Mesoscale polymer arrays: high aspect ratio surface structures and their digital reconstruction.

Inspired by adhesive bio-filamentous structure, such as bacterial pili, this work details the methods used to fabricate and characterize a surface-anchored array of thin, flexible and shape-responsive mesoscale polymer ribbons with a length-to-thickness aspect ratio of up to 100 000. The resulting structures exhibit geometrically complex and dynamic morphologies consistent with elastocapillary bending that experience an increase in curvature over hours of aging due to creep. We develop a computational image analysis framework to generate 3D reconstructions of these densely crowded geometries and extract quantitative descriptors to demonstrate morphological changes due to aging. We demonstrate the robustness of this quantitative method by characterizing the creep-induced change in an aging ribbon array's shape and develop a scaling relationship to describe the importance of ribbon thickness for shape and dynamical observations. These methods demonstrate an essential baseline to probe morphology-property relationships of mesoscale polymer ribbon arrays fabricated from a variety of materials in numerous environments. Through the introduction of perfluorodecalin droplets, we illustrate the potential of these ribbon arrays towards applications in adhesive, microrobotic, and biomedical devices.

Thermal preconditioning of membrane stress to control the shapes of ultrathin crystals.

We employ the phospholipid bilayer membranes of giant unilamellar vesicles as a free-standing environment for the growth of membrane-integrated ultrathin phospholipid crystals possessing a variety of shapes with 6-fold symmetry. Crystal growth within vesicle membranes, where more elaborate shapes grow on larger vesicles is dominated by the bending energy of the membrane itself, creating a means to manipulate crystal morphology. Here we demonstrate how cooling rate preconditions the membrane tension before nucleation, in turn regulating nucleation and growth, and directing the morphology of crystals by the time they are large enough to be visualized. The crystals retain their shapes during further growth through the two phase region. Experiments demonstrate this behavior for single crystals growing within the membrane of each vesicle, ultimately comprising up to 13% of the vesicle area and length scales of up to 50 microns. A model for stress evolution, employing only physical property data, reveals how the competition between thermal membrane contraction and water diffusion from tensed vesicles produces a size- and time-dependence of the membrane tension as a result of cooling history. The tension, critical in the contribution of bending energy in the fluid membrane regions, in turn selects for crystal shape for vesicles of a given size. The model reveals unanticipated behaviors including a low steady state tension on small vesicles that allows compact domains to develop, rapid tension development on large vesicles producing flower-shaped domains, and a stress relaxation through water diffusion across the membrane with a time constant scaling as the square of the vesicle radius, consistent with measurable tensions only in the largest vesicles.

Shape equilibria of vesicles with rigid planar inclusions.

Motivated by recent studies of two-phase lipid vesicles possessing 2D solid domains integrated within a fluid bilayer phase, we study the shape equilibria of closed vesicles possessing a single planar, circular inclusion. While 2D solid elasticity tends to expel Gaussian curvature, topology requires closed vesicles to maintain an average, non-zero Gaussian curvature leading to an elementary mechanism of shape frustration that increases with inclusion size. We study elastic ground states of the Helfrich model of the fluid-planar composite vesicles, analytically and computationally, as a function of planar fraction and reduced volume. Notably, we show that incorporation of a planar inclusion of only a few percent dramatically shifts the ground state shapes of vesicles from predominantly prolate to oblate, and moreover, shifts the optimal surface-to-volume ratio far from spherical shapes. We show that for sufficiently small planar inclusions, the elastic ground states break symmetry via a complex variety of asymmetric oblate, prolate, and triaxial shapes, while inclusion sizes above about 8% drive composite vesicles to adopt axisymmetric oblate shapes. These predictions cast useful light on the emergent shape and mechanical responses of fluid-solid composite vesicles.

Economical routes to size-specific assembly of self-closing structures.

