Complex tiling patterns in liquid crystals

Coarse-Grained Molecular Simulation of Bolapolyphiles with a Multident Lateral Chain: Formation and Structural Analysis of Cubic Network Phases

Bolapolyphiles constitute a versatile class of materials with a demonstrated potential to form a wide variety of complex ordered mesophases. In particular, cubic network phases (like the gyroid, primitive, and diamond phases) have been a target of many studies for their ability to create percolating 3D nanosized channels. In this study, molecular simulations are used to explore the phase behavior of bolapolyphiles containing a rigid rodlike core, associating hydrophilic core ends and a hydrophobic side chain with a multident architecture, i.e., where the branching pattern can vary from bident (two branches) to hexadent (six branches). Upon network phase formation, its skeleton is made up of "nodes" populated by the core ends and "struts" populated by the cores. It is shown that, by varying the side chain length, branching pattern, and attachment point to the core, one can alter the crowding around the cores and hence tune the nodal size and nodal valence (i.e., number of connecting struts) which lead to different types of network morphologies. For example, for a fixed total side chain length, having more branches generates a stronger crowding around the molecular core, driving them to form bundlelike domains with curvier interfaces that result in thinner struts. Also, attaching the lateral chain closer to one core end breaks the symmetry between the environments around the two core ends, leading to networks with bimodal nodal sizes. Importantly, since the characterization of (ordered or partially ordered) network phases is challenging given the potential incompatibilities between the simulation box size with the structure's space group periodic symmetry and the effect of morphological defects, a detailed framework is presented to analyze and fully characterize the unit cell parameters and structure factor of such systems.

Understanding the Role of Trapezoids in Honeycomb Self-Assembly-Pathways between a Columnar Liquid Quasicrystal and its Liquid-Crystalline Approximants

Quasiperiodic patterns and crystals-having long range order without translational symmetry-have fascinated researchers since their discovery. In this study, we report on new p-terphenyl-based T-shaped facial polyphiles with two alkyl end chains and a glycerol-based hydrogen-bonded side group that self-assemble into an aperiodic columnar liquid quasicrystal with 12-fold symmetry and its periodic liquid-crystalline approximants with complex superstructures. All represent honeycombs formed by the self-assembly of the p-terphenyls, dividing space into prismatic cells with polygonal cross-sections. In the perspective of tiling patterns, the presence of unique trapezoidal tiles, consisting of three rigid sides formed by the p-terphenyls and one shorter, incommensurate, and adjustable side by the alkyl end chains, plays a crucial role for these phases. A delicate temperature-dependent balance between conformational, entropic and space-filling effects determines the role of the alkyl chains, either as network nodes or trapezoid walls, thus resulting in the order-disorder transitions associated with emergence of quasiperiodicity. In-depth analysis suggests a change from a quasiperiodic tiling involving trapezoids to a modified one with a contribution of trapezoid pair fusion. This work paves the way for understanding quasiperiodicity emergence and develops fundamental concepts for its generation by chemical design of non-spherical molecules, aggregates, and frameworks based on dynamic reticular chemistry.

Highly branched bolapolyphilic liquid crystals with a cubic A15 network at the triangle-square transition

Rod-like bolapolyphiles with highly branched carbosilane-based side-chains self-assemble into several honeycomb structures if the oligo(p-phenylene ethynylene) core is polyfluorinated, whereas for the non-fluorinated series an A15 type cubic network of rod-bundles was observed instead, suggesting a brand new pathway for the transition between triangular and square honeycomb phases.

A columnar liquid quasicrystal with a honeycomb structure that consists of triangular, square and trapezoidal cells

Quasicrystals are intriguing structures that have long-range positional correlations but no periodicity in real space, and typically with rotational symmetries that are 'forbidden' in conventional periodic crystals. Here, we present a two-dimensional columnar liquid quasicrystal with dodecagonal symmetry. Unlike previous dodecagonal quasicrystals based on random tiling, a honeycomb structure based on a strictly quasiperiodic tessellation of tiles is observed. The structure consists of dodecagonal clusters made up of triangular, square and trapezoidal cells that are optimal for local packing. To maximize the presence of such dodecagonal clusters, the system abandons periodicity but adopts a quasiperiodic structure that follows strict packing rules. The stability of random-tiling dodecagonal quasicrystals is often attributed to the entropy of disordering when strict tiling rules are broken, at the sacrifice of the long-range positional order. However, our results demonstrate that quasicrystal stability may rest on energy minimization alone, or with only minimal entropic intervention.

