Interfacial properties of the Mie n−6 fluid: Molecular simulationsand gradient theory results

Guillaume Galliero,1,a� Manuel M. Piñeiro,2 Bruno Mendiboure,1 Christelle Miqueu,1

Thomas Lafitte,3 and David Bessieres1

1Laboratoire des Fluides Complexes (UMR5150 with CNRS), Université de Pau et des Pays de l’Adour,B.P. 1155, 64013 Pau Cedex, France2Departamento de Física Aplicada, Facultade de Ciencias, Universidade de Vigo, E-36310, Vigo, Spain3Department of Chemical Engineering, Imperial College London, South Kensington Campus, SW7 2AZ,United Kingdom

�Received 7 October 2008; accepted 29 January 2009; published online 10 March 2009�

In a first part, interfacial properties of a pure monoatomic fluid interacting through the Mie n−6potential �n=8, 10, 12, and 20� have been studied using extensive molecular simulations. MonteCarlo and molecular dynamics simulations have been employed, using, respectively, the test areaapproach and the mechanic route. In order to yield reference values, simulations have beenperformed with a cutoff radius equal to 10�, which is shown to be sufficient to avoid long rangecorrections. It is shown that both approaches provide results consistent with each other. Using themolecular simulations results, it is demonstrated that a unique scaling law is able to provide anaccurate estimation of the surface tension whatever the repulsive exponent n, even far from thecritical point. Furthermore, it is shown that the surface tension of the Mie n−6 fluid is as wellaccurately described by a unique Parachor’s law. Density profiles are shown to be well representedby the tanh mean field profile, with slight deviations for the lowest temperatures and the smallest n.In addition, the interfacial width is shown to increase when n decreases �for a given reducedtemperature� and to follow the usual scaling behavior for not too low temperature. In a second part,interfacial properties of the Mie n−6 fluid computed by the gradient theory, coupled with anequation of state based on the Barker–Henderson perturbation theory, have been compared withthose obtained by molecular simulations. It is demonstrated that, even far from the critical point, thegradient theory is efficient to compute surface tensions and density profiles of this model fluid,provided the equation of state accurately model the phase behavior of the fluid involved �which isnot the case for n=8 in this study�. © 2009 American Institute of Physics.�DOI: 10.1063/1.3085716�

I. INTRODUCTION

An accurate knowledge of surface and interfacial prop-erties is fundamental for a large variety of practicalapplications.1–3 In particular, surface tension between a liq-uid and its vapor needs to be properly described because itcontrols many processes in reservoir and chemical engineer-ing. In addition, inhomogeneous fluids are by far more com-plex to deal with than homogeneous ones and so deservedincreasing interest from a more fundamental point of view.1–3

Among the tools to tackle the problem of interfacialproperties, the molecular simulations �MS� techniques can beoutlined. Monte Carlo �MC� and molecular dynamics �MD�are able to provide “exact” results for model fluids.4–6 Theseapproaches can be used both to test theories based on thesemodel fluids1,7–9 and to provide quasiexperimental results�limited by the quality of the molecular model� when experi-ments are hard to achieve.10 Apart from the difficulties en-countered to compute properly these interfacialproperties,5,6,11 the main limitation of such approaches comes

from the computational needs to obtain accurate values, es-pecially for realistic systems. Therefore, these MS tech-niques are still difficult to use from the engineer point ofview, despite an obvious growing interest. In fact, engineer-ing requires simple and quick approaches to estimate inter-facial properties. Among them, the gradient theory �GT� ofinhomogeneous fluid1,12,13 is one of the most promisingphysically based approaches to estimate surface tensions anddensity profiles through fluid interfaces. Nevertheless, theGT should be coupled with efficient equation of state �EoS�in order to provide accurate results on real systems.13

In this frame, the “molecular” based EoS such as thoseof the SAFT family14 could be valuable when coupled withthe GT in order to predict interfacial properties. Amongthem, we may cite the SAFT-VR Mie EoS,15 which hasshown to exhibit interesting features especially on secondderivative properties compared to more usual SAFT-like EoSbased on square well and Lennard-Jones �LJ� potentials.15,16

This EoS is based on a monomer term aiming at represent-ing, using the Barker-Henderson perturbation theory,17 thethree parameters Mie n−6 fluid model �where n is the repul-sive exponent and for n�12�. By Mie n−6 fluid model, itmeant nonassociative monomers in which intermolecular in-

a�Author to whom correspondence should be addressed. Electronic mail:guillaume.galliero@univ-pau.fr. Tel.: �33 5 59 40 7704. FAX: � 33 5 5940 7695.

