Smectic layer dilation

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. 

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.)...

The deformations in the smectic A phase liquid crystals are the bending of the smectic layer (accordingly to the splay of the directors) and the dilation or compression of the layers. The energy is thus... 

N in SI system). Modulus B found for a liquid crystal 80CB at temperature 60°C is B = 8TO erg/cm (or 8TO J/m in the SI system) [18]. In that experiment, the compression-dilatation distortion of smectic layers was induced by an external force from a piezoelectric driver. 

For discussion of dynamics of lamellar smectic phases it is important to include another variable, the layer displacement u (r) [3] or, more generally, the phase of the density wave [4]. This variable is also hydrodynamic for a weak compression or dilatation of a very thick stack of smectic layers (L oo) the relaxation would require infinite time. On the other hand, the director in the smectic A phase is no longer independent variable because it must always be perpendicular to the smectic layers. Therefore, total number of hydrodynamic variables for a SmA is six. For the smectic C phase, the director acquires a degree of freedom for rotation about the normal to the layers and the number of variables again becomes seven. 

In the experiment, it is possible to create a dilatation of the smectic layers with piezoelectric drivers. Evidently, an increase of the interlayer distance would cost a lot of energy. Instead, at a certain critical dilatation Xc = Tiks/d, where Xs =, a wave-like or undulation distortion is observed as illustrated by... 

Smectic liquid crystals possess partial positional orders besides the orientational order exhibited in nematic and cholesteric liquid crystals. Here we only consider the simplest case smectic-A. The elastic energy of the deformation of the liquid crystal director in smectic-A is the same as in nematic. In addition, the dilatation (compression) of the smectic layer also costs energy, which is given by [23]... 

Upon stretching along the layer normal, at relatively small strain on a rubbery scale of about 5—7%, smectic elastomers may break up into stripes leading macro-scopically to a cloudy appearance [148]. The striped texture corresponds to a local layer inclination (rotation) relative the average direction of the layer normal. The system prefers the layers to rotate in order to relieve any layer extension deformation in favor of lower-cost rubber distortions at constant layer spacing. This reaction is the rubbery equivalent of the classical instability to avoid layer dilation in low-molecular-mass smectics, described in [149]. However, this type of behavior is not universal because other samples show isotropic rubber behavior [142, 144, 145]. 

If the amplitude of the complex order parameter ij/ is constant, (10.8) describes changes of the free energy which are due to compression or dilation of the layers (i.e., deviations of Q from Qo ) or due to phase shifts of ij/, i.e., displacements of the smectic layers. In contrast to the gradient term appearing in the Ginzburg-Landau ansatz [44], two coefficients, C and C , appear because of the anisotropy of the liquid crystal. In a one-constant approximation, C = C = C, (10.8) is reduced to... 

The smectic layers may themselves compress or dilate in a full general theory of deformations in smectic liquid crystals. These effects have not been considered in the theory presented above because, in many instances, it may be assumed that layer compression or dilation may be neglected in basic planar layered geometries of SmC liquid crystals. Nevertheless, for the sake of completeness, we record for the interested reader some details from the literature. The additional term for smectic layer compression in SmA liquid crystals, which also serves as a first approximation to the compression term in SmC for the planar geometry pictured in Fig. 6.3, is of the form mentioned by de Gennes [106] and de Gennes and Frost [110, pp.345-346]... 

The role of permeation has not been mentioned in this Chapter. This effect occurs when there is a mass transport through the structure [110, p.413]. At this stage, it would appear that an additional equation or term is perhaps needed as a supplement to the theory presented here in order to describe this phenomenon. Such a term for smectics was first discussed by Helfrich [123] and later by de Gennes [108], and some details can be found in de Gennes and Frost [110, pp.435-445] for the case of SmA liquid crystals. The modelling of dynamics of layer undulations has also been carried out by some authors. Ben-Abraham and Oswald [14] and Chen and Jasnow [39] have examined dynamic aspects of SmA undulations using models based on the static theory described in Section 6.2.6 which incorporate flow and the influence of permeation. Experimental observations of a boundary layer in permeative flow of SmA around an obstacle have been reported by Clark [48]. Some more recent experimental and theoretical results involving permeation with compression and dilation of the smectic layers in a flow problem around a solid obstacle where there is a transition from SmA to SmC have been presented by Walton, Stewart and Towler [277] and Towler et al [269]. 

G. McKay and F.M. Leslie, A continuum theory for smectic liquid crystals allowing layer dilation and compression, Euro. Jnl of Applied Mathematics 8, 273-280 (1997). 

SmA layers is a very rare event (except for large K). A possible solution is that the nucleation starts from a nonuniform state. The main idea is simple a bulk dust particle or surface irregularity dilates (or compresses) the smectic layers and thus the initial state is characterized by some nonzero dilation energy. The nucleation of the TFCD means the substitution of the dilation by curvature deformations which generally have less energy. The... 

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. 

When the layers of a lamellar block copolymer are distorted, the free energy density is augmented by a distortional term that can, like the smectic-A phase, be described as the sum of layer compression/dilation and layer-bending energies ... 

A noteworthy aspect of the above solution is that it does not involve either lattice dilatation du/dz or layer undulation V n. Therefore, within the approximations of the linear theory considered here, screw dislocations in smectic A have no self energy (apart from the core), nor do they interact amongst themselves. In this respect they are entirely different from screw dislocations in crystals. 

Fig. 9.12 Undulation or wave-like instability in the smectic A layer subjected to a dilatation-compression distortion...

Vertical disclination lines normal to horizontal layers in smectic C phases also form nuclei. Polarizing microscopy shows that these nuclei have four branches, and when examined in projection onto the layer plane the observed patterns correspond to + withi= l.Ithasbeendemonstrated that the tilt angle of molecules with respect to the normal to layers is variable, but decreases to zero in the vicinity of the disclination core [103]. This also resembles an escape in the third dimension , and is mainly due to the low value of the dilatation modulus B. 

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