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.

Tetrahedral Liquid-Crystalline Networks: An A15-Like Frank-Kasper Phase Based on Rod-Packing

The Pm 3 ‾ n cubic and other low-symmetry Frank-Kasper phases are known to be formed by soft spheres, ranging from metals to block copolymer micelles and colloidal nanoparticles. Here, we report a series of X-shaped polyphiles composed of sticky rods and two non-symmetric branched side-chains, which self-assemble into the first example of a cubic liquid-crystalline phase representing a tetrahedral network of rods with a Pm 3 ‾ n lattice. It is the topological dual to the Weaire-Phelan foam, being the Voronoi tessellation of the A15 sphere packing, from which this network is obtained by Delaunay triangulation.

Intercluster Exchange-Stabilized Novel Complex Colloidal χc Phase

Designing complex cluster crystals with a specific function using simple colloidal building blocks remains a challenge in materials science. Herein, we propose a conceptually new design strategy for constructing complex cluster crystals via hierarchical self-assembly of simple soft Janus colloids. A novel and previously unreported colloidal cluster-χ (χ_c) phase, which resembles the essential structural features of α-manganese but at a larger length scale, is obtained through molecular dynamics simulations. The formation of the χ_c phase undergoes a remarkable two-step self-assembly process, that is, the self-assembly of clusters with specific size dispersity from Janus colloids, followed by the highly ordered organization of these clusters. More importantly, the dynamic exchange of particles between these clusters plays a critical role in stabilizing the χ_c phase. Such a conceptual design framework based on intercluster exchange has the potential to effectively construct novel complex cluster crystals by hierarchical self-assembly of colloidal building blocks.

Emergence of uniform tilt and π-stacking in triangular liquid crystalline honeycombs

The synclinic tilted organization of specifically designed polyphilic oligo(p-phenylene ethynylene) rods in cylindrical shells around triangular prismatic cells on the <5 nm scale leads to a new kind of liquid crystalline honeycomb composed of helical shells with alternating helix sense. Core fluorination at the outer ring modifies the core-core interactions, thus resulting in triangular arrays with face-to-face π-stacking along the honeycomb.

Different Modes of Deformation of Soft Triangular Honeycombs at the Sub-5 nm Scale

Patterning on the sub-5 nm length scale is a contemporary challenge for further miniaturization of microelectronic circuits. Here, the first soft self-assembled triangular patterns are reported showing transitions between regular and two different kinds of isosceles (acute and obtuse angled) triangles on this length scale, formed by liquid crystalline honeycombs of polyphilic block molecules involving a fluorinated oligo(para-phenylene ethynylene) core. The type of formed triangular pattern depends on the degree and position of fluorination and on temperature. They are the first soft honeycombs combining tilted and nontilted organizations in a uniform nanostructure, where the tilted molecules in only one or two sides of the triangular prismatic cells dominate the shape and the size of the morphology.

A self-assembled liquid crystal honeycomb of highly stretched (3-1-1)-hexagons

A new liquid crystalline honeycomb phase is reported, containing highly stretched giant hexagonal cells with two opposing walls spanned by three consecutive end-to-end H-bonded rods, the (3-1-1) hexagons.

Soft self-assembled sub-5 nm scale chessboard and snub-square tilings with oligo(para-phenyleneethynylene) rods

X-Shaped bolapolyphiles, comprising a linear polyaromatic core with glycerol groups at each end and two chemically different and incompatible chains, fixed to it at opposite sides, were synthesized and found to self-assemble into honeycomb-type liquid crystalline phases with square symmetry. The polyaromatic π-conjugated rods form the cell walls and the resulting prismatic cells of sub-5 nm size are alternatively filled with perfluorocarbon (RF) and the carbosilane chains (RSi). The resulting structures can be represented as either a two-colour snub-square tiling with triangular and square cells or as a chessboard tiling of squares.

