Modeling the actuation and sensing behavior of an IPMC within the framework of the Theory of Porous Media

Abstract

Electroactive polymers (EAPs) are considered as smart materials. Therein, a differentiation between an electric and an ionic EAP will be made. In this contribution, in particular the ionic polymer-metal composites (IPMCs) are investigated which belong to the class of the ionic EAPs. These kind of materials act as an actuator and a sensor in variety of applications. The actuation application is given by applying an electrical voltage to the electrodes which instantaneously leads to the relocation of the mobile cations towards the cathode. The sensor application is performed by loading the IPMC mechanically which generates an electrical field. For the prediction of the behavior of this material, the theoretical framework of electrodynamics in combination with the Theory of Porous Media has been taken into account, see [1][2][3]. A numerical example for the 2D sensing behavior is investigated. Theoretical modelingIn general, the structure of an IPMC can be divided into a solid phase ϕ S and an overall fluid phase ϕ F . The solid phase contains the polymer network ϕ P and the anions ϕ − . The overall fluid phase is defined by the constituents of the liquid ϕ L and the mobile cations ϕ + . Attention is paid to the motion functions of both phases, i.e., the solid phase is given by one single motion function (χ S = χ P = χ − ) due to the attachment of the anions to the polymer network. The constituents of the overall fluid phase possess their individual motion function (χ L = χ + ). In order to derive the constitutive relations and the evolution equations, the entropy inequality given in [4] has been considered. The evaluation of the entropy inequality has been performed by neglecting the dynamic effects (x ′′ α = 0) and an isothermal process (Θ α = Θ = const.) is considered. All constituents are assumed to be incompressible (ρ αR = const.). Production terms such asρ α (mass production),ρ α e (charge production),κ α e (electric flux continuity) andm α (moment of momentum) are excluded. The direct production terms of the liquid and the cations are taken into account in order to evaluate the Darcy and the Nernst-Planck equations. The local electric potentials of the constituents are equal (φ α e = φ e ). The influence of the higher order terms such as the gradients of the deformation gradients are neglected. With the given assumptions, the set of field equations and the corresponding constitutive relations are given as below:• Balance of momentum of the mixture • Overall Cauchy stressGauss law for the mixture • Constitutive relation for the electric displacement div d SF e = ρ e d SF e = −ǫ 0 ǫ r grad φ e • Darcy equation • Overall electrical body forceNernst-Planck equation • Overall charge density n F c + m w +S = − D + R Θ { R Θ grad c + m + c + m V + m grad P ρ e = n F F { z + c + m + z − c − m } − c + m M + m b + + z + F c + m grad φ e } • Concentration of the anions