Key-lock colloids in a nematic liquid crystal

The Landau–de Gennes free energy is used to study theoretically the effective interaction of spherical “key” and anisotropic “lock” colloidal particles. We assume identical anchoring properties of the surfaces of the key and of the lock particles, and we consider planar degenerate and perpendicular anchoring conditions separately. The lock particle is modeled as a spherical particle with a spherical dimple. When such a particle is introduced into a nematic liquid crystal, it orients its dimple at an oblique angle ${\theta }_{\text{eq}}$ with respect to the far field director ${\mathbf{n}}_{\infty }$. This angle depends on the depth of the dimple. Minimization results show that the free energy of a pair of key and lock particles exhibits a global minimum for the configuration when the key particle is facing the dimple of the lock colloidal particle. The preferred orientation ${\varphi }_{\text{eq}}$ of the key-lock composite doublet relative to ${\mathbf{n}}_{\infty }$ is robust against thermal fluctuations. The preferred orientation ${\theta }_{\text{eq}}^{\left(2\right)}$ of the dimple particle in the doublet is different from the isolated situation. This is related to the “direct” interaction of defects accompanying the key particle with the edge of the dimple. We propose that this nematic-amplified key-lock interaction can play an important role in self-organization and clustering of mixtures of colloidal particles with dimple colloids present.

Figure 1

Sketch of the formation of a dimple particle. The red particle is the seed. The yellow particle, of radius $R$, corresponds to the dimple particle. When removing the volume occupied by the red particle, the yellow particle is left with a dimple of depth $A$, measured from the original spherical surface.

Figure 2

Schematic representation of a dimple colloidal particle with the radius $R$. The orientation of the particle is described by the vector $\mathbf{p}$, which makes an angle $\theta$ with respect to the far field director ${\mathbf{n}}_{\infty }$.

Figure 3

Landau–de Gennes free energy of an isolated dimple particle as a function of the orientation of the dimple relative to the far field director ${\mathbf{n}}_{\infty }$, for several dimple depths $A/R=0.25$, 0.5, and 0.75. (a) Free energy for a dimple particle with homeotropic anchoring. (b) Free energy for a dimple particle with planar degenerate anchoring. The reference state is taken for $\theta =0^{\circ }$. The panels represent the nematic configurations around dimple particles close to their equilibrium orientation ${\theta }_{\text{eq}}$. (c)–(e) For homeotropic anchoring ${60}^{\circ }\lesssim {\theta }_{\text{eq}}\lesssim {66}^{\circ }$. (h)–(j) For planar degenerate anchoring $40\lesssim {\theta }_{\text{eq}}\lesssim {56}^{\circ }$. The dimples have depths (c) and (h) $A=0.25R$, (d) and (i) $A=0.5R$, and (e)–(g) and (j) $A=0.75R$. For homeotropic anchoring, particles with large dimples exhibit a local minimum at $\theta =0^{\circ }$. (f) Configuration of a homeotropic dimple particle with $A=0.75R$ oriented at $\theta =4^{\circ }$. (g) Metastable configuration of a homeotropic dimple particle with $A=0.75R$ for $\theta =3^{\circ }$. The topological defects are represented as green isosurfaces of the scalar order parameter. The black bars represent the director field $\mathbf{n}$. $R=0.1\mu \text{m}$.

Figure 4

Schematic illustration of a pair of a dimple lock particle and a spherical key particle, both with the radius $R$. The center-to-center vector $\mathbf{d}$ makes an angle $\varphi$ with respect to the far field director ${\mathbf{n}}_{\infty }$. Upon varying $\mathbf{d}$ and $\varphi$, the orientation of the dimple particle is kept fixed.

Figure 5

Landau–de Gennes free energy $F$ for a pair of key and lock particles as a function of the angle $\varphi$ between the far field director ${\mathbf{n}}_{\infty }$ and the center-to-center vector $\mathbf{d}$, and the distance $d$ between particles; see Fig. 4. (a) Homeotropic and (b) planar degenerate anchoring conditions. In both cases, the orientation of the dimpled particle is fixed at $\theta ={\theta }_{\text{eq}}$; see Fig. 3. Here $2.1R\le d\le 4.5R$. At $d=2.1R$, the free-energy profiles exhibit four minima, and the corresponding nematic configurations are show in panels (c) $\varphi ={280}^{\circ }$, (d) $\varphi ={258}^{\circ }$, (e) $\varphi ={120}^{\circ }$, and (f) $\varphi ={60}^{\circ }$ for homeotropic anchoring; and in (g) $\varphi ={330}^{\circ }$, (h) $\varphi ={210}^{\circ }$, (i) $\varphi ={150}^{\circ }$, and (j) $\varphi ={56}^{\circ }$ for planar degenerate anchoring. The topological defects are shown as green isosurfaces of the scalar order parameter $Q$, and the director field $\mathbf{n}$ is represented by the black bars. Particles have equal radii, $R=0.1\mu \text{m}$.

Figure 6

Landau–de Gennes free energy $F$ for a pair of key and lock particles as a function of the distance $d$ at $A=0.75R$. The particles have equal radii, $R=0.1\mu \text{m}$. The orientation of the dimple particle is fixed at $\theta ={\theta }_{\text{eq}}$ (${\theta }_{\text{eq}}\simeq {60}^{\circ }$ for homeotropic anchoring, and ${\theta }_{\text{eq}}\simeq {56}^{\circ }$ for planar anchoring; see Fig. 3). (a) Homeotropic anchoring, $\varphi ={60}^{\circ }$; (b) planar degenerate anchoring, $\varphi ={56}^{\circ }$. (c) and (d) Nematic configuration corresponding to the left branch of $F$ in (a) at distances (c) $d=1.45R$ and (d) $d=1.99R$. (e) Nematic configuration corresponding to the right branch of the free energy in (a) at $d=2.02R$. (f) Alternative view of the entangled colloids showing how the ring defect is detached from the edge of the dimple. (g) Colloidal particles with planar degenerate anchoring at $d=1.45R$. The inset shows the bottom boojum of the key particle occupying the region of high nematic deformation produced by the edge of the dimple. The topological defects are shown as green isosurfaces of the order parameter $Q$, and the black bars represent the director field $\mathbf{n}$.

Figure 7

Landau–de Gennes free energy $F$ as a function of the orientation $\theta$ of the dimple particle (see Fig. 2), for (a) homeotropic and (b) planar degenerate anchoring conditions. In both cases $d=2.1R$, and $\varphi ={\varphi }_{\text{eq}}\simeq {60}^{\circ }$ for homeotropic anchoring and $\varphi ={\varphi }_{\text{eq}}\simeq {56}^{\circ }$ for planar degenerate anchoring conditions. $d$ and $\varphi$ are defined in Fig. 4. We take as a reference the local minimum of the free energy $F(\theta ={\theta }_{\text{eq}}^{\left(2\right)})$, where ${\theta }_{\text{eq}}^{\left(2\right)}\simeq {68}^{\circ }$ for homeotropic anchoring and ${\theta }_{\text{eq}}^{\left(2\right)}\simeq {48}^{\circ }$ for planar degenerate anchoring conditions. The particles have equal radii, $R=0.1\mu \text{m}$. The panels show the nematic configuration around homeotropic colloidal particles for dimple orientation $\theta ={68}^{\circ }$ for (c) the black (circles) energy branch and for (d) the red (squares) energy branch shown in (a). (e) Nematic configuration around particles with planar anchoring for dimple orientation $\theta ={48}^{\circ }$. The topological defects are shown as green isosurfaces of the order parameter $Q$, and the black bars represent the director field $\mathbf{n}$.