Towards frustration of freezing transition in a binary hard-disk mixture
The freezing mechanism, recently suggested for a monodisperse hard-disk fluid [Huerta et al, Phys. Rev. E, 2006, 74, 061106] is extended here to an equimolar binary hard-disk mixtures. We are showing that for diameter ratios, smaller than 1.15 the global orientational order parameter of the binary m...
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| Cite this: | Towards frustration of freezing transition in a binary hard-disk mixture / A. Huerta, V. Carrasco-Fadanelli, A. Trokhymchuk // Condensed Matter Physics. — 2012. — Т. 15, № 4. — С. 43604:1-9. — Бібліогр.: 13 назв. — англ. |
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Huerta, A. Carrasco-Fadanelli, V. Trokhymchuk, A. 2017-06-11T15:04:32Z 2017-06-11T15:04:32Z 2012 Towards frustration of freezing transition in a binary hard-disk mixture / A. Huerta, V. Carrasco-Fadanelli, A. Trokhymchuk // Condensed Matter Physics. — 2012. — Т. 15, № 4. — С. 43604:1-9. — Бібліогр.: 13 назв. — англ. PACS: 64.60.Fr, 68.35.Rh DOI:10.5488/CMP.15.43604 arXiv:1212.6359 https://nasplib.isofts.kiev.ua/handle/123456789/120309 The freezing mechanism, recently suggested for a monodisperse hard-disk fluid [Huerta et al, Phys. Rev. E, 2006, 74, 061106] is extended here to an equimolar binary hard-disk mixtures. We are showing that for diameter ratios, smaller than 1.15 the global orientational order parameter of the binary mixture behaves like in the case of a monodisperse fluid. Namely, by increasing the disk number density there is a tendency to form a crystalline-like phase. However, for diameter ratios larger than 1.15 the binary mixtures behave like a disordered fluid. We use some of the structural and thermodynamic properties to compare and discuss the behavior as a function of diameter ratio and packing fraction. Механiзм замерзання, який був запропонований недавно для однокомпонентного флюїду твердих дискiв [Huerta et al, Phys. Rev. E, 74, 2006, 061106], узагальнено на випадок еквiмолярної бiнарної сумiшi твердих дискiв. Ми показуємо, що у випадку, коли вiдношення дiаметрiв є меншим за 1.15, то поведiнка параметра глобального орiєнтацiйного порядку бiнарної сумiшi є подiбною до випадку однокомпонентного флюїду. А саме, при збiльшеннi густини дискiв спостерiгається тенденцiя до утворення кристалоподiбної фази. Однак, для вiдношень дiаметрiв бiльших нiж 1.15 вiдмiчається змiна цiєї тенденцiї на поведiнку, характерну для невпорядкованого флюїду. Ми використовуємо окремi структурнi та термодинамiчнi властивостi для порiвняння та обговорення їх поведiнки як функцiї вiдношення дiаметрiв та параметра упаковки. This work is supported by the CONACYT of Mexico under the project 152431 and through the Red Temática de la Materia Condensada Blanda. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Towards frustration of freezing transition in a binary hard-disk mixture Про зникнення фазового переходу замерзання у бiнарнiй сумiшi твердих дискiв Article published earlier |
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Towards frustration of freezing transition in a binary hard-disk mixture |
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Towards frustration of freezing transition in a binary hard-disk mixture Huerta, A. Carrasco-Fadanelli, V. Trokhymchuk, A. |
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Towards frustration of freezing transition in a binary hard-disk mixture |
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Towards frustration of freezing transition in a binary hard-disk mixture |
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Towards frustration of freezing transition in a binary hard-disk mixture |
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Towards frustration of freezing transition in a binary hard-disk mixture |
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towards frustration of freezing transition in a binary hard-disk mixture |
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Huerta, A. Carrasco-Fadanelli, V. Trokhymchuk, A. |
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Huerta, A. Carrasco-Fadanelli, V. Trokhymchuk, A. |
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Інститут фізики конденсованих систем НАН України |
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Про зникнення фазового переходу замерзання у бiнарнiй сумiшi твердих дискiв |
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The freezing mechanism, recently suggested for a monodisperse hard-disk fluid [Huerta et al, Phys. Rev. E, 2006, 74, 061106] is extended here to an equimolar binary hard-disk mixtures. We are showing that for diameter ratios, smaller than 1.15 the global orientational order parameter of the binary mixture behaves like in the case of a monodisperse fluid. Namely, by increasing the disk number density there is a tendency to form a crystalline-like phase. However, for diameter ratios larger than 1.15 the binary mixtures behave like a disordered fluid. We use some of the structural and thermodynamic properties to compare and discuss the behavior as a function of diameter ratio and packing fraction.
