Colloidal smectics in button-like confinements: Experiment and theory

Liquid crystals can self-organize into a layered smectic phase. While the smectic layers are typically straight, forming a lamellar pattern in bulk, external confinement may drastically distort the layers due to the boundary conditions imposed on the orientational director field. Resolving this distortion leads to complex structures with topological defects. Here, we explore the configurations adopted by two-dimensional colloidal smectics made from nearly hard rod-like particles in complex confinements, characterized by a button-like structure with two internal boundaries (inclusions): a two-holed disk and a double annulus. The topology of the confinement generates new structures which we classify in reference to previous work as generalized laminar and generalized Shubnikov states. To explore these configurations, we combine particle-resolved experiments on colloidal rods with three complementary theoretical approaches: Monte Carlo simulation, first-principles density functional theory, and phenomenological $\mathbf{Q}$-tensor modeling. This yields a consistent and comprehensive description of the structural details. In particular, we characterize a nontrivial tilt angle between the direction of the layers and symmetry axes of the confinement.

Figure 1

Schematic depiction of the button-like confining geometries investigated in this work generalizing an annulus (middle) with inclusion size ratio $b=R_{\text{in}}/R_{\text{out}}$, where $R_{\text{out}}$ is the radius of the outer confining wall and $R_{\text{in}}$ is the radius of the circular wall in the interior, called inclusion [61]. Here, we consider two inclusions at distance $C$, introducing the inclusion distance ratio $c=C/R_{\text{out}}$. In detail, we define the two-holed-disk geometry (top), where the outer wall is always circular, and the double-annulus geometry (bottom), where the outer wall is defined via two intersecting circles at the same distance $C$. The outer radius $R_{\text{out}}$ is kept at a fixed value throughout the paper, which corresponds to about five smectic layers. Further details are provided in Sec. 2a.

Figure 2

Example structures in the two confining geometries shown in Fig. 1 with two inclusions. The left and right columns depict the two-holed-disk geometry for $b\approx 0.25$ and $c\approx 1.0$ and the double-annulus geometry for $b\approx 0.3$ and $c\approx 1.2$, respectively. In both geometries we show (from top to bottom) typical particle configurations in colloidal experiments, Monte Carlo simulations of rod-like particles, density profiles and director fields from hard-rod density functional theory (DFT), and $\mathbf{Q}$-tensor theory with suitably adapted parameters (see text).

Figure 3

Illustrative overview of topological structures. The rows show different realizations of generalized laminar ($\mathcal{L}$, two $+1/2$ and four $-1/2$ charges), composite ($\mathcal{C}$, one $+1/2$ and three $-1/2$ charges), and generalized Shubnikov ($\mathcal{S}$, two $-1/2$ charges) states. The first three rows display the structures with a continuous central domain which is tilted by the angle $\alpha$ relative to the axis that connects the two inclusion centers, as annotated (the definition of $\alpha$ is included in the first illustration of the second row). The fourth row displays the structures for $\alpha =\pi /2$ with the central domain interrupted by a few layers spanning between the two inclusions, which we refer to as an inclusion tunnel. The illustrations depict the typical appearance of these generalized states in the two-holed-disk geometry (where we also speak of dual laminar and stretched Shubnikov states), but the same general classification also holds for the double-annulus geometry. Further details are provided in Sec. 2c.

Figure 4

DFT state diagram in the two-holed-disk geometry indicating the stable laminar (square symbols) or Shubnikov (round symbols) states for different inclusion distance ratios $c$ and inclusion size ratios $b$. The data for $c=0$, indicating the special case of annular confinement, are taken from Ref. [61]. The color denotes the relative free energy difference $\mathrm{\Delta }{\mathcal{F}}_{\text{rel}}:=({\mathcal{F}}_{\text{0}}^{\left(\mathcal{L}\right)}-{\mathcal{F}}_{\text{0}}^{\left(\mathcal{S}\right)})/{\mathcal{F}}_0$ between the optimal laminar $\left(\mathcal{L}\right)$ and Shubnikov $\left(\mathcal{S}\right)$ states, where ${\mathcal{F}}_0:=min\{{\mathcal{F}}_{\text{0}}^{\left(\mathcal{L}\right)},{\mathcal{F}}_{\text{0}}^{\left(\mathcal{S}\right)}\}$ is the free energy of the overall optimal state. For $b=0.3$ and $c=1.4$ the inclusions are in contact with the outer wall, such that no Shubnikov state can exist. The snapshots in the bottom panel depict four representative examples of optimal structures: laminar structures with tilt angles $\alpha =0$ or $\alpha =\pi /2$ (intermediate values of $\alpha$ are examined below in Fig. 7) and Shubnikov structures without and with an inclusion tunnel.

