Geometric Percolation of Spherically Symmetric Fractal Aggregates

The connectedness percolation threshold ( ϕ_c ) for spherically symmetric, randomly distributed fractal aggregates is investigated as a function of the fractal dimension ( d_F ) of the aggregates through a mean-field approach. A pair of aggregates (each of radius R ) are considered to be connected if a pair of primary particles (each of diameter δ ), one from each assembly, 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 fully penetrable spheres to calculate ϕ_c . For sufficiently large aggregates, our analysis reveals the existence of two regimes for the dependence of ϕ_c upon R/δ namely: (i) when d_F > 1.5 aggregates form contacts near to tangency, and ϕ_c≈( R/δ)^d_F - 3 , whereas (ii) when d_F < 1.5 deeper interpenetration of the aggregates is required to achieve contact formation, and ϕ_c≈( R/δ)^ - d_F . For a fixed (large) value of R/δ , a minimum for ϕ_c as a function of d_F occurs when d_F = 1.5. Taken together, these dependencies consistently describe behaviors observed over the domain 1 ≤ d_F≤ 3 , ranging from compact spheres to rigid rod-like particles.