Curvature Moduli

The resistance to bending is expressed in terms of two moduli the mean curvature modulus, kc, and the Gaussian curvature modulus. Kg, which are both defined via the energy, k, required for bending the sheet (43) ... 

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 solid films Le., films with a shear modulus), there is an additional resistance to bending arising from the resistance to shear deformations. This results in a nonzero curvature modulus even for a system which is elastically isotropic in the bulk. As shown below, this is not the case for systems with zero shear modulus isotropic fluids show no resistance to bending deformations. 

The role of the mean curvature modulus K is easy to understand. Its contribution to the ground state energy levels is zero, the mean curvature being everywhere zero in these states. It only comes in if we leave these states and consider thermal fluctuations in the mean curvature, such as we would expect to occur spontaneously at any non-zero temperature. However, these fluctuations do not act symmetrically on the two structures. In the lamellar phase Lq., smectic order is maintained by steric interaction, whereas in the sponge phase L3, cubic periodicity in the ground state is lost. Since the sponge phase exhibits liquid type disorder, whilst maintains smectic order, we may presume that stability of L3 is favoured at low K and that of at high K. 

Fig. 5.23, the mean curvature must be as close as possible to cq < 0, and indeed, it is symmetrically negative. When co < 0, the saddle surface is therefore preferred in comparison with the plane surface. These intuitive geometrical considerations can be re-expressed in terms of the Gaussian curvature modulus K of the bilayer. The curvature elasticity of the bilayer (Kbiiayer, biiayer and Co, bilayer = 0) Can bc simply calculated in terms of the curvature elasticity of the monolayers (i monoiayer) -f monoiayer and Cq). The problem is as simple as for a bimetallic strip. We obtain... 

Collecting together all the ideas in this section, we can now make a rather simple general picture. Starting out with a system in which the true curvature modulus Kq of the membranes is greater than there are two concentration ranges ... 

Hu M, Briguglio JJ, Desemo M (2012) Determining the Gaussian curvature modulus of lipid membranes in simulations. Biophys J 102(6) 1403-1410... 

Templer RH, Khoo BJ, Seddon JM (1998) Gaussian curvature modulus of an amphiphile monolayer. Langmuir 14(26) 7427-7434... 

Since the curvature modulus is negative, a saddle configuration is energetically unfavorable in this system. If one assumes that the width of the bilayer is proportional to... 

YatciUa et al. investigated the conversion of a mixtime of cationic and anionic surfactants to form vesicles. In this case, the reaction was veiy slow and a kinetic phase was associated with the evolution and growth of vesicles over a period of weeks (see Figure 6.12) to a final vesicular system, which, as already mentioned, was thought by the authors to be thermodynamically stable. A subsequent study explored the system CTAB mixed with sodium perfiuorooctanoate. Cylinders, disks, and spherical uni-lamellar vesicles were found to coexist at equilibrium by cryo-TEM. This observation confirms the importance of structural confirmation by cryo-TEM when this technique can be applied. Erom their analysis of the data, the mean curvature modulus, the Gaussian curvature modulus, and the spontaneous curvature could all be evaluated. 

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