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 Interface Langmuir, Vol. 21, No. 9, 2005
indication of even the expected sign of τ, and while it
suggests that τ might diverge at the azeotrope, it may be
thatΓ2vanishes there also, which would negate the basis
for the asymptotic form. A similar analysis applied to eq
τ ∼ -(N -N1
to gauge the behavior of the first term in parentheses,
which captures the surface effects.
The study of simple molecular models can provide
qualitative insight regarding the molecular origins of
how features of the surface connect to the way vapor-
liquid surface tension varies with pressure for two-
component mixtures. With the molecular model, we can
In this case, we observe that the pressure derivative is
positive, in agreement with experimental observations
that use helium as a pressurizing gas. The addition of
modest attraction between the gas and the solvent finds
that the pressure derivative takes negative values and
may be at points of azeotropy, at which we again observe
that this is a general feature of azeotropic systems.
The present study permits us to examine the relation
of the pressure derivative to surface-excess properties.
We find that the connection between the derivative and
the Gibbs surface excess is in place for the limit in which
the solute is insoluble in the solvent. In the more general
case, the general surface thermodynamics of Hansen
We show that the pressure derivative established using
this formalism is in good agreement with the observed
variation of the surface tension with pressure.
Acknowledgment. This work has been supported by
the U.S. National Science Foundation, grants CTS-
been provided by the University at Buffalo Center for
R)(v?- vR)δ-1+ O(δ0)(18)
Langmuir, Vol. 21, No. 9, 2005Singh and Kofke