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

Abstract

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 (R² = 0.998 for CNT and 0.996 for CB). Inverse reconstruction of the Bruggeman equation yields a concentration-dependent mixing function $\mathrm{\alpha }\left(\varphi \right)$ , 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 $\varphi \left(\mathrm{x},\mathrm{y},\mathrm{z}\right)$ with an adjustable fluctuation amplitude parameter $K_{\text{uni}}=0.01$ -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 ( $\varphi$ ≈ 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.