Phase Transitions in Two-Dimensional Monolayer Films on the (100

We present the results of Monte Carlo simulation of two-dimensional films formed on the (100) face of a face centered cubic crystal. Systems with diff...
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1036

Langmuir 1997, 13, 1036-1046

Phase Transitions in Two-Dimensional Monolayer Films on the (100) Face Centered Cubic Crystal Surface† A. Patrykiejew,* S. Sokołowski, and T. Zientarski Faculty of Chemistry, MCS University, 20031 Lublin, Poland Received October 2, 1995. In Final Form: February 21, 1996X We present the results of Monte Carlo simulation of two-dimensional films formed on the (100) face of a face centered cubic crystal. Systems with different corrugation of the gas-solid potential are studied and the results are compared with a similar study (J. Chem. Phys. 1995, 102, 8221) performed for threedimensional systems. The inner structure of the solid phase is studied using the bond-orientational order parameters and the radial distribution functions. It is shown that the systems characterized by the strong corrugation of the gas-solid potential and exhibiting the registered ordered phases at low temperatures undergo a continuous disordering into the fluid phase. This fluid phase remains partially ordered over a wide temperature range. In the case of weakly corrugated surfaces, exhibiting the hexagonal closed packed solid phase, the melting transition is preceded either by the reorientation of the adsorbate lattice relative to the surface lattice or by the formation of a uniaxially ordered phase. The subsequent melting occurs via the first-order transition or according to the mechanism predicted by the theory of Nelson and Halperin, depending on the system properties.

1. Introduction Intensive and systematic study of phase transitions in adsorbed films began in late 1960s,1-3 after the development of technologies for the preparation of the graphite substrate with a highly uniform surface.4,5 Already very early experiments1-3 revealed that monolayers formed on graphite may have very different structure and properties, depending on the fine details of the adsorption system geometry and molecular interactions. In particular, it was found that argon, krypton, and xenon films exhibit quite different behavior.5-7 It was also pointed out that monolayer films resemble, in many respects, the behavior of bulk matter, exhibiting the gas, liquid, and solid phases. Further studies revealed a number of different twodimensional solid phases.8-10 In particular, it was found that the solid monolayer films may be commensurate or incommensurate with the substrate lattice and that there are transitions between these phases under certain thermodynamic conditions. It soon became clear that one of the most important factors determining the structure of two-dimensional phases is the periodic corrugation of the gas-solid potential.11-15 Although the effects of the * To whom all correspondence should be addressed: e-mail, [email protected]. † 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. X Abstract published in Advance ACS Abstracts, May 1, 1996. (1) Thomy, A.; Duval, X. Colloq. Int. CNRS 1965, 132, 81. (2) Thomy, A.; Duval, X. J. Chim. Phys. 1969, 66, 1966. (3) Thomy, A.; Duval, X. J. Chim. Phys. 1969, 67, 286, 1101. (4) Thomy, A.; Duval, X. C. R. Hebd. Seances Acad. Sci. 1964, 259, 4007. (5) Thomy, A.; Duval, X.; Regnier, J. Surf. Sci. Rep. 1981, 1, 1. (6) Nielsen, M.; McTague, J. P.; Passell, L. In Phase Transitions in Surface Films; Dash, J. G., Ruvalds, J., Eds.; Plenum Press: New York, 1980. (7) Thomy, A.; Duval, X. Surf. Sci. 1994, 299/300, 415. (8) Zhang, Q. M.; Kim, H. K.; Chan, M. W. H. Phys. Rev. 1986, B33, 5149. (9) Vilches, O. E. Ann. Rev. Phys. Chem. 1980, 31, 463. (10) Meichel, T.; Suzanne, J.; Girard, C.; Girardet, C. Phys. Rev. 1988, B38, 3781. (11) Steele, W. A. Surf. Sci. 1973, 36, 317. (12) Doll, J. J.; Steele, W. A. Surf. Sci. 1974, 44, 449. (13) Gooding, R. J.; Joos, B.; Bergersen, B. Phys. Rev. 1983, B27, 7669. (14) Houlrik, J. M.; Landau, D. P. Phys. Rev. 1991, B44, 8962. (15) Kim, H.-Y.; Steele, W. A. Phys. Rev. 1992 B45, 6226.

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gas-solid potential corrugation were considered much earlier,16 a full appreciation of their importance came with the discovery of different structures in the films formed on graphite, lamellar dihalides, metals, and other crystalline solids. The calculations of the gas-solid potential for various gases in contact with the graphite surface demonstrated,11-15 however, that the corrugation is quite small in the considered systems. It is obvious that the properties and structure of adsorbed films are determined by the combined effects due to the gas-gas and gas-solid interactions. Of course, the film density, temperature and pressure are also important parameters influencing the state of the adsorbed layer. In general, the gas-gas interaction tends to enforce the formation of the hexagonal close-packed (hcp) solid phase in the monolayer film at low temperatures, while the corrugated gas-solid potential favors the formation of registered structures. In the case of graphite, both the commensurate and the incommensurate solid phases have the same hexagonal symmetry and differ only by the spacing between neighboring atoms and by the orientation of adsorbed layer lattice relative to the surface lattice.5,17,18 The situation looks quite different for substrates with other than the hexagonal symmetry. Numerous experimental studies of low temperature adsorption of noble gases on the low index faces of various metals19-23 have demonstrated that the competition between the gas-gas and the gas-solid interactions gives rise to the formation of various registered superstructures, ordinary hexagonal close-packed phase as well as to the appearence of uniaxially ordered phases. The term “uniaxially ordered phase” used in this paper refers to a hexagonal phase exhibiting uniaxial registry along one axis of the rectangular lattice. Recently, we have performed extensive three-dimensional24 and two-dimensional25 Monte Carlo studies of (16) Hill, T. L. J. Chem. Phys. 1946, 14, 441. (17) Novaco, A. D.; McTague, J. P. Phys. Rev. Lett. 1977, 38, 1286. (18) McTague, J. P.; Novaco, A. D. Phys. Rev. 1979, B19, 5299. (19) Dickey, J. M.; Farrell, H. H.; Strongin, M. Surf. Sci. 1970, 23, 448. (20) Palmberg, P. W. Surf. Sci. 1971, 25, 598. (21) Chesters, M. A.; Hussain, M.; Pritchard, J. Surf. Sci. 1971, 28, 460. (22) Glachant, A.; Jaubert, M.; Bienfait, M.; Boato, G. Surf. Sci. 1981, 115, 219. (23) Bibe´rian, J. P.; Huber, M. Surf. Sci. 1976, 55, 259.

