Mean spherical approximation for the Lennard-Jones-like two Yukawa model: Comparison against Monte Carlo data
Monte Carlo simulation studies are performed for the Lennard-Jones like two Yukawa (LJ2Y) potential to show how properties of this model fluid depend on the replacement of the soft repulsion by the hard-core repulsion. Different distances for the positioning of hard core have been explored. We have...
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nasplib_isofts_kiev_ua-123456789-1200202025-02-10T01:38:55Z Mean spherical approximation for the Lennard-Jones-like two Yukawa model: Comparison against Monte Carlo data Середньосферичне наближення для Леннард-Джонс-подiбного плину, змодельованого сумою двох потенцiалiв Юкави: порiвняння з результатами методу Монте Карло Krejčí, J. Nezbeda, I. Melnyk, R. Trokhymchuk, A. Monte Carlo simulation studies are performed for the Lennard-Jones like two Yukawa (LJ2Y) potential to show how properties of this model fluid depend on the replacement of the soft repulsion by the hard-core repulsion. Different distances for the positioning of hard core have been explored. We have found, that for temperatures that are slightly lower and slightly higher of the critical point temperature for the Lennard-Jones fluid, placing the hard core at distances that are shorter than zero-potential energy is well justified by thermodynamic properties that are practically the same as in original LJ2Y model without hard core. However, going to extreme conditions with the high temperature one should be careful since presence of the hard core provokes changes in the properties of the system. The later is extremely important when the mean spherical approximation (MSA) theory is applied to treat the Lennard-Jones-like fluid. Методом Монте Карло проведено дослiдження Леннард-Джонс-подiбного плину, взаємодiя мiж частинками якого задається сумою двох потенцiалiв Юкави (ЛД2Ю). Метою дослiджень є продемонструвати вплив, якому пiддаються властивостi моделi при замiнi “м’якого” вiдштовхування “твердим” кором. Розглянуто рiзнi вiдстанi для розмiщення твердого кору. Виявлено, що при температурах, трохи нижчих i трохи вищих за критичну температуру леннард-джонсiвського плину, перемiщення твердого кору на вiдстанi, коротшi за вiдстань з нульовою потенцiальною енергiєю, практично не змiнює термодинамiчних властивостей моделi, значення яких є дуже близькими до термодинамiчних характеристик ЛД2Ю моделi без твердого кору. Однак, при переходi в зону екстремально високих температур слiд бути обережним, оскiльки присутнiсть твердого кору провокує суттєвi змiни властивостей системи. Це застереження є дуже важливим при застосуваннi методу середньосферичного наближення до опису Леннард-Джонс-подiбного плину. This work was supported by the Grant Agency of the Academy of Sciences of the Czech Republic (Grant No. IAA400720710) and the Czech-Ukrainian Bilateral Cooperative Program. 2011 Article Mean spherical approximation for the Lennard-Jones-like two Yukawa model: Comparison against Monte Carlo data/ J. Krejčí, I. Nezbeda, R. Melnyk, A. Trokhymchuk // Condensed Matter Physics. — 2011. — Т. 14, № 3. — С. 33005: 1-11. — Бібліогр.: 13 назв. — англ. 1607-324X PACS: 01.65Q DOI:10.5488/CMP.14.33005 arXiv:1202.4262 https://nasplib.isofts.kiev.ua/handle/123456789/120020 en Condensed Matter Physics application/pdf Інститут фізики конденсованих систем НАН України |
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Monte Carlo simulation studies are performed for the Lennard-Jones like two Yukawa (LJ2Y) potential to show how properties of this model fluid depend on the replacement of the soft repulsion by the hard-core repulsion. Different distances for the positioning of hard core have been explored. We have found, that for temperatures that are slightly lower and slightly higher of the critical point temperature for the Lennard-Jones fluid, placing the hard core at distances that are shorter than zero-potential energy is well justified by thermodynamic properties that are practically the same as in original LJ2Y model without hard core. However, going to extreme conditions with the high temperature one should be careful since presence of the hard core provokes changes in the properties of the system. The later is extremely important when the mean spherical approximation (MSA) theory is applied to treat the Lennard-Jones-like fluid. |
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Article |
| author |
Krejčí, J. Nezbeda, I. Melnyk, R. Trokhymchuk, A. |
| spellingShingle |
Krejčí, J. Nezbeda, I. Melnyk, R. Trokhymchuk, A. Mean spherical approximation for the Lennard-Jones-like two Yukawa model: Comparison against Monte Carlo data Condensed Matter Physics |
| author_facet |
Krejčí, J. Nezbeda, I. Melnyk, R. Trokhymchuk, A. |
| author_sort |
Krejčí, J. |
| title |
Mean spherical approximation for the Lennard-Jones-like two Yukawa model: Comparison against Monte Carlo data |
| title_short |
Mean spherical approximation for the Lennard-Jones-like two Yukawa model: Comparison against Monte Carlo data |
| title_full |
Mean spherical approximation for the Lennard-Jones-like two Yukawa model: Comparison against Monte Carlo data |
| title_fullStr |
Mean spherical approximation for the Lennard-Jones-like two Yukawa model: Comparison against Monte Carlo data |
| title_full_unstemmed |
Mean spherical approximation for the Lennard-Jones-like two Yukawa model: Comparison against Monte Carlo data |
| title_sort |
mean spherical approximation for the lennard-jones-like two yukawa model: comparison against monte carlo data |
