche, C.; Nunney, T. S. Surf. Sci. 2002, 512, 165. (13) Tripa, C. E.; Zubkov, T. S.; Yates, J. T., Jr.; Mavrikakis, M.; Nørskov, J. K. J. Chem. Phys. 1999, 111, 8651. (14) Tripa, C. E.; Zubkov, T. S.; Yates, J. T., Jr. J. Phys. Chem. B 2001, 105, 3724. (15) Zubkov, T. S.; Tripa, C. E.; Yates, J. T., Jr. J. Phys. Chem. B 2001, 105, 3733. (16) Janssen, N. M. H.; Cobden, P. D.; Nieuwenhuis, B. E. J. Phys.: Condens. Matter 1997, 9, 1889. (17) Wang, H.; Tobin, R. G.; DiMaggio, C. L.; Fisher, G. B.; Lambert, D. K. J. Chem. Phys. 1997, 107, 9569. (18) Backus, E. H. G.; Eichler, A.; Grecea, M. L.; Kleyn, A. W.; Bonn, M. J. Chem. Phys. 2004, 121, 7946. (19) Skelton, D. C.; Wang, H.; Tobin, R. G.; Lambert, D. K.; DiMaggio, C. L.; Fisher, G. B. J. Phys. Chem B 2001, 105, 204. (20) Prinz, H.; Strehblow, H-H. Electrochim. Acta 2002, 47, 3093.

The general technique for dealing with surface adsorption, whether on terraces or on nanotubes with the same lattice geometry, has been recently presented in ref 21 and applied to equilateral triangular terraces22-25 and nanotubes.26 The geometry of fcc (112) and fcc (335) surfaces belongs to the family of infinitely long equilateral triangular terraces, any number M of atomic sites in width, separated by single (100) steps. The sites at the bottom or top of a step are first-neighbors. We refer to such terraces as “armchair” equilateral triangular, as opposed to the “zigzag” case in which edge sites are second-neighbors.21-26 Thus fcc (112) and fcc (335) are armchair equilateral triangular lattices with M ) 3 and 4, respectively. Figure 1 illustrates a section of a general armchair terrace as viewed from the top. The following is a summary of the notation used in this study. The surface is exposed to a gas at a pressure under which the chemical potential energy of a particle is µ′. The system is at thermodynamic equilibrium and absolute temperature T. Figure 1 indicates that the right edge sites of the terrace are at the top of a step-down, to which we refer as “step” sites; the remaining sites are “bulk” sites. There is often preferential adsorption either on the step-down edge sites or on the remaining (bulk) sites. Thus the adsorbate-substrate interaction energy on step sites, Vs, is, in general, different from the adsorbate-substrate interaction energy on bulk sites, Vb, as shown in Figure 1. Our investigation considers the commonly occurring situation of a repulsive first-neighbor adsorbate-adsorbate interaction energy V, with arbitrary second-neighbor interaction energy W. It is convenient to introduce the shifted chemical potential energy µ, (21) Phares, A. J.; Grumbine, D. W., Jr. ; Wunderlich, F. J. Langmuir 2007, 23, 558. (22) Phares, A. J.; Grumbine, D. W., Jr. ; Wunderlich, F. J. Langmuir 2006, 22, 7646. (23) Phares, A. J.; Grumbine, D. W., Jr. ; Wunderlich, F. J. Langmuir 2007, 23, 1928. (24) Phares, A. J.; Grumbine, D. W., Jr. ; Wunderlich, F. J. Langmuir 2008, 24, 124. (25) Phares, A. J.; Grumbine, D. W., Jr. ; Wunderlich, F. J. Phys. Lett. A 2007, 366, 497. (26) Phares, A. J.; Grumbine, D. W., Jr.; Wunderlich, F. J. Langmuir, 2008, 24, 11722.

