Curvature-driven stability of defects in nematic textures over spherical disks

Stabilizing defects in liquid-crystal systems is crucial for many physical processes and applications ranging from functionalizing liquid-crystal textures to recently reported command of chaotic behaviors of active matters. In this work, we perform analytical calculations to study the curvature-driven stability mechanism of defects based on the isotropic nematic disk model that is free of any topological constraint. We show that in a growing spherical disk covering a sphere the accumulation of curvature effect can prevent typical +1 and +1/2 defects from forming boojum textures where the defects are repelled to the boundary of the disk. Our calculations reveal that the movement of the equilibrium position of the +1 defect from the boundary to the center of the spherical disk occurs in a very narrow window of the disk area, exhibiting the first-order phase-transition-like behavior. For the pair of +1/2 defects by splitting a +1 defect, we find the curvature-driven alternating repulsive and attractive interactions between the two defects. With the growth of the spherical disk these two defects tend to approach and finally recombine towards a +1 defect texture. The sensitive response of defects to curvature and the curvature-driven stability mechanism demonstrated in this work in nematic disk systems may have implications towards versatile control and engineering of liquid-crystal textures in various applications.

Figure 1

The configurations and energetics of (a), (b) a single +1 defect and (c), (d) a pair of +1/2 defects on a planar disk. The defects are represented by red dots. The pair of +1/2 defects in (c) are constructed out of a +1 defect by inserting a uniform direct field between (indicated by green lines). The negative derivative of the Frank free energy $F\left(c\right)$ with respect to $c$ (the position of the defects) indicates that defects tend to slide to the boundary of a planar disk.

Figure 2

Spiral +1 defect patterns subject to typical pinning boundary conditions in a planar nematic disk.

Figure 3

Schematic plot of a +1 defect over a spherical cap. The relative position the defect is characterized by the ratio $r_d^{\prime }/r_s^{\prime }$.

Figure 4

Stability analysis of a +1 defect in a spherical nematic disk. (a)–(c) show the derivative of the Frank free energy $F_{+1,s}^{\prime }\left(c\right)$ versus $c$ at typical values for $r_s^{\prime }/R$. The curve starts to develop a stable zero point for the +1 defect with the increase of the disk size $r_s^{\prime }$. (a) $r_s^{\prime }/R=1.0$, (b) 1.414180, and (c) 1.414187. (d) shows the rapid movement of the equilibrium position of the +1 defect from the boundary $\left(r_d^{\prime }/r_s^{\prime }=1\right)$ to the center $\left(r_d^{\prime }/r_s^{\prime }=0\right)$ of the disk with the increase of the disk size $r_s^{\prime }/R$.

Figure 5

Stability analysis of a pair of +1/2 defects in a spherical nematic disk. (a) is the schematic plot of the defect pair over the spherical disk in gray. (b)–(d) show the derivative of the Frank free energy $F_{+1/2,s}^{\prime }\left(c\right)$ versus $c$ at typical values for $b/R$. $b$ is the radius of the circular boundary of the spherical cap. A pair of zero points appear in the curve with the increase of the disk size $b$. (b) $b/R=0.80000$, (c) 0.92933, and (d) 0.94000. The curvature-driven alternating repulsive and attractive regimes in the $F_{+1/2,s}^{\prime }\left(c\right)$ curve are indicated by the arrows in (d). In (e), we plot the variation of the equilibrium location of the defect pair versus the disk size $b/R$. $r_d^{\prime }$ is the Euclidean distance from one of the two defects to the center of the disk. The defect pair merge to form a +1 defect $\left(r_d^{\prime }/r_s^{\prime }\to 0\right)$ in the half sphere limit $\left(b/R\to 1\right)$.