Directed Self-Assembly of Diamond Networks in Triblock Terpolymer Films on Patterned Substrates

Block copolymers (BCPs) are particularly effective in creating soft nanostructured templates for transferring complex 3D network structures into inorganic materials that are difficult to fabricate by other methods. However, achieving control of the local ordering within these 3D networks over large areas remains a significant obstacle to advancing material properties. Here, we address this challenge by directing the self-assembly of a 3D alternating diamond morphology by solvent vapor annealing of a triblock terpolymer film on a chemically patterned substrate. The hexagonal substrate patterns were designed to match a (111) plane of the diamond lattice. Commensurability between the sparse substrate pattern and the BCP lattice produced a uniformly ordered diamond network within the polymer film, as confirmed by a combination of atomic force microscopy and cross-sectional imaging using focused ion beam scanning electron microscopy. The successful replication of the complex and well-ordered 3D network structure in gold promises to advance optical metamaterials and has potential applications in nanophotonics.

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.

Recent advances in versatile inverse lyotropic liquid crystals

Owing to the rapid and significant progress in advanced materials and life sciences, nanotechnology is increasingly gaining in popularity. Among numerous bio-mimicking carriers, inverse lyotropic liquid crystals are known for their unique properties. These carriers make accommodation of molecules with varied characteristics achievable due to their complicated topologies. Besides, versatile symmetries of inverse LCNPs (lyotropic crystalline nanoparticles) and their aggregating bulk phases allow them to be applied in a wide range of fields including drug delivery, food, cosmetics, material sciences etc. In this review, in-depth summary, discussion and outlook for inverse lyotropic liquid crystals are provided.

Medial packing and elastic asymmetry stabilize the double-gyroid in block copolymers

Triply-periodic networks are among the most complex and functionally valuable self-assembled morphologies, yet they form in nearly every class of biological and synthetic soft matter building blocks. In contrast to simpler assembly motifs – spheres, cylinders, layers – networks require molecules to occupy variable local environments, confounding attempts to understand their formation. Here, we examine the double-gyroid network phase by using a geometric formulation of the strong stretching theory of block copolymer melts, a prototypical soft self-assembly system. The theory establishes the direct link between molecular packing, assembly thermodynamics and the medial map, a generic measure of the geometric center of complex shapes. We show that “medial packing” is essential for stability of double-gyroid in strongly-segregated melts, reconciling a long-standing contradiction between infinite- and finite-segregation theories. Additionally, we find a previously unrecognized non-monotonic dependence of network stability on the relative entropic elastic stiffness of matrix-forming to tubular-network forming blocks. The composition window of stable double-gyroid widens for both large and small elastic asymmetry, contradicting intuitive notions that packing frustration is localized to the tubular domains. This study demonstrates the utility of optimized medial tessellations for understanding soft-molecular assembly and packing frustration via an approach that is readily generalizable far beyond gyroids in neat block copolymers.

Double-gyroid networks assemble in diverse soft materials, yet the molecular packing that underlies their complex structure remains obscure. Here, authors advance a theory that resolves a long-standing puzzle about their formation in block copolymers.

Handlebody decompositions of three-manifolds and polycontinuous patterns

We introduce the concept of a handlebody decomposition of a three-manifold, a generalization of a Heegaard splitting, or a trisection. We show that two handlebody decompositions of a closed orientable three-manifold are stably equivalent. As an application to materials science, we consider a mathematical model of polycontinuous patterns and discuss a topological study of microphase separation of a block copolymer melt.

Aqueous Lyotropic Mesophase Behavior of Gemini Dicarboxylate Surfactants Swollen with n-Decane