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Інститут фізики конденсованих систем НАН України |
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2011 |
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https://nasplib.isofts.kiev.ua/handle/123456789/120020 |
| citation_txt |
Mean spherical approximation for the Lennard-Jones-like two Yukawa model: Comparison against Monte Carlo data/ J. Krejčí, I. Nezbeda, R. Melnyk, A. Trokhymchuk // Condensed Matter Physics. — 2011. — Т. 14, № 3. — С. 33005: 1-11. — Бібліогр.: 13 назв. — англ. |
| series |
Condensed Matter Physics |
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Condensed Matter Physics, 2011, Vol. 14, No 3, 33005: 1–11
DOI: 10.5488/CMP.14.33005
http://www.icmp.lviv.ua/journal
Mean-spherical approximation for the
Lennard-Jones-like two Yukawa model: Comparison
against Monte Carlo data
J. Krejčí1, I. Nezbeda1,2, R. Melnyk3, A. Trokhymchuk3,4
1 Faculty of Science, J. E. Purkinje University, 400 96 Ústí n. Lab., Czech Republic
2 E. Hála Laboratory of Thermodynamics, ICPF, Acad. Sci., 165 02 Prague 6 – Suchdol, Czech Republic
3 Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine,
1 Svientsitskii Str., 79011 Lviv, Ukraine
4 Department of Chemistry and Biochemistry, Brigham Young University, Provo, UT 84602, USA
Received August 1, 2011
Monte Carlo simulation studies are performed for the Lennard-Jones-like two Yukawa (LJ2Y) potential to show
how properties of this model fluid depend on the replacement of soft repulsion by hard-core repulsion. Different
distances for the positioning of hard-core have been explored. We have found that for temperatures slightly
lower and slightly higher than the critical point temperature for the Lennard-Jones fluid, the placement of the
hard-core at distances shorter than zero-potential energy is well justified by thermodynamic properties that are
practically the same as in the original LJ2Y model without hard-core. However, going to extreme conditions
with the high temperature one should be careful since the presence of the hard-core provokes changes in
the properties of the system. The later is extremely important when the mean-spherical approximation (MSA)
theory is applied to the treatment of the Lennard-Jones-like fluid.
Key words: two Yukawa potential, Lennard-Jones fluid, mean-spherical approximation, Monte Carlo
simulations
PACS: 01.65.Q
1. Introduction
The so-called simple fluid models cannot be always applied to the study of realistic systems.
However, this type of models is very important in basic science while studying fundamental prob-
lems in the liquid state theory. One of these problems refers to the role played by repulsive and
attractive forces. While comprehension of the repulsive forces is due to the progress reached in the
theoretical and computer modeling of hard sphere fluids, the simplest nontrivial model that makes
it possible to study the liquid phase of a matter as well as the vapor-liquid coexistence seems to
be the one that consists of a hard-core plus attractive Yukawa (HCAY) potential
u1Y(r)/ǫ = ∞ for r < R
= −
R
r
exp[κ(r −R)] for r > R, (1)
where ǫ is the depth of the potential energy well, R is a hard-core diameter, κ−1 is a measure of
the range of the attractive tail.
An important feature due to which the HCAY model is intensively employed in the studies
of simple fluids is that approximate analytical and semianalytical solutions for this model are
available owing to the pioneering paper by Waisman [1] on the analytic solution of the Ornstein-
Zernike equation using the mean-spherical approximation (MSA). These were Henderson et al. [2]
who by using this MSA solution suggested that having R = σ and κR = 1.8, the HCAY fluid
is qualitatively similar to argon for the densities and temperatures of the liquid in equilibrium
with its vapour and with the potential parameters approximately the same as the Lennard-Jones
c© J. Krejčí, I. Nezbeda, R. Melnyk, A. Trokhymchuk, 2011 33005-1
http://dx.doi.org/10.5488/CMP.14.33005
http://www.icmp.lviv.ua/journal
I. Nezbeda et al.
potential (i.e. ǫ/k 120 K,σ ≈ 3.4 Å). Figure 1 (a) shows this Lennard-Jones-like HCAY fluid in
comparison with the original Lennard-Jones (LJ) fluid.
1,0 1,5 2,0 2,5
-1,0
-0,6
-0,2
0,2
0,6
1,0
1,4
R = σ
distance, r/σ
pa
ir
in
te
ra
ct
io
n,
u
/ε
1,0 1,5 2,0
0
5
10
15
20
R = 0.9σ
R = σ
distance, r/σ
pa
ir
in
te
ra
ct
io
n,
u
/ε
1 2 3 4 5
-0,10
-0,05
0,00
0,05
0,10
Figure 1. The Lennard-Jones potential in comparison with the hard-core attractive Yukawa
(HCAY) potential (left part) and with the Lennard-Jones-like two Yukawa (LJ2Y) potential
with different positions of the hard-core (right part).
Recently, Kadiri et al. [3] have extended the mapping of the LJ system into the HCAY system
over a wide domain of a phase diagram. Specifically, at every density and temperature the ther-
modynamics of the LJ fluid was reproduced from the equation of state of the HCAY fluid with
suitable values of its two parameters R and κ. An advantage of such an approach is that it provides
analytical equations for the thermodynamics of the LJ system with only two parameters that are
of precision comparable to that of the simulation data. It can be compared to the fully empirical
equations of state by Johnson et al. [4] that uses 33 parameters. However, one can see from figure 1
that the shapes of two potential profiles, LJ and 1Y, are rather different and one cannot be sure
that the thermodynamics predicted by two models will be always the same. Thus, this approach
should be used with great care while exploring the thermodynamic states outside of the mapping
domain.
