Phase Transitions in Adsorbed Layers Formed on the (100) Plane of

938

Langmuir 2001, 17, 938-947

Phase Transitions in Adsorbed Layers Formed on the (100) Plane of Face Centered Cubic Crystals A. Patrykiejew* and S. Sokołowski Faculty of Chemistry, MCS University, 20031 Lublin, Poland Received July 18, 2000. In Final Form: October 31, 2000 The Monte Carlo simulation method is used to study the structure and phase transitions in adsorbed films formed on the (100) plane of the model face-centered cubic crystals characterized by the different corrugation of the surface potential. The systems consisting of the adsorbate atoms that tend to form the c(2 × 2) registered phase are studied. It is shown that the actual structure of the dense monolayer films depends on the corrugation of the surface potential and on the density of the adsorbed layer and the temperature. The mechanism of melting transition in submonolayer, monolayer, and bilayer films is discussed. In particular, it is demonstrated that the submonolayer (incommensurate and registered) films retain monolayer character upon melting, whereas melting of a dense (incommensurate) two-dimensional solid is accompanied by the promotion of the second layer. The phase diagrams for a series of systems are determined. It is shown that the location of the critical point for the first-layer condensation is affected considerably by the surface corrugation. In the second layer formed on weakly corrugated surfaces and built on top of the dense incommensurate first layer, the effects due to the surface potential corrugation seem to be negligible. One observes the strong influence of the first-layer roughness (induced by the surface potential corrugation) on the critical temperature of the second-layer condensation only in the strongly corrugated surfaces.

Introduction The adsorption of simple gases on well-defined crystal surfaces has attracted a great deal of interest during the past three decades.1-8 In particular, the discovery that the monolayer films may exhibit different phases, which resemble the gas, liquid, and solid phases so well-known in the three-dimensional bulk matter, inspired the intensive experimental and theoretical studies. It soon became clear that many of the phenomena observed in adsorbed films do not have simple counterparts in bulk matter. In surface systems, the interaction between the adsorbed particles usually competes with the effects exerted on the adsorbate by the corrugated surface potential. These competing interactions lead to the formation of various structures in adsorbed layers that do not occur in uniform systems. Here, we can distinguish different commensurate and incommensurate phases, axially ordered phases, and other forms of ordering specific to adsorbed layers. Numerous experimental observations and achievements, in particular for adsorption on the graphite basal plane,1,8-12 have strongly stimulated theoretical studies * Corresponding author. E-mail: [email protected]. (1) Thomy, A.; Duval, X.; Regnier, J. Surf. Sci. Rep. 1981, 1, 1. (2) Dash, J. G. Films on Solid Surfaces; Academic Press: New York, 1975. (3) Nielson, M.; McTague, J. P.; Passell, L. In Phase Transitions in Surface Films; Dash, J. G., Ruvalds, J., Eds.; Plenum Press: New York, 1980. (4) Vilches, O. E. Annu. Rev. Phys. Chem. 1980, 31, 436. (5) Ordering in Two Dimensions; Sinha, S. K., Ed.; North-Holand, Amsterdam, 1980. (6) Larher Y. In Surface Properties of Layered Structured; Benedek, G. Ed.; Kluwer: Netherlands, 1992; p 261. (7) Patrykiejew, A.; Sokołowski, S.; Binder, K. Surf. Sci. Rep. 2000, 37, 207. (8) Phase Transitions in Surface Films 2; Taub, H., Torzo, G., Lauter, H. J., Fain, S. C., Eds.; Plenum: New York, 1991. (9) Thomy, A.; Duval, X. Surf. Sci. 1994, 299/300, 415. (10) Marx, D.; Wiechert, H. Adv. Chem. Phys. 1996, 95, 213 and references therein. (11) Bak, P. Rep. Prog. Phys. 1982, 45, 587. (12) Marx, R. Phys. Rep. 1985, 125, 1.

and greatly contributed to the development of theories for two-dimensional melting,13-16 commensurate-incommensurate transitions,17-20 the theory of wetting phenomena,21-23 and the theory of critical phenomena in systems of low dimensionality.24,25 Our present understanding of all the above-mentioned phenomena rests heavily on the vast use of computer simulation methods.7,26-31 Computer simulations allow for precise determination of the interplay between different forces operating in adsorption systems, their effect on the structure of adsorbed films and on the transformations they undergo. In particular, computer simulations are well suited to studying the changes in the behavior, structure, and properties of adsorption system resulting from the con(13) Kosterlitz, M.; Thouless, P. J. J. Phys. C 1983, 6, 1181. (14) Halperin, B. I.; Nelson, R. D. Phys. Rev. Lett. 1978 41, 121; Nelson, R. D.; Halperin, B. I. Phys. Rev. 1979, B19, 2457. (15) Young, A. P. In Ordering in Strongly Fluctuating Condensed Matter Systems; Riste, T., Ed., Plenum Press: New York, 1980; p 271. (16) Strandburg, K. J. Rev. Mod. Phys. 1988, 60, 161. (17) den Nijs, M. In Phase Transitions and Critical Phenomena; Domb, C., Lebowitz, J. L., Eds.; Academic Press: London, 1988; Vol. 12, p 219. (18) Villain J. Surf. Sci. 1980, 97, 219. (19) Selke, W. Phys. Rep. 1988, 170, 213. (20) Haldane, F. D. M.; Villain, J. J. Phys. (Paris) 1981, 42, 1673. (21) Cahn, J. W. J. Chem. Phys. 1977, 66, 3667. (22) Dietrich, S. In Phase Transitions and Critical Phenomena; Domb, C., Lebowitz, J. L., Eds.; Academic Press: London, 1988; Vol. 12, p 1. (23) Weeks, I. W. In Ordering in Strongly Fluctuating Condensed Matter Systems; Riste, T, Ed.; Plenum Press: New York, 1980; p 293. (24) Fisher, M. E. Rev. Mod. Phys. 1974, 46, 587. (25) Cardy, J. L. In Phase Transitions and Critical Phenomena; Domb, C., Lebowitz, J. L., Eds.; Academic Press: London, 1987; Vol. 11, p 55. (26) The Monte Carlo Method in Statistical Physics; Binder, K., Ed.; Springer: Berlin, 1979. (27) Applications of the Monte Carlo Method in Statistical Physics; Binder, K. Ed.; Springer: Berlin, 1984. (28) Allan, M. P.; Tildesley, D. J. Computer Simulation of Liquids; Clarendon Press: Oxford, 1987. (29) Abraham, F. F. Phys. Rep. 1981, 80, 339. (30) Landau, D. P. In Monte Carlo and Molecular Dynamics of Condensed Matter Systems; Binder, K., Ciccotti, G. Eds.; Italian Physical Society: Bologna, 1996. (31) Binder, K.; Landau, D. P. In Advances in Chemical Physics: Molecule-Surface Interactions; Lawley, K. P., Ed.; Wiley: New York, 1989.