© 1997 American Chemical Society

Phase Transitions in Monolayer Films

Langmuir, Vol. 13, No. 5, 1997 1037

melting and disordering for monolayer films formed on the (100) face of face centered cubic (fcc) crystals characterized by different corrugation of the gas-solid potential. It has been shown that depending on the height of the potential barrier for translation and the temperature, the film may exhibit different structure and thermodynamic behavior. Unfortunately, we have not been able to draw any conclusions concerning the order of the observed phase transitions from the threedimensional simulations. This has resulted from the fact that rather small systems have been considered and the effects due to the out-of-plane motion and desorption phenomena may have contributed considerably to the obtained results. In order to clarify this question, we have performed the Monte Carlo study for strictly twodimensional systems,25 but only for one size of adsorbed atoms and at the density corresponding to the completely filled 1 × 1 registered layer. It has been shown24,25 that for sufficiently high corrugation of the gas-solid interaction potential the registered film disorders gradually as the temperature in the system increases. On the other hand, the melting of the hcp phase, which is stable at low temperatures for weakly corrugated surfaces, may occur either via the first-order transition or exhibit the mechanism predicted by the theory of Nelson, Halperin, and Young.26-28 In this paper we discuss systems of particles of different size, forming either the 1 × 1 or x2 × x2 registered phases, as well as consider changes in the mechanism of melting and disordering upon the changes in the film density. In the section 2 we present the model, specified by the potential functions representing the gas-gas and the gassolid interactions. Then, in section 3 we define the bondorientational parameters and their higher moments, and briefly discuss their relevance to the problems considered in this paper. Section 4 presents some details of the simulation method used. The introductory and technical part of the paper concludes in section 5, where we consider basic ground state properties of the model. Our results are presented and discussed in the section 6. 2. The Model We consider two-dimensional monolayer films formed on the (100) face of a simple fcc crystal. The adsorbed atoms are assumed to interact one with another via pairwise additive forces only, with the pair potential given by the (12,6) Lennard-Jones function, u(r), truncated at a certain distance, rmax

u(r) )

{

4[(σ/r)12 - (σ/r)6], r e rmax r > rmax 0,

(1)

V3D(z,τ) ) V03D(z) +

∑q Vq3D(z)fq(τ)

(2)

In the above equation the summation runs over all reciprocal surface lattice vectors q. The Fourier coefq (z) can be readily calculated using the folficients V3D lowing expressions given by Steele

V03D(z)

) *gs

2πA6 a*s

∞

∑ k)0

[

2A6

]

1 10

5(z* + k∆z*)

(z* + k∆z*)4 (3)

and

Vq3D(z) ) *gs

[ ( )

2πA6 A6 q*k a*s 30 2z*

5

K5(q*kz*) -

( )

2

]

q*k 2 K2(q*kz*) (4) 2z*

In the above *gs ) gs/ measures the strength of the interaction between a gas atom and a single atom of the solid, A ) (1 + σ*)/2 (σ* ) σ/a), a* s ) 1.0, z* ) z/a, ∆z* ) 1/x2, q* k is the reduced length of the reciprocal lattice vector qk and K5(x) and K2(x) are the modified Bessel functions of the second kind and the fifth and second order, respectively. The functions fq(τ) are expressed by the cosine functions of the product qτ and can be readily derived for the specified set of reciprocal lattice vectors q. In this work we take into account only the first five nonzero vectors q. It has been shown by Steele11 that in the case of the (100) face of an fcc crystal it gives quite satisfactory agreement between the potential calculated using the above eq 2 and the potential obtained by a direct summation of the atom-atom interactions. In order to make the above expression (2) more flexible with respect to the effects due to the corrugation of the gas-solid potential, we introduce, following Kim and Steele,15 an adjustable parameter Vb and rewrite eq 2 in the form

∑q Vq3D(z)fq(τ)

V3D(z,τ) ) V03D(z) + Vb

(5)

Thus, by changing the parameter Vb it is possible to change the magnitude of the periodic variations of the gas-solid potential, and when Vb ) 1.0 one recovers the original expression due to Steele given by eq 2. Now, we assume that in our two-dimensional systems the gas-solid potential takes the following form25

V2D(τ) ) min[V3D(z,τ)]

(6)

z

Throughout this paper we assume that the cut-off distance is set at rmax ) 2.5σ, as it was also assumed in our earlier works.24,25 The adsorbed atoms are also subject to the surface potential, V2D(τ), being a function of the two-dimensional vector τ ) (x,y), which specifies the position of an atom relative to the surface lattice. The surface potential is periodic and in the three-dimensional system can be calculated using the Fourier expansion developed by Steele11 (24) Patrykiejew, A.; Sokołowski, S.; Zientarski, T.; Binder, K. J. Chem. Phys. 1995, 102, 8221. (25) Patrykiejew, A.; Zientarski, T.; Binder, K. Submitted to Acta Phys. Pol. (26) Halperin, B. I.; Nelson, D. R. Phys. Rev. Lett. 1978, 41, 121. (27) Nelson, D. R.; Halperin, B. I. Phys. Rev. 1979, B19, 2457. (28) Young, A. P. Phys. Rev. 1979, B19, 1855.

