$K_{13}$ term and effective boundary condition for the nematic director

We consider the problem of including the divergence term $K_{13}\mathbf{\nabla }\left[\mathbf{n}\right(\mathbf{\nabla }\mathbf{n}\left)\right]$ in the macroscopic theory of a nematic liquid crystal. The orientation of the bulk director is shown to be determined by the standard Euler-Lagrange equation with an effective boundary condition which assumes a smooth vanishing of the nematic density at the surface and incorporates additional subsurface deformations. This boundary condition implies that, in three dimensions, the $K_{13}$ term does not reduce to an anchoring term.