We investigate the structure of defects in nematic liquid crystals confined in spherical droplets and subject to radial strong anchoring. Equilibrium configurations of the order-parameter tensor field in a Landau-de Gennes free energy are numerically modeled using a finite-element package. Within the class of axially symmetric fields, we find three distinct solutions: the familiar radial hedgehog, the small ring (or loop) disclination predicted by Penzenstadler and Trebin, and a solution that consists of a short disclination line segment along the rotational symmetry axis terminating in isotropic end points. Phase and bifurcation diagrams are constructed to illustrate how the three competing configurations are related. They confirm that the transition from the hedgehog to the ring structure is first order. The third configuration is metastable (in our symmetry class) and forms an alternate solution branch bifurcating off the radial hedgehog branch at the temperature below which the hedgehog ceases to be metastable. Dependence on temperature, droplet size, and elastic constants is investigated, and comparisons with other studies are made.

We study the phase diagram of director structures in cholesteric liquid crystals of negative dielectric anisotropy in homeotropic cells of thickness d which is smaller than the cholesteric pitch p. The basic control parameters are the frustration ratio d/p and the applied voltage U. Upon increasing U, the direct transition from completely unwound homeotropic structure to the translationally invariant configuration (TIC) with uniform in-plane twist is observed at small d/p < or = 0.5. Cholesteric fingers that can be either isolated or arranged periodically occur at 0.5 < or = d/p<1 and at the intermediate U between the homeotropic unwound and TIC structures. The phase boundaries are also shifted by (1) rubbing of homeotropic substrates that produces small deviations from the vertical alignment; (2) particles that become nucleation centers for cholesteric fingers; (3) voltage driving schemes. A novel reentrant behavior of TIC is observed in the rubbed cells with frustration ratios 0.6 < or = d/p < or = 0.75, which disappears with adding nucleation sites or using modulated voltages. In addition, fluorescence confocal polarizing microscopy (FCPM) allows us to directly and unambiguously determine the three-dimensional director structures. For the cells with strictly vertical alignment, FCPM confirms the director models of the vertical cross sections of four types of fingers previously either obtained by computer simulations or proposed using symmetry considerations. For rubbed homeotropic substrates, only two types of fingers are observed, which tend to align along the rubbing direction. Finally, the new means of control are of importance for potential applications of the cholesteric structures, such as switchable gratings based on periodically arranged fingers and eyewear with tunable transparency based on TIC.

To estimate numerically multidimensional director configurations in a liquid crystal cell, it is important to use the Q tensor representation of the strain free energy because it solves the problem of the difference between the directors, n and -n, in the Frank-Oseen free energy representation. In this paper, we discuss the numerical methods for calculating the multidimensional director configurations, using Berreman's Q tensor representation. Numerical issues discussed include the relaxation method for the director calculation, the liquid crystal (LC)/glass interface problem, the boundary conditions for the electric potential, and the possible ways to obtain faster convergence. We compare the calculated results obtained from the Frank-Oseen and Q tensor representations. By considering a π cell with patterned electrodes, we show the consistency of the model used with experimental observations. The calculated data explain well the position shift of the defects that appear in the test π cell.

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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.