arXiv subjects

Gregory M. Grason

Publications and source records attributed to Gregory M. Grason.

At least 55 records · Page 3Linked to original sources

arXiv2021

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 {\it 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: {\it self-closing assemblies} and {\it 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.

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arXiv2019

Chiral and achiral mechanisms of self-limiting, twisted bundle assembly

A generalized theory of the self-limiting assembly of twisted bundles of filaments and columns is presented. Bundles and fibers form in a broad variety of supramolecular systems, from biological to synthetic materials. A widely-invoked mechanism to explain their finite diameter relies on chirality transfer from the molecular constituents to collective twist of the assembly, the effect of which frustrates the lateral assembly and can select equilibrium, finite diameters of bundles. In this article, the thermodynamics of twisted-bundle assembly is analyzed to understand if chirality transfer is necessary for self-limitation, or instead, if spontaneously-twisting, achiral bundles also exhibit self-limited assembly. A generalized description is invoked for the elastic costs imposed by twist for bundles of various states of intra-bundle order from nematic to crystalline, as well as a generic mechanism for generating twist, classified both by {\it chirality} but also the {\it twist susceptibility} of inter-filament alignment. The theory provides a comprehensive set of predictions for the equilibrium twist and size of bundles as a function of surface energy as well as chirality, twist susceptibility, and elasticity of bundles. Moreover, it shows that while spontaneous twist can lead to self-limitation, assembly of twisted {\it achiral} bundles can be distinguished qualitatively in terms of their range of equilibrium sizes and thermodynamic stability relative to bulk (untwisted) states.

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arXiv2019

All twist and no bend makes raft edges splay: Spontaneous curvature of domain edges in colloidal membranes

Using a combination of theory and experiments we study the interface between two immiscible domains in a colloidal membrane composed of rigid rods of different lengths. Geometric considerations of rigid rod packing imply that a domain of sufficiently short rods in a background membrane of long rods is more susceptible to twist than the inverse structure, a long-rod domain in a short-rod membrane background. The tilt at the inter-domain edge forces splay, which in turn manifests as a spontaneous edge curvature whose energetics are controlled by the length asymmetry of constituent rods. A thermodynamic model of such tilt-curvature coupling at inter-domain edges explains a number of experimental observations, including a non-monotonic dependence of the edge twist on the domain radius, and annularly shaped domains of long rods. Our work shows how coupling between orientational and compositional degrees of freedom in two-dimensional fluids give rise to complex shapes and thermodynamics of domains, analogous to shape transitions in 3D fluid vesicles.

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arXiv2019

Ground States of Crystalline Caps: Generalized Jellium on Curved Space

We study the structure and elastic energy of the ground states of crystalline caps conforming to a spherical surface. These ground states consist of positive disclination defects in structures spanning from flat and weakly curved crystals to closed shells. We compare two different continuum theories and one discrete-lattice approach for the elastic energy of defective caps. We first investigate the transition between defect-free caps to single-disclination ground states and show that it is characterized by continuous symmetry-breaking transition. Further, we show that ground states with icosahedral subgroup symmetries in caps arise across a range of curvatures, even far from the closure point of complete shells. Thus, while superficially similar to other models of 2D `jellium' (e.g. superconducting disks and 2D Wigner crystals), the interplay between the free edge of crystalline caps and the non-Euclidean geometry of its embedding leads to non trivial ground state behaviors, without counterpart in planar jellium models.

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arXiv2019

Constant spacing in filament bundles

Assemblies of filaments appear in a wide range of systems: from biopolymer bundles, columnar liquid crystals, and superconductor vortex arrays; to familiar macroscopic materials, like ropes, cables and textiles. Interactions between the constituent filaments in such systems are most sensitive to the {\it distance of closest approach} between the central curves which approximate their configuration, subjecting these distinct assemblies to common geometric constraints. In this paper, we consider two distinct notions of constant spacing in multi-filament packings in $\mathbb{R}^3$: {\it equidistance}, where the distance of closest approach is constant along the length of filament pairs; and {\it isometry}, where the distances of closest approach between all neighboring filaments are constant and equal. We show that, although any smooth curve in $\mathbb{R}^3$ permits one dimensional families of collinear equidistant curves belonging to a ruled surface, there are only two families of tangent fields with mutually equidistant integral curves in $\mathbb{R}^3$. The relative shapes and configurations of curves in these families are highly constrained: they must be either (isometric) developable domains, which can bend, but not twist; or (non-isometric) constant-pitch helical bundles, which can twist, but not bend. Thus, simultaneously bent and twisted filament textures, such as twisted toroids of condensed DNA plasmids or wire ropes, are doubly frustrated: twist frustrates constant neighbor spacing in the cross-section, while non-equidistance requires longitudinal variations of inter-filament spacing. To illustrate the consequences of the failure of equidistance, we compare spacing in three "almost equidistant" ansatzes for twisted toroidal bundles and use our formulation of equidistance to construct upper bounds on the growth of longitudinal variations of spacing with bundle thickness.

