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On the phase behavior of mixed Ar–Xe submonolayer ﬁlms on graphite

arXiv:1207.2995v1 [cond-mat.mtrl-sci] 12 Jul 2012

A. Patrykiejew Department for the Modelling of Physico-Chemical Processes, MCS University Lublin, 20031 Lublin, Poland

Received December 31, 2011, in ﬁnal form February 28, 2012 Using Monte Carlo simulation methods in the canonical and grand canonical ensembles, we discuss the melting and the formation of ordered structures of mixed Ar–Xe submonolayer ﬁlms on graphite. The calculations have been performed using two- as well as three-dimensional models of the systems studied. It is demonstrated that out-of plane motion does not affect the properties of the adsorbed ﬁlms as long as the total density is not close to the monolayer completion. On the other hand, close to the monolayer completion, the promotion of particles to the second layer considerably affects the properties of mixed ﬁlms. It has been shown that the mixture exhibits complete mixing in the liquid phase and freezes into solid phases of the structure depending upon the ﬁlm composition. For submonolayer densities, the melting temperature exhibits non-monotonous changes with the ﬁlm composition. In particular, the melting temperature initially increases when the xenon concentration increases up to about 20%, then it decreases and reaches minimum for the xenon concentration of about 40%. For still higher xenon concentrations, the melting point gradually increases to the temperature corresponding to pure xenon ﬁlm. It has been also demonstrated that the topology of phase diagrams of mixed ﬁlms is sensitive to the composition of adsorbed layers. Key words: adsorption of mixtures, phase transitions, computer simulation, melting PACS: 68.35.Rh, 68.43.De, 68.43.Fg, 64.75.St

1. Introduction It is now well known [1–8] that monolayer ﬁlms of both argon and xenon on graphite form incommensurate two-dimensional solid phases at low temperatures. However, the melting transition of these two solid phases has a different mechanism [9]. Experimental data [10–14] as well as computer simulations [15, 16] have demonstrated that submonolayer ﬁlms of argon melt via continuous phase transition at the triple point temperature equal to Tt ≈ 49.7 K [17]. At the temperatures well below the melting point, the solid-like argon submonolayer ﬁlm is rotated by about 2–3 degrees with respect to the R30◦ axis of p p ◦ the commensurate ( 3× 3)R30 structure Since the argon atoms are rather small, the solid p[16, 18–20]. p phase is compressed with respect to the ( 3 × 3)R30◦ commensurate structure, and the monolayer density is about 25% higher than the density of a perfect commensurate structure. The melting of incommensurate submonolayer ﬁlms of xenon is of ﬁrst order [21, 22], and occurs at the triple point temperature of about 100 K. However, the xenonpatoms pare larger so that the incommensurate solid phase is dilated with respect to the commensurate ( 3 × 3)R30◦ structure. The differences in the structure of low temperature phases and the phase behavior of argon and xenon submonolayer ﬁlms on graphite lead to a quite complex phase behavior of mixed ﬁlms. The Xray scattering studies of the Ar–Xe mixture adsorbed on graphite [23–25] have demonstrated that the structure of submonolayer and monolayer solid-like phases strongly depends upon the ﬁlm composition. Three different solid-like phases have been found. Apart from the compressed argon-like and dilated Xelike incommensurate phases, being p stable p for suﬃciently low and high xenon mole fraction, respectively, the formation of krypton-like ( 3 × 3)R30◦ commensurate structure has been found over a rather wide range of the mixture composition. The calorimetric study of Ma et al. [26] has shown that even very small amounts of xenon, about 1.5%, added into submonolayer argon ﬁlms, lead to the disappearance of

© A. Patrykiejew, 2012

23601-1

A. Patrykiejew

the heat capacity peak attributed to the orientational transition in the argon-like incommensurate solid phase [16, 27]. A vast majority of theoretical studies of mixed adsorbed layers has been based on lattice gas models [28–33], which do not constitute a good basis for the discussion of the incommensurate-commensurate transitions in the ﬁlms of rare gases on graphite. Such models cannot properly describe the incommensurate ﬂoating solid. However, there have also been some attempts to construct theoretical models for the commensurate-incommensurate transitions in monolayer ﬁlms of rare gas mixtures on graphite [34– 36]. The primary aim of the models proposed by Marti et al. [34, 36], was to explain the anomaly in the phase behavior of Ar–Xe and Kr–Xe submonolayer ﬁlms on graphite. Namely, experiments have demonstrated [34] that less krypton than argon is needed to induce the formation of commensurate phase. On the other hand, a rather general mean-ﬁeld model of Villain and Moreira [35] requires the introduction of several approximations in order to make the resulting equations numerically tractable. Consequently, the agreement with experimental data is rather poor. Nevertheless, these authors have derived qualitative phase diagrams for the Ar–Xe mixture adsorbed on graphite. The theory predicts the existence of incommensurate, Ar-like and Xe-like structures at low temperatures, and the formation of the commensurate structure at higher temperatures over a rather limited range of the Xe mole fraction. In this work, we present and discuss the results of rather extensive Monte Carlo simulations of mixed Ar–Xe ﬁlms on graphite. Our main goal has been to investigate the structure of low temperature solid phases as well as to determine the changes of the melting temperature with the mixture composition. However, we also discuss the evolution of phase diagrams resulting from the changes in the ﬁlm composition. The paper is organized as follows. In the next section we present the model used and describe the Monte Carlo method used to determine the properties of mixed submonolayer ﬁlms. Then, in section 3 we brieﬂy discuss the behavior of pure Ar and Xe ﬁlms. The last section 4 is devoted to the presentation of the results for the mixed Ar–Xe ﬁlms.

