Helfrich curvature

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

Phases made up of flexible surfactant films can be understood in terms of the Helfrich curvature free energy ... 

The interest in vesicles as models for cell biomembranes has led to much work on the interactions within and between lipid layers. The primary contributions to vesicle stability and curvature include those familiar to us already, the electrostatic interactions between charged head groups (Chapter V) and the van der Waals interaction between layers (Chapter VI). An additional force due to thermal fluctuations in membranes produces a steric repulsion between membranes known as the Helfrich or undulation interaction. This force has been quantified by Sackmann and co-workers using reflection interference contrast microscopy to monitor vesicles weakly adhering to a solid substrate [78]. Membrane fluctuation forces may influence the interactions between proteins embedded in them [79]. Finally, in balance with these forces, bending elasticity helps determine shape transitions [80], interactions between inclusions [81], aggregation of membrane junctions [82], and unbinding of pinched membranes [83]. Specific interactions between membrane embedded receptors add an additional complication to biomembrane behavior. These have been stud-... 

Helfrich has shown [18] that the surface tension of a curved interface can be expressed as a Taylor series up to second order in the radius of curvature ... 

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

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 ... 

Andelman, D., Kozlov, M.M., Helfrich, W. Phase transitions between vesicles and micelles driven by competing curvature. Europhys. Lett. 1994, 25, 231-6. 

Davis summarized the concepts about HLB, PIT, and Windsor s ternary phase diagrams for the case of microemulsions and reported topologically ordered models connected with the Helfrich membrane bending energy. Because the curvature of surfactant lamellas plays a major role in determining the patterns of phase behavior in microemulsions, it is important to reveal how the optimal microemulsion state is affected by the surface forces determining the curvature... 

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... 

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. 

Eq. (40) with the spectrum of Helfrich s curvature Hamiltonian for membranes [97],... 

It is clear that the two polar vectors respect the apolar nature of n. There is also an obvious analogy of the above mechanism with the orientation polarization of a liquid dielectric, which was used by Helfrich to relate the two flexocoefficients with molecular properties. The intrinsic splay or bend can be related to an appropriate angle and molecular dimensions. The relevant component of the electric dipole moment and the curvature elastic constant, viz., and the splay constant Ki or /ux and the bend constant It s figure in the estimation of the flexocoefficients. Nematic liquid crystals made of banana-shaped molecules have been studied only recently, and a comparison of the experimental measurements with the Helfrich formula leads to interesting inferences, as will be mentioned later in this chapter, and covered more thoroughly in the companion Chapter 3 by Jakli et al. ... 

D. Schmidt, M. Schadt and W. Helfrich, Liquid-crystalline curvature electricity The bending mode of MBBA, Z. Naturforsch. 27A(2), 277-280, (1971). 

The first tension-like quantity in planar membranes is the lateral mechanical stress in the membrane, as discussed above. If the stress is imposed by a boundary condition, such as, for instance, a craistraint on the lateral (projected) area of the membrane, it is an internal property of the membrane system that depends, among other parameters, on the area compressibility [36] and the curvature elasticity [154—161]. Alternatively, mechanical stress can be imposed externally. In that case, the projected area fluctuates, and the appropriate thermodynamic potential can be introduced into the Helfrich Hamiltonian, Eq. (1), in a straightforward manner ... 

Deuling H, Helfrich W (1976) The curvature elasticity of fluid membranes a catalogue of vesicle shapes. J Phys (France) 37 1335-1345... 

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. 

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