Page 1

Molecular Simulation Study of the Effect of Pressure on

the Vapor-Liquid Interface of the Square-Well Fluid

Jayant K. Singh and David A. Kofke*

Department of Chemical and Biological Engineering, University at Buffalo,

The State University of New York, Buffalo, New York 14260-4200

Received November 15, 2004. In Final Form: February 28, 2005

We examine a model system to study the effect of pressure on the surface tension of a vapor-liquid

interface. The system is a two-component mixture of spheres interacting with the square-well (A-A) and

hard-sphere (B-B) potentials and with unlike (A-B) interactions ranging (for different cases) from hard

sphere to strongly attractive square well. The bulk-phase and interfacial properties are measured by

molecular dynamics simulation for coexisting vapor-liquid phases for various mixture compositions,

pressures, and temperatures. The variation of the surface tension with pressure compares well to values

givenbysurface-excessformulasderivedfromthermodynamicconsiderations.Wefindthatsurfacetension

increases with pressure only for the case of an inert solute (hard-sphere A-B interactions) and that the

presenceofA-Battractionsstronglypromotesadecreaseofsurfacetensionwithpressure.Anexamination

ofdensityandcompositionprofilesismadetoexplaintheseeffectsintermsofsurface-adsorptionarguments.

I. Introduction

The effect of pressure, p, on interfacial tension, γ, is an

issueoflongstandinginterest.1-7Thebehavioriscaptured

by the derivative

where a change that occurs along the saturation curve

(subscript σ) at constant temperature, T, and interfacial

area, A, is indicated. From the phase rule, there is only

one degree of freedom for two coexisting phases of a pure

substance and thus one cannot vary saturation pressure

at a fixed temperature; for a pure substance, τ is not

defined. To proceed, it is necessary to consider a two-

component system, for which the phase rule permits

isothermal variation of the pressure while maintaining

the presence of two phases. However, in this case, one

still does not get a description of the purely mechanical

effects that pressure has on surface tension. It is not

possible to effect the change in pressure without also

changingthespeciescompositionofthecoexistingphases,

which in turn can modify the composition and structure

of the interfacial region. Thus, the effect of pressure on

surface tension, when measured this way, is necessarily

a result of the combined mechanical (pressure) and

chemical(composition)effects.Inthebestcase,an“inert”

gas (insoluble in the liquid) is added to pressurize the

system, which then produces changes in the vapor-phase

composition only.

A Maxwell relation provides some insight that can be

used to predict and understand the effect of pressure for

a two-component system containing N1and N2molecules

of species 1 and 2, respectively:

(

The right-hand side describes the change in total volume

that results from a change in the amount of interfacial

area between the phases, keeping the overall mole

numbers fixed. Rice5has discussed the effects giving rise

to the change of volume. On one hand, movement of

material from the bulk liquid to form the new surface

(wherethedensityisless)willresultinanincreaseinthe

volume and tend to make the derivative positive. On the

otherhand,ifvapor-phasemoleculesadsorbtosomedegree

onthesurface,thenasnewsurfaceforms,itadsorbsmore

material from the vapor, causing the volume there to

decrease and thus tend to make the derivative negative.

In practice, both positive and negative values of τ have

beenobservedinexperimentsinvolvingthepressurization

of a vapor-liquid interface using an inert gas, although

negative values are much more prevalent.7

Hansen8presented a general formulation of interfacial

thermodynamics, developed such that the pressure re-

mainsarelevantindependentvariable,whilemakingboth

species chemical potentials into dependent variables.

Turkevich and Mann6also showed how Hansen’s con-

struction could be used to determined τ strictly in terms

of the volume and moles of the two-phase system and the

densities of the bulk phases. Considering henceforth a

mixture of two species only, a Gibbs-Duhem equation

canbewrittenforthecompositeliquid+vapor+interface

system

where S, V, and Niare the total entropy, volume, and

number of moles of species i in the two-phase system,

respectively,andµiisthechemicalpotentialofcomponent

i. To maintain equilibrium between the phases, an

isothermal change in pressure must be accompanied by

changes in the chemical potentials that permit them to

(1) Gibbs,J.W.CollectedWorks(YaleUniversityPress: NewHaven,

1906); Dover: New York, 1961; Vol. 1, p 236.

