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Mono-sized Spheres

The results of the monosized sphere experiments are shown in Tables 2--5. In each table, the results are divided between constant number density (n) and constant sphere diameter experiments, with one pair of values in common for both. The constant number density experiments have 20 voids per cubic millimeter. The first constant number density experiment consists of voids with zero diameter. This is equivalent to placing 20 points per cubic millimeter. Since each point has no volume, the air content is zero. However, both the paste-void and void-void spacing distributions are still well-defined.

  

Diameter n A K'(0) F50 KA 50 KK' 50 EV 50 pv50
(mm) (mm−3) (mm) (mm) (mm) (mm) (mm) (mm)
0.000 20 0.0000 0.0000 0.202 0.202 0.202 0.202 0.205 ± .006
0.025 20 0.0002 0.0002 0.190 0.190 0.190 0.190 0.191 ± .003
0.075 20 0.0044 0.0044 0.165 0.164 0.164 0.164 0.165 ± .003
0.150 20 0.035 0.035 0.130 0.124 0.124 0.125 0.124 ± .003
0.225 20 0.12 0.11 0.099 0.078 0.079 0.087 0.086 ± .003
0.300 20 0.28 0.25 0.072 0.024 0.028 0.053 0.052 ± .002
0.150 10 0.018 0.018 0.182 0.178 0.178 0.178 0.179 ± .003
0.150 20 0.035 0.035 0.130 0.124 0.124 0.125 0.124 ± .003
0.150 50 0.088 0.085 0.079 0.068 0.068 0.072 0.071 ± .002
0.150 100 0.18 0.16 0.051 0.033 0.034 0.042 0.041 ± .001

Diameter n A K'(0) L F95 KA 95 KK' 95 EV 95 pv95
(mm) (mm−3) (mm) (mm) (mm) (mm) (mm) (mm)
0.000 20 0.0000 0.0000 0.320 0.330 0.330 0.330 0.330 0.330 ± .006
0.025 20 0.0002 0.0002 0.307 0.317 0.317 0.317 0.317 0.318 ± .005
0.075 20 0.0044 0.0044 0.282 0.292 0.289 0.289 0.290 0.292 ± .005
0.150 20 0.035 0.035 0.245 0.255 0.234 0.235 0.244 0.243 ± .005
0.225 20 0.12 0.11 0.207 0.219 0.166 0.168 0.190 0.186 ± .004
0.300 20 0.28 0.25 0.127 0.183 0.089 0.097 0.130 0.120 ± .003
0.150 10 0.018 0.018 0.328 0.341 0.326 0.326 0.333 0.335 ± .006
0.150 20 0.035 0.035 0.245 0.255 0.234 0.235 0.244 0.243 ± .005
0.150 50 0.088 0.085 0.161 0.169 0.138 0.139 0.152 0.149 ± .003
0.150 100 0.18 0.16 0.112 0.119 0.079 0.081 0.097 0.092 ± .002


Table 2: Estimates of the paste-void spacing percentiles for monosized spheres. 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.

  

Diameter n A t tG EP 50 vv50 EP 95 vv95
(mm) (mm−3) (mm) (mm) (mm) (mm) (mm) (mm)
0.000 20 0.0000 ∞ 0.000 0.202 0.211 ± .004 0.330 0.378 ± .007
0.025 20 0.0002 50.913 0.060 0.177 0.178 ± .003 0.304 0.320 ± .006
0.075 20 0.0044 5.609 0.177 0.130 0.130 ± .003 0.254 0.260 ± .006
0.150 20 0.035 1.316 0.333 0.072 0.072 ± .002 0.178 0.179 ± .005
0.225 20 0.12 0.488 0.417 0.034 0.036 ± .001 0.107 0.109 ± .004
0.300 20 0.28 0.182 0.182 0.013 0.014 ± .001 0.049 0.051 ± .002
0.150 10 0.018 2.730 0.346 0.117 0.118 ± .003 0.264 0.269 ± .006
0.150 20 0.035 1.316 0.333 0.072 0.072 ± .002 0.178 0.179 ± .005
0.150 50 0.088 0.470 0.298 0.032 0.033 ± .001 0.093 0.094 ± .003
0.150 100 0.18 0.192 0.192 0.014 0.015 ± .001 0.047 0.049 ± .002

Table 3: Estimates of the void-void spacing percentiles for monosized spheres. The estimates include results from Attiogbe (t) and (tG); and Lu and Torquato (EP). The measured quantities are labeled vv and have the one standard deviation uncertainties shown. The suffix 50 indicates the percentile.

  

Diameter n A λ t tG lP
(mm) (mm−3) (mm) (mm) (mm) (mm) (mm)
0.000 20 0.0000 ∞ ∞ 0.000 0.204 0.219 ± .003
0.025 20 0.0002 101.843 50.913 0.060 0.179 0.183 ± .002
0.075 20 0.0044 11.268 5.609 0.177 0.133 0.135 ± .002
0.150 20 0.035 2.729 1.316 0.333 0.079 0.080 ± .002
0.225 20 0.12 1.108 0.488 0.417 0.042 0.043 ± .001
0.300 20 0.28 0.507 0.182 0.182 0.018 0.018 ± .001
0.150 10 0.018 5.559 2.730 0.346 0.125 0.127 ± .002
0.150 20 0.035 2.729 1.316 0.333 0.079 0.080 ± .002
0.150 50 0.088 1.032 0.470 0.298 0.038 0.039 ± .001
0.150 100 0.18 0.466 0.192 0.192 0.018 0.019 ± .001

Table 4: Estimates of the average void-void spacing and the mean free path (λ) for monosized spheres. The estimates include results from Attiogbe (t) and (tG); and Lu and Torquato (lP). The measured values are labeled and have the one standard deviation uncertainties shown.

  

Diameter n A p/A t tG G EV (t) EV (tG)
(mm) (mm−3) (mm) (mm)
0.000 20 0.0000 ∞ ∞ 0.000 0.000 1.000 0.000
0.025 20 0.0002 6111. 50.913 0.060 0.001 1.000 0.031
0.075 20 0.0044 225.4 5.609 0.177 0.032 1.000 0.566
0.150 20 0.035 27.29 1.316 0.333 0.253 1.000 0.998
0.225 20 0.12 7.383 0.488 0.417 0.854 1.000 1.000
0.300 20 0.28 2.537 0.182 0.182 1.000 1.000 1.000
0.150 10 0.018 55.59 2.730 0.346 0.125 1.000 0.963
0.150 20 0.035 27.29 1.316 0.333 0.079 1.000 0.998
0.150 50 0.088 10.32 0.470 0.298 0.038 1.000 1.000
0.150 100 0.18 4.659 0.192 0.192 0.018 1.000 1.000

Table 5: Estimates of the fraction of paste within either t or tG of an air void surface for a monosized 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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