The Elastic Energy for Smectic

This energy density can also be written in an equivalent Cartesian component form when surface terms which can be written as a divergence are neglected (cf. the comments after equation (2.50) on page 21). This form is often useful in calculations and can be expressed as [172] 

Notice that the constraint (6.4) is equivalent to the condition aij = Uj i in Cartesian component form. Other equivalent forms for the energy in terms of any two of the vectors a, b and c are available [172] and these allow comparisons with earlier results obtained by other workers, particularly the Orsay Group [213], Rapini [228], Dahl and Lager wall [62] and Nakagawa [209]. It is worth remarking here that three surface terms have been identified for the SmC phase, namely [172], 

The equality in (6.19) arises in Section 6.4.1 below and its derivation requires de- 

The above elastic constants are those used by the Orsay Group, except that for later notational convenience we have set and Ci = -Cf 

As we shall see below, the constants A12, A21 and An are related to bending of the smectic layers (cf. de Gennes and Frost [110, p.346]) while the constants Bi, B2, B3 and Bi3, originally introduced by Saupe [239] in an earlier description of smectics, are related to the reorientation of the c-director within or across layers. The constants Ci and C2 are related to various couplings of these deformations. The formulation of the energy in (6.24) is not entirely convenient for calculations and is relevant only for small distortions, which is why alternative nonlinear vector formulations have been sought. 

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