2. The model and Monte Carlo methods The interaction between adsorbate atoms is assumed to be represented by the (12,6) Lennard-Jones potential

ui,j (r i j ) = 4εi,j

h¡

σi,j /r i j

¢12

¡ ¢6 i , − σi,j /r i j

(2.1)

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

Table 1. Lennard-Jones parameters for Ar and Xe used in this work.

i, j

σi,j

Ar,Ar Kr,Kr Ar,Kr

Å 3.4 4.1 3.5

εi,j K 120.0 221.0 162.85

representing the Ar–Xe interaction, also given in table 1, have been obtained using the usual LorentzBertholot combining rules:

σi,j =

¢ 1¡ σi,i + σ j ,j 2

and

εi,j =

p

εi,i · ε j ,j .

(2.2)

The potential (2.1) has been cut at the distance 3σi,j . We are aware of some drawbacks that the assumption of the LJ potential has got, and that other authors have used different potentials to reproduce experimental data for adsorbed ﬁlms of pure rare 23601-2

Ar–Xe mixed ﬁlm on graphite

gases on graphite [37]. Also, we have not taken into account the surface mediated interactions, which are known to affect the strength of adsorbate-adsorbate interaction in the vicinity of solid substrates [7]. The interaction of rare gas atoms with the graphite basal plane can be represented by the potential proposed by Steele [38]

i h X v i (x, y, z) = εgs,i v 0,i (z) + v k,i (z) f k (x, y) ,

i = Ar or Kr.

k

(2.3)

In the above, the ﬁrst term in the square brackets is the ﬂuid-solid potential averaged over the entire surface, while the second term represents the corrugation part of the ﬂuid-solid potential. Assuming that the interaction between an adsorbate atom and the carbon atom of the graphite substrate is also represented by the Lennard-Jones potential, the explicit expressions for v 0,i (z), the Fourier components v k,i (z) and the functions f k (x, y) are given by the following equations:

v 0,i (z) =

v k,i (z) =

∞ 4πA 6i X

as

2πA 6i as

"

n=0

"

2A 6i (z + n∆z)10

A 6i ³ q k ´5

30 2z

1 − (z + n∆z)4

K 5 (q k z) − 2

³q ´ k

2z

#

(2.4)

,

K 2 (q k z)

#

(2.5)

and

f k (x, y) =

X

exp[iqk,l τ],

(2.6)

τ = (x, y)

l

with the sum running over all graphite reciprocal lattice vectors of the length q k . In the above equations A i = σi,C /a1 , where a1 = 2.46 Å is the graphite lattice constant, i.e., the distance between the centers of adjacent carbon hexagons, the values of σi,C and εgs,i (i = Ar, Xe) are given in table 2, ∆z = 3.4 Å is the spacing between graphite planes, a s = 5.24 Å2 is the area of the graphite unit cell, K 2 and K 5 are the modiﬁed Bessel functions of the second kind and of the second and ﬁfth order respectively, and q k ’s are the lengths of the graphite basal plane reciprocal lattice vectors.

Table 2. The parameters describing the Ar-graphite and Xe-graphite interaction, obtained using the Lorentz-Bertholot combining rules [given by equation (2.2)] and assuming that εC,C = 28 K and σC,C = 3.4 Å.

i

εgs,i

Ai

Ar Kr

K 58.00 78.66

Å 3.40 3.75

In the case of only partially ﬁlled monolayer ﬁlms and at suﬃciently low temperatures, the promotion of the second layer is likely to be negligibly small. This allows us to consider a simple strictly twodimensional model with the external ﬁeld of the form

© ª v(x, y) = Vb,i f 1 (x, y) = −Vb,i cos(q1 r) + cos(q2 r) + cos([q1 − q2 ]r) ,

(2.7)

where the parameter Vb,i (i = Ar, Kr) determines the amplitude of the corrugation potential. The magnitudes of Vb,i (Vb,Ar = 0.07 and Vb,Xe = 0.08) have been adjusted in such a way that the results for each component are more or less consistent with the full 3D calculations. Simulations have been performed using the Monte Carlo method in the canonical and grand canonical ensembles p [39, 40]. In the case of two-dimensional model, the rectangular simulation cell of the size La1 × L 3a1 /2, with L = 60 and with the standard periodic boundary conditions has been used. In three dimensional calculations, the simulation cell has been the rectangular parallelepiped of the size 60a 1 × p 60a1 3/2×10a1 , with the periodic boundary conditions applied in the directions parallel to the substrate surface and with the reﬂecting hard wall located at z = 10a 1 . 23601-3

A. Patrykiejew

The quantities recorded included the average potential energy, 〈e〉, the contributions to the potential energy due to the ﬂuid-ﬂuid interaction, 〈e gg 〉 and the contributions due to the ﬂuid-solid interaction for each component 〈e gs,i 〉 and the heat capacity obtained from the ﬂuctuations of the potential energy,

CV =

¢ N ¡ 2 〈e 〉 − 〈e〉2 . 2 kT

(2.8)

In order to monitor the structure of solid phases we have used radial distribution functions, g i j (r ), for different pairs of species i and j , and appropriate order parameters. The formation of hexagonally ordered phases has been monitored using the bond-orientational order parameters [41, 42]

