Equilibrium properties of the lattice system with SALR interaction potential on a square lattice: quasi-chemical approximation versus Monte Carlo simulation
The lattice system with competing interactions that models biological objects (colloids, ensembles of protein molecules, etc.) is considered. This system is the lattice fluid on a square lattice with attractive interaction between nearest neighbours and repulsive interaction between next-next-near...
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irk-123456789-1574652019-06-21T01:28:44Z Equilibrium properties of the lattice system with SALR interaction potential on a square lattice: quasi-chemical approximation versus Monte Carlo simulation Groda, Ya.G. Vikhrenko, V.S. D. di Caprio The lattice system with competing interactions that models biological objects (colloids, ensembles of protein molecules, etc.) is considered. This system is the lattice fluid on a square lattice with attractive interaction between nearest neighbours and repulsive interaction between next-next-nearest neighbours. The geometric order parameter is introduced for describing the ordered phases in this system. The critical value of the order parameter is estimated and the phase diagram of the system is constructed. The simple quasi-chemical approximation (QChA) is proposed for the system under consideration. The data of Monte Carlo simulation of equilibrium properties of the model are compared with the results of QChA. It is shown that QChA provides reasonable semiquantitative results for the systems studied and can be used as the basis for next order approximations. Розглядається ґраткова система з конкурентними взаємодiями, що моделює бiологiчнi об’єкти (колоїди, ансамблi протеїнових молекул i т.д.). Ця система представляється ґратковим плином на квадратнiй ґратцi з притягальною взаємодiєю мiж найближчими сусiдами i вiдштовхувальною взаємодiєю мiж наступними за наступними до найближчих сусiдiв. Для опису впорядкованої фази в такiй системi вводиться геометричний параметр порядку. Зроблена оцiнка критичного значення параметра порядку i побудована фазова дiаграма системи. Для системи, що розглядається, запропоновано просте квазiхiмiчне наближення (КХН). Данi симуляцiй Монте Карло рiвноважних властивостей моделi порiвнюються з результатами КХН. Показано, що КХН забезпечує прийнятнi напiвкiлькiснi результати для системи, що вивчається, i може використовуватися як базис для наступних наближень. 2018 Article Equilibrium properties of the lattice system with SALR interaction potential on a square lattice: quasi-chemical approximation versus Monte Carlo simulation / Ya.G. Groda, V.S. Vikhrenko, D. di Caprio // Condensed Matter Physics. — 2018. — Т. 21, № 4. — С. 43002: 1–10. — Бібліогр.: 16 назв. — англ. 1607-324X PACS: 05.10.Ln, 64.60.De, 64.60.Cn DOI:10.5488/CMP.21.43002 arXiv:1812.08526 http://dspace.nbuv.gov.ua/handle/123456789/157465 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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English |
| description |
The lattice system with competing interactions that models biological objects (colloids, ensembles of protein
molecules, etc.) is considered. This system is the lattice fluid on a square lattice with attractive interaction
between nearest neighbours and repulsive interaction between next-next-nearest neighbours. The geometric
order parameter is introduced for describing the ordered phases in this system. The critical value of the order parameter is estimated and the phase diagram of the system is constructed. The simple quasi-chemical
approximation (QChA) is proposed for the system under consideration. The data of Monte Carlo simulation of
equilibrium properties of the model are compared with the results of QChA. It is shown that QChA provides
reasonable semiquantitative results for the systems studied and can be used as the basis for next order approximations. |
| format |
Article |
| author |
Groda, Ya.G. Vikhrenko, V.S. D. di Caprio |
| spellingShingle |
Groda, Ya.G. Vikhrenko, V.S. D. di Caprio Equilibrium properties of the lattice system with SALR interaction potential on a square lattice: quasi-chemical approximation versus Monte Carlo simulation Condensed Matter Physics |
| author_facet |
Groda, Ya.G. Vikhrenko, V.S. D. di Caprio |
| author_sort |
Groda, Ya.G. |
| title |
Equilibrium properties of the lattice system with SALR interaction potential on a square lattice: quasi-chemical approximation versus Monte Carlo simulation |
