^{1,a)}, Vincent K. Shen

^{2}and Jeffrey R. Errington

^{3}

### Abstract

Computer simulations are employed to obtain subcritical isotherms of small finite sized systems inside the coexistence region. For all temperatures considered, ranging from the triple point up to the critical point, the isotherms gradually developed a sequence of sharp discontinuities as the system size increased from to molecular diameters. For the smallest system sizes, and more so close to the critical point, the isotherms appeared smooth, resembling the continuous van der Waals loop obtained from extrapolation of an analytic equation of state outside the coexistence region. As the system size was increased, isotherms in the chemical potential-density plane developed first two, then four, and finally six discontinuities. Visual inspection of selected snapshots revealed that the observed discontinuities are related to structural transitions between droplets (on the vapor side) and bubbles (on the liquid side) of spherical, cylindrical, and tetragonal shapes. A capillary drop model was developed to qualitatively rationalize these observations. Analytic results were obtained and found to be in full agreement with the computer simulation results. The analysis shows that the shape of the subcritical isotherms is dictated by a single characteristic volume (or length scale), which depends on the surface tension, compressibility, and coexistence densities. For small reduced system volumes, the model predicts that a homogeneous fluid is stable across the whole coexistence region, thus explaining the continuous van der Waals isotherms observed in the simulations. When the liquid and vapor free energies are described by means of an accurate mean-fieldequation of state and surface tensions from simulation are employed, the capillary model is found to describe the simulated isotherms accurately, especially for large systems (i.e., larger than about 15 molecular diameters) at low temperature (lower than about 0.85 times the critical temperature). This implies that the Laplace pressure differences can be predicted for drops as small as five molecular diameters, and as few as about . The theoretical study also shows that the extrema or apparent spinodal points of the finite size loops are more closely related to (finite system size) bubble and dew points than to classical spinodals. Our results are of relevance to phase transitions in nanopores and show that first order corrections to nucleation energies in finite closed systems are power laws of the inverse volume.

One of the authors (L.G.M.) would like to thank P. Virnau, M. Müller, and K. Binder for helpful discussions (Ref. 61) and joint work on a related project (Ref. 26) and D. Frenkel for pointing out Ref. 6, and also wishes to thank the Universidad Complutense de Madrid and the Spanish Ministerio de Ciencia y Tecnologia (MCYT) for the award of a Ramon y Cajal fellowship and for financial support under Grant Nos. FIS2004-06227-C02-02 (MCYT) and S-0505/ESP/0299 (CAM). Another author (J.R.E.) gratefully acknowledges financial support from the National Science Foundation under Grant No. CTS-0238772. A portion of this study utilized the high-performance computational capabilities of the Biowulf PC/Linux cluster at the National Institute of Health, Bethesda, MD (http://biowulf.nih.gov).

I. INTRODUCTION

II. THEORETICAL ANALYSIS

A. General solution

1. Preliminary definitions

2. Solution of the Laplace equation

3. Free energy

B. Spherical bubble

1. Solutions of the Laplace equation

2. Free energy

C. Cylindrical bubble

D. Tetragonal bubble (slab)

E. Crossover regimes

F. Numerical calculations

III. MODEL AND SIMULATIONS

IV. COMPARISON BETWEEN THEORY AND SIMULATION

A. Qualitative test of the theory

B. Quantitative test of the capillary model

C. Significance of the finite size transition densities

V. CONCLUSIONS

### Key Topics

- Fluid drops
- 36.0
- Free energy
- 29.0
- Mean field theory
- 27.0
- Chemical potential
- 24.0
- Equations of state
- 24.0

## Figures

Pressure vs chemical potential isotherms for systems of different sizes at . The chemical potential and pressure are expressed as and , where and is the chemical potential employed during the grand canonical simulations. Full line, ; dashed line, , and dot-dashed line, . The arrow points in the direction of increasing system size.

Pressure vs chemical potential isotherms for systems of different sizes at . The chemical potential and pressure are expressed as and , where and is the chemical potential employed during the grand canonical simulations. Full line, ; dashed line, , and dot-dashed line, . The arrow points in the direction of increasing system size.

Chemical potential vs density isotherms for systems of different sizes at . The chemical potential is measured relative to the coexistence chemical potential. The full lines are simulation results, the dashed lines are results from the many-state model, and the dot-dashed line is the mean-field parametric equation of state. The inset shows the inverse susceptibility for the two smallest system sizes. Units are arbitrary and results have been shifted vertically for clarity. The system sizes studied are , , , and . The arrows point in the direction of increasing system size.

Chemical potential vs density isotherms for systems of different sizes at . The chemical potential is measured relative to the coexistence chemical potential. The full lines are simulation results, the dashed lines are results from the many-state model, and the dot-dashed line is the mean-field parametric equation of state. The inset shows the inverse susceptibility for the two smallest system sizes. Units are arbitrary and results have been shifted vertically for clarity. The system sizes studied are , , , and . The arrows point in the direction of increasing system size.

A series of snapshots at and , corresponding to states of increasing density.

A series of snapshots at and , corresponding to states of increasing density.

