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Langmuir 1997, 13, 1162-1167
Models for Site Adsorption of an Associating Fluid on Crystalline Surfaces† Douglas Henderson,‡ Stefan Sokołowski,*,§ and Orest Pizio| Department of Chemistry and Biochemistry, Brigham Young University, Provo, Utah 84602, Faculty of Chemistry, Maria Curie-Sklodoska University, Lublin 20013, Poland, and Instituto de Quimica de la UNAM, Coyoacan 04510, Mexico D.F., Mexico Received December 8, 1995. In Final Form: June 4, 1996X A model for a site adsorption of an associating fluid on the crystalline surfaces is studied. The (100) and (111) lattice surfaces are considered. An integral equation method is used to obtain the density profiles and investigate adsorption. It is shown that the associative interaction between the bulk fluid particles and reactive atoms of the crystalline surface leads to a formation of surface complexes. They include monomeric and dimeric species of the bulk fluid and the atoms of the exposed solid plane. Adsorption on the on-top, 2-fold bridging site and the 4-fold hollow site positions is studied for the case of the (100) lattice surface. The complexes, which include one, two, and three surface atoms and bulk species, are investigated for (111) lattice surface.
1. Introduction The adsorption of a chemically reacting fluid on a crystalline surface is a preliminary step in heterogeneous catalysis on crystals.1,2 We seek a detailed microscopic description of this phenomenon. It can be obtained by using the theory of inhomogeneous chemically associating (reacting) fluids, which has been initiated recently3-16 as a natural extension of the theory of association in the bulk fluids. Two methods are most commonly used to describe homogeneous associating fluids (AFs), namely that of Stell and collaborators17-21 and that of Wertheim.22-25 The † Presented at the Second International Symposium on Effects of Surface Heterogeneity in Adsorption and Catalysis on Solids, held in Poland/Slovakia, September 4-10, 1995. ‡ Brigham Young University. § Maria Curie-Sklodowska University. | Instituto de Quimica de la UNAM. X Abstract published in Advance ACS Abstracts, September 15, 1996.
(1) The Chemical Physics of Solid Surfaces and Heterogeneous Catalysis; King, D. A., Woodruff, D. P., Eds.; Elsevier: Amsterdam, 1983; Vol. 2. (2) The Chemical Physics of Solid Surfaces and Heterogeneous Catalysis; King, D. A., Woodruff, D. P., Eds.; Elsevier: Amsterdam, 1982; Vol. 4. (3) Kierlik, E.; Rosinberg, M. L. J. Chem. Phys. 1994, 100, 1716. (4) Kierlik, E.; Rosinberg, M. L. J. Chem. Phys. 1994, 100, 3181. (5) Phan, S.; Kierlik, E.; Rosinberg, M. L.; Yethiraj, A.; Dickman, R. J. Chem. Phys. 1995, 102, 2141. (6) Jamnik, A.; Bratko, D. Phys. Rev. E 1994, 50, 1151. (7) Holovko, M. F.; Vakarin, E. V. Mol. Phys. 1995, 84, 1057. (8) Holovko, M. F.; Vakarin, E. V.; Duda, Yu.Ya. Chem. Phys. Lett. 1995, 233, 420. (9) Pizio, O.; Henderson, D.; Sokołowski, S. J. Phys. Chem. 1995, 99, 2408. (10) Pizio, O.; Henderson, D.; Sokołowski, S. J. Colloid Interface Sci. 1995, 173, 254. (11) Pizio, O.; Henderson, D.; Sokołowski, S. Mol. Phys. 1995, 85, 407. (12) Henderson, D.; Sokołowski, S; Pizio, O. J. Chem. Phys. 1995, 102, 9048. (13) Kalyuzhnyi, Yu. V.; Pizio, O.; Sokołowski, S. Chem. Phys. Lett. 1995, 242, 297. (14) Henderson, D.; Sokołowski, S.; Trokhymchuk, A.; Pizio, O. Physica A 1995, 220, 24. (15) Pizio, O.; Trokhymchuk, A.; Sokołowski, S. Mol. Phys. 1995, 86, 649. (16) Trokhymchuk, A.; Pizio, O.; Henderson, D.; Sokołowski, S. Mol. Phys., in press. (17) Cummings, P. T.; Stell, G. Mol. Phys.1994, 51, 253. (18) Stell, G.; Zhou, Y. J. Chem. Phys. 1989, 91, 3618. (19) Zhou, Y.; Stell, G., J. Chem. Phys. 1992, 96, 1504, 1507. (20) Kalyuzhnyi, Yu.V.; Stell, G. Mol. Phys. 1993, 78, 1247.
