Equilibrium state of a cylindrical particle with flat ends in nematic liquid crystals

A continuum theory is employed to numerically study the equilibrium orientation and defect structures of a circular cylindrical particle with flat ends under a homeotropic anchoring condition in a uniform nematic medium. Different aspect ratios of this colloidal geometry from thin discotic to long rodlike shapes and several colloidal length scales ranging from mesoscale to nanoscale are investigated. We show that the equilibrium state of this colloidal geometry is sensitive to the two geometrical parameters: aspect ratio and length scale of the particle. For a large enough mesoscopic particle, there is a specific asymptotic equilibrium angle associated to each aspect ratio. Upon reducing the particle size to nanoscale, the equilibrium angle follows a descending or ascending trend in such a way that the equilibrium angle of a particle with the aspect ratio bigger than 1:1 (a discotic particle) goes to a parallel alignment with respect to the far-field nematic, whereas the equilibrium angle for a particle with the aspect ratio 1:1 and smaller (a rodlike particle) tends toward a perpendicular alignment to the uniform nematic direction. The discrepancy between the equilibrium angles of the mesoscopic and nanoscopic particles originates from the significant differences between their defect structures. The possible defect structures related to mesoscopic and nanoscopic colloidal particles of this geometry are also introduced.

Figure 1

Schematic illustration of the studied system.

Figure 2

Defect structures of a mesoscopic cylindrical particle with $R=160\mathrm{nm}$ and aspect ratio 1:1 for $W={10}^{-2}\mathrm{J}/{\mathrm{m}}^2$. Black dashed lines are a 2D view of the director field in the $y-z$ plane. Topological defect cores are visualized by the isosurface $S_q=0.5$ of the rescaled bulk scalar order parameter. [(a)–(c)] Defect structures for $\theta =0$ in descending order of stability: (a) top-ring, (b) chairlike, and (c) mid-ring-0. [(d)–(f)] Defect structures for $\theta =90$ in descending order of stability: (a) chairlike, (b) boatlike, and (c) mid-ring-90. The insets of (a) and (c) illustrate the director field across the defect cores of the corresponding defect lines.

Figure 3

(a) Total free energy of the cylindrical particles with the aspect ratios from 8:1 to 1:6 as a function of $\theta$ for the homeotropic anchoring constant $W={10}^{-2}\mathrm{J}/{\mathrm{m}}^2$. Solid squares are related to the chairlike structure and open circles represent the top-ring structure. All the particles have equal surface areas with a particle of the aspect ratio 1:2 and $R=160\mathrm{nm}$. The angular resolution is $5^{\circ }$ for the chairlike structure and $2^{\circ }$ for the top-ring structure. (b) Equilibrium angle as a function of aspect ratio. The equilibrium defect structures for the aspect ratios 1:4 and 6:1 (as typical examples respectively for rodlike and discotic particles) are shown in the insets of (b) in which black dashed lines are 2D view of the director field in the $y-z$ plane and the topological defect cores are visualized by the isosurface $S_q=0.5$.

Figure 4

Normalized free energy as a function of $\theta$ when the particle radius changes from mesoscale to nanoscale considering $W=5\times {10}^{-3}\mathrm{J}/{\mathrm{m}}^2$ for (a) the aspect ratio 2:1 and (b) the aspect ratio 1:2.

Figure 5

Equilibrium angle as a function of the particle radius for the anchoring constant $W=5\times {10}^{-3}\mathrm{J}/{\mathrm{m}}^2$. Small nanoparticles with $R=10\mathrm{nm}$ ($w=2$) to mesoscopic particles with $R=320\mathrm{nm}$ ($w=64$) are considered. The angular resolution for the equilibrium angles is $5^{\circ }$.

Figure 6

Equilibrium defect structures around colloidal particles with the aspect ratio 1:1 for [(a) and (c)] $R=240\mathrm{nm}$ and [(b) and (d)] $R=40\mathrm{nm}$ and $W=5\times {10}^{-3}\mathrm{J}/{\mathrm{m}}^2$. Black dashed lines in (a) and (b) are a 2D view of the director field in the $y-z$ plane. Topological defect cores are visualized by the isosurface $S_q=0.5$.

Figure 7

[(a)–(c)] Defect structures around a colloidal particle with the aspect ratio 1:2 and $R=10\mathrm{nm}$ at $0^{\circ }$, ${45}^{\circ }$ and ${90}^{\circ }$. [(d)–(f)] Defect structures around a colloidal particle with the aspect ratio 2:1 and $R=20\mathrm{nm}$ at $0^{\circ }$, ${45}^{\circ }$ and ${90}^{\circ }$. The anchoring constant is $W=5\times {10}^{-3}\mathrm{J}/{\mathrm{m}}^2$. Black dashed lines are a 2D view of the director field in the $y-z$ plane. Topological defect cores are visualized by the isosurface $S_q=0.5$. Based on the results of Fig. 5, snapshot (c) illustrates the equilibrium defect structure related to the aspect ratio 1:2 and (d) the equilibrium defect structure related to the aspect ratio 2:1.

Figure 8

Director field around a nanoparticle with $R=10\mathrm{nm}$ and aspect ratio 1:2 for the three orientations (a) $0^{\circ }$, (b) ${45}^{\circ }$, and (c) ${90}^{\circ }$ and the anchoring constant $W=5\times {10}^{-3}\mathrm{J}/{\mathrm{m}}^2$. Black dashed lines are a 2D view of the director field in the $y-z$ plane. The colored background represents the contour plot of the rescaled scalar order parameter in the bulk.

Figure 9

[(a)–(c)] Schematic representation of a colloidal particle in (a) a cubic box where the symmetry axis of the particle is in the $y-z$ plane, (b) a cubic box in which the symmetry axis of the particle is in a common plane with one of the space diagonals of the box, and (c) a cylindrical simulation box. [(d)–(f)] Free energy as a function of $\theta$ for a particle with $W=5\times {10}^{-3}\mathrm{J}/{\mathrm{m}}^2$, with the aspect ratio 1:2, and (d) $R=20$ nm, (e) $R=80$ nm, and (f) $R=120$ nm in the three setups.

Figure 10

[(a)–(c)] Schematic representation of a colloidal particle in a cubic box. [(a) and (b)] The symmetry axis of the particle is in the $y-z$ plane and the particle is positioned (a) at the box center and (b) 500 nm along the $z$ axis out of the center of the box. (c) The symmetry axis of the particle is in a common plane with one of the space diagonals of the box and the particle is positioned $\sqrt{2}500$ nm along the mentioned space diagonal out of the center of the box. (d) Free energy as a function of $\theta$ for a particle with $W=5\times {10}^{-3}\mathrm{J}/{\mathrm{m}}^2$, with the aspect ratio 1:2 and $R=120\mathrm{nm}$ in the three setups.