Piezoelectricity of Cholesteric Elastomers

We consider theoretically the properties of piezoelectricity in cholesteric elastomers. We deduce using symmetry considerations the piezoelectric contributions to the free energy in the context of a coarse-grained description of the material. In contrast to previous work we find that compressions or elongations of the material along the pitch axis do not produce a piezoelectric response, in agreement with fundamental symmetry considerations. Rather only suitable shear strains or local rotations produce a polarization. We propose some molecular mechanisms to explain these effects. Piezoelectric materials are characterized by the appearance of an electric polarization when a suitable mechanical stress is applied [1]. Symmetry considerations require that these materials be non-centrosymmetric, i.e. not invariant under inversion. Chiral liquid crystals such as a cholesteric or smectic C satisfy this requirement but their fluid nature will not support a static shear. However, chiral liquid crystalline elastomers which consist of a cholesteric liquid crystal homogeneously embedded in a polymer gel can support static stresses, including shear, due to the presence of the underlying gel structure. Thus, they are candidates for the observation of true piezoelectricity in a liquid crystalline system. In this paper we consider theoretically the nature of piezoelectricity in a cholesteric elastomer. We show that a shearing of the elastomer along the pitch axis causes a piezoelectric response. Local rotations of the elastic medium can also in principle produce a polarization. In contrast to previous work [2, 3], we find that compressions or elongations of the elastomer along the pitch axis cannot produce a polarization. We also propose some molecular arguments to explain the mechanism of piezoelectricity in these materials.Finally we discuss the relationship of our work to previous experimental studies [4, 5]. Our starting point is a hydrodynamic description of cholesterics in terms of a pitch vector due to Lubensky [6]. While it is possible to develop a description in terms of the director, all macroscopic quantities such as the polarization require a coarse-graining, i.e. an averaging of the director field over the pitch length. The description in terms of the pitch vector is already coarse-grained. We demonstrate below that our theory is equivalent to Terentjev’s theory of piezoelectricity [3] which is phrased in terms of the director. However, the coarse-graining built into the pitch vector description is more convenient in determining the elastic strains associated with piezoelectricity. The director n in a cholesteric has the following form [6]: n(r) = n0 cosψ(r) + p× n0 sinψ(r), (1) where p is a unit vector along the pitch axis, n0 is a unit vector in the plane perpendicular to p, and ψ is the phase angle of the director. The latter quantity may be expressed as ψ = 2π λ p · r + φ, where λ is the pitch of the helix, and φ is a phase factor. We can define a wavevector q0 for the helix via the relation, q0 = 2π λ . It is important to note that a helix is not a polar object, i.e. it looks the same whether viewed from the top or the bottom. Mathematically speaking this nonpolarity arises from the