Shear, director orientations instabilities

Experiments by Muller et al. [17] on the lamellar phase of a lyotropic system (an LMW surfactant) under shear suggest that multilamellar vesicles develop via an intermediate state for which one finds a distribution of director orientations in the plane perpendicular to the flow direction. These results are compatible with an undulation instability of the type proposed here, since undulations lead to such a distribution of director orientations. Furthermore, Noirez [25] found in shear experiment on a smectic A liquid crystalline polymer in a cone-plate geometry that the layer thickness reduces slightly with increasing shear. This result is compatible with the model presented here as well. 

Experimentally, both small molecule liquid crystals and LCP s display all sorts of director orientation patterns at high shear rates, and the existence of these Is of obvious Importance to control of orientation In processing (see Page 656 of Reference 2 for some discussion and references). It is conjectured that these may correspond to instabilities of the solutions of L-E theory. If feasible, a study of these instabilities - classification of patterns, prediction of conditions for occurrence, etc. - would be useful to relate to experimental observations. 

The strategy is as follows. We start by rewriting the equations in cylindrical coordinates (r, ,z). The variables we consider are the layer displacement u (now in the radial direction) from the cylindrical state, the director n, and the fluid velocity v. The central part of the cylinder, r < Ri, containing a line defect, is not included. It is not expected to be relevant for the shear-induced instability. We write down linearized equations for layer displacement, director, and velocity perturbations for a multilamellar (smectic) cylinder oriented in the flow direction (z axis). We are interested in perturbations with the wave vector in the z direction as this is the relevant direction for the hypothetical break-up of the cylinder into onions. The unperturbed configuration in the presence of shear flow (the ground state) depends on r and 0 and is determined numerically. The perturbations, of course, depend on all three coordinates. We take into account translational symmetry of the ground state in the z direction and use a plane wave ansatz in that direction. Thus, our ansatze for the perturbed variables are... 

The parameter controlling the instabilities is N=XYo)/D where D is the diffusivity for the nematic orientation. The experiments are performed for increasing values of N keeping the ellipticity E =Xq / Y constant. At low shear rates, the director moves at frequency o) on an elliptic cone the axis of which is perpendicular to the plates but the system is homogeneous in a... 

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