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Local metrics of surface shape. (A) Schematic of the local curvature measurement from a 2D surface, shown here as a saddle-like surface, with discs highlighting the curvatures κ 1 and κ 2 measured in two perpendicular directions in the tangent plane of the surface. The orthogonal directions corresponding to the maximal and minimal values of curvature are known as principal curvature directions, and the corresponding curvatures are the principal curvatures of the surface at that point. (B) "Curvature space" of possible 2D surfaces in terms of the mean curvature and Gaussian curvature (eq 1). The dashed lines highlight characteristic classes of shape: spherical, cylindrical, and minimal saddles (H = 0 and negative K G ). No shapes exists in the gray parabolic region bound by spherical geometry.
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Block copolymer (BCP) melts are a paradigm for pluripotent molecular assembly, yielding a complex and expanding array of variable domain shapes and symmetries from a fairly simple and highly expandable class of molecular designs. This Perspective addresses recent advances in the ability to model and measure features of domain morphology that go bey...
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... can be shown that for every direction there is a special set of directions, i.e., a choice of v̂ and û , such that the curvatures κ v̂ and κ û are respectively maximal and minimal. Those directions are known as the principal curvature directions, and the curvatures, which we denote as κ 1 and κ 2 , are known as the principal curvatures (see Figure 2A). The principal curvatures and directions fully characterize the local shape of the surface, and these can be easily computed via a range of numerical techniques, given either an analytical representation of an IMDS shape or instead a discrete approximation, e.g., a triangulated mesh of the ϕ A (S d ) = ϕ B (S d ) = 1/2 isocontour from, for example, an experimental 3D reconstruction of the IMDS. ...
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... we take the intuitive definition such that H > 0 for cylinders and spheres. Examples of the local surface shape can be mapped on 2D plane spanned by H and K G , where standard surface shapes, like planes (H = K G = 0), cylinders (H > 0, K G = 0), and spheres (H 2 = K G ), can be visualized by locus of points in "curvature space" (see Figure 2B). Bicontinuous network shapes like DG and DD are characterized by regions of saddle-like surface shapes, corresponding to principal curvatures of opposite sign and K G < 0. Note that no surfaces exists where Gaussian curvature exceeds the spherical limit, i.e., ...
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... better visualize the complex curvatures of IMDS shapes for complex morphologies, and following the experimental analysis of Jinnai, Spontak, Hashimoto, and co-workers 77 (see discussion of IMDS shape characterization below), it is useful to analyze the distributions of mean and Gaussian curvatures plotted in the H and K G plane, as shown in Figure 2A. Notably, although curvature distributions for IMDSs from complex bicontinuous networks are necessarily variable, even in an ideal equilibrium structure, the two heuristic models of CMC vs CMT shapes, which represent optimal geometries for minimal IMDS areas vs uniform matrix thickness, provide a simple way to frame and interpret the curvature distributions. ...
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... length scales of the domain dimensions and domain periodicities in the range of tens to hundreds of nanometers, optical microscopy could only detect the presence of birefringence in the supposed hexagonal cylinder and lamellar systems (and its absence in the hypothesized cubic sphere systems) due to the optical anisotropy of the domains and presumably also in part to the anisotropic orientation of the chains within these domains. It was not until 1966 that Vanzo 102 working at Dow employed TEM of chromium shadowed silicon monoxide 2 stage poly(vinyl alcohol) surface replicas to demonstrate direct evidence of a lamellar structure in a high-MW polystyrene−polybutadiene (PS−PB) diblock (see images in Figure 12). However, since the technique used could only image the alternating layers at the sample surface and at unknown viewing angles, it was not possible to positively determine the individual layer thicknesses or the identity of a particular layer. ...
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... sampling is limited since a volume typically containing only tens to hundreds of unit cells is reconstructed. A comparison of different experimental techniques used to measure the supraand subunit cell structural information (curvature of the IMDS) to geometric models of DG (i.e., CMT vs CMC shapes) is discussed in the next section (see Figure 21). III.C.3. ...
