Defect generation and deconfinement on corrugated topographies

We investigate topography-driven generation of defects in liquid crystal films coating frozen surfaces of spatially varying Gaussian curvature whose topology does not automatically require defects in the ground state. We study in particular disclination-unbinding transitions with increasing aspect ratio for a surface shaped as a Gaussian bump with a hexatic phase draped over it. The instability of a smooth ground state texture to the generation of a single defect is also discussed. Free boundary conditions for a single bump are considered as well as periodic arrays of bumps. Finally, we argue that defects on a bump encircled by an aligning wall undergo sharp deconfinement transitions as the aspect ratio of the surface is lowered.

Figure 1

(Color online) (a) The vector field $\mathbf{n}$ is confined to a surface shaped as a Gaussian. (b) Top view of (a) showing a schematic representation of the positive and negative Gaussian curvature as a background “charge” distribution that switches sign at $r=r_0$. Note that, according to the electrostatic analogy, a positive (negative) distribution of Gaussian curvature corresponds to negative (positive) topological charge density.

Figure 2

Plot of $l(r∕r_0)$ as a function of the dimensionless radial coordinate $r∕r_0$ for $\alpha =1,2,3,4$. The arrow is oriented in the direction of increasing $\alpha$.

Figure 3

Projected ground state texture for an $XY$ model on the bump, with the boundary condition that the vector field is parallel to the $y$ axis at infinity. The two insets show the defect pairs suggested by two regions of large frustration, which lie close to a circle of radius $r_0$.

Figure 4

Effect of changing the aspect ratio of the bump on the work needed to pull apart two oppositely charged defects. Positive and negative defects are represented by open circles with their sign printed. The line elements corresponding to the projected length $dr$ for the two aspect ratios are shown.

Figure 5

Geometric potential $V(r∕r_0)$ as a function of the dimensionless ratio of $r$ and $r_0$ for $\alpha =1$, 2, 3, 4, 5. Note that $\mid V\left(r\right)\mid ⪡\mid V\left(0\right)\mid$ for $r\gtrsim r_0$. The arrow points in the direction of increasing $\alpha$.

Figure 6

Plot of the geometric potential evaluated at the center of the bump, $V\left(0\right)$, for aspect ratios $\alpha$ between 0 and 5. The continuous line is plotted using the exact form of $V\left(r\right)$ while the dashed line is obtained from the low-$\alpha$ expansion given in Eq. (26).

Figure 7

Plot of $\Delta F∕K_A$ versus $x_1$ and $x_2$, the positions of the negative and positive defects, respectively, in units of $r_0$, for $\alpha =1.2$. A constant energy offset equal to $-(q^2∕2\pi )\ln (r_0∕a^{\prime })$ has been neglected. The metastable minimum at $x_1\simeq 1.3r_0$ and $x_2\simeq 0.2r_0$ corresponds to an unbound pair [see Fig. 8a]. As $\alpha$ is decreased further the energy barrier that separates this minimum from the smooth texture solution (corresponding to $x_1$ approaching $x_2$) disappears and the two opposite defects annihilate.

Figure 8

The equilibrium defects positions are illustrated schematically in the case of one (a) and two (b) dipoles. We assume free boundary conditions at infinity, as in Fig. 3, so that the effect of image charges can be neglected.

Figure 9

Plot of $f\left(\alpha \right)$ versus the aspect ratio $\alpha$ obtained by minimizing over the single-dipole defect configuration represented schematically in Fig. 8a. As discussed in the text, the first unbinding transition occurs when $f\left({\alpha }_{c_1}\right)$ is equal to $-q^2∕2\pi \ln (r_0∕a^{\prime })$. The value ${\alpha }_{c1}$ is indicated by the dashed line for $r_0∕a^{\prime }={10}^4$ and $q=2\pi ∕6$. Note that no minimum (corresponding to an unbound pair) exists for $\alpha$ less than 1 (approximately); hence the curve cannot be continued to the origin (see Fig. 7).

Figure 10

Plot of $f\left(\alpha \right)$ versus the aspect ratio $\alpha$ obtained by minimizing the two-dipole defect configuration represented schematically in Fig. 8b. The second unbinding transition occurs when $f\left({\alpha }_{c_2}\right)$ is equal to $-(q^2∕\pi )\ln \left(r_0{a}^{\prime }\right)$. The value ${\alpha }_{c_2}$ is indicated by the dashed line for $r_0∕a^{\prime }={10}^4$ and $q=2\pi ∕6$ (compare with Fig. 9).

Figure 11

Plot of $\Delta F\left(\alpha \right)∕K_A$ versus $\alpha$ corresponding to a single dipole (continuous line) and two dipoles (dotted line) for $r_0∕a^{\prime }={10}^4$. The critical aspect ratios ${\alpha }_{c_1}$ and ${\alpha }_{c_2}$ are indicated by dashed lines. Note that the aspect ratio for which the two-dipole configuration becomes energetically favored occurs for $\alpha >4.2$.

