Triply periodic surfaces and multiply continuous structures from the Landau model of microemulsions

Surface-Constrained Metropolis Monte Carlo: Simulation of Reactions on Triply Periodic Minimal Surfaces

Triply periodic minimal surfaces (TPMS) inspired by nature serve as a foundation for developing novel nanomaterials, such as templated silicas, graphene sponges, and schwarzites, with customizable optical, poroelastic, adsorptive, catalytic, and other properties. Computer simulations of reactions on TPMS using reactive intermolecular potentials hold great promise for constructing and screening potential TPMS with the desired properties. Here, we developed an off-lattice, surface-constrained Metropolis Monte Carlo (SC-MMC) algorithm that utilized a temperature quench process. The presented SC-MMC algorithm was used to investigate the process of graphitization reactions on the Schwarz primitive, Schwarz diamond, and Schoen gyroid TPMS, all with a cubic lattice parameter of 8 nm. We show that the optimized carbon TPMS exhibits a low energy, approximately -7.1 eV/atom, comparable to that of graphite and diamond crystals, along with a variety of topological defects. Furthermore, these structures showcase extensive and smooth surfaces characterized by a negative discrete Gaussian curvature, a distinctive feature indicative of an interconnected morphology. They possess specific surface areas of ∼2700 m2/g, comparable to graphene, and exhibit a significant porosity of around 90%. The theoretical X-ray correlation functions and nitrogen adsorption isotherms confirm that the constructed TPMS exhibit remarkably similar surface properties, although the pore space topology varies significantly.

Design and Analysis of Biomedical Scaffolds Using TPMS-Based Porous Structures Inspired from Additive Manufacturing

Gyroid (G) and primitive (P) porous structures have multiple application areas, ranging from thermal to mechanical, and fall in the complex triply periodic minimal surface (TPMS) category. Such intricate bioinspired constructs are gaining attention because they meet both biological and mechanical requirements for osseous reconstruction. The study aimed to develop G and P structures with varying porosity levels from 40% to 80% by modulating the strut thickness to proportionally resemble the stiffness of host tissue. The performance characteristics were evaluated using Ti6Al4V and important relationships between feature dimension, strut thickness, porosity, and stiffness were established. Numerical results showed that the studied porous structures could decrease stiffness from 107 GPa (stiffness of Ti6Al4V) to the range between 4.21 GPa to 29.63 GPa of varying porosities, which matches the human bone stiffness range. Furthermore, using this foundation, a subject-specific scaffold (made of P unit cells with an 80% porosity) was developed to reconstruct segmental bone defect (SBD) of the human femur, demonstrating a significant decrease in the stress shielding effect. Stress transfer on the bone surrounded by a P scaffold was compared with a solid implant which showed a net increase of stress transfer of 76% with the use of P scaffold. In the conclusion, future concerns and recommendations are suggested.

Confinement of Colloids with Competing Interactions in Ordered Porous Materials

In this work, we explore the possibility of promoting the formation of ordered microphases by confinement of colloids with competing interactions in ordered porous materials. For that aim, we consider three families of porous materials modeled as cubic primitive, diamond, and gyroid bicontinuous phases. The structure of the confined colloids is investigated by means of grand canonical Monte Carlo simulations in thermodynamic conditions at which either a cluster crystal or a cylindrical phase is stable in bulk. We find that by tuning the size of the unit cell of these porous materials, numerous novel ordered microphases can be produced, including cluster crystals arranged into close packed and open lattices as well as nonparallel cylindrical phases.

3D Analysis of Ordered Porous Polymeric Particles using Complementary Electron Microscopy Methods

Highly porous particles with internal triply periodic minimal surfaces were investigated for sorption of proteins. The visualization of the complex ordered morphology requires complementary advanced methods of electron microscopy for 3D imaging, instead of a simple 2D projection: transmission electron microscopy (TEM) tomography, slice-and-view focused ion beam (FIB) and serial block face (SBF) scanning electron microscopy (SEM). The capability of each method of 3D image reconstruction was demonstrated and their potential of application to other synthetic polymeric systems was discussed. TEM has high resolution for details even smaller than 1 nm, but the imaged volume is relatively restricted (2.5 μm)³. The samples are pre-sliced in an ultramicrotome. FIB and SBF are coupled to a SEM. The sample sectioning is done in situ, respectively by an ion beam or an ultramicrotome, SBF, a method so far mostly applied only to biological systems, was particularly highly informative to reproduce the ordered morphology of block copolymer particles with 32–54 nm nanopores and sampling volume (20 μm)³.

