Frustration of freezing in a two dimensional hard-core fluid due to particle shape anisotropy
The freezing mechanism suggested for a fluid composed of hard disks [Huerta et al., Phys. Rev. E, 2006, 74, 061106] is used here to probe the fluid-to-solid transition in a hard-dumbbell fluid composed of overlapping hard disks with a variable length between disk centers. Analyzing the trends in the...
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| Опубліковано в: : | Condensed Matter Physics |
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| Дата: | 2016 |
| Автори: | , , , |
| Формат: | Стаття |
| Мова: | Англійська |
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Інститут фізики конденсованих систем НАН України
2016
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| Цитувати: | Frustration of freezing in a two dimensional hard-core fluid due to particle shape anisotropy / A. Huerta, D. Tejeda, D. Henderson, A. Trokhymchuk // Condensed Matter Physics. — 2016. — Т. 19, № 2. — С. 23605: 1–9 . — Бібліогр.: 19 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859640723142344704 |
|---|---|
| author | Huerta, A. Tejeda, D. Henderson, D. Trokhymchuk, A. |
| author_facet | Huerta, A. Tejeda, D. Henderson, D. Trokhymchuk, A. |
| citation_txt | Frustration of freezing in a two dimensional hard-core fluid due to particle shape anisotropy / A. Huerta, D. Tejeda, D. Henderson, A. Trokhymchuk // Condensed Matter Physics. — 2016. — Т. 19, № 2. — С. 23605: 1–9 . — Бібліогр.: 19 назв. — англ. |
| collection | DSpace DC |
| container_title | Condensed Matter Physics |
| description | The freezing mechanism suggested for a fluid composed of hard disks [Huerta et al., Phys. Rev. E, 2006, 74, 061106] is used here to probe the fluid-to-solid transition in a hard-dumbbell fluid composed of overlapping hard disks with a variable length between disk centers. Analyzing the trends in the shape of second maximum of the radial distribution function of the planar hard-dumbbell fluid it has been found that the type of transition could be sensitive to the length of hard-dumbbell molecules. From the NpT Monte Carlo simulations data we show that if a hard-dumbbell length does not exceed 15% the fluid-to-solid transition scenario follows the case of a hard-disk fluid, i.e., the isotropic hard-dumbbell fluid experiences freezing. However, for a hard-dumbbell length larger than 15% fluid-to-solid transition may change to continuous transition, i.e., such an isotropic hard-dumbbell fluid will avoid freezing.
Механiзм замерзання, який був запропонований для плину жорстких дискiв [Huerta A. et al., Phys. Rev. E,
2006, 74, 061106], використовується тут для дослiдження фазового переходу плин-тверде тiло у системi,
що складається з жорстких гантелеподiбних молекул, утворених двома жорсткими дисками зi змiнною
вiдстанню мiж їх центрами. З аналiзу тенденцiй змiни форми другого максимуму радiальної функцiї розподiлу зроблено висновок, що тип фазового переходу плин-тверде тiло може бути чутливий до видовження гантелеподiбних молекул. На основi даних комп’ютерного експерименту Монте Карло при фiксованому тиску було знайдено, що коли видовження молекул не перевищує 15% дiаметра жорсткого диска,
то фазовий перехiд плин-тверде тiло вiдбувається за тим же сценарiєм що i у системi жорстких дискiв,
тобто плин гантелеподiбних молекул замерзає. У випадку, коли видовження молекул перевищує 15% дiаметра жорсткого диска, то є пiдстави стверджувати, що фазовий перехiд плин-тверде тiло проходить
неперервно, тобто плин гантелеподiбних молекул у цьому випадку не замерзає.
|
| first_indexed | 2025-12-07T13:21:36Z |
| format | Article |
| fulltext |
Condensed Matter Physics, 2016, Vol. 19, No 2, 23605: 1–9
DOI: 10.5488/CMP.19.23605
http://www.icmp.lviv.ua/journal
Frustration of freezing in a two dimensional
hard-core fluid due to particle shape anisotropy
∗
A. Huerta1, D. Tejeda1, D. Henderson2, A. Trokhymchuk3,4
1 Facultad de Física, Universidad Veracruzana, Circuito Gonzálo Aguirre Beltrán s/n Zona Universitaria,
Xalapa, Veracruz, C.P. 91000, México
2 Department of Chemistry and Biochemistry, Brigham Young University, Provo, UT 84602, USA
3 Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine,
1 Svientsitskii St., 79011 Lviv, Ukraine
4 Institute of Applied Mathematics and Fundamental Sciences, Lviv Polytechnic National University,
79013 Lviv, Ukraine
Received February 10, 2016, in final form February 25, 2016
The freezing mechanism suggested for a fluid composed of hard disks [Huerta et al., Phys. Rev. E, 2006, 74,
061106] is used here to probe the fluid-to-solid transition in a hard-dumbbell fluid composed of overlapping
hard disks with a variable length between disk centers. Analyzing the trends in the shape of second maximum
of the radial distribution function of the planar hard-dumbbell fluid it has been found that the type of transition
could be sensitive to the length of hard-dumbbell molecules. From the N pT Monte Carlo simulations data we
show that if a hard-dumbbell length does not exceed 15% of the disk diameter, the fluid-to-solid transition sce-
nario follows the case of a hard-disk fluid, i.e., the isotropic hard-dumbbell fluid experiences freezing. However,
for a hard-dumbbell length larger than 15% of disk diameter, there is evidence that fluid-to-solid transition may
change to continuous transition, i.e., such an isotropic hard-dumbbell fluid will avoid freezing.
