Melting and Low-Temperature Structures of Mixed ArâKr Monolayer

ARTICLE pubs.acs.org/JPCC

Melting and Low-Temperature Structures of Mixed ArKr Monolayer Films on Graphite A. Patrykiejew,* W. R_zysko, and S. Sokozowski Department for the Modelling of Physico-Chemical Processes, Faculty of Chemistry, MCS University, 20031 Lublin, Poland ABSTRACT: Using Monte Carlo simulation methods in the canonical and grand canonical ensembles, we study the melting and the structure of low-temperature phases of mixed ArKr submonolayer films on graphite. It is shown that such films exhibit a complete mixing in the liquid phase and freeze into a mixed solid phase, independently of the composition. The structure of the solid phase, however, depends upon the film composition, its total density, and the temperature. For submonolayer coverages, when the mole fraction of Kr is lower than about 0.1, the mixture freezes into the incommensurate, argonlike phase. For the√higher√mole fractions of Kr, the freezing leads to the formation of a mixed commensurate ( 3 3)R30 phase. The lowering of temperature leads to structural phase transitions in the solid. When the krypton mole fraction is lower than about 0.88, the transition leads to the formation of domain-wall structures, in which the commensurate domains are made of krypton atoms, while the composition of walls depends upon the Kr mole fraction. It is shown that even rather small concentrations of argon atoms can trigger the commensurateincommensurate transition. For still higher Kr mole fractions, exceeding about 0.88, the commensurate, krypton-like, solid phase is stable at any temperatures below the melting point. At sufficiently low temperatures, the phase separation takes place and argon atoms are removed from the film interior to the peripheries of submonolayer. In the case of films with the total density close to the monolayer completion, the commensurate structure shows much higher stability. It is demonstrated, however, that it is an artifact of the simulation methods used and, in particular, of the periodic boundary conditions applied, rather than a real phenomenon. It is also demonstrated that the phase diagram topology of monolayer films changes with the film composition. In particular, the vaporliquid critical point appears only when krypton concentration is lower than about 0.45.

I. INTRODUCTION The adsorption of argon and krypton on graphite has been extensively studied for several decades using various experimental methods,113 computer simulations,1423 and theory.2430 The phase diagrams of monolayer films of Ar and Kr are now rather well-established. In particular, it is known that argon submonolayer films on graphite exhibit two-dimensional gas, liquid, and solid phases. The critical temperature of the gasliquid transition is located at Tc = 57 ( 2 K,8,31 and the triple point temperature is equal to Tt ≈ 49.7 K.32 It is commonly accepted that submonolayer argon films freeze into the hexagonally ordered two-dimensional floating solid phase being incommensurate with the graphite lattice (IC phase). Both experiment2,3,8,9,33 and computer simulation16,23 provide rather convincing evidence that the melting of submonolayer argon films on graphite is a continuous phase transition. Low energy electron diffraction (LEED)34 and X-ray scattering32,35 measurements as well as Monte Carlo simulation22,23 have also demonstrated that the solid argon submonolayer films are rotated by about √ 23 √ with respect to the R30 axis of the commensurate ( 3 3)R30 structure. This effect, known as epitaxial rotation, was for the first time predicted by the theory of Novaco and McTague.24,25 On the other hand, the submonolayer films of krypton formed √ on the √ graphite basal plane order into the commensurate ( 3 3)R30 structure (C phase). The phase diagram of r 2011 American Chemical Society

krypton on graphite does not exhibit the usual vaporliquid transition, and hence the vaporliquid critical point does not exist.10,11,35,36 Instead, the low-density fluid phase is partially ordered due to rather strong effects of periodic variations of the kryptongraphite interaction potential and small natural misfit (about 6%) between krypton atoms and the array of adsorption sites on the graphite surface. The fluid undergoes a transition into the C (solidlike) phase. The melting of commensurate krypton films seems to be quite consistent with the model assuming the increases existence of incipient triple point.37 When √ the density √ beyond that corresponding to a perfect ( 3 3)R30 phase, the C phase undergoes a phase transition, leading to the formation of incommensurate phase consisting of commensurate domains separated by superheavy walls, which carry the excess density.19,35,36,38 The thickness of those walls gradually increases with the film density, and ultimately the film forms an incommensurate phase, being a triangular lattice with a lattice constant close to that of bulk krypton.17 The differences in the structure and the phase behavior of argon and krypton monolayer films on graphite are expected to considerably affect the phase behavior of mixed films. Unlike the mixed ArXe and KrXe films on graphite,3943 the ArKr Received: August 29, 2011 Revised: November 9, 2011 Published: November 17, 2011 753

dx.doi.org/10.1021/jp208323b | J. Phys. Chem. C 2012, 116, 753–763

The Journal of Physical Chemistry C

ARTICLE

Table 1. Lennard-Jones Parameters for Ar and Kr Used in This Work i,j

σi,j (Å)

εi,j (K)

Ar,Ar

3.4

120.0

Kr,Kr Ar,Kr

3.6 3.5

171.0 142.83

Table 2. Parameters Describing the ArGraphite and Kr Graphite Interaction, Obtained Using the LorentzBertholot Combining Rules and Assuming that εc,c = 28 K and σc,c = 3.4 Å

mixed films on graphite have not been experimentally studied yet. The only experimental study of submonolayers formed by ArKr mixture on well-defined solid surface was performed by Zeppenfeld et al.44 Using the He diffraction method they investigated the structure of ArKr monolayer films on Pt(111) surface over a range of temperatures as well as for different composition of the adsorbed layer. One interesting finding was the observation that krypton dissolves rather easily in argon patches, while dissolution of argon in krypton patches is considerably hindered. It was also shown that the average lattice parameter of the mixed film varies linearly with the film composition. Moreover, the demixing transition has been observed at sufficiently low temperatures. One should note, however, that the behavior of pure argon and krypton on Pt(111) surface is considerably different than that observed on the graphite basal plane. Namely, on the Pt(111) surface, both adsorbates were found to form hexagonally ordered incommensurate phases,45 and argon was also shown46 to exhibit different high-order commensurate structures. The solidlike submonolayer films of ArKr mixture adsorbed on the graphite basal plane are expected to show strong effects due to the competing tendencies to form the incommensurate (Ar) and commensurate (Kr) structures. In particular, one expects that the increasing krypton concentration should lead to the incommensurate (IC)commensurate (C) transition. Some aspects of the melting transition of a mixed ArKr monolayer on graphite have already been studied by the molecular dynamics method by Roth,47 and we shall briefly comment on that work later in section IV. In this work we report on the results of extensive Monte Carlo simulation of submonolayer and monolayer mixed films of argon and krypton on graphite. Our main goal has been to investigate the structure of low-temperature solid phases and to determine the changes in the melting temperature with the mixture composition; nevertheless, we have also investigated the evolution of the phase diagram topology with the film composition. The paper is organized as follows. In the next section (II) we present the model and describe the Monte Carlo methods used to determine the properties of mixed films. Then, in section III we briefly discuss the behavior of pure Ar and Kr submonolayer film. The results for mixed films are presented and discussed in section IV. The final section V contains some concluding remarks.

