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The equation for G given in Eqn. 13 for the sphere distribution of Eqn. 11 can also be used to approximate the correct value of G for the zeroth-order logarithmic distribution used here. The parameters a and b in Eqn. 11 are chosen to yield a sphere diameter distribution with the same modal diameter Dm and specific surface as the logarithmic distribution used here. The specific surface and the modal diameter are related to the parameters from the following equations:
The corresponding values of the parameters are
The corresponding equation for G is
Note again that the value of the numerator differs very little from that in Eqn. 10. Since the distribution proposed by Attiogbe is based upon the zeroth-order logarithmic function used here, it is reasonable to expect that a corrected form for Eqn. 13 which is based the zeroth-order logarithmic distribution would differ very little from the above result. It is this equation for G that is used in Tables 6--8.
As in the case of monosized spheres, the Attiogbe equation does not appear to accurately estimate any reported statistic of the void-void spacing distribution. In the data shown in Table 7, the Attiogbe equation t is nearly an order of magnitude greater than both the 50 th and the 95 th percentiles. As the paste air fraction increases from 0.02 to 0.07, the value of tG only decreases by 10%, where as the measured values decrease by 50%.
Table 8 shows the performance of G in estimating the fraction of paste within either t or tG of an air void surface for lognormally distributed radii. Again, since the Lu and Torquato equation performs so well for the lognormally distributed spheres, it is treated as the true value. As in the case of monosized spheres, the parameter G does not provide a useful estimate of the paste volume fraction within either t or tG of an air void surface. The value of G for the two greatest air contents, although correct, is unremarkable since a cursory analysis would predict that the volume fraction of paste within one half the mean free path should be nearly unity.