Механiзм замерзання, який був запропонований недавно для однокомпонентного флюїду твердих дискiв [Huerta et al, Phys. Rev. E, 74, 2006, 061106], узагальнено на випадок еквiмолярної бiнарної сумiшi твердих дискiв. Ми показуємо, що у випадку, коли вiдношення дiаметрiв є меншим за 1.15, то поведiнка параметра глобального орiєнтацiйного порядку бiнарної сумiшi є подiбною до випадку однокомпонентного флюїду. А саме, при збiльшеннi густини дискiв спостерiгається тенденцiя до утворення кристалоподiбної фази. Однак, для вiдношень дiаметрiв бiльших нiж 1.15 вiдмiчається змiна цiєї тенденцiї на поведiнку, характерну для невпорядкованого флюїду. Ми використовуємо окремi структурнi та термодинамiчнi властивостi для порiвняння та обговорення їх поведiнки як функцiї вiдношення дiаметрiв та параметра упаковки.
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https://nasplib.isofts.kiev.ua/handle/123456789/120309 |
| citation_txt |
Towards frustration of freezing transition in a binary hard-disk mixture / A. Huerta, V. Carrasco-Fadanelli, A. Trokhymchuk // Condensed Matter Physics. — 2012. — Т. 15, № 4. — С. 43604:1-9. — Бібліогр.: 13 назв. — англ. |
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| fulltext |
Condensed Matter Physics, 2012, Vol. 15, No 4, 43604: 1–9
DOI: 10.5488/CMP.15.43604
http://www.icmp.lviv.ua/journal
Towards frustration of freezing transition in a binary
hard-disk mixture
A. Huerta1, V. Carrasco-Fadanelli1, A. Trokhymchuk2
1 Facultad de Física e Inteligencia Artificial, Depto. de Física, Universidad Veracruzana,
Circuito Gonzálo Aguirre Beltrán s/n Zona Universitaria Xalapa, Ver. C.P. 91000, México.
2 Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine,
Department of the Theory of Solutions, 1 Svientsitskii St., 79011 Lviv, Ukraine
Received September 27, 2012
The freezing mechanism, recently suggested for a monodisperse hard-disk fluid [Huerta et al, Phys. Rev. E, 2006,
74, 061106] is extended here to an equimolar binary hard-disk mixtures. We are showing that for diameter
ratios, smaller than 1.15 the global orientational order parameter of the binary mixture behaves like in the case
of a monodisperse fluid. Namely, by increasing the disk number density there is a tendency to form a crystalline-
like phase. However, for diameter ratios larger than 1.15 the binary mixtures behave like a disordered fluid. We
use some of the structural and thermodynamic properties to compare and discuss the behavior as a function
of diameter ratio and packing fraction.
Key words: hard-disk fluid, freezing transitions, binary equimolar hard-disk mixture
PACS: 64.60.Fr, 68.35.Rh
1. Introduction
One-component or monodisperse system of hard spheres is the simplest and the most popular model
system in condensed matter physics. The only feature that distinguishes a hard-sphere model from the
ideal system is a non-zero hard-core diameter. Because of this feature the hard-sphere model already
exhibits a set of fundamental properties observed for a real condensed matter such as liquid-like short-
range ordering and liquid-to-solid transition. A number of papers and textbooks can be found in the
literature that are devoted to the discussion of the hard-sphere model and its properties [1, 2].
Amuch simpler andmore transparent system seems to be a two-dimensional counterpart of the hard-
sphere model, i.e., just one layer of hard spheres or, which in this case is the same, an assembly of hard
disks spread out in the plane. In contrast to the ideal system and similarly to the hard-sphere model,
by increasing the number of disks, this system undergoes transformation from a disordered fluid into
a perfect two-dimensional crystal. This transformation is accompanied by a freezing transition which
was first evidenced by Alder and Wainwright [3] fifty years ago. By means of the molecular dynamics
simulations Alder and Wainwright have shown that a system of hard disks has a density region where
its pressure isotherm exhibits a van der Waals-like loop. Since then, a number of papers have been pub-
lished [4–6] discussing various aspects of this phenomenon, such as the origin of the freezing transition,
the mechanisms of freezing, the criteria necessary for the freezing transition to occur, etc. To date, there
is a consensus that the freezing transition in a hard-disk system is driven by entropy, and precise locations
of the freezing and melting densities are detected [4–6].
