317 research outputs found
Basic Understanding of Condensed Phases of Matter via Packing Models
Packing problems have been a source of fascination for millenia and their
study has produced a rich literature that spans numerous disciplines.
Investigations of hard-particle packing models have provided basic insights
into the structure and bulk properties of condensed phases of matter, including
low-temperature states (e.g., molecular and colloidal liquids, crystals and
glasses), multiphase heterogeneous media, granular media, and biological
systems. The densest packings are of great interest in pure mathematics,
including discrete geometry and number theory. This perspective reviews
pertinent theoretical and computational literature concerning the equilibrium,
metastable and nonequilibrium packings of hard-particle packings in various
Euclidean space dimensions. In the case of jammed packings, emphasis will be
placed on the "geometric-structure" approach, which provides a powerful and
unified means to quantitatively characterize individual packings via jamming
categories and "order" maps. It incorporates extremal jammed states, including
the densest packings, maximally random jammed states, and lowest-density jammed
structures. Packings of identical spheres, spheres with a size distribution,
and nonspherical particles are also surveyed. We close this review by
identifying challenges and open questions for future research.Comment: 33 pages, 20 figures, Invited "Perspective" submitted to the Journal
of Chemical Physics. arXiv admin note: text overlap with arXiv:1008.298
A Quasiphysical and Dynamic Adjustment Approach for Packing the Orthogonal Unequal Rectangles in a Circle with a Mass Balance: Satellite Payload Packing
Packing orthogonal unequal rectangles in a circle with a mass balance (BCOURP) is a typical combinational optimization problem with the NP-hard nature. This paper proposes an effective quasiphysical and dynamic adjustment approach (QPDAA). Two embedded degree functions between two orthogonal rectangles and between an orthogonal rectangle and the container are defined, respectively, and the extruded potential energy function and extruded resultant force formula are constructed based on them. By an elimination of the extruded resultant force, the dynamic rectangle adjustment, and an iteration of the translation, the potential energy and static imbalance of the system can be quickly decreased to minima. The continuity and monotony of two embedded degree functions are proved to ensure the compactness of the optimal solution. Numerical experiments show that the proposed QPDAA is superior to existing approaches in performance
Self-Templating Assembly of Soft Microparticles into Complex Tessellations
Self-assembled monolayers of microparticles encoding Archimedean and
non-regular tessellations promise unprecedented structure-property
relationships for a wide spectrum of applications in fields ranging from
optoelectronics to surface technology. Yet, despite numerous computational
studies predicting the emergence of exotic structures from simple interparticle
interactions, the experimental realization of non-hexagonal patterns remains
challenging. Not only kinetic limitations often hinder structural relaxation,
but also programming the inteparticle interactions during assembly, and hence
the target structure, remains an elusive task. Here, we demonstrate how a
single type of soft polymeric microparticle (microgels) can be assembled into a
wide array of complex structures as a result of simple pairwise interactions.
We first let microgels self-assemble at a water-oil interface into a
hexagonally packed monolayer, which we then compress to varying degrees and
deposit onto a solid substrate. By repeating this process twice, we find that
the resultant structure is not the mere stacking of two hexagonal patterns. The
first monolayer retains its hexagonal structure and acts as a template into
which the particles of the second monolayer rearrange to occupy interstitial
positions. The frustration between the two lattices generates new symmetries.
By simply varying the packing fraction of the two monolayers, we obtain not
only low-coordination structures such as rectangular and honeycomb lattices,
but also rhomboidal, hexagonal, and herringbone superlattices which display
non-regular tessellations. Molecular dynamics simulations show that these
structures are thermodynamically stable and develop from short-ranged repulsive
interactions, making them easy to predict, and thus opening new avenues to the
rational design of complex patterns
Sampling High-Dimensional Bandlimited Fields on Low-Dimensional Manifolds
Consider the task of sampling and reconstructing a bandlimited spatial field
in using moving sensors that take measurements along their path. It is
inexpensive to increase the sampling rate along the paths of the sensors but
more expensive to increase the total distance traveled by the sensors per unit
area, which we call the \emph{path density}. In this paper we introduce the
problem of designing sensor trajectories that are minimal in path density
subject to the condition that the measurements of the field on these
trajectories admit perfect reconstruction of bandlimited fields. We study
various possible designs of sampling trajectories. Generalizing some ideas from
the classical theory of sampling on lattices, we obtain necessary and
sufficient conditions on the trajectories for perfect reconstruction. We show
that a single set of equispaced parallel lines has the lowest path density from
certain restricted classes of trajectories that admit perfect reconstruction.