10.1021/la802800y CCC: $40.75  2009 American Chemical Society Published on Web 12/16/2008

Monomer Adsorption on fcc (335) Surfaces

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µ ) µ ′ + Vb

of four occupational characteristics: the coverage θ0, the number per site of adsorbate first-neighbors θ, the number per site of adsorbate second-neighbors β, and the number per site of adsorbates on step sites γ. These are the following:

(1)

and the difference U between adsorbate-substrate interaction energies at step and bulk sites,

U ) Vs - Vb

(2)

θ0 )

The activities x, y, z, and s associated with µ, V, W, and U are

x ) exp[µ/kT],

y ) exp[V/kT],

V ) -µ/V,

w ) -W/V

θ)

1 ∂Z ; Z ∂y

β)

1 ∂Z ; Z ∂z

γ)

1 ∂Z (5) Z ∂s

A state of the system is specified by its occupational characteristics (θ0, θ, β, γ), and its average energy per site is

z ) exp[W/kT], s ) exp[U/kT] (3)

) µθ0 + Vθ + Wβ + Uγ

where k is Boltzmann’s constant and we have used the convention that negative energies correspond to repulsive forces. In this article, we consider only V < 0 and scale the other energies relative to the positive quantity, -V. This introduces dimensionless energy parameters u, V, and w, associated with U, µ, and W, as

u ) -U/V,

1 ∂Z ; Z ∂x

(6)

In turn, the entropy per site divided by Boltzmann constant for this state is given by

S ) ln Z - /kT

(7)

Our study of adsorption on fcc (112) terraces 3-atoms wide2 is now extended to fcc (335) terraces, 4-atoms wide. For completeness, section 2 summarizes the results and computational method of the general technique of ref 21 as it applies to armchair equilateral triangular terraces M atomic sites in width. Section 3 presents the systematic construction of the low temperature phase diagram. Sections 4, 5, and 6 apply the model to the adsorption of CO, O, and N2 on Pt(335), and section 7 is the summary and discussion.

(4)

When u is negative (positive), adsorption occurs preferentially on bulk (step) sites. A change in V indicates a change in chemical potential energy or gas pressure. A negative (positive) w corresponds to repulsive (attractive) second-neighbors. The partition function of the system, Z, expressed in terms of the four activities x, y, z, and s, provides the statistical average

2. Partition Function and Computational Method The partition function of the adsorption system under consideration is derived from a transfer matrix as shown in ref 21. This matrix is constructed recursively from two sets of square matrices, called AN(a, b, c, d; c′ d′) and BN(c, d, e, f; f′). Index N refers to the rank of these matrices, which is 4N. The arguments may take on the values 0 or 1. Therefore, for any N, there are 26 A-type matrices and 25 B-type matrices. They satisfy (26 + 25) initial conditions,

A0(a, b, c, d;c′, d′) ) 1,

B0(c, d, e, f;f′) ) 1

(8)

and are recursively related according to the same set of matrices of lower rank, as follows:

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

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

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

xyc + d za+d+c ×BN-1(c, d, 1, 0;0)

zc ×BN-1(c, d, 0, 0;1)

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

zc ×BN-1(c, d, 0, 0;1)

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

AN(a, b, c, d;c ′ , d ′ ))

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

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

xyc+dzb+d ×BN-1(c, d, 0, 1;0)

xyc+dzb+c+d ×BN-1(c, d, 0, 1;1)

xyc+dzb+c+d ×BN-1(c, d, 0, 1;1)

xye+fzc+f′

x2y2+2c+d+d za+b+c+d+c +d ×BN-1(c, d, 1, 1;1)

x2y2+2c+d+d za+b+c+d+c +d ×BN-1(c, d, 1, 1;1)

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

×AN-2(e, f, 1, 0;0, 0)

xyfzd+e ×AN-2(e, f, 0, 1;0, 0)

e

z ×AN-2(e, f, 0, 0;1, 0)

xye+fzc+e+f′ ×AN-2(e, f, 1, 0;1, 0)

xyfzd+2e ×AN-2(e, f, 0, 1;1, 0)

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

yezf ×AN-2(e, f, 0, 0;0, 1)

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

xye+fzd+e+f ×AN-2(e, f, 0, 1;0, 1)

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

yeze+f ×AN-2(e, f, 0, 0;1, 1)

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

xye+fzd+2e+f ×AN-2(e, f, 0, 1;1, 1)

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

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

BN-1(c, d, e, f;f ′ ) )

xyc+dzb+d ×BN-1(c, d, 0, 1;0)

(9)

(10)

If the terrace is M atomic sites in width and if the step effect is neglected (s ) 1), the transfer matrix is AM(0, 0, 0, 0; 0, 0). To account for the step effect, a number of elements in this latter matrix are modified as follows: • a factor s multiplies all the elements of AM in the columns numbered (2 + 4p) and (3 + 4p), with p ) 0, 1,..., 4M-1; and • a factor s2 multiplies all the elements of AM in the columns numbered (4p), with p ) 1,..., 4M-1.

