Macroscopic dynamics of the ferroelectric smectic \(A_F\) phase with \(C_{\infty v} \) symmetry

We present the macroscopic dynamics of ferroelectric smectic A, smectic \(A_F\), liquid crystals reported recently experimentally by three groups. In this fluid and orthogonal smectic phase, the macroscopic polarization, \({\textbf{P}}\), is parallel to the layer normal thus giving rise to \(C_{\infty v}\) overall symmetry for this phase in the spatially homogeneous limit. A combination of linear irreversible thermodynamics and symmetry arguments is used to derive the resulting dynamic equations applicable at sufficiently low frequencies and sufficiently long wavelengths. Compared to non-polar smectic A phases, we find a static cross-coupling between compression of the layering and bending of the layers, which does not lead to elastic forces, but to elastic stresses. In addition, it turns out that a reversible cross-coupling between flow and the magnitude of the polarization modifies the velocities of both, first and second sound. At the same time, the relaxation of the polarization gives rise to dissipative effects for second sound at the same order of the wavevector as for the sound velocity. We also analyze reversible cross-coupling terms between elongational flow and electric fields as well as temperature and concentration gradients, which lend themselves to experimental detection. Apparently this type of terms has never been considered before for smectic phases. The question how the linear \({{\textbf{P}} \cdot \textbf{E}}\) coupling in the energy alters the macroscopic response behavior when compared to usual non-polar smectic A phases is also addressed.