We then generalize some of our results to higher dimensions. We first obtain
results on designing sampling trajectories in higher dimensional fields.
Further, interpreting trajectories as 1-dimensional manifolds, we extend some
of our ideas to higher dimensional sampling manifolds. We formulate the problem
of designing -dimensional sampling manifolds for -dimensional
spatial fields that are minimal in \emph{manifold density}, a natural
generalization of the path density. We show that our results on sampling
trajectories for fields in can be generalized to analogous results on
-dimensional sampling manifolds for -dimensional spatial fields.Comment: Submitted to IEEE Transactions on Information Theory, Nov 2011;
revised July 2012; accepted Oct 201
Introduction to Louis Michel’s lattice geometry through group action
Group action analysis developed and applied mainly by Louis Michel to the study of N-dimensional periodic lattices is the central subject of the book. Di¬fferent basic mathematical tools currently used for the description of lattice geometry are introduced and illustrated through applications to crystal structures in two- and three-dimensional space, to abstract multi-dimensional lattices and to lattices associated with integrable dynamical systems. Starting from general Delone sets the authors turn to di¬fferent symmetry and topological classifications including explicit construction of orbifolds for two- and three-dimensional point and space groups
Thermodynamic and Structural Phase Behavior of Colloidal and Nanoparticle Systems.
We design and implement a scalable hard particle Monte Carlo simulation toolkit (HPMC), and release it open source. Common thermodynamic ensembles can be run in two dimensional or three dimensional triclinic boxes. We developed an efficient scheme for hard particle pressure measurement based on volume perturbation.
We demonstrate the effectiveness of low order virial coefficients in describing the compressibility factor of fluids of hard polyhedra. The second virial coefficient is obtained analytically from particle asphericity and can be used to define an effective sphere with similar low-density behavior. Higher-order virial coefficients --- efficiently calculated with Mayer Sampling Monte Carlo --- are used to define an exponential approximant that exhibits the best known semi-analytic characterization of hard polyhedron fluid state functions.
We present a general method for the exact calculation of convex polyhedron diffraction form factors that is more easily applied to common shape data structures than the techniques typically presented in literature. A proof of concept user interface illustrates how a researcher might investigate the role of particle form factor in the diffraction patterns of different particles in known structures.
We present a square-triangle dodecagonal quasicrystal (DQC) in a binary mixture of nanocrystals (NCs). We demonstrate how the decoration of the square and triangle tiles naturally gives rise to partial matching rules via symmetry breaking in layers perpendicular to the dodecagonal axis. We analyze the geometry of the experimental tiling and, following the ``cut and project'' theory, lift the square and triangle tiling pattern to four dimensional space to perform phason analysis historically applied only in simulation and atomic systems. Hard particle models are unsuccessful at explaining the stability of the binary nanoparticle super lattice. However, with a simple isotropic soft particle model, we are able to demonstrate seeded growth of the experimental structure in simulation. These simulations indicate that the most important stabilizing properties of the short range structure are the size ratio of the particles and an A--B particle attraction.PhDMaterials Science and EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/120906/1/eirrgang_1.pd
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Self-Limiting Morphologies in Geometrically Frustrated Assemblies
Geometrically frustrated assembly, where locally preferred motifs are incompatible with constraints on global ordering of the assembly, may result in a super-extensive energy penalty to assembly growth and self-limitation of the assembly size. Using theory and simulation, we study how this mechanism may also shape the assembly\u27s boundary and its interior packing, which are distinct morphological changes. In Chapter 1, we provide some background and a theoretical framework for understanding self-limiting behavior due to geometric frustration. Three distinct projects are detailed in the subsequent chapters: original numerical results are presented on competing responses to frustration in helical bundles made of chiral filaments in Chapter 2, a novel tilt-curvature coupling and its consequences in microphase separated chiral rod membranes are illustrated by both numerical and theoretical results in Chapter 3, and a new particle model for simulation of saddle-wedge membrane assemblies with new theoretical advancement of a continuum model and new numerical results are presented in Chapter 4. In Chapter 5, we conclude by considering lessons from this work for future engineering of self-limiting materials
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