More possibilities are offered by the MSA solution that has been obtained by Blum and Hoye [5]
for a linear superposition of the Yukawa tails
∑
ǫi exp(−κir)/r. This was quite an important
step since superposition of the attractive and repulsive Yukawa tails makes it possible to mimic
practically any potential profile. It is natural that this MSA solution has been applied to Lennard-
Jones fluid represented by the hard core and sum of two Yukawas (HC2Y) [6, 7],
u2Y(r)/ǫ = ∞ for r < R
=
R
r
exp[−κ1(r −R)]−
R
r
exp[−κ2(r −R)] for r > R, (2)
where again R = σ determines the separation of zero energy, u(σ) = 0, and ǫ is, as usual, the
depth of the potential minimum. The detailed course of the 2Y potential (with a specific set of
the decay parameters κi which will be discussed in the following section 2) is shown in figure 1 (b)
where it is also compared with the Lennard-Jones (LJ) potential. Indeed, two Yukawa terms allow
us to significantly improve the mimicking of LJ potential at short and intermediate distances. This
is a rather important achievement since in this case the differences in the properties of these two
model systems can be attributed to the long-ranged asymptote of the LJ interaction only. Another
important issue is that in general, the LJ2Y potential (like the parent LJ potential) does not
require the presence of hard-core, e.g., for computer simulation studies; the hard-core is necessary
33005-2
MSA for the Lennard-Jones-like two Yukawa model
if one intends to apply the MSA theory to the treatment of the problem. However, even in this
case placing the hard-core at R = σ (as it usually is assumed) seems not to be always appropriate
and justified since it may modify the properties of the original model. To clarify this issue, we will
report here the Monte Carlo simulation data for the original LJ2Y potential without hard-core
as well as for the LJ2Y potential with a few different choices of the hard-core diameter R in the
range 0.8 < R/σ < 1. One more question that we are seeking to answer in this study concerns the
performance of MSA theory for different choices of the hard-core diameters 0.8 < R/σ < 1. The
MSA studies reported so far for the LJ2Y model are dealing mainly with the case R = σ.
The structure of this paper is as follows. In section 2 we outline the way the parameters of
the LJ2Y potential have been chosen and present a brief description of the Monte Carlo computer
simulations and MSA computations that we have carried out for the LJ2Y model. The results are
collected and discussed in section 3 while section 4 contains conclusions.
2. The potential models and computational details
Originally, the LJ2Y potential is a model made up of two Yukawa tails without any hard-core,
u2Y (r)/ǫ = ǫ1
rm
r
exp[−κ1r]− ǫ2
rm
r
exp[−κ2r] , (3)
where rm is the position of the potential energy well, ǫ1 > 0 and ǫ2 > 0 are the strengths of the
repulsive and attractive contributions, respectively, while κ−1
1
and κ−1
2
are the measures of the
range of the corresponding tails. Superposition of two Yukawa tails in the form of equation (3)
to mimic the LJ potential has been already used by other authors [8–10]. However, all previous
studies have been concerned with the medium and long distances without paying much attention
to short separations governed by repulsive forces.
To map the properties of the LJ fluid onto those of a LJ2Y fluid, the above four parameters
of potential function (3) must be determined. Focussing first on the course of the LJ function
itself, the following three conditions seem evident: the coincidence of the location and depth of the
potential minimum of the LJ and LJ2Y functions,
u2Y(r = rmin) = uLJ(r = rmin) = −ǫ , (4)
du2Y(r)
dr
|r=rmin
=
duLJ(r)
dr
|r=rmin
= 0 , (5)
and the location of the potential zero,
u2Y(r = σ) = uLJ(r = σ) = 0 . (6)
Then there remains to impose one more condition on the 2Y potential parameters to complete
the set of equations. Some authors attempt to follow the (12–6) LJ curve to certain intermediate
distance r∗ by equalizing the integrals from two potential functions in the interval from rmin to r∗.
By contrast, in this study we will set the condition that both potentials attain the same value at
the location of the inflection point r = rinf = 1.244455σ that follows immediately after the position
of the potential well of the LJ potential, i.e.,
u2Y(r = rinf) = uLJ(r = rinf) . (7)
Proceeding in this way we obtain a 2Y potential that very accurately reproduces the LJ potential
profile at distances up to inflection point, but showing some discrepancies after inflection point
due to an exponential decay at large distances. The resulting parameters are as follows: ǫ1 =
1954325.046σ, ǫ2 = 50.26984765σ, κ1σ = 13.66462 and κ2σ = 3.10147. These values are consistent
with those in equation (2) and in figure 1 (b).
Figure 2 shows the vapour-liquid phase diagram for the LJ2Y potential model. To determine
the vapour-liquid envelope we used the common Gibbs ensemble with the total number of particles
N = 512 and applied the long-range correction to truncate the potential at rc =
3
√
N/ρ∗. For the
33005-3
I. Nezbeda et al.
0,0 0,2 0,4 0,6 0,8 1,0
0,8
1,0
1,2
1,4
two Yukawas
Lennard-Jones
density, ρ*
te
m
pe
ra
tu
re
, T
*
Figure 2. The vapour-liquid coexistence envelope for the Lennard-Jones fluid [11] and for the
Lennard-Jones-like two Yukawa (LJ2Y) fluid.
purpose of comparison we are presenting computer simulation data for the liquid-vapor coexistence
in a parent LJ fluid. The differences that are observed in figure 2 concern the lowering of the critical
point temperature and are caused by the differences between two potential models, LJ and LJ2Y,
at distances r larger than the position of inflection point rinf . These differences are consistent with
what has been already learned from the computer simulation studies of the HCAY model, namely,
the shortening of the range of attraction brings about a decrease of the critical point temperature.
The corresponding MC data are collected in table 1.
Table 1. Monte Carlo data for vapour-liquid coexistence in the Lennard-Jones-like 2Y (LJ2Y)
fluid.
