15 research outputs found
A Zador-Like Formula for Quantizers Based on Periodic Tilings
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
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
EThOS - Electronic Theses Online ServiceGBUnited Kingdo
Structure formation and identification in geometrically driven soft matter systems
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
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MODELING CHAIN PACKING IN COMPLEX PHASES OF SELF-ASSEMBLED BLOCK COPOLYMERS
Block copolymer (BCP) melts undergo microphase seperation and form ordered soft matter crystals with varying domain shapes and symmetries. We study the con- nection between diblock copolymer molecular designs and thermodynamic selection of ordered crystals by modeling features of variable sub-domain geometry filled with individual blocks within non-canonical sphere-like and network phases that together with layered, cylindrical and canonical spherical phases forms “natural forms” of self- assembled amphiphilic soft matter at large. First, we present a model to revise our understanding of optimal Frank-Kasper sphere-like morphologies by advancing the- ory to account for varying domain volumes. We then develop generic approaches to quantify local changes to domain thickness or packing frustration using medial sets and show its application to morphologies with arbitrary domain topologies and sym- metries in both theoretical models and experimental data. We further use medial sets as a proxy for terminal boundaries of blocks within different domains and revise thermodynamic models of BCP assembly in the strong segregation limit. Finally, we use this revised model to study effect of elastic stiffness asymmetry on relaxing packing frustration experienced by BCPs in tubular and matrix domains leading to equilibrium double gyroid network morphology in diblock copolymers
Modelling and characterisation of porous materials
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