Continuum percolation for randomly oriented soft-core prisms

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, 〈v_ex〉. 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〈v_ex〉, 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〈v_ex〉≈2.3 and n_c〈v_ex〉≈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〈v_ex〉≈2.79, is reached for cubes, a value close to n_c〈v_ex〉=2.8 for the most equant shape, a sphere. Similarly, cubes yield a maximum critical object volume fraction of φ_c=0.22. φ_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 φ_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=〈v_ex̅ 〉_e/〈v_ex̅ 〉_p. This ratio is at its minimum of R=0.758 for spheres and cubes, where φ_c(sphere)=0.2896 may be related to φ_c(cube)=0.22 by φ_c(cube)=1-[1-φ_c(sphere)]^R=0.23. With respect to biaxial prisms, triaxial prisms show increased normalized average excluded volumes, 〈v_ex̅ 〉, due to increased shape anisotropies, resulting in reduced values of φ_c. We confirm that B_c=n_c〈v_ex〉=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.