Helfrich mechanism

Table 1. Eight different combinations (labelled A to H) of initial director alignments and the sign of anisotropies a, ffa- The EC pattern species are characterized in the last column CH stands for patterns, which are compatible with the Carr-Helfrich mechanism, in contrast to the remaining, nonstandard ones (ns-EC).

A distinguishing feature of this model is the absence of a positive feedback between the magnitude of the space charge and the orientation of the director, which is required for the Carr-Helfrich mechanism. These physical parameters are independent of each other here. 

Thus, in this model ion accumulation effects are included wdiereas ionic diffusion is neglected as in the SM. The charge accumulation counteracts the standard (Helfrich) mechanism of generation of space charges. If r c is sufficiently slow one can find an oscillatory behaviour of the system at threshold, i.e. a Hopf bifurcation (Section 13.5). 

This value is significantly lower than that measured for most lipidic membrane systems and is precisely why fluctuations on short wavelength < d and long wavelength d scales become important for these dilute membranes. Therefore, since the Helfrich mechanism scales as kc i we expect this interaction Fu(Eq. (10)), which ranges between 0.12kgT/d2 and 0.46 kgT/d, to dominate the van der Waals interaction FvdW = -0.03kBT/d2 (Eq. (11)). 

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. 

The link from lipid properties to mechanical properties of the bilayers is now feasible within the SCF approach. The next step is to understand the phase behaviour of the lipid systems. It is likely that large-scale (3D) SCF-type calculations are needed to prove the conjectures in the field that particular values of the Helfrich parameters are needed for processes like vesicle fusion, etc. In this context, it may also be extremely interesting to see what happens with the mechanical parameters when the system is molecularly complex (i.e. when the system contains many different types of molecules). Then there will be some hope that novel and deep insights may be obtained into the very basic questions behind nature s choice for the enormous molecular complexity in membrane systems. 

Although the dominant means to create a space-charge field within the interference pattern in liquid crystals is given by Eq. (2), it has been shown that there are other mechanisms to create a space-charge field. One is derived from the conductivity anisotropy and is known as the Carr-Helfrich effect [43,49] ... 

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... 

Helfrich proposed a simple physical mechanism, which he called... 

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

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 mentioned above, the first expressions for the flexoelectric coefficients were obtained by Helfrich and Petrov and Derzhanski while a systematic molecular-statistical theory was developed later by Straley. The results of these two approaches were compared by Marcerou and Prost who concluded that the theories of Helfrich and Petrov and Derzhanski and of Straley describe different mechanisms for the dipolar flexoelectric effect because Straley s theory 5nelds values for the flexocoefficients that are two orders of magnitude smaller than the experimental ones, and which therefore can be neglected. 

It is clear that the two polar vectors respect the apolar nature of n. There is also an obvious analogy of the above mechanism with the orientation polarization of a liquid dielectric, which was used by Helfrich to relate the two flexocoefficients with molecular properties. The intrinsic splay or bend can be related to an appropriate angle and molecular dimensions. The relevant component of the electric dipole moment and the curvature elastic constant, viz., and the splay constant Ki or /ux and the bend constant It s figure in the estimation of the flexocoefficients. Nematic liquid crystals made of banana-shaped molecules have been studied only recently, and a comparison of the experimental measurements with the Helfrich formula leads to interesting inferences, as will be mentioned later in this chapter, and covered more thoroughly in the companion Chapter 3 by Jakli et al. ... 

The dipolar mechanism is sensitive to the molecular shape. By dimensional considerations one can estimate the flexocoefficients due to dipolar mechanism as ei, 63 < /Xe/a, where /Xg 1-5 debye (1 D = 3.3x 10 Cm) is the molecular dipole moment and a 2-4 nm is the typical molecular dimension for a low molecular weight liquid crystal. This means that e and 63 are expected to be of the order of pCm. Assuming a random three-dimensional distribution of the centre of masses of the constituent bent-core (banana-shaped) molecules, Helfrich and Derzhanski and Petrov derived a more precise expression for the macroscopically testable bend fiexo-electric coefficient ... 

The physical mechanism of the instability is related to several coupled phenomena discussed by Helfrich [42]. His elegant calculation of the instability threshold is reproduced here for the simplest steady state one-dimensional model shown in Fig. 11.33a. A homogeneously aligned nematic liquid crystal layer of thickness d is stabilised by the rubbed surfaces of the limiting glasses. The dielectric torque is considered negligible (s = 0). At first, a small director fluctuation 9(x) with a period dh postulated ... 

The Helfrich Hamiltonian, Eq. (1), does not include a surface tension contribution. Free membrane patches can relax and adjust their area such that they are stress-free. In many situations, however, membranes do experience mechanical stress. For example, an osmotic pressure difference between the inside and the outside of a lipid vesicle generates stress in the vesicle membrane. Stress also occurs in supported bilayer systems, or in model membranes patched to a frame. In contrast to other quantities discussed earlier (bending stiffness etc.), and also in contrast to the surface tension of demixed fluid phases, membrane stress is not a material parameter. Rather, it is akin to a (mechanical or thermodynamic) control parameter, which can be imposed through boundary conditions. 

The first tension-like quantity in planar membranes is the lateral mechanical stress in the membrane, as discussed above. If the stress is imposed by a boundary condition, such as, for instance, a craistraint on the lateral (projected) area of the membrane, it is an internal property of the membrane system that depends, among other parameters, on the area compressibility [36] and the curvature elasticity [154—161]. Alternatively, mechanical stress can be imposed externally. In that case, the projected area fluctuates, and the appropriate thermodynamic potential can be introduced into the Helfrich Hamiltonian, Eq. (1), in a straightforward manner ... 

Although not seen experimentally, the Helfrich-Hurault instability [1,30,31,79,80] induced in smectic A liquid crystals by an electric field or uniaxial extension is expected to occur in block copolymers as well (see Fig. 13). In the case of mechanical extension, above a critical elongational strain of... 

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