S0743-7463(95)01519-8 CCC: $14.00
theory of Stell is based on a standard Ornstein-Zernike (OZ) relation with standard liquid state closures, modified to provide an adequate treatment of strong, short-ranged, spherically symmetric associative interactions. Particular examples of this theory for the pair correlation functions are the extended mean spherical approximation (EMSA) and the “site-site” EMSA (SSEMSA),21,26,27 developed for dimerizing fluids. The theory of Wertheim is constructed in a different manner. It introduces the densities of bonded and nonbonded particles and an OZ-like equation which contains these densities explicitely. The closure relations are given in terms of the partial correlation functions, which correspond to the bonded and nonbonded species. Two levels of description of inhomogeneous simple fluids by means of the integral equation method have been developed, namely, singlet and pair level equations.28,29 In the singlet level theory the density profile (DP) of the fluid particle i against the surface w is defined as
Fwi(r) ) Figwi(r) ) Fi[hwi(r) + 1]
(1)
and corresponds to the pair distribution function gwi(r) ) hwi(r) + 1, of the bulk fluid containing species i and w in the limit of infinite dilution of w particles (Fw f 0 ) under the condition that the diameter of w particles tends to infinity (σw f ∞ ); this is the so-called wall limit.28 The DP results from the solution of the OZ equation for either hwi(r) or hiw(r) ) hiw(r)
hwi(r12) - cwi(r12) )
∑l Fl ∫ dr3 hwl(r13) cli(r32)
(21) Kalyuzhnyi, Yu.V.; Stell, G.; Llano-Restrepo, M. L.; Chapman, W.; Holovko, M. F. J. Chem. Phys. 1994, 101, 7939. (22) Wertheim, M. S. J. Stat. Phys. 1984, 35, 19, 35. (23) Wertheim, M. S. J. Stat. Phys. 1986, 42, 459, 477. (24) Wertheim, M. S. J. Chem. Phys. 1986, 85, 2929. (25) Wertheim, M. S. J. Chem. Phys. 1987, 87, 7323. (26) Stell, G.; Zhou, Y. J. Chem. Phys. 1989, 91, 4861, 4869. (27) Stell, G. Condensed Matter Phys. (Acad.Sci. Ukraine) 1993, 2, 4. (28) Henderson, D.; Abraham F. F.; Barker, J. A. Mol. Phys. 1976, 31, 1291. (29) Fundamentals of Inhomogeneous Fluids; Henderson, D., Ed.; Marcel Dekker: New York, 1992. (30) Henderson, D.; Sokołowski, S.; Trokhymchuk, A. J. Chem. Phys., in press.
© 1997 American Chemical Society
Models for Site Adsorption
hiw(r12) - ciw(r12) )
∑l Fl ∫ dr3 hil(r13) clw(r32)
Langmuir, Vol. 13, No. 5, 1997 1163
(2)
under the appropriate approximation or closure for the direct correlation functions involving the w species. The correlation functions of the bulk fluid, either hij(r) or cij(r), serve as input into the procedure. Only the DPs are available from the singlet level theory. However, they are sufficient to investigate the structural and thermodynamic aspects of adsorption of an AF on different types of surfaces. From eq 2 it is clear how the bulk fluid description, in terms of particle-particle correlation functions may be combined with the procedure of determination of the DPs. Thus, the application of the theory of Stell is straightforward. We have recently focused our attention on the adsorption of AFs on crystalline surfaces. Periodicity of the surface introduces additional difficulties for the theory (cf. ref 31). Our study of this problem has been initiated in refs 11 and 12, where it was assumed that there was no association between the bulk fluid and the surface. We refer to this model as nonassociative adsorption of the AFs. A more sophisticated model, which permits associative adsorption on crystalline lattices, has been introduced in our preliminary work.32 Associative adsorption is characterized by a high degree of localization of the adsorbed monomers and dimers as a consequence of a strong bonding between the bulk species and the surface atoms. It was shown that for the inhomogeneous model, A + B h AB,32 in which there is an association A + Sa h ASa (Sa denotes surface atoms), one can generate the formation of surface complexes consisting of the bulk species and surface atoms of the type: A-Sa, BA-Sa, A-2Sa, BA-2Sa, A-4Sa, BA4Sa [for the (100) lattice], which correspond to the on-top site, the 2-fold bridging site, and the 4-fold hollow site adsorption, respectively. Highly localized adsorption (site adsorption) and the formation of surface complexes result in an adlayer structure which is very different to that obtained for nonassociative adsorption. In this communication, we investigate site adsorption on two lattice surfaces, namely, the (100) and (111) plane of the face-centered cubic (fcc) crystal. Our attention is directed mainly to the geometrical aspects of the site adsorption (SA)sthe issue which permits the use the liquid state closures (PY1 and HNC1). The energetic aspects of the SA on these crystalline lattices are described less accurately but should be qualitatively correct. To proceed, we describe a model of inhomogeneous AFs on crystalline lattices which provides SA. 2. A Model for Site Adsorption of an Associating Fluid on Crystalline Surfaces Consider an inhomogeneous associating fluid which is characterized by thefollowing potential energy
Upot )
∑ ∑ Umn(rij) + m)A,B ∑ ∑j Um(rj)
(3)
m,n)A,B i < j
The first term is the potential energy of interaction between the particles of the bulk subsystem. The second term gives the fluid-solid interaction energy. The model for the bulk fluid is that of Cummings and Stell16 proposed for heterogeneous association of the particles of species A and B, i.e., A + B h AB. The interactions defining the bulk model are (31) Steele, W. A. Surf. Sci. 1973, 36, 317. (32) Pizio, O.; Sokołowski, S. Phys. Rev. E, in press.
