Minkowski Tensors of Anisotropic Spatial Structure

This article describes the theoretical foundation of and explicit algorithms for a novel approach to morphology and anisotropy analysis of complex spatial structure using tensor-valued Minkowski functionals, the so-called Minkowski tensors. Minkowski tensors are generalisations of the well-known scalar Minkowski functionals and are explicitly sensitive to anisotropic aspects of morphology, relevant for example for elastic moduli or permeability of microstructured materials. Here we derive explicit linear-time algorithms to compute these tensorial measures for three-dimensional shapes. These apply to representations of any object that can be represented by a triangulation of its bounding surface; their application is illustrated for the polyhedral Voronoi cellular complexes of jammed sphere configurations, and for triangulations of a biopolymer fibre network obtained by confocal microscopy. The article further bridges the substantial notational and conceptual gap between the different but equivalent approaches to scalar or tensorial Minkowski functionals in mathematics and in physics, hence making the mathematical measure theoretic method more readily accessible for future application in the physical sciences

Morphological Thermodynamics of Fluids: Shape Dependence of Free Energies

We examine the dependence of a thermodynamic potential of a fluid on the geometry of its container. If motion invariance, continuity, and additivity of the potential are fulfilled, only four morphometric measures are needed to describe fully the influence of an arbitrarily shaped container on the fluid. These three constraints can be understood as a more precise definition for the conventional term "extensive" and have as a consequence that the surface tension and other thermodynamic quantities contain, beside a constant term, only contributions linear in the mean and Gaussian curvature of the container and not an infinite number of curvatures as generally assumed before. We verify this numerically in the entropic system of hard spheres bounded by a curved wall.Comment: 4 pages, 3 figures, accepted for publication in PR

Local Anisotropy of Fluids using Minkowski Tensors

Statistics of the free volume available to individual particles have previously been studied for simple and complex fluids, granular matter, amorphous solids, and structural glasses. Minkowski tensors provide a set of shape measures that are based on strong mathematical theorems and easily computed for polygonal and polyhedral bodies such as free volume cells (Voronoi cells). They characterize the local structure beyond the two-point correlation function and are suitable to define indices $0\leq \beta_\nu^{a,b}\leq 1$ of local anisotropy. Here, we analyze the statistics of Minkowski tensors for configurations of simple liquid models, including the ideal gas (Poisson point process), the hard disks and hard spheres ensemble, and the Lennard-Jones fluid. We show that Minkowski tensors provide a robust characterization of local anisotropy, which ranges from $\beta_\nu^{a,b}\approx 0.3$ for vapor phases to $\beta_\nu^{a,b}\to 1$ for ordered solids. We find that for fluids, local anisotropy decreases monotonously with increasing free volume and randomness of particle positions. Furthermore, the local anisotropy indices $\beta_\nu^{a,b}$ are sensitive to structural transitions in these simple fluids, as has been previously shown in granular systems for the transition from loose to jammed bead packs

Virus shapes and buckling transitions in spherical shells

We show that the icosahedral packings of protein capsomeres proposed by Caspar and Klug for spherical viruses become unstable to faceting for sufficiently large virus size, in analogy with the buckling instability of disclinations in two-dimensional crystals. Our model, based on the nonlinear physics of thin elastic shells, produces excellent one parameter fits in real space to the full three-dimensional shape of large spherical viruses. The faceted shape depends only on the dimensionless Foppl-von Karman number \gamma=YR^2/\kappa, where Y is the two-dimensional Young's modulus of the protein shell, \kappa is its bending rigidity and R is the mean virus radius. The shape can be parameterized more quantitatively in terms of a spherical harmonic expansion. We also investigate elastic shell theory for extremely large \gamma, 10^3 < \gamma < 10^8, and find results applicable to icosahedral shapes of large vesicles studied with freeze fracture and electron microscopy.Comment: 11 pages, 12 figure

