The Analytical Solutions to a Cation–Water Coupled Multiphysics Model of IPMC Sensors

Ionic polymer-metal composite (IPMC) sensors generate voltages or currents when subjected to deformation. The magnitude and time constant of the electrical response vary significantly with ambient humidity and water content. However, most conventional physical models focus solely on cation dynamics and do not consider water dynamics. In addition to cation dynamics, Zhu's model explicitly incorporates the dynamics of water. Consequently, Zhu's model is considered one of the most promising approaches for physical modeling of IPMC sensors. This paper presents exact analytical solutions to Zhu's model of IPMC sensors for the first time. The derivation method transforms Zhu's model into the frequency domain using Laplace transform-based analysis together with linear approximation, and subsequently solves it as a boundary value problem of a set of linear ordinary differential equations. The resulting solution is expressed as a transfer function. The input variable is the applied bending deformation, and the output variables include the open-circuit voltage or short-circuit current at the sensor terminals, as well as the distributions of cations, water molecules, and electric potential within the polymer. The obtained transfer functions are represented by irrational functions, which typically arise as solutions to a system of partial differential equations. Furthermore, this paper presents analytical approximations of the step response of the sensor voltage or current by approximating the obtained transfer functions. The steady-state and maximum values of the time response are derived from these analytical approximations. Additionally, the relaxation behavior of the sensor voltage is characterized by a key parameter newly derived from the analytical approximation presented in this paper.