Vapour/liquid coexistence in long-range Yukawa fluids determined by means of an augmented van der...

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Vapour/liquid coexistence in long-range Yukawa fluids determined bymeans of an augmented van der Waals approachRoman Melnyka; Ivo Nezbedabc; Andrij Trokhymchukad

a Institute for Condensed Matter Physics, National Academy of Sciences of Ukraine, Lviv, 79011,Ukraine b Faculty of Science, J.E. Purkinje University, 400 96 Ústí nad Labem, Czech Republic c E. HalaLab. of Thermodynamics, Institute of Chemical Process Fundamentals, Academy of Sciences, 165 02Prague 6, Czech Republic d Department of Chemistry and Biochemistry, Brigham Young University,Provo, UT 84602, USA

Online publication date: 24 January 2011

To cite this Article Melnyk, Roman , Nezbeda, Ivo and Trokhymchuk, Andrij(2011) 'Vapour/liquid coexistence in long-range Yukawa fluids determined by means of an augmented van der Waals approach', Molecular Physics, 109: 1, 113 —121To link to this Article: DOI: 10.1080/00268976.2010.542034URL: http://dx.doi.org/10.1080/00268976.2010.542034

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Molecular PhysicsVol. 109, No. 1, 10 January 2011, 113–121

INVITED ARTICLE

Vapour/liquid coexistence in long-range Yukawa fluids determined

by means of an augmented van der Waals approach

Roman Melnyka, Ivo Nezbedabc and Andrij Trokhymchukad*

aInstitute for Condensed Matter Physics, National Academy of Sciences of Ukraine, Lviv,79011, Ukraine; bFaculty of Science, J.E. Purkinje University, 400 96 Ustı nad Labem,

Czech Republic; cE. Hala Lab. of Thermodynamics, Institute of Chemical Process Fundamentals,Academy of Sciences, 165 02 Prague 6, Czech Republic; dDepartment of Chemistry and Biochemistry,

Brigham Young University, Provo, UT 84602, USA

(Received 8 August 2010; final version received 15 November 2010)

The vapour–liquid phase diagram of long-range Yukawa fluids with the decay parameter z�� 1 is calculated bymeans of an augmented van der Waals theory based on the reference system that includes both the repulsive andshort-range attractive interaction. Comparison with Monte Carlo simulation data, as well as with the traditionalvan der Waals approach based on a purely repulsive hard-sphere reference system and other semi-analyticaltheoretical approaches, is made.

Keywords: Yukawa fluid; perturbation theory; augmented van der Waals approach; vapour–liquid coexistence

1. Introduction

Because of its versatility the attractive hard-sphere

Yukawa (AHSY) potential has become one of the most

popular fluid models studied in the last decade

intensively both by molecular simulations [1–3] and

theory [4–9]. In their recent paper [10] Caillol et al.

have reported an extensive computer simulation

study of the vapour–liquid (VL) coexistence in the

long-range attractive Yukawa fluids. In the same

study these authors have also shown that two

rather sophisticated theoretical approaches, namely,

the self-consistent Ornstein–Zernike equation

(SCOZA) and the hierarchical reference theory

(HRT) perform extremely well in reproducing these

computer simulation data while an optimized mean

field theory (OMF), while yielding the exact result in

the limit of an infinite range of the potential, still

deteriorates when the interaction is long ranged but yet

finite. Very recently the same systems were studied [11]

also by means of the numerical solution of the

Ornstein–Zernike integral equation using Sarkisov’s

closure [12].As an alternative to the above numerically demand-

ing methods we have recently proposed a transparent

perturbation theory approach [13,14] built upon the

Yukawa (Y) models as a sort of reference and which

makes use of the availability of both approximate

analytic [4] and accurate empirical [7–9] results for the

Y fluid. The use of the Y reference considerably

improves accuracy of the perturbation methods: the

more features are captured by the reference, the less

important become perturbation correction terms and,

consequently, all considerations are confined to a

first-order expansion or further even to an augmented

van der Waals (the simplified first-order expansion)

theory only [15]. This approach can be viewed as a

counterpart of the common augmented van der Waals

(vdW) theory based on the purely repulsive hard-

sphere (HS) reference; we will refer to these two

theories as the vdW/Y and vdW/HS approaches,

respectively. The vdW/Y approach has been applied

already to the medium range (Lennard-Jones-like)

attractive hard-core Yukawa potential [14] and to the

Sutherland potential [15] made up of a hard-core

repulsion and a power law attractive tail, while

potentially it is applicable to any simple fluid.

