Competition brings out the best: modelling the frustration between curvature energy and chain stretching energy of lyotropic liquid crystals in bicontinuous cubic phases

Stabilizing Differential Interfacial Curvatures by Mismatched Molecular Geometries: Toward Polymers with Percolating 1 nm Channels of Gyroid Minimal Surfaces

Soft materials with self-assembled networks possess saddle-shaped interfaces with distributed negative Gaussian curvatures. The ability to stabilize such a geometry is critically important for various applications but can be challenging due to the possibly "deficient" packing of the building blocks. This nontrivial challenge has been manifested, for example, by the limited availability of cross-linkable bicontinuous cubic (Q) liquid crystals (LCs), which can be utilized to fabricate compelling polymers with networked nanochannels uniformly sized at ∼1 nm. Here, we devise a facile approach to stabilizing cross-linkable Q mesophases by leveraging the synergistic self-assembly from pairs of scalably synthesized polymerizable amphiphiles. Hybridization of the molecular geometries by mixing significantly increases the propensity of the local deviations in the interfacial curvature specifically required for Q assemblies. "Normal" (type 1) double gyroid LCs possessing 1 nm ionic channels conforming to minimal surfaces can be formulated by simultaneous hydration of the amphiphile mixtures, as opposed to the formation of hexagonal or lamellar mesophases exhibited by the single-amphiphile systems, respectively. Fixation of the bicontinuous network in polymers via radical polymerization has been efficaciously facilitated by the presence of the bifunctional polymerizable groups in one of the employed amphiphiles. High-fidelity lock-in of the ordered continuous 1 nm channels has been unambiguously confirmed by the observation of single-crystal-like diffraction patterns from synchrotron small-angle X-ray scattering and large-area periodicities by transmission electron microscopy. The produced polymeric materials exhibit the required mechanical integrity as well as chemical robustness in a variety of organic solvents that benefit their practical applications for selective transport of ions and molecules.

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.

Polar-Nonpolar Interfaces of Normal Bicontinuous Cubic Phases in Nonionic Surfactant/Water Systems Are Parallel to the Gyroid Surface

We investigated the structures of normal (type I) bicontinuous cubic phases in hexa-, hepta-, and octaethylene glycol dodecyl ether/water mixtures by small-angle X-ray crystallography of single-crystal domains. Reconstructed electron densities showed that the hydrophilic chains with high electron density are confined to a film centered on the surface of the Gyroid (a triply periodic minimal surface), while hydrophobic chains with low electron density are distributed within the pair of interwoven labyrinths carved out by the Gyroid. Further, the local minimum within the high electron density region, due to bulk water, coincides precisely with the Gyroid. This minimum is less pronounced in mixtures with longer ethylene glycol chains, consistent with their decreased water content. Our analysis clearly shows that the polar-nonpolar interfaces are parallel to the Gyroid surface in all mixtures. The repulsive hydration or overlapping force between the pair of facing monolayers of ethylene glycol chains on either side of the Gyroid surface is the likely origin of the parallel interfaces.

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.

Crystal twinning of bicontinuous cubic structures

Bicontinuous cubic structures in soft matter consist of two intertwining labyrinths separated by a partitioning layer. Combining experiments, numerical modelling and techniques in differential geometry, we investigate twinning defects in bicontinuous cubic structures. We first demonstrate that a twin boundary is most likely to occur at a plane that cuts the partitioning layer almost perpendicularly, so that the perturbation caused by twinning remains minimal. This principle can be used as a criterion to identify potential twin boundaries, as demonstrated through detailed investigations of mesoporous silica crystals characterized by diamond and gyroid surfaces. We then discuss that a twin boundary can result from a stacking fault in the arrangement of inter-lamellar attachments at an early stage of structure formation. It is further shown that enhanced curvature fluctuations near the twin boundary would cost energy because of geometrical frustration, which would be eased by a crystal distortion that is experimentally observed.

Bicontinuous cubic phases in biological and artificial self-assembled systems

Nature has created innumerable life forms with miraculous hierarchical structures and morphologies that are optimized for different life events through evolution over billions of years. Bicontinuous cubic structures, which are often described by triply periodic minimal surfaces (TPMSs) and their constant mean curvature (CMC)/parallel surface companions, are of special interest to various research fields because of their complex form with unique physical functionalities. This has prompted the scientific community to fully understand the formation, structure, and properties of these materials. In this review, we summarize and discuss the formation mechanism and relationships of the relevant biological structures and the artificial self-assembly systems. These structures can be formed through biological processes with amazing regulation across a great length scales; nevertheless, artificial construction normally produces the structure corresponding to the molecular size and shape. Notably, the block copolymeric system is considered to be an applicable and attractive model system for the study of biological systems due to their versatile design and rich phase behavior. Some of the phenomena found in these two systems are compared and discussed, and this information may provide new ideas for a comprehensive understanding of the relationship between molecular shape and resulting interface curvature and the self-assembly process in living organisms. We argue that the co-polymeric system may serve as a model to understand these biological systems and could encourage additional studies of artificial self-assembly and the creation of new functional materials.

Polar-Nonpolar Interfaces of Inverse Bicontinuous Cubic Phases in Phytantriol/Water System are Parallel to Triply Periodic Minimal Surfaces

We investigated two distinct lyotropic liquid crystal inverse bicontinuous cubic phases of phytantriol/water mixtures by small-angle X-ray crystallography of the single-crystal regions. Reconstructed electron density maps revealed hydrophilic head and hydrophobic tail regions of the phytantriol bilayer membranes and water regions. The bilayer membranes are shown to be located on the D and gyroid triply periodic minimal surfaces. To investigate the structures of the polar-nonpolar interfaces, we optimized two models: a parallel surface model and a constant mean curvature surface model. The parallel surface model agreed well with the X-ray data, and the R factors, which show the degree of agreement between those structural models and the data, were less than 0.04. In stark contrast, the constant mean curvature surface model deviated significantly from the data, and the R factors were around 0.15. We therefore conclude that the polar-nonpolar interface of the inverse bicontinuous cubic phase of the phytantriol/water system is close to a parallel surface to a triply periodic minimal surface.

Micellar Mimicry of Intermetallic C14 and C15 Laves Phases by Aqueous Lyotropic Self-Assembly

Concentration-dependent supramolecular self-assembly of amphiphilic molecules in water furnishes a variety of nanostructured lyotropic liquid crystals (LLCs), which typically display high symmetry bicontinuous network and discontinuous micellar morphologies. Aqueous dispersions of soft spherical micelles derived from small molecule amphiphile hydration typically pack into exemplary body-centered cubic and closest-packed LLCs. However, investigations of hydrated mixtures of the ionic surfactant tetramethylammonium decanoate loaded with 40 wt % n-decane (TMADec-40) revealed the formation of a high symmetry bicontinuous double diamond LLC, as well as cubic C15 and hexagonal C14 Laves LLC phases that mirror the MgCu₂ and MgZn₂ intermetallic structure types, respectively. Detailed small-angle X-ray scattering analyses demonstrate that the complex C15 and C14 LLCs exhibit large unit cells, in which 12 or more ∼3-4 nm diameter micelles of multiple discrete sizes arrange into tetrahedral close packing arrangements with exceptional long-range translational order. The symmetry breaking that drives self-assembly into these low-symmetry LLC phases is rationalized in terms of a frustrated balance between maximizing counterion-mediated micellar cohesion within the ensemble of oil-swollen particles, while simultaneously optimizing local spherical particle symmetry to minimize molecular-level variations in surfactant solvation.

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.