MICHAEL MANGA: Recent Refereed Publications
Abstracts
Saar, M.O. and Manga, M. (2002)
Continuum Percolation for Randomly Oriented Soft-core Prisms,
Physical Review E, vol. 29, paper number
10.1029/2002GL01485.
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 6 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, $\left\langle v_{ex} \right\rangle$. For large shape anisotropies
we find close agreement between prisms and most of the above mentioned
shapes for the critical total average excluded volume, $n_c\left\langle
v_{ex} \right\rangle$, where $n_c$ is the critical number density of
objects at the percolation threshold. In the extreme oblate and prolate
limits simulations yield $n_c\left\langle v_{ex} \right\rangle \approx
2.3$ and $n_c\left\langle v_{ex} \right\rangle \approx 1.3$, respectively.
Cubes exhibit the lowest shape anisotropy of prisms minimizing the
importance of randomness in orientation. As a result, the maximum prism
value, $n_c\left\langle v_{ex} \right\rangle \approx 2.79$, is reached for
cubes, a value close to $n_c\left\langle v_{ex} \right\rangle = 2.8$ 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, $R
= \left\langle \overline{v_{ex}} \right\rangle_e / \left\langle
\overline{v_{ex}} \right\rangle_p$. 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, $\left\langle
\overline{v_{ex}} \right\rangle$, due to increased shape anisotropies,
resulting in reduced values of $\phi_c$.
We confirm that $B_c = n_c\left\langle v_{ex} \right\rangle = 2C_c$
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.
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