Scaling Laws for Charge Transport in Isotropic Bulks and Solutions of Conjugated Polymers

Conjugated polymers comprise alternating single and double bonds along the backbones, which ensures their electrical conductivity. Herein, charge transport in them is modeled as a diffusion process, with two types of mechanisms involved: intra- and interchain hopping characterized by τ₁ and τ₂ times, respectively. Charge carriers’ diffusivity in isotropic melts and semidilute solutions of conjugated polymers under a weak electric field is calculated. By extending de Gennes’ approach [Phys. A: Stat. Mech. Appl. 1986, 138, 206–219], we identify three universal regimes of charge transport. In the free regime, charges rapidly hop between different chains so that the charge carrier diffusivity is D_F ∼ τ₁^–1N⁰. In the semifree regime, interchain hopping is slow compared to intrachain hopping, which makes diffusivity dependent on both characteristic times, D_SF ∼ (τ₁τ₂)^−1/2N⁰. Finally, in the captive regime, interchain hopping is strongly hindered, causing the charge carriers to traverse the entire chain before hopping to another. For flexible polymers, this leads to diffusivity increasing linearly with the chain length N, D_C ∼ τ₂^–1N¹. These laws also hold for semiflexible polymers and/or semidilute solutions, where the pairwise contact probability depends on the polymer volume fraction. Weakly and strongly fluctuating solutions are considered within the mean-field and scaling approaches, respectively. In addition to the diffusion (charge hopping) model, the conductivity of polymer systems is quantitatively described by considering them as finite resistor ladders. The charge carrier’s diffusivity and the resulting conductivity in alternating current (AC) are predicted as a function of its frequency. We believe that our findings will facilitate the rational design of polymer-based flexible and stretchable electronic devices.