Limiting Laws for Charge Transport in Isotropic Systems of Conjugated Polymers

Conjugated polymers have alternating single and double bonds along their backbones, which ensures their electrical conductivity and makes them promising candidates for electrically active components in flexible, large-area electronic devices. We model charge transport in these polymer systems as a diffusion process of charge carriers, involving two types of movements: intra-chain hopping and inter-chain hopping, characterized by the characteristic hopping times τ₁ and τ₂, respectively. A scaling approach is proposed to the diffusivity of the charge carriers D and the resulting macroscopic conductance of the polymer melt or semidilute solution. The effect of the polymer length and stiffness as well as solution concentration and solvent quality on the electronic properties of the system is considered. Following de Gennes's nomenclature, three universal transport regimes are identified. In the free regime, charges rapidly hop between different chains so that D ∝ 1/τ₁. In the semi-free regime, inter-chain hopping is slow and diminishes the charge diffusivity, D ∝ 1/√(τ₁τ₂). Finally, in the captive regime, the charge carriers change the chain only after completely exploring it, which leads to diffusivity increasing with the chain length, D ∝ N/τ₂. These results are also derived by considering the polymer melt/solution as a resistor ladder. The framework's applicability to alternating current is demonstrated. Our findings are important for the rational design of conducting polymer-based materials.