An improved first-order mean spherical approximation theory for thesquare-shoulder fluidS. P. Hlushak, P. A. Hlushak, and A. Trokhymchuk Citation: J. Chem. Phys. 138, 164107 (2013); doi: 10.1063/1.4801659 View online: http://dx.doi.org/10.1063/1.4801659 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v138/i16 Published by the American Institute of Physics. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors

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THE JOURNAL OF CHEMICAL PHYSICS 138, 164107 (2013)

An improved first-order mean spherical approximation theoryfor the square-shoulder fluid

S. P. Hlushak,1,2,a) P. A. Hlushak,2 and A. Trokhymchuk2

1Department of Chemical and Biomolecular Engineering, Vanderbilt University, Nashville,Tennessee 37235, USA2Institute for Condensed Matter Physics, National Academy of Sciences of Ukraine, Svientsitskoho 1,79011 Lviv, Ukraine

(Received 18 February 2013; accepted 29 March 2013; published online 23 April 2013)

The theory, which utilizes an exponential enhancement of the first-order mean spherical approxima-tion (FMSA) for the radial distribution functions of the hard-core plus square-well fluid, is adoptedto study the properties of the simplest model of the core-softened fluids, i.e., the hard spheres witha square-shoulder interaction. The results for structure and thermodynamic properties are reportedand compared against both the Monte Carlo simulation data as well as with those obtained withinthe conventional FMSA theory. We found that in the region of low densities and low temperatures,where the conventional FMSA theory fails, the exponential-based FMSA theory besides being qual-itatively correct also provides with a notable quantitative improvement of the theoretical description.© 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4801659]

I. INTRODUCTION

The simplest non-trivial discrete potential fluid, com-posed of species interacting via the hard-core plus square-well (HCSW) potential, plays an important role in the modernliquid-state theory as well as in various chemical engineer-ing applications.1–3 While the hard-core part of this discretepotential fluid is the most widely used as a reference systemin perturbation theory approaches, the square-well interactionconstitutes the simplest extension to introduce an extra at-traction, promoting the vapour-liquid phase transition. Con-sequently, the HCSW discrete potential is very often used asa model to study the vapour-liquid coexistence in simple andcomplex fluids.4, 5

Recent increased interest6–23 to the HCSW and to thediscrete potential model fluids in general, could be mainlylinked to the liquid-liquid transition that takes place in aone-component fluid system composed of species interact-ing via discrete potential consisting of the hard-core plusboth the square-shoulder repulsive and square-well attractiveinteractions.24–27 The similar liquid-liquid transition is ob-served in a number of important substances like water,28–30

carbon,31, 32 and silica.33 Thus, there is a possibility that phys-ical properties of these substances as well as some other realsystems might be understood by the simple coarse-grainedmodels represented by the discrete potential fluids. The lat-ter were employed recently to model adsorption of com-plex fluids into nanoporous materials,17–19 and to developequations of state for fluids interacting with complex pairpotentials.20–23

In present work we are focusing on the second varietyof the simplest discrete potential model system, namely, onthe hard-core plus square-shoulder (HCSS) fluid. The sys-tems composed of species interacting via this purely repul-

a)stepan.hlushak@gmail.com.

sive model potential were actively studied recently4, 8, 11, 34–41

mainly because of the interesting phase behaviour resultingin the formation of a rich variety of crystal structures.34–37

Majority of these studies have been performed by meansof Monte Carlo (MC) and molecular dynamics computersimulation techniques35, 37 and/or numerical solutions of theOrnstein-Zernike integral equation.4, 38 Because of founda-tion, the analytical or semianalytical approaches within thethermodynamic perturbation theory are limited to high tem-peratures only and their straightforward application to theHCSS fluid has been shown to result in some artificialpredictions.38 Nevertheless, only few years ago the first at-tempts were made to describe analytically the direct corre-lation function11, 39 and radial distribution function41 of theHCSS fluid. Therefore, we will look to contribute further tothis shortage of analytical approaches to the core-softened flu-ids by introducing a theory that is based on the solution of thefirst-order mean-spherical approximation (FMSA).

The FMSA is a theory due to Tang and Lu42 and con-sists of an analytical solution of the Ornstein-Zernike (OZ)equation,

h (k) = c (k) + ρh (k) c (k) (1)

for the total, h, and direct, c, correlation functions, respec-tively, by employing the first-order perturbation expansion

h (k) = h0 (k) + h1 (k) , c (k) = c0 (k) + c1 (k) , (2)

around the Percus-Yevick OZ solution (h0 and c0) for the fluidof hard spheres. The first-order correction terms, h1 and c1,are then forced to satisfy the mean-spherical approximation(MSA) type of closure condition,

c1 (r) = −βu (r) , for r ≥ σ, and(3)

h1 (r) = 0, for r < σ,

0021-9606/2013/138(16)/164107/13/$30.00 © 2013 AIP Publishing LLC138, 164107-1

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164107-2 Hlushak, Hlushak, and Trokhymchuk J. Chem. Phys. 138, 164107 (2013)

where ρ is the number density, β = 1/kBT (kB is theBoltzmann constant and T is the temperature), σ is the diame-ter of a hard core and u(r) is the pair interaction potential. Forsome functional forms of the u(r), including that of Yukawaand both the HCSW and HCSS interactions, the solution ofa perturbative OZ equation (1)–(3) can be performed analyti-cally. The FMSA theory was later shown43 to be identical tothe inverse temperature expansion of the full MSA solution.44