| title_short |
Equilibrium properties of the lattice system with SALR interaction potential on a square lattice: quasi-chemical approximation versus Monte Carlo simulation |
| title_full |
Equilibrium properties of the lattice system with SALR interaction potential on a square lattice: quasi-chemical approximation versus Monte Carlo simulation |
| title_fullStr |
Equilibrium properties of the lattice system with SALR interaction potential on a square lattice: quasi-chemical approximation versus Monte Carlo simulation |
| title_full_unstemmed |
Equilibrium properties of the lattice system with SALR interaction potential on a square lattice: quasi-chemical approximation versus Monte Carlo simulation |
| title_sort |
equilibrium properties of the lattice system with salr interaction potential on a square lattice: quasi-chemical approximation versus monte carlo simulation |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2018 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/157465 |
| citation_txt |
Equilibrium properties of the lattice system with SALR interaction potential on a square lattice: quasi-chemical approximation versus Monte Carlo simulation / Ya.G. Groda, V.S. Vikhrenko, D. di Caprio // Condensed Matter Physics. — 2018. — Т. 21, № 4. — С. 43002: 1–10. — Бібліогр.: 16 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT grodayag equilibriumpropertiesofthelatticesystemwithsalrinteractionpotentialonasquarelatticequasichemicalapproximationversusmontecarlosimulation AT vikhrenkovs equilibriumpropertiesofthelatticesystemwithsalrinteractionpotentialonasquarelatticequasichemicalapproximationversusmontecarlosimulation AT ddicaprio equilibriumpropertiesofthelatticesystemwithsalrinteractionpotentialonasquarelatticequasichemicalapproximationversusmontecarlosimulation |
| first_indexed |
2025-07-14T09:53:35Z |
| last_indexed |
2025-07-14T09:53:35Z |
| _version_ |
1837615615772196864 |
| fulltext |
Condensed Matter Physics, 2018, Vol. 21, No 4, 43002: 1–10
DOI: 10.5488/CMP.21.43002
http://www.icmp.lviv.ua/journal
Equilibrium properties of the lattice system with
SALR interaction potential on a square lattice:
quasi-chemical approximation versus Monte Carlo
simulation
Ya.G. Groda1, V.S. Vikhrenko1, D. di Caprio2
1 Belarusian State Technological University, 13a Sverdlova St., 220006 Minsk, Belarus
2 Chimie ParisTech, PSL Research University, CNRS, Institute of Research of Chimie Paris (IRCP),
11 Pierre and Marie Curie St., 75231 Paris, France
Received November 2, 2018, in final form November 12, 2018
The lattice system with competing interactions that models biological objects (colloids, ensembles of protein
molecules, etc.) is considered. This system is the lattice fluid on a square lattice with attractive interaction
between nearest neighbours and repulsive interaction between next-next-nearest neighbours. The geometric
order parameter is introduced for describing the ordered phases in this system. The critical value of the or-
der parameter is estimated and the phase diagram of the system is constructed. The simple quasi-chemical
approximation (QChA) is proposed for the system under consideration. The data of Monte Carlo simulation of
equilibrium properties of the model are compared with the results of QChA. It is shown that QChA provides
reasonable semiquantitative results for the systems studied and can be used as the basis for next order approx-
imations.
Key words: lattice fluid model, competing interaction, order-disorder phase transition, Monte Carlo
simulation, quasi-chemical approximation, phase diagram
PACS: 05.10.Ln, 64.60.De, 64.60.Cn
1. Introduction
At present, there is a great interest in studying the processes of self-organization and self-assembly
in systems of a nanoscale range. As the elements of such systems are supramolecular formations with a
sufficiently large molecular mass, this leads to low velocities of their thermal motion and to large, on the
molecular scale, characteristic times of the processes within the system. At the same time, the interaction
between these elements is very complex. Despite their rather large dimensions, the interactions remain
of the same order as the thermal energy. This leads to a large variety of possibilities for various phase
transitions in such systems at room temperature. Examples are solutions of protein molecules [1], clays
and soil suspensions [2], ecosystems [3], etc.
