Decomposition of strongly charged topological defects

We study decomposition of geometrically enforced nematic topological defects bearing relatively large defect strengths $m$ in effectively two-dimensional planar systems. Theoretically, defect cores are analyzed within the mesoscopic Landau–de Gennes approach in terms of the tensor nematic order parameter. We demonstrate a robust tendency of defect decomposition into elementary units where two qualitatively different scenarios imposing total defect strengths on a nematic region are employed. Some theoretical predictions are verified experimentally, where arrays of defects bearing charges $m=\pm 1$ and even $m=\pm 2$ are enforced within a plane-parallel nematic cell using an atomic force microscopy scribing method.

Figure 1

The phase space $(s,\psi )$ revealing possible nematic states. Solid lines: Positively uniaxial states $(S>0);\vec{n}[\psi =0]={\vec{e}}_1,\vec{n}[\psi =2\pi /3]={\vec{e}}_2,\vec{n}[\psi =-2\pi /3]={\vec{e}}_3$. Dashed lines: Negatively uniaxial states ($S<0$); $\vec{n}[\psi =\pi ]={\vec{e}}_1,\vec{n}[\psi =-\pi /3]={\vec{e}}_2,\vec{n}[\psi =\pi /3]={\vec{e}}_3$. Dotted lines: States with maximal degree of biaxiality ${\beta }^2=1.$ The parameter $s$ attains its maximum value on the circle, and the center of the circle corresponds to the isotropic phase, $s=0$. The dashed red line indicates a trajectory in the order parameter space $\{s,\psi \}$ joining the points A and B (uniaxial states) of the $m=1/2$ defect core structure depicted in Fig. 2.

Figure 2

A characteristic degree of biaxiality ${\beta }^2(x,y)$ plot of an $m_0=1/2$ topological defect: (a) top view with the color code of ${\beta }^2\in [0,1],$ (b) side view. The dashed line indicates a path joining states $\vec{n}={\vec{e}}_1$ with positive uniaxiality (point B) and $\vec{n}={\vec{e}}_2$ with negative uniaxiality (point A). The corresponding path in the order parameter space {$s,\psi$} is depicted in Fig. 1 with a dashed red line.

Figure 3

${\beta }^2(x,y)$ plots of configurations of TDs where we impose via BAC (a) $m=1$, (b) $m=2$, (c) $m=3$, and (d) $m=4$. In all cases only TDs exhibiting unit charge $m_0=1/2$ exist. $A_e=8, \tau =4$.

Figure 4

${\beta }^2(x,y)$ plots of configurations of TDs where we impose via FAC a single $m=1$ defect. (a) $A_s=0.3$, (b) $A_s=0.44$, (c) $A_s=0.45$. $A_e=10, \tau =4$. In (c) we indicate a trajectory that transverses the core of TD. Here B indicates a point where nematic ordering exhibits essentially positive uniaxiality, and A marks a point where the order displays negative uniaxiality. The corresponding trajectory in the order parameter space {$s,\psi$} is schematically sketched in Fig. 1 by a dashed red line.

Figure 5

${\beta }^2(x,y)$ plots of configurations of TDs where we impose via FAC a single $m=2$ defect. (a) $A_s=0.8$, (b) $A_s=1.2$, (c) $A_s=1.3$. $A_e=10, \tau =4$. In (c) we indicate a trajectory which transverses the core of TD. Here B indicates a point where nematic ordering exhibits essentially positive uniaxiality, and A marks a point where the order displays negative uniaxiality. The corresponding trajectory in the order parameter space {$s,\psi$} is schematically sketched in Fig. 1 by a dashed red line.

Figure 6

${\beta }^2(x,y)$ plots of configurations of TDs where we impose via the FAC an $m=2$ defect centered at ($x=0,y=0$). The defect is decomposed into four $m=1/2$ daughter TDs. The spatial orientation of their cores is affected by different local perturbations. (a) $A_s^{\left(p\right)}=0$; (b) $A_s^{\left(p\right)}=1, {\phi }_s^{\left(p\right)}=0$; (c) $A_s^{\left(p\right)}=1, {\phi }_s^{\left(p\right)}=\pi /2$; (d) $A_s^{\left(p\right)}=0.1, {\phi }_s^{\left(p\right)}=\pi /2$. In all figures we set $A_e=30, A_s=1, \tau =4$. The center of the perturbation patch of size $h_p={\xi }_b/2$ was imposed at coordinates $(x_p={\xi }_b/2,y_p={\xi }_b/2)$.

Figure 7

(a) Schematic representation of the “easy axis” director pattern of a $3\times 3$ array of $m=\pm 1$ topological defects. Filled circles correspond to positive defects and open circles to negative defects. (b) Extinction resulting from the scribed director pattern. Image taken by polarized microscopy. The polarizer and analyzer are crossed and oriented parallel/perpendicular to the horizontal. (c) A bright field microscopy image of the defect cores from (b). Each core is visible as a dark spot due to light scattering. The rows of three spots at the top and bottom of the image, partially obscured by the yellow square, are due to director discontinuities at the boundaries between the scribed square and outer region. The yellow boxes nominally denote the boundaries of the AFM lithography on each image. Scale: 30 $\mu \mathrm{m}$ rectilinear distance between scribed defect cores.

Figure 8

Same as Fig. 7 except for $m=\pm 2$ topological defects.

Figure 9

Scribed surface topography as scanned by an Agilent 5500 AFM and a TAP300 noncontact stylus. Noncontact mode was used to avoid damaging the topography. Scale bar = 250 nm.

Figure 10

${\beta }^2(x,y)$ plots of configurations of TDs calculated using the FAC. The defect of the strength $m=2$ is enforced at the center of the figure, where ($x=0,y=0$). We chose typical length scales which were roughly used in the experimental setup shown in Fig. 8; consequently $A_e=h/{\xi }_b^{\left(0\right)}=100$. The center of the perturbation patch of size $h_p={\xi }_b$ was imposed at coordinates $(x_p,y_p)$. (a) The reference pattern with $A_s^{\left(p\right)}=0$; (b) $(x_p=h/10,y_p=h/10), {\phi }_s^{\left(p\right)}=\pi /4, A_s^{\left(p\right)}=0.1$; (c) $(x_p=h/5,y_p=h/5), {\phi }_s^{\left(p\right)}=0, A_s^{\left(p\right)}=0.1$; (d) $(x_p=h/5,y_p=h/5), {\phi }_s^{\left(p\right)}=\pi /4, A_s^{\left(p\right)}=0.1$. In all cases $A_s=0.1, \tau =4$.