Helfrich Hamiltonian

Commonly the bending rigidity of the membrane can be calculated by comparing the simulation results to the prediction of the Helfrich-Hamiltonian [44,141],... 

Within this continuum description, the membrane is conceived as a thin elastic sheet. If one assumes that the membrane fluctuations are small, its surface can be parameterized within a Monge representation via a height function, h(y,z), describing its position over some reference plane. Then, the Helfrich-Hamiltonian takes the form... 

The foregoing expression (Eq. [27]) is known as the Helfrich Hamiltonian.At equilibrium, the vesicle will adopt the shape of the surface 5 that minimizes Eg, subject to two restrictions a constant enclosed volume V and a... 

To provide a basis for both the theoretical ideas and the computational techniques that we will discuss in this chapter, we start by reminding the reader of some essential concepts. Section 2.1 reviews some basic aspects of the Helfrich Hamiltonian. Section 2.2 introduces three coarse-grained membrane models that will be used in the remainder of this chapter. In Sects. 2.3 and 2.4, we discuss the bending moduli and the surface tension of membranes in more detail, and finally comment on multicomponent membranes in Sect. 2.5. 

The Helfrich Hamiltonian, Eq. (1), does not include a surface tension contribution. Free membrane patches can relax and adjust their area such that they are stress-free. In many situations, however, membranes do experience mechanical stress. For example, an osmotic pressure difference between the inside and the outside of a lipid vesicle generates stress in the vesicle membrane. Stress also occurs in supported bilayer systems, or in model membranes patched to a frame. In contrast to other quantities discussed earlier (bending stiffness etc.), and also in contrast to the surface tension of demixed fluid phases, membrane stress is not a material parameter. Rather, it is akin to a (mechanical or thermodynamic) control parameter, which can be imposed through boundary conditions. 

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 ... 

Under these conditions, we derive the Helfrich Hamiltonian [6] for a bilayer of vesicle membrane. We consider the length scale such that the width of membrane is negligible compared to the size of the vesicle. 

The coefficients of the Helfrich Hamiltonian are readily obtained from Eqs. (9) and (10) ... 

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

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

This assumption relies on scale-separation, and it is at the root of the phenomenological Hamiltonian proposed by Canham in 1970, Helfrich in 1973, and Evans in 1975, which identifies stretching and bending as the two main energy components of membranes ... 

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