In general, structure elements of such systems attract each other at small distances due, for example,
to the Van der Waals attraction, and repulse on longer separations because of the electrostatic interactions
(SALR systems, short-range attractive and long-range repulsive interaction) [4, 5]. In the case of biological
molecules, repulsion can also be caused by elastic deformations of the lipid membranes. In any case, the
attraction between the structural elements of the system ensures the phase separation, and repulsion —
the formation of clusters.
One of the simplest methods for investigating the general properties of SALR systems is to consider
their lattice models. These models are simple enough and allow one to make a detailed analysis both by
This work is licensed under a Creative Commons Attribution 4.0 International License . Further distribution
of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
43002-1
https://doi.org/10.5488/CMP.21.43002
http://www.icmp.lviv.ua/journal
http://creativecommons.org/licenses/by/4.0/
Ya.G. Groda, V.S. Vikhrenko, D. di Caprio
analytical methods and computer simulation, using the Monte Carlo method. A large number of common
properties of such systems can be obtained within these frameworks.
For example, in papers [6, 7], a lattice fluid was studied with attraction of nearest neighbours and
repulsion of the third neighbours on a plane triangular lattice. Possible configurations of the ensemble of
particles were investigated. A phase diagram of the system in the mean field approximation and byMonte
Carlo simulation was constructed. It revealed the existence of several phase transitions in the system.
In [8] the generalized quasi-chemical approximation (QChA) was proposed for lattice systems with
SALR interaction potential on a triangular lattice. This approximation has demonstrated its applicability
for estimating the equilibrium properties of the model in the disordered phase.
In this paper, we present the results for a similar model of the lattice fluid on a square lattice and
propose a geometric order parameter, which makes it possible to investigate the ordered phases in the
system.
2. The model and its order parameter
We consider the lattice fluid consisting of n particles on a square lattice of N lattice sites. Multiple
occupation of sites is forbidden. Particles that occupy nearest lattice sites and sites that are neighbours of
the third order interact with each other. The interaction energies are equal to J1 and J3, respectively. It is
assumed that J1 = −J < 0, and J3 = 3J > 0, which corresponds to the attraction of the nearest neighbours
and the repulsion of the third ones. The second, forth and more distant neighbours are considered as
noninteracting ones.
The simulation of the equilibrium characteristics of the system under consideration in the grand
canonical ensemble using the Monte Carlo method is performed within the framework of the Metropolis
algorithm [9]. For simulation, we used a lattice containing 214 lattice sites with periodic boundary
conditions. The total length of the simulation procedure consisted of 70000 steps of the Monte Carlo
algorithm (MCS). The first 20000 MCSs were used to equilibrate the system and were not taken into
account at subsequent averaging.
A preliminary simulation on the lattice containing 210 lattice sites has shown that two different types
of ordered phases are formed in the system at low temperatures (below the critical temperature Tc). The
both types of ordered phases are shown in figure 1.
On smaller lattices, such as those shown in figure 1, the phases appear sporadically on different
trajectories. On larger lattices (e.g., of 216 lattice sites) the system breaks up into domains with different
types of the ordered structures. The ground state of the model is degenerated with the energy −J per
0 4 8 12 16 20 24 28 32
0
4
8
12
16
20
24
28
32
0 4 8 12 16 20 24 28 32
0
4
8
12
16
20
24
28
32
Figure 1. The final screenshot of the system at µ = 4J and J/kBT = 1.0. The left-hand and right-hand
parts of the figure correspond to different launches of the MC-simulation program.
43002-2
Equilibrium properties of the lattice system with SALR interaction potential
particle for both configurations. Subsequently, with an increase in temperature, both states are realized
with approximately equal probabilities.