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arXiv2019

Conformational switching of chiral colloidal rafts regulates raft-raft attractions and repulsions

Membrane-mediated particle interactions depend both on the properties of the particles themselves and the membrane environment in which they are suspended. Experiments have shown that chiral rod-like inclusions dissolved in a colloidal membrane of opposite handedness assemble into colloidal rafts, which are finite-sized reconfigurable droplets consisting of a large but precisely defined number of rods. Here, we systematically tune the chirality of the background membrane, and find that in the achiral limit colloidal rafts acquire fundamentally new structural properties and interactions. In particular, rafts can switch between two chiral states of opposite handedness, which dramatically alters the nature of the membrane-mediated raft-raft interactions. Rafts with the same chirality have long-ranged repulsions, while those with opposite chirality acquire attractions with a well-defined minimum. Both attractive and repulsive interactions are explained by a continuum model that accounts for the coupling between the membrane thickness and the local tilt of the constituent rods. These switchable interactions enable assembly of colloidal rafts into intricate higher-order architectures with unusual symmetries, including stable tetrameric clusters and "ionic crystallites" of counter-twisting domains organized on a binary square lattice. Furthermore, the properties of individual rafts, such as their sizes, are controlled by their complexation with other rafts. The emergence of these complex behaviors can be rationalized purely in terms of generic couplings between compositional and orientational order of fluids of rod-like elements. Thus, the uncovered principles might have relevance for conventional lipid bilayers, in which the assembly of higher-order structures is also mediated by complex membrane-mediated interactions.

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arXiv2019

Geometrically incompatible confinement of solids

The complex morphologies exhibited by spatially confined thin objects have long challenged human efforts to understand and manipulate them, from the representation of patterns in draped fabric in Renaissance art to current day efforts to engineer flexible sensors that conform to the human body. We introduce a theoretical principle, broadly generalizing Euler's {\emph{elastica}} -- a core concept of continuum mechanics that invokes the energetic preference of bending over straining a thin solid object and has been widely applied to classical and modern studies of beams and rods. We define a class of {\emph{geometrically incompatible confinement}} problems, whereby the topography imposed on a thin solid body is incompatible with its intrinsic ("target") metric and, as a consequence of Gauss' {\emph{Theorema Egregium}}, induces strain. Focusing on a prototypical example of a sheet attached to a spherical substrate, numerical simulations and analytical study demonstrate that the mechanics is governed by a principle, which we call the "Gauss-Euler {\emph{elastica}}". This emergent rule states that - despite the unavoidable strain in such an incompatible confinement - the ratio between the energies stored in straining and bending the solid may be arbitrarily small. The Gauss-Euler {\emph{elastica}} underlies a theoretical framework that greatly simplifies the daunting task of solving the highly nonlinear equations that describe thin solids at mechanical equilibrium. This development thus opens new possibilities for attacking a broad class of phenomena governed by the coupling of geometry and mechanics.

