Fluid Phase Equilibria, 75 (1992) 185-196 Elsevier Science Publishers B.V., Amsterdam

185

A Critical Vapor Inter s

tudy of Simulations of the Lennard-Jones Liquid- ace

C. D. Holcomb, P. Clancy, S. M. Thompson, and J. A. Zollweg

School of Chemical Engineering, Olin Hall, Cornell University, Ithaca, NY 14853 (USA)

Keywords: Molecular Dynamics, Lennard-Jones Interface, Interfacial Tension, Liquid-Vapor Interface

ABSTRACT

The surface tension and li and two- component Lennard- P

uid-gas density profile through the interface of one-

its simulation techniques. ones fluids were calculated using Molecular Dynam-

The system size, film thickness, interface area, inter- molecular potential cut-off, composition, and temperature were varied. For the one- component system, the results were compared to previous work and some discre - ancies of the past work were resolved. By combining this work with correct resu ts P from and t R

revious authors, the minimum system size, film thickness, equilibration time, e trade-off between cut-off and corn uter time were determined. Using

configurations calculated for moderate cut-o ffp s, the surface tension was extrapolated to the full potential value by using a tail correction and the results compared to simulations performed with the longer cut-offs. The results showed the possibility of obtaining estimates of the surface tension for large cut-off simulattons from moderate cut-off simulations provided that the density profile does not change significantly with increase in cut-off. Using the cntenon for equrhbratron deter- mined from the one-component systems, two-component systems at varying compo- sitions and temperatures were simulated and the tail correction applied.

INTRODUCTION

Although the liquid-vapor interface of a Lennard-Jones model was simulated nearly 15 years ago, there have been a number of later publications calculating the surface tension in which the (erroneous) suggestion has been made that the ex eri- mental value for argon can be achieved from simulations of the Lennard-Jones K uid if only the “correct” cut-off value is used. The results for the surface tension available in the literature are quite scattered. Since our interest is measuring the surface tension of mixtures and then testing the potential models via simulations, we

Ee erformed a thorough investigation of the evaluation of the surface tension of the

nnard-Jones fluid in order to lay to rest some of the apparently existing miscon- ceptions concerning the relationship between the experiential measurements for the surface tension of real argon and the simulated values for the Lennard-Jones fluid.

Although the principal techniques for simulatin an interface are the same as simulating homogeneous phases, the time require % to calculate key interfacial properties such as the density profile and the surface tension take significantly

0376-3612/92/$05.00 0 1992 Elsevier Science Publishers B.V. All rights reserved

186

longer than the equilibration R”

riod in a bulk fluid. Also the thickness of the liquid film and the area of the inte ace become important in obtaining the correct profile through the interface and in minimizin the effects of periodicity. By carefully

F erforming several simulations to estab tsh a recipe for obtaining accurate results P; or the surface tension and by combinin these results with correct results of the

K ast, the criteria for obtaining good simu ation results for the liquid-vapor interface lg ave been determined as a prelude to subsequent mixture studies.

It should be obvious that changing the cut-off used for the Lennard-Jones model alters the potential model and hence the surface tension which will result from the simulation. The interfacial pro erties are more sensitive to the cut-off than the bulk properties. Because the inte rF ace profile changes rapidly within a few layer of atoms, the size of cut-off (typically 2.50) is close to the len th scale of this than e and therefore strongly affects the profile. The sensitivity o surface tension to e P L cut-off size has been shown by Nijmeijer et al. (1988) where the increase in surface tension between 2.50 and 7.330 at a reduced temperature( T* = kT/& ) of 0.92 was 160%. He compared his results to the experimental value of noble gases and concluded that the difference was in part due to the truncation of the attractive tail. From the work here it can be shown that this conclusion is somewhat in error. At T*=0.92 the surface tension calculated with a full Lennard-Jones potential is close to the surface tension of argon. However, at lower temperatures there is a much lar er difference between the surface tension of the full Lennard-Jones and that of rea f argonsuggesting that the agreement at 0.92 is fortuitous. This is completely understandable in terms of the increasing importance of the inaccuracies in the

