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 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 .
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.
Et1/2 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, v n (z), of density distribution when expressed as a series in powers of the bulk density.
Et1/2 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 .
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.
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 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.
Expansion coefficients , , and W n .
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