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Since all of the elements of this modified transfer matrix are real and non-negative, its eigenvalue of largest modulus, R(x, y, z, s), is real and positive. As shown in ref 21 when the terrace is M atomic sites in width and infinitely long, R is the only contribution to the partition function of the system, which is then

Z ) R1/(2M)

(11)

In ref 22, the calculations were carried out for M ) 3. Here, M ) 4 and the occupational characteristics of the system and its entropy are obtained by computing R and its partial derivatives with respect to the four activities as follows from eqs 5-7. The computations were carried out with long double precision arithmetic, ensuring at least 10 significant figures.

3. Systematic Construction of the Low Temperature Energy Phase Diagram The dimensionless energy phase diagram is viewed as a righthanded 3D space (u, V, w). The (uV)-plane is horizontal and the w-axis is vertical. Thus a horizontal plane corresponds to a fixed value of the energy parameter w. A value of u is then chosen in such a plane and a numerical scan is performed by changing V. A sequence of phases is observed, starting from empty and reaching full coverage. A u-region is defined as the range of values of u for which the phases encountered, and the sequence in which they appear, are the same. The boundaries of a u-region depend on the chosen value of w. For the chosen value of w, all of the u-regions are then identified and a 2D (u, V) energy phase diagram is generated which is the intersection of the full 3D phase diagram with the chosen w-plane. When the transition between phases is second-order, with differential characteristics ∆θ0, ∆θ, ∆β, and ∆γ, the entropy has a local maximum when

V(∆θ0) - (∆θ) + w(∆β) + u(∆γ) ) 0

this point is undefined. In the 3D phase diagram, whether the transition between phases is first- or second-order, the boundary between two phases is defined by eq 12 and is therefore planar. Since the 2D phase diagram mentioned above is a horizontal section of the 3D diagram, a boundary between two phases in this 2D plot is a straight line, and boundary lines intersect at critical points as shown in Figures 2 and 3. These figures are horizontal sections for w in the range (-1/17, - 1/18) and for w ) 0, respectively. A w-region is defined as the range of values of w for which the u-regions are the same. There are 120 w-regions; 43 with

(12)

This follows from eqs 6 and 7, as derived and verified numerically in our previous adsorption studies. When the transition is firstorder, there is a discontinuity in the plot of coverage versus V which occurs at the value of V given by eq 12. The entropy at

Figure 2. 2D plot of a horizontal section of the 3D phase diagram in the w-region -1/17 < w < -1/18 at w ) -0.17/3.

Figure 1. Top view of an armchair terrace with associated notations.

Figure 3. 2D plot of a horizontal section of the 3D phase diagram at w ) 0.