T ∗ ρ∗l ρ∗v
0.900 0.0278±0.0084 0.7031±0.0133
1.000 0.0579±0.0145 0.6385±0.0157
1.100 0.1204±0.0278 0.5560±0.0169
1.120 0.1324±0.0364 0.5259±0.0150
1.125 0.1442±0.0373 0.5138±0.0169
1.130 0.1448±0.0364 0.5074±0.0247
1.135 0.1459±0.0493 0.4949±0.0267
1.140 0.1568±0.0609 0.4792±0.0228
To discuss the role played by repulsive forces of the hard-core origin we employed both Monte
Carlo (MC) computer simulations and MSA approaches. First, we performed MC computer sim-
ulations for the original LJ2Y model given by equation (3) for a set temperature and density
conditions that are representative for LJ fluid model. Then similar MC computer simulations at
the same thermodynamic states have been repeated for the LJ2Y model with an artificially em-
bedded hard-core at different interparticle separations smaller than LJ diameter σ.
In all these studies, the standard Monte Carlo simulations for the LJ2Y model fluids were
carried out in an NVT ensemble with N = 512 particles using the largest possible cutoff for each
isotherm and with appropriate long range corrections applied [12]. In addition to the common
thermodynamic properties, the internal energy and pressure (evaluated by the virtual volume
change), we also determined, for all thermodynamic state conditions considered, the excess chemical
33005-4
MSA for the Lennard-Jones-like two Yukawa model
potential using the standard Widom’s particle insertion method [12]. Our MSA study is based on
the general solution obtained by Blum and Hoye [5] for a mixture of hard spheres interaction via
pair potential that is the sum of Yukawa tails with different decay parameters. The numerical
solution of the resulting set of nonlinear algebraic equation was obtained using a relatively simple
and effective iterative procedure proposed recently by Kalyuzhnyi and Cummings [13].
Firstly, we are expecting that the placement of a hard-core at different distances may modify
the properties set up for the initial LJ2Y potential function given by equation (3). This issue can
be resolved by analyzing the MC simulation data. Secondly, the MSA performance could depend
on the position of the hard-core, and this can be concluded by comparing with computer simulation
data.
3. Results and discussions
The studies that we are reporting here have been performed for three reduced temperatures,
T ∗ = 1.25, 1.45 and 4.85. These three temperatures reflect three important temperature conditions
of the parent LJ fluid, namely, subcritical, supercritical and extreme. This can be illustrated by
Table 2. Monte Carlo data for thermodynamics of the Lennard-Jones-like 2Y fluid (LJ2Y) with-
out any hard-core.
σHS T ∗ ρ∗ U∗ P ∗ Z
0.00 0.81 0.8645 –5.5392 ± 0.0012 1.5908 ± 0.0009 2.2718 ± 0.0009
0.00 1.25 0.70 –4.3084 ± 0.0014 0.9776 ± 0.0004 1.1638 ± 0.0004
0.00 1.25 0.80 –4.8443 ± 0.0015 2.3691 ± 0.0007 2.4678 ± 0.0007
0.00 1.25 0.85 –5.0608 ± 0.0017 3.5213 ± 0.0008 3.4523 ± 0.0008
0.00 1.25 0.90 –5.2355 ± 0.0018 5.0848 ± 0.0012 4.7081 ± 0.0012
0.00 1.45 0.30 –1.8953 ± 0.0013 0.2560 ± 0.0001 0.5885 ± 0.0001
0.00 1.45 0.40 –2.4559 ± 0.0013 0.3389 ± 0.0001 0.5842 ± 0.0001
0.00 1.45 0.50 –3.0183 ± 0.0013 0.5048 ± 0.0002 0.6963 ± 0.0002
0.00 1.45 0.70 –4.1552 ± 0.0015 1.7492 ± 0.0005 1.7233 ± 0.0005
0.00 1.45 0.95 –5.0474 ± 0.0023 8.8882 ± 0.0016 6.4524 ± 0.0016
0.00 4.85 0.20 –0.8883 ± 0.0018 1.1520 ± 0.0001 1.1876 ± 0.0001
0.00 4.85 0.40 –1.7065 ± 0.0029 3.0689 ± 0.0003 1.5819 ± 0.0003
0.00 4.85 0.60 –2.3346 ± 0.0038 6.8935 ± 0.0006 2.3689 ± 0.0006
0.00 4.85 0.80 –2.4590 ± 0.0051 14.9992 ± 0.0012 3.8658 ± 0.0012
0.00 4.85 1.00 –1.5713 ± 0.0062 31.7007 ± 0.0025 6.5362 ± 0.0025
the phase diagram shown in figure 2 with a reminder that critical point temperature of the LJ fluid
is around T ∗
c,LJ ≈ 1.35 [11]. The distances that have been explored for the position of hard-core
diameter are: R = σ, 0.95σ, 0.9σ, 0.85σ and 0.8σ. The values of the internal energy U∗, pressure
P ∗ and compressibility factor Z = P/ρkT for each hard-core position and at three temperature
conditions are collected in tables from 3 to 7. Additionally, the density dependencies of the pressure
P ∗ and internal energy U∗ are illustrated in figures 3 and 4, respectively. Typical radial distribution
functions are presented in figures 5 and 6. Tables and figures show both computer simulation data
and the results obtained from the MSA theory. For the sake of comparison between different parts,
figures 3 and 4 are using the same scale which is not always convenient for distinguishing the
data within the same part of the figure. For these purposes we recommend to look for the data in
tables 2–7.
For the lowest temperature, T ∗ = 1.25, which is slightly below the critical temperature for the
LJ fluid one can see (upper parts in figures 3 and 4) that MC simulation data indicate very weak
dependence of the thermodynamic data on the replacement of soft repulsion by stiff hard-core
repulsion in the entire range of distances r 6 σ. This is more evident for pressure, when only the
limiting case R = σ (filled triangles in figures from 3 to 6) indicates the tendency to be separated
from the rest of data; in the case of internal energy, similar tendency is seen for the case R = 0.95σ.
33005-5
I. Nezbeda et al.
Table 3. Monte Carlo data and MSA results (in parenthesis) for thermodynamics of the Lennard-
Jones-like 2Y fluid (LJ2Y) with a hard-core located at R = 0.8σ.