Amphiphilic Dimers at Liquid-Liquid Interfaces: A Density Functional Approach

We apply density functional theory to study the structure of dimers at the interface between two partially miscible symmetric liquids. The dimers are built of two tangentially jointed spheres and do not solve the coexisting liquids. The interactions in the system are modeled using Lennard-Jones potentials with different interactions between segments of the dimers and the liquid components. We study how asymmetry of the interactions between dimers and molecules of the liquid, i.e., the degree of dimer amphiphilicity, influences the interfacial structure. Two unexpected phenomena have been found. First, for some systems, the liquid-liquid interface is able to accommodate only a finite amount of dimers. If the amount of added dimers is larger than a threshold value, a part or all of the dimers move to the interior one of the coexisting phase, forming an insoluble sheet inside it, or the initial interface splits into separate parts. The second is a peculiar behavior of the dependence of the interfacial width with an increase of the amount of added dimers. In this case, we observe a discontinuous jump that is connected with reorientation of dimers with respect to the interface.

Assembly of Helical Structures in Systems with Competing Interactions under Cylindrical Confinement

The behavior under confinement of nanoparticles interacting with the short-range attraction and long-range repulsion potential is studied by means of Monte Carlo simulations in the grand canonical ensemble. The study is performed at thermodynamic conditions at which a hexagonal cylindrical phase is the most stable phase in bulk. In these conditions, cylindrical confinement promotes the formation of helical structures whose morphology depends upon both the pore radius and boundary conditions. As the pore radius increases, the fluid undergoes a series of structural transitions going from single to multiple intertwined helices to concentric helical structures. When the pore ends are closed by planar walls, ring and toroidal clusters are formed next to these walls. Dependent upon the cylinder length, molecules away from the pore edges can either keep growing into ring and toroidal aggregates or arrange into helical structures. It is demonstrated that the system behaves in cylindrical confinement in the same way as the block copolymer systems. Such behavior has not been observed for the colloidal systems in cylindrical confinement with only repulsive interactions.

Self-consistent theory for systems with mesoscopic fluctuations

We have developed a theory for inhomogeneous systems that allows for the incorporation of the effects of mesoscopic fluctuations. A hierarchy of equations relating the correlation and direct correlation functions for the local excess [Formula: see text] of the volume fraction of particles ζ has been obtained, and an approximation leading to a closed set of equations for the two-point functions has been introduced for the disordered inhomogeneous phase. We have numerically solved the self-consistent equations for one-dimensional (1D) and three-dimensional (3D) models with short-range attraction and long-range repulsion. Predictions for all of the qualitative properties of the 1D model agree with the exact results, but only semi-quantitative agreement is obtained in the simplest version of the theory. The effects of fluctuations in the two 3D models considered are significantly different, despite the very similar properties of these models in the mean-field approximation. In both cases we obtain the sequence of large-small-large compressibility for increasing ζ. The very small compressibility is accompanied by the oscillatory decay of correlations with correlation lengths that are orders of magnitude larger than the size of particles. In one of the two models considered, the small compressibility becomes very small and the large compressibility becomes very large with decreasing temperature, and eventually van der Waals loops appear. Further studies are necessary in order to determine the nature of the strongly inhomogeneous phase present for intermediate volume fractions in 3D.

Equilibrium Phase Behavior of a Continuous-Space Microphase Former

Periodic microphases universally emerge in systems for which short-range interparticle attraction is frustrated by long-range repulsion. The morphological richness of these phases makes them desirable material targets, but our relatively coarse understanding of even simple models hinders controlling their assembly. We report here the solution of the equilibrium phase behavior of a microscopic microphase former through specialized Monte Carlo simulations. The results for cluster crystal, cylindrical, double gyroid, and lamellar ordering qualitatively agree with a Landau-type free energy description and reveal the nontrivial interplay between cluster, gel, and microphase formation.

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.

Predicting the orientation of lipid cubic phase films

Lipid cubic phase films are of increasingly widespread importance, both in the analysis of the cubic phases themselves by techniques including microscopy and X-ray scattering, and in their applications, especially as electrode coatings for electrochemical sensors and for templates for the electrodeposition of nanostructured metal. In this work we demonstrate that the crystallographic orientation adopted by these films is governed by minimization of interfacial energy. This is shown by the agreement between experimental data obtained using grazing-incidence small-angle X-ray scattering (GI-SAXS), and the predicted lowest energy orientation determined using a theoretical approach we have recently developed. GI-SAXS data show a high degree of orientation for films of both the double diamond phase and the gyroid phase, with the [111] and [110] directions respectively perpendicular to the planar substrate. In each case, this matches the lowest energy facet calculated for that particular phase.

