15 research outputs found

    A Zador-Like Formula for Quantizers Based on Periodic Tilings

    Full text link
    We consider Zador's asymptotic formula for the distortion-rate function for a variable-rate vector quantizer in the high-rate case. This formula involves the differential entropy of the source, the rate of the quantizer in bits per sample, and a coefficient G which depends on the geometry of the quantizer but is independent of the source. We give an explicit formula for G in the case when the quantizing regions form a periodic tiling of n-dimensional space, in terms of the volumes and second moments of the Voronoi cells. As an application we show, extending earlier work of Kashyap and Neuhoff, that even a variable-rate three-dimensional quantizer based on the ``A15'' structure is still inferior to a quantizer based on the body-centered cubic lattice. We also determine the smallest covering radius of such a structure.Comment: 8 page

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

    Full text link
    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 {\sigma} 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.Comment: 40 pages, 22 figure

    Foam geometry and structural design of porous material

    Get PDF
    EThOS - Electronic Theses Online ServiceGBUnited Kingdo

    Quantizing Using Lattice Intersections

    Get PDF

    Structure formation and identification in geometrically driven soft matter systems

    Get PDF
    Subdividing space through interfaces leads to many space partitions that are relevant to soft matter self-assembly. Prominent examples include cellular media, e.g. soap froths, which are bubbles of air separated by interfaces of soap and water, but also more complex partitions such as bicontinuous minimal surfaces. Using computer simulations, this thesis analyses soft matter systems in terms of the relationship between the physical forces between the system’s constituents and the structure of the resulting interfaces or partitions. The focus is on two systems, copolymeric self-assembly and the so-called Quantizer problem, where the driving force of structure formation, the minimisation of the free-energy, is an interplay of surface area minimisation and stretching contributions, favouring cells of uniform thickness. In the first part of the thesis we address copolymeric phase formation with sharp interfaces. We analyse a columnar copolymer system “forced” to assemble on a spherical surface, where the perfect solution, the hexagonal tiling, is topologically prohibited. For a system of three-armed copolymers, the resulting structure is described by solutions of the so-called Thomson problem, the search of minimal energy configurations of repelling charges on a sphere. We find three intertwined Thomson problem solutions on a single sphere, occurring at a probability depending on the radius of the substrate. We then investigate the formation of amorphous and crystalline structures in the Quantizer system, a particulate model with an energy functional without surface tension that favours spherical cells of equal size. We find that quasi-static equilibrium cooling allows the Quantizer system to crystallise into a BCC ground state, whereas quenching and non-equilibrium cooling, i.e. cooling at slower rates then quenching, leads to an approximately hyperuniform, amorphous state. The assumed universality of the latter, i.e. independence of energy minimisation method or initial configuration, is strengthened by our results. We expand the Quantizer system by introducing interface tension, creating a model that we find to mimic polymeric micelle systems: An order-disorder phase transition is observed with a stable Frank-Caspar phase. The second part considers bicontinuous partitions of space into two network-like domains, and introduces an open-source tool for the identification of structures in electron microscopy images. We expand a method of matching experimentally accessible projections with computed projections of potential structures, introduced by Deng and Mieczkowski (1998). The computed structures are modelled using nodal representations of constant-mean-curvature surfaces. A case study conducted on etioplast cell membranes in chloroplast precursors establishes the double Diamond surface structure to be dominant in these plant cells. We automate the matching process employing deep-learning methods, which manage to identify structures with excellent accuracy

    Modelling and characterisation of porous materials

    Get PDF
    Porous materials possessing random microstructures exist in both organic (e.g. polymer foam, bone) and in-organic (e.g. silica aerogels) forms. Foams and aerogels are two such materials with numerous engineering and scientific applications such as light-weight cores in sandwich structures, packaging, impact and crash structures, filters, catalysts and thermal and electrical insulators. As such, design and manufacture using these materials is an important task that can benefit significantly from the use of computer aided engineering tools. With the increase in computational power, multi-scale modelling is fast becoming a powerful and increasingly relevant computational technique. Ultimately, the aim is to employ this technique to decrease the time and cost of experimental mechanical characterisation and also to optimise material microstructures. Both these goals can be achieved through the use of multi-scale modelling to predict the macro-mechanical behaviour of porous materials from their microstructural morphologies, and the constituent materials from which they are made. The aim of this work is to create novel software capable of generating realistic randomly micro-structured material models, for convenient import into commercial finite element software. An important aspect is computational efficiency and all techniques are developed paying close attention to the computation time required by the final finite element simulations. Existing methods are reviewed and where required, new techniques are devised. The research extensively employs the concept of the Representative Volume Element (RVE), and a Periodic Boundary Condition (PBC) is used in conjunction with the RVEs to obtain a volume-averaged mechanical response of the bulk material from the micro-scale. Numerical methods such as Voronoi, Voronoi-Laguerre and Diffusion Limited Cluster-Cluster Aggregation are all employed in generating the microstructures, and where necessary, enhanced in order to create a wide variety of realistic microstructural morphologies, including mono-disperse, polydisperse and isotropic microstructures (relevant to gas-expanded foam materials) as well as diffusion-based microstructures (relevant for aerogels). Methods of performing large strain simulations of foams microstructures, up to and beyond the onset strain of densification are developed and the dependence of mechanical response on the size of an RVE is considered. Both mechanical and morphological analysis of the RVEs is performed in order to investigate the relationship between mechanical response and internal microstructural morphology of the RVE. The majority of the investigation is limited to 2-d models though the work culminates in extending the methods to consider 3-d microstructures
    corecore