THE JOURNAL OF CHEMICAL PHYSICS 130, 104704 �2009�

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teractions are described by a Mie n−6 potential. It is be-lieved that the good results obtained by the SAFT-VR MieEoS mainly stem from the use of this three parameters fluidmodel as a basis as recently emphasized.15,18–20 In the fol-lowing, the EoS restricted to the nonassociative monomerterm will be simply named Mie EoS. It is so expected thatthe SAFT VR Mie EoS, coupled with the GT, could be ofvaluable interest for estimating interfacial properties of realfluids, especially in associated systems. Nevertheless, beforeapplying the GT coupled with the SAFT-VR Mie EoS on realfluids, it is important to better understand the interfacialproperties of the Mie n−6 fluid through MS. In addition, itseems crucial as well to better quantify the limits of theresults provided by the GT+Mie EoS approach with thosededuced from MS on the Mie n−6 fluid.

In that context, using systematic MS of the Mie n−6fluid along the liquid-vapor coexistence line, the aim of thiswork is threefold:

�1� To provide reference interfacial properties values forthe Mie n−6 fluid �n=8, 10, 12 and 20� using both MDand MC simulations based on different techniques.

�2� To analyze the MS results obtained in the frame ofsome usual approaches employed to estimate theseproperties in simple real fluids �scaling laws, Parachor,corresponding states�.

�3� To compare the MS results with those provided byGT+Mie EoS in order to better quantify the limitationsof such a theoretical approach.

MC and MD simulations are performed using two dif-ferent approaches �the test-area �TA� method4 and the moreclassical mechanical route1 �MR�� to obtain the surface ten-sion and the density profiles of the Mie n−6 fluid.

In Sec. II, the fluid model employed in this work is de-scribed, followed by a brief description of the way in whichinterfacial properties are computed using MD and MC simu-lations with corresponding numerical details. Then, a concisedescription of the GT and the Mie EoS, is given. In Sec. III,all results obtained are provided, analyzed, and discussed.Finally, Sec, IV provides a summary of the findings.

II. THEORY AND MODELS

A. Fluid model

All interactions between fluid particles �isolated spheres�have been represented by the Mie n−6 potential, U, whichcan be expressed as

U =n

n − 6�n

6�6/n−6

����

r�n

− ��

r�6� . �1�

r is the intermolecular distance, � is the potential depth, � isthe position at which the potential is equal to zero �the so-called atomic diameter�, and n is the stiffness of the repulsiveslope of the potential.

Concerning MS, we used a truncated version of this in-teraction potential, i.e., UMS=U when r�rc and UMS=0elsewhere. The cutoff radius rc has been taken equal to 10�,except when stated. No long range corrections have beenincluded. This particularly large cutoff radius is used in order

to avoid as much troubles due to the cutoff distance and toprovide reference values without any long range correctionsthat are difficult to deal with when nonhomogeneous systemsare simulated.11,21,22

B. Molecular simulation of interfacial properties

1. Mechanic route approach

By employing the usual MR �Ref. 1�, the interfacialproperties of model fluids can be computed thanks to MS�both MD and MC�. In that frame, the surface tension of aliquid-vapor interface is estimated during diphasic simula-tions, with periodic boundary conditions in all directions,thanks to1

� =1

2

0

Lz

�PN�z� − PT�z��dz , �2�

where PN�z� and PT�z� are �relatively to the interface� thenormal and tangential components of the pressure, respec-tively, at position z and Lz is the simulation box width. Thismethod has been widely employed in the past1 and, even ifsome care should be taken in some cases,21,23 this approachis rather easy to implement.

2. Test-area method

As an alternative to the usual MR described previously,the recently proposed TA method4 aiming at determining theliquid-vapor interfacial properties using MS is receiving con-siderable attention. This new TA technique proposes an alter-native thermodynamic route, whose basis is the calculationof the interfacial tension through a free energy perturbation,�A, related to a change of the explicit interfacial area withina MS box, �S, according to the definition

� = lim�S→0

��A

�S�

NVT. �3�

This leads to the calculation of interfacial tension through theequation

� = −kT

�Slnexp�− �U

kT�� , �4�

where �U represents the variation in the configurational en-ergy associated with the total surface area variation �S. Theevaluation of this difference is done, according to the recom-mendations of the authors of Ref. 4, following a two stepprocedure, increasing and decreasing the surface area andusing a central difference calculation for the interfacialtension determination.