Controlling the Miscibility of X-Shaped Bolapolyphiles in Lipid Membranes by Varying the Chemical Structure and Size of the Polyphile Polar Headgroup

Bolaamphiphiles are well-known naturally occurring structures that can increase the thermal and mechanical stability of the phospholipid membrane by incorporation in a transmembrane manner. Modifications of bolaamphiphiles to introduce particular structural elements such as a conjugated aromatic backbone and lateral side chains in the hydrophobic region lead to bolapolyphiles (BPs). We investigated the ability of BPs to form lyotropic phases in water. The BPs had an identical backbone and side chains, but different headgroup structures, leading to different abilities to act as hydrogen bond donors and acceptors. BPs with hydrophilic headgroups capable of acting as hydrogen bond donors as well as acceptors did not form lyotropic phases and were insoluble in water, independent of whether the polar groups were small or large. The extended lipophilic core structure and the multiple intermolecular hydrogen bonds between the headgroups prevented the formation of well-hydrated lyotropic aggregates. A BP with two large hydrophilic headgroups of several ethylene oxide moieties terminated by methyl groups formed sheet- and vesicle-like aggregates in water. These headgroups act only as hydrogen bond acceptors and cannot form hydrogen bonds in the absence of water. The miscibility of BPs with vesicles of 1,2-dipalmitoyl- sn-glycero-3-phosphocholine (DPPC) in water and the resulting aggregate structures were also investigated. For BPs with headgroups acting as donors and acceptors of hydrogen bonds, macroscopic phase separation occurred in the mixed membranes, and two different membrane domains, a DPPC-rich one containing only little polyphile and a BP-rich one containing varying amounts of lipid, were formed. For headgroups without the ability to act as hydrogen bond donors, small BP aggregates were formed that were homogeneously distributed over the membrane. The lateral organization of BPs in lipid membranes can thus be controlled by the nature of the BP headgroup.

Lamellar Liquid Crystals of In-Plane Lying Rod-Like Mesogens with Designer Side-Chains: The Case of Sliding versus Locked Layers

The dimensionality of self-assembled nanostructures plays an essential role for their properties and applications. Herein, an understanding of the transition from weakly to strongly coupled layers in soft matter systems is provided involving in-plane organized π-conjugated rods. For this purpose, bolaamphiphilic triblock molecules consisting of a rigid biphenyl core, polar glycerol groups at the ends, and a branched (swallow-tail) or linear alkyl or semiperfluoroalkyl chain in lateral position have been synthesized and investigated. Besides weakly coupled lamellar isotropic (Lam_Iso ), lamellar nematic (Lam_N ) and sliding lamellar smectic phases (Lam_Sm ), a sequence of three distinct types of strongly coupled (correlated) lamellar smectic phases with either centered (c2mm) or non-centered rectangular (p2mm) lattice and an intermediate oblique lattices (p2) were observed depending on chain length, chain branching and degree of chain fluorination. This new sequence is explained by the strengthening of the layer coupling and the competition between energetic packing constraints and the entropic contribution of either longitudinal or tangential fluctuations. This example of directed side chain engineering of small generic model compounds provides general clues for morphological design of two-dimensional and three-dimensionally coupled lamellar systems involving larger π-conjugated molecular rods and molecular or supramolecular polymers, being of actual interest in organic electronics and nanotechnology.

Transition between tangential and co-axial liquid crystalline honeycombs in the self-assembly of Y-shaped bolapolyphiles

A new liquid crystalline honeycomb with an organization of π-conjugated rods parallel to the honeycomb cells is formed by molecular self-assembly.

Molecular dynamics simulation of thermotropic bolaamphiphiles with a swallow-tail lateral chain: formation of cubic network phases

T-shaped bolaamphiphiles (TBA) with a swallow-tail lateral chain have been found to provide a fertile platform to produce complex liquid crystalline phases that are accessible through changes of temperature and lateral chain length and design. In this work, we use molecular simulations of a simple coarse-grained model to map out the phase behavior of this type of molecules. This model is based on the premise that the crucial details of the fluid structure stem from close range repulsions and the strong directional forces typical of hydrogen bonds. Our simulations confirm that TBAs exhibit a rich phase behavior upon increasing the length of their lateral chain. The simulations detect a double gyroid phase and an axial-bundle columnar phase which bear some structural resemblance to those found in the experiment. In addition, simulations predict two cocontinuous phases with 3D-periodicity: the "single" diamond and the "single" plumber's nightmare phase. Our analysis of energetic and entropic contributions to the free energy of phases formed by TBA with either swallow-tail or linear side-chains suggest that the 3D-periodic network phases formed by the former are stabilized by the large conformation entropy of the side-chains.

