Numerical simulation of the twist-grain-boundary phase of chiral liquid crystals

We study the structure of the twist-grain-boundary phase of chiral liquid crystals by numerically minimizing the Landau–de Gennes free energy. We analyze the morphology of layers at the grain boundary, to better understand the mechanism of frustration between the smectic layer order and chirality. As the chirality increases, the layer compression energy strongly increases while the effective layer bending rigidity is reduced due to unlocking of the layer orientation and the director. This results in large deviation of the layer morphology from that of Scherk’s first minimal surface and linear stack of screw dislocations.

Figure 1

Single twist-grain-boundary structure. The broken and gray lines indicate the parallel screw dislocations and the smectic layers in the grain boundary plane $(z=0)$. The solid and dotted lines are the layers at $z\to \pm \infty$, respectively.

Figure 2

Snapshot of the grain boundary structure at $\tau =-0.02$, $g=1$, $B=0.2$, $K=0.02$, and $\alpha =50°$. Plotted are the isosurface $\mathrm{Re}\Psi =0$ and the director $\mathit{n}$ (arrows).

Figure 3

The spatial average of the squared and dimensionless mean curvature $⟨{\left(Hd\right)}^2⟩$ versus the twist angle $\alpha$ at $\tau =(\times )$ $-0.005$, (◇) $-0.02$, and (●) $-0.05$.

Figure 4

(a) Twist-angle dependence of (●) the dimensionless locking energy $f_l=(g∕\mid \tau \mid ){\left\{\mid \Psi \mid \mid \nabla \Phi \mid (m-n)d\right\}}^2$ and (◇) the dimensionless layer compression energy $f_c=(g∕\mid \tau \mid ){\left\{\mid {\Psi }_0\mid (\mid \nabla \Phi \mid -q_0)d\right\}}^2$. (b) The layer compression factor $f_{c0}={\{\cos (\alpha ∕2)-1\}}^2$ (see text).

Figure 5

The squared and dimensionless order parameter ${\mid \Psi \mid }^2∕(\mid \tau \mid ∕g)$ in the cross section $y=\mathrm{const}$ at $\tau =-0.005$ and $\alpha =\left(\mathrm{a}\right)$ 30°, (b) 60°, and (c) 90°. The contour lines are drawn at 0.1, 0.2,…, 1.0. For large $\alpha$, a significant melting of the smectic order is observed even far from the dislocation cores, while it is not seen at lower temperature.

Figure 6

Deviation of the layer normal from that of (●) Scherk’s first surface and (◇) LSD. Each data set is smoothed by averaging over three consecutive points.