Anatomy of triply-periodic network assemblies: characterizing skeletal and inter-domain surface geometry of block copolymer gyroids

Nonaffinity of Liquid Networks and Bicontinuous Mesophases

Amphiphiles self-assemble into a variety of bicontinuous mesophases whose equilibrium structures take the form of high-symmetry cubic networks. Here, we show that the symmetry-breaking distortions in these systems give rise to anomalously large, nonaffine collective deformations, which we argue to be a generic consequence of "mass equilibration" within deformed networks. We propose and study a minimal "liquid network" model of bicontinuous networks, in which acubic distortions are modeled by the relaxation of residually stressed mechanical networks with constant-tension bonds. We show that nonaffinity is strongly dependent on the valency of the network as well as the degree of strain-softening or strain-stiffening tension in the bonds. Taking diblock copolymer melts as a model system, liquid network theory captures quantitative features of two bicontinuous phases based on comparison with self-consistent field theory predictions and direct experimental characterization of acubic distortions, which are likely to be pronounced in soft amphiphilic systems more generally.

Revealing the 3D Structure of Block Copolymers with Electron Microscopy: Current Status and Future Directions

Block copolymers (BCPs) are considered model systems for understanding and utilizing self-assembly in soft matter. Their tunable nanometric structure and composition enable comprehensive studies of self-assembly processes as well as make them relevant materials in diverse applications. A key step in developing and controlling BCP nanostructures is a full understanding of their three-dimensional (3D) structure and how this structure is affected by the BCP chemistry, confinement, boundary conditions, and the self-assembly evolution and dynamics. Electron microscopy (EM) is a leading method in BCP 3D characterization owing to its high resolution in imaging nanosized structures. Here we discuss the two main 3D EM methods: namely, transmission EM tomography and slice and view scanning EM tomography. We present each method's principles, examine their strengths and weaknesses, and discuss ways researchers have devised to overcome some of the challenges in BCP 3D characterization with EM- from specimen preparation to imaging radiation-sensitive materials. Importantly, we review current and new cutting-edge EM methods such as direct electron detectors, energy dispersive X-ray spectroscopy of soft matter, high temporal rate imaging, and single-particle analysis that have great potential for expanding the BCP understanding through EM in the future.

Well-Ordered Nanonetwork Metamaterials from Block Copolymer Templated Syntheses

ConspectusThrough the morphological evolution to give highly optimized complex architectures at different length scales, fine-tuned textures for specific functions in living organisms can be achieved in nature such as a bone core with very complicated porous architecture to attain a significant structural efficiency attributed to delicately structured ligaments and density gradients. As inspired by nature, materials with periodic network structures (i.e., well-defined porous textures) in the nanoscale are appealing and promising for innovative properties. Biomimicking from nature, organic and/or inorganic nanonetworks can be synthetically fabricated, giving broadness and effectiveness when tuning the desired properties. Metamaterials are materials whose effective properties do not result from the bulk behavior of the constituent materials but rather mainly from their deliberate structuring. The performances of fabricating metamaterials will depend on the control of size, shape, order, and orientation of the forming textures. One of the appealing textures for the deliberate structuring is network architecture. Network materials possess self-supporting frameworks, open-cell character, high porosity, and large specific surface area, giving specific functions and complexity for diverse applications. As demonstrated by recent studies, exceptional mechanical performances such as negative thermal expansion, negative Poisson's ratio, and twisting under uniaxial forces can be achieved by the effect of the deliberate structuring with nanonetwork textures. In contrast to a top-down approach, a bottom-up approach is cost-effective, and also it can overcome the size limitation to reach nanoscale fabrication. It can be foreseen that network metamaterials with a feature size of tens of nanometers (referred as nanonetwork metamaterials) may provide new comprehension of the structure and property relationships for various materials. The self-assembly of block copolymers (BCPs) is one of the most used methods to build up well-ordered nanostructured phases from a bottom-up approach with precise control of size, shape, and orientation in the thin films for realistic applications. In this account, we summarize recent advancements in the fabrication of nanohybrids and nanoporous materials with well-ordered nanonetwork textures even with controlled helicity by combining block copolymer self-assembly and templated syntheses for mechanical and optical applications with superior properties beyond nature as metamaterials as well as chiral metamaterials with new properties for chiroptic applications such as chiral plasmonics, beam splitter, and negative refraction. The description of the fundamental facets of a nonconventional structure-property relationship with the characters of metamaterials and the state-of-the-art methodologies to fabricate nanonetworks using block copolymer self-assembly will stimulate research activities for the development of nanonetwork metamaterials with exceptional individual and multifunctional properties for futuristic devices.