Effects of Lateral and Terminal Chains of X-Shaped Bolapolyphiles with Oligo(phenylene ethynylene) Cores on Self-Assembly Behaviour. Part 1: Transition between Amphiphilic and Polyphilic Self-Assembly in the Bulk

Polyphilic self-assembly leads to compartmentalization of space and development of complex structures in soft matter on different length scales, reaching from the morphologies of block copolymers to the liquid crystalline (LC) phases of small molecules. Whereas block copolymers are known to form membranes and interact with phospholipid bilayers, liquid crystals have been less investigated in this respect. Here, series of bolapolyphilic X-shaped molecules were synthesized and investigated with respect to the effect of molecular structural parameters on the formation of LC phases (part 1), and on domain formation in phospholipid bilayer membranes (part 2). The investigated bolapolyphiles are based on a rod-like π-conjugated oligo(phenylene ethynylene) (OPE) core with two glycerol groups being either directly attached or separated by additional ethylene oxide (EO) units to both ends. The X-shape is provided by two lateral alkyl chains attached at opposite sides of the OPE core, being either linear, branched, or semiperfluorinated. In this report, the focus is on the transition from polyphilic (triphilic or tetraphilic) to binary amphiphilic self-assembly. Polyphilic self-assembly, i.e., segregation of all three or four incorporated units into separate nano-compartments, leads to the formation of hexagonal columnar LC phases, representing triangular honeycombs. A continuous transition from the well-defined triangular honeycomb structures to simple hexagonal columnar phases, dominated by the arrangement of polar columns on a hexagonal lattice in a mixed continuum formed by the lipophilic chains and the OPE rods, i.e., to amphiphilic self-assembly, was observed by reducing the length and volume of the lateral alkyl chains. A similar transition was found upon increasing the length of the EO units involved in the polar groups. If the lateral alkyl chains are enlarged or replaced by semiperfluorinated chains, then the segregation of lateral chains and rod-like cores is retained, even for enlarged polar groups, i.e., the transition from polyphilic to amphiphilic self-assembly is suppressed.

Transition from nematic to gyroid-type cubic soft self-assembly by side-chain engineering of π-conjugated sticky rods

A sequence of liquid crystalline phases, involving cybotactic nematics, a lamellar phase, bicontinuous cubics and triangular honeycombs, was observed for oligo(phenylene ethynylene) based X-shaped bolapolyphiles with two long lateral alkyl chains and sticky ends provided by glycerol groups. In the cubic phase with Ia3[combining macron]d lattice - which is tailored by alkyl chain engineering - the aromatic cores are organized on the gyroid minimal surface in 3D curved layers of almost parallel aligned π-conjugated rods. It is shown that this type of cubic phase is a general mode of soft self-assembly of X-shaped bolapolyphiles at the cross-over from the (long or short range) lamellar to the triangular honeycomb-like organization. Cubic phase formation is found only in a narrow range with respect to temperature and chain-length for the non-fluorinated compounds and in much wider ranges for related core-fluorinated molecules.

Formation of a Double Diamond Cubic Phase by Thermotropic Liquid Crystalline Self-Assembly of Bundled Bolaamphiphiles

A quaternary amphiphile with swallow-tail side groups displays a new bicontinuous thermotropic cubic phase with symmetry Pn3‾ m and formed by two interpenetrating networks where cylindrical segments are linked by H bonds at tetrahedral junctions. Each network segment contains two bundles, each containing 12 rod-like mesogens, lying along the segment axis. This assembly leads to the first thermotropic structure of the "double diamond" type. A quantitative geometric model is proposed to explain the occurrence of this rare phase.