The above definition of the surface potential ensures that the corrugation effects are exactly the same as in the threedimensional model considered earlier,24 providing that all parameters are the same. One can expect, however, that for a given corrugation parameter, Vb, the twodimensional systems should exhibit slightly higher tendency toward localized adsorption than their threedimensional counterparts. This is so, because by supressing the possibility of out-of-plane motion we eliminate the possibility of a better accommodation of the hexagonal phase. In three dimensions some particles may stay at larger distance from the surface, but the interparticle distances in the adsorbed layer may still remain practically unchanged. In this work we consider systems of adatoms with different size given by σ* ) 0.9 and 1.2, and of different

1038 Langmuir, Vol. 13, No. 5, 1997

density. The first system (with σ* ) 0.9) forms a simple 1 × 1 ordered state and hence the number density of the completely filled registered layer is equal to Fc ) 1.0. The second system (with σ* ) 1.2) orders into the x2 × x2 phase and has the number density of the registered phase equal to Fc ) 0.5. Thus, the reduced densities, F* ) Fcσ*2, for the completely filled registered films for these two systems are equal to 0.81 (for σ* ) 0.9) and 0.72 (for σ* ) 1.2). Then, we assume that *gs ) 2.0, in agreement with the assumption made in refs 24 and 25. One should note that unlike in the studies of two-dimensional systems on noncorrugated surfaces29-31 and in lattice gas models,32 where the strength of the gas-solid potential is irrelevant, here it influences the height of potential barriers for surface diffusion. As it follows from our definition of the gas-solid potential (cf. eqs 6 and 4), the corrugation term is proportional to *gs. For each size of the adsorbed atoms we then consider a series of systems characterized by different values of the gas-solid potential corrugation parameter, Vb. In practice, we take into account the values of Vb not exceeding 1.0. From the simple ground state calculations24 it follows that for sufficiently low values of Vb, the systems considered here form the hcp solid phase at low temperatures. On the other hand, when Vb exceeds a certain limiting value, Vb,lim(σ*), the stable low-temperature structure corresponds to the registered phase. The observations made for three-dimensional systems have clearly demonstrated that for intermediate values of Vb the uniaxial ordering appears. This possibility has not been taken into account in the ground state calculations of ref 24 and we could not determine the limits of the gas-solid potential corrugation parameter Vb embracing the region of stability for this phase. The predictions based on the simple geometrical argument as proposed by Bruch and Venables33 may be helpful in determining whether such a phase is likely to appear for a given system. 3. Methodology Since our aim here is to study subtle changes in the system structure, we need to use appropriate tools. The usual calculations of such thermodynamic properties like the gas-gas and the gas-solid contributions to the total energy of the system and the heat capacity are very useful in determining the appearance of different phase transitions, but they do not give precise information concerning the microscopic changes in the system structure. Therefore, apart from those quantities mentioned above we also consider the radial distribution function, g(r), the distribution of particles relative to the lattice unit cell, n(τ*), reduced to a single lattice cell, as well as the appropriate bond-orientational order parameters.25-27,34-37 For the systems considered here we define three different bondorientational order parameters: ψ4, ψ6, and ψe6 given by the following expressions: (29) Abraham, F. F. Phys. Rev. Lett. 1980, 44, 463. (30) Phillips, J. M.; Bruch, L. W.; Murphy, R. D. J. Chem. Phys. 1981, 75, 5097. (31) Naidoo, K. J.; Schnitker, J.; Weeks, J. D. Mol. Phys. 1993, 80, 1. (32) Patrykiejew, A.; Borowski, P. Thin Solid Films 1991, 195, 367. (33) Bruch, L. W.; Venables, J. A. Surf. Sci. 1984, 148, 167. (34) Steinhardt, P. J.; Nelson, D. R.; Ronchetti, M. Phys. Rev. 1983, B28, 784. (35) Strandburg, K. J. Rev. Mod. Phys. 1988, 60, 161. (36) Strandburg, K. J. In Bond-Orientational Order in Condensed Matter System; Strandburg, K. J., Ed.; Springer: Berlin, 1992. (37) Weber, H.; Marx, D.; Binder, K. Phys. Rev. 1995, B51, 14636.

Patrykiejew et al.

ψk )

|

1

Nb

|

∑i ∑j exp[kiφij] , k ) 4, 6

and

ψe6 )

|

1

|

∑i ∑j cos[6φij]

Nb

(7)

(8)

In the above, the first sums runs over all particles in the system, the second sum runs over all nearest neighbors of the particle i, φij is the angle between the bond joining particles i and j and a chosen reference axis (assumed here to be the x-axis of the surface lattice), and Nb is the number of bonds in the system. Basically, in an ideal situation, when only pure phases are present, we can readily determine their nature by looking at the behavior of the bond-orientational order parameters. Namely, in the disordered, liquid or gas phase, all these parameters should be equal to zero since all possible mutual orientations of bonds appear in the system with the same probability. In the registered phase of a square symmetry, either the 1 × 1 or the x2 × x2, we expect that ψ4 ≈ 1 and both ψ6 and ψe6 should be equal to zero. In the hexagonal close packed phase, we have ψ4 ≈ 0 and ψ6 ≈ 1.0. The bond-orientational order parameter ψe6 may assume different values depending on the relative orientation of the adsorbate (hexagonal) lattice and the surface (square) lattice of the solid substrate.17,18,25 In the case of the uniaxially ordered hexagonal phase we may either have the same behavior of the bondorientational order parameters as in the hcp phase or all of them may take different values. The first situation occurs when the uniaxially ordered phase is just a simple hexagonal structure with all interparticle distances the same (no modulations due to gas-solid potential corrugation), and it only exhibits alignment along one of the surface symmetry axes x or y. When the hexagonal arrangement is not perfect, however, and the angle between the adsorbate phase lattice unit vectors is different than π/3, we expect nontrivial behavior of the bond-orientational order parameters. We also defines the corresponding susceptibilities associated with the introduced above bond-orientational order parameters

