A Statistical Approach to the Filtration of Rods

We investigate the efficacy of a square-grid mesh at filtering rods from solution. The volume fraction $\phi$ is kept low, reducing the chance of rods cooperatively jamming at the mesh. For round particles, filtering at low $\phi$ is trivially determined by the ratio of particle diameter to mesh size. Because rods have two length scales, filtering is non-trivial for meshes larger than the rod width but smaller than the length, a potentially very large range. We have measured experimentally the probability for a rod to be filtered as a function of mesh size, particle length, and aspect ratio. Results are compared with a theoretical extension of the Buffon-Laplace Needle problem that accounts for finite rod width and an isotropic distribution in the zenith angle. The solution is the probability that a sphero-cylinder in three dimensions makes contact with a 2D sieve-like mesh, a necessary but not sufficient condition for filtration. Comparison of experiment and theory is then suggestive of what conditions are both necessary and sufficient.