Ψ6,i

¯ ¯ ¯ 1 XX ¡ ¢¯¯ ¯ exp i6φm,n ¯ =¯ ¯ ¯ Nb,i mi ni

(2.9)

measured separately for each adsorbate (i = Ar or Xe). In the above, the ﬁrst sum runs over all atoms of the i -th 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 pairs of the like atoms. Also, we have calculated the total bond-orientational order parameter

¯ ¯ ¯ 1 XX ¡ ¢¯ exp i6φm,n ¯¯ , Ψ6 = ¯¯ Nb m n

(2.10)

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 make it possible to detect the hexagonally ordered structures, but are not suitable to distinguish between the commensurate and incommensurate phases. In the commensurate phase, the atoms are localized over the centers of carbon hexagons, and the appropriate order parameter allowing to monitor such localized structures can be deﬁned as [43]

¯ ¯ ¯ 1 XX 6 ¡ ¢¯¯ ¯ Φi = ¯ exp iqn rm,i ¯ . ¯ 6Ni m n=1 ¯

(2.11)

The ﬁrst sum is taken over all atoms of the i -th component, while the second sum runs over the six reciprocal lattice vectors qn of the graphite substrate and rm,i is the position of the m -th atom of component i . The above deﬁned order parameters have been supplemented by the corresponding susceptibilities

χop =

Lx L y £ kT

¤ 〈op2 〉 − 〈op〉2 ,

(2.12)

where ‘op’ stands for any of the above given order parameters. When grand canonical simulations have been carried out, we have also recorded the adsorptiondesorption isotherms. Throughout this paper, we use reduced quantities, assuming that the graphite lattice constant a 1 is the unit of length, and the Lennard-Jones parameter εAr,Ar is the unit of energy. We have decided, however, to give the temperature in Kelvins, as it allows for easier comparison of our simulation results with experimental data. All the densities are expressed in commensurate monolayers.

3. The results for pure Ar and Xe ﬁlms The phase diagram of xenon monolayer on graphite exhibits the vapor-liquid critical point and the triple point (see parts (a) and (b) of ﬁgure 1). The triple point temperature, equal to Ttr ≈ 89 K, agrees quite well with some experimental data [44], yet it is lower than the value of about 99 K stemming from other experiments [22, 45] and from the recent Monte Carlo results of Przydrozny and Kuchta [37]. The underestimation of the triple point temperature is associated with our choice of the interaction potential and its parameters. Przydrozny and Kuchta applied a semi-empirical potential proposed by Aziz and Slaman [46], while we have used a simple Lennard-Jones potential. We should mention that in the 23601-4

Ar–Xe mixed ﬁlm on graphite

-15.0

a

-9

c

-9.2 µ∗Ar -9.4

-16.0 µ∗Xe -17.0

-9.6 -18.0 -9.8 b

d

0.8

1.0 0.8 ρc

0.6 ρc

0.6 0.4 0.4 0.2

0.2

0

80

100 T [K]

120 48 50 52 54 56 58 T [K]

0.0

Figure 1. The phase diagrams for pure Xe (parts (a) and (b)) and pure Ar (parts (c) and (d)) monolayer ﬁlms on graphite derived from grand canonical Monte Carlo simulation. Parts (a) and (c) show the temperature-chemical potential projections, while parts (b) and (d) show the temperature-density projections.

earlier molecular dynamics simulation studies of the melting transition of xenon on graphite [47, 48], also based on the Lennard-Jones potential, but with slightly different values of the parameters εXe,Xe and σXe , the triple point temperature was found to be located at Ttr ≈ 0.4εXe,Xe /k , while our result is Ttr ≈ 0.402εXe,Xe /k . Also, the critical point temperature Tcr ≈ 109 K is lower than the experimental value by about 18 K [45]. Nevertheless, the qualitative agreement with the available experimental data is good enough to assume that the results for the mixed ﬁlms are also qualitatively correct. In the case of argon, the phase diagram derived from our grand canonical simulation (see parts (c) and (d) of ﬁgure 1) agrees very well with experiment. In particular, the triple point temperature, equal to 49.5±0.5 K, is practically the same as the experimental value of 49.7 K [11]. Also, the critical temperature agrees very well with experiment [49]. The freezing of submonolayer xenon and argon ﬁlms leads to the formation of incommensurate structures. The xenon incommensurate solid attains the p of about 0.85 at the monolayer completion and p density is expended with respect to the commensurate ( 3 × 3)R30 structure. Figure 2 shows the temperature 1 a

Ψ6,Xe, ΦXe

0.8 ρc=0.5 ρc=0.7 ρc=0.8 ρc=0.9

0.6 0.4 0.2 0

0

50

100 T [K]

150

Figure 2. The temperature changes of the order parameters Ψ6,Xe and ΦXe for pure Xe ﬁlms on graphite at different densities (given in the ﬁgure). The results for ρ c = 0.5 and 0.7 have been obtained using a two-dimensional model, while those for ρ c = 0.8 and 0.9 using a three-dimensional model. Filled and open circles mark the bond-orientational and positional order parameters, respectively.

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A. Patrykiejew

changes of the order parameters Ψ6,Xe and ΦXe for xenon ﬁlms of different total densities and one sees that the order parameter ΦXe is quite low even at very low temperatures, indicating the lack of localization of adatoms over the minima of the graphite lattice. On the other hand, the bond-orientational order parameter Ψ6 demonstrates the formation of hexagonally ordered phase below the freezing point.

