Colloidal Surfaces with Boundaries, Apex Boojums, and Nested Elastic Self-Assembly of Nematic Colloids

Self-assembly of colloidal particles is poised to become a powerful composite material fabrication technique, but remains challenged by a limited control over the ensuing structures. We develop a new breed of nematic colloids that are physical analogs of a mathematical surface with boundary, interacting with the molecular alignment field without inducing defects when flat. However, made from a thin nanofoil, they can be shaped to prompt formation of self-compensating defects that drive preprogramed elastic interactions mediated by the nematic host. To show this, we wrap the nanofoil on all triangular side faces of a pyramid, except its square base. The ensuing pyramidal cones induce point defects with fractional hedgehog charges of opposite signs, spontaneously align with respect to the far-field director to form elastic dipoles and nested assemblies with tunable spacing. Nanofoils shaped into octahedrons interact as elastic quadrupoles. Our findings may drive realization of low-symmetry colloidal phases.

Figure 1

Nematic colloidal analogs of mathematical surfaces (a), (c) with and (b), (d) without boundary. (a), (b) Optical micrographs of discs made of (a) thick foil with $h_f\approx 1\mu \mathrm{m}$ and (b) thin foil with $h_f\approx 100\mathrm{nm}$ obtained in brightfield (left) and polarizing imaging modes without (middle) and with (right) a 530 nm phase retardation plate with a fast axis $\gamma$ inserted between the crossed polarizer ($P$) and analyzer ($A$); red arrows indicate boojums. (c), (d) Schematic illustrations of $\mathbf{n}(\mathbf{r})$ around the (c) thick and (d) thin discs, with the boojums in (c) depicted as red hemispheres.

Figure 2

CPCs in a nematic LC. (a) Schematic of a CPC. (b) Scanning electron microscopy of CPCs made from gold foil with (from top left to bottom right) $h_f\approx 100\mathrm{nm}$, $h_f\approx 200\mathrm{nm}$, $h_f\approx 300\mathrm{nm}$, $h_f\approx 1\mu \mathrm{m}$, $h_f\approx 1.1\mu \mathrm{m}$. (c) Polarizing optical micrograph and (d) schematic of $\mathbf{n}(\mathbf{r})$ and defects around particles that orient with $\mathbf{b}\perp {\mathbf{n}}_0$. (e) Polarizing optical micrograph and (f) schematic of $\mathbf{n}(\mathbf{r})$ and defects around CPCs with $\mathbf{b}\parallel {\mathbf{n}}_0$. Complementary red fragments of spheres in (d), (f) show fractional boojums with undefined $\mathbf{n}(\mathbf{r})$. Orientations of crossed polarizer ($P$) and analyzer ($A$) and ${\mathbf{n}}_0$ are shown using double arrows. In (c), (e), $h_f\approx 100\mathrm{nm}$ and red arrows indicate boojums.

Figure 3

Self-assembly of CPCs with $\mathbf{b}\perp {\mathbf{n}}_0$. (a) Experimental (squares) $r_{cc}(t)$ for CPCs with coaligned ${\mathbf{d}}_e$ vectors and its fit (red line) shown for $r_0=7.7\mu \mathrm{m}$ and ${\alpha }_d=456.9\mu {\mathrm{m}}^5/\mathrm{s}$. (b) Color-coded time trajectories of two CPCs with antiparallel ${\mathbf{d}}_e$ released simultaneously from optical traps in a homeotropic cell, demonstrating that particles rotate around the vertical ${\mathbf{n}}_0$ (orthogonal to the image plane and glass substrates) and translate to always attract and form a nested self-assembly despite being released with antiparallel initial orientations of ${\mathbf{d}}_e$. The trajectories were constructed from videomicroscopy data, such as the ones shown in the Supplemental Material video S1 [11]. Relative positions of the particles released from laser traps are shown with filled black circles atop of CPCs. (c) Brightfield (left), polarizing (middle), and reflection-mode (right) optical micrographs of a nested assembly of CPCs in a planar cell. The equilibrium interbase distance within the nested assembly is $\approx 0.9\mu \mathrm{m}$. (d) Schematic of $\mathbf{n}(\mathbf{r})$ around a nested colloidal assembly of CPCs. $h_f\approx 100\mathrm{nm}$.

Figure 4

Pair interactions and self-assembly of CPCs with $\mathbf{b}\parallel {\mathbf{n}}_0$. (a) Experimental $r_{cc}(t)$ for CPCs with antiparallel $\mathbf{b}$, repelling at ${\mathbf{r}}_{cc}\parallel {\mathbf{n}}_0$ (blue curve) and attracting at ${\mathbf{r}}_{cc}\perp {\mathbf{n}}_0$ (red curve). (b) Experimental $r_{cc}(t)$ for CPCs with parallel $\mathbf{b}$, attracting at ${\mathbf{r}}_{cc}\parallel {\mathbf{n}}_0$ (red curve) and repelling at ${\mathbf{r}}_{cc}\perp {\mathbf{n}}_0$ (blue curve). Black arrows in (a), (b) indicate directions of elastic forces. (c) Color-coded (according to the color scheme shown at the top of the figure part) time trajectories of CPCs released from optical traps at different angles between ${\mathbf{r}}_{cc}$ and ${\mathbf{n}}_0$, as constructed from videomicroscopy data such as the ones shown in the Supplemental Material video S2 [11]. (d) Schematic and (e) brightfield (left), polarizing (middle), and reflection (right) optical micrographs of a nested assembly of CPCs. The CPC assemblies with ${\mathbf{d}}_e\parallel {\mathbf{n}}_0$ have an average interbase distance of $\approx 1.0\mu \mathrm{m}$. $h_f\approx 100\mathrm{nm}$.

Figure 5

Nematic colloidal octahedrons. (a) Brightfield (left), polarizing (middle), and reflection (right) micrographs showing an octahedron in a nematic LC, with the particle-induced distortion of $\mathbf{n}(\mathbf{r})$ and surface defects depicted in (b). (c) Brightfield (left) and polarizing (right) micrographs of an octahedron and (d) the corresponding $\mathbf{n}(\mathbf{r})$ and defects. (e), (f) $r_{cc}(t)$ for two colloidal octahedrons with ${\mathbf{r}}_{cc}$ at $\approx 45°$ to ${\mathbf{n}}_0$ (e) and along it (f) obtained from the videomicroscopy data such as the ones shown in the Supplemental Material video S3 [11]. The red curve is a fit with $r_{cc}(t)={(r_0^7-7{\alpha }_{q}t)}^{1/7}$ expected for quadrupoles, yielding $r_0=9.3\mu \mathrm{m}$ and ${\alpha }_q=2.6\times {10}^4\mu {\mathrm{m}}^7/\mathrm{s}$. Insets in (e) and (f) are brightfield micrographs of the interacting octahedrons extracted as frames from the same videos at the elapsed times indicated by arrows. $h_f\approx 100\mathrm{nm}$.