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

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=d₂K_PK_T and e=e₂K_PK_S, where d₂ and e₂ are the piezoelectric strain and stress constants of phase 2, respectively, and K_P is a function of the dielectric constants of phases 1, 2, and 3. K_T and K_S 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 ε_i and σ_i 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 K_P on frequency implies the occurrence of a relaxation process. Conversely, when τ₁<τ₂, a retardation process is observed.