T ∗ ρ∗ U∗ P ∗ Z
1.25 0.70 –4.2994 ± 0.0419 (–16.3395) 0.9725 ± 0.0031 (3.2445) 1.1578 ± 0.0031 (3.7080)
1.25 0.80 –4.7962 ± 0.0239 (–15.9032) 2.3756 ± 0.0333 (6.6936) 2.4746 ± 0.0333 (6.6936)
1.25 0.85 –5.0644 ± 0.0280 (–15.4926) 3.4930 ± 0.0042 (8.9445) 3.4245 ± 0.0042 (8.4181)
1.25 0.90 –5.2403 ± 0.0051 (–14.9595) 5.0984 ± 0.0180 (11.5884) 4.7207 ± 0.0180 (10.3008)
1.45 0.30 –1.9247 ± 0.0328 (–26.8528) 0.2564 ± 0.0004 (–2.0151) 0.5895 ± 0.0004 (–4.6421)
1.45 0.40 –2.4288 ± 0.0316 (–31.5121) 0.3387 ± 0.0003 (–2.2172) 0.5839 ± 0.0003 (–3.8314)
1.45 0.50 –3.0109 ± 0.0123 (–34.5365) 0.5109 ± 0.0044 (–1.3935) 0.7046 ± 0.0044 (–1.9267)
1.45 0.70 –4.1595 ± 0.0107 (–36.5587) 1.7429 ± 0.0049 (–4.9188) 1.7171 ± 0.0049 (–4.8570)
1.45 0.95 –5.0264 ± 0.0324 (–32.8354) 8.8919 ± 0.0253 (–26.0895) 6.4551 ± 0.0253 (–18.9368)
4.85 0.20 –0.8916 ± 0.0176 (–5.8499) 1.1504 ± 0.0012 (0.8452) 1.1860 ± 0.0012 (0.8729)
4.85 0.40 –1.8134 ± 0.0401 (–9.4717) 3.0687 ± 0.0006 (2.7345) 1.5818 ± 0.0006 (1.4126)
4.85 0.60 –2.2961 ± 0.0130 (–10.6515) 6.8818 ± 0.0021 (8.0328) 2.3649 ± 0.0021 (2.7667)
4.85 0.80 –2.6086 ± 0.0897 (–9.3413) 15.0001 ± 0.0335 (19.6659) 3.8660 ± 0.0335 (5.0803)
4.85 1.00 –1.6613 ± 0.1132 (–5.4406) 31.7864 ± 0.0342 (41.6984) 6.5539 ± 0.0342 (8.5976)
Quite similar dependencies on the hard-core position are shown by MC computer simulation data
for the higher temperature, T ∗ = 1.45 (see the middle parts in figures 3 and 4). Indeed, we can
see that the case R = σ is rather special for these temperature conditions. Interestingly, this is
also true for the MSA theory (thick solid lines in figures from 3 to 6). We also note that being
compared with MC data for the case of a hard-core placed at zero-potential energy separation
distance R = σ, the MSA theory performs quite well for the thermodynamics of the system.
The MSA curves calculated with the hard-core placed at shorter distances, R = 0.95σ, for both
pressure and energy (dashed lines in figures 3 and 4) are clearly separated from the MSA results
obtained with the hard-core at R = σ and are very close to the MC computer simulation data.
Table 4. The same as in table 3 but with a hard-core located at R = 0.85σ.
T ∗ ρ∗ U∗ P ∗ Z
1.25 0.70 –4.3081 ± 0.0226 (–7.9827) 0.9914 ± 0.0081 (2.4220) 1.1802 ± 0.0081 (2.7680)
1.25 0.80 –4.7836 ± 0.0109 (–7.8873) 2.3767 ± 0.0104 (4.8594) 2.4757 ± 0.0104 (4.8594)
1.25 0.85 –5.1362 ± 0.0261 (–7.7041) 3.5104 ± 0.0043 (6.5203) 3.4415 ± 0.0043 (6.1366)
1.25 0.90 –5.1569 ± 0.0355 (–7.4408) 5.1130 ± 0.0144 (8.5280) 4.7342 ± 0.0144 (7.5805)
1.45 0.30 –1.8984 ± 0.0207 (–6.7843) 0.2577 ± 0.0015 (–0.1523) 0.5924 ± 0.0015 (–0.3509)
1.45 0.40 –2.4270 ± 0.0301 (–8.3220) 0.3377 ± 0.0016 (–0.0368) 0.5822 ± 0.0016 (–0.0637)
1.45 0.50 –3.0328 ± 0.0127 (–9.4128) 0.5029 ± 0.0012 (0.5125) 0.6936 ± 0.0012 (0.7083)
1.45 0.70 –4.1860 ± 0.0193 (–10.1948) 1.7520 ± 0.0059 (3.9614) 1.7262 ± 0.0059 (3.9117)
1.45 0.95 –5.0685 ± 0.0069 (–8.4980) 8.8548 ± 0.0213 (16.2394) 6.4282 ± 0.0213 (11.7872)
4.85 0.20 –0.9473 ± 0.0343 (–1.5582) 1.1596 ± 0.0115 (1.1249) 1.1955 ± 0.0115 (1.1618)
4.85 0.40 –1.7224 ± 0.0409 (–2.6245) 3.0919 ± 0.0218 (3.1618) 1.5937 ± 0.0218 (1.6333)
4.85 0.60 –2.4364 ± 0.0392 (–2.9535) 6.9479 ± 0.0715 (7.5399) 2.3876 ± 0.0715 (2.5969)
4.85 0.80 –2.6936 ± 0.0909 (–2.2996) 15.1613 ± 0.0870 (16.6089) 3.9076 ± 0.0870 (4.2906)
4.85 1.00 –1.7806 ± 0.1141 (–0.3807) 31.7812 ± 0.4630 (34.3812) 6.5528 ± 0.4630 (7.0889)
At the highest temperature, T ∗ = 4.85, (see the bottom parts in figures 3 and 4) that we are
referring to as the extremely high for the LJ fluid (more than three times higher than its critical
point temperature), simulation data show that fixing a hard-core at distances r 6 0.9σ produces,
for all the calculated thermodynamic properties, the values that are very close to those for the
initial LJ2Y model without a hard-core. At the same time, hard-core R = 0.9σ seems to be the
smallest one when the MSA theory performs well. Although this is not the case of the pressure
33005-6
MSA for the Lennard-Jones-like two Yukawa model
0,2 0,4 0,6 0,8 1,0
0
5
10
15
20
0.9σ
0.95σ
R=σ
0.85σ P
*
ρ*
0.8σ
T*=1.25
0,2 0,4 0,6 0,8 1,0
0
5
10
15
20
R=σ
0.9σ
0.85σ
P
*
ρ*
0.95σ
T*=1.45
0,2 0,4 0,6 0,8 1,0
0
5
10
15
20
0.8σ
0.85σ
R=σ
0.9σ
0.8σ
P
*
ρ*
0.95σ
T*=4.85
Figure 3. The pressure of the LJ2Y fluid obtained from Monte Carlo computer simulations
(symbols) and in compliance with the MSA theory (lines). The meaning of the symbols: filled
triangles – R = σ, filled squares – R = 0.95σ, crossed squares – R = 0.9σ, crossed circles –
R = 0.85σ, filled circles – R = 0.8σ, crossed triangles – without hard-core. The notes in the
figures indicate the position of hard-core in MSA calculations.
