A continuum theory of organic mixed ionic-electronic conductors of phase separation

Organic mixed ionic-electronic conductors (OMIECs) are the core functioning component in the emerging flexible, bio-, and optoelectronics owning to their unique capability of mixed conduction. Of all types, two-phase OMIECs exhibit exceptional performance due to their high stretchability and balanced ionic-electronic conduction. However, the electron-conducting phase may segregate from the ion-conducting phase in a two-phase OMIEC, changing the conducting path and eventually leading to degraded performance and dysfunction of the devices. In this work, we formulate a continuum theory following the thermodynamics framework of a two-phase OMIEC undergoing phase separation. The free energy consists of contributions from the deformation of the polymer chains, the mixing of the polymer with salts and solvents, the electrostatic field, and the two-phase interfaces. The equilibrium conditions and kinetics equations are derived with the constraint of mass conservation, thermodynamics laws, and electrostatics. We implement the theory into a finite element model and study the mechanics and electrochemistry of the OMIEC channel in an electrolyte-gated organic electrochemical transistor (OECT) device. The computational model captures the concurrent transport of charge carriers, mechanical swelling, and phase separation in the OMIEC and replicates the transfer curves of an OECT which agree well with the experiments. More specifically, we reveal the origin of the volumetric capacitance as the accumulation of charge carriers at the two-phase interfaces. We examine the parametric space to elucidate experimental observations such as molecular size-dependent conductivity and substrate-dependent phase separation. The swelling behavior and the transfer curves of OECTs under stretched, free, and constrained states are compared, demonstrating the effects of deformation on the phase dynamics and the electron-conducting behavior. We show that, for volumetric swelling and the electrochemical transfer curves, the effect of stress-transport coupling dominates while the effect of the Maxwell stress is negligible. This work provides a theoretical basis for the mechanics and electrochemistry of two-phase OMIECs.