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... filtered SVSEM data can be used to analyze subdomain morphological metrics of IMDS shape and domain thickness. Figure 20 shows analyses of the DG morphology obtained as described in ref 78. Notably, this method does not require any prior assumptions about the space group, and in particular, it was observed that the PS−PDMS DG studied in ref 78 exhibits a triclinic variant of the idealized cubic DG morphology, which ...
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... was attributed to stresses encountered during the nonequilibrium process of domain ordering from solution casting. Figure 20A shows the IMDS from multiple views of four different 3-valent nodal regions from the triclinic DG in Figure 20B. SVSEM reconstruction captures both the alteration of the periodicity at the unit cell repeats and local rearrangements of pubs.acs.org/Macromolecules ...
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... was attributed to stresses encountered during the nonequilibrium process of domain ordering from solution casting. Figure 20A shows the IMDS from multiple views of four different 3-valent nodal regions from the triclinic DG in Figure 20B. SVSEM reconstruction captures both the alteration of the periodicity at the unit cell repeats and local rearrangements of pubs.acs.org/Macromolecules ...
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... the perfect cubic DG, which contains two enantiomeric sets of equivalent nodal regions associated with the Wyckoff 16b positions, in the experimental triclinic DG nodal regions are no longer equivalent and exhibit distinct and distorted shapes. Medial thickness analysis ( Figures 20C,D) shows variation in block stretching in both PS and PDMS domains and spatial variations in curvature ( Figure 20E). While the gross patterns of negative curvature and local thickness are similar to the idealized cubic models in Figures 3 and 7, nodal regions of PS−PDMS DG exhibit spatial fluctuations of shape far from the ideal . ...
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... the perfect cubic DG, which contains two enantiomeric sets of equivalent nodal regions associated with the Wyckoff 16b positions, in the experimental triclinic DG nodal regions are no longer equivalent and exhibit distinct and distorted shapes. Medial thickness analysis ( Figures 20C,D) shows variation in block stretching in both PS and PDMS domains and spatial variations in curvature ( Figure 20E). While the gross patterns of negative curvature and local thickness are similar to the idealized cubic models in Figures 3 and 7, nodal regions of PS−PDMS DG exhibit spatial fluctuations of shape far from the ideal . ...
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... Figure 21, we compare the distributions of mean and Gaussian curvatures that can be measured from BCP DGs via two different tomographic techniques: TEMT in (A) and SVSEM in (B). Although there are differences in the experimental conditions and polymer chemistries for the two physical samples (i.e., the TEMT is from a OsO 4 stained, microtomed PS−PI−PS triblock, MW = 83 kg/mol with 0.32 volume fraction PI while the SVSEM sample is from a PS− PDMS with MW = 72.5 kg/mol having 0.40 volume fraction PDMS), normalization of the curvatures relative the unit cell parameters provides a useful basis of comparison for curvature measurement from two examples of nominally the same experimental DG morphology. ...
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... an increase in the small-angle scattering signals the onset of structure formation, little can be inferred about the details of the structure(s) present and how these entities combine and evolve with the extent of the transformation from homogeneous solution to micellar fluid to the long-range ordered (i.e., crystalline) solid. A useful way to appreciate the various possible structural pathways is to view the "block copolymer solution phase cube" 137 that features three orthogonal axes for the equilibrium temperature T (or χN) vs BCP composition f vs BCP volume fractions ϕ, in the binary solvent-BCP phase diagram, as shown in Figure 22. ...
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... second challenge derives from adapting the definition and associated key "degrees of freedom" with the subdomain volumes themselves to account for alterations of the molecular designs and corresponding assembled morphologies. In Figure 23, we show schematically the classes of subdomain "types" that have been explored for distinct BCP architectures thus far. The primary focus of this Perspective has been on the simplest class of subdomains, which extend in straight trajectories from one "inner" terminal boundary through the IMDS(s) to an "outer" terminal boundary. ...
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... details associated with variation along the subdomain, we note that introducing multiblock architectures, such as ABC terpolymers, may lead to morphologies with "nonsimple" subdomains. For example, as shown in Figure 23, alternating domains morphology of ABC triblocks have been modeled by using "compound subdomains", 58 while Archimedean tiling phases (and even so-called "tricontinuous" networks 148 ) of ABC stars have been described by "vertex associated" subdomains, 149−151 which junctions localized to lines of three-phase contact. These examples illustrate that notions of subdomain shapes and thicknesses are inextricably linked to constraints imposed by the chain connectivity, interdomain contact, and constant density of the melt. ...