The determining feature for such a claim is the

flexibility of the Y model to mimic short-range

interactions of various kinds that usually are achieved

by employing several Yukawa terms (e.g. see equations

that have been developed by Guerin [16] for the sum of

two Yukawa (2Y) potentials and by Tang [17] for the

multi-Yukawa potentials). Quite recently [18] we

reported 2Y representation of the short-range inter-

action in the EXP6 fluids and examined it using

molecular simulations.

*Corresponding author. Email: [email protected]

ISSN 0026–8976 print/ISSN 1362–3028 online

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The aim of the present paper is twofold. First, we

will show that the simple vdW-like theory may perform

as well as the advanced theoretical approaches like

SCOZA and HRT. Secondly, we will emphasize that a

natural way to improve the traditional vdW theory lies

in an improvement of the reference system contribu-

tion rather than in a modernizing the perturbation

contribution. We proceed presenting first the model

and theoretical basis of the vdW/Y approach in the

following section. The original results obtained for

selected model long-range Yukawa fluids are then

shown and discussed in Section 3.

2. Model and theory

The attractive hard sphere Yukawa (AHSY) potential

is defined by

uðrÞ ¼1, if r5 �,

��ð�=rÞ exp �zðr� �Þ� �

, if r � �,

�ð1Þ

where r is the distance between two particles, � is the

hard-core diameter of the particles and � is the potentialdepth, both set to unity henceforth. The exponent z

determines the range of the interaction and in this

study will consider it within the range 0.1� z�� 1.

Specifically, the long-range AHSY potential functions

for the case of z�¼ 1, 0.5 and 0.1 are shown in

Figure 1. We have chosen these values of parameter z

for the possibility to compare the results with extensive

Monte Carlo (MC) computer simulation data for the

VL coexistence reported by Caillol et al. [10].The augmented vdW theory employed in this study

results as a simplified byproduct when deriving a

general perturbation theory based on the attractive

Yukawa (Y) reference system [14]. Specificity of such a

perturbation theory is that it is built upon the reference

fluid that, besides the HS repulsion, includes also some

portion of the attraction within the spirit of a unified

view of fluids [19]. To be more specific, the basic idea is

that in the (dense) fluid phase the pair interaction u can

be separated into two parts,

u ¼ u0 þ Du, ð2Þ

where u0, referred to as a reference, is responsible for

the local ordering in the system while Du plays the role

of a perturbation that has only little effect on this local

structure. The important point is that the reference

part need not be only repulsive, as it is usually thought,

but may (should) incorporate also a piece of the

attractive interactions at short interparticle separa-

tions. A route toward the thermodynamic properties of

the original u-fluid at hand goes then via the expansion

of the Helmholtz free energy,

�A ¼ �A0 � 2p��NZ

g0ðrÞDuðrÞr2 drþ . . . , ð3Þ

where � ¼ 1=kBT, kB is Boltzmann’s constant, N is the

number of molecules, �¼N/V is the number density,

and g0 is the radial distribution function of the

–1.0

–0.8

–0.6

–0.4

–0.2

0.0

0.2

0.4

Pai

r in

tera

ctio

n, u

(r)/

Distance, r/s

(a)

zs=0.1

zs=0.5

zs=1

0 1 2 3 4 5 6 7 8 9 10 1 2 3 4

–1.0

–0.8

–0.6

–0.4

–0.2

0.0

0.2

0.4

Distance, r/s

(b)

Perturbation Δu

AHSY fluid (zs=1)

Reference Y–fluid (z0s=3)

Pai

r in

tera

ctio

n, u

(r)/

Figure 1. Part (a): the long-range AHSY potential at different values of the range parameter, z, as shown in the figure. Part (b):the decomposition of the long-range Yukawa potential with the range parameter z�¼ 1 into the reference and perturbationcontributions.

114 R. Melnyk et al.


reference system. The degree of similarity of the

short-range structure of the parent u-fluid and that of

the reference u0-fluid determines then the magnitude

of the difference A�A0 and governs the convergence of

expansion (3).For the approach based on expansion (3) to be of a

practical use it is necessary that it converges fast and

can be considered only with the first-order vdW-like

correction,

aðT, �Þ ¼ �2pZ

g0ðrÞDuðrÞr2 dr: ð4Þ

Higher order corrections are both only approximate

and time consuming, and defy any analytic treatment.

