Shape Energy of Fluid Membranes: Analytic Expressions for the Fourth-Order Variation of the Bending Energy

The shape energy of a fluid membrane was formulated previously in terms of the socalled Helfrich Hamiltonian [W. Helfrich, Z. Naturforsch. C 28 (1973), 693; H. J. Deuling and W. Helfrich, J. de Phys. 37 (1976), 1335]. Although the shape deformation might be minor, it should play an important role in the case of a membrane in the proximity of instabilities. Here, we extend the work of Ou-Yang and Helfrich [Ou-Yang Zhong-Can and Wolfgang Helfrich, Phys. Rev. A 39 (1989), 5280; Ou-Yang Zhong-Can, Liu Ji-Xing and Xie Yu-Zhang, Geometric Methods in the Elastic Theory of Membranes in Liquid Crystal Phases (World Scientific, Singapore, 1999)], expanding the bending energy up to fourth order in the deformation function with the help of theoretical tools from differential geometry. After obtaining the fourth-order correction for the bending energy, we investigate its role in the bending deformation by considering two special cases - those of spherical and cylindrical surfaces. In the case of a sphere, we restrict our consideration to two deformed shapes, prolate and oblate. The results of our analysis indicate that, taking account of fourth-order corrections, the critical value determining the oblate-to-prolate transition behaves in a more complicated way than in the case that only corrections up to third order are considered. We also compare the present results will those obtained previously. In the case of a cylinder, two special surface deformations are considered, and conditions for the stability of the surface under such deformations are found. The formation of possible surface shapes deviating slightly from a cylinder with fourth-order corrections is found, even though no such stable state is found in the calculation up to third order. We also apply the variation of the energy up to third order to the case of a deformed torus.