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As an estimate of the 50 * th* and 95 * th* percentiles,
the Lu and Torquato
equation performs better for lognormally distributed sphere radii than
for monosized spheres.
For the lognormally distributed air void radii used here,
the maximum associated errors are 2% and 1% for the
50 * th* and 95 * th* percentiles of the paste-void spacing distribution,
respectively. The void-void spacing percentile estimates for the
lognormally distributed sphere radii have approximately
the same performance as for the monosized spheres.

In addition to estimating the percentiles of the void-void spacing
distribution, the Lu and Torquato equation is used to estimate
the average void-void spacing as a function of sphere radius. The
results for particle densities of 20 mm^{−3} and
240 mm^{−3}
are shown in Figs. 5 and 6, respectively. The
measured data (solid circles) are the average of 100 system iterations.
As can be seen in the figures, the Lu and Torquato equation
*l _{P} (r)* is
accurate for paste air contents of nearly 20%.

**Figure 5:** Mean void-void spacing
(*l _{P}*) for lognormally distributed
sphere radii with a density of 20 mm

**Figure 6:** Mean void-void spacing (*l _{P}*) for lognormally distributed
sphere radii with a density of 240 mm