While the FMSA is a simpler theory, it is only slightly lessaccurate than the full MSA solution, but its analyticity makesit a good candidate for employment in different related ap-proaches such as the density functional,45–47 renormalizationgroup,48 and modern perturbation theories.49 Moreover, struc-tural properties predicted by the FMSA are linear with respectto the interaction potential. This seems to allow the straight-forward application of the FMSA description to the model flu-ids with complex interaction potentials like superposition ofthe attractive and repulsive Yukawa tails50 or combination ofthe discrete square-well and square-barrier potentials.10 How-ever, looking for such an application of the FMSA theory tothe description of the HCSS fluid, one should be aware thatin contrast to the case of simple fluids, the MSA-type of theo-ries are not well suited in the case of the core-softened modelpotentials.4, 51 This fact also makes questionable a straight-forward employment of the FMSA theory in other theoreticalapproaches to study the core-softened fluids as it was alreadydiscussed45–49 for simple potentials.

Therefore, from all purposes it is tempting to develop asimple reliable theory for the HCSS fluid that will be basedon the FMSA solution. In the current work we will try to dothis by improving the FMSA radial distribution function us-ing the exponential approximation52, 54, 55 or more correctlythe simplified exponential (SEXP) approximation.56 It wasalready shown that exponential approximation agrees wellwith computer simulation data in the case of the attractiveYukawa55 and Lennard-Jones57 fluids. Thus, we will try tocomplement these studies by assessing the performance of theSEXP/FMSA theory in the description of the HCSS fluid. Tothis end we also report the Monte Carlo simulation data onthe structure and thermodynamic properties of the HCSS fluidthat will be used to test the theoretical predictions.

II. GENERAL CONSIDERATION

This section begins by introducing the discrete poten-tial fluid model of interest, namely, the HCSS fluid, and dis-cusses briefly the drawbacks of its description within theconventional FMSA theory. To proceed further, we outlinethen the details of Monte Carlo computer simulations ofthe HCSS fluid to obtain the structure and thermodynamicsdata that we will need to discuss more rigorously our argu-ments. Right after this we recall all necessary equations of theFMSA solutions towards the radial distribution function ofthe HCSS fluid. By comparing the results of the FMSA andSEXP/FMSA theories with simulation data we are looking tojustify the usage of the exponential approximation for the ra-dial distribution function in the case of HCSS fluid. Our nextstep then will be to present the equations for the SEXP/FMSAthermodynamics of the HCSS fluid derived within the energy

route2 that utilizes an exponentially improved radial distribu-tion function. We will do this by evaluation of the internalenergy E using its definition through the radial distributionfunction,

βE

N= 2πρβ

∫ ∞

0drr2g (r) u (r) , (4)

where N is the number of species (particles or molecules) inthe system. By employing the exponential (SEXP) approx-imation for the g(r), Eq. (4) represents in fact the founda-tion of the theory that here is referred to as the SEXP/FMSAtheory.

A. Towards a model and its theoretical treatment

We consider the model fluid composed of sphericalspecies that interact with each other via the HCSS potential

u(r) =

⎧⎪⎨⎪⎩

∞, r < σ

ε, σ < r < λσ

0, r > λσ

, (5)

where σ is the hard-core diameter, ε > 0 is the height ofthe square shoulder, while parameter λ controls the widthof the shoulder. In this exploratory study we limit ourselvesby the values of λ that do not exceed 2, i.e., the square-shoulder width parameter will vary in the range 1 < λ ≤ 2.

The HCSS potential (5) is the discrete version of a moregeneral fully repulsive model potential in the statistical theoryof complex fluids – the hard-core plus continuous repulsiveYukawa potential that is widely used to mimic the interac-tion between like-charged species such as charged particlesin colloidal dispersions, etc. The main defect of the MSA-like theories for these so-called core-softened fluids consistsin an appearance of the negative portions on the radial dis-tribution function for the range of distances associated withthe repulsive shoulder.4, 51 This reflects itself, in particular, inthe change of the sign of the theoretically predicted internalenergy of the core-softened fluid. Namely, the MSA-like the-ories are predicting the negative values for internal energy ofthe repulsive Yukawa fluid model51 that is a nonsense for thesystem with a purely repulsive interaction.

That example with the repulsive Yukawa model is impor-tant since there is a general solution of the FMSA theory16, 58

for the potential model (5) and this theory already was shownto perform well for the HCSW fluid model, i.e., for the casewhen energy parameter in Eq. (5) is negative, ε < 0. Thus,even if it has not been documented yet, we are pretty sureand it will be shown later on in this section, that straight-forward application of the conventional FMSA theory to de-scribe the HCSS fluid will suffer of the similar drawbacksalready discussed for the repulsive Yukawa fluid. Therefore,certain modifications within the FMSA solution are requiredto have this theory be suited for the studies of the HCSS fluid.

B. Computer simulation details

To justify the SEXP/FMSA theory which will be out-lined in Subsection II D, we have performed both canonical

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164107-3 Hlushak, Hlushak, and Trokhymchuk J. Chem. Phys. 138, 164107 (2013)

constant volume (NVT) and constant pressure (NPT) MonteCarlo simulations. For these purposes the slightly modifiedcodes taken from the book of Frenkel and Smit,59 were used.

The radial distribution functions (RDF) were obtainedfrom the constant volume MC simulations for N = 1000particles in a cubic simulation box with periodic boundaryconditions. Simulations were performed during 5 × 103 equi-libration steps and the next 5 × 103 steps were used to collectthe averages. Each step consisted of 104 attempts to displaceeach particle.