Besides the pressure calculations that were originally employed by Alder andWainwright [3], another
common way to illustrate the transition from a disordered fluid to an ordered solid in two-dimensional
monodisperse system is the density evolution of the global-bond orientational order parameter [4, 7].
The global orientational order parameter quantifies the degree of deviation of the nearest neighbors
arrangement of each particle in the system from the perfect hexagonal arrangement. The highest value
© A. Huerta, V. Carrasco-Fadanelli, A. Trokhymchuk, 2012 43604-1
http://dx.doi.org/10.5488/CMP.15.43604
http://www.icmp.lviv.ua/journal
A. Huerta, V. Carrasco-Fadanelli, A. Trokhymchuk
1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8
0
2
4
6
C
B
∆
> 2σ
< 2σ
D
A
> 2σ< 2σ
< 2σ
η = 0.686
η = 0.723
ra
di
al
d
is
tr
ib
ut
io
n
fu
nc
tio
n
centre-to-centre distance, r/σ
(a) (b)
Figure 1. Monodisperse disk system. Part a) Global bond-orientational order parameter ψ6 as a function
of packing fraction η. Part b) Disk-disk radial distribution function g (r ) at two packing fractions, namely,
η = 0.686 that is just before freezing transition and η = 0.723 which is slightly higher than the melting
density in this system [6]. The inset shows a schematic of the central disk (the hollow disk A at the center)
and its first (the disks filled with black color and hollow disks with dash line) neighbors and the second
(the disks filled with gray color and hollow disks with dashed line) neighbors, or the first and the second
coordination shells, respectively.
of this parameter equals unity and corresponds to a perfect hexagonal crystal. The case of amonodisperse
hard-disk system is illustrated in figure 1 (a). The formation of local quasiregular hexagonal orderingwith
increasing density in a hard disk system [see the inset in figure 1 (b)] implies that the average center-to-
center distance between any disk chosen as central and its closer second coordination shell neighbors
[the open, solid line disks in the inset in figure 1 (a)] becomes shorter than two disk diameters. This fact is
consistent with the shape of the disk-disk radial distribution function, reflecting itself in the appearance
of the shoulder on its second maximum.
On the other hand, within the hexagonal arrangement, each central disk and its closer second coordi-
nation shell neighbors simultaneously serve as the alternating nearest neighbors of the common neigh-
boring disk. When the centre-to-centre distance between alternating nearest neighbors becomes shorter
than two disk diameters, this common neighboring disk becomes caged.
Based on this observation, we have recently reported a simple mechanism for the freezing of hard
disks [6]. This mechanism considers that by taking only alternating nearest neighbors into account it is
sufficient to describe the fluid and solid phases as well as the transition between them. The fluid becomes
unstable when the average centre-to-centre distance between alternating nearest neighbors becomes
shorter than two disk diameters and the resulting gap between them is shorter than hard-core diameter
and does not allow for the central disk to wander. Such a caging concept allows for both the quantitative
and qualitative description of the thermodynamics of freezing transition in monodisperse hard-disk fluid
and has been already utilized to discuss percolation [8].
We also note, that the appearance of a shoulder on the second peak of the radial distribution function
has been already suggested as a structural precursor to the freezing transition in the hard disk system [5]
and is considered among others as empirical criteria for identification of the freezing transition not only
in the hard-disk system but more generally in the two-dimensional condensed matter [9].
Such a detailed understanding of the freezing in a monodisperse hard-disk system implies that now
one can try to move further and see how the freezing scenario that occurs here will be extended and/or
modified going to more complex two-dimensional systems. There are different possibilities to complicate
the hard-disk model, e.g., by modifying the disk-disk interaction. In this paper we will consider the sim-
plest non-trivial complication of the monodisperse hard-disk system by introducing the difference in the
43604-2
Towards frustration of freezing transition in a binary hard-disk mixture
disk diameters. Our goal is to investigate how the ordering behavior in a hard-disk system will change
the transition from a monodisperse hard-disk system to an equimolar binary mixture of hard disks of
different diameters.
For a binary hard-disk mixture, it is already established that a progressive increase of the hard-core
diameter for one half of the disks (equimolar mixture) does substantially change the properties of this
system, especially, in the vicinity of the density region that corresponds to the freezing/melting transitions
in a monodisperse system. In particular, following the results due to Speedy [10], the equimolar binary
hard-disk mixtures exhibit a freezing transition and proceed to a mixed crystal only for disk diameter
ratios that do not exceed 1.2. When the disk diameter ratio is exactly 1.2, the transition region shrinks,
tending to disappear, while equilibrium freezing point was not precisely located in that study. In the same
paper [10], it is also shown that starting from the disk diameter ratio 1.3, the equimolar binary mixtures
pass into an amorphous state and it was suggested to consider them to be good glass-formers.
