Session N57: 50 Years of Gyroid Structures in Materials and Tissues - in Honor of Alan Schoen I

Functional Block Copolymer Gyroidal Hybrid Nanomaterials

This talk will discuss block copolymer self-assembly directed hybrid materials with double and single gyroid morphologies. The focus will be on preparation of functional materials in which the co-continuous cubic gyroid structures with nanoscopic confinement imparts novel properties on the final materials. To that end, diblock copolymers and triblock terpolymers will be used to structure direct inorganic components typically in the form of nanoparticles. Fundamental design criteria for successful periodic cubic gyroidal lattice formation of the resulting block copolymer-nanoparticle hybrids will be discussed. In addition to the characterization of resulting bulk hybrids, the talk will present the formation of mesoporous inorganic materials with gyroidal morphology resulting from further thermal processing. It will be demonstrated how fundamental understanding of block copolymer co-assembly with inorganic nanoparticles allows these approaches to be generalized from amorphous oxides and high-temperature non-oxides to highly crystalline transition metal oxides, metals, and seminconductors all the way to superconductors with applications ranging from separation technologies to energy conversion and storage all the way to catalysis. Examples will be provided of how the resulting periodically structured and mesoporous inorganic materials with double or single gyroid morphologies control properties beyond the intrinsic atomic lattice structures, resulting in metamaterials.    

Stabilizing gyroid structures by blending or asymmetry of high-χ oligomers

Molecular dynamics simulations of block oligomers containing polar CHxOH and nonpolar CHy repeat units are used to explore features that stabilize gyroid phases with domain pitches under 3 nm. In their pure form, AB diblock and AB₂ miktoarm triblock amphiphiles can self-assemble into ordered lamellar (LAM) and cylindrical (CYL) structures, respectively.   However, their AB -rich blend (0.2 ≤ x ≤ 0.4) forms a double gyroid (DG) network structure. Structural analyses reveal that the non-uniform interfacial curvature of the DG structure is supported by local composition variations of the LAM and CYL forming amphiphiles.  Interestingly, introducing large shape asymmetry into miktoarm multiblock amphiphiles by coupling of blocks of different length can also yield DG structures where the volume fraction of polar and nonpolar blocks can be varied over a wide range. This work provides molecular-level insights into how blending and asymmetry of shape-filling molecular architectures enables network phase formation with extremely small feature sizes over a wide composition and volume fraction ranges.   

Gyroid Materials: triply periodic minimal surfaces, skeletal graphs, mesoatoms and crystallographic defects

Triply periodic minimal surfaces (TPMS) have zero mean curvature and their associated constant mean curvature families are all quite good structural models for many self-assembled, bicontinuous amphiphilic soft materials, ranging from water and cation-soap systems to block copolymers. The G or gyroid TPMS discovered by Schoen in 1970 subdivides space into two intertwined labyrinths. These labyrinths can be reduced to a pair of enantiomorphic 3-connected skeletal graphs. Structural aspects of crystalline materials have been depicted with respect to the networks. Laves first used network graphs to describe silicate structures in 1932. In his 1954 paper on 3 connected nets, Wells described this same periodic graph, denoted the 10-3a, indicating the smallest closed loop contained 10 segments, each of which was 3 connected. Luzzati and Spegt used small angle xray scattering to solve the structure of a self-assembled strontium soap in 1967. The polar groups formed the two interpenetrating networks of 3-connected rods embedded in the hydrocarbon matrix having the cubic Ia3d space group. Schoen's 1970 NASA report debuted the gyroid TPMS and referred to the Laves' periodic graph. Only in 1994 did researchers find that block copolymers could also form a gyroid structure. Here the G TPMS bisects the majority A block domain while the two minority B domains surround the pair of skeletal graphs. In soft matter, the structural units correspond not to atoms as in hard matter but to large groups of molecules aggregating into mesoatoms. In the case of block copolymers, a mesoatom can contain millions of atoms. Mesoatoms exhibit point group symmetries, shapes and sizes that satisfy space group symmetry and pack together to fill the unit cell. As with all crystalline materials, symmetry breaking crystallographic defects occur during growth and from mechanical forces. Point, line and surface defects induce alteration of the malleable mesoatoms near the defects. Because the bicontinuous gyroid presents interesting topology as well as geometry, the structure and its various defects afford multiple opportunities for technological applications ranging from photonics to batteries.   

Conformational and topological correlations in the gyroid morphology formed by non-frustrated triblock copolymers with homopolymers

The gyroid morphology is often only stable in a narrow region of the phase diagram, in part due to the high packing frustration, where blocks are stretched to fill the volume of the networks. Here, we investigate whether the stability of the gyroid phase can be improved by relieving the packing frustration using homopolymers. The effect of topology on the chain conformation is studied using dissipative particle dynamics (DPD) simulations. In a non-frustrated ABC, the A-blocks stretch to fill deep within the A-rich region. To relieve this stretching, A-selective homopolymers of different lengths were co-assembled with the ABC copolymer at several compositions. Topological analysis showed that homopolymers with lengths shorter than the A-block length filled the middle of the A-networks as hypothesized. The block copolymer chains exhibited some stretching, but the homopolymers did not. Since entropy is additive, the effect of the stretched block copolymer decreases by adding more homopolymers. Thus, entropic benefits can be tuned by choosing the amount and size of the homopolymer. From a macroscopic perspective, the domain size of the network can also be tuned using different lengths and amounts of added homopolymers, offering greater control over the final stable phase and bridging two separate regions in the phase diagram, making it more practical. Broadly, our method of analyzing the topology can be applied to other mixtures and blends to analyze questions pertaining to chain residence within the gyroid.   

