Equimolar Radius Research Articles

Variation of Surface Propensity of Halides with Droplet Size and Temperature: The Planar Interface Limit.

The radial number density profiles of halide and alkali ions in aqueous clusters with equimolar radius ≲1.4 nm, which correspond to ≲255 H2O molecules, have been extensively studied by computations. However, the surface abundance of Cl-, Br-, and I- relative to the bulk interior in these smaller clusters may not be representative of the larger systems. Indeed, here we show that the larger the cluster is, the lower the relative surface abundance of chaotropic halides is. In droplets with an equimolar radius of ≈2.45 nm, which corresponds to ≈2000 H2O molecules, the polarizable halides show a clear number density maximum in the droplet's bulk-like interior. A similar pattern is observed in simulations of the aqueous planar interface with halide salts at room temperature. At elevated temperature the surface propensity of Cl- decreases gradually, while that of I- is partially preserved. The change in the chaotropic halide location at higher temperatures than the room temperature may considerably affect photochemical reactivity in atmospheric aerosols, vapor-liquid nucleation and growth mechanisms, and salt crystallization via solvent evaporation. We argue that the commonly used approach of nullifying parameters in a force field in order to find the factors that determine the ion location does not provide transferable insight into other force fields.

Molecular droplets vs bubbles: Effect of curvature on surface tension and Tolman length

Droplets and bubbles are thought to be two sides of the same coin; this work determines how true this is at the molecular scale. Stable cylindrical nanodroplets and nanobubbles are obtained in Molecular Dynamics (MD) simulations with three-phase contact lines pinned by alternate hydrophobic and hydrophilic patterns. The surface tension and Tolman length for both types of curved interfaces are obtained with the Kirkwood–Buff method, based on the difference between normal and tangential pressure components. Both bubble and droplet cases are compared to the flat interface case for reference. Results show that the surface tension decreases linearly while the Tolman length increases linearly with the gas/liquid density ratio. By running a careful parameter study of the flat interface over a range of densities, the effect of the density ratio can be corrected isolating the effects of curvature on the surface tension and Tolman length. It is found that such effects start to be seen when the equimolar curvature radius goes down to 20 reduced Lennard–Jones (LJ) units. They have the same magnitude but act with opposite signs for nanodroplet and nanobubble interfaces. Considering effects of the density ratio and curvature, a fitted Tolman equation was obtained, which predicts the surface tension of a curved interface. Results obtained by the fitted Tolman equation agree well with those obtained by the MD simulations except at very small curvature radius (<10 reduced LJ units) due to the accumulation of the curvature dependence of the Tolman length.

Gibbs-Tolman approach to the curved interface effects in asymmetric nuclei

We redefine the surface tension coefficient and the surface symmetry energy for an asymmetric nuclear Fermi-liquid drop with a finite diffuse layer. Considering a two-component charged Fermi-liquid drop and following the Gibbs-Tolman concept, we introduce the equimolar radius ${R}_{e}$ of a sharp-surface droplet at which the surface tension is applied and the radius of the tension surface ${R}_{s}$ (Laplace radius) which provides the minimum of the surface tension coefficient $\ensuremath{\sigma}$. We have shown that the nuclear Tolman length $\ensuremath{\xi}$ is negative and the modulus of $\ensuremath{\xi}$ grows quadratically with the asymmetry parameter $X=(N\ensuremath{-}Z)/(N+Z)$.

About applicability of thermodynamic parameters to small drops and clusters

In our paper the results of molecular dynamics calculations of nanodrops are presented. The density profiles, the Irving-Kirkwood pressure tensors, the chemical potentials of the systems, the equimolar radii of the drops and the radii of tension, the mechanical and thermodynamic surface tensions have been calculated.It is shown that both the mechanical and thermodynamic surface tensions decrease with the decrease of the equimolar radius of the drop and reach zero at the same R0 depending on temperature. With the further equimolar radius decrease the surface tension becomes negative. It means that such droplet is metastable and that the notion of the macroscopic surface tension cannot be applied to such small drops. The dependence of the ratio of the drop surface tension to the surface tension of the flat liquid-vapor interface on the ratio of the drop equimolar radius to R0 is a universal function.

Curved and diffuse interface effects on the nuclear surface tension

We redefine the surface tension coefficient for a nuclear Fermi-liquid drop with a finite diffuse layer. Following Gibbs-Tolman concept, we introduce the equimolar radius R_e of sharp surface droplet at which the surface tension is applied and the radius of tension surface R_s which provides the minimum of the surface tension coefficient \sigma. This procedure allows us to derive both the surface tension and the corresponding curvature correction (Tolman length) correctly for the curved and diffuse interface. We point out that the curvature correction depends significantly on the finite diffuse interface. This fact is missed in traditional nuclear considerations of curvature correction to the surface tension. We show that Tolman's length \xi is negative for nuclear Fermi-liquid drop. The value of the Tolman length is only slightly sensitive to the Skyrme force parametrization and equals \xi=-0.36 fm.