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... The same group employed the medial strong segregation theory to reevaluate the thermodynamics of cubic network formation in strongly segregated diblock melts, relating chain packing environments to the medial geometry of tubular network surfaces [96,97]. Medial packing motifs are crucial for stabilizing complex morphologies with high-packing frustration, such as DD and DP structures. ...
The exploration of nanoscale morphologies with three-dimensional spatial arrangements in block copolymers involves the manipulation of compositional fluctuations at interfaces, induction of conformational asymmetry, and design of complex architectures. Despite the abundance of triply periodic minimal surfaces identified to date, the experimental demonstration of thermodynamically stable complex nanostructures with high-packing frustration remains limited. This research update focuses on the importance of molecular interactions for stabilizing complex nanostructures and proposes the use of end-group chemistry as a versatile method for realizing thermodynamically stable network structures with high-packing frustration in simple linear diblock copolymers. The proposed approach substantially alters phase diagrams and reveals unprecedented network structures in block copolymers. Ongoing research has the potential to generate even more diverse and stable nanostructures, thereby advancing nanotechnology and expanding our understanding of polymers and materials science.
... The main bicontinuous network phase is double-gyroid (DG), of which each node is branched with three struts [16][17][18][19]. The DG phase has more uniform interfacial curvature or smaller variation of domain thickness than the other competing network phases, such as double-diamond (DD) and double-primitive (DP) [18,20]. The fundamental knowledge achieved from the self-assembly of AB diblock copolymers serves as the foundation for understanding the self-assembly behavior of more complex systems. ...
Block copolymers (BCPs) provide a versatile platform for the formation of various nanostructures. Among them, ABC-type BCPs have great potential to form complex structures, which may have promising applications in nanotechnology. However, the self-assembly behavior of ABC-type BCPs is still not well understood because of its complexity, especially about the formation of complex structures that are difficult to be assumed from AB-type BCPs. In this paper, we propose a useful rule to systematically assume ordered structures possibly formed by ABC-type BCPs. We hypothetically change an AB-type BCP to an ABC-type architecture by replacing the ending portion of each free B block with a C block, to see how the C blocks separate from the ordered structure preformed by the AB-type BCP. We propose that the C domains firstly aggregate at the vertices of the Voronoi cell of the precursory AB-type structures to form spheres, then along the edges to form struts, and finally on the faces to form layers. Accordingly, we have obtained a large number of ABC-type structures, some of which are very complex and have not been reported before. The validity of the assumption rule is testified by using self-consistent field theory to determine the stability of some assumed structures. This rule can serve as a useful guide for the exploration of interesting novel structures in ABC-type BCPs.
... Through the directed self- assembly of such block copolymers, tailored and nanoscopically ordered morphologies, like lamellae, cylinders, spheres and bicontinuously interconnected gyroids can be produced ( Fig. 10) [79]. A large amount of research to date has shown that there is a strong correlation between the mechanical properties of these block copolymers and the unique and differing morphologies that result from differences in volume fraction, geometries, and topologies of the domains of these block copolymers, which vary depending on the composition, molecular weight, and block number [80]. ...
... One of them is the bicontinuous, double diamond (DD) phase (see Fig. 1(b)) which forms via thermodynamically driven microphase separation taking place in block copolymer systems. 4,5 Recently, Chang et al. 6 reported that controlled annealing protocols enable sampling of metastable phases and observation of order-order transitions from double primitive to double diamond and then to double gyroid (DG) for simple diblock copolymers. Interestingly, both diamond and gyroid structures (see animation 1 and 2 in the ESI †) were also discovered in naturally developed scaffolds, rst in the case of the exoskeleton of the Lamprocyphus augustus beetle 7 and the second in the case of the Lycaenid buttery's wing structure. ...