Key words: hard disk fluid, hard-dumbbell fluid, radial distribution function, freezing transition
PACS: 64.60.Fr, 68.35.Rh
1. Introduction
Since Alder and Wainwright [1] in 1962 showed by molecular dynamics simulations that two-dimen-
sional (2D) fluid of circular hard-core species— hard disks— can freeze, a number of papers have been
published studying the properties and physics behind such a phenomenon. The hard-disk fluid is one of
the simplest interaction models in soft condensed matter science. This model is far from being trivial as
might be thought from a first look and should be treated in the same way as some other (classical) two-
dimensional models such as the Isingmodel, X Y model, and other lattice models. Species of the hard-disk
model (e.g., atoms, molecules or particles) do not experience any other interaction except being prohib-
ited from mutual overlap. This apparent simplicity as well as a desire to understand the mechanism that
enables the system with no thermal activation to become unstable, undergoing a freezing transition, are
the main driving forces of the interest to this model fluid. The most recent advances in understanding
the physics behind the freezing transition in a hard-disk fluid concern the large scale computer simula-
tion data reported by Zollweg and Chester [2], Mak [3] and Bernard and Krauth [4]. According to these
simulation studies, the hard-disk fluid becomes solid in two steps: (i) by means of the first-order type of
transition from an isotropic fluid phase to a hexatic phase and (ii) continuously from a hexatic phase to a
solid.
∗
The authors are pleased to dedicate this article to our good friend and collaborator, Anthony (Tony) Haymet on the occasion of
his 60th birthday. He made a number of valuable contributions to the physics and chemistry of the condensed state. He has also
held a number of important administrative positions. We wish him well.
© A. Huerta, D. Tejeda, D. Henderson, A. Trokhymchuk, 2016 23605-1
http://dx.doi.org/10.5488/CMP.19.23605
http://www.icmp.lviv.ua/journal
A. Huerta, D. Tejeda, D. Henderson, A. Trokhymchuk
The monolayers of colloidal particles that very often serve as a prototype of two-dimensional and
quasi-two-dimensional systems (e.g., see review article [5] and references therein) still in various ways
(the size polydispersity, shape anisotropy, extra interactions, etc) differ from the “ideal” hard-disk system.
Some of these features, namely, the extra out-of-core interaction between disks as well as the disk size
bidispersity, have been investigated [6, 7] and their impact on freezing transition has been revealed. In
particular, it has been shown [7] that in a binary equimolar mixture of hard disks, there is a limiting
large-to-small disk diameter ratio of around 1.2 when freezing transition is still localized, while the same
binary hard-disk mixture having the diameter ratio of around 1.4 does not exhibit the freezing behavior.
In the present study we aim to explore how the freezing transition in a hard-disk fluid might be
affected by the shape anisotropy imposed on hard-disk species. In the following section 2 we start with
the introduction of the shape anisotropy into a hard-disk model and the description of the necessary
information regarding the details of computer simulations. The full set of the results for the present
study and discussions are collected in section 3, while conclusions are presented in a final section 4.
2. Modelling and details of simulations
A simple way to introduce shape anisotropy to the planar hard-disk species is to consider the planar
hard dumbbell-like objects that consist of two fused hard disks of diameter σ and their centers at a dis-
tance dσ. Such an approach permits to have the hard-core species elongated in one direction with aspect
ratio (σ+dσ)/σ= 1+d , where d plays the role of the anisotropy parameter, assuming values 0 É d É 1.
When d = 0, the hard-disk fluid is recovered, while the largest anisotropy that can be reached for this
hard-core model, d = 1, corresponds to the case of tangent hard disks and will be referred to as a hard-
dimer fluid. Besides the anisotropy parameter, the system is characterized by packing fraction η defined
as η= ρAd, where ρ = N /A and A is the area of the system, Ad is the area of a hard-dumbbell molecule
that depends on the anisotropy parameter d , while N is the number of hard-dumbbell species.
