Computer simulations of liquid/vapor interface in …web.mst.edu/~hale/argonMC/References/JCP.1999.Trokhymchuk.LJ.liq... · Computer simulations of liquid/vapor interface in Lennard-Jones

Computer simulations of liquid/vapor interface in Lennard-Jones fluids:Some questions and answers

Andrij Trokhymchuka)

Instituto de Quımica de la UNAM, Circuito Exterior, Coyoacan 04510, Mexico D.F., Mexico

Jose Alejandreb)

Departamento de Quımica, Universidad Autonoma Metropolitana, Iztapalapa, Apdo. Postal 55-534, 09340,Mexico D.F., Mexico

�Received 23 February 1999; accepted 11 August 1999�

Canonical molecular dynamics �MD� and Monte Carlo �MC� simulations for liquid/vaporequilibrium in truncated Lennard-Jones fluid have been carried out. Different results for coexistenceproperties �orthobaric densities, normal and tangential pressure profiles, and surface tension� havebeen reported in each method. These differences are attributed in literature to different set upconditions, e.g., size of simulation cell, number of particles, cut-off radius, time of simulations, etc.,applied by different authors. In the present study we show that observed disagreement betweensimulation results is due to the fact that different authors inadvertently simulated different modelfluids. The origin of the problem lies in details of truncation procedure used in simulation studies.Care has to be exercised in doing the comparison between both methods because in MC calculationsone deals with the truncated potential, while in MD calculations one uses the truncated forces, i.e.,derivative of the potential. The truncated force does not uniquely define the primordial potential. Itresults in MD and MC simulations being performed for different potential models. No differencesin the coexistence properties obtained from MD and MC simulations are found when the samepotential model is used. An additional force due to the discontinuity of the truncated potential atcut-off distance becomes crucial for inhomogeneous fluids and has to be included into the virialcalculations in MC and MD, and into the computation of trajectories in MD simulations. The normalpressure profile for the truncated potential is constant through the interface and both vapor andliquid regions only when this contribution is taken into account, and ignoring it results in incorrectvalue of surface tension. © 1999 American Institute of Physics. �S0021-9606�99�52441-0�

I. INTRODUCTION

Computer simulations are one of the most powerful toolsof the modern statistical mechanical theory of condensedmatter. Generally, it is assumed that computer simulationsproduce exact data for a given potential model. Two mainfactors which can affect the accuracy of simulation data arecaused by computational limitations, i.e., system size andtruncation of interactions. Both of them have been discussedin literature and for the case of single-phase homogeneoussystems, reasonable criteria have been established.1 In thecase of two-phase systems, particularly when densities arevarying through the simulation cell, as in the case of liquid/liquid or liquid/vapor phase coexistence, it becomes morecomplicated.2

Two methodologies are usually used in direct computersimulations �coexisting phases are in physical contact andinterface region is presented� of phase coexistence influids:3–43 conventional canonical Monte Carlo �MC� andmolecular dynamics �MD�. Chapela et al.8 have pointed out

that in MC calculations one deals directly with the pair po-tential, while in MD calculations one uses the pair forces,i.e., derivative of the potential. Truncation of the potential isperformed in MC and truncation of the forces in MD algo-rithm. MC creates configurations according to energy criteriawhile MD uses force route. The truncated force does notdefine uniquely the primordial potential. It results in the MDand MC simulations being performed for different potentialmodels: MC for spherically truncated �ST� potential whileMD for spherically truncated and shifted �STS� potential.

The truncation of interactions has different consequencesdependent on the physical nature of the system under mod-eling, i.e., whether it is simple �nonpolar� or complex �polar,ionic, etc.� fluids. Electrostatic interactions �Coulombic, di-polar, and higher multipole� determine the peculiar proper-ties of a such systems �conductivity, dielectric permittivity,etc.� caused by charge and polarization fluctuations. Nonad-equate �not large enough� truncation of full interactionsmight change the physics of the original system. Examplesare water and aqueous solutions where application of ad-equate truncation procedure for electrostatic forces in inho-mogeneous simulations becomes crucial to preserve a physi-cally correct microscopic model, especially whenelectrostatic information is to be obtained. In particular, ithas been shown by Spohr44 that the use of truncated interac-

a�Permanent address: Institute for Condensed Matter Physics, NationalAcademy of Sciences of the Ukraine, Lviv 11, Ukraine.

b�Also at Departmento de Simulacion Molecular, Instituto Mexicana delPetroleo, Eje central Lazaro C’ardenas 152, Apdo. Postal 14-805, 07730,Mexico D.F., Mexico.

JOURNAL OF CHEMICAL PHYSICS VOLUME 111, NUMBER 18 8 NOVEMBER 1999

85100021-9606/99/111(18)/8510/14/$15.00 © 1999 American Institute of Physics

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tions might lead to unphysical results for electrostatic poten-tials. In contrast, for simple fluids, where the long-range in-teractions between molecules are dominated by dispersionforces, truncation does not change the physics of originalsystem, but slightly modifies original model due to thechanges in potential energy. Additionally, for the sphericallytruncated potential, an ‘‘impulsive’’ contribution to the pres-sure, due to the discontinuous change of the potential at cut-off distance, appears. In the modeling of single-phase simplefluids the changes in original model, introduced by trunca-tion procedure, have been discussed in literature �see, e.g.,Refs. 1,2� and found as not of great importance for the ma-jority of bulk properties. Is it true for a two-phase system?The review of the literature sources on the subject of thephase coexistence in simple fluids seems to indicate that itmay not be valid any longer.

To obtain a conclusive answer on the above question isindeed the main objective of the present article. To accom-plish this objective we have chosen a classical pure atomicLennard-Jones �LJ� 12–6 fluid. This model itself is of im-portance and is by far the best studied continuous potentialbecause it provides moderately well a description of liquid/vapor coexistence properties of nonpolar real fluids ofspherical �argon, krypton, xenon� and homonuclear diatomic�oxygen, nitrogen� molecules.3–22,45–47 LJ model also iswidely employed in modeling of phase coexistence in mix-tures of simple fluids23–27 as well as for more complex sys-tems, like liquid/vapor coexistence in nonpolar �chlorine,hexane, alkanes� and polar �water, methanol� molecularfluids,28–36 liquid/liquid interfaces in aqueous solutions �eth-anol, octane, hydrocarbon, benzene, dichloroethane�,37–41 bi-layers and monolayers in water,42,43 etc., where the disper-sion or atom–atom interactions are described by LJ potential.Therefore, it becomes important to have correct and unam-biguous computer modeling of phase coexistence in LJ fluid.

Despite a number of articles on liquid/vapor coexistencein LJ fluid published in the past, the situation is far fromclear. In particular, Smit and Frenkel48,49 using Gibbs en-semble MC �GEMC� simulations �in the context of the prob-lems discussed in the present study, GEMC simulations canbe considered as homogeneous or bulk, since coexistingphases are not in physical contact and no interface is pre-sented� have shown that in the case of LJ fluid differenttemperature–density coexisting envelopes are produced de-pending on how the potential is truncated, i.e., whether it isST or STS. In direct canonical simulations of phase equilib-ria, where the phases coexist through the interface, only theeffect of the value of cut-off distance on the properties ofliquid/vapor interface has been analyzed so far;12,15,17–19 theway the potential is truncated �ST or STS� has not been takeninto account. However, different truncation procedureschange the phase diagram and can by no means be ignoredwhen coexisting densities and properties of interfacial regionare to be obtained, and when different studies are to be com-pared. Nevertheless, this issue continues not to be suffi-ciently addressed by many authors �see, e.g., Refs. 12,15–17� and a high scattering of the results on coexistingdensities and surface tension in LJ fluid may be found be-cause �i� applications of canonical MC or MD simulations to

different potential models have not been distinguished, and�ii� the impulsive contribution from the derivative of thetruncated potential at cut-off distance to the pressure andforces has not been taken into account. As a consequence,the results obtained for liquid/vapor coexistence in LJ fluidare confusing and some questionable conclusions have beendrawn.

