Transformation between Inverse Bicontinuous Cubic Phases of a Lipid from Diamond to Gyroid

Gradient Transformation of the Double Gyroid to the Double Diamond in Soft Matter

Transitions between gyroid and diamond intercatenated double network phases occur in many types of soft matter, but to date, the structural pathway and the crystallographic relationships remain unclear. Slice and view scanning electron microscopy tomography of a diblock copolymer affords monitoring of the evolving shape of the intermaterial dividing surface, allowing structural characterization of both the majority and minority domains. Two trihedral malleable mesoatoms combine to form a single tetrahedral mesoatom in a volume additive manner while preserving network topology, as the types of loops, the number of mesoatoms in a loop, minority domain strut lengths, and directions that connect a given mesoatom to its neighbors evolve across a 150 nm wide transition zone (TZ). The [111]DD direction is coincident with the [110]DG direction so that the (111)DD and (110)DG planes define the boundaries of the TZ. Selection of the particular crystal orientations and direction and width of the transition zone is to minimize the cost of morphing the mesoatoms from one structure to the other, by maximizing like-block continuity and minimizing the variation of the surface curvature and thickness of the domains across the TZ. Such coherent continuity of the independent, intercatenated networks across the transition zone is critical for applications such as graded mechanical trusses where the pair of different networks are joined to provide different mechanical properties for adjacent grains or could serve as a nanoscale anode/cathode allowing super charging and discharging provided the networks are continuous and rigorously separate.

Network Phases with Multiple-Junction Geometries at the Gyroid-Diamond Transition

A novel phase sequence for the transition from the double diamond to the double gyroid cubic phases via two non-cubic intermediate phases, an orthorhombic Fmmm (O⁶⁹) phase and a hexagonal P6₃/m (H¹⁷⁶) phase, is reported for specifically designed bolapolyphiles composed of a linear rod-like bistolane core with sticky glycerol ends and two branched central and two linear peripheral side chains. These liquid crystalline (LC) phases represent members of a new class of unicontinuous network phases, formed by longitudinal rod bundles with polar spheres acting as junctions and the alkyl chains forming the continuum around them. In contrast to previously known bicontinuous cubic networks, they combine different junctions with different angles in a common structure, and one of them even represents a triple network instead of the usually found double networks. This provides new perspectives for the design of soft network phases with enhanced structural complexity, inspiring the search for new supramolecular networks, nano-particle arrays, and photonic band-gap materials.

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

We investigated the structures of normal (type I) bicontinuous cubic phases in hexa-, hepta-, and octaethylene glycol dodecyl ether/water mixtures by small-angle X-ray crystallography of single-crystal domains. Reconstructed electron densities showed that the hydrophilic chains with high electron density are confined to a film centered on the surface of the Gyroid (a triply periodic minimal surface), while hydrophobic chains with low electron density are distributed within the pair of interwoven labyrinths carved out by the Gyroid. Further, the local minimum within the high electron density region, due to bulk water, coincides precisely with the Gyroid. This minimum is less pronounced in mixtures with longer ethylene glycol chains, consistent with their decreased water content. Our analysis clearly shows that the polar-nonpolar interfaces are parallel to the Gyroid surface in all mixtures. The repulsive hydration or overlapping force between the pair of facing monolayers of ethylene glycol chains on either side of the Gyroid surface is the likely origin of the parallel interfaces.

Bicontinuous cubic phases in biological and artificial self-assembled systems

Nature has created innumerable life forms with miraculous hierarchical structures and morphologies that are optimized for different life events through evolution over billions of years. Bicontinuous cubic structures, which are often described by triply periodic minimal surfaces (TPMSs) and their constant mean curvature (CMC)/parallel surface companions, are of special interest to various research fields because of their complex form with unique physical functionalities. This has prompted the scientific community to fully understand the formation, structure, and properties of these materials. In this review, we summarize and discuss the formation mechanism and relationships of the relevant biological structures and the artificial self-assembly systems. These structures can be formed through biological processes with amazing regulation across a great length scales; nevertheless, artificial construction normally produces the structure corresponding to the molecular size and shape. Notably, the block copolymeric system is considered to be an applicable and attractive model system for the study of biological systems due to their versatile design and rich phase behavior. Some of the phenomena found in these two systems are compared and discussed, and this information may provide new ideas for a comprehensive understanding of the relationship between molecular shape and resulting interface curvature and the self-assembly process in living organisms. We argue that the co-polymeric system may serve as a model to understand these biological systems and could encourage additional studies of artificial self-assembly and the creation of new functional materials.

