On the phase behavior of mixed Ar-Xe submonolayer films on graphite
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 studie...
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irk-123456789-1202812017-06-12T03:03:36Z On the phase behavior of mixed Ar-Xe submonolayer films on graphite Patrykiejew, A. 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. 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. Так 2012 Article On the phase behavior of mixed Ar-Xe submonolayer films on graphite / A. Patrykiejew // Condensed Matter Physics. — 2012. — Т. 15, № 2. — С. 23601:1-16. — Бібліогр.: 50 назв. — англ. 1607-324X PACS: 68.35.Rh, 68.43.De, 68.43.Fg, 64.75.St DOI:10.5488/CMP.15.23601 arXiv:1207.2995 http://dspace.nbuv.gov.ua/handle/123456789/120281 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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| description |
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. |
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Patrykiejew, A. |
| spellingShingle |
Patrykiejew, A. On the phase behavior of mixed Ar-Xe submonolayer films on graphite Condensed Matter Physics |
| author_facet |
Patrykiejew, A. |
| author_sort |
Patrykiejew, A. |
| title |
On the phase behavior of mixed Ar-Xe submonolayer films on graphite |
| title_short |
On the phase behavior of mixed Ar-Xe submonolayer films on graphite |
| title_full |
On the phase behavior of mixed Ar-Xe submonolayer films on graphite |
| title_fullStr |
On the phase behavior of mixed Ar-Xe submonolayer films on graphite |
| title_full_unstemmed |
On the phase behavior of mixed Ar-Xe submonolayer films on graphite |
| title_sort |
on the phase behavior of mixed ar-xe submonolayer films on graphite |
| publisher |
Інститут фізики конденсованих систем НАН України |
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2012 |
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http://dspace.nbuv.gov.ua/handle/123456789/120281 |
| citation_txt |
On the phase behavior of mixed Ar-Xe submonolayer films on graphite / A. Patrykiejew // Condensed Matter Physics. — 2012. — Т. 15, № 2. — С. 23601:1-16. — Бібліогр.: 50 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT patrykiejewa onthephasebehaviorofmixedarxesubmonolayerfilmsongraphite |
| first_indexed |
2025-07-08T17:35:33Z |
| last_indexed |
2025-07-08T17:35:33Z |
| _version_ |
1837101097770024960 |
| fulltext |
Condensed Matter Physics, 2012, Vol. 15, No 2, 23601: 1–16
DOI: 10.5488/CMP.15.23601
http://www.icmp.lviv.ua/journal
On the phase behavior of mixed Ar–Xe submonolayer
films on graphite
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 incom-
mensurate 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 simula-
tions [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 themelting point,
the solid-like argon submonolayer film is rotated by about 2–3 degrees with respect to the R30◦ axis of
the commensurate (
p
3×
p
3)R30◦ structure [16, 18–20]. Since the argon atoms are rather small, the solid
phase is compressed with respect to the (
p
3×
p
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 xenon atoms are larger so that the incommen-
surate solid phase is dilated with respect to the commensurate (
p
3×
p
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 X-
ray 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 Xe-
like incommensurate phases, being stable for sufficiently low and high xenon mole fraction, respectively,
the formation of krypton-like (
p
3 ×
p
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
http://dx.doi.org/10.5488/CMP.15.23601
http://www.icmp.lviv.ua/journal
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 incommen-
surate 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 demon-
strated [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 andMoreira [35] requires the introduction of
several approximations in order tomake 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 incom-
mensurate, 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 compo-
sition.
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 (ri j ) = 4εi , j
[
(
σi , j /ri j
)12 −
(
σi , j /ri j
)6
]
, (2.1)
where ri 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 εi , j
Å K
Ar,Ar 3.4 120.0
Kr,Kr 4.1 221.0
Ar,Kr 3.5 162.85
representing the Ar–Xe interaction, also given in table 1, have been obtained using the usual Lorentz-
Bertholot combining rules:
σi , j =
1
2
(
σi ,i +σ j , j
)
and εi , j =
√
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]
vi (x, y, z) = εgs,i
[
v0,i (z)+
∑
k
vk ,i (z) fk (x, y)
]
, i =Ar or Kr. (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 v0,i (z), the Fourier components
vk ,i (z) and the functions fk (x, y) are given by the following equations:
v0,i (z) =
4πA6
i
as
∞
∑
n=0
[
2A6
i
(z +n∆z)10
−
1
(z +n∆z)4
]
, (2.4)
vk ,i (z)=
2πA6
i
as
[
A6
i
30
( qk
2z
)5
K5(qk z)−2
( qk
2z
)
K2(qk z)
]
(2.5)
and
fk (x, y) =
∑
l
exp[iqk ,lτ], τ= (x, y) (2.6)
with the sum running over all graphite reciprocal lattice vectors of the length qk . In the above equations
Ai = 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, as = 5.24 Å2 is the area of the graphite unit cell, K2 and K5 are the
modified Bessel functions of the second kind and of the second and fifth order respectively, and qk ’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
K Å
Ar 58.00 3.40
Kr 78.66 3.75
In the case of only partially filled monolayer films and at sufficiently low temperatures, the promo-
tion 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
v(x, y) =Vb,i f1(x, y) =−Vb,i
{
cos(q1r)+cos(q2r)+cos([q1 −q2]r)
}
, (2.7)
where the parameter Vb,i (i = Ar, Kr) determines the amplitude of the corrugation potential. The mag-
nitudes 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 theMonte Carlomethod in the canonical and grand canonical
ensembles [39, 40]. In the case of two-dimensional model, the rectangular simulation cell of the size
La1 ×L
p
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 60a1 ×
60a1
p
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 = 10a1.