10.1021/la001017y CCC: $20.00 © 2001 American Chemical Society Published on Web 01/04/2001

Films on Face-centered Cubic Crystals

trolled changes of interaction potentials and the structure of both the adsorbate and the adsorbent. One of the problems that is not yet fully understood is the influence of the surface potential corrugation on the stability of the different phases in adsorbed films and on the phase transitions that occur in such films. Experimental work has concentrated on adsorption on weakly corrugated surfaces of graphite,1,8-12 lamellar dihalides,6,32-34 and closed packed planes of metal crystals,35-40 which usually exhibit the hexagonal symmetry of the surface lattice. The hexagonal symmetry is also preferred by the adsorbed monolayer, although the intrinsic lattice constant for the film is usually different from the distance between the adjacent minima of the potential wells at different surface cells. This gives rise to a misfit that may be responsible for stabilization of different structures in adsorbed layers.41,42 A more complicated situation occurs in adsorption on surfaces characterized by a square or rectangular symmetry. In such cases, the stability of commensurate ordered phases is often limited to the narrow range of the temperature and the film density (or gas pressure), because the potential barriers between adjacent sites are usually much lower than the adsorption energy, and the arrangement of adsorbed atoms in the stable state may be quite different from that imposed by the symmetry of the surface. Numerous experimental studies of simple gases adsorbed on fcc metal crystals with predominantly exposed (100) and (110) faces43-47 and on other crystalline solids, like MgO48-52 and alkali halides,53-56 have demonstrated that the formation of epitaxial films occurs rarely. (32) McTague, J. P.; Nielsen, M.; Passell, L. In Ordering in Strongly Fluctuating Condensed Matter Systems; Riste, T., Ed.; Plenum Press: New York, 1980; p 195. (33) Volkmann, U. G.; Knorr, K. Surf. Sci. 1989, 221, 379. (34) Chan, M. H. W. In Phase Transitions in Surface Films 2; Taub, H., Torzo, G., Lauter, H. J., Fain, S. C., Eds.; Plenum: New York, 1991; p 1. (35) Kern, K.; Comsa, G. In Phase Transitions in Surface Films 2; Taub, H., Torzo, G., Lauter, H. J., Fain, S. C., Eds.; Plenum: New York, 1991; p 41. (36) Kern, K.; David, R.; Zeppenfeld, P.; Palmer, R. L.; Comsa, G. Solid State Commun. 1987, 62, 361. (37) Bruch, L. W.; Wei, M. S. Surf. Sci. 1980, 100, 481. (38) Wei, M. S.; Bruch, L. W. J. Chem. Phys. 1981, 75, 4130. (39) Gibson, K. D.; Sibener, S. J. J. Chem. Phys. 1988, 88, 7893. (40) Itakura, A.; Arakawa, I. J. Vac. Sci. Technol. 1991, A 9, 1779. (41) Kern, K.; Comsa, G. In Advances in Chemical Physics; Lawley, K. P., Ed.; Wiley: Chichester, 1989. (42) Binder, K. In Cohesion and Structure of Surfaces; de Boer, F. R., Pettifor, D. G., Eds.; Elsevier: Amsterdam, 1995; p 121. (43) Cohen, P. I.; Unguris, J.; Webb, M. B. Surf. Sci. 1976, 58, 429. (44) Unguris, J.; Bruch, L. W.; Moog, E. R.; Webb, M. B. Surf. Sci. 1979, 87, 415. (45) Palmberg, P. W. Surf. Sci. 1971, 25, 598. (46) Ku¨ppers, J.; Nitchke´, F.; Wandelt, K.; Ertl, G. Surf. Sci. 1979, 87, 295. (47) Glachant, A.; Bardi, W. Surf. Sci. 1976, 87, 259. (48) Coulomb, J. P.; Sullivan, T. S.; Vilches, O. E. Phys. Rev. 1984, B 30, 4753. (49) Jordan, J. L.; McTague, J. P.; Hastings, J. B.; Passel, L. Surf. Sci. Lett. 1985, 150, L82. (50) Sidoumon, M.; Angot, T.; Suzanne, J. Surf. Sci. 1992, 272, 347. (51) Coulomb, J. P.; Vilches, O. E. J. Phys. (Paris) 1984, 45, 1381. (52) Coulomb, J. P. In Phase Transitions in Surface Films 2; Taub, H., Torzo, G., Lauter, H. J., Fain, S. C., Eds.; Plenum: New York, 1991; p 113. (53) Takaishi, T.; Mohri, M. J. Chem. Soc., Faraday Trans. 1 1972, 68, 1921. (54) Klekamp, A.; Umbach, E. Surf. Sci. 1993, 284, 291. (55) Pfnu¨r, H.; Schwennicke, C.; Schimmelpfennig, J. In Adsorption on Ordered Surfaces of Ionic Solids and Thin Films; Freund, H.-J., Umbach, E., Eds.; Springer: Berlin, 1993; p 24. (56) Klekamp, A.; Reissner, R.; Umbach E. In Adsorption on Ordered Surfaces of Ionic Solids and Thin Films; Freund, H.-J., Umbach, E., Eds.; Springer: Berlin, 1993; p 35.