There are a number of exceptional examples indicating the unique position of tetrahedral symmetry in the vast landscape of different spatial organization pathways which can be sampled by matter. This work shows that the design and analysis of relatively simple tetrahedron clusters can lead to the formulation of a new type of dendritic structure together with unique periodic frameworks resembling clathrates and foams. A simple sequential protocol leading from regular tetrahedron clusters to more complex structural motifs can be employed to determine interesting repetitive building units. Accordingly, four different hierarchical superstructures are introduced, in which the dominant population of nodes is based on tetrahedral symmetry. The introduced architectures could be of particular interest for the field of regenerative medicine and metamaterial engineering.
... Nonetheless, a few examples include a detailed study of different M w BCP thin films self-assembly after SIS, 38 membrane pore structure analysis, 115 and in-depth understanding of bulk assembly for complex structures like double gyroid 30,109,113,114 and double diamond, 163 as well as measurement of such structures' interfacial curvature 23,164−166 through both measurement and simulation, as recently reviewed by Thomas and Grason. 167 ...
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.
... 22 However, as resolution of SCFT algorithms improved sufficiently to address very strong segregation regimes, calculations showed that DG phase retains a finite, albeit narrowing window of equilibrium up to at least χN ≳ 100, 23 which is also consistent with experimental studies of this regime. 24 These original SST models of chain packing in network phases are based on what might be called a "skeletal ansatz," which assumes that the 1D skeletal graph that connects the nodal centers of the tubular network domains constitutes what has been dubbed a terminal boundary, 25 i.e. the region of maximal extension of chain trajectories away from the IMDS within a brush-like subdomain composed of one polymer block. While the skeletal ansatz is a seemingly intuitive proxy for the tubular network regions, 26 it has recently been understood that this approximation severely overestimates the entropic penalty of stretching in those domains. ...
... In short, the medial map provides the shortest-distance map of points within a volume onto points on its bounding surface as well as a corresponding medial set, which is the set of maximally-distant points at the general "center" of a domain of arbitrary shape. 25,28 Importantly, we showed that the stability of the DG phase relies on the ability of chain ends to spread out over a web-like surface (shown in Fig. 1) in each tubular domain, rather than being forced to stretch to the 1D skeletal graph, thus lowering the entropic penalty for tubular-domain filling to the point where the free-energy gap between competitor morphologies is eliminated. As a result, we predicted DG stability windows, between Hex and Lam phases for diblocks, that open up and widen as the conformational asymmetry between blocks is increased. ...
... On one hand, packing frustration is associated with the variability of BCP packing and the resulting entropic and enthalpic costs. 7,12,25,32 Alternatively, packing frustration is often connected to certain locations in a morphology that can be identified as the "sources" of frustration. Such a perspective focuses on where is frustration coming from in the packing, and in turn, how this relates to especially costly "hot spots" in the resulting morphology. ...
Self-consistent field theory (SCFT) has established that for cubic network phases in diblock copolymer melts, the double-gyroid (DG) is thermodynamically stable relative to the competitor double-diamond (DD) and double-primitive (DP) phases, and exhibits a window of stability intermediate to the classical lamellar and columnar phases. This competition is widely thought to be controlled by "packing frustration" -- the incompatibility of uniformly filling melts with a locally preferred chain packing motif. Here, we reassess the thermodynamics of cubic network formation in strongly-segregated diblock melts, based on a recently developed medial strong segregation theory ("mSST") approach that directly connects the shape and thermodynamics of chain packing environments to the medial geometry of tubular network surfaces. We first show that medial packing significantly relaxes prior SST upper bounds on the free energy of network phases, which we attribute to the spreading of terminal chain ends within network nodal regions. Exploring geometric and thermodynamic metrics of chain packing in network phases, we show that mSST reproduces effects dependent on the elastic asymmetry of the blocks that are consistent with SCFT at large $\chi N$. We then characterize geometric frustration in terms of the spatially-variant distributions of local entropic and enthalpic costs throughout the morphologies, extracted from mSST predictions. We find that the DG morphology, due to its unique medial geometry in the nodal regions, is stabilized by the incorporation of favorable, quasi-lamellar packing over much of its morphology, motifs which are inaccessible to DD and DP morphologies due to "interior corners" in their medial geometries. Finally, we use our results to analyze "hot spots" of chain stretching and discuss implications for network susceptibility to the uptake of guest molecules.