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, 〈egg〉 and the contributions due to the fluid-solid interaction for
each component 〈egs,i 〉 and the heat capacity obtained from the fluctuations of the potential energy,
CV =
N
kT 2
(
〈e2〉−〈e〉2
)
. (2.8)
In order to monitor the structure of solid phases we have used radial distribution functions, gi 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
Nb,i
∑
mi
∑
ni
exp
(
i6φm,n
)
∣
∣
∣
∣
∣
(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
Ψ6 =
∣
∣
∣
∣
1
Nb
∑
m
∑
n
exp
(
i6φm,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]
Φi =
∣
∣
∣
∣
∣
1
6Ni
∑
m
6
∑
n=1
exp
(
iqnrm,i
)
∣
∣
∣
∣
∣
. (2.11)
The first sum is taken over all atoms of the i -th component, while the second sum runs over the six recip-
rocal 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 Ly
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 adsorption-
desorption isotherms.
Throughout this paper, we use reduced quantities, assuming that the graphite lattice constant a1
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 poten-
tial 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
-18.0
-17.0
-16.0
-15.0
µ∗
Xe
a
80 100 120
T [K]
0
0.2
0.4
0.6
0.8
ρ
c
b
-9.8
-9.6
-9.4
-9.2
-9
µ∗
Ar
c
48 50 52 54 56 58
T [K]
0.0
0.2
0.4
0.6
0.8
1.0
ρ
c
d
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 pro-
jections.
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 struc-
tures. The xenon incommensurate solid attains the density of about 0.85 at the monolayer completion and
is expended with respect to the commensurate (
p
3×
p
3)R30 structure. Figure 2 shows the temperature
0 50 100 150
T [K]
0
0.2
0.4
0.6
0.8
1
Ψ
6,
X
e, Φ
X
e
a
ρ
c
=0.5
ρ
c
=0.7
ρ
c
=0.8
ρ
c
=0.9
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 ρc = 0.5 and 0.7 have been obtained using a
two-dimensional model, while those for ρc = 0.8 and 0.9 using a three-dimensional model. Filled and
open circles mark the bond-orientational and positional order parameters, respectively.
23601-5
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 localiza-
tion of adatoms over the minima of the graphite lattice. On the other hand, the bond-orientational order
parameterΨ6 demonstrates the formation of hexagonally ordered phase below the freezing point.
α=3ο
Figure 3. The snapshot of configuration for pure argon film recorded for ρc = 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 a well developed hexagonal symmetry, but it is con-
tracted with respect to the commensurate (
p
3×
p
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] re-
sults.
The results of Monte Carlo simulation for pure argon and xenon films given above will serve as ref-
erence 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 susceptibil-
ity, χΨ6 . Of course, one also expects that the melting transition is manifested by sudden changes of the po-
tential energy and the heat capacity anomalies. Figure 4 gives an example of our results, obtained for the
submonolayer film of the total density ρc = 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 incom-
mensurate 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
-0.15
-0.10
-0.05
u
gs
Ar
Xe
0
5
10
15
20
C
V
-3.5
-3.0
-2.5
-2.0
-1.5
u
0.0 20.0 40.0 60.0 80.0
T [K]
0
0.2
0.4
0.6
0.8
Ψ6
0
10
20
30
40
50
χΨ6
a
b
c
Figure 4. The temperature changes of the heat
capacity (part (a)), the total potential energy
and contributions to the potential energy due
to for the mixed submonolayer film of ρc =
0.4 and xXe = 0.1667, obtained using two-
dimensional model.