χk ) LxLy[〈ψ2k〉 - 〈|ψk|〉2]/T*, k ) 4, 6

(9)

and e 2 χe6 ) LxLy[〈ψe2 6 〉 - 〈|ψ6|〉 ]/T*

(10)

and the fourth-order cumulants

Uk,L ) 1 -

〈ψ4k〉

, k ) 4, 6 3 〈ψ2k〉2

(11)

and e U6,L

)1-

〈ψe4 6 〉 2 3 〈ψe2 6 〉

(12)

Numerous computer simulation studies25,37-40 have demonstrated that the behavior of the fourth-order (38) Binder, K.; Landau, D. P. Phys. Rev. 1984, B30, 1477. (39) Vollmayr, K.; Reger, J. D.; Scheucher, M.; Binder, K. Z. Phys. B 1993, 91, 113. (40) Challa, M. S. S.; Landau, D. P. Phys. Rev. B 1986, 33, 437.

Phase Transitions in Monolayer Films

cumulant of the order parameter for systems of different size is closely related to the nature of possible phase transitions. In particular, one expects that for the firstorder transition the cumulants for different system sizes have a common intersection point at the temperature of the phase transition. Besides, the value of the cumulant at this intersection, U*, is fixed for a given universality class. In particular, for the two-dimensional Ising model U* ≈ 0.61.41,42 For systems undergoing higher-order transition, the cumulants do not cross. At the temperatures below the transition point, the cumulants assume the same values for systems of different size, while different values are taken for higher temperatures. Thus, the transition point corresponds to the temperature at which the cumulants for systems of different size meet. 4. Monte Carlo Method The simulation has been performed using a standard Monte Carlo method in the canonical ensemble.43,44 Systems of different number of particles and different size of the simulation cell (with standard periodic boundary conditions in both directions) have been used. In general, we have started the simulation runs at very low temperatures, using the starting configurations corresponding to the various possible ordered states. The choice of the starting configuration for any particular system has been guided by the results of the ground state calculations, discussed in the following section. The final equilibrium configuration generated at one temperature has been then used as the starting configuration for the subsequent run at higher temperature. In some cases we have also performed freezing experiments starting the simulation at high temperature with the disordered (fluid) phase used as the initial state. In order to increase the efficiency of computations the gas-solid interaction energies have been evaluated for x* ) x/a and y* ) y/a ranging from 0 to 1 at the intervals of 0.02 and stored in an array that was the input to the simulation program. The gas-gas interactions have been calculated by a direct summation of the pair interactions within the circle of the radius rmax. To speed up the calculation we have used lists of neighbors, which have been updated every 5 to 10 Monte Carlo steps (per particle). The radius of the circle used to construct the neighbor lists has been larger than rmax by about 10 times the maximum allowed jump length. The jump length itself has been adjusted during the run to maintain the acceptance ratio of about 0.5 of the attempted moves. The number of Monte Carlo steps used in the calculation of averages ranged from 2 × 107 to 5 × 108 (per particle) and initially 2 × 107-108 Monte Carlo steps (per particle) have been used for equilibrating the system. 5. Ground State Properties Knowledge about the structure and properties of adsorbed films in the ground state (T* ) 0) is of interest by itself13,14,45,46 as well as it provides important information for the efficient use of computer simulation at finite temperatures. In particular, it allows construction of the (41) Binder, K. Z. Phys. B 1981, 43, 119. (42) Bruce, A. D. J. Phys. A 1985, 18, 873. Landau, D. P.; Stauffer, D. J. Phys. (Paris) 1989, 50, 509. (43) Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids; Clarendon Press: Oxford, 1987. (44) Allen, M. P.; Tildesley, D. J. In Computer Simulation in Chemical Physics, Allen, M. P., Tildesley, D. J., Eds.; Kluwer Academic Publishers: Dortrecht, 1993. (45) Bruch, L. W. Surf. Sci. 1985, 150, 503. (46) Borowski, P.; Partykiejew, A.; Sokołowski, S. Thin Solid Films 1989, 177, 333.

Langmuir, Vol. 13, No. 5, 1997 1039

starting configurations that are close to the equilibrium configuration at low temperatures. In the case of a flat (noncorrugated) surface, characterized by Vb ) 0, the problem of the ground state structure is trivial. The film forms the hcp solid phase and the energy (per particle) is given by the following expression:

e*0(σ*) )

1

u*(r0j) + V0* ∑ 2D(σ*) 2r ∈Ω

(13)

0j

where Ω is the circular region of the radius rmax, r0j is the distance between a chosen particle and the particle j, while 0* V2D (σ*) is the value of V2D(τ) for Vb ) 0 and a given value 0 of σ*. The superscript * at u and V2D refer to the reduced energy values with  used as a unit. Similarly, we define the reduced temperature T* ) kT/. The summation in the above eq 13 can be readily performed and we obtain 0 (σ*/r*0)12 - C06(σ*/r*0)6] + V0* e*0(σ*) ) 12[C12 2D(σ*)