w < 0, and 77 with w > 0. For convenience, the horizontal w-plane boundaries of these regions in the 3D phase diagram are listed here: -1, -3/4, -2/3, -5/8, -3/5, -4/7, -5/9, -6/11, -7/13, -8/15, -1/2, -3/7, -2/5, -1/3, -2/7, -1/4, -1/5, -2/11, -1/ 6, -2/13, -1/7, -2/15, -5/39, -1/8, -1/9, -2/19, -1/10, -5/ 53, -1/11, -1/12, -1/13, -1/14, -1/15, -1/16, -1/17, -1/18, -1/19, -1/21, -1/22, -1/23, -1/25, -1/31, 0, 2/37, 3/41, 2/25, 3/37, 1/12, 1/11, 1/10, 1/9, 5/43, 2/17, 3/25, 1/8, 2/15, 1/7, 2/13, 3/19, 2/11, 12/61, 1/5, 12/55, 8/33, 1/4, 4/15, 5/18, 2/7, 5/17, 3/10, 6/19, 8/25, 1/3, 11/32, 6/17, 4/11, 11/30, 3/8, 8/21, 5/13, 2/5, 3/7, 4/9, 14/31, 6/13, 1/2, 18/35, 8/15, 22/41, 17/31, 16/29, 5/9, 9/16, 4/7, 3/5, 12/19, 2/3, 17/25, 16/23, 7/10, 19/27, 5/7, 17/23, 4/5, 8/9, 1, 47/37, 18/13, 2, 19/9, 5, 79/13, 59/9, 11, 12, 15, 214/13, 21, 36. These w-regions are ordered sequentially from w1 (w < -1) to w120 (36 < w). There are 84 phases in addition to empty and full coverage. Their occupational characteristics are listed in Table and are ordered first by increasing coverage (θ0), then by increasing numbers of first neighbors (θ), followed by increasing numbers of second neighbors (β), and finally by increasing number of occupied step-sites (γ). Following this order, they are labeled pi with i ranging from 1 to 84. Fourteen of these phases are partially ordered, corresponding to i ) 11, 21, 23, 24, 29, 30, 31, 32, 48, 49, 52, 62, 64, and 65. The nonzero entropies of these phases are listed in the table as closed form expressions which agree with the values obtained numerically, using long double precision arithmetic, to better than 10 significant figures. Occupational configurations of the phases show a pattern that repeats after a specified number of terrace rows, each containing four atomic sites, represented by circles in Figures 4-9. An occupied site is a shaded circle. Figures 4-9 provide the occupational configurations of 60 of the 84 phases, limited to the portion of the terrace that repeats. In that sample, there are a number of nonunique occupational configurations. This is the case of p11 shown in Figure 5. An arc linking two sites, one vacant and the other occupied, refers to the possible exchange of the state of occupancy of these sites without affecting the occupational characteristics of the phase. The portion of the terrace that repeats consists of three rows where each portion has two possible occupational configurations. There are 4n sites in n rows, and (n/3) sections with the same pattern, for a total of 2(4n/3) configurations with the same occupational characteristics. Thus, the entropy per site of this state divided by Boltzmann’s constant, S, is the natural logarithm of the number of configurations divided by the number of sites. This is (1/12) ln(2) which is independent of n and is valid for the infinitely long terrace, as numerically obtained. This provides a consistency check of our numerical computations. The 120 w-regions may be divided into 40 subgroups. The w-regions containing the same phases form a subgroup. One such subgroup consists of six regions in the range 0 < w < 1/11. There are 43 pi-phases in this subgroup corresponding to i ) 1, 4, 5, 6, 9, 11, 17, 19, 20, 21, 23, 24, 25, 31, 32, 33, 35, 36, 37, 39, 41, 44, 48, 49, 51, 52, 53, 55, 60, 61, 62, 65, 66, 68, 69, 72, 73, 75, 78, 79, 81, 82, and 84. A partially ordered phase with an occupational configuration very close to that of a p(2 × 2) is p24 shown in Figure 6. In the 3D phase diagram, we have determined that this phase is surrounded by 14 pi-phases, with i ) 6, 9, 11, 15, 22, 23, 25, 30, 31, 32, 33, 35, 38, and 66. The planar boundaries of the region of the 3D phase diagram containing phase p24 may be analytically determined using eq 12, thus obtaining bounds on the interaction energies. In the following sections, we present

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Figure 4. Occupational configurations p1 to p9.

some of the cases for which the available experimental data is sufficient and appropriate for analysis with our model.

4. Adsorption of CO on Pt(335) Experimental observation of CO on stepped platinum surfaces shows preferential adsorption on the steps, as well as first-neighbor CO-CO repulsion.27 In addition, Xu, Henriksen, and Yates, Jr., made the following observations on Pt(335) using infrared reflection absorption spectroscopy (IRAS):1 At low coverages... the band at ≈2072 cm-1 is assigned to CO terminally bonded (27) Henderson, M. A.; Szabo`, A.; Yates, J. T., Jr. Chem. Phys. Lett. 1990, 168, 51.