We report detailed small-angle X-ray scattering (SAXS) studies of the impact of variable n-decane loadings on the lyotropic liquid crystalline (LLC) phase behaviors of homologous bis(tetramethylammonium) gemini didecanoate surfactants TMA-7x, which derive from dimerizing decanoic acid through its α-carbon with hydrocarbyl linkers -(CH₂)_x- where x = 3, 4, 5, and 6. TMA-7x amphiphiles with x = 3 or 5 exhibit a strong propensity to form normal double gyroid (G) LLC network mesophases over wide surfactant hydration ranges, as compared to homologues with x = 4 or 6. On swelling aqueous TMA-7x LLC mesophases with up to 35 wt % n-decane, we demonstrate that odd-carbon linked surfactants (x = 3 or 5) form G and normal double diamond (D) phases over wide water concentration windows with T = 22-100 °C. Complementary studies of decane-swollen TMA-7x (x = 4 or 6) aqueous LLCs instead demonstrate significantly diminished network phase stability, in favor of hexagonally-packed cylinder phases and a zoo of complex quasispherical micelle packings, which include micellar C14 and C15 Laves phases (P6₃/mmc and Fd3(-)m symmetries, respectively) and high-symmetry hexagonally close packed (HCP) and body-centered cubic (BCC) arrangements. These rich phase behaviors are rationalized in terms of linker length parity-dependent surfactant conformations and the delicate free energy balance that guides the packing of these geometrically anisotropic amphiphiles by minimizing unfavorable water-hydrophobic contacts, maximizing ionic surfactant-headgroup counterion solvation with minimal local variations, and maximizing electrostatic cohesion within these supramolecular assemblies.

Anatomy of triply-periodic network assemblies: characterizing skeletal and inter-domain surface geometry of block copolymer gyroids

Triply-periodic networks (TPNs), like the well-known gyroid and diamond network phases, abound in soft matter assemblies, from block copolymers (BCPs), lyotropic liquid crystals and surfactants to functional architectures in biology. While TPNs are, in reality, volume-filling patterns of spatially-varying molecular composition, physical and structural models most often reduce their structure to lower-dimensional geometric objects: the 2D interfaces between chemical domains; and the 1D skeletons that thread through inter-connected, tubular domains. These lower-dimensional structures provide a useful basis of comparison to idealized geometries based on triply-periodic minimal, or constant-mean curvature surfaces, and shed important light on the spatially heterogeneous packing of molecular constituents that form the networks. Here, we propose a simple, efficient and flexible method to extract a 1D skeleton from 3D volume composition data of self-assembled networks. We apply this method to both self-consistent field theory predictions as well as experimental electron microtomography reconstructions of the double-gyroid phase of an ABA triblock copolymer. We further demonstrate how the analysis of 1D skeleton, 2D inter-domain surfaces, and combinations therefore, provide physical and structural insight into TPNs, across multiple length scales. Specifically, we propose and compare simple measures of network chirality as well as domain thickness, and analyze their spatial and statistical distributions in both ideal (theoretical) and non-ideal (experimental) double gyroid assemblies.

Optimal packings of three-arm star polyphiles: from tricontinuous to quasi-uniformly striped bicontinuous forms

Star-shaped molecules with three mutually immiscible arms self-assemble to form a variety of novel structures, with conformations that attempt to minimize interfacial area between the domains composed of the different arms. The geometric frustration caused by the joining of these arms at a common centre limits the size and shape of each domain, encouraging the creation of complex and interesting solutions. Some solutions are tricontinuous, and these solutions (and others) share aspects of bicontinuous structures with amphiphilic assemblies as similar molecular segregation factors are at work. We describe both highly symmetric and balanced structures, as well as unbalanced solutions that take the form of intricately striped amphiphilic membranes. All these patterns can result in chiral assemblies with multiple networks.

Purely entropic self-assembly of the bicontinuous Ia3d gyroid phase in equilibrium hard-pear systems