Excess equimolar radius of liquid drops

The curvature dependence of the surface tension is related to the excess equimolar radius of liquid drops, i.e., the deviation of the equimolar radius from the radius defined by the macroscopic capillarity approximation. Based on the Tolman [J. Chem. Phys. 17, 333 (1949)] approach and its interpretation by Nijmeijer et al. [J. Chem. Phys. 96, 565 (1991)], the surface tension of spherical interfaces is analyzed in terms of the pressure difference due to curvature. In the present study, the excess equimolar radius, which can be obtained directly from the density profile, is used instead of the Tolman length. Liquid drops of the truncated and shifted Lennard-Jones fluid are investigated by molecular dynamics simulation in the canonical ensemble, with equimolar radii ranging from 4 to 33 times the Lennard-Jones size parameter σ. In these simulations, the magnitude of the excess equimolar radius is shown to be smaller than σ/2. This suggests that the surface tension of liquid drops at the nanometer length scale is much closer to that of the planar vapor-liquid interface than reported in studies based on the mechanical route.

Numerical approaches to determine the interface tension of curved interfaces from free energy calculations

A recently proposed method to obtain the surface free energy σ(R) of spherical droplets and bubbles of fluids, using a thermodynamic analysis of two-phase coexistence in finite boxes at fixed total density, is reconsidered and extended. Building on a comprehensive review of the basic thermodynamic theory, it is shown that from this analysis one can extract both the equimolar radius R(e) as well as the radius R(s) of the surface of tension. Hence the free energy barrier that needs to be overcome in nucleation events where critical droplets and bubbles are formed can be reliably estimated for the range of radii that is of physical interest. It is found that the conventional theory of nucleation, where the interface tension of planar liquid-vapor interfaces is used to predict nucleation barriers, leads to a significant overestimation, and this failure is particularly large for bubbles. Furthermore, different routes to estimate the effective radius-dependent Tolman length δ(R(s)) from simulations in the canonical ensemble are discussed. Thus we obtain an instructive exemplification of the basic quantities and relations of the thermodynamic theory of metastable droplets/bubbles using simulations. However, the simulation results for δ(R(s)) employing a truncated Lennard-Jones system suffer to some extent from unexplained finite size effects, while no such finite size effects are found in corresponding density functional calculations. The numerical results are compatible with the expectation that δ(R(s) → ∞) is slightly negative and of the order of one tenth of a Lennard-Jones diameter, but much larger systems need to be simulated to allow more precise estimates of δ(R(s) → ∞).

Positive Tolman Length in a Lattice Gas with Three-Body Interactions

We present a new method to determine the curvature dependence of the interface tension between coexisting phases in a finite volume from free energies obtained by Monte Carlo simulations. For the example of a lattice gas on a 3D fcc lattice with nearest neighbor three-body interactions, we demonstrate how to calculate the equimolar radius R(e) as well as the radius R(s) of the surface of tension and thus the Tolman length δ(R(s))=R(e)-R(s). Within the physically relevant range of radii, δ(R(s)) shows a pronounced R(s) dependence, such that the simple Tolman parametrization for the interface tension is refutable. For the present model, extrapolation of δ(R(s)) to R(s)→∞ by various methods clearly indicates a positive limiting value.

Molecular dynamics calculations of small drops surface tension

In our paper the results of molecular dynamics calculations of nanodrops are presented. The density profiles, the Irving–Kirkwood pressure tensors, the chemical potentials of the systems, the equimolar radii of the drops and the radii of tension, the mechanical and thermodynamic surface tensions have been calculated. It is shown that both the mechanical and thermodynamic surface tension decrease with the decrease of the equimolar radius of the drop, and reach zero at the same R 0 depending on temperature. The radii of tension surfaces also reach zero. The notion of surface tension which is used in macroscopic theory cannot be applied to small droplets with equimolar radius less than R 0.

A thermodynamically consistent determination of surface tension of small Lennard-Jones clusters from simulation and theory

We have determined the surface tension of small Lennard-Jones clusters using molecular dynamics and Monte Carlo simulation methods as well as density functional theory calculations. For the two simulation methods the surface tension is calculated via a rigorous thermodynamic route using simulation data as input. The capillary approximation of the classical nucleation theory, where the surface tension of a planar surface is used for cluster surface, is found to be quite reasonable even when the cluster size is as small as 100-150 atoms. For smaller cluster sizes the cluster surface tension is considerably lower than the planar value. We have also obtained an approximative value for the Tolman length by extrapolating to the planar limit the difference between the equimolar radius and the radius of the surface of tension. A negative Tolman length is suggested by all the methods used.

