Pattern templates

The overall aim of this work is to determine whether proper pattern symmetries exist that can lead to formation of a single-crystal, ordered BP (BPI or BPII) having a specific crystallographic orientation. A first line of thinking could be to construct patterns that follow the preferred molecular alignment of the liquid crystal at different crystallographic planes of a BP in the bulk (see Supplementary Fig. 1). Such an alignment, however, would depend on where the crystallographic plane is cut (see Supplementary Fig. 2). The design of surface patterns based on this information would introduce additional strain on the BP structure if the film’s thickness was not commensurate with the BP-lattice constant. Moreover, confining the BP into a film would break the symmetry in the vicinity of the boundaries, even for weak anchoring conditions12. To address these challenges, we therefore begin by analysing how a BP responds to confinement.

The BPII crystallographic planes are denoted by (hkl), where h, k and l are the Miller indices. In this work BPII (hkl) denotes a BPII oriented with the (hkl) plane parallel to the surface. For light that is incident normal to a (hkl) plane, the reflected light has wavelength

where n is the diffraction index and a is the unit cell size. Experimentally, Bukusoglu et al.11 found that a BPII film with lattice size of 150 nm, under uniform planar or homeotropic conditions, is polycrystalline and is dominated by BPII (111) domains inter-dispersed by a few BPII (110) and BPII (100) platelets9. Such monodomains reflect light with λ (111)= 260 nm, λ (110) =318 nm and λ (100) =450 nm. The structural stability of a BP depends on the interplay between surface and bulk contributions to the free energy; identifying conditions that favour a given BPII-lattice orientation with respect to a surface therefore requires an analysis of the system’s response to confinement. To that end, building on results reported in the literature11,12, we carried out numerical simulations of a typical BPII with unit cell size a BPII =150 nm, confined into a 2 μm-thick channel with homeotropic anchoring at the top and bottom surfaces (additional details are provided in the Methods section).

Through careful choice of the initial conditions, one can generate three monocrystalline reference states, namely BPII (100), BPII (110) and BPII (111) (see the Methods section for additional details). Figure 1 shows the corresponding topological defects and the average molecular orientation in the immediate vicinity of the homeotropic surface, along with a two-dimensional map of the behaviour of the scalar order parameter (S), which measures the local degree of molecular ordering, evaluated immediate vicinity of the surface. For each of the reference states, BPII (100), BPII (110) and BPII (111) , there is a correlation between the symmetry of the S-maps and the preferred molecular alignment: a value of S=1 corresponds to a material that is perfectly homeotropic at the surface, whereas S≈0 corresponds to an isotropic, disordered region. The S-map allows one to identify which regions near the surface undergo costly elastic distortions as a result of the tendency of a material to adopt an average orientation that is different from that imposed by the surface anchoring. For instance, topological defects—which result from abrupt changes of the local molecular order and where there is no preferred molecular orientation—appear in the S-map as regions where an abrupt change of colour occurs; in the bulk, such abrupt changes correspond to topological line defects that can be represented as the blue isosurfaces with S=0.35 shown in Fig. 1. Each BPII presents a near-surface S-map that is characteristic for a particular crystallographic orientation. By recognizing that, to first order, surface-order maps provide an indication of the local strain (and therefore energetic cost) associated with presenting a particular crystallographic plane onto a surface, such maps provide a blueprint for creating patterns of planar and homeotropic regions whose aim is to relieve the elastic distortions induced by the surface. Moreover, the symmetry of the S-map appears to be essentially unaffected when the channel’s thickness varies (see Supplementary Figs 3–5). Building on this idea, in Fig. 2 we show the particular designs that we hypothesize will direct the BPII orientation along specific Miller indices.

Figure 1: Defect structure and scalar order parameter maps for BPII. (a) Defect structure of the BPII with different lattice orientations. (b) Close-up of the BPII-topological line defects and molecular orientation in the proximity of uniform homeotropic interfaces. (c) Scalar order parameter maps at the corresponding interfaces. To represent low and high values of S we use colour maps that go from blue (disordered) to red (ordered). In the case of the local director, the colour corresponds to the projection of the molecular orientation onto the surface normal vector; as a result, blue directors are parallel to the surface, while red directors are perpendicular to the surface. Full size image

Figure 2: Pattern templates for BPII with different lattice orientations. (a) Stripe pattern for BPII (100) ; (b) rectangular-array pattern for BPII (110) ; (c) circular-array pattern for BPII (111) . Insets correspond to the S-map, the symmetry and spatial dimensions of these maps are simplified into binary patterns where red and blue sections correspond to homeotropic and planar regions, respectively. Colour map as in Fig. 1. Full size image