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to step sitessCO(S). As coverage increases... two bands are observed in this region with the higher frequency component being assigned to CO terminally bonded to terrace sitessCO(T). [Here, they cite refs 28, 29 and 30.] Upon further increasing the coverage, the CO(T) band absorbance continues to increase while that of CO(S) decreases due to mode coupling between the species. [Here, they cite ref 31.] This stepwise site occupancy of terminal CO is in good agreement with other studies. [Here, they cite refs 28-30 and 32.] To avoid confusion, we have mentioned earlier that the sites in a 4-atom wide terrace are divided into step-sites (28) Hayden, B. E.; Kretzschmar, K.; Bradshaw, A. M.; Greenler, R. G. Surf. Sci. 1985, 149, 394. (29) Reutt-Robey, J. E.; Doren, D. J.; Chabal, Y. J.; Christman, S. B. J. Chem. Phys. 1990, 93, 9113. (30) Henriksen, P.; Xu, J.; Yates, J. T., Jr., in preparation. These are refs 2 and 3.


and bulk-sites (terrace sites). Thus, the terminology CO(T) of ref 1 refers to CO terminally bonded to bulk-sites, and we interpret the above observation as providing the evidence that the sequence of phases with increasing coverage begins with p1 ) {1/8, 0, 0, 1/8}. According to our phenomenological model, this phase has the lowest coverage where CO molecules are terminally bonded to every other step-site. As the coverage increases, the experimental evidence indicates that there is a decrease in step-site occupancy and an increase in bulk-site occupancy. In the 3D phase diagram, p1 has six adjacent phases: E ) {0, 0, 0, 0}, p3 ) {1/4, 0, 0, 1/8}, p5 ) {1/4, 1/4, 0, 1/4}, p6 ) {7/24, 0, 5/24,


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0 < w < (3/25) with (8/3)w < u < (39/4)w (3/25) < w < (2/7) with (8/3)w < u < 1 + 5w (2/7) < w with (8/3)w < u < (8/3)w + (5/3)

(13) (14) (15)

This shows the specific ranges of the interaction energies for which adsorbate-adsorbate second-neighbors are attractive, while the adsorbate-substrate interaction energy at step-sites is greater than at bulk-sites. In addition, the model shows that the sequence E f p1 f p9 f p24 is also found in the same energy range, where p24 is the partially ordered phase (5/12, 1/6, 2/3, 1/6). As compared to the occupational configuration of p9, that of p24 has an additional CO molecule which occupies one of the two vacant step-sites located between any two occupied step-sites of p9. This is exhibited in Figure 5 and explains why p24 is a partially ordered


1/8}, p9 ) {1/3, 0, 2/3, 1/12}, and p11 ) {1/3, 1/12, 1/4, 1/6}. The model suggests that the only phase following p1 that has the experimentally observed properties is p9 ) {1/3, 0, 2/3, 1/12}, shown in Figure 4. In this figure, the CO molecules are terminally bonded to every third step-site and form a (3 × 3) R30° configuration with the CO molecules terminally bonded to bulksites. It is then reasonable to interpret the observations of ref 1 as evidence of the existence of the sequence E f p1 f p9. According to our model, this sequence is found in the regions of the 3D energy phase diagram delimited by


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information on the CO migration from bulk- to step-sites: “The experiment was carried out as follows: First the step sites are completely filled with 13C18O. Then about 1/4 monolayer coverage of the terrace sites is produced by adsorbing 12C16O at a very low crystal temperature (73 K).” Our model does not make a distinction among isotopes. The filling of the step-sites is 1/4-coverage and the filling of the bulk-sites is 1/4-coverage for a total of 1/2-coverage, thus leading us to believe that this corresponds to the p38-phase. In other words, we conclude that the sequence E f p1 f p9 f p24 f p38 must hold for CO absorbed on Pt(335). In this case, the energy ranges of (13)-(15) are reduced to