α=3

ο

Figure 3. The snapshot of conﬁguration for pure argon ﬁlm recorded for ρ c = 1.0 at T = 10 K. The dashed lines show that the ﬁlm is rotated with respect to the symmetry axis of the commensurate phase by the angle α ≈ 3◦ .

The argon incommensurate solid also shows p pa well developed hexagonal symmetry, but it is contracted with respect to the commensurate ( 3 × 3)R30 structure and attains the density of about 1.25. At suﬃciently low temperatures, the ﬁlm exhibits epitaxial rotation of about 3 degrees (see ﬁgure 3). Again, this result agrees very well with earlier theoretical [18, 19] and computer simulation [16, 27] results. The results of Monte Carlo simulation for pure argon and xenon ﬁlms given above will serve as reference data for the study of mixed ﬁlms.

4. The results for mixed ﬁlms We begin with the presentation of canonical ensemble Monte Carlo simulation results aiming at the determination of the melting temperature and the structure of solid phases in submonolayer mixed ﬁlms. Since the solid phases (commensurate and incommensurate) exhibit hexagonal symmetry, the location of the melting point can be estimated using the bond-orientational order parameter, Ψ6 , and its susceptibility, χΨ6 . Of course, one also expects that the melting transition is manifested by sudden changes of the potential energy and the heat capacity anomalies. Figure 4 gives an example of our results, obtained for the submonolayer ﬁlm of the total density ρ c = 0.4 and the xenon mole fraction equal to xXe = 0.1667. Part (a) of ﬁgure 4 shows the heat capacity curve and one sees a sharp peak at the melting point at T ≈ 54 K. At the same temperature, the total potential energy u and the contributions to the potential energy due to Ar-graphite and Xe-graphite interactions exhibit sudden drops (see part (b) of ﬁgure 4). Finally, part (c) of ﬁgure 4, which shows the temperature changes of the bond-orientational order parameter Ψ6 , and its susceptibility χΨ6 demonstrates that the melting transition is accompanied by the loss of hexagonal ordering. It should be emphasized that a large increase of the Ar-graphite and Xe-graphite interaction energies accompanying the freezing transition marks a sudden increase of localization of the adsorbed argon and xenon in the solid phase. Upon a decrease of temperature, the localization of xenon gradually increases, while argon exhibits a decrease of localization at temperatures below T ≈ 30 K. This behavior can be attributed to the transition between the commensurate phase, stable at T > 30 K, and the incommensurate phase, stable at T < 30 K. Note that the transition is not accompanied by any changes in the 23601-6

Ar–Xe mixed ﬁlm on graphite

20

a

15 CV 10 5 0

b

-1.5

Ar Xe

-0.05

-2.0

ugs

u

-2.5

-0.10

-3.0 -0.15

-3.5 c 50

0.8

40

0.6 Ψ6 0.4

30 20

0.2 0 0.0

10 20.0

40.0 T [K]

60.0

0 80.0

χΨ

6

behavior of the bond-orientational order parameter, but produces a well seen heat capacity anomaly. The inspection snapshots of conﬁgurations recorded during the simulation runs have shown that the commensurate phase is mixed, while the incommensurate phase consists of argon only. In the case of small xenon mole fraction, as in the system considered now, we expect to observe only a partially developed commensurate phase. Indeed, the snapshot given in ﬁgure 5 (a) shows that the ﬁlm peripheries are predominantly occupied by argon atoms, which also show a rather high degree of incommensuration. At the temperature below commensurate-incommensurate transition, we ﬁnd coexisting domains of mixed commensurate and argon-like incommensurate phases (see ﬁgure 5 (b)). In the snapshots given in ﬁgure 5, we have assigned the atoms to commensurate and incommensurate positions using the following order parameter [50]: (4.1)

φ(r) = cos(q1 r) + cos(q2 r) + cos([q1 − q2 ]r) ,

and assuming that the atom is commensurate (incommensurate) when φ > 0 (φ É 0).One should note that Figure 4. The temperature changes of the heat even in a rather small system used, consisting of only capacity (part (a)), the total potential energy 480 atoms, the argon-like incommensurate domain exand contributions to the potential energy due hibits epitaxial rotation, just the same as observed for to for the mixed submonolayer ﬁlm of ρ c = pure argon ﬁlms. 0.4 and xXe = 0.1667, obtained using twodimensional model. In order to determine the locations of the commensurate-incommensurate transition, we have monitored the behavior of the order parameters ΦAr and ΦXe , deﬁned by the equation (2.11). Figure 6 shows the changes of these two order parameters with the xenon mole fraction at two different temperatures in

Figure 5. The snapshots obtained for the ﬁlm of the total density ρ c = 0.4 and xXe = 0.1667 at T = 36 (left panel) and 24 K (right panel). Black dots mark the centers of graphite cells, open circles with thin and thick lines represent argon atoms being commensurate and incommensurate with the graphite lattice, while larger light shaded and dark shaded circles are the xenon atoms being commensurate and incommensurate with the graphite lattice. The dashed line in part (b) shows that the argon-like incommensurate phase exhibits epitaxial rotation.

