Change in Stripes for Cholesteric Shells via Anchoring in Moderation

Chirality, ubiquitous in complex biological systems, can be controlled and quantified in synthetic materials such as cholesteric liquid crystal (CLC) systems. In this work, we study spherical shells of CLC under weak anchoring conditions. We induce anchoring transitions at the inner and outer boundaries using two independent methods: by changing the surfactant concentration or by raising the temperature close to the clearing point. The shell confinement leads to new states and associated surface structures: a state where large stripes on the shell can be filled with smaller, perpendicular substripes, and a focal conic domain (FCD) state, where thin stripes wrap into at least two, topologically required, double spirals. Focusing on the latter state, we use a Landau–de Gennes model of the CLC to simulate its detailed configurations as a function of anchoring strength. By abruptly changing the topological constraints on the shell, we are able to study the interconversion between director defects and pitch defects, a phenomenon usually restricted by the complexity of the cholesteric phase. This work extends the knowledge of cholesteric patterns, structures that not only have potential for use as intricate, self-assembly blueprints but are also pervasive in biological systems.

Popular Summary

Liquid crystals are widely known for their use in flat-panel electronic displays found in many gadgets, from digital watches to large televisions. These liquid-crystal displays (LCDs) make use of the unique way rodlike molecules in the liquid crystal interact with light and align with one another. Altering their organization can modify both of these properties, facilitating their use for other technologies. Here, we develop and exploit a platform that lets us produce and transform a wealth of intricate structures including both new and previously reported states, making the dynamic study of transitions between these patterns accessible for the first time.

We focus on cholesteric liquid crystals, a phase where the molecules are arranged in a helical fashion. We confine the cholesteric within onionlike spherical shells, sandwiched between two layers of water, produced using microfluidics. In this arrangement, topological defects—areas where the cholesteric order is ill defined—are required. Using two independent mechanisms, we precisely manipulate both the thickness of the shells and the orientation of the cholesteric molecules at the interfaces, therefore altering the topology of the system. This results in a series of complex states, where stripes form organized, hierarchical patterns on the shell surfaces. By shining light on this structure, we can study transitions between these patterns.

Our methods can be used in systems with differing topologies, such as droplets or tori, to further investigate the nature of cholesteric defects. Such a comprehensive study could also lay the groundwork for using topological defects in liquid crystals to design nanostructures.

Figure 1

Stripe patterns emerge on a CLC shell as the anchoring is changed from homeotropic to planar. Throughout, the pitch is $5\mu \mathrm{m}$ (2.8% CB15). (a) Time series of a CLC shell losing surfactant from its interface from (i) to (iv) (see Video 1 in Ref. [32]). As the surfactant surface concentration is diluted from 10 mM SDS with time, the homeotropic anchoring strength weakens. Four distinct patterns, from strongest to weakest homeotropic anchoring, are identifiable: subsurface defect lines (i), double-spiral domains (ii), thick planar stripes (iii), and planar anchoring (iv). (b) Schematic of surfactants on the CLC-water interface. The surfactant tails cause the CLC molecules to prefer perpendicular alignment to the interface. (c) Time series of a heated CLC shell, with temperature increasing from (i) to (iv) (see Video 2 in Ref. [32]). A few tenths of degrees below the clearing point, an isotropic layer nucleates at the CLC-water interface, inducing weak out-of-plane anchoring of the remaining CLC. As temperature is increased, the isotropic layer grows inwards, confining the remaining CLC until the entire thickness turns isotropic. Four patterns are also identified: planar anchoring (i), thick planar stripes (ii), thin stripes (iii), and (iv) complete untwisting. (d) Schematic of the isotropic layer growing between the bulk CLC and the PVA-stabilized interface between water and liquid crystal. The radius ($R$) and thickness ($h$) of the shells are $R\sim 150\mu \mathrm{m}$ and $h\sim 50\mu \mathrm{m}$ in (a) and $R\sim 75\mu \mathrm{m}$ and $h\sim 15\mu \mathrm{m}$ in (c).

Figure 2

Focal conic domains (FCDs) form on inner shell surface. A CLC shell has 5 mM SDS, 1 M NaCl, and 1% PVA in the inner phase water phase and 5 mM SDS, 0.1 M NaCl, and 1% PVA in the outer phase. FCDs can only be created on the inner shell surface alone when an extreme concentration of salt (1 M NaCl) is added to the inner water phase. (a) Before the shell completes osmotically swelling, FCDs, formed from the inner shell, are seen (zoom-in, right). The high salt concentration screens the SDS surfactant head group, allowing the surfactant to pack more densely on the interface. (b) After one day, the shell completes its swelling. The surfactant and salt concentrations are decreased, and the FCDs disappear as the inner salt concentration matches the outer salt concentration. The shell has mostly planar anchoring on its surface, apparent from the defect required from concentric cholesteric layers (zoom-in, right).

