Transformation from walls to disclination lines: Statics and dynamics of the pincement transition

We present an experimental and theoretical study of the pincement phenomenon—transformation of a wall associated with the Fréedericksz transition into a pair of disclination lines. We measure the velocity of the boundary (front) between the two states as a function of the voltage. Experimental results are recovered by numerical simulations based on the nematic tensor order parameter, which also reveal the detailed three-dimensional structure of the front. By introducing reduced models we obtain approximate expressions for the two-state coexistence voltage and the front velocity. We find a bifurcation scenario incorporating a pair of saddle nodes at which the wall and disclination solutions appear or disappear.

Figure 1

Schematic director field presentation of the two possible transition regions between two equivalent Fréedericksz states: (a) domain wall and (b) a pair of disclination lines (indicated by the dots).

Figure 2

Two disclination lines (upper part) in coexistence with the domain wall state (lower part) at $f=300\mathrm{Hz}$ and $U_{\mathit{rms}}=10.19\mathrm{V}$. The front between the two states moves upward with a velocity $v=13.40\mu \mathrm{m}∕\mathrm{s}$. One border of the channel (see Sec. 2A) can be seen in the left part of the picture. The height and width of the image are 0.42 and 0.55 mm, respectively.

Figure 3

Dependence of the front velocity on the voltage for a mixture with ${\epsilon }_a=0.17$. Positive velocities represent an advancing disclination state. The velocity is zero at $U_{\mathit{eq}}=18.85\mathrm{V}$. The slope of the linear part at $U_{\mathit{eq}}$ is $k=2.30\mu \mathrm{m}∕\mathrm{s}\mathrm{V}$. Note that the velocity is curving up slightly with increasing voltage. Inset: fit with the prediction of the saddle-node model (32) of Sec. 4D.

Figure 4

Measured dependence of the equilibrium voltage $U_{\mathit{eq}}$ on the inverse square root of the dielectric anisotropy for three different mixtures (circles). The experimental points are fit by the line with a slope $d{U}_{\mathit{eq}}∕d\left({\epsilon }_a^{-1∕2}\right)=7.94\mathrm{V}$, which is forced to go through the origin.

Figure 5

(Color online) Simulated twisted Fréedericksz wall in a schematic presentation. Long axes of the boxes correspond to the director. The two opposite Fréedericksz states continue to the left and right, respectively.

Figure 6

Twisted Fréederikcsz wall observed between crossed polarizers. The zig and zag regions correspond to the two possible twist directions. The height and width of the image are 1.19 and 1.58 mm, respectively.

Figure 7

Detection of the twisted Fréedericksz wall. A strong change of the transmitted intensity determines the threshold voltage at $U_{\mathit{\text{twist}}}=13.08\mathrm{V}$.

Figure 8

(Color) Middle part of the cross section through the front: the defect structure (left) transforms to the wall (right) via a $1∕2$ twist disclination line (middle) normal to the substrates. The nematic $\mathsf{Q}$ tensor field is represented by the boxes; the lengths of the edges correspond to the eigenvalues (a constant is added to make them nonnegative). The distance between the $xz$ slices has been increased for clarity; in reality it is comparable to the distance between the boxes in the slice.

Figure 9

(Color) A view through the front (from left to right in Fig. 8), revealing the $\pi$ twist rotation around the $1∕2$ twist disclination line in the middle. Note that it is required topologically since the director in the center of the Fréedericksz wall points in the $x$ direction, whereas the central director of the disclination configuration is parallel to $z$. The color coding is consistent with the one in Fig. 8 and helps to distinguish between near (red) and distant (green) regions.

Figure 10

Numerically determined pincement threshold voltage $U_{\mathit{eq}}$ vs cell thickness $d$, fitted using the estimate (18). For $d=19\mu \mathrm{m}$ we get $U_{\mathit{eq}}=16.5\mathrm{V}$; the experimental value is $U_{\mathit{eq}}=18.85\mathrm{V}$.

Figure 11

The front velocity-voltage coefficient, $v=k(U-U_{\mathit{eq}})$, vs cell thickness, fitted by the model (28). The inset shows this function magnified in the region of experimentally relevant thicknesses; at $d=19\mu \mathrm{m}$ it reads $0.8\mu \mathrm{m}∕\mathrm{s}\mathrm{V}$.

Figure 12

A bifurcation diagram indicating the wall (circles) and disclination (squares) solutions and the unphysical unstable branch (dotted line). The data presented as circles and squares are calculated in the tensor simulation for a system with thickness $d=0.64\mu \mathrm{m}$, whereas the unstable branch connecting the two saddle nodes is added by hand and is only schematic. As a convenient order parameter we choose the ${\mathsf{Q}}_{zz}$ component of the nematic order parameter in the middle of the wall, measuring the ordering with respect to the $z$ axis (see Sec. 4D).