The effect of antagonistic salt on a confined near-critical mixture

We consider a near-critical binary mixture with addition of antagonistic salt (hydrophilic cations and hydrophobic anions) confined between weakly charged and selective surfaces. A mesoscopic functional for this system is developed from a microscopic description by a systematic coarse-graining procedure. The functional reduces to the Landau-Brazovskii functional for amphiphilic systems for a sufficiently large ratio between the correlation length in the critical binary mixture and the screening length. Our theoretical result agrees with the experimental observation [Sadakane et al., J. Chem. Phys., 2013, 139, 234905] that the antagonistic salt and the surfactant both lead to a similar mesoscopic structure. For very low salt concentration ρion the Casimir potential is the same as in the presence of inorganic salt. For larger ρion the Casimir potential takes a minimum followed by a maximum for separations of order of tens of nanometers, and exhibits an oscillatory decay very close to the critical point. For separations of tens of nanometers the potential between surfaces with a linear size of hundreds of nanometers can be of order of kBT. We have verified that in the experimentally studied samples [Sadakane et al., J. Chem. Phys., 2013, 139, 234905, Leys et al., Soft Matter, 2013, 9, 9326] the decay length is too small compared to the period of oscillations of the Casimir potential, but the oscillatory force could be observed closer to the critical point.

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.

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.

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.

A look through 'lens' cubic mitochondria

Cell membranes may fold up into three-dimensional nanoperiodic cubic structures in biological systems. Similar geometries are well studied in other disciplines such as mathematics, physics and polymer chemistry. The fundamental function of cubic membranes in biological systems has not been uncovered yet; however, their appearance in specialized cell types indicates a role as structural templates or perhaps direct physical entities with specialized biophysical properties. The mitochondria located at the inner segment of the retinal cones of tree shrew (Tupaia glis and Tupaia belangeri) contain unique patterns of concentric cristae with a highly ordered membrane arrangement in three dimensions similar to the photonic nanostructures observed in butterfly wing scales. Using a direct template matching method, we show that the inner mitochondrial membrane folds into multi-layered (8 to 12 layers) gyroid cubic membrane arrangements in the photoreceptor cells. Three-dimensional simulation data demonstrate that such multi-layer gyroid membrane arrangements in the retinal cones of a tree shrew's eye can potentially function as: (i) multi-focal lens; (ii) angle-independent interference filters to block UV light; and (iii) a waveguide photonic crystal. These theoretical results highlight for the first time the significance of multi-layer cubic membrane arrangements to achieve near-quasi-photonic crystal properties through the simple and reversible biological process of continuous membrane folding.

Microphase-separated multicontinuous phase in low-molecular-mass thermotropic liquid crystals

Extensive application of chemical thermodynamics to exotic aggregation formed in thermotropic liquid crystals is briefly described. Through thermodynamic analyses and considerations of experimental results on liquid crystals, the unexpected sharing of common properties by thermo- and lyotropic liquid crystals is demonstrated. In some thermotropic liquid crystals, the terminal alkyl chain attached to the molecular core is highly disordered, as indicated by the magnitude of configurational entropy. The molten chain serves as intramolecular solvent (self-solvent), as evidenced by the close similarity between phase diagrams against chain length and composition in the binary system with n-alkane. These facts lead to the quasi-binary picture of thermotropic liquid crystals. Consideration of the thermodynamic potential expanded in terms of density fluctuation gives a new insight into the multicontinuous phases formed in simple systems consisting of anisotropic, rodlike particles.

Effects of lipid confinement on insulin stability and amyloid formation

We report on a study of insulin incorporation into cubic phases of mono-olein (MO), using synchrotron small-angle X-ray scattering and FT-IR spectroscopy. We studied the thermal stability and aggregation scenario of insulin as a function of protein concentration in the narrow water channels of the cubic lipid matrix and compared it with data for insulin unfolding and fibrillation in bulk water solutions. The concomitant effect of insulin entrapment on the structure and phase behavior of the lipid matrix itself was also examined. We show that the protein's unfolding behavior and stability are influenced by confinement due to geometrical limitations, and vice versa, the topological properties of the lipid matrix change as well. The addition of insulin already at concentrations as low as 0.1 wt % significantly alters the phase behavior of MO. Surprisingly, new cubic structures are induced by insulin incorporation into the lipid matrix. When insulin begins to partially unfold at higher temperatures, the structure of the new cubic phase changes and finally disappears around 60 degrees C, where the aggregation process sets in. The aggregation in cubo proceeds much faster and leads to the formation of medium-sized oligomers or clusters, while the formation of large fibrillar agglomerates, as observed for bulk insulin aggregation, is largely prohibited. Hence, the results yield valuable information about the use of cubic mesoporous lipid systems as a medium for long-term storage of insulin and aggregation-prone proteins in general. Furthermore, the results provide new insights into the effects of soft-matter confinement on protein aggregation and fibrillation, a situation usually met in natural cell environments.

