Structure of the elastic free energy for chiral nematic liquid crystals

In Landau–de Gennes theory, the free energy f of liquid crystals is expanded into powers of a symmetric, traceless tensor order parameter $Q_{\alpha \beta }$ and its derivatives $Q_{\alpha \beta }$,γ. The expansion is subject to the condition that f is a scalar, i.e., invariant under all rotations of the group SO(3). Using the method of integrity basis, we have established the most general SO(3)-invariant free-energy density up to all powers in $Q_{\alpha \beta }$ and up to second order in $Q_{\alpha \beta }$,γ. It turns out that this free-energy density is composed of 39 invariants, which are multiplied by arbitrary polynomials in Tr$Q^2$ and Tr$Q^3$. On the other hand, these 39 invariants can be expressed as polynomials of 33 so-called irreducible invariants. Interestingly, among the irreducible invariants there are only three chiral terms (i.e., linear in $Q_{\alpha \beta }$,γ). They locally give rise to three independent helix modes in chiral, biaxial liquid crystals. This conclusion generalizes results of Trebin [J. Phys. (Paris) 42, 1573 (1981)] and Govers and Vertogen [Phys. Rev. A 31, 1957 (1985); 34, 2520 (1986)] and contradicts a statement of Pleiner and Brand [Phys. Rev. A 24, 2777 (1981); 34, 2528 (1986)], according to which only one twist term is supposed to exist.