(bottom part in figure 3), but this is already seen from the behavior of the internal energy (bottom
part in figure 4) and, especially, from the comparison with MC data for the radial distribution
function (figures 5 and 6) when extra oscillations around the first minima start to appear.
4. Conclusions
In this study we report the Monte Carlo (MC) simulation studies of the thermodynamics and
radial distribution functions of the Lennard-Jones-like two Yukawa (LJ2Y) fluid. This fluid model
has been defined to nearly exactly mimic the pair interaction at short and intermediate distances
in the Lennard-Jones (12–6) fluid. In particular, the parameters of the LJ2Y have been defined
from the condition that zero-potential energy distance and the slope of the potential profile at
the zero-potential distance are the same. Moreover, the magnitude and position of the potential
minima in both models coincide. The only differences that two potential models experience are
attributed to large distances. Exactly these differences have caused quite pronounced differences in
the vapour-liquid coexistence occurring in two models. However, the purpose of the present study
is not to describe the LJ fluid by means of the two Yukawa model. Our goal is to find how the
mean-spherical approximation (MSA) performs when it is applied to the fluid model whose soft
repulsion is the same or quite similar to that of the LJ fluid. The task we set forth is twofold. First,
33005-7
I. Nezbeda et al.
0,0 0,2 0,4 0,6 0,8 1,0
-6
-5
-4
-3
-2
-1
0
0.9σ 0.95σ
R=σ
0.85σ
U
*
ρ*
T*=1.25
0,2 0,4 0,6 0,8 1,0
-6
-5
-4
-3
-2
-1
0
R=σ
0.9σ
0.85σ
U
*
ρ*
0.95σ
T*=1.45
0,2 0,4 0,6 0,8 1,0
-6
-5
-4
-3
-2
-1
0
0.85σ
R=σ
0.9σ U
*
ρ*
0.95σ
T*=4.85
Figure 4. The internal energy of the LJ2Y fluid obtained from Monte Carlo computer simulations
(symbols) and in compliance with the MSA theory (lines). The meaning of the symbols is the
same as in figure 3.
1,0 1,5 2,0 2,5
0,0
0,5
1,0
1,5
2,0 0.85σ
R=σ
g(
r)
r/σ
T*=1.45
ρ∗=0.3
1,0 1,5 2,0 2,5
0,0
0,5
1,0
1,5
2,0
2,5 0.85σ
R=σ
g(
r)
r/σ
T*=1.45
ρ∗ = 0.7
Figure 5. The radial distribution functions of the LJ2Y fluid obtained from Monte Carlo com-
puter simulations (symbols) and in compliance with the MSA theory (lines) at a slightly sub-
critical temperature T ∗
= 1.45 and two densities specified in the figure. The meaning of the
symbols is the same as in figure 3.
the application of the MSA theory requires cutting-off the soft repulsion at some interparticle
separation and placing the hard-core R at this distance. However, only such a modification of the
initial potential may intervene and change the properties of the system. Without this knowledge it
33005-8
MSA for the Lennard-Jones-like two Yukawa model
Table 5. The same as in table 3 but with a hard-core located at R = 0.9σ.
