Damping rate of the undulation mode

5 Damping rate of the undulation mode We write the energy density in a more general form to include volume dilatation d  

For static isothermal deformations, F may be minimized with respect to 0 to give 

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). 

As far as the highly damped undulation modes are concerned, the volume dilatation can justifiably be neglected and the isothermal approximation is probably satisfactory. The equation of motion then reduces to 

For small q, the last term may be neglected (except very near the boundary, but we shall ignore the boundary layer). This results in a purely damped mode whose relaxation rate is 

---

Fig. 5.3.9. Damping rate of the undulation mode in the smectic A phase of CBOOA determined by laser beat spectroscopy for two different sample thicknesses, 200 and 800 nm. Solid lines represent the theoretical curves calculated from (5.3.40) and (5.3.41). (After Ribotta, Salin and Durand. )...

---