(2) Lewis, G. N.; Randall, M. Thermodynamics and the Free Energy

of chemical substances; McGraw-Hill: New York, 1923; Chapter 21.

(3) Bridgman, P. W. The Physics of High Pressure; Beel: London,

1952.

(4) Defay, R.; Prigogine, I.; Bellemans, A.; Everett, D. H. Surface

Tension and Adsorption; Wiley: New York, 1966; p 89.

(5) Rice, O. K. J. Chem. Phys. 1947, 15, 333.

(6) Turkevich, L. A.; Mann, J. A. Langmuir 1990, 6, 445.

(7) Turkevich, L. A.; Mann, J. A. Langmuir 1990, 6, 457.

(8) Hansen, R. S. J. Phys. Chem. 1962, 66, 410.

τ ≡(

∂γ

∂p)σ,T,A

(1)

∂γ

∂p)T,A,N1,N2)(

∂V

∂A)T,p,N1,N2

(2)

-S dT + V dp - N1dµ1- N2dµ2- A dγ ) 0 (3)

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Langmuir 2005, 21, 4218-4226

10.1021/la0471947 CCC: $30.25 © 2005 American Chemical Society

Published on Web 03/30/2005

Page 2

remain equal between the phases. These changes can be

describedbyGibbs-Duhemequationswrittenseparately

for the two phases

with the phases indicated by the superscript (R or ?).

Alternatively,

whereniisthemolardensityofspeciesiinthebulkphase

and s is the entropy density. By subtracting each of the

equationsgivenineq5fromeq3anddividingbythearea,

we obtain a Gibbs-Duhem equation that amplifies the

effects of the interface

where s ˆ is the surface-excess entropy. The surface excess

of species i is defined as

At the level of detail where the interface is significant,

thevolumesVRandV?areambiguous.Typically,theyare

definedtomaketwoofthetermsineq6vanish.Themost

common definition is that due to Gibbs, for which Γ1) 0

and V - VR- V?) 0. This definition has the advantage

of conserving the total volume and permits the identifica-

tion of a single plane that separates the two phases. In

thepresentcontext,however,thisdefinitionisnothelpful,

as it obscures the influence of pressure on the other

quantities. Instead, it is more useful to define the two

molar excess properties to be zero: Γ1) 0 and Γ2) 0.

With the volumes VRand V?defined this way, the

derivative, τ, defined in eq 1 is precisely (V - VR- V?)/A.

More specifically,6

Here, the partition coefficients are defined as κ1) n1

and κ2) n2

dividing plane, but Turkevich and Mann have described

how it can be connected to the Motomura two-plane

approach.9

Equation 8 is subtle. It appears to describe a surface

property in terms of purely bulk-phase properties, but

thisisnotthecase.Werewetoignoretheinterfacialeffects

andtreatthephasesashomogeneouswithdensitiesgiven

everywhere by their bulk values, even very near the

interface, we could write N1) V(n1

) V(n2

volume occupied by phase R. Substitution of these expres-

sions in eq 8 yields, incorrectly, τ ) 0. When applied

without this approximation, eq 8 yields a nonzero value

of τ that includes elements that describe how the overall

?/n1

R

R/n2

?. This approach makes no reference to a

Rφ + n1

?(1 - φ)) and N2

Rφ + n2

?(1 - φ)), where φ is the fraction of the total

composition differs from a simple average of the bulk

compositions.