The constant pressure MC simulations were used to ex-plore the pressure and internal energy dependence on the den-sity. Simulations were performed for N = 500 particles using5 × 103 equilibration steps and 5 × 103 production steps.Each step consisted of 105 attempts to displace each particleand, additionally, 102 attempts to change the box volume.

To obtain the data for chemical potential, the grandcanonical MC simulations were performed. The size of sim-ulation box was fixed at the volume (8σ )3 and standard peri-odic boundary conditions were applied. The MC step consistsof an attempt to insert the molecule, an attempt to remove themolecule, or an attempt to change the position of a randomlyselected molecule. We used 107 MC steps for both the equili-bration and production runs.

Five different square-shoulder width parameter values,λ = 1.1. 1.2, 1.3, 1.5, and 1.8 were considered for the HCSSfluid at the reduced temperature T∗ = kBT/ε = 0.5. The tem-perature was chosen to be such a low in order to observeclearly the deficiencies of the theories.4, 51

C. FMSA theory for the radial distribution functionof the HCSS fluid

Here we will recall the results of previous studies58 thatconcern analytical equations derived within the FMSA theoryfor the radial distribution function and that have been used todescribe the structure of the HCSW fluid and in particular, forthe case of an interaction range parameter values of λ < 2.Following conventional FMSA theory, the radial distributionfunction gFMSA(r) reads

gFMSA (r; β) = g0 (r) − βεg1 (r) , (6)

where g0(r) is the radial distribution function of the hard-sphere fluid while g1(r) is the FMSA first-order correctionterm that can be written as

g1 (r) = g11 (r) + g12 (r) , (7)

consisting of two parts given by

g11 (r) = σ

r

2∑i=0

(Aif

Ai (r) + Bif

Bi (r)

), (8)

g12 (r) = − (1 − η)8 σ

r

{2∑

i=0

[T (ti) (r/σ − λ)

S21 (ti) S2 (−ti)

− S2 (ti) T (ti)

S31 (ti) S2 (−ti)

+ T1 (ti) S (−ti) + 2T (ti) S1 (−ti)

S21 (ti) S3 (−ti)

]eti (r/σ−λ)

+2∑

i=0

[T (−ti) (r/σ − λ)

S21 (ti) S2 (−ti)

+ S2 (ti) T (−ti)

S31 (ti) S2 (−ti)

+ T1 (−ti) S (−ti) − 2T (−ti) S1 (−ti)

S21 (ti) S3 (−ti)

]e−ti (r/σ−λ)

}H (r/σ − λ) ,

(9)

where

Ai = − (1 − η)8

S21 (ti)

(λt5

i − t4i

)e(λ−1)ti ,

Bi = (1 − η)8

S31 (ti)

([λ2S1 (ti) − λS1 (ti) − λS2 (ti)]t

5i (10)

+ [S1 (ti) + S2 (ti) + 4λS1 (ti)] t4i − 4S1 (ti) t3

i

)e(λ−1)ti ,

f Ai (r) =

[t6i

S2 (−ti)(r/σ − 1) − 6t5

i

S2 (−ti)− 2t6

i S1 (−ti)

S3 (−ti)

]e−ti (r/σ−1)

+2∑

j=0

[t6j (r/σ − 1)

S21 (tj )(ti + tj )2

− t6j S2(tj )

S31 (tj )(ti + tj )2

+ 4t6j + 6ti t

5j

S21 (tj )(ti + tj )3

]etj (r/σ−1), (11)

f Bi (r) = t6

i

S2 (−ti)e−ti (r/σ−1) +

2∑j=0

[t6j (r/σ − 1)

S21 (tj )(ti + tj )

− t6j S2(tj )

S31 (tj )(ti + tj )

+ 5t6j + 6ti t

5j

S21 (tj )(ti + tj )2

]etj (r/σ−1).

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164107-4 Hlushak, Hlushak, and Trokhymchuk J. Chem. Phys. 138, 164107 (2013)

The function H(x) in Eq. (9) is the Heaviside step functionand because of its definition, H(x) = 1 only for x > 0 and H(x)= 0 in all other cases, it follows that the second term, g12(r),in Eq. (7) is nonzero for r > λσ only. Here we also introducednotation T1(t) = T′(t), where T(t) = λt11 + t10, while

tj = (−2η + (2ηf )1/3(y+cj + y−c−j ))

1 − η, j = 0, 1, 2

(12)with

f = 3 + 3η − η2, y± =(

1 ±(

1 + 2η4

f 2

)1/2)1/3

,

c = e2πi/3, i2 = −1,

are three roots of the equation

S (t) = (1 − η)2 t3 + 6η (1 − η) t2 + 18η2t

− 12η (1 + 2η) = 0.

In the above equations η = πρσ 3/6 and is the so-calledpacking fraction parameter, typical for the systems with ahard core. The other yet undefined quantities, introduced inEqs. (8)–(12) are as follows

S1 (t) = 3 (1 − η)2 t2 + 12η (1 − η) t + 18η2,

S2 (t) = 6 (1 − η)2 t + 12η (1 − η) .

For the radial distribution function g0(r) of the hard-sphere fluid in the range of distances σ ≤ r ≤ 2σ we have

g0 (r) = σ

r

2∑i=0

L (ti) tie(r−σ )ti /σ

S1 (ti)(13)

with

L (t) =(

1 + η

2

)t + 1 + 2η.

