59th Annual Report on Research 2014

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,

     ,                                                                 (1)

where s₀ 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 second year of the award period focused on determining the constant d for a system of Lennard-Jones particles.

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. In the first year of the award, we 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.

In the final year of the award, 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 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.