(14)

and 0 r*0 ) σ*(2C12 /C06)1/6

(15)

0 and C06 depend on the assumed cutThe parameters C12 off distance, rmax, and for our choice (rmax ) 2.5σ) are equal to

0 ) 1.001632882 and C06 ) 1.058492141 (16) C12

and hence

r*0 ) 1.11218σ*

(17)

e*0(σ*) ) -3.355742387 + V0* 2D(σ*)

(18)

and

For the systems considered in this paper with *gs ) 2.0 and for the particles of size σ* ) 0.9 and 1.2, the values 0* (σ*) are equal to -8.538 and -12.207, respectively. of V2D Another trivial situation corresponds to the fully registered phases, in which all the adsorbed particles occupy centers of adsorption sites. In this case the energy (per particle) is given by r r e*r,R(σ*,Vb) ) 8[C12,R σ*12 - C6,R σ*6] + V*2D(τs)

(19)

r where τ* s ) (0.5,0.5). Presently, the parameters C12,R and r C6,R depend not only upon the cut-off distance, rmax, but also on the form of the ordered state, labeled by R. For the system with σ* ) 0.9 this registered state is just the 1 × 1 phase and

r r C12,1×1 ) 1.015997141, C6,1×1 ) 1.156625 (20)

while for the system with σ* ) 1.2 it corresponds to the x2 × x2 phase with r r C12, x2×x2 ) 0.01587295532, C6,x2×x2 ) 0.142578125 (21) r and Cr6,R into eq 19 we Inserting the above values of C12,R obtain

e*r,1×1(0.9,Vb) ) -2.62184276 + V*2D(τ*s)

(22)

1040 Langmuir, Vol. 13, No. 5, 1997

Patrykiejew et al.

and

e*r,x2×x2(1.2,Vb) ) -2.273689088 + V*2D(τ*s) (23) respectively. In the case of the hexagonal solid phase formed on the corrugated square substrate we also have to take into account that the film may exhibit the lowest energy for the nearest neighbor distances slightly different than in the case of a flat (noncorrugated) surface. Moreover, this phase may be rotated with respect to the substrate lattice by a certain angle θ. Of course, other distortions from the ideal hexagonal arrangement may appear as well. In particular, the film may be uniaxially ordered. For simplicity, we assume here that all the nearest neighbor distances are the same and equal to, say, r* min in the state of minimum energy e*min. This minimum is then defined as

[∑

1 N V*2D(τ* e*min ) lim min min i) + egg(r* NN) Nf∞ θ r* Ni)1 NN

]

(24)

where r*NN is the nearest neighbor distance and N is the number of particles in the system. Our calculations have shown that the regions of stability of the registered and hcp phases are only very slightly affected by the orientational effects. Also, taking into account that the ground state calculations have only aimed at the determination of appropriate low temperature starting configurations for the Monte Carlo simulation, we have assumed that θ ) 0. With this assumption, we have found that the hcp phase is stable for the corrugation parameter Vb lower than about 0.45 and 0.55 for σ* ) 0.9 and 1.2, respectively.

Figure 1. Heat capacities for the films of particles with σ* ) 1.2 at the density F* ) 0.72 and for different values of the gas-solid potential corrugation parameter Vb and the different system size.

6. Results for Finite Temperatures A. σ* ) 1.2. We begin the presentation of results for the systems with particles of σ* ) 1.2 and the surface corrugation high enough to stabilize the registered x2 × x2 phase at low temperatures. From the ground state calculations, presented in the previous section, it follows that for the systems of densities not exceeding the density of fully filled registered layer and equal to F* ) 0.72, the registered phase is stable at T* ) 0 for Vb g 0.55. Here we consider three systems characterized by the values of the parameter Vb equal to 0.6, 0.8, and 1.0. The basic results for these systems are presented in Figures 1 and 2. It appears that the qualitative behavior of all these systems is quite the same. The heat capacities exhibit finite peaks with maxima located at practically the same temperatures as observed for the corresponding threedimensional systems discussed in ref 24. From the temperature changes of the bond-orientational order parameter ψ4, shown in Figure 2, it clearly follows that the systems considered here undergo quite abrupt, though continuous, disordering as the temperature increases. It is noteworthy that the results presented in Figure 2, obtained for the systems of different size, are the same. The absence of finite size effects is clear evidence that no long range fluctuations develop. The finite size effects are only manifested in the behavior of the residual hexagonal orientational order shown in Figure 3, but the observed decrease of ψ6 with the increase of the system size does not result from the presence of long range fluctuations, but merely reflects the fact that we are gradually approaching the thermodynamic limit. Thus, the conclusion concerning the disordering of the registered x2 × x2 phase as resulting from the transition similar to the order-disorder transition in the Ising model, which we have presented in ref 24 seems to be incorrect. The

Figure 2. Bond-orientational order parameters ψ4 for the same systems as in Figure 1.

present results suggest that the destruction of the registered phase is a gradual process bearing some resemblance to the localized-to-mobile transition. It should be noted that the properties of the presently discussed two-dimensional systems are only slightly different from the properties of three-dimensional systems considered in ref 24. Figure 4 presents the gas-gas and the gas-solid contributions to the total energy for the two- and three-dimensional systems of different surface corrugation. The effects of the out-of-plane motion are manifested by a considerably more rapid decrease of the gas-solid interaction energy in three-dimensional systems, while the gas-gas interaction is practically the same in the considered temperature range. One might expect that in two dimensions the disordering transition should be sharper than in three dimensions. This is so because the suppression of the out-of-plane motion makes the disordering of the registered structures look more like

