Towards Template-Assisted Assembly of Nematic Colloids

Colloidal crystals belong to a new class of materials with unusual properties in which the big challenge is to grow large-scale structures of a given symmetry in a well-controlled and inexpensive way. Recently, template-assisted crystallization was successfully exploited experimentally in the case of colloidal particles dispersed in isotropic fluids. In liquid crystal (LC) colloids, particles are subjected to long-range anisotropic elastic forces originating from the anisotropic deformation of the underlying order parameter. These effective interactions are easily tunable by external electric or magnetic fields, light, temperature, or confinement and, thus, provide additional handles for better control of colloidal assembly. Here we use the coupling between microsculptured bounding surfaces and LC elasticity in order to guide the self-assembly of large-scale colloidal structures. We present explicit numerical calculations of the free energy landscape of colloidal particles in the presence of convex protrusions modeled as squared pyramids comparable to the size of the particles. We show the existence of strong trapping potentials that are able to efficiently localize the colloidal particles and withstand thermal fluctuations. Three-dimensional optical imaging experiments support the theoretical predictions.

Figure 1

Schematic illustrations of a spherical colloidal particle of the radius $R$ in the vicinity of a convex protrusion modeled as a pyramid with the square base of length $2R$ and height $2R$. (a) Homeotropic anchoring at the particle’s surface (SP in the experiments) and tangential degenerate anchoring at the substrate. The far-field director ${\mathbf{n}}_0$ is parallel to the $x$ axis. (b) Tangential degenerate anchoring at the surface of the colloidal particle (MR in the experiments) and homeotropic anchoring at the substrate. ${\mathbf{n}}_0$ is along the $z$ axis.

Figure 2

Experimental textures of particles elastically trapped at the tip of the pyramids patterned at the glass substrate: unpolarized bright field (a) and 3PEF-PM in-plane $xy$ (b) and cross sectional $zx$ (d) textures of a substrate patterned with a pyramid array in a planar liquid crystal cell. (c) SP with homeotropic anchoring and a Saturn ring defect elastically trapped on the tip of a pyramid with planar anchoring in the thin (e) and thick (f) nematic cell. (g) MR particle with tangential degenerate anchoring elastically trapped on the tip of a pyramid with homeotropic anchoring. White double arrow and crossed circle show a far-field director ${\mathbf{n}}_0$, respectively, in plane and out of plane of the image. Dark red double arrow shows the direction of the polarization of excitation beam. Scale bar is $5\mu \mathrm{m}$.

Figure 3

Colloidal particles trapped at the tips of pyramidlike protrusions: the centers of the colloidal particles and the pyramid tips are aligned in the vertical direction. The surface-to-surface distance $d$ from the tips to the particles $d=0.1R$. (a),(b) Strong ($w_{\text{col}}=22$) planar degenerate anchoring is assumed at the surface of the particle with homeotropic anchoring at the patterned substrate. (c),(d) Strong homeotropic anchoring at the particle surface and planar degenerate anchoring at the substrate. Regimes of a neutral ($w_{\text{sub}}=0$), (a),(c), and strong ($w_{\text{sub}}=22$) anchoring (b),(d) substrates are presented. Numerically calculated $\mathbf{n}(\mathbf{r})$ is shown by short black lines. Isosurfaces of reduced scalar order parameter $Q=0.5Q_b$ are shown in blue color. The sharp edges of the pyramids also induce a reduction of the local orientational order but correspond to regions where $Q>0.5Q_b$; hence, they are not represented here.

Figure 4

Landau–de Gennes free energy $F$ as a function of the lateral Cartesian coordinates ($x_{\text{col}}$, $y_{\text{col}}$) of the center of the particle with radius $R$, with planar degenerate anchoring ($w_{\text{col}}=22$) at a pyramid-decorated substrate with homeotropic anchoring ($w_{\text{sub}}=22$). The $z$ coordinate of the colloid’s center $z_{\text{col}}$ is fixed $z_{\text{col}}=z_{\text{tip}}+1.1R$. The Cartesian coordinates of the pyramid tip $(x_{\text{tip}},y_{\text{tip}},z_{\text{tip}})=(0,0,2R)$ and $F_{\text{tip}}\equiv F(0,0)$.

Figure 5

Binding potential (a) as a function of $x_{\text{col}}$ at fixed $z_{\text{col}}=z_{\text{tip}}+1.1R$ and $y_{\text{col}}=y_{\text{tip}}$. $F_{\max }\equiv F(x_{\text{col}}-x_{\text{tip}}=2R)$. (b) As a function of the surface-to-surface distance $d=z_{\text{col}}-z_{\text{tip}}-R$ to the tip of the pyramid at fixed $(x_{\text{col}},y_{\text{col}})=(x_{\text{tip}},y_{\text{tip}})$ for several values of $w_{\text{sub}}$ at the homeotropic substrate, $F_{\max }\equiv F(d=2.1R)$. Inset in (b) shows the vertical component of the trapping force ${\mathrm{\Phi }}_z$ exerted on the particle in the trapped state, $(x_{\text{col}},y_{\text{col}})=(x_{\text{tip}},y_{\text{tip}})$, $d=0.1R$ as a function of the anchoring strength $w_{\text{sub}}$. The particle has planar anchoring with $w_{\text{col}}=22$.