Dynamic Theory of Smectic

The above analysis assumes that the undulations of the layers vanish at the boundaries. Ishikawa and Lavrentovich [131] have modelled a lamellar system of cholesteric liquid crystal which allows layer undulations near the boundaries. This was achieved by adding a finite surface anchoring energy to a bulk energy that is essentially of the same form as that stated above for SmA. This idea could perhaps also be modified to model general SmA or SmC liquid crystals. The results in [131] seem to indicate that the incorporation of finite surface anchoring leads to a decrease in the theoretical threshold for the onset of the Helfrich-Hurault transition. 

Some comments on the dynamics of undulations, not considered here, are mentioned on page 319. It should also be emphasised that the numerical thresholds given above are speculative and depend on the values of physical parameters, many of which are currently unknown for most materials. Nevertheless, these results demonstrate the methodology which is used by numerous authors and are in the spirit and general flavour of many of the articles in the literature. 

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The constitutive hydrodynamic equations for uniaxial nematic calamitic and nematic discotic liquid crystals are identical. In comparison to nematic phases the hydro-dynamic theory of smectic phases and its experimental verification is by far less elaborated. Martin et al. [17] have developed a hydrodynamic theory (MPP theory) covering all smectic phases but only for small deformations of the director and the smectic layers, respectively. The theories of Schiller [18] and Leslie et al. [19, 20] for SmC-phases are direct continuations of the theory of Leslie and Ericksen for nematic phases. The Leslie theory is still valid in the case of deformations of the smectic layers and the director alignment whereas the theory of Schiller assumes undeformed layers. The discussion of smectic phases will be restricted to some flow phenomena observed in SmA, SmC, and SmC phases. 

I.W. Stewart and R.J. Atkin, Derivation of a dynamic theory for smectic C liquid crystals, to appear. 

The results and applications in Chapters 2 to 5 for nematic liquid crystals are given in fairly full mathematical detail. It has been my experience that the stumbling block for many people comes at the first attempts at the actual calculations here I will reveal many details and more explanation than is usually given in articles and common texts, in the hope that readers will gain confidence in how to apply the main results from continuum theory to practical problems. These Chapters contain extensive derivations of the static and dynamic nematic theory and applications. Chapter 6, on the other hand, does not give as many detailed computations as those presented in the earlier Chapters it is my intention that it introduces the reader to a continuum theory of smectic C liquid crystals and it is probably written more in the style of an introductory review. This is partly because some of the calculations are similar for both nematic and smectic C materials, but with different physical parameters and some different physical interpretations. However, despite some of these similarities, smectic liquid crystals have some uniquely different mathematical problems, and these can only be touched upon within the remit of a book such as this. 

This paper presents summaries of unique new static and dynamic theories for backbone liquid crystalline polymers (LCPs), side-chain LCPs, and combined LCPs [including the first super-strong (SS) LCPs] in multiple smectic-A (SA) LC phases, the nematic (N) phase, and the isotropic (I) liquid phase. These theories are used to predict and explain new results ... 

In smectic C materials, the relative twist of the planes is uncoupled to the layer thickness, giving rise to strong, nematic-like scattering [106]. There are relatively few light scattering studies of either achiral or chiral SmC phases, despite the technological importance of the ferroelectric SmC phase. This is in part due to the few discussions of the elastic theories of these phases, and in particular descriptions that include dynamic behaviour. Indeed it is only relatively recently that Leslie et al. [107] de-... 

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]. 

Figure 19.3. The time evolution of the morphology of an A3B7 surfactant melt, as predicted by dissipative particle dynamics [84], Mean field theory predicts the correct equilibrium hexagonal phase but does not provide any insights into how this equilibrium morphology is reached. The simulations showed that the dynamics of ordering is determined by the percolation of tubes, subsequently destabilized by a nematic or smectic phase transition and also that hydrodynamic effects are important in reaching equilibrium for this system. See the insert showing the colored figures for a better view.

Chapter 6 contains the most advanced mathematics in the book, and introduces a particular nonlinear static and dynamic continuum model for smectic C liquid crystals. This continuum theory is a natural extension of nematic theory, originating from ideas that are familiar from the continuum description of nematics. 

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