Phase Transitions in Monolayer Films

Langmuir, Vol. 13, No. 5, 1997 1041

Figure 3. Bond-orientational order parameter ψ6 for the systems of different size and characterized by σ* ) 1.2 and Vb ) 0.6.

the order-disorder transition in the lattice gas model.47 The observed behavior of the discussed here twodimensional systems and the properties of the earlier considered24 three-dimensional systems, together with the aforementioned lack of the finite size effects, support the view that disordering is a gradual (continuous) process rather than a phase transition of any kind. Thus, our interpretation of the observed heat capacity maxima for three-dimensional systems as marks of the Ising-like order-disorder phase transition24 has been not correct. The situation changes considerably for lower corrugation of the gas-solid potential, when the stable solid phase at low temperatures has the incommensurate hexagonal close packed structure. Figure 5 presents the heat capacities obtained for systems of different size and characterized by Vb ) 0.2 and 0.4. In both cases we find very sharp peaks with the maxima slightly shifted toward higher temperatures and considerably higher when the size of the system increases. The observed transition is the melting of the hexagonal solid phase into a more-orless isotropic liquid phase. This statement is well supported by the changes in the behavior of the radical distribution function for the system with Vb ) 0.2, shown in Figure 6. In the case of Vb ) 0.4 we observe very similar changes in the behavior of g(r) with temperature. The melting temperature for the system with Vb ) 0.2 is equal to T*m ≈ 0.385, while for the system with Vb ) 0.4 it is slightly lower and equal to T*m ≈ 0.375. This result agrees very well with the changes in the melting temperature observed in the studied earlier three-dimensional systems.24 Although the melting transition seems to have the same mechanism in both systems considered here, there are important differences in the inner structure of the respective solid phases. We first consider the case of Vb ) 0.2. Figure 7 presents the behavior of the bondorientational order parameters ψ6, ψe6, and ψ4 and demonstrates that ψe6 exhibits a more sudden drop at lower temperatures than ψ6. This indicates that the solid phase undergoes a rotation relative to the substrate lattice prior melting. The simulation runs performed for increasing and decreasing temperature have shown that the observed rotation is reversible with respect to the direction of the (47) Patrykiejew, A.; Borowski, P. Thin Solid Films 1991, 195, 367.

Figure 4. Gas-gas (a) and the gas-solid (b) contributions to the total energy for the same systems (with N ) 200) as in Figure 1 obtained for the two- and three-dimensional models.

Figure 5. Heat capacities for the systems of different size, with σ* ) 1.2 and Vb ) 0.2 (a) and 0.4 (b).

temperature changes. Of course, to assure the reversibility of the observed effect requires sufficiently long runs. In our simulations, with 108 Monte Carlo steps (per particle) we have not observed any appreciable effects due to metastability. At very low temperatures the solid phase appears to be uniaxially registered but without true

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Figure 6. Radial distribution function for the system of 450 particles with σ* ) 1.2 and Vb ) 0.2 at different temperatures (shown in the figure).

Patrykiejew et al.

Figure 8. Snap-shots of equilibrium configurations for the system of 450 particles with σ* ) 1.2 and Vb ) 0.2 at different temperatures: T* ) 0.275 (a), 0.3 (b), 0.375 (c), and 0.4 (d).

Figure 9. An example of equilibrium configuration for the system of 450 particles with σ ) 1.2 and Vb ) 0.4 at T* ) 0.175. Figure 7. Bond-orientational order parameters ψ6, ψe6, and ψ4 for the systems of 450 particles with σ* ) 1.2 and Vb ) 0.2.

site adsorption, while at higher temperatures, just below the melting point, it loses this uniaxial order. The radial distribution functions obtained for the temperatures below (T* ) 0.35) and above (T* ) 0.375) the rotation temperature (see Figure 6) demonstrate that the inner structure of the solid phase is only slightly affected by the film rotation. Direct evidence of the film rotation is provided by the examples of equilibrium configurations shown in Figure 8. We find that already at T* ) 0.3 the film is slightly rotated. In the case of the equilibrium configuration shown in Figure 8b, the estimated angle of the film rotation is equal to about 1.9°. It changes slightly as the temperature increases and jumps to about 16.7 ( 0.2° at T* ) 0.375. The configuration at T* ) 0.4 already corresponds to a liquid phase. Figure 7 shows also that the localization of adsorbed particles over the centers of adsorption sites increases slightly upon melting. This is also supported by the calculations of the particle distribution function n(τ*), which exhibits a pronaunced maximum at the center of the surface lattice unit cell above the melting temperature. The system characterized by Vb ) 0.4 also exhibits uniaxially registered hexagonal solid phase at low tem-

peratures. In this case, however, the allignment occurs along the line x ) y, and hence the adsorbed layer rotation angle equals 45°. This is demonstrated in Figure 9, which presents an example of the equilibrium configuration at low temperature equal to T* ) 0.175. For a hexagonal solid phase rotated by 45°, the bond-orientational order parameters ψe6 and ψ4 should be close to zero. Indeed, the results presented in Figure 10 indicate that at sufficiently low temperatures these two order parameters do approach zero. On the other hand, the order parameter ψ6 is quite high, as expected for the hexagonal arrangement of adsorbed particles. The particle distribution function n(τ*), calculated for the temperatures below the melting point confirm the above picture, as well as demonstrate that, on the average, the adsorbed particles occupy positions far from the centers of the surface lattice unit cell (see Figure 11a). A considerable increase in ψ4 upon melting is observed, however. This indicates that the liquid phase exhibits partial localization over the gassolid potential minima. This is clearly demonstrated in Figure 11b, which shows the particle distribution function n(τ*) obtained at T* ) 0.40, i.e., above the melting point located at T* ≈ 0.36. It should be noted that the results presented in Figure 10 have been obtained from two independent runs corresponding to the increasing and decreasing temperatures. The first run has been started

Phase Transitions in Monolayer Films

Figure 10. Bond-orientational order parameters ψ6, ψe6, and ψ4 for the systems of 450 particles with σ* ) 1.2 and Vb ) 0.4.

