Bend elastic properties

In order to proceed further, a thermodynamic formalism is needed that quantitatively relates the bending elastic quantities mentioned above to measurable properties of microemulsion systems (such as average droplet size or interfacial tension of the flat interface between the microemulsion phase and the excess oil or water phase). As a second step it is necessary to relate experimentally adjustable quantities such as the salt concentration and the nature of the surfactant to the bending elastic properties by means of molecular models. The state of the art is presented in the following sections. 

These three theories predict the global phase behavior of microemulsion systems in terms of bending elastic properties (first approach), molecular interaction parameters (second approach), and expansion coefficients of order parameter fields (third approach). In this review, we try to fill the gap between the early years approach and item 1, above. 

J.G. Guo, Y.P. Zhao, The size-dependent bending elastic properties of nanobeams with surface effects. Nanotechnology 18(29), 295701 (2007)... 

This equation may be utilised to give elastic properties, strains, curvatures, etc. It is much more general than the approach in the previous section and can accommodate bending as well as plane stresses. Its use is illustrated in the following Examples. 

As it can be seen from the above data, by introducing 4—15% of monomeric units of ISP into the macromolecules of the AN copolymer, the elastic properties of PAN fibres and, especially, their resistance to abrasion and double bends can considerably be improved. 

Arbabi, S Sahimi, M, Elastic Properties of Three-Dimensional Percolation Networks with Stretching and Bond-Bending Eorces, Physical Review B 38, 7173, 1988. 

For coarse-grained models of linear biopolymers—such as DNA or chromatin— two types of interactions play a role. The connectivity of the chain implies stretching, bending, and torsional potentials, which exist only between directly adjacent subunits and are harmonic for small deviations from equilibrium. As mentioned above, these potentials can be directly derived from the experimentally known persistence length or by directly measuring bulk elastic properties of the chain. 

The elastic properties of the substrate can be determined more accurately than those of the film. Hence, in principle, the magnetoelastic coupling parameter By-2 can be determined more accurately than the magnetostriction coefficient ky-2. An analogous conclusion can be drawn with respect to bending experiments (see e.g. section 3.1 the magnetoelastic cantilever method). 

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. 

A theoretical basis for different shapes of microemulsions (even for small W/O or O/W volume fractions) has been established on the basis of the relationship between shape and interfacial curvature [350,351]. It is reasonable to expect that the relevant properties of the surfactant film are represented by a bending elasticity with a spontaneous curvature, Co (as was demonstrated for binary systems). If the elastic modulii k, ksT, the fluctuations in curvature of the film are very small, and the entropy associated with them can be neglected. The actual morphology is the result of the competition between the tendency to minimize the bending free energy (which prefers spheres of optimal radius of curvature, = l/c ) and the necessity to use up all of the water, oil, and surfactant... 

It is a well-known experimental fact that incorporation of cholesterol in lipid bilayers affects the mechanical and transport properties of membranes. This includes their increased bending elasticity (22) and reduced passive permeability to small molecules (23, 24). Despite a great deal of theoretical and experimental research, no definitive microscopic understanding of phospholipid-cholesterol interactions has been proposed yet. 

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