Two-dimensional core-softened model with water like properties. Study by thermodynamic perturbation theory
Thermodynamic properties of the particles interacting through smooth version of Stell-Hemmer interaction were studied using Wertheim's thermodynamic perturbation theory. The temperature dependence of molar volume, heat capacity, isothermal compressibility and thermal expansion coefficient at co...
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irk-123456789-1208562017-06-14T03:04:15Z Two-dimensional core-softened model with water like properties. Study by thermodynamic perturbation theory Urbic, T. Thermodynamic properties of the particles interacting through smooth version of Stell-Hemmer interaction were studied using Wertheim's thermodynamic perturbation theory. The temperature dependence of molar volume, heat capacity, isothermal compressibility and thermal expansion coefficient at constant pressure for different number of bonding sites on particle were evaluated. The model showed water-like anomalies for all evaluated quantities, but thermodynamic perturbation theory does not properly predict dependence of these properties at fixed number of bonding points. Термодинамiчнi властивостi системи частинок, що взаємодiють за допомогою зм’якшеного потенцiалу типу Стелла-Хеммера дослiджено в рамках термодинамiчної теорiї збурень Вертхайма. Розраховано температурну залежнiсть молярного об’єму, теплоємностi, iзотермiчної стисливостi i коефiцiєнта термiчного розширення при сталому тиску для рiзного числа зв’язкiв для заданої частинки. Модель характеризується аномальною поведiнкою властивостей, подiбною до води, але термодинамiчна теорiя збурень незадовiльно описує цi властивостi, якщо число зв’язкiв в розрахунку на частинку є зафiксоване. 2013 Article Two-dimensional core-softened model with water like properties. Study by thermodynamic perturbation theory / T. Urbic // Condensed Matter Physics. — 2013. — Т. 16, № 4. — С. 43605:1-7 . — Бібліогр.: 29 назв. — англ. 1607-324X PACS: 64.70.Ja, 61.20.Ja DOI:10.5488/CMP.16.43605 arXiv:1312.4559 http://dspace.nbuv.gov.ua/handle/123456789/120856 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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Thermodynamic properties of the particles interacting through smooth version of Stell-Hemmer interaction were studied using Wertheim's thermodynamic perturbation theory. The temperature dependence of molar volume, heat capacity, isothermal compressibility and thermal expansion coefficient at constant pressure for different number of bonding sites on particle were evaluated. The model showed water-like anomalies for all evaluated quantities, but thermodynamic perturbation theory does not properly predict dependence of these properties at fixed number of bonding points. |
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Urbic, T. Two-dimensional core-softened model with water like properties. Study by thermodynamic perturbation theory Condensed Matter Physics |
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Urbic, T. |
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Urbic, T. |
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Two-dimensional core-softened model with water like properties. Study by thermodynamic perturbation theory |
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Two-dimensional core-softened model with water like properties. Study by thermodynamic perturbation theory |
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Two-dimensional core-softened model with water like properties. Study by thermodynamic perturbation theory |
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Two-dimensional core-softened model with water like properties. Study by thermodynamic perturbation theory |
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Two-dimensional core-softened model with water like properties. Study by thermodynamic perturbation theory |
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two-dimensional core-softened model with water like properties. study by thermodynamic perturbation theory |
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Інститут фізики конденсованих систем НАН України |
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2013 |
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Two-dimensional core-softened model with water like properties. Study by thermodynamic perturbation theory / T. Urbic // Condensed Matter Physics. — 2013. — Т. 16, № 4. — С. 43605:1-7 . — Бібліогр.: 29 назв. — англ. |
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Condensed Matter Physics |
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AT urbict twodimensionalcoresoftenedmodelwithwaterlikepropertiesstudybythermodynamicperturbationtheory |
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2025-07-08T18:44:23Z |
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2025-07-08T18:44:23Z |
| _version_ |
1837105435047362560 |
| fulltext |
Condensed Matter Physics, 2013, Vol. 16, No 4, 43605: 1–7
DOI: 10.5488/CMP.16.43605
http://www.icmp.lviv.ua/journal
Two-dimensional core-softened model with water
like properties. Study by thermodynamic
perturbation theory¤
T. Urbic
Faculty of Chemistry and Chemical Technology, University of Ljubljana, Askerceva 5, 1000 Ljubljana, Slovenia
Received August 1, 2013, in final form August 28, 2013
Thermodynamic properties of the particles interacting through smooth version of Stell-Hemmer interaction
were studied using Wertheim’s thermodynamic perturbation theory. The temperature dependence of molar
volume, heat capacity, isothermal compressibility and thermal expansion coefficient at constant pressure for
different number of bonding sites on particle were evaluated. The model showed water-like anomalies for all
evaluated quantities, but thermodynamic perturbation theory does not properly predict the dependence of
these properties at a fixed number of bonding points.