Figure 1 (the left-hand-side frame) shows the hard-dumbbell modelling as well as the phase diagram
in anisotropy-density coordinates that was suggested by Wojciechowski [8] based on heuristic reasoning
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
Figure 1. (Color online) Left: Phase diagram of the hard-dumbbell system suggested byWojciechowski [8],
and schematic of a single hard-diskmolecule of diameterσ, and hard-dumbbell molecules that consists of
two fused hard disks of diameter σ and centers at a distance dσ. Right: Monte Carlo simulations data for
the radial distribution function of a hard-disk system at two packing fractions η: before freezing (dashed
line) and before melting (solid line). Inset: Sketch of the disk regular hexagonal configurations where
four disks filled with black and two empty disks drawn by the dashed line are the first coordination shell
neighbors of the central disk A, while those six filled with gray and those six empty disks drawn by the
solid line belong to the second shell neighbors.
23605-2
Frustration of freezing in a two dimensional hard-core fluid
using free-volume arguments. According to this diagram, any hard-dumbbell systemwill always undergo
a first-order transition from an isotropic fluid phase to a solid phase. The latter could be of various types
such as an ordinary crystal, a plastic crystal (PC) or a disordered crystal (DC) depending on themagnitude
of the anisotropy parameter d . As it has been already mentioned in the Introduction, a scenario that
involves first-order transition indeed takes place in the limit d = 0 case, i.e., for a hard-disk fluid [2–4].
The transition with two coexisting phases— isotropic fluid and disordered crystal phase— has been also
confirmed for another limiting case d = 1, i.e., for a hard-dimer system [9], as well as for a hard-dumbbell
fluid with d < 1, namely, in the case of d = 0.924 [10].
In the present study, we consider the features of fluid-to-solid transition in a hard-dumbbell fluid that
is characterized by a small deviation of molecule shape anisotropy from a hard-disk fluid, by assuming
the values of the anisotropy parameter within the range 0 É d < 0.28. The latter was chosen by means
of the anisotropy-density phase diagram in figure 1 that associates this range of a hard-dumbbell fluid
anisotropy with the coexisting plastic crystal solid.
We have used Monte Carlo simulations technique to study the properties related to freezing tran-
sition in a hard-dumbbell fluid. All simulations were performed in a rectangular box with an aspect
ratio of
p
3/2 with the usual periodic boundary conditions. Two different thermodynamic ensembles
have been employed. Namely, the constant-density NV T ensemble was used to evaluate the radial dis-
tribution function and global bond-orientational order parameter of the hard-dumbbell fluid, while the
constant-pressure N pT ensemble was used to discriminate between a first-order and continuous transi-
tion in the same system. The symbols N ,V ,T and p specify the number of particles, volume (area in two
dimensions), temperature and pressure used in the simulations, respectively.
In the course of our simulations, the single molecular move (in what follows is referred to as the
iterative step) consists of a random positional displacement of molecular mass center accompanied by
a rotation of molecular axes. For denser states, we also attempted a π/3 orientational displacement of
molecular axes that may be commensurable with a crystalline solid phase. The N pT simulations addi-
tionally consist in the change of the area of the simulation box. The acceptance ratio was maintained
between 20% and 30% by adjusting the maximal size of positional displacement and maximal amplitude
for the rotation in the case of NV T ensemble, and between 20% and 50% by adjusting the maximal size
of positional displacement, maximal amplitude for the rotation and maximal variation of the box size in
the case of N pT ensemble.
A standard Metropolis algorithm was used to obtain the ensemble averages. For NV T simulations,
each equilibration run was relaxed for at least 104
iterative steps. The resulting data for radial distribu-
tion function and global bond-orientational order parameter were obtained by averaging at least over
600 configurations, each being relaxed by 103
iterative steps. Whereas for the N pT simulations, each
equilibration run was relaxed for at least 106
iterative steps, the distribution of densities was accumu-
lated after at least 2×107
iterative steps to obtain a good sampling.
Various sizes of system were investigated, ranging from 100 to 1000 molecules. Similar to the case
of a hard-disk fluid [11, 12], no systematic system size effects were observed in the NV T data for the
second maximum of the radial distribution function, and subsequent calculations were made using N =
400 hard-dumbbell molecules. These simulations have been used to collect data for the global bond-
orientational order parameter. However, in this case it is well documented [13] that N = 400 is too
small to identify the boundary of freezing and melting densities. Therefore, the data for global bond-
orientational order parameter could be used for qualitative purposes only.
The Monte Carlo simulations in the N pT ensemble are efficient at identifying the type of fluid-to-
solid transition. According to the studies by Lee and Strandburg [14] on the application of the isobaric
Monte Carlo simulations, the system size 100 É N É 400 is sufficiently large to identify the fluid-to-solid
transition in a hard-disk system as first order transition. Therefore, the N pT calculations of the equation
of state were made using N = 100 hard-dumbbell molecules. The consequences of such a system size for
pressure magnitude of the hard-disk fluid in the vicinity of transition region are already discussed in the
literature [15].
