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ABSTRACT: Cohesive assemblies of filaments are a common structural motif found in
diverse contexts, ranging from biological materials such as fibrous proteins,
to artificial materials such as carbon nanotube ropes and micropatterned
filament arrays. In this paper, we analyze the complex dependence of cohesive
energy on twist, a key structural parameter of both self-assembled and
fabricated filament bundles. Based on the analysis of simulated ground states
of cohesive bundles, we show that the non-linear influence of twist derives
from two distinct geometric features of twisted bundles: (i) the geometrical
frustration of inter-filament packing in the bundle cross-section; and (ii) the
evolution of the surface geometry of bundles with twist, which dictates the
cohesive cost of non-contacting filaments at the surface. Packing frustration
in the bundle core gives rise to the appearance of a universal sequence of
topological defects, excess 5-fold disclinations, with increasing twist, while
the evolution of filament contact at the surface of the bundle generically
favors twisted geometries for sufficiently long filaments. Our analysis of both
continuum and discrete models of filament bundles shows that, even in the
absence of external torque or intrinsic chirality, cohesive energy universally
favors twisted ground states above a critical (length/radius) aspect ratio and
below a critical filament stiffness threshold.
03/2013;
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ABSTRACT: We study the effects of chirality at the segment scale on the thermodynamics of block copolymer melts using self-consistent field theory. In linear diblock melts where segments of one block prefer a twisted, or cholesteric, texture, we show that melt assembly is critically sensitive to the ratio of random coil size to the preferred pitch of cholesteric twist. For weakly chiral melts (large pitch), mesophases remain achiral, while below a critical value of pitch, two mesoscopically chiral phases are stable: an undulated lamellar phase and a phase of hexagonally ordered helices. We show that the nonlinear sensitivity of mesoscale chiral order to preferred pitch derives specifically from the geometric and thermodynamic coupling of the helical mesodomain shape to the twisted packing of chiral segments within the core, giving rise to a second-order cylinder-to-helix transition.
Physical Review Letters 02/2013; 110(5):058301. · 7.37 Impact Factor
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ABSTRACT: We study the effects of chirality at the segment scale on the thermodynamics
of block copolymer melts using self consistent field theory. In linear diblock
melts where segments of one block prefer a twisted, or cholesteric, texture, we
show that melt assembly is critically sensitive to the ratio of random coil
size to the preferred pitch of cholesteric twist. For weakly-chiral melts
(large pitch), mesophases remain achiral, while below a critical value of
pitch, two mesocopically chiral phases are stable: an undulated lamellar phase;
and a phase of hexagonally-ordered helices. We show that the non-linear
sensitivity of meso-scale chiral order to preferred pitch derives specifically
from the geometric and thermodynamic coupling of the helical mesodomain shape
to the twisted packing of chiral segments within the core, giving rise to a
second-order cylinder-to-helix transition.
10/2012;
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ABSTRACT: We study the phase behavior of diblock copolymer melts with one block possessing orientation-dependent segmental interactions using self-consistent field theory. A generalized coarse-grained description is introduced based on the local (polar) orientational order parameter and K, an effective Frank elastic constant for orientational gradients. To explore the role played by orientational interactions in assembly thermodynamics, we apply the theory to two-dimensional melt morphologies for a range of K. As microphase segregation necessarily introduces splay deformations of the segment orientation, we find that increasing the stiffness K raises the critical χN at the onset of microphase separation. Furthermore, we find that strong orientational interactions in one block give rise to highly asymmetric phase diagrams due to the large penalty for high-splay morphologies, such as the cylindrical phase. Finally, we analyze the costs of inter-segmental splay as well as the size dependence of domain spacing on K based on a strong-segregation picture of morphologies.