In what follows we will probe the behavior of the equimolar binary hard-disk mixture by means
of the global orientational order parameter and the disk-disk radial distribution functions. In the next
section 2, we describe the model and outline the details of the computer simulation studies performed by
us. The results and their discussion are presented in section 3, while section 4 presents the summary and
conclusions.
2. Model definition and simulation details
We consider a two-component (A and B) system of disks, at equimolar conditions, i.e., ρA = ρB = ρ/2,
where ρ is the total number density of disks. The disks that belong to components A and B are of different
diameters, namely, σA =σ+δ and σB =σ−δ. The disk-disk pair interactionsUi j (r ), where i and j stand
for the species A and B , are given by
UA A(r ) =
{
∞ if r < σ+δ ,
0 if r > σ+δ ,
(2.1)
UBB (r )=
{
∞ if r < σ−δ ,
0 if r > σ−δ ,
(2.2)
and
UAB (r )=UB A(r )=
{
∞ if r < σ ,
0 if r > σ ,
(2.3)
where r is the center-to-center separation. Useful quantities to characterise the system will be the total
packing fraction given by η = (π/4)ρ(σ2
+δ2) and the disk diameter ratio R = (σ+δ)/(σ−δ). In what
follows we will vary the parameter δ from δ = 0, which corresponds to a monodisperse system, up to
δ= 0.166 which corresponds to a binary disk mixture having size ratio R = 1.4. The parameter σ will be
used as a unit length throughout the paper.
Under certain pressure conditions the system of hard disks exhibits a fluid-to-solid transition. To
simulate such a system, a number N of hard disks were placed into a squared area of the size A = L ×L
and periodic boundary conditions in both directions were applied. A standard Monte Carlo simulation
technique using Metropolis algorithm was used to obtain ensemble averages of the dense equilibrium
hard disk system.
Two types of computer simulations have been carried out. To evaluate the structural properties we
have used an NVT ensemble with 400 particles. Each equilibration run was relaxed for at least 107 iter-
ations per particle. For productive runs, we have averaged over at least 20000 different configurations,
each being relaxed by 1000 iterations per particle. The acceptance ratio has been fixed between 20% and
30% and was controlled by choosing the maximum displacement of at least 13%.
To calculate the thermodynamical properties we have employed a NPT ensemble which consists in
the translation and change of the volume (area). The Monte Carlo simulations in the NPT ensemble were
aimed at localizing the fluid-solid transition region. To obtain an ensemble averaging of each state, we
have to proceed in the following way. Starting with fluid densities at equilibrium, obtained from the NVT
ensemble, we fix a fluid pressure attempting to translate each particle for at least 1000, followed by an
43604-3
A. Huerta, V. Carrasco-Fadanelli, A. Trokhymchuk
attempt to change the simulation box size. Translation and box size acceptance ratio were controlled
choosing the maximum displacements of particles and box size, for at least 13% and 30%, respectively.
Averages were stored for at least 30000 iterations. We have also verified that the attempts to interchange
the particle identity do not change the results at the higher packing fractions studied.
3. Results and discussion
Weexplore the effect of the disk diameter ratio on the hexagonal ordering phenomenon in the equimo-
lar binary hard-disk mixture by performing Monte Carlo simulation studies of a set of six values of the
size ratio parameter δ = 0.02, 0.05, 0.07, 0.08, 0.09 and 0.166 that correspond to the diameter ratios
R = 1.04, 1.11, 1.15, 1.17, 1.2 and 1.4, respectively. We restricted the present study to the maximal size
ratio R = 1.4, since it was already suggested by Speedy [10] that such a binary hard-disk mixture can be
considered as a glassforming fluid. While analyzing the obtained results we found that the equimolar
binary hard-disk mixtures having size ratios that do not exceed R = 1.15, behave qualitatively similarly
to a monodisperse hard-disk system. Thus, in what follows we will discard from the discussion the results
for the diameter size ratios R = 1.04 by presenting only the results for the case R = 1.11 as being typical
of this class of the equimolar binary hard-disk mixtures. Similarly, we will use mainly the case of R = 1.2
to discus the mixtures having diameter size ratios that exceed the value R = 1.15.
