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
increases with pressure only for the case of an inert solute (hard-sphere A-B interactions) and that the
The effect of pressure, p, on interfacial tension, γ, is an
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
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
gas (insoluble in the liquid) is added to pressurize the
system, which then produces changes in the vapor-phase
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
volume and tend to make the derivative positive. On the
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
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-
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
where S, V, and Niare the total entropy, volume, and
number of moles of species i in the two-phase system,
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,
(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.
-S dT + V dp - N1dµ1- N2dµ2- A dγ ) 0 (3)
Langmuir 2005, 21, 4218-4226
10.1021/la0471947 CCC: $30.25 © 2005 American Chemical Society
Published on Web 03/30/2005
remain equal between the phases. These changes can be
for the two phases
with the phases indicated by the superscript (R or ?).
and s is the entropy density. By subtracting each of the
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,
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
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.
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
Equation 8 is subtle. It appears to describe a surface
property in terms of purely bulk-phase properties, but
everywhere by their bulk values, even very near the
interface, we could write N1) V(n1
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
?. 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
for the pressure derivative of the surface tension, its
connection to the equally rigorous eq 2 is not clear. The
solute at the surface, and this quantity is more naturally
captured by the Gibbs definition of Γ2of eq 7. One can
too. Using the Gibbs convention for VRand V?, we write
τ ) -Γ2(
The derivative here can be given in terms of the bulk-
to the Clapeyron equation.10The result is
where the approximate equality is based on the assump-
tions that the amount of solute in the liquid is negligible
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
vapor promotes the decrease of surface tension with
pressure, as argued in the context of eq 2.
it involves the effect of solutes on surface behavior,
the surface tension, as shown by Lee et al.13Interesting
behavior was also noticed by Lee et al. for a low ratio of
the interface, under such conditions, surface tension
increases with an increase in the composition of solute.
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
(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,
(15) Salomons, E.; Mareschal, M. J. Phys.: Condens. Matter 1991,
(16) Salomons, E.; Mareschal, M. J. Phys.: Condens. Matter 1991,
(17) Mecke, M.; Winkelmann, J.; Fischer, J. J. Chem. Phys. 1999,
-SRdT + VRdp - N1
-S?dT + V?dp - N1
VR(-sRdT + dp - n1
Rdµ2) ) 0
V?(-s?dT + dp - n1
?dµ2) ) 0 (5)
-s ˆ dT +1
A(V - VR- V?) dp -
Γ1dµ1- Γ2dµ2- dγ ) 0 (6)
Γi) (Ni- ni
τA ) V - [(1 - κ2)N1/n1
R+ (1 - κ1)N2/n2
?]/(1 - κ1κ2)
Effect of Pressure on the Vapor-Liquid InterfaceLangmuir, Vol. 21, No. 9, 2005
square-well (SW) model, in the presence of solutes of
varying degree of attraction for the SW solvent. We
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
is not as easily implemented using experimental data.
The rest of this paper is organized as follows. In the next
performed in this study. Then, in section 3, we present
and discuss the results. We conclude in section 4.
II. Model and Methods
the simplest model that incorporates both repulsive and
attractive forces between molecules. It is defined by the
pair energy, 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
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
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
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
phases is changed at constant volume, V,
can be expressed in terms of components of the pressure
tensor for the slab based geometry
length of the box.
Pressure-tensor components can be obtained from the
virial.30For pairwise-additive potentials, the expression
1/2 multiplying the average accounts for the
rijis the vector between the center of mass of molecules
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
in eq 14
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
(which occurs when any two particles reach a separation
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 +
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
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,
(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,
(28) Zhou, Y.; Karplus, M.; Ball, K. D.; Berry, R. S. J. Chem. Phys.
2002, 116, 2323.
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
pR?) FkT +
pR?) FkT +
Langmuir, Vol. 21, No. 9, 2005 Singh and Kofke
The phase coexistence point was adjusted by varying
the overall composition of the system. A fixed number of
(from 300 to 0) were introduced, and the vapor and liquid
densities and compositions were permitted to adopt their
by averaging the virial.
III. Results and Discussion
We consider first a system in which the solute has no
solvent interactions are pure hard sphere. This system is
a true example of the inert gas that is sometimes
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
is as expected, with the surface tension decreasing
in which surface tension increases with pressure. This
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
above as eq 8. The small lines on each data point indicate
the slope expected from the formula, and they show an
this way are presented in Table 1. This quantity derives
its value from an imbalance between the number of
of the bulk densities, and consequently, its evaluation
requires precise knowledge of the bulk-phase densities
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
(31) Singh, J. K.; Kofke, D. A.; Errington, J. R. J. Chem. Phys. 2003,
Figure 1. Vapor-liquid coexistence diagram for the ?AB) 0
data for different temperatures: 1.0 (bottom), 1.05, 1.1, and
Figure 2. Surface tension versus saturation pressure for the
?AB ) 0 mixture. Temperatures, kT/?AA, are indicated in the
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
0.7 1.0 0.36(1)1.24
0.7 1.05 0.29(1)1.64
0.7 1.1 0.22(1) 1.65
0.7 1.150.16(1) 0.67
0.8 1.0 0.33(1)1.24
0.8 1.05 0.25(2)1.23
0.95 1.00.30(2) 1.27
aThe data are given for different temperatures, T (in units of
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
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
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
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
systems (i.e., any not involving a helium solute) exhibit
a negative τ value; it simply does not take much solute-
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.
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
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
equal to τ, as given in eq 8.
Langmuir, Vol. 21, No. 9, 2005 Singh and Kofke