The differences between ST and STS models in simula-tions of liquid/vapor coexistence in complex fluids,28–43

where LJ interactions are part of total Hamiltonian, have notbeen addressed as well. Results of prior works on nonpolarmolecular systems show that LJ-type contribution to the co-existence properties might be significant,32–34,36,39 and there-fore a correct handling of truncated LJ interactions is impor-tant. In the simulations of liquid/vapor coexistence in highlypolar fluids, like water, the contribution of electrostaticforces prevails31 though care has to be taken since besidesthe LJ interactions, there is contribution of so-called ‘‘real’’part of electrostatic interactions separated from the full elec-trostatic interactions if Ewald summation technique is ap-plied. The real part has a complementary function form andits contribution depends on the parameters employed inEwald scheme.

Additional questions arise regarding simulation ofliquid/vapor interface in LJ fluids. Is the LJ potential an ad-equate model to describe the liquid/vapor coexistence andinterfacial properties in a real nonpolar fluid?4,6,46,47 To an-swer this question the simulations of the untruncated LJ po-tential are required. For coexisting densities and pressure,simulations of full LJ potential have been done using theGEMC method.50,51 For interfacial properties �e.g., surfacetension� it can be done either in the way similar to the single-phase case, applying the long-range corrections �LRC� to the‘‘reference’’ results obtained in direct canonical simulationswith truncated potential,5,8,15,21 or including the LRC schemeexplicitly in the direct canonical simulations. The first way isclearer and simpler but results obtained depend on the prop-erties of the reference system. In particular, it has beenshown by Blokhuis et al.21 that, corrected in such manner,surface tension is different for reference calculations per-formed with different cut-off radius. To apply LRC explicitlyin inhomogeneous simulations is not an easy task. Differentschemes to include LRC during the direct canonical simula-tions of the interface have been proposed recently17,18 anddistinct results have been obtained. Guo and Lu17 have simu-lated the liquid/vapor interface in LJ fluid using canonicalMC and performing LRC of the potential energy, pressuretensor, and surface tension. These authors have concludedthat without taking LRC into account, constant normal pres-sure profile through the interface and the vapor and liquidbulk regions �that is a necessary condition of the mechanicalequilibrium in canonical simulations� could not be reached insimulations with a cut-off radius smaller than 3.1�. We dis-agree with this conclusion. The MD simulations results re-ported in the literature indicate �see, e.g., Refs. 7,10� thatnormal pressure profile is constant for a cut-off radius assmall as 2.5�.

It is our primary interest to clarify these subtle points inthe comparison of canonical MD and MC simulations of

8511J. Chem. Phys., Vol. 111, No. 18, 8 November 1999 Liquid/vapor interface in Lennard-Jones fluids


liquid/vapor coexistence in LJ fluid using the same potentialmodel �ST or STS� and the same setup conditions in bothsimulation procedures. We will analyze how the differencesin the truncation of interactions enter the calculation algo-rithm for liquid/vapor coexistence, and how they affect theresults so obtained. We will show that the source of theproblems arising in MC simulations with truncated LJ poten-tial is not the small cut-off radius, as stated by Guo and Lu,17

but an additional impulsive contribution to the pressure dueto the discontinuous change of truncated potential at cut-offdistance which, in contrast to the single-phase system,1,2 can-not be ignored in phase coexistence calculations. As for theMD simulations, discontinuity of truncated potential contrib-utes to the forces as well. Taking these findings into accountpermitted us to reconcile most of the results existing in theliterature on MC and MD studies: MC and MD produce thesame results when they are properly applied to the samepotential models. To verify our approach, we have per-formed simulations in the wide window of thermodynamicstates scanning the one used in Gibbs and grand canonicalensemble MC simulations already reported in the literature.To get a conclusive answer on the question regarding thesize of truncation sphere large enough to achieve the fullpotential results, the simulations for cut-off radius as large as4.4� and 5.5� have been performed. Finally, the critical pa-rameters of LJ fluids and relation of LJ potential model to theliquid/vapor coexistence in real systems, like noble gases,are discussed.

The presentation is organized as follows: We describe inSec. II, the potential models. Section III consists of the simu-lation details and definitions of properties. In Sec. IV, a com-parison between typical results taken from literature andthose obtained by us using MD and MC methods for theliquid/vapor interface are presented. The new results for co-existing densities, components of the pressure tensor profiles,and surface tension are discussed in Sec. V. Then, conclu-sions and references are given.

II. POTENTIAL MODELS

The full Lennard-Jones potential, ULJ(r), is given by

ULJ�r ��4��

r � 12

��

r � 6� , �1�

where � and � are related to the diameter of LJ atoms and thestrength of the interparticle interaction, respectively. The ma-jority of reported MC simulations on the liquid/vapor inter-face are done for spherically truncated �ST� potential

UST�r ��ULJ�r ��R�r �� ULJ�r �, r�R

0, r�R, �2�

while most MD simulations are developed for the sphericallytruncated force

F�r ��dULJ�r �

dr�� 24�

r �2� �

r � 12

��

r � 6� , r�R

0, r�R

,

�3�

where �(x) is the unit step function: �(x)�0 when x�0 and�(x)�1 when x�0, and R is the cut-off radius. By integrat-ing Eq. �3� one obtains the primordial potential

USTS�r ��ULJ�r ��ULJ�R ��R�r �

�� ULJ�r ��ULJ�R �, r�R

0, r�R, �4�

which corresponds to the spherically truncated and shifted�STS� potential. Thus, the MD simulations with sphericallytruncated force corresponds to the STS potential modelrather than that of ST potential model. This fact has to betaken in consideration when MC and MD results are com-pared. The truncated force, which has to be used in MDsimulations to correspond to the ST potential model, is

F�r ��dULJ�r �

dr

�� 24�

r �2� �

r � 12

��

r � 6��ULJ�R ��r�R �, r�R

0, r�R

,

�5�

where �(r�R) is a delta function. It means, that atomscrossing in or out of the cut-off radius, R, drawn around anyother fixed atoms would feel in this case an additional im-pulse. This �-function impulsive contribution is inconvenientin MD simulations and usually they are performed with forcedefined by Eq. �3� that corresponds to the STS potentialmodel. One way to handle �-function in a computationalprocedure is to use its approximate representation52

��r�R ��r�R ��r�R�r �

r, as r→0, �6�

where r is a fixed parameter.

III. SIMULATION DETAILS

To study the liquid/vapor interfacial properties in LJfluid we have performed canonical MD and MC simulationsfor the same potential models, i.e., ST or STS, under thesame setup conditions, i.e., the same number of particles,box size, initial configuration, temperature region.