εgs,i (K)

σi,c (Å)

Ar

58.0

3.4

Kr

69.2

3.5

parameters representing the ArKr interaction, also given in Table 1, have been obtained using the usual LorentzBertholot combining rules, and the potential (eq 1) has been cut at the distance 3σi,j. The interaction of rare gas atoms with the graphite basal plane can be represented by the potential48 vi ðx, y, zÞ ¼ εgs, i ½vo, i ðzÞ þ

∑k vk, i ðzÞ fk ðx, yÞ,

i ¼ Ar or Kr

ð2Þ

The first term in the square brackets is the average over the entire surface fluidsolid potential, while the second term represents the periodic, corrugation, part of the fluidsolid potential. The explicit expressions for vo,i(z), the Fourier components vk,i(z), and the functions fk(x,y) are given in ref 48. The values of parameters entering the above potential function are given in Table 2. Throughout this paper we use reduced quantities, assuming that the graphite lattice constant, a1 = 2.46 Å, is the unit of length, and the Lennard-Jones parameter εAr,Ar is the unit of energy. In this work we consider only the films of the total density not exceeding one monolayer at rather low temperatures, so that the promotion of the second layer is likely to be negligibly small. This allows us to consider a simple, strictly two-dimensional model with the external field of the form vðx, yÞ ¼ Vb, i f1 ðx, yÞ=2 ¼ Vb, i ½cosðq1 rÞ þ cosðq2 rÞ

þ cosððq1 q2 ÞrÞ

ð3Þ

where q1 and q2 are the reciprocal lattice vectors of the graphite surface and the parameter Vb,i (i = Ar, Kr) determines the amplitude of the corrugation potential. The magnitudes of Vb,i have been adjusted in such a way that the results for pure Ar and Kr films were consistent with those obtained using the full 3D gassolid potential. Figure 1a presents a comparison of the bond-orientational order parameter, Ψ6 (see eq 7 below), and its susceptibility (given in the inset to Figure 1a) for the argon submonolayer film on graphite at the density Fc = 0.65 (the density Fc = 1.0 corresponds to the perfect commensurate phase, in which one adsorbate atom occupies one of every three carbon hexagons) obtained using the surface potentials given by eqs 2 and 3. The results for the threedimensional model have been obtained using different number of Fourier terms (k = 5 and 1) in the expansion (eq 2), while in the two-dimensional model we have used different values of Vb,Ar. It is evident that for Vb,Ar = 0.07, also used in ref 15, the twodimensional approximation with only the first leading Fourier term works quite well. Of course, the choice of Vb,Ar = 0 corresponds to a uniform two-dimensional Lennard-Jones fluid, which for the density used melts at the triple point located at T ≈ 0.4. In the case of krypton, we have found that Vb,Kr = 0.08 leads to a very good agreement with three-dimensional calculations (see Figure 1b). However, the earlier study has demonstrated,38 and the same has been confirmed by our calculations, that the potential (eq 2) underestimates the corrugation part of the

II. THE MODEL AND MONTE CARLO METHODS We assume that the interaction between adsorbate atoms is represented by the (12,6) Lennard-Jones potential ui, j ðrij Þ ¼ 4εi, j ½ðσ i, j =rij Þ12 ðσi, j =rij Þ6

i

ð1Þ

where rij is the distance between a pair of atoms and i and j mark the species, Ar and Kr. The values of the parameters εi,i and σi,i used in this work are given in Table 1. The corresponding 754

dx.doi.org/10.1021/jp208323b |J. Phys. Chem. C 2012, 116, 753–763


ARTICLE

Figure 1. (a) Temperature changes of the bond-orientational order parameter, Ψ6, for submonolayer argon film of the density Fc = 0.65 obtained using the three-dimensional model with the liquidsolid potential given by eq 2 with k = 5 (open circles) and k = 1 (open squares) and using the twodimensional model with the fluidsolid potential given by eq 3 with different values of the corrugation parameter Vb,Ar (given in the figure). The inset shows the plots of χΨ6 for the three-dimensional models with five and one Fourier terms in eq 2 and for the two-dimensional model with Vb = 0.07. (b) Temperature changes of the bond-orientational order parameter,Ψ6, for submonolayer krypton film of the density Fc = 0.40 obtained using the three-dimensional model with the fluidsolid potential given by eq 4 with k = 1 and with VD,Kr = 1.0 (open squares) and 1.5 (filled diamonds) and using the two-dimensional model with the fluidsolid potential given by eq 3 with the corrugation parameter Vb,Kr = 0.08 and 0.12. The inset shows the plots of χΨ6 for the three-dimensional model and for the two-dimensional model.

fluidsolid interaction and leads to the formation of incommensurate submonolayer films, which exhibit the domain-wall structure. This contradicts experimental data indicating that such structures appear only for Fc > 1.0. In order to properly describe the behavior of submonolayer krypton films it is necessary to modify the gassolid potential (eq 2) to the form49 vi ðx, y, zÞ ¼ εgs, i ½vo, i ðzÞ þ VD, i

∑k vk, i ðzÞ fk ðx, yÞ

and appropriate order parameters. The formation of hexagonally ordered phases has been monitored using the bond-orientational order parameters53,54 " # 1 Ψ6, i ¼ exp i6ϕm, n ð6Þ Nb, i mi ni

∑∑

ð4Þ

measured separately for each adsorbate (i = Ar or Kr). In the above, the first sum runs over all atoms of the ith component, the second sum runs over all nearest neighbors of the same type, ϕm,n is the angle between the bond joining the atoms m and n and an arbitrary reference axis, chosen here to be the x-axis of the simulation cell, and Nb,i is the number of bonds between two like atoms. Also, we have monitored the total bond-orientational order parameter " # 1 ð7Þ exp i6ϕm, n Ψ6 ¼ Nb m n

and put VD,Kr = 1.5.50 Also, in the two-dimensional model, the amplitude of the surface potential, Vb,Kr must be properly adjusted, and the results given in Figure 1b demonstrate that the choice of Vb,Kr = 0.12 leads to a very good agreement of twoand three-dimensional models. Simulations have been carried out using Monte Carlo methods in the canonical as well as grand canonical ensembles.51,52 In the case of the two-dimensional model, a rectangular simulation cell of the size √ La1 L 3a1/2, with L = 60, 90, and 120 and with the standard periodic boundary conditions, has been used. Three dimensional calculations have been performed for the simulation cell being a √ rectangular parallelepiped of the size 60a1 60a1 3/2 10a1, with periodic boundary conditions in the directions parallel to the substrate surface and the reflecting hard wall located at z = 10a1. The quantities recorded included the average potential energy, Æeæ; the contribution to the potential energy due to the fluid fluid interaction, Æeggæ; and the contributions due to the fluid solid interaction for each component, Æega,iæ, i = Ar, Kr; and the heat capacity

