Nematic topological defects positionally controlled by geometry and external fields

Abstract

Using a Landau-de Gennes approach, we study the impact of confinement topology, geometry and external fields on the spatial positioning of nematic topological defects (TDs). In quasi two-dimensional systems we demonstrate that a confinement-enforced total topological charge of m > 1/2 decays into elementary TDs bearing a charge of m = 1/2. These assemble close to the bounding substrate to enable essentially bulk-like uniform nematic ordering in the central part of a system. This effect is reminiscent of the Faraday cavity phenomenon in electrostatics. We observe that in certain confinement geometries, varying the correlation length size of the order parameter could trigger a global rotation of an assembly of TDs. Finally, we show that an external electric field could be used to drag the boojum fingertip towards the interior of the confinement cell. Assemblies of TDs could be exploited as traps for appropriate nanoparticles, opening several opportunities for the development of functional nanodevices.

Keywords: nanoparticles; nematic liquid crystals; topological charge; topological defects.

Figure 1

Geometry of cells used in simulations. (a) In the “Cartesian” cells we enforce at the top “master” plate uniaxial nematic structures defined by Equation 10 and assume . (b) In the “cylindrical” cells we impose a boojum topological defect at the top plate and assume .

Figure 2

(a) Typical scribed surface topography enforcing the m = 2 topological defect. (b) Schematic representation showing the AFM-scribed director pattern of four topological defects (m = 2: blue circles, m = −2: gray squares), part of a larger square array of such defects. The separation between neighboring defects is roughly 30 μm, h ≈ 3 μm. (c) Darkfield microscopy image of a nematic cell the master plate of which enforces a square array of m = ±2 TDs; this image shows a rare example of a double integer defect that decomposes into a pair of half integer defects plus one integer defect. (d) Typical polarizing microscopy pattern image of this defect array. The scale bars are (a) 500 nm and (b,c,d) 5 μm.

Figure 3

Plots of β²(x,y) for different imposed total topological charges using BAC: (a) m = 1, (b) m = 2, (c) m = 4, (d) m = 6. TDs are decomposed into elementary units bearing the charge m₀ = 1/2, which assemble close to the bounding circle. In all panels we set R/ξ_b = 30 and τ = −8. The corresponding director field is depicted in Figure 4.

Figure 4

2D Plot of the eigenvectors of with the largest positive eigenvalue, corresponding to the 2D biaxiality profiles β²(x,y) plotted in Figure 3. (a) m = 1, (b) m = 2, (c) m = 4, (d) m = 6. In all panels we set R/ξ_b = 30 and τ = −8.

Figure 5

Plots of β²(x,y) of the configuration of TDs with increasing ratio η = R/ξ_b: (a) η = 14, (b) η = 17, (c) η = 21, (d) η = 25. In practice this could be achieved by decreasing the temperature of the sample. The trapezoid boundary enforces a total topological charge of m = 3, which splits into six elementary charges m₀ = 1/2.

Figure 6

Changes in the nematic director field with increasing ratio η = R/ξ_b: (a) η = 14, (b) η = 17, (c) η = 21, (d) η = 25. The corresponding biaxiality profiles are depicted in Figure 5.

Figure 7

Cross section through a cylindrically symmetric boojum core structure. The biaxial shell exhibiting maximal biaxiality joins the melted fingertip with the top plate. In the given case, the anchoring strength at the top plate is finite. The negative uniaxial region is indicated by a dashed white line. The fingertip is marked with a circle. The color bar indicates the values of β².

Figure 8

Radial spatial variation of the director field (thin black lines, ρ ≡ r, ρ₀ ≡ ξ_b; θ₀ = π/2), and degree of uniaxial order along the symmetry axis (thick red line, z ≡ ρ, ρ₀ ≡ h; S₀ = |S(r = 0, z = h)|, r = 0). In the simulations we establish a strong anchoring condition at the top plate, h/ξ_b = 8, τ = −8.

Figure 9

Extension of the fingertip with an increasing external field. Because of the cylindrical symmetry, we plot only the right-hand part of the boojum core structure (see Figure 8): (a) h/ξ_E = 0, (b) h/ξ_E = 8, (c) h/ξ_E = 10, (d) h/ξ_E = 12; h = 10ξ_b, h/d_e = 100, τ = −8.