The versatility and ease of implementation of thismethod is remarkable, as well as its wide applicability, as itmay be combined with either MC or MD and may also beused for instance in interfaces involving solid phases. TheTA method has been applied so far to calculate the interfacialtension of fully flexible Lennard-Jones chains,24 differentwater models,25 or linear alkanes.11 Nevertheless, despite thesimplicity and low computing time requirements of this

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method, special care must be taken when considering thearea displacement magnitude, because it might lead to inac-curacies, as shown by Errington and Kofke.5

3. Simulations details

In this work, we employed the MR approach when per-forming MD simulations as in Ref. 26 and the TA approachwhen using MC simulations. Concerning MD simulations, anoncubic simulation box �Lz�Lx=Ly =2rc=20�� containinga planar liquid slab surrounded by its vapor has been em-ployed. In all situations, it has been verified that both liquidand vapor slabs are larger than 2rc. By doing so, we limitfinite size effects and ensure that no interactions occur be-tween the two liquid-vapor interfaces, i.e., a particle locatedin the first interface does not interact with another particlethat may interact with a third particle located in the secondinterface. To integrate the equation of motion, the velocityVerlet algorithm is used.27 A time step, �t, equal to0.003� / �� /M�1/2 has been employed. To maintain the de-sired temperature during simulations, a Berendsen28 thermo-stat with a large time constant �equal to 1000�t� has beenutilized for the coupling with the external bath at constanttemperature. In order to construct the initial system, the pro-cedure described in Ref. 23 has been employed. Then, afterequilibration �at least 105 time steps�, data have been col-lected during 4105 time steps. In order to estimate errorson the variables computed, the subblocks average methodhas been applied.27

Concerning MC simulations, first, a cubic simulationbox containing 3000 particles has been equilibrated usingNPT MC at close to equilibrium conditions in the liquidregion. After that, two empty cubic boxes were added at bothends of this initial system in the z direction, and this biphasicparalellepipedic box was allowed to evolve then in NVTMC. The formation of the liquid vapor interfaces was con-trolled by monitoring the density profiles along the z axis.Due to the large cutoff radius value, systems containing 6000molecules were used for systems where the final distancebetween surfaces in the biphasic simulation box used in thesecond stage of the calculation was smaller than 20�. Oncethe interface was stabilized, the calculation of interfacial ten-sion through the TA method was performed. A typical run ofthis stage of the biphasic box simulation consisted of 5105 stabilization cycles followed by 5105 productioncycles, and each cycle includes nP displacement moves,where nP stands for the number of molecules considered.The estimation of the uncertainties was obtained by the usualblock averaging method. The final interfacial tension valueobtained at each temperature was compared with that ob-tained using the MR, using Eq. �2�, where the pressure com-ponent profiles were computed according to de Miguel andJackson.29 The agreement between both methods was excel-lent.

C. Theoretical approach to interfacial propertiesestimation

1. The Gradient theory

The GT has been described extensively by severalauthors,1,12,13 therefore we present only its most significant

part for a pure fluid. In a pure fluid, the GT is based on theassumption that the density gradient in the interface is smallcompared to the reciprocal of the intermolecular distance r.On this assumption, the Helmholtz free energy can be ex-panded in a Taylor series around the uniform state withoutthe interface. For a plane interface between the two bulkphases of a pure fluid, the expansion is expressed as, keepingthe second order term,

A = S−

� f0���z�� +1

2c�d�

dz�2

dz� , �5�

where S is the interface area and c is the so-called influenceparameter. This parameter carries the information on the mo-lecular structure of the interface and essentially determinesthe density gradients response to the local deviations of thechemical potentials from their bulk value. c has a moleculartheoretical definition,7,12 but is usually not exactly calculable.

According to the Helmholtz free energy criterion appliedto Eq. �5� and assuming that the density dependence of theinfluence parameter can be neglected, the equilibrium den-sity must satisfy the following Euler–Lagrange equation,

cd2�

dz2 = � ��

���

T, , �6�

with the grand thermodynamic potential per unit volume

��T,�, � = a0��� − � . �7�

Multiplying by d� /dz and integrating over z gives

c

2�d�

dz�2

= �� = � − �b = � + P , �8�

where �b=−P is the equilibrium pressure. The boundaryconditions for the planar interface are ��z→−�=�v and��z→ +�=�l, where �v and �l are the equilibrium densities,respectively, in the vapor and liquid bulk phases. The surfacetension � is thus given by

� = −

c�d�

dz�2

dz = 2−

�����dz

= �2c�v

�l ������d� . �9�

The density profile, ��z�, in the direction perpendicular to theinterface can be determined by integration of dz=��c /2������d� deduced from Eq. �8� that gives

z = z0 + �0

� � c

2��d� , �10�

where z0 and �0 being arbitrarily chosen origin and density.Thus once the vapor-liquid equilibrium �VLE� has been

obtained, the only inputs of the GT are the Helmholtz freeenergy density �A /V� of the homogeneous fluid and the in-fluence parameter of the inhomogeneous fluid. The Mie EoSis used to model the VLE and the free energy density in theinterface.