Figure 3

Unbending transition under increasing confinement. (a) Micrographs show a CLC shell with initially planar anchoring (i) heated towards the isotropic phase transition temperature with concomitant thinning of the cholesteric shell and introduction of weak out-of-plane anchoring. As the isotropic layer expands, thick planar stripes form initially (ii) and continuously decrease to the size of the pitch (iii), at which point, the stripe narrows discontinuously to the size of a half-pitch (iv), until the entire shell transitions to the isotropic phase (see Video 3 in Ref. [32]). The pitch is $5\mu \mathrm{m}$; the radius ($R$) and thickness ($h$) of the shell are $R\sim 200\mu \mathrm{m}$ and $h\sim 1\mu \mathrm{m}$. (b) Stripe periodicity over pitch versus temperature difference to the phase transition temperature. A continuous decrease followed by an abrupt halving is clearly seen. (c) Grayscale values of stripes in (a-iv) before (red line, 1) and after (red line, 2) the front of the unbending transition, showing the halving of the stripe periodicity. (d) Simulation of a $0.81\times 0.81\times 0.54\text{−}\mu \mathrm{m}$ CLC slab with a $0.28\text{−}\mu \mathrm{m}$ pitch, moderate homeotropic anchoring ($W_0\approx 8\times {10}^{-5}\mathrm{J}/{\mathrm{m}}^2$) on the top and bottom, and periodic boundary conditions in the horizontal directions. The outermost cholesteric layers are bent, yielding a stripe periodicity greater than the pitch. Alternating ${\lambda }^{+}$ and ${\lambda }^{-}$ pitch defect lines are labeled. The director color indicates the magnitude of the director along the vertical direction, $\widehat{\mathbf{z}}$. (e) When the slab thickness decreases from 0.54 to $0.13\mu \mathrm{m}$ (around half-pitch), the CLC unbends and runs parallel to the interface, creating half-pitch-wide stripes.

Figure 4

Ordering focal conic domains on a sphere. CLC shells with 7-mM SDS in the outer phase, without (a) and with (b) 7-mM SDS in the inner phase, have thin stripes that form double-spiral domains. The left micrographs focus on the outer surface of the shell; the right micrographs focus on the inner surface. The pitch is $5\mu \mathrm{m}$. In panel (c), blue lines represent the edges of FCDs on the outer surface of (b), while purple lines represent the edges of FCDs on the inner surface. The cholesteric polygonal texture comes from the staggered packing of FCDs. (d) A simulated CLC shell ($2.2\text{−}\mu \mathrm{m}$ diameter, $0.70\text{−}\mu \mathrm{m}$ thickness, and $0.42\text{−}\mu \mathrm{m}$ pitch) with matching homeotropic anchoring conditions ($W_0\approx 1.5\times {10}^{-5}\mathrm{J}/{\mathrm{m}}^2$) on the inner and outer surfaces. The top view shows the double-spiral pattern of the outer shell surface, while the cross section additionally shows the patterning of the inner shell surface. The shell has no defect in $\mathbf{n}$, only two defects in the pitch axis, located underneath the double spirals. The coloring indicates the magnitude of $\mathbf{n}$ along the radial direction $\widehat{\mathbf{r}}$ (i.e., perpendicular to the shell surfaces).

Figure 5

Double spirals are signatures of FCDs. Layers of concentric circles (dashed lines), representative of cholesteric layers, collide on a focal line (green). In red, we depict the surfaces of the shell confining the cholesteric, separating physical areas of the circular sections (black dashed lines) from virtual areas (gray dashed lines). If we rotate the two-dimensional texture around the vertical axis (blue), we get a texture between two red spheres with a focal plane dividing the top and bottom concentric CLC spheres. Topology, however, creates two charge $+1$ nematic defects along the rotation axis. Though a $+1$ disclination in a uniaxial nematic can “escape into the third dimension” [46], the biaxial-like cholesteric texture cannot [27]. As a result, the nematic disclinations become pitch defects, located along the rotation axis. Alternatively, if we rotate the two-dimensional texture around the horizontal axis (green), we generate a toroidal focal conic [45], with one focal line connecting the centers of the concentric CLC circles and another focal line along the axis of rotation. Pitch defects, at the center of double spirals, are located along the green axis of rotation. The director configuration within the shell is similar for either of these rotation axes. The structure of the FCDs becomes more complex as the number of double-spiral domains increases.

Figure 6

The bent state can coexist with the FCD state in the CLC shell with heterogeneous thickness. A CLC shell with 0 mM SDS, 1 M NaCl, and 1% PVA in the inner water phase begins to swell in an outer water solution with 7 mM SDS, 0.1 M NaCl, and 1% PVA. Thickness variation in the CLC shell shows that thinner regions have stripes from bent cholesteric layers (left panel), expressing the bent state, while thicker regions in the back of the drop have FCDs (right panel).