3.1. Equation of state
To present our results, we start with the Monte Carlo simulation data obtained for the equation of
state of the hard-disk systems under consideration. One of the purposes of the pressure calculations was
to make a reference to the work by Speedy [10] who calculated the equation of state for hard-disk mix-
tures by means of molecular dynamics simulations. It was suggested [10] that a freezing transition, ob-
served for a monodisperse hard-disk system, can be frustrated in a binary hard-disk mixture by increas-
ing the ratio of disk diameters. The highest disk size asymmetry that still allows for a freezing transition
5
10
15
20
25
30
0.6 0.65 0.7 0.75 0.8
βp
η
Figure 2. Equation of state of the equimolar binary hard-disk mixtures with diameter size ratios R = 1.4,
1.2, 1.15, 1.11 (from the top to the bottom) in comparison against a monodisperse, R = 1, hard-disk system
(the dotted line from the bottom). The two thin vertical lines, from left to right,mark the packing fractions,
ηf = 0.69 and ηm = 0.723, that correspond to the freezing and melting points, respectively, in the case of
a monodisperse hard-disk system; the density region between these two lines is the fluid-solid transition
region and the results shown by symbols are taken from the extensive computer simulation study due to
Mak [11]. For all other systems the fluid-solid transition regions correspond to the empty space between
fluid and solid pressure curves. In the case of the mixture with diameter size ratio R = 1.4 transition
region was not detected.
43604-4
Towards frustration of freezing transition in a binary hard-disk mixture
Table 1. Pressure and packing fraction at the freezing and melting points of the equimolar binary hard-
disk mixtures with different ratios of diameters. The data for a monodisperse disk system (R = 1) are
taken from [6] and are given for comparison.
R =σA/σB 1 1.04 1.11 1.15 1.17 1.2
Freezing
P/kT 9.0 9.0 10.0 12.0 13.0 14.0
η 0.692 0.692 0.707 0.727 0.735 0.740
Melting
P/kT 10.0 10.0 11.0 13.0 14.0 15.0
η 0.724 0.724 0.730 0.742 0.747 0.752
in a binary hard-disk mixture to occur was found to be 1.2:1, while for the size ratio 1.4:1 the glassy state
can be formed by detouring the freezing transition.
Figure 2 shows the pressure versus density data obtained for four equimolar binary hard-disk mix-
tures having diameter ratios R = 1.11, 1.15, 1.2 and 1.4. For comparison and for better understanding,
the corresponding results for the case of a monodisperse hard-disk system, R = 1, taken from our earlier
study [6], are shown in figure 2 as well. The two thin vertical lines in this figure, from left to right, mark
the packing fractions, ηf = 0.69 and ηm = 0.723, that correspond to the freezing and melting points, re-
spectively, in the case of a monodisperse hard-disk system. The density region between these two lines
refers to the fluid-solid transition region for hard disks of the same diameters. From the results presented
in figure 2 it follows that the disk diameter asymmetry does not affect the pressure for the densities that
precede this transition region. Namely, at η < ηf , all hard-disk systems, considered in the present study,
exhibit nearly the same pressure profile, showing smooth increment with an increase of density. It is only
at densities η> ηf that notable differences in the pressure caused by disk size asymmetry are observed.
To locate the possible freezing and melting points and to determine the transition region in the case
of a hard-disk mixture, we carried out a series of constant pressure (NPT) Monte Carlo simulation runs
for each of the binary systems under study. By smoothly incrementing a fixed pressure value, we were
trying to find the pressure P at which the disk density, that was also changing smoothly, makes a jump to
a notably higher value. These values of the pressure and density might be referred to as the upper bounds
for the pressure and density at melting point, Pm and ηm, respectively. The pressure P that preceded such
a density jump, and corresponding to this pressure density, might be referred to as the bottom bounds
for the pressure and density at freezing point, Pf and ηf , respectively. By proceeding in this way, we
were able to locate the transition regions in all binary hard-disk mixtures considered, except the one
that corresponds to the disk diameter ratio R = 1.4. In figure 2, these transition regions can be seen as
an empty space between the corresponding fluid pressure and solid pressure curves. A complete set of
data, that include freezing and melting densities and pressures for all systems considered in this study,
are collected in table 1.
From the results presented in table 1 one can see that both the freezing and themelting point densities
increase with an increase of a disk size asymmetry. Consequently, the transition regions (see figure 2 for
details) are also shifted to higher densities, with clear tendency to shrink and, finally, to disappear in
the case of the highest disk size asymmetry considered in the present study, i.e., at R = 1.4. We note that
pressure calculations for the same hard-disk systems were also performed from molecular dynamics
simulations. This has been done by taking the final disk configurations generated during the NPT Monte
Carlo runs. The results obtained from both techniques are identical within the margin of error.