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arXiv2018

Defect-driven shape instabilities of bundles

Topological defects are crucial to the thermodynamics and structure of condensed matter systems. For instance, when incorporated into crystalline membranes like graphene, disclinations with positive and negative topological charge elastically buckle the material into conical and saddle-like shapes respectively. A recently uncovered mapping between the inter-element spacing in 2D columnar structures and the metric properties of curved surfaces motivates basic questions about the interplay between defects in the cross section of a columnar bundle and its 3D shape. Such questions are critical to the structure of a broad class of filamentous materials, from biological assemblies like protein fibers to nano- or micro-structured synthetic materials like carbon nanotube bundles. Here, we explore the buckling behavior for elementary disclinations in hexagonal bundles using a combination of continuum elasticity theory and numerical simulations of discrete-filaments. We show that shape instabilities are controlled by a single material-dependent parameter that characterizes the ratio of inter-filament to intra-filament elastic energies. Along with a host of previously unknown shape equilibria---the filamentous analogs to the conical and saddle-like shapes of defective membranes---we find a profoundly asymmetric response to positive and negative topologically charged defects in the infinite length limit that is without parallel to the membrane analog. The highly non-linear dependence on the sign of the disclination charge is shown to have a purely geometric origin, stemming the from the distinct compatibility (or incompatibility) of effectively positive- (or negative-) curvature geometries with lengthwise-constant filament spacing.

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arXiv2018

Stable Frank-Kasper phases of self-assembled, soft matter spheres

Single molecular species can self-assemble into Frank Kasper (FK) phases, finite approximants of dodecagonal quasicrystals, defying intuitive notions that thermodynamic ground states are maximally symmetric. FK phases are speculated to emerge as the minimal-distortional packings of space-filling spherical domains, but a precise quantitation of this distortion and how it affects assembly thermodynamics remains ambiguous. We use two complementary approaches to demonstrate that the principles driving FK lattice formation in diblock copolymers emerge directly from the strong-stretching theory of spherical domains, in which minimal inter-block area competes with minimal stretching of space-filling chains. The relative stability of FK lattices is studied first using a diblock foam model with unconstrained particle volumes and shapes, which correctly predicts not only the equilibrium σ lattice, but also the unequal volumes of the equilibrium domains. We then provide a molecular interpretation for these results via self-consistent field theory, illuminating how molecular stiffness regulates the coupling between intra-domain chain configurations and the asymmetry of local packing. These findings shed new light on the role of volume exchange on the formation of distinct FK phases in copolymers, and suggest a paradigm for formation of FK phases in soft matter systems in which unequal domain volumes are selected by the thermodynamic competition between distinct measures of shape asymmetry.

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arXiv2018

Intradomain phase transitions in flexible block copolymers with self-aligning segments

We study a model of flexible block copolymers (BCPs) in which there is an enlthalpic preference for local alignment, among like-block segments. We describe a generalization of the self-consistent field theory (SCFT) of flexible BCPs to include inter-segment orientational interactions via a Landau-DeGennes free energy associated with a polar or nematic order parameter for segments of one component of a diblock copolymer. We study the equilibrium states of this model numerically, using a pseudo-spectral approach to solve for chain conformation statistics in the presence of a self-consistent torque generated by inter-segment alignment forces. Applying this theory to the structure of lamellar domains composed of symmetric diblocks possessing a single block of "self-aligning", polar segments, we show the emergence of spatially complex segment order parameters (segment director fields) within a given lamellar domain. Because BCP phase separation gives rise to spatially inhomogeneous orientation order of segments even in the absence of explicit intra-segment aligning forces, the director fields of BCPs, as well as thermodynamics of lamellar domain formation, exhibit a highly non-linear dependence on both the inter-block segregation ($χN$) and the enthalpy of alignment ($\varepsilon$). Specifically, we predict the stability of new phases of lamellar order in which distinct regions of alignment coexist within a single mesodomain, and which spontaneously break the symmetry of the lamella pattern of composition in the melt via in-plane tilt of the director in the centers of the like-composition domains. We show further that, in analogy to a Freedericksz transition in confined nematics, that the elastic costs to reorient segments within the domain, as described by Frank elasticity, increase the threshold value $\varepsilon$ needed to induce this intra-domain phase transition.

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arXiv2018

Persistence of perfect packing in twisted bundles of elastic filaments

It is generally understood that geometric frustration prevents maximal hexagonal packings in uniform filament bundles upon twist. We demonstrate that a hexagonal packed elastic filament bundle can preserve its order over a wide range of twist due to a subtle counteraction of geometric expansion with elastic contraction. Using x-ray scanning and by locating each filament in the bundle, we show the remarkable persistence of order even as the twist is increased well above 360 degrees, by measuring the spatial correlation function across the bundle crosssection. We introduce a model which analyzes the combined effects of elasticity including filament stretching, and radial and hoop compression necessary to explain this generic preservation of order observed with Hookean filaments.