In the case of two-phase liquid/vapor system the choiceof simulation conditions such as number of particles, N, andsize of simulation cell, Lx�Ly�Lz , is very important sincethey may influence the obtained results �system size effect�.These factors have been analyzed in detail by Chapelaet al.,8 Holcomb et al.,15 and Chen.16 To avoid the depen-dences of the properties on system size it has been recom-mended to perform simulations with at least N�1000 par-ticles and with system dimensions Lx , Ly not smaller than10�. To eliminate the system size effects, simulations re-ported in the present study are performed with a parallelepi-ped cell of dimensions Lx�Ly�13.41� and Lz�39.81� ,applying periodic boundary conditions in x, y, and z direc-tions. The simulations are done using the STS and ST poten-tial models defined by Eqs. �2� and �4�, respectively, with

8512 J. Chem. Phys., Vol. 111, No. 18, 8 November 1999 A. Trokhymchuk and J. Alejandre


cut-off radius, R�2.5� . Additional MD simulations withcut-off radius of 4.4� and 5.5� were developed for ST po-tential model.

The simulation procedure was quite similar to that usu-ally used for LJ fluid with the constraint proposed by Leeet al.4 to prevent excessive translation motion of the center-of-mass. In our case, this constraint includes rejection of anytrial movement which leads to interchange of particles fromboth vapor phase regions only. We have investigated theliquid/vapor coexistence for a set of isotherms, T*�kT/� inthe range in between T*�0.70 and T*�1.127. These tem-peratures scan the range between the triple point and thecritical temperature for truncated LJ model. To prepare theinitial configuration for each temperature, we place a denseslab of N�2048 molecules in the middle of a simulation cellalong z direction. Such setting up allows us to obtain twovapor regions with a liquid slab in between. Therefore, wehave two free liquid/vapor interfaces perpendicular to the zaxis. Equilibration in MC was performed from 8�107 trialsto displace each particle and at least 108 trials for obtainingthe ensemble averages. MD equilibration was followed after50 000 time steps �*�0.005� and at least 150 000 timesteps for time averages. The density and pressure profileswere updated after every 2000 configurations in MC andevery 50 time steps in MD.

The density profiles, obtained at the end of each produc-tion run have been fitted to a tangent hyperbolic function todetermine the liquid and vapor coexisting densities. We alsocalculated the normal and tangential pressure profiles to-gether with the surface tension. The profiles of componentsof the pressure tensor were obtained using the Irving andKirkwood10,45 definition. The tangential component is

PT�z ��z � kT

�1

A � �i

�j�i

�xi j2 �yi j

2 �

2ri j

dULJ�ri j�

dri j

1

�zi j�

�� z�zi

zi j� �� z j�z

zi j� � . �7�

In this equation �(z) is the density profile along the z direc-tion, k is the Boltzman’s constant, T denotes the absolutetemperature, A�Lx�Ly is the surface area of one interface,xi j , yi j , zi j , and ri j are the separations of the centers-of-mass of particles i and j, and �¯ denotes a canonical en-semble average. The distance zi j is divided into Ns slabs ofwidth z , and the particles i and j contribute to the pressureif the slab contains the line connecting them. Each slab has1/Ns fraction of the total contribution from the i� j pairs.The normal component, PN(z), is given by an expression ofthe form of Eq. �7� but with zi j

2 instead of (xi j2 �yi j

2 )/2. In thecalculations of the components PN(z) and PT(z) one mustuse the derivative, dULJ(r)/dr , defined by Eq. �3� for STSpotential model, and by Eq. �5� in the case of ST potentialmodel, in both MC and MD simulations. The advantage ofcalculating the components of pressure tensor as a functionsof z is that we can establish that the system is at equilibriumby checking that PN(z) is constant and PN(z)�PT(z) awayfrom the interfaces.

The surface tension is calculated from the profiles of thecomponents of pressure tensor by using the mechanicaldefinition10,45

��1

2 0

Lz�PN�z ��PT�z ��dz . �8�

Due to the way of preparing and aging the simulation systemdescribed above, we will have two symmetric interfaces, thatis taken into account by the extra division by 2 in Eq. �8� toevaluate the surface tension.

IV. COMPARISON WITH PREVIOUS RESULTS

There were two prime reasons that pushed us to go intothis work. First, the analysis of the orthobaric densities ob-tained by us and those available from the literature on theMC and MD simulations of the liquid/vapor coexistence inLJ fluid. Figure 1�a� shows typical density profiles, �*(z*)��3�(z/�), obtained using canonical MC and MD simula-tions for reduced temperature, T*�0.92, with cut-off dis-tance, R�2.5� . One can see that the coexisting densities,

FIG. 1. Local density distribution, �*(z*), normal, PN*(z*), tangential,PT*(z*), and difference, PN*(z*)�PT*(z*), pressure profiles for truncatedLJ potential defined by Eq. �2�. The cut-off radius, R�2.5� , reduced tem-perature, T*�0.92. The solid and short-dashed lines correspond to MC andMD data, respectively. In both simulations derivative, dULJ(r)/dr , has beencalculated according to Eq. �3�.



resulting from MC and MD simulations, are not the same:liquid density from MC is higher than that from MD, whileMC vapor density is lower. The same thermodynamic statehas been simulated by other authors using both MC and MDmethods. Table I gives the orthobaric densities, �V* , �L* , andsurface tensions, �*��2/� , from those and our simula-tions. Rao and Berne,9 using MC, have obtained almost thesame densities as MD results of both our and Nijmeijeret al.12 calculations, while MC simulation results of Chapelaet al.8 are remarkably higher and close to the MC data of thiswork. Why are the densities so different? The second reasonwas the unphysical behavior of the normal component of thepressure tensor, PN(z), discussed by Guo and Lu17 from MCsimulations. This problem is summarized in parts �b� and �c�of Fig. 1 which show the profiles of the normal, PN*(z*)��3PN(z/�)/� , and tangential, PT*(z*)��3PT(z/�)/� ,profiles of pressure tensor obtained using MD and MC meth-ods. We can see that PN(z) and PT(z) components in thebulk phases are the same within MC or MD techniques butare different when MD and MC are compared: from MCsimulation the normal component is not constant through theinterface �exactly the same behavior has been obtained inMC simulations by Guo and Lu�17 but it is almost straightconstant when MD technique is used; PN(z) in MD is lowerthan in MC simulations. Why is PN(z) not constant in MCsimulations and different within MC and MD?

The answers to these questions are found to be in termsof the potential function used in MD and MC procedures.The results of Chapela et al.8 and MC of this work are for ST

potential model, while MC simulations of Rao and Berne,9

MD simulations of Nijmeijer et al.,12 and MD simulations ofthis work are for STS potential model. It has been shown bySmit and Frenkel48,49 from Gibbs ensemble MC simulationsthat these potential models have different phase diagrams. Inparticular, the temperature–density coexisting envelop isshrunk, i.e., orthobaric densities will be higher for the vaporpart of the phase diagram and lower for the liquid one, goingfrom the STS to ST potential model. That is exactly what weobserve for coexisting densities from Table I.

The normal and tangential pressure profiles resultingfrom MC and MD simulations discussed in Figs. 1�b� and1�c� have been calculated according to the definition �7� withdULJ(r)/dr given by Eq. �3�. This expression corresponds tothe STS potential model and is consistent with MD simula-tions for STS potential model. As for MC simulations, wheremolecule configurations have been generated for ST poten-tial model, dULJ(r)/dr should be calculated using Eq. �5�which consists of an additional contribution from discontinu-ity of ST potential. In direct simulations of phase equilib-rium, this term in Eq. �5�, to our knowledge, has been omit-ted in all MC simulations for ST potential model andtherefore, both profiles of the components of pressure tensorare not correct. Particularly, from Figs. 1�b� and 1�c� weobserve that the tangential component has different valuesfor the vapor and liquid regions, and the normal componentis not constant through the interface. This fact introduceserrors in the calculation of the surface tension. Figure 1�d�shows a comparison between the difference, PN(z)�PT(z). The surface tension, that according to Eq. �8� is thearea under these curves, will be higher and incorrect fromMC than that obtained from MD �what we observe fromresults labeled ‘‘this work’’ in Table I�. Table I contains alsoMC and MD results reported by other authors at T*�0.92and 0.7. The coexisting densities and surface tension followthe same trend, except MC from Rao and Berne9 which cor-responds to STS potential model.

