Defect absorption and emission for $p$-atic liquid crystals on cones

We investigate the ground-state configurations of two-dimensional liquid crystals with $p$-fold rotational symmetry ($p$-atics) on fixed curved surfaces. We focus on the intrinsic geometry and show that isothermal coordinates are particularly convenient as they explicitly encode a geometric contribution to the elastic potential. In the special case of a cone with half-angle $\beta$, the apex develops an effective topological charge of $-\chi$, where $2\pi \chi =2\pi (1-sin\beta )$ is the deficit angle of the cone, and a topological defect of charge $\sigma$ behaves as if it had an effective topological charge $Q_{\mathrm{eff}}=(\sigma -{\sigma }^2/2)$ when interacting with the apex. The effective charge of the apex leads to defect absorption and emission at the cone apex as the deficit angle of the cone is varied. For total topological defect charge 1, e.g., imposed by tangential boundary conditions at the edge, we find that for a disk the ground-state configuration consists of $p$ defects each of charge $+1/p$ lying equally spaced on a concentric ring of radius $d={\left(\frac{p-1}{3p-1}\right)}^{\frac{1}{2p}}R$, where $R$ is the radius of the disk. In the case of a cone with tangential boundary conditions at the base, we find three types of ground-state configurations as a function of cone angle: (i) for sharp cones, all of the $+1/p$ defects are absorbed by the apex; (ii) at intermediate cone angles, some of the $+1/p$ defects are absorbed by the apex and the rest lie equally spaced along a concentric ring on the flank; and (iii) for nearly flat cones, all of the $+1/p$ defects lie equally spaced along a concentric ring on the flank. Here the defect positions and the absorption transitions depend intricately on $p$ and the deficit angle, which we analytically compute. We check these results with numerical simulations for a set of commensurate cone angles and find excellent agreement.

Figure 1

Schematic of the three coordinate systems for cone used in this paper: (a) Our preferred isothermal coordinates $z=r{e}^{i\varphi }$, which can be viewed as the result of squashing a cone into a plane, in a way that preserves the azimuthal angle $\varphi , 0\le \varphi <2\pi$. Here $R$ is the maximum radius down the cone flanks in our isothermal coordinate system. (b) The also useful $\tilde{z}$ coordinates, the result of isometrically cutting open and unrolling a cone into a plane, so that the resulting azimuthal angle is $\tilde{\varphi }, 0\le \tilde{\varphi }<2\pi (1-\chi )$. (c) Three-dimensional Cartesian coordinates $x_i$, where $\beta$ is the cone half-angle.

Figure 2

Schematic of $\tilde{z}$ geometry for $sin\beta =1/n$, for $n=10$. The conical singularity is represented by the star at the origin, the topological defect is represented by the black dot in the black wedge, and $n-1$ image charges are represented by white circles in the dashed wedges.

Figure 3

Schematic illustration of ground-state defect configurations on a disk and a cone. The positive Gaussian curvature at the cone apex gives rise to a geometrical background charge that attracts like-signed defects in the $p$-atic liquid crystal. (a) A hexatic ($p=6$) liquid crystal on a flat disk has six defects with charge $+1/6$ distributed evenly along the angular direction at positions given by Eq. (80) with $p=6$. (b) A cone with angle $sin\beta =4/6$ absorbs two of those defects onto the apex, leaving four defects on the flanks, again evenly distributed along the azimuthal direction. Defects are depicted by black dots.

Figure 4

Ground-state numerical textures for a nematic ($p=2$) liquid crystal with tangential boundary conditions for various values of $\chi =1-sin\beta$, where $\beta$ is the cone half-angle. (a) On a flat disk ($\chi =0$), there are two $+1/2$ defects, labeled with green dots, at positions given by Eq. (80) with $p=2$. (b) On the surface of a cone corresponding to $\chi =1/3$, there is one $+1/2$ defect on the flank and another at the cone apex. (c) On a cone corresponding to $\chi =2/3$, there are two $+1/2$ defects at the cone apex, leaving none on the flanks. For (b) and (c), we show both top and perspective views of the cone.