T ∗ ρ∗ U∗ P ∗ Z
1.25 0.70 –4.3027 ± 0.0225 (–4.8682) 0.9804 ± 0.0068 (1.6581) 1.1672 ± 0.0068 (1.8706)
1.25 0.80 –4.8347 ± 0.0078 (–5.0781) 2.3805 ± 0.0212 (3.4166) 2.4796 ± 0.0212 (3.4166)
1.25 0.85 –5.1022 ± 0.0361 (–5.1068) 3.5282 ± 0.0153 (4.6974) 3.4590 ± 0.0153 (4.4210)
1.25 0.90 –5.1948 ± 0.0299 (–5.0802) 5.1086 ± 0.0262 (6.3142) 4.7302 ± 0.0262 (5.6126)
1.45 0.30 –1.8667 ± 0.0218 (–2.3832) 0.2576 ± 0.0015 (0.2301) 0.5922 ± 0.0015 (0.5301)
1.45 0.40 –2.4916 ± 0.0246 (–3.0631) 0.3329 ± 0.0011 (0.3595) 0.5740 ± 0.0011 (0.6212)
1.45 0.50 –3.0062 ± 0.0212 (–3.6535) 0.5020 ± 0.0108 (0.6795) 0.6924 ± 0.0108 (0.9393)
1.45 0.70 –4.1425 ± 0.0353 (–4.4270) 1.7299 ± 0.0155 (2.6462) 1.7043 ± 0.0155 (2.6130)
1.45 0.95 –5.0493 ± 0.0142 (–4.2580) 9.0221 ± 0.0269 (10.8535) 6.5496 ± 0.0269 (7.8779)
4.85 0.20 –1.0006 ± 0.0140 (–0.9999) 1.1575 ± 0.0174 (1.1613) 1.1933 ± 0.0174 (1.1994)
4.85 0.40 –1.8827 ± 0.0278 (–1.8953) 3.0517 ± 0.0853 (3.1656) 1.5731 ± 0.0853 (1.6353)
4.85 0.60 –2.8002 ± 0.0563 (–2.5607) 6.9707 ± 0.1030 (7.2761) 2.3954 ± 0.1030 (2.5061)
4.85 0.80 –3.1223 ± 0.0609 (–2.7985) 15.1610 ± 0.5335 (16.0293) 3.9075 ± 0.5335 (4.1409)
Table 6. The same as in table 3 but with a hard-core located at R = 0.95σ.
T ∗ ρ∗ U∗ P ∗ Z
1.25 0.70 –4.3871 ± 0.0092 (–4.3674) 0.9627 ± 0.0137 (1.2917) 1.1461 ± 0.0137 (1.4763)
1.25 0.80 –4.9808 ± 0.0317 (–4.8345) 2.4198 ± 0.0559 (2.8369) 2.5206 ± 0.0559 (2.8369)
1.25 0.85 –5.1823 ± 0.0216 (–5.0266) 3.6847 ± 0.0610 (4.0519) 3.6124 ± 0.0610 (3.8134)
1.45 0.30 –1.9166 ± 0.0214 (–1.8592) 0.2739 ± 0.0162 (0.2652) 0.6297 ± 0.0162 (0.6108)
1.45 0.40 –2.5016 ± 0.0075 (–2.4347) 0.3106 ± 0.0138 (0.3645) 0.5355 ± 0.0138 (0.6299)
1.45 0.50 –3.1156 ± 0.0266 (–3.0047) 0.5126 ± 0.0249 (0.5765) 0.7071 ± 0.0249 (0.7969)
1.45 0.70 –4.2619 ± 0.0286 (–4.0737) 1.7885 ± 0.1040 (2.0262) 1.7621 ± 0.1040 (2.0008)
4.85 0.20 –1.1121 ± 0.0170 (–1.0698) 1.1870 ± 0.0396 (1.1943) 1.2237 ± 0.0396 (1.2335)
4.85 0.40 –2.2148 ± 0.0142 (–2.1748) 3.1669 ± 0.0449 (3.3415) 1.6324 ± 0.0449 (1.7261)
4.85 0.60 –3.3309 ± 0.0147 (–3.2611) 7.4178 ± 0.2012 (7.9735) 2.5491 ± 0.2012 (2.7463)
4.85 0.80 –4.3372 ± 0.0376 (–4.1799) 19.1784 ± 0.5030 (18.6856) 4.9429 ± 0.5030 (4.8271)
1,0 1,5 2,0 2,5
0,0
0,5
1,0
1,5
0.8σ
R=σ
g(
r)
r/σ
T*=4.85
ρ∗=0.2
1,0 1,5 2,0 2,5
0,0
0,5
1,0
1,5
2,0
2,5
0.8σ0.95σ
0.95σ
g(
r)
r/σ
T*=4.85
ρ∗=0.8
Figure 6. The radial distribution functions of the LJ2Y fluid obtained from Monte Carlo com-
puter simulations (symbols) and in compliance with the MSA theory (lines) at high temperature
T ∗
= 4.85 and two densities specified in the figure. The meaning of the symbols is the same as
in figure 3.
will be hard to judge about the MSA performance. To fulfill this task, we performed MC computer
simulations for both the initial LJ2Y model defined according to equation (3) and for hard-core
modified potential (2) – the form that is usually utilized by the MSA theory. The positions of
the hard-core that have been implemented in computer simulations include R = σ; 0.95σ; 0.9σ;
0.85σ and 0.8σ. Having compared this with the data obtained in computer simulations with initial
33005-9
I. Nezbeda et al.
Table 7. The same as in table 3 but with a hard-core located at R = σ.
T ∗ ρ∗ U∗ P ∗ Z
1.25 0.70 –4.6285 ± 0.0201 (–4.5755) 1.4418 ± 0.1111 (1.5986) 1.7164 ± 0.1111 (1.8270)
1.45 0.30 –1.9693 ± 0.0031 (–1.8995) 0.3043 ± 0.0362 (0.2881) 0.6996 ± 0.0362 (0.6636)
1.45 0.40 –2.5870 ± 0.0119 (–2.5222) 0.3813 ± 0.0258 (0.4013) 0.6575 ± 0.0258 (0.6934)
1.45 0.50 –3.2332 ± 0.0154 (–3.1744) 0.5695 ± 0.0126 (0.6383) 0.7855 ± 0.0126 (0.8824)
1.45 0.70 –4.6221 ± 0.0079 (–4.5500) 2.2816 ± 0.0362 (2.3683) 2.2479 ± 0.0362 (2.3387)
4.85 0.20 –1.1368 ± 0.0137 (–1.1383) 1.1796 ± 0.0193 (1.2640) 1.2161 ± 0.0193 (1.3056)
4.85 0.40 –2.4041 ± 0.0068 (–2.4050) 4.1509 ± 0.2956 (3.7915) 2.1396 ± 0.2956 (1.9586)
4.85 0.60 –3.7846 ± 0.0061 (–3.7873) 10.0931 ± 0.4284 (9.8521) 3.4684 ± 0.4284 (3.3933)
LJ2Y potential, i.e., without hard-core, we conclude that the insertion of the hard-core into a LJ-
like models is extremely sensitive to the temperature conditions. Namely, at normal temperature
conditions of the order and around the critical point temperature, an insertion of the hard-core at
separations r 6 0.95σ practically does not change the thermodynamics of the initial system, while
the case R = σ already shows the tendency to exhibit slightly different thermodynamic properties.
