Modeling percolation transition in carbon black/CNT conductive composites using a probability-reconstructed Bruggeman approach

A probability-informed reconstructed Bruggeman effective medium framework is developed to model the percolation transition in carbon black (CB), carbon nanotube (CNT), and hybrid CB/CNT conductive systems. In contrast to deterministic volume-fraction weighting, the formulation introduces statistically defined overlap, contact, and tunneling probabilities derived from a Poisson double-coverage assumption. Preliminary compaction experiments, on dried fillers provide conductivity–volume fraction relations that are accurately described by sigmoid functions (R 2 = 0.998 for CNT and 0.996 for CB). Inverse reconstruction of the Bruggeman equation yields a concentration-dependent mixing function α ( ϕ ) , demonstrating that the macroscopic percolation transition corresponds to a rapid redistribution of conductive-event dominance from tunneling-controlled to contact-controlled transport. Finite-element simulations incorporating a continuous spatial volume-fraction field ϕ ( x , y , z ) with an adjustable fluctuation amplitude parameter K uni = 0.01 -0.09, further reveal that increasing fluctuation amplitude, promotes an earlier onset of percolation behavior, occurring before the equivalence of contact probabilities and tunneling probabilities ( ϕ ≈ 0.10 for CNT and 0.13 for CB). In hybrid systems under parallel coupling, increasing CNT loading (0–0.04) raises baseline conductivity by ∼2 orders of magnitude at low CB content (0–0.6), while convergence occurs at high CB fractions where CB-dominated transport saturates at 1 S/m. The proposed framework provides a computationally efficient and experimentally calibrated predictive tool for designing conductive polymer composites, with direct applicability to flexible electronics, EMI shielding, and antistatic coatings.