Diszkrét és kombinatórikus geometriai kutatások = Topics in discrete and combinatorial geometry

A most lezárult OTKA grant, 8 résztvevő diszkrét geometriai kutatását támogatta. Itt a témák ilusztrálására kiemelünk néhányat az elért 72 publikációból. 1. Jelentős eredmények születtek (8 cikk) gráfok síkba rajzolhatóságáról, például az úgynevezett metszési számról. 2. Többek között sikerült igazolni Katchalski és Lewis 20 éves sejtését, mely szerint diszjunkt egységkörökből álló rendszereknél ha bármely három körnek van közös metsző egyenese akkor van olyan egyenes, amely legfeljebb 2 kör kivételével valamennyit metsz. 3. Littlewood (1964) problémájaként ismert volt az a kérdés, hogy hány henger érintheti kölcsönösen egymást? Viszonylag alacsony felső korlátot találtunk és egy régóta ismert elhelyzés valótlanságát igazoltuk. 4. Többszörös fedések egyszerű fedésekre való szétbontását vizsgáltuk és értünk el lényeges előrelépést. 5. A Borsuk-féle darabolási problémanak azt a variánsát vizsgáltuk, amelyben a darabolást u. n. hengeres darabolásra korlátozták. 6. Bebizonyítottuk, hogy ''nem nagyon elnyúlt'' ellipszisek esetében a sík legritkább fedésének meghatározásánál el lehet tekinteni az u.n. nem-keresztezési feltételtől. 7. A sejtetthez nagyon közeli korlátot találtunk arra a problémára, hogy az n-dimenziós térben legfeljebb hány homotetikus konvex test helyezhető el úgy, hogy bármely kettő érintse egymást. | Discrete geometry in Hungary flourished since the sixties as a result of the work of László Fejes Tóth. The supported research of 8 participant also belongs to this area. Here we illustrate the achieved 72 publications by mentioning a few results. 1. Important theorems (8 papers) were proved concerning graph drawing. 2. Among others, a 20 year old problem of Katchalsky was proved, stating that in a packing of congruent circles, if any three has a common transversal, then there is a line, which avoids at most two of the circles. 3. Concerning a conjecture of Littlewood we found a small upper bound for the number of infinite cylinders which mutually touch each other. 4. We studied decomposability of multiple coverings into single coverings. 5. We studied that variant of the famous Borsuk problem where the partitions are restricted to cylindrical partitions. 6. We proved that in case of ellipses which are not ''too long'' at determining the thinnest covering one can omit the usually needed noncrossing condition. 7. A bound close to the conjectured bound was found concerning the number of n-dimensional homothetic convex solids which mutually touch each other

A Study of the Thermodynamics of Small Systems and Phase Transition in Bulk Square Well-Hard Disk Binary Mixture

Under the umbrella of statistical mechanics and particle-based simulations, two distinct problems have been discussed in this study. The first part included systems of finite clusters of three and 13 particles, where the particles are interacting via Lennard Jones potential. A machine learning technique, Diffusion Maps (DMap), has been employed to the large datasets of thermodynamically small systems from Monte Carlo simulations in order to identify the structural and energetic changes in these systems. DMap suggests at most three dimensions are required to describe and identify the systems with 9 (N = 3) and 39 (N = 13) dimensions. At the end of the study, a model has been proposed to functionalize the potential energy in terms of geometric variables that are identified with a heuristic screening. Investigation of the thermodynamics of bulk systems was another major focus of this thesis. The phase diagrams of the pure square-well solids and binary mixture of square-well and hard-disk particles, under the assumption of a pseudo-single- component model, have been constructed, and the phase equilibria behaviors were discussed. The datasets were also created in Monte Carlo simulations. The results showed isostructural solid-solid phase transition, which was previously identified that the pure square-well system with a very short range of attraction undergoes, also occurs in the presence of additional hard-disk components, namely for the binary mixture of square-well and hard-disk systems

