Wound-Ribbon Helix Structure of Tilted Chiral Lipid Bilayers

With the equation of helicoid, it is found that for wound-ribbon helix of chiral lipid bilayers the edge line of the helix can only be parallel to the azimuthal direction of the director. This clears the dilemma of the existence of both the parallel and the perpendicular case under the treatment of a chiral bilayer around a circular cylinder by Ou-Yang and Liu Phys. Rev. Lett. 65 (1990) 1679; Phys. Rev. A43 (1991) 6826. The present calculation also explains the self-transformation of the wound-ribbon helix into a prolate tube and the prolate tube into a multilayer soda straw structure.