3.2. Orientational ordering
Following the case of amonodisperse disk system shown in figure 1, the freezing transition in a binary
hard-disk mixture can be probed by mapping the density dependence of the orientational ordering of the
disks. A commonly used quantitative measure of the orientational order in two-dimensional systems is
the global bond-orientational order parameter [4, 7]. This parameter was evaluated during the Monte
43604-5
A. Huerta, V. Carrasco-Fadanelli, A. Trokhymchuk
Carlo simulation runs using the following definition,
ψ6 =
∣
∣
∣
∣
∣
1
Nnn
∑
j
∑
k
e6iθ j k
∣
∣
∣
∣
∣
, (3.1)
where j runs over all disks in the system, k runs over all “geometric” nearest neighbours (nn) of disk
j , each obtained through the Voronoi analysis, and Nnn is the total number of such nearest neighbours
in the system. The angle θ j k is defined between some fixed reference axis in the system and the vectors
(“bonds”) connecting the nearest neighbours j and k.
0
0.2
0.4
0.6
0.8
1
0.5 0.6 0.7 0.8
ψ
6
η
0
0.5
1
1 1.1 1.2 1.3 1.4
R
Figure 3. Global bond-orientational order parameter ψ6 of equimolar binary hard-disk mixtures having
diameter size ratios R = 1.4, 1.2, 1.15, 1.11 (from the bottom to the top) in comparison with a monodis-
perse, R = 1, hard-disk system (the dotted line from the top). The two thin vertical lines mark the same as
in figure 2, while open circles mark the magnitude ofψ6 at freezing packing fraction for a corresponding
system. The inset shows ψ6 as a function of the disk diameter ratio R for the selected packing fractions
from the bottom to the top η= 0.62, 0.69, 0.70, 0.723, 0.74 and 0.80.
In figure 3 we show the density dependence of the global bond-orientational order parameter ψ6
for binary hard-disk mixtures having diameter size ratios R = 1.11, 1.15, 1.2 and 1.4. The correspond-
ing results for a monodisperse hard-disk systems, R = 1, are presented in figure 3 as well. As expected,
at low densities and independently of the disk diameter ratio, all hard-disk systems behave like a two-
dimensional isotropic fluid. As the density of disks increases, both the monodisperse hard-disk system
and the binary hard-disk mixture tend to form a hexagonal ordering. However, in the case of a binary
mixture having the disk diameter ratio R = 1.4, this tendency is notably weaker, including the highest
density, η= 0.8, that was probed in this study.
Overall, by comparing the results for ψ6 in the case of binary hard-disk mixtures at the same packing
fraction with the results for the monodisperse disk system we can conclude that an increase of the disk
diameter asymmetry promotes a decrease of the hexagonal ordering in the system. However, the rate of
this decrease depends on the disk diameter ratio. At the same time, determining the magnitude of the
hexagonal ordering in hard-disk systems at the freezing point (open circles in figure 3) one can see that
ψ6 initially increases with an increase of the disk size asymmetry. However, for the diameter size ratios
larger than R = 1.15, the hexagonal ordering at the freezing point starts to decline.
In the inset of figure 3, the global orientational order parameter is plotted against the disk diameter
ratio R at several fixed packing fractions. One can see that at small disk diameter asymmetry, the rate of a
decline of the hexagonal ordering is very slow, but the values ofψ6 significantly drop as the disk diameter
ratio R increases. We notice that similarly to the results for the pressure infigure 2, the crossover between
two different behaviors of the global bond-orientational order parameter ψ6 takes place around the disk
diameter ratio R = 1.15.
