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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 lP (r) is accurate for paste air contents of nearly 20%.
Figure 5: Mean void-void spacing (lP) for lognormally distributed sphere radii with a density of 20 mm−3. Measured values are shown as solid circles, the solid line is the estimate by Lu and Torquato.
Figure 6: Mean void-void spacing (lP) for lognormally distributed sphere radii with a density of 240 mm−3. Measured values are shown as solid circles, the solid line is the estimate by Lu and Torquato.