**Alan E. van Giessen, PhD**, Mount Holyoke College

Since the start of award period, we have made progress on two fronts: determining the Tolman length for the intrinsic liquid-vapor surface and developing a more general formulation of density functional theory for curved surfaces.

The standard approach in describing the free energy
of a curved surface is to expand the radius-dependent surface free energy
density *s*(R) in powers of the
curvature (1/*R*). For a spherical
drop, this takes the form,

where *s*_{0} is the surface tension of a planar interface, *d* is the Tolman length, and *k* and`*k* are known collectively as the rigidity constants. The work that we have
accomplished during the first year of the award period focused on determining
the constants *d*, *k*, and`*k* for a system of Lennard-Jones particles.

Firstly, we have performed simulations of a planar,
liquid-vapor interface. The density profile of this interface – how the
density varies as one goes from deep in the liquid phase to deep in the vapor
phase – is thought to be composed of an intrinsic density profile that moves
back and forth due to the effects of capillary waves. The capillary waved in
effect broaden or smear out the density profile, and it is this broadened
density profile that is typically reported in Molecular Dynamics simulations.
One goal of this work is to eliminate the effects of capillary waves in
broadening the liquid-vapor interface, examine the resulting intrinsic density
profile, and determine how the Tolman length depends
on this profile. To do so, we have developed criteria for robustly and uniquely
locating the Gibbs dividing surface in the interfacial region. As described in
the original proposal, we formally divide up the simulation box into a series
of columns, each column spans the length of the box
perpendicular to the interface (the *z*-direction)
but is only a molecular diameter wide in the other directions. We determine the
density profile *r*(*z*) within each column and apply a crossing constraint
– when the density profile crosses a value of *r*_{c} = 0.5(*r*_{l}
– *r*_{v}) – to determine the Gibbs dividing surface. *r*_{l} and *r*_{v} are
bulk liquid and vapor densities, respectively. The resulting intrinsic density profile
is shown below (solid line) with the capillary-wave-broadened profile (dashed
line) for comparison.

As expected, the capillary-broadened interface has a
significantly thicker interfacial region. The intrinsic profile shows oscillations
in the density reflecting a layered structure near the interface. The existence
of these oscillations in the liquid phase have been seen before, but their extension into the vapor phase is new. The decrease in
density immediately on the vapor side of the interface represents a depletion
layer. This is due to particles in the region feeling a stronger attraction
from the higher-density liquid phase than lower-density vapor phase. Curiously,
there is a small peak in the density profile in this region at about *z* = 0.8. We do not yet fully understand
this peak, but suspect it is due to particles on the vapor side of the
interface being temporarily prevented from being absorbed into the liquid phase
by the presence of a liquid particle immediately beneath it.

In addition to determining the intrinsic density
profile, we have also calculated the Tolman length, *d*, using the intrinsic interface. The Tolman length is related to the first-order coefficient in
an expansion of the surface free energy in power of the curvature, as shown in
Eq. (1). It is now generally agreed upon that the Tolman
length is negative and is equal to *d* = -0.1* *in units of the molecular diameter. Molecular
Dynamics simulations of planar interfaces have reported positive values for the
Tolman length and the source of this discrepancy
remains unclear. We have applied a similar methodology to the calculation of
the Tolman length as to that of determining the
intrinsic density profile and the resulting values for* d* are indeed negative and range from -0.2* *to -0.07. However, this calculation is very sensitive
to the details of the methodology and we do not yet have a precise value for
the Tolman length.

Secondly, we have developed new expressions for the
bending rigidity *k* and Gaussian
rigidity`*k*. Both of these constants are necessary to fully
describe the surface free energy of a curved surface. These expressions feature
a non-local integral term that accounts for the interaction between molecules.
We have used these expressions to investigate the influence of the choice of
the Gibbs dividing surface for a one-component fluid. The location of the
dividing surface is a key element in our calculations of the Tolman length described above. The value of the Tolman length is independent of the choice of dividing
surface. However, the value of the rigidity constants is not.

We have shown that when one chooses the equimolar dividing surface for the Gibbs dividing surface,
the values of the rigidity constants are each at an extremum:
the bending rigidity *k* is at its maximum
and the Gaussian rigidity`*k* is at its minimum.

We have explicitly calculated both *k* and`*k* using a short-ranged potential and determined that *k* is negative and has a value around minus 0.5-1.0 *k*_{B}*T* and`*k *is positive and has a value that is a bit more than
half the magnitude of *k*.

In the coming year, we plan to finalize the
methodology for calculating the Tolman length from our
simulations of planar interfaces and then explore the temperature dependence of
*d*. We will also use this methodology to explore the
discrepancy between the known negative value for *d* and the positive values that have been calculated using Molecular
Dynamics simulations. We hope to have an understanding of this origin of this
discrepancy by the end of the second year of the award. We will also introduce
a second species into our simulations and begin to investigate the behavior of
the surface free energy in systems that contain surfactant or nanoparticles.

Copyright © 2014 American Chemical Society