Nonlinear macroscopic description of liquid crystalline elastomers in external fields
We concentrate on a continuum characterization of the macroscopic behavior of side-chain liquid single crystal elastomers (SCLSCEs). These materials consist of chemically crosslinked polymer backbones, to which mesogenic units are attached as side-groups. Due to specific routes of synthesis SCLSCEs feature a monodomain of the liquid crystalline order in the ground state. Their macroscopic coupling of liquid crystalline order and elastic mechanical deformations makes them unique. In particular, we investigate the macroscopic behavior of cholesteric and nematic SCLSCEs in external electric and mechanical fields. We characterize the respective liquid crystalline state using the director field and describe the current state of mechanical distortion by a displacement field. The specific coupling between these two components is taken into account explicitly by additional macroscopic variables. These are the relative rotations between the director orientation and the polymer network. Using this kind of description, we first study the influence of an external electric field on the state of a cholesteric SCLSCE. For this purpose, the field direction is chosen to be parallel to the helical axis of the cholesteric mesogen alignment. Director reorientation and mechanical distortions are analyzed to linear order. In the case of low electric field amplitudes, we find an effect that has been termed rotatoelectric. Here, with increasing electric field amplitude, the director arrangement rotates around the helical axis, relative to the polymer network. This effect is specific for cholesteric SCLSCEs. We discuss several aspects important for an experimental observation of this effect. Next, we investigate the dielectric instabilities of a cholesteric SCLSCE in a Frederiks splay geometry at higher electric field amplitudes. On the one hand, we find a scenario that corresponds to the Frederiks transition in conventional low molecular weight liquid crystals. Here, the director reorientation arises in a way that is spatially homogeneous in the directions perpendicular to the cholesteric helical axis. On the other hand, however, we also find a qualitatively different instability. The latter is characterized by spatial undulations of the director reorientation, which occur in at least one direction perpendicular to the cholesteric helical axis. We recover the same results in the case of an external magnetic field. Besides, we discuss elastic mechanical compressions or dilations of a cholesteric SCLSCE in the directions parallel and perpendicular to the cholesteric helical axis. Here, small amplitudes of deformation lead to a distortion of the cholesteric helical structure. In the simplest case, we obtain an elongation or compression of the cholesteric helix along its axis. Furthermore, we propose ways to experimentally access so-far unknown values of the material parameters involved. We proceed by developing a model to characterize the nonlinear macroscopic behavior of the materials. For this purpose, we identify two coupled preferred directions in nematic and cholesteric SCLSCEs. One of them is imprinted into the polymer network during the process of synthesis to align the mesogens in a liquid crystalline monodomain. On the other hand, the actual average mesogen orientation may deviate from this imprinted direction and is described by the director field. We derive expressions characterizing nonlinear relative rotations between these two coupled preferred directions and we include them as macroscopic variables into our description. Using our model, we first investigate the shear deformation of a nematic SCLSCE. If the shear plane contains the director, the latter will be reoriented due to the mechanical deformation. In addition, however, we find as a nonlinear effect that the director reorientation acts back onto the elastic mechanical distortion of the material. This leads to compressive and dilative strain deformations. Finally, we study the specific stress-strain behavior of nematic SCLSCEs. It has been found for nematic SCLSCEs stretched perpendicularly to the initial director orientation that their director reorients towards the stretching direction. This reorientation of the director sets in above a critical threshold strain. In the strain regime where the director reorientation occurs, the slope of the corresponding stress-strain curve is significantly decreased. We demonstrate that our model describes this nonlinear behavior. Furthermore, we compare the predictions of our model with experimental data. As a result, we find that nonlinear relative rotations play the central role in the macroscopic characterization of the behavior of the materials. However, we also conclude that the macroscopic stress-strain behavior can be qualitatively influenced by those contributions to the elastic response that are not connected to the director reorientation and relative rotations.
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