Key words: Monte Carlo, thermodynamic perturbation theory, core softened fluid
PACS: 64.70.Ja, 61.20.Ja
1. Introduction
They were Stell and Hemmer who first proposed core-softened potentials in 1970 [1]. In their early
work, they stressed that negative curvature in interaction potential might lead to a second critical point
in addition to a standard liquid-gas critical point. In different works [2, 3] it has been shown that core-
softened potentials and similar shouldered potentials can reproduce various fluid anomalies that are
typical of water and other substances with angular dependent interactions, such as silica [3], silicon [4],
BeF
2
[5]. Core-softened potentials were also used to study single-component liquid metal systems [6–
10] and as solvent for studying ions [11]. Poole et al. [12] proposed liquid-liquid phase transition as an
explanation for anomalous properties of water. After that there was an increased interest to the studies of
these liquid-liquid phase transitions. Franseze et al. [13] suggested that the liquid-liquid phase transition
and its critical point might be caused by the potential with two characteristic distances (hard core and
soft core). In their work, they reported the existence of the low-density liquid phase and the high-density
liquid phase obtained for 3D model using molecular dynamics (MD) simulations. On the other hand,
2D MD produced only a density anomaly but no liquid-liquid phase transition [14, 15]. Scala et al. [16]
carried out MD simulations of 2D discrete and smoothed version of potential to study liquid anomalies.
These studies were continued by Buldyrev et al. [17] to explore liquid-liquid phase transition for 2D and
3D version of potentials and by Almudallal et al. [18]. They both produced phase diagrams for a discrete
version of potential with liquid anomalies, and no liquid-liquid critical point in stable liquid region was
obtained.
Our aim here is to apply Wertheim’s thermodynamic perturbation theory (TPT) to capture the physics
of the model of 2D molecules interacting by Stell-Hemmer potential. In recent years, a theory has been
developed for fluids comprised of molecules that associate into dimers and higher clusters due to the
presence of highly directional attractive forces [19–21]. In the present work, we apply the thermodynamic
perturbation theory (TPT) [19–22] to central symmetric attractive potential.
¤Dedicated to Professor Myroslav Holovko on the occasion of his 70th birthday.
© T. Urbic, 2013 43605-1
http://dx.doi.org/10.5488/CMP.16.43605
http://www.icmp.lviv.ua/journal
T. Urbic
2. Model
The smooth version of the core-softened potential proposed by Scala et al. [16] is used in this work.
The interaction potentialU (r ) is a sum of a Gaussian well and the Lennard-Jones (LJ) part of the potential
U (r )ÆULJ(r )ÅUa
(r ), (2.1)
whereULJ(r ) is standard Lennard-Jones potential
ULJ(r )Æ 4"
·
³
¾
r
´
12
¡
³
¾
r
´
6
¸
. (2.2)
" is here the well-depth and ¾ is the distance, where LJ part of the potential is zero. The Gaussian part of
the interaction is as follows:
U
a
(r )Æ¡¸"exp
·
¡a
³
r ¡ r
0
¾
´
2
¸
. (2.3)
This part of potential is stronger than the LJ part and is the reason that the particles make strong as-
sociation. We also refer to this part of potential as association potential. We apply the units and values
of model parameters as used before by Scala et al. [16] as " Æ 1.0, ¾ Æ 1.0, ¸ Æ 1.7, a Æ 25.0, r
0
Æ 1.5¾.
Figure 1 shows the shape of the smooth version of the core-softened potential used here.
Figure 1. (Color online) The core-softened potential U (r ) (solid line) with both contributions (LJ — long
dashed line and Gaussian part — dashed line).
3. Monte Carlo simulation details
We performed Monte Carlo simulations in the isothermal-isobaric (NpT) ensemble to obtain thermo-
dynamic properties of the model. At each step, the displacements in the x, y coordinates were chosen ran-
domly. We used periodic boundary conditions and the minimum image convention to mimic an infinite
system of particles. The starting configurations were selected at random. Every 10 moves of particles an
attempt is made to scale the dimensions of the box and all of its component particles in order to hold the
pressure constant. 5£104 moves per particle were needed to equilibrate the system. The statistics were
gathered over the next 1£106 moves to obtain well converged results. All simulations were performed
with N Æ200 or N Æ400molecules. The maximum change of dimensions of the box was calibrated during
equilibration simulations. The physical properties of the system such as enthalpy and volume were calcu-
lated as the statistical averages of these quantities over the course of simulations [24]. The heat capacity,
C
p
, the isothermal compressibility, ·, and the thermal expansion coefficient, ® are computed from the
fluctuations [25] of enthalpy, H , and volume, V .
