Membranes curvature energy

Here, and // denote respectively the local mole fraction and local electrochemical potential of the charged lipid species in that particular leaflet, g is the metric tensor defined on the leaflet surface, and Di p represents the diffusion coefficient of charged lipids. Note that Diip should not affect the equilibrium state. The local electrochemical potentials, in turn, are derived from the free energy functional that itself depends on local lipid component densities membrane curvature. This property results in a self-consistent formulation, which remains as the main computational task. 

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 ... 

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... 

Flexible, solid membranes, are also of interest. However, they are experimentally much less prevalent and are somewhat more complicated to treat since in addition to the membrane shape one must include the effects of shear. Their curvature energy is discussed in the problems at the end of this chapter. Another type of system that has received much theoretical attention is that of a tethered membrane which may describe polymerized, but not crystalline sheets. While a single fluid membrane that is unconstrained by walls or other membranes is strongly affected by thermal fluctuations ( crumpled ), solid membranes, particularly if self-avoidance of the membrane is included, tend to be more weakly affected by fluctuations and are hence flattef . 

This form for the free energy per unit area was discussed by Helfirich and states that the mean curvature which minimizes the free energy has a value Co, termed the spontaneous curvature of the membrane. The energy cost of deviating from the spontaneous curvature is the bending or curvature modulus, k. The parameter k, known as the saddle-splay modulus, measures the energy cost of saddlelike deformations. 

For small curvatures, Eq. (6.15) shows that the curvature energy of a thin film is characterized by the three parameters k, k, and cq. The qualitative behavior of any system, including such properties such as the equilibrium shape, magnitude of thermal fluctuations, and any phase transitions, can of course be calculated as a function of these constants. However, the physics of the system can be radically different depending on the physical parameters e.g., a change in cq can induce shape changes in the system. It is thus of interest to relate the bending elastic moduli and the spontaneous curvature to the physics of the particular system of interest. This section first shows how these parameters are related to the pressure distribution in the membrane and then presents a simple but instructive microscopic model that relates k, and Co to more molecular properties. 

Consider a membrane whose physics is described by a Hamiltonian consisting of the curvature energy with zero spontaneous curvature, which is constrained to lie between two hard walls separated by a distance D. Describe the conformation of the membrane by its height, h(x,y), relative to the plane at z = 0, located at the midplane between the two walls. Assume that the effect of the walls adds to the Hamiltonian an effective harmonic potential of the form which acts on the membrane in addition to the bending energy... 

Harmandaris and Desemo [113] proposed a method that relies on simulating cylindrical membranes. Imagine a membrane of area A that is curved into a cylinder of curvature radius R. Its length L satisfies 2%RL = A, and the curvature energy per area of this membrane is ... 

Membrane Elasticity. For a vesicle with fixed volume V = Vo. area A = Aq, and genus the curvature energy of the bilayer membrane reduces to the sum of the remaining two terms of equation 9... 

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... 

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