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 incommensu-
rate 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 com-
mensurate phase. Indeed, the snapshot given in fig-
ure 5 (a) shows that the film peripheries are predom-
inantly occupied by argon atoms, which also show a
rather high degree of incommensuration. At the tem-
perature below commensurate-incommensurate tran-
sition, we find coexisting domains of mixed commen-
surate and argon-like incommensurate phases (see fig-
ure 5 (b)). In the snapshots given in figure 5, we have
assigned the atoms to commensurate and incommensu-
rate positions using the following order parameter [50]:
φ(r)= cos(q1r)+cos(q2r)+cos([q1 −q2]r) , (4.1)
and assuming that the atom is commensurate (incom-
mensurate) when φ > 0 (φ É 0).One should note that
even in a rather small system used, consisting of only
480 atoms, the argon-like incommensurate domain ex-
hibits epitaxial rotation, just the same as observed for
pure argon films.
In order to determine the locations of the commen-
surate-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 ρc = 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 incom-
mensuratewith the graphite lattice. The dashed line in part (b) shows that the argon-like incommensurate
phase exhibits epitaxial rotation.
23601-7
A. Patrykiejew
0 0.2 0.4 0.6 0.8
x
Xe
0
0.2
0.4
0.6
0.8
Φ
k
Ar, T=12 K
Xe, T=12 K
Ar, T=36 K
Xe, T=36 K
Figure 6. The order parameters ΦAr and ΦXe versus the xenon mole fraction in submonolayer film of the
density ρc = 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 ρc = 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 parametersΦAr andΦXe, it follows that the increase of the xenon concentration leads to a se-
quence of changes in the film structure. For small xXe we find that xenon is highly localized, while the de-
gree 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 commensu-
rate 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 ρc = 0.4 and 0.667 have been
obtained using a two-dimensional model, while those for ρc = 0.8 have been obtained within a more re-
alistic 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 ρc = 0.4 and 0.667, but not for ρc = 0.8. This suggests that the den-
sity ρc = 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 ρc = 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
0 0.2 0.4 0.6 0.8 1
x
Xe
0
20
40
60
80
100
120
T
[
K
]
(Ar+Xe)
C
+Xe
IC
A
r IC
+
(A
r+
X
e)
C
Xe
IC
liquid
(Ar+Xe)
C
a
0 0,2 0,4 0,6 0,8 1
x
Xe
0
20
40
60
80
100
120
140
T
[
K
]
(A
r+
X
e)
C
+
X
e IC
Xe
IC
liquid
A
r IC
+
(A
r+
X
e)
C (Ar+Xe)
C
b
(Ar+Xe)
IC
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 ρc = 0.4 and 0.5 have been obtained using two-dimensional model, while
those for ρc = 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 ρc = 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 diagramobtained. 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 con-
siderably 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 in-
commensurate 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
0
1
2
3
4
5
C
V
a
20 40 60 80 100 120
0.0
0.2
0.4
0.6
0.8
op
b
Ψ6
Φ
Ar
Φ
Xe
Figure 8. The temperature changes of the heat ca-
pacity (part (a)) and of different order parame-
ters (part (b)) obtained for the system with ρc =
1.0 and xXe = 0.7 using three-dimensional model.
The vertical dashed lines mark the freezing and
commensurate-incommensurate transitions.
of snapshots has shown that the incommensurate
phase is mixed and the argon is partially pro-
moted to the second layer. The second heat ca-
pacity anomaly, at T ≈ 70 K, is due to the onset
of the transition accompanied by a rather large
increase of the order parameters ΦAr and ΦXe,
indicating the increase of localization of Ar and
Xe upon the lowering of temperature. Figure 9
shows the snapshots recorded at 72 K and 12 K,
which demonstrate that the transition observed
leads to the formation of domains consisting of
the mixed commensurate and demixed Xe-like in-
commensurate phases. The snapshot recorded at
12 K also demonstrates that argon is partially pro-
moted to the second layer, and forms a compact
island of a solid-like phase. It is also noteworthy
that the solid-like patch of argon in the second
layer is located over the demixed xenon domain
rather than over the domain formed by the mixed
commensurate phase. This can be readily under-
stood by taking into account the magnitudes of
Ar–Ar and Ar–Xe interaction energies, measured
by the Lennard-Jones potential parameters εAr,Ar
and εAr,Xe, and of course εAr,Xe is considerably
larger than εAr,Ar (cf. table 1). When the argon
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 ρc = 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.
23601-10
Ar–Xe mixed film on graphite
Figure 10. The examples of adsorption isotherms at T = 60 K, obtained for µ∗
Xe
= −19.0 (part (a)) and
−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 canon-
ical 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
Figure 11. The phase diagram for the system with
µXe = −20.0 (filled symbols) and of pure argon film
(open symbols). Parts (a) and (b) show the temper-
ature – argon chemical potential and the tempera-
ture – total density projections, respectively.
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 par-
ticular, the melting transition appears to be con-
tinuous and the solid phase is an incommensurate
argon-like phase. However, we find that even very
small amounts of xenon shift the locations of the
triple and critical points towards higher tempera-
tures.