Whileeq8hasthebenefitofbeingarigorousexpression

for the pressure derivative of the surface tension, its

connection to the equally rigorous eq 2 is not clear. The

argumentsgiveninconnectiontoeq2relatetotheexcess

solute at the surface, and this quantity is more naturally

captured by the Gibbs definition of Γ2of eq 7. One can

proceedlesseasilybutstillrigorouslyfromthisstandpoint

too. Using the Gibbs convention for VRand V?, we write

τ ) -Γ2(

The derivative here can be given in terms of the bulk-

phasepropertiesusingananalysissimilartothatleading

to the Clapeyron equation.10The result is

(

n1

≈1

n2

where the approximate equality is based on the assump-

tions that the amount of solute in the liquid is negligible

(n2

greater than its vapor-phase density (n1

indication is that the derivative is positive. Combination

of eqs 9 and 10 indicates that the slope, τ, is of opposite

sign to the surface excess of species 2. Thus, adsorption

ofsoluteonthesurface(indicatedbypositiveΓ2)fromthe

vapor promotes the decrease of surface tension with

pressure, as argued in the context of eq 2.

Molecularsimulationshavebeenappliedtounderstand

surfacebehaviorinavarietyofcontexts.11,12Suchstudies

canbeusefulinprobingmolecular-levelaspectsofsurface

phenomena.Modelingstudiesarealsoofinterestfortheir

abilitytoexaminesystematicallyhowqualitativefeatures

ofmolecularinteractionsinfluencesurfaceproperties.As

it involves the effect of solutes on surface behavior,

adsorptionofavolatilecomponentcansignificantlyreduce

the surface tension, as shown by Lee et al.13Interesting

behavior was also noticed by Lee et al. for a low ratio of

solute/solventdiameter.Astheadsorptionisverylittleat

the interface, under such conditions, surface tension

increases with an increase in the composition of solute.

Severalothers14-17alsostudiedthevapor-liquidinterface

of binary mixtures. However, the issue of the effect of

pressure on surface tension has not been explored previ-

ously with these techniques.

In this work, we study the effect of pressure on surface

tension for some model binary systems. In particular, we

examine the vapor-liquid interfacial properties for the

R≈ 0) and that the liquid-phase density of solvent is

R. n1

?); the

(9) Motomura, K.; Aratano, M. Langmuir 1987, 3, 304.

(10) Denbigh, K. Principles of Chemical Equilibrium, 4th ed.;

Cambridge University Press: Cambridge, U.K., 1971.

(11) Rowlinson, J. S.; Widom, B. Molecular Theory of Capillarity;

Oxford University Press: Oxford, U.K., 1982.

(12) Croxton, C. A. Statistical Mechanics of the Liquid Surface;

Wiley: New York, 1980.

(13) Lee, D. J.; daGamma, M. M. T.; Gubbins, K. E. J. Phys. Chem.

1985, 89, 1514.

(14) Lee, D. J.; daGamma, M. M. T.; Gubbins, K. E. Mol. Phys. 1984,

53, 1113.

(15) Salomons, E.; Mareschal, M. J. Phys.: Condens. Matter 1991,

3, 3645.

(16) Salomons, E.; Mareschal, M. J. Phys.: Condens. Matter 1991,

3, 9215.

(17) Mecke, M.; Winkelmann, J.; Fischer, J. J. Chem. Phys. 1999,

110, 1188.

-SRdT + VRdp - N1

Rdµ1- N2

Rdµ2) 0

-S?dT + V?dp - N1

?dµ1- N2

?dµ2) 0(4)

VR(-sRdT + dp - n1

Rdµ1- n2

Rdµ2) ) 0

V?(-s?dT + dp - n1

?dµ1- n2

?dµ2) ) 0 (5)

-s ˆ dT +1

A(V - VR- V?) dp -

Γ1dµ1- Γ2dµ2- dγ ) 0 (6)

Γi) (Ni- ni

RVR- ni

?V?)/A

(7)

τA ) V - [(1 - κ2)N1/n1

R+ (1 - κ1)N2/n2

?]/(1 - κ1κ2)

(8)

∂µ2

∂p)σ,T

(9)

∂µ2

∂p)T,σ)

n1

?n2

?- n1

R- n1

R

Rn2

?

?