E otential at lower tern eratures. The outer wall of the attractive well of the ennard-Jones potentia P is seriously is error in corn arison to the “true” potential

function for ar on in so far as that can be determine . f

cf A more e ficient use of computing resources would be to use methods to

extrapolate the results of truncated potential simulations to larger cut-offs or to the full Lennard-Jones potential as necessary. The use of a tail correction to deduce the surface tensions for an infinite cut-off has already been demonstrated by Chapela et a1.(1975). Their values for the surface tension, calculated for a 2.50 cut-off system, were modified using a tail correction to predict those for a full Lennard-Jones potential. Their extra olated value at T*=0.92 was very close to Nijmeijer et aZ.(1988)‘s for a cut-o R of 7.330. Cha ela et a1.(1975) stated that a concern in their treatment was the change in the bulk B ensities for larger cut-off systems. We shall show here that usin a tail correction on results. obtained with a cut-off sufficiently larst;hat the bulk B ensttres are no longer stgmfrcantly changmg does produce better

There are two common ways of setting up the simulation cell. The simulation cell itself can either be a infinite liquid film in the center of the box with a vapor on each side and periodic boundaries m the x and y directions or a bulk liquid abutting a molecularly-homogeneous wall with a vapor phase against another wall. The simulation can either be performed usin Molecular Dynamics or Monte Carlo methods. Fi ure 1 shows a plot of the re uced surface tension versus the reduced

P %

temperature or the past results. The lot is very scattered and shows very little repeatability between authors. This P ack of repeatability warrants a thorou h investigation in order to determine which results are correct and the cause of t e a discrepancies in the others. A common misconception was that the larger the cut- off the closer the results would be to the real noble gas results. It will be shown that although some of the results appear to match the experimental results, their simula- tions were not equilibrated long enough or were too small to obtain accurate results. The fact remains that a Lennard-Jones otential will never match the real noble gas results, except fortuitously, since the Ee nnard-Jones potential does not accurately reproduce the true potential for the inert gas. The real potential produces higher surface tensions than the Lennard-Jones potential and the difference becomes larger at lower reduced temperatures.

187

2.5 1 I I I

I.

2.0 -

1.5 - 0 F

1.0 -

w 0.5 -

0.0 ’ 0.6 0.8

T* 1.0 1.2

Figure 1: Summary of the reduced surface tension versus reduced temperature of previous scientists. + Experiment, ( n r, = 2.5 CT, l r, = 7.330 Nijmeijer (1988), (A n = 255 . n=1020

n=4080 r, = 2.5 o) Capela (1977). q Liu (1974), * Lee (1974). X r, = 2.5 G Chapela (1975). q r, = 2.5 (T Miyazaki (1976), 0 r, = 2.5 (T Rao (1976). A rc = 2.5 (T Rao (1979), q r, = 2.25 Q Gii (1981), EJ r, = 3.0 0 Walton (1983), and + r, = 4.4 CT Matsumoto (1988).

SIMULATION METHODOLOGY

In order to evaluate the past results and avoid size-dependent surface tensions, some Molecular Dynamics stmulations were run with a varie of choices for the

film from bein

The liquid film was positioned in the center of the simulation cell with periodic boundaries in the x- and y- directions. The z- axis was extended to produce a large vapor space and reflective walls positioned at the two ends. The film was allowed to equilibrate creating a vapor space on either side between the liquid and the reflec- tive walls. The center of mass was kept in the middle of the box by allowing the reference frame to move. The velocities were then scaled to maintain the desired reduced temperature.

The systems were equilibrated for 70,000 time steps. A production phase of an additional 180,000 time ste evaluated. The criteria c R

s was then performed durmg which the pro rties were osen for equilibration were three-fold: K e running

188

average for the surface tension and density profile must not change si nificantly (this usually occurs after 120,000 time steps), the block average over 3 ,000 steps t? must have the same value as the running average, and the inte ral of the difference in the normal and tangential pressure components through the t ulk phases must not contribute significantly to the surface tension calculation. Figures 2 and 3 show the progression to equilibrium of the surface tension averages and the interface profiles of a typical run. Figure 2 shows the running and block averages for the surface tension after the mitral vapor phase production. Figure 3 shows the progression of the mte system %

ml of the difference m the ressure components. It can be seen that the id not reach equilibrium be ore 120,000 trme steps because the bulk liquid P

was contributing to the surface tension integral.