Breaking Mirror Symmetry of Double Gyroids via Self-Assembly of Chiral Block Copolymers

Significant enhancement of segment-scale chirality, as measured by vibrational circular dichroism (VCD), is observed in the helical phase (H*) of polylactide-based chiral block copolymers (BCPs*) due to the mesoscale chirality of the microphase-separated domains. Here, we report a weaker, yet meaningful, enhancement on the VCD signal of a double gyroid phase (DG) as compared to a double diamond phase (DD) and disordered phase from the same diblock BCPs*. Residual VCD enhancement indicates a weak degree of chiral symmetry breaking, implying the formation of a chiral double gyroid (DG*) instead of the canonical achiral form. Calculations on the basis of orientational self-consistent field theory, comparing coupling between the segmental-scale preference of an intradomain twist and morphological chirality, show that a transition between DG and DG* takes place above the critical chiral strength, driving a weak volume asymmetry between the two enantiomeric single networks of DG*. The formation of nanostructures with controllable mesoscale chiral asymmetry indicates a pathway for the amplification of optical activity driven by self-assembly.

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.

The emergence of valency in colloidal crystals through electron equivalents

Colloidal crystal engineering of complex, low-symmetry architectures is challenging when isotropic building blocks are assembled. Here we describe an approach to generating such structures based upon programmable atom equivalents (nanoparticles functionalized with many DNA strands) and mobile electron equivalents (small particles functionalized with a low number of DNA strands complementary to the programmable atom equivalents). Under appropriate conditions, the spatial distribution of the electron equivalents breaks the symmetry of isotropic programmable atom equivalents, akin to the anisotropic distribution of valence electrons or coordination sites around a metal atom, leading to a set of well-defined coordination geometries and access to three new low-symmetry crystalline phases. All three phases represent the first examples of colloidal crystals, with two of them having elemental analogues (body-centred tetragonal and high-pressure gallium), while the third (triple double-gyroid structure) has no known natural equivalent. This approach enables the creation of complex, low-symmetry colloidal crystals that might find use in various technologies.

Computational Reverse Engineering Analysis for Scattering Experiments (CREASE) on Vesicles Assembled from Amphiphilic Macromolecular Solutions

In this paper we present the development and validation of the "Computational Reverse-Engineering Analysis for Scattering Experiments" (CREASE) method for analyzing scattering results from vesicle structures that are commonly found upon assembly of synthetic, biomimetic, or bioderived amphiphilic copolymers in solution. The two-step CREASE method takes the amphiphilic polymer chemistry and small-angle scattering intensity profile, I exp(q), as input and determines the vesicles' structural features on multiple length scales ranging from assembled vesicle wall's individual layer thicknesses to the monomer-level packing and distribution of polymer conformations. In the first step of CREASE, a genetic algorithm (GA) is used to determine the relevant vesicle dimensions from the input macromolecular solution information and I exp(q) by identifying the structure whose computed scattering profile best matches the input I exp(q). Then in the second step, the GA-determined dimensions are used for molecular reconstruction of the vesicle structure. To validate CREASE for vesicles, we test CREASE on input scattering intensity profiles generated mathematically (termed as in silico I exp(q) vs q) from a variety of vesicle sizes with known dimensions. We also test CREASE on in silico I exp(q) vs q generated from vesicles with dispersity in all relevant dimensions, resembling real experiments. After successful validation of CREASE, we compare the CREASE-determined dimensions against those obtained from the traditional approach of fitting the scattering intensity profile to relevant analytical model in SASVIEW package. We show that CREASE performs better than or as well as the core-multishell analytical model's fitting in SASVIEW in determining vesicle dimensions with dispersity. We also show that CREASE provides structural information beyond those possible from traditional scattering analysis using the core-multishell model, such as the distribution of solvophilic monomers between the vesicle wall's inner and outer layers in the vesicle wall and the chain-level packing within each vesicle layer.

