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Percolation theory can be used to study the flow-related properties of various porous systems. In particular, recently developed membranes from silica nanoparticles with surface grafted polymer brushes represent a quintessential hard-sphere soft-shell system for which fluid-flow behavior can be illuminated via a percolation framework. However, a critical parameter in membrane design involves the maximum pass-through size of particles. While percolation theory considers path connectedness of a system, little explicit consideration is given to the size of the paths that traverse the space. This paper employs a hard-sphere soft-shell percolation model to investigate maximum particle pass-through size of membranes. A pixelated (as opposed to continuous) representation of the geometry is created, and combined with readily available homology software to analyze percolation behavior. The model is validated against previously published results. For a given sphere volume fraction, the maximum diameter of a percolating path is determined by applying iterative dilations to the spheres until the percolation threshold is reached. A simple approximate relationship between maximum particle size and sphere volume fraction is derived for application to membrane design. Experimental particle cutoff size results for the polymer modified silica nanoparticle membranes were used as a partial verification of the model created in this paper. The presence of a distribution of sphere sizes (naturally created by the manufacturing process) is found to have negligible effect, compared to results for a single sphere size.

Percolation analysis for estimating the maximum size of particles passing through nanosphere membranes