∑∑

ð5Þ

where the first sum runs over all atoms in the system and the second over all nearest neighbors. The above-defined bond-orientational order parameters allow one to detect hexagonally ordered structures, but they are not suitable to distinguish the commensurate and incommensurate phases. In the commensurate phase, the atoms are localized close to the centers of carbon hexagons, and the appropriate order parameter allowing to monitor such localized structures can be defined as47 " # 1 6 Φi ¼ exp iqn rm, i ð8Þ 6Ni m n ¼ 1

In order to study the structure of solid phases we have used radial distribution functions, gij(r), for ij = ArAr, ArKr, and KrKr,

The first sum is taken over all atoms of the ith component, while the second sum runs over the six reciprocal lattice vectors qn of

CV ¼

N ½Æe2 æ Æeæ2 kT 2

∑∑

755



ARTICLE

Figure 2. Temperature changes of the liquidsolid contribution to the potential energy (a), the heat capacity (b), the bond-orientational order parameter Ψ6 (c), and its susceptibility (d) for submonolayer films of argon (filled circles) and krypton (squares) of the density Fc = 0.4. In the case of krypton, we have used two different values of Vb,Kr equal to 0.08 (filled squares) and 0.12 (open squares). The inset to part b shows the low-temperature behavior of the potential energy of krypton film in the region of the CIC transition observed for Vb,Kr = 0.08.

Figure 3. The phase diagram, in the (T*,xKr) plane for the films of different total density and obtained using two and three-dimensional model. Filled black circles, filled black squares, and filled black diamond mark the melting points, the transition between the mixed commensurate phase (Cm) and the domain-wall incommensurate phase (ICAr + CKr), and the onset of the phase separation leading to the demixed commensurate phase Cd, respectively, obtained using the two-dimensional model, with Vb,Kr = 0.12, for the submonolayer of the total density Fc = 0.4. Open black circles and squares correspond to the melting and the Cm ICAr + CKr transition, respectively obtained for the film of the total density Fc = 0.5 within the three-dimensional model with VD,Kr = 1.5. Dark (light) shaded circles and squares correspond to the melting and the Cm ICAr + CKr transition respectively obtained for the film of the total density Fc = 0.8 (Fc = 1.0) within the three-dimensional model with VD,Kr = 1.5. Filled triangles are the locations of the vaporliquid critical temperatures obtained from the grand canonical simulation, and the open triangle marks the experimental value of the argon critical temperature. The dashed lines are only a guide for the eye.

the graphite substrate and rm,i is the position of the mth atom of component i. The above-defined order parameters have been supplemented by the corresponding susceptibilities χop ¼

Lx Ly ½Æop2 æ Æopæ2 kT

ð9Þ

where op stands for any of the above given order parameters. In the case of grand canonical simulation, we have also recorded the adsorption isotherms and the local density profiles, averaged over the substrate surface, F(z).

Our simulations have also shown that pure argon film forms incommensurate (floating) solid phase, which exhibits epitaxial rotation.2325 Part a of Figure 2 demonstrates that the adsorbate substrate interaction energy for the Argraphite system reaches a shallow minimum just below the melting point,and remains quite small even at very low temperatures. One should note that our model predicts that in the perfect commensurate phase ugs = 3Vb, and for the assumed value of Vb = 0.07 it would approach the value close to 0.21 at T = 0. On the other hand, the floating incommensurate solid should give the values of ugs close to zero. Epitaxial rotation causes that ugs for argon attains values well below zero. We have not performed any systematic calculations of the monolayer rotation angle; nevertheless, the inspection of individual snapshots demonstrated that the film is rotated by the angle of about 3 at low temperatures. It should be noted, however, that for low-density films the observed angles of rotation are somewhat reduced due to the finite size effects and the periodic boundary conditions, which favor the formation of elongated patches spanning the simulation cell along the y-axis, which is shorter than the cell size in the x direction. On the other hand, the submonolayer films of krypton form the commensurate structure, so that the average fluidsolid interaction energy becomes quite close to the value predicted for the fully localized system, which is equal to 0.36. The fact that our data converge to a smaller value of about 0.32 can be readily

III. PURE COMPONENTS Both the experiment3 and the earlier computer simulation23 demonstrated that the melting temperature of argon submonolayer films of the density Fc e 1 is independent of the film density and is located at the triple point temperature of T* ≈ 0.414. Our results obtained for Fc = 0.4 and 2/3 agree quite well with those earlier findings. On the other hand, the melting point of pure krypton submonolayer films increases slightly with the density. This difference in the behavior of pure components results from the fact that the phase diagram of krypton monolayer on graphite is different than that of argon. In particular, krypton exhibits the incipient triple point and hence the phase diagram of the swan neck topology without a critical point of the gasliquid transition.55,56 Figure 2 shows the temperature changes of the adsorbatesubstrate potential energies (part a), the heat capacity (part b), the bond-orientational order parameter (part c), and its susceptibility (part d) for pure submonolayer films of Ar and Kr obtained at the density Fc = 0.4. Part c clearly shows that both adsorbates form hexagonally ordered phases at the temperatures below the melting point. The melting point has been located at the temperature at which the heat capacity and the susceptibility of the bond-orientational order parameter reach maximum values. 756



ARTICLE

Figure 4. The plots of the bond-orientational order parameter susceptibility (a) and of the heat capacity (b) versus temperature for submonolayer ArKr films of the total density Fc = 0.4 and different composition (see the legend given in part a).

explained. In the regime of submonolayer densities the system exhibits a vaporsolid interface, and the krypton atoms located close to the solid-phase boundaries exhibit rather large displacements from registry positions. These boundary effects are nonnegligible in finite systems, as used in our simulation.

