Theories Helfrich model

In the experimentally typical cases R -500 pm and P 1 pm, which gives the experimentally observed high viscosity at low pressures. Later it was shown that the essential features of the Helfrich model can be explained also on the basis of the Ericksen-Leslie theory. °... 

We shall now show that the essential features of Helfrich s model can be derived on the basis of the Ericksen-Leslie theory. ... 

The force g normal to the layers will be associated with permeation effects. The idea of permeation was put forward originally by Helfrich to explain the very high viscosity coefficients of cholesteric and smectic liquid crystals at low shear rates (see figs. 4.5.1 and 5.3.7). In cholesterics, permeation falls conceptually within the framework of the Ericksen-Leslie theory > (see 4.5.1), but in the case of smectics, it invokes an entirely new mechanism reminiscent of the drift of charge carriers in the hopping model for electrical conduction (fig. 5.3.8). 

The first three terms on the right follow from Helfrich s free energy expansion in the curvature [13] as discussed in Sec. Ill of this chapter and are identical to the right-hand side of Eq. (4). The last term quantifies the finite size effect as mentioned at the end of Sec. V. A. It was introduced by Fisher [35] in his treatment of condensation and is widely used in phenomenological theories of nucleation (see, e.g.. Refs. 36 and 37). r has an estimated value on the order of 1. We note that the calculation of z from a model is far from trivial see, for example. Ref 38 for a discussion of the relatively simple case of on average flat interfaces. 

A theoretical investigation of the stability of nematic liquid crystals with homeotropic orientation requires a three-dimensional approach. Helfrich s one-dimensional theory predicts the dependence of the threshold of the instability on the magnitude of Ae, as shown by curve 2 in Fig. 5.8, according to which the electrohydrodynamic instability should be observed when either Ae < 0 (and consequently the bend Frederiks effect reorientation will not take place), or when small Ae > 0. In Helfirich s model the destabilizing torque as dvzjdx is responsible for this instability, which replaces the destabilizing torque a dvzjdx in the equation for the director rotations (5.27). Although the torque is small ( a3 -C o 2 ) it is not compensated for (e.g., when Ae = 0) by anything else apart from the elastic torque. 

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]. 

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