In summary of this section: first, all orthobaric densitiesof liquid/vapor coexistence in LJ fluid �at T*�0.92 and0.70� collected in Table I which look rather scattered at thefirst glance, in reality are consistent when they are classifiedin accordance with the model employed in simulations; itdoes not matter whether MC or MD are used. Second, in allMC simulations performed so far for the ST potential model,the coexisting densities are correct, but the component of thepressure tensor profiles, and as consequence the surface ten-sion, unfortunately, are not. Third, to our best knowledge,any simulations for ST potential model of LJ fluid, using MDmethod, have been performed in the past; even ifauthors7,12,15,16 stated that ST potential is employed in theirMD studies, but since forces have been calculated accordingto Eq. �3�, their results are therefore for STS potential model.Fourth, apart from differences in potential models, we drawattention that different definitions of surface tension are usedin the literature �see footnotes and references given in TableI�; it complicates the analysis of results and introduces extraconfusion.

TABLE I. Liquid/vapor coexistence densities, �L* , �V* , and surface ten-sions, �*, obtained from computer simulation for LJ potential with cut-offradius, R�2.5� . MC and MD results of ‘‘this work’’ at reduced tempera-ture, T*�0.92, correspond to those shown on Fig. 1.

T* Method �L* �V* �* Model Source

0.918 MC 0.722 0.017 0.77a ST Chapela et al.b

0.92 MC 0.7065 0.0294 0.310 ST this work0.922 MC 0.65 0.05 0.42c STS Rao and Berned

0.92 MD 0.649 0.063 0.24 STS Nijmeijer et al.e

0.92 MD 0.6478 0.0532 0.197 STS this work

0.70 MC 0.812 0.0029 0.640 ST this work0.701 MC 0.807 0.004 1.28a ST Chapela et al.b

0.704 MD 0.790 0.005 0.94f STS Rao and Levesqueg

0.701 MD 0.788 0.010 1.12a STS Chapela et al.b

0.70 MD 0.7852 0.0067 0.610 STS this work0.70 MD 0.7861 0.007 0.65h STS Mecke et al.i

0.70 MD 0.787 0.006 0.57 STS Adams and Hendersonj

aThese values are the contributions to surface tension evaluated directlyfrom the simulations in that work, but apparently a factor of 1/2 was omit-ted in Eq. �A1�.

bReference 8.cIt seems that surface tension calculated from Eq. �7� of that work is for twosurfaces.

dReference 9.eReference 12.fEquation �4� of that work for surface tension missed a factor of 1/2, i.e.,result is for two surfaces.

gReference 7.hSurface tension was calculated from Eq. �4� of that work and is affectedslightly by long-range correction applied to the dynamics.

iReference 18.jReference 22.



V. RESULTS AND DISCUSSION

In order to shed more light on the interpretation of theexisting data for coexistence properties and surface tensionin LJ fluid, and to compare the results of MC and MD simu-lations performed for the same potential models, we havecarried out MC simulations using interaction functions givenby Eqs. �2� and �4�, while MD simulations have been runusing the force functions defined by Eqs. �3� and �5�, respec-tively. The second term in Eq. �5�, related to the discontinu-ity of the potential at r�R , has been computed according toEq. �6� as

F��U�R ��r�R �

��U�R �

r��r�R ��r�R�r �� . �9�

The parameter r in this expression determines the size ofthe spherical shell between two spheres centered on the par-ticle i and of the radii R and R�r . It means that all par-ticles, j�i , localized in such shell, will contribute to thevalue U(R)�(r�R). The numerical value of parameter rhas to be fixed by computational conditions �number of par-ticles in simulations, time of the runs, etc.� and could notinfluence obtained results. It follows that increasing/decreasing r will decrease/increase U(R)/r value, butsimultaneously it will increase/decrease, respectively, thenumber of pairs (i� j) which will contribute. We find that inour case a value of r�0.01� is optimal. This term wasincluded in the computational scheme to calculate the forcesin MD and the virial contribution to the pressure, in both MDand MC procedures, for the ST potential model.

A. Classical cut-off radius R�2.5�

The results of our calculations for local density distribu-tion resulting from MC and MD simulations at reduced tem-perature, T*�0.92, are shown in Fig. 2. The profiles of the

normal and tangential components of the pressure and thesurface tension profiles, for the same isotherm, are plotted inFigs. 3 and 4 for STS and ST potential models, respectively.From Figs. 2–4 it is seen that for the same potential model,within the error bars in the calculations, profiles of localdensity distributions, profiles of components of the pressuretensor, and surface tension, calculated using MD and MCtechniques are the same. In particular, in contrast to Fig. 1�b�the normal pressure profile, PN(z), resulting from MC simu-lations is constant through the interface as it is imposed bythe condition of hydrostatic equilibrium.

The orthobaric densities, �V* and �L* , vapor pressure,PV* , and surface tension, �*, from our calculations as well asthose available from the literature for a set of isotherms inthe range from T*�0.7 to T*�1.127, obtained with cut-offradius, R�2.5� , are collected in Tables II and III for theSTS and ST potential models, respectively. The results ofboth MC and MD simulations coincide when comparison areperformed for the same potential, namely, STS or ST. Nocontradictions are observed between our results and thosereported in the literature for coexisting densities if the priorresults are classified in accordance with model employed insimulations. It ensures that expression �9� provides the effi-cient way to evaluate the ‘‘impulsive’’ contribution, F , inboth MC and MD simulations.

At the same time, significant differences for all listedproperties between ST and STS potential models are ob-served. As for the coexisting densities, the differences be-

FIG. 2. Local density distribution, �*(z*), of the LJ fluid within the STS�part �a�� and ST �part �b�� potential models defined by Eqs. �4� and �2�,respectively. The cut-off radius, R�2.5� , reduced temperature, T*�0.92.Notations of the lines are the same as in Fig. 1. In MD simulations with STpotential, derivative dULJ(r)/dr , has been calculated according to Eq. �5�.

FIG. 3. Normal, PN*(z*), tangential, PT*(z*), and difference, PN*(z*)�PT*(z*), pressure profiles for the LJ fluid within the STS potential modeldefined by Eq. �4�. The cut-off radius, R�2.5� , reduced temperature, T*�0.92. Notations of the lines are the same as in Fig. 1. In both simulationsderivative, dULJ(r)/dr , has been calculated according to Eq. �3�.



tween both potential models have been already analyzed byPowles53 from Nicolas et al.54 empirical equation of state ofLJ fluid, and by Smit49 from the GEMC simulations. Toensure the global validity of the results of canonical simula-tions we build up the density–temperature coexistencecurves for both potentials which are compared in Figs. 5�a�and 6�a� with the data from Gibbs and grand canonical en-semble techniques reported by Smit49 for STS and by Finnand Monson,55 Panagiotopoulos,56 and Wilding57 for ST po-tential model, respectively. From Figs. 5�a� and 6�a� is seenthat ST and STS potential models truncated at 2.5� producedifferent density–temperature coexistence lines with criticalpoint shifted to low temperatures going from ST to STSpotential. The differences between the two models are moresignificant for liquid coexisting densities and decrease whenone goes to low temperatures. Observed agreement of theresults from different statistical ensembles confirms that be-low the critical temperature, direct simulations of phase equi-libria by canonical MC and MD techniques are equivalent toGibbs ensemble simulation and grand canonical ensemblesimulations combined with histogram reweighting tech-niques.