IV. MIXED FILMS Now, we proceed to the discussion of mixed films. The results of canonical ensemble MC simulation demonstrate that the twodimensional liquid phase consisting of krypton and argon is mixed, independently of the film composition. The freezing also leads to the formation of a mixed solid, but the film undergoes a gradual demixing at sufficiently low temperatures. Also, the structure of a solid phase depends upon the mixture composition and temperature. Figure 3 presents the central result of our study, i.e., the phase diagram, in the T*xKr plane for the films of different total density and obtained using a strictly two-dimensional and a full three-dimensional models. First, we discuss the results of calculations for submonolayer films of the total density Fc = 0.4 and 0.5, performed using twoand three-dimensional models, respectively. It is evident that the melting temperature of such films gradually increases with xKr from the melting point of pure argon to the melting point of pure krypton. At the melting transition the system loses hexagonal order, so that the melting points have been located using the susceptibility of the bond-orientational order parameter. This quantity exhibits a pronounced maximum at the melting point (see part a of Figure 4). Also, the heat capacity exhibits a maximum at the melting point (see part b of Figure 4), but sharp peaks have been obtained only for the krypton concentrations equal to and higher than about xKr = 0.4. For small krypton concentrations the heat capacity shows a rather weak anomaly, very similar to that observed for pure argon film.8 Figure 3 demonstrates that for very low concentrations of Kr, not exceeding about 0.1, the solid is an incommensurate, Ar-like phase. The inspection of configurations has shown that the film may exhibit epitaxial rotation at low temperatures, although one would rather expect the disappearance of epitaxial rotation at all,

Figure 5. Snapshots of configurations of the mixed submonolayer with xKr = 0.3 and Fc = 0.4 recorded at the temperatures T* = 0.10 (a) and 0.02 (b). Small filled circles represent the centers of carbon hexagons (registry positions). The argon atoms assigned to the graphite lattice (with ϕ > 0) are marked by open circles with thin lines, while those with ϕ < 0 are marked by open circles with heavy lines. The krypton atoms assigned to the graphite lattice (with ϕ > 0) are marked by light shaded circles, and those with ϕ < 0 are marked by dark shaded circles.

even for very low concentrations of krypton. This expectation is supported by the results of heat capacity measurements for submonolayer films of argon containing even very small amounts of xenon impurities.42 The heat capacity of pure argon submonolayer films exhibits a rather sharp maximum at the temperature about 2K below the melting point, and the Monte Carlo study of Flenner and Etters23 has clearly demonstrated that that maximum is associated with a disappearance of epitaxial rotation prior to melting. The disappearance of epitaxial rotation in mixed films can be explained by the fact that krypton atoms more or less randomly distributed and located very close to registry positions somehow stiffen the entire film along the directions of the commensurate phase symmetry axes. We recall that on the graphite basal plane krypton atoms dissolve easily in argon-rich film, just the same as 757



ARTICLE

Figure 6. (a) Heat capacity curve for the mixed submonolayer with xKr = 0.3 and Fc = 0.4, and (b) temperature changes of the order parameters Ψ6,k and Φk (k = Ar, Kr) as well as of Ψ6. (c and d) Temperature dependences of the potential energy and of the order parameters Ψ6, ΦAr, and ΦKr, respectively, for the submonolayer film with Fc = 0.4 and xKr = 0.4.

found in the case of argonkrypton mixture adsorbed on Pt(111).44 This explains the observed lack of epitaxial rotation below the melting point temperature. Upon the decrease of temperature, krypton atoms √ tend √ to form small compact clusters of the commensurate ( 3 3)R30 structure. The clustering of krypton also influences the positions of nearby argon atoms, “pushing” them into registry. This effect considerably influences the structure of the remaining incommensurate argon, but for small krypton concentrations, there are large regions of pure argon film, and the system may reduce its free energy by epitaxial rotation at low temperatures. In fact, the inspection of numerous configurations confirms this picture quite well. For higher krypton concentrations, exceeding 0.1, we find a different behavior below the melting temperature. The calculations have shown √ the √formation of partially ordered mixed commensurate ( 3 3)R30 phase at the temperatures just below the melting point. This phase undergoes a transition at lower temperatures, depending on the krypton concentration (see the heat capacity peak at T* ≈ 0.4 in Figure 4b, marked with the vertical arrow, for the film with xKr = 0.2), to the phase consisting of commensurate domains, separated by incommensurate walls predominantly built of argon atoms. This is illustrated by the snapshots shown in parts a and b of Figure 5, obtained for the film of the total density Fc = 0.4 with xKr = 0.3 at the temperatures T* = 0.10 and 0.02, respectively. In the snapshots presented, we have assigned a given atom as being commensurate or incommensurate using the following local order parameter19 ϕðrÞ ¼ cosðq1 rÞ þ cosðq2 rÞ þ cosððq1 q2 ÞrÞ

Figure 7. Snapshots of configurations of the mixed submonolayer of the density Fc = 0.4 recorded at the reduced temperature T* = 0.02 and different mole fraction of krypton, equal to 0.5 (part a) and 0.8 (part b). Small filled circles represent the centers of carbon hexagons (registry positions). The argon atoms assigned to the graphite lattice (with ϕ > 0) are marked by open circles with thin lines, while those with ϕ < 0 are marked by open circles with heavy lines. The krypton atoms assigned to the graphite lattice (with ϕ > 0) are marked by light shaded circles, and those with ϕ < 0 are marked by dark shaded circles.

the temperature T* ≈ 0.395. The transition is accompanied by a decrease of the order parameters ΦAr and ΦKr. On the other hand, the bond-orientational order parameters Ψ6,Ar, Ψ6,Kr, and Ψ6 do not show any anomalies (Figure 6b), indicating that the film is hexagonally ordered at any temperature below the melting point. The behavior of the order parameters ΦAr and ΦKr also shows a gradual separation of argon and krypton as the temperature decreases. Namely, ΦAr decreases due to the removal of argon atoms from the commensurate domains. On the other hand, the increasing clustering and localization of krypton atoms over graphite cells leads to a gradual increase of ΦKr when the temperature becomes lower. Qualitatively similar behavior, i.e., formation of krypton commensurate domains separated by incommensurate walls,

ð10Þ

and assuming that the atom is commensurate (incommensurate) when ϕ > 0 (ϕ e 0). The heat capacity curve for this system, depicted in part a of Figure 6, exhibits a small but quite well developed maximum at 758



