Bending elastic energy

Tilt varies throughout the cell thickness In twisted cells due to a competition between the splay and bend elastic energies and the twist elastic energy. For small tilts and twists, 90 , this variation Is calculated to be small. Doping the liquid crystal with a cholesteric compound, as Is commonly done In TN-LCDs, stabilizes the twisted structure, thereby decreasing tilt variation. 

When the tilt angle is small, the bend elastic energy dominates, and the cell geometry is called bend geometry. The electric energy is negative and is approximately given by... 

Besides the four droplet configurations discussed above, other droplet configurations could exist under appropriate conditions. For example, there is a twisted bipolar droplet where the splay and bend elastic energy is reduced by introducing twist deformation. The point defect in the radial droplet could escape away from the center in an effort to reduce the total free energy. 

Two different bending elastic moduli exist A f, when the exchange of lipid molecules between the monolayers of the bilayer is free, and when it is blocked. When the exchange is forbidden, the number of the molecules in each monolayer of the bilayer is constant. At free flip-flop, the bending elasticity energy is lower because it has been minimized with respect to the difference between the number of molecules in each monolayer and, consequently, k < kf. For all phenomena related to the out-of-plane fluctuations of membranes, the relevant quantity is k [6-8]. These phenomena include the thermal fluctuations of quasispherical vesicles [9,10], as well as vesicle suction in micropipettes at very low suction pressures [1]. 

The theoretical approach to describe all these effects was formulated long ago [7], at the time purely on the basis of Hookean deformations that eontribute to the bending elastic energy density of fluid membranes ... 

In the molten state, molecules may difluse within their monolayer therein the membrane resembles a two-dimensional liquid. Fluid membranes are soft and therefore easily deformable, whereas solid membranes are stiff and rather break on tearing than to adjust into a new shape. Nevertheless bending of fluid bilayers requires energy. The final equilibrium shape of an object that is confined by a fluid membrane is given by the minima of the bending elastic energy [8,9]. 

Liposome is a closed vesicle with the lipid-bilayer membrane, which takes a variety of shapes such as biconcave discocytes, cup-shaped stomatocytes and prolate and oblate ellipsoids, depending on the temperature and the osmotic conditions [1]. Among various attempts made to explain these shapes [2,3], Helfrich has discussed the bending elastic energy [4] of fluid membranes formed by lipids as... 

The twist of these aggregates determines the resulting structure. When ribbons stack into fibrils, and fibrils into fibres, packing considerations cause these twisted aggregates to bend and modify their twist with an elastic energy penalty8eiast- For fibrils, it can be shown that the afibrii per peptide in a fibril is ... 

Example 12.9. We consider a film with zero spontaneous curvature (6 0 = 0). What is the elastic energy for bending such a film to a sphere of radius R1 With C = 1/R and C2 = 1/R we obtain... 

Elastic energy of a surfactant film. Please estimate the bending energy per unit area for a surfactant film with a bending rigidity of 10kbT and zero spontaneous curvature, which is at the interface of a drop of radius 5,20, and 100 nm. 

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