Key-lock mechanism in nematic colloidal dispersions

We consider the interaction between two-dimensional nematic colloids and planar or sculpted walls. The elastic interaction between colloidal disks and flat walls, with homeotropic boundary conditions, is always repulsive. These repulsions may be turned into strong attractions at structured or sculpted walls, with cavities that match closely the shape and size of the colloids. This key-lock mechanism is analyzed in detail for colloidal disks and spherocylindrical cavities of various length to depth ratios, by minimizing the Landau–de Gennes free energy functional of the nematic orientational order parameter. We find that the attractions occur only for walls with cavities within a small range of the colloidal size and a narrow range of orientations with respect to the cavity’s symmetry axis.

Figure 1

(Color online) Reduced interaction energy as a function of the distance $R$ from the center of the colloid, of radius $a$, to the wall. $\overline{F}=F∕K$, where $K=2L{Q}_{\mathit{bulk}}$ is the Frank elastic constant, and $F_0$ is the Landau–de Gennes free energy of the system without colloid. The inset illustrates the order parameter maps at different $R$. (a) $R∕a=1.1$; (b) $R∕a=2.0$; (c) $R∕a=4.0$. The nematic order parameter varies between $Q=Q_{\mathit{bulk}}$ (white regions) and $Q=0$ (colored regions).

Figure 2

Defect distance $r_d$ from the center of the disk (continuous line) and defects orientation $\delta \theta$ (dash-dotted line), with respect to their orientation in isolated disks, as a function of $R$.

Figure 3

Structured wall: geometry and notation.

Figure 4

Reduced interaction energy as a function of the distance $R$ of the center of the colloid to the midpoint of the cavity’s entrance for several widths $\left(r∕a=0.25,0.50,0.75,1.00,1.25\right)$. The depth is $d∕a=0.01$.

Figure 5

(Color online) Order parameter maps for a cavity with $r∕a=1.1$ and $d∕a=0.01$, and different colloidal separations $R$. (a) $R∕a=3.0$; (b) $R∕a=0.6$; (c) $R∕a=-0.32$. The nematic order parameter varies between $Q=Q_{\mathit{bulk}}$ (white regions) and $Q=0$ (colored regions).

Figure 6

Minimal interaction energy (left) and equilibrium position $R_{min}$ (right) as functions of the radius of the cavity $r∕a$.

Figure 7

Left: interaction energy as a function of the separation $R$ for several cavities $\left(d∕a=0.01,0.10,0.50,1.00\right)$ with radius $r∕a=1.5$. Right: interaction energy as a function of the separation $R$ for several depths $\left(d∕a=0.01,1.00\right)$. The radius of the cavity is $r∕a=3.0$. The lines in bold correspond to the minimal energy configurations. Thinner lines correspond to metastable solutions.

Figure 8

(Color online) Critical depth $d_c$ as a function of the radius of the cavity. The continuous line is the fit $d_c∕a=0.48+1.14r∕a$. Inset: Order parameter map for a nematic filled cavity of depth $d⪢d_c$. The nematic order parameter varies between $Q=Q_{\mathit{bulk}}$ (white regions) and $Q=0$ (colored regions).

Figure 9

Reduced interaction energy along four lines parallel to the wall $\left(R∕a=2.1,3.1,4.1,9.1\right)$ as a function of the lateral distance from the cavity $x∕a$. The radius and depth of the cavity are $r∕a=1.5$ and $d∕a=1.00$, respectively.