The Problem of Surface Tension Definition of Nanodrops

The dependence of the surface tension of Lennard-Jones liquid small drops on the radius and temperature has been calculated by the molecular dynamics method. It is found that the mechanical surface tension is different from the Gibbs's surface tension for drops of 50 — 2000 molecules and is equal to it for drops of more than 2000 molecules. It is shown that both the mechanical and Gibbs's surface tensions decrease with the decrease of the equimolar radius of the drop and reach zero at the same R0 depending on temperature. The radii of tension also reach zero at the same R0. The dependence of the ratio of the mechanical surface tension of the drop to the surface tension of the flat surface liquid—vapor (σ/σ∞) on the ratio of the equimolar radius of the drop to R0 is a universal function. The limit of applicability of the surface tension concept to the small radii droplets changes from 50 to 300 molecules depending on temperature. [DOI: 10.1380/ejssnt.2010.197]

Direct determination of the Tolman length from the bulk pressures of liquid drops via molecular dynamics simulations

An expression for the difference in pressure between a liquid drop in equilibrium with its vapor Deltap=p([script-l])-p(v) is derived from previous expressions for the components of the Irving-Kirkwood pressure tensor. This expression, as well as the bulk values of the pressure tensor, is then evaluated via molecular dynamics simulations of particles interacting through a truncated Lennard-Jones potential. We determine the Tolman length delta from the dependence of Deltap on the equimolar radius. We determine the Tolman length to be delta=-0.10+/-0.02 in units of the particle diameter. This is the first determination of the Tolman length for liquid droplets via the pressure tensor route through computer simulation that is negative, in contrast to all previous results from simulation, but in agreement with results from density functional theory. In addition, we study the planar liquid-vapor interface and observe a dependence of the physical properties of the system on the system size, as measured by the surface area.

Surface Tension of the Vapor-Liquid Interface with Finite Curvature

Based on the division of particles into internal and surface particles, the expression is derived closing the system of equations of classical thermodynamics for curvature-dependent surface tension, equimolar radius, and radius of tension surface. A solution to this system allows one to find the surface tension of new phase nucleus of any size (including minimal) and any sign of surface curvature. The obtained results indicate the weak size dependence of thermodynamic parameters that are the functions of surface tension; it is shown that Tolman's length cannot be determined using experimental determination of these parameters. It is shown that the work of nucleus formation strongly depends on its size and is the function of effective rather than true surface tension. Numerical simulation of clusters by the molecular dynamics method indicates that the pressure inside a fairly small cluster is described by Laplace's formula with the coefficient of surface tension for the plane surface of a liquid that agrees with the proposed theory.

Density functional theory for spherical drops

We have applied Sullivan's model to study the interfacial properties of a spherical drop embedded in a one-component fluid using the mean field approximation. We have examined the effect of drop size (expressed by the equimolar radius Re) on the density profile, pressure tensor, pressure difference Delta p across the interface, surface tension, Tolman's length, and density, pressure, and normal component of pressure tensor at the centre of the drop. Delta p is found to be an ill defined quantity in the sense that it can be defined in various ways, whose results coincide for large drops. The surface tension is a non-monotonic function of Re; it increases slowly from its flat interface value as Re decreases until a maximum value is attained, then it decreases rapidly. For small supersaturations, the drops are under tension and compression while at large supersaturations they are only under tension. The results of this theory are compared qualitatively with previous molecular dynamics simulations and theoretical calculations for drops.

A molecular dynamics study of liquid drops

We report molecular dynamics studies of small liquid drops (41–2004 molecules) in which the atoms interact with a Lennard-Jones intermolecular potential cutoff at 2.5σ and shifted by the potential at cutoff. We calculate the density profiles ρ(r) and the normal and tangential components of the pressure tensor pN(r) and pT(r), using both the Irving–Kirkwood and Harasima definitions of p. From these functions we calculate the surface thickness, the equimolar radius Re and surface of tension Rs, the surface tension γs referred to Rs, the length δ that appears in Tolman’s equation for γs, the pressure change across the drop, and the densities and pressures of the liquid at the drop center and of the gas. The variation of these properties with both surface curvature and temperature is studied, and the results are used to discuss the validity of Laplace’s equation for the pressure change, Tolman’s equation for the effect of curvature on surface tension, and Kelvin’s equation for the vapor pressure. We also make a qualitative comparison with previous theoretical calculations for drops using density gradient and integral equation theory.