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arXiv2018

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

Triply-periodic networks (TPNs), like the well-known gyroid and diamond network phases, abound in soft matter assemblies, from block copolymers (BCPs), lyotropic liquid crystals and surfactants to functional architectures in biology. While TPNs are, in reality, volume-filling patterns of spatially-varying molecular composition, physical and structural models most often reduce their structure to lower-dimensional geometric objects: the {\it 2D interfaces} between chemical domains; and the {\it 1D skeletons} that thread through inter-connected, tubular domains. These lower-dimensional structures provide a useful basis of comparison to idealized geometries based on triply-periodic minimal, or constant-mean curvature surfaces, and shed important light on the spatially heterogeneous packing of molecular constituents that form the networks. Here, we propose a simple, efficient and flexible method to extract a 1D skeleton from 3D volume composition data of self-assembled networks. We apply this method to both self-consistent field theory predictions as well as experimental electron microtomography reconstructions of the double-gyroid phase of an ABA triblock copolymer. We further demonstrate how the analysis of 1D skeleton, 2D inter-domain surfaces, and combinations therefore, provide physical and structural insight into TPNs, across multiple length scales. Specifically, we propose and compare simple measures of {\it network chirality} as well as {\it domain thickness}, and analyze their spatial and statistical distributions in both ideal (theoretical) and non-ideal (experimental) double gyroid assemblies.

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arXiv2017

Sub-Jamming Transition in Binary Sphere Mixtures

We study the influence of particle size asymmetry on structural evolution of randomly jammed binary sphere mixtures with varying large-sphere/small-sphere composition. Simulations of jammed packings are used to assess the transition from large-sphere dominant to small-sphere dominant mixtures. For weakly asymmetric particle sizes, packing properties evolve smoothly, but not monotonically, with increasing small sphere composition, $f$. Our simulations reveal that at high values of ratio $α$ of large to small sphere radii, ($α\geq α_c \approx 5.75$) evolution of structural properties such as packing density, fraction of jammed spheres and contact statistics with $f$ exhibit features that suggest a sharp transition, either through discontinuities in structural measures or their derivatives. We argue that this behavior is related to the singular, composition dependence of close-packing fraction predicted in infinite aspect ratio mixtures $α\rightarrow\infty$ by the Furnas model, but occurring for finite values range of $α$ above a critical value, $α_c \approx 5.75$. The existence of a sharp transition from small- to large-$f$ values for $α\geq α_c$ can be attributed to the existence of a {\it sub-jamming transition} of small spheres in the interstices of jammed large spheres along the line of compositions $f_{sub}(α)$. We argue that the critical value of finite size asymmetry $α_c \simeq 5.75$ is consistent with the geometric criterion for the transmission of small sphere contacts between neighboring tetrahedrally close packed interstices of large spheres, facilitating a cooperative sub-jamming transition of small spheres confined within the disjoint volumes.

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arXiv2017

Measuring geometric frustration in twisted inextensible filament bundles

We investigate with experiments and novel mapping the structure of a hexagonally ordered filament bundle that is held near its ends and progressively twisted around its central axis. The filaments are free to slide relative to each other and are further held under tension-free boundary conditions. Measuring the bundle packing with micro x-ray imaging, we find that the filaments develop the helical rotation $Ω$ imposed at the boundaries. We then show that the observed structure is consistent with a mapping of the filament positions to disks packed on a dual non-Euclidean surface with a Gaussian curvature which increases with twist. We further demonstrate that the mean inter-filament distance is minimal on the surface which can be approximated by a hemisphere with an effective curvature $K_{eff} = 3Ω^2$. Examining the packing on the dual surface, we analyze the geometric frustration of packing in twisted bundles and find the core to remain relatively hexagonally ordered with inter-filament strains growing from the bundle center, driving the formation of defects at the exterior of highly twisted bundles.