Figure 11. Distribution function n(x,y) for the system of 450 particles with σ* ) 1.2, Vb ) 0.4 at T* ) 0.325 (a) and 0.4 (b).

at T* ) 0.05 with the rotated by 45° hcp phase, while the starting configuration for the second run has been the disordered fluid phase at T* ) 0.70. No effects due to metastable states have been found and the bond-orientational order parameters as well as the thermodynamic properties indicate that the observed transition is a reversible process.

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Figure 12. Changes in the melting (disordering) temperature with the film density for systems of 450 particles with σ* ) 1.2 and different values of the corrugation parameter Vb (shown in the figure).

The calculations performed for the systems of different densities are summarized in Figure 12. From our results it follows that the disordering of the registered phase of a square symmetry occurs at lower temperatures when the density of the film decreases with respect to the density corresponding to the completely filled layer with all adsorption sites occupied. For the systems with the hexagonal solid phase, the decrease in the density does not lead to any shift of the melting point. In this case, however, the transition observed in our simulation becomes sharper as the film density decreases. On the other hand, when the film density is higher than the density of the fully filled registered layer, the melting of the hexagonal phase is considerably shifted toward higher temperatures and appears to be continuous. In the systems considered here the film density slightly exceeds the density of the triple point for the two-dimensional Lennard-Jones system on a noncorrugated surface. A sharp increase in the melting point temperature indicates that the triple point density for the systems considered here with weakly corrugated gas-solid potential is located about F* ) 0.8. We have not attempted to determine the shift in the triple point density resulting from the presence of periodic potential, but it seems that this effect is not high for weakly corrugated surfaces. In the case of registered films formed on highly corrugated surfaces, the points shown in Figure 12 correspond to the temperatures at which the heat capacity reaches maximum. For comparison, we have also included points at which the heat capacity for a single particle reaches maximum. This maximum corresponds to the localized-to-mobile transition discussed by Doll and Steele.12 It is seen that for highly corrugated surfaces the temperature at which the heat capacity reaches maximum changes smoothly with the film density. On the other hand, the behavior of weakly corrugated surfaces resembles the behavior of a twodimensional Lennard-Jones system on the flat surface. The melting temperature remains constant over a wide range of densities, and this corresponds to the melting at the triple point temperature. The fact that isolated atoms adsorbed on such weakly corrugated surfaces exhibit the localized-to-mobile transition at considerably lower temperatures demonstrates that the stability of the hcp phase results from the mutual attractive interaction between the adsorbed particles. The system characterized by the value of the gas-solid corrugation parameter Vb equal to 0.6 shows somewhat peculiar behavior. Namely, the

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Figure 14. Bond-orientational order parameters ψ4 and ψ6 for the system characterized by σ* ) 0.9, Vb ) 0.6, and N ) 224.

Figure 13. Bond-orientational order parameters ψ4 (a) and ψ6 (b) for the systems characterized by σ* ) 0.9, different value of Vb (shown in the figure) and different size (O, N ) 224; N ) 504; ], N ) 896).

location of the heat capacity maximum appears at practically the same temperature over a wide density range, as in the systems with only weakly corrugated surfaces. On the other hand, the behavior of the bond-orientational parameters, their higher moments, and the heat capacity suggest that the film disorders gradually, quite similar to that found for highly corrugated surfaces. B. σ* ) 0.9. In general the results obtained for systems exhibiting the registered phase at low temperatures and the density corresponding to the completely filled layer are quite similar to the results obtained for the previously discussed systems of particles with σ* ) 1.2. Thus, we do not find any phase transitions but only gradual disordering. Thus, the present results are consistent with the earlier finding for three-dimensional systems presented in ref 24. In three dimensions, however, the lack of any trace of a phase transition for the completely filled registered films for particles with σ* ) 0.9 resulted from the desorption of adsorbed particles at quite low temperatures. This effect produced broad heat capacity maxima and it could be suspected that the effects due to disordering of the monolayer were masked. In two dimensions, however, desorption cannot take place and the observed changes in the thermodynamic properties can be attributed to the changes in the state of adsorbed film. Figure 13 presents the behavior of the bondorientational parameters ψ6 and ψ4 for the systems characterized by the values of Vb equal to 0.6, 0.8, and 1.0, and obtained for systems of different size. Similarly as for the registered films of particles with σ* ) 1.2, we do not observe any finite size effects and the films exhibit gradual disordering. Comparing the results for Vb ) 0.8 and 1.0, presented in Figure 13, with the corresponding results for σ* ) 1.2, given in Figure 1, we find that the disordering of registered films of particles with σ* ) 0.9 occurs over a considerably wider temperature range. This

