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

Curved Voronoi diagrams

Voronoi diagrams are fundamental data structures that have been extensively studied in Computational Geometry. A Voronoi diagram can be defined as the minimization diagram of a finite set of continuous functions. Usually, each of those functions is interpreted as the distance function to an object. The as- sociated Voronoi diagram subdivides the embedding space into regions, each region consisting of the points that are closer to a given object than to the others. We may define many variants of Voronoi diagrams depending on the class of objects, the distance functions and the embedding space. Affine di- agrams, i.e. diagrams whose cells are convex polytopes, are well understood. Their properties can be deduced from the properties of polytopes and they can be constructed efficiently. The situation is very different for Voronoi dia- grams with curved regions. Curved Voronoi diagrams arise in various contexts where the objects are not punctual or the distance is not the Euclidean dis- tance. We survey the main results on curved Voronoi diagrams. We describe in some detail two general mechanisms to obtain effective algorithms for some classes of curved Voronoi diagrams. The first one consists in linearizing the diagram and applies, in particular, to diagrams whose bisectors are algebraic hypersurfaces. The second one is a randomized incremental paradigm that can construct affine and several planar non-affine diagrams. We finally introduce the concept of Medial Axis which generalizes the concept of Voronoi diagram to infinite sets. Interestingly, it is possible to efficiently construct a certified approximation of the medial axis of a bounded set from the Voronoi diagram of a sample of points on the boundary of the set

A Micromechanically-Informed Model of Thermal Spallation with Application to Propulsive Landing

During the propulsive landing of spacecraft, the retrorocket exhaust plume introduces the landing site surface to significant pressure and heating. Landing site materials include concrete on Earth and bedrock on other bodies, two highly brittle materials. During a landing event, defects and voids in the material grow due to thermal expansion and coalesce, causing the surface to disaggregate or spall. After a spall is freed from the surface, the material beneath it is exposed to the pressure and heat load until it spalls, continuing the cycle until engine shutdown. Spalls and debris entrained in the exhaust plume risk damaging the lander or nearby assets- a risk that increases for larger engines. The purpose of this work is to develop a micromechanically-informed model of thermal spallation to improve understanding of this process, in the context of propulsive landing. A preliminary simulation of landing site spallation, utilizing an empirical thermal spallation model, indicates that spallation may occur for human-scale Mars landers. This model, however, was developed for drilling through granite, which has a fundamentally different microstructure compared to typical landing sites, necessitating a more general approach. To that end, highly-detailed simulations of thermomechanical loading, applied to representative microstructures, inform a functional relationship between applied heat flux and spallation rate. These representative microstructures can be generated using an algorithm that has been validated for a wide variety of materials, including basalt from Gusev Crater, Mars.Ph.D

Minkowski Sum Construction and other Applications of Arrangements of Geodesic Arcs on the Sphere

We present two exact implementations of efficient output-sensitive algorithms that compute Minkowski sums of two convex polyhedra in 3D. We do not assume general position. Namely, we handle degenerate input, and produce exact results. We provide a tight bound on the exact maximum complexity of Minkowski sums of polytopes in 3D in terms of the number of facets of the summand polytopes. The algorithms employ variants of a data structure that represents arrangements embedded on two-dimensional parametric surfaces in 3D, and they make use of many operations applied to arrangements in these representations. We have developed software components that support the arrangement data-structure variants and the operations applied to them. These software components are generic, as they can be instantiated with any number type. However, our algorithms require only (exact) rational arithmetic. These software components together with exact rational-arithmetic enable a robust, efficient, and elegant implementation of the Minkowski-sum constructions and the related applications. These software components are provided through a package of the Computational Geometry Algorithm Library (CGAL) called Arrangement_on_surface_2. We also present exact implementations of other applications that exploit arrangements of arcs of great circles embedded on the sphere. We use them as basic blocks in an exact implementation of an efficient algorithm that partitions an assembly of polyhedra in 3D with two hands using infinite translations. This application distinctly shows the importance of exact computation, as imprecise computation might result with dismissal of valid partitioning-motions.Comment: A Ph.D. thesis carried out at the Tel-Aviv university. 134 pages long. The advisor was Prof. Dan Halperi

The structure of random ellipsoid packings

Disordered packings of ellipsoidal particles are an important model for disordered granular matter and can shed light on geometric features and structural transitions in granular matter. In this thesis, the structure of experimental ellipsoid packings is analyzed in terms of contact numbers and measures from mathematical morphometry to characterize of Voronoi cell shapes. Jammed ellipsoid packings are prepared by vertical shaking of loose configurations in a cylindrical container. For approximately 50 realizations with packing fractions between 0.54 and 0.70 and aspect ratios from 0.40 to 0.97, tomographic images are recorded, from which positions and orientations of the ellipsoids are reconstructed. Contact numbers as well as discrete approximations of generalized Voronoi diagrams are extracted. The shape of the Voronoi cells is quantified by isotropy indexes b,r,s,n based on Minkowski tensors. In terms of the Voronoi cells, the behavior for jammed ellipsoids differs from that of spheres; the Voronoi Cells of spheres become isotropic with increasing packing fraction, whereas the shape of the Voronoi Cells of ellipsoids with high aspect ratio remains approximately constant. Contact numbers are discussed in the context of the jamming paradigm and it is found that the frictional ellipsoid packings are hyperstatic, i.e. have more contacts than are required for mechanical stability. It is observed, that the contact numbers of jammed ellipsoid packings predominantly depend on the packing fraction, but also a weaker dependence on the aspect ratio and the friction coefficient is found. The achieved packing fractions in the experiments lie within upper and lower limits expected from DEM simulations of jammed ellipsoid packings. Finally, the results are compared to Monte Carlo and Molecular Dynamics data of unjammed equilibrium ellipsoid ensembles. The Voronoi cell shapes of equilibrium ensembles of ellipsoidal particles with a low aspect ratio become more anisotropic by increasing the packing fraction, while the cell shape of particles with large aspect ratios does the opposite. The experimental jammed packings are always more anisotropic than the corresponding densest equilibrium configuration

Clifford wavelets for fetal ECG extraction

Analysis of the fetal heart rate during pregnancy is essential for monitoring the proper development of the fetus. Current fetal heart monitoring techniques lack the accuracy in fetal heart rate monitoring and features acquisition, resulting in diagnostic medical issues. The challenge lies in the extraction of the fetal ECG from the mother's ECG during pregnancy. This approach has the advantage of being a reliable and non-invasive technique. For this aim, we propose in this paper a wavelet/multi-wavelet method allowing to extract perfectly the feta ECG parameters from the abdominal mother ECG. The method is essentially due to the exploitation of Clifford wavelets as recent variants in the field. We prove that these wavelets are more efficient and performing against classical ones. The experimental results are therefore due to two basic classes of wavelets and multi-wavelets. A first-class is the classical Haar Schauder, and a second one is due to Clifford valued wavelets and multi-wavelets. These results showed that wavelets/multiwavelets are already good bases for the FECG processing, provided that Clifford ones are the best.Comment: 21 pages, 8 figures, 1 tabl