Figure 5

State diagram from $\mathbf{Q}$-tensor theory in the two-holed-disk geometry for different values of the elastic parameter $K$ and the wall anchoring parameter $w$. The inclusion size ratio $b=0.2$ and the inclusion distance ratio $c=0.6$ are kept fixed. We distinguish between four different structures: laminar ($\mathcal{L}$), composite ($\mathcal{C}$), and Shubnikov without ($\mathcal{S}$) and with an inclusion tunnel (${\mathcal{S}}^{*}$). The bottom panel depicts one representative snapshot for each case (parameters according to the connected stars).

Figure 6

Global orientational distribution in the two-holed-disk geometry for the inclusion size ratio $b=0.25$. As illustrated at the top right, the relative frequency of individual rod orientations (blue) reflects the preferred tilt angle $\alpha$ of the layers (gray line) relative to the axis that connects the two inclusions (dotted line) (see also Fig. 3). We compare experimental data for different inclusion distance ratios $c\approx 0.8$, $c\approx 1.0$, and $c\approx 1.2$, averaged over 21, 24, and 15 available structures, respectively, and Monte Carlo data for $c=1.0$, sampled from 1800 independent simulation runs. For symmetry reasons we map orientations with $\alpha >\pi /2$ onto $\pi -\alpha$ and only consider $0\le \alpha \le \pi /2$. The bottom panel depicts one particular snapshot for each case, where the rods are colored according to their orientation.

Figure 7

Energy landscape for different structures depending on the tilt angle $\alpha$ of the central domain in the two-holed-disk geometry for the inclusion size ratio $b=0.25$ and the inclusion distance ratio $c=1.0$. According to the legend, we compare different laminar DFT structures ${\mathcal{L}}_{ab}$, where the indices $a$ and $b$ denote the number of layers in the central domain and perpendicular to it, respectively, and several minimizers of the energy functional from $\mathbf{Q}$-tensor theory with $K=1$ and $w=5$. Exemplary snapshots are shown in the bottom panel. As only the energy difference is relevant for the stability, the vertical axis depicts the rescaled difference to the global minimum (indicated by the dotted line), calculated separately for DFT and $\mathbf{Q}$-tensor results in arbitrary units. Since the minima for large angles $\alpha$ are generally deeper, it is more likely to find such structures, consistent with the observation in Fig. 6.

Figure 8

Relative frequencies of structures with an inclusion tunnel (cf. the second row in Fig. 3) in the two-holed-disk geometry for the inclusion size ratio $b=0.25$. We compare experimental data (green bars) for different inclusion distance ratios $c\approx 0.6$, $c\approx 0.8$, $c\approx 1.0$, and $c\approx 1.2$, averaged over 44, 21, 24, and 15 available structures, respectively, and Monte Carlo data (black crosses) averaged over 20 simulations for each selected $c$. The dotted line serves as a guide to the eye, illustrating how the fraction of inclusion tunnels decreases with increasing inclusion distance. Regarding the occurrence of structures with inclusion tunnel in DFT and $\mathbf{Q}$-tensor theory, please refer to the bottom-right snapshot in Fig. 4 and the state diagram in Fig. 5, respectively.

Figure 9

Structures in the double-annulus geometry for different inclusion size ratios $b$ and inclusion distance ratios $c$. While we generally observe layering of the Shubnikov type in the annular arcs for the parameters considered, we distinguish three states by the layer arrangement in the intersection region: inclusion tunnel (red), double-Shubnikov state (yellow), and diagonal tunnel (blue). Left: State diagram indicating the relative frequencies of the three structures between the two inclusions. The experimental and Monte Carlo results are represented by pie charts and background pixels with proportional color mixing, respectively. For $c\lesssim 2b$ no rods fit in between the inclusions and there is no distinction (black pixels). Right: Observed structures for three pairs of $b$ and $c$ corresponding to the state points indicated by the arrows. We depict experimental snapshots (top), solution profiles of the $\mathbf{Q}$-tensor model with for $K=1.0$ and $w=10$ (middle), and the orientational order parameter $S\left(\mathbf{r}\right)$ averaged over ${10}^3$ independent Monte Carlo simulation runs per parameter pair, revealing the typical location of the topological defects through the darker shades (bottom).