The Journal of chemical physics 09/2012; 137(10):104911. · 3.09 Impact Factor
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ABSTRACT: Densely packed and twisted assemblies of filaments are crucial structural motifs in macroscopic materials (cables, ropes, and textiles) as well as synthetic and biological nanomaterials (fibrous proteins). We study the unique and nontrivial packing geometry of this universal material design from two perspectives. First, we show that the problem of twisted bundle packing can be mapped exactly onto the problem of disc packing on a curved surface, the geometry of which has a positive, spherical curvature close to the center of rotation and approaches the intrinsically flat geometry of a cylinder far from the bundle center. From this mapping, we find the packing of any twisted bundle is geometrically frustrated, as it makes the sixfold geometry of filament close packing impossible at the core of the fiber. This geometrical equivalence leads to a spectrum of close-packed fiber geometries, whose low symmetry (five-, four-, three-, and twofold) reflect non-euclidean packing constraints at the bundle core. Second, we explore the ground-state structure of twisted filament assemblies formed under the influence of adhesive interactions by a computational model. Here, we find that the underlying non-euclidean geometry of twisted fiber packing disrupts the regular lattice packing of filaments above a critical radius, proportional to the helical pitch. Above this critical radius, the ground-state packing includes the presence of between one and six excess fivefold disclinations in the cross-sectional order.
Proceedings of the National Academy of Sciences 06/2012; 109(27):10781-6. · 9.68 Impact Factor
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ABSTRACT: Twisted and ropelike assemblies of filamentous molecules are common and vital structural elements in cells and tissues of living organisms. We study the intrinsic frustration occurring in these materials between the two-dimensional organization of filaments in cross section and out-of-plane interfilament twist in bundles. Using nonlinear continuum elasticity theory of columnar materials, we study the favorable coupling of twist-induced stresses to the presence of edge dislocations in the lattice packing of bundles, which leads to a restructuring of the ground-state order of these materials at intermediate twist. The stability of dislocations increases as both the degree of twist and lateral bundle size grow. We show that in ground states of large bundles, multiple dislocations pile up into linear arrays, radial grain boundaries, whose number and length grows with bundle twist, giving rise to a rich class of "polycrystalline" packings.
Physical Review E 03/2012; 85(3 Pt 1):031604. · 2.26 Impact Factor
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ABSTRACT: We study, numerically and theoretically, defects in an anisotropic liquid that couple to the extrinsic geometry of a surface. Though the intrinsic geometry tends to confine topological defects to regions of large Gaussian curvature, extrinsic couplings tend to orient the order along the local direction of maximum or minimum bending. This additional frustration is generically unavoidable, and leads to complex ground-state thermodynamics. Using the catenoid as a prototype, we show, in contradistinction to the well-known effects of intrinsic geometry, that extrinsic curvature expels disclinations from the region of maximum curvature above a critical coupling threshold. On catenoids lacking an "inside-outside" symmetry, defects are expelled altogether above a critical neck size.
Physical Review Letters 01/2012; 108(1):017801. · 7.37 Impact Factor
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ABSTRACT: Inspired by the complex influence of the globular crosslinking proteins on the formation of biofilament bundles in living organisms, we study and analyze a theoretical model for the structure and thermodynamics of bundles of helical filaments assembled in the presence of crosslinking molecules. The helical structure of filaments, a universal feature of biopolymers such as filamentous actin, is shown to generically frustrate the geometry of crosslinking between the "grooves" of two neighboring filaments. We develop a coarse-grained model to investigate the interplay between the geometry of binding and mechanics of both linker and filament distortion, and we show that crosslinking in parallel bundles of helical filaments generates intrinsic torques, of the type that tend to wind the bundle superhelically about its central axis. Crosslinking mediates a non-linear competition between the preference for bundle twist and the size-dependent mechanical cost of filament bending, which in turn gives rise to feedback between the global twist of self-assembled bundles and their lateral size. Finally, we demonstrate that above a critical density of bound crosslinkers, twisted bundles form with a thermodynamically preferred radius that, in turn, increases with a further increase in crosslinking bonds. We identify the stiffness of crosslinking bonds as a key parameter governing the sensitivity of bundle structure and assembly to the availability and affinity of crosslinkers.
The Journal of chemical physics 07/2011; 135(3):035104. · 3.09 Impact Factor
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ABSTRACT: Polymer vesicles made from poly(isoprene-b-styrene-b-2-vinyl pyridine) (PI-b-PS-b-P2VP) triblock copolymer confined within the nanopores of an anodic aluminum oxide (AAO) membrane are studied. It was found that these vesicles have well-defined, nanoscopic size, and complex microphase-separated hydrophobic membranes, comprised of the PS and PI blocks, while the coronas are formed by the P2VP block. Vesicle formation was tracked using both transmission and scanning electron microscopy. A mesh-like morphology formed in the membrane at a well-defined composition of the three blocks that can be tuned by changing the copolymer composition. The nanoscale confinement, copolymer composition, and subtle molecular interactions contribute to the generation of these vesicles with such unusual morphologies.
