Revisiting Tilt in Classical Curvature Elastic Theories for Membranes

Abstract

The classical continuum description of a fluid lipid membrane is the curvature elastic Hamiltonian due to Helfrich, in which surface geometry is the only degree of freedom. This model does not consider the deviation of a lipid's orientation away from the local membrane normal, which is another degree of freedom, called lipid tilt. Hamm and Kozlov [Eur. Phys. J. E 3, 323 (2000)] have shown how this variable emerges naturally during the dimensional reduction of a thin but still three-dimensional fluid sheet described by a functional quadratic in the permissible strains. They show that the tilt divergence acts as a locally varying spontaneous curvature, the tilt's magnitude is penalized by a quadratic term proportional to a new tilt modulus, and the Gaussian curvature gets entangled with derivatives of the tilt. We revisit and extend the Hamm-Kozlov theory and show that disentangling Gaussian curvature and tilt leads to a curvature-dependent modification of tilt modulus, which matters for strongly curved membranes. Moreover, by consistently collecting all curvature-tilt couplings, we also find that the effective tilt modulus becomes anisotropic, implying that in general the price for tilt depends on the tilt direction. If we assume that this quadratic continuum theory remains valid at a membrane edge, the corrections are large and significantly modify other observables, such as the edge tension.