At this point it also worth noting that the simple

vdW-like term correction in the expansion for A is also

a prerequisite for obtaining other thermodynamic

functions in a tractable form. For instance, by deriving

the equation for pressure, P, we have to deal with the

density derivative @a/@�. To obtain the equation of

state (EOS) in a closed form, an approximation

aðT, �Þ ! a0 ¼ �2pZ

DuðrÞr2 dr, ð5Þ

is then very often used. Following the traditional

approach, i.e. to operate with the HS reference fluid,

u0ðrÞ � uHSðrÞ ¼1, if r5 �,

0, if r � �,

�ð6Þ

it is easy to show that such an approximation is too

crude. This is discernible from Figure 2: the vdW

correction term a being evaluated via definition,Equation (4), with the HS radial distribution functionfor g0, strongly depends on density and significantlydiffers from the constant a0 given by approximation,Equation (5). Consequently, the density derivative@a/@� cannot be neglected in the case of the HSreference fluid. On the other hand, following oursuggestion [13,14] and using as the reference the short-range (z0�¼ 3) attractive Y fluid,

u0ðrÞ � uYðrÞ ¼1, if r5 �,

��ð�=rÞ exp �z0ðr� �Þ� �

, if r � �,

�

ð7Þ

we see in Figure 2 that a(T, �) practically does notchange with a density having a magnitude nearly thatof a0. It means that using the Y reference system theassumption (5) becomes fully justified and leads thento the augmented vdW theory that reads as

�A

N¼�A0

N� ��a0,

�P

�¼�P0

�� a0,

�U

N¼�U0

N� ��a0: ð8Þ

To understand the phenomenon of the Y reference wefirst note that for both reference systems, HS and Y,the radial distribution functions g0� gHS or g0� gY arevery similar and both depend on density. But it is theperturbation term, Du, that differs significantly withdependence on the choice of the reference system.Indeed, when the HS fluid is employed as the reference

0

2

4

6

8(a)

T*=1.5

T*=2

T*=1

HS reference system

vdW

coe

ffici

ent,

ba

Density, rs30.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

0

2

4

6

8(b)

T*=2

T*=1.5

T*=1

Y reference system

vdW

coe

ffici

ent,

ba

Density, rs3

Figure 2. An example of the vdW correction term �a(T, �) evaluated for the Lennard-Jones-like hard-core Yukawa fluidwith z�¼ 1.8. Part (a) shows the results for the case of a traditional HS reference system while part (b) shows the results in thecase of a proposed Y reference system characterized by z0�¼ 3. The thick solid lines in both parts correspond to a0 given byEquation (5); the thin lines represent a(T, �) from Equation (4) using for the radial distribution function g0 of the HS and Yreference systems the PY and MSA results, respectively; the symbols in part (b) have the same meaning but obtained by using forthe radial distribution function g0 the ‘exact’ computer simulation data.

Molecular Physics 115


system then the perturbation term Du(r) incorporatesthe entire interaction energy u(r) which is the strongest

at the hard-core contact r¼ �. However, hard-core

contact distance is the most density sensitive region in

the HS radial distribution function gHS. Conversely, as

we can see from Figure 1b, inclusion into the reference

part of the short-range attraction of the same energy

minimum � as in the parent fluid, makes the pertur-

bation term Du(r) very weak in the most important

region of small separations between the molecules.

In particular, the perturbation term is exactly zero at

the hard-core contact, i.e. Du(r¼ �)¼ 0. It means that

in the case of the Y reference system the contribution

to a(T, �) from the range of small r in g0� gY is

practically killed by the value of Du, and an approx-

imation g0� 1 within the framework of the augmented

vdW/Y theory becomes then well grounded.The Y reference system defined by Equation (7) is

characterized by a so far unspecified range parameter z0.