The results for the radial distribution functions gFMSA(r)of the HCSS fluid that follow from Eq. (6) are presented inFigs. 1–4 (see the dashed lines) where they can be verifiedagainst the MC simulation data. Without going into much de-tails we wish to point out that as expected, at temperatures aslow as T∗ = kBT/ε = 0.5 and at low densities, the radial distri-bution function gFMSA(r) becomes partially negative, particu-larly, in the regions of distances r associated with the range ofthe square-shoulder repulsion. This flaw of the conventionalFMSA theory will be fixed by employing the exponential ap-proximation.

D. SEXP/FMSA theory for the thermodynamicsof the HCSS fluid

Within exponential approximation the radial distributionfunction of the HCSS fluid assumes the form

gSEXP(r; β) = g0 (r) exp[−βεg1 (r)]. (14)

By comparing Eqs. (14) and (6), we note that gSEXP(r) is al-ways positive, that is due to the FMSA correction residingin the exponent. Additionally, the radial distribution functionthat is defined by Eq. (14) is asymptotically correct in the low

density limit, where gSEXP(r) ≈ exp [− βu(r)]. The results forthe radial distribution functions gSEXP(r) of the HCSS fluidthat follow from the SEXP/FMSA theory are presented inFigs. 1–4 as well (see the solid lines). Although some quan-titative inaccuracies in the shape of resulting radial distribu-tion function still remain, one can see significant qualitativeimprovement in gSEXP(r) in comparison with that of the con-ventional FMSA theory and against the computer simulationdata.

All these improved features of the exponential approxi-mation also are preserving within the exponential approxima-tion based on the conventional MSA theory as well. The mainadvantage of usage of the FMSA theory is that it is much sim-pler than that of the MSA solution at nearly the same levelof accuracy. To illustrate this claim we are pointing out thatin contrast to the MSA theory, the correction function g1(r)given by Eq. (7) does not depend on the temperature. Becauseof this, there is very simple linear temperature dependence ofthe FMSA correction for the radial distribution function. Thisfeature will be exploited in the following to write down thethermodynamics within the SEXP/FMSA theory.

To evaluate the equations for thermodynamic propertieswithin the SEXP/FMSA theory, we employ the energy routedefined according to Eq. (4). In the case of the HCSS fluid itcan be simplified as follows:

βE

N= 2πρβ

∫ ∞

0drr2g(r; β)u (r)

= 12ηβε

σ 3

∫ λσ

σ

drr2g(r; β). (15)

Exactly this equation was a starting point for Tang and Lu58

in their derivation of the thermodynamics of the HCSW fluid(ε < 0) within the FMSA theory where radial distributionfunction g(r; β) assumes the form given by Eq. (6). In thatcase, the distance integral in Eq. (15) can be evaluated analyt-ically that allows to express all FMSA thermodynamics in theform of analytical equations as well.

In the present case of exponential approximation for theradial distribution function, Eq. (14), the equations for ther-modynamic properties will not be fully analytical expressionsbut will contain integration that has to be performed numeri-cally. To derive the expression for Helmholtz free energy weobtain

β (A − A0)

N=

∫ β

0dβ ′ E

N

= 12ηε

σ 3

∫ λσ

σ

drr2∫ β

0dβ ′gSEXP(r; β ′), (16)

where A0 is the Helmholtz free energy of the reference systemof hard spheres and g(r; β) is given by Eq. (14). Taking intoaccount that g0(r) and g1(r) in Eq. (14) both are independentof the temperature, the inverse temperature integral in Eq. (16)could be evaluated analytically

1

β

∫ β

0dβ ′gSEXP(r; β ′) = gSEXP(r; β)ϕ0(−βεg1(r)), (17)

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164107-5 Hlushak, Hlushak, and Trokhymchuk J. Chem. Phys. 138, 164107 (2013)

FIG. 1. Radial distribution functions of the HCSS fluid with square-shoulder width parameter λ = 1.1 at the reduced temperature T∗ = kBT/ε = 0.5 and forseveral densities ρ∗ = ρσ 3 = 0.1, 0.2, 0.4, 0.6, and 0.8 as it is specified in the figure. The dashed (black) lines denote the results of the conventional FMSAtheory, while the solid (red) lines mark the results of the SEXP/FMSA theory. Symbols correspond to the MC simulations data. Insets show radial distributionfunctions of the MC simulation for larger distances from the centre.

where ϕ0(x) = (1 − exp ( − x))/x. This results in the followingequation for the Helmholtz free energy

β (A − A0)

N= 12ηβε

σ 3

∫ λσ

σ

drr2gSEXP(r; β)ϕ0(−βεg1(r)),

(18)

where the remaining integral over r is evaluated numerically,using the Gaussian quadratures. The latter is possible, since,by construction, the radial distribution function gSEXP(r) iscontinuous inside r ∈ [σ , λσ ]. Nonetheless, to be strict, weshould note, that gSEXP(r) is not a polynomial inside the shoul-der region, and Gaussian integration of the radial distributionfunction still is an approximation.

The chemical potential and pressure are computed usingstandard thermodynamic relations

βμ = ∂

∂ρ

(βA

V

), (19)

βP = βμρ − βA. (20)

Although the numerical integration over distance will remainin the equation for chemical potential, the density derivativestill can be performed analytically. To do this one needs totake the density derivative of the integrand in Eq. (18). Al-though such derivative is straightforward, it is still quite cum-bersome. Therefore, it will be presented in the Appendix.