We investigate a model of hard pear-shaped particles which forms the bicontinuous Ia[Formula: see text]d structure by entropic self-assembly, extending the previous observations of Barmes et al. (2003 Phys. Rev. E68, 021708. (doi:10.1103/PhysRevE.68.021708)) and Ellison et al. (2006 Phys. Rev. Lett.97, 237801. (doi:10.1103/PhysRevLett.97.237801)). We specifically provide the complete phase diagram of this system, with global density and particle shape as the two variable parameters, incorporating the gyroid phase as well as disordered isotropic, smectic and nematic phases. The phase diagram is obtained by two methods, one being a compression-decompression study and the other being a continuous change of the particle shape parameter at constant density. Additionally, we probe the mechanism by which interdigitating sheets of pears in these systems create surfaces with negative Gauss curvature, which is needed to form the gyroid minimal surface. This is achieved by the use of Voronoi tessellation, whereby both the shape and volume of Voronoi cells can be assessed in regard to the local Gauss curvature of the gyroid minimal surface. Through this, we show that the mechanisms prevalent in this entropy-driven system differ from those found in systems which form gyroid structures in nature (lipid bilayers) and from synthesized materials (di-block copolymers) and where the formation of the gyroid is enthalpically driven. We further argue that the gyroid phase formed in these systems is a realization of a modulated splay-bend phase in which the conventional nematic has been predicted to be destabilized at the mesoscale due to molecular-scale coupling of polar and orientational degrees of freedom.

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.

Two sides of the coin. Part 1. Lipid and surfactant self-assembly revisited

Hofmeister, specific ion effects, hydration and van der Waals forces at and between interfaces are factors that determine curvature and microstructure in self assembled aggregates of surfactants and lipids; and in microemulsions. Lipid and surfactant head group interactions and between aggregates vary enormously and are highly specific. They act on the hydrophilic side of a bilayer, micelle or other self assembled aggregate. It is only over the last three decades that the origin of Hofmeister effects has become generally understood. Knowledge of their systematics now provides much flexibility in designing nanostructured fluids. The other side of the coin involves equally specific forces. These (opposing) forces work on the hydrophobic side of amphiphilic interfaces. They are due to the interaction of hydrocarbons and other "oils" with hydrophobic tails of surfactants and lipids. The specificity of oleophilic solutes in microemulsions and lipid membranes provides a counterpoint to Hofmeister effects and hydration. Together with global packing constraints these effects determine microstructure. Another factor that has hardly been recognised is the role of dissolved gas. This introduces further, qualitative changes in forces that prescribe microstructure. The systematics of these effects and their interplay are elucidated. Awareness of these competing factors facilitates formulation of self assembled nanostructured fluids. New and predictable geometries that emerge naturally provide insights into a variety of biological phenomena like anaesthetic and pheromone action and transmission of the nervous impulse (see Part 2).

Bragg-mirror-like circular dichroism in bio-inspired quadruple-gyroid 4srs nanostructures

The smooth and tailorable spectral response of Bragg mirrors has driven their pervasive use in optical systems requiring customizable spectral control of beam propagation. However, the simple nature of Bragg mirror reflection prevents their application to the control of important polarization states such as circular polarization. While helical and gyroid-based nanostructures exhibiting circular dichroism have been developed extensively to address this limitation, they are often restricted by the spectral inconsistency of their optical response. Here we present the fabrication and characterization of quadruple-gyroid 4srs nanostructures exhibiting bio-inspired Bragg-mirror-like circular dichroism: a smooth and uniform band of circular dichroism reminiscent of the spectrum of a simple multilayer Bragg-mirror. Furthermore, we demonstrate that the circular dichroism produced by 4srs nanostructures are robust to changes in incident angle and beam collimation, providing a new platform to create and engineer circular dichroism for functional circular polarization manipulation.

Comparative study of the inverse versus normal bicontinuous cubic phases of the β-d-glucopyranoside water-driven self-assemblies using fluorescent probes

This work investigates the head group region of the inverse and normal bicontinuous cubic phases (Ia3d space group) of the glucopyranoside/water system using 2-(2′-hydroxyphenyl)benzoxazole and its derivatives as fluorescent probes.

Periodic entanglement III: tangled degree-3 finite and layer net intergrowths from rare forests

Entanglements of two-dimensional honeycomb nets are constructed from free tilings of the hyperbolic plane (H2) on triply periodic minimal surfaces. The 2-periodic nets that comprise the structures are guaranteed by considering regular, rare free tilings in H2. This paper catalogues an array of entanglements that are both beautiful and challenging for current classification techniques, including examples that are realized in metal-organic materials. The compactification of these structures to the genus-3 torus is considered as a preliminary method for generating entanglements of finite θ-graphs, potentially useful for gaining insight into the entanglement of the periodic structure. This work builds on previous structural enumerations given in Periodic entanglement Parts I and II [Evans et al. (2013). Acta Cryst. A69, 241-261; Evans et al. (2013). Acta Cryst. A69, 262-275].