23605-3
A. Huerta, D. Tejeda, D. Henderson, A. Trokhymchuk
3. Results and discussions
Following the observation by Truskett et al. [11], the fluid composed of planar hard disks (the case of
the anisotropy parameter d = 0) exhibits a structural precursor to freezing transition that manifests itself
as a shoulder in the second maximum of the disk-disk radial distribution function. In fact, an increase
of the density in a planar array of N hard disks of diameter σ, that are uniformly spread on area A, is
accompanied by formation of the local quasi-regular hexagonal arrangement. By quasi-regular hexagons
we mean the sixfold configurations (see the right-hand side frame of figure 1 adopted for details from
reference [12]) that, at certain disk density or packing fraction η = πNσ2/(4A), is characterized by a
constant gap width
∆
σ
=
(
ηcp
η
)1/2
−1
between each pair of the neighbor disks [12]. Here, ηcp = π/(2
p
3) ≈ 0.907 is the disk close packing (CP)
fraction. Such a sixfold formation implies that center-to-center distances r between disks and, conse-
quently, the gap width ∆ shortens under the density increasing.
By analyzing the schematic drawing in the inset of figure 1, Huerta et al. [12] pointed out that there
exists a packing fraction ηcage = π
p
3/8 = 0.680 and, consequently, a gap width ∆/σ= 2/
p
3−1 ≈ 0.155,
which in transparent way demonstrates the existence of two kinds of the second shell neighbors dis-
cerned by means of the distance criterion. Namely, the second shell neighbors on the distances (e.g.,
AD, . . . ) that are always larger than 2σ, and those on the distances (e.g., AB, AC, . . . ) that could be shorter
than 2σ. In fact, the existence of these two kinds (larger and shorter) of the second shell neighbors is the
origin, initially of a shoulder and then, with an increase of the density, a split of the second peak of the
disk-disk radial distribution function g (r ) for distances around r /σ= 2.
On the other hand, it implies that the center-to-center distances r between the first-shell alternating
neighbors of any disk in a fluid system become, on average, shorter than 2σ, i.e., the set of only three first-
shell alternating neighbors is capable of forming a cage, preventing the central disk to wander. Thus,
the hard-disk fluid will exhibit a tendency to freeze. The corresponding hard-disk fluid caging density,
ηcage = 0.680, is lower than the freezing transition density, ηf = 0.700, that follows from large scale com-
puter simulation data [4]. Actually it means that in practice disks are not spread uniformly and some
of the center-to-center distances (AB, AC, . . . ) become shorter than 2σ already for the disk packing frac-
tions η Ê 0.6, initiating deformation of the smooth second maximum of the disk-disk radial distribution
function g (r ).
Therefore, a starting point to discuss the effect of shape anisotropy on the freezing transition in a
fluid composed of planar hard dumbbells will be to analyze the trends in the second maximum of the
center-center radial distribution functions shown in figure 1 for packing fractions η = 0.686 and 0.723
upon increasing the value of anisotropy parameter d .
Figures 2–4 show a set of NV T Monte Carlo simulation data for the center-center radial distribution
functions g (r ) of the hard dumbbell fluids characterized by three values of the shape anisotropy param-
eter d = 0.05, 0.15 and 0.27. that are explicitly compared against the case of hard disk fluid with d = 0.
In each case, the radial distribution functions are shown at two values of packing fraction, η = 0.686
(left-hand side panel) and η= 0.723 (right-hand side panel). These particular values of packing fractions
correspond to the close proximity of the freezing and melting transitions, respectively, in the case of
hard-disk fluid (d = 0). To facilitate the comparison and discussion, in all cases that are presented in fig-
ures 2–4, the data for hard-dumbbell fluid are explicitly compared against the corresponding data for
hard-disk fluid.
As expected, and in agreement with a prediction of the density-anisotropy phase diagram in figure 1,
for small values of the anisotropy parameter, d = 0.05, the freezing scenario seems to be quite similar to
the case of a hard-disk fluid. Namely, for the case of d = 0.05we are observing practically the same shape
of the secondmaxima for both densities with a tiny shift in the positions to slightly larger center-to-center
distances r .
As the shape anisotropy increases to d = 0.15, the shift in the positions of maxima and minima of
the g (r ) becomes more pronounced. Although for packing fraction η= 0.686, the shoulder in the second
maxima is still noticed but it is less evident. However, since the split of the second maxima is definitely
23605-4
Frustration of freezing in a two dimensional hard-core fluid
0
1
2
3
4
5
6
7
1 1.5 2 2.5 3 3.5 4 4.5 5
r/σ
0
1
2
3
4
5
6
7
1 1.5 2 2.5 3 3.5 4 4.5 5
r/σ
Figure 2. (Color online) The NV T Monte Carlo simulations data for the center-center radial distribution
function g (r ) of the hard-dumbbell fluid with anisotropy parameter d = 0.05 at two values of packing
fraction: η= 0.686, associated with an isotropic fluid phase (the left-hand-side frame) and η= 0.723, asso-
ciated with an ordered solid phase (the right-hand-side frame). Both frames show a comparison against
the corresponding radial distribution function of the hard-disk, d = 0, fluid (the dotted line).
present, one could expect that the freezing mechanism that was discussed for a hard-disk fluid is still at
work.