Controlling interfacial curvature in nanoporous silica films formed by evaporation-induced self-assembly from nonionic surfactants. II. Effect of processing parameters on film structure

The double-gyroid phase of nanoporous silica films has been shown to possess facile mass-transport properties and may be used as a mold to fabricate a variety of highly ordered inverse double-gyroid metal and semiconductor films. This phase exists only over a very small region of the binary phase diagram for most surfactants, and it has been very difficult to synthesize metal oxide films with this structure by evaporation-induced self-assembly (EISA). Here, we show the interplay of the key parameters needed to synthesize these structures reproducibly and show that the interfacial curvature may be systematically controlled. Grazing angle of incidence small-angle X-ray scattering (GISAXS) is used to determine the structure and orientation of nanostructured silica films formed by EISA from dilute silica/(poly(ethylene oxide)-b-poly(propylene oxide)-b-alkyl) surfactant solutions. Four different highly ordered film structures are observed by changing only the concentration of the surfactant, the relative humidity during dip-coating, and the aging time of the solution prior to coating. The highly ordered films progress from rhombohedral (Rm) to 2D rectangular (c2m) to double-gyroid (distorted Iad) to lamellar systematically as interfacial curvature decreases. Under all experimental conditions investigated, increasing the aging time of the coating solution was found to decrease the interfacial curvature. In particular, this parameter was critical to being able to synthesize highly ordered, pure-phase double-gyroid films. The key role of the aging time is shown via processing diagrams that map out the interplay between the aging time, composition, and relative humidity. 29Si nuclear magnetic resonance (NMR) spectroscopy and solution-phase small-angle X-ray scattering (SAXS) of the aged coating solutions presented in part I of this series are then used to interpret the effects of aging prior to dip-coating. Specifically, it was found that a predictive model based on volume fractions and the silica cluster stoichiometry obtained from 29Si NMR qualitatively explains the trends observed with composition and aging. However, apart from the effects of relative humidity, a quantitative comparison of the predicted phase with the experimental processing diagram suggests that less-condensed silica clusters are more effective at swelling the EO blocks at early aging times. This enhanced swelling decreases with aging time and results in lower-curvature nanostructures such as the double-gyroid. The decrease in swelling is attributed to the decreased thermodynamic driving force for the more-condensed silica clusters to mix with the EO block of the surfactant.

Coarsening of bicontinuous structures via nonconserved and conserved dynamics

Coarsening subsequent to phase separations occurs in many two-phase mixtures. While unique scaled particle size distributions have been determined for highly asymmetric mixtures in which spherical particles form in a matrix, it is not known if a unique scaled structure exists for symmetric mixtures, which yield bicontinuous structures having intricately interpenetrating phase domains. Using large-scale simulations, we have established that unique scaled microstructures exist in symmetric mixtures evolving via nonconserved and conserved dynamics. We characterize their morphologies by the interfacial shape distribution, a counterpart to the particle size distribution, and their topologies by the genus. We find that the two dynamics result in unique, but different, scaled interfacial shape distributions, with conserved dynamics yielding a narrower distribution around zero mean curvature. In contrast, the two scaled structures are topologically similar, having nearly equal values of the scaled genus.

Inverse lyotropic phases of lipids and membrane curvature

In recent years it has become evident that many biological functions and processes are associated with the adoption by cellular membranes of complex geometries, at least locally. In this paper, we initially discuss the range of self-assembled structures that lipids, the building blocks of biological membranes, may form, focusing specifically on the inverse lyotropic phases of negative interfacial mean curvature. We describe the roles of curvature elasticity and packing frustration in controlling the stability of these inverse phases, and the experimental determination of the spontaneous curvature and the curvature elastic parameters. We discuss how the lyotropic phase behaviour can be tuned by the addition of compounds such as long-chain alkanes, which can relieve packing frustration. The latter section of the paper elaborates further on the structure, geometric properties, and stability of the inverse bicontinuous cubic phases.