Figure 7

Equilibrium thin stripes. (a) A CLC shell with 7-mM SDS in the water phases equilibrates after one month, leaving only two, topologically required FCDs. The inset shows a side view of a FCD. (b) Schematic showing how two spirals can form on a sphere from a line connecting the two poles. For two spirals of opposite handedness, a point (black) must exist where the handedness is not defined, playing the role of the stripe dislocation in the right of (a) and in the red inset of (c). (c) Two FCDs in an equilibrated CLC shell, having double spirals of the opposite handedness and requiring a stripe dislocation (red inset). When the surfactant is washed away, the stripes rotate as they become wider and unwind, which is evident from the opposite rotation of FCD poles (blue arrows) with respect to the stripe dislocation (red arrows). The stripes widen first at the poles [(i) and (ii)] until an instability occurs [(iii) and (iv)]. The stripe dislocation becomes the planar defect. (d) Schematic of cholesteric layers in a shell for planar, hybrid, and homeotropic anchoring. Blue boxes highlight the fact that layers do not rearrange much to accommodate the anchoring change near a FCD. The opposite is true away from a FCD, highlighted by red boxes.

Figure 8

Fabrication of CLC shells. CLC double emulsions are produced using either a microfluidic device with two nested coaxial capillaries, in the case of surfactant experiments (a), or a microfluidic device with two tapered capillaries facing opposite directions, for temperature experiments (b). The inner and outer phases are water with 1% wt PVA. The middle phase is CLC. The shells are collected into a vial containing water and 1% wt PVA (c). For surfactant experiments, the shells are then pipetted into a vial containing water, 1% wt PVA, 0.1-M NaCl, and varying concentrations of SDS. After about 10 minutes, the CLC shells are pipetted into a viewing chamber that is then sealed to prevent evaporation (d).

Figure 9

Interface undulations at an isotropic-CLC interface. We simulate an isotropic-CLC interface in a slab geometry ($24\times 24\mathrm{nm}$ in the $x$ and $y$ directions and 18 nm wide in the vertical direction) by tuning our Landau–de Gennes potential to the coexistence point, which corresponds to constants $A=5.2311$, $B=-11.2612$, and $C=0.8889$. In this case, the CLC and isotropic phases are equally favored energetically. We plot just the CLC region, showing the local orientation of the molecules according to the $\mathbf{Q}$ tensor. By starting with an initial condition that is isotropic on the top of the slab and a uniform CLC on the bottom, we make an interface where the two regions initially meet in the middle of the slab. The anchoring conditions are strongly planar on the bottom of the slab (to force the pitch to align along the $z$ axis in the bulk of the slab) and free on the top. There are periodic boundary conditions in the $x$ and $y$ directions. After minimizing the Landau–de Gennes free energy, we see that the isotropic-CLC interface undulates (with hills and valleys indicated) and forms stripes with a periodicity equal to one pitch. This illustrates that a free CLC interface naturally undulates in response to the variation of the molecular orientation at the interface. We use the two-constant approximation with $L_1=L_2=2.32$ and $L_{24}=L_2/2$. The pitch is 6 nm in the CLC region.

Figure 10

Subsurface defect lines with strong homeotropic anchoring. CLC shells with 0.1 M NaCl in the water phases and 10 mM SDS in both the inner and outer phases (a) and only in the outer phase (b) have subsurface defect lines. Pictures are focused either on the inner surface (i) or on the outer surface (ii). Scale bars are all $50\mu \mathrm{m}$. (a,c) Because the defect lines (black lines in the simulations) form from both surfaces, the defect lines align to reduce the distortion. (b,d) Because the pitch axis must distort away from being parallel to the surface with hybrid anchoring, double spirals delineated by the defect line are less tightly wound. In the simulations in (c) and (d), the one-constant elastic constant approximation is used. The shells have a radius of $0.42\mu \mathrm{m}$ and a thickness of $0.31\mu \mathrm{m}$. In this case, the smallness of the simulated shell is necessitated by the presence of the defect lines at the surface, which have a size governed by the nematic correlation length, on the order of a few nanometers [39]. The anchoring is strong, with $W_0\approx 4\times {10}^{-3}\mathrm{J}/{\mathrm{m}}^2$ for the homeotropic surfaces and $W_1\approx 4\times {10}^{-3}\mathrm{J}/{\mathrm{m}}^2$ for the planar ones. The pitch is $0.3\mu \mathrm{m}$. The coloring indicates the magnitude of the director along the radial $\widehat{\mathbf{r}}$ direction, perpendicular to the shell surfaces. The simulation insets show that the stripes in the hybrid anchoring case (d) are less tightly wound.