To describe the ordered phases, the initial square lattice was divided into a system of 8 identical
sublattices rotated by 45◦ with the spacing 2a
√
2, where a is the lattice spacing of the initial lattice. In the
case of complete ordering of the system at the lattice concentration c = 0.5 and at low temperatures, four
sublattices are completely filled (p-sublattices) and four sublattices are completely vacant (v-sublattices).
This makes it possible to determine the system order parameter δc as the difference between the particle
concentrations on the sublattices
δc =
cp − cv
2
. (2.1)
If the sublattices are numbered, we can consider the order parameter matrix
δci j =
|ci − cj |
2
, (2.2)
where ci(j) is the concentration of particles on the sublattice i( j).
This is a symmetric matrix with the diagonal elements equal to zero and 28 independent non-diagonal
elements equal to one or zero for the ordered structures. It is possible to distinguish the ordered phases
by the order of filled sublattices or by the structure of the matrix; e.g., even numbers (two or four) of 1 or
0 appear in the sequences in rows and columns of the matrix for squares, while odd numbers (three) are
characteristic of stripes. Importantly, for large lattices when both structures are present simultaneously,
the scalar order parameter equation (2.1) distinguishes ordered states from disordered ones.
The order parameter characterizes the strength of the ordering of the ordered state and is equal
to zero in a disordered state. The total lattice concentration c and the sublattices concentrations cp
and cv are connected by the expressions (the subscripts 1 and 0 are related to particles and vacancies,
correspondently)
cp = cp1 = c + δc, cp0 = 1 − cp1 = 1 − c − δc, (2.3)
cv = cv1 = c − δc, cv0 = 1 − cv1 = 1 − c + δc, (2.4)
c = c1 =
n
N
=
cp1 + cv1
2
, c0 = 1 − c. (2.5)
The MC simulation shows (see figure 2) that the order parameter increases sharply at J/kBTc = 0.655 for
the chemical potential µ = 4J which corresponds to the system with the average concentration c = 0.5.
0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
J k
B
T
c
Figure 2. The order parameter versus the inverse temperature at µ = 4J (c = 0.5).
43002-3
Ya.G. Groda, V.S. Vikhrenko, D. di Caprio
Such an increase of the order parameter corresponds to the order-disorder phase transition, which is
of the second order like in the system with the nearest neighbour repulsive interaction [10]. In this case,
the value 0.655 can be interpreted as a critical parameter of the system.
Thus, the value of the order parameter can be used to localize points of the structural phase transitions
and to construct the phase diagram of the model.
3. The quasi-chemical approximation
The free energy of the system can be represented as a sum of the free energy of the reference system
Fr and the diagrammatic part Fd [11, 12]:
F = Fr + Fd. (3.1)
The reference system is characterized by the mean potentials φ βj (n
α
i ) describing the interaction of a
particle (nαi = 1) or vacancy (nαi = 0) on the site i of α-sublattice with site j of β-sublattice. Equation (3.1)
is an identity and the free energy should not be dependent on the choice of the mean potentials. Therefore,
the mean potentials can be found from the minimal susceptibility principle [13]
∂F
∂φαi
= 0. (3.2)
The free energy is a function of the sublattice concentrations. It is useful to represent it as a function
of the lattice concentration c and the order parameter δc. The latter can be determined from the extremity
condition
∂F
∂δc
= 0, (3.3)
which is equivalent to the requirement that the chemical potentials on all the sublattices are equal.
As a first step, one can consider a quasi-chemical approximation when the diagrammatic part of free
energy contains the two-vertex graph contribution only
F =
kBT
2
v∑
α=p
1∑
i=0
cαi
[
ln cαi −
∑
k
zk ln Xα(k)
i
]
−
kBT
2
∑
k
zk
2
v∑
α,β=p
1∑
i, j=0
cα(k)i c β(k)j
W (k)i j
Xα(k)
i Xβ(k)
j
− 1
,
(3.4)
where
Xα(k)
i = exp
[
−βφk(nαi )
]
, (3.5)
zk is the coordination number for neighbours of k-order (z1 = z3 = 4 for a square lattice).
