Bending energy density

The bending energy density related to a fold is thus given by... 

The invariants are needed to express the bending energy of membranes. This is because the bending energy density of a fluid, laterally isotropic membrane can consist only of terms that do not depend on the orientation of the xy cross. Multiplying each of them with a coefficient leads to one of the standard formulas for the bending energy surface density. 

Both formulas for the bending energy density are regarded as expansions up to quadratic order about the. o/ state of the membrane. This elasticity is considered to be Hookean, despite the possible presence of a linear term. Only for purely cylindrical curvature (e.g. 62 s 0) would the first term of eq. (7) suffice to describe an expansion about a spontaneously curved state c, = cq. However, both k and co could depend on the origin of the expansion. 

However, the neutral surface cannot be the surface of equal molecular density if the membrane curvature is nonuniform [6]. This is because in mechanical equilibrium the monolayer lateral tension contains the bending energy density extra... 

Only the even orders can occur in the bending energy density of a symmetric bilayer. The second order is therefore followed by the fourth. A complete set of quartic invariants under rotation of the xy cross of local Cartesian coordinates has been given in partial tensor notation by Mitov [13]. They are... 

The same primitive model using no more than two of the five relevant fourth-order invariants listed above seems to be sufficient to understand both graininess and knotted sticks. Its bilayer bending energy density is expressed by... 

Squaring Eqs. (AlO-2) and (AlO-3) and dotting Eq. (AlO-4) into itself give the splay, twist, and bend free energy densities, which, using Eq. (10-7), add up to... 

When the layers of a lamellar block copolymer are distorted, the free energy density is augmented by a distortional term that can, like the smectic-A phase, be described as the sum of layer compression/dilation and layer-bending energies ... 

The second energy contribution stems from the fact that a wall with a finite thickness will resist bending because bending implies compression of the wall on one side and extension on the other. The energy density of this bending is... 

Since the flexoelectric effect is associated with curvature distortions of the director field it seems natural to expect that the splay and bend elastic constants themselves may have contributions from flexoelectricity. The shape polarity of the molecules invoked by Meyer will have a direct mechanical influence independently of flexoelectricity and can be expected to lower the relevant elastic constants.The flexoelectric polarization will generate an electrostatic self-energy and hence make an independent contribution to the elastic constants. In the absence of any external field, the electric displacement D = 0 and the flexoelectric polarization generates an internal field E = —P/eo, where eq is the vacuum dielectric constant. Considering only a director deformation confined to a plane, and described by a polar angle 9 z), and in the absence of ionic screening, the energy density due to a splay-bend deformation reads as ... 

Why does the free energy density acquire this particular form First, in the curvature term with modulus Ku, we must use the second derivatives because the first derivatives correspond to a pure rotation of all the layers that does not cost energy. The higher derivatives are ignored for small distortions. For the compressibility term, the first derivative (du/dz) is sufficient. Second, both the compressibility and the curvature terms must be squared due to head-to-tail symmetry and parabolic form of the density increment gdisrgo as a function of distortion (Hooke s law). However, the question arises why is only splay modulus taken into account in (8.44) and not the other two Frank moduli K22 and 33. Considering the splay and bend distortions of the SmA phase in Fig. 8.24 we can see that only the splay distortion is allowed because it leaves the interlayer distance and the... 

For = 0> the free energy density in the bulk includes only the flexoelectric and the elastic (bend) terms ... 

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