Shape stability and bundling of ultrathin nanowires

Ultrathin nanowires are promising nanoscale materials. They can reach length-to-diameter aspect ratios exceeding 1000, making them suitable building blocks for optoelectronic devices such as transparent conducting films. An organic ligand shell surrounds their inorganic core, provides colloidal stability, and guides their one-dimensional growth. Two unresolved issues limit their application. Nanowires can agglomerate into elongated bundles, but efficient use of this superstructure is difficult since we do not yet understand the bundling mechanisms. Furthermore, nanowires are prone to the Plateau-Rayleigh instability: thin wires tend to fragment into discrete spheroidal particles to reduce their surface energy, limiting their lifetime and reliability. This thesis investigates superstructure formation and nanowire stability and the link between both topics. Bundles are shown to emerge in non-polar solvents for entropic reasons. Solvent or unbound ligand molecules align in proximity to the ligand shell, thus losing entropy. Bundling decreases this loss in entropy by reducing contact with the bulk solvent. The structural stability of nanowires is enhanced or degraded by the ligand shell, depending on the relationship between free energy and local surface curvature. Kinetic barriers in ad- and desorbing ligands and rearrangement of surface atoms slow down the break-up. Bundling further stabilizes the wires by confining the space available to them.Ultradünne Nanodrähte, bestehend aus einem anorganischen Kern und einer organischen Ligandenhülle, können Aspektverhältnisse von über 1:1000 erreichen und sind potenzielle Materialien für optoelektronische Technologien wie transparente Elektroden. Einer breiteren Anwendung stehen zwei Herausforderungen entgegen. Nanodrähte können zu Bündeln agglomerieren, aber eine effiziente Nutzung dieser Superstruktur ist schwierig, da unser Verständnis der zugrundeliegenden Bündelungsmechanismen unvollständig ist. Zudem sind Nanodrähte instabil: gemäß der Plateau-Rayleigh-Instabilität zerbrechen sie zur Reduktion ihrer Oberflächenenergie in kleinere Nanopartikel, was ihre Langzeit-Anwendung verhindert. Sowohl Formstabilität und Superstruktur als auch der Zusammenhang zwischen beiden Themen wurden in dieser Dissertation untersucht. Bündel entstehen in unpolaren Lösemitteln, weil sich Lösemittelmoleküle oder freie Liganden parallel zur Ligandenhülle ausrichten und dabei Entropie verlieren. Durch die Anordnung in Bündeln wird der Kontakt zum Lösemittel reduziert, sodass der Entropieverlust geringer ausfällt. Die Formstabilität von Nanodrähten wird von der Ligandenhülle verbessert oder verschlechtert, je nach Zusammenhang zwischen freier Energie und Oberflächenkrümmung. Kinetische Barrieren in der Ad- und Desorption von Liganden und der Reorganisation der Oberfläche verlangsamen den Zerfall. Bündel verbessern die Formstabilität, indem sie den für die Nanodrähte verfügbaren Raum begrenzen

The role of clay mineral diagenesis in overpressure generation and compaction of siliciclastic mudstones

Clay mineral diagenesis has a considerable effect on the physical properties of siliciclastic mudstones, with important implications for pore pressure prediction. The dominant clay mineral reaction, the conversion of smectite to illite, involves a series of dissolution and reprecipitation reactions which results in a significant change in the orientation of the clay mineral fabric. Unloading is a direct result of clay mineral diagenesis and concomitant fabric destabilisation, due to the local transfer of load from dissolving detrital clay grains to fluid. Pore pressure is then a function of the rate at which it is generated by clay mineral diagenesis (and other mechanisms such as disequilibrium compaction) and the rate at which it is dissipated by compaction and fluid flow. Clear evidence has been found for chemical compaction (porosity loss/sediment volume reduction) associated with illitization of smectite in Miocene mudstones in the Central Malay Basin, in Cretaceous mudstones at Haltenbanken, offshore mid-Norway, in Cretaceous to Tertiary mudstones in the Sergipe-Alagoas Basin, offshore Brazil and in Triassic mudstones in the North Sea Central Graben from measured physical, textural, and mineralogical properties, and from log responses. In addition to this diagenetically mature, illitized mudstones continue to compact mechanically with increasing effective stress. The newly presented data have been interpreted to discriminate between two models for the chemical compaction of diagenetically altered mudstones proposed by previous researchers: (effective) stress-independent chemical compaction and chemically-enhanced mechanical compaction. Key evidence in favour of the chemically-enhanced mechanical compaction model comes from density logs of Cretaceous mudstones at the Halten Terrace, offshore mid-Norway in association with the pore pressure history inferred by previous pore pressure analysis. This model is also consistent with the petrographic evidence that clay-rich siliciclastic mudstones have a clay-supported matrix both before and after illitization. Established methods of pore pressure estimation do not correctly account for the mechanical and chemical contributions to mudstone compaction, except empirically or in favourable circumstances where use can be made of data from offset wells with similar lithology, burial history and temperature history