In this approximation, the diagrammatic part of free energy is equal to zero and the mean potentials
for nearest-neighbours do not depend on the sublattice structure. The free energy in the QChA reads
F(QChA)(c, δc) =
kBT
2
∑
i
ci
[
ln(c2
i − δc2) − 2z1 ln Xi
]
−
kBT
2
z3
(
ln Zp
0 Z v0 + c ln ξvξp
)
+
kBT
2
δc
(∑
k
ln
ci + δc
ci − δc
− z3 ln
ξp
ξv
)
, (3.6)
where
W = exp
(
−
J1
kBT
)
; Ω = exp
(
−
J3
kBT
)
; (3.7)
43002-4
Equilibrium properties of the lattice system with SALR interaction potential
η = −
c1 − c0
2c0
+
√(
c1 − c0
2c0
)2
+
c1
c0
W ; (3.8)
X0 =
√
c0 +
c1
η
, X1 = ηX0; (3.9)
ξp(v) = −
c1 − c0 ± 2Ωδc
2(c0 ∓ δc)
+
√[
c1 − c0 ± 2Ωδc
2(c0 ∓ δc)
]2
+
c1 ± δc
c0 ∓ δc
Ω; (3.10)
Z v0 Zp
0 = cv0 +
cv1
ξv
= cp0 +
cp1
ξp
. (3.11)
All the thermodynamic characteristics can be investigated with equation (3.6) for the free energy.
Thus, the chemical potential µ, the thermodynamic factor χT and the correlation function gk(1; 1) for two
nearest- and next-next-nearest neighbours (at k = 1 and 3, respectively) are determined by the expressions
βµ =
(
∂(βF)
∂c
)
T
, (3.12)
χT =
∂(βµ)
∂ ln c
, (3.13)
gk(1; 1) =
2
zkc2
(
∂F
∂Jk
)
T
. (3.14)
4. Calculation and simulation results
The most important structural feature of an ordered state is the order parameter. The comparisons of
the calculation and simulation results for the order parameter are shown in figure 3.
The order parameter is used to determine the phase transition curve. The corresponding phase diagram
is represented in figure 4. One can note that QChA leads to a wider area of the ordered phase in the system
as compared to the Monte Carlo simulation results. In addition, the critical temperature is overestimated
by approximately 30% in this approximation. These deviations are known to be typical of QChA [14, 15].
It should be noted that the phase diagram of the system with the competing interaction on a square
lattice is much simpler than that of the system on a triangular lattice [7] which is probably the consequence
of a close-packed structure of the latter.
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
c
c
1
2
Figure 3. The order parameter versus concentration at βJ = 0.8. The solid line 1 represents the QChA
results, the full circles 2 are the MC simulation data.
43002-5
Ya.G. Groda, V.S. Vikhrenko, D. di Caprio
0.0 0.2 0.4 0.6 0.8 1.0
0.7
0.8
0.9
1.0
O
rd
er
ed
p
ha
se
Disordered
phase
c
T/T
c
Disordered
phase
2
1
Figure 4. The order-disorder phase transition curves. The solid line 1 represents the QChA results, the
full circles 2 are the MC simulation data.
0.0 0.2 0.4 0.6 0.8 1.0
-30
-20
-10
0
10
20
30
40
50
6
5
4
3
J
c
1
2
Figure 5. (Colour online) The chemical potential (in units of the nearest neighbour interaction energy J)
versus concentration at βJ = 0.3 (1); 0.5 (2); 0.6 (3); 0.7 (4); 0.8 (5) and 1.0 (6). The solid lines represent
the QChA results, the full circles are the MC simulation data. Each group of curves is shifted down by
10 units along the µ axis with respect to the previous one for better visibility. The unshifted curve (3) is
characterized by µ/J = 4 at c = 0.5, and this point is the same for all the temperatures. Thus, the groups
of curves (1) and (2) are shifted up from their true position, while (4), (5) and (6) are shifted down. The
same curves numbers are kept in figures 6–8.