ACS Nano 01/2011; 5(1):486-92. · 10.77 Impact Factor
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ABSTRACT: We construct a coarse-grained model of parallel actin bundles crosslinked by compact globular bundling proteins, such as fascin and espin, necessary components of filopodial and mechanosensory bundles. Consistent with structural observations of bundles, we find that the optimal geometry for crosslinking is overtwisted, requiring a coherent structural change of the helical geometry of the filaments. We study the linker-dependent thermodynamic transition of bundled actin filaments from their native state to the overtwisted state and map out the "twist-state" phase diagram in terms of the availability as well as the flexibility of crosslinker proteins. We predict that the transition from the uncrosslinked to fully crosslinked state is highly sensitive to linker flexibility: flexible crosslinking smoothly distorts the twist state of bundled filaments, while rigidly crosslinked bundles undergo a phase transition, rapidly overtwisting filaments over a narrow range of free crosslinker concentrations. Additionally, we predict a rich spectrum of intermediate structures, composed of alternating domains of sparsely bound (untwisted) and strongly bound (overtwisted) filaments. This model reveals that subtle differences in crosslinking agents themselves modify not only the detailed structure of parallel actin bundles, but also the thermodynamic pathway by which they form.
Physical Review E 11/2010; 82(5 Pt 1):051919. · 2.26 Impact Factor
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Gregory M Grason
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ABSTRACT: Twisted assemblies of filaments in ropes, cables, and bundles are essential structural elements in both macroscopic materials and living organisms. We develop the unique, nonlinear elastic properties of twisted filament bundles that derive from generic properties of two-dimensional line-ordered materials. Continuum elasticity reveals a formal equivalence between the elastic stresses induced by bundle twist and those induced by the positive curvature in thin, elastic sheets. These geometrically induced stresses are screened by fivefold disclination defects in the lattice packing, and we predict a discrete spectrum of elastic-energy ground states associated with integer numbers of disclinations in cylindrical bundles. Finally, we show that elastic-energy ground states are extremely sensitive to the defect position in the cross section, with off-center disclinations driving the entire bundle to buckle and writhe.
Physical Review Letters 07/2010; 105(4):045502. · 7.37 Impact Factor
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ABSTRACT: We examine the mechanism of bundling of cytoskeletal actin filaments by two representative bundling proteins, fascin and espin. Small-angle x-ray studies show that increased binding from linkers drives a systematic overtwist of actin filaments from their native state, which occurs in a linker-dependent fashion. Fascin bundles actin into a continuous spectrum of intermediate twist states, while espin only allows for untwisted actin filaments and fully overtwisted bundles. Based on a coarse-grained, statistical model of protein binding, we show that the interplay between binding geometry and the intrinsic flexibility of linkers mediates cooperative binding in the bundle. We attribute the respective continuous (discontinuous) bundling mechanisms of fascin (espin) to difference in the stiffness of linker bonds themselves.
Physical Review Letters 12/2009; 103(23):238102. · 7.37 Impact Factor
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ABSTRACT: Soft spheres interacting via a hard core and range of attractive and repulsive "soft-shoulder" potentials self-assemble into clusters forming a variety of mesophases. We combine a mean field theory developed from a lattice model with a level surface analysis of the periodic structures of soft-sphere aggregates to study stable morphologies for all clustering potentials. We develop a systematic approach to the thermodynamics of mesophase assembly in the low-temperature, strong-segregation and predict a generic sequence of phases including lamella, hexagonal-columnar and body-center cubic phases, as well as the associated inverse structures. We discuss the finite-temperature corrections to strong segregation theory in terms of Sommerfeld-like expansion and how these corrections affect the thermodynamic stability of bicontinuous mesophase structures, such as gyroid. Finally, we explore the opposite limit of weakly-segregated particles, and predict the generic stability of a bicontinuous cluster morphology within the mean-field phase diagram. Comment: 11 pages, 7 figures, 1 table
02/2009;
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Gregory M Grason
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ABSTRACT: We demonstrate that the phase transition from columnar-hexagonal liquid crystal to hexagonal-crystalline solid falls into an unusual universality class, which in three dimensions allows for both discontinuous transitions as well as continuous transitions, characterized by a single set of exponents. We show by a renormalization group calculation (to first order in =4-d) that the critical exponents of the continuous transition are precisely those of the XY model, giving rise to a continuous evolution of elastic moduli. Although the fixed points of the present model are found to be identical to the XY model, the elastic compliance to deformations in the plane of hexagonal order, mu, is nonetheless shown to critically influence the crystallization transition, with the continuous transition being driven to first order by fluctuations as the in-plane response grows weaker, micro-->0.