Langmuir, Vol. 17, No. 3, 2001 939

In a recent series of articles,57-61 we presented the results of the extensive Monte Carlo simulations for adsorbed films formed on the (100) plane of simple face-centered cubic crystals. Most of the results have been obtained for the films formed by rather small atoms, which form a simple (1 × 1) registered phase. We have concentrated on the phase transitions, such as melting and disordering, in monolayer and bilayer films formed on surfaces with different corrugation of the gas-solid potential. Although the results obtained there have allowed us to draw some important conclusions concerning the effects of surface potential corrugation on the behavior of adsorbed films, the assumed size of adsorbate atoms (with respect to the size of the surface lattice unit cell) was much smaller than in most experimental systems. Therefore, we decided to extend the scope of the study and consider the systems in which the registered phase exhibits the c(2 × 2) structure. The preliminary results of Monte Carlo simulation for such systems have already been presented in one of our earlier articles,57 where we discussed the temperaturedriven changes in the properties of monolayer films at a constant density equal to the density of the ordered c(2 × 2) phase. Depending on the height of the potential barrier between adjacent minima, the low-temperature stable phase may be either the above-mentioned c(2 × 2) registered structure or the hexagonally packed incommensurate structure. Of course, we have not been able to observe a possible commensurate-incommensurate transition because of the assumed, rather low adsorbate density. In this article we discuss the results of the Monte Carlo simulations performed in the canonical and grand canonical ensembles. The calculations, performed over a wide range of temperature, film density, and chemical potential, have allowed the study of various phenomena that occur in the monolayer and in the bilayer films. Our primary goal was to elucidate the influence of the surface potential corrugation on the phase behavior of the adsorbed films and to compare the results obtained for the systems that tend to form different registered phases. This article is organized as follows. In section 2, we briefly recall the model used, which is the same as considered in our previous works,57,61 and specify the systems to be studied. Then, in section 3, we present a description of the simulation methods and provide some details concerning the methodology applied in our study. Section 4 presents of our results and a general discussion. The article concludes with section 5, which contains the discussion of the differences and similarities in the behavior of the systems considered in this work and those which order into the (1 × 1) registered structure. Section 5 also presents some final remarks and indicates some questions that still need to be answered. The Model and Monte Carlo Method Here, we apply the same model as used in our previous works.57,61 Thus, the solid substrate is assumed to be a simple fcc regular crystal, and its surface exposed toward the gas phase is a single (100) plane of the size Lx × Ly located at z ) 0. Here, (57) Patrykiejew, A.; Sokołowski, S.; Zientarski, T.; Binder, K. J. Chem. Phys. 1995, 102, 8221. (58) Patrykiejew, A.; Zientarski, T.; Binder, K. Acta Phys. Polon. 1996, 89, 735. (59) Patrykiejew, A.; Sokołowski, S.; Zientarski, T. Langmuir 1997, 13, 1036. (60) Patrykiejew, A.; Sokołowski, S.; Zientarski, T.; Binder, K. Surf. Sci. 1999, 421, 308. (61) Patrykiejew, A.; Sokołowski, S.; Zientarski, T.; Binder, K. Langmuir 1999, 15, 3642.

940

Langmuir, Vol. 17, No. 3, 2001

Patrykiejew and Sokołowski

Table 1. Locations and Values of the Gas-Solid Interaction Potential Minima for the Adsorbed Atoms Placed over the Adsorption Center [τ* ) (0.5,0.5)] and over the Saddle Point [τ* ) (0.5,0.0)] and the Potential Barriers for Diffusion V/D for the Systems Considered in This Article Vb

τ*

z/min(τ*)

V*(z/min,τ*)

1.00 1.00 0.80 0.80 0.60 0.60 0.40 0.40

(0.5,0.0) (0.5,0.5) (0.5,0.0) (0.5,0.5) (0.5,0.0) (0.5,0.5) (0.5,0.0) (0.5,0.5)

1.065 0.975 1.069 1.015 1.073 1.039 1.077 1.057

-12.694 -15.225 -12.584 -14.144 -12.481 -13.450 -12.384 -12.940

V/D 2.531 1.560 0.969 0.556

all distances and lengths are expressed in units of the solid lattice unit vector length a. The gas-gas interaction is assumed to be pairwise additive and represented by a truncated (12,6) Lennard-Jones potential, with the cutoff distance, equal to 3.0σ. Here, we consider adsorbate atoms of the diameter σ* ) 1.20. Such atoms are too large to form a simple (1 × 1) ordered phase, and the densest possible registered phase has the c(2 × 2) structure and the surface number density equal to Fn ) 0.5. We define Fn as the ratio N/M, where N stands for the number of adsorbed atoms and M is the number of surface unit cells equal to LxLy. The interaction of a gas particle with the crystalline substrate is represented by the modified Steele potential57,62,63 in the form

∑v (z)f (τ)]

v(z,τ) ) /gs[vo(z) + Vb

g

g

(1)

g

where τ is the two-dimensional vector which represents the position of an adsorbed atom over the surface plane and the sum runs over all nonzero reciprocal lattice vectors g. The parameter Vb controls the surface potential corrugation, and /gs ) gs/ measures the strength of the interaction between a gas atom and a single atom of the solid. The properties of the gas-solid interaction potential are determined primarily by the parameters /gs and Vb, as well as by the size of adsorbed atoms σ. As in our earlier works,57-61 we assumed that /gs ) 2.0. With this choice of /gs, the surface potential felt by the adsorbate atoms is highly attractive and the adsorbed film is very stable throughout the wide range of temperature and confined to the narrow space above the surface, even at the elevated temperatures well above the melting point.57 This is a convenient situation, because the out-of-plane effects that accompany the adsorption process can be attributed to the structural changes in the film rather than to a simple thermal desorption. Of course, desorption phenomena are also observed, but only at high temperatures. To study the effects of the surface potential corrugation on the properties of adsorbed films, we considered several choices of the parameter Vb, ranging from 0.4 to 1.0. According to the approximate ground-state calculations of ref 57; there exists a certain limiting value (Vb,lim) of the corrugation parameter Vb, such that for Vb < Vb,lim the monolayer film tends to form a hexagonally packed dense structure at T ) 0, whereas for Vb exceeding Vb,lim the stable phase at the ground state is the c(2 × 2) registered structure. For the system with σ* ) 1.2, the value of Vb,lim has been estimated to be about 0.4, although the finite temperature simulation results have demonstrated57 that the actual value of Vb,lim is higher, and slightly exceeds 0.5. The potential barriers for diffusion, VD, defined as the difference between the gas-solid potential minima corresponding to the saddle point and to the unit cell center, do not scale linearly with Vb. In Table 1, we summarize the values of VD for the systems considered in this article. The results were obtained by a standard Monte Carlo method in the canonical and grand canonical ensembles.28 The description of the algorithms and technical details can be found in ref 61. (62) Steele, W. A. Surf. Sci. 1973, 36, 317. (63) Kim, H.-Y.; Steele, W. A. Phys. Rev. 1992, B45, 6226.