1.0. I

0.8 - rc = 2.5 (T

0.6 - ,J _,.--

r;h * --........ . . . . . .."...__..... . . . -I-

0.4 -

0.2 -

0.0 I

0 100 200 Number of Steps (Thousands)

Figure 2: Running and Block averages of the surface tension after a vapor phase production run of 70000 steps for a run at T* = 0.72 with 2048 particles for 2.50 cut-off.

By waiting until the block and runuing averages had settled down to the same value, we were confident that the final configurations accurately represented the slowly-e uilibrating surface tension. The final density profiles were very smooth compare 8 to some of the previous work. Table 1 presents the results for simulations

TABLE 1:

Molecular Dynamics Simulation Results.

#Particles r, T* PI+ pg + y*

2048 2.5 0.72 0.777 0.009 0.549 1000 4.4 0.72 0.825 0.004 0.908 2048 0.72 0.825 0.004 0.909 2048 1:: 0.92 0.729 0.020 0.530 2048 1.127 0.596 0.081 0.170 3072 44.: 2048 6:3

0.72 0.825 0.003 0.917 0.92 0.739 0.018 0.605

Dimensions x*xz*

Liquid film L*

13.41x39.81 14.41 9.40x26.43 13.65 13.41x39.81 13.62 13.41x39.81 14.97 13.41x39.81 16.16 13.68x47.36 19.79 13.41x39.81 14.81

with differences in the number of reduced temperature, and cut-off. P

articles, liquid film thickness, interface area, t can be seen that no size or area dependence

was present. The models with different cut-offs represent three quite different

189

potential models and have correspondingly different values for the surface tension and to a lesser extent different values for the coexisting densities.

1

P- W

0 ,‘J 2

0 Time Steps -1

I , I . I .

-20 -10 0 10 20 2+

120000 Time Steps

-1 . ’ * ’ * ’ . -20 -10 0 10 20

21

2

1

&

0

-1

80000 Time Steps I . I . I ,

-20 -10 0 10 20

r t r’

1

-1

180000 Time Steps I . I . I .

-20 -10 0 10 20 2*

Figure 3: Progression of the integral of PN-PT over a production run at T* = 0.72 with 2048 particles and a 4.40 cut-off.

COMPARISON TO PREVIOUS WORK

Corn l?

aring this work to revious calculations, it can be seen that much of the past wor was not fully equi ‘brated or suffered from initial setup c

I? roblems. For

example, we repeated a simulation performed by Matsumoto and ataoka (1988), whose reported results were very close to the experimental values. The size of the cell, number of particles, and initial configuration were the same for the T*=0.92 simulation performed b Matsumoto and Kataoka. When the initial bulk liquid density is low compare B to the equilibrated bulk liquid value as was the case in Matsumoto and Kataoka’s work, the surface tension will initially have a value above the equilibrium value. It sta ed at this hi h value for near1 vapor phase was created an (Y the liquid f &n condensed to x

1200 steps while the

the surface tension tended downward until the e e correct density. Then

uilibrium value was reached. Matsumoto and Kataoka terminated the simulation %e fore the e uilibrium surface tension was obtained and an artificially high surface tension resu ts P were reported.

190

For simulations where the liquid film was started off with a higher density than the equilibrated value, the opposite trend was observed. This reason for these trends can be seen more clearly m Figure 4. The two lots show the inte ral to calculate the surface tension across the equilibrium cell a ter an initial 70,O P & steps to set u the vapor phase and an additional 60,000 steps into the production run. The bu 1R phases should not contribute to the inte ral if the system 1s in equilibrium. For the case where the liquid density was initia # y higher than the e uilibrium value the bulk liquid adds a ne ative contribution to the surface tension.

t!l YIh e opposite is true for

the case where e li uid density is initialized too low. Therefore the condition in which the bulk liqui % phase_ no longer contributes to the surface tension integral shou~h~e;ac;t;;~9~) y?libr

rJme1 er et aZ.(1988), and Walton et aL(1983) performed accurate simulations whose resu ts were consistent with this work. By combining those results and the results presented here, the effects of number of particles, cut- off, and tail correction validity can be examined.

