Energy curvature

For lipid bilayers, equation (4) can be simplified. Above we have seen that the flat unsupported bilayer is without tension, i.e. y(0, 0) = 0, and therefore the first two terms must cancel y0 = — kcj. As argued above, JQ = 0, and thus also the third term drops out. The remaining two terms are proportional to the curvature to the power two. For a cylindrical geometry only, the term proportional to J2 is present. For spherical vesicles, the two combine into one ( kc + k)J2. The curvature energy of a homogeneously curved bilayer is found by integrating the surface tension over the available area ... 

The total curvature energy of a spherical vesicle is given by 4tt(2/cc + k). As all experimental data on phospholipids indicate that kc is not small, one is inclined to conclude that the vesicles are thermodynamically unstable the reduction of the number of vesicles, e.g. by vesicle fusion or by Ostwald ripening, will reduce the overall curvature energy. However, such lines of thought overlook the possibility that k is sufficiently negative to allow the overall curvature free energy of vesicles to remain small. 

For sake of simplicity and to make our present discussion more transparent, in the previous expression (5) for the droplet potential energy, we neglected the terms connected with the electrostatic energy and with the so-called curvature energy. The inclusion of these terms will not modifies the conclusions of the present study (Iida Sato 1997 Bombaci et al. 2003). 

Tensions of non-relaxed interfaces are sometimes known by the adjective dynamic dynamic surface tension or dynamic inteifacial tension. The term dynamic is not absolute. It depends on De (i.e. on the time scale of the measurement as compared with that of the relaxation process). Some interfacial processes have a long relaxation time (polymer adsorption-desorption), so that for certain purposes (say the measurement of y] they may be considered as being in a state of frozen equilibrium. This last notion was introduced at the end of sec. 1.2.3. Unless otherwise stated, we shall consider static tensions and interfaces which are so weakly curved that curvature energies, bending moments, etc. may be neglected. 

Figure 4. Wall profile as calculated for needles with large curvature energy and no lattice relaxation (see Salje and Ishibashi 1996). The coordinate systems used in the text is shown. For single needles y = P = all, for forked needles we observed p < y where p is the interior angle of a forked pair.

Aveyard R, Clint JH, Horozov TS (2003) Aspects of the stabilisation of emulsions by solid particles effects of line tension and monolayer curvature energy. Phys Chem Chem Phys 5(ll) 2398-2409... 

The solid curves in Fig. 5 are the results of a thermodynamic analysis based on a Flory-Huggins model for the interaction between the monomer and polymer and a phenomenological curvature energy model to describe the chemical potential of the monomer in the micelle [30]. 

In addition the jellium model for a planar surfare can be employed to calculate a, [21]. However, Genzken obtained a, from a plot of the slope of E N)/N — gj,) versus to suppress the shell oscillations for large values of N. Finally the curvature energy was fixed in a similar way by the slope of a plot of E N) - gfcN — versus. ... 

Here a, a and y are the volume, surface and curvature energies respectively. The chemical potential... 

Adsorption of nonionic surfactants on porous solids has been studied by Huinink et al. in a series of p ers [ 149,150]. They elaborated a thermodynamic approach that accounts for the major features of experimental adsorption isotherms. It is a very well known fact that during the adsorption of nonionic surfactants there is a sharp step in the isotherm. This step is interpreted as a change from monomer adsorption to a regime where micelle adsorption takes place. Different surfactants produce the step in a different concentration range. The step is more or less vertical depending on the adsorbate. The thermodynamic analysis made by Huinink et al. is based on the assumption that the step could be treated as a pseudo first order transition. Their final equation is a Kelvin-like one, which shows that the change in chemical potential of the phase transition is proportional to the curvature constant (Helmholtz curvature energy of the surface). 

The monomer consumption from the intetfacial layer (cosurfactant effect) modifies the film curvature energy. The formation of spherical polymo- particles dispersed in the oily phase corresponds to the minimum free energy of the system. 

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 . 

One can also discuss the curvature energy using symmetry considerations and relate it to the models analyzed previously. The most general form of the curvature free energy, fc, per unit area, up to quadratic order in the two curvatures, k and K2 can be written in terms of the mean and Gaussian curvatures defined in Eqs. (6.3,6.4). One can write ... 

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. 

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