Hookean bending elasticity

The following is a brief introduction to fluid membrane bending elasticity. The emphasis is on some basic ideas and not on particular models or applications. The theory of vesicle shapes is treated in Chapter 7. A subject to be included in the following is non-Hookean bending elasticity, i.e. energy terms of higher than... 

In order to define the curvatures of a membrane, it is convenient and often sufficient to think of a mathematical surface. In a purely formal way, a formula for the Hookean bending energy of such a surface is derived, after finding suitable expressions that are linear or quadratic in the principal curvatures. These preparations are followed by three-dimensional descriptions of monolayers and bilayers. Subsequently, stress profiles and thermal undulations are discussed in terms of Hookean bending elasticity. 

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 ... 

Viscoelastic behavior can be divided into five subclassifications (Figure 14.4). From 1 to 2, the material behaves as a viscous glass or Hookean elastic or glass, where chain segmental motion is quite restricted and involves mainly only bond bending and bond angle deformation. 

In general, the behavior of all classes of polymer behavior is Hookean before the yield point. The reversible recoverable elongation before the yield point, called the elastic range, is primarily the result of bending and stretching of covalent bonds in the polymer backbone. This useful portion of the stress-strain curve may also include some recoverable uncoiling of polymer chains. Irreversible slippage of polymer chains is the predominant mechanism after the yield point. 

It is important to note that the behavior of polymers below the yield point is Hookean and essentially reversible for short-term service. Thus this range, which is associated with stretching and bending of covalent bonds, is called the elastic range. The area under the stress-strain curve is a measure of toughness. 

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. 

Any dilation of the bilayer bending surface due to Gaussian curvature is proportional to C C2 and therefore does not affect Hookean elasticity. Substituting (z - z ) +Zj, for z in eq. (30) leads to... 

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