ARTICLE

has been observed for xKr up to about 0.88. The only effect of the increasing krypton concentration results in a gradual increase of the commensurate domains of krypton at low temperatures, while the walls made of argon atoms become thinner (see part a of Figure 7). The CIC transition becomes sharper and our simulation results show that for xKr exceeding about 0.4 it looks like a first-order transition. In particular, the potential energy, as well as the order parameters ΦAr and ΦKr, exhibit discontinuous jumps at the transition point (see parts c and d of Figure 6). Of course, we cannot exclude the possibility that the transition is continuous. The simulations have been performed for rather small systems, consisting of 480 (F = 0.4) and 600 (F = 0.5) atoms. In such small systems and at low temperatures, one may expects large effects due to metastable states. On the other hand, in small simulation systems the rounding and smearing of the transition region may also be a problem and one would need to apply the finite size scaling theory57 to determine the order of the transition. The observed low temperature domain-wall structure results from the tendency of krypton to form the commensurate phase, while argon prefers to be incommensurate with the graphite lattice. It is also of importance that the presence of a large number of argonkrypton pairs is energetically favored, so the system tries to maximize the length of the interface separating krypton argon domains. When the krypton concentration becomes higher than about 60%, the low-temperature structure of domain walls starts to change and we have observed a gradual increase of the number of Kr atoms incorporated into the walls and a gradual displacement of argon atoms to the peripheries of submonolayer upon the decrease of temperature. Figure 7b shows the snapshot recorded at T* = 0.02 and for xKr = 0.8 and corresponds to the situation in which the walls are nearly entirely formed by krypton atoms, while the argon atoms are located at the film peripheries. It shows that the presence of even a small amount of argon within krypton submonolayer film, is sufficient to destabilize the commensurate structure and trigger the CIC transition at sufficiently low temperatures. For the krypton concentration of xKr = 0.9 and higher the behavior of theadsorbed layer becomes qualitatively the same as observed in pure krypton film. The liquid freezes into the commensurate phase, which remains stable up to very low temperatures, while argon atoms are predominantly located at the submonolayer boundaries. It should be emphasized that the behavior of submonolayer films resulting from the use of two- as well as three-dimensional models is practically the same. Of course, in the three-dimensional model the krypton atoms assume larger average distances from the surface than the smaller argon atoms, as confirmed by the inspection of density profiles (not shown here for the sake of brevity), but even at the temperatures above the melting point, we have not observed any effects due to desorption or the promotion of adatoms to the second layer. One can expect, however, that such phenomena may be of importance for the films of higher densities, very close to the monolayer completion. Thee-dimensional calculations performed for the submonolayer films of the total density Fc = 0.8 and different krypton concentrations (see the green symbols in Figure 3) have shown that the increase of the krypton concentration causes a gradual increase of the melting temperature with respect to the values obtained for Fc = 0.4 and 0.5. This suggests that the phase diagram topology for such films is the same as observed for pure

krypton, i.e., with the vaporliquid transition preempted by the fluidsolid transition, and this question will be discussed later. Now, we turn to the discussion of the behavior of monolayer films of the total density F = 1.0, which have been studied within the three-dimensional model only. For very low krypton concentrations (xKr up to about 0.1), the high-density film shows the same behavior as the previously discussed submonolayer films. Here, one should note that experimental7 as well as computer simulation23 studies of pure argon films show that the melting temperature remains practically unchanged for the surface densities up to Fc slightly above 1.0. The mixed ArKr films with a very small concentration of krypton exhibit a similar behavior. Then, the increase of krypton concentration beyond xKr = 0.1 causes a gradual increase of the melting temperature with respect to the values obtained for submonolayer films of the same composition (see red circles in Figure 3). The inspection of density profiles allows one to state that the promotion of the second layer is of no importance at the temperatures used and for the entire range of krypton concentrations. Small amounts of argon appear in the second layer only at the temperatures well above the melting point. The increase of melting temperature with the krypton mole fraction, over the values observed in submonolayer films, can be attributed to a gradual increase of the commensurate phase stability. The calculations have shown that already for xKr g 0.4 the film remains commensurate even at the temperatures below the demixing transition, which has been observed to occur at a rather low temperature of about 0.06. Only for still lower krypton concentrations we have observed the commensurate incommensurate transition, at temperatures quite close to, though a little lower than, the transition temperature in submonolayer films (cf. Figure 3). The observed high stability of the commensurate phase at very low temperatures has to be discussed in some detail, since it is quite likely that it is only an artifact of the simulation. In particular, a finite size of the simulation cell and periodic boundary conditions may be responsible for a vast overestimation of the commensurate phase stability. The monolayer film of the density Fc = 1 freezes into a mixed commensurate solid phase, just the same as observed in the case of submonolayer films. As long as the film is mixed the commensurate phase forms a very well ordered and completely filled hexagonal lattice. As long as the film is mixed, small clusters of commensurate krypton distributed over the entire film pin the argon atoms to registry positions. Only at the temperatures below the demixing transition may the appearing large commensurate argon patches be destabilized, due to a large misfit between the size of argon atoms and the commensurate phase lattice. In small systems, the strain exercised by the argon commensurate patch is considerably reduced by the periodic boundary conditions applied and may appear to be not high enough to trigger the commensurateincommensurate transition. We have performed some additional simulations using smaller (with Lx = 36) and larger (with Lx = 90 and 120) simulation cells. In the case of Lx = 36, we have found that even the system containing only about 10% of krypton forms a commensurate structure at the temperatures down to 0.01, i.e., the lowest temperature used, well below the demixing transition. Only for still lower krypton concentrations, the formation of incommensurate phase could be observed at the temperatures just below the demixing transition. On the other hand, when the simulation cell has been made larger, the incommensurate phase has been 759



ARTICLE

observed for the mixtures of considerably higher krypton concentration. In the case of Lx = 90 and 120, the incommensurate phase has been observed in the films containing up to 50% and 65% krypton, respectively, and the transition occurs at the temperature at which the demixing transition occurs. It does not mean, however, that the incommensurate phase is stable only at such low temperatures. When the simulation run starts from the configuration corresponding to the low-temperature incommensurate phase, an increase of temperature above the demixing transition temperature does not lead to the recovery of the commensurate structure. The transition to the commensurate phase occurs at considerably higher temperatures, just the same as in submonolayer films. The locations of the commensurate incommensurate phase transition for the films of the total density Fc = 1 and the krypton mole fraction equal to 0.5 and 0.6 (see Figure 4) have been estimated from the simulations with Lx = 90 and 120, respectively. One can anticipate that the upper limit of the krypton mole fraction at which this transition occurs is not much different than in submonolayer films. These results confirm our prediction that finite size effects and periodic boundary conditions are responsible for the overestimation of the commensurate phase stability at low temperatures. In small systems, the argon atoms cannot leave the registry positions, even at the temperatures below demixing transition, due to rather large effects resulting from the presence of boundaries between argon and krypton patches. An increase of the simulation cell causes the boundary effects to become less important, while argon patches become larger. This allows for the CIC transition to occur, even in the krypton-rich films. One should note that in real adsorption systems even large graphite sheets have free boundaries so that the film is strained and exercises large stress at those boundaries. This causes that the stability of the commensurate phase is not artificially enhanced by the periodic boundary conditions. We have then performed a simple test, by making the simulation box a little larger in the x exercises, by setting Lx = 63 instead of 60, by running the simulation for the mixture with xKr = 0.5 at the temperature of T = 0.2, and by starting with the configuration recorded at the same temperature but for Lx = 60. After a short time we could observe the development of dense walls consisting mostly of argon atoms. This confirms that the observed formation of phaseseparated commensurate structure is only an artifact of the simulation. Nevertheless, one can expect a higher stability of the mixed commensurate phase in monolayer films, with respect to submonolayer films. Another problem concerning the influence of argon impurity on the melting of commensurate phase appears. Namely, it has already been mentioned that the phase diagram of krypton on graphite exhibits the incipient triple point, as convincingly proved in the ref 37. Our canonical ensemble simulation results do not allow one to say much about the changes of phase diagram topology with the film composition. To address this issue one rather needs to perform simulations in the grand canonical ensemble. We have carried out such simulation for the pure krypton and argon films as well as for the argonkrypton mixtures. The phase diagram for pure krypton obtained, shown in parts a and b of Figure 8, is consistent with the experimental data and the incipient triple point concept. In particular, we do not observe any trace of the vaporliquid coexistence. It should be emphasized that the phase diagram depicted in Figure 8 agrees quite well with experimental data, showing that the parameters of the interaction potentials have been properly chosen. Similarly,

