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.

Additive lattice kirigami.

Kirigami uses bending, folding, cutting, and pasting to create complex three-dimensional (3D) structures from a flat sheet. In the case of lattice kirigami, this cutting and rejoining introduces defects into an underlying 2D lattice in the form of points of nonzero Gaussian curvature. A set of simple rules was previously used to generate a wide variety of stepped structures; we now pare back these rules to their minimum. This allows us to describe a set of techniques that unify a wide variety of cut-and-paste actions under the rubric of lattice kirigami, including adding new material and rejoining material across arbitrary cuts in the sheet. We also explore the use of more complex lattices and the different structures that consequently arise. Regardless of the choice of lattice, creating complex structures may require multiple overlapping kirigami cuts, where subsequent cuts are not performed on a locally flat lattice. Our additive kirigami method describes such cuts, providing a simple methodology and a set of techniques to build a huge variety of complex 3D shapes.

Algorithmic lattice kirigami: A route to pluripotent materials.

We use a regular arrangement of kirigami elements to demonstrate an inverse design paradigm for folding a flat surface into complex target configurations. We first present a scheme using arrays of disclination defect pairs on the dual to the honeycomb lattice; by arranging these defect pairs properly with respect to each other and choosing an appropriate fold pattern a target stepped surface can be designed. We then present a more general method that specifies a fixed lattice of kirigami cuts to be performed on a flat sheet. This single pluripotent lattice of cuts permits a wide variety of target surfaces to be programmed into the sheet by varying the folding directions.

Making the cut: lattice kirigami rules.

In this Letter we explore and develop a simple set of rules that apply to cutting, pasting, and folding honeycomb lattices. We consider origami-like structures that are extrinsically flat away from zero-dimensional sources of Gaussian curvature and one-dimensional sources of mean curvature, and our cutting and pasting rules maintain the intrinsic bond lengths on both the lattice and its dual lattice. We find that a small set of rules is allowed providing a framework for exploring and building kirigami­folding, cutting, and pasting the edges of paper.

Trading spaces: building three-dimensional nets from two-dimensional tilings.

We construct some examples of finite and infinite crystalline three-dimensional nets derived from symmetric reticulations of homogeneous two-dimensional spaces: elliptic (S (2)), Euclidean (E (2)) and hyperbolic (H (2)) space. Those reticulations are edges and vertices of simple spherical, planar and hyperbolic tilings. We show that various projections of the simplest symmetric tilings of those spaces into three-dimensional Euclidean space lead to topologically and geometrically complex patterns, including multiple interwoven nets and tangled nets that are otherwise difficult to generate ab initio in three dimensions.

The role of curvature in silica mesoporous crystals.

Silica mesoporous crystals (SMCs) offer a unique opportunity to study micellar mesophases. Replication of non-equilibrium mesophases into porous silica structures allows the characterization of surfactant phases under a variety of chemical and physical perturbations, through methods not typically accessible to liquid crystal chemists. A poignant example is the use of electron microscopy and crystallography, as discussed herein, for the purpose of determining the fundamental role of amphiphile curvature, namely mean curvature and Gaussian curvature, which have been extensively studied in various fields such as polymer, liquid crystal, biological membrane, etc. The present work aims to highlight some current studies devoted to the interface curvature on SMCs, in which electron microscopy and electron crystallography (EC) are used to understand the geometry of silica wall surface in bicontinuous and cage-type mesostructures through the investigation of electrostatic potential maps. Additionally, we show that by altering the synthesis conditions during the preparation of SMCs, it is possible to isolate particles during micellar mesophase transformations in the cubic bicontinuous system, allowing us to view and study epitaxial relations under the specific synthesis conditions. By studying the relationship between mesoporous structure, interface curvature and micellar mesophases using electron microscopy and EC, we hope to bring new insights into the formation mechanism of these unique materials but also contribute a new way of understanding periodic liquid crystal systems.

Bicontinuous cubic mesoporous materials with biphasic structures.

The replication of amphiphilic systems within an inorganic silica matrix allows the study of the fundamental properties of mesostructural changes, that is, kinetic and structural parameters. Herein we report a detailed study of the transition between cubic bicontinuous mesostructure with space groups Ia ̅3d and Pn ̅3m symmetry, which are associated with the minimal G and D surfaces, respectively. The transition may be induced through micellar swelling of the anionic amphiphilic surfactant N-lauroyl alanine by trimethylbenzene. Rich kinetic behaviour is observed and has been exploited to prepare particles with biphasic structures. Transmission electron microscopy evidence indicates that there is epitaxial growth from one mesostructure to the other involving the [111] and [110] orientations of the Ia ̅3d and Pn ̅3m symmetry structures, respectively. From kinetic studies, we show that the formation of the Ia ̅3d mesophase is preceded by a hexagonal phase (plane group p6mm) and an epitaxial relationship has been observed involving the sixfold or ̅3 axis orientations of both structures. Our data suggests that the Pn ̅3m mesostructure is kinetically stable at low temperatures whereas the Ia ̅3d mesostructure is the more stable structure after prolonged periods of hydrothermal treatment. We present evidence from transmission electron microscopy and small-angle X-ray diffractograms and also electron crystallography modelling of the unit cells at particular points in the structural change.

Shape- and size-controlled synthesis in hard templates: sophisticated chemical reduction for mesoporous monocrystalline platinum nanoparticles.

Here we report a novel hard-templating strategy for the synthesis of mesoporous monocrystalline Pt nanoparticles (NPs) with uniform shapes and sizes. Mesoporous Pt NPs were successfully prepared through controlled chemical reduction using ascorbic acid by employing 3D bicontinuous mesoporous silica (KIT-6) and 2D mesoporous silica (SBA-15) as a hard template. The particle size could be controlled by changing the reduction time. Interestingly, the Pt replicas prepared from KIT-6 showed polyhedral morphology. The single crystallinity of the Pt fcc structure coherently extended over the whole particle.