Structure of twin boundaries of cholesteric blue phase I

We investigate numerically the structure of twin boundaries of cholesteric blue phases. Our study is based on the Landau–de Gennes continuum theory describing the orientational order of the liquid crystal by a second-rank tensor. We pay particular attention to blue phase I (BP I) with body-centered-cubic symmetry and consider twin boundaries between BP I lattices in which their (110) planes are shared and the ($\overline{1}12$) plane of one lattice is parallel to the ($1\overline{1}2$) plane of the other as observed in previous experiments [Jin et al., Sci. Adv. 6, eaay5986 (2020); Zhang et al., ACS Appl. Mater. Interf. 13, 36130 (2021)]. We discuss two plausible cases in which the twin boundaries are parallel to the $\left\{112\right\}$ planes or the $\left\{111\right\}$ planes. In the former, disclination lines of obtusely bent form penetrate the twin boundaries, and in the latter straight disclination lines as well as bent ones are found at the twin boundaries. The former twin boundaries are energetically less costly, consistent with previous experimental identifications. From our numerical results the free energy of a twin boundary per unit area is estimated to be $\simeq 4\times {10}^{-6}{\mathrm{J}\mathrm{m}}^{-2}$, which indeed indicates that the formation of twin boundaries is not prohibitively costly.

Figure 1

(a) Schematic illustration of a double-twist cylinder. Short rods represent the orientation profile in a plane perpendicular to the cylinder axis. [(b) and (c)] Arrangement of double-twist cylinders (left) and disclination lines (right) in (b) BP I and (c) BP II.

Figure 2

[(a) and (b)] Numerical systems for (a) the $\left\{112\right\}$ case and (b) the $\left\{111\right\}$ case. In both cases, the (110) lattice plane is parallel to the plane of the figure. The region in which numerical calculations are carried out is depicted by a rectangle with thick black edges, whose size in the $xy$ plane is (a) $2\sqrt{6}a\times \sqrt{3}a$ and (b) $\sqrt{6}a\times 2\sqrt{3}a$. Thick black edges represent periodic boundaries. Green and blue regions (lighter-shaded and darker-shaded regions in the print version) are filled with BP I lattices with their (110) plane parallel to the plain of the figure. Different lattice orientation in each region is specified by the schematic illustrations with double-twist cylinders on the top of panel (a). Apparently separated green (lighter-shaded in print) regions are in fact identical because of the periodic boundaries. Twin boundaries, depicted by thick red dashed lines, are parallel to (a) the $yz$ plane and (b) the $xz$ plane. Black dots depict the “lattice points” shared by lattices of different orientation. Black dashed horizontal lines in (a) emphasize the periodicity in the $y$ direction. (c) “Lattice points” of body-centered-cubic BP I (black dots) and the lattice constant $a$.

Figure 3

Illustration of how the lattice in the green region is shifted from the “reference initial profile” (top) by $(d_x,d_y,d_z)$. The lattice in the blue region is now shown for clarity. Note again that the the lattices in the green regions are identical under periodic boundary. Refer to the caption of Fig. 2 for colors.

Figure 4

The profile of BP I containing twin boundaries relaxed from the reference initial condition for the $\left\{112\right\}$ case. Top two and bottom two panels show disclination lines and axes of double-twist cylinders, respectively, viewed from different directions specified by the coordinate axes. The location of the twin boundaries is highlighted by dot-dashed lines. Here and in the following, the profiles are shown with orthographic projection, and in Supplemental Material [25], they are presented in perspective view.

Figure 5

The same as Fig. 4, except that the left part of the initial condition (connected to the right part by a periodic boundary) is shifted by $(d_x,d_y,d_z)=(\sqrt{6}a/4,0,\sqrt{2}a/4)$ from the reference initial condition. Solid and dotted circles highlight the bending of disclination lines whose bending angle is ${cos}^{-1}(-7/9)\simeq {141}^{\circ }$ and ${cos}^{-1}(-1/3)\simeq {109}^{\circ }$, respectively (disclination lines at dotted circles continue through the periodic boundaries).

Figure 6

Detailed structure of a twin boundary in Fig. 5 over the thickness $(\sqrt{6}/5)\times 4\pi \simeq 0.49a$. Disclination lines (red, marked by white lines) and axes of double-twist cylinders (blue) are shown simultaneously. The structure of one twin boundary there is the same as that of the other.

Figure 7

The profile of BP I containing twin boundaries relaxed from the reference initial condition for the $\left\{111\right\}$ case. Top two and bottom two panels show disclination lines and axes of double-twist cylinders, respectively, viewed from different directions specified by the coordinate axes.

Figure 8

The same as Fig. 7, except that the upper part of the initial condition (connected to the lower part by a periodic boundary) is shifted by $(d_x,d_y,d_z)=(\sqrt{6}a/4,3\sqrt{3}a/32,\sqrt{2}a/4)$ from the reference initial condition. Here the profiles seen from the $y$ direction are also shown to emphasize straight disclination lines along the $y$ direction. Circles highlight the bending of disclination lines with the angle of ${cos}^{-1}(-5/9)\simeq {124}^{\circ }$ at the twin boundaries.

Figure 9

Detailed structures of a twin boundary in Fig. 8, each over the thickness $(3\sqrt{3}/4)\times 4\pi \simeq 0.65a$. Disclination lines (red, marked by white lines) and axes of double-twist cylinders (blue) are shown simultaneously. The structure of one twin boundary there is the same as that of the other.