Figure 5

Top row: plots of number of flank charges as a function of $\chi =1-sin\beta$, where $\beta$ is the cone half-angle. Purple markers are from numerical energy minimization, and blue line is theoretical prediction for defect absorption transitions [Eq. (83)]. Bottom row: plots of flank defect positions in the ground states of $p$-atics on cones as a function of $\chi$. Purple markers are from numerical energy minimization, and blue curve is theoretical prediction [Eq. (85)].

Figure 6

Nematic textures from numerical energy minimizations of $p=2$ liquid crystals on truncated $\chi =1/6$ cones with an inner truncation of $r_{\mathrm{inner}}=3$ lattice constants and $r_{\mathrm{outer}}=10$, with (a) tangential BC at the bottom rim and free BC at the top rim (b) tangential BC at both the top and bottom rims. Green circles indicate $\sigma =+1/2$ defects.

Figure 7

(a) A flat two-dimensional crystal with a five-fold disclination at the origin can lower its energy by forming five grain boundaries to screen the central disclination charge. Each grain boundary itself is a row of dislocations. (b) If allowed to buckle into the third dimension, the crystal with the disclination can lower its energy further without needing to form any grain boundaries. Adapted from Refs. [54, 61].

Figure 8

Electric field lines (without arrows) of two positive charges inside a disk and their negative image charges outside the disk.

Figure 9

$\omega$ denotes the angle of the director field $\widehat{n}$ of the liquid crystal molecule (blue arrow) relative to the local orthogonal axes on the cone surface. $\mathrm{\Omega }$ indicates the angle of the director field $\widehat{n}$ on the squashed isothermal cone, relative to the real axis of the $2d$ complex plane.

Figure 10

Defect configurations for a nematic liquid crystal with cone angle $\chi =1/6$ and inner truncation radii $r_{\mathrm{inner}}=2,4,6$. When the truncation radius of the cone is sufficiently small, a truncated cone with free boundary conditions on the inner rim retains qualitatively the same distribution of defect charges on the flank and the apex as that for the untruncated cones studied in this paper. When the cone is truncated too sufficiently close to the outer rim, flank defects get absorbed to the center of the inner rim.

Figure 11

Ground state textures from numerical energy minimizations for various values of $(p,\chi )$. Simulations for $\chi =0/6$ and $1/6$ are performed on a triangular lattice, and simulations for $\chi =0/4$ and $1/4$ are performed on a square lattice. Red dots denote the presence of flank defects.

Figure 12

Ground state textures from numerical energy minimizations for various values of $(p,\chi )$. Simulations for $\chi =2/6,3/6$ and $4/6$ are performed on a triangular lattice, and simulations for $\chi =2/4$ are performed on a square lattice. Red dots denote the presence of flank defects.

Figure 13

Ground state textures from numerical energy minimizations for various values of $(p,\chi )$. Simulations for $\chi =5/6$ are performed on a triangular lattice, and simulations for $\chi =3/4$ are performed on a square lattice. Red dots denote the presence of flank defects.

Figure 14

Ground state textures from numerical energy minimizations for various values of $(p,\chi )$. Simulations for $\chi =0/6$ and $1/6$ are performed on a triangular lattice, and simulations for $\chi =0/4$ and $1/4$ are performed on a square lattice. Red dots denote the presence of flank defects.

Figure 15

Ground state textures from numerical energy minimizations for various values of $(p,\chi )$. Simulations for $\chi =2/6,3/6$ and $4/6$ are performed on a triangular lattice, and simulations for $\chi =2/4$ are performed on a square lattice. Red dots denote the presence of flank defects.

Figure 16

Ground state textures from numerical energy minimizations for various values of $(p,\chi )$. Simulations for $\chi =5/6$ are performed on a triangular lattice, and simulations for $\chi =3/4$ are performed on a square lattice. Red dots denote the presence of flank defects.