System size is one of the most crucial parameters in eliminating size dependent affects. Figure 5 shows the combined results for systems with a 2.50 cut-off with varying numbers of particles. The highest and lowest temperature points for 256 particles do not agree with the rest of the data which show no significant size de en- dence on the reduced surface tension versus reduced tern erature. At the hig est

B K

temperature the interface is thicker than the cut-off an equilibration times are much longer which may account for some for the error. By examining the results against density difference, we observed the results a ainst the distance from the critical point. This analysis shows that most of the 2 s 6 particle system results are significantly different from the larger systems.

This system size dependence can also be seen by comparing interfacial thickness for the systems. The approximate surface thickness was calculated b Cha ela et aL(1977) for their systems by fitting a hyperbolic tangent equation of t e fol owing K f form to the interface profile.

p(z)=0.5(~~+~~)-0.5(~~-~~)tanh[2(z-z,)/dl (1)

d = - (P, - pg 1 [ dp(z)/dzl& (2)

p(z,)=os(p~+p,) (3) where p = density where subscripts ‘1’ and ‘g’ refer to liquid and

gas respectively, z = position or height in the simulation cell in units of (3, d = an approximate interfacial thickne.ss, and z, = the position in the interface where the density is the

average of those for the bulk liquid and bulk gas.

Cha ela et al. (1977) noted that for T* = 0.699 the surface thickness, d*, changed 199k)between 255 and 1020 article size s stems and only 4.3% between 1020 and 4080 particle size systems. R s the reduce IK temperature increases the effect became worse. Table 2 reports the interface thickness for Chapela et aL(1977) and this work. Figure 6 shows the interfacial thickness versus reduced temperature for various cut-offs, system sizes, and interfacial areas. The interfacial thickness is most strong1 affected by cut-off size. The larger the cut-off the more attracttve the potential an the smaller the interfacial thickness. It can also be seen that. larger d mterfacial areas cause the interfacial thickness to become larger without srgmficant- ly affecting the surface tension. To avoid this dependence of mterfacial thickness on system size es eciall at high reduced temperature, it is recommended that the SYS- tern size be at east 1 00 particles and the film thickness be at least twice the cut-off. P 8

191

Although in this case the effect was not seen in the surface tension, for sys@ms with more complex molecules, and hence intermolecular otential function, this depen- dence of interfacial thickness on film thickness shoul B be considered.

60000 Time Steps

--113.50-6.75 I . 0 00 I .

.

h* 6.75 I 13.50

60000 Time Steps -1 - ’ . ’ . ’ .

-20 -10 ,a 10 20

Figure 4: Comparison of initial liquid density choices on the integral to calculate the surface tension through the bulk phases. Both systems are at T * = 0.72 with a cut-off of 4.40 where the p* at equilibrium = 0.825.

0.6 0.8 T+

1.0 1.2

Figure 5: Reduced surface tension versus sauced temperature for 2.50 cut-off with a varying number of particles. _ Experiment, 0 n=255, + n=1020, q n=2048, l n=4080, . n=7619, 0 n=7968, and A n=10390. (Data from this work and refs. Nijmeijer (1988). Chapela (1977), and

Walton (1983))

192

0.8 T+ 1.0

4

* 3 ‘ct

2 I

rc = 2.5 CJ

1.2

1- 0.6 0.8 T* 1.0 1.2

Figure 6: Interfacial thickness versus system size for different reduced temperatures and cut-off values. p r,=2.50 X,Y=So, l rC=2.50X,Y=10a, + rC=2.50X,Y=200, . rC=2.50X,Y=13.410. 0 rC=4.40X,Y=13.41a, m rC=4.40X,Y=9.40, A rC=4.40 X,Y=13.680, and A r,=6.3o X,Y=13.410.