Figure 12

(Color online) (Top) Ground states for square (left) and triangular (right) arrays of bumps. The first and second rows correspond to moderate values of the aspect ratio $\alpha$, respectively. For simplicity, we assume that $r_0$, the bump width, is comparable to the lattice spacing. Positive defects (red dots) “screen” regions of positive Gaussian curvature while negative ones (blue dots) are located on the saddles of the “hilly” landscape.

Figure 13

(a) The free energy for a nematic (double headed vector field) living on a Gaussian bump surrounded by an aligning circular wall is plotted for $\alpha =2$ as a function of the scaled radial coordinates $x_1$ and $x_2$ of the two disclinations. The radial coordinates have been scaled by $r_0$ and the size of the system is $R=7r_0$. Note that the energy plot is symmetric with respect to the line $x_1=x_2$. (b) Schematic illustration of the positions of the two disclinations (black dots) corresponding to the deep energy minima at positions $x_1=0.04$ and $x_2=4.9$ (or vice versa) and to a shallow minimum at $x_1=x_2=4.7$. The two defects are on opposite sides of the bump. The continuous line corresponds to the circular boundary and the dashed one to the circle of zero Gaussian curvature and radius $r_0$ (drawing not to scale).

Figure 14

Plot of the free energy of a nematic (double headed vector field) on a Gaussian bump encircled by an aligning wall as a function of $\alpha$. The dotted line represents the energy of the fully deconfined configuration in Fig. 13b top panel while the continuous line corresponds to the defect pattern illustrated in the bottom panel of Fig. 13b. The energy of the fully deconfined configuration is approximately independent of $\alpha$ because the two disclinations are far away from the bump.

Figure 15

Plot of the free energy of a hexatic phase (draped on the Gaussian bump encircled by a wall) as a function of $\alpha$. The dotted line represents the energy of the hexagonal configuration illustrated in the top panel on the left while the continuous line corresponds to the pentagonal arrangement in the bottom panel. The outer defect rings in both configurations are approximately 90% of $R$. The critical aspect ratio ${\alpha }_D$ corresponding to the deconfinement transition discussed in the text is indicated by the dashed line.

Figure 16

Graphic construction of the stereographic projection. Regions close to the north pole have larger images in the conformal plane than regions of equal areas close to the south pole. The stereographic projection preserves the topology of the surface provided all points at infinity are identified with the north pole.

Figure 17

(a) Defects with fixed core size $a$ on a Gaussian bump encircled by a circular boundary of radius $R$ denoted by $B$. (b) The size of the vortex cores varies with position when plotted in the conformal plane. One can avoid the singularities associated with the defects cores by puncturing the conformal plane. This introduces circular boundaries $C_i$ of varying radius at the position of each defect in the conformal plane, reflecting corresponding constant core radii on the Gaussian bump.

Figure 18

(a) Schematic illustration of the boundary director texture corresponding to free boundary conditions. The vector order parameter orientation close to the edge of the sample does not vary appreciably as one moves along the radial direction and it is parallel to itself at every point on the boundary. (b) Tangential boundary conditions. The vector order parameter is locally aligned to a wall located at the edge of the sample.

Figure 19

Schematic illustration of the method of images. The image defect is of the same sign for free boundary conditions (a) and opposite for fixed boundary conditions (b). Defects closer to the center of the circle have images further away from it.

Figure 20

A topological defect located at position $P$ in a circular domain of radius $\overline{OQ}$ in flat space. Fixed boundary conditions are obtained by placing an image defect of the same sign at a distance $\overline{O{P}^{\prime }}$ from the center such that $\overline{OP}\overline{O{P}^{\prime }}={\overline{OQ}}^2$. The two triangles $\Delta OQP$ and $\Delta OQ{P}^{\prime }$ are similar and $\angle OQP=\angle O{P}^{\prime }Q=\pi -{\theta }^{\prime }$. By the theorem of the external angle, we conclude that ${\theta }^{\prime }+\theta =\varphi +\pi$ as long as $Q$ lies on the circumference $B$. This is equivalent to the boundary condition in Eq. (D2) if a nonrotating vector basis is used. Similarly, for free boundary conditions the symmetric Green’s function is constant on the boundary if the image defect is negative. Since $\overline{PQ}∕\overline{P^{\prime }Q}=\overline{OP}∕\overline{OQ}$, the potential $\ln \left(\overline{PQ}\right)-\ln \left(\overline{P^{\prime }Q}\right)$ (generated by the defect at distance $\overline{OP}$ and its image) is constant on the circumference of radius $\overline{OQ}$.