A general phenomenological relation for the subdiffusive exponent of anomalous diffusion in disordered media

Abstract

This work numerically investigates the diffusion of finite inert tracer particles in different types of fixed gels. The mean square displacement (MSD) of the tracers reveals a transition to subdiffusive motion MSD ∼ t^α as soon as the accessible volume fraction p in the gel decreases from unity. Individual tracer dynamics reveals two types of particles in the gels: mobile tracers cross the system through percolating pores following subdiffusive dynamics MSD_mob ∼ t^α_mob, while a fraction p_trap(p) of the particles remain trapped in finite pores. Below the void percolation threshold p < p_c all the particles get trapped and α → 0. By separately studying both populations we find a simple phenomenological law for the mobile tracers α_mob(p) ≈ a ln p + c where c ≈ 1 and a ∼ 0.2 depends on the gel type. On the other hand, a cluster-analysis of the gel accessible volume reveals a power law for the trapping probability p_trap ∼ (p/p_c)^−γ, with γ ≃ 2.9. This yields a prediction for the ensemble averaged subdiffusion exponent α = α_mob(1 − p_trap). Our predictions are successfully validated against the different gels studied here and against numerical and experimental results in the literature (silica gels, polyacrylamide gels, flexible F-actin networks and in different random obstacles). Notably, the parameter a ∼ 0.2 presents small differences amongst all these cases, indicating the robustness of the proposed relation.