Supramolecular modification of ABC triblock terpolymers in confinement assembly

The self-assembly of AB diblock copolymers in three-dimensional (3D) soft confinement of nanoemulsions has recently become an attractive bottom up route to prepare colloids with controlled inner morphologies. In that regard, ABC triblock terpolymers show a more complex morphological behavior and could thus give access to extensive libraries of multicompartment microparticles. However, knowledge about their self-assembly in confinement is very limited thus far. Here, we investigated the confinement assembly of polystyrene-block-poly(4-vinylpyridine)-block-poly(tert-butyl methacrylate) (PS-b-P4VP-b-PT or SVT) triblock terpolymers in nanoemulsion droplets. Depending on the block weight fractions, we found spherical microparticles with concentric lamella–sphere (ls) morphology, i.e., PS/PT lamella intercalated with P4VP spheres, or unusual conic microparticles with concentric lamella–cylinder (lc) morphology. We further described how these morphologies can be modified through supramolecular additives, such as hydrogen bond (HB) and halogen bond (XB) donors. We bound donors to the 4VP units and analyzed changes in the morphology depending on the binding strength and the length of the alkyl tail. The interaction with the weaker donors resulted in an increase in volume of the P4VP domains, which depends upon the molar fraction of the added donor. For donors with a high tendency of intermolecular packing, a visible change in the morphology was observed. This ultimately caused a shape change in the microparticle. Knowledge about how to control inner morphologies of multicompartment microparticles could lead to novel carbon supports for catalysis, nanoparticles with unprecedented topologies, and potentially, reversible shape changes by light actuation

Compaction of Quasi One-Dimensional Elastoplastic Materials

Insight in the crumpling or compaction of one-dimensional objects is of great importance for understanding biopolymer packaging and designing innovative technological devices. By compacting various types of wires in rigid confinements and characterizing the morphology of the resulting crumpled structures, here we report how friction, plasticity, and torsion enhance disorder, leading to a transition from coiled to folded morphologies. In the latter case, where folding dominates the crumpling process, we find that reducing the relative wire thickness counter-intuitively causes the maximum packing density to decrease. The segment-size distribution gradually becomes more asymmetric during compaction, reflecting an increase of spatial correlations. We introduce a self-avoiding random walk model and verify that the cumulative injected wire length follows a universal dependence on segment size, allowing for the prediction of the efficiency of compaction as a function of material properties, container size, and injection force.Comment: 7 pages, 6 figure

Finite Element Simulation of Dense Wire Packings

A finite element program is presented to simulate the process of packing and coiling elastic wires in two- and three-dimensional confining cavities. The wire is represented by third order beam elements and embedded into a corotational formulation to capture the geometric nonlinearity resulting from large rotations and deformations. The hyperbolic equations of motion are integrated in time using two different integration methods from the Newmark family: an implicit iterative Newton-Raphson line search solver, and an explicit predictor-corrector scheme, both with adaptive time stepping. These two approaches reveal fundamentally different suitability for the problem of strongly self-interacting bodies found in densely packed cavities. Generalizing the spherical confinement symmetry investigated in recent studies, the packing of a wire in hard ellipsoidal cavities is simulated in the frictionless elastic limit. Evidence is given that packings in oblate spheroids and scalene ellipsoids are energetically preferred to spheres.Comment: 17 pages, 7 figures, 1 tabl

Orientation and symmetry control of inverse sphere magnetic nanoarrays by guided self-assembly

Inverse sphere shaped Ni arrays were fabricated by electrodeposition on Si through the guided self-assembly of polystyrene latex spheres in Si/SiO2 patterns. It is shown that the size commensurability of the etched tracks is critical for the long range ordering of the spheres. Moreover, noncommensurate guiding results in the reproducible periodic triangular distortion of the close packed self-assembly. Magnetoresistance measurements on the Ni arrays were performed showing room temperature anisotropic magnetoresistance of 0.85%. These results are promising for self-assembled patterned storage media and magnetoresistance devices