Chemical potential vs density isotherms for systems of different sizes at . The chemical potential is measured relative to the coexistence chemical potential. The full lines are simulation results, the dashed lines are results from the two-state model, and the dot-dashed line is the mean-field parametric equation of state. The system sizes studied are , , , and . The arrows point in the direction of decreasing system size. The inset shows the size of those domains that are stable at the given density in the system. , , and denote the radii and the width of spherical (full lines), cylindrical (dashed lines), and tetragonal (dot-dashed lines) domains, respectively.

Chemical potential vs density isotherms for systems of different sizes at . The chemical potential is measured relative to the coexistence chemical potential. The full lines are simulation results, the dashed lines are results from the two-state model, and the dot-dashed line is the mean-field parametric equation of state. The system sizes studied are , , , and . The arrows point in the direction of decreasing system size. The inset shows the size of those domains that are stable at the given density in the system. , , and denote the radii and the width of spherical (full lines), cylindrical (dashed lines), and tetragonal (dot-dashed lines) domains, respectively.

Chemical potential vs density isotherms for systems of different sizes at . The chemical potential is measured relative to the coexistence chemical potential. The full lines are simulation results, the dashed lines are results from the many-state model, and the dot-dashed line is the mean-field parametric equation of state. The system sizes studied are , , and . The arrows point in the direction of decreasing system size.

Chemical potential vs density isotherms for systems of different sizes at . The chemical potential is measured relative to the coexistence chemical potential. The full lines are simulation results, the dashed lines are results from the many-state model, and the dot-dashed line is the mean-field parametric equation of state. The system sizes studied are , , and . The arrows point in the direction of decreasing system size.

Temperature dependence of some relevant volume dimensions. The empty symbols refer to the characteristic volume of spherical domain formation, (left ordinate axis). The full symbols refer to “spinodal” volumes, , as explained in the text. The circles refer to the condensation transition and the squares to the cavitation transition. is calculated using MBWR equation of state data, together with interpolated surface tensions as obtained from simulation. is calculated using Eq. (15), with coexistence and spinodal points as determined from the MBWR equation of state.

Temperature dependence of some relevant volume dimensions. The empty symbols refer to the characteristic volume of spherical domain formation, (left ordinate axis). The full symbols refer to “spinodal” volumes, , as explained in the text. The circles refer to the condensation transition and the squares to the cavitation transition. is calculated using MBWR equation of state data, together with interpolated surface tensions as obtained from simulation. is calculated using Eq. (15), with coexistence and spinodal points as determined from the MBWR equation of state.

A series of isotherms obtained for system size and different temperatures. The density is shifted by an amount and then normalized by . The chemical potential is expressed relative to the coexistence chemical potential and then normalized by , with , the (vapor) mean-field spinodal point as determined by the MBWR equation of state. The arrows point in the direction of increasing temperature.

A series of isotherms obtained for system size and different temperatures. The density is shifted by an amount and then normalized by . The chemical potential is expressed relative to the coexistence chemical potential and then normalized by , with , the (vapor) mean-field spinodal point as determined by the MBWR equation of state. The arrows point in the direction of increasing temperature.

Chemical potential vs density isotherms for large system sizes as predicted by the MSCD model. The isotherms were calculated for and several system sizes up to . Because of numerical reasons, the calculations for the liquid side and large system sizes become difficult. For this reason, we have calculated the isotherms on the vapor side only. The full loop was obtained assuming the antisymmetric property of the chemical potential. The arrows point in the direction of decreasing system size.

Chemical potential vs density isotherms for large system sizes as predicted by the MSCD model. The isotherms were calculated for and several system sizes up to . Because of numerical reasons, the calculations for the liquid side and large system sizes become difficult. For this reason, we have calculated the isotherms on the vapor side only. The full loop was obtained assuming the antisymmetric property of the chemical potential. The arrows point in the direction of decreasing system size.

## Tables

Table showing the different possible stable states that can be found as a function of system size (hom, homogeneous state; sph, spherical bubble state; cyl, cylindrical bubble state; slb, slablike state). The first column indicates the range of volumes where each sequence of transitions may be observed, and the third column illustrates the actual sequence observed in that range. Assuming bubble formation, the arrows indicate decreasing system density. Note that the crossover from one regime to the other occurs for volumes actually several orders of magnitude larger than , as explicitly indicated in the second column.

Table showing the different possible stable states that can be found as a function of system size (hom, homogeneous state; sph, spherical bubble state; cyl, cylindrical bubble state; slb, slablike state). The first column indicates the range of volumes where each sequence of transitions may be observed, and the third column illustrates the actual sequence observed in that range. Assuming bubble formation, the arrows indicate decreasing system density. Note that the crossover from one regime to the other occurs for volumes actually several orders of magnitude larger than , as explicitly indicated in the second column.

Coexistence chemical potentials as obtained from the equal area rule and the pressure-chemical potential intercept for the different system sizes and temperatures considered in this work. First column: system size; second column: temperature; third column: coexistence chemical potential from equal area rule; fourth column: coexistence chemical potential from pressure-chemical potential intercept.

Coexistence chemical potentials as obtained from the equal area rule and the pressure-chemical potential intercept for the different system sizes and temperatures considered in this work. First column: system size; second column: temperature; third column: coexistence chemical potential from equal area rule; fourth column: coexistence chemical potential from pressure-chemical potential intercept.

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