Nodal surfaces of Fourier series: Fundamental invariants of structured matter

Mechanobiological optimization of scaffolds for bone tissue engineering

Synthetic bone graft scaffolds aim to generate new bone tissue and alleviate the limitations of autografts and allografts. To meet that aim, it is essential to have a design approach able to generate scaffold architectures that will promote bone formation. Here, we present a topology-varying design optimization method, the "mixed-topology" approach, that generates new designs from a set of starting structures. This approach was used with objective functions focusing on improving the scaffold's local mechanical microenvironments to mechanobiologically promote bone formation within the scaffold and constraints to ensure manufacturability and achieve desired macroscale properties. The results demonstrate that this approach can successfully generate scaffold designs with improved microenvironments, taking into account different combinations of relevant stimuli and constraints.

Catalog of triply periodic minimal surfaces, equation-based lattice structures, and their homogenized property data

The use of lattice structure in the Design for Additive Manufacturing (DfAM) engineering practice offers the ability to tailor the properties (and therefore the response) of an engineered component independent of the material and overall geometry. The selection of a lattice topology is critical in maximizing the value of the lattice structure and its unique properties for the intended application. To support this, we have compiled a catalog of lattice structures from the literature that includes all Triply Periodic Minimal Surfaces (TPMS) for which a low-order Fourier series fit is known (so that they can be modeled and manufactured). We also include equations that do not directly correspond to known TPMS but do produce a triply periodic structure without sharp corners that would give rise to stress concentrations. This catalog includes images, elastic mechanical property data, and CAD models useful for the visualization, selection, and implementation of these lattice structures for any engineered structure.

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.

Bayesian modeling of pattern formation from one snapshot of pattern

Partial differential equations (PDEs) have been widely used to reproduce patterns in nature and to give insight into the mechanism underlying pattern formation. Although many PDE models have been proposed, they rely on the pre-request knowledge of physical laws and symmetries, and developing a model to reproduce a given desired pattern remains difficult. We propose a method, referred to as Bayesian modeling of PDEs (BM-PDEs), to estimate the best dynamical PDE for one snapshot of a objective pattern under the stationary state without ground truth. We apply BM-PDEs to nontrivial patterns, such as quasicrystals (QCs), a double gyroid, and Frank-Kasper structures. We also generate three-dimensional dodecagonal QCs from a PDE model. This is done by using the estimated parameters for the Frank-Kasper A15 structure, which closely approximates the local structures of QCs. Our method works for noisy patterns and the pattern synthesized without the ground-truth parameters, which are required for the application toward experimental data.

Polarization-dependent reflection of I-WP minimal-surface-based photonic crystal

Brilliantly colored butterflies and weevils are known to utilize photonic crystals for their coloration. Interestingly, the morphology of such crystals made of cuticle is based on triply periodic minimal surfaces such as gyroid and diamond surfaces. Recently, a different minimal-surface-based photonic crystal, the I-WP surface, was discovered inside the scale of a longhorn beetle. The letter I is derived from expressing the body center symmetry and WP is derived from a wrapped package. It was reported that the brilliant green color is produced by the photonic band gap existing along the [110] direction of this crystal. In this study, the polarization dependence of the reflection from this photonic crystal was investigated. A peculiar reflectance spectrum with two peaks was observed under the crossed polarizers. This characteristic is theoretically reproduced by calculating the reflectance from a finite-sized photonic crystal, and the spectral shape is explained based on the symmetry of the electromagnetic modes. In addition, inspired by this longhorn beetle, a photonic crystal structure consisting of colloidal particles is proposed, which has a similar polarization effect.