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arXiv2016

Intradomain Textures in Block Copolymers: Multizone Alignment and Biaxiality

Block copolymer (BCP) melt assembly has been the subject of decades of study, with focus largely on self-organized spatial patterns of periodically-ordered segment density. In this study, we demonstrate that underlying these otherwise well-known composition profiles (i.e. ordered lamella, cylinders, spheres and networks) are generic and heterogeneous patterns of segment orientation that couple strongly to morphology, even in the absence of specific factors that promote intra- or inter-chain segment alignment. We employ a combination of self-consistent field theory and coarse-grained simulation methods to measure polar and nematic order parameters of segments in a freely-jointed chain model of diblock melts. We show that BCP morphologies are generically characterized by a {\it multizone} texture, with segments predominantly aligned normal and parallel to inter-domain interfaces in the respective brush and interfacial regions of the microdomain. Further, morphologies with anisotropically-curved interfaces (i.e. cylinders and networks) exhibit biaxial order that is aligned to the principal curvature axes of the interface. Understanding these generic features of intra-domain texture provide new opportunities for manipulating multi-scale structure and functional properties of BCP assembly.

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arXiv2016

How geometric frustration shapes twisted fibers, inside and out: Competing morphologies of chiral filament assembly

Chirality frustrates and shapes the assembly of flexible filaments in rope-like, twisted bundles and fibers by introducing gradients of both filament shape (i.e. curvature) and packing throughout the structure. Previous models of chiral filament bundle formation have shown that this frustration gives rise to several distinct morphological responses, including self-limiting bundle widths, anisotropic domain (tape-like) formation and topological defects in the lateral inter-filament order. In this paper, we employ a combination of continuum elasticity theory and discrete filament bundle simulations to explore how these distinct morphological responses compete in the broader phase diagram of chiral filament assembly. We show that the most generic model of bundle formation exhibits at least four classes of equilibrium structure -- finite-width, twisted bundles with isotropic and anisotropic shapes, with and without topological defects, as well as bulk phases of untwisted, columnar assembly (i.e. "frustration escape"). These competing equilibrium morphologies are selected by only a relatively small number of parameters describing filament assembly: bundle surface energy, preferred chiral twist and stiffness of chiral filament interactions, and mechanical stiffness of filaments and their lateral interactions. Discrete filament bundle simulations test and verify continuum theory predictions for dependence of bundle structure (shape, size and packing defects of 2D cross section) on these key parameters.

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arXiv2016

Perspective: Geometrically-Frustrated Assemblies

This perspective will overview an emerging paradigm for self-organized soft materials, {\it geometrically-frustrated assemblies}, where interactions between self-assembling elements (e.g. particles, macromolecules, proteins) favor local packing motifs that are incompatible with uniform global order in the assembly. This classification applies to a broad range of material assemblies including self-twisting protein filament bundles, amyloid fibers, chiral smectics and membranes, particle-coated droplets, curved protein shells and phase-separated lipid vesicles. In assemblies, geometric frustration leads to a host of anomalous structural and thermodynamic properties, including heterogeneous and internally-stressed equilibrium structures, self-limiting assembly and topological defects in the equilibrium assembly structures. The purpose of this perspective is to 1) highlight the unifying principles and consequences of geometric frustration in soft matter assemblies, 2) to classify the known distinct "modes" of frustration and review corresponding experimental examples, and 3) to describe outstanding questions not yet addressed about the unique properties and behaviors of this broad class of systems.

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arXiv2016

Neutral versus charged defect patterns in curved crystals

Characterizing the complex spectrum of topological defects in ground states of curved crystals is a long-standing problem with wide implications, from the mathematical Thomson problem to diverse physical realizations, including fullerenes and particle-coated droplets. While the excess number of "topologically-charged" 5-fold disclinations in a closed, spherical crystal is fixed, here, we study the elementary transition from defect-free, flat crystals to curved-crystals possessing an excess of "charged" disclinations in their bulk. Specifically, we consider the impact of topologically-neutral patterns of defects -- in the form of multi-dislocation chains or "scars" stable for small lattice spacing -- on the transition from neutral to charged ground-state patterns of a crystalline cap bound to a spherical surface. Based on the asymptotic theory of caps in continuum limit of vanishing lattice spacing, we derive the morphological phase diagram of ground state defect patterns, spanned by surface coverage of the sphere and forces at the cap edge. For the singular limit of zero edge forces, we find that scars reduce (by half) the threshold surface coverage for excess disclinations. Even more significant, scars flatten the geometric dependence of excess disinclination number on Gaussian curvature, leading to a transition between stable "charged" and "neutral" patterns that is, instead, critically sensitive to the compressive vs. tensile nature of boundary forces on the cap.

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