can be attributed to the fact that the density of the completely filled layer is higher in the present case. This considerably hinders diffusion of the adsorbed particles. The calculations performed for systems of lower density have shown that disordering becomes sharper and occurs at considerably lower temperatures. In the case of Vb ) 0.6, the disordering of the registered film starts at very low temperatures, however, and is accompanied by the formation of uniaxially ordered film. This process is continuous and results from a gradual increase of the thermal energy of adsorbed particles. The stability of uniaxial ordering is limited to a rather narrow temperature range and this structure disorders gradually as the temperature increases. For densities lower than the density corresponding to the completely filled registered layer, the system with Vb ) 0.6 forms a uniaxially ordered state at still lower temperatures. This follows from the direct examination of the equilibrium configurations as well as from the behavior of the bond-orientational order parameters shown in Figure 14. The disordering of this structure is again a continuous process. The systems will still lower corrugation, characterized by Vb ) 0.3 and 0.1 form the hexagonal closed packed solid phase at low temperatures. In the case of Vb ) 0.3 we find a quite sharp first-order melting transition located at T* ≈ 0.3. Although the bondorientational order parameter ψ6 does not exhibit a discontinuity at the transition point (see Figure 15a), the smooth behavior results from the strong finite size effects. Convincing proof of the first-order character of melting in this system comes from the behavior of the fourth-order cumulants U4(L) and U6(L). In the case of first-order transition the cumulants for systems of different size should exhibit a common intersection point at the transition temperature.38-41 The results presented in Figure 16 show that both U4(L) and U6(L) possess common intersection points at the reduced temperature T* ≈ 0.3. It is noteworthy, that the bond-orientational order parameter ψ4 increases considerably upon melting (see Figure 15b). Thus, the liquid phase appears to be partially ordered due to the effects of the surface potential. Again, this result is similar to the results obtained for the system of particles with σ* ) 1.2 and Vb ) 0.3. On the other hand, the bond-orientational order parameter ψe6 (not presented here) is practically the same as ψ6 over the entire temperature range. This shows that the adsorbate layer orientation coincides with one of the symmetry axes of the substrate surface.

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Figure 17. Bond-orientational order parameters ψ6 and ψe6 for the system with Vb ) 0.1 and the size of the simulation cell 32 × 28 containing 894 particles as well as the fourth-order cumulants U6(L) for the same system but for different sizes of the simulation cell.

Figure 15. Bond-orientational order parameters ψ6 (a) and ψ4 (b) for the systems with V6 ) 0.3 and different size of the simulation cell.

Figure 18. Radial distribution functions for the system of 896 particles of the size σ* ) 0.9 adsorbed on the surface characterized by the corrugation parameter Vb ) 0.10 at three different temperatures, shown in the figure.

Figure 16. Fourth-order cumulants U4(L) (a) and U6(L) (b) for the systems with Vb ) 0.3 and different size of the simulation cell.

For the lowest, considered here, corrugation of the surface potential with Vb ) 0.1, the mechanism of melting changes again and appears to be a two-stage process. Figure 17 presents the behavior of the bond-orientational order parameters ψ6 and ψe6 for the largest systems considered here. The results are quite consistent with

the predictions of the KTHNY theory.26-28 At T* ≈ 0.4 we observe a sudden, though small, drop of the bondorientational parameters. Thus, the system retains a considerable orientational order at temperatures above this transition. Then, the second transition occurs at T* ≈ 0.485, which is connected with the loss of orientational order in the system. The KTHNY theory predicts that the melting of a two-dimensional film on a weakly corrugated square substrate should be a two-stage process. The first transition, connected with the dissociation of dislocation pairs, transforms the solid phase into a hexatic phase, quite similar to the case of a smooth substrate. The subsequent, disclination-unbinding transition predicted for the smooth substrate should be replaced by an Ising-like transition. Indeed, from the behavior of the cumulants U6(L) obtained for systems of different size (see Figure 17) we find that a common intersection point U* occurs at T* ≈ 0.485 ( 0.005 and U* 6 ≈ 0.61 has the value corresponding to the universality class of the twodimensional Ising model.42 One should also note that the cumulants U6(L) are practically size independent for the temperatures around the first transition observed at T* ≈ 0.4. Only when the temperature exceeds about 0.45 do the size effects become visible. This is consistent with

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the prediction starting that U6(L) is size independent over the temperature range corresponding to the KosterlitzThouless phase. This point requires further study, however. In particular, simulation for still larger systems might considerably help to resolve the question of the mechanism of the observed phase transitions. Figure 18 shows the radial distribution functions calculated for three temperatures corresponding to different regimes. It is evident that already the first transition, occurring at T* ≈ 0.4, destroys the quasi-long-range order characterizing the two-dimensional solid phase. This finding, presented in ref 25, is of interest as it provides a clear hint that the mechanism of melting on a weakly corrugated square substrate agrees very well with the predictions made by Nelson and Halperin.26,27 Also, the observed first-order melting on the surface with slightly higher corrugation (Vb ) 0.3) is consistent with the prediction of that theory. Preliminary results obtained for the film formed on the surface with the corrugation parameter Vb equal 0.1, but at the lower density equal to F* ) 0.70875 suggest that the mechanism of melting changes and becomes a firstorder transition. This transition occurs at considerably lower temperature (T* ≈ 0.36 ( 0.01) than the first transition in the film of the density F* ) 0.81. Moreover,

Patrykiejew et al.

the low temperature hcp solid phase appears to be slightly rotated relative to the substrate lattice. It is of interest to note that the observed behavior of the low density twodimensional film appears to be very similar to the behavior of the high density (F* ) 0.81) film in a three-dimensional system.24 In the latter case we have also found a sharp transition at low temperature and observed gradual rotation of the solid phase as the temperature was raised. This similarity can be understood by considering the effects due to out-of-plane motion of adsorbed particles in the three-dimensional system. As the temperature increases, the adsorbed particles gain higher thermal energy and the adsorbed layer loses its compactness. Some of the particles are displaced toward larger distances from the solid surface. This phenomenon produces the apparent lowering of the film density. Besides, the film density may be lowered due to desorption of adsorbed particles. Acknowledgment. A grant for computer time from the ICM (Warsaw University, Poland) is gratefully acknowledged. LA9508203