Figure 8. The phase diagrams for pure krypton (parts a and b) and pure argon (parts c and d) monolayers obtained using grand canonical simulation. Parts a and c (b and d) give the chemical potential temperature (densitytemperature) projections.

Figure 9. The phase diagrams for the argonkrypton mixtures obtained for the fixed chemical potential of argon, equal to μ*Ar = 10.0 (a and b), 9.8 (c and d), and 9.7 (e and f) . Parts a, c, and e give the total densitytemperature projections, while parts b, d and f show the krypton mole fractiontemperature projections of the phase diagrams.

the phase diagram for pure argon monolayer, given in parts c and d of Figure 8, is also quite consistent with experimental data. In order to study the effects of film composition on the phase behavior, we have performed the simulations in the grand canonical ensemble, under the condition of a fixed chemical potential of either argon or krypton. The adsorption isotherms measured at different temperatures have been used to construct the phase diagrams. The calculations have been done for three choices of the argon chemical potential, μ*Ar = μAr/εArAr = 10.0, 9.8, and 9.7. For the first choice of μ*Ar = 10.0, pure argon film is a gaslike, dilute phase at the temperatures down to about 0.36, while for μ*Ar = 9.8 it exhibits the gassolid 760



ARTICLE

Figure 10. (a) Adsorption isotherm obtained at T* = 0.52 for the system with the chemical potential of krypton equal to μ* = 13.7. The vertical dashed line shows the location of the fluidsolid transition. (b) Radial distribution functions gij(r), with ij = ArAr, ArKr, and KrKr at T* = 0.52 and for the argon chemical potential corresponding to the liquid (μ*Ar = 8.4) and the solid μ*Ar = 8.2) phase. The vertical dashed lines show √ the √ lengths of the first five vectors of a perfect commensurate ( 3 3)R30 structure.

Figure 11. (a) Adsorption isotherm obtained at T* = 0.52 for the system with the chemical potential of krypton equal to μ* = 13.6. The vertical dashed line shows the location of the fluidsolid transition. (b) Radial distribution functions gij(r), with ij = ArAr, ArKr, and KrKr at T* = 0.52 and for the argon chemical potential corresponding to the liquid (μ*Ar = 8.5) and the solid (μ*Ar = 8.7) phase. The vertical dashed lines √ show √ the lengths of the first five vectors of a perfect commensurate ( 3 S)R30 structure.

transition at T* ≈ 0.4 and remains in a gas phase at higher temperatures. When, however, μ*Ar = 9.7, the pure argon film is a gas at the temperatures down to about 0.44. Below T ≈ 0.44, it forms a liquid phase, and at the temperature of about 0.415, the liquid freezes into the incommensurate solid. Our calculations have been carried out at temperatures high enough to ensure that pure argon is in a dilute, gaslike phase. Parts a and b of Figure 9 present the phase diagram obtained for μ*Ar = 10.0, which is qualitatively the same as found for the pure krypton. Part a of Figure 9 give the FtT projections, while part b shows the changes of the krypton mole fraction at the phase boundaries. Although the krypton concentration in the dilute fluid is rather low, it is quite high in the condensed, commensurate solid phase. In the case of μAr = 9.8, the phase diagram topology changes (see parts c and d of Figure 9). In particular, we find the vapor liquid coexistence, which ends at the critical point, located at T ≈ 0.56 and the triple point at T ≈ 0.55. At the temperature range considered here, the solid is a mixed commensurate phase, and the fluidsolid transition is a first-order transition. One should note that the krypton concentration in the solid is now lower than in the case of μAr = 10.0, but it is sufficiently high to lead to the first-order fluidsolid transition, just the same as in pure krypton film. The situation changes when the chemical potential of argon is further increased to μ*Ar = 9.7. The calculations have demonstrated that the nature of the solid phase and the order of the fluidsolid transition both change with the temperature. At temperatures lower than the threshold temperature, Ts ≈ 0.515, the fluidsolid transition is continuous, while for the temperatures exceeding Ts the fluidsolid transition becomes discontinuous (see parts e and f of Figure 9). This crossover can be attributed to the changes of the krypton mole fraction

in the solid phase when the temperature changes, and our simulation data indicate that for xKr lower than about 0.2 the fluid solid transition is continuous, while for higher concentrations of krypton the transition becomes discontinuous. This result agrees quite well with the predictions stemming from canonical ensemble simulations. The heat capacity curves (shown in part b of Figure 4) have suggested that the crossover between the continuous and discontinuous melting occurs for xKr below 0.4. It is quite likely, however, that it appears for lower krypton mole fractions. In the case of the chemical potential of krypton fixed at μ*Kr = 13.7, the krypton concentration in the condensed phases, liquid and solid, is quite low. Only at the vaporliquid condensation we observe the effect of enhanced adsorption58,59 of krypton (see part a of Figure 10). The krypton mole fraction in the film attains the maximum value of about 0.24, at the liquid side of the vaporliquid coexistence. As the chemical potential of argon increases, the krypton concentration becomes lower and lower. In particular, the fluidsolid transition has been found to be of the same nature as in the pure argon monolayer and leads to the formation of incommensurate solid (see part b of Figure 10) of the density well above Fc = 1.0. The radial distribution functions given in part b of Figure 10 demonstrate that the locations of subsequent peaks are not related to those predicted for the commensurate phase. Quite different behavior is observed when a slightly higher chemical potential of krypton (μ*Kr = 13.6) has been used. Figure 11a gives the adsorption isotherm at T* = 0.52, which demonstrates that the system undergoes a sharp first-order vaporcommensurate solid phase transition at μ*Ar = 9.80 ( 0.04, and the krypton mole fraction reaches the value of about xKr = 0.45 in the solid phase. Upon the increase of the argon 761



