^{1}, Subir K. Das

^{1,2}, Martin Oettel

^{1,3,a)}, Peter Virnau

^{1}and Kurt Binder

^{1}

### Abstract

We study the excess free energy due to phase coexistence of fluids by Monte Carlo simulations using successive umbrella sampling in finite boxes with periodic boundary conditions. Both the vapor-liquid phase coexistence of a simple Lennard-Jones fluid and the coexistence between A-rich and B-rich phases of a symmetric binary (AB) Lennard-Jones mixture are studied, varying the density in the simple fluid or the relative concentration of in the binary mixture, respectively. The character of phase coexistence changes from a spherical droplet (or bubble) of the minority phase (near the coexistence curve) to a cylindrical droplet (or bubble) and finally (in the center of the miscibility gap) to a slablike configuration of two parallel flat interfaces. Extending the analysis of Schrader *et al.*, [Phys. Rev. E79, 061104 (2009)], we extract the surfacefree energy of both spherical and cylindrical droplets and bubbles in the vapor-liquid case and present evidence that for the leading order (Tolman) correction for droplets has sign opposite to the case of bubbles, consistent with the Tolman length being independent on the sign of curvature. For the symmetric binary mixture, the expected nonexistence of the Tolman length is confirmed. In all cases and for a range of radii relevant for nucleationtheory, deviates strongly from which can be accounted for by a term of order . Our results for the simple Lennard-Jones fluid are also compared to results from density functional theory, and we find qualitative agreement in the behavior of as well as in the sign and magnitude of the Tolman length.

This work was supported in part by the Deutsche Forschungsgemeinschaft (DFG) through the Collaborative Research Centre SFB-TR6 under Grant Nos. No. TR6/A5 and TR6/N01 and through the Priority Program SPP 1296 under Grant Nos. Bi 314/19 and Schi 853/2. S.K.D. is grateful to the Institut für Physik (Mainz) for the hospitality during his extended visits. We are also grateful to the Zentrum für Datenverarbeitung (ZDV) Mainz and the Jülich Supercomoputer Centre (JSC) for computer time. One of us (K.B.) is grateful to C. Dellago, D. Frenkel, G. Jackson, and E.A. Müller for stimulating discussions.

I. INTRODUCTION

II. THE CURVATURE-DEPENDENT SURFACE TENSION IN A SYMMETRIC BINARY LENNARD-JONES MIXTURE

III. DROPLETS VERSUS BUBBLES IN THE SINGLE-COMPONENT LENNARD-JONES FLUID

IV. RESULTS FROM DENSITY FUNCTIONAL THEORY

V. CONCLUSIONS

### Key Topics

- Fluid drops
- 79.0
- Free energy
- 25.0
- Surface tension
- 25.0
- Density functional theory
- 21.0
- Mean field theory
- 17.0

## Figures

Phase diagram of the symmetric binary (A,B) Lennard-Jones mixture with , cf. Eqs. (7)–(10), in the plane of variables and relative concentration of A particles (; ) for fixed . The cross shows the critical point, as obtained previously (Ref. 48). The horizontal broken line means that phase coexistence is studied at .

Phase diagram of the symmetric binary (A,B) Lennard-Jones mixture with , cf. Eqs. (7)–(10), in the plane of variables and relative concentration of A particles (; ) for fixed . The cross shows the critical point, as obtained previously (Ref. 48). The horizontal broken line means that phase coexistence is studied at .

Effective free energy of finite-size boxes of linear dimension with plotted vs at for the model of Fig. 1. The estimation of the size-dependent interfacial tension is indicated.

Effective free energy of finite-size boxes of linear dimension with plotted vs at for the model of Fig. 1. The estimation of the size-dependent interfacial tension is indicated.

Extrapolation of as function of , cf. Eq. (12), in order to estimate .

Extrapolation of as function of , cf. Eq. (12), in order to estimate .

Effective free energy plotted vs at for , as indicated.

Effective free energy plotted vs at for , as indicated.

(a) Snapshot of a spherical droplet configuration formed by A particles in the background of B-particles (not shown) for , , and . (b) Same as (a), but for a cylindrical droplet, choosing . Note that our method does not at all suppress statistical fluctuations in the size and shape of these droplets, which therefore have spherical or cylindrical symmetry on the average only.

(a) Snapshot of a spherical droplet configuration formed by A particles in the background of B-particles (not shown) for , , and . (b) Same as (a), but for a cylindrical droplet, choosing . Note that our method does not at all suppress statistical fluctuations in the size and shape of these droplets, which therefore have spherical or cylindrical symmetry on the average only.