104704-3 Interfacial properties of the Mie n−6 fluid J. Chem. Phys. 130, 104704 �2009�

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So, once the VLE is computed and adjusted, we deter-mined the influence parameters suitable for fitting the surfacetension data by writing the third definition of � under theform

c =1

2� �

�v

�l ���d�� . �11�

2. Mie equation of state

To construct a Mie EoS for nonassociative monomers,the residual Helmholtz energy is obtained from the Barkerand Henderson theory17 for soft-core segments. In thisframework the properties of the Mie n−6 fluid can be de-scribed by considering an equivalent potential UBH with ahard-core temperature-dependant diameter �BH,

UBH = � if r � �BH�T�

n

n − 6�n

6� 6

n−6����BH

r�n

− ��BH

r�6� if r � �BH�T� � , �12�

where

�BH�T� = 0

� �1 − exp�− U�r�kT

��dr . �13�

Thus, the Helmholtz free energy can be obtained as ahigh-temperature expansion with the hard spheres �HS� ofdiameter �BH representing the reference system

aM = aHS + �a1 + �2a2. �14�

In this expression we define a=A /NkbT as the reducedHelmholtz free energy per molecule and a1 and a2 are thefirst two perturbation terms associated with the attractive partof the Mie potential. The HS residual free energy is obtainedfrom the Carnahan and Starling30 expression,

aHS =4� − 3�2

�1 − ��2 , �15�

where �=���BH3 /6 is the packing fraction of the system.

Given the fact that the Mie n−6 potentials represent asum of a repulsive and attractive Sutherland potential, themean-attractive energy can thus be expressed as the sum oftwo Sutherland a1

s terms as

a1 = C�a1S�� = n� − a1

S�� = 6�� , �16�

where the mean-attractive energy a1s��� is given by the fol-

lowing expression:

a1s��� = 2���

�BH

r2��BH

r��

gHS�r�dr . �17�

Here the radial distribution function of the HS gHS is calcu-lated using the Malijevsky and Labik closure.31

The same treatment is used for the second-order pertur-bation term a2, which is evaluated by summing the fluctua-tion terms of two Sutherland systems,

a2 � C�a2S�� = n� − a2

S�� = 6�� . �18�

Note that while this decomposition is exact for the first order,it constitutes an approximation for the second-order term.Furthermore, noticing that a2

s is very small for large values of

�, Gil-Villegas et al.32 assumed that a2 could be approxi-mated as

a2 � C�− a2S�� = 6�� , �19�

with a2s calculated with the local compressibility approxima-

tion �LCA�

a2 � − ��KHS �

���

�BH

gHS�r���BH

r�2

r2dr . �20�

In this equation KHS represents the isothermal compress-ibility of HS evaluated by the Percus–Yevick expression,33

KHS =�1 − ��4

1 + 4� + 4�2 . �21�

The reader is referred to Ref. 15 for the detailed expressionsused to calculate the integrals in Eqs. �17� and �20�. It shouldbe noted that, in the present form, this Mie EoS is restrictedto cases for which n�12.

III. RESULTS

In this section, we make extensive use of dimensionlessunits, using � and � as the energy and the length scales. Alldimensionless variables are noted with a star as superscript.

A. Preliminary results

1. Cutoff radius impact

In a first step, MD simulations coupled with the MRapproach have been performed in order to quantify the ex-pected high impact of the cutoff radius on the interfacialproperties results. It should be reminded that no long rangecorrections have been included in our computations. In orderto avoid size effects and to focus only on the impact of thecutoff radius value, all simulations have been performed witha constant number of particles �=8000�. rc

� has been variedfrom 2.5 to 10 and n has been taken equal to 8, 12, and 20.For each Mie n−6 fluid, only one temperature has been stud-ied, T�=1.3 for n=8, T�=1 for n=12, and T�=0.8 for n=20.

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As expected, results provided on Fig. 1 clearly demon-strate the crucial influence of the cutoff radius on the valueof the surface tension, e.g., for rc

�=3, �� is roughly two timessmaller than for rc

�=10! This decrease in �� with rc� is con-

sistent with previous findings8,23,34 and is mainly related tothe reduction of the critical temperature with rc

� when using atruncated potential.34 In addition, when n is small, the reduc-tion in �� with rc

� is more pronounced than when n is large.This result is consistent with the fact that �U��rc

��� decreaseswhen n increases. Trends on equilibrium densities with thecutoff radius, see Fig. 2, are consistent with those noticed onthe surface tensions, i.e., a very strong impact of rc

� on thevalues, this effect increasing when n decreases.

In addition, as can be seen from Figs. 1 and 2, it seemsthat for a sufficiently large cutoff radius, the interfacial prop-erties of the tested Mie n−6 �for n�8� fluid reach a plateau,this plateau being attained earlier when n increases. As a ruleof thumb, the plateau is attained when �U��rc

���� �210−5.In the following, we used a large cutoff radius equal to 10�and we considered that such a cutoff radius provides a suf-ficiently good approximation of the full Mie n−6 potential.

