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- Hydrodynamische Gleichungen (1) (remove)

- Instabilities in layered liquids induced by external fields (2003)
- In this thesis, we have shown that the inclusion of a nematic degree of freedom in the macroscopic hydrodynamic description of smectic-A-like liquids leads to a number of interesting results. While the director and the layer normal are coupled such that they are parallel in equilibrium, in non-equilibrium situations, the director needs not be parallel to the smectic layer normal. This is in contrast to standard smectic-A hydrodynamics. Using irreversible thermodynamics and symmetry arguments, we derived a complete set of macroscopic hydrodynamic equations for the director variables, the layer displacement, the velocity field, and the moduli of the nematic and smectic order parameters. Recent experiments find that the parallel orientation of smectic-A- like liquids is destabilized by an applied shear. After destabilization, two typical scenarios are observed in a steady state situation: i) The layers are oriented perpendicular to the vorticity direction of the flow, i.e., they lie in the plane spanned by the velocity and the gradient direction (`perpendicular' orientation). ii) Closed multi-lamellar vesicles (`onions') form. A number of experiments indicate that the onset of this reorientation is controlled by the applied shear rate. In contrast to standard smectic-A hydrodynamics where shear in the parallel orientation has no effect on the layers, this destabilizing effect comes out naturally from our extended smectic-A hydrodynamics. The argumentation goes along the following lines. The shear field exerts a torque on the director that must be balanced by the coupling to the layer normal. In the limit of small angles, balancing these torques leads, in the steady state, to a shear-induced director tilt proportional to the shear rate. The preferred thickness of a smectic layer is directly connected to the projection of the averaged molecular axes on the layer normal, or, in terms of our model, the thickness is proportional to the projection of the director on the layer normal. If the director is tilted, this projection is shorter. This decrease of the projection is equivalent to an effective dilation, because the actual layer thickness is larger than the preferred layer thickness. Similar to the case of low molecular weight smectic-A liquid crystals under a dilative strain, this effective dilation leads to an undulation instability. To investigate the stability of the parallel alignment, we performed a linear and weakly non-linear analysis of the governing equations. The initial state is the above described spatially homogeneous director tilt with the smectic layers in the parallel orientation. The linear stability analysis showed an undulation instability which sets in above a critical tilt angle (or equivalently, a critical shear rate). This critical tilt angle turned out to depend strongly on the material parameters. For a typical low molecular weight thermotropic liquid crystal, we estimated the critical tilt angle to be on the order of a few degrees. The linear stability analysis also revealed that the nematic and smectic order is modulated close to the boundaries. Since the probability for the formation of defects is larger in regions with a decreased modulus of the order parameter, these variations in the modulus of the order parameter open the way for a destabilization of the layered structure. We note that a detailed investigation of this point is beyond the scope of the present work. Finally, we could exclude an oscillatory instability for all physically reasonable regions in parameter space. The weakly non-linear analysis shows that the bifurcation is supercritical for most physically relevant regions in the parameter space. A detailed comparison to an independent approach was undertaken in a collaboration with simulation physicists from the Max-Planck- Institute for Polymer Research in Mainz. In a molecular dynamics simulation, a model layered liquid consisting of chains of four particles (AABB) was considered. The interaction potential of particles not connected by springs is attractive for like particles and repulsive for particles of a different nature. The simulation demonstrated the two main predictions of our analytic theory: The director tilts in the flow direction and, above a critical shear rate, the layers show stationary undulations with a wave vector in the vorticity direction. Besides this good qualitative agreement, a reasonable quantitative agreement for the critical shear rate was found.