Physical Review Letters 09/2008; 101(10):105702. · 7.37 Impact Factor
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Gregory M Grason
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ABSTRACT: The authors propose a mean-field model to explore the equilibrium coupling between micelle aggregation and lattice choice in neutral copolymer and selective solvent mixtures. They find both thermotropic and lyotropic transitions from face-centered cubic to body-centered cubic ordered phases of spherical micelles, in agreement with experimental observations of these systems over a broad range of conditions. The stability of the nonclosed packed phase can be attributed to two physical mechanisms: the large entropy of lattice phonons near crystal melting and the preference of the intermicelle repulsions for the body-centered cubic structure when the lattice becomes sufficiently dense at higher solution concentrations. Both mechanisms are controlled by the decrease of micelle aggregation and subsequent increase of lattice density as solvent selectivity is reduced. These results shed new light on the relationship between micelle structure--"crewcut" or "hairy"--and long-range order in micelle solutions.
The Journal of Chemical Physics 04/2007; 126(11):114904. · 3.33 Impact Factor
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10/2006;
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Gregory M. Grason
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ABSTRACT: We propose a mean-field model to explore the equilibrium coupling between micelle aggregation and lattice choice in neutral copolymer and selective solvent mixtures. We find both thermotropic and lyotropic transitions from face-centered cubic to body-centered cubic ordered phases of spherical micelles, in agreement with experimental observations of these systems over a broad range of conditions. Stability of the non-closed packed phase can be attributed to two physical mechanisms: the large entropy of lattice phonons near crystal melting and the preference of the inter-micelle repulsions for the body-centered cubic structure when the lattice becomes sufficiently dense at higher solution concentrations. Both mechanisms are controlled by the decrease of micelle aggregation and subsequent increase of lattice density as solvent selectivity is reduced. These results shed new light on the relationship between micelle structure -- "crewcut" or "hairy" -- and long-range order in micelle solutions.
08/2005;
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ABSTRACT: We present a numerical algorithm to evaluate the self-consistent field theory for melts composed of block copolymers with multiply branched architecture. We present results for the case of branched copolymers with doubly functional groups for multiple-branching generations. We discuss the stability of the cubic phase of spherical micelles, the A15 phase, as a consequence of the tendency of the AB interfaces to conform to the polyhedral environment of the Voronoi cell of the micelle lattice.
Physical Review E 06/2005; 71(5 Pt 1):051801. · 2.26 Impact Factor
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ABSTRACT: We study AB$_n$ miktoarm star block copolymers in the strong segregation limit, focussing on the role that the AB interface plays in determining the phase behavior. We develop an extension of the kinked-path approach which allows us to explore the energetic dependence on interfacial shape. We consider a one-parameter family of interfaces to study the columnar to lamellar transition in asymmetric stars. We compare with recent experimental results. We discuss the stability of the A15 lattice of sphere-like micelles in the context of interfacial energy minimization. We corroborate our theory by implementing a numerically exact self-consistent field theory to probe the phase diagram and the shape of the AB interface.
05/2004;
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ABSTRACT: We analyze the energetics of spherelike micellar phases in diblock copolymers in terms of well-studied, geometric quantities for their lattices. We argue that the A15 lattice with Pm3;n symmetry should be favored as the blocks become more symmetric and corroborate this through a self-consistent field theory. Because phases with columnar or bicontinuous topologies intervene, the A15 phase, though metastable, is not an equilibrium phase of symmetric diblocks. We investigate the phase diagram of branched diblocks and find that the A15 phase is stable.
Physical Review Letters 09/2003; 91(5):058304. · 7.37 Impact Factor