2. Results versus literature

As a validation test, our surface tensions simulation re-sults, using both MD and MC with rc

�=10, have been com-pared with recent literature data21,22,35 for the LJ fluid. Itshould be noted that these literature data take into accountlong-range corrections using various schemes and that theyexhibit non-negligible deviations between other values, seeFig. 3.

It can be seen on Fig. 3 that there is good agreementbetween our results without long range corrections and thosefrom the literature for the whole range of temperature. Inaddition, it is interesting to note that both MD and MCschemes used in this work provide close results for the LJfluid. Nevertheless, it should be noted that the �� values pro-vided by our MD simulations are slightly higher than thoseprovided by our MC computations of a quantity around 0.03,see Table I.

B. Molecular simulations results

1. MD versus MC

In order to confirm the agreement found for the LJ fluidbetween MD and MC results, surface tensions and equilib-rium densities have been evaluated using the two differenttechniques for n=8 and 10. All data are provided inTable I.

As shown on Fig. 4, MD and MC results on equilibriumdensities �both liquid and vapor� are in good agreement for

FIG. 1. Impact of the cutoff radius on the surface tension of three Mien−6 fluids.

FIG. 2. Influence of the cutoff radius on the equilibrium densities of threeMie n−6 fluids. �a� Vapor branch. �b� Liquid branch.

FIG. 3. Surface tension of the LJ fluid. Circles: MD. Down triangles: MC.Squares: Ref. 22. Diamonds: Ref. 35. Up triangles: Ref. 21. Error bars are ofthe order of the symbol sizes.

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all n values tested. Nevertheless, when approaching the criti-cal point from the liquid branch, it appears that the MC val-ues of �l

�, for a given T�, are slightly smaller than the MDones, a fact that might be attributed to the difference in sizeof the simulation box used for each technique. This impliesthat the critical temperature estimated from MC data isslightly lower than the one using MD data.

Concerning surface tensions, see Fig. 4, MC and MDvalues are in excellent agreement, whatever n, taking intoaccount the fact that the technique used are different �TAmethod for MC, MR approach for MD�. However, �� valuesprovided by the MC simulations are systematically slightlylower �of an amount of �0.03� than those yielded by MDsimulations. This trend is consistent with the probable shiftof the critical temperature �Tc

MD�TcMC� noticed on equilib-

rium densities values, see Fig. 4.

2. Scaling law and Parachor

In this section, the MD results on surface tensions havebeen analyzed in the frame of two usual approaches to pre-

dict this property in simple pure real fluids: scaling law andParachor.2 In order to cover a wider range of repulsive expo-nent to perform this study, a new set of MD simulations hasbeen realized for n=20. Data are provided in Table I.

From a theoretical point of view, in a region close to thevapor-liquid critical point, the thermodynamic properties ofpure fluids are described by scaling laws.36 In particular sur-face tension follows:

�� = �0��1 −

T�

Tc��

, �22�

where �0� is a constant of the fluid studied, Tc

� is its criticaltemperature, and is a universal critical exponent which isequal37 to 1.26. The ratio T� /Tc

� corresponds to the so calledreduced temperature. In addition, as first postulated by vander Waals based on simple dimension consideration, one canwrite that

TABLE I. MS results on interfacial properties of the Mie n−6 fluid. Values in parentheses represent theuncertainty on the last digit.

n T�

MD MC

�� �v� �l

� �� �v� �l

�

8 1 1.467�18� 0.0034�4� 0.8111�24� 1.428�23� 0.0038�2� 0.8012�87�8 1.1 1.241�17� 0.0075�5� 0.778�26� 1.206�17� 0.0088�4� 0.7773�87�8 1.2 1.024�28� 0.0136�8� 0.7443�21� 0.993�17� 0.0139�3� 0.7434�89�8 1.3 0.818�21� 0.0241�11� 0.708�23� 0.780�16� 0.0237�6� 0.7012�94�8 1.4 0.606�18� 0.0375�10� 0.6685�29� 0.577�13� 0.0388�6� 0.6579�112�8 1.5 0.421�22� 0.0568�15� 0.6243�36� 0.412�8� 0.0568�16� 0.6186�34�8 1.6 0.264�26� 0.083�36� 0.5761�24� 0.229�17� 0.0876�20� 0.5687�17�8 1.7 0.118�17� 0.1293�27� 0.5081�35� 0.079�13� 0.1324�53� 0.4753�9�8 1.75 0.044�26� 0.1585�69� 0.4734�88�

10 0.8 1.230�19� 0.0021�4� 0.8247�35� 1.189�10� 0.0023�2� 0.8197�24�10 0.9 1.006�21� 0.0062�8� 0.7865�21� 0.981�20� 0.0065�3� 0.7820�35�10 1 0.794�19� 0.0128�7� 0.7461�27� 0.783�14� 0.0130�7� 0.7389�31�10 1.1 0.604�25� 0.0256�18� 0.7022�30� 0.578�15� 0.0252�9� 0.7016�12�10 1.2 0.416�28� 0.0461�23� 0.6504�28� 0.379�9� 0.0460�13� 0.6489�15�10 1.3 0.241�23� 0.0749�22� 0.5945�27� 0.209�13� 0.0754�23� 0.5835�11�10 1.4 0.084�17� 0.1244�23� 0.5122�22� 0.079�9� 0.1297�51� 0.4831�13�10 1.45 0.033�25� 0.1641�97� 0.4478�39�12 0.7 1.111�22� 0.0015�11� 0.8419�44� 1.073�29� 0.0026�3� 0.8364�39�12 0.75 1.007�15� 0.0031�5� 0.8202�28�12 0.8 0.898�19� 0.0055�4� 0.7983�37� 0.860�23� 0.0072�7� 0.7920�15�12 0.85 0.791�26� 0.0089�6� 0.7751�22�12 0.9 0.681�22� 0.0136�8� 0.7517�27� 0.656�23� 0.0145�4� 0.7459�6�12 0.95 0.579�21� 0.0215�9� 0.7264�25�12 1 0.479�19� 0.0297�11� 0.6998�29� 0.470�20� 0.0294�17� 0.6906�9�12 1.05 0.383�28� 0.0405�12� 0.672�32�12 1.1 0.295�18� 0.0568�21� 0.6383�34� 0.285�12� 0.0518�10� 0.6337�19�12 1.15 0.209�21� 0.0771�32� 0.6019�24�12 1.2 0.132�21� 0.1013�38� 0.5602�27� 0.113�11� 0.108�13� 0.5352�13�12 1.25 0.059�24� 0.1285�19� 0.5148�34�12 1.3 0.011�20� 0.2010�120� 0.426�11�20 0.75 0.475�18� 0.0198�8� 0.7718�28�20 0.8 0.38�25� 0.0298�9� 0.7347�27�20 0.85 0.282�16� 0.0448�12� 0.695�31�20 0.9 0.186�26� 0.0768�19� 0.6427�33�20 0.95 0.102�21� 0.0977�33� 0.5914�27�20 1 0.038�23� 0.1380�54� 0.5054�57�

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�0� = CTc

�1/3Pc�2/3, �23�

where Pc� is the critical pressure and C is a universal constant

independent of the fluid model. So, in that correspondingstates frame, the surface tension should follow whatever thefluid model,

�� = CTc�1/3Pc

�2/3�1 −T�

Tc��1.26

. �24�

Figure 5 clearly shows that the Mie n−6 fluid�n=8–20� surface tension respects very well this “universal”scaling law, Eq. �24�, for the whole range of temperatures

tested, simply by taking C=10.7. It should be noted that thecritical temperature and pressure needed in Eq. �24� havebeen taken from Ref. 38.

In addition to the scaling law, one of the most popularmethods for estimating surface tension is based on the so-called Parachor approach.2 This method, first proposed byMacleod,39 postulates, on the basis of experimental observa-tions, that there is a relation between � and the differencebetween the coexisting liquid and vapor densities, ��=�l

−�v,

�� = �Pa�������, �25�

where �Pa� is the Parachor and � the Parachor exponent. �Pa�is temperature independent and � ranges from 3, emergingfrom mean-field theory,1 to39 4. It is worth noticing that,using a simple dimension analysis, one can rewrite Eq. �25�as

��

Tc�1/3Pc

�2/3 = �Pa������

�c� ��

. �26�

It appears from Fig. 6 that the surface tension of the Mien−6 fluid is obviously related to the difference between thecoexisting vapor and liquid densities. Moreover, using theproposed scaling, all values lie on the same “universal” line.In fact, the Mie n−6 surface tensions can be very well esti-mated using the Parachor relation, Eq. �26�, taking �=3.55and �Pa��=0.11, see Fig. 6. It is interesting to note that thevalue obtained here for � is very close to the widely usedeffective value of 11/3 first proposed by Guggenheim40 andthe effective value in the range of 3.6 suggested by a recentreview of experimental data for several fluids.37

An interesting feature that emerges from these results isthat the surface tension of the Mie n−6 fluid follows a qua-siperfect corresponding states behavior versus n. Thus, thesurface tension cannot be used alone to discriminate betweenthe most appropriate repulsive exponent n, when modelingsimple real fluids with the Mie n−6 potential. This argumentfinds special relevance when a model including this type ofintermolecular potential among its theoretical foundations, as

FIG. 4. Equilibrium densities, left figure, and surface tensions, right figure,of the Mie n−6 fluids for various temperatures. Legends are the same forboth figures.

FIG. 5. Reduced surface tension of the Mie n−6 fluids. Critical values havebeen taken from Ref. 38.