However, the case of the shape anisotropy parameter d = 0.27 seems to be crucial from the point
of view of the freezing transition. The resulting radial distribution function in such a hard-dumbbell
fluid, shown in figure 4, is totally different from the one of the hard-disk fluid for both packing fractions
considered. Most important is that there are no indications either of a shoulder at lower packing fraction
or of a split of the secondmaximum at higher packing fraction.When discussing figure 1, we have already
pointed out that besides serving as a precursor towards approaching the freezing transition, the presence
of these features is signaling about the formation of the local quasi-regular hexagonal arrangement.
A quantitative measure of hexagonal order in the system is provided by the so-called global bond-
orientational order parameter Ψ6, that was evaluated during the NV T Monte Carlo simulation runs
using definition
Ψ6 =
∣∣∣∣ 1
Nnn
N∑
j=1
Nnn∑
k=1
ei6θ j k
∣∣∣∣,
where j runs over all disks in the system, k runs over all geometric nearest neighbors (nn) of disk j ,
each obtained through the Voronoi analysis and Nnn is the total number of such nearest neighbors in the
system. The angle θ j k is defined between some fixed reference axis in the system and the vectors bonds
connecting nearest neighbors j and k. As one can see from figure 5, indeed, the global bond-orientational
0
1
2
3
4
5
6
7
1 1.5 2 2.5 3 3.5 4 4.5 5
r/σ
0
1
2
3
4
5
6
7
1 1.5 2 2.5 3 3.5 4 4.5 5
r/σ
Figure 3. (Color online) The same as in figure 2 but for the anisotropy parameter d = 0.15.
23605-5
A. Huerta, D. Tejeda, D. Henderson, A. Trokhymchuk
0
1
2
3
4
5
6
7
1 1.5 2 2.5 3 3.5 4 4.5 5
r/σ
0
1
2
3
4
5
6
7
1 1.5 2 2.5 3 3.5 4 4.5 5
r/σ
Figure 4. (Color online) The same as in figure 2 but for the anisotropy parameter d = 0.27.
order parameter ψ6 of the hard-dumbbell fluid with anisotropy parameter d = 0.27 being very small at
low packing fraction, continuously grows under density increasing towards a denser phase.
Analyzing the trends in the secondmaximum of the radial distribution function in figures 2–4, as well
as the trends in profile of the global bond-orientational order parameter in figure 5, one can expect that
the hard-dumbbell fluid with anisotropy parameter d > 0.15 could behave differently than the hard-disk
fluid is doing while approaching the fluid-to-solid transition region. To obtain a better insight into the
features of fluid-to-solid transition in the system composed of hard-dumbbell molecules, in figure 6 we
show the N pT Monte Carlo data for pressure βp , where β = 1/kBT and kB is the Boltzmann constant,
in a wide range of packing fraction η (the left-hand-side frame of figure 6). The hard-dumbbell systems
presented in figure 6 are the same as have been discussed in figures 2–5 to learn about the trends in the
features of the second maximum of the radial distribution function. As expected, those features that are
sensitive to the shape anisotropy in the density region that precedes fluid-to-solid transition; for better
visualization, this region is shown separately on the right-hand-side frame of figure 6. On the other hand,
from the results in the left-hand-side frame of figure 6 it follows that in the limiting case of low (η< 0.3) as
well as at high (η≈ 0.8) packing fractions, all pressure isotherms tend to coincide, showing no dependence
on the anisotropy parameter.
Although quantitatively the results for pressure p and for the freezing and melting transition densi-
ties could be affected by the system size effects, which is rather evident in the case of a hard-disk fluid,
0
0.2
0.4
0.6
0.8
1
0.6 0.65 0.7 0.75 0.8
ψ
6
η
Figure 5. (Color online) The NV T Monte Carlo data for global bond-orientational order parameter ψ6 of
a hard-dumbbell fluid with anisotropy parameter values d = 0.27 and 0.15 (from the bottom to the top)
in comparison against a hard-disk (d = 0) fluid system (the solid line at the top). The vertical dashed lines
mark the fluid-to-solid transition region that corresponds to the system with N = 400.