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A. Patrykiejew

0.8

Ar, T=12 K Xe, T=12 K Ar, T=36 K Xe, T=36 K

Φk

0.6

0.4

0.2

0

0

0.2

0.4

xXe

0.6

0.8

Figure 6. The order parameters ΦAr and ΦXe versus the xenon mole fraction in submonolayer ﬁlm of the density ρ c = 0.4 at two different temperatures, shown in the ﬁgure. The dotted vertical lines mark the regions of xXe over which the pure mixed commensurate phase appears, and the vertical dash-dotted line marks the xenon mole fraction at which the domain of commensurate phase disappears.

submonolayer ﬁlms of the total density ρ c = 0.4. Quite similar results have been obtained for the ﬁlms of different total densities and using two- as well as three-dimensional models. From the observed changes of the order parameters ΦAr and ΦXe , it follows that the increase of the xenon concentration leads to a sequence of changes in the ﬁlm structure. For small xXe we ﬁnd that xenon is highly localized, while the degree of localization of argon increases with xXe . In this region, the ﬁlm consists of two coexisting phases: one being the incommensurate argon-like solid and the second being the mixed commensurate solid. A gradual increase of the xenon mole fraction causes the commensurate domain to become larger and the size of incommensurate domain to shrink gradually. Then, there is a region of xenon concentration over which both adsorbates are highly localized. This corresponds to the presence of pure mixed commensurate phase and terminates at the xenon mole fraction close to about 0.4. Then, both order parameters gradually decrease when xXe increases up to about 0.75. In this region, the mixed commensurate phase coexists with the demixed xenon-like incommensurate phase. Finally, for xXe exceeding about 0.75, the ﬁlm consists of xenon-like incommensurate phase with the argon atoms located at its peripheries. This has been conﬁrmed by the inspection of snapshots and radial distribution functions. The central result of the canonical ensemble Monte Carlo study is given in ﬁgure 7, which contains the phase diagrams showing the locations of the melting transition and the regions of stability of different solid phases in the ﬁlms of different total densities. Part (a) of ﬁgure 7 gives the results for submonolayer ﬁlms of different total densities, equal to 0.4, 0.667 and 0.8. The results for ρ c = 0.4 and 0.667 have been obtained using a two-dimensional model, while those for ρ c = 0.8 have been obtained within a more realistic three-dimensional model. The locations of the melting point are more or less the same over a wide range of xXe between 0 and about 0.8. The independence of the melting temperature of the total density indicates that the melting occurs at the triple point temperature. For the xenon concentration higher than 0.8 the triple point melting occurs for ρ c = 0.4 and 0.667, but not for ρ c = 0.8. This suggests that the density ρ c = 0.8 is higher than the liquid density at the triple point. A rather sharp increase of the melting temperature with the xenon mole fraction, for xXe above 0.8, results from the fact that the increase of xenon concentration brings the ﬁlm closer to the monolayer completion. We should emphasize that even for xXe close to unity there is no trace of the promotion of adsorbed argon and xenon to the second layer, even at the temperatures above the melting point. Thus, the ﬁlms remain practically two-dimensional. Although we have not performed any simulation at ρ c = 0.8 using the two-dimensional model, it can be anticipated that the results should be quite the same as those obtained with the three-dimensional model. The two-dimensional approximation is expected to fail when the ﬁlm density starts to exceed the monolayer capacity. 23601-8

Ar–Xe mixed ﬁlm on graphite

b

a 120

140

100

120 liquid 100 T [K]

T [K]

80 liquid 60

80

(Ar+Xe)IC XeIC

20

0

0

XeIC

(Ar+Xe)C

40

(Ar+Xe)C+XeIC

20

0.2

0.4

xXe

0.6

0.8

1

0

0

(Ar+Xe)C

0,2

(Ar+Xe)C+XeIC

40

ArIC+(Ar+Xe)C

ArIC+(Ar+Xe)C

60

0,4

0,6

0,8

1

xXe

Figure 7. The phase diagrams derived from the canonical ensemble simulations for submonolayer (part (a)) and monolayer (part (b)) mixed ﬁlms of argon and xenon. In part (a), the ﬁlled, shaded and open circles show the melting temperatures for the ﬁlms of different total density equal to 0.4, 0.5 and 0.8, respectively. The results for ρ c = 0.4 and 0.5 have been obtained using two-dimensional model, while those for ρ c = 0.8 have been obtained using three-dimensional model. The ﬁlled squares, diamonds and triangles mark the stability regions of differently ordered solid phases. In part (b) the ﬁlled and open circles mark the melting points obtained using three- and two-dimensional models, respectively. Open squares give the onset of the second layer promotion and the ﬁlled diamonds show the limit of stability of the mixed incommensurate phase. Dotted lines show the approximate locations of stability limits of different solid phases.