The calculations have been performed for the simulation box of two different sizes, with Lx × Ly × Lz ) 20 × 20 × 10 and 40 × 40 × 10. The fundamental quantities recorded, in both cases of canonical Monte Carlo (CMC) and grand canonical Monte Carlo (GCMC) simulations, have been the average adsorbate-adsorbate, 〈e/gg〉, and the adsorbate-solid, 〈e/gs〉, interaction energies (per particle), the heat capacity (from the fluctuation theorem), and the density profiles of the adsorbate n(z*), averaged across the entire solid surface. We also performed the analysis of Voronoi polygons using the algorithm developed by Tanemura et al.64

Results and Discussion At first, we recall the predictions stemming from the results of the ground-state calculations57 and concerning the effect of the surface corrugation on the behavior of monolayer films. One can define two different regimes, corresponding to the weak and the strong corrugation of the surface potential. In the first regime, the monolayer at T* ) 0 forms the incommensurate solid phase of hexagonal symmetry, called the IC phase. The details of its structure, that is, the lattice spacing, possible uniaxial ordering, and the orientation of the film relative to the surface lattice, depend on the particular value of the potential barrier for translation (V/D, defined as the difference of the gas-solid potential minimum over the center and the saddle point of the unit lattice cell) and hence on the corrugation parameter Vb, the temperature, and the density. In fact, we have observed57 the formation of uniaxially ordered and rotated solid phases. The weak corrugation regime ends at a certain value of Vb ) Vb,lim. In the presently discussed systems of Lennard-Jones (LJ) particles of the diameter σ* ) 1.20, the ground-state prediction was that Vb,lim ≈ 0.42. The finite temperature calculations57 performed for the film of a constant density Fn ) 0.5 have clearly demonstrated that, even for Vb ) 0.5, the monolayer forms a hexagonally ordered solid phase. The considered density (equal to the density of an epitaxial c(2 × 2) phase) is lower than the upper limit of the monolayer density corresponding to the IC solid phase. Therefore, the stability of the commensurate phase may be enhanced at the number density of 0.5. Under such conditions, the patch of the dense IC phase coexists with a dilute gas phase and, already strong, surface corrugation may easily destroy hexagonal ordering in the film. Consequently, the epitaxial order wins. The discussion above shows that our simple ground-state calculations underestimate the value of Vb,lim. The same was found in the systems that form a simple (1 × 1) ordered state.61 When the surface corrugation exceeds Vb,lim, the film exhibits epitaxial ordering and, in the large atoms of σ* ) 1.2, the structure of this epitaxial film corresponds to the c(2 × 2) phase. Of course, the formation of the registered film occurs only under suitable conditions. When the film density grows (e.g., because of the increase of the gas pressure), the formation of still denser incommensurate phases is possible before the adsorption in the second layer sets in. Now, we turn to the discussion of adsorption on the surface with the rather weakly corrugated surface potential, characterized by the parameter Vb equal to 0.4, and the potential barrier for translation equal to V/D ) 0.5556. Because the value of Vb is less than (Vb,lim, this system is expected to exhibit behavior qualitatively similar to that corresponding to adsorption on a flat surface. Figure (64) Tanemura, M.; Ogawa, T.; Ogita, N. J. Comput. Phys. 1983, 51, 191.


Langmuir, Vol. 17, No. 3, 2001 941

Figure 1. Adsorption isotherms obtained from GCMC for the system characterized by the corrugation parameter Vb ) 0.4 at different temperatures (shown in the figure). In all cases the simulation box size was equal to 40 × 40 × 10.

1 shows a series of adsorption isotherms recorded at different temperatures and in the range of the chemical potential µ*, which corresponds only to the monolayer adsorption. The isotherm at T* ) 0.35 indicates the presence of the first-order transition between a dilute 2Dgas and a dense 2D-solid phase, whereas the isotherm at T* ) 0.40 exhibits two discontinuous jumps, which result from the 2d-gas-2d-liquid and the 2d-liquid-2d-solid transitions. At still higher temperatures (T* ) 0.5, 0.55, and 0.65) only the transition between the 2d-liquid and the 2d-solid occurs. The adsorption isotherms in Figure 1 do not show any trace of multilayer adsorption and are practically flat for µ* above the phase-transition point. At low temperatures, the density of the 2d-solid phase is equal / . A two-dimensional character of the solid phase to Fn,2ds has been verified by the inspection of density profiles recorded at the both sides of the 2d-fluid-2d-solid transition. A strictly two-dimensional system, at the constant density equal to Fn ) 0.5 and characterized by the same value of the parameter Vb, has already been studied by the canonical ensemble Monte Carlo simulation method.59 The 2D system exhibits the melting transition at the temperature T* ) 0.37. In light of the present results, the observed transition should be interpreted as triple-point melting. Indeed, the phase diagram derived from the GCMC and CMC simulations (see Figure 2) confirms the location of the triple point in the first layer at T* ) 0.37. The mechanism of melting transition in the first adsorbed layer strongly depends on the film density. In the submonolayer regime, when the density does not exceed F/n ) 0.6, the results of CMC calculations have shown that the melting does not change the two-dimensional character of the film. In particular, neither desorption nor the promotion of the second layer has been found.57 On the other hand, when the total film density increases, the strong effect of the second-layer promotion is observed. The same mechanism of melting transition has also been found in the systems considered in ref 61. In particular, the temperature changes of the densities in layers 1 and 2 are qualitatively the same as observed previously (cf. Figure 9b of ref 61). The transition between two condensed phases observed near the monolayer completion leads to a hexagonally

