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Pleau and Pigeon[6] have recently proposed a spacing equation for the paste-void spacing distribution. Their approach considered both the air void radii distribution and the distribution of distances between a random point in the paste and the nearest air void center. Let h(x) represent the probability density function of the distance between a random point in the system and the center of the nearest air void. As defined before, let f(r) represent the probability density function of air void radii. The joint probability[16] that this random point is a distance s from the surface of an air void with radius r is
As an approximation for h(x), Pleau and Pigeon use the probability density function for the nearest neighbor distance between random points,
which is the Hertz distribution [18] used by Philleo. However, the centers of air voids are not entirely random since air voids do not overlap one another. The consequence of this choice for h(x) is discussed subsequently.
The joint probability density function β(s,r) depends upon h(s+r). If a point chosen at random throughout the entire system lies at a distance x from the center of a sphere, the quantity s is defined as x-r. Therefore, if the random point lies within the sphere, the quantity s will be negative, but the argument x of h(x) will be either zero or some positive number.
The parameter r may be eliminated from the joint probability β (s,r) by integrating over the possible radii:
where the Heaviside function Θ (r + s) [22] insures that the argument of the function h remains positive. This equation is the fundamental equation of Pleau and Pigeon. The cumulative distribution function is
and corresponds to the volume fraction of the entire system within s of an air void center. The volume fraction of the entire system that would lie within an air void is K′ (0), and corresponds to an estimate of the air void volume fraction. The volume fraction of paste within s of an air void surface would then be
where Q normalizes the result by the volume fraction of paste.
The normalization factor Q should equal 1-A, or the paste volume fraction. By the authors' development, this is equivalent to
which the authors use in their derivation. However, as demonstrated previously [23], for monosized spheres the quantity K′(0) corresponds to the air volume fraction for a system of overlapping spheres. This is a consequence of using the Hertz distribution for h(x).
In the subsequent numerical experiment, two results will be reported for the Pleau and Pigeon equation corresponding to the normalization factors 1 - K′(0) and 1-A.