The conclusions change when we explored the high temperature conditions, namely, T ∗ = 4.85.
At these temperature conditions, the presence of the hard-core at distances as short as r 6 0.85σ
may be considered not to modify the initial system. All other cases, R = σ; 0.95σ and 0.9σ should
be treated as separate model fluids that are different from the initial LJ2Y fluid model. These
observations are valid for both thermodynamics and radial distribution functions.
Having taken these considerations into account we were able to fulfill the second part of our
task, namely, to judge the MSA performance for the LJ2Y fluid model. First of all, we found that
MSA correctly reflects the temperature conditions when the case of a hard-core R = σ starts to
deviate from the other hard-core modified models. Secondly, the MSA performs rather satisfactorily
for this case R = σ at all considered temperature conditions. The following conclusion seems to be
rather important and reads: at normal temperature conditions (of the order and around the critical
point temperature for the LJ fluid), an optimal position of the hard-core R for the MSA to be a
reasonable theoretical approach is R = 0.95σ. This means that placing the hard-core at R = 0.95σ
and using the MSA theory to describe the LJ-like two Yukawa model will give you the results that
are of reasonable accuracy and are very close to those of the initial model. By contrast, using the
hard-core for MSA description as small as R 6 0.9σ will result in incorrect values for initial LJ2Y
fluid model and even for the LJ-like model with an embedded hard-core.
The above conclusion is valid when you are seeking the possibility to use the MSA theory at
high temperatures, T ∗ = 4.85. The only thing that changes in this case is the limiting value of
the hard-core. Namely, at this high temperature you may use the hard-core position as small as
R = 0.9σ and still continue to apply the MSA theory and obtain reasonably accurate description
of the LJ2Y fluid model. On the contrary, using the hard-core at R = 0.85σ and 0.8σ leads to an
increased inaccuracy in both thermodynamics and the radial distribution functions. We also note
that at this high temperature, the hard-core modified LJ2Y models with R = σ and R = 0.95σ
significantly differ from the initial LJ2Y model without hard-core and, very importantly, the MSA
theory performs quite well for both these cases.
Acknowledgement
This work was supported by the Grant Agency of the Academy of Sciences of the Czech Republic
(Grant No. IAA400720710) and the Czech-Ukrainian Bilateral Cooperative Program.
33005-10
MSA for the Lennard-Jones-like two Yukawa model
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Середньосферичне наближення для Леннард-Джонс-
подiбного плину, змодельованого сумою двох потенцiалiв
Юкави: порiвняння з результатами методу Монте Карло
Я. Крейчi1, I. Незбеда1,2, Р. Мельник3, А. Трохимчук3,4
1 Факультет природничих наук, Унiверситет Я.Е. Пуркiнйє, Устi над Лабем, Чеська Республiка
2 Iнститут фундаментальних основ хiмiчних процесiв, Академiя наук, Прага, Чеська Республiка
3 Iнститут фiзики конденсованих систем НАН України, вул. I. Свєнцiцького, 1, 79011 Львiв, Україна
4 Факультет хiмiї та бiохiмiї, Унiверситет Бригам Янг, Прово, США
Методом Монте Карло проведено дослiдження Леннард-Джонс-подiбного плину, взаємодiя мiж ча-
стинками якого задається сумою двох потенцiалiв Юкави (ЛД2Ю). Метою дослiджень є продемон-
струвати вплив, якому пiддаються властивостi моделi при замiнi “м’якого” вiдштовхування “твердим”
кором. Розглянуто рiзнi вiдстанi для розмiщення твердого кору. Виявлено, що при температурах,
трохи нижчих i трохи вищих за критичну температуру леннард-джонсiвського плину, перемiщення
твердого кору на вiдстанi, коротшi за вiдстань з нульовою потенцiальною енергiєю, практично не
змiнює термодинамiчних властивостей моделi, значення яких є дуже близькими до термодинамi-
чних характеристик ЛД2Ю моделi без твердого кору. Однак, при переходi в зону екстремально ви-
соких температур слiд бути обережним, оскiльки присутнiсть твердого кору провокує суттєвi змiни
властивостей системи. Це застереження є дуже важливим при застосуваннi методу середньосфе-
ричного наближення до опису Леннард-Джонс-подiбного плину.
Ключовi слова: 2-Юкава потенцiал, леннард-джонсiвський плин, середньосферичне наближення,
метод Монте Карло
33005-11
http://dx.doi.org/10.1080/00268977300100061
http://dx.doi.org/10.1080/00268977800100181
http://dx.doi.org/10.1016/j.chemphys.2008.05.018
http://dx.doi.org/10.1080/00268979300100411
http://dx.doi.org/10.1007/BF01011750
http://dx.doi.org/10.1080/00268979650026929
http://dx.doi.org/10.1016/S0378-3812(97)00049-6
http://dx.doi.org/10.1016/0009-2614(79)85016-2
http://dx.doi.org/10.1063/1.442469
http://dx.doi.org/10.1080/00268979200102111
http://dx.doi.org/10.1063/1.2176677
Introduction
The potential models and computational details
Results and discussions
Conclusions
|