On the validity of the elastic expansion of the free energy of nematic liquid crystals

Abstract

The elastic free energy F el of nematic liquid crystals is a truncated power expansion of the free energy F in the gradients of the director field n(r) . In principle, the cut-off distortion length for F el could be estimated from the direct comparison between the elastic terms and the higher order ones. However, in the case of dispersion interactions, the higher order elastic constants diverge and this comparison is not possible. The divergence of the higher order constants poses some doubt about the internal consistency of the elastic expansion itself. In this paper, we use a molecular model of a nematic liquid crystal to calculate the total free energy and its elastic approximation. The two free energies are calculated numerically for some director configurations as a function of the characteristic distortion length L dis. A good agreement between the two free energies is found for slow director fields, and large deviations are observed only if L dis ≈ a, where a is the molecular length. This result demonstrates the validity of the elastic expansion and shows that the higher order terms affect only the behavior at very short length scales.