Helfrichs theory

The basic mechanism for the electric-field-induced instabilities is now quite well understood. The current carriers in the nematic phase are ions whose mobility is greater along the preferred axis of the molecules than perpendicular to it. The ratio of the conductivities is usually about. Because of this anisotropy, space charge can be formed by ion segregation in the liquid crystal itself, as was first pointed out by Carr. The manner in which the space charge can build up due to a bend fluctuation is shown 

With a DC field, there may be injection of charge carriers at the solid-liquid interface but its role in the electrohydrodynamics of the nematic phase is not yet fully understood. However, as remarked earlier, a frequency of about 10 Hz is enough to suppress charge injection. We shall therefore neglect it in the present discussion. 

We shall now outline the theory of electrohydrodynamic instabilities proposed by Helfrich and extended by Dubois-Violette, de Gennes and Parodi and Smith et We consider a nematic film of thickness d lying in the xy plane and subjected to an electric field along z. Let the initial unperturbed orientation of the director be along x, and let there also be a stabilizing magnetic field along the same direction. We consider a bend 

Due to the anisotropy of conductivity, space charges will develop as indicated in fig. 3.10.7 till the transverse electric field stops the transverse current. The local transverse field in the steady state is easily seen to be 

The space charge per unit area produced on any plane normal to the x axis is 

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Some other experiments, however, indicated that the separation between membranes remained finite even in the presence of excess water.1,12 A possible explanation could be that the Helfrich theory, which involves in the calculation of the entropic confinement only a hard-wall (steric) repulsion, does not account for other forces between membranes. 

The Helfrich theory is valid for hard-wall interaction between membranes. However, at small separation additional interactions have to be taken into account. 

The mechanical properties of a Upid bilayer can be described by the Helfrich theory (Safran 2003), which treats the bilayer as a smooth, undulating surface. The free energy F of a lipid bilayer is given by... 

The period X does not follow from the Carr-Helfrich theory. Experimental observations show that X is nearly equal to the sample thickness d. Thus is approximately independent from the sample thickness the threshold is given by a voltage and not by a field strength. 

The Helfrich theory was extended by Dubois-Violette, de Gennes, and Parodi [17] to cover the AC field. The geometrical conditions are still the same. In the distorted state, the molecules are deflected by a small angle 0 in the xz plane. The most important parameter from the point of view of charge accumulation is not exactly 0, but rather the curvature v = 30/3x of the molecular pattern y/ and the charge density, q, are used as fundamental variables. 

The Carr-Helfrich theory and its extension to AC field by the Orsay group have been verified experimentally by several workers and they found good agreement with the theory [8-12, 16-28, 33, 34, 39-41]. Besides the above-mentioned theories, attempts have been made by other workers [41-43]. Some of these are beautifully reviewed by Goosen [8], Dubois-Violette et al. [9], Chandrasekhar [11], and Blinov [28]. 

Helfrich [49] was first to propose electro-hydrodynamic instability in cholesterics with negative dielectric anisotropy. Harault [56], combining Helfrich theory with time-dependent formalism, calculated a voltage frequency relationship similar to that observed for Williams domains. The existence of conduction and dielectric regimes was experimentally verified. The domain periodicity is proportional to wherepo... 

We plot in Fig. 9 f i(d) resulting from fits to the profile at the first harmonic as a function of d. The solid line, which agrees well with the experimental data, is a plot of the predicted value for 71 (d) (Eq.(18)) of the Helfrich theory. Here, we have taken the effective water thickness 5-29 A to include the known excluded volume effects [25] of the surfactant tails in the oil. This then provides compelling evidence that in this SDS multimembrane system swollen by dodecane the intermembrane interactions are dominated by the Helfrich mechanism of entropicallv driven undulation forces. 

Fig. 15. Power-law exponent i as a function of (1-S/d) for three dilution systems. The solid squares are for the DMPC-pentanol -water system while the open circles and squares are for the SDS-pentanol membranes along brine and oil dilution lines respectively. The solid line is the prediction of the Helfrich theory of undulation interactions.

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