For the pressure and surface tension the differences be-tween STS and ST models are discussed for the first time. Inthe case of pressure we plotted function ln PV* �Fig. 7�a��. Weobserve that simulation data lie perfectly on the two differentstraight lines, i.e., two models have significantly differentcoexisting pressures. The surface tensions for ST and STSpotential models are compared in Fig. 8�a�. At any fixed

temperature, the STS model systematically has a lower valueof �* than the ST one; this difference does not change withtemperature. The results of Mecke et al.,18 obtained for thesame cut-off radius R�2.5� , but with the LRC applied tothe dynamics during MD simulations are also shown in Fig.8�a�. We see that the correction applied at this cut-off radiusdoes not improve quantitatively the results: they are lying inbetween those for ST and STS potential models obtainedwithout any correction, and at low temperature are closer toSTS while at high temperature coincide with ST model.

How different are ST and STS potential models trun-cated at R�2.5� from the full LJ potential? The deviation ofdensity–temperature and pressure–temperature coexistingcurves for both models from the results of untruncated LJpotential can be estimated from the comparison with the dataobtained by Panagiotopoulos et al.51 for full potential usingthe GEMC �Figs. 5�a� and 6�a��. The similar comparison forsurface tension can be performed in Fig. 7�a� with respect tothe result of Nijmeijer et al.12 obtained with R�7.33� .What is clearly seen that the results of ST potential modelare significantly closer to the results of full potential.

B. Dependence on cut-off radius, R�4.4�

It is obvious, that the differences between two potentialmodels will decrease with increasing cut-off distance since inthe limit R→� both potentials will approach the original fullLJ potential, ULJ(r), given by Eq. �1�. We note, that theways of approaching this limit are different, since STS pro-cedure changes the potential depth, while ST neglects thepotential tail only.

In order to obtain more accurate insight into the varia-tions of the coexistence properties with increasing of cut-offradius, we calculated for the ST model the coexisting densi-ties, vapor pressure, and surface tension at cut-off radius R�4.4� using MD simulations with forces computed accord-ing to Eq. �5�. These results together with results for STSpotential at larger R obtained by other authors are collectedin Table IV and shown in parts �b� of Figs. 5–8.

Comparing our results with those of Holcomb et al.15 forthe STS model we can conclude that in contrast to Fig. 5�a�the differences in coexisting densities between ST and STSpotential models at R�4.4� are significantly smaller �of or-der of 1% for liquid density at the highest temperature�. Asfor the pressure, we have results for both models for onetemperature, T*�0.92 �see Table IV� only, at which they arepractically identical. As can be seen from Table IV and Figs.5�b�–7�b� for coexisting densities and vapor pressures, thedifference between truncated and full potential results alsodecreases, and cut-off radius of at least 4.4� seems almostenough to approach the full LJ potential when simulationsare carried out using the ST potential model. At the sametime it is less valid for the surface tension �Fig. 8�b�� whereboth differences between STS and ST models as well asbetween ST and full system results are still pronounced �bothare of order of 10%�, i.e., surface tension is more sensitive tothe tail of the potential. This fact has been pointed out firstby Nijmeijer et al.12 and discussed in the literature �see, forexample, Blokhuis et al.�21 but with respect to the STS

FIG. 4. The same as in Fig. 3 but for ST potential model. In MC and MDsimulations, derivative dULJ(r)/dr , has been calculated according to Eq.�5�.



model. Particularly, Nijmeijer et al.12 observed that trunca-tion of the potential at large distances increases the surfacetension by a factor of 2.8 (T*�0.92), namely, �*�0.24 forR�2.5� and �*�0.63�0.02 for R�7.33� , and such effectwas attributed completely to the omission of the tail. In thiswork we show that for the ST model at R�2.5� , surfacetension is 0.375, i.e., increased by a factor of 1.7 only goingto R�7.33� �see Fig. 9 for more details�. The results ofNijmeijer et al.12 additionally have been affected by thechange in potential depth because inadvertently comparisonof results with full potential have been performed for STSpotential model but not for ST.

C. R�5.5� and extrapolation to the full potential

An important problem for inhomogeneous fluid is theextrapolation of the results for truncated potential simula-tions to larger distances to achieve the full potential result.This task is not so easy and the way to realize it is not soevident, in contrast to homogeneous case. The first attempt toperform such extrapolation in the computer simulations ofthe surface tension of LJ fluid has been proposed by Chapelaet al.8 and later improved by Holcomb et al.15 and Blokhuis

et al.21 These authors, similarly as has been done in the ho-mogeneous case for pressure, employed the LRC of the sur-face tension at the end of the simulations for cut-off dis-tances R�2.5� �Ref. 8� and 4.4�.15,21 The disadvantage ofthis procedure even in the improved version21 is that thecoexisting densities and width of the interface, which enterthe correction contribution, are dependent on cut-off dis-tance, R. An attractive possibility is to included LRC duringthe simulations and has been considered by Guo and Lu17 inthe framework of MC and by Lotfi et al.19 and Mecke et al.18

in MD techniques. The approach of Guo and Lu17 is basedon the assumption of local dependence of thermodynamicproperties i.e., they are uniform within each local element.Lotfi et al.19 and Mecke et al.18 have included LRC to bothdynamics, by adding additional force contribution, and sur-face tension, by using the virial route in conjunction withlocal density distribution. The only way to test different LRCschemes is to compare obtained results with direct simula-tions at large enough cut-off radius. But which cut-off dis-tance is enough for surface tension calculations? Let us dis-cuss the results which follow from our MD simulationsperformed with cut-off distance, R�5.5� .

TABLE II. Liquid/vapor coexistence densities, �L* , �V* , vapor pressure, PV* , and surface tension, �*, for STSpotential model from MC and MD computer simulations. The cut-off radius, R�2.5� .

T* Method �L* �V* PV* �* Source

0.72 MCa 0.7722 0.0086 0.0062 0.560�0.002 this work0.72 MDa 0.7749 0.0088 0.0062 0.544�0.018 this work0.72 MDa 0.777 0.009 ¯ 0.55 Holcomb et al.b

0.72 MDc 0.776 0.0093 ¯ 0.552�0.009 Chend

0.72 MDe 0.776 0.0092 ¯ 0.548�0.006 Chend

0.72 MDf 0.7764 0.0093 ¯ 0.55�0.01 Nijmeijer et al.g

0.80 MCa 0.7288 0.0174 0.0127 0.405�0.003 this work0.80 MDa 0.7306 0.0196 0.0140 0.408�0.018 this work0.80 MDf 0.7315 0.0195 ¯ 0.39�0.01 Nijmeijer et al.g

0.80 MDh 0.7287 0.0200 ¯ 0.388�0.004 Haye and Bruini

0.80 MDj 0.7310 0.020 0.0143 0.39�0.02 Adams and Hendersonk

0.90 MCa 0.6668 0.0414 0.0298 0.230�0.004 this work0.90 MDa 0.6644 0.0444 0.0314 0.227�0.016 this work0.90 MDl 0.662 0.0439 ¯ 0.224�0.009 Nijmeijer et al.g