Diagenetic controls on the phyllosilicate fabric of mudstones

PhD ThesisChanges to the orientation of phyllosilicates in mudstones in the Podhale Basin and the Northern North Sea have been quantified using High Resolution X-ray Texture Goniometry (HRXTG). Samples were studied from four wells: Chocholów PIG-1 and Bukowina Tatrzanska PIG-1 from the Podhale Basin, and Magnus 211/12-2 and Rhum 3/29a4 from the Northern North Sea. The samples cover a temperature range of ~60 to 170°C. Porosity and pore size distributions were quantified by mercury intrusion porosimetry. Changes in mineralogy with depth have been addressed by Quantitative Xray Diffraction (XRD). Analytical Transmission Electron Microscopy (ATEM) was performed on illite-smectite from selected samples from the Podhale Basin. All wells were subjected to 1-D maturity modelling using Genesis 4.8®. Maximum burial temperatures of the Podhale samples range from ~60 to ~170°C (3 and a km). XRD reveals that the sample sets straddle part of the smectite to illite RO to R1 ordering transition zone and that illitization terminates at ~8O% illite layers. An increase in chlorite and a decrease in both kaolinite and K-feldspar occur over the same depth/temperature interval. Quartz contents increase with increasing temperature. At the top of the sequence, porosities are. close to 10%, indicating very substantial compaction. However, HRXTG shows that there is only a modestly preferred alignment of both iIIite-smectite and chlorite-kaolinite. A strongly aligned fabric is developed through the smectite to illite transition in Chocholów, with a less marked but readily measurable increase beyond the transition in Bukowina Tatrzanska. Samples from Chocholów document a strong relationship between millimetre-scale fabric and clay mineral transformations, but the relationship is more complex in Bukowina Tatrzanska. The trend in %1 in illite-smectite shows a gradual increase with depth from ~37% layers to ~75% layers in Chochol6w, with a fabric change from ~3.4 m.r.d. to 5.2 m.r.d. The trend in the %1 in illite-smectite is constant at ~75% of layers in Bukowina Tatrzanska, but with a fabric increase from ~4.8 m.r.d. to ~6.2 m.r.d. ATEM has been performed on end member samples to determine chemical variation in illite-smectite. ATEM reveals significant differences in the octahedral sheet Fe concentrations of samples from Bukowina Tatrzanska with increasing temperature, pointing to continued crystallite change beyond the termination of smectite illitization. A relationship between SADP and structural formulae has been found. Single crystal patterns dominate where structural formulae reveal little or no substitution in the octahedral sheet. Samples from Magnus (~3200m; ~120°C) and Rhum (~4800m; ~150°C) consist of Tertiary, Cretaceous and Jurassic sediments of varying thicknesses and maturity. Porosity reduction in both wells is from relatively uncompacted sediments (~45%) to tight muds (~5%). Preferred alignment of phyllosilicates is weak in shallow samples from both wells (m.r.d. ~2.3) and only shows a modest increase in Magnus as compaction and burial diagenesis (smectite illitization) proceed. Stronger phyllosilicate alignments are developed at the base of Rhum (6.1 m.r.d.). Generally, a high correlation between the two mineral peaks exists, the exception being the Jurassic of Rhum where low kaolinite and chlorite abundance as measured by XRD explains the low fabric development by HRXTG. XRD also reveals an increase in the %1 in illite-smectite, 60% in Magnus and 65% in Rhum, with increasing depth and temperature. XRD reveals differences in mineralogy between the Jurassic, Cretaceous and Tertiary reflecting differences in depositional environment and sediment input. 1-D maturity modelling has established a correlation between, vitrinite reflectance, %1 in illite-smectite and the development of preferred alignment of phyllosilicates. Little change in preferred alignment at porosities >15% has been noted, with the major changes occurring between ~15% and 5% porosity; this coincides with the change between RO and R1 ordering in I-S and sits in a temperature range between 75°C and 125°C. The mechanism for preferred orientation change is therefore postulated to be dependent on the dissolution-precipitation reaction mechanism in smectite to illite formation in a high effective stress environment