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164107-6 Hlushak, Hlushak, and Trokhymchuk J. Chem. Phys. 138, 164107 (2013)

FIG. 2. Radial distribution functions of the HCSS fluid with square-shoulder width parameter λ = 1.2 at the reduced temperature T∗ = kBT/ε = 0.5 and forseveral densities ρ∗ = ρσ 3 = 0.1, 0.2, 0.4, 0.6, 0.8, and 0.7639 (η = 0.4) as it is specified in the figure. On the subplots for ρ∗ = ρσ 3 = 0.1, 0.2, 0.4, 0.6, 0.8,the dashed (black) lines denote the results of the conventional FMSA theory, while the solid (red) lines mark the results of the SEXP/FMSA theory. Symbolscorrespond to the MC simulations data. Insets show radial distribution functions of the MC simulation for larger distances from the centre. The bottom-rightsubplot for (ρ∗ = 0.7639 or η = 0.4) compares the radial distribution functions predicted by the FMSA and SEXP/FMSA theories with the results of theoptimized random phase approximation (ORPA) and optimized cluster theory (OCT).4 Thick solid lines of the bottom-right subplot denote the results of theMonte Carlo simulation. ORPA, OCT, and MC results for this subplot are from Ref. 4.

III. RESULTS AND DISCUSSION

As we already mentioned, the five values of the square-shoulder width parameter λ have been considered in thisstudy, namely, λ = 1.1, 1.2, 1.3, 1.5, and 1.8. The temperatureis fixed at the value T∗ = kBT/ε = 0.5 while the density ρσ 3

is changing in the range from 0.1 to 0.8. The correspondingresults for the radial distribution functions of the HCSS fluidthat include both the conventional FMSA and SEXP/FMSAtheoretical predictions as well as the MC simulations data, areshown in Figs. 1–4, and have been already briefly discussedin Sec. II. In addition, Table I presents the MC simulation data

for the characteristic values of g(r) at the distances of discon-tinuities, r = σ , r = λσ +, and r = λσ −. The results for ther-modynamic properties of the HCSS fluid are shown in Figs. 5and 6. In these figures the internal energy βE, compressibilityfactor Z = PV/NkBT , and chemical potential βμ that arepredicted by the SEXP/FMSA theory are compared againstthe results obtained within the conventional FMSA theory andMC simulation data.

The common feature for all considered values of thesquare-shoulder width parameter λ is the nonphysical neg-ative values of the radial distribution function gFMSA(r) inthe region of distances 1 ≤ r/σ ≤ λ at low and intermediate

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164107-7 Hlushak, Hlushak, and Trokhymchuk J. Chem. Phys. 138, 164107 (2013)

FIG. 3. Radial distribution functions of the HCSS fluid with square-shoulder width parameter λ = 1.5. The layout and notations are the same as in Fig. 2.

densities up to ρσ 3 ≈ 0.3. As expected, this results in the neg-ative values of the internal energy E predicted by the FMSAtheory in this range of densities. Surprisingly, other calculatedthermodynamic properties of the HCSS fluid (the compress-ibility factor Z and chemical potential μ) are not so sensitiveto that failure of gFMSA(r).

With respect of this, the mostly notable improvement dueto the SEXP/FMSA theory is obtained for the HCSS fluidwith shorter width of square shoulder. In our case, this couldbe observed for the fluid system with square-shoulder widthparameter values λ = 1.1 and 1.2 (see Figs. 1, 2, and 5). TheSEXP/FMSA theory not only corrects the nonphysical valuesof gFMSA(r) at low densities but, in general, produces resultsfor the HCSS fluid with these values of parameter λ, both forthe radial distribution function and thermodynamics, that arein good quantitative agreement with MC simulation data forall considered densities.

Although radial distribution function gSEXP(r), predictedby the SEXP/FMSA theory, continues to be rather accuratefor the larger width of square shoulder λ = 1.3 and even λ

= 1.5 (see Fig. 3), the corresponding thermodynamic prop-erties begin to deviate systematically in the region of higherdensities (ρσ 3 > 0.6, in the case of λ = 1.3 and ρσ 3 > 0.4,in the case of λ = 1.5). By analyzing the corresponding pro-files of the radial distribution functions, it seems that inaccu-racies in the shape of gSEXP(r) in the range of distances asso-ciated with the first coordination shell, 1 < r/σ < λ, mightbe the origin for these discrepancies; both the value of the ra-dial distribution function at the contact r/σ = 1, as well asthe values at the discontinuity r/σ = λ seem to be reasonablyaccurate.

The case with the HCSS fluid characterized by the largestconsidered value of the square shoulder width parameter,λ = 1.8, is not so clear. The radial distribution functions g(r)

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164107-8 Hlushak, Hlushak, and Trokhymchuk J. Chem. Phys. 138, 164107 (2013)

FIG. 4. Radial distribution functions of the HCSS fluid with square-shoulder width parameter λ = 1.8. The layout and notations are the same as in Fig. 1.

that follow from the MC simulations are rather nontrivial, es-pecially, at larger densities (see Figs. 4). In particular, for thefirst time in this study we can see that going from densityρσ 3 = 0.6 to higher density ρσ 3 = 0.8, the radial distributionfunction contact value g(r = σ ) is decreasing. This resultsthat g(r) for the density ρσ 3 = 0.8 exhibits unusual maxi-mum at the distances r ≈ 1.1σ just before the contact r = σ .Despite that both these features are rather well shaped by theSEXP/FMSA theory, this does not seem to result in the accu-rate prediction of the thermodynamics (see Fig. 6). In partic-ular, the SEXP/FMSA theory still overestimates the values ofthe internal energy and chemical potential at high densities.At the same time, the similar values for the compressibilityfactor seem to be underestimated.