SPIRE-a software tool for bicontinuous phase recognition: application for plastid cubic membranes

Bicontinuous membranes in cell organelles epitomize nature's ability to create complex functional nanostructures. Like their synthetic counterparts, these membranes are characterized by continuous membrane sheets draped onto topologically complex saddle-shaped surfaces with a periodic network-like structure. Their structure sizes, (around 50-500 nm), and fluid nature make transmission electron microscopy (TEM) the analysis method of choice to decipher their nanostructural features. Here we present a tool, Surface Projection Image Recognition Environment (SPIRE), to identify bicontinuous structures from TEM sections through interactive identification by comparison to mathematical "nodal surface" models. The prolamellar body (PLB) of plant etioplasts is a bicontinuous membrane structure with a key physiological role in chloroplast biogenesis. However, the determination of its spatial structural features has been held back by the lack of tools enabling the identification and quantitative analysis of symmetric membrane conformations. Using our SPIRE tool, we achieved a robust identification of the bicontinuous diamond surface as the dominant PLB geometry in angiosperm etioplasts in contrast to earlier long-standing assertions in the literature. Our data also provide insights into membrane storage capacities of PLBs with different volume proportions and hint at the limited role of a plastid ribosome localization directly inside the PLB grid for its proper functioning. This represents an important step in understanding their as yet elusive structure-function relationship.

Discovery of I-WP minimal-surface-based photonic crystal in the scale of a longhorn beetle

The structural colours of certain insects are produced by three-dimensional periodic cuticle networks. The topology of the cuticle network is known to be based on the mathematically well-defined triply periodic minimal surface. In this paper, we report the discovery of an I-WP minimal-surface-based photonic crystal on the scale of a longhorn beetle. In contrast to gyroid or diamond surfaces, which are found in butterfly and weevil scales, respectively, the I-WP surface is an unbalanced minimal surface, wherein two subspaces separated by the surface are different in terms of shape and volume fraction. Furthermore, adjacent photonic crystal domains were observed to share a particular crystal plane as their domain boundary, indicating that they were developed as twin crystals. These structural features pose certain new questions regarding the development of biological photonic crystals. We also performed an optical analysis of the structural colour of the longhorn beetle and successfully explained the wavelength of reflection by the photonic bandgap of the I-WP photonic crystal.

Double-Level Energy Absorption of 3D Printed TPMS Cellular Structures via Wall Thickness Gradient Design

This paper investigates the deformation mechanism and energy absorption behaviour of 316 L triply periodic minimal surface (TPMS) structures with uniform and graded wall thicknesses fabricated by the selective laser melting technique. The uniform P-surface TPMS structure presents a single-level stress plateau for energy absorption and a localized diagonal shear cell failure. A graded strategy was employed to break such localized geometrical deformation to improve the overall energy absorption and to provide a double-level function. Two segments with different wall thicknesses separated by a barrier layer were designed along the compression direction while keeping the same relative density as the uniform structure. The results show that the crushing of the cells of the graded P-surface TPMS structure occurs first within the thin segment and then propagates to the thick segment. The stress–strain response shows apparent double stress plateaus. The stress level and length of each plateau can be adjusted by changing the wall thickness and position of the barrier layer between the two segments. The total energy absorption of the gradient TPMS structure was also found slightly higher than that of the uniform TPMS counterparts. The gradient design of TPMS structures may find applications where the energy absorption requires a double-level feature or a warning function.

Programmable Multistable Perforated Shellular

Developing bistable metamaterials has recently offered a new design paradigm for deployable structures and reusable dampers. While most bistable mechanisms possess inclined/curved struts, a new 3D multistable shellular metamaterial is developed by introducing delicate perforations on the surface of Schwarz's Primitive shellular, integrating the unique properties of shellular materials such as high surface area, stiffness, and energy absorption with the multistability concept. Denoting the fundamental snapping part by motif, certain shellular motifs with elliptical perforations exhibit mechanical bistability. To bring the concept of multistability to a single motif, multistable shellular motifs are developed by introducing multilayer staggered perforations that form hinges and facilitate local instability. Adopting an n-layer staggered perforation (n hinges) design leads to a maximum 2^n-1 stable states within one shellular motif during loading and unloading. Three-directional multistable shellulars are attained by extending the perforation design in three orthogonal directions. Harnessing snap-through and snap-back behaviors and self-contact, the introduced multistable perforated shellulars exhibit strong rigidity both in loading and unloading, and enhanced energy dissipation. The introduced design strategy opens up new horizons for creating multidirectional multistable metamaterials with load bearing capabilities for applications in soft robotics, shape-morphing architectures, and reusable and deployable energy absorbers/dampers.

