^{1}, Andrew J. Schultz

^{1}, Jeffrey R. Errington

^{1}and David A. Kofke

^{1,a)}

### Abstract

We examine the suitability of cluster expansion methods for the description of inhomogeneous fluids. In particular, we apply these methods to characterize the density profile, surface tension, and excess adsorption for a hard-sphere fluid near a hard wall. Coefficients for these series up to seventh order are evaluated by the Mayer-sampling Monte Carlo method. Comparison of the series to Monte Carlo simulations of these systems finds very good agreement up to bulk densities approaching the freezing point. This work indicates that knowledge of surface cluster integrals of inhomogeneous systems can be at least as useful as the bulk-phase virial expansions.

This work is financially supported by the U.S. National Science Foundation (NSF), Grant Nos. CBET-0854340, CHE-0626305, and CHE-1012356.

I. INTRODUCTION

II. CLUSTER EXPANSION FOR BEHAVIOR NEAR A SURFACE

A. Formalism

B. Activity expansions

C. Density expansions

III. MODEL AND METHODS

A. Mayer-sampling algorithm

B. Simulation details

IV. RESULTS AND DISCUSSION

A. Density profile

B. Surface tension and adsorption

V. CONCLUSIONS

### Key Topics

- Surface tension
- 16.0
- Adsorption
- 10.0
- Monte Carlo methods
- 9.0
- Density functional theory
- 8.0
- External field
- 8.0

## Figures

Cluster diagrams in the density expansion of the singlet density, Eq. (18) . The first row shows diagrams in w 2(r) and w 3(r), respectively. The second row shows diagrams in w 4(r). The remainder of the rows show diagrams in w 5(r), which consists of 58 distinct clusters. Open and filled circles represent the root point and field points, respectively, with weight g(r), which for the hard-sphere/hard-wall system precludes overlap with the wall. The black and shaded squares represent field points with respective weights g(r) − 1 (requires overlap with the wall) and (requires one or more wall overlaps among the points with this shading). ‡ denotes that shaded squares represent (gg − 1)(gg − 1) not (gggg − 1). Points are joined by Mayer f-bonds.

Cluster diagrams in the density expansion of the singlet density, Eq. (18) . The first row shows diagrams in w 2(r) and w 3(r), respectively. The second row shows diagrams in w 4(r). The remainder of the rows show diagrams in w 5(r), which consists of 58 distinct clusters. Open and filled circles represent the root point and field points, respectively, with weight g(r), which for the hard-sphere/hard-wall system precludes overlap with the wall. The black and shaded squares represent field points with respective weights g(r) − 1 (requires overlap with the wall) and (requires one or more wall overlaps among the points with this shading). ‡ denotes that shaded squares represent (gg − 1)(gg − 1) not (gggg − 1). Points are joined by Mayer f-bonds.

A comparison of cluster diagrams with different field points. (a) Homogeneous , (b) inhomogeneous b 3 (r), (c) inhomogeneous w 3 (r).

A comparison of cluster diagrams with different field points. (a) Homogeneous , (b) inhomogeneous b 3 (r), (c) inhomogeneous w 3 (r).

A schematic section of hard-spherical particles near a wall.

A schematic section of hard-spherical particles near a wall.

A schematic example for the integral range of length. Shading of points has the same meaning as in Figure 2 .

A schematic example for the integral range of length. Shading of points has the same meaning as in Figure 2 .

The phase space schematic of appropriate intermediate system: (a) direct-sampling and (b) umbrella and overlap-sampling. The arrows indicate perturbing from one system to another.

The phase space schematic of appropriate intermediate system: (a) direct-sampling and (b) umbrella and overlap-sampling. The arrows indicate perturbing from one system to another.

The second- to seventh-order coefficients of the excess singlet density in a series in powers of activity.

The second- to seventh-order coefficients of the excess singlet density in a series in powers of activity.

Et^{1/2} presents the difficulty of calculation for each coefficient, a n (z) and w n (z). Curves proceed in sequence from n = 2 to 7 as indicated on plot.

Et^{1/2} presents the difficulty of calculation for each coefficient, a n (z) and w n (z). Curves proceed in sequence from n = 2 to 7 as indicated on plot.