43605-2
Two-dimensional core-softened model
C
p
Æ
C
p
kB
Æ
hH
2
i¡hHi
2
NT
2
,
·Æ
hV
2
i¡hV i
2
T hV i
,
®Æ
hV Hi¡hV ihHi
T
2
hV i
, (3.1)
T is temperature of the system and N number of particles.
4. Thermodynamic perturbation theory
The Helmholtz free energy of the system is the key quantity of the thermodynamic perturbation the-
ory [19, 20]. In case of the model studied in this work, this quantity is the sum of two terms
A
NkBT
Æ
ALJ
NkBT
Å
A
a
NkBT
. (4.1)
N is the number of molecules, T is temperature and kB is Boltzmann’s constant. The Helmholtz free
energy of Lennard-Jones system, ALJ, is calculated using the Barker-Henderson perturbation theory [26]
ALJ
NkBT
Æ
AHD
NkBT
Å
½
2kBT
1
Z
¾
gHD(r,´)ULJ(r )dr . (4.2)
AHD is the hard-disk contribution to the Helmholtz free energy, gHD(r,´) is pair correlation function for
hard disks at packing fraction ´ Æ
1
4
¼d
2
½ and ½ is the number density of molecules. d is the hard-disk
diameter calculated using Barker-Henderson approximation as
d Æ
¾
Z
0
·
1¡exp
µ
¡
ULJ
kBT
¶¸
dr. (4.3)
We used the procedure by Scalise et al. [27] to calculate the HD term of the Helmholtz free energy
AHD¡ Aideal
NkBT
Æ¡1.10865¡0.8678ln (1¡´)¡0.0157(1¡´)Å
1.1322
1¡´
¡
0.00785
(1¡´)
2
. (4.4)
For gHD(r ), the expression of Gonzalez et al. [28] was used.
The association contribution to Helmholtz free energy, A
a
, was calculated by [19, 20, 29]
A
a
NkBT
ÆN
a
µ
logx¡
x
2
Å
1
2
¶
, (4.5)
where x is the fraction of molecules not bonded at particular interaction site and is obtained from the
mass-action law [19, 20] in the form
x Æ
1
1ÅN
a
½x¢
. (4.6)
½ is the total number density. Finally, ¢ is defined by [19, 20, 29]
¢Æ 2¼
Z
gLJ(r ) fa(r )rdr . (4.7)
f
a
(r ) is a Mayer function for the association potential
f
a
(r )Æ exp
·
¡
U
a
(r )
kBT
¸
¡1. (4.8)
43605-3
T. Urbic
The pair distribution function gLJ(r ) is obtained by solving the Percus-Yevick equation for Lennard-Jones
disks. N
a
is the number of association points on the particles. Particles have a spherically symmetric
association potential. They do not have a varied number of bonding points as is usually the case where
Wertheim’s theory is used. We made approximation that each particle can have N
a
association points in
the center of a particle interacting with associating potential. We used a different number of interaction
sites, from 1 to 6, the last being a coordination number of particle in a perfect hexagonal crystal. Once
the Helmholtz free energy is known, other thermodynamic quantities may be calculated from standard
thermodynamic relations [26]
p Æ
½
2
N
µ
�A
�½
¶
T
, (4.9)
(·
T
)
¡1
Æ ½
µ
�P
�½
¶
T
, (4.10)
®Æ·
T
µ
�P
�T
¶
½
, (4.11)
C
P
ÆC
V
Å
®
½
·
P ¡½
2
µ
�U
�½
¶
T
¸
. (4.12)
5. Results
All the results are given in reduced units; the excess internal energy and temperature are normal-
ized to the LJ interaction parameter " (E¤ Æ E/", T ¤ Æ kBT /") and all the distances are scaled to the
characteristic length ¾ (r¤ Æ r /¾).