We have carried out the canonical ensemble calculations assuming that ρ c = 1.0. This density is lower than monolayer capacity of pure argon ﬁlm, but it is well above the monolayer capacity of pure xenon ﬁlm. Part (b) of ﬁgure 7 shows the phase diagram obtained. In this case, the two-dimensional model works well only in the region of the xenon mole fraction not greater than about 0.2, and starts to overestimate the stability of the solid phase for higher xenon concentrations. Three-dimensional calculations have shown that argon is partially promoted to the second layer when the temperature becomes high enough. The temperature at which the second layer promotion begins depends upon the ﬁlm composition and it is higher than the melting temperature only for xXe not exceeding about 0.2 and lies below the melting temperature for higher xenon mole fractions. It is therefore not surprising that the two-dimensional model works well only for xXe É 0.2. One of the consequences of the promotion of argon atoms to the second layer is the increase of xenon concentration in the ﬁrst layer with respect to the nominal xenon concentration in the simulation cell. The locations of phase transitions in ﬁgure 7 (b) have been plotted for the values of xXe corresponding to the actual xenon concentration in the ﬁrst layer. One sees that the xenon mole fraction range over which the commensurate phase is stable is considerably wider than in the previously discussed ﬁlms of lower total density (cf. ﬁgure 7 (a)). Moreover, we ﬁnd that over a certain range of xenon concentrations, between about 0.48 and 0.83, the mixed incommensurate phase appears at the temperatures just below the freezing transition, whereas no trace of such a phase has been found in submonolayer ﬁlms. Upon the decrease of temperature, this phase transforms either into the a commensurate phase, when xXe is lower than 0.6, or into the coexisting commensurate and xenon-like incommensurate phase, when xXe is higher than 0.6. Figure 8 shows the changes of the heat capacity (part (a)) and the order parameters (part (b)) in the case of the ﬁlm with xXe = 0.7. The heat capacity has two pronounced anomalies. The ﬁrst one, at T ≈ 108.5 K, is the signature of freezing transition, accompanied by the development of hexagonal order in the ﬁlm (see the behavior of the bond-orientational order parameter Ψ6 in ﬁgure 8 (b)). At the temperatures between the freezing point and the second heat capacity anomaly at T ≈ 70 K, the solid phase is incommensurate and the order parameters ΦAr and ΦXe remain very small, as expected for the incommensurate phase. The inspection 23601-9

A. Patrykiejew

op

CV

of snapshots has shown that the incommensurate a phase is mixed and the argon is partially pro5 moted to the second layer. The second heat ca4 pacity anomaly, at T ≈ 70 K, is due to the onset of the transition accompanied by a rather large 3 increase of the order parameters ΦAr and ΦXe , 2 indicating the increase of localization of Ar and Xe upon the lowering of temperature. Figure 9 1 shows the snapshots recorded at 72 K and 12 K, 0 which demonstrate that the transition observed b leads to the formation of domains consisting of 0.8 the mixed commensurate and demixed Xe-like incommensurate phases. The snapshot recorded at 0.6 Ψ6 12 K also demonstrates that argon is partially proΦAr moted to the second layer, and forms a compact 0.4 ΦXe island of a solid-like phase. It is also noteworthy that the solid-like patch of argon in the second 0.2 layer is located over the demixed xenon domain 0.0 rather than over the domain formed by the mixed 20 40 80 100 120 60 commensurate phase. This can be readily underFigure 8. The temperature changes of the heat castood by taking into account the magnitudes of pacity (part (a)) and of different order parameAr–Ar and Ar–Xe interaction energies, measured ters (part (b)) obtained for the system with ρ c = by the Lennard-Jones potential parameters εAr,Ar 1.0 and xXe = 0.7 using three-dimensional model. and εAr,Xe , and of course εAr,Xe is considerably The vertical dashed lines mark the freezing and larger than εAr,Ar (cf. table 1). When the argon commensurate-incommensurate transitions. atoms from the second layer are located over the pure xenon patch, each of them has three xenon atoms from the ﬁrst layer as nearest neighbors. On the other hand, if the argon patch were located over the mixed commensurate patch then some nearest neighbors from the ﬁrst layer would be argon atoms, and this situation is energetically less favorable. When the xenon concentration exceeds about 0.83, the ﬁrst layer consists only of xenon, while all argon atoms are promoted to the second layer. Of course, when the amount of xenon in the ﬁlm exceeds the monolayer capacity of pure xenon ﬁlm, then the excess of xenon is also located in the second layer.

Figure 9. The snapshots for the mixed ﬁlm of the total density ρ c = 1.0 and xXe = 0.7 at T = 72 K (left panel) and 12 K (right panel). Black dots mark the centers of graphite cells, open circles represent argon atoms being commensurate with the graphite lattice, while larger light shaded and dark shaded circles are the xenon atoms being commensurate and incommensurate with the graphite lattice. Filled circles stand for argon atoms located in the second layer.

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Ar–Xe mixed ﬁlm on graphite

Figure 10. The examples of adsorption isotherms at T = 60 K, obtained for µ∗ = −19.0 (part (a)) and Xe −18.6 (part (b)). Circles show the total adsorption, while squares and diamonds denote the adsorption of argon and xenon, respectively.

We now proceed to the discussion of the changes in the phase behavior resulting from grand canonical Monte Carlo simulation. The calculations have been performed under the condition of the ﬁxed chemical potential of xenon, so that the xenon concentration in the ﬁlm was not conserved. Along the adsorption isotherms obtained by changing the chemical potential of argon, the amounts of xenon change as well. Figure 10 shows two examples of adsorption isotherms, both recorded at T = 60 K, but with different values of the xenon chemical potential. It is quite evident that the gas-liquid transition is accompanied by a sudden increase of the xenon density and that this effect is much stronger when the chemical potential of xenon is higher. We have determined the phase diagrams for a series of systems with different values of the xenon chemical potential, µXe = −20.0, −19.5, −19.0 and −18.6. In the case of µXe = −20.0, the amounts of xenon in the ﬁlm are very low, with xXe < 0.01, over the entire range of temperatures and ﬁlm densities studied. It is, therefore, not surprising that the phase diagram obtained is very similar to that of pure argon ﬁlm (see ﬁgure 11). In particular, the melting transition appears to be continuous and the solid phase is an incommensurate Figure 11. The phase diagram for the system with argon-like phase. However, we ﬁnd that even very µXe = −20.0 (ﬁlled symbols) and of pure argon ﬁlm small amounts of xenon shift the locations of the (open symbols). Parts (a) and (b) show the tempertriple and critical points towards higher temperaature – argon chemical potential and the temperatures. ture – total density projections, respectively.