Soap film inspired mechanical metamaterials approaching theoretical bound of stiffness across full density range

Structural mechanical metamaterials, with their mass-efficient architectures and unprecedented mechanical properties, are in critical demand for high-performance applications. However, finding the optimal 3D geometries towards a particular property, such as reaching the stiffness upper bound, usually demands high volume of calculations or numerical optimizations. Here we generate structured mechanical metamaterials by imitating the natural occupation of periodic volume by inflated soap films. Our strategy of occupying volume between two periodic constant mean curvature (CMC) surfaces generates a series of mechanical metamaterials of varied relative densities ranging from 0 to 1. The mechanically isotropic ISO-CMC structures exhibit bulk moduli over 94% of the theoretical limit. Using finite element models, we reveal the fundamental mechanical behaviors of the structures that lead to ideal performances. These phenomena are found to be in close relation to the curvature-driven design of our metamaterial structures. These structures are compared to other reported mechanical metamaterials, such as closed-cell plate structures and triply periodic minimal surface (TPMS) structures. The unique curvature-driven thickening strategy of our method renders structures that outperform their peers in terms of bulk moduli and relative density coverage. The CMC structures present a new class of easily 3D printable, permeable and stiff mechanical metamaterials. The design methodology also could serve in future development of novel mechanical metamaterials powered by advanced computational tools.

Functionally Graded Scaffolds with Programmable Pore Size Distribution Based on Triply Periodic Minimal Surface Fabricated by Selective Laser Melting

Functional graded materials are gaining increasing attention in tissue engineering (TE) due to their superior mechanical properties and high biocompatibility. Triply periodic minimal surface (TPMS) has the capability to produce smooth surfaces and interconnectivity, which are very essential for bone scaffolds. To further enhance the versatility of TPMS, a parametric design method for functionally graded scaffold (FGS) with programmable pore size distribution is proposed in this study. Combining the relative density and unit cell size, the effect of design parameters on the pore size was also considered to effectively govern the distribution of pores in generating FGS. We made use of Gyroid to generate different types of FGS, which were then fabricated using selective laser melting (SLM), followed by investigation and comparison of their structural characteristics and mechanical properties. Their morphological features could be effectively controlled, indicating that TPMS was an effective way to achieve functional gradients which had bone-mimicking architectures. In terms of mechanical performance, the proposed FGS could achieve similar mechanical response under compression tests compared to the reference FGS with the same range of density gradient. The proposed method with control over pore size allows for effectively generating porous scaffolds with tailored properties which are potentially adopted in various fields.

Design and Characterization of Sheet-Based Gyroid Porous Structures with Bioinspired Functional Gradients

A new type of sheet porous structures with functionally gradients based on triply periodic minimal surfaces (TPMS) is proposed for designing bone scaffolds. The graded structures were generated by constructing branched features with different number of sheets. The design of the structure was formulated mathematically and five types of porous structure with different structural features were used for investigation. The relative density (RD) and surface area to volume (SA/V) ratio of the samples were analyzed using a slice-based approach to confirm their relationships with design parameters. All samples were additively manufactured using selective laser melting (SLM), and their physical morphologies were observed and compared with the designed models. Compression tests were adopted to study the mechanical properties of the proposed structure from the obtained stress-strain curves. The results reveal that the proposed branched-sheet structures could enhance and diversify the physical and mechanical properties, indicating that it is a potential method to tune the biomechanical properties of porous scaffolds for bone tissue engineering (TE).

