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Void-Void Proximity Distribution

The approach used by Lu and Torquato[41] for the void-void proximity is similar to that for the paste-void proximity. Given that a point is located at the center of an air void with radius R, the probability that the nearest air void surface is farther away than w is [41]

Lu and Torquato refer to this as the particle exclusion probability.

The probability that the nearest air void surface is within a distance w from the center of an air void with radius R is

Let s represent the shortest surface-surface distance between two air voids. The probability that the nearest air void surface is within s of the surface of the void with radius R is

The function EP (s,R) is equivalent to the void-void spacing cumulative distribution function.

The most important feature of Eqn. 34 is that EP (s,R) depends upon the size of the sphere one starts from. For monodispersed sphere diameters, R is simply a constant. However, for a system composed of polydispersed sphere diameters, EP (s,R) is a continuous function of R. Since a continuous distribution of sphere diameters would have an infinite number of possible diameters, there would exist an infinite number of possible EP (s,R) distributions. This complicates an evaluation of void-void spacing distributions for systems composed of polydispersed sphere radii.

One possible remedy is to simply calculate an ensemble average. Ensemble averages can be calculated based on either number density or volume density. This bulk value can then be compared to measured values. Here, the number density ensemble average was chosen:

For a system of polydispersed sphere diameters one can also calculate the mean nearest surface-surface distance[41]:

which gives the average distance to the nearest air void surface when starting from spheres of radius R. The quantity lP (R) decreases as R increases. Therefore, on average, the larger the sphere one starts from, the shorter the distance one travels to reach the surface of the nearest air void. A technique is described to measure this quantity, and the results are given along with this prediction.

Next: Numerical Experiment Up: Lu and Torquato Previous: Paste-Void Proximity Distribution