Static Theory of Smectic

in the absence of any singularities or defects in the smectic layers the unit layer normal a is additionally subject to the constraint 

The static (and dynamic) theory of SmC will be developed using the vectors a and c which are subject to the constraints contained in equations (6.3) and (6.4). These constraints will lead to four Lagrange multipliers in the theory three scalar function multipliers arising from the three constraints in (6.3) and one vector function multiplier arising from the vector constraint in (6.4). Knowledge of the behaviour of a and c is sufficient to derive the orientation of the usual director n through the relation (6.1). 

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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. 

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]. 

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 ... 

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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