Tessellation of a stripe

This paper describes enumeration of a class of topologically distinct periodic divisions of a stripe. Optimization of the geometry of these periodic tilings--optimization that yields minimum total perimeter of the tiles--gives a set of physically plausible periodic structures of monodisperse, two-dimensional foams bounded by two parallel walls. Evaluation of the minimum total perimeters of the lattices that we enumerated suggests two possible lower bounds for the mean perimeter of tiles forming periodic coverings of a stripe.

Fluid permeabilities of triply periodic minimal surfaces

It has recently been shown that triply periodic two-phase bicontinuous composites with interfaces that are the Schwartz primitive (P) and diamond (D) minimal surfaces are not only geometrically extremal but extremal for simultaneous transport of heat and electricity. The multifunctionality of such two-phase systems has been further established by demonstrating that they are also extremal when a competition is set up between the effective bulk modulus and electrical (or thermal) conductivity of the bicontinuous composite. Here we compute the fluid permeabilities of these and other triply periodic bicontinuous structures at a porosity using the immersed-boundary finite-volume method. The other triply periodic porous media that we study include the Schoen gyroid (G) minimal surface, two different pore-channel models, and an array of spherical obstacles arranged on the sites of a simple cubic lattice. We find that the Schwartz P porous medium has the largest fluid permeability among all of the six triply periodic porous media considered in this paper. The fluid permeabilities are shown to be inversely proportional to the corresponding specific surfaces for these structures. This leads to the conjecture that the maximal fluid permeability for a triply periodic porous medium with a simply connected pore space at a porosity is achieved by the structure that globally minimizes the specific surface.

Some features of soft matter systems

Pierre Gilles de Gennes was awarded a Nobel prize in physics "for discovering that methods developed for studying order phenomena in simple systems can be generalized to more complex forms of matter, in particular to liquid crystals and polymers". Thanks to his works "soft matter" became a new legitimate discipline in physics. Soft matter includes a vast range of materials, which cannot be classified as simple liquids or solids. Many soft matter systems exhibit partially broken translational and/or rotational symmetry. In others we observe mesoscopic self-assembling into supramolecular structures leading to viscoelastic behavior. The partial ordering with viscoelastic properties, topological and geometrical complexity, and long relaxations associated with broken symmetries and/or supramolecular assembling are the main features of these systems. Among them we find liquid crystals, gels, biological membranes, colloidal suspensions, polymer solutions and polymer melts and blends, surfactant solutions . Typical models used in soft matter theory are based on statistical mechanics and classical thermodynamics, supplemented by the theory of elasticity, hydrodynamics and thermodynamics of irreversible processes and also some elements of the field theory. In this short overview I would like to discuss three theoretical issues related to soft matter systems: interactions, the role of the entropy, and finally the order parameter description.

A triple-network tricontinuous cubic liquid crystal

Soft matter such as surfactant-water systems, block copolymers or liquid crystals can form periodic structures on nanometre to micrometre scales. This property can be used for templating nanoporous ceramics, surface patterning for electronic devices, or generation of photonic materials. Much attention has been paid to structures appearing between the layer and cylinder phases, the three so-called bicontinuous cubic phases. These are formed by two continuous interpenetrating networks of channels. In this article we describe a related phase, which has the first reported structure consisting of three interpenetrating infinite networks. It is a thermotropic (solvent-free) liquid crystal of cubic symmetry Im3m. The structure is one of the most complex in liquid crystals, and is determined by direct Fourier reconstruction of electron density. We discuss the possible rationale for the existence of such a phase, its structural relationship with the bicontinuous phases, and its position in the phase diagram.

Modeling liquid crystal bilayer structures with minimal surfaces

This paper describes a new convenient and accurate method of calculating x-ray diffraction integrated intensities from detailed cubic bilayer structures. The method is employed to investigate the structure of a particular surfactant system (didodecyldimethylammonium bromide in a solution of oil and heavy water), for which single-crystal experimental data have recently been collected. The diffracted peak intensities correlate well with theoretical structures based on mathematical minimal surfaces. Optimized electron density profiles of the bilayer are presented, providing new insight into key features of the bilayer structure.

Smectic blue phases: layered systems with high intrinsic curvature

We report on a construction for smectic blue phases, which have quasi-long-range smectic translational order as well as three-dimensional crystalline order. Our proposed structures fill space by adding layers on top of a minimal surface, introducing either curvature or edge defects as necessary. We find that for the right range of material parameters, the favorable saddle-splay energy of these structures can stabilize them against uniform layered structures. We also consider the nature of curvature frustration between mean curvature and saddle splay.

