Comments on Theories of Smectics

The forms of the equation and solution given by (6.335) and (6.337) also arise in the theory of non-chiral SmC liquid crystals when an applied electric field is tilted with respect to the planes of the smectic layers, as discussed by Schiller et al. [246]. They are also relevant when both electric and magnetic fields are applied parallel to the smectic layers across a sample of SmC, as envisaged by van Saarloos, van Hecke and Holyst [236]. In both these circumstances the coefficients a and b must obviously be different due to the tilt of the field or the inclusion of an additional magnetic field details may be found in References [236, 246]. Some further elementary stability analysis has been performed by van Saarloos et al [236] and Stewart [262]. 

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]. 

A splay or bending distortion may create a polarisation in liquid crystals. This phenomenon is called the flexoelectric effect and was first studied theoretically by Meyer [197] in the context of nematic liquid crystals convenient summaries can be found in Chandrasekhar [38, pp.205-212] or de Gennes and Frost [110, pp.l35-139]. The flexoelectric effect for SmC liquid crystals has been investigated by Dahl and Lagerwall [62] and a brief development can be found in [110, pp.347-349]. Flexoelectric effects in smectics have also been discussed by Lagerwall [158]. 

It should also be mentioned that an earlier attempt at SmC d3mamic theory was made by Schiller [245], whose approach differs from that outlined here, and results in fewer viscosity coefficients. Among much more recent theoretical descriptions which include nonlinear hydrodynamics and smectic layer compression effects in SmC and SmC liquid crystals is the work by Pleiner and Brand [224] the reader is referred to their article and its references for further details. 

In this Appendix we establish some relations for constraints which are required in Section 2.4.2. We can employ the result in (2.125) with / given by / (x,p(x)) and Ui replaced by p(x) to find that 

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