Percolation thresholds for randomly distributed spherical fractal aggregates

The connectedness percolation threshold (ϕc) for spherically symmetric, randomly distributed fractal aggregates is investigated as a function of the fractal dimension (dF) of the aggregates through a mean-field approach. A pair of aggregates (each of radius R) are considered to be connected if the centers of a pair of primary particles (each of hard core diameter δ), one from each aggregate, are located within a prescribed distance of each other. An estimate for the number of such contacts between primary particles for a pair of aggregates is combined with a mapping onto the model for penetrable spheres with finite non-zero hard core diameters to calculate ϕc. Effective values for the apparent diameters for the impenetrable cores and connectedness shells are estimated from this analogy. For sufficiently large aggregates, our analysis reveals the existence of two regimes for the dependence of ϕc upon R/δ namely: (i) when dF > 1.5 aggregates form contacts near to tangency and ϕc≈R/δdF−3, whereas (ii) when dF < 1.5 deeper interpenetration of the aggregates is required to achieve contact formation and ϕc≈R/δ−dF . For a fixed (large) value of R/δ, a minimum for ϕc as a function of dF occurs when dF≈ 1.5. Taken together, these dependencies consistently describe behaviors observed over the domain 1≤dF≤3, ranging from compact spheres to rigid rod-like particles.