Finite particle size drives defect-mediated domain structures in strongly confined colloidal liquid crystals

Abstract

When liquid crystals are confined to finite volumes, the competition between the surface anchoring imposed by the boundaries and the intrinsic orientational symmetry-breaking of these materials gives rise to a host of intriguing phenomena involving topological defect structures. For synthetic molecular mesogens, like the ones used in liquid-crystal displays, these defect structures are independent of the size of the molecules and well described by continuum theories. In contrast, colloidal systems such as carbon nanotubes and biopolymers have micron-sized lengths, so continuum descriptions are expected to break down under strong confinement conditions. Here, we show, by a combination of computer simulations and experiments with virus particles in tailor-made disk- and annulus-shaped microchambers, that strong confinement of colloidal liquid crystals leads to novel defect-stabilized symmetrical domain structures. These finite-size effects point to a potential for designing optically active microstructures, exploiting the as yet unexplored regime of highly confined liquid crystals.

Figure 1. Patterns of rod-like particles in circular confinement.

Overview of packing structures of rod-like particles confined in shallow chambers with a circular geometry. Top row: schematic showing classification by the location of the disclination points: (a) type B_i, (b) type B_b, (c) type B_o, (d) type B_∞. Second row: snapshots showing particle positions and their orientations with respect to the vertical (mod π) (see colour bar on the right). Third row: orientation patterns averaged over >10³ configurations. Fourth row: value of the scalar order parameter S∈[0,1], scale bar on the right. For a discussion of the error estimate in this quantity see the Supplementary Note 8 and Supplementary Fig. 10 Note the characteristic dips at the location of the defects. Fifth row: normalized angular deficit parameter, which peaks at the centre of the defects. Simulation parameters: (e,i,m,q) L/D=15 and η=0.16; (f,j,n,r) L/D=15 and η=0.20; (g,k,o,s) L/D=20 and η=0.20; (h,l,p,t) L/D=25 and η=0.20.

Figure 2. Patterns of rod-like particles in annular confinement.

Overview of packing structures of rod-like particles confined in annulus-shaped chambers, showing threefold (left column), fourfold (middle column) and fivefold (right column) symmetry. Top row: snapshots of simulations. Second row: particle orientations averaged over averaged over >10³ configurations, labelled by colour bar on the right. Third row: scalar order parameter S∈[0,1]. Fourth row: normalized angular deficit parameter. Simulation parameters for each column (from left to right): (a,d,g,j) H=6, L/D=15, η=0.20 and R_inner=7.5; (b,e,h,k) H=6, L/D=25, η=0.20 and R_inner=15; (c,f,i,l) H=3, L/D=25, η=0.20 and R_inner=15. For error calculations see Supplementary Note 8 and Supplementary Fig. 10.

Figure 3. Number of nematic domains in different geometries.

The symmetry number n of the patterns observed in simulations as a function of the inner hole radius R_inner for rods with aspect ratio L/D=15 in an annular geometry with R_outer=40D, for chamber heights H=D (the strictly 2D case) and H=3D, 6D at a number of different packing fractions. Each coloured symbol represents a different pattern, the chosen shape indicating the symmetry as explained in the legend at the bottom. The dotted vertical lines mark the predicted boundaries between the different patterns based on the geometrical rule discussed in the main text.

Figure 4. Experimentally observed patterns.

Structures observed in colloidal liquid crystals confined to disk-shaped and annulus-shaped microchambers. (a–c) Representative images of three types of director-field patterns. Hue corresponds to average orientation (Supplementary Note 4; Supplementary Figs 3 and 4) according to legend (left). Brightness corresponds to a maximum-time projection over 2,000 acquired frames of fluorescently labelled fd-virus particles. (a) D₂: director field exhibits twofold symmetry and two singularities. (b) D₃: director field exhibits threefold symmetry and three singularities. (c) D_∞: director field exhibits full rotational symmetry and no singularities. Scale bars, 5 μm. (d) Probability P_pattern that a given pattern occurs as a function of inner diameter R_inner (in units of outer diameter R_outer). D₂ (green diamonds) is most likely for R_inner=0; D₃ (red triangles) is most likely for R_inner/R_outer=0.2; D_∞ (blue circles) is most likely for R_inner/R_outer=0.7. Lines are guides to the eye. Other patterns were also found (Supplementary Note 5; Supplementary Figs 5 and 6).