Ring Defects in a Strongly Confined Chiral Liquid Crystal

We find numerically that a regular array of isolated ring defects can exist as a stable state in a highly chiral liquid crystal confined in a thin cell imposing fixed planar anchoring at the parallel confining surfaces. This peculiar defect structure can be stable when the cell thickness $d$ is around $3/4$ of the helical pitch $p$. A cell of thickness $3p/4$ with parallel surface anchoring is incompatible with helical alignment that favors $d=mp/2$ (with $m$ being an integer). Formation of ring defects can thus be regarded as a result of frustrations between the helical alignment with a specific pitch and the confining surfaces that prevent it.

Figure 1

Profiles of ring defects for $d/p=0.756$ and $\tau =-0.1$ from three different viewpoints. The defects with weaker orientational order are depicted by isosurfaces $\mathrm{Tr}{\chi }^2=0.4$. Thin cylinders describe the orientational order along the $x$ direction at the confining surfaces (shaded planes). Here we show $2\times 2$ numerical cells connected by periodic boundary conditions along the $x$ and $y$ directions.

Figure 2

Free energy per unit area $\mathcal{F}$ as a function of $d/p$ for $\tau =$ (a) $-0.3$, (b) $-0.1$, and (c) 0.1.

Figure 3

Profile of the director $\mathit{n}$ in the vicinity of a ring defect (shown in translucence) at (a) a $yz$ plane ($x=\mathrm{const}$), (b) an $xz$ plane, and (c) an $xy$ plane for $d/p=0.756$ and $\tau =-0.1$. (d) and (e) give magnified views in the vicinity of defects in (b) and (c), respectively. In (c), the $x$ direction is along the vertical direction. In (a), (b), and (d), parts of confining surfaces are shown as shaded planes. Director profile $\mathit{n}$ is determined as the direction of the eigenvector of ${\chi }_{ij}$ with the largest eigenvalue.