Theory of helicoids and skyrmions in confined cholesteric liquid crystals

Cholesteric liquid crystals experience geometric frustration when they are confined between surfaces with anchoring conditions that are incompatible with the cholesteric twist. Because of this frustration, they develop complex topological defect structures, which may be helicoids or skyrmions. We develop a theory for these structures, which extends previous theoretical research by deriving exact solutions for helicoids with the assumption of constant azimuth, calculating numerical solutions for helicoids and skyrmions with varying azimuth, and interpreting the results in terms of competition between terms in the free energy.

Figure 1

Single helicoid in the director field of a cholesteric liquid crystal at $x=0$, calculated with the assumption of constant azimuth. The picture shows the cross section at $y=0$; the director field is extended uniformly forward and backward in $y$. The symbol D represents disclinations, which are also extended uniformly forward and backward in $y$.

Figure 2

Prediction for the helicoid lattice spacing $\lambda$ (scaled by the cell thickness $d$), as a function of the natural cholesteric twist $q_0$ above the critical twist $q_H$ (also scaled by $d$).

Figure 3

Lattice of helicoids at $x=-\frac{1}{2}\lambda , \frac{1}{2}\lambda , \frac{3}{2}\lambda ,...$, in the case where $\lambda =d$. The picture shows the cross sections at $y=0$; the director field is extended uniformly forward and backward in $y$.

Figure 4

Single helicoid at $x=0$, calculated numerically without the assumption of constant azimuth. The picture shows the cross section at $y=0$; the director field is extended uniformly forward and backward in $y$.

Figure 5

Numerical calculation of the free energy of a single helicoid as a function of the natural twist, without the assumption of constant azimuth, using the maximum free-energy density $f_{\text{max}}=100K/d^2$.

Figure 6

Numerical calculation of the average free energy per volume of a helicoid lattice, as a function of the periodicity $\lambda$, using the natural twist $q_0=3/d$ and the maximum free-energy density $f_{\text{max}}=100K/d^2$, without the assumption of constant azimuth.

Figure 7

Lattice of helicoids at $x=-\lambda$, 0, $\lambda ,...$, calculated numerically without the assumption of constant azimuth. This example is constructed using the natural twist $q_0=3/d$ and the maximum free-energy density $f_{\text{max}}=100K/d^2$, and hence the periodicity is $\lambda \approx 2d$. The picture shows the cross section at $y=0$; the director field is extended uniformly forward and backward in $y$.

Figure 8

Director field of a single skyrmion, calculated numerically for natural twist $q_0=3/d$. Images show the horizontal cross section at $z=0$ and the vertical cross section at $y=0$. The symbol D represents point defects on the top and bottom surfaces.

Figure 9

Free energy of a single skyrmion as a function of natural twist $q_0$.

Figure 10

Hexagonal lattice of skyrmions. The hexagonal unit cell is approximated by a circle of radius ${\rho }_{\text{max}}$.

Figure 11

Average free energy per unit volume of the skyrmion lattice $F/\left(\pi {\rho }_{\text{max}}^{2}d\right)$ (scaled by $K/d^2$), as a function of the unit cell radius ${\rho }_{\text{max}}$ (scaled by $d$), using the natural twist $q_0=4/d$.

Figure 12

Phase diagram indicating the uniform vertical, skyrmion lattice, and helicoid lattice phases, as functions of the natural twist $q_0$ and maximum free-energy density $f_{\text{max}}$. The symbols $V, S$, and $H$ indicate numerical calculations of the lowest free-energy structure, while the lines are guides to the eye. As discussed in the text, the vertical axis can also be interpreted as the ratio of the cell thickness $d$ to the disclination core radius $a$.