Curvature energy microscopic model

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. 

Although the balance equations are linear, in the absence of bulk convection, the unknown shape of the melt-crystal interface and the dependence of the melting temperature on the energy and curvature of the surface make the model for microscopic interface shape rich in nonlinear structure. For a particular value of the spatial wavelength, a family of cellular interfaces evolves from the critical growth rate VC(X) when the velocity is increased. 

A subtle refutation of the simple spontaneous curvature model without the bilayer aspect follows from the observation of vesicles of non-spherical topology, i.e. vesicles with holes (like anchor rings) or with handles. For vesicles with at least two holes or handles, the shape of lowest total energy is not unique but rather one fold continuously degenerate due to conformal invariance. This theoretical finding led to the prediction that such vesicles should permanently change their shape, a phenomenon termed conformal diffusion [21], This was indeed verified experimentally somewhat later [22]. Apart from the esthetic pleasure when visualizing conformal transformations under the microscope, this observation also shows the spontaneous curvature model to be incomplete with out the additional term. 

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