43604-6
Towards frustration of freezing transition in a binary hard-disk mixture
R=1.11 R=1.2 R=1.4
0
2
4
6
8
1 2 3 4 5
g A
A
(r
)
r/σ
0
1
2
1.5 2 2.5
0
2
4
6
8
1 2 3 4 5
g A
A
(r
)
r/σ
0
1
2
1.5 2 2.5
0
2
4
6
8
1 2 3 4 5
g A
A
(r
)
r/σ
0
1
2
1.5 2 2.5
0
2
4
6
8
1 2 3 4 5
g A
B
(r
)
r/σ
0
1
2
1.5 2 2.5
0
2
4
6
8
1 2 3 4 5
g A
B
(r
)
r/σ
0
1
2
1.5 2 2.5
0
2
4
6
8
1 2 3 4 5
g A
B
(r
)
r/σ
0
1
2
1.5 2 2.5
0
2
4
6
8
1 2 3 4 5
g B
B
(r
)
r/σ
0
1
2
1.5 2 2.5
0
2
4
6
8
1 2 3 4 5
g B
B
(r
)
r/σ
0
1
2
1.5 2 2.5
0
2
4
6
8
1 2 3 4 5
g B
B
(r
)
r/σ
0
1
2
1.5 2 2.5
Figure 4. Radial distribution functions gi j (r ) of equimolar binary hard-disk mixtures having disk diame-
ter ratios R = 1.11, 1.2 and 1.4. The dotted lines correspond to the radial distribution function g (r ) of the
monodisperse hard-disk system at η= 0.692 that is the freezing point packing fraction of a monodisperse
disk system, while thin solid lines correspond to the packing fractions at the freezing point of particular
binary hard-disk mixture, namely: η = 0.702 in the case of R = 1.11, η = 0.740 in the case of R = 1.2; in
the case of R = 1.4, there is no freezing point and we show the results for η= 0.740. All the insets show in
detail the region of the second peak of the corresponding radial distribution function.
3.3. Radial distribution functions
The fact that pressure isotherms for hard-disk mixtures in figure 2 is quite similar to that of the
monodisperse disk system at low packing fractions, but drastically changes when the disk packing frac-
tion η exceeds the value ηf = 0.692 at freezing point of a monodisperse disk system, may indicate that
the freezing mechanism that was recently suggested [6] for a monodisperse hard-disk system undergoes
important changes caused by the disk size asymmetry. The principal ingredient of the mechanism of a
monodisperse hard disk freezing is the caging phenomenon that occurs in the system of hard disks of
equal diameters as a result of the entropy driven hexagonal ordering. As it was already mentioned in
the Introduction, this mechanism is quite understandable and can be explained by means of the radial
distribution function of a monodisperse hard-disk system in the way shown in figure 1 (b).
To shed some light on the structural rearrangements that take place in binary hard-disk mixtures, in
figure 4 we present a set of radial distribution functions, g A A (r ), gBB (r ), and g AB (r ) for three equimolar
binary hard-disk mixtures characterized by the disk diameter ratios, R = 1.11, 1.2 and 1.4. The first one,
R = 1.11, is a representative of the type of binary hard-disk mixtures for which the freezing and post-
freezing features are similar to those of a monodisperse hard-disk system. On the other hand, the second
hard-disk systemwith R = 1.2 represents the type of binary hard-diskmixtures that still exhibit a freezing
transition, but their properties seem to be different from those of monodisperse hard-disk systems [12].
Finally, the third disk diameter ratio, R = 1.4, corresponds to the binary hard-disk mixture that does not
43604-7
A. Huerta, V. Carrasco-Fadanelli, A. Trokhymchuk
experience a freezing transition, representing a glass-forming system [10].
The radial distribution functions shown in figure 4 are calculated at the packing fraction values that
correspond to the freezing point packing fraction for each hard-disk mixture listed in table 1. Namely,
η= 0.707 in the case of hard-disk mixture having the disk diameter ratio R = 1.11, and η= 0.74 in the case
of R = 1.2. Since there is no freezing transition in the case of a hard-diskmixture having the disk diameter
ratio R = 1.4, the radial distribution functions for this system are obtained at a packing fraction η= 0.74.
To make a reference to the case of all disks being of the same diameter, the radial distribution function
g AB (r ) for each binary hard-disk system is explicitly compared with the radial distribution function g (r )
of the monodisperse hard-disk system at the corresponding freezing point packing fraction, ηf = 0.692,
for a monodisperse system.
First we note that the disk size asymmetry being introduced results in an increase of the values of
the radial distribution functions gi j (r ) at contact distances, r =σi j . However, since the so-called contact
value uniquely determines the pressure P in both themonodisperse and in the binary hard-disk systems,
from the results presented in figure 4 it follows that binary hard-disk mixtures require higher pressures
for the freezing transition to occur. This is consistent with the results already discussed for the equation
of state in figure 2.