(10)

Effect of Pressure on the Vapor-Liquid InterfaceLangmuir, Vol. 21, No. 9, 2005

4219

Page 3

square-well (SW) model, in the presence of solutes of

varying degree of attraction for the SW solvent. We

demonstratetheeffectsdiscussedabove,namely,thatthe

surface tension increases with pressure for a truly inert

solute but that even a modest degree of attraction can

cause τ to become negative. We also use the detailed

informationprovidedbysimulationtotesteq8forτ,which

is not as easily implemented using experimental data.

The rest of this paper is organized as follows. In the next

section,wegivedetailsaboutthemodelsandsimulations

performed in this study. Then, in section 3, we present

and discuss the results. We conclude in section 4.

II. Model and Methods

Themodelchoseninthisstudyisaliquid-vaporsystem

ofsquare-well(SW)particles.TheSWpotentialisarguably

the simplest model that incorporates both repulsive and

attractive forces between molecules. It is defined by the

pair energy, u(r):

u(r) ){

where λσ is the potential-well diameter, ? is the depth of

the well, and σ is the diameter of the hard core. Because

ofitssimplicityandanalytictractability,theSWpotential

hasbeenappliedasamodelofsimpleatomicsystems,18-20

colloidal particles,21-24heterochain molecules,25,26and

complex systems,27-29among others.

In the present study, all systems employed N ) 1000

particles of which the xswfraction of the molecules were

square-well solvent species (A) and the rest were solute

particles(B).Allhard-corediametersweresetequal,that

is, σAA) σAB) σBB≡ σ. The solute particles interacted

only as hard spheres (?BB ) 0, no attraction), and the

solvent SW parameters were the same for all systems

studied here. A range of values for the solute-solvent

interaction was examined, from purely repulsive with no

attraction (?AB) 0) to highly attractive (?AB) 2.0). In this

and all that follows, properties are given in units such

that σ and ?AAare unity.

A common method to calculate surface tension by

molecular simulation is by placing a slab of fluid in a

rectangularsimulationcellwithperiodicboundaries,such

that the fluid spans the short (x, y) dimensions of the

simulation volume.30The z-axis is extended to produce

the vapor phase, and the system is allowed to equilibrate

to create a vapor space before taking the averages of the

properties of interest. We adopted this approach for the

present study. The thermodynamic definition of the

surface tension, γ, expresses it in terms of the change in

freeenergy,F,astheinterfacialarea,A,oftwocoexisting

phases is changed at constant volume, V,

Fromthisdefinition,wecanshowthatthesurfacetension

can be expressed in terms of components of the pressure

tensor for the slab based geometry

wherepRRistheRRcomponentofthepressuretensor.The

factor of

presenceoftwointerfacesinthesystem.Lzistheextended

length of the box.

Pressure-tensor components can be obtained from the

virial.30For pairwise-additive potentials, the expression

is

1/2 multiplying the average accounts for the

whereNisthetotalnumberofmolecules,Fisthenumber

density,kistheBoltzmannconstant,Tisthetemperature,

rijis the vector between the center of mass of molecules

iandj,andfij)-∇uijistheforcebetweenthem;theangle

brackets indicate an ensemble or time average. For hard

potentials such as those used in this study, the forces are

impulsive, having infinite magnitude but acting for an

infinitesimal time. When integrated over time, each

collisioncontributesawell-definedamounttotheaverage

in eq 14

wheretsimisthetotalsimulationtimeandthesumisover

all collisions occurring in this time; ∆pijis the impulse

associated with the collision between atoms i and j. The

simulation proceeds in the usual manner for impulsive

potentials:30solveforthetimewhenthenextpaircollides

(which occurs when any two particles reach a separation

equaltothehard-coreorsquare-welldiameters),advance

eachparticletothattimeviafree-flightkinematics,process

the dynamics of the colliding pair, and move on to the

next collision to repeat the process. With each collision,

a contribution to the pressure-tensor averages is made in

accordance with eq 15.