Programmable self-assembly has seen an explosion in the diversity of synthetic crystalline materials, but developing strategies that target "self-limiting" assemblies has remained a challenge. Among these, self-closing structures, in which the local curvature defines the finite global size, are prone to polymorphism due to thermal bending fluctuations, a problem that worsens with increasing target size. Here, we show that assembly complexity can be used to eliminate this source of polymorphism in the assembly of tubules. Using many distinct components, we prune the local density of off-target geometries, increasing the selectivity of the tubule width and helicity to nearly 100%. We further show that by reducing the design constraints to target either the pitch or the width alone, fewer components are needed to reach complete selectivity. Combining experiments with theory, we reveal an economical limit, which determines the minimum number of components required to create arbitrary assembly sizes with full selectivity.

Symmetry-Guided Inverse Design of Self-Assembling Multiscale DNA Origami Tilings.

Recent advances enable the creation of nanoscale building blocks with complex geometries and interaction specificities for self-assembly. This nearly boundless design space necessitates design principles for defining the mutual interactions between multiple particle species to target a user-specified complex structure or pattern. In this article, we develop a symmetry-based method to generate the interaction matrices that specify the assembly of two-dimensional tilings, which we illustrate using equilateral triangles. By exploiting the allowed 2D symmetries, we develop an algorithmic approach by which any periodic 2D tiling can be generated from an arbitrarily large number of subunit species, notably addressing an unmet challenge of engineering 2D crystals with periodicities that can be arbitrarily larger than the subunit size. To demonstrate the utility of our design approach, we encode specific interactions between triangular subunits synthesized by DNA origami and show that we can guide their self-assembly into tilings with a wide variety of symmetries, using up to 12 unique species of triangles. By conjugating specific triangles with gold nanoparticles, we fabricate gold-nanoparticle supracrystals whose lattice parameter spans up to 300 nm. Finally, to generate economical design rules, we compare the design economy of various tilings. In particular, we show that (1) higher symmetries allow assembly of larger unit cells with fewer subunits and (2) linear supracrystals can be designed more economically using linear primitive unit cells. This work provides a simple algorithmic approach to designing periodic assemblies, aiding in the multiscale assembly of supracrystals of nanostructured "meta-atoms" with engineered plasmonic functions.

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.

Limits of economy and fidelity for programmable assembly of size-controlled triply periodic polyhedra.

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 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.

Flower-shaped 2D crystals grown in curved fluid vesicle membranes.

The morphologies of two-dimensional (2D) crystals, nucleated, grown, and integrated within 2D elastic fluids, for instance in giant vesicle membranes, are dictated by an interplay of mechanics, permeability, and thermal contraction. Mitigation of solid strain drives the formation of crystals with vanishing Gaussian curvature (i.e., developable domain shapes) and, correspondingly, enhanced Gaussian curvature in the surrounding 2D fluid. However, upon cooling to grow the crystals, large vesicles sustain greater inflation and tension because their small area-to-volume ratio slows water permeation. As a result, more elaborate shapes, for instance, flowers with bendable but inextensible petals, form on large vesicles despite their more gradual curvature, while small vesicles harbor compact planar crystals. This size dependence runs counter to the known cumulative growth of strain energy of 2D colloidal crystals on rigid spherical templates. This interplay of intra-membrane mechanics and processing points to the scalable production of flexible molecular crystals of controllable complex shape.

Reimagining Hair Science: A New Approach to Classify Curly Hair Phenotypes via New Quantitative Geometric and Structural Mechanical Parameters.

