Manifold configurations of the director field formed by topological defects in free and confined geometry in smectic films

We study the topology of the $\mathbf{c}$-director field near topological defects with point core and with a droplet in the core of the defect in nonpolar smectic-$C$ and ferroelectric smectic-$C*$ freestanding films using polarized optical microscopy. Free and confined geometry of topological defects and droplets with strong outer boundary condition are compared. The $\mathbf{c}$-director field can be remarkably different around a point defect and a droplet with the same topological charge $S=+1$. In ferroelectric films, splay deformation of the $\mathbf{c}$-director transforms into bend deformation after droplet nucleation. Heating a ferroelectric film with an $S=+1$ droplet leads to a dramatic change of the $\mathbf{c}$-director topology from bend to splay. In confined geometry we found spiral structures in which the $\mathbf{c}$-director has opposite direction of rotation along the inner and outer boundaries of the island. Our observations are discussed on the basis of theories taking into account both the influence of polarity and of confined geometry on elasticity and topology of the $\mathbf{c}$-director field.

Figure 1

A nonpolar $\mathrm{Sm}C$ freestanding film. (a) A point topological defect with topological charge $S=+1,T=112$°С. The topological defect (c) after nucleation of a nematic droplet in the core of the defect, $T=115$°С, DOBA. The photographs were made under crossed polarizers (P, A). A schematic representation of the $\mathbf{c}$-director configuration around the topological defect (b) and the droplet (d).

Figure 2

A ferroelectric $\mathrm{Sm}C*$ freestanding film. (a) A point defect with topological charge $S=+1$. The topological defect (c) after nucleation of a cholesteric droplet in the core of the defect on heating, $T=109$°С, 11BSMHOB. The photographs were made under slightly decrossed polarizers (P, A). A schematic representation of the $\mathbf{c}$-director configuration around the topological defect (b) and the droplet (d).

Figure 3

Modification of the $\mathbf{c}$-director field near a cholesteric droplet with topological charge $S=+1$ in a ferroelectric film. At low temperature tangential boundary conditions on the droplet boundary as in Fig. 2 are observed. At high temperature (a) the director at the droplet boundary rotates by 90°, the boundary conditions become radial (b), $T=107.6$ °C. The photograph was made under slightly decrossed polarizers (P, A). The white line in (a) indicates the $\mathbf{c}$-director orientation in dark regions. 9BSMHOB.

Figure 4

(a) A nonpolar $\mathrm{Sm}C$ island with an $S=+1$ topological defect. $T=52.7$ °C, HOPDOB. (b) A schematic representation of the pure bend $\mathbf{c}$-director configuration. (c) A ferroelectric $\mathrm{Sm}C*$ island with an $S=+1$ topological defect. Black dots show the calculated spiral structure for orientations of the $\mathbf{c}$-director $\phi =\pi /4$, 3π/4, 5π/4, and 7π/4 and $K_B/K_S=3.8$. $T=101\hspace{0.17em}°\mathrm{C}$, 9BSMHOB. The dark areas around the islands (a, c) are two-layer films. The photographs are taken in reflection with crossed polarizers (P, A). (d) A schematic representation of the spiral with $\pi /2$ rotation of the $\mathbf{c}$-director.

Figure 5

Dots show the orientation of the $\mathbf{c}$-director (angle $\psi$) as a function of the distance from the island center obtained using Fig. 4. Curves are the calculated dependences of $\psi \left(r\right)$ obtained for different values of the ratio $K_B/K_S$ : curve 1 for $K_B/K_S=3.8$, curve 2 for $K_B/K_S=2.5$, curve 3 for $K_B/K_S=7$. $R_i=37$ μm, $r_0=0.5\mu \mathrm{m}$.

Figure 6

(a) A ferroelectric $\mathrm{Sm}C*$ island with a cholesteric droplet corresponding to the topological charge $S=+1$. $T=104.8$ °C, 9BSMHOB. Reflection image under slightly decrossed polarizers (P, A). Points are the calculated spiral structure for orientations of the $\mathbf{c}$-director $\phi =\pi /4$, 3π/4, 5π/4, and 7π/4 and $K_B/K_S=3.8$. (b) A schematic representation of the spiral with rotation of the $\mathbf{c}$-director by $\pi$.