Modeling, Assessment, and Design of Porous Cells Based on Schwartz Primitive Surface for Bone Scaffolds

The design of bone scaffolds for tissue regeneration is a topic of great interest, which involves different issues related to geometry of architectures, mechanical behavior, and biological requirements, whose optimal combination determines the success of an implant. Additive manufacturing (AM) has widened the capability to produce structures with complex geometries, which should potentially satisfy the different requirements. These architectures can be obtained by means of refined methods and have to be assessed in terms of geometrical and mechanical properties. In this paper a triply periodic minimal surface (TPMS), the Schwarz's Primitive surface (P-surface), has been considered as scaffold unit cell and conveniently parameterized in order to investigate the effect of modulation of analytical parameters on the P-cell geometry and on its properties. Several are the cell properties, which can affect the scaffold performance. Due to the important biofunctional role that the surface curvature plays in mechanisms of cellular proliferation and differentiation, in this paper, in addition to properties considering the cell geometry in its whole (such as volume fraction or pore size), new properties were proposed. These properties involve, particularly, the evaluation of local geometrical-differential properties of the P-surface. The results of this P-cell comprehensive characterization are very useful for the design of customized bone scaffolds able to satisfy both biological and mechanical requirements. A numerical structural evaluation, by means of finite element method (FEM), was performed in order to assess the stiffness of solid P-cells as a function of the changes of the analytical parameters of outer surface and the thickness of cell. Finally, the relationship between stiffness and porosity has been analyzed, given the relevance that this property has for bone scaffolds design.

Improved Mechanical Properties and Energy Absorption of BCC Lattice Structures with Triply Periodic Minimal Surfaces Fabricated by SLM

The triply periodic minimal surface (TPMS) method is a novel approach for lattice design in a range of fields, such as impact protection and structural lightweighting. In this paper, we used the TPMS formula to rapidly and accurately generate the most common lattice structure, named the body centered cubic (BCC) structure, with certain volume fractions. TPMS-based and computer aided design (CAD) based BCC lattice structures with volume fractions in the range of 10⁻30% were fabricated by selective laser melting (SLM) technology with Ti⁻6Al⁻4V and subjected to compressive tests. The results demonstrated that local geometric features changed the volume and stress distributions, revealing that the TPMS-based samples were superior to the CAD-based ones, with elastic modulus, yield strength and compression strength increasing in the ranges of 18.9⁻42.2%, 19.2⁻29.5%, and 2⁻36.6%, respectively. The failure mechanism of the TPMS-based samples with a high volume fraction changed to brittle failure observed by scanning electron microscope (SEM), as their struts were more affected by the axial force and fractured on struts. It was also found that the TPMS-based samples have a favorable capacity to absorb energy, particularly with a 30% volume fraction, the energy absorbed up to 50% strain was approximately three times higher than that of the CAD-based sample with an equal volume fraction. Furthermore, the theoretic Gibson⁻Ashby mode was established in order to predict and design the mechanical properties of the lattice structures. In summary, these results can be used to rapidly create BCC lattice structures with superior compressive properties for engineering applications.

Nonequilibrium phase transitions of sheared colloidal microphases: Results from dynamical density functional theory

By means of classical density functional theory and its dynamical extension, we consider a colloidal fluid with spherically symmetric competing interactions, which are well known to exhibit a rich bulk phase behavior. This includes complex three-dimensional periodically ordered cluster phases such as lamellae, two-dimensional hexagonally packed cylinders, gyroid structures, or spherical micelles. While the bulk phase behavior has been studied extensively in earlier work, in this paper we focus on such structures confined between planar repulsive walls under shear flow. For sufficiently high shear rates, we observe that microphase separation can become fully suppressed. For lower shear rates, however, we find that, e.g., the gyroid structure undergoes a kinetic phase transition to a hexagonally packed cylindrical phase, which is found experimentally and theoretically in amphiphilic block copolymer systems. As such, besides the known similarities between the latter and colloidal systems regarding the equilibrium phase behavior, our work reveals further intriguing nonequilibrium relations between copolymer melts and colloidal fluids with competing interactions.

An "on-off" switchable cubic phase with exceptional thermal stability and water sorption capacity

We report on the phase behaviour of a wedge-shaped mesogen, which can exist in two different states at room temperature, a stable columnar and a metastable cubic gyroid phase. The latter reveals exceptional stability and remarkable water sorption capacity accounted for by the locally-ordered peripheral alkyl chains.