0.90 MDh 0.6619 0.0454 ¯ 0.223�0.003 Haye and Bruini

0.90 MDj 0.671 0.040 0.0322 0.23�0.01 Adams and Hendersonk

0.92 MCa 0.6444 0.0500 0.0352 0.201�0.004 this work0.92 MDa 0.6505 0.0518 0.0363 0.197�0.016 this work0.92 MDm 0.649 0.063 ¯ 0.24�0.02 Nijmeijer et al.g

0.918 MCm 0.65 0.050 ¯ 0.42n Rao and Berneo

1.0 MCa 0.5794 0.1030 0.0626 0.095�0.027 this work1.0 MDa 0.5764 0.1002 0.0615 0.082�0.008 this work1.0 MDp 0.565 0.103 ¯ 0.088�0.007 Nijmeijer et al.g

1.0 MDq 0.5694 0.0987 ¯ 0.076�0.002 Haye and Bruini

aN�2048, Lx�Lz�13.41��39.81� .bReference 15.cN�2048, Lx�Lz�10.00��53.944� .dReference 16.eN�2048, Lx�Lz�13.00��31.9195� .fN�10390, Lx�Lz�29.1��29.1� .gReference 12.hN�12432, Lx�Lz�33.01��33.01� .iReference 20.jN�512, Lx�Lz�7.0��28.0� .

kReference 22.lN�7968, Lx�Lz�29.1��29.1� .mN�2048, Lx�Lz�14.66��25.1� .nApparently this value of surface tension has beencalculated for two surfaces, and for comparisonwith data from another source should be dividedby 2.

oReference 9.pN�7619, Lx�Lz�29.1��29.1� .qN�10160, Lx�Lz�25.0��48.0� .



The density–temperature coexisting curves obtained forR�5.5� are shown in Figs. 5�c� and 6�c� and compared withthose for the full LJ potential obtained by Panagiotopouloset al.51 using the GEMC. In these figures the results ofMecke et al.18 for R�5� and those of Nijmeijer et al.12 withR�7.33� which correspond to STS potential model are alsodisplayed. The agreement between our orthobaric densitiesand those from Refs. 12,18,51 is rather good. The same con-clusion is valid for vapor pressure, PV* , as we can see fromFig. 7�c� where our data for the ST potential with R�5.5�are compared with those obtained for the full LJ potentialusing GEMC.51 Our estimations show that by going fromR�4.4� to 5.5�, the changes in orthobaric densities withinST model are less than 1.5% and less than 4% for the vaporpressure. At the same time, for R�5.5� , the deviations ob-served both within the ST and STS models and with respectto the results for full potential are even smaller and lie in therange of the errors allowed in simulations.

In Sec. V B, it was pointed out that surface tension ismore sensitive than orthobaric densities and pressure to thetail of the potential. We estimated from Table IV that whenR increases from 4.4� to 5.5� surface tension still changesup to 10% �less than 1.5% for coexisting densities�. Never-theless, comparing these results with those calculated atlarger cut-off radii, R�6.3� by Holcomb et al.15 and R�7.33� by Nijmeijer et al.12 �both have been calculated forisotherm, T*�0.92) we see that difference practically van-ishes �see Fig. 8�c��, and within the simulation uncertaintiesit might correspond to the full LJ potential.

The above mentioned LRC procedures are inconsistentin the case of surface tension �those results are also collectedin part �c� of Fig. 8�. The results of Blokhuis et al.21 practi-cally repeat at T*�0.92 the surface tension value ofNijmeijer et al.12 and are slightly higher �as we suppose it

has to be� than results of this work for ST potential modelwith cut-off radius, R�5.5� . The results of Mecke et al.18

slightly overestimate both of these results though the tem-perature dependence is quite similar. Very probably, the re-sults of Mecke et al.18 are influenced by the double countingof the tail contribution, i.e., in the dynamics and in the sur-face tension calculations. The most unexpected results �forboth the surface tension values and its change with tempera-ture�, as it follows from Fig. 8�c�, have been obtained byGuo and Lu17 and look, at least, questionable.

Thus, it can be concluded that final density–temperatureand pressure–temperature coexisting curves and surface ten-sion for LJ fluid can be achieved within the ST model byemploying cut-off radius, R�5.5� .

D. Critical parameters of Lennard-Jones fluid

The density–temperature–pressure coexistence data ob-tained in our canonical simulations �see Tables II–IV� have

FIG. 5. Vapor density–temperature coexisting curves of the STS �squares�,ST �triangles�, and full LJ �solid circles� potential models. Cut-off radius,R�2.5� , 4.4�, 5.5� employed in simulations are written in the figure. Theopen squares and triangles at all cut-offs resulting from our canonical MDsimulations �MC and MD are indistinguishable on the scale of the figure�. Inpart �a�, solid squares result from GEMC of Smit �Ref. 49� for STS model,while solid triangles indicate data for ST model obtained from GEMC byFinn and Monson �Ref. 55� and Panagiotopoulos �Ref. 56�, and from grandcanonical MC by Wilding �Ref. 57�. Solid diamonds in part �c� result fromcanonical MD simulations of Mecke et al. �Ref. 18� with cut-off, R�5� ,and LRC. Results for the full LJ potential, in parts �a�, �b�, and �c� areobtained within GEMC simulation by Panagiotopoulos et al. �Ref. 51�.

TABLE III. Liquid/vapor coexistence densities, �L* , �V* , vapor pressure,PV* , and surface tension, �*, for ST potential model with �-correction fromMC and MD computer simulations of this work.a The cut-off radius, R�2.5� .

T* Method �L* �V* PV* �*

0.70 MC 0.8122 0.0040 0.0027 0.806�0.016

0.72 MC 0.8025 0.0045 0.0037 0.743�0.0030.72 MD 0.8043 0.0042 0.0028 0.748�0.012

0.80 MC 0.7657 0.0128 0.0075 0.618�0.0090.80 MD 0.7653 0.0122 0.0080 0.623�0.028

0.90 MC 0.7157 0.0254 0.0199 0.438�0.0040.90 MD 0.7130 0.0237 0.0203 0.418�0.021

0.92 MC 0.7065 0.0294 0.0207 0.374�0.0060.92 MD 0.7027 0.0279 0.0220 0.375�0.030

1.0 MC 0.6542 0.0439 0.0336 0.244�0.0051.0 MD 0.6507 0.0500 0.0380 0.235�0.0071.0 GEMCb 0.650 0.049 ¯ ¯1.127 MC 0.5378 0.1108 0.0769 0.049�0.0041.127 MD 0.5400 0.1223 0.0790 0.049�0.006

aN�2048, Lx�Lz�13.41��39.81� .bPanagiotopoulos �Ref. 56� for a system size L�9� .



been used to estimate the critical parameters of LJ fluid forthe different cut-offs. These results in conjunction with dataobtained by others are collected in Table V.

To estimate the critical density, �C* , and critical tem-perature, TC* , we have fitted our results with scaling law forthe density and law of rectilinear diameters using the firstthree terms of a Wegner expansion58–60 in the form

��*��C*�C2�t��1

2�B0�t��B1�t��1/2�, �10�

where t�1�T*/TC* . The subcritical simulation data havebeen extrapolated to the critical point corresponding to uni-versal critical exponent value determined from RG theory,��0.325.

