Curvature Instability of Membranes near Rigid Inclusions

In multicomponent membranes, internal scalar fields may couple to membrane curvature, thus renormalizing the membrane elastic constants and destabilizing the flat membranes. Here, a general elasticity theory of membranes is considered that employs a quartic curvature expansion. The shape of the membrane and its deformation energy near a long rod-like inclusion are studied analytically. In the limit where one can neglect the end-effects, the nonlinear response of the membrane to such inclusions is found in exact form. Notably, new shape solutions are found when the membrane is curvature unstable, manifested by a negative rigidity. Near the instability point (i.e. at vanishing rigidity), the membrane is stabilized by the quartic term, giving rise to a new length scale and new scale exponents for the shape and its energy profile. The contact angle induced by an applied force at the inclusion provides a method to experimentally determine the quartic curvature modulus.