The effect of cut-off on surface tension can be seen more clear1 by combinin the results of references Nijmeijer (1988), Chapela (1977), and TV alton (1983 . 7 Figure 7 shows the effect of the cut-off on reduced surface tension versus reduced temperature. The potentials with different cut-off sizes clearly have different surface tensions. If the results were plotted against density difference, the effect of cut-off size is less since the critical point were the density difference and surface tension are zero is fixed. This shows that the cut-off size cannot easily be scaled to collapse all the data and a more complicated method to extrapolate the data must be employed. It should be noted that the bulk densities change less than 1.5% when the cut-off is increased beyond 4.4~~. Although at each temperature the surface tension had a ower law de endence on the cut-off, the exponent varies with temperature. & erefore, ano tl! er more theoretically based method of extrapolating the surface tension to infinite cut-off must be exammed.

193

0.6 0.8 T* 1.0 1.2

Figure. 7: Reduced surface tension versus reduced temperature for varying cut-offs. _ Experiment, q r, = 2.50, + r, = 3.00, 9 rc = 4.40 ,+ rc = 6.30, and .rc = 7.330. (Data from this work and refs. Nijmeijer (1988), Chapela (1977), and Walton (1983))

TAIL CORRECTIONS FOR THE SURFACE TENSION

One method of extrapolating the surface tension for the truncated cut-off simulations to the surface tension of the full Lennard-Jones potential was include a tail correction. This was first successfully shown by Chapela et aZ.(1977) for the data already presented. The tail correction equation used was

yTai’ = 12q=JlJ- p(zl) p(zZ) ( 1-3s’ ) r4 dr ds dzl E

(4)

where s = ( z1 - z2 )Jr and z1 and z2 are the positions of molecules 1 and 2.

Assuming p(z) can be represented by a hyperbolic tangent, the equation becomes

tanh (2rdd) (3$ - s ) 1-j dr d.s

Figure 8 shows how equation 3 corrected Cha ela et al.‘s(1977) data with a cut-off of 2.50 and its comparison to runs performe B with larger cut-offs. The correction brings their calculated surface tensions close to the results obtained with cut-offs of 6.30 and 7.330. Chapela et al. (1977) had expressed concern in their paper over the fact that the bulk densities and interfacial profile would be different if the simula- tion had been performed with a larger cut-off. These differences were not taken into accoun in the correction, so the results are not exact.

Since the bulk densities and interfacial profiles are significant1 closer to the full potential values when a 4.40 cut-off is used, the tail correction o f Chapela et al. was ap Table . $

lied to the results in this work. The results are presented in Figure 8 and

The tail correction e tension at T*=0.92 to 7. ?3

uation was further tested by correcting the 4.40 surface 30. The corrected surface tension was 0.616 and it

corn ared favorably to the 7.330 cut-off simulation by Ni’meijer et aZ.(1988) where yO=t63. The 4.40 data at T*=O.92 was corrected to 6. 3l <T as well in order to test our own data internally. The corrected 4.40 surface tension was 0.601 which compared very well to the actual 6.3~ surface tension of 0.603. The difference

194

between the 6.30 corrected result and the 4.40 result at T*= 0.92 was not signifi- cantly large to warrant running simulations with lar er cut-offs. The results for the corrected 4.40 simulations were on average 5% hig a er than corrected 2.50 simula- tions while there was no significant difference between the corrected 4.4~~ and 6.3a surface tensions. It seems reasonable that this difference is mainly due to the differences in the bulk densities and inter-facial profile. use the tail correction

Therefore it is important to only when the bulk properties are no longer changing

significantly with cut-off.

TABLE 2:

Approximate interfacial thickness for systems with different sizes and cut-offs.