23605-6
Frustration of freezing in a two dimensional hard-core fluid
0
4
8
12
16
20
0.2 0.3 0.4 0.5 0.6 0.7 0.8
βp
η
7
8
9
10
0.64 0.68 0.72 0.76
βp
η
Figure 6. (Color online) The N pT Monte Carlo data for pressure βp of a hard-dumbbell fluid with
anisotropy parameter values d = 0.27, 0.15 and 0.05 (from the bottom to the top for η < 0.7) in com-
parison against a hard-disk (d = 0) fluid (at the top). The symbols correspond to Monte Carlo simulation
data while lines are shown to guide the eyes. The left-hand-side frame presents the full density range,
while the right-hand-side panel shows the region nearby the fluid-to-solid transition only. In the right-
hand-side panel: the vertical dashed lines mark the fluid-to-solid transition region according to Bernard
and Krauth [4]; the filled squares at the top represent computer simulation data for a hard-disk fluid that
are corrected for finite-size effects [2, 15, 16].
the qualitative picture must be correct. Namely, like in the previous N pT Monte Carlo simulations [14]
as well as in quite recent large-scale NV T Monte Carlo simulations [4, 17, 18] of the freezing transition
in a hard-disk system, our N pT Monte Carlo data for a hard-disk (d = 0) fluid show a discontinuous first-
order transition from an isotropic fluid phase to a denser and highly ordered phase. The same scenario
persists for hard-dumbbell fluids with anisotropy parameter values d = 0.05 and 0.15. However, in the
case of a hard-dumbbell fluid with anisotropy parameter d = 0.27, a continuous transition is obtained.
Regarding the issue of the system size effects in the case of a hard-dumbbell fluid, we note that contrary
to a hard-disk system, all the existing computer simulation data for a hard-dumbbell system have been
obtained with N = 112molecules [8–10].
In fact, the statement regarding the frustration of freezing in a hard-dumbbell fluid with the aniso-
tropy parameter d = 0.27 is consistent with the information that follows from the data presented in
figure 7 where density distributions obtained in the N pT Monte Carlo simulation are shown. Each curve
0
10
20
30
40
0.64 0.66 0.68 0.7 0.72 0.74 0.76
η
0
10
20
30
40
0.64 0.66 0.68 0.7 0.72 0.74 0.76
η
0
10
20
30
40
0.64 0.66 0.68 0.7 0.72 0.74 0.76
η
Figure 7. (Color online) Density distributions obtained in the N pT Monte Carlo simulations of the system
of N = 100 hard-dumbbell molecules with three different values of the anisotropy parameter. Namely,
d = 0, i.e., for a hard-disk fluid and for two hard-dumbbell fluids with d = 0.15 and 0.27 from the left to
the right. Each curve for a particular system corresponds to a constant pressure p shown by symbols in
figure 6.
23605-7
A. Huerta, D. Tejeda, D. Henderson, A. Trokhymchuk
for a particular system in figure 7 corresponds to a constant pressure p , and the position of its maximum
determines the density (packing fraction η) of the stable phase of this system in this thermodynamic
state. The curves with two maxima in the case of d = 0 and d = 0.15 each tells us that at pressure p ,
that corresponds to this curve, there is coexistence of two phases and, consequently, the fluid-to-solid
transition is discontinuous or of first order. By contrast, in the third case of hard-dumbbell fluid with d =
0.27, all curves show only one maximum, and under pressure increasing, the position of this maximum
is moving towards higher densities (packing fractions η). Thus, the fluid-to-solid transition in this case is
continuous.
4. Conclusion
Contrary to the hard-disk system, where fluid-to-solid transition has been the subject of a long-stan-
ding debate [19], the hard-dumbbell system has not been explored in full yet. This fact looks rather
strange in view that almost three decades ago the theoretically predicted phase diagram of the hard-
dumbbell system in full range of the anisotropy parameter, 0 É d É 1, was reported based on heuristic
reasoning and using free-volume arguments [8]. The main conclusion of that study was that in the full
range of the anisotropy parameter values, fluid-to-solid transition is of the first order. Up to date, this
result was confirmed by computer simulations for two particular values of the anisotropy parameter,
i.e., d = 1 [9] and 0.924 [10]. Obviously, theoretical predictions are valid for the case of d = 0, i.e., for a
hard-disk fluid [3, 4, 13, 17, 18].
In this study, we turned our attention to the case of anisotropy parameter values range 0 É d É 0.28
that was indicated in that theoretical analysis [8] as a unique one where the isotropic fluid phase freezes
by coexisting with the plastic crystal solid. First of all, using the NV T Monte Carlo simulations data we
have shown that such a feature of the radial distribution function as the shoulder in its secondmaximum,
that was suggested by Truskett et al. [11] as the structural precursor to freezing transition in the case of
a hard-disk fluid, is still preserved for a hard-dumbbell fluid with anisotropy parameter values up to
d É 0.15. However, in the case of a hard-dumbbell system with anisotropy parameter d = 0.27, the radial
distribution functions do not show any sign of the shoulder in the range of distances that correspond
to the second maximum. Secondly, by performing the N pT Monte Carlo simulations we obtained that
indeed first-order freezing transition from an isotropic fluid to denser highly ordered phase persists in
the case of a hard-dumbbell fluid with an anisotropy parameter d É 0.15 while two-phase coexistence is
absent in the case of d = 0.27.
