Within the Landau-de Gennes theory of liquid crystals, we study the equilibrium configurations of a nematic cell with twist boundary conditions. Under the assumption that the order tensor Q be uniaxial on both bounding plates, we find three separate classes of solutions, one of which contains the absolute energy minimizer, a twistlike solution that exists for all values of the distance d between the plates. The solutions in the remaining two classes exist only if d exceeds a critical value d(c). One class consists of metastable, twistlike solutions, while the other consists of unstable, exchangelike solutions, where the eigenvalues of Q are exchanged across the cell. When d=d(c), the metastable solution relaxes back to the absolute energy minimizer, undergoing an order reconstruction somewhere within the cell. The critical distance d(c) equals, in general, a few biaxial coherence lengths. This scenario applies to all the values of the boundary twist but pi/2, which thus appears as a very special case, though it is the one more studied in the literature. In fact, when the directors prescribed on the two plates are at right angles, two symmetric twistlike solutions merge continuously into an exchangelike solution at the critical value of d where the latter becomes unstable. Our analysis shows how the classical bifurcation associated with this phenomenon is unfolded by perturbing the boundary conditions.

We study a class of quadratic Hamiltonians which describe both fully attractive and partly repulsive molecular interactions, characteristic of biaxial liquid crystal molecules. To treat the partly repulsive interactions we establish a minimax principle for the associated mean-field free energy. We show that the phase diagram described by Sonnet [Phys. Rev. E 67, 061701 (2003)] is universal. Our predictions are in good agreement with the recent observations on both V-shaped and tetrapodal molecules.

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