Continuum percolation for randomly oriented soft-core prisms.

Abstract

We study continuum percolation of three-dimensional randomly oriented soft-core polyhedra (prisms). The prisms are biaxial or triaxial and range in aspect ratio over six orders of magnitude. Results for prisms are compared with studies for ellipsoids, rods, ellipses, and polygons and differences are explained using the concept of the average excluded volume, . For large-shape anisotropies we find close agreement between prisms and most of the above-mentioned shapes for the critical total average excluded volume, n(c) , where n(c) is the critical number density of objects at the percolation threshold. In the extreme oblate and prolate limits simulations yield n(c) approximately 2.3 and n(c) approximately 1.3, respectively. Cubes exhibit the lowest-shape anisotropy of prisms minimizing the importance of randomness in orientation. As a result, the maximum prism value, n(c) approximately 2.79, is reached for cubes, a value close to n(c) =2.8 for the most equant shape, a sphere. Similarly, cubes yield a maximum critical object volume fraction of phi(c)=0.22. phi(c) decreases for more prolate and oblate prisms and reaches a linear relationship with respect to aspect ratio for aspect ratios greater than about 50. Curves of phi(c) as a function of aspect ratio for prisms and ellipsoids are offset at low-shape anisotropies but converge in the extreme oblate and prolate limits. The offset appears to be a function of the ratio of the normalized average excluded volume for ellipsoids over that for prisms, R= (e)/ (p). This ratio is at its minimum of R=0.758 for spheres and cubes, where phi(c(sphere))=0.2896 may be related to phi c(cube))=0.22 by phi(c(cube))=1-[1-phi(c(sphere))](R)=0.23. With respect to biaxial prisms, triaxial prisms show increased normalized average excluded volumes, , due to increased shape anisotropies, resulting in reduced values of phi(c). We confirm that B(c)=n(c) =2C(c) applies to prisms, where B(c) and C(c) are the average number of bonds per object and average number of connections per object, respectively.