FIG. 6. Reduced surface tension of the Mie n−6 fluids vs relative differencein equilibrium densities. Critical values have been taken from Ref. 38.

104704-7 Interfacial properties of the Mie n−6 fluid J. Chem. Phys. 130, 104704 �2009�

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for instance in the SAFT VR Mie EoS, is considered with theaim to characterize the thermophysical behavior of real fluid.

3. Density profiles

Usually, resulting from the original mean field van derWaals theory,1 the density profile across the interface isdescribed by

���z� =��v

� + �l��

2−

��v� − �l

��2

tanh�2z�

D� � , �27�

where D� is the interfacial thickness. It should be noted thatother relations have been proposed, but none seems perfectfor all states.41 In addition, close to the critical point, theinterfacial thickness36 follows a universal scaling law,

D� � �1 − T�/Tc��−0.63. �28�

As expected, the density profile provided by Eq. �27� isperfectly consistent with the one obtained by MD simula-tions for temperature close to the critical ones, whatever n,see Fig. 7, corresponding to thermodynamic conditions forwhich the density profile varies slowly across the interface.Nevertheless, when approaching the triple point temperature,the evolution of the density profiles across the interface ismore abrupt and so the mean field approach is less appropri-ated, see Fig. 7. As a result, the tanh form is not as adequateas for higher temperature to describe the MD density pro-files. Furthermore, this weak discrepancy is more pro-nounced when n decreases, see Fig. 7.

When analyzing the interfacial width, D�, deduced fromthe MD density profiles and Eq. �27�, different features ap-pear, see Fig. 8. When �1-T� /Tc

�� is smaller than �0.2, thescaling behavior, Eq. �28�, is well respected for the Mie n−6 fluid whatever n, but this is not the case for lower tem-perature. This is probably partly due to the fact that, at lowreduced temperature �T� /Tc

��, the tanh profile is not perfectlyadequate, see Fig. 7, and so the interfacial width deducedfrom Eq. �27� is not the “true” one.

Another interesting feature is that, generally, for a givenreduced temperature, the larger n, the smaller D�. This meansthat for a given reduced temperature the density profile

across the interface varies more abruptly when n is larger. Inaddition and in contrast with what has been found to occurfor surface tension, this implies that a perfect correspondingstates behavior on the density profile does not exist for Mien−6 fluid with different n.

C. Comparison between gradient theoryand MD results

One interesting characteristic of the surface tension isthat it allows exploring the phase diagram below the coex-istence curve �binodal curve�. In fact, when using modelsbased on EoS to estimate surface tension, as the GT, becauseof this “exploration” of the van der Waals loop, it is a severetest for the EoS in an area not accessible elsewhere �at leastbelow the spinodal curve�.

In the literature, there exist a lot of accurate correlations/theories for estimating the Helmholtz energy of the LJ fluidover the entire fluid region,42–44 apart from the one used inthis work. Duque et al.8 employed one of these EoS �Ref. 45�in order to combine it with the GT. By doing so, they esti-mated the ability of the GT to provide surface tension con-sistent with those obtained by MD simulations on the LJchain fluid model. They found a reasonable agreement with aconstant influence parameter c in Eq. �5� for short chains nottoo close from the triple point temperature.

1. Phase diagrams of the Mie n−6 fluid

In a first step, we computed the phase equilibrium curvesof the Mie n−6 �n=8, 10, and 12� fluids using the Mie EoSand compared them with MD results. As shown on Fig. 9,the EoS is able to provide excellent results for the Mie 10−6 equilibrium curve, good results for the LJ 12−6 fluid andonly reasonable results for the Mie 8−6. These results arenot surprising and consistent with previous findings in vari-ous one-phase thermodynamic conditions.19 In fact, the fail-ure of the Mie EoS for the softest potential was expectable asthe deviation from a HS potential dictates the convergencerate of the expansion employed. In addition, it should benoted that the second-order term in the Helmholtz free en-ergy expansion is not calculated exactly here as this would

FIG. 7. Density profiles along vapor-liquid interfaces for two Mie n−6fluids and different temperatures.

FIG. 8. Interfacial width for different Mie n−6 fluids and differenttemperatures.

104704-8 Galliero et al. J. Chem. Phys. 130, 104704 �2009�

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involve the knowledge of the third- and fourth-order distri-bution functions of the unperturbed HS system. We proposea simpler approach here by using the LCA in order to derivean analytical expression for the Helmholtz free energy. Wewish to draw attention to this since LCA is not an integralpart of the Barker and Henderson theory and its use might aswell explain the discrepancies found for very soft potentialswhere this approximation is probably less reliable.

2. Interfacial properties

To analyze the results provided by the GT+Mie EoS oninterfacial properties of the Mie n−6 fluid, the influence pa-rameter has been adjusted for each temperature so that thetheory provides the MD values of surface tension.