Lipidic Mesophases as Novel Nanoreactor Scaffolds for Organocatalysts: Heterogeneously Catalyzed Asymmetric Aldol Reactions in Confined Water

The unique molecular architecture of lipidic cubic phases (LCPs) and their cubosome dispersions comprise a well-defined, curved bilayer that spans the entire three-dimensional (3-D) material space, encompassing a network of two periodic, curved, and nonintersecting 3-D aqueous channels. The ensuing large lipid/water interfacial area makes these biomaterials an interesting matrix for the lateral immobilization of organocatalysts to catalyze organic reactions in confined water. Herein, we report for the first time the design, synthesis, assembly, and characterization of catalytically active LCPs and cubosomes and demonstrate their applicability as self-assembled, biomimetic, and recyclable nanoreactor scaffolds. Small-angle X-ray scattering, cryo-transmission electron microscopy, and dynamic light scattering were applied for the characterization of the mesophases. These mesophases can be recycled and enable efficient catalytic activity as well as modulation of the diastereo- and enantioselectivity for the aldol reaction of several benzaldehyde derivatives and cyclohexanone in water.

The [BN2]3− anion – a carbon dioxide isosteric building unit for a large family of complex nitridoborate structures

The crystal chemistry of nitridoborates with the CO2 analogous [BN2]3− ion is reviewed. Such nitridoborates form with the alkali and alkaline earth metals as well as with divalent europium. Also quaternary compounds with mixed cations along with nitridoborate nitrides, oxides, halides and hydrides are discussed. The spectroscopic (IR, Raman, solid state NMR and Mössbauer spectroscopy) and magnetic behavior as well as optical properties are discussed in the light of structure-property relationships.

Phase behavior and bulk structural properties of a microphase former with anisotropic competing interactions: A density functional theory study

Using classical density functional theory, we investigate systems exhibiting interactions where a short-range anisotropic attractive force competes with a long-range spherically symmetric repulsive force. The former is modelled within Wertheim's first-order perturbation theory for patchy particles, and the repulsive part is assumed to be a Yukawa potential which is taken into account via a mean-field approximation. From previous studies of systems with spherically symmetric competing interactions, it is well known that such systems can exhibit stable bulk cluster phases (microphase separation) provided that the repulsion is sufficiently weak compared to the attraction. For the present model system, we find rich phase diagrams including both reentrant clustering and liquid-gas binodals. In particular, the model predicts inhomogeneous bulk phases at extremely low packing fractions, which cannot be observed in systems with isotropic competing interactions.

Harmonic distortions enhance circular dichroism of dielectric single gyroids

The departure from strict periodic order in two-phase dielectric materials can offer properties that are otherwise inaccessible to perfectly ordered photonic crystals. Herewith, we investigate the circular dichroism of the single gyroid photonic crystal in the presence of spatial distortions. FDTD simulations and microwave transmission measurements on 3D-printed replicas show that certain harmonic long-wavelength spatial distortions ("sinusoidal chirp") nearly doubles the imbalance of the circular polarisation reflectances, as well as significantly strengthens polarisation-incoherent reflectance. The observed changes are partially rationalised by comparison with simpler distortion models (linear chirp and tetragonal deformation) of the Gyroid.

Morphology of symmetric ABCD tetrablock quaterpolymers studied by Monte Carlo simulation

Morphology of symmetric ABCD tetrablock quaterpolymers in melt was studied by the Monte Carlo (MC) simulation, where the volume fractions of the block chains, f, kept the relationships of f_A=f_D and f_B=f_C, and the volume fraction of the two mid-blocks φ was defined as φ=f_B+f_C. Previous self-consistent field theory for ABCD reported morphological change including several structures; however, the scope was limited within a two-dimensional system. To the contrary, in this paper, MC simulations were carried out in three dimensions with changing the φ value finely, which resulted in finding a tetracontinuous structure in the range of 0.625≤φ≤0.75. Moreover the tetracontinuous structure has been found to be the gyroid structure, and the mean curvature of the B/C interface is nearly zero. We concluded that the B/C interface must be the Schoen gyroid surface, one of three-dimensional periodic minimal surfaces. The geometrical nature of the A/B interface should be equivalent to that of the C/D interface, and they stand as level surfaces to the Schoen gyroid surface.