Our molecular dynamics (MD) simulations were per-

formed in a canonical (NVT) ensemble, that is, at a

prescribed total particle number (liquid + vapor +

interface),totalvolume,andtemperature.Thesimulation

was started from a face-centered-cubic lattice configura-

tion in a cubic periodic box. The initial overall density

was fixed at Fσ3) 0.84, from which we created a vacuum

by expanding the box in one dimension, such that the

final dimension of the box had Lx) Ly) 10σ and Lz) 4

× Lx. The temperature was kept constant by simple

momentum scaling. The simulations were equilibrated

for 1.3 million time steps, and averages were taken for

around 400 000 time steps (where the time step is ∆t )

0.02σAA?m/?AA, with m being the particle mass; this step

is given only as a convenient measure of the length of the

simulation,andithasnoeffectonthedynamics.Typically,

multiple collisions will occur in a single ∆t).

(18) Del Rio, F.; Delonngi, D. A. Mol. Phys. 1985, 56, 691.

(19) Vega, L.; de Miguel, E.; Rull, L. F.; Jackson, G.; McLure, I. A.

J. Chem. Phys. 1992, 96, 2296.

(20) Chang, J.; Sandlar, S. I. Mol. Phys. 1994, 81, 745.

(21) Bolhuis, P.; Frenkel, D. Phys. Rev. Lett. 1994, 72, 2211.

(22) Asherie, N.; Lomakin, A.; Benedek, G. B. Phys. Rev. Lett. 1996,

77, 4832.

(23) Noro, M. G.; Frenkel, D. J. Chem. Phys. 2000, 113, 2941.

(24) Zaccarelli, E.; Foffi, G.; Dawson, K. A.; Sciortino, F.; Tartaglia,

P. Phys. Rev. E 2001, 63, 031501.

(25) Cui, J.; Elliot, J. R. J. Chem. Phys. 2001, 114, 7283.

(26) McCabe, C.; Gil-Villegas, A.; Jackson, G.; Del Rio, F. Mol. Phys.

1999, 97, 551.

(27) Lomakin, A.; Asherie, N.; Benedek, G. B. J. Chem. Phys. 1996,

104, 1646.

(28) Zhou, Y.; Karplus, M.; Ball, K. D.; Berry, R. S. J. Chem. Phys.

2002, 116, 2323.

(29) Zhou,Y.;Karplus,M.;Wichert,J.M.;Hall,C.K.J.Chem.Phys.

1997, 107, 10691.

(30) Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids;

Oxford University Press: Oxford, U.K., 1987.

∞, 0 < r < σ

-?, σ er < λσ

0, λσ er

(11)

γ )(

∂F

∂A)T,V,N

(12)

γ )Lz

2〈pzz-1

2(pxx+ pyy)〉

(13)

pR?) FkT +

1

V〈∑

i)1

N-1

∑

j>i

N

(rij)R(fij)?〉

(14)

pR?) FkT +

1

Vtsim∑

collisions

(rij)R(∆pij)?

(15)

4220

Langmuir, Vol. 21, No. 9, 2005 Singh and Kofke

Page 4

The phase coexistence point was adjusted by varying

the overall composition of the system. A fixed number of

solventatoms(rangingfrom700to1000)andsoluteatoms

(from 300 to 0) were introduced, and the vapor and liquid

densities and compositions were permitted to adopt their

equilibriumvalues.Theresultingpressurewasmeasured

by averaging the virial.

III. Results and Discussion

We consider first a system in which the solute has no

attractiontothesolvent,sothatsolute-soluteandsolute-

solvent interactions are pure hard sphere. This system is

a true example of the inert gas that is sometimes

approximatedexperimentallywhenfocusingontheeffect

ofpressureonsurfacetension.Weexaminedthebehavior

for four temperaturessT (in units of ?AA/k) ) 1.0, 1.05,

1.1, and 1.15sand with solvent-solvent square-well

parameters λ ) 1.5 and 2.0 (in separate studies); we

present detailed results for the λ ) 1.5 case only, as the

results for λ ) 2.0 are qualitatively similar. Figure 1

presents a portion of a pressure-composition (p-xy)

coexistence diagram as determined using the vapor +

liquid simulations described above. The amount of hard-

sphere solute in the liquid phase is negligibly small.