What is surprising, is that the conventional FMSA theoryis rather accurate at high densities, even when the correspond-ing radial distribution functions gFMSA(r) are not (Fig. 3).

Thus, there seems to be some cancellation of errors within theconventional FMSA/MSA theory at higher densities, whichreveals itself for the HCSS fluid systems with larger values ofthe square-shoulder width parameter λ, starting from λ ≈ 1.3.

The SEXP/FMSA theory, on the other hand, is more ac-curate at lower densities, where, for T∗ < 1.0, conventionaltheories predict negative regions of the radial distributionfunction. The density region, where the conventional theoriesfail, tends to widen with the increase of interaction strength(or decrease of temperature).4 This makes exponential ap-proximation attractive tool for description of such stronglyinteracting systems.

The accuracy of the SEXP/FMSA theory is limited bythe accuracy of the FMSA theory, which is a first-order termof the perturbation expansion around the reference systemof hard spheres. Therefore, the FMSA theory is most accu-rate for systems which are close to the reference system of

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164107-9 Hlushak, Hlushak, and Trokhymchuk J. Chem. Phys. 138, 164107 (2013)

TABLE I. MC simulation data for the values of the radial distribution function at the distances of discontinuities, i.e., r = σ and r = λσ of the HCSS fluidwith different values of the width of repulsive square shoulder. Several densities ρσ 3 are given in the table and temperature is fixed at T∗ = 0.5.

λ = 1.1 λ = 1.2 λ = 1.3 λ = 1.5 λ = 1.8

ρσ 3 σ λσ− λσ+ σ λσ− λσ+ σ λσ− λσ+ σ λσ− λσ+ σ λσ− λσ+

0.1 0.169 0.160 1.18 0.185 0.167 1.23 0.203 0.177 1.29 0.279 0.198 1.45 0.578 0.232 1.730.2 0.204 0.194 1.43 0.256 0.205 1.56 0.340 0.228 1.72 0.710 0.268 1.98 1.60 0.267 1.950.4 0.358 0.287 2.14 0.619 0.333 2.53 1.18 0.353 2.65 2.56 0.291 2.15 2.73 0.257 1.900.6 0.758 0.461 3.38 1.94 0.482 3.59 3.37 0.386 2.80 4.32 0.255 1.88 3.03 0.258 1.880.8 2.02 0.737 5.40 5.05 0.505 3.70 6.31 0.304 2.27 6.00 0.212 1.57 2.65 0.263 1.94

FIG. 5. Internal energy βE, compressibility factor Z = βP/ρ, and chemical potential βμ of the HCSS fluids with the square-shoulder width parameter λ = 1.1(left column) and λ = 1.2 (right column) at the reduced temperature T∗ = 0.5. The dashed (black) lines denote the results of the conventional FMSA theory,while the solid (red) lines mark the results of the SEXP/FMSA theory. Symbols correspond to the MC simulations data and insets show low density regions ofthe plots.

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164107-10 Hlushak, Hlushak, and Trokhymchuk J. Chem. Phys. 138, 164107 (2013)

FIG. 6. Internal energy βE, compressibility factor Z = βP/ρ, and chemical potential βμ of the HCSS fluids with the square-shoulder width parameter λ = 1.5(left column) and λ = 1.8 (right column) at the reduced temperature T∗ = 0.5. The dashed (black) lines denote the results of the conventional FMSA theory,while the solid (red) lines mark the results of the SEXP/FMSA theory. Symbols correspond to the MC simulations data. The filled triangles correspond to theMC simulations data of Zhou and Solana.38 Insets show low density regions of the plots.

hard spheres, i.e., for weakly and shortly interacting hard-coreparticles.

When compared to more advanced theories (see lastsubplot of Figs. 2 and 3) like the optimized random-phaseapproximation (ORPA)4 and the optimized-cluster theory(OCT),53 which are similar to the FMSA and SEXP/FMSAtheories, respectively, the latter two seem to be slightly lessaccurate. This is because the FMSA theory is a first-orderterm of the expansion of the full MSA theory, whereas theORPA theory is comparable in accuracy to the MSA theory.Similarly, the OCT theory, based on ORPA solution, seems tobe more accurate than the SEXP/FMSA theory. The formertends to overestimate the contact values of the RDF, while the

SEXP/FMSA theory tends to underestimate it. This is proba-bly connected to the fact that the ORPA theory predicts higherand more accurate values of the contact value than the FMSAtheory, while both of the theories underestimate the contactvalue of the radial distribution function.

IV. CONCLUSIONS

In this paper an exponential enhancement of the FMSAtheory for the non-trivial HCSS model of the so-called core-softened fluids is presented. What we are referring to theSEXP/FMSA theory is built up on the analytical solutionof the FMSA theory for the radial distribution function due

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164107-11 Hlushak, Hlushak, and Trokhymchuk J. Chem. Phys. 138, 164107 (2013)

to Tang and Lu.58 The main weakness of the conventionalFMSA theory itself for the considered HCSS fluid con-cerns the nonphysical negative values of the radial distribu-tion function gFMSA(r) (in the range of distances associatedwith the radius of the square-shoulder repulsive interaction)when the temperature is getting lower. Obviously, this arti-fact of the conventional FMSA theory causes the problemsfor the FMSA thermodynamics of the square-shoulder fluidas well.