Figure 2. (Color online) Temperature dependence of the molar volume at P¤ Æ 0.75 as obtained by the
Monte Carlo simulation (symbols), the thermodynamic perturbation theory for a different number of
associating points (N
a
Æ 1 full red line, N
a
Æ 2 long dashed green line, N
a
Æ 3 dashed blue line, N
a
Æ 4
dotted pink line, N
a
Æ 5 long dashed-dotted light blue line and N
a
Æ 6 dashed-dotted grey line.
In figure 2, we compare the molar volume or volume per particle, V ¤/N , obtained from the Monte
Carlo simulations, with the results of the thermodynamic perturbation theory for a different number of
bonding sites on particle (N
a
Æ 1¡6). The calculations were performed at a reduced pressure of P¤ Æ 0.75.
We find out that the TPT does not properly capture the results for simulations as well as it does not
predict the maxima in density or minima in molar volume. In order to obtain an agreement, N
a
should
be dynamically varied. N
a
should be varied with temperature and density in order to get an agreement
between theoretical and simulation results. We can see from figures that we have good agreement of TPT
with simulation for N
a
Æ 2 at high temperature T ¤ Æ 2.0, then 3 at T ¤ Æ 1, etc.
43605-4
Two-dimensional core-softened model
The remaining figures show the temperature dependencies of the other thermodynamic quantities of
interest: the isothermal compressibility, ·¤
T
(figure 3), the thermal expansion coefficient, ®¤ (figure 4), the
heat capacity, C¤
p
(figure 5), and the excess chemical potential ¹¤ex (figure 6). The TPT for a fixed number
of associating points is not in agreement with the Monte Carlo simulation data for all quantities. From
the results we can also see that the model behaves in a way that the number of associating points is not
fixed, but it changes at a constant pressure with temperature. We calculated free energy as a function of
the number of boding sites, N
a
, and the free energy decreases within the whole range.
Figure 3. (Color online) Temperature dependence of
the isothermal compressibility at P¤ Æ 0.75; legend
as for figure 2.
Figure 4. (Color online) Temperature dependence of
the thermal expansion coefficient at P¤ Æ 0.75; leg-
end as for figure 2.
Figure 5. (Color online) Temperature dependence of
the heat capacity at P¤ Æ 0.75; legend as for figure 2.
Figure 6. (Color online) Temperature dependence of
the excess chemical potential at P¤ Æ 0.75; legend as
for figure 2.
6. Conclusion
The thermodynamic perturbation theory was used to study the thermodynamics of the particles in-
teracting through a smooth version of Stell-Hemmer interaction. The results for the molar volume, the
isothermal compressibility, the thermal expansion coefficient, the heat capacity and the excess chemi-
cal potential obtained by the TPT theory for a fixed number of bonding sites on the particles are not in
agreement with the computer simulation results for all the parameters studied. This is caused by the fact
that the interaction is spherical symmetric, and for TPT we have to make approximations. It is crucial to
43605-5
T. Urbic
change the spherical symmetric potential into a directional one and to have a different number of inter-
action points with directional forces. We cannot obtain correct thermodynamic properties with a fixed
number of association points. Proper thermodynamics could be obtained if the number of association
points varied with temperature and pressure.
7. Acknowledgements
We appreciate the support by the Slovenian Research Agency (P1 0103-0201 and J1 4148) and NIH
Grant GM063592.
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http://dx.doi.org/10.1080/00268978800100821
Two-dimensional core-softened model
Двовимiрна модель потенцiалу з м’яким кором з
властивостями подiбними до води. Дослiдження методом
термодинамiчної теорiї збурень
Т. Урбiч
Факультет хiмiї i хiмiчної технологiї, Унiверситет м. Любляна, 1000 м. Любляна, Словенiя
Термодинамiчнi властивостi системи частинок, що взаємодiють за допомогою зм’якшеного потенцiалу
типу Стелла-Хеммера дослiджено в рамках термодинамiчної теорiї збурень Вертхайма. Розраховано тем-
пературну залежнiсть молярного об’єму, теплоємностi, iзотермiчної стисливостi i коефiцiєнта термiчного
розширення при сталому тиску для рiзного числа зв’язкiв для заданої частинки. Модель характеризується
аномальною поведiнкою властивостей, подiбною до води, але термодинамiчна теорiя збурень незадо-
вiльно описує цi властивостi, якщо число зв’язкiв в розрахунку на частинку є зафiксоване.
Ключовi слова: Монте Карло, термодинамiчна теорiя збурень, плин з м’яким кором
43605-7
Introduction
Model
Monte Carlo simulation details
Thermodynamic perturbation theory
Results
Conclusion
Acknowledgements
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