... However, the identification of networks formed by the self-assembly of block copolymers remains challenging. Unlike their natural counterparts, for example in butterflies 21 , they often exhibit unique symmetrybreaking distortions 22 that are difficult to assess using conventional characterisation techniques such as X-ray scattering 23 . The situation is exacerbated by the transition from diblock copolymer to triblock terpolymerderived materials, as reported herein, which exhibit even more complex phase behaviour, including multiple network structures 24 . ...
... For the unambiguous morphological assignment of 3D network structures, tomographic imaging and 3D reconstruction of the structure are essential. Established 3D imaging using TEM tomography typically produces volumes of 10s to 100s of unit cells, allowing quantitative structural analysis 29 , while more recent FIB-SEM tomography extends this to 1000s of unit cells 22,23 . The ptychographic X-ray computed tomography (PXCT) 3,4,12 reported here allows imaging of about 70,000 unit cells of a 3D network structure (as detailed below) with an estimated resolution of approximately 11 nm, i.e. substantially larger sample volumes than with any previous technique. ...
Block copolymers are recognised as a valuable platform for creating nanostructured materials with unique properties. Morphologies formed by block copolymer self-assembly can be transferred into a wide range of inorganic materials, enabling applications including energy storage and metamaterials. However, imaging of the underlying, often complex, nanostructures in large volumes has remained a challenge, limiting progress in materials development. Taking advantage of recent advances in X-ray nanotomography, we non-invasively imaged exceptionally large volumes of nanostructured soft materials at high resolution, revealing a single diamond morphology in a triblock terpolymer composite network. This morphology, which is ubiquitous in nature, has so far remained elusive in block copolymers, despite its potential to create materials with large photonic bandgaps. The discovery was made possible by the precise analysis of distortions in a large volume of the self-assembled diamond network, which are difficult to unambiguously assess using traditional characterisation tools. We anticipate that high-resolution X-ray nanotomography, which allows imaging of much larger sample volumes than electron-based tomography, will become a powerful tool for the quantitative analysis of complex nanostructures and that structures such as the triblock terpolymer-directed single diamond will enable the generation of advanced multicomponent composites with hitherto unknown property profiles.
... Notably, the first two of these three descriptors can be measured by careful experimentation (except for more realistic deviations from idealized crystallographic order), while the last is currently essentially invisible to experimental characterization. Hence, the expected chain trajectories rely on information from computational models as well as heuristic assumptions about likely chain trajectories (as discussed in detail in Ref. [25]). To begin with, we clarify our objective in terms of the arguably much more intuitive case of crystals of spherical or cylindrical domains. ...
... As we describe below, the notion of mesoatoms is intimately connected to the concept of a domain in the BCP melts, following the topological definitions introduced in Ref. [25]. Simply put, a domain corresponds to the volumes occupied by chains that have their junctions located on or associated with a particular IMDS. ...
... Colloquially, the terminal boundary can be thought of as the contact surface between two opposing brushlike subdomains, each of which stems from the IMDS of distinct domains. Given this notion, it has been proposed that the terminal boundary is well approximated by medial sets of the IMDSs [25,31,32], which are loci of midpoints within a region of the A or B component between distinct IMDS regions. For the DG network, the outer terminal surface that separates the two networks closely approximates the gyroid minimal surface, and the corresponding boundary for DD and DP closely match the Schwarz D and P minimal surfaces, respectively [33]. ...
We propose and describe a framework to understand 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 spherelike or cylinderlike (micellelike) domains, analogous notions of natural and 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, we propose generic rules for decomposing TPN crystals into a unique set of mesoatomic building blocks. Based on physically motivated principles, the combination of 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 spherelike and cylinderlike mesoatoms. We analyze the shapes of these mesoatoms, their internal structure, and importantly, their local packing with neighbor mesoatomic units. Importantly, we hypothesize that mesoatoms are kinetically favored intermediate structures whose local shapes and packing template network crystal assembly on long time scales. We propose and study a minimal energetic model of mesoatom assembly for three different cubic double-network crystals, based on local shape packing, which predicts a detailed picture for kinetics of intercatenation and surface growth. Based on these analyses, we discuss several possible extensions and elaborations of the mesoatomic description of supramolecular soft matter network crystals, most notably the implications of mesoatomic malleability, a feature that distinguishes soft matter from hard matter crystals. We describe experimental observations of malleable mesoatomic units in the precursor sponge phase as well as in ordered cubic networks and suggest possibilities for observing mesoatoms in primordial, precrystalline states.
