Topology of Orientational Defects in Confined Smectic Liquid Crystals

We propose a general formalism to characterize orientational frustration of smectic liquid crystals in confinement by interpreting the emerging networks of grain boundaries as objects with a topological charge. In a formal idealization, this charge is distributed in pointlike units of quarter-integer magnitude, which we identify with tetratic disclinations located at the end points and nodes. This coexisting nematic and tetratic order is analyzed with the help of extensive Monte Carlo simulations for a broad range of two-dimensional confining geometries as well as colloidal experiments, showing how the observed defect networks can be universally reconstructed from simple building blocks. We further find that the curvature of the confining wall determines the anchoring behavior of grain boundaries, such that the number of nodes in the emerging networks and the location of their end points can be tuned by changing the number and smoothness of corners, respectively.

Figure 1

Topological characterization of grain boundaries in confined smectic liquid crystals. Left: coarse-grained topological structure with an idealized nematic disclination of half-integer charge $Q$ induced by the presumed planar alignment with the nearby wall. Middle: particle-resolved simulation snapshots of hard rods with highlighted grain boundaries in the form of a line close to a circular wall (top row) and a network induced by corners (bottom row). Right: continuum model with quarter-integer tetratic point charges $q$ at the end points and nodes, which can be interpreted as the distribution of the spatially extended charge $Q$.

Figure 2

Survey of grain boundaries with different connectivities in smectic liquid crystals at a convex confining wall. Top: composition of the two fundamental building blocks with total nematic charge $Q=1/2$ (type A, red line) and $Q=0$ (type B, orange line), determined by the tetratic quarter charges $q$ (dots) at the end points. The illustration of bulk orientational ordering depicts a typical arrangement of the surrounding rods and a schematic continuum picture with idealized straight grain boundaries separating regions of perpendicular smectic layers (grey lines). The closed contour $\mathcal{C}$ (green line) highlights the contribution of the tetratic end points to $Q=\sum q$. Bottom: relation between the geometry-dependent manifestations of $Q=+1/2$ grain-boundary networks (red) in smectics confined to polygons. The simplest structure $(A_m)$ only contains one type-$A$ defect with $m\in \{0,1,2\}$ of its end points attached to a corner of the confining wall. In general, complex networks $(A+nB)$ can form, which amounts to adding $n$ building blocks of type $B$. Approaching the limit $n\to \infty$, the network detaches from the increasingly smooth corners, gradually reverting to ($A_0$), as the tetratic defects in the type-$B$ branches annihilate [83].

Figure 3

Topological defect structure of representative simulations for hard rods with aspect ratio $p=15$ in different convex confining geometries. Top row: particle snapshots with superimposed networks of grain boundaries and isolated tetratic point defects, compare Fig. 2. The color of the rods highlights individual domains according to cluster analysis. Second row: nematic order parameter field $S(\mathbf{r})$. Third row: tetratic order parameter field $T(\mathbf{r})$. Point defects at the confining wall are not visible. Bottom row: idealized continuum interpretation of the depicted snapshots, as detailed in the Supplemental Material [83].

Figure 4

Two selected sets of experimental reference data in hexagonal confinement, presented as in Fig. 3. Shown are bare bright-field microscopy images with $N=1400\pm 150$ colloidal rods of effective hard-rod aspect ratio $p_{\mathrm{eff}}=10.6$ in the field of view and the extracted order parameters.