Between the scales: water from different perspectives

In der vorliegenden Arbeit werden verschiedene Wassermodelle in sogenannten Multiskalen-Computersimulationen mit zwei Auflösungen untersucht, in atomistischer Auflösung und in einer vergröberten Auflösung, die als "coarse-grained" bezeichnet wird. In der atomistischen Auflösung wird ein Wassermolekül, entsprechend seiner chemischen Struktur, durch drei Atome beschrieben, im Gegensatz dazu wird ein Molekül in der coarse-grained Auflösung durch eine Kugel dargestellt.rnrnDie coarse-grained Modelle, die in dieser Arbeit vorgestellt werden, werden mit verschiedenen coarse-graining Methoden entwickelt. Hierbei kommen hauptsächlich die "iterative Boltzmann Inversion" und die "iterative Monte Carlo Inversion" zum Einsatz. Beides sind struktur-basierte Ansätze, die darauf abzielen bestimmte strukturelle Eigenschaften, wie etwa die Paarverteilungsfunktionen, des zugrundeliegenden atomistischen Systems zu reproduzieren. Zur automatisierten Anwendung dieser Methoden wurde das Softwarepaket "Versatile Object-oriented Toolkit for Coarse-Graining Applications" (VOTCA) entwickelt.rnrnEs wird untersucht, in welchem Maße coarse-grained Modelle mehrere Eigenschaftenrndes zugrundeliegenden atomistischen Modells gleichzeitig reproduzieren können, z.B. thermodynamische Eigenschaften wie Druck und Kompressibilität oder strukturelle Eigenschaften, die nicht zur Modellbildung verwendet wurden, z.B. das tetraedrische Packungsverhalten, welches für viele spezielle Eigenschaft von Wasser verantwortlich ist.rnrnMit Hilfe des "Adaptive Resolution Schemes" werden beide Auflösungen in einer Simulation kombiniert. Dabei profitiert man von den Vorteilen beider Modelle:rnVon der detaillierten Darstellung eines räumlich kleinen Bereichs in atomistischer Auflösung und von der rechnerischen Effizienz des coarse-grained Modells, die den Bereich simulierbarer Zeit- und Längenskalen vergrössert.rnrnIn diesen Simulationen kann der Einfluss des Wasserstoffbrückenbindungsnetzwerks auf die Hydration von Fullerenen untersucht werden. Es zeigt sich, dass die Struktur der Wassermoleküle an der Oberfläche hauptsächlich von der Art der Wechselwirkung zwischen dem Fulleren und Wasser und weniger von dem Wasserstoffbrückenbindungsnetzwerk dominiert wird.rnWater is one of the most frequently studied fluids on earth. In this thesis, water was investigated at two resolutions using multi-scale computer simulation techniques. First, the atomistic and coarse-grained resolutions were studied separately. In the atomistic resolution, a~water molecule is described chemically by three atoms, while in the coarse-grained case, a~molecule is modeled by a~sphere.rnrnIn this work, various coarse-grained models have been developed using different coarse-graining techniques, mainly iterative Boltzmann inversion and iterative inverse Monte Carlo, which are structure-based approaches that aim to reproduce distributions, such as the pair distribution functions, of the atomistic model. In this context the Versatile Object-oriented Toolkit for Coarse-graining applications (VOTCA) was developed.rnrnIt was studied to which extent the coarse-grained models can simultaneously reproduce several properties of the underlying atomistic model, such as thermodynamic properties like pressure and compressibility or structural properties, which have not been used in the coarse-graining process, e.g. the tetrahedral packing behavior, which is responsible for many special properties of water.rnrnSubsequently, these two resolutions were combined using the adaptive resolution scheme, which combines the advantage of atomistic details in a~small cavity of high resolution with the computational efficiency of the coarse-grained model in order to access larger time and length scales. In this scheme, the introduced coarse-grained models were used to study the influence of the hydrogen bonds on the hydration of small fullerenes. It was found that the interface structure is more dependent on the nature of the interaction between the solute and water molecules than on the presence of the hydrogen bond network.r

Program and Abstracts for Clay Minerals Society 28th Annual Meeting

This volume contains abstracts that were accepted for presentation at the annual meeting. Some of the main topics covered include: (1) fundamental properties of minerals and methods of mineral analysis; (2) surface chemistry; (3) extraterrestrial clay minerals; (4) geothermometers and geochronometers; (5) smectite, vermiculite, illite, and related reactions; (6) soils and clays in environmental research; (7) kaolinite, halloysite, iron oxides, and mineral transformations; and (8) clays in lakes, basins, and reservoirs