The chemical potential isotherms are shown in figure 5. The ordered phase exists at temperatures
below critical (βJ = 0.6, 0.7, 0.8 and 1.0) where a steep increase of the chemical potential with an
increase of the particle concentration is observed. The concentration derivative of the chemical potential
or the thermodynamic factor (3.13) indicates the second-order phase transition by discontinuities at the
concentrations that correspond to the phase transition points. The strong peaks at c = 0.5 correspond to
the most ordered states of the system at corresponding temperatures (see figure 6).
The thermodynamic factor is inverse to the concentration fluctuations. The latter grow immediately
after the transition from a disordered to an ordered state and then decrease systematically until concen-
tration 0.5 is reached when the most ordered state is possible. Physically, this means that concentration
fluctuations are suppressed in the ordered states.
43002-6
Equilibrium properties of the lattice system with SALR interaction potential
0.0 0.2 0.4 0.6 0.8 1.0
0
10
20
30
40
50
60
0.3 0.4 0.5 0.6 0.7
0
2
4
6
8
10
4
1
T
1
2
4
5
c
Figure 6. (Colour online) The thermodynamic factor versus concentration at βJ = 0.3 (1, square); 0.5
(2, circle); 0.7 (4, down triangle) and 0.8 (5, diamond). The solid lines represent the QChA results, the
symbols are the MC simulation data.
In QChA, the chemical potential and the thermodynamic factor were calculated by numerical differ-
entiating of the free energy expression (3.6). However, such a differentiation of the chemical potential
extracted fromMC simulations may not be used. In this case, the thermodynamic factor can be calculated
as the value inversely proportional to the mean square concentration fluctuations
χT =
〈n〉
〈(n − 〈n〉)2〉
. (4.1)
The parameter χT plays an important role in describing the diffusion process in lattice fluids [16].
The calculation and simulation results for the chemical potential and thermodynamic factor are in a
good agreement except for those at close vicinity of the second order phase transition curve where the
QChA curves show abrupt jumps.
0.0 0.2 0.4 0.6 0.8 1.0
1.0
1.5
2.0
2.5
g 1(1
;1
)
1
3
5
6
c
Figure 7. (Colour online) The correlation functions for nearest neighbours versus concentration at βJ =
0.3 (1, square); 0.6 (3, up triangle); 0.8 (5, diamond) and 1.0 (6, left-hand triangle). The solid lines
represent the QChA results, the symbols are the MC simulation data.
43002-7
Ya.G. Groda, V.S. Vikhrenko, D. di Caprio
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.5
1.0
g 3
(1
;1
)
1
3
5
6
c
Figure 8. (Colour online) The correlation functions for the next-next-nearest neighbours versus concen-
tration at βJ = 0.3 (1, square); 0.6 (3, up triangle); 0.8 (5, diamond) and 1.0 (6, left-hand triangle). The
solid lines represent the QChA results, the symbols are the MC simulation data.
A short range ordering is characterized by the correlation functions (3.14). These functions are more
informative objects than the distribution functions because they represent the deviation of the short range
correlations in interacting systems from the case of noninteracting (Langmuir) lattice gases where they
are equal to one.
The correlation functions for nearest and next-next-nearest neighbours are show in figures 7 and 8,
respectively.
The long range ordering manifests itself in a short range structure as well. At temperatures T < Tc
(see curves 3, 5 and 6) the probability to find two next-next-nearest neighbour sites occupied by particles
becomes very low. Of course, the higher is the temperature, the less pronounced difference from the
Langmuir gas behaviour is observed.
The simulation and analytical results satisfactorily match each other in the disordered phase. They
significantly differ at intermediate concentrations and at low temperatures due to the problems concerning
the determination of the order parameter and the critical temperature in QChA. The most significant
differences appear for the correlations on the nearest neighbour lattice sites.
