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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.
Research on monolayers of amphiphilic lipids on aqueous solution is of basic importance in surface science. Due to the applicability of a variety of surface sensitive techniques, floating insoluble monolayers are very suitable model systems for the study of order, structure formation and material transport in two dimensions or the interactions of molecules at the interface with ions or molecules in the bulk (headword 'molecular recognition'). From the behavior of monolayers conclusions can be drawn on the properties of lipid layers on solid substrates or in biological membranes. This work deals with specific and fundamental interactions in monolayers both on the molecular and on the microscopic scale and with their relation to the lattice structure, morphology and thermodynamic behavior of monolayers at the air-water interface. As model system especially monolayers of long chain fatty acids are used, since there the molecular interactions can be gradually adjusted by varying the degree of dissociation by means of the suphase pH value. For manipulating the molecular interactions besides the subphase composition also temperature and monolayer composition are systematically varied. The change in the monolayer properties as a function of an external parameter is analyzed by means of isotherm and surface potential measurements, Brewster-angle microscopy, X-ray diffraction at grazing incidence and polarization modulated infrared reflection absorption spectroscopy. For this a quantitative measure for the molecular interactions and for the chain conformational order is derived from the X-ray data. The most interesting results of this work are the elucidation of the origin of regular polygonal and dendritic domain shapes, the various effects of cholesterol on molecular packing and lattice order of long chain amphiphiles, as well as the detection of an abrupt change in the head group bonding interactions, the chain conformational order and the phase transition pressure between tilted phases in fatty acid monolayers near pH 9. For the interpretation of the latter point a model of the head group bonding structure in fatty acid monolayers as a function of the pH value is developed.