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-8.0

a

c

b

d

-8.5 µAr

-9.0 -9.5 -10.0

1.0

ρ

0.8 0.6 0.4 0.2 0.0

50

55 60 T [K]

65

50

55 60 T [K]

65

Figure 12. The phase diagrams for the systems with µXe = −19.5 (parts (a) and (b)) and −19.0 (parts (c) and (d)). Parts (a) and (c) show the temperature – argon chemical potential projections and parts (b) and (d) the temperature – total density projections, respectively. In parts (b) and (d), ﬁlled circles show the total density and open circles show the argon density. The vertical dash-dotted lines mark the temperature above which the commensurate solid looses stability.

An increase of µXe to −19.5 leads to some qualitative changes in the phase diagram topology (see parts (a) and (b) of ﬁgure 12). In particular, at the temperatures below about 50.5 K, the two-dimensional gas condenses into the commensurate krypton-like structure, which undergoes a transition into the incommensurate phase when the argon chemical potential becomes high enough. This commensurate-

6

a

Ar-Ar Ar-Xe

5

gi-j(r)

4

µAr = -9.3

3 µAr = -9.9

2 1 0 6

b

gAr-Ar(r)

5 4

µAr = -8.8

3 2

µAr = -9.0

1 0

0

2

4

6 r

8

10

12

Figure 13. The argon-argon radial distribution functions for the system with µXe = −19.0, recorded at T = 51 K (part (a)) and T = 57 (part (b)) for the values of the argon chemical potential given in the ﬁgure. The vertical dotted lines mark the locations of subsequent maxima in a perfectly ordered commensurate phase. Part (a) also shows the argon-xenon radial distribution function obtained for µAr = −9.9.

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Ar–Xe mixed ﬁlm on graphite

ρ

µAr

incommensurate transition is continuous. One should note that the xenon concentration in the commensurate phase is rather low ( xXe < 0.2), and becomes still lower in the incommensurate phase. For still higher value of µXe equal to −19.0 the phase diagram topology remains the same and only the stability region of the commensurate phase becomes wider and extents up to T ≈ 57 K. Also, the commensurateincommensurate transition occurs at higher values of the argon chemical potential. At the temperature between the upper limit of the commensurate phase, i.e., T ≈ 57 K, and the critical point, the gas phase condenses into the liquid phase. The picture presented above is very well conﬁrmed by the behavior of radial distribution functions. Part (a) of ﬁgure 13 gives the argon-argon distribution functions recorded at the temperature of 51 K and for the argon chemical potentials below and above the commensurateincommensurate transition. It is evident that the maxima, apart from the ﬁrst one, coincide very nicely with the locations of subsequent neighbors in the commensurate phase. The stability of commensurate phase is due to the presence of argon-xenon nearest neighbors, and the argon-xenon p distribution function (also shown in ﬁgure 13 as a dashed line) exhibits the ﬁrst maximum very close to 3a 1 , as expected for the commensurate phase. At the temperature of 58 K, i.e, above the upper limit of the commensurate phase stability, the argonargon distribution function recorded at µAr = −9.0 demonstrates the presence of a liquid phase, while at µAr = −8.8 it corresponds to an incommensurate solid phase. The liquid is of course partially ordered due to the effects of periodic corrugation potential. One should also note a gradual increase of the critical temperature resulting from the ina crease of xenon concentration. The phase behavIC -8.5 ior changes when the chemical potential of xenon is increased to −18.6. Figure 14 shows that the -9.0 gas condenses into a liquid phase of rather high L C xenon concentration, ranging between xXe ≈ 0.67 -9.5 at T = 56 K and xXe ≈ 0.44 at T = 66 K. When the argon chemical potential increases, we observe G -10.0 the transition between the liquid and commensuIC b rate phases. This transition, quite well illustrated 1.0 C L by the change in the behavior of the argon-argon 0.8 radial distribution function given in ﬁgure 15, occurs only at the temperatures lower than 61 K. 0.6 A further increase of the argon chemical poten0.4 tial does not lead to the transition between the commensurate and incommensurate solid phases, 0.2 as in the previously considered cases, but rather G 0.0 again to the liquid phase. The liquid undergoes 56 58 60 62 64 66 68 T [K] a transition into the incommensurate solid-like phase at still higher values of the argon chemiFigure 14. The phase diagram for the system with cal potential (cf. ﬁgure 15). This re-entrant behavµXe = −18.6. Parts (a) and (b) show the temperaior can be understood by taking into account that ture – argon chemical potential and the temperature – total density projections, respectively. The abthe upper limit of the ﬁlm density in the commenbreviations G, L, C and IC stand for the gas, liquid, surate phase is equal to 1.0, while the transition commensurate solid and incommensurate solid, reinto the incommensurate solid in the argon rich spectively. The vertical dash-dotted line marks the ﬁlm and at the temperatures used occurs at the temperature above which the commensurate phase densities well above unity. An increasing chemidoes not appear. cal potential leads to a gradual removal of xenon, so that the dense ﬁlm becomes more and more argon-like. Note that the liquid-incommensurate solid transition in pure argon ﬁlm at T = 56 K occurs at the density of about 1.07 (cf. ﬁgure 1 (b)), and at still higher densities at higher temperatures. Concluding, we would like to present the comparison of our results with the available experimental data for submonolayer ﬁlms of the total density equal to ρ c = 0.4. One readily notes a qualitative agreement between Monte Carlo and experimental results. However, the present simulation predicts a considerably narrower range of xenon concentrations corresponding to the stability region of the com23601-13