Hair is a natural polymeric composite primarily composed of tight macrobundles of keratin proteins, which are highly responsive to external stimuli, similarly to the hydrogels and other natural fibrous gel systems like collagen and fibrin.Hair and its appearance play a significant role in human society. As a highly complex biocomposite system, it has been traditionally challenging to characterize and thus develop personal care products. Over the last few decades, a significant societal paradigm shift occurred among those with curly hair, accepting the natural morphological shape of their curls and styling their hair according to its innate, distinct, and unique material properties, which has given rise to the development of new hair classification systems, beyond the traditional and highly limited race-based distinction (Caucasian, Mongolian, and African). L'Oréal developed a hair typing taxonomy based on quantitative geometric parameters among the four key patterns─straight, wavy, curly, and kinky, but it fails to capture the complex diversity of curly and kinky hair. Acclaimed celebrity hair stylist Andre Walker developed a classification system that is the existing gold standard for classifying curly and kinky hair, but it relies upon qualitative classification measures, making the system vague and ambiguous of phenotypic differences. The goal of this research is to use quantitative methods to identify new geometric parameters more representative of curly and kinky hair curl patterns, therefore providing more information on the kinds of personal care products that will resonate best with them and thus maximize desired appearance and health, and to correlate these new parameters with its mechanical properties. This was accomplished by identifying new geometric and mechanical parameters from several types of human hair samples.Geometric properties were measured using scanning electron microscopy (SEM), photogrammetry, and optical microscopy. Mechanical properties were measured under tensile extension using a texture analyzer (TA) and a dynamic mechanical analyzer (DMA), which bears similarity to the common act of brushing or combing. Both instruments measure force as a function of applied displacement, thus allowing the relationship between stress and applied stretch ratio to be measured as a hair strand uncurls and stretches to the point of fracture. From the resulting data, correlations were made between fiber geometry and mechanical performance. This data will be used to draw more conclusions on the contribution that fiber morphology has on hair fiber mechanics and will promote cultural inclusion among researchers and consumers possessing curly and kinky hair.

Self-spinning filaments for autonomously linked microfibers.

Filamentous bundles are ubiquitous in Nature, achieving highly adaptive functions and structural integrity from assembly of diverse mesoscale supramolecular elements. Engineering routes to synthetic, topologically integrated analogs demands precisely coordinated control of multiple filaments' shapes and positions, a major challenge when performed without complex machinery or labor-intensive processing. Here, we demonstrate a photocreasing design that encodes local curvature and twist into mesoscale polymer filaments, enabling their programmed transformation into target 3-dimensional geometries. Importantly, patterned photocreasing of filament arrays drives autonomous spinning to form linked filament bundles that are highly entangled and structurally robust. In individual filaments, photocreases unlock paths to arbitrary, 3-dimensional curves in space. Collectively, photocrease-mediated bundling establishes a transformative paradigm enabling smart, self-assembled mesostructures that mimic performance-differentiating structures in Nature (e.g., tendon and muscle fiber) and the macro-engineered world (e.g., rope).

Building blocks of non-Euclidean ribbons: size-controlled self-assembly via discrete frustrated particles.

Geometric frustration offers a pathway to soft matter self-assembly with controllable finite sizes. While the understanding of frustration in soft matter assembly derives almost exclusively from continuum elastic descriptions, a current challenge is to understand the connection between microscopic physical properties of misfitting "building blocks" and emergent assembly behavior at the mesoscale. We present and analyze a particle-based description of what is arguably the best studied example for frustrated soft matter assembly, negative-curvature ribbon assembly, observed in both assemblies of chiral surfactants and shape-frustrated nanoparticles. Based on our particle model, known as saddle wedge monomers, we numerically test the connection between microscopic shape and interactions of the misfitting subunits and the emergent behavior at the supra-particle scale, specifically focussing on the propagation and relaxation of inter-particle strains, the emergent role of extrinsic shape on frustrated ribbons and the equilibrium regime of finite width selection. Beyond the intuitive role of shape misfit, we show that self-limitation is critically dependent on the finite range of cohesive interactions, with larger size finite assemblies requiring increasing short-range interparticle forces. Additionally, we demonstrate that non-linearities arising from discrete particle interactions alter self-limiting behavior due to both strain-softening in shape-flattened assembly and partial yielding of highly strained bonds, which in turn may give rise to states of hierarchical, multidomain assembly. Tracing the regimes of frustration-limited assembly to the specific microscopic features of misfitting particle shapes and interactions provides necessary guidance for translating the theory of size-programmable assembly into design of intentionally-frustrated colloidal particles.

Geometrically programmed self-limited assembly of tubules using DNA origami colloids.

