Excess equimolar radius of liquid drops

Martin Horsch, Hans Hasse, Alexander K. Shchekin, Animesh Agarwal, Stefan Eckelsbach, Jadran Vrabec, Erich A. Müller, and George Jackson

Phys. Rev. E 85, 031605 – Published 26 March 2012

Abstract

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.

Received 26 September 2011

DOI:https://doi.org/10.1103/PhysRevE.85.031605

Authors & Affiliations

Martin Horsch^* and Hans Hasse

Lehrstuhl für Thermodynamik, Fachbereich Maschinenbau und Verfahrenstechnik, Technische Universität Kaiserslautern, Erwin-Schrödinger-Str. 44, 67663 Kaiserslautern, Germany

Alexander K. Shchekin

Department of Statistical Physics, Faculty of Physics, Saint Petersburg State University, ulica Ulyanovskaya, Petrodvoretz, 198504 Saint Petersburg, Russia

Animesh Agarwal, Stefan Eckelsbach, and Jadran Vrabec

Lehrstuhl für Thermodynamik und Energietechnik, Institut für Verfahrenstechnik, Universität Paderborn, Warburger Str. 100, 33098 Paderborn, Germany

Erich A. Müller and George Jackson

Molecular Systems Engineering Group, Department of Chemical Engineering, Imperial College London, London SW7 2AZ, United Kingdom

^*Corresponding author: Also at Imperial College, London, UK; martin.horsch@mv.uni-kl.de

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Vol. 85, Iss. 3 — March 2012

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Images

Figure 1
Diagram of van Giessen and Blokhuis [26], showing $ϕ R_{ρ}$ as a function of the equimolar curvature $1 R_{ρ}$ for liquid drops of the truncated-shifted Lennard-Jones fluid at a reduced temperature of $k T ɛ$ $=$ $0.9$ , where $R_{ρ}$ is determined from the density profiles and $ϕ$ from the difference between the values of the normal component of the Irving-Kirkwood pressure tensor in the homogeneous regions inside the liquid drop as well as outside, i.e., in the homogeneous supersaturated vapor. In addition to the results of van Giessen and Blokhuis ( $□$ ), the data of Vrabec et al. [27] ( $\circ$ ), which were obtained using the same method, are included here along with a data point ( $▵$ ) where $ϕ$ is determined by MD simulation of the homogeneous fluid. The data for the planar surface tension $γ_{\infty}$ are taken from simulations of Vrabec et al. ( $•$ ) and van Giessen and Blokhuis ( $▪$ ) as well as the correlation of Vrabec et al. ( $▴$ ). The continuous lines are guides to the eye: In the planar limit, a positive slope corresponds to a negative Tolman length and vice versa; cf. Eq. (12).Reuse & Permissions
Figure 2
Density profiles from canonical MD simulations of TSLJ liquid drops at the reduced temperatures of $k T ɛ$ $=$ $0.65$ and $0.95$ , showing the average densities from simulation ( $•$ ) and exponential approximants (– –). The steeper profile corresponds to the lower temperature.Reuse & Permissions
Figure 3
Density profiles from canonical MD simulations of TSLJ liquid drops at $k T ɛ$ $=$ $0.75$ with equimolar radii of $R_{ρ}$ $=$ $9.977$ $\pm$ $0.001$ ( $\cdot$ — $\cdot$ ), $12.029$ $\pm$ $0.003$ (– –), $13.974$ $\pm$ $0.002$ $σ$ ( $\dots$ ) and $15.967$ $\pm$ $0.001$ $σ$ (—); cf. Tables and .Reuse & Permissions
Figure 4
Density profiles from a single canonical MD simulation of a TSLJ liquid drop, showing the average densities from simulation ( $\circ$ ) and exponential approximants (curves) corresponding to the sampling intervals 2000–3000 ( $\dots$ ; green), 3000–4000 ( $\cdot$ – $\cdot$ ; red), 4000–5000 (––; blue), and 5000–6000 time units (—; black) after the onset of the simulation. The standard deviation between the densities at an infinite distance from the interface, according to the exponential fits for all sampling intervals of a single MD simulation, is used to determine the error of the bulk densities here.Reuse & Permissions
Figure 5
Equimolar radius $R_{ρ}$ as a function of the capillarity radius $R_{κ}$ for TSLJ liquid drops, from density profiles and bulk pressures determined with canonical MD simulations at reduced temperatures of $k T ɛ$ $=$ $0.75$ ( $□$ ) and $0.85$ ( $\circ$ ), in comparison with results from the previous work of Vrabec et al. [27] at $k T ɛ$ $=$ $0.75$ ( $▪$ ) and $0.85$ ( $•$ ), using pressure differences based on evaluating the IK tensor in the (approximately) homogeneous regions inside and outside the liquid drop. The continuous diagonal line denotes $R_{ρ} = R_{κ}$ and thus corresponds to an excess equimolar radius of $η = 0$ , while the dotted lines correspond to $η = \pm 0.5$ $σ$ .Reuse & Permissions

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