ARTICLE

chemical potential, we observe a well seen decrease of the film total density at μ*Ar ≈ 9.59. The inspection of radial distribution functions obtained for values of the chemical potentials of argon lower and higher than 9.59 (see part b of Figure 11) demonstrate that there is a transition from the commensurate solid to a dense liquid phase. Although we have not performed the simulations for the argon chemical potential higher than 9.2, one expects to observe the liquid-to-incommensurate (Ar-like) solid at sufficiently high chemical potential of argon. The isotherms obtained at still lower temperatures of 0.5 and 0.47 have shown that the vaporcommensurate solid transition is accompanied by increasing effects due to enhanced adsorption of krypton, whose mole fraction becomes on the order of about 0.9 at the transition point at T* = 0.47. The evaluated phase diagrams obtained under different conditions of fixed μ*Ar and μ*Kr allowed us to estimate the locations of the vaporliquid critical points, which are included in Figure 3 (filled triangles). The data suggest that the vaporliquid transition occurs only when the krypton concentration is lower than about xKr = 0.46. For higher krypton concentrations, the vapor liquid transition is predicted to be preempted by the vapor commensurate solid transition, leading to the krypton-like phase diagram with the incipient triple point.

V. SUMMARY AND FINAL REMARKS We have performed an extensive Monte Carlo study of the melting and low-temperature behavior of mixed submonolayer films of argon and krypton on graphite. It has been found that both adsorbates form a mixed liquid phase for any composition and the liquid freezes into a mixed solid. However, the solid phase undergoes a gradual phase separation at low temperatures. In the region of small krypton concentration, when xKr is lower than about 0.1, the solid is an argon-like incommensurate floating phase, while for higher krypton concentrations the film freezes into the mixed commensurate phase. Upon the lowering of temperature, this phase undergoes a first-order transition to the domain-wall phase. The composition of commensurate domains and the thickness and the composition of walls change with the krypton concentration and temperature. The decrease of temperature leads to the removal of argon from the commensurate domains to the walls, so at very low temperatures the domains are made only of krypton atoms. However, the composition and the thickness of domain walls depend again upon the temperature and the film composition. At very low temperature, below the demixing transition, and for the krypton concentrations up to about 0.65, the walls consist of argon atoms only, and the walls become thinner as the krypton concentration increases. When, however, the krypton concentration exceeds xKr = 0.65, the walls are composed of both argon and krypton atoms, and the concentration of krypton within the walls gradually increases when the total concentration of krypton becomes higher. Also, the transition temperature between the commensurate and the domain-wall phases gradually decreases with xKr and seems to go down to zero for xKr ≈ 0.88. Then, for still higher krypton concentrations, the picture changes and the solid becomes a krypton-like commensurate phase. At sufficiently low temperatures, all argon atoms are removed from the film interior to the peripheries of the submonolayer patch of krypton. At this point, we should also make some comments about the observed domain-wall structure in mixed submonolayers. The snapshots given in Figures 5 and 7 show that the domains are

rather small and the walls are straight. The question is whether it is a real property of the system or just an artifact resulting from the finite size effects. We have also performed simulation for the system with Ft = 0.4 and xKr = 0.5 using a larger simulation cell, with Lx/a1 = 120, and observed quite similar sizes of commensurate domains. The increase of the system size caused only the appearance of a larger number of commensurate domains. One should note that the snapshots presented have been recorded at extremely low temperatures. At higher temperatures, just below the temperature at which the transition to the domain-wall phase occurs, the domains of varying size appear and the walls are not that well localized and straight. This indicates that the systems exhibit the meandering and breathing entropies, just the same as observed in many other incommensurate phases.60 These entropic effects stabilize the domain-wall phase. One should also note that the formation of a hexagonal network of domain walls demonstrates that the energy due to wall crossings must be negative,61 otherwise the striped phase with parallel walls would appear. Theory predicts that the commensurate incommensurate transition may be of first-order when the domain walls form a hexagonal network, while it is continuous when the incommensurate phase has a striped structure with parallel walls.61 This supports our observation that the commensurateincommensurate transition is discontinuous. Grand canonical simulation predicts the changes of the phase diagram topology when the krypton concentration increases. In particular, we have found the crossover between the argon-like and the krypton-like phase diagrams. Finally, we would like to comment on the molecular dynamics simulation of mixed ArKr monolayer films performed by Roth.47 He performed the canonical ensemble MD simulations for a rather special situation, assuming that the film of the total density Fc = 1.0 consisted of a finite patch made of one type of particle, surrounded by the atoms of the second component. Allowing only for the translational moves, he observed a high stability of such patches, even at rather high temperatures above the melting point. As a consequence, the melting temperature was found to depend not only upon the film composition but also on the choice of the adsorbate forming the central “impurity” patch. In our Monte Carlo simulation, also performed in the canonical ensemble, apart from the translational moves, also the moves allowing for the exchange of identity of adatoms have been included into the algorithm. This allowed us to study the formation of mixed as well as phase-separated phases, and this seems to be closer to reality. It should be emphasized that the phase separation in the films of the total density Fc = 1.0 has been estimated to occur at a very low temperature of about T* = 0.06, i.e, at about T = 7 K.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected].

’ ACKNOWLEDGMENT This work was supported by the Polish Ministry of Science under the grant No. N N202 046137. ’ REFERENCES (1) Thomy, A.; Duval, X. J. Chim. Phys. Chim. Biol. 1969, 66 (286), 1101. 762



ARTICLE

(46) Ramseyer, C.; Hoang, P. N. M.; Girardet, C. Phys. Rev. B 1994, 49, 2861. (47) Roth, M. W. Langmuir 1999, 15, 2169. (48) Steele, W. A. Surf. Sci. 1973, 36, 317. (49) Kim, H.-Y.; Steele, W. A. Phys. Rev. B 1992, 45, 6226. (50) Amat, M. A.; Arienti, M.; Fonoberov, V. A.; Kevrekidis, I. G.; Maroudas, D. J. Chem. Phys. 2008, 129, 184106. (51) Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids; Oxford University Press: Oxford, 1987. (52) Landau, D. P.; Binder, K. A Guide to Monte Carlo Simulation in Statistical Physics; Cambridge University Press: Cambridge, UK, 2000. (53) Strandburg, K. J. Rev. Mod. Phys. 1988, 60, 161. (54) Patrykiejew, A.; Sokol owski, S.; Binder, K. Surf. Sci. Rep. 2000, 37, 207. (55) Butler, D. M.; Litzinger, J. A.; Stewart, G. A.; Griffiths, R. B. Phys. Rev. Lett. 1979, 42, 1289. (56) Butler, D. M.; Litzinger, J. A.; Stewart, G. A. Phys. Rev. Lett. 1980, 44, 466. (57) Landau, D. P. In Finite Size Scaling and Numerical Simulation of Statistical Systems; World Scientific: Singapore, 1990; p 223. (58) Hommeril, F.; Mutaftschiev, B. Phys. Rev. B 1989, 38, 556. (59) Kruk, M.; Patrykiejew, A.; Sokolowski, S. Surf. Sci. 1995, 340, 179. (60) den Nijs, M. in Phase Transitions and Critical Phenomena; Academic Press: London, 1988; Vol. 12, p 219. (61) Bak, P.; Mukamel, D.; Villain, J.; Wentowska, K. Phys. Rev. B 1979, 19, 1610. (62) Patrykiejew, A.; Sokolowski, S; Binder, K. Surf. Sci. 2002, 512, 1.