Figure 2. The phase diagram for the system characterized by Vb ) 0.40. Part a presents the phase diagram in the (T*,Fn) plane. Filled points are the results of GCMC calculations whereas open circles correspond to the results obtained from the three-dimensional and two-dimensional CMC calculations. Triple line of the solid-fluid-gas coexistence in the first layer and the melting line in the second layer are shown as broken vertical lines. Part b is the phase diagram in the (T*,µ*) plane obtained from the GCMC calculations. Open circles correspond to the 2d-gas-2d-liquid transition points, the filled circles are for the 2d-gas-2d-solid, open squares represent the fluidincommensurate solid transition points in the first layer, whereas filled diamonds correspond to the condensation of the second layer.

ordered phase, as demonstrated in Figure 3. Figure 3b shows that at the temperatures above the triple point, the liquid phase is highly disordered and characterized by larger average nearest-neighbor distance than the solid phase. From the canonical ensemble calculations performed for the systems with the density exceeding monolayer capacity, we have also obtained information concerning the melting transition in the second layer. The heat capacity curves, presented in Figure 4, exhibit rather broad peaks with the maxima at the temperature of about T* ) 0.485. The sharp peaks, which occur at still higher temperatures, are due to the melting transition of the dense first layer. This transition is accompanied by the promotion of the second layer as already discussed. In general, the monolayer behavior of the system considered above is similar to the behavior of the twodimensional uniform Lennard-Jones system.65,66 The only effect of surface corrugation is the lowering of the triplepoint and critical-point temperatures with respect of the uniform system. In the last case, the triple-point temperature is equal to about 0.4, whereas in a monolayer formed on the corrugated surface the triple point occurs at T* ) 0.37. The critical point of the condensation in the first layer, estimated to be equal to T/c (1) ≈ 0.48, is also shifted toward the lower temperature with respect to the (65) Barker, J. A.; Henderson, D.; Abraham, F. F. Physica 1981, 106A, 226. (66) Tabochnik, J.; Chester, G. V. Phys. Rev. 1982, B25, 6778.

942



Figure 4. Temperature changes of the heat capacity for the system characterized by Vb ) 0.40 obtained from the CMC simulation at three different densities (shown in the figure).

Figure 3. The contributions of different Voronoi polygons [pk(T*)], k ) 4, 5, and 6 (a) and the average nearest-neighbor distance between the adsorbed particles (r/nn) (b) in the monolayer film formed on the surface characterized by the corrugation parameter Vb ) 0.40, plotted against the chemical potential at the different temperatures (shown in the figure).

uniform 2d-system, characterized by the critical temper/ ) 0.53 ( 0.03.65,67 The second-layer condenature Tc,2d sation critical point is located at the considerably higher temperature of T/c (2) ≈ 0.54. Here, the effects of surface corrugation are expected to be negligible, because the periodic variations of the surface potential decay with z* as exp[-Rz*],64 where R is a positive constant. The next system, characterized by the corrugation parameter of 0.6, exhibits considerably different properties. In particular, we do not observe the first-order dense fluid-solid transition in the monolayer regime. The adsorption-desorption isotherms depicted in Figure 5, obtained from the GCMC simulation, show a smooth change of the film density with the chemical potential and only the jumps caused by the condensation of the first layer are found. The melting of both the monolayer and of the bilayer film has been studied with the help of CMC simulation. In particular, the submonolayer films of the density not exceeding 0.5 have melted at a constant (67) Rovere, M.; Heermann, D. W.; Binder, K. J. Phys. Condens. Matter 1990, 2, 7009.

Figure 5. The examples of the adsorption-desorption isotherms in the region of monolayer formation obtained for the system with Vb ) 0.60 and different temperatures (shown in the figure).

temperature of T* ≈ 0.44, in agreement with the results obtained for a strictly two-dimensional system.59 That is not surprising, because the melting of low-density films is not accompanied by any effects of desorption and/or the promotion of the second layer. On the contrary, the melting of the first layer in the dense monolayer and bilayer films leads to a sudden transfer of the adsorbate particles to the second layer and then the melting temperature gradually increases when the total film density becomes higher. The results obtained for this system are summarized in Figure 6, which shows the phase diagrams in the (µ*,T*) and (F/n,T*) planes. The results of the CMC simulations allow location of the melting of the monolayer solid above the triple point, whereas the adsorption isotherms obtained from the GCMC simulations do not show any visible traces of that transition. Also, the recorded changes of the total potential energy (〈e/tot〉) and the heat capacity of the adsorbed film along the isotherms do not show any anomalies. On the other hand, the contributions to the


Figure 6. The phase diagram for the system characterized by Vb ) 0.60. Part a presents the phase diagram in the (T*,µ*) plane obtained from the GCMC simulations. Open circles represent the 2d-gas-c(2 × 2) transition, filled circles are for the 2d-gas-2d-incommensurate solid transition, filled diamonds correspond to the commensurate-incommensurate transition, open squares represent the location of the second layer condensation, and open diamond represents the location of the triple point in the first layer. Part b is the phase diagram in the (T*,Fn) plane. Filled circles and filled diamonds are the results obtained from the GCMC calculations, open circles (filled squares) correspond to the results obtained from the threedimensional (two-dimensional) CMC calculations. The triple line of the solid-fluid-gas coexistence in the first layer and the melting line in the second layer are shown as dashed vertical lines.

total energy, associated with the adsorbate-adsorbate (〈e/gg〉) and the adsorbate-substrate (〈e/gs〉) interactions, exhibit well-seen jumps indicating the melting transition, as illustrated by Figure 7. It is evident that the adsorbateadsorbate and the adsorbate-substrate energies change

