Dynamic fractal structure of flocs under competing aggregation and breakup

Flocculation is governed by the competing processes of aggregation and breakup, and together these mechanisms shape the emergent rheology and settling behavior of cohesive suspensions in natural and engineered systems. Classical flocculation models often assume constant floc yield strength and fractal dimension, limiting their ability to represent structural hysteresis and path dependence under variable shear rate. This study develops a dynamic flocculation framework that couples a physically based, time-varying yield strength to a new structural law in which the floc fractal dimension n_f(t) relaxes toward process-specific targets for aggregation and breakup. Exact, state-dependent time scales for aggregation and breakup are derived directly from the governing growth and breakup rates, avoiding linearized, near-equilibrium approximations. The model yields a closed-form expression for the equilibrium fractal dimension n_f,eq(G), which transitions sigmoidal-like from diffusion-limited to shear-limited structures with increasing shear rate. Validation against three widely used datasets including activated sludge, polystyrene particles, cohesive sediments in flume tests shows that allowing n_f to evolve in time improves the prediction of transient compaction and steady-state sizes relative to constant-n_f formulations, as quantified by higher correlation and lower errors. The framework also clarifies how shear history modulates structural hysteresis under unsteady forcing and provides interpretable links from microstructure to bulk properties. These advances support more accurate and transferable parameterizations for water and wastewater treatment, mining separations, and hydromorphodynamic modeling in coastal and estuarine environments.