One obvious restriction imposed on z0 is that the rangeof the attraction for the reference fluid pair potential

u0(r) must be shorter than that of u(r). Moreover, the

perturbation theory may be intuitively expected to

perform the better the closer the reference fluid is to theparent fluid. Before attempting to be more specific one

should always realize reasons why the HS reference has

been so successful. Not only because availability of

results for both the structure and thermodynamics but

primarily because of its flexibility. The lack of thecritical point in the HS fluid makes it possible to change

the HS diameter (and hence the packing density) over a

large range of values without imposing serious restric-

tions on the properties of the considered system and

thus tune the results to the wished outcome. Recentcomputer simulation studies [1–3,6] of the short-rangeY

fluid showed that: (i) the VL critical temperature

decreases in this system and is approaching the triple

point temperature when the range of attraction

2.0

2.1

2.2

2.3

2.4

2.5 zs=1

vdW/HS

vdW/Y

Density, rs3

Tem

pera

ture

, Tk B

/

(a)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

6.0

6.5

7.0zs=0.5

Density, rs3

Tem

pera

ture

, Tk B

/

(b)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

110

115

120

125 zs=0.1

Density, rs3

Tem

pera

ture

, Tk B

/

(c)

Figure 3. Vapour–liquid coexistence curve for the long-range AHSY fluids with different values of the range parameter(a) z�¼ 1, (b) 0.5 and (c) 0.1 obtained by means of augmented vdW theory with HS (dashed line) and with Y (solid line) referencesystems. The symbols stand for MC computer simulation data (filled spheres) and the hierarchical reference theory (triangle)results, both due to Caillol et al. [10].



Table 1. Coexistence densities, internal energy and chemical potential at vapour–liquidcoexistence in the long-range AHSY fluid with z�¼ 1 obtained by means of augmented vdWtheory with two different reference systems and those from MC computer simulations due toCaillol et al. [10].

kBT/� �v�3 �l�

3 (U/NkBT )v (U/NkBT )l �/kBT

2.460 MC 0.183(2) 0.395(10) �1.025(50) �2.19(5) �2.77(1)vdW/Y 0.178 0.391 �0.951 �2.13 �2.73

2.392 MC 0.135(2) 0.439(5) �0.78(2) �2.52(2) �2.87(1)vdW/Y 0.139 0.448 �0.76 �2.52 �2.82

2.338 MC 0.112(2) 0.474(4) �0.67(1) �2.79(2) �2.95(1)vdW/Y 0.117 0.484 �0.652 �2.80 �2.90

2.283 MC 0.096(2) 0.508(5) �0.59(2) �3.07(3) �3.04(2)vdW/Y 0.099 0.515 �0.565 �3.068 �2.98

2.229 MC 0.085(3) 0.543(6) �0.54(2) �3.37(3) �3.11(2)vdW/Y 0.084 0.544 �0.492 �3.332 �3.071vdW/HS 0.177 0.330 �1.000 �1.860 �2.835

2.175 MC 0.070(2) 0.564(4) �0.447(15) �3.60(3) �3.21(1)vdW/Y 0.071 0.572 �0.428 �3.598 �3.169vdW/HS 0.139 0.383 �0.804 �2.213 �2.908

2.120 MC 0.062(2) 0.594(6) �0.420(15) �3.89(3) �3.30(2)vdW/Y 0.060 0.597 �0.373 �3.868 �3.274vdW/HS 0.114 0.423 �0.677 �2.507 �2.986

2.066 MC 0.050(2) 0.612(3) �0.344(5) �4.120(15) �3.43(1)vdW/Y 0.051 0.622 �0.324 �4.146 �3.387vdW/HS 0.095 0.457 �0.579 �2.782 �3.071

2.012 MC 0.042(2) 0.634(4) �0.30(1) �4.440(25) �3.55(1)vdW/Y 0.043 0.645 �0.280 �4.432 �3.510vdW/HS 0.080 0.488 �0.498 �3.050 �3.163

Table 2. The same as in Table 1 but with z�¼ 0.5.

kBT/� �v�3 �l�


6.892 MC 0.169(8) 0.355(8) �0.93(3) �1.99(2) �2.82(1)vdW/Y 0.171 0.360 �0.945 �2.01 �2.81

6.859 MC 0.160(8) 0.370(8) �0.89(4) �2.05(4) �2.84(1)vdW/Y 0.163 0.371 �0.903 �2.08 �2.82

6.727 MC 0.130(6) 0.400(6) �0.75(2) �2.31(4) �2.90(1)vdW/Y 0.137 0.408 �0.775 �2.34 �2.88vdW/HS 0.192 0.312 �1.077 �1.75 �2.82