Results provided on Fig. 10 are consistent with the factthat, if the EoS is able to predict correctly the phase diagramof one substance, the influence parameter should be weaklydependent on the temperature �a slight increase with T��.Nevertheless, if the EoS is not sufficiently accurate, as forthe Mie 8−6 fluid, see Fig. 9, the influence parameter be-comes largely variable with the temperature and takes abnor-mally high value.

These results support the fact that using the GT with aninfluence parameter slowly increasing with temperature �orconstant in a first approximation� is able to provide a verygood estimation of the surface tension46,47 in simple fluids aslong as the EoS used models accurately the phase behaviorof the fluid involved. As an example, using c�=6.3 for theMie 10−6 fluid, the relative deviations between GT and MDsimulations surface tension values stay always smaller than4% for T� varying from 0.9 to 1.3.

Concerning the density profiles deduced from the GT,they are consistent with the MD ones when the equilibriumdensities are correctly predicted by the EoS, see Fig. 11. Thisconfirms that the GT is a thermodynamical model efficient todescribe the microstructure of simple fluids liquid-vapor in-terfaces even far from the critical point, i.e., even in caseswhere the density profile across the interface is not varyingvery slowly.

IV. CONCLUSIONS

In this work, the interfacial properties �surface tensionsand density profiles� of the pure Mie n−6 fluid �n=8, 10, 12,and 20� have been studied using MS and the GT coupledwith a Mie EoS based on a Barker and Henderson perturba-tion scheme. To perform the MS, both MC and MD ap-proaches have been employed, using the TA method and theMR, respectively.

In a preliminary step, using MS on truncated potentialswithout long-range corrections, the impact of the cutoff ra-dius on the results has been estimated �rc

� varying from 2.5 to10�. As expected, it has been noticed that a strong impact ofthe cutoff radius on the interfacial properties influence thatincreases when n decreases. Nevertheless, results haveshown that a plateau is reached for a sufficiently large cutoffradius whatever n. Consequently, in this study, we employeda 10� cutoff radius in order to limit as much as possible theeffects of the truncation. For the LJ fluid, it has been noticedthat such a cutoff radius allows obtaining results in agree-ment with those from the literature including long-range cor-

FIG. 9. Comparison between the phase diagrams obtained by MD simula-tions and the Mie EoS for three Mie n−6 fluids.

FIG. 10. Influence parameters vs temperature for three Mie n−6 potential.

FIG. 11. Density profiles along vapor-liquid interfaces for the Mie 10−6fluids using MD simulations and the GT+Mie EoS.

104704-9 Interfacial properties of the Mie n−6 fluid J. Chem. Phys. 130, 104704 �2009�

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rections. Additionally, MC and MD simulations have shownto provide results consistent with each other for the Mie n−6 fluid interfacial properties.

Then, using an adequate scaling on the MD results, it hasbeen demonstrated that a simple “universal” scaling law isable to provide a good estimation of the surface tensionwhatever the repulsive exponent n. In addition, results haveshown that the surface tension of the Mie n−6 fluid followsa universal Parachor’s law with an exponent of 3.55. Bothresults imply that the surface tension of the Mie n−6 fluidfollows a corresponding states behavior relatively to n. Thus,when modeling simple real fluids with the Mie n−6 poten-tial, this result indicates that the surface tension cannot beused alone to discriminate between the most appropriaterepulsive exponent n.

Concerning the density profiles obtained by MS, it hasbeen shown that the tanh mean field profiles fit well MDresults for all n values with slight deviations for the lowesttemperature and the smallest n. In addition, it appears thatthe interfacial width follows the usual scaling law, whatevern, for not too low temperature. Moreover, results have shownthat the lower n, the larger the interfacial width for a givenreduced temperature.

Finally, interfacial properties of the Mie n−6 fluid modelhave been estimated using the GT coupled with a Mie EoS.The results on both density profiles and surface tensions,using an influence parameter slowly increasing with tem-perature �or constant in a first approximation�, have shown tobe consistent with those obtained by MS on the Mie n−6fluid, except for n=8 for which the Mie EoS yields non-negligible errors. This result confirms that the GT coupledwith an accurate EoS is a thermodynamical model efficientto describe the microstructure of simple fluids liquid-vaporinterfaces even far from the critical point.

ACKNOWLEDGMENTS

One of us �G.G.� gratefully acknowledges computationalfacilities provided by TREFLE laboratory, whose supercom-puter was financially supported by the Conseil Régionald’Aquitaine. M.M.P. acknowledges CESGA �Santiago deCompostela, Spain� for providing access to computing facili-ties. This work was carried out in the framework of the co-operation project Acción Integrada HF2006-0230, �Ministe-rio de Educación y Ciencia, Spain�, and Programme desActions Integrées Picasso, �Egide, France�.

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