Undulation versus Frederiks instability in nematic elastomers in an external electric field

We study the behaviour of a nematic side-chain elastomer under the influence of an external static electric field for a specific geometry. For this investigation, the nematic elastomer is considered to be a perfect insulator. On the basis of a macroscopic description we generalize the classical Frederiks transition in a low-molecular-weight (LMW) nematic liquid crystal to the elastomeric case. We predict, using a linear stability analysis, that the onset of the instability can be qualitatively different from the LMW case: in liquid crystalline elastomers an undulation instability can arise at onset. Whether the analogue of a Frederiks instability or an undulation instability occurs first depends on the sample thickness as well as on the material parameters. It turns out that the parameter which describes the coupling between the deformations of the elastomer and the relative rotations between the elastomer and the director field of the nematic phase is most important for the predicted response of the system. Furthermore, we find that the magnitude of the critical electric field is much higher in the elastomeric than in the low-molecular-weight case.