In order to estimate the critical pressure, PC* , we havefitted the vapor pressure data obtained from the simulation toan equation of the Clausis–Clapeyron form,

ln PV*�A�B

T*. �11�

The value of PC* is calculated as that corresponding to thevalue of TC* obtained from Eq. �10�. The amplitude termsC2 , B0 , B1 , and A ,B , obtained from the expressions �10�and �11�, are listed in Table V for each cut-off radius.

We observe from Table V that critical parameters of LJfluid, obtained within different statistical ensembles, at leastmoderately agree well when data are classified in accordancewith models �ST or STS at the same cut-off distance� used insimulations. The critical temperature, TC* , and critical pres-sure, PC* , increase going from small to large cut-offs andfrom STS to ST potential. There is no clear trend in thecritical densities for the different cut-offs as well as it is notobserved within STS and ST potential. The cut-off radius,R�5�5.5� seems to be enough to reproduce the full LJpotential critical parameters.

E. Lennard-Jones model and real simple fluids

In some studies performed on the liquid/vapor coexist-ence in LJ model fluid the obtained results have been com-pared with experimental data for real simple fluid �see forexample Refs. 3,12� and degree of coincidence has been usedas criteria of modeling performance. In different studiespositive and negative deviations from measured data, depen-dent on the methods employed and set up conditions used,have been observed. Miyazaki et al.6 and Sung andChandler47 have suggested that disagreement may be causedby inadequacies in the LJ potential as a model for effectivepairwise interactions in liquid noble gases. In Fig. 10 weshow the comparison of temperature–density, pressure–

FIG. 6. The same as in Fig. 5 but for liquid density–temperature coexistingcurves.

FIG. 7. The same as in Fig. 5 but for vapor pressure–temperature coexistingcurves.



density coexisting envelops, and surface tension vs. tempera-ture curves obtained for LJ model at different cut-off dis-tances as well as for full potential with experimental data forliquid argon. For all considered coexistence properties thefull LJ results overestimate observable data. The effectivepairwise interactions in liquid argon seem to be less long-ranged. Manipulating by cut-off radius, it is possible toachieve the experimental values �cut-off distance of order of�4� might be an appropriate choice, as it can be estimatedfrom Fig. 10�. A more sophisticated way to model the trueinteractions in real simple fluids is to include the effect ofthree-body interactions. It has been shown already by pertur-bation theory calculations4 that contribution of three-bodyinteractions to surface tension is negative and leads to favor-able agreement with experimental results.

VI. CONCLUSIONS

In the present study, for the first time, the differencesbetween spherically truncated �ST� and spherically truncatedand shifted �STS� Lennard-Jones potential models have beentaken into account in direct canonical MC and MD simula-tions of liquid/vapor coexistence. We have performed simu-lations for both models with the same setup conditions and in

FIG. 8. The same as in Fig. 5 but for surface tension, �*, as function oftemperature. Solid triangles correspond to MD simulations results of Meckeet al. �Ref. 18� with LRC to the dynamics and tail correction to the surfacetension applied to cut-offs R�2.5� �part �a�� and R�5� �part �c��. Solidsquares in part �b� correspond to canonical MD simulations of Blokhuiset al. �Ref. 21�. Open and solid diamonds in part �c� correspond to MDsimulations results of Blokhuis et al. �Ref. 21� with R�4.4� and tail cor-rection to the surface tension and to MC with LRC during simulationsreported by Guo and Lu �Ref. 17�, respectively. Solid circles shown in �a�,�b�, and �c�, at T*�0.92, denote results obtained from MD simulations byHolcomb et al. �Ref. 15� with R�6.3� , and by Nijmeijer et al. �Ref. 12�with R�7.33� , respectively; we assume these data as full LJ potentialresults.

FIG. 9. Relative changes in the liquid coexisting densities, �L* , and surfacetension, �*, for STS �squares� and ST �triangles� models with respect to thefull LJ potential results vs. cut-off radius, R, at reduced temperature, T*�0.92.

TABLE IV. Liquid/vapor coexistence densities, �L* and �V* , surface ten-

sion, �*, and vapor pressure, PV* , from MD simulations of this work with

ST potential model at different cut-off radii, R�4.4� and 5.5�.

T* �L* �V* PV* �* Model

R�4.4�0.72 0.8245 0.0018 0.0018 0.992 ST0.72a 0.825 0.004 ¯ 0.91 STS0.8 0.7895 0.0060 0.0050 0.798 ST0.9 0.7438 0.0157 0.0135 0.607 ST0.92 0.7341 0.0189 0.0161 0.580 ST0.92b 0.7347 0.0192 0.0171 0.579 ST0.92a 0.729 0.020 ¯ 0.53 STS0.92c 0.7255 0.0198 0.0161 0.530 STS1.0 0.6913 0.0310 0.0267 0.407 ST1.0d 0.684 0.035 ¯ 0.41 STS1.127 0.6063 0.0696 0.0566 0.200 ST1.127a 0.596 0.081 ¯ 0.18 STS

R�5.5�0.72 0.8254 0.0019 0.0017 1.042 ST0.8 0.7927 0.0052 0.0046 0.854 ST0.9 0.7458 0.0147 0.0120 0.679 ST0.92 0.7358 0.0208 0.0151 0.629 ST0.92b 0.7362 0.0159 0.0132 0.616�0.003 STS1.0 0.6940 0.0309 0.0257 0.450 ST1.127 0.6147 0.0663 0.0545 0.224 ST

R�6.3�0.92a 0.739 0.018 ¯ 0.61 STS

R�7.33�0.92e 0.740 0.018 ¯ 0.63�0.02 STS

aMD simulations of Holcomb et al. �Ref. 15�.bMC simulation of this work.cMC simulation of this work.dMD simulation of Matsumoto and Kataoka �Ref. 11� with cut-off radius,R�4.41� .

eMD simulation of Nijmeijer et al. �Ref. 12�.



the wide temperature window. To calculate the forces in MDsimulations and to calculate the pressure in both MC andMD simulations for ST potential model, the simple expres-sion to compute an ‘‘impulsive’’ contribution, F , due tothe discontinuous change of the potential functions at cut-offdistance has been proposed �Eq. �9��. It has been shownquantitatively, that in the simulations of two-phase systemsthis contribution, which is not so important in the case ofsingle-phase systems, becomes crucial and has to be includedinto the simulation algorithms.

The MC simulations for the ST potential model, inwhich the contribution of F in computing the virial is ne-glected, result in unphysical behavior of the normal compo-nent of the pressure tensor: normal pressure profile is notconstant through the interface and the vapor and liquid bulkregions; both normal and tangential components of pressuretensor and resulting surface tension have incorrect values�see Fig. 1�.

From another perspective, if discontinuity of the ST po-tential is not included into the calculations of the forces, thenthe moving trajectories generated in MD simulations corre-spond to the STS potential model �it follows from Eqs. �3�and �4��. As a consequence, all the MD simulations reportedso far in the literature for truncated LJ potential, withouttaking this force contribution into account, have generatedthe results for the STS potential models and have to be com-pared with MC data obtained using STS potential model.Any MD simulations for liquid/vapor coexistence with STpotential model have been reported in the literature.