(a) and (b) The second- to seventh-order coefficients of the density distribution when expressed as a series in powers of the bulk density.

(a) and (b) The second- to seventh-order coefficients of the density distribution when expressed as a series in powers of the bulk density.

(a) and (b) The second- to seventh-order coefficients, v n (z), of density distribution when expressed as a series in powers of the bulk density.

(a) and (b) The second- to seventh-order coefficients, v n (z), of density distribution when expressed as a series in powers of the bulk density.

Et^{1/2} presents effort needed for each coefficient, v n (z). Curves proceed in sequence from v 2(z) to v 7(z) as indicated on plot.

Et^{1/2} presents effort needed for each coefficient, v n (z). Curves proceed in sequence from v 2(z) to v 7(z) as indicated on plot.

(a) and (b) Comparison of w n (z) and v n (z) series of the density distribution for hard-spheres near a hard wall. The system is in equilibrium with a bulk hard-sphere fluid of .

(a) and (b) Comparison of w n (z) and v n (z) series of the density distribution for hard-spheres near a hard wall. The system is in equilibrium with a bulk hard-sphere fluid of .

Same as Fig. 10 , but expanding on w n (z) by adding which is a product of w 2(z).

Plots of density distribution for hard-spheres near a hard wall. The open squares correspond to MC data and the curves labeled 2, 3, 4, etc., correspond to truncated virial expansions, truncated after w 2(z) (black), w 3(z) (red), w 4(z) (blue), w 5(z) (cyan), w 6(z) (magenta), and w 7(z) (gold) terms, respectively. The system is in equilibrium with a bulk hard-sphere fluid of density of (a) = 0.206, (b) = 0.296, (c) = 0.397, (d) = 0.498, (e) = 0.608, (f) = 0.699. Dashed-dotted lines are from lower-order series, and solid lines from highest-order plotted in each case. Inset figures indicate the difference obtained upon subtracting each truncated virial series from MC data.

Plots of density distribution for hard-spheres near a hard wall. The open squares correspond to MC data and the curves labeled 2, 3, 4, etc., correspond to truncated virial expansions, truncated after w 2(z) (black), w 3(z) (red), w 4(z) (blue), w 5(z) (cyan), w 6(z) (magenta), and w 7(z) (gold) terms, respectively. The system is in equilibrium with a bulk hard-sphere fluid of density of (a) = 0.206, (b) = 0.296, (c) = 0.397, (d) = 0.498, (e) = 0.608, (f) = 0.699. Dashed-dotted lines are from lower-order series, and solid lines from highest-order plotted in each case. Inset figures indicate the difference obtained upon subtracting each truncated virial series from MC data.

Residual surface tension at a hard-sphere/hard-wall interface for different truncated series. Curves proceed in sequence from W 2 to W 7 as indicated on plot. Activity expansion at 7th order is plotted as a function of activity, α (top axis). The error bars in MC data are smaller than the symbol size.

Residual surface tension at a hard-sphere/hard-wall interface for different truncated series. Curves proceed in sequence from W 2 to W 7 as indicated on plot. Activity expansion at 7th order is plotted as a function of activity, α (top axis). The error bars in MC data are smaller than the symbol size.

The exponential approximants, [J/0]^{∞}, are compared against the surface virial expansion truncated at W 7 and MC simulation data. The error bars in MC data are smaller than the symbol size.

The exponential approximants, [J/0]^{∞}, are compared against the surface virial expansion truncated at W 7 and MC simulation data. The error bars in MC data are smaller than the symbol size.

The excess adsorption of a hard-sphere fluid at a planar hard wall as a function of the bulk density. The curves represent the virial results for different truncated series proceed in sequence from W 2 to W 7 as indicated on plot, while squares denote MC data. The error bars in MC data are smaller than the symbol size.

The excess adsorption of a hard-sphere fluid at a planar hard wall as a function of the bulk density. The curves represent the virial results for different truncated series proceed in sequence from W 2 to W 7 as indicated on plot, while squares denote MC data. The error bars in MC data are smaller than the symbol size.

## Tables

Expansion coefficients , , and W n .

Expansion coefficients , , and W n .

Article metrics loading...

Full text loading...

Commenting has been disabled for this content