The surface tension for the hard-sphere solute system

ispresentedasafunctionofpressureandtemperaturein

Figure2,alongwithdataforthepuresquare-wellsystem

takenfrompreviouswork.31Thetemperaturedependence

is as expected, with the surface tension decreasing

smoothlywithincreasingtemperature.Asforthepressure

dependence,thissystemexhibitstheuncommonbehavior

in which surface tension increases with pressure. This

outcomecouldbeanticipatedonthebasisofthearguments

reviewed above, relating the pressure dependence to the

volume change accompanying the creation of interfacial

area (eq 2). For this system, an increase in area results

in an increase in volume (at fixed pressure), because the

new area takes liquid and moves it to the lower-density

interface. There is no competing effect of adsorption of

solute at the interface, because the solute is completely

inert. It would seem that this behavior is described well

through examination of the density and composition

profiles, which are presented in Figure 3 for T ) 1.0. It

is evident from the figure that there is indeed a marked

depletion of solute at the interface. The solvent mole

fraction in the vapor is significantly enhanced in the

vicinity of the interface (as defined by the density

variation). However, we will revisit this issue below and

find that the picture presented by Figure 3 gives an

inadequate representation of the relevant effects.

Figure 2 gives an indication of the slope τ ) (∂γ/∂p)T

accordingtotheHansen-Turkevich-Mannformulagiven

above as eq 8. The small lines on each data point indicate

the slope expected from the formula, and they show an

imperfectbutstillsatisfactoryagreementwiththeoverall

behavioroftheγversuspcurves.Thevaluesofτcomputed

this way are presented in Table 1. This quantity derives

its value from an imbalance between the number of

moleculesinthesystemversusthenumbergiveninterms

of the bulk densities, and consequently, its evaluation

requires precise knowledge of the bulk-phase densities

andthenumbersofmoleculesofeachspeciesinthephase.

Ofcourseinasimulation,themoleculenumbersareknown

exactly and the densities are given rather precisely also.

One might consider in this context how well this calcula-

tion could be completed using experimentally obtained

data. To aid in this evaluation, we have performed a

sensitivity analysis, in which we computed ∂ ln τ/∂ ln x,

where x is any of the quantities appearing in eq 8. This

derivative describes, roughly, what percent change in τ

can be expected from a 1% change (or error) in each

quantity. We find that this derivative is of the order of 20

orso(i.e.,τchangesby20%fora1%changeinthequantity)

(31) Singh, J. K.; Kofke, D. A.; Errington, J. R. J. Chem. Phys. 2003,

119, 3405.

Figure 1. Vapor-liquid coexistence diagram for the ?AB) 0

(hard-sphere)mixture,showingcoexistingliquid,x,andvapor,

y,solventmolefractionsforvariouspressures.Thelinesdescribe

data for different temperatures: 1.0 (bottom), 1.05, 1.1, and

1.15 (top).

Figure 2. Surface tension versus saturation pressure for the

?AB ) 0 mixture. Temperatures, kT/?AA, are indicated in the

legend.Theshortslantedlinesthrougheachpointhaveaslope

equal to τ, as given in eq 8.

Table 1. Surface Tension, γ, Its Pressure Derivative, τ,

According to eq 8, and the Gibbs Surface Excess, Γ2a

?AB) 0.0

xA

T

γτ

0.7 1.0 0.36(1)1.24

-0.10

0.7 1.05 0.29(1)1.64

-0.14

0.7 1.1 0.22(1) 1.65

-0.14

0.7 1.150.16(1) 0.67

-0.059

0.8 1.0 0.33(1)1.24

-0.073

0.8 1.05 0.25(2)1.23

-0.073

0.81.10.20(1)1.61

-0.098

0.81.15 0.13(2)0.90

-0.055

0.91.0 0.30(2)1.11

-0.034

0.9 1.050.24(1)1.38

-0.044

0.91.1 0.18(2)1.02

-0.032

0.91.150.11(1)0.94

-0.031

0.95 1.00.30(2) 1.27

-0.020

0.951.05 0.22(1)0.48

-0.0075

0.95 1.10.16(2)0.98

-0.016

0.951.150.091(13)0.92

-0.015

?AB) 0.5

Γ2

γτ

Γ2

0.19(1)