Gyroid phase of fluids with spherically symmetric competing interactions

We study the phase diagram of a fluid with spherically symmetric competing pair interactions that consist of a short-ranged attraction and a longer-ranged repulsion in addition to a hard core. To this end we perform free minimizations of three-dimensional triple periodic structures within the framework of classical density functional theory. We compare our results to those from Landau theory. Our main finding is that the double gyroid phase can exist as a thermodynamically stable phase.

Cubosome Topologies at Various Particle Sizes and Crystallographic Symmetries

The nanoparticles built of bicontinuos lyotropic phases of cubic symmetry are studied within the framework of the Landau-Brazovskii functional that correctly predicts the structure of soft monocrystals and thin films of bicontinuos lyotropic phases. A detailed description of the geometry and topology of cubosomes is presented. This level of description of the internal structure of cubosomes is not easily accessible by experimental techniques. I show that the internal structure of the cubosomes may be extremely rich.

Direct visualization of dispersed lipid bicontinuous cubic phases by cryo-electron tomography

Bulk and dispersed cubic liquid crystalline phases (cubosomes), present in the body and in living cell membranes, are believed to play an essential role in biological phenomena. Moreover, their biocompatibility is attractive for nutrient or drug delivery system applications. Here the three-dimensional organization of dispersed cubic lipid self-assembled phases is fully revealed by cryo-electron tomography and compared with simulated structures. It is demonstrated that the interior is constituted of a perfect bicontinuous cubic phase, while the outside shows interlamellar attachments, which represent a transition state between the liquid crystalline interior phase and the outside vesicular structure. Therefore, compositional gradients within cubosomes are inferred, with a lipid bilayer separating at least one water channel set from the external aqueous phase. This is crucial to understand and enhance controlled release of target molecules and calls for a revision of postulated transport mechanisms from cubosomes to the aqueous phase.

Probing the self-assembled nanostructures of functional polymers with synchrotron grazing incidence X-ray scattering

For advanced functional polymers such as biopolymers, biomimic polymers, brush polymers, star polymers, dendritic polymers, and block copolymers, information about their surface structures, morphologies, and atomic structures is essential for understanding their properties and investigating their potential applications. Grazing incidence X-ray scattering (GIXS) is established for the last 15 years as the most powerful, versatile, and nondestructive tool for determining these structural details when performed with the aid of an advanced third-generation synchrotron radiation source with high flux, high energy resolution, energy tunability, and small beam size. One particular merit of this technique is that GIXS data can be obtained facilely for material specimens of any size, type, or shape. However, GIXS data analysis requires an understanding of GIXS theory and of refraction and reflection effects, and for any given material specimen, the best methods for extracting the form factor and the structure factor from the data need to be established. GIXS theory is reviewed here from the perspective of practical GIXS measurements and quantitative data analysis. In addition, schemes are discussed for the detailed analysis of GIXS data for the various self-assembled nanostructures of functional homopolymers, brush, star, and dendritic polymers, and block copolymers. Moreover, enhancements to the GIXS technique are discussed that can significantly improve its structure analysis by using the new synchrotron radiation sources such as third-generation X-ray sources with picosecond pulses and partial coherence and fourth-generation X-ray laser sources with femtosecond pulses and full coherence.

WinCSD: software package for crystallographic calculations (Version 4)

The fourth version of the program packageWinCSDis multi-purpose computer software for crystallographic calculations using single-crystal and powder X-ray and neutron diffraction data. The software environment and the graphical user interface are built using the platform of the Microsoft .NET Framework, which grants independence from changing Windows operating systems and allows for transferring to other operating systems. Graphic applications use the three-dimensional OpenGL graphics language.WinCSDcovers the complete spectrum of crystallographic calculations, including powder diffraction pattern deconvolution, crystal structure solution and refinement in 3 + dspace, refinement of the multipole model and electron density studies from diffraction data, and graphical representation of crystallographic information.

