Nonlinear actuator model for flexoelectric membranes

Abstract

Synthetic and biological flexoelectric membranes are actuators that bend under the action of external electric fields, a phenomenon of interest to the development of emerging adaptive materials as well as biological mechano-transduction. This paper presents an actuator model of flexoelectric membranes based on a Helmholtz free energy that incorporates tension, bending, and torsion as well as polarization and dielectric energies. The electro-elastic components of the membrane tension, moment tensor and tensor are derived and used to construct an actuator model that includes dissipation due to viscous fluids in contact with the membrane. The actuator model is expressed by a balance between the externally imposed electric forces, the viscous dissipation of the contacting fluid phases, and the elastic storage of the membrane. The nonlinearity is shown to originate in the viscous dissipation. The model is analyzed for externally imposed oscillating electric fields. The Deborah number De given by the ratio of driving frequency and the resonant frequency is shown to control the viscoelastic response. The key findings are: (i) for De « 1 the response is purely elastic and the electric energy is stored in the elastic deformations of the membrane; (ii) at larger De, the response is anharmonic and viscoelastic; (iii) due to the nature of the viscous nonlinearity only even harmonics are generated in the response; and (iv) secondary resonant frequencies appear at lower driving frequencies. These finding contribute towards the emerging understanding of flexoeletricity in biological membranes, pioneered by Petrov and co-workers (Petrov, A.G., The Lyotropic State of Matter, Gordon and Breach Science Publishers: Amsterdam, 1999).