0.12(1)

0.011(7)

-1.81

-1.21

-0.73

0.13

0.087

0.033

0.21(2)

0.16(1)

0.089(9)

-1.83

-2.69

-1.75

0.092

0.13

0.086

0.25(1)

0.18(1)

0.12(1)

-2.90

-1.75

-2.06

0.073

0.046

0.051

0.27(1)

0.20(1)

0.15(1)

-3.07

-2.44

-1.97

0.040

0.033

0.026

aThe data are given for different temperatures, T (in units of

?AA/k),andoverall(liquid+vapor+interface)solventmolefraction,

xA. All other quantities are given in units such that σAAand ?AAare

unity. The numbers in parentheses indicate the 67% confidence

limits of the last digit(s) of the tabled value.

Effect of Pressure on the Vapor-Liquid InterfaceLangmuir, Vol. 21, No. 9, 2005

4221

Page 5

foralmostalloftheindependentvariables,indicatingthat

the τ value calculated this way is indeed sensitive to the

quality of these data. The noise in the slopes depicted in

Figure 2 is consistent with this observation.

To examine the effect of solvent (A)-solute (B) interac-

tion on the surface tension, we repeated the calculations

describedabovefortheSW-HSmixturebutwithanA-B

interaction that is mildly attractive (?AB) 0.5). The B-B

interaction remains purely repulsive. Figure 4 presents

a portion of the phase diagram for the ?AB) 0.5 system.

Perhaps its most notable feature is that it is not quali-

tatively different from the SW-HS behavior shown in

Figure1.Themolefractionofsoluteintheliquidisslightly

more, but overall, the form of the coexistence envelopes

is very similar to that seen in Figure 1.

In light of the great similarities in the phase diagrams

for this and the SW-HS systems, it is interesting to

observe qualitatively different behaviors in the surface

tension versus pressure. Figure 5 presents these data.

The slope has changed sign, and the system now exhibits

the more common behavior of decreasing surface tension

with pressure. The small attraction added to the solute-

solvent interactions is quite sufficient to change the

phenomenology.Thisoutcomeexplainswellwhymostreal

systems (i.e., any not involving a helium solute) exhibit

a negative τ value; it simply does not take much solute-

solventattractiontocausethesurfacetensiontodecrease

withpressure.Slopesaccordingtoeq8areagainindicated

on the figure and agree acceptably with the shape of the

curves. The sensitivities of the slopes to the parameters

of the equation are listed in Table 2. Roughly the same

degree of sensitivity is seen as in the previous example.

Giventhequalitativechangeinthepressuredependence

of γ, one might expect to see a stark change in the

concentration and density profiles. However, we do not,

as shown in Figure 6. There remains a significant layer

of solvent enhancement in the vapor near the interface,

which itself is not surprising given the stronger affinity

ofthesolventmoleculesforthe(solvent-dominated)liquid

phase.Amoreilluminatingpictureexaminesthebehavior

ofthesolventandsoluteprofilestogether.Thesebehaviors

canconnecttothesurfaceexcess,Γ2,whichwasshownby

eqs 9 and 10 to relate simply to τ for the type of system

presently under study. Figures 7 and 8 are the relevant

Figure 3. Profiles of solvent mole fraction (solid line) and total molar density (dashed line) for various compositions of the ?AB

) 0 mixture at T ) 1.0. The overall mole fraction (vapor + liquid + interface) for square-well (A) species is given in the inside box.

Figure 4. Vapor-liquid coexistence diagram for the ?AB) 0.5

mixture. The lines describe data for different temperatures:

1.0 (bottom), 1.05, and 1.1 (top).

Figure 5. Surface tension versus saturation pressure for the

?AB) 0.5 mixture. Temperatures, kT/?AA, are indicated in the

legend.Theshortslantedlinesthrougheachpointhaveaslope

equal to τ, as given in eq 8.

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Langmuir, Vol. 21, No. 9, 2005 Singh and Kofke