Keywords: piezoelectric, polymer, ceramic, free energy, poling, finite element modelling.

Polymer semiconductors and their composite variants with significant piezoelectric effect are considered as promising candidates to compete with the silicon based technology. Piezoelectric copolymer composite films, such as ultra-thin Langmuir-Boldgett films of PVDF and its copolymer with trifluoroethylene (P(VDF-TrFE)), ionic polymer, and electrostrictive elastomers have several potential applications, especially in photonics, bio-medical devices, macromolecular assembly and smart structures. The category of polymer composites we are interested in this paper is the one in which the piezoelectricity, the electrical charge and the polymer dispersion interplay to produce a desired functionality. In the solid state, the dynamics of the piezoelectric copolymer (thin film) is governed by the deformation of the polymer chains which also undergo electrical polarization and transport of charge. Due to this coupled dynamics, the charge transport with dynamical changes in the conduction band of the polymeric network is of importance in photonics related applications. On the other hand, electrostriction induced mechanical motion and flexibility of the piezoelectric copolymer film is of importance for smart structural applications. Molecular level modelling of the dynamics and related observations have been reported in the literature. The macroscopic effect of linear piezoelectricity and electrostriction in piezoelectric composite structures have been studied widely. However, in polymer composites, the study of the effects which are important for small scale applications (e.g. flexible thin film device) has not been well reported in literature.

In this paper, a mesoscopic model and its finite element implementation are developed to study the dynamic visco-piezoelastic behaviour of a composite thin film made of PVDF copolymer and PZT ceramic particulate. Piezoelectric copolymers and blends we are interested in, such as PVDF-PTrFE with appropriate doping for controlled mechanical and electrical properties, are of substantial interest in microelectronics and other engineering applications. Ferroelectric ceramic particles, which can be used for doping a copolymer or a blend, have high dielectric, piezo and pyroelectric coefficients and high electromechanical coupling. However, they have poor mechanical strength, low reliability and high acoustic impedance. On the other hand, the PVDF based copolymers and reported blends have high relative compliance and flexibility and low density compared to the piezoelectric ceramics. Therefore, an appropriate compositional structure of the above two can be useful in various applications. For processing a device-grade material, most often in the form of thin film, it is essential to polarize the polymer composite under strain. Here, the effect of the compositional variation on the effective properties are difficult to clearly identify from experiments alone. The reported studies in literature are either focused on the molecular dynamic aspect of the small-scale dynamics, as mentioned above, [1], or they are based on the linear model of piezoelasticity [2]. In addition, there are several other factors, such as the interfacial stress due to the film substrate, the free-surface energy and the dipole orientation and size dependent inhomogeneity, which can play important role in the overall structure and functions.

In order to develop a mathematical model including the above effects, first the free energy is constructed as a function of the strain, the effective polarization, and the assumed distribution functions of the polymer chains and ceramic particles in a cell. These distribution functions generally can vary over time and they are governed by the Smoluchowski equation for dispersed system [3]. Corresponding driving force includes the spatial derivative of the potential energy density. The potential energy density is expressed in terms of the bonded and nonbonded energy of molecular interaction in the cell, whereas their spatial gradient is transformed into the homogenized strain energy of the cell in the present model. Conservation laws are derived for the composite, which results in the thermomechanical equilibrium equation and the Maxwell equation of electrostatics. A coupled finite element model is developed with an element containing a number of cells. The continuum displacement, the temperature and the electric potential are determined by considering the quasi-three dimensional geometry of the thin film along with the substrate. A case of poling under controlled thermomechanical distribution over a randomly dispersed film is considered for numerical analysis. Various parametric effects of the resulting piezoelectric properties of the film under a number of poling cycles are analyzed and computational aspects of the developed code are discussed.

1

J.A. Young, B.L. Farmer and J.A. Hinkley, "Molecular modeling of the poling of piezoelectric polyamides", Polymer, 40, 2787-2795, 1999. doi:10.1016/S0032-3861(98)00474-1

2

T.E. Gomez, F.M. Espinosa, F. Levassort, M. Lethiecq, A. James, E. Ringgard, C.E. Miller and P. Hawkins, "Ceramic powder-polymer piezocomposites for electroacoustic transduction: modeling and design", Springer-Verlag, NY, 2000.

3

T.S. Chow, "Mesoscopic Physics of Complex Materials", Springer-Verlag, NY, 2000.

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