Langmuir, Vol. 17, No. 3, 2001 943

in the opposite directions upon melting. Because these changes are practically the same in magnitude, the total energy does not change at all. The locations of the liquidsolid transition (shown in Figure 6a) correspond to the chemical potential values at which the above-discussed anomalies in the behavior of the adsorbate-adsorbate and the adsorbate-substrate energies occur. Although both 〈e/gg〉 and 〈e/gs〉 change discontinuously at the transition point, the melting transition seems continuous. The influence of the surface corrugation on the structure of the monolayer solid phase is much stronger than in the previously discussed system of Vb ) 0.4. In particular, the distributions of the different Voronoi polygons exhibit quite pronounced contributions because of the tetragons and pentagons, even at low temperatures. The solid phase in the first layer assumes a well-developed hexagonal structure only in the region of rather high chemical potential, when the second layer is formed. The further increase of the corrugation parameter Vb to 0.8 and 1.0 leads to a different phase behavior than in the both systems discussed above. First, the first step of the monolayer condensation, which takes place at low temperatures, leads to the formation of the registered c(2 × 2) phase of the density Fn ) 0.5. Upon further increase of the chemical potential, the registered phase undergoes a sharp, first-order transition, which results in the formation of a more dense incommensurate solid phase. When the temperature is high enough, however, the transition between the commensurate and incommensurate phases seems to be continuous. Thus, one expects to find a tricritical point for this system. It is illustrated in Figure 8, which presents the examples of the adsorption isotherms obtained for the system with Vb ) 0.8. In Vb ) 1.0, the qualitative picture is the same. The phase diagrams evaluated for those two systems are also qualitatively the same (see Figure 9). From the phase diagrams (for the systems with Vb ) 0.8) it follows that the tricritical-point temperature is located somewhere between 0.6 and 0.7. In the system with Vb ) 1.0 even at the temperature T* ) 0.8, the fluid-solid transition exhibits the first-order character. In the regime of strong corrugation of the surface potential, the critical temperature of the 2D-gas-registered c(2 × 2) phase transition increases when the potential barrier for diffusion becomes higher. It is connected with the gradual approach to the

Figure 7. The energies 〈e/gg〉, 〈e/tot〉, and 〈e/gs〉 versus µ* for the system with Vb ) 0.80 at the temperature T* ) 0.45 (open circles) and 0.50 (filled circles).

944



Figure 8. The examples of the adsorption-desorption isotherms in the region of monolayer and bilayer formation obtained for the system with Vb ) 0.80 and different temperatures (shown in the figure).

Figure 9. The phase diagrams (obtained from the GCMC calculations) for the system characterized by Vb ) 0.80. (a) The phase diagram in the (T*,Fn) plane; (b) the phase diagram in the (T*,µ*) plane. In b, the 2d-gas-c(2 × 2) phase-transition points are represented by filled circles: the c(2 × 2)-2dincommensurate solids are represented by open circles and the second-layer condensation points are represented by filled squares.

lattice gaslike behavior. The estimated critical temperature for the two-dimensional lattice gas of LJ particles, with σ* ) 1.2 and the cutoff set at 3.5σ*, is equal to about 0.66. From the results obtained it follows that the critical temperatures for the order-disorder transition in the first layer of the systems with Vb ) 0.8 and 1.0 are equal to 0.555 and 0.615, respectively. In the systems that are characterized by the highly corrugated surface potential, the dense IC monolayer solid phase exhibits a domain-wall structure. Figure 10 shows two examples of the snapshots of the configurations recorded at both sides of the commensurate-incommensurate transition. In the configuration corresponding

Figure 10. The examples of configurations recorded at both sides of the commensurate-incommensurate transition in the monolayer film formed on the surface characterized by the corrugation parameter Vb ) 0.80 at T* ) 0.45 and the chemical potential µ* ) -11.35. Part a shows the configuration of the commensurate c(2 × 2) phase, and part b shows the configuration of the incommensurate phase.

to the IC phase, there are patches of the commensurate phase separated by the walls of the hexagonally packed particles. Analysis of the Voronoi polygons found in the monolayer film provides a much more convincing proof of the domainwall structure of the incommensurate phase. One expects that, as long as the film is the commensurate c(2 × 2) phase, only tetragons should be found. On the contrary, in the more-or-less perfect hexagonal phase, the distribution of the Voronoi polygons should be vastly dominated by hexagons. When the film exhibits the domain-wall structure, however, a remarkable contribution by the pentagons should be found. The presence of such penta-


Figure 11. The contributions of different Voronoi polygons [pk(T*)], k ) 4, 5, and 6, in the monolayer film formed on the surface characterized by the corrugation parameter Vb ) 1.00, plotted against the chemical potential at the different temperatures (shown in the figure).

gons marks the occurrence of the boundary regions between the paths of the commensurate phase and the walls, characterized by the hexagonal packing of the adsorbed particles. The structure of the incommensurate monolayer has been stable throughout a wide range of the chemical potential and temperature. Thus, the estimated average nearest-neighbor distance in the solid phase is constant and equal to about 1.30, when Vb ) 0.8 and to about 1.33, when Vb ) 1.0. Figure 11 shows the changes in the contributions caused by the different polygons along the isotherms at T* ) 0.3 and 0.45, for the system with Vb ) 1.0, in the region of the commensurate-incommensurate transition. Mostly tetragons and pentagons are found in agreement with the suggested domain-wall structure of the film. Quite similar results have been obtained for the system characterized by the corrugation parameter equal to 0.8. The increase of the contribution caused by the hexagons, at the expense of the reduction of the contribution caused by tetragons, is connected with the widening of the walls separating the domains of the commensurate phase. The contribution of pentagons remains practically unchanged, which suggests that the total length of the wall does not change when the temperature increases from 0.3 to 0.45. The widening of the walls with temperature can be related to the enhancement of the kinetic energy of adsorbed atoms and the increase of their average distance from the surface. Thus, the surface potential corrugation felt by the adsorbed atoms is then reduced, and the effects of the adsorbateadsorbate interaction become more important. We recall that the system characterized by the parameter Vb ) 0.4 does not show the domain-wall structures, and the incommensurate phase possesses a rather uniform hexagonal order. However, in the system with the corrugation parameter equal to Vb ) 0.6, the domain-wall structures have already been observed. In the IC phase formed on the highly corrugated surfaces, the locations of the adsorbed particles exhibit larger deviations from planarity than in the commensurate phase and, on the average, the particles assume a longer distance from the surface. That effect is more pronounced in the Vb ) 1.0 than in the Vb ) 0.8 (see Figure 12). We emphasize the observed out-of-plane modulation of the

Langmuir, Vol. 17, No. 3, 2001 945

Figure 12. Density profiles for the commensurate c(2 × 2) and the IC monolayer films, formed on the surfaces characterized by the corrugation parameter Vb ) 0.80 (solid lines) and 1.0 (dashed lines) at T* ) 0.45 and the chemical potential µ* ) -11.35 (Vb ) 0.8) and µ* ) -9.65 (Vb ) 1.0).