(3/25) < w < (1/4) with 1 < u < 1 + 5w (16) (1/4) < w < (2/7) with (8/3)w + (1/3) < u < 1 + 5w (17) (2/7) < w with (8/3)w + (1/3) < u < (8/3)w + (5/3) (18) Consider the four transitions (E f p1), (p1 f p9), (p9 f p24), and (p24 f p38) which occur at pressures corresponding to chemical potential energies µ1′, µ2′, µ3′, and µ4′. At these values of the chemical potential, the energy parameter V takes on the values V1, V2, V3, and V4, respectively. In the definitions stated in (1), (2), and (4), we make the assumption that interaction energies do not vary with coverage, at least in the range of values considered. Taking this into account, as well as (12) that holds at every transition, we arrive at the following conditions, listed in the order in which the transitions appear: • V1 ) - u, leading to µ1′ + Vb ) -(Vs - Vb), or

Vs ) -µ1′

(19)

• V2 ) (1/5)u - (16/5)w, leading to 5(µ2′ + Vb) ) -(Vs - Vb) - 16W, which, combined with (19), gives

4Vb + 16W ) µ1′ - 5µ2′

(20)

• V3 ) -u + 2, leading to µ3′ + Vb ) -(Vs - Vb) - 2V, which, combined with (19), yields

V ) (µ1′ - µ3′)/2

(21)

• V4 ) -u + 3, leading to µ4′ + Vb ) -(Vs - Vb) - 3V. This equation combined with (21) does not provide any additional information but can be used as a consistency check which verifies that

µ4′ ) (3µ3′ - µ1′)/2


phase. This may also explain why this phase is difficult to observe experimentally. In any case, the prediction of the model is that, as coverage is increased beyond the p9-phase, the additional CO molecules are adsorbed on step-sites. Further predictions are made as the CO coverage continues to increase. The model suggests two possibilities for the adsorption sites of the additional CO molecules: either CO(T) reaching p66 ) {2/3, 5/6, 4/3, 1/6}, or CO(S) reaching p38 ) {1/2, 5/12, 2/3, 1/4}. The site-occupancy of p66 is a honeycomb structure and that of p38 is (3 × 3) R30° in the bulk of the terrace, with all step-sites occupied. We believe that this latter phase has been observed in ref 2, where CO isotopes have been used to provide

(22)

From the experimental measurement of the chemical potential energies at the first three transitions, µ1′, µ2′, and µ3′, the model predicts the value of the adsorbate-substrate interaction energy at step-sites Vs, (19), the first-neighbor adsorbate-adsorbate interactionenergyV,(21),andarelationshipbetweenadsorbate-substrate energyatbulk-site,Vb,andthesecond-neighboradsorbate-adsorbate interaction energy, W, (20). The numerical values of the occupational characteristics at the first transition fit, to better than 10 significant figures, closed form expressions in terms of the golden ratio, φ ) (1 + 5)/2, namely,

θ0 ) γ ) 1/4(φ + 2),

θ ) β ) 0,

S ) (1/4) ln(φ) (23)

5. Adsorption of Atomic Oxygen on Pt(335) Gee and Hayden9 have reported that atomic oxygen on Pt(335) is adsorbed first on the step, occupying every other site. As mentioned earlier, this is phase p1 ) {1/8, 0, 0, 1/8}. As coverage increases, atomic oxygen adsorbs on bulk-sites to “form a (2 × 2) ordered overlayer with oxygen in threefold sites.” They conclude: “The saturation coverages of atomic oxygen at step


Langmuir, Vol. 25, No. 2, 2009 951

and terrace have been calculated from the relative peak areas of the recombinative desorption peaks β1 and β2 giving θsat(step) ) 0.12 ML and θsat(terrace) ) 0.13 ML. The β2 saturation of 0.12 ML is equivalent to filling approximately half the step sites with oxygen adatoms.” This is phase p3 ) {1/4, 0, 0, 1/8}. The sequence E f p1 f p3 occurs in the following region:

