Polar-Nonpolar Interfaces of Normal Bicontinuous Cubic Phases in Nonionic Surfactant/Water Systems Are Parallel to the Gyroid Surface

Mapping hyperbolic order in curved materials

Nature employs an impressive range of topologically complex ordered nanostructures that occur in various forms in both natural and synthetic materials. A particular class of these exhibits negative curvature and forms periodic saddle-shaped surfaces in three dimensions. Unlike pattern formation on flat or positively curved surfaces like spherical systems, the understanding of patterning on such surfaces is highly complicated due to the structures being intrinsically intertwined in three dimensions. We present a new method for visualisation and analysis of patterns on triply periodic negatively curved surfaces by mapping to two-dimensional hyperbolic space analogous to spherical projections in cartography thus effectively creating a more accessible "hyperbolic map" of the pattern. Specifically, we exemplify the method via the simplest triply periodic minimal surfaces: the Primitive, Diamond, and Gyroid in their universal cover along with decorations from a soft materials, whose structures involve decorations of soft matter on negatively curved surfaces, not necessarily minimal.

A phase retrieval algorithm for triply periodic minimal surface like structures

A method to solve the crystallographic phase problem of materials with triply periodic minimal surface like structures, such as lyotropic liquid crystal bicontinuous cubic phases, is reported. In triply periodic minimal surface like structures, the difference between the maximum and minimum electron densities tends to be the smallest for the true phase combination among the possible combinations [Oka (2022). Acta Cryst. A78, 430-436]. Using this feature, a new iterative phase retrieval algorithm for structure determination was developed. The algorithm modifies electron densities outside the upper and lower thresholds in the iterative Fourier transformation process with fixed amplitudes for the structure factors, and efficiently searches for the structure with the smallest difference between the maximum and minimum electron densities. The proper structure was determined by this algorithm for all tested data for lyotropic liquid crystal bicontinuous cubic phases and mesoporous silicas. Although some cases required constraints such as the volume fraction for structure determination, more than half could be determined without any constraints, including space groups.

Stabilizing Differential Interfacial Curvatures by Mismatched Molecular Geometries: Toward Polymers with Percolating 1 nm Channels of Gyroid Minimal Surfaces

Soft materials with self-assembled networks possess saddle-shaped interfaces with distributed negative Gaussian curvatures. The ability to stabilize such a geometry is critically important for various applications but can be challenging due to the possibly "deficient" packing of the building blocks. This nontrivial challenge has been manifested, for example, by the limited availability of cross-linkable bicontinuous cubic (Q) liquid crystals (LCs), which can be utilized to fabricate compelling polymers with networked nanochannels uniformly sized at ∼1 nm. Here, we devise a facile approach to stabilizing cross-linkable Q mesophases by leveraging the synergistic self-assembly from pairs of scalably synthesized polymerizable amphiphiles. Hybridization of the molecular geometries by mixing significantly increases the propensity of the local deviations in the interfacial curvature specifically required for Q assemblies. "Normal" (type 1) double gyroid LCs possessing 1 nm ionic channels conforming to minimal surfaces can be formulated by simultaneous hydration of the amphiphile mixtures, as opposed to the formation of hexagonal or lamellar mesophases exhibited by the single-amphiphile systems, respectively. Fixation of the bicontinuous network in polymers via radical polymerization has been efficaciously facilitated by the presence of the bifunctional polymerizable groups in one of the employed amphiphiles. High-fidelity lock-in of the ordered continuous 1 nm channels has been unambiguously confirmed by the observation of single-crystal-like diffraction patterns from synchrotron small-angle X-ray scattering and large-area periodicities by transmission electron microscopy. The produced polymeric materials exhibit the required mechanical integrity as well as chemical robustness in a variety of organic solvents that benefit their practical applications for selective transport of ions and molecules.

Optical Reflection from Unforbidden Diffraction of Block Copolymer Templated Gyroid Films

We present material substitutions and optical characterization of block copolymer (BCP)-templated gyroid structures that are obtained from a volume-asymmetric polystyrene-b-poly(methyl methacrylate) (PS-b-PMMA). In addition to the structural analyses reported earlier, we elucidate the optical responses to the nonaffine gyroid planes, in which the PMMA channels are complexed with Al₂O₃ by sequential infiltration synthesis and the organic components are further eliminated to produce an inorganic air-Al₂O₃ gyroid film. Grazing-incidence small-angle X-ray scattering measurements show that three-dimensional gyroid lattices are retained in both in-plane and out-of-plane directions through these material substitution processes. Our BCP-templated gyroid films respond to the middle UV wavelength from 200 to 300 nm, and peculiar optical reflectance peaks correlate with the unforbidden {110} diffraction spots. Together with the red- and blue-shifts of the reflectance peaks by the component substitutions, the air-Al₂O₃ gyroid structure reveals the high-amplitude spectrum due to the large refractive-index difference between channel and matrix.

Visualizing the double-gyroid twin

Periodic gyroid network materials have many interesting properties (band gaps, topologically protected modes, superior charge and mass transport, and outstanding mechanical properties) due to the space-group symmetries and their multichannel triply continuous morphology. The three-dimensional structure of a twin boundary in a self-assembled polystyrene-b-polydimethylsiloxane (PS-PDMS) double-gyroid (DG) forming diblock copolymer is directly visualized using dual-beam scanning microscopy. The reconstruction clearly shows that the intermaterial dividing surface (IMDS) is smooth and continuous across the boundary plane as the pairs of chiral PDMS networks suddenly change their handedness. The boundary plane therefore acts as a topological mirror. The morphology of the normally chiral nodes and strut loops within the networks is altered in the twin-boundary plane with the formation of three new types of achiral nodes and the appearance of two new classes of achiral loops. The boundary region shares a very similar surface/volume ratio and distribution of the mean and Gaussian curvatures of the IMDS as the adjacent ordered DG grain regions, suggesting the twin is a low-energy boundary.