Topological defect transformation and structural transition of two-dimensional colloidal crystals across the nematic to smectic-$A$ phase transition

We observe that topological defects in nematic colloids are strongly influenced by the elasticity and onset of smectic layering across the nematic $\left(N\right)$ to smectic-$A \left(\mathrm{Sm}A\right)$ phase transition. When approaching the $\mathrm{Sm}A$ phase from above, the nematic hyperbolic hedgehog defect that accompanies a spherical colloidal inclusion is transformed into a focal conic line in the $\mathrm{Sm}A$ phase. This phase transformation has a strong influence on the pairwise colloidal interaction and is responsible for a structural transition of two-dimensional colloidal crystals. The pretransitional behavior of the point defect is supported by Landau–de Gennes $Q$-tensor modeling accounting for the increasing elastic anisotropy.

Figure 1

(a),(c),(e) Experimental optical micrographs of $7\mu \mathrm{m}$ homeotropic microspheres in a thick planar 8CB cell between crossed polarizers in the nematic phase, (a) just before the $N\text{−}\mathrm{Sm}A$ transition (c) and after the transition (e). Cell thickness is $15\mu \mathrm{m}$. The hyperbolic hedgehog defect in the nematic phase is marked with an arrow (a). (b), (d) Simulated micrographs of the $N$ phase for (b) equal Frank elastic constants $K_{33}=K_{11}$ and for (d) increased bend elastic constant $\left(K_{33}=8K_{11}\right)$ which pushes the hedgehog towards the particle's surface. (f) The sketch of a reconstructed cross section of smectic layer structure in the vicinity of the microsphere.

Figure 2

(a) Variation of the center to center separation $\left(D\right)$ of two homeotropic microspheres across the $N\text{−}\mathrm{Sm}A$ transition in 8CB. See SM (Video-2). The microsphere diameter is $5.2\mu \mathrm{m}$, and the cell thickness is $7.7\mu \mathrm{m}$. Insets show experimental images of a pair of dipolar microspheres before and after $N\text{−}\mathrm{Sm}A$ transition. (b) Variation of $D/R_0$ above the $N\text{−}\mathrm{Sm}A$ transition as a function of the elastic constant anisotropy $\left(1-K_{11}/K_{33}\right)$. The red line is the best fit to Eq. (1) with a slope $0.06$. The scale on the top shows the temperature range that corresponds to the anisotropy range on the principal axis. (c) Variation of the separation with reduced temperature below the transition (logarithmic scale). The red line shows the best fit to the power law $(D-D_0)\sim {\left|\chi \right|}^{\alpha }$ with $\alpha =0.38$.

Figure 3

Optical photomicrograph of a 2D dipolar crystal of particles with diameter $5.2\mu \mathrm{m}$ in the $N$ phase $\left(35.8{}^{\circ }\mathrm{C}\right)$ and $\mathrm{Sm}A$ phase $\left(32.7{}^{\circ }\mathrm{C}\right)$ of 8CB. (a),(b) Between crossed polarizers; (c),(d) without polarizers. See SM video-4. (e) Temperature variation of the lattice parameters $a$ and $b$ across the $N\text{−}\mathrm{Sm}A$ transition. Inset: Temperature variation of $b$ with the best fit (red line) to $b=b_0+C{\left|\chi \right|}^{-\beta }$ with $\beta =0.8$ (logarithmic scale). Diameter of the microspheres is $5.2\mu \mathrm{m}$ and cell thickness is $9.3\mu \mathrm{m}$.