Existence of a bending rigidity for a hard-sphere liquid near a curved hard wall: Validity of the Hadwiger theorem

In the context of Rosenfeld's fundamental measure theory, we show that the bending rigidity is not equal to zero for a hard-sphere fluid in contact with a curved hard wall. The implication is that the Hadwiger theorem does not hold in this case and the surface free energy is given by the Helfrich expansion instead. The value obtained for the bending rigidity (i) is an order of magnitude smaller than the bending constant associated with Gaussian curvature, (ii) changes sign as a function of the fluid volume fraction, and (iii) is independent of the choice of the location of the hard wall.