Periodic minimal surface organizations of the lipid bilayer at the lung surface and in cubic cytomembrane assemblies

The existence of infinite periodic lipid bilayer structures in biological systems was first demonstrated in cell membrane assemblies. Such periodicity is only possible in symmetric bilayers, and their occurrence is discussed here in relation to the asymmetry of cell membranes in vivo. A periodic membrane conformation in the prolamellar body of plants corresponds to a dormant state without photosynthesis. A similar reversible formation of a dormant state has also been observed in the mitochondria of the amoeba Chaos. In these cases the energy production has become insufficient to maintain the membrane asymmetry. Formation of membranes that are symmetric over the bilayer is proposed to be a principal mechanism behind formation of cubic membrane systems. Another type of bicontinuous minimal surface structure is considered to form the alveolar lining of mammals at normal breathing conditions. The CLP surface corresponds to such a tetragonal surface phase. It is also a symmetric bilayer and in a state of zero energy expenditure. Structural alternatives of the bilayer conformation in this latter system are also discussed here.

Facets of lyotropic liquid crystals

The bicontinuous lyotropic liquid phases surrounded by the isotropic phase form monocrystals with well-developed facets. We investigate the structure and stability of the facets formed by the bicontinuous phase of Pn3̅m symmetry, at three preferred directions, which are developed on a spherical droplet of Pn3̅m phase surrounded by the isotropic phase. The structure of the facets is obtained by minimization of the Landau-Brazovskii functional with one scalar order parameter.

Symmetry, topology and faceting in bicontinuous lyotropic crystals

Phase diagrams of phytantriol/ethanol/water and phytantriol/DSPG/ethanol/water systems are explored and experiments on facetings of Pn3m-in-L1 and Im3m-in-L1 crystals are performed. Observed crystal habits do not agree with the Friedel-Donnay-Harker rules. We argue that this paradox can be explained in terms of constraints imposed on Pn3m/L1 and Im3m/L1 interfaces by the bicontinuous topology of the cubic phases. We point out that when free edges of the surfactant bilayer are prohibited at these interfaces, the two labyrinthes separated by the bilayer cannot anymore be equivalent. The corresponding [Formula: see text] and [Formula: see text] symmetry breakings are unveiled by the abnormal facetings.

Earliest stage of the tetrahedral nanochannel formation in cubosome particles from unilamellar nanovesicles

Studies of nonequilibrium lipid polymorphism at the nanoscale contribute to the in-depth understanding of the structural pathways for formation of aqueous channels and emerging of channels-network ordering in liquid-crystalline (LC) nanovehicles. We present experimental structural evidence for the smallest tetrahedral-type lipid membrane aggregate, which involves completely formed nanochannels and occurs as an early intermediate state during the bilayer vesicle-to-cubosome particle transition. Nanovehicles are generated from a self-assembled lipid mixture and studied by means of high-resolution cryogenic transmission electron microscopy (cryo-TEM) and synchrotron radiation small-angle X-ray scattering (SAXS). The investigated lipid membrane composition allows for the stabilization of long-lived intermediates throughout the unilamellar vesicle-to-cubosome nanoparticle (NP) transformation at ambient temperature. The observed small cubosomic particles, with well-defined water channels, appear to be precursors of larger cubic membrane structures, thus confirming the theoretical modeling of nanochannel-network growth in diamond-type cubic lipid particles. The reported structural findings, highlighting that bilayer vesicle membrane packing and fusion are required for nanochanneled cubosome particle formation, are anticipated to advance the engineering of small lipid NPs with controllable channels for biomolecular loading and release.

Three-dimensional periodic complex structures in soft matter: investigation using scattering methods

Three-dimensional periodic complex structures are encountered in various soft matter systems such as liquid crystals, block-copolymer phases and the related nano-structured materials. Here, we review several well-defined topologies: two-dimensional hexagonal phase, three-dimensional packing of spheres, tetrahedral close packing (tcp) bi-continuous and tri-continuous cubic phases. We illustrate how small-angle X-ray scattering experiments help us to investigate these different structures and introduce the main available structural models based on both direct and inverse methods.

