Helfrich model

Taking into account flexoelectricity, it is possible to explain the appearance of a certain angle a, which the Kapustin-Williams domains form in some cases with the y-axis (the usual domain strips are parallel to the 2/-axis, Fig. 5.5). This oblique roll motion was observed in [90] and cannot be explained within the framework of the usual three-dimensional Carr-Helfrich model with strong anchoring at the boundaries [91]. The angle of the domain pattern a was shown [88, 89] to depend on the flexoelectric moduli eii, 633, the dielectric Ae, and the conductive Aa anisotropy. In certain intervals of the en — 633 and 611/633 values the angle A = 0 (the usual Kapustin-Williams domains) or A = 7t/2 (the longitudinal domains, also seen in experiment near the nematic-smectic A transition [91]). 

At high frequencies, with a reduction in temperature, the threshold field of the normal prechevron domains increases smoothly, not displaying any peculiarity when the temperature is T, where the anisotropy of the electrical conductivity disappears. Without doubt, the high-frequency electrohydrodynamic mode is caused by the isotropic mechanism of destabilization, since when cr /crx = 1, the Carr-Helfrich model does not hold. Analysis shows [123] that the new low-frequency mode (longitudinal domains) is also caused by the isotropic mechanism. 

The mathematical treatment of the Carr-Helfrich model leads to the following expression for the threshold voltage for Williams domains [64] ... 

The basic mechanism for electric field induced instabilities is now very well understood in terms of the Carr-Helfrich model based on field induced space charges due to conductivity and dielectric anisotropies [16, 31], Helfrich [16] made derivations for only DC fields, which were further extended to AC fields by Dubois-Violette and co-... 

Figure 31. Schematic diagram of the Helfrich model of permeation in a chiral nematic. The flow is along the helix axis (z) at low shear rates and maintains the z direction distribution (see text).

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. °... 

The interest in vesicles as models for cell biomembranes has led to much work on the interactions within and between lipid layers. The primary contributions to vesicle stability and curvature include those familiar to us already, the electrostatic interactions between charged head groups (Chapter V) and the van der Waals interaction between layers (Chapter VI). An additional force due to thermal fluctuations in membranes produces a steric repulsion between membranes known as the Helfrich or undulation interaction. This force has been quantified by Sackmann and co-workers using reflection interference contrast microscopy to monitor vesicles weakly adhering to a solid substrate [78]. Membrane fluctuation forces may influence the interactions between proteins embedded in them [79]. Finally, in balance with these forces, bending elasticity helps determine shape transitions [80], interactions between inclusions [81], aggregation of membrane junctions [82], and unbinding of pinched membranes [83]. Specific interactions between membrane embedded receptors add an additional complication to biomembrane behavior. These have been stud-... 

The simplest model of tubule formation based on chiral elastic properties was developed by Helfrich and Prost.180 They considered the elastic free energy of... 

The Helfrich-Prost model was extended in a pair of papers by Ou-Yang and Liu.181182 These authors draw an explicit analogy between tilted chiral lipid bilayers and cholesteric liquid crystals. The main significance of this analogy is that the two-dimensional membrane elastic constants of Eq. (5) can be interpreted in terms of the three-dimensional Frank constants of a liquid crystal. In particular, the kHp term that favors membrane twist in Eq. (5) corresponds to the term in the Frank free energy that favors a helical pitch in a cholesteric liquid crystal. Consistent with this analogy, the authors point out that the typical radius of lipid tubules and helical ribbons is similar to the typical pitch of cholesteric liquid crystals. In addition, they use the three-dimensional liquid crystal approach to derive the structure of helical ribbons in mathematical detail. Their results are consistent with the three conclusions from the Helfrich-Prost model outlined above. 

B. J. Kroesen, W. Helfrich, G. Molema, and L. de Leij, Bispecific antibodies for treatment of cancer in experimental animal models and man, Adv. Drug Delivery Rev. 31 105-129 (1998). 

Davis summarized the concepts about HLB, PIT, and Windsor s ternary phase diagrams for the case of microemulsions and reported topologically ordered models connected with the Helfrich membrane bending energy. Because the curvature of surfactant lamellas plays a major role in determining the patterns of phase behavior in microemulsions, it is important to reveal how the optimal microemulsion state is affected by the surface forces determining the curvature... 

Figure 3.5.3 Chiral liquid crystals without a chromophore may reflect colors because of light-scattering effects in helices of 400-700 nm pitches. Models of the cholesteric phase and the Schadt-Helfrich cell for liquid crystal displays are given. Two perpendicular polarization filters let light pass only if its direction of polarization has been rotated by the liquid crystals. If the hquid crystals are destroyed by an electric field, no light is transmitted, because the crossed polarizers quench it.

Electrically driven convection in nematic liquid crystals [6,7,16] represents an alternative system with particular features listed in the Introduction. At onset, EC represents typically a regular array of convection rolls associated with a spatially periodic modulation of the director and the space charge distribution. Depending on the experimental conditions, the nature of the roll patterns changes, which is particularly reflected in the wide range of possible wavelengths A found. In many cases A scales with the thickness d of the nematic layer, and therefore, it is convenient to introduce a dimensionless wavenumber as q = that will be used throughout the paper. Most of the patterns can be understood in terms of the Carr-Helfrich (CH) mechanism [17, 18] to be discussed below, from which the standard model (SM) has been derived... 

Fig. 4.5.4. Helfrich s model of permeation in a cholesteric liquid crystal. At low shear rates flow takes place along the helical axis without the helical structure itself...

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). 

Fig. 5.3.8. Helfrich s model of permeation in smectic liquid crystals. The flow takes place normal to the layers in a manner similar to the drift of charge carriers in the hopping model for electrical conduction.

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