Polar-Nonpolar Interfaces of Inverse Bicontinuous Cubic Phases in Phytantriol/Water System are Parallel to Triply Periodic Minimal Surfaces

We investigated two distinct lyotropic liquid crystal inverse bicontinuous cubic phases of phytantriol/water mixtures by small-angle X-ray crystallography of the single-crystal regions. Reconstructed electron density maps revealed hydrophilic head and hydrophobic tail regions of the phytantriol bilayer membranes and water regions. The bilayer membranes are shown to be located on the D and gyroid triply periodic minimal surfaces. To investigate the structures of the polar-nonpolar interfaces, we optimized two models: a parallel surface model and a constant mean curvature surface model. The parallel surface model agreed well with the X-ray data, and the R factors, which show the degree of agreement between those structural models and the data, were less than 0.04. In stark contrast, the constant mean curvature surface model deviated significantly from the data, and the R factors were around 0.15. We therefore conclude that the polar-nonpolar interface of the inverse bicontinuous cubic phase of the phytantriol/water system is close to a parallel surface to a triply periodic minimal surface.

Small-Angle X-ray Crystallography on Single-Crystal Regions of Inverse Bicontinuous Cubic Phases: Lipid Bilayer Structures and Gaussian Curvature-Dependent Fluctuations

I report for the first time an X-ray crystallographic study on single-crystal regions of cubic phases of a lyotropic liquid crystal. The single-crystal regions of three inverse bicontinuous cubic phases of a lipid, monoolein, diffracted X-ray only in small-angle regions, but amplitudes of structure factors were determined from the small-angle X-ray diffraction data with high accuracy. Structure factors from lipid bilayer models with constant thickness were optimized to amplitudes obtained from the X-ray data. By using amplitudes of the structure factors from X-ray data and phases from the models, electron density maps of three cubic phases were reconstructed. Lipid bilayer membranes, consisting of high density head regions and low density tail regions, were clearly distinguished in the electron density maps. Water regions had slightly lower density than that of the lipid head regions and were clearly visible for two of the cubic phases. Centers of bilayer membranes were located on the corresponding triply periodic minimal surfaces in the maps. Electron density data indicated Gaussian curvature-dependent fluctuations of bilayer membranes: the smaller the Gaussian curvature is, the larger the fluctuation becomes. The technique described in this report is expected to bring new knowledge in the structural research of lyotropic liquid crystals.

Silica Scaffold with Shifted "Plumber's Nightmare" Networks and their Interconversion into Diamond Networks

Bicontinuous structures with hyperbolic surfaces have been found in a variety of natural and synthetic systems. Herein, we present the synthesis and structural study of the shifted double-primitive networks, which is known as the rare "plumber's nightmare", and its interconversion into diamond networks. The scaffold was prepared by self-assembly of an amphiphilic triblock terpolymer and silica precursors. Electron crystallography indicates that the structure consists of two sets of hollow primitive networks shifted along 0.75b and 0.25c axes (2pcu(38 63), space group Cmcm). The "side-by-side" epitaxial relationship of the primitive and diamond networks with unit cell ratio of about 1.30 has been directly observed with the intermediate surface related to the rPD family. These results bring new insights to previous theoretical studies.

Emergence of ordered network mesophases in kinetic pathways of order-order transition for linear ABC triblock terpolymers

Applying the string method to the self-consistent field theory (SCFT) of ABC linear triblock copolymers, we developed a new strategy to design kinetic pathways for the formation of stable or metastable network mesophases in order-order transition (OOT) processes. The design principle regarding the kinetic pathways between distinct mesophases is based on the matching relationships of both domain spacing and dominant Fourier components of the density distributions. The results suggest that complex ordered network mesophases, such as alternating diamond (D^A) and alternating plumber's nightmare (P^A) could be obtained in kinetic pathways between simple phases covering lamellae, cylinders and spheres. By virtue of the minimal free energy pathway (MEP) obtained, we could acquire the epitaxial relationship and phase transition mechanism. Furthermore, we managed to regulate the MEP by changing the block composition to adjust packing frustration. Two new metastable networks, core-shell five-pronged and six-pronged morphologies, were found in the kinetic pathways, further demonstrating the regulating mechanism. The results will contribute to a better understanding of the kinetic relationship between simple phases and complex networks, thus providing a platform for soft materials design via the OOT route and guiding experimental procedures to fabricate ordered network mesophases.

Two Distinct Cylinder Arrangements in Monodomains of a Lyotropic Liquid Crystalline Hexagonal II Phase: Monodomains with Straight Cylinders and Ringed Cylinders in Capillaries

We report a method to produce two different monodomains of an inverse hexagonal II (HII) phase in capillaries. Capillaries filled with glyceryl monooleyl ether (GME) in an inverted micellar phase were soaked in water. After a week, a monodomain of the HII phase with straight cylinders was observed in a capillary with a diameter of 1.0 mm. The axis of the straight cylinders was almost parallel to the capillary axis, and the cylinders were slightly undulated. The lattice constant of the HII phase was 5.85 nm, which indicated the monodomain was fully hydrated. Another monodomain with ringed cylinders was observed in a 0.2 mm diameter capillary. The ringed cylinders aligned to the round capillary wall, where one of the ⟨10⟩ directions in the hexagonal lattice always faced the wall. The lattice constant was 4.89 nm, from which the estimated water content of the monodomain was almost the lowest reported for the HII phase. The monodomain with ringed cylinders is stabilized by the capillary wall and the low water content. This method to produce specific monodomains is expected to be of benefit for basic and applied research on the HII phase.