Helfrich free energy

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 ... 

Seifert and Lipowsky [47,48] were the first to apply the Helfrich free energy for the description of the shape and free energy of such systems adhered to a solid substrate (see Fig. 19). In their analysis they calculated the phase diagram for the unbinding of a vesicle adsorbed to a substrate. In this section, we discuss the analysis by Seifert and Lipowsky [47]. First, we introduce the expression for the curvature free energy by Helfrich and derive the shape equations that minimize the Helfrich free energy for a vesicle in contact with a solid substrate. Since these shape equations cannot, in general, be solved... 

The Helfrich free energy [45] describes the surface free energy associated with bending in terms of the radius of spontaneous curvature, the rigidity constant associated with bending, k, and the rigidity constant associated with Gaussian curvature, k. 

The total free energy is the sum of the Helfrich free energy Fu and substrate interaction energy F ... 

The simplest model of tubule formation based on chiral elastic properties was developed by Helfrich and Prost.180 They considered the elastic free energy of... 

To find the optimum tubule structure, Helfrich and Prost minimized the free energy over the radius r and the tilt direction . This minimization... 

The Helfrich-Prost model was extended in a pair of papers by Ou-Yang and Liu.181182 These authors draw an explicit analogy between tilted chiral lipid bilayers and cholesteric liquid crystals. The main significance of this analogy is that the two-dimensional membrane elastic constants of Eq. (5) can be interpreted in terms of the three-dimensional Frank constants of a liquid crystal. In particular, the kHp term that favors membrane twist in Eq. (5) corresponds to the term in the Frank free energy that favors a helical pitch in a cholesteric liquid crystal. Consistent with this analogy, the authors point out that the typical radius of lipid tubules and helical ribbons is similar to the typical pitch of cholesteric liquid crystals. In addition, they use the three-dimensional liquid crystal approach to derive the structure of helical ribbons in mathematical detail. Their results are consistent with the three conclusions from the Helfrich-Prost model outlined above. 

Following Helfrich we can express the Gibbs free energy of curvature by an integral over the area considered [557] ... 

In the case of a linear interaction between neighboring lipid bilayers, Helfrich has demonstrated that the repulsive free energy due to confinement is inversely proportional to (7b2. While this result is strictly valid for a harmonic interaction potential (linear force), we assume that it can be extended to any interaction. We will examine later under what conditions this approximation is accurate. 

Following Helfrich,8,18 one can compute the free energy of confinement, starting from the Hamiltonian of the confined bilayer... 

The entropic term of the free energy (pci unit area) due to the confinement is obtained by subtracting the interaction energy per unit area from Eq. (B.10), a result which is essentially due to Helfrich [4] ... 

The type of structure observed is closely related to the spontaneous curvature Co of the surfactant assemblies [7]. By using an analogy with liquid crystals, which can also adopt layered structures, Helfrich [8] introduced the concept of the elastic-free energy associated with thermally excited deviations from the spontaneous curvature of the microstructures. This elastic-free energy per unit area is given by... 

Now, the modulus B is related to the second derivative of the free energy with respect to the average layer spacing i.e., imagine a uniform expansion or compression of the system. TTie restoring force is just the effective value of B which is proportional to the macroscopic compressibility of the system. Thus, following Helfrich we can obtain a self-consistent equation to determine B from... 

Derive an expression for the curvature elastic free energy of a bilayer in terms of the curvature elastic constants of the monolayer. Treat the case where the two monolayers are equivalent and noninterpenetrating, so that one adds their curvature energies, but note that the monolayers each have a finite thickness, which makes their curvatures inequivalent. Compare with the Helfrich form and comment on the effective saddle curvature as a function of the spontaneous curvature of each monolayer. 

From the above reasoning we expect that each composition of the interface has its own curvature at which the interface forms most easily and thus has the lowest interfacial tension (this interfacial tension, a, of the droplet interface should not be confused with that of the macroscopic interface, y). This consideration was made more quantitative by Helfrich [13], who presented an expression for the curvature free energy. 

The first three terms on the right follow from Helfrich s free energy expansion in the curvature [13] as discussed in Sec. Ill of this chapter and are identical to the right-hand side of Eq. (4). The last term quantifies the finite size effect as mentioned at the end of Sec. V. A. It was introduced by Fisher [35] in his treatment of condensation and is widely used in phenomenological theories of nucleation (see, e.g.. Refs. 36 and 37). r has an estimated value on the order of 1. We note that the calculation of z from a model is far from trivial see, for example. Ref 38 for a discussion of the relatively simple case of on average flat interfaces. 

If we intend to calculate precisely the threshold field for the two-dimensional distortion we should write the Frank free energy with the director compcments (12.34) and the field term (Ea/47t)(En) and then make minimization of the free energy with respect to the two variables cp and 9 [18]. For a qualitative estimation of the threshold we prefer to follow the simple arguments by Helfrich [17]. We consider a one-dimensional (in layer plane xy) periodic distortion of a cholesteric... 

Here, the average values of = = 1/2 are used. As the cell thickness is assumed to be large, d w Helfrich discarded the splay term and the elastic free energy density is reduced to the form... 

The free energy per unit area arising from the bending of the interfacial mono-layer was introduced by Helfrich [61] and applied to amphiphilic mixtures by various authors [15,19,23,30] ... 

The mechanical properties of a Upid bilayer can be described by the Helfrich theory (Safran 2003), which treats the bilayer as a smooth, undulating surface. The free energy F of a lipid bilayer is given by... 

Thermodynamics of Curved Interfaces in Relation to the Helfrich Curvature Free Energy Approach Jan Christer Eriksson and Stig LJunggren... 

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