Helfrich deformation

Fig. 5.3.4. The Helfrich deformation in a smectic A film subjected to a mechanical...

Fig. 5.3.5. Dependence of the wavevector of the Helfrich deformation in the smectic A phase of CBOOA on the sample thickness d. The slope yields a value of A = 22 3 A. (After Durand. )...

Figure 10.26 (a) Microphotograph of HeUrich deformation, (b) Schematic diagram showing the strucmre of Helfrich deformation in a plane perpendicular to the cell surface. 

In the Helfrich deformation shown in Figure 10.26, the cell thickness of the cell is h. The cholesteric liquid crystal has the pitch P and dielectric anisotropy A (> 0). For small undulation, calculate the field threshold Eueijnch and the wavelength A of the undulation. 

W. Helfrich, Deformation of Cholesteric Liquid Crystals with Low Threshold Voltage, Appl. Rhys. Lett., 17, p. 531 (1970). 

Kummrov, M., and Helfrich, W. (1991). Deformation of giant lipid vesicles by electric fields... 

The membrane bending energy in Eq. (2) is the sum of local elastic energies associated with deformations of individual membrane leaflets away from their spontaneous curvatures, as described by the Helfrich free energy ... 

Vesicles exposed to electric fields deform. The response of GUVs to electric fields has been the subject of extensive i rives tiga tion. When exposed to AC electric fields, as a stationary state they attain elhpsoidal shapes-prolate or oblate-depending on the field frequency and media conduchvity [42, 64]. Initiated by tlie seminal work of Winterhalter and Helfrich ]65], this effect has been considered theoretically [66-74] and experimentally ]42, 64, 67, 75-77], whereby intereshng dynamics and flows in the membrane and in the surrounding medium were observed... 

The undulation force arises from the configurational confinement related to the bending mode of deformation of two fluid bilayers. This mode consists in undulation of the bilayer at constant bilayer area and thickness (Figure 5.30a). Helfrich et al. doi established that two such bilayers, apart at a mean distance h, experience a repulsive disjoining pressure given by the expression ... 

Fig. 4.6.3. Deformation of a planar structure due to a magnetic field acting along the helical axis of cholesteric liquid crystal composed of molecules of positive diamagnetic anisotropy. A similar deformation superposed in an orthogonal direction results in the square-grid pattern (see fig. 4.6.4). (Helfrich. )...

Helfrich, W., Deformation of cholesteric liquid crystals with low threshold voltage, Appl. Phy.s. Lett., 17, 531-532 (1970). 

This problem can be considered in the framework of the Helfrich approach to the nematic, though we have to take into account the specific viscoelastic properties of smectics and a proper sign of the conductivity anisotropy. First of all, it makes sense to consider only the onset of a splay deformation in a homeotropic structure for smectic A, since => 00. This approach is developed in [121], where the following expression for the threshold field of an instability is derived ... 

The stability (minimum) condition with respect to a deformation mode as described by a spherical harmonics Tim has been derived by Ou-Yang and Helfrich [29]. It has the form ... 

Since the mid-1970s, much work has been done to determine the role of these mechanical properties on the overall morphological behavior of cell membranes. Canham [1], Helfrich [2] and Evans [3] have identified the different characteristics responsible for the spontaneous shape of a membrane or its resistance to deformation. Using phenomenological parameters, they gave descriptions of the membrane deformation valid on a scale where membranes can be considered as a continuum material (i.e. on a scale larger than the membrane thickness). Following this idea... 

A wide variety of shape transformations of fluid membranes has been extensively studied theoretically in the past two decades using a bending elasticity model proposed by Canham and Helfrich [1]. The model has succeeded in explaining equilibrium shapes of the erythrocyte. However, much attention has recently been paid to shape deformations induced by internal degrees of freedom of membranes. For example, the bending elasticity model cannot explain the deformation from the biconcave shape of the erythrocyte to the crenated one (echinocytosis) [2, 3]. It is pointed out [3] that a local asymmetry in the composition between two halves of the bilayer plays an important role in the crenated shape. It has been observed [4] that a lateral phase separation occurs on an artificial two-component membrane where domains prefer local curvatures depending on the composition. In order to study the shape deformation accompanied by the intramembrane phase separation, we consider a two-component membrane as the simplest case of real biomembranes composed of several kinds of amphiphiles. 

A liquid crystal (LC) in which the electric dipoles point in the same direction as the respective LC directors should exhibit not only a nonuniform strain but also a piezoelectric response when it undergoes one or more of the three nonuniform deformation modes that are identified as splay, bend, and twist. Accordingly, three different modes of piezoelectricity from nonuniform strain distributions were postulated for liquid crystals (Meyer 1969), but it was not clear whether the resulting piezoelectric effects were large enough to be observed in real experiments (Helfrich 1971). In the meantime, since the early concepts, a whole new field - flexoelectricity in liquid crystals (Buka and Eber 2013) - has developed from the pioneering work of Meyer and Helfrich on splay and bend deformation in liquid crystals. 

Hebbian learning, of presynaptic and postsynaptic activation, 12-3 Heetderks, W.J., 34-3 Helfrich, W., 62-4 Helicotrema, 5-4 Hellebrandt, F.A., 76-20 Heller, L, 37-7 Helmholtz, H.L.F., 27-4 Helmholtz, H., 4-7-4-S Hematocytes, 1 -2 constituents, 1-2 fundamentals, 60-2 mechanics and deformability, 60-1-60-11 red cells, 60-3-60-7 stresses and strains in, 60-2 Hematocytopoiesis, 1-2 Hemidecussation, 4-3 Hemmert, W 63-9 Henson cells, in cochlear... 

In general, B and (as well as Ci and Ca) are curvature dependent. To determine the latter dependence, one has to specify the flexural rheology of the interface. A frequently employed model of Helfrich [202] assumes that the work of flexural deformation can be written in the form... 

In calculating the threshold voltage, Hel-frich assumed that the spatial periodicity of the fluid deformation was proportional to the thickness of the cell. Penz and Ford [19-21] solved the boundary value problem associated with the electrohydrodynamic flow process. They reproduced Helfrich s results and showed several other possible solutions that may account for the higher order instabilities causing turbulent fluid flow. 

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