255 255 255 255 255 1020 1020 1020 4080

This work 2048 1000 2048 2048 2048 3072 2048

#Particles r, Tc x* cf* Y* P” y + piI

Chapela et aL(1977) 255 ‘2.5 ’ 0.701

0.708 0.759 0.823 0.918 1.127 0.699 0.785 0.836 0.701

1.54 1.62 1.74 1.85 2.22 3.05 1.84 2.02 2.42 1.92

0.78 0.58 0.44 0.37 0.25 0.04 0.60 0.39 0.33 0.60

0.52 0.49 0.46 0.41 0.42 0.31 0.50 0.44 0.41 0.50

1.30 1 .r)7 0.90 0.78 0.67 0.35 1.10 0.83 0.74 1.10

0.72 0.72 0.72 0.92 1.127 0.72 0.92

13.41 9.4 13.41 1.55 13.41 2.70 13.41 4.41 13.68 1.61 13.41 2.40

1.75 1.43

0.549 0.908 0.909 0.530 0.179 0.917 0.605

0.483 1.032 0.169 1.077 0.169 0.136 0.007 0.170 0.066

1.078

8% 1:087 0.671

MIXTURE RESULTS

Only one previous study has attempted to simulate the liquid-vapor interface of a mixture of Lennard-Jones particles resembing (Lee 1984). Unfortunately this simulation was

Argon+Kry ton r-formed with less than 100 8

2.50 cut-off, and at a temperature tz low particles, with a

Using what we had learned from the the experimental triple point of Krypton.

ure fluid work, we simulated three mixtures of Lemiard-Jones particles at T *=l. 8 mole fraction of Ar

0 with compositions of 0.25, 0.5, and 0.75 on and a cut-off of 4.40. The 0.5 mole fraction run was also

simulated with a 6. 4 cr cut-off in order to test the validity of the tail correction for mixtures. The conditions, simulation results, and experimental results are presented in Table 3. It can be seen that the tail correction works ade because the 4.40 cut-off results extra

tl! olate to the same v 19

uately for the mixtures ue as the 6.30 cut-off

result. This gives us confidence that e results for the full Lennard-Jones potential results can be determined from the truncated cut-off results. It is apparent that the estimated full Lermard-Jones do not agree with the ex erimental results. More detailed potentials are required to adequately represent & e surface tension of real mixtures. Thus it seems that while the Lennard-Jones mixtures may adequately

195

model the bulk thermodynamic properties of inert mixtures, they are less able the reproduce the interfacial properties.

1.0 -

F

0.5 -

0.0 0.6 0.8 T* 1.0 1.2

Figure 8: Reduced surface tension versus reduced temperature for 2.50 and 4.40 cut-off data and 2.50 and 4.40 cut-off plus tail corrections compared with larger cut-off simulations._ Experiment, q r, = 2.50, q r, = 4.40,+ rc = 6.3cr, n rc = 1.33 (T, q rc = 2.50 + tail correction, and A rc = 4.40 + tail correction (Data from this work and refs. Nijmeijer (1988), Chapela (1977), and Walton (1983))

TABLE 3:

Jnterfacial tensions for binary mixtures of Lennard-Jones Atoms at T*=l.OO.

011=1.0000000 El 1=1.0000000 X,Y =14.3020 #Particles=2048 qa=1.0330396 &r2=1.1668058 Z =42.6050 022=1.0660793 a22=1.3614357 T*=l.OO

:4 414 xatart 0.25 0.50

d * “Itail

2.09 2.33 1;.21 0.44 8.69 0.87 ;i 0:73 82 0.12 0.12 ,*9+4Ytai1 0:86 YExperimentt 0.96 1.21

t:; 0.50 0.75 2.45 2.36 0.44 0.70 0.94 0.88 0.80 0.57 0.06 0.12 0.86 0.69 0.72 0.96

rEz- xpernnen resu ts mterpo ereta. .

SUMMARY AND CONCLUSIONS

There are several subtle points to remember in et-forming computer simula- tions of the Lennard-Jones li in this work are appropriate 9

uid-vapor interface. ATthough the criteria produced

appear in more corn or a pure atomic fluid, these same effects will also

future simulations. h licated systems and should be considered anew in settin up

ese criteria can be grouped into size effects, cut-off ef ects, fg and e uilibration determination.