23601-11
A. Patrykiejew
-10.0
-9.5
-9.0
-8.5
-8.0
µ A
r
a c
50 55 60 65
T [K]
0.0
0.2
0.4
0.6
0.8
1.0
ρ
b
50 55 60 65
T [K]
d
Figure 12. The phase diagrams for the systemswith 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 in-
commensurate phase when the argon chemical potential becomes high enough. This commensurate-
0
1
2
3
4
5
6
g i-
j(r
)
µ
Ar
= -9.9
µ
Ar
= -9.3
a
µ
Ar
= -8.8
µ
Ar
= -9.0
Ar-Ar
Ar-Xe
0 2 4 6 8 10 12
r
0
1
2
3
4
5
6
g A
r-
A
r(r
)
b
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.
23601-12
Ar–Xe mixed film on graphite
incommensurate transition is continuous. One should note that the xenon concentration in the commen-
surate 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 commensurate-
incommensurate 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 commensurate-
incommensurate 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 distribution func-
tion (also shown in figure 13 as a dashed line) exhibits the first maximum very close to
p
3a1, as expected
for the commensurate phase.
At the temperature of 58 K, i.e, above the upper limit of the commensurate phase stability, the argon-
argon 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.
-10.0
-9.5
-9.0
-8.5
µ A
r
a
b
G
L
C
IC
56 58 60 62 64 66 68
T [K]
0.0
0.2
0.4
0.6
0.8
1.0
ρ
G
LC
IC
Figure 14. The phase diagram for the system with
µXe = −18.6. Parts (a) and (b) show the tempera-
ture – argon chemical potential and the tempera-
ture – total density projections, respectively. The ab-
breviations G, L, C and IC stand for the gas, liquid,
commensurate solid and incommensurate solid, re-
spectively. The vertical dash-dotted line marks the
temperature above which the commensurate phase
does not appear.
One should also note a gradual increase of
the critical temperature resulting from the in-
crease of xenon concentration. The phase behav-
ior changes when the chemical potential of xenon
is increased to −18.6. Figure 14 shows that the
gas condenses into a liquid phase of rather high
xenon concentration, ranging between xXe ≈ 0.67
at T = 56 K and xXe ≈ 0.44 at T = 66 K. When the
argon chemical potential increases, we observe
the transition between the liquid and commensu-
rate phases. This transition, quite well illustrated
by the change in the behavior of the argon-argon
radial distribution function given in figure 15, oc-
curs only at the temperatures lower than 61 K.
A further increase of the argon chemical poten-
tial does not lead to the transition between the
commensurate and incommensurate solid phases,
as in the previously considered cases, but rather
again to the liquid phase. The liquid undergoes
a transition into the incommensurate solid-like
phase at still higher values of the argon chemi-
cal potential (cf. figure 15). This re-entrant behav-
ior can be understood by taking into account that
the upper limit of the film density in the commen-
surate phase is equal to 1.0, while the transition
into the incommensurate solid in the argon rich
film and at the temperatures used occurs at the
densities well above unity. An increasing chemi-
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 experimen-
tal data for submonolayer films of the total density equal to ρc = 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 com-
23601-13
A. Patrykiejew
0 5 10 15 20 25
r
0
2
4
6
8
10
g A
r-
A
r(r
)
µ
Ar
= -9.86
µ
Ar
= -9.60
µ
Ar
= -9.00
µ
Ar
= -8.60
Figure 15. The argon-argon radial distribution functions for the system with µ∗
Xe
= −18.6, recorded at
T = 57 K and different values of µ∗
Xe
(shown in the figure). The vertical dotted lines mark the locations of
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 commen-
surate (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.
0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0
x
Xe
IC
Ar
+C
C
C
C+IC
Xe
T = 45 K
T = 11 K
IC
Ar
IC
Ar
+C
C C+IC
Xe
IC
Xe
IC
Ar C C+IC
Xe
IC
Xe
IC
Xe
IC
Xe
C+IC
Xe
MC simulation
exp.
MC simulation
exp.
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].
23601-14
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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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 вла-
стивостi 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 ксено-
ну 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
http://dx.doi.org/10.1103/PhysRevB.40.4797
http://dx.doi.org/10.1016/0167-5729(81)90004-2
http://dx.doi.org/10.1080/00268978600100591
http://dx.doi.org/10.1103/PhysRevLett.50.978
http://dx.doi.org/10.1103/PhysRevB.28.7338
http://dx.doi.org/10.1051/jphyslet:019790040010900
http://dx.doi.org/10.1103/PhysRevB.44.8962
Introduction
The model and Monte Carlo methods
The results for pure Ar and Xe films
The results for mixed films
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