Metastable Network Phases from Controlled Self-Assembly of High-χ Block Copolymers for Biomimetic Materials

Well-ordered nanonetwork materials are appealing and promising for innovative properties such as optical and mechanical metamaterials as inspired by nature (photonic property from wing structure of butterfly and high impact property from dactyl club of mantis shrimp). Here, this work aims to suggest a facile method for acquiring network phases from self-assembly. By taking advantage of controlled self-assembly for high-χ polystyrene-block-polydimethylsiloxane block copolymers (BCPs), it is feasible to acquire metastable network phases from the use of selective solvent for self-assembly under controlled evaporation of the solvent. A variety of kinetically trapped network phases with high degree of ordering can be obtained from a single-composition lamellar phase, giving an easy method to acquire network phases even with large packing frustration (entropic penalty). Furthermore, the windows for the formation of metastable network phases can be even expanded through controlled self-assembly of star-block BCPs as compared to diblocks. The easy fabrication of network phases from the bottom-up approach gives rise to the feasibility for biomimicking the outstanding properties of butterfly wing structure and mantis shrimp dactyl club.   

How does your gyroid grow? A mesoatomic perspective on supramolecular, soft matter network crystals

I describe a proposed framework for understanding the structure of supramolecular network crystals formed in soft matter in terms of mesoatomic building blocks, collective groupings of amphiphilic molecules that play a role analogous to atomic or molecular subunits of hard matter crystals. While the concept of mesoatoms is intuitive and widely invoked in crystalline arrangements of spherical or cylindrical (micellelike) domains, analogous notions physically meaningful building blocks of triply periodic network (TPN) crystals, like the double-gyroid or double-diamond structures are obscured by the complex, bicontinuous domain shapes and intercatenated topologies of the double networks. Focusing on the specific example of diblock copolymer melts, I describe generic rules for decomposing TPN crystals into a unique set of mesoatomic building blocks. The combination of simple physical principles for assembly as well as symmetries and topologies of these structures point to mesoatomic elements associated with the nodal connections, leading to mesoatomic volumes that are nonconvex and bound by smoothly curved faces, unlike the more familiar Voronoi polyhedral shapes associated with spherical/cylindrical mesoatoms. I describe the shapes of these mesoatoms, their internal structure, and importantly, their local packing. We further hypothesize that mesoatoms may be kinetically favored intermediate structures whose local shapes and packing template network crystal assembly on long time scales and study a minimal energetic model of mesoatom assembly for three different cubic double-network crystals. Based on these analyses, I discuss several possible elaborations of the mesoatomic description of TPN assemblies, most notably the implications of mesoatomic malleability, a feature that distinguishes soft matter from hard matter crystals.   

Programmable Self-Assembly of Nanoplates into Bicontinuous Nanostructures

Self-assembly is the process by which individual components arrange themselves into an ordered structure by changing shapes, components, and interactions. It has enabled us to construct a remarkable range of geometric forms at many length scales. However, the potential of two-dimensional polygonal nanoplates to self-assemble into extended three-dimensional structures with compartments and corridors has not been explored. In this presentation, we demonstrate coarse-grained Monte Carlo simulations showing self-assembly of hexagonal/triangular nanoplates via complementary interactions into facetted, sponge-like "bicontinuous polyhedra" whose flat walls partition space into a pair of mutually interpenetrating labyrinths. Two bicontinuous polyhedra can be self-assembled: the regular Petrie-Coxeter infinite polyhedron (denoted {6,4|4}) and the semi-regular Hart "gyrangle". The latter structure is chiral and has both left- and right-handed version. We show that the Petrie-Coxeter assembly is constructed from two complementary populations of hexagonal nanoplates. Remarkably, we find that the 3D chiral Hart gyrangle can be assembled from identical achiral triangular nanoplates decorated with regioselective complementary interaction sites. The assembled Petrie-Coxeter and Hart polyhedra are facetted versions of two of the simplest triply-periodic minimal surfaces, namely Schwarz' Primitive and Schoen's Gyroid surfaces respectively. These findings offer new ways to create those bicontinuous nanostructures, which are prevalent in synthetic and biological materials.   

