Helicoids in chiral liquid crystals under external fields

Cholesteric liquid crystals, subject to externally applied magnetic fields and confined between two parallel planar surfaces with strong homeotropic anchoring conditions, are found to undergo transitions to different types of helicoidal configurations with disclinations. Analytical and numerical studies are performed in order to characterize their properties. In particular, we produce a phase diagram for the transitions from the nematic state to the helicoidal phases in terms of the molecular chirality and the strength of the applied magnetic field.

Figure 1

Distribution of $\mathbf{n}\left(\mathbf{r}\right)$ for the $2\pi$ helicoid, given in (19) for $n=0$. The picture shows a cross section of the configuration at $y=0$ for $\mathrm{\Lambda }L=1$. The disclination is indicated by the letter D.

Figure 2

Contour plot of ${\theta }_0$ around the disclination located at $x=0,z=-L/2$.

Figure 3

Distribution of the $\mathbf{n}(x,z)$ field for the $\pi$ helicoid. The picture shows a cross section of the configuration at $y=0$ for $\mathrm{\Lambda }L=1$. The disclination is indicated by the letter D.

Figure 4

Comparison of the numerical and approximated $\pi$-helicoid solutions for different values of the external field: (a) $\mathrm{\Lambda }L=0$, (b) $\mathrm{\Lambda }L=1$, (c) $\mathrm{\Lambda }L=3$, and (d) $\mathrm{\Lambda }L=5$

Figure 5

Phase diagram on the plane $L{q}_0,L\mathrm{\Lambda }$, representing the nematic (blue), $\pi$-helicoid (orange) and $2\pi$-helicoid (ochre) phases. The dashed black line corresponds to the best fit to (40).

Figure 6

Integration path for the Riemann-Hilbert problem (A4, A5, A6).