Concave and convex shapes of the Pn3m/L1 interface

Shapes of the interface between the L1 and cubic Pn3 m phases in the mixture C(12)EO(2)/water are studied. The concave and convex variants of the interface are realised using Pn3 m crystals surrounded by the L1 phase and L1 inclusions on surfaces and in the bulk of the Pn3 m phase. It is shown that both variants of the Pn3 m/L1 interface contain the (111)-type facets in coexistence with everywhere else rough surfaces. The matching between facets and curved parts of the interface is angular. In the vicinity of the upper limit of the L1 + Pn3 m coexistence domain, additional (200)-type facets appear on the interface. The influence of the contact angle at glass walls on shapes of crystals and of inclusions is discussed.

Multifunctional composites: optimizing microstructures for simultaneous transport of heat and electricity

Composite materials are ideally suited to achieve multifunctionality since the best features of different materials can be combined to form a new material that has a broad spectrum of desired properties. Nature's ultimate multifunctional composites are biological materials. There are presently no simple examples that rigorously demonstrate the effect of competing property demands on composite microstructures. To illustrate the fascinating types of microstructures that can arise in multifunctional optimization, we maximize the simultaneous transport of heat and electricity in three-dimensional, two-phase composites using rigorous optimization techniques. Interestingly, we discover that the optimal three-dimensional structures are bicontinuous triply periodic minimal surfaces.

Morphological changes during the order-disorder transition in the two- and three-dimensional systems of scalar nonconserved order parameters

The order-disorder transition is studied in a system of a scalar nonconserved order parameter. We use this well studied system to show that the application of the methods of topology and geometry reveals that our knowledge of the kinetic pathways by which the order-disorder transition proceeds is far from being complete. We show that in two-dimensional (2D) and 3D systems there are three dynamical regimes in the evolution of the system: early, intermediate, and late. In the intermediate regime two length scales govern the behavior of the system, whereas in the early and intermediate regime there is only one length scale. The size distribution of the domain area indicates the pathway by which the domains change their size. There are only two types of domains in a 2D system: circular and elongated with well defined characteristics (scaling of the area with the contour length) which in the late regime do not depend on time after rescaling by the average area and contour in the system. The elongated domains continuously change into circular domains reducing in this way the overall dissipation in the system. In order to reach a Lifshitz-Cahn-Allen (LCA) late stage regime the number of elongated domains must be strongly reduced. In the intermediate regime the number of elongated domains is large and simple LCA scaling does not hold. In a 3D symmetric system we always have a bicontinuous structure that evolves by cutting small connections. The late stage regime seems to be associated with the appearance of the preferred nonzero mean curvature. The early-intermediate regime crossover is associated with the saturation of the order parameter inside the domains, while the intermediate-late stage regime crossover is related to the global breaking of the +/- order parameter symmetry (marked by the appearance of the nonzero mean curvature but still zero average magnetization). The times for the occurrence of these crossovers do not depend on the size of the system.

Lattice-Boltzmann simulations of self-assembly of a binary water-surfactant system into ordered bicontinuous cubic and lamellar phases

We used our recently developed mesoscale amphiphilic lattice-Boltzmann method (Nekovee, M.; Coveney, P. V.; Chen, H.; Boghosian, B. M. Phys. Rev. E 2000, 62, 8282-8894) to investigate the dynamics of self-assembly of the bicontinuous cubic phase in a binary water-surfactant system, and the transition from the lamellar structure to a bicontinuous cubic phase. Our study provides insight into how such structures emerge as a result of competing molecular interactions between water and amphiphiles and among amphiphilic molecules themselves, and represents the first application of any lattice-Boltzmann model to amphiphilic systems in three dimensions.

Periodic surfaces of simple and complex topology: comparison of scattering patterns

We compute scattering patterns for six triply periodic minimal surfaces formed in oil/surfactant/water solutions: Three surfaces of a simple topology, Schwarz P (Im3m), Schwarz D-diamond (Pn3m), and Schoen G-gyroid (Ia3d), and three surfaces of a complex topology, SCN1 (Im3m), CD (Pn3m), and GX6 (Ia3d). We show that in the case of the complex structures, scattering intensity is shifted towards the higher hkl peaks. This might cause their misidentification and wrong estimates about the cell size of the structure.