One-elastic-constant nematic free energy

In the one-elastic-constant approximation, slow director distortions have a quadratic gradient energy with one modulus. For a planar angle field it reduces to $F=(K/2)\int |\nabla \theta {|}^2$ up to the convention-dependent effective modulus.

For a planar nematic director $\mathbf{n}=(\cos \chi ,\sin \chi )$ and constant amplitude ${\lambda }_0$, the nematic order parameter isDifferentiation gives $\nabla \cdot Q={\lambda }_0(-\sin 2\chi {\chi }_x+\cos 2\chi {\chi }_y,\cos 2\chi {\chi }_x+\sin 2\chi {\chi }_y)$. The coefficient matrix is orthogonal, proving the identity. Hence $K|\nabla \cdot Q{|}^2/2$ is the angle-gradient energy with effective modulus $K{\lambda }_0^2$. The full tensor-gradient norm instead obeys ${\sum }_{ijk}({\partial }_k{Q}_{ij}{)}^2=2{\lambda }_0^2|\nabla \chi {|}^2$ in this normalization; the two local densities must not be confused.