Figure 1

Schematic representation of the perturbations described by the base tensors (3) for a uniaxial distribution with a positive degree of order (dashed). The $\mathsf{Q}$ tensor eigensystem is represented by the box, the length of the edges corresponds to the eigenvalue (plus a constant). ${\mathsf{T}}_0$ describes a perturbation of the degree of order, ${\mathsf{T}}_1$ describes a biaxial perturbation, ${\mathsf{T}}_{-1}$, ${\mathsf{T}}_2$, ${\mathsf{T}}_{-2}$ represent rotations of the eigensystem. The interpretation of the perturbations varies according to which of the axes has been identified with the director. Irrespective of this, the perturbations given by ${\mathsf{T}}_{-1}$, ${\mathsf{T}}_2$, and ${\mathsf{T}}_{-2}$ possess Goldstone modes, while those given by ${\mathsf{T}}_0$ and ${\mathsf{T}}_1$ are massive.Reuse & Permissions

Figure 2

Radial eigenfunctions of the splitting fluctuation, $s=1$, $m=2$, $\lambda =-0.22$. The length unit is $\xi =2.11\mathrm{nm}$, the time unit is $\tau =32.6\mathrm{ns}$. For comparison, the eigenvalue of the fastest escape mode is $\lambda =-0.0042$.Reuse & Permissions

Figure 3

(Color online) Cross section through the $s=1$ radial disclination line: the ground state (gray/dark) is perturbed by the splitting mode (red/light), leading to two $1∕2$ disclinations on the $x$ axis. Note the two uniaxial regions of the perturbed structure corresponding to the centers of the $1∕2$ disclinations.Reuse & Permissions

Figure 4

A ninefold $(m=9)$ splitting of the $s=3$ defect. The lower diagram depicts a $-3∕2$ defect in the center and nine $1∕2$ defects around it. The shading denotes regions of reduced degree of order.Reuse & Permissions

Figure 5

Growth rates of the fastest splitting modes with the angular eigenvalue $m$ for different winding numbers. The splitting into $1∕2$ defects $(m=2s)$ is always the fastest.Reuse & Permissions

Figure 6

Radial eigenfunctions of the first two growing modes (not normalized), responsible for the escape of the $\pm 1$ defect $(m=0)$. These modes are quite extensive, yet localized [Eq. (33)]; the range of the second function is $\approx 1000\xi$.Reuse & Permissions

Figure 7

(Color online) Cross section through the $s=1$ radial disclination line: the ground state (gray/dark) is perturbed by the escape mode (red/light). The mode is quite extensive, only the very central part is depicted here (before the maximum of $R_{2,0,1}$ in Fig. 6) to point out the vanishing perturbation in the center.Reuse & Permissions

Figure 8

Radial eigenfunctions of the splitting modes with $m=4$ for the $s=2$ defect: (a) $\lambda =-0.40$ and (b) $\lambda =-0.059$.Reuse & Permissions

Figure 9

Splitting of the $s=2$ defect into four $1∕2$ defects, $m=4$; the corresponding radial eigenfunctions are depicted in Fig. 8. The $m=4$ modes are numerically boosted due to the $C_4$ symmetry of the computational grid.Reuse & Permissions