Undulation mode smectic

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. )...

This surprising result prompted Mazenko, Ramaswamy and Toner to examine the anharmonic fluctuation effects in the hydrodynamics of smectics. We have already shown that the undulation modes are purely dissipative with a relaxation rate given by (5.3.39). To calculate the effect of these slow, thermally excited modes on the viscosities, we recall that a distortion u results in a force normal to the layers given by (5.3.32). This is the divergence of a stress, which, from (5.3.53), contains the non-linear term 0,(Vj uf. Thus, there is a non-linear contribution (Vj uf to the stress. Now the viscosity at frequency co is the Fourier transform of a stress autocorrelation function, so that At (co), the contribution of the undulations to the viscosity, can be evaluated. It was shown by Mazenko et that Atj(co) 1 /co. In other words, the damping of first and second sounds in smectics, which should go as >/(oo)oo , will now vary linearly as co at low frequencies. 

It is important to note that the lamellar phase is thus stabilized by the balance of a negative interfacial tension (of the free oil/water interface covered by an amphiphilic monolayer), which tends to increase the internal area, and a repulsive interaction between interfaces. The result, Eq. (48), indicates that the scattering intensity in a lamellar phase, with wave vector q parallel to the membranes, should have a peak at nonzero q for d > d due to the negative coefficient of the q term in the spectrum of Eq. (40). just as in the microemulsion phase. This effect should be very small for strongly swollen lamellar phases (in coexistence with excess oil and excess water), as both very small [96]. Very similar behavior has been observed in smectic liquid crystals (Helfrich-Hurault effect) [122]. Experimentally, the lamellar phase under an external tension can be studied with the surface-force apparatus [123,124] simultaneous scattering experiments have to be performed to detect the undulation modes. 

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