Why GPCRs behave differently in cubic and lamellar lipidic mesophases

Recent successes in the crystallographic determination of structures of transmembrane proteins in the G protein-coupled receptor (GPCR) family have established the lipidic cubic phase (LCP) environment as the medium of choice for growing structure-grade crystals by the method termed "in meso". The understanding of in meso crystallogenesis is currently at a descriptive level. To enable an eventual quantitative, energy-based description of the nucleation and crystallization mechanism, we have examined the properties of the lipidic cubic phase system and the dynamics of the GPCR rhodopsin reconstituted into the LCP with coarse-grained molecular dynamics simulations with the Martini force-field. Quantifying the differences in the hydrophobic/hydrophilic exposure of the GPCR to lipids in the cubic and lamellar phases, we found that the highly curved geometry of the cubic phase provides more efficient shielding of the protein from unfavorable hydrophobic exposure, which leads to a lesser hydrophobic mismatch and less unfavorable hydrophobic-hydrophilic interactions between the protein and lipid-water interface in the LCP, compared to the lamellar phase. Since hydrophobic mismatch is considered a driving force for oligomerization, the differences in exposure mismatch energies between the LCP and the lamellar structures suggest that the latter provide a more favorable setting in which GPCRs can oligomerize as a prelude to nucleation and crystal growth. These new findings lay the foundation for future investigations of in meso crystallization mechanisms related to the transition from the LCP to the lamellar phase and studies aimed at an improved rational approach for generating structure-quality crystals of membrane proteins.

Understanding the nature of "superhard graphite"

Numerous experiments showed that on cold compression graphite transforms into a new superhard and transparent allotrope. Several structures with different topologies have been proposed for this phase. While experimental data are compatible with most of these models, the only way to solve this puzzle is to find which structure is kinetically easiest to form. Using state-of-the-art molecular-dynamics transition path sampling simulations, we investigate kinetic pathways of the pressure-induced transformation of graphite to various superhard candidate structures. Unlike hitherto applied methods for elucidating nature of superhard graphite, transition path sampling realistically models nucleation events necessary for physically meaningful transformation kinetics. We demonstrate that nucleation mechanism and kinetics lead to M-carbon as the final product. W-carbon, initially competitor to M-carbon, is ruled out by phase growth. Bct-C₄ structure is not expected to be produced by cold compression due to less probable nucleation and higher barrier of formation.

Tensorial Minkowski functionals of triply periodic minimal surfaces

A fundamental understanding of the formation and properties of a complex spatial structure relies on robust quantitative tools to characterize morphology. A systematic approach to the characterization of average properties of anisotropic complex interfacial geometries is provided by integral geometry which furnishes a family of morphological descriptors known as tensorial Minkowski functionals. These functionals are curvature-weighted integrals of tensor products of position vectors and surface normal vectors over the interfacial surface. We here demonstrate their use by application to non-cubic triply periodic minimal surface model geometries, whose Weierstrass parametrizations allow for accurate numerical computation of the Minkowski tensors.

The three dimensionality of cell membranes: lamellar to cubic membrane transition as investigated by electron microscopy

Biological membranes are generally perceived as phospholipid bilayer structures that delineate in a lamellar form the cell surface and intracellular organelles. However, much more complex and highly convoluted membrane organizations are ubiquitously present in many cell types under certain types of stress, states of disease, or in the course of viral infections. Their occurrence under pathological conditions make such three-dimensionally (3D) folded and highly ordered membranes attractive biomarkers. They have also stimulated great biomedical interest in understanding the molecular basis of their formation. Currently, the analysis of such membrane arrangements, which include tubulo-reticular structures (TRS) or cubic membranes of various subtypes, is restricted to electron microscopic methods, including tomography. Preservation of membrane structures during sample preparation is the key to understand their true 3D nature. This chapter discusses methods for appropriate sample preparations to successfully examine and analyze well-preserved highly ordered membranes by electron microscopy. Processing methods and analysis conditions for green algae (Zygnema sp.) and amoeba (Chaos carolinense), mammalian cells in culture and primary tissue cells are described. We also discuss methods to identify cubic membranes by transmission electron microscopy (TEM) with the aid of a direct template matching method and by computer simulation. A 3D analysis of cubic cell membrane topology by electron tomography is described as well as scanning electron microscopy (SEM) to investigate surface contours of isolated mitochondria with cubic membrane arrangement.