Fluctuation route to the bending rigidity

Abstract

A comparison is made of two routes for the calculation of the rigidity constant of bending: the fluctuation route and the equilibrium route. In the fluctuation route the rigidity constant is determined from the calculation of the curvature energy of the fluctuating planar interface, while in the equilibrium route the free energy of curved surfaces in equilibrium is considered. It is known that the expressions for the surface tension of a planar interface in the fluctuation and equilibrium routes, leading to the Triezenberg—Zwanzig and Kirkwood—Buff formula, respectively, are indeed equivalent. Within the context of a squared-gradient density functional theory, we find that, although the form of the expressions for the rigidity constant of bending are similar following both routes, the thermodynamic conditions necessary to bend the interface are different, leading to different values for the rigidity constant of bending.