Ideal geometry of periodic entanglements

Three-dimensional entanglements, including knots, knotted graphs, periodic arrays of woven filaments and interpenetrating nets, form an integral part of structure analysis because they influence various physical properties. Ideal embeddings of these entanglements give insight into identification and classification of the geometry and physically relevant configurations in vivo . This paper introduces an algorithm for the tightening of finite, periodic and branched entanglements to a least energy form. Our algorithm draws inspiration from the Shrink-On-No-Overlaps (SONO) (Pieranski 1998 In Ideal knots (eds A Stasiak, V Katritch, LH Kauffman), vol. 19, pp. 20–41.) algorithm for the tightening of knots and links: we call it Periodic-Branched Shrink-On-No-Overlaps (PB-SONO). We reproduce published results for ideal configurations of knots using PB-SONO. We then examine ideal geometry for finite entangled graphs, including θ -graphs and entangled tetrahedron- and cube-graphs. Finally, we compute ideal conformations of periodic weavings and entangled nets. The resulting ideal geometry is intriguing: we see spontaneous symmetrisation in some cases, breaking of symmetry in others, as well as configurations reminiscent of biological and chemical structures in nature.

Identification of large channels in cationic PEGylated cubosome nanoparticles by synchrotron radiation SAXS and Cryo-TEM imaging

Extra-large nanochannel formation in the internal structure of cationic cubosome nanoparticles results from the interplay between charge repulsion and steric stabilization of the lipid membrane interfaces and is evidenced by cryogenic transmission electron microscopy (Cryo-TEM) and synchrotron radiation small-angle X-ray scattering (SAXS). The swollen cubic symmetry of the lipid nanoparticles emerges through a shaping transition of onion bilayer vesicle intermediates containing a fusogenic nonlamellar lipid. Cationic amphiphile cubosome particles, thanks to the advantages of their liquid crystalline soft porous nanoarchitecture and capability for multi-drug nanoencapsulation, appear to be of interest for the design of mitochondrial targeting devices in anti-cancer therapies and as siRNA nanocarriers for gene silencing.

Comment on "Discovery of a tetracontinuous, aqueous lyotropic network phase with unusual 3D-hexagonal symmetry" by M. Mahanthappa, G. Sorenson and A. Schmitt

The article by Sorenson et al. (Soft Matter10, 8229, 2014) reports a novel phase formed by gemini surfactants in water, of symmetry P63/mcm and based on a triple intergrowth of three identical degree-three networks, known as 3etc(193). This phase is the first lyotropic liquid crystalline phase based on the intergrowth of a triplet of network- or labyrinth-like hydrophobic domains. We provide here results from self-consistent field theory that demonstrate that the same morphology is almost stable in standard AB diblock copolymer melts; at the phase transition between the double gyroid phase and the hexagonal columnar phase, the 3etc(193) morphology only incurs a marginal free energy penalty compared to the equilibrium phases. Interestingly, the ratio of lattice parameters c/a = 0.955 of the 3etc(193) as a diblock morphology is very close to that of the gemini surfactant phase and of the related IBN-9 mesoporous silicate phase (Han et al., Nat. Chem.1, 123, 2009). Based on the combination of these results, we hypothesise that the 3etc(193) morphology is likely a generic phase in soft materials, rather than an oddity.

Reply to the 'Comment on "Discovery of a tetracontinuous, aqueous lyotropic network phase with unusual 3D-hexagonal symmetry"' by G. Schröder-Turk, M. Fischer and S. Hyde

A new calculation by Fischer et al. suggests that the 3etc network phase with 3D-hexagonal symmetry (space group #193: P63/mcm) may be a generic structure adopted by self-assembling soft materials, thereby broadening the context for our recent report of its spontaneous formation in lyotropic liquid crystals. The experimental observation of the 3etc phase further validates previous theoretical models used to predict its stability, provocatively suggesting that other polycontinuous network phases predicted by these methods may be discovered in the future. While these network phase morphologies are often mathematically described in terms of their underlying triply periodic minimal surfaces (TPMS), the numerous potential applications of these functional nanostructured soft materials require the development of a concise, consistent, and unambiguous nomenclature for their complete description. In this comment, we propose adoption of a nomenclature that describes each mesophase more generally in terms of the total number of non-intersecting domains into which three-dimensional space is partitioned.