Although in the present study we did not consider this question, the coexisting denser phase in the
case of a hard-dumbbell fluid, with anisotropy parameter d É 0.15, could be associated with a plastic
crystal, as it was predicted [8]. We remind the reader that the plastic crystal phase represents one where
densely packed molecules are still free to rotate. In fact, close examination of the hard-disk fluid config-
urations in the vicinity of freezing transition revealed the formation of a local quasi-regular hexagonal
arrangement of hard disks [12]. Under increasing density, the average center-to-center distance between
neighboring disks continuously shortens and, at the point of freezing. This results in a disk spacing of ap-
proximately 15% of disk diameter. Evidently, any disturbance that will tend to violate such a spacing, e.g.,
due to the attractive or repulsive interaction between disks, or the degree of disk size polydispersity, etc.,
could potentially result in the frustration of the freezing phenomenon. This is what we have observed in
the present study: if the disk elongation starts to exceed 15% of the disk diameter (the anisotropy param-
eter d = 0.27), the freezing transition cannot survive and under an increase of the density, the disordered
isotropic hard-dumbbell fluid undergoes a continuous transition to a denser and more ordered phase.
Such a mechanism has already proved to work in the case of hard disks with square-well attraction [6]
and for an equimolar binary hard-disk mixture [7].
Acknowledgements
This work is supported by the CONACYT under the project 152431, Red Temática de la Materia Con-
densada Blanda and Promep of México.
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Frustration of freezing in a two dimensional hard-core fluid
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Уникнення замерзання двовимiрного плину жорстких
молекул через анiзотропiю їх форми
А. Уерта1, Д. Техеда1, Д. Гедерсон2, А. Трохимчук3,4
1 Унiверситет Веракрузана, факультет фiзики та iнженерiї, кафедра фiзики,
Халапа, Веракруз, СР 91000,Мексика
2 Факультет хiмiї та бiохiмiї, Унiверситет Брiгама Янга, Прово,Юта 84602, США
3 Iнститут фiзики конденсованих систем НАН України, вул. I. Свєнцiцького, 1, 79011 Львiв, Україна
4 Iнститут прикладної математики та фундаментальних наук, Нацiональний унiверситет “Львiвська
Полiтехнiка”, 79013 Львiв, Україна
Механiзм замерзання, який був запропонований для плину жорстких дискiв [Huerta A. et al., Phys. Rev. E,
2006, 74, 061106], використовується тут для дослiдження фазового переходу плин-тверде тiло у системi,
що складається з жорстких гантелеподiбних молекул, утворених двома жорсткими дисками зi змiнною
вiдстанню мiж їх центрами. З аналiзу тенденцiй змiни форми другого максимуму радiальної функцiї роз-
подiлу зроблено висновок,що тип фазового переходу плин-тверде тiло може бути чутливий до видовже-
ння гантелеподiбних молекул. На основi даних комп’ютерного експерименту Монте Карло при фiксова-
ному тиску було знайдено, що коли видовження молекул не перевищує 15% дiаметра жорсткого диска,
то фазовий перехiд плин-тверде тiло вiдбувається за тим же сценарiєм що i у системi жорстких дискiв,
тобто плин гантелеподiбних молекул замерзає. У випадку, коли видовження молекул перевищує 15% дi-
аметра жорсткого диска, то є пiдстави стверджувати, що фазовий перехiд плин-тверде тiло проходить
неперервно, тобто плин гантелеподiбних молекул у цьому випадку не замерзає.