Self-assembly is one of the most promising strategies for making functional materials at the nanoscale, yet new design principles for making self-limiting architectures, rather than spatially unlimited periodic lattice structures, are needed. To address this challenge, we explore the tradeoffs between addressable assembly and self-closing assembly of a specific class of self-limiting structures: cylindrical tubules. We make triangular subunits using DNA origami that have specific, valence-limited interactions and designed binding angles, and we study their assembly into tubules that have a self-limited width that is much larger than the size of an individual subunit. In the simplest case, the tubules are assembled from a single component by geometrically programming the dihedral angles between neighboring subunits. We show that the tubules can reach many micrometers in length and that their average width can be prescribed through the dihedral angles. We find that there is a distribution in the width and the chirality of the tubules, which we rationalize by developing a model that considers the finite bending rigidity of the assembled structure as well as the mechanism of self-closure. Finally, we demonstrate that the distributions of tubules can be further sculpted by increasing the number of subunit species, thereby increasing the assembly complexity, and demonstrate that using two subunit species successfully reduces the number of available end states by half. These results help to shed light on the roles of assembly complexity and geometry in self-limited assembly and could be extended to other self-limiting architectures, such as shells, toroids, or triply periodic frameworks.

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.

Thermodynamic Size Control in Curvature-Frustrated Tubules: Self-Limitation with Open Boundaries.

We use computational modeling to investigate the assembly thermodynamics of a particle-based model for geometrically frustrated assembly, in which the local packing geometry of subunits is incompatible with uniform, strain-free large-scale assembly. The model considers discrete triangular subunits that drive assembly toward a closed, hexagonal-ordered tubule, but have geometries that locally favor negative Gaussian curvature. We use dynamical Monte Carlo simulations and enhanced sampling methods to compute the free energy landscape and corresponding self-assembly behavior as a function of experimentally accessible parameters that control assembly driving forces and the magnitude of frustration. The results determine the parameter range where finite-temperature self-limiting assembly occurs, in which the equilibrium assembly size distribution is sharply peaked around a well-defined finite size. The simulations also identify two mechanisms by which the system can escape frustration and assemble to unlimited size, and determine the particle-scale properties of subunits that suppress unbounded growth.

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.

Mechanics of Metric Frustration in Contorted Filament Bundles: From Local Symmetry to Columnar Elasticity.

Bundles of filaments are subject to geometric frustration: certain deformations (e.g., bending while twisted) require longitudinal variations in spacing between filaments. While bundles are common-from protein fibers to yarns-the mechanical consequences of longitudinal frustration are unknown. We derive a geometrically nonlinear formalism for bundle mechanics, using a gaugelike symmetry under reptations along filament backbones. We relate force balance to orientational geometry and assess the elastic cost of frustration in twisted-toroidal bundles.

End-exclusion zones in strongly stretched, molten polymer brushes of arbitrary shape.

Theories of strongly stretched polymer brushes, particularly the parabolic brush theory, are valuable for providing analytically tractable predictions for the thermodynamic behavior of surface-grafted polymers in a wide range of settings. However, the parabolic brush limit fails to describe polymers grafted to convex curved substrates, such as the surfaces of spherical nanoparticles or the interfaces of strongly segregated block copolymers. It has previously been shown that strongly stretched curved brushes require a boundary layer devoid of free chain ends, requiring modifications of the theoretical analysis. While this "end-exclusion zone" has been successfully incorporated into the descriptions of brushes grafted onto the outer surfaces of cylinders and spheres, the behavior of brushes on surfaces of arbitrary curvature has not yet been studied. We present a formulation of the strong-stretching theory for molten brushes on the surfaces of arbitrary curvature and identify four distinct regimes of interest for which brushes are predicted to possess end-exclusion zones, notably including regimes of positive mean curvature but negative Gaussian curvature. Through numerical solutions of the strong-stretching brush equations, we report predicted scaling of the size of the end-exclusion zone, the chain end distribution, the chain polarization, and the free energy of stretching with mean and Gaussian surface curvatures. Through these results, we present a comprehensive picture of how the brush geometry influences the end-exclusion zones and exact strong-stretching free energies, which can be applied, for example, to model the full spectrum of brush geometries encountered in block copolymer melt assembly.