9

(2) Taub, H.; Carneiro, K.; Kjems, J. K.; Passell, L.; McTague, J. P. Phys. Rev. B 1977, 16, 4551. (3) Chung, T. T. Surf. Sci. 1979, 87, 348. (4) Millot, F. J. Phys. (Paris) 1979, 40, L–9. (5) Thomy, A.; Duval, X.; Regnier, J. Surf. Sci. Rep. 1981, 1, 1. (6) Tiby, C.; Lauter, H. J. Surf. Sci. 1982, 117, 277. (7) McTague, J. P.; Als-Nielsen, J.; Bohr, J.; Nielsen, M. Phys. Rev. B 1982, 25, 7765. (8) Migone, A. D.; Li, Z. R.; Chan, M. H. W. Phys. Rev. Lett. 1984, 53, 810. (9) Zhang, Q. M.; Larese, J. Z. Phys. Rev. B 1991, 43, 938. (10) Stephens, P. W.; Heiney, P. A.; Birgeneau, R. J.; Horn, P. M.; Moncton, D. E.; Brown, G. S. Phys. Rev. B 1984, 29, 3512. (11) Specht, E. D.; Mak, A.; Peters, C.; Sutton, M.; Birgeneau, R. J.; D’Amico, K. L.; Moncton, D. E.; Nagler, S. E.; Horn, P. M. Z. Phys. B—Condens. Matter 1987, 69, 347. (12) Johnson, D. E.; Phillips, J. M.; Larese, J. Z. Phys. Rev. B 1997, 56, 6462. (13) Bruch, L. W.; Diehl, R. D.; Venables, J. A. Rev. Mod. Phys. 2007, 79, 1381. (14) Nicholson, D.; Rowley, L. A.; Parsonage, N. G. J. Phys. (Paris) 1977, 38, 69. (15) Hanson, F. E.; Mandell, M. J.; McTague, J. P. J. Phys. (Paris) 1977, 38, 76. (16) Abraham, F. F. Phys. Rev. B 1983, 28, 7338. (17) Abraham, F. F.; Rudge, W. E.; Auerbach, D. J.; Koch, S. W. Phys. Rev. Lett. 1984, 52, 445. (18) Nicholson, D.; Parsonage, N. G. J. Chem. Soc., Faraday Trans. 2 1986, 82, 1657. (19) Houlrik, J. M.; Landau, D. P. Phys. Rev. B 1991, 44, 8962. (20) Roth, M. W. Phys. Rev. B 1998, 57, 12520. (21) Roth, M. W.; Salazar, M. Surf. Sci. 1999, 441, 270. (22) Flenner, E.; Etters, R. D. Phys. Rev. Lett. 2002, 88, 106101. (23) Flenner, E.; Etters, R. D. Phys. Rev. B 2006, 73, 125419. (24) Novaco, A. D.; McTague, J. P. Phys. Rev. Lett. 1977, 38, 1286. (25) McTague, J. P.; Novaco, A. D. Phys. Rev. B 1979, 19, 5299. (26) Price, G. L.; Venables, J. A. Surf. Sci. 1976, 59, 509. (27) Venables, J. A.; Schabes-Retchkiman, P. S. Surf. Sci. 1978, 71, 27. (28) Bruch, L. W.; Hansen, F. Y. Phys. Rev. B 1997, 55, 1782. (29) Gordon, M. B.; Villain, J. J. Phys. C 1985, 18, 3919. (30) Bruch, L. W.; Cole, M. W.; Zaremba, E. Physical Adsorption: Forces and Phenomena; Clarendon Press: Oxford, 1977. (31) Larher, Y.; Gilquin, B. Phys. Rev. A 1979, 20, 1599. (32) D’Amico, K. L.; Bohr, J.; Moncton, D. E.; Gibbs R. D., D. Phys. Rev. B 1990, 41, 4368. (33) Larher, Y.; Terlain, A. J. Chem. Phys. 1980, 72, 1052. (34) Shaw, C. G.; Fain, S. C.; Chinn, M. D. Phys. Rev. Lett. 1978, 41, 955. (35) Nielsen, M.; Als-Nielsen, J.; Bohr, J.; McTague, J. P.; Moncton, D. E.; Stephens, P. W. Phys. Rev. B 1987, 35, 1419. (36) Butler, D. M.; Litzinger, J. A.; Stewart, G. A.; Griffiths, R. B. Phys. Rev. Lett. 1979, 42, 1289. (37) Niskanen, K. J; Griffiths, R. B. Phys. Rev. B 1985, 32, 5858. (38) Gooding, R. J.; Joos, B.; Bergersen, B. Phys. Rev. B 1983, 27, 7669. (39) Regnier, J.; Bockel, G.; Dupont-Pavlovsky, N. Surf. Sci. 1981, 119, L770. (40) Bohr, J.; Nielsen, M.; McTague, J. P.; Als-Nielsen, J.; Kjaer, K. Surf. Sci. 1983, 125, 181. (41) De Beauvais, C. Ph.D. Thesis, Universite Paris 7, 1986. (42) Ma, J.; Carter, E. D.; Kleinberg, H. B. Phys. Rev. B 1998, 57, 9270. (43) Stephens, P. W.; Goldman, A. I.; Heiney, P. A.; Bancel, P. A. Phys. Rev. B 1986, 33, 655. (44) Zeppenfeld, P.; Becher, U.; Kern, K.; David, R.; Comsa, G. Surf. Sci. Lett. 1993, 297, L141. (45) Bruch, L. W.; Graham, A. P.; Toennies, J. P. J. Chem. Phys. 2000, 112, 3314. 763


Melting and Low-Temperature Structures of Mixed ArâKr Monolayer

Recommend Documents

Melting and Low-Temperature Structures of Mixed ArâKr Monolayer