More insight into the local structural rearrangement that presumably takes place in a binary hard-
disk mixture in the density region close to the freezing transition can be achieved by analyzing the shape
of the second maximum of the radial distribution functions. It is quite evident that radial distribution
functions gij(r ) for each of three hard-disk mixtures show a quite different and specific behavior, which
is also different from the radial distribution functions g (r ) of the monodisperse hard-disk system. Nev-
ertheless, we noticed that for a binary mixture having the disk diameter ratio R = 1.11 (the first column
in figure 4), all three radial distribution functions , g A A(r ), gBB (r ) and g AB (r ), exhibit nearly the same
shape of the second maximum. Moreover, this shape obeys the shoulder, i.e., the feature that is discussed
in figure 1 (b) and is responsible for the occurrence of the freezing transition in amonodisperse hard-disk
system [6]. However, speaking more generally we note that this suggestion by Truskett et al [5] towards
the shoulder in the second maximum of the radial distribution function as the structural precursor of
the freezing transition does not seem to be general. In particular, it could not be extended to the binary
hard-disk mixtures with the disk diameter ratios R > 1.2, where both the shoulder and splitting of the
second maximum are extremely pronounced but freezing transition does not occur.
4. Conclusions
A binary equimolar mixture of hard disks having diameter ratios R = 1.04, 1.11, 1.15, 1.17, 1.2 and
1.4 have been considered by means of computer simulations. The main purpose of this study was to
probe these binary mixtures on the freezing transition which normally occurs in a system of hard disks
with the same diameters. By calculating the equation of state we found that the size asymmetry being
introduced alters the pressure in the systems. We showed that a freezing transition shifts to higher disk
packing fractions as the diameter asymmetry increases while the width of transition region shrinks. The
largest diameter ratio is R = 1.2 when the freezing transition is still localized in hard-diskmixtures, while
the binary hard-disk mixture having diameter ratio R = 1.4 does not exhibit the freezing behavior. We
consider that the change of the behavior towards the frustration of hexagonal order is caused by the
change of the maximum number of neighbors that each particle can have. This is similar to the crossover
observed earlier by introducing the short-ranged square-well attraction [13] for the disk-disk interaction.
Acknowledgement
This work is supported by the CONACYT of Mexico under the project 152431 and through the Red
Temática de la Materia Condensada Blanda.
43604-8
Towards frustration of freezing transition in a binary hard-disk mixture
References
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2. Hansen J.-P., MacDonalds I.R., Theory of Simple Liquids, 3rd ed., Academic Press, 2006.
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Про зникнення фазового переходу замерзання у бiнарнiй
сумiшi твердих дискiв
А. Уерта1, В. Караско-Фаданеї1, А. Трохимчук2
1 Унiверситет Веракрузана, Факультет фiзики та iнженерiї, Кафедра фiзики,
Халапа, Веракруз, СР 91000, Мексика
2 Iнститут фiзики конденсованих систем НАН України, Вiддiл теорiї розчинiв,
вул. Свєнцiцького, 1, 79011 Львiв, Україна
Механiзм замерзання, який був запропонований недавно для однокомпонентного флюїду твердих дискiв
[Huerta et al, Phys. Rev. E, 74, 2006, 061106], узагальнено на випадок еквiмолярної бiнарної сумiшi твердих
дискiв. Ми показуємо, що у випадку, коли вiдношення дiаметрiв є меншим за 1.15, то поведiнка пара-
метра глобального орiєнтацiйного порядку бiнарної сумiшi є подiбною до випадку однокомпонентного
флюїду. А саме, при збiльшеннi густини дискiв спостерiгається тенденцiя до утворення кристалоподiбної
фази. Однак, для вiдношень дiаметрiв бiльших нiж 1.15 вiдмiчається змiна цiєї тенденцiї на поведiнку,
характерну для невпорядкованого флюїду. Ми використовуємо окремi структурнi та термодинамiчнi вла-
стивостi для порiвняння та обговорення їх поведiнки як функцiї вiдношення дiаметрiв та параметра упа-
ковки.
Ключовi слова: флюїд твердих дискiв, переходи замерзання, еквiмолярна бiнарна сумiш твердих дискiв
43604-9
http://dx.doi.org/10.1103/RevModPhys.48.587
http://dx.doi.org/10.1103/PhysRev.127.359
http://dx.doi.org/10.1103/PhysRevB.51.14636
http://dx.doi.org/10.1103/PhysRevE.58.3083
http://dx.doi.org/10.1103/PhysRevE.74.061106
http://dx.doi.org/10.1063/1.1289238
http://dx.doi.org/10.1063/1.3545967
http://dx.doi.org/10.1063/1.3372618
http://dx.doi.org/10.1063/1.478337
http://dx.doi.org/10.1103/PhysRevE.73.065104
http://dx.doi.org/10.1063/1.1632893
Introduction
Model definition and simulation details
Results and discussion
Equation of state
Orientational ordering
Radial distribution functions
Conclusions
|