Piezoelectricity in three-phase systems: Effect of the boundary phase

Abstract

The influence of electrical conductivity on the piezoelectricity of polymers was studied using a three-phase model. The model consists of a continuous nonpiezoelectric phase (phase 1) surrounding a spherical piezoelectric phase (phase 2), which is itself bounded by an electrically conductive shell (phase 3). The piezoelectric constants for the system can be expressed as d=d2KPKT and e=e2KPKS, where d2 and e2 are the piezoelectric strain and stress constants of phase 2, respectively, and KP is a function of the dielectric constants of phases 1, 2, and 3. KT and KS are expressed as functions of the elastic constant of each phase. The piezoelectric constants d and e decrease with decreasing elastic constant of phase 3 and with increasing dielectric constant of phase 3. The three-phase system shows a Maxwell–Wagner piezoelectric relaxation with a relaxation time given by τ=(2ε1+ε2+ηε3)/(2σ1+σ2+ησ3), where η=2δ/a. In these expressions εi and σi are the dielectric constant and the electrical conductivity of phase i, respectively, a is the radius of the sphere (piezoelectric phase), and δ is the thickness of the electrically conductive shell around the piezoelectric phase. When τ1(=ε1/σ1)>τ2[=(ε2+ηε3)/(σ2+ησ3)], the functional dependence of KP on frequency implies the occurrence of a relaxation process. Conversely, when τ1<τ2, a retardation process is observed.