Conformational and topological correlations in non-frustated triblock copolymers with homopolymers

The phase behavior of non-frustrated ABC block copolymers polymers, modeling poly(isoprene-b-styrene-b-ethylene oxide) (ISO), is studied using dissipative particle dynamic (DPD) simulations. The phase diagram showed a wide composition range for the alternating gyroid morphology, which can be transformed to a chiral metamaterial. A quantitative analysis of topology was developed, that correlates the location of a block relative to the interface with the block's end-to-end distance. This analysis showed that the A-blocks stretched as they were located deeper in the A-rich region. To further expand the stability of the alternating gyroid phase, 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 or equal to the A-block length filled the middle of the networks, decreasing packing frustration and stabilizing them, while longer homopolymers stretched across the network but allowed for the formation of stable, novel morphologies. Adding homopolymers to triblock copolymer melts increases tunability of the network, offering greater control over the final stable phase and bridging two separate regions in the phase diagram.

Networks with controlled chirality via self-assembly of chiral triblock terpolymers

Nanonetwork-structured materials can be found in nature and synthetic materials. A double gyroid (DG) with a pair of chiral networks but opposite chirality can be formed from the self-assembly of diblock copolymers. For triblock terpolymers, an alternating gyroid (G^A) with two chiral networks from distinct end blocks can be formed; however, the network chirality could be positive or negative arbitrarily, giving an achiral phase. Here, by taking advantage of chirality transfer at different length scales, G^A with controlled chirality can be achieved through the self-assembly of a chiral triblock terpolymer. With the homochiral evolution from monomer to multichain domain morphology through self-assembly, the triblock terpolymer composed of a chiral end block with a single-handed helical polymer chain gives the chiral network from the chiral end block having a particular handed network. Our real-space analyses reveal the preferred chiral sense of the network in the G^A, leading to a chiral phase.

Open-source code for self-consistent field theory calculations of block polymer phase behavior on graphics processing units

Self-consistent field theory (SCFT) is a powerful approach for computing the phase behavior of block polymers. We describe a fast version of the open-source Polymer Self-Consistent Field (PSCF) code that takes advantage of the massive parallelization provided by a graphical processing unit (GPU). Benchmarking double-precision calculations indicate up to 30× reduction in time to converge SCFT calculations of various diblock copolymer phases when compared to the Fortran CPU version of PSCF using the same algorithms, with the speed-up increasing with increasing unit cell size for the diblock polymer problems examined here. Where double-precision accuracy is not needed, single-precision calculations can provide speed-up of up to 60× in convergence time. These improvements in speed within an open-source format open up new vistas for SCFT-driven block polymer materials discovery by the community at large.

Mesochiral phases from the self-assembly of chiral block copolymers

Self-assembly of block copolymers with chiral sense gives mesochiral phases possessing helical sense. With the controlled chirality of the helical cylinder and chiral network, it is appealing to fabricate chiral materials for applications.

Diffusion in Lamellae, Cylinders, and Double Gyroid Block Copolymer Nanostructures

We study transport of penetrants through nanoscale morphologies motivated by common block copolymer morphologies, using confined random walks and coarse-grained simulations. Diffusion through randomly oriented grains is 1/3 for cylinder and 2/3 for lamellar morphologies versus an unconstrained (homopolymer) system, as previously understood. Diffusion in the double gyroid structure depends on the volume fraction and is 0.47-0.55 through the minority phase at 30-50 vol % and 0.73-0.80 through the majority at 50-70 vol %. Thus, among randomly oriented standard minority phase structures with no grain boundary effects, lamellae is preferable for transport.