Ionic Liquid Crystals: Versatile Materials

This Review covers the recent developments (2005-2015) in the design, synthesis, characterization, and application of thermotropic ionic liquid crystals. It was designed to give a comprehensive overview of the "state-of-the-art" in the field. The discussion is focused on low molar mass and dendrimeric thermotropic ionic mesogens, as well as selected metal-containing compounds (metallomesogens), but some references to polymeric and/or lyotropic ionic liquid crystals and particularly to ionic liquids will also be provided. Although zwitterionic and mesoionic mesogens are also treated to some extent, emphasis will be directed toward liquid-crystalline materials consisting of organic cations and organic/inorganic anions that are not covalently bound but interact via electrostatic and other noncovalent interactions.

Zeolite-like liquid crystals

Zeolites represent inorganic solid-state materials with porous structures of fascinating complexity. Recently, significant progress was made by reticular synthesis of related organic solid-state materials, such as metal-organic or covalent organic frameworks. Herein we go a step further and report the first example of a fluid honeycomb mimicking a zeolitic framework. In this unique self-assembled liquid crystalline structure, transverse-lying π-conjugated rod-like molecules form pentagonal channels, encircling larger octagonal channels, a structural motif also found in some zeolites. Additional bundles of coaxial molecules penetrate the centres of the larger channels, unreachable by chains attached to the honeycomb framework. This creates a unique fluid hybrid structure combining positive and negative anisotropies, providing the potential for tuning the directionality of anisotropic optical, electrical and magnetic properties. This work also demonstrates a new approach to complex soft-matter self-assembly, by using frustration between space filling and the entropic penalty of chain extension.

Zeolites with regular porous structures are widely used as gas adsorbents and scaffolding for catalysts. Poppe et al. report a liquid crystal with zeolite-like structure by self-assembly of polyphilic molecules with π-conjugated rod-like cores into a honeycomb formed by pentagonal/octagonal channels.

Dendritic domains with hexagonal symmetry formed by x-shaped bolapolyphiles in lipid membranes

A novel class of bolapolyphile (BP) molecules are shown to integrate into phospholipid bilayers and self-assemble into unique sixfold symmetric domains of snowflake-like dendritic shapes. The BPs comprise three philicities: a lipophilic, rigid, π-π stacking core; two flexible lipophilic side chains; and two hydrophilic, hydrogen-bonding head groups. Confocal microscopy, differential scanning calorimetry, XRD, and solid-state NMR spectroscopy confirm BP-rich domains with transmembrane-oriented BPs and three to four lipid molecules per BP. Both species remain well organized even above the main 1,2-dipalmitoyl-sn-glycero-3-phosphocholine transition. The BP molecules only dissolve in the fluid membrane above 70 °C. Structural variations of the BP demonstrate that head-group hydrogen bonding is a prerequisite for domain formation. Independent of the head group, the BPs reduce membrane corrugation. In conclusion, the BPs form nanofilaments by π stacking of aromatic cores, which reduce membrane corrugation and possibly fuse into a hexagonal network in the dendritic domains.

Temperature-dependent in-plane structure formation of an X-shaped bolapolyphile within lipid bilayers

Polyphilic compound B12 is an X-shaped molecule with a stiff aromatic core, flexible aliphatic side chains, and hydrophilic end groups. Forming a thermotropic triangular honeycomb phase in the bulk between 177 and 182 °C but no lyotropic phases, it is designed to fit into DPPC or DMPC lipid bilayers, in which it phase separates at room temperature, as observed in giant unilamellar vesicles (GUVs) by fluorescence microscopy. TEM investigations of bilayer aggregates support the incorporation of B12 into intact membranes. The temperature-dependent behavior of the mixed samples was followed by differential scanning calorimetry (DSC), FT-IR spectroscopy, fluorescence spectroscopy, and X-ray scattering. DSC results support in-membrane phase separation, where a reduced main transition and new B12-related transitions indicate the incorporation of lipids into the B12-rich phase. The phase separation was confirmed by X-ray scattering, where two different lamellar repeat distances are visible over a wide temperature range. Polarized ATR-FTIR and fluorescence anisotropy experiments support the transmembrane orientation of B12, and FT-IR spectra further prove a stepwise "melting" of the lipid chains. The data suggest that in the B12-rich domains the DPPC chains are still rigid and the B12 molecules interact with each other via π-π interactions. All results obtained at temperatures above 75 °C confirm the formation of a single, homogeneously mixed phase with freely mobile B12 molecules.