Figure 13. The plots of the critical temperatures of the firstand second-layer condensation [T/c (k), k ) 1, 2) against the potential barrier for diffusion. The dash-dotted line at the righthand side of the figure marks the critical temperature of the 2D lattice gas model (V/D f ∞).

dense monolayer films formed on the highly corrugated surfaces because it may help to explain the lowering of the critical temperature of the second-layer condensation, as compared with the adsorption on the weakly corrugated surfaces. Figure 13 shows the plots of the critical temperatures of the first- and second-layer condensations against the potential barrier between adjacent minima. It is clearly that the second-layer condensation [T/c (2)] is lower in strongly corrugated surfaces. A plausible reason for that effect is the enhanced surface roughness of the first layer. Therefore, the adsorption of the second layer occurs on a rather nonuniform surface. The lowering of the 2D condensation critical point due to the weak surface heterogeneity is a well-known phenomenon.68 One can apply the same argument to explain the great lowering of T/c (1) for the adsorbed layer formed on the surface with (68) Patrykiejew, A.; Boro´wko, M. In Computational Methods in Surface and Colloid Science; Boro´wko, M., Ed.; Marcel Dekker: New York, 2000; p 245.

946


Vb ) 0.4 with respect to the value obtained for strictly uniform two-dimensional Lennard-Jones fluid (cf. Figure 13). Of course, a further increase of the surface corrugation is bound to lead to a new increase of the critical temperature, because of gradual approach toward the lattice gas regime. Summary and Final Remarks In this article we have discussed the phase behavior of adsorbed layers formed by the Lennard-Jones particles on the (100) surface of model fcc crystal. The assumed size of the adsorbate atoms was chosen big enough (σ* ) 1.20) to exclude the mutual occupation of adjacent minima of the gas-solid potential. This potential has been represented in the form of Fourier series with the adjustable corrugation parameter, which determines the potential barrier for surface diffusion, V/D. The magnitude of V/D is one of the most important parameters that determine the properties of adsorbed monolayers, and the two main regimes can be identified: the weak and the strong surface corrugation regime. In the weak corrugation regime the adsorbed monolayer exhibits behavior qualitatively similar to the twodimensional uniform Lennard-Jones system. In particular, the film does not form epitaxial structures and the solid phase is incommensurate with the substrate. This phase can be identified as the “floating solid”.14 The melting of that solid phase has a different mechanism, depending on the film density. Thus, submonolayer films retain twodimensional character upon melting, whereas the melting of dense monolayer films is accompanied by the promotion of the second layer. The same was found in systems of the smaller Lennard-Jones atom of σ* ) 0.8 studied previously.61 In general, the behavior of monolayer films on weakly corrugated surfaces has been insensitive to the size of the adsorbate atoms. The situation looks quite different when the potential barrier for diffusion places the system in the strong surface corrugation regime. In such cases the adsorbed monolayer may form the commensurate phase. In the small adsorbate atoms (see ref 61), the commensurate phase has a simple (1 × 1) structure, whereas the larger atoms of σ* ) 1.2 form into the c(2 × 2) structure. The most important consequence of the difference in the structure of the commensurate films formed by small and large atoms is the formation of incommensurate solid phases of different structures. In the small atoms of σ* ) 0.8 the incommensurate solid phase has a well-developed and uniform hexagonal structure. On the contrary, the structure of the incommensurate phase formed by larger atoms is consistent with the concept of domain walls.17,18,20 The validity of the domain-wall theory is limited to the socalled weakly incommensurate systems with degenerated commensurate ground state. Therefore, the systems that form into the c(2 × 2) commensurate structure are expected to exhibit the domain-wall structure. Our results confirm that prediction very well. We have been able to identify the existence of domain walls in the incommensurate phase, although our simulation cell was rather small. It would be desirable to perform the simulation for still larger systems. It might then be possible to estimate how the shape and size of the domains, and the thickness of the walls between the domains and their orientation with respect to the surface lattice, change with temperature and the film density. The case of Vb ) 0.6 represents a special situation. At low temperatures (below the triple point) the condensed monolayer is the incommensurate solid. Above the triple


