Undulation instability

Smectic A liquid crystals are known to be rather sensitive to dilatations of the layers. As shown in [34, 35], a relative dilatation of less than 10-4 parallel to the layer normal suffices to cause an undulation instability of the smectic layers. Above this very small, but finite, critical dilatation the liquid crystal develops undulations of the layers to reduce the strain locally. Later on Oswald and Ben-Abraham considered dilated smectic A under shear [36], When a shear flow is applied (with a parallel orientation of the layers), the onset for undulations is unchanged only if the wave vector of the undulations points in the vorticity direction (a similar situation was later considered in [37]). Whenever this wave vector has a component in the flow direction, the onset of the undulation instability is increased by a portion proportional to the applied shear rate. 

Throughout this chapter we focus on the extended hydrodynamic description for smectic A-type systems presented in [42,43], We discuss the possibility of an undulation instability of the layers under shear flow keeping the layer thickness and the total number of layers constant. In contrast to previous approaches, Auernhammer et al. derived the set of macroscopic dynamic equations within the framework of irreversible thermodynamics (which allows the inclusion of dissipative as well as reversible effects) and performed a linear stability analysis of these equations. The key point in this model is to take into account both the layer displacement u and the director field ft. The director ft is coupled elastically to the layer normal p = in such a way that ft and p are parallel in equilibrium z is the coordinate perpendicular to the plates. 

Consequently, a parallel alignment of smectic layers is linearly stable against undulations even if the perpendicular alignment might be more preferable due to some thermodynamic considerations. As we have shown in Fig. 8, this rigorous result of standard smectic A hydrodynamics is weakened in our extended formulation of smectic A hydrodynamics. When the director can show independent dynamics, an appropriate anisotropy of the viscosity tensor can indeed reduce the threshold values of an undulation instability. 

In the previous sections we have shown that the inclusion of the director of the underlying nematic order in the description of a smectic A like system leads to some important new features. In general, the behavior of the director under external fields differs from the behavior of the layer normal. In this chapter we have only discussed the effect of a velocity gradient, but the effects presented here seem to be of a more general nature and can also be applied to other fields. The key results of our theoretical treatment are a tilt of the director, which is proportional to the shear rate, and an undulation instability which sets in above a threshold value of the tilt angle (or equivalently the shear rate). 

We start with the ground state (°), fi(° defined by the simple shear flow y(°), Fig. 17. The principal effect is, as expected, the appearance of a small tilt of the director from the layer normal (flow alignment), predominantly in z direction (Fig. 18). Note that the configuration of layers is also modified by the shear (Figs. 19 and 20), i.e., the cylindrical symmetry is lost. This is analogous to the shear-flow-induced undulation instability of planar layers (wave vector of undulations in the... 

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. 

Alternatively, if one dilates a smectic stack by increasing its thickness by an amount Sh > 27t X, then the sample will prefer to bend the layers in an undulational instability (Rosenblatt et al. 1977 Ostwald and Allain 1985) in order to restore the lamellar spacing to its preferred value (see Fig. 10-29d). Note that the increase in thickness 8h required to produce this instability is independent of the initial thickness h of the stack. Hence for a macroscopic sample of thickness, say, h = 60 p,m, the strain Sh/h required to induce the undulational instability is extremely small, 8h/h IzrXfh 10-4. Thus smectic monodomains are extremely delicate and can easily be disrupted by mechanical deformation. 

Figure 10.29 Response of an aligned smectic to layer dilation, (a) Initial equilibrium sample, (b) For a very small dilation Sh < Ink, the layer spacing simply increases, (c) A uniform rotation of the layers decreases the spacing toward that of equilibrium, but doesn t satisfy the boundary conditions, (d) Hence, the sample undergoes an mdulational instability, which also narrows the layer spacing while satisfying homeotropic boundary conditions, (e) For a large enough dilation, the undulation instability leads to formation of parabolic focal conic defects. (From Rosenblatt et al. 1977, with permission from EDP Sciences.)...

Sens, P. and Isambert, H. (2002) Undulation instability of lipid membranes under an electric field. Physical Review Letters, 88 (12), 128102. 

Fig. 5. A two-dimensional illustration of the undulation instability caused by a sudden temperature deciease within the LAM phase.

Figure 28. Undulation instability in a columnar liquid crystal subjected to a mechanical dilatation normal to the columns.

S.I. Ben-Abraham and P. Oswald, Dynamic Aspects of the Undulation Instability in Smectic A Liquid Crystals, Mol. Cryst. Liq. Cryst, 94, 383-399 (1983). 

R. Ribotta, Experimental Study of the Elasticity of Smectic Liquid Crystals Undulation Instability of Layers, in Molecular Fluids, R. Balian and G. Weill (Eds.), 353-371, Gordon and Breach, London, 1976. 

It is worth taking note that Hsat for a modest Wa is significantly smaller than the thresholds of the well-laiown Helfrich (1970) undulation instability and Parodi s (1972) dislocation instability. These two instabilities are defined by the balance of the field energy... 

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