5. Conclusion
The lattice system with an attractive interaction between nearest neighbours and repulsive interaction
between next-next-nearest neighbours has been studied.
It is shown that the competing interactions lead to the order-disorder phase transitions. The order
parameter δc is used as the indicator of the second order phase transitions. With this parameter, it was
established that the critical value of the interaction parameter is equal to |J1 |/kBTc = 0.655 ± 0.005, and
the phase diagram of the system was constructed.
The quasi-chemical approximation is found to be self-consistently constructed through the mean
potentials that describe the interaction of a particle or a vacancy with its nearest and next-next-nearest
neighbour lattice sites. The chemical potential, thermodynamic factor and correlation functions are de-
termined both in the QChA and in the Monte Carlo simulations. The chemical potential demonstrates
irregular behaviour in the phase transition region, while the thermodynamic factor indicates strong sup-
pressing of fluctuations that are inherent to the ordered states of the system. The complicated behaviour of
the correlation functions that reflects structural peculiarities of the system demonstrate a great importance
of competing interactions.
The order parameter of the system δc is determined in the QChA with significant errors. This leads
43002-8
Equilibrium properties of the lattice system with SALR interaction potential
to errors in determining the critical temperature of the system, which is overestimated by approximately
30%. As a result, the quasichemical approximation fails to reproduce the structural characteristics of the
system with competing interactions. At the same time, the thermodynamic characteristics such as the
chemical potential and the thermodynamic factor are determined in the quasi-chemical approximation
with a sufficiently high accuracy.
Thus, the developed approach allows us to correctly describe the qualitative features of the structural
properties of the systems with competing interactions, and can be used to quantify the thermodynamic
characteristics of these systems.
Acknowledgements
The project has received funding from the European Union’s Horizon 2020 research and innovation
programme under theMarie Skłodowska-Curie grant agreementNo 73427, Institute for Nuclear Problems
of Belarusian State University (agreement No 209/103) and the Ministry of Education of Belarus.
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Ya.G. Groda, V.S. Vikhrenko, D. di Caprio
Рiвноважнi властивостi ґраткової системи з потенцiалом
взаємодiї типу SALR на квадратнiй ґратцi: квазiхiмiчне
наближення у порiвняннi з симуляцiями Монте Карло
Я.Г. Грода1, В.С. Вiхренко1, Д. дi Капрiо2
1 Бiлоруський державний технологiчний унiверситет, вул. Свєрдлова, 13a, 220006Мiнськ, Бiлорусь
2 Дослiдницький унiверситет науки та лiтератури Парижу, ChimieParisTech— CNRS, Iнститут хiмiчних
дослiджень Парижу, Париж, Францiя
Розглядається ґраткова система з конкурентними взаємодiями, що моделює бiологiчнi об’єкти (колоїди,
ансамблi протеїнових молекул i т.д.). Ця система представляється ґратковим плином на квадратнiй ґратцi
з притягальною взаємодiєю мiж найближчими сусiдами i вiдштовхувальною взаємодiєю мiж наступними
за наступними до найближчих сусiдiв. Для опису впорядкованої фази в такiй системi вводиться геоме-
тричний параметр порядку. Зроблена оцiнка критичного значення параметра порядку i побудована фа-
зова дiаграма системи. Для системи, що розглядається, запропоновано просте квазiхiмiчне наближення
(КХН). Данi симуляцiй Монте Карло рiвноважних властивостей моделi порiвнюються з результатами КХН.
Показано, що КХН забезпечує прийнятнi напiвкiлькiснi результати для системи, що вивчається, i може
використовуватися як базис для наступних наближень.
Ключовi слова: ґраткова модель плину, конкуруюча взаємодiя, фазовий перехiд порядок-безлад,
симуляцiї Монте Карло, квазiхiмiчне наближення, фазова дiаграма
43002-10
Introduction
The model and its order parameter
The quasi-chemical approximation
Calculation and simulation results
Conclusion
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