point, however, we have found two different condensed phases. The phase of lower density exhibits a high degree of commensurability and can be identified as the weakly incommensurate phase of the domain-wall structure. On the other hand, the phase of higher density has the structure of the floating solid phase. A rather peculiar melting behavior of the above-discussed system demonstrates that the competing adsorbate-adsorbate and adsorbate-substrate interactions may considerably affect the properties of the adsorption system. Of course, to verify the nature of the melting transition in this system, one would need to perform simulations in larger systems and perform the finite-size scaling analysis. The properties of the second layer have been affected only slightly by the corrugated surface potential. From the phase diagrams presented it follows that the coexistence curves, corresponding to the condensation of the / second layer [µcoex,2 (T*)] nearly coincide over the entire range of temperature, up to the critical point. The only effect of surface potential corrugation on the second layer is the shift in the location of the critical temperature. The source of that shift is the roughness of the dense incommensurate first layer. The same effect has been observed previously in bilayer films of smaller atoms,61 but at that time we had not identified the source of the observed changes in the second-layer critical point. At low temperatures, the condensed second layer was packed hexagonally in all systems discussed. Even in the highly corrugated surfaces, with the highest corrugation parameter equal to Vb ) 1.0, we have not observed the formation of commensurate bilayers found in the films formed by smaller atoms of σ* ) 0.80.61 However, the potential barrier between adjacent sites is considerably smaller for larger atoms of σ* ) 1.2 (V/D ) 2.531, see Table 1), than in the smaller atoms of σ* ) 0.8 (V/D ) 3.262). Although the model systems discussed in this article do not mimic any real situation, it would be interesting to compare the results obtained with relevant experimental data. Adsorption of noble gases on the (100) plane of fcc metal crystals45,47,69,70 usually leads to the formation of hexagonally packed dense films and such behavior is attributed to rather low corrugation of the substrate potential. More complex behavior has been found for CO molecules adsorbed on Ni(100) and Cu(100) surfaces,71 studied by low-energy electron diffraction, Auger, and work-function measurements. Namely, different condensed phases exist: the low-density commensurate c(2 × 2) phase and the incommensurate hexagonal phase of higher density. A recent infrared reflection-absorption study72 also suggested the presence of high-order (3 × 5) commensurate structure. The same commensurate structure has been found in the Kr adsorbed on the Ir(100) surface.73 A RHEED study of selenium adsorbed on Ni(100)74 has shown the existence of the c(2 × 2) and the p(2 × 2) ordered commensurate structures in addition to various disordered phases. Symmetry arguments indicated that the phase diagram of that system may belong to the universality class of the Ashkin-Teller model.75 (69) Chesters, M.; Hussain, M.; Pritchard, J. Surf. Sci. 1971, 28, 460. (70) Moog, E. R.; Webb, M. B. Surf. Sci. 1984, 148, 338. (71) Tracy, J. C. J. Chem. Phys. 1972, 56, 2736. (72) Grossmann, A.; Erley, W.; Ibach, H. Surf. Sci. Lett. 1995, 330, L646. (73) Legg, K. O.; Jona, F.; Jepsen, D. W.; Marcus, P. M. Phys. Rev. 1977, B16, 5271. (74) Bak, P.; Kleban, P.; Unertl, W. N.; Ochab, J.; Bertelt, N. C.; Einstein, T. L. Phys. Rev. Lett. 1985, 54, 1559. (75) Ashkin, J.; Teller, E. Phys. Rev. 1943, 64, 178.


Thermodynamic and structural studies of the adsorption of noble gases (Ar, Kr, and Xe) and CH4 on highly homogeneous MgO substrate48,52,76 have also demonstrated the formation of various high-order commensurate phases in addition to the c(2 × 2) and hexagonal structures. The adsorption isotherms for noble gases on MgO are strikingly similar to the isotherms depicted in Figures 1 and 8. Theoretical calculations and LEED measurements77 have shown, however, that the c(2 × 2) phase may be stable only in the case of Kr films. In general, more complex (2 × n) (n > 2) phases have been found. Here, we note that the relative size of Kr atoms and the MgO unit cell are equal to σ* ≈ 1.21 and are very close to the value of σ* assumed in our model study. Of course, in the Kr/MgO system, the properties of the adsorbate-substrate potential are different than in our model because of the ionic structure of MgO crystal. Therefore, we cannot make any direct comparison with experimental data. Nevertheless, we find that our calculations are in a reasonable qualitative agreement with the experimental data. Experimental studies of the CH4/MgO system52,78-80 have shown that only one condensed phase exists and it corresponds to the commensurate c(2 × 2) phase. The stability of that structure is connected with good agreement between the surface lattice structure, with the second nearest-neighbor distance equal to 4.21 Å and the methane molecule diameter equal to 4.17 Å. We believe that the application of our model to such a situation would also lead to similar results. The results presented in this article are a part of the project aiming at the systematic study of adsorption on surfaces of square and rectangular symmetry. (For review see ref 7.) In such systems the competing adsorbateadsorbate and adsorbate-substrate interactions lead to the formation of the new types of ordering in adsorbed layers. The mechanisms of phase transitions between different types of surface phases are quite complex and (76) Bienfait, M.; Coulomb, J. P.; Palmari, J. P. Surf. Sci. 1987, 182, 557. (77) Meichel, T.; Suzanne, J.; Girard, C.; Girardet, C. Phys. Rev. 1988, B15, 3781. (78) Coulomb, J. P.; Madih, K.; Croset, B.; Lauter, H. J. Phys. Rev. Lett. 1985, 54, 1536 (79) Miechel, T.; Suzanne, J.; Gay, J. M. C. R. Acad. Sci. Paris 1988, 11, 989. (80) Jung, D. R.; Cui, J.; Frankl, D. R.; Ihm, G.; Kim, H. Y.; Cole, M. C. Phys. Rev. 1989, B40, 11893.

Langmuir, Vol. 17, No. 3, 2001 947

not as well understood as in the hexagonal surfaces. Until now we have focused on the effects of the surface corrugation on the behavior of adsorbed films formed on strongly adsorbing surfaces and discussed the structure of monolayer and bilayer films and the mechanism of various phase transitions that occur in adsorbed layers. The results of our study have been compared with other theoretical works and with experimental data (cf. ref 7 and the references therein). There are several unresolved problems, however, that have not been considered yet. For example, there is a question about how the behavior of adsorption system changes when the strength of the substrate potential is varied. A rather surprising result in this article is the observed domain structure of dense monolayer films formed on highly corrugated surfaces. In such cases one expects the formation of epitaxial bilayer film rather than a dense IC phase. It is of interest whether it is only a consequence of the modest size of our simulation system or an intrinsic property of the system. If the latter is true then the high potential barrier for diffusion is not a sole factor that guarantees the stability of the commensurate phases. The energy loss due to the formation of the second layer may be greater than the energy change resulting from the placement of a certain number of particles in less favorable positions (not directly above the adsorption sites) in the first layer. It should be remembered that the surface potential decays with the distance from the surface as z*-4, whereas the periodic part of that potential decays exponentially with z*. Therefore, it would be interesting to investigate the behavior of films formed on surfaces characterized by different strengths of the adsorbing potential. Another closely related question is the stability of adsorbed film and the crossover from complete to partial wetting when the substrate potential becomes weak enough.81,82 The above-mentioned problems will be discussed in our future articles. Acknowledgment. This work has been supported by KBN (Poland) under grant no. 3 T09A 161 18. LA001017Y (81) Phillips, J. M. Phys. Rev. 1995, B 51, 7186. (82) Rowlinson, J. S.; Widom, B. Molecular Theory of Capillarity; Clarendon Press: Oxford, 1982.

Phase Transitions in Adsorbed Layers Formed on the (100) Plane of

Recommend Documents