Tolane-based bent bolaamphiphiles forming liquid crystalline hexagonal honeycombs with trigonal symmetry

Fluorescent bent bolaamphiphiles self-assemble into non-centrosymmetric honeycombs with p3m1 symmetry, which can be solvent-stabilized to form lyotropic LC phases.

Engineering entropy in soft matter: the bad, the ugly and the good

The role of entropic interactions, often subtle and sometimes crucial, on the structure and properties of soft matter has a well-recognized place in the classic and modern scientific literature. However, the lessons learned from many of those studies do not always form part of the standard arsenal of strategies that are taught or used for de novo studies relevant to the engineering of new materials. Fortunately, a growing number of examples exist where entropic effects have been designed a priori to achieve a desired or new outcome. This tutorial review describes some recent such examples, selected to illustrate the potential benefits of a more pro-active approach to harnessing the often overlooked power of entropy.

Development of structural complexity by liquid-crystal self-assembly

Since the discovery of the liquid-crystalline state of matter 125 years ago, this field has developed into a scientific area with many facets. This Review presents recent developments in the molecular design and self-assembly of liquid crystals. The focus is on new exciting soft-matter structures distinct from the usually observed nematic, smectic, and columnar phases. These new structures have enhanced complexity, including multicompartment and cellular structures, periodic and quasiperiodic arrays of spheres, and new emergent properties, such as ferroelctricity and spontaneous achiral symmetry-breaking. Comparisons are made with developments in related fields, such as self-assembled monolayers, multiblock copolymers, and nanoparticle arrays. Measures of structural complexity used herein are the size of the lattice, the number of distinct compartments, the dimensionality, and the logic depth of the resulting supramolecular structures.

Local frustration determines molecular and macroscopic helix structures

Decorative domains force amyloid fibers to adopt spiral ribbon morphologies, as opposed to the more common twisted ribbon. We model the effect of decorating domains as a perturbation to the relative orientation of β strands in a bilayered extended β-sheet. The model consists of minimal energy assemblies of rigid building blocks containing two anisotropic interacting ellipsoids. The relative orientation of the ellipsoids dictates the morphology of the resulting assembly. Amyloid structures derived from experiment are consistent with our model, and we use magnets to demonstrate that the frustration principle is scale and system independent. In contrast to other models of amyloid, our model isolates the effect of frustration from the fundamental interactions between building blocks to reveal the frustration rather than dependence of morphology on the physical interactions. Consequently, amyloid is viewed as a discrete molecular version of the more general macroscopic frustrated bilayer that is exemplified by Bauhinia seedpods. The model supports the idea that the interactions arising from an arbitrary peptide sequence can support an amyloid structure if a bilayer can form first, which suggests that supplementary protein sequences, such as chaperones or decorative domains, could play a significant role in stabilizing such bilayers and therefore in selecting morphology during nucleation. Our model provides a foundation for exploring the effects of frustration on higher-order superstructural polymorphic assemblies that may exhibit complex functional behavior. Two outstanding examples are the systematic kinking of decorated fibers and the nested frustration of the Bauhinia seedpod.

Striped networks and other hierarchical structures in AmBmCn (2m+n)-miktoarm star terpolymer melts

Using dissipative particle dynamics simulations we give numerical evidence of the formation of "striped" (or AB alternating) diamond and gyroid network structures and other hierarchical morphologies in A(m)B(m)C(n) (2m+n)-miktoarm star terpolymers where the main variable is the ratio x=n/m with m,n being the number of equal length polymer arms of A and B and C, respectively. The formed networks are purely a result of the star topology, as clearly shown by direct comparison with parallel ABC miktoarm star terpolymer simulations with matching overall composition. Progressively changing x, the system adopts the following phase sequence: three-colored lamellae, C spheres embedded in AB lamellae, C spheres decorating AB lamellae, three-colored [6.6.6] tiling, AB striped diamond network, AB striped gyroid network, AB striped hexagonally arranged cylinders, and finally AB striped globular aggregates. The striped gyroid is particularly interesting as it constitutes an inherently chiral structure made from achiral building blocks.