Solution of the Oldano-Barbero paradox

The appearance of the splay-bend elastic constant K-13 in the theory of elasticity of nematic liquid crystals causes a serious problem. This problem is known as the Oldano-Barbero paradox. The approaches of Barbero et al., Faetti, and Pergamenshchik for solving this paradox are analyzed in view of two starting points of the continuum theory. The analysis clearly shows the imperfections of their treatments. In order to remedy these imperfections a proposal is made to eliminate the paradox in accordance with the assumptions of the continuum theory. The solution requires a modification of the surface free energy density, namely, the derivatives of the director field normal to the surface must be excluded.