Plot of vs at for . Data refer to a single run at each size to illustrate the typical noise level (200 Monte Carlo steps per particles have been used for each window of the successive umbrella sampling). For the final analysis, five such runs were averaged over.

Plot of vs at for . Data refer to a single run at each size to illustrate the typical noise level (200 Monte Carlo steps per particles have been used for each window of the successive umbrella sampling). For the final analysis, five such runs were averaged over.

Schematic explanation of how the estimation of the functions and together allows the estimation of the concentration difference and free energy difference due to a droplet.

Schematic explanation of how the estimation of the functions and together allows the estimation of the concentration difference and free energy difference due to a droplet.

Plot of of spherical A-rich droplet at for the binary symmetric LJ mixture of Fig. 1. The description in terms of the capillarity approximation of CNT is shown as a broken curve, using as obtained in Fig. 3. The full curve is a superposition of independent simulation results for , 16, 18, 20, 22, and 24, where a running averaging was done using the combined data set.

Plot of of spherical A-rich droplet at for the binary symmetric LJ mixture of Fig. 1. The description in terms of the capillarity approximation of CNT is shown as a broken curve, using as obtained in Fig. 3. The full curve is a superposition of independent simulation results for , 16, 18, 20, 22, and 24, where a running averaging was done using the combined data set.

Plot of vs . Here is taken from Fig. 3, while is estimated using Eq. (14). Ideally the estimates obtained from different values of should superimpose on a single curve. The scatter between the curves for different values of is due to residual statistical errors. The thin straight line is a fit function giving .

Plot of vs . Here is taken from Fig. 3, while is estimated using Eq. (14). Ideally the estimates obtained from different values of should superimpose on a single curve. The scatter between the curves for different values of is due to residual statistical errors. The thin straight line is a fit function giving .

Effective free energy density of the single-component Lennard-Jones fluid at plotted vs density for three values of , as indicated in the figure.

Effective free energy density of the single-component Lennard-Jones fluid at plotted vs density for three values of , as indicated in the figure.

Plot of for the one-component Lennard-Jones fluid as a function of , at , for three values of , as indicated.

Plot of for the one-component Lennard-Jones fluid as a function of , at , for three values of , as indicated.

Extrapolation of as a function of for the simple LJ fluid at , giving . Similar exercise at gives .

Extrapolation of as a function of for the simple LJ fluid at , giving . Similar exercise at gives .

Plots of of spherical droplets and bubbles for the one-component LJ fluid, at , as a function of sphere radius . The capillarity approximation (CNT), is included, using the estimate of from Fig. 12.

Plots of of spherical droplets and bubbles for the one-component LJ fluid, at , as a function of sphere radius . The capillarity approximation (CNT), is included, using the estimate of from Fig. 12.

Same as Fig. 13, but for cylindrical droplets and bubbles. Note that here the -axis corresponds to the surface free energy per unit height of the cylinder.

Same as Fig. 13, but for cylindrical droplets and bubbles. Note that here the -axis corresponds to the surface free energy per unit height of the cylinder.

Plots of vs for spherical droplets and bubbles for the LJ fluid at (a) and (b) . Fits to functional forms (20) are included.

Plots of vs for spherical droplets and bubbles for the LJ fluid at (a) and (b) . Fits to functional forms (20) are included.

Same as Fig. 15, but for cylindrical droplets and bubbles.

Same as Fig. 15, but for cylindrical droplets and bubbles.

(a) Radius-dependent Tolman length for bubbles and droplets with a corresponding linear fit in the range . (b) Surface tension ratio as function of the equimolar radius for bubbles and droplets with a corresponding quadratic fit in the range . All results are for the temperature .

(a) Radius-dependent Tolman length for bubbles and droplets with a corresponding linear fit in the range . (b) Surface tension ratio as function of the equimolar radius for bubbles and droplets with a corresponding quadratic fit in the range . All results are for the temperature .

(a) Radius-dependent Tolman length for bubbles and droplets with a corresponding linear fit in the range . (b) Surface tension ratio as function of the equimolar radius for bubbles and droplets with a corresponding quadratic fit in the range . All results are for the temperature .

## Tables

Comparison between DFT and simulation results for the coexistence densities and liquid-vapor surface tension at the two investigated temperatures.

Comparison between DFT and simulation results for the coexistence densities and liquid-vapor surface tension at the two investigated temperatures.

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