General Description of Piezoelectricity in a Polymer Film

A polymer film with the thickness l is assumed to be uniform in the plane perpendicular to the x-axis (thickness direction) and all the quantities are functions of x alone (Fig. 2). The electric displacement (x-component) D is related to the electric field E and polarization P (both x-components) as follows  

When the film is regarded as an assembly of charged particles, 

When an electric field E is applied, Eq changes by an amount proportional to E, as usually expressed by (a — 1) E/4n, where e is the dielectric constant. When a strain is applied to the film, changes in and hence changes in polarization are divided into two components 1. displacement equal to the macroscopic displacement and 2. residual displacement. The latter is the internal strain and causes the intrinsic piezoelectricity. The effect of internal strain on P is expressed by eu, where e is the intrinsic piezoelectric constant and u is the elongational strain along the x-axis. The electric displacement D can therefore be written 

From Eq. (15), the potential V of the upper electrode with respect to the lower one is 

Dielectric constant and piezoelectric constant are assumed to depend on 

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