Optimal pattern designs

The stripe-like pattern of Fig. 2a was conceived to produce monocrystalline BPII (100) domains that reflect visible light with λ (100) =450 nm. We found from simulations using a Landau-de Gennes formalism (see Methods section) that the periodicity of the stripes is equal to the BPII-lattice constant; therefore, for the stripe-pattern design, the sum of the planar, P, and homeotropic, H, stripe widths, must be equal to the BP lattice size (as indicated in Fig. 2a). We use the homeotropic ratio, HR=H/(P+H), to determine the optimal proportion of planar and homeotropic contributions from the pattern. Figure 3a shows the free energy density difference between the confined system and its value in the bulk, Δf, as a function of HR for P+H=a BPII =150 nm. Although BPII (111) is the most favourable configuration for either uniform planar (HR=0) or uniform homeotropic (HR=1) anchoring, the Stripe pattern reverses that trend by reducing the free energy of the BPII (100) , which becomes the most stable orientation for 0.2<HR<0.8. A value of HR=0.5 is optimal for Stripe patterns.

Figure 3: Free-energy densities of BPII on different patterned surfaces. (a–c) Free-energy density difference, Δf=f−f Bulk , and (d–f) surface free energy density, f S , as a function of different pattern parameters, for different BPII-lattice orientations: BPII (100) (circles), BPII (110) (squares) and BPII (111) (triangles). Consistent with experimental observations, BPII (111) is obtained for uniform homeotropic and planar anchoring (HR=1 and HR=0, of the stripe-like pattern). These results show the parameter intervals over which each pattern favours a particular lattice orientation; thus BPII (100) is favoured when HR≈0.5, BPII (110) when W≈75 nm and BPII (111) for all values of r. Full size image

The patterns that we propose to stabilize BPII (110) and BPII (111) consist of hexagonal arrays of rectangles or circles of homeotropic anchoring over a planar background. In what follows we refer to them as rectangular and circular patterns, respectively (Fig. 2b,c). We find that the parameters associated with the spatial distribution of these homeotropic regions depend on the size of the BPII unit cell, a BPII , and the lattice orientation (see Fig. 2). For these nano-patterns, the area of the rectangles is 2W × W and the radius of the circular domains is denoted by r; by changing W and r in a systematic manner, one can determine the optimal conditions to produce single crystals of BPII (110) and BPII (111) , respectively. Figure 3b, shows the W-range over which BPII (110) becomes the stable configuration when the blue-phase is confined into a film on a rectangular patterned surface. In the case of the circular pattern, BPII (111) is the stable state for all values of r considered here, as shown in Fig. 3c. It is important to note that, due to the strong anchoring conditions imposed by the pattern regions, the surface energies associated with different BP orientations are similar for all three cases considered above (Fig. 3d–f); the overall behaviour of the free energy and the changes induced by the patterns are primarily due to elastic distortions. As seen in Fig. 3, the energetic cost associated with such distortions is minimal and represents only a small fraction of the overall energy of the system. It is, however, sufficient to influence the orientation of the entire material over macroscopic regions. The main role of the patterned surface is therefore that of seeding the correct crystallographic orientation in the proximity of the interface, and favour the (100), (110) and (111) lattice orientations with stripes, rectangles and circles, respectively; these shapes and the corresponding surface anchoring induce a local deformation that prevents the material from adopting other, more unfavourable orientations (see Fig. 4). Based on these theoretical results, it is reasonable to infer that the patterned surfaces shown in Fig. 2 will direct the orientation of a BPII film. These predictions are examined in the following section.

Figure 4: BPII topological defects in the proximity of patterned surfaces. Close up of the defects and molecular orientation in the proximity of the stripe (a–c) , rectangular (d–f) and circular (g–i) patterned surfaces for (100), (110) and (111)-BPII lattice orientations. Colour map as in Fig. 1. Full size image

Experimental results

To summarize our design strategy, binary homeotropic/planar patterns are first designed from continuum simulations of a BPII (hkl) under uniform homeotropic interfacial conditions. The S-maps at the interface are correlated with the preferred molecular alignment: BP molecules in regions with the highest order parameter (red sections) show a preferred perpendicular alignment at the interface. In the other regions, the preferred molecular alignment deviates slightly from that imposed by the interface and is associated with a preference for planar alignment, as indicated by the behaviour of the director field above the interface. The S-maps are simplified into a binary pattern consisting of planar and homeotropic regions; the symmetry of the patterns is described in terms of the BP lattice constant. Landau-de Gennes calculations are then performed to determine the optimal dimensions of the homeotropic and planar regions (that is, the optimal values of HR, W and r of the striped, rectangular and circular patterns, respectively). Once a pattern is optimized, the resulting information can be used to experimentally prepare each pattern of interest (see Supplementary Fig. 6).