UAA(r) ) UBB(r) )
{
UAB(r) ) UBA(r) ) ∞, -b, m, 0,
{
∞, r < σf 0, r > σf
r < Lb - 0.5w Lb - 0.5w < r < Lb + 0.5w (4) Lb + 0.5w < r < σf r > σf
where σf is the diameter of the fluid particles. The associative interaction between different species is characterized by the bonding length Lb and width of the square well w, b is the strength of associative interaction, and m is the height of the square mound, which satisfies the condition [-βm] ≈ 0. We restrict ourselves to bulk dimerization only, i.e. Lb + 0.5w < 0.5σf. The bulk associating fluid is in contact with a crystalline surface. We assume that the interaction between each of the atoms of the crystalline solid and fluid particles contains a nonassociative and an associative term. Thus,
Um(r) ) Umn(z) + U(a) m (r)
(5)
{
(6)
where
Umn(z) )
∞, z < 0 0, z > 0
is the nonassociative term. The surface atoms Sa are located in the exposed plane at z ) 0, and the diameter of the surface atoms is denoted as σs. The analytical form of the nonassociative energy, Umn, means that the solid is impenetrable; this also means that the associative interaction is only possible between a fluid particle and a solid atom located in the top layer (i.e., at z ) 0) of the solid. Any possibility of association occurring “inside the solid” is excluded. For simplicity and transparency in the following analysis, consider the case of equal sizes, σf ) σs ) σ ) 1. The associative interaction U(a) m (r) is the periodic function in the lattice plane, i.e. U(a) (r) ) U(a) m (r + l), where l ) l1a1 m + l2a2; a1 and a2 are the two-dimensional unit lattice vectors. It is chosen to have the form
U(a) m (r)
{
) δmA
∞, -s, m, 0,
|r| < Ls - 0.5w Ls - 0.5w < |r| < Ls + 0.5w Ls + 0.5w < |r| < 1 |r| > 1
(7)
It is worth commenting on this interaction in more detail. In the bulk, both species are indistinguishable; however, they differ with respect to theone-particle potential U(a) m (r), only A particles react with the surface atoms. The energetic aspects of U(a) m (r) are described by the depth of the square well s, whereas the geometry of the interaction is given by the bonding length Ls and width of the well w. We have not made yet the choice of the crystalline surface. Different surfaces will associatively adsorb monomeric and dimeric bulk species differently. Besides the effects of the crystalline symmetry, the value of the bonding length parameter Ls is important. It will also determine the complexes which are formed in the adsorbate layer. We have simplified the model by the assumption of equal sizes of the bulk fluid particles and solid atoms; if their ratio would differ from unity, the adlayer structure will change, compared with the case in hand. 3. Theoretical Procedure In this study, we combine the theory of the bulk associating fluids of Stell et al. and the singlet level integral
1164 Langmuir, Vol. 13, No. 5, 1997
Henderson et al.
equations for the density profiles of particles. The singlet Percus-Yevick (PY1) approximation is used to calculate the profiles. It reads
ym(ri) ) 1 +
∑ Fn ∫ drj cmn(|ri - rj|)[gm(rj) - 1]
(8)
HNC1 equation for crystalline surfaces have been given in ourrecent publications11,12 and, for sake of brevity, are not presented. The site adsorption is characterized quantitatively by the parameter Λs, which is defined as follows
n)A,B
where gm(r) ) Fm(r)/Fm ) ym(r) exp(-βUm(r)] is the normalized density profile and ym(r) is the one-particle cavity distribution function. The functions cmn(r) are the direct correlation functions of the bulk fluid and are the only input necessary for the singlet theory. We calculate the direct correlation functions from the SSEMSA approximation (see ref 21) for detailed description) which consists of the standard OZ equation
hij(r12) - cij(r12) )
∑ Fl ∫ dr3 hil(r13) clj(r32)
(9)
complemented by the closure conditions
ωij(r) , xFiFj
r