High-symmetry embeddings of interpenetrating periodic nets. Essential rings and patterns of catenation

Symmetrical embeddings are given for multiply intergrown sets of some commonly occurring nets such asdia(diamond),qtz(quartz),pcu(net of primitive cubic lattice) andsrs(labyrinth net of theGminimal surface). Data are also given for all known pairs of nets which have edge-transitive self-dual tilings. Examples are given for symmetrical polycatenation of the 2-periodic netssql(square lattice) andhcb(honeycomb). The idea that the rings that are the faces of natural tilings form a complete basis set (essential rings) is explored and patterns of catenation of such rings described.

Discovery of a tetracontinuous, aqueous lyotropic network phase with unusual 3D-hexagonal symmetry

Network phase aqueous lyotropic liquid crystals (LLCs) are technologically useful materials with myriad applications in chemistry, biology, and materials science, which stem from their structurally periodic aqueous and hydrophobic nanodomains (∼0.7-5.0 nm in diameter) that are lined with well-defined chemical functionalities. The exclusive observation of bicontinuous cubic network phase LLCs (e.g., double gyroid, double diamond, and primitive phases) has fueled speculations that all stable LLC network phases must exhibit cubic symmetry. Herein, we describe the self-assembly behavior of a simple aliphatic gemini surfactant that forms the first example of a triply periodic network phase LLC with the 3D-hexagonal symmetry P63/mcm (space group #193). This normal, tetracontinuous 3D-hexagonal network LLC phase HI(193) partitions space into four continuous and interpenetrating, yet non-intersecting volumes. This discovery directly demonstrates that the gemini amphiphile platform furnishes a rational strategy for discovering and stabilizing new, three-dimensionally periodic multiply continuous network phase LLCs with variable symmetries and potentially new applications.

Phase behavior and complex crystal structures of self-assembled tethered nanoparticle telechelics

Motivated by growing interest in the self-assembly of nanoparticles for applications such as photonics, organic photovoltaics, and DNA-assisted designer crystals, we explore the phase behavior of tethered spherical nanoparticles. Here, a polymer tether is used to geometrically constrain a pair of nanoparticles creating a tethered nanoparticle "telechelic". Using simulation, we examine how varying architectural features, such as the size ratio of the two end-group nanospheres and the length of the flexible tether, affects the self-assembled morphologies. We demonstrate not only that this hybrid building block maintains the same phase diversity as linear triblock copolymers, allowing for a variety of nanoparticle materials to replace polymer blocks, but also that new structures not previously reported are accessible. Our findings imply a robust underlying ordering mechanism is common among these systems, thus allowing flexibility in synthesis approaches to achieve a target morphology.

Introductory lecture: basic quantities in model biomembranes

One of the many aspects of membrane biophysics dealt with in this Faraday Discussion regards the material moduli that describe energies at a supramolecular level. This introductory lecture first critically reviews differences in reported numerical values of the bending modulus K(C), which is a central property for the biologically important flexibility of membranes. It is speculated that there may be a reason that the shape analysis method tends to give larger values of K(C) than the micromechanical manipulation method or the more recent X-ray method that agree very well with each other. Another theme of membrane biophysics is the use of simulations to provide exquisite detail of structures and processes. This lecture critically reviews the application of atomic level simulations to the quantitative structure of simple single component lipid bilayers and diagnostics are introduced to evaluate simulations. Another theme of this Faraday Discussion was lateral heterogeneity in biomembranes with many different lipids. Coarse grained simulations and analytical theories promise to synergistically enhance experimental studies when their interaction parameters are tuned to agree with experimental data, such as the slopes of experimental tie lines in ternary phase diagrams. Finally, attention is called to contributions that add relevant biological molecules to bilayers and to contributions that study the exciting shape changes and different non-bilayer structures with different lipids.