Ключовi слова: плин жорстких дискiв, гантелеподiбнi молекули, радiальна функцiя розподiлу,
фазовий перехiд замерзання
23605-9
http://dx.doi.org/10.1103/PhysRev.127.359
http://dx.doi.org/10.1103/PhysRevB.46.11186
http://dx.doi.org/10.1103/PhysRevE.73.065104
http://dx.doi.org/10.1103/PhysRevLett.107.155704
http://dx.doi.org/10.1016/j.cplett.2009.07.059
http://dx.doi.org/10.1063/1.1632893
http://dx.doi.org/10.5488/CMP.15.43604
http://dx.doi.org/10.1016/0375-9601(87)90846-2
http://dx.doi.org/10.1103/PhysRevLett.66.3168
http://dx.doi.org/10.1103/PhysRevB.46.26
http://dx.doi.org/10.1103/PhysRevE.58.3083
http://dx.doi.org/10.1103/PhysRevE.74.061106
http://dx.doi.org/10.1103/PhysRevB.51.14636
http://dx.doi.org/10.1103/PhysRevB.46.11190
http://dx.doi.org/10.1080/00268970600967963
http://dx.doi.org/10.1063/1.1670653
http://dx.doi.org/10.1103/PhysRevE.87.042134
http://dx.doi.org/10.1063/1.4931731
http://dx.doi.org/10.1103/RevModPhys.60.161
Introduction
Modelling and details of simulations
Results and discussions
Conclusion
|
| id | nasplib_isofts_kiev_ua-123456789-155803 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1607-324X |
| language | English |
| last_indexed | 2025-12-07T13:21:36Z |
| publishDate | 2016 |
| publisher | Інститут фізики конденсованих систем НАН України |
| record_format | dspace |
| spelling | Huerta, A. Tejeda, D. Henderson, D. Trokhymchuk, A. 2019-06-17T12:53:42Z 2019-06-17T12:53:42Z 2016 Frustration of freezing in a two dimensional hard-core fluid due to particle shape anisotropy / A. Huerta, D. Tejeda, D. Henderson, A. Trokhymchuk // Condensed Matter Physics. — 2016. — Т. 19, № 2. — С. 23605: 1–9 . — Бібліогр.: 19 назв. — англ. 1607-324X PACS: 64.60.Fr, 68.35.Rh DOI:10.5488/CMP.19.23605 arXiv:1603.07147 https://nasplib.isofts.kiev.ua/handle/123456789/155803 The freezing mechanism suggested for a fluid composed of hard disks [Huerta et al., Phys. Rev. E, 2006, 74, 061106] is used here to probe the fluid-to-solid transition in a hard-dumbbell fluid composed of overlapping hard disks with a variable length between disk centers. Analyzing the trends in the shape of second maximum of the radial distribution function of the planar hard-dumbbell fluid it has been found that the type of transition could be sensitive to the length of hard-dumbbell molecules. From the NpT Monte Carlo simulations data we show that if a hard-dumbbell length does not exceed 15% the fluid-to-solid transition scenario follows the case of a hard-disk fluid, i.e., the isotropic hard-dumbbell fluid experiences freezing. However, for a hard-dumbbell length larger than 15% fluid-to-solid transition may change to continuous transition, i.e., such an isotropic hard-dumbbell fluid will avoid freezing. Механiзм замерзання, який був запропонований для плину жорстких дискiв [Huerta A. et al., Phys. Rev. E, 2006, 74, 061106], використовується тут для дослiдження фазового переходу плин-тверде тiло у системi, що складається з жорстких гантелеподiбних молекул, утворених двома жорсткими дисками зi змiнною вiдстанню мiж їх центрами. З аналiзу тенденцiй змiни форми другого максимуму радiальної функцiї розподiлу зроблено висновок, що тип фазового переходу плин-тверде тiло може бути чутливий до видовження гантелеподiбних молекул. На основi даних комп’ютерного експерименту Монте Карло при фiксованому тиску було знайдено, що коли видовження молекул не перевищує 15% дiаметра жорсткого диска, то фазовий перехiд плин-тверде тiло вiдбувається за тим же сценарiєм що i у системi жорстких дискiв, тобто плин гантелеподiбних молекул замерзає. У випадку, коли видовження молекул перевищує 15% дiаметра жорсткого диска, то є пiдстави стверджувати, що фазовий перехiд плин-тверде тiло проходить неперервно, тобто плин гантелеподiбних молекул у цьому випадку не замерзає. The authors are pleased to dedicate this article to our good friend and collaborator, Anthony (Tony) Haymet on the occasion of his 60th birthday. He made a number of valuable contributions to the physics and chemistry of the condensed state. He has also held a number of important administrative positions. We wish him well. This work is supported by the CONACYT under the project 152431, Red Temática de la Materia Condensada Blanda and Promep of México. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Frustration of freezing in a two dimensional hard-core fluid due to particle shape anisotropy Уникнення замерзання двовимiрного плину жорстких молекул через анiзотропiю їх форми Article published earlier |
| spellingShingle | Frustration of freezing in a two dimensional hard-core fluid due to particle shape anisotropy Huerta, A. Tejeda, D. Henderson, D. Trokhymchuk, A. |
| title | Frustration of freezing in a two dimensional hard-core fluid due to particle shape anisotropy |
| title_alt | Уникнення замерзання двовимiрного плину жорстких молекул через анiзотропiю їх форми |
| title_full | Frustration of freezing in a two dimensional hard-core fluid due to particle shape anisotropy |
| title_fullStr | Frustration of freezing in a two dimensional hard-core fluid due to particle shape anisotropy |
| title_full_unstemmed | Frustration of freezing in a two dimensional hard-core fluid due to particle shape anisotropy |
| title_short | Frustration of freezing in a two dimensional hard-core fluid due to particle shape anisotropy |
| title_sort | frustration of freezing in a two dimensional hard-core fluid due to particle shape anisotropy |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/155803 |
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