w < 0 and 0 < u < 2

6. Molecular Nitrogen Chemisorption on Pt(335) Here the experimental evidence13-15 is that molecular Nitrogen “chemisorbs on stepped Pt(335) and Pt(779) surfaces at 88 K with exclusive occupancy of step sites.”14 At this temperature, ref 14 mentions: “The saturation coverage on the steps of the Pt(335) surface is 0.38 N2/Pt atom; on Pt(779), the saturation coverage is 0.52 N2/Pt atom.” This article also states: “At a higher partial pressure of nitrogen, the N2 saturation coverage on both stepped surfaces can be increased due to the availability of empty step sites.” Thus, while N2 partial pressure is increased, starting from zero coverage, our model predicts that the low temperature phases, having only step-sites occupied, are p1 ) {1/8, 0, 0, 1/8} and p5 ) {1/4, 1/4, 0, 1/4}. The regions of the 3D energy phase diagram where the sequence (E f p1 f p5) occurs are the following: • w < 0 and u > 2: this is repulsive second-neighbors with

(25)

• w > 0 and u > 2 + (3/2)w: this is attractive second-neighbors with

Vs - Vb > -2V + (3/2)W

(26)

Let µ1′ and µ2′ be the chemical potential energies at the first and second transition, respectively. The first transition (E f p1) has already been analyzed in the previous examples and (19) holds again. The second transition (p1 f p5) occurs at V2 ) -u + 2, which is a condition discussed earlier, leading to an expression similar to (21):

(27)

The occupational characteristics at the second transition have also been obtained in terms of the golden ratio, namely:

θ0 ) γ ) (φ + 1)/4(φ + 2),

θ ) φ/4(φ + 2), β ) 0, S ) (1/4) ln(φ) (28)

7. Summary and Discussion

(24)

This situation is drastically different than CO-adsorption, as it requires second-neighbors to be repulsive. The first transition, (E f p1), is the same as before and the knowledge of µ1′ at this transition provides the value of the adsorbate-substrate interaction energy at step-sites, given by (19). However, as follows from (12), the second transition, (p1 f p3), occurs at V2 ) 0 ) (µ2′ + Vb). In this case, the knowledge of the chemical potential energy at the transition provides the adsorbate-substrate interaction energy at bulk-sites. A lower bound on first-neighbor repulsion energy is derived from (24), namely, 0 < (Vs - Vb) < -2V, or 0 < (µ2′ - µ1′) < -2V.

Vs - Vb > -2V

V ) (µ1′ - µ2′)

We have demonstrated that our phenomenological model providesconstraintsontheadsorbate-substrateandadsorbate-adsorbate interaction energies based on experimental observations. The three cases considered were selected based on the amount of experimental information available and their diversity to properly exhibit the applications of the model. In the case of CO/Pt(335), the binding energy of CO(T) has been reported to be about 55 kJ/mol lower than that of CO(S),2,33 or Vs - Vb ) 55 kJ/mol. The value of Vs is determined from the pressure (or equivalently, the chemical potential energy µ1′) at which the transition (E f p1) occurs, as noted in (19). In turn, this gives the value of Vb. Subsequently, from the knowledge of Vs and Vb, the measurement of the chemical potential energy µ2′ at the transition (p1 f p9) provides the value of the secondneighbor interaction energy W using (20). The structure of the phase reported to follow p1 was qualitatively described as a depletion of CO(S) and an increase in CO(T). Following p1, our model shows that p9 is the only phase that has these features, and explicitly provides the (3 × 3) R30° structure. Following p9, the model also predicts the existence of the partially ordered phase p24, which is accompanied by an increase in CO(S) with no changes in CO(T). Should this phase be observed and the chemical potential energy µ3′ at the transition (p9 f p24) be measured, then the first-neighbor interaction energy V would follow from (21). The knowledge of the chemical potential energies at the transitions between the phases identified experimentally in the adsorption of O and N2 on Pt(335) is again crucial in determining some of the interaction energies and in obtaining bound on the others. 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 Cray XT3 BigBen at the Pittsburgh Supercomputing Center. LA802800Y (31) Leibsle, F. M.; Sorbello, R. S.; Greenler, R. G. Surf. Sci. 1987, 179, 101. (32) Henderson, M. A.; Szabo`, A.; Yates, J. T., Jr. J. Chem. Phys. 1989, 91, 7255. (33) McClellan, M. R.; Gland, J. L.; McFeeley, F. R. Surf. Sci. 1981, 112, 63.

Phenomenological Study of Monomer Adsorption on fcc - American

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