Symmetry breaking of nematic umbilical defects through an amplitude equation

The existence, stability properties, and bifurcation diagram of the nematic umbilical defects is studied. Close to the Fréedericksz transition of nematic liquid crystals with negative anisotropic dielectric constant and homeotropic anchoring, an anisotropic Ginzburg-Landau equation for the amplitude of the tilt of the director away from the vertical axis is derived by taking the three-dimensional (3D) to 2D limit of the Frank-Oseen model. The anisotropic Ginzburg-Landau equation allows us to reveal the mechanism of symmetry breaking of nematic umbilical defects. The positive defect is fully characterized as a function of the anisotropy, while the negative defect is characterized perturbatively. Numerical simulations show quite good agreement with the analytical results.

Figure 1

Nematic umbilical defects. (a) Schematic representation of the system under study, the rods describe the orientation of the director and the gray rods (green rods) stand for the vortex position. (b) Three-dimensional representation of the nematic umbilical defect, where arrows stand for the position of the defect. (c) Experimental image of umbilical defects. (d) Bifurcation diagram of a degenerate pitchfork bifurcation with $O\left(2\right)$ symmetry.

Figure 2

Vortex solution of Ginzburg-Landau equation (7) with ${\mu }_0=1$ (from numerical simulations). Structure of the magnitude (a) and phase of the positive vortex (b). Structure of the magnitude (c) and phase of the negative vortex (d). (e) Nullcline field, $\psi (x,y,t)=\text{Re}\left(A\right)\text{Im}\left(A\right)$ at given time. This field is equivalent to the light intensity observed when one considers crossed polarizers on an experimental setup. (f) Modulus of the amplitude $A$ at given time.

Figure 3

Vortex solution of the anisotropic Ginzburg-Landau equation (3) with ${\mu }_0=1$ and $\delta =0.7$ (from numerical simulations). Structure of the magnitude (a) and phase of the positive vortex (b). Structure of the magnitude (c) and phase of the negative vortex (d). (e) Colormap of nullcline field $\psi (x,y,t)=\text{Re}\left(A\right)\text{Im}\left(A\right)$ and (f) modulus of the amplitude $A$ at given time.

Figure 4

Vortex solution with positive topological charge of the anisotropic Ginzburg-Landau equation (3) with ${\mu }_0=1$ and $\delta =0.1$. Magnitude of the vortex with positive (a) and negative (b) anisotropy, respectively. The dashed curve stands for the magnitude of the vortex for isotropic systems $\left(\delta =0\right)$. (c) and (d) schematic representation of the orientation field $A(r,\theta ,\left\{{\phi }_0\right\})$ for different values of ${\phi }_0$: $R_v^{-}$ $\left({\phi }_0=\pi /2\right)$ and $R_v^{+}$ $\left({\phi }_0=0\right)$. (e) Schematic representation of the orientation amplitude field for the negative topological charge.

Figure 5

Energy of the positive vortex solutions for different jump phase ${\phi }_0$ as function of $\delta$. Numerical results obtained from vortex solutions of Eq. (3) are shown by the geometrical symbols (circles and diamonds) and the theoretical result obtained from expression (16) by a continuous and dashed line. The continuous and dashed line indicate, respectively, the stable and unstable vortex solution with positive topological charge. The bottom panel schematically illustrates the bifurcation diagram for the phase jump ${\phi }_0$, which correspond to a degenerate transcritical bifurcation. The dark and white circles account for stable and unstable vortices solutions.

Figure 6

Vortex solution with negative topological charge of anisotropic Ginzburg-Landau equation (3) with ${\mu }_0=1$ and $\delta =0.7$. (a) left panel magnitude of amplitude $\left|A\right|$ and right panels different radial profiles. (b) Numerical coefficients of the modal expansion (25).

Figure 7

Energy of the vortex solutions as function of $\delta$. The star symbols account for the free energy $\mathcal{E}$ obtained numerically using a vortex with negative topological charge and formula (5). The solid and dashed lines, drawn to guide the eye, show the evolution of the free energy of the vortices with topological negative and positive charge, respectively, as function of the anisotropy.