... In this article we describe the mechanisms of chirality transfer to triply periodic network phases of soft, selfassembling molecules, focusing on a particular example of block copolymer (BCP) melts [16]. On one hand, BCPs provide a prototypical system for understanding self-assembly in a much broader class of supramolecular systems [17,18]. Beyond this, the chemical versatility of BCPs [19,20] enables numerous routes to engineering nanostructured materials with controlled functions that rely on their complex self-assembled structures. ...
... The tubular domains -of distinct chemical composition from the matrix region -connect in multi-valent junctions (e.g. 3-and 4-valent connections), with each single constituent network interlinking with its partner to form an intercatened double network [18,26]. The unique combination of nanoscopic dimensions, polycontinuous topology, and triply periodic symmetries of selfassembled double networks imbue them with remarkable functionality [27], from photonic architectures appearing in birds, beetles, and butterflies [28,29], to hybrid plasmonic or photonic metamaterials [30,31]. ...
Chirality transfer from the level of molecular structure up to mesoscopic lengthscales of supramolecular morphologies is a broad and persistent theme in self-assembled soft materials, from biological to synthetic matter. Here, we analyze the mechanism of chirality transfer in a prototypical self-assembly system, block copolymers (BCPs), in particular, its impact on one of the most complex and functionally vital phases: the cubic, triply-periodic, gyroid network. Motivated by recent experimental studies, we consider a self-consistent field model of ABC* triblock copolymers possessing an end-block of chain chemistry and examine the interplay between chirality at the scale of networks, in alternating double network phases, and the patterns of segmental order within tubular network domains. We show that while segments in gyroids exhibit twist in both polar and nematic segmental order parameters, the magnitude of net nematic twist is generically much larger than polar twist, and more surprising, {\it reverses handedness} relative to the sense of polar order as well as the sense of dihedral twist of the network. Careful analysis of the intra-domain nematic order reveals that this unique chirality transfer mechanism relies on the strongly biaxial nature of segmental order in BCP networks and relates the {\it biaxial twist} to complex patterns of frame rotation of the principal directors in the intra-domain texture. Finally, we show that this mechanism of twist reversal leads to chirality selection of alternating gyroid networks in ABC* triblocks, in the limit of very weak chirality .
... While there are strict requirements that both blocks are required to fill space as a melt, there are only mild restrictions with the localization of the A block-B block junction at the IMDS. Here, a tetrapodal minority network subdomain, surrounded by the outer majority matrix subdomain, forms a basic motif in the DD networks, designated as a "mesoatom" (SI Appendix, Fig. S1) (16). Due to the relatively weak physical interactions between the molecules, block copolymer "mesoatoms" as well as their associated microdomain IMDS shapes are much more malleable than their atomic counterparts and can adjust their dimensions and shapes in response to the presence of defects or other sources of stress (17,18). ...
A twin boundary (TB) is a common low energy planar defect in crystals including those with the atomic diamond structure (C, Si, Ge, etc.). We study twins in a self-assembled soft matter block copolymer (BCP) supramolecular crystal having the double diamond (DD) structure, consisting of two translationally shifted, interpenetrating diamond networks of the minority polydimethyl siloxane block embedded in a polystyrene block matrix. The coherent, low energy, mirror-symmetric double tubular network twin has one minority block network with its nodes offset from the (222) TB plane, while nodes of the second network lie in the plane of the boundary. The offset network, although at a scale about a factor of 103 larger, has precisely the same geometry and symmetry as a (111) twin in atomic single diamond where the tetrahedral units spanning the TB retain nearly the same strut (bond) lengths and strut (bond) angles as in the normal unit cell. In DD, the second network undergoes a dramatic restructuring-the tetrahedral nodes transform into two new types of mirror-symmetric nodes (pentahedral and trihedral) which alternate and link to form a hexagonal mesh in the plane of the TB. The collective reorganization of the supramolecular packing highlights the hierarchical structure of ordered BCP phases and emphasizes the remarkable malleability of soft matter.