Defects in conformal crystals: Discrete versus continuous disclination models.

We study the relationship between topological defect formation and ground-state 2D packings in a model of repulsions in external confining potentials. Specifically we consider screened 2D Coulombic repulsions, which conveniently parameterizes the effects of interaction range, but also serves as simple physical model of confined, parallel arrays of polyelectrolyte filaments or vortices in type II superconductors. The countervailing tendencies of repulsions and confinement to, respectively, spread and concentrate particle density leads to an energetic preference for nonuniform densities in the clusters. Ground states in such systems have previously been modeled as conformal crystals, which are composed of locally equitriangular packings whose local areal densities exhibit long-range gradients. Here we assess two theoretical models that connect the preference for nonuniform density to the formation of disclination defects, one of which assumes a continuum distributions of defects, while the second considers the quantized and localized nature of disclinations in hexagonal conformal crystals. Comparing both theoretical descriptions to numerical simulations of discrete particles clusters, we study the influence of interaction range and confining potential on the topological charge, number, and distribution of defects in ground states. We show that treating disclinations as continuously distributable well captures the number of topological defects in the ground state in the regime of long-range interactions, while as interactions become shorter range, it dramatically overpredicts the growth in total defect charge. Detailed analysis of the discretized defect theory suggests that that failure of the continuous defect theory in this limit can be attributed to the asymmetry in the preferred placement of positive vs negative disclinations in the conformal crystal ground states, as well as a strongly asymmetric dependence of self-energy of disclinations on sign of topological charge.

Switchable positioning of plate-like inclusions in lipid membranes: Elastically mediated interactions of planar colloids in 2D fluids.

We demonstrate how manipulating curvature in an elastic fluid lamella enables the reversible relative positioning of flat, rigid, plate-like micrometer-scale inclusions, with spacings from about a micrometer to tens of micrometers. In an experimental model comprising giant unilamellar vesicles containing solid domain pairs coexisting in a fluid membrane, we adjusted vesicle inflation to manipulate membrane curvature and mapped the interdomain separation. A two-dimensional model of the pair potential predicts the salient experimental observations and reveals both attractions and repulsions, producing a potential minimum entirely a result of the solid domain rigidity and bending energy in the fluid membrane. The impact of vesicle inflation on domain separation in vesicles containing two solid domains was qualitatively consistent with observations in vesicles containing many domains. The behavior differs qualitatively from the pure repulsions between fluid membrane domains or interactions between nanoscopic inclusions whose repulsive or attractive character is not switchable.

Equilibrium mechanisms of self-limiting assembly.

Self-assembly is a ubiquitous process in synthetic and biological systems, broadly defined as the spontaneous organization of multiple subunits (e.g. macromolecules, particles) into ordered multi-unit structures. The vast majority of equilibrium assembly processes give rise to two states: one consisting of dispersed disassociated subunits, and the other, a bulk-condensed state of unlimited size. This review focuses on the more specialized class of self-limiting assembly, which describes equilibrium assembly processes resulting in finite-size structures. These systems pose a generic and basic question, how do thermodynamic processes involving non-covalent interactions between identical subunits "measure" and select the size of assembled structures? In this review, we begin with an introduction to the basic statistical mechanical framework for assembly thermodynamics, and use this to highlight the key physical ingredients that ensure equilibrium assembly will terminate at finite dimensions. Then, we introduce examples of self-limiting assembly systems, and classify them within this framework based on two broad categories: self-closing assemblies and open-boundary assemblies. These include well-known cases in biology and synthetic soft matter - micellization of amphiphiles and shell/tubule formation of tapered subunits - as well as less widely known classes of assemblies, such as short-range attractive/long-range repulsive systems and geometrically-frustrated assemblies. For each of these self-limiting mechanisms, we describe the physical mechanisms that select equilibrium assembly size, as well as potential limitations of finite-size selection. Finally, we discuss alternative mechanisms for finite-size assemblies, and draw contrasts with the size-control that these can achieve relative to self-limitation in equilibrium, single-species assemblies.