The ST and STS potential models truncated at the mostpopular cut-off distance, R�2.5� , produce significantly dif-ferent liquid/vapor coexistence curves in both temperature–density �Fig. 10�a�� and pressure–density �Fig. 10�b�� planes:

both coexisting envelops are shrunken �more pronouns forthe liquid side� and critical temperature and pressure areshifted to lower values when one goes from ST to STS po-tential. This shift of the phase diagram affects significantlythe surface tensions, producing the shift for the whole tem-perature range ��ST��STS�0.2 �more than 30%� be-tween both models. For all coexistence properties, the STmodel gives results closer to the full LJ potential. No con-tradictions between our data and that obtained earlier in theliterature for coexisting densities have been found when theresults are classified in accordance with models employed insimulations �see Tables II and III�. By comparing these datawith the results of Gibbs ensemble and grand canonical MCsimulations, for the first time it has been shown in detail thatconventional canonical simulations �MC or MD� are appli-cable for the subcritical region.

Different dependence of the coexistence properties onthe cut-off distance for STS and ST models have been found�see Fig. 9�. Particularly, going from cut-off radius R�2.5� to 4.4� the orthobaric densities change less than 10%in the case of ST model and more than 15% for STS model.The cut-off radius, R�4.4� , can be assumed as one whereboth models become indistinguishable with respect of coex-isting densities. The same conclusion is valid for the coex-isting pressures. Both coexisting densities and pressure, re-sulting from the truncated simulations with cut-off radius R�5.5� , differ from the full LJ results within the error bars.

At the same time, the surface tension calculated directlyfrom the simulations is more sensitive to the potential tail. Inthe case of the ST model, �* changes of order of 35% and5% when R increases from R�2.5� to 4.4� and from 4.4�to 5.5�, respectively. For the same increase of R in the caseof the STS model, �* changes more than 60% and more than

TABLE V. Critical properties of Lennard-Jones fluid.

Cut-off, R Method �C* TC* PC* Source

STS2.5� MD�NVT�a 0.323 1.073 0.0908 this work2.5� GEMC 0.3176 1.0855 ¯ Smitb

ST2� GEMC 0.321 1.0615 ¯ Panagiotopoulosc

2.5� GEMC 0.33�0.01 1.176�0.008 ¯ Panagiotopoulosc

2.5� MC��VT�c 0.31974 1.18763 ¯ Wildingd

2.5� MD�NVT�e 0.319 1.186 0.1098 this work4.4� MD�NVT�f 0.307 1.295 0.1297 this work5� GEMC 0.321 1.2815 ¯ Panagiotopoulosc

5.5� MD�NVT�g 0.307 1.309 0.1303 this work�5.7� MD�NPT�test� 0.314 1.310 0.126 Lotfi et al.h

none GEMC 0.3046 1.3163 0.1311i Smitb

none MC��VT� 0.333 1.302 0.132 Adamsj

aB0�1.175, B1��0.132, C2�0.207, A�3.08362, B�5.88239.bReference 49.cReference 56.dReference 57.eB0�1.125, B1��0.067, C2�0.220, A�3.34685, B�6.5895.fB0�1.016, B1�0.089, C2�0.239, A�3.24112, B�6.84260.gB0�1.010, B1�0.093, C2�0.240, A�3.24157, B�6.91117.hReference 61.iObtained from the vapor pressures reported by Panagiotopoulos et al.51

jReference 62.



10% �see Fig. 9�. The simulations with R�5.5� for the STmodel produce the results which are similar to those obtainedwith cut-off radius R�6.3� and 7.33� already reported inthe literature. In summary, we can conclude that in the directcalculations of surface tension the cut-off radius, R�5.5� , ispractically enough to approach the full LJ potential results;the long range correction to the surface tension, in the spiritof Blokhius et al.,21 is applicable to the STS model startingfrom the cut-off radius as large as 4.4�.

The critical parameters of LJ fluid extrapolated from co-existing data generated in canonical simulations are consis-tent with literature estimates, obtained within different statis-tical ensembles and different methodologies, for the same

models �STS or ST� and the same cut-offs. Comparing thetemperature–density and pressure–density phase diagramsand surface tension vs. temperature curves for the full LJpotential with measurements for liquid argon �see Fig. 10�we confirmed that the LJ model is not well suited to describethe coexistence and interfacial properties of real simple flu-ids; observable data cannot be used as criteria of simulationperformance for the LJ model.

The conclusions listed above have been drawn for trun-cation in the LJ model but the procedure described in thiswork can be applied to any other spherically truncated po-tential models �Yukawa, Kihara, etc.� of dispersion forces.The method of handling the truncation, proposed in thiswork, does not matter if the total potential model includespolarizability or Coulombic interactions. When the Hamil-tonian of more complex fluid models includes, as a part, theLJ-type interactions, the proposed scheme has to be appliedexactly in the same way as it is described in this work, whileelectrostatic interactions have to be handled in an appropriateway,44 e.g., using Ewald summation method, etc. In the caseof inhomogeneous systems this is particularly important be-cause, in contrast to electrostatic forces, the long range cor-rection of dispersion contribution is not easily applied indirect simulations.

The problems of truncated interactions are of the sameimportance for mixtures of LJ fluids �where, besides theliquid/vapor coexistence, the liquid/liquid interfaces are al-lowed�, complex systems, such as nonpolar molecular fluids,and their mixtures32–36 defined by a united atom modelwhere the dispersion interactions are usually described by LJpotential and contribute significantly to the coexistence prop-erties of entire systems. This is the case of water,28–31 aque-ous solutions,37–41 monolayer/water,42 and bilayer/water43 in-terfaces, etc. In all these studies, dispersion contributions tothe coexistence properties have been incorporated within theSTS model; in some of them23 it has been determined andrealized, while in others24,26–28,31–37 it was used inadvertentlybecause evaluation of impulsive contribution to the forceshas not been included into the MD algorithm. The quantita-tive significance of the correct handling of LJ-type contribu-tion to the coexisting and interfacial properties of complexsystem, is expected to be different depending on the physicsof the system under modeling. It is important in the case ofnonpolar molecular fluids but appears less important forhighly polar fluid where phase coexistence is observed athigher temperatures and electrostatic contribution dominates.Particularly, Alejandre et al.31 have found that if electrostaticforces are accounted within Ewald method, coexisting den-sities and surface tension of water are essentially unchangedon variations of cut-off distance for dispersion forces. How-ever, we wish to turn our attention to that, when Ewaldmethod is applied, care has to be exercised with the realspace term of electrostatic interactions which becomes a partof dispersion forces. This term has the same amplitude coef-ficient as the main electrostatic contribution with decay de-fined by error function. Depending on the damping param-eters chosen in Ewald scheme, the real part contributionmight be significant and has to be treated adequately.

FIG. 10. Temperature–density �part �a��, pressure–density �part �b�� phaseenvelopes, and surface tension vs. temperature curves �part �c�� for LJ po-tential models and liquid argon. The experimental data for temperature–density curves have been adopted from Ref. 3 and for pressure–density fromRef. 47 while the solid lines have been calculated according to Eqs. �10� and�11�, respectively; temperature dependence of surface tension are taken fromRefs. 4,63. All experimental data are scaled with �/k�119.8 K and��3.405 Å. Dotted line corresponds to the calculations of Lee et al. �Ref.4� including the three-body interactions.



ACKNOWLEDGMENTS

This work was partially supported by the National Coun-cil for Science and Technology �CONACyT� of Mexico un-der Grant Nos. 25301-E and L0080-E, and DGAPA of theUNAM under Grant IN111597. We are grateful to D. Kofke,J. de Pablo, and D. J. Tildesley for fruitful discussions andcomments. We also thank computer centers of the MexicanNational University �DGSCA UNAM� and Mexican Petro-leum Institute �IMP� for a generous allocation of computertime.

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