6.595 MC 0.118(4) 0.439(4) �0.689(15) �2.58(2) �2.96(1)vdW/Y 0.118 0.440 �0.678 �2.570 �2.94vdW/HS 0.153 0.362 �0.876 �2.07 �2.88

6.463 MC 0.103(4) 0.467(4) �0.62(1) �2.74(2) �3.02(1)vdW/Y 0.102 0.467 �0.598 �2.79 �3.01vdW/HS 0.129 0.399 �0.752 �2.33 �2.94

6.331 MC 0.085(4) 0.485(4) �0.52(1) �2.99(1) �3.11(1)vdW/Y 0.089 0.493 �0.531 �3.01 �3.08vdW/HS 0.110 0.430 �0.657 �2.56 �3.00

6.199 MC 0.076(2) 0.512(2) �0.468(5) �3.21(1) �3.17(2)vdW/Y 0.077 0.517 �0.472 �3.22 �3.16vdW/HS 0.095 0.457 �0.579 �2.78 �3.07

6.067 MC 0.068(2) 0.539(3) �0.43(1) �3.46(2) �3.27(3)vdW/Y 0.067 0.539 �0.420 �3.44 �3.24vdW/HS 0.083 0.482 �0.513 �3.00 �3.14

5.935 MC 0.058(2) 0.556(3) �0.375(10) �3.64(2) �3.34(1)vdW/Y 0.058 0.561 �0.374 �3.66 �3.33vdW/HS 0.072 0.506 �0.456 �3.21 �3.22



decreases, and (ii) the short-range Y fluid with z0�¼ 3has its critical temperature nearly the same asthe triple point temperature of the Lennard-Jonesfluid. If one chooses, for instance, z0 larger than butclose to 3, then for the whole family of the medium-and long-range fluids such a defined Y referencesystem will be always closer to the parent fluid thanany HS reference, and yet it will enjoy the sameflexibility.

As regards available equations for the properties ofthe short-range Y reference fluids, they are comparablewith those for the HS reference fluid. An analyticaltreatment of the Y model is available within the meanspherical approximation (MSA) [4] that can be viewedas a counterpart of the common Percus–Yevick theoryin the case of the HS reference system. There are alsoanalytic results developed within the high temperatureexpansions (HTE) [7–9] that make it possibleto evaluate the thermodynamic properties for the Yreference fluid. By comparison with computer simula-tion data for a set of the range parameter valuesranging from z0�¼ 3 up to 6, which are typical for theY reference system, it has been already shown [9] thatthe HTE equations work well over wide density andtemperature windows.

3. Results

The main result of this study concerns the vapour–liquid (VL) phase diagram of the long-range attractivehard-sphere Yukawa (AHSY) fluid that follows fromthe augmented vdW approach built on the Y referencefluid and is given by Figure 3. Independently of thereference system, the VL coexistence within the vdWapproach is obtained by finding the intersection of theP�� surfaces for both the liquid and vapour branchesalong the same isotherm. For this purpose the chemicalpotential � was obtained from

�� ¼�A0

Nþ�P0

�� 2��a: ð9Þ

Figure 3 shows the VL coexistence curves for allthree cases of the long-range AHSY fluid considered inthis study, namely, z�¼ 1, 0.5 and 0.1. For each case theresults of the augmented vdW theory based on bothreference systems, HS and Y, are shown and comparedagainst MC computer simulation data due to Caillolet al. [10]. All the vdW/Y calculations were performedusing for the Y reference fluid the range parametervalue, z0�¼ 3. The equations to evaluate the thermo-dynamic properties A0, P0 and U0 of the Y reference

Table 3. The same as in Table 1 but with z�¼ 0.1.

kBT/� �v�3 �l�


121.57 MC 0.17(1) 0.340(15) �0.96(2) �1.96(5) �2.86(2)vdW/Y 0.159 0.356 �0.904 �2.03 �2.86vdW/HS 0.162 0.351 �0.920 �1.99 �2.86

120.46 MC 0.145(10) 0.360(15) �0.82(1) �2.05(1) �2.89(1)vdW/Y 0.146 0.374 �0.838 �2.15 �2.89vdW/HS 0.148 0.370 �0.851 �2.12 �2.89