A. Patrykiejew

gAr-Ar(r)

10

8

µAr = -8.60

6

µAr = -9.00

4

µAr = -9.60

2

µAr = -9.86

0

0

10

5

20

15

r

25

Figure 15. The argon-argon radial distribution functions for the system with µ∗ = −18.6, recorded at Xe T = 57 K and different values of µ∗ (shown in the ﬁgure). The vertical dotted lines mark the locations of Xe subsequent maxima in a perfectly ordered commensurate phase.

mensurate phase. Unfortunately, we cannot propose any reasonable explanation for the underestimation of the commensurate phase stability by computer simulation. One can speculate that our model based on Lennard-Jones potential and standard mixing rules overestimates the tendency towards demixing in submonolayer ﬁlms. The commensurate phase is mixed, while the incommensurate xenon-like phase is demixed. It is also possible, however, that x-ray diffraction data overestimate the range of xenon mole fractions corresponding to the commensurate phase. Note that within the region of coexisting commensurate (C) and xenon-like incommensurate (IXXe ) phases the paches of incommensurate phase may be quite small and hence escape detection. We recall that Villain and Moreira [35] have also questioned the reliability of experimental results given in reference [23] and suggested that the results were affected by metastability effects. ICAr+C

MC simulation

C+ICXe

C

ICXe T = 45 K

exp.

ICAr

MC simulation

ICAr+C

C+ICXe

C

ICXe

C+ICXe

C

ICXe T = 11 K

exp.

ICAr

0,0

0,1

0,2

0,3

0,4

ICXe

C+ICXe

C

0,5 xXe

0,6

0,7

0,8

0,9

1,0

Figure 16. A comparison of the phase diagrams of Ar–Xe submonolayer ﬁlms at two temperatures (given in the ﬁgure) resulting from the present Monte Carlo simulation and from the experiment [23].

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Ar–Xe mixed ﬁlm on graphite

Acknowledgements This work has been supported by the Polish Ministry of Science under the grant No. N N202 046137.

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44. Hong H., Petters C.J., Mak A., Birgeneau R.J., Horn P.M., Suematsu H., Phys. Rev. B, 1989, 40, 4797; doi:10.1103/PhysRevB.40.4797. 45. Thomy A., Duval X., Regnier J., Surf. Sci. Rep., 1981, 1, 1; doi:10.1016/0167-5729(81)90004-2. 46. Aziz R.A., Slaman M.J., Molec. Phys., 1986, 57, 825; doi:10.1080/00268978600100591. 47. Abraham F.F., Phys. Rev. Lett., 1983, 50, 978; doi:10.1103/PhysRevLett.50.978. 48. Abraham F.F., Phys. Rev. B, 1983, 28, 7338; doi:10.1103/PhysRevB.28.7338. 49. Millot F., J. Phys. Lett. (Paris), 1979, 40, L-9; doi:10.1051/jphyslet:019790040010900. 50. Houlrik J. M., Landau D.P., Phys. Rev. B, 1991, 44, 8962; doi:10.1103/PhysRevB.44.8962.

До фазової поведiнки змiшаних субмоношарових плiвок Ar–Xe на графiтi А. Патрикєєв Вiддiл моделювання фiзико-хiмiчних процесiв, унiверситет Марiї Кюрi-Склодовської, 20031 Люблiн, Польща Використовуючи методи комп’ютерного моделювання методом Монте Карло у канонiчному i великому канонiчному ансамблях, ми обговорюємо плавлення i формування впорядкованих структур змiшаних Ar–Xe субмоношарових плiвок на графiтi. Обчислення виконуються з використанням дво- i тривимiрних модельних систем. Показано, що позаплощинний рух не впливає на властивостi адсорбованої плiвки до тих пiр, поки загальна густина не стає близькою до моношарового завершення. З iншого боку, близько до моношарового завершення, просування частинок до другого шару значною мiрою впливає на властивостi змiшаних плiвок. Показано, що сумiш повнiстю змiшується в рiдкiй фазi i заморожується у твердi фази зi структурою, що залежить вiд складу плiвки. Для субмоношарових густин, температура плавлення змiнюється немонотонно зi змiною складу плiвки. Зокрема, температура плавлення спочатку зростає з ростом концентрацiї ксенону близько 20%, потiм зменшується i досягає мiнiмуму для концентрацiї ксенону близько 40%. Для вищих концентрацiй ксенону точка плавлення поступово зростає до температур, що вiдповiдають плiвцi чистого ксенону. Також показано, що топологiя фазових дiаграм змiшаних плiвок є чутливою до складу адсорбованих шарiв. Ключовi слова: адсорбцiя сумiшей, фазовi переходи, комп’ютерне моделювання, плавлення

23601-16