Experiments were carried out on a silicone substrate with an approximately 5 nm-thick synthetic grafted polymer brush that imposes homeotropic anchoring on the planar substrate. Following Li et al.35, nanopatterns were produced through a lithographic process using e-beam on a polymer-covered surface (see Supplementary Fig. 7). The technique of Li et al.35 is particularly helpful in that it enables preparation of flat patterned surfaces that are devoid of micrometre-scale topographic steps. The chiral liquid crystals considered here were prepared by mixing the mesogen MLC 2142 with 36.3 wt% of the chiral dopant 4-(1-methylheptyloxycarbonyl)phenyl-4-hexyloxybenzonate (S-811). This mixture produces a BPI and a BPII with lattice sizes a BPI ≈255 nm and a BPII ≈150 nm, respectively11.

This liquid crystal mixture was confined into 3.5 μm-thick slits with homeotropic anchoring on the top surface and a 0.25 mm2 patterned area on the bottom surface. To provide a reference for the influence of the patterns, in all samples the patterned area was surrounded by a region of uniform homeotropic anchoring.

Following the predictions outlined above, for the Stripe patterns we use HR=0.5 and P+H=a BPII ; for the rectangular and circular patterns we use 2W=2r=a BPII . In this way, all the patterns are produced in terms of the unit cell size, a feature that will be useful for extending the results presented here to systems having different chirality.

Figure 5a shows optical micrographs of the experimental systems. As one can see from the figure, the patterned surfaces do not significantly affect the structural and thermal behaviour of the Chol and BPI phases. For BPII, however, our results demonstrate that patterns induce the formation of a single-crystal BPII specimen over the entire patterned area, with no platelet-like domains or grain boundaries. In all cases, the micrographs were taken for light normally incident in the reflection mode of a cross polarizer. As predicted, the stripe pattern (Fig. 5a top) induces a BPII (100) single crystal, which reflects light with wavelength λ (100) ≈450 nm; the rectangular and circular patterns (Fig. 5a centre and bottom) give rise to BPII (110) and BPII (111) single crystals, respectively. They appear black in the images because the reflected light in this case is outside the visible spectrum (λ (110) ≈318 nm, λ (111) ≈260 nm). Kossel diagrams were obtained using monochromatic light with λ=405 nm; the lines shown in these diagrams correspond to light reflected by the (100) planes and reveal the lattice orientation of the BP. Our measurements are consistent with theoretically and experimentally determined Kossel diagrams36,37,38, and confirm the existence of the (100) and the (110) lattice orientations on the stripe and rectangular patterns, respectively. For the circular pattern, the symmetry of the diagram can be explained by analysing the lattice structure of the BPII (111) in the proximity of the patterned surface. We find that such a structure depends on the channel thickness and is consistent with the formation of a hexagonal BP layer at the wall, as revealed by the symmetry of line defects (see Supplementary Fig. 8). As the name indicates, the hexagonal BP consists of a hexagonal array of double twist cylinders; the corresponding Kossel diagram agrees with that shown in Fig. 5 for the circular pattern39,40. Our simulation results, shown in the Supplementary Fig. 8, show that the circular pattern produces a BPII (111) monodomain where the lattice structure adopts a hexagonal symmetry in the vicinity of the patterned surface, and this structure is revealed experimentally by the Kossel diagram.

Figure 5: Experimental confirmation of directed BP-lattice orientation by patterned surfaces. (a) Scanning electron microscopy images of three different patterned surfaces, along with the corresponding micrographs of the cholesteric and BPs I and II and the Kossel diagrams of the BPII in the patterned area. The Chol-BPI and BPI-BPII transition temperatures for each case are also indicated. As predicted by our theoretical results, each pattern stabilizes the BPII orientation for that it was designed to stabilize. The resulting single-crystal monodomains are as large as the patterned area. (b,c) Reflected light optical images of the BP cell under crossed polarizers: (b) single crystalline domain on stripe patterned surface; (c) monodomain adjacent to the stripe pattern area. The Kossel diagrams correspond to the right and left sides of the zoomed-in images. Full size image

Additional experimental evidence for the single-crystal characteristic of the domains produced by the process outlined here can also be obtained from the Kossel diagrams. In a single crystal, Kossel diagrams obtained from different regions of a sample should be identical. In contrast, in a polycrystalline monodomain, platelets having different x–y orientations would produce Kossel diagrams that differ in their relative orientation. This is exactly what is found in our experiments, as shown in Fig. 5b,c. Kossel diagrams extracted from different regions of the single crystals produced here have the same orientation, whereas those extracted from polycrystalline monodomains do not (additional results can be found in the Supplementary Fig. 9).