119.36 MC 0.136(5) 0.380(5) �0.79(2) �2.21(2) �2.92(2)vdW/Y 0.135 0.391 �0.781 �2.27 �2.92vdW/HS 0.137 0.387 �0.793 �2.24 �2.91

118.25 MC 0.126(2) 0.393(3) �0.73(1) �2.32(2) �2.95(1)vdW/Y 0.125 0.406 �0.732 �2.38 �2.95vdW/HS 0.127 0.402 �0.742 �2.35 �2.94

117.15 MC 0.121(4) 0.416(4) �0.716(15) �2.46(2) �2.97(1)vdW/Y 0.117 0.420 �0.688 �2.48 �2.98vdW/HS 0.118 0.416 �0.697 �2.46 �2.97

116.05 MC 0.109(2) 0.425(3) �0.65(1) �2.54(2) �3.000(15)vdW/Y 0.109 0.434 �0.648 �2.59 �3.01vdW/HS 0.110 0.430 �0.656 �2.56 �3.00

114.94 MC 0.102(1) 0.439(2) �0.641(5) �2.64(1) �3.04(1)vdW/Y 0.102 0.446 �0.612 �2.69 �3.04vdW/HS 0.103 0.443 �0.619 �2.66 �3.03

112.73 MC 0.091(1) 0.466(2) �0.555(5) �2.86(1) �3.10(1)vdW/Y 0.089 0.470 �0.546 �2.89 �3.10vdW/HS 0.090 0.467 �0.555 �2.88 �3.10

110.52 MC 0.077(3) 0.483(5) �0.48(1) �3.028(25) �3.17(1)vdW/Y 0.078 0.493 �0.490 �3.09 �3.17vdW/HS 0.079 0.489 �0.498 �3.08 �3.17



fluid are taken from [15]. One can see that over the rangeof z values considered, the vdW/Y theory reproducesthe MC data for the VL coexistence extremely well.Conversely, the vdW/HS theory can be applicable onlyin the case of very small values of z�� 0.1. This isexpected, since increasing the range of attraction(decreasing the values of z) results in an increase ofthe critical point temperature (e.g. T �c for z�¼ 0.1 isapproximately of two orders larger than that for z�¼ 1)and role of the attraction becomes rather small. Forz�4 0.1 the critical temperature in the AHSY fluid isgetting lower and the vdW/HS theory rapidly deterio-rates: the coexistence curve of vdW/HS is found to benarrower and the critical point significantly lower thanthat of vdW/Y theory. The study due to Hendersonet al. [5] has shown that to describe the VL coexistencein the AHSY fluid with z�¼ 1.8 by using the HSreference one needs to take into account the secondcorrection term in expansion (3).

The vapour and liquid coexistence density, �vand �l, that result from the augmented vdW/Y theoryand those due to Caillol et al. [10] from MC computer

simulations are for convenience collected in Tables 1to 3 for z�¼ 1, 0.5 and 0.1, respectively. The sametables also consist of the theoretical results andcomputer simulation data for the coexistence internalenergies and excess chemical potential along thecoexistence curve. For the sake of clarity, the compar-ison of the coexistence internal energies and excesschemical potential that result from vdW/Y, vdW/HStheories and from MC computer simulations areshown in Figures 4 and 5. The tendency is quite thesame as we already have seen for the VL phasediagrams presented in Figure 3.

As can be noted from Figure 3, there are no MCcomputer simulation data for the critical point param-eters since those were not computed by Caillol et al.[10]. Instead, these authors reported for the criticalpoint parameters the results of the SCOZA and HRTtheories. The comparison between different theoreticalpredictions for the critical point parameters can bemade from Table 4. It can be seen that the agree-ment between the vdW/Y and the two theories isquite good.

2.0

2.1

2.2

2.3

2.4

2.5

2.6

zs=1

vdW/HS

vdW/Y

Internal energy, U/NkBT

Tem

pera

ture

, Tk B

/

(a)

6.0

6.4

6.8

zs=0.5


Tem

pera

ture

, Tk B

/

(b)

–4 –3 –2 –1 0 –4 –3 –2 –1 0

–4 –3 –2 –1 0

110

115

120

125zs=0.1


Tem

pera

ture

, Tk B

/

(c)

Figure 4. Internal energy U/NkBT along the vapour–liquid coexistence curve for the long-range AHSY fluids with (a) z�¼ 1,(b) 0.5 and (c) 0.1. The symbols are MC computer simulation data of Caillol et al. [10].



