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Lognormally Distributed Spheres

The results of the lognormally distributed sphere radii are shown in Tables 6--8. The results for the average void-void spacing lP as a function of radius are shown in Figs. 5 and 6, and include both the Lu and Torquato estimate and the measured values.

  

n A K'(0) F50 KA 50 KK' 50 EV 50 pv50
(mm−3) (mm) (mm) (mm) (mm) (mm)
20 0.016 0.012 0.146 0.172 0.173 0.163 0.162 ± .003
40 0.033 0.022 0.105 0.131 0.131 0.120 0.120 ± .002
80 0.066 0.039 0.073 0.098 0.100 0.085 0.085 ± .002
160 0.131 0.065 0.048 0.071 0.075 0.056 0.057 ± .002
240 0.197 0.082 0.037 0.058 0.063 0.042 0.043 ± .001
n A K'(0) L F95 KA 95 KK' 95 EV 95 pv95
(mm−3) (mm) (mm) (mm) (mm) (mm) (mm)
20 0.016 0.012 0.450 0.272 0.302 0.304 0.290 0.290 ± .005
40 0.033 0.022 0.337 0.204 0.231 0.236 0.220 0.219 ± .004
80 0.066 0.039 0.247 0.150 0.173 0.185 0.162 0.162 ± .003
160 0.131 0.065 0.175 0.108 0.125 0.143 0.114 0.114 ± .002
240 0.197 0.082 0.136 0.087 0.099 0.123 0.089 0.090 ± .002

Table 6: Estimates of the paste-void distribution percentiles for lognormally distributed sphere radii. The estimates include results from Philleo (F); Pleau and Pigeon (KA) and (KK') using the normalization factors of 1-A and 1-K'(0), respectively; Lu and Torquato (EV); and Powers (). The measured values are labeled pv and have the one standard deviation uncertainties shown. The suffixes 50 and 95 indicate the percentile.

  

n A t tG EP 50 vv50 EP 95 vv95
(mm−3) (mm) (mm) (mm) (mm) (mm) (mm) (mm)
20 0.016 7.969 3.919 0.642 0.134 0.134 ± .003 0.263 0.270 ± .006
40 0.033 3.918 1.895 0.621 0.092 0.093 ± .002 0.194 0.196 ± .005
80 0.066 1.892 0.884 0.580 0.060 0.060 ± .002 0.138 0.138 ± .003
160 0.131 0.880 0.382 0.382 0.035 0.035 ± .001 0.092 0.093 ± .002
240 0.197 0.542 0.218 0.218 0.024 0.024 ± .001 0.069 0.069 ± .002

Table 7: Estimates of the ensemble average (Eqn. 35) void-void distribution percentiles for lognormally distributed sphere radii. The estimates include results from Attiogbe (t) and (tG); and Lu and Torquato (EP). The measured values are labeled vv and have the one standard deviation uncertainties shown. The suffixes 50 and 95 indicate the percentile.

  

n A p/A t tG G EV (t) EV (tG)
(mm−3) (mm) (mm)
20 0.016 59.840 3.919 0.642 0.164 1.000 1.000
40 0.033 29.420 1.895 0.621 0.328 1.000 1.000
80 0.066 14.210 0.884 0.580 0.656 1.000 1.000
160 0.131 6.605 0.382 0.382 1.000 1.000 1.000
240 0.197 4.070 0.218 0.218 1.000 1.000 1.000

Table 8: Estimates of the fraction of paste within either t or tG of an air void surface for a lognormal air void distribution. The estimates are based on the Attiogbe equation for G and on the Lu and Torquato equation for EV.



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