In contrast, the radial distribution function gSEXP(r) re-sulting from the SEXP/FMSA theory is always positive and,additionally, provides with correct asymptotic in the low den-sity limit, where gSEXP(r) ≈ exp [− βu(r)]. From the resultsobtained in this study we can conclude that the SEXP/FMSAtheory, in general, is the most successful for the HCSS fluidcharacterized by shorter ranges of the repulsive square shoul-der, λ < 1.3. For the larger values of the square-shoulderwidth parameter, λ > 1.3, the SEXP/FMSA theory tends tooverestimate the impact of the repulsion at higher densities,ρσ 3 > 0.4, but still performs reasonably well in the range oflow densities.

We also wish to point out that thermodynamics predictedby the conventional FMSA theory behaves in quite differentmanner. As expected, for all values of the square-shoulderwidth parameter λ it gives the nonphysical negative values ofthe internal energy at low densities, although the results forthe compressibility factor and chemical potential are quitesatisfactory. At the same time, the thermodynamics obtainedwithin the FMSA theory for larger values of the square-shoulder width parameter, λ > 1.3, and higher densities,ρσ 3 > 0.4, is remarkably accurate. The most probable ex-planation of this fact is due to some sort of error cancellationoccurs. Therefore, a possibility to combine these two rather

simple and quite similar theories to provide an accurate de-scription of the HCSS fluid in the wide range of model pa-rameters and thermodynamic states look very promising.

ACKNOWLEDGMENTS

S.H. was supported by the U.S. Department of Energy(DOE), Office of Basic Energy Sciences, Geoscience Re-search Program, through Grant No. ERKCC72 to Oak RidgeNational Laboratory, which is managed for DOE by UTBattelle, LLC under Contract No. DE-AC05-00OR22725.

APPENDIX: CHEMICAL POTENTIAL OF THE HCSSFLUID WITHIN THE SEXP/FMSA APPROXIMATION

Expressions for the excess chemical potential is derivedby taking analytically the density derivative of the Helmholtzexcess free energy,

β�μ = ∂

∂ρ

[β (A − A0)

V

]

= −4πρβεσ 3∫ λ

1g0 (x) eβεg1(x)ϕ0 (βεg1 (x)) x2dx

− 2πρ2βεσ 3∫ λ

1

[∂g0 (x)

∂ρeβεg1(x)ϕ0 (βεg1 (x))

− g0 (x) eβεg1(x)ϕ1 (βεg1 (x)) βε∂g1 (x)

∂ρ

]x2dx,

(A1)

where ∂ϕ0(y)/∂y = −ϕ1(y) − ϕ0(y), ϕ1(y) = (1 − y − e−y)/y2.The density derivative ∂g0(x)/∂ρ of the radial distribution

function of the hard-sphere fluid given by Eq. (13), reads

∂g0 (x)

∂ρ= ∂ti

∂ρ

L (ti) ti(x − 1)

S1 (ti) xeti (x−1) +

[∂ti

∂ρ

(L1 (ti) ti

S1 (ti)+ L (ti)

S1− L (ti) S2 (ti) ti

S21

)1

x

+ πσ 3

6

(∂L (ti)

∂η

xti − 1

S1 (ti) x2− ∂S1 (ti)

∂η

L (ti) ti

S1 (ti)2 x

)]eti (x−1). (A2)

Next, since during the evaluation of thermodynamics we use radial distribution functions only in the 1 ≤ x ≤ λ region, thedensity derivative of the first-order contribution term, g1(x), includes contribution from the first term, g11(x), in Eq. (6) only.Thus, we have

∂g1 (x)

∂ρ= ∂g11 (x)

∂ρ= ∂Ai

∂ρf A

i (x) + Ai

∂f Ai (x)

∂ρ+ ∂Bi

∂ρf B

i (x) + Bi

∂f Bi (x)

∂ρ, (A3)

where Ai, Bi, f Ai (x), and f B

i (x) are given by Eqs. (11) and (12). The corresponding derivatives are bulky but quite straightfor-ward. It might be more convenient to calculate them numerically, but we have chosen to implemented them using followinganalytical expressions:

∂Ai

∂ρ=

[4πσ 3

3

(λt2

i − ti) − (1 − η)

[5λti − 4 + (

λt2i − ti

)(λ − 1)

]∂ti

∂ρ

+ 2 (1 − η)(λt2

i − ti)

S1 (ti)

(S2 (ti)

∂ti

∂ρ+ πσ 3

6

∂S1 (ti)

∂η

)](1 − η)7

S21 (ti)

t3i e(λ−1)ti , (A4)

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164107-12 Hlushak, Hlushak, and Trokhymchuk J. Chem. Phys. 138, 164107 (2013)

∂Bi

∂ρ=

{3 (1 − η)

4πσ 3

[λ ((λ − 1)S2 (ti) − S3) t5

i + (λ − 1) (5λS1 (ti) − S2 (ti) + S3) t4i

+ 4(1 + 4λ)S1 (ti) t3i − 12S1 (ti) t2

i

] ∂ti

∂ρ+ [

λ ((1 − λ)S1 (ti) + S2 (ti)) t5i

− ((1 + 4λ)S1 (ti) + S2 (ti)) t4i + 4S1 (ti) t3

i

] +[λ

((λ − 1)

∂S1 (ti)

∂η− ∂S2 (ti)

∂η

)t5i

+(

(1 + 4λ)∂S1 (ti)

∂η+ ∂S2 (ti)

∂η

)t4i − 4

∂S1 (ti)

∂ηt3i

]1 − η

8

}4πσ 3

3

(1 − η)7

S31 (ti)

e(λ−1)ti

− 3Bi

S1 (ti)

(S2 (ti)

∂ti

∂ρ+ πσ 3

6

∂S1 (ti)