4. Conclusion

We have shown that recently proposed augmentedvdW/Y theory, built upon the short-range attractiveYukawa (Y) reference system, being applied to thelong-range AHSY fluid agrees well with computersimulation data due to Caillol et al. [10] and iscomparable with advanced theoretical approaches,namely, the SCOZA and HRT. At first glance thisresult may look rather trivial. However, being com-pared with the traditional augmented vdW theory thatuses the HS reference system (vdW/HS theory)it highlights the importance of the inclusion of theshort-range correlations and, consequently, the short-range attraction between molecules into the referencesystem. In particular, we note that the presence of theshort-range attraction in a reference system is a naturalway to provide the systematic and self-consistentimprovement of the traditional vdW theory. Indeed,the inclusion of the short-range attraction intoa reference fluid modifies both the contribution ofthe perturbation term and the contribution of thereference system. This novel reference system becomes

2.0

2.1

2.2

2.3

2.4

2.5

2.6

zσ=1

vdW/HS

vdW/Y

Tem

pera

ture

, Tk B

/ε

Chemical potential, m/kBT

(a)

6.0

6.2

6.4

6.6

6.8

7.0

–3.6 –3.4 –3.2 –3.0 –2.8 –2.6 –3.4 –3.2 –3.0 –2.8

zσ=0.5

Tem

pera

ture

, Tk B

/ε


(b)

110

115

120

125

–3.4 –3.2 –3.0 –2.8

zσ=0.1

Tem

pera

ture

, Tk B

/ε


(c)

Figure 5. Chemical potential �/kBT along the vapour–liquid coexistence curve for the long-range AHSY fluids with (a) z�¼ 1,(b) 0.5 and (c) 0.1. The meaning of symbols are the same as in Figure 4.

Table 4. Critical point temperature T�c ¼ kBTc=� and den-sity ��c ¼ �c�

3 of the long-range AHSY fluid with differentvalues of the range parameter z�¼ 1, 0.5 and 0.1 as resultingfrom the SCOSA and HRT both due to Caillol et al. [10] andfrom augmented vdW theory with two different referencesystems.

z� T�c ��c

0.1 SCOZA 124.82 0.2495HRT 124.77 0.250vdW/Y 124.76 0.248vdW/HS 124.52 0.248

0.5 SCOZA 7.068 0.259HRT 7.076 0.260vdW/Y 7.047 0.258vdW/HS 6.795 0.247

1 SCOZA 2.518 0.279HRT 2.514 0.279vdW/Y 2.521 0.275vdW/HS 2.263 0.246



quantitatively and, what is even more important,qualitatively very different from the HS referencesystem exhibiting important new features, like depen-dence on the temperature, etc. Importantly, despitesuch progress in the formulation of the referencesystem, the contribution of the Y reference system toboth the thermodynamics as well as the structure is stillavailable in the form of closed analytical equations(within the MSA or HTE approximations).

To evaluate the perturbation interaction contribu-tion a(T, �), we are using a simplified expression (5) thatusually is obtained from an exact expression (4) byassuming that the radial distribution function of thereference system is uniform, i.e. g0(r)¼ 1. We haveshown (see Figure 2) that the assumption aðT, �Þ ! a0,being a rather crude approximation in the case of theHS reference system, is quite accurate in the case of theY reference system. The small discrepancies that stillcould be observed between a(T, �) and a0 are negligiblein describing the VL coexistence, being comparable tothe accuracy of the description of the radial distributionfunction gY(r) of the Y reference system (the MSAtheory) and to the accuracy of the description ofthe thermodynamics of the reference system itself(the MSA or HTE approximations). Summarizing, theproposed Y reference system may play a role similar tothat played in the liquid state theories by the HS system.Although in the present study this has been illustratedusing the long-range AHSY fluid, it can be implementedwith any other kind of long-range interaction potential.

Acknowledgements

This work was supported by the Grant Agency of theAcademy of Sciences of the Czech Republic (Grant No.

IAA400720710) and Czech–Ukrainian Bilateral CooperativeProgram.

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Vapour/liquid coexistence in long-range Yukawa fluids determined by means of an augmented van der...

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