7% e size effects are related to the number of particles, the area of the interface and the thickness of the slab. The number of particles in the system must be large enough so the surface tension, density difference, and interfacial thickness are not being significantly affected. The interfacial thickness is the most sensitive para- meter to system size. For simple Lennard-Jones atoms, systems should contain at least 1000 particles. The area of the interface in the system must be large enough so the surface tension, density difference, and interfacial thickness are not being

196

significantly effected. Also the area must be at least 4 times the square of the cut- off to avoid periodic boundary effects. The interfacial thickness is the most sensitive parameter to area. The thickness of the liquid slab must be at least two times the cut-off to avoid the interface from being affected by another interface or wall. It must also be thick enough to produce a representative bulk liquid phase. For simple Lennard-Jones atoms the slab should be at least 13.60 thick.

In order to use the tail correction reported by Chapela et al. (1977) to extrapo- late to the full Lennard-Jones potential values, the cut-off must be large enough so the bulk densities are no longer changing significantly. A cut-off of 4.40 is appro- priate for simple Lennard-Jones systems.

To determine when equilibration has been reached the running averages, block averages, and integral of PN-PT are important. The running average surface tension and density profile must no longer be changing significantly. The running and bulk average surface tension must have converged to the same value. The contribution to the calculation of the surface tension from the bulk liquid must be zero.

The initial system setup affects the direction of approach and time to equili- brium. The initial bulk liquid density should be as near to the equilibrium value as possible. Artificially low or high liquid densities cause contributions to the surface tension from the bulk liquid for many time steps as the vapor phase is produced and the liquid slab either collapses or expands. This can cause the running and block average surface tensions as well as the density profiles to appear to be at equilibri- um. However plots of the integral over the cell of the normal minus the tangential pressure component will reveal the bulk liquid contribution and is a more sensitive test for equilibrium. By understanding and accounting for the potential problems in simulating interfaces, reproducible results can be obtained and the structure of the interface studied more thoroughly.

REFERENCES

Chapela, G. A., Saville, G.. and Rowlinson, J. S.,1975. Computer simulation of the gas/Iiquid surface. Faraday Dis. Chem. Sot., 59: 22-28. Chapela, G. A., Saville, G., Thompson, S. M., and Rowlinson. J. S., 1977. Computer simulation of a Gas-Liquid Surface. J. Chem. Sec. Faraday Trans. 2,8: 1133-l 144. Giro, A., Conzalez, J. M., Padro, J. A., and Torra, V., 1981. Molecular dynamics: An analysis of initial conditions and integration methods. Anales de Fisica Se& A, 77: 57-62. Lee, J. K., Barker, J. A., and Pound, G. M., 1974. Surface structure and surface tension: Perturba- tion theory and Monte Carlo calculation. J. Chem. Phys., 60: 1976-1980, Liu, K. S., 1974. Phase separation of Lennard-Jones systems: A film in equilibrium with a vapor. J. Chem. Phys. 60: 4226-4230. Matsumoto, M., and Kataoka, Y.. 1988. Study on liquid-vapor interface of water. I. Simulational results of thermodynamic properties and orlentational structure. J. Chem. Phys., 88: 3233-3245. Lee, D. J., Telo da Gama, M. M., and Gubbins, K. E., 1984. The vapour-liquid interface for a Lennard-Jones model of argon-krypton mixtures. Molec. Phys., 53: 1113-1130. Miyazaki, J., Barker, J. A., and Pound, G. M., 1976. A new Monte Carlo method for calculating surface tension. J. Chem. Phys, 64: 3364-3369. Nadler, K. C., Zollweg, J. A., Streett, W. B.. and McLure, I. A., 1988. Argon+krypton form 120 to 200 K. J. Colloid andInterface Sci., 122: 530-536. Nijmeijer, M. J. P., Baker, A. F., and Bruin, C., 1988. A molecular dynamics simulation of the Lennard-Jones liquid-vapor interface. J. Chem. Phys., 89: 3789-3792. Rao, M., and Levesque, D., 1976. Surface structure of a liquid film. J. Chem. Phys., 65: 3233- 3236. Rae. M., and Beme, B. J.. 1979. On the location of surface of tension in the planar interface between liquid and vapour. Molec. Phys., 37: 455-461. Walton, J. P. R. B. , Tildesley, D. J., and Rowlinson, J. S., 1983. The pressure tensor at the planar surface of a liquid. Molec. Phys.. 48: 1357-1368.