Spherical microparticles with Saturn ring defects and their self-assembly across the nematic to smectic-$A$ phase transition

We report experimental studies on the Saturn ring defect associated with a spherical microparticle across the nematic $\left(N\right)$ to smectic-$A$ $\left(\mathrm{Sm}A\right)$ phase transition. We observe that the director distortion around the microparticle changes rapidly with temperature. The equilibrium interparticle separation and the angle between two quadrupolar particles in the $N$ phase are larger than those of the $\mathrm{Sm}A$ phase. They are almost independent of the temperature in both phases, except for a discontinuous jump at the transition. We assembled a few particles using a laser tweezer to form a two-dimensional colloidal crystal in the $N$ phase. The lattice structure of the crystal dissolves irreversibly across the $N-\mathrm{Sm}A$ phase transition. The results on the pretransitional behavior of the defect are supported by the Landau–de Gennes $Q$-tensor modeling.

Figure 1

(a)–(d) Optical photomicrographs (obtained from polarizing optical microscopy) of a quadrupolar colloidal particle of diameter $8\mu \mathrm{m}$ in an 8CB filled cell of thickness $11\mu \mathrm{m}$ under crossed polarizers at various temperatures across the $N-\mathrm{Sm}A$ phase transition. See Supplemental Material, video 1 (f)–(i) [26]. Corresponding images with the red wave-plate (green line denotes the slow axis) between the analyzer and the sample. Note that the blue and yellow colors have exchanged their positions around the microsphere in the $\mathrm{Sm}A$ phase [in Fig. 1] due to higher-order effects. (e) The streamlines of the director above the transition. (j) In the smectic phase, the boundary condition is incompatible with the layer structure at the equator, where the Saturn ring was present in the nematic phase. Note that the layer thickness is greatly exaggerated.

Figure 2

(a)–(c) Optical photomicrographs just before the $N-\mathrm{Sm}A$ phase transition of 8-$\mu \mathrm{m}$ homeotropic microsphere in a planar cell. Cell thickness: $11\mu \mathrm{m}$. (d)–(f) Photomicrographs with red wave-plate between the sample and the analyzer in the $N$ phase. (See Supplemental Material, video 2 [26].) (g)–(f) Simulated images with the wave plate at various ratios of $K_{33}/K_{11}$.

Figure 3

(a) Variation of center-to-center separation ($D$) between a pair of colloidal particles across the $N-\mathrm{Sm}A$ phase transition. The red and black color corresponds to particle diameters of 8 and $5.2\mu \mathrm{m}$, respectively. (b) Variation of $\theta$ with temperature across the $N-\mathrm{Sm}A$ phase transition.

Figure 4

Variation of the free energy with the angle ($\theta$) at two different ratios of $K_{33}/K_{11}$ obtained by Landau–de Gennes modeling for the nematic phase. The free energy minimum is around ${79}^{\circ }$ for both ratios.

Figure 5

Optical photomicrographs of a pair of colloidal particles in the $\mathrm{Sm}A$ phase (at $T-T_{NA}$ = $-1{}^{\circ }\mathrm{C}$), without (a) and with (b) the wave plate. The blue and yellow colors around the microsphere in the $\mathrm{Sm}A$ phase are second-order colors. Particle diameter: $8\mu \mathrm{m}$. (c) A sketch of the smectic layer arrangement with marked focal lines that are visible in the photomicrographs.

Figure 6

Optical photomicrographs of a 2D quadrupolar crystal in the (a, c) $N$ phase ($36{}^{\circ }\mathrm{C}$) and (b, d) $\mathrm{Sm}A$ phase ($32.5{}^{\circ }\mathrm{C}$) of 8CB liquid crystal; (a,b) without polarizers and (c, d) with crossed polarizers. Particle diameter $5.2\mu \mathrm{m}$, and cell thickness $8\mu \mathrm{m}$. (e) Temperature variation of lattice parameters $a$ and $b$ across the $N-\mathrm{Sm}A$ transition (see Supplemental Material, video 3 [26]).