∂η

)+ (λ − 1) Bi

∂ti

∂ρ, (A5)

∂f Ai (x)

∂ρ=

{(6t5

i (x − 1)

S2 (−ti)+ 2t6

i S1 (−ti) (x − 1)

S3 (−ti)− 30t4

i

S2 (−ti)− 24t5

i S1 (−ti)

S3 (−ti)

+ 2t6i S2 (−ti)

S3 (−ti)− 6t6

i S1 (−ti)2

S4 (−ti)

)∂ti

∂ρ− (x − 1)t5

i

S2 (−ti)

[ti(x − 1) − 6 − 2tiS1 (−ti)

S (−ti)

]∂ti

∂ρ

− πσ 3t5i

3S3 (−ti)

([ti(x − 1) − 6 − 3tiS1 (−ti)

S (−ti)

]∂S (−ti)

∂η+ ti

∂S1 (−ti)

∂η

)}e−ti (x−1)

+2∑

j=0

[2t5

j (x − 1)

S21 (tj )(ti + tj )2

((3 − S2(tj )

S1(tj )tj

)∂tj

∂ρ− tj

(ti + tj )

(∂ti

∂ρ+ ∂tj

∂ρ

))

+ t4j

S21 (tj )(ti + tj )2

(−6tj S2(tj ) + t2

j S3

S1(tj )+ 3t2

j S22 (tj )

S21 (tj )

+ 24tj + 30ti

(ti + tj )

)∂tj

∂ρ

+ 6t5j

S21 (tj )(ti + tj )3

∂ti

∂ρ− 2(4t6

j + 6ti t5j )S2(tj )

S31 (tj )(ti + tj )3

∂tj

∂ρ+ 2t6

j S2(tj )

S31 (tj )(ti + tj )3

(∂ti

∂ρ+ ∂tj

∂ρ

)

− 3(4t6

j + 6ti t5j

)S2

1 (tj )(ti + tj )4

(∂tj

∂ρ+ ∂ti

∂ρ

)− πσ 3

6

(2t6

j (x − 1)

S31 (tj )(ti + tj )2

∂S1(tj )

∂η

− t6j

S31 (tj )(ti + tj )2

∂S2(tj )

∂η− 3t6

j S2(tj )

S41 (tj )(ti + tj )2

∂S1(tj )

∂η+ 8t6

j + 12ti t5j

S31 (tj )(ti + tj )3

∂S1(tj )

∂η

)

+(

t6j (x − 1)

S21 (tj )(ti + tj )2

− t6j S2(tj )

S31 (tj )(ti + tj )2

+ 4t6j + 6ti t

5j

S21 (tj )(ti + tj )3

)(x − 1)

∂tj

∂ρ

]etj (x−1), (A6)

∂f Bi (x)

∂ρ=

[(6t5

i

S (−ti)2 + 2t6

i S1 (−ti)

S (−ti)3 − t6

i (x − 1)

S2 (−ti)

)∂ti

∂ρ− 2πσ 3t6

i

6S (−ti)3

∂S (−ti)

∂η

]e−ti (x−1)

+2∑

j=0

[6t5

j (x − 1)

S1(tj )2(ti + tj )

∂tj

∂ρ− 2t6

j S2(tj )(x − 1)

S1(tj )3(ti + tj )

∂tj

∂ρ− t6

j (x − 1)

S1(tj )2(ti + tj )2

(∂ti

∂ρ+ ∂tj

∂ρ

)

− 6t5j S2(tj ) + t6

j S3 (ti)

S1(tj )3(ti + tj )

∂tj

∂ρ+ 3t6

j S2(tj )2

S1(tj )4(ti + tj )

∂tj

∂ρ+ t6

j S2(tj )

S1(tj )3(ti + tj )2

(∂ti

∂ρ+ ∂tj

∂ρ

)

+ 30t5j + 30ti t

4j

S1(tj )2(ti + tj )2

∂tj

∂ρ+ 6t5

j

S1(tj )2(ti + tj )2

∂ti

∂ρ− 2

(5t6

j + 6ti t5j

)S2(tj )

S1(tj )3(ti + tj )2

∂tj

∂ρ

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164107-13 Hlushak, Hlushak, and Trokhymchuk J. Chem. Phys. 138, 164107 (2013)

− 2(5t6

j + 6ti t5j

)S1(tj )2(ti + tj )3

(∂tj

∂ρ+ ∂ti

∂ρ

)− πσ 3

6S1(tj )3(ti + tj )

(2t6

j (x − 1)∂S1(tj )

∂η

+ t6j

∂S2(tj )

∂η− 3t6

j S2(tj )

S1(tj )

∂S1(tj )

∂η+ 2

(5t6

j + 6ti t5j

)ti + tj

∂S1(tj )

∂η

)

+ (x − 1)

S21 (tj )(ti + tj )

(t6j (x − 1) − t6

j S2(tj )

S1(tj )+ 5t6

j + 6ti t5j

(ti + tj )

)∂tj

∂ρ

]etj (x−1). (A7)

In the above equations S3 = 6(1 − η)2 with η = πρσ 3/6, while tj (j = 0, 1, and 2) are given by Eq. (12). Correspondingdensity derivatives, ∂tj/∂ρ, read

∂tj

∂ρ= π

6

[tj

1 − η+ 1

1 − η

(−2 +

(2f + 2η

∂f

∂η

)y+cj + y−c−j

3 (2ηf )23

+ (2ηf )13

(∂y+∂η

cj + ∂y−∂η

c−j

))]. (A8)

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