Curvature elastic energy

Under this assumption, the bending energy Eg can be represented in terms of the membrane s curvature. For this reason. Eg is also referred to as the curvature elastic energy. The curvature of smooth surfaces is characterized by two functions that depend on the local canonical curvatures, h t) and h lr), in a surface element dS centered at r. These functions are the mean curvature, H = ( 1 + hi) , and the Gaussian curvature, K = hih2- In general, H and K change with the point r. 

Siegel DP (2008) The Gaussian curvature elastic energy of intermediates in membrane fusion. Biophys J 95(11) 5200-5215... 

If the tension-free membrane is bent, its shape can be characterized locally by its principal curvatures, C and C2- Then the density of the curvature elastic energy g is given by [5]... 

The first term represents the curvature elastic energy of the membrane with the bending rigidity k. c, C2i are two principal curvatures at a joint i written as... 

The competition between the polar and steric dipoles of molecules may also lead to internal frustration. In this case, the local energetically ideal configuration cannot be extended to the whole space, but tends to be accomodated by the appearance of a periodic array of defects. For example, the presence of the strong steric dipole at the head of a molecule forming bilayers will induce local curvature. As the size of the curved areas increases, an increase in the corresponding elastic energy makes energetically preferable the... 

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. 

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 ... 

Electric or magnetic fields acting on the anisotropy of the electric or magnetic susceptibility exert torques within a liquid crystal which may compete with the elastic torques determining its internal structure (55). Equations w ich describe the liquid crystalline structure can be derived from molecularly uniaxial liquid crystals on the basis of the curvature-elasticity theory (54). In doing so, tl structure is determined so as to minimize the total free energy of the system, and this method is applied to the cholesteric structure (55, 55). 

Vn is the normal velocity of the surface caused by the atoms redistribution and ps is the surface density of atoms. The surface chemical potential is typically determined by the film elastic energy and surface energy, and as such it is a function of the him local thickness as well as its slope, curvature, and may be higher spahal derivahves (see below). For very thin hlms (a few atomic layers) wetting interactions between the him and the substrate can also become important. These interachons are somewhat similar to wetting interactions between a liquid him and a solid substrate. They are responsible for the presence of an ultra-thin wetting layer of the him material between the islands resulting from the him instabihty and depend on the him thickness and its slope. Naturally, this dependence decays rapidly with the increase of the him thickness. 

At the molecular level, it is obvious that curvature elasticity is a consequence of the orientational order in the liquid crystal. A quantitative relationship between them was established by Saupe using the mean field theory. From (2.3.6) is it seen that the dipole-dipole part of the dispersion energy of interaction of a molecule i in the average field due to its neighbours k is given by... 

The stratified structure of a smectic liquid crystal imposes certain restrictions on the types of deformation that can take place in it. A compression of the layers requires considerable energy - very much more than for a curvature elastic distortion in a nematic - and therefore only those deformations are easily possible that tend to preserve the interlayer spacing. Consider the smectic A structure in which each layer is, in effect, a two-dimensional fluid with the director n normal to its surface. Assuming the layers to be incompressible, the integral... 

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