"Inverting" Caspar-Klug design rules for programmable assembly of size-controlled minimal-surface assemblies

As first suggested by Caspar and Klug, many viruses assemble icosahedral shells (capsids) because the high symmetry of the icosahedron enables economical assembly – enclosing a large volume with relatively few distinct protein subunit types. We generalize this design principle to triply-periodic polyhedra, mesoporous structures approximating cubic minimal surfaces. We propose and investigate an extension of the Caspar-Klug symmetry principles for viral capsid assembly to the programmable assembly of size-controlled triply-periodic polyhedra, discrete variants of the Primitive, Diamond, and Gyroid cubic minimal surfaces. Inspired by a recent class of programmable DNA origami triangular colloids, we demonstrate that the economy of design in these crystalline assemblies -- in terms of the growth of the number of distinct particle species required with the increased size-scale (e.g. periodicity) -- is comparable to viral shells. We further test the role of geometric specificity in these assemblies via dynamical assembly simulations, which show that conditions for simultaneously efficient and high-fidelity assembly require an intermediate degree of flexibility of local angles and lengths in programmed assembly. Off-target misassembly occurs via incorporation of a variant of disclination defects, generalized to the case of hyperbolic crystals. The possibility of these topological defects is a direct consequence of the very same symmetry principles that underlie the economical design, exposing a basic tradeoff between design economy and fidelity of programmable, size controlled assembly.   

A Multiscale Molecular Simulation Approach to Designing DNA Decorated Colloids for Double Gyroid Self-Assembly

Submitted for “ 50 years of Gyroid structures in materials and tissues - in honor of Alan Schoen.”

The gyroid phase is one of the most interesting and feature rich mesophases, with potential applications in photonic crystals, solar cells, and separation and catalytic membranes. Gyroidal sructures are of interest for different applications, which has fueled interest in developing different types of building blocks to achieve them. Inspired by previous modeling studies, in this simulation work we focus on developing colloidal particles as building blocks and take advantage of hybridizing DNA strands grafted to nanoparticle surfaces to design effective interactions that lead to the self-assembly of the double gyroid phase. We achieve this in a two-component system through judicious control of DNA patterning designs, DNA chain lengths and relative contact interaction distances. Our approach is iterative and entails multiscale models wherein for each proposed design: (i) a detailed model is used to obtain effective pair-particle potentials, (ii) a highly coarse-grained, many-particle system is simulated to characterize its self-assembly behavior, and (iii) the stability of gyroid-forming systems are validated through detailed many-particle simulations. The results are fed into a machine learning model to aid the exploration of the design space of our colloids. Through this process we demonstrate the self-assembly of the double gyroid phase and delineate the design rules needed to induce its formation.   

Oral: Role of Interaction Range on the Microstructure and Dynamics of Attractive Colloidal Systems

Colloidal gels, a class of complex materials, occupy a pivotal position at the interface of soft matter physics, materials science, and nanotechnology. They play a crucial role in various natural and industrial settings, from biological tissues to advanced materials. These gels form as colloidal particles form reversible bonds in a liquid medium, eventually creating a network structure that exhibits a remarkable interplay of solid and fluid-like properties. The colloid gelation phase diagram has been traditionally characterized using three key factors: particle volume fraction, strength of attraction, and range of attraction. While there's a rich body of literature on the role of attraction strength and particle volume fraction, majority of studies have been limited to short range interactions. Using molecular simulations, we explored the effect that the range of attractions has on the microstructure and dynamics of both weakly and strongly attractive colloidal systems. Although gelation occurs significantly faster at high attraction strength, by an order of magnitude compared to low strength, we did not observe any clear trend in gelation-rate with respect to a change in the range of interaction. However, as the attraction range increases in both systems, the final structure undergoes a transition from a fractal configuration to a fluid of dense clusters. In a very dilute system with very-long range attraction, we observed only a fluid of clusters, irrespective of the attraction strength.   

Theoretical results for the Gyroid wire system

We discuss the classical and non-commutative geometry of wire systems which are

the complements of triply periodic surfaces. Our predominant example is the

gyroid surface and its complementary wire network which can be fabricated on a

nanoscale, but we also worked on wire networks stemming from other CMC

surfaces. For all these geometries, we study the Harper Hamiltonian and its band

structure in the commutative and non-commutative cases.  In the latter, an

ambient magnetic field destroys the commutativity of the relevant algebras. We

have successfully applied methods from singularity theory, representation theory

and topology to these materials. In this setting the gyroid geometry can be seen as

the 3d generalization of graphene. We will discuss the most important results of our work.   

Disconnectivity Graphs of Spin Glass Systems on Snub Archimedean ($3^2$, 4, 3, 4 ) Lattices.

We use augmented disconnectivity graphs to visualize and analyze the energy landscapes of small spin systems confined to Snub Archimedean lattices of type ($3^2$,4,3,4). The disconnectivity graphs show how the different types of energy minima are connected in a two-dimensional representation. They visualize the types of minima, their lowest energy barriers, and their respective sizes. We distinguish between different minimum energy structures, namely regular minima and dales. Additionally, to the disconnectivity graphs, we will discuss the distribution of the types of minima, their average size for the respective energy levels, and barrier heights. The disconnectivity graphs for each model have distinctive features that give valuable insight into their complexity concerning state transitions and difficulties that optimization procedures could face.