Domain coarsening and interface kinetics in the Ising model

In this thesis, I investigate in detail two basic problems in nonequilibrium statistical mechanics. First, if a spin system such as a kinetic Ising model or a kinetic Potts model is quenched from supercritical temperature to subcritical temperature, how does the system coarsen, and what complexities arise as the system descends in energy toward one of its equilibrium states? Second, if a kinetic Ising model is evolved from a deterministic initial condition at zero temperature, how do the domain interfaces evolve in time? I first study the nonconserved coarsening of the kinetic spin systems mentioned above. The coarsening of a 2d ferromagnet can be described exactly by an intriguing connection with continuum critical percolation. Furthermore, careful simulations of phase ordering in the 3d Ising model at zero temperature reveal strange nonstatic final states and anomalously slow relaxation modes, which we explain in detail. I find similarly rich phenomena in the zero-temperature evolution of a kinetic Potts model in 2d, where glassy behavior is again manifest. We also find large-scale avalanches in which clusters merge and dramatically expand beyond their original convex hulls at late times in the dynamics. Next, I study the geometrically simpler problem of the evolution of a single corner interface in the Ising model. We extend prior work by investigating the Ising Hamiltonian with longer interaction range. We solve exactly the limiting shapes of the corner interface in 2d for several interaction ranges. In 3d, where analytical treatments are notoriously difficult, we develop novel methods for studying corner interface growth. I conjecture a growth equation for the interface that agrees quite well with simulation data, and I discuss the interface's surprising geometrical features. In the summary, I discuss the broader implications of our findings and offer some thoughts on possible directions for future work

Accurate Prediction of Core Properties for Chiral Molecules using Pseudo Potentials

Pseudo potentials (PPs) constitute perhaps the most common way to treat relativity, often in a formally non-relativistic framework, and reduce the electronic structure to the chemically relevant part. The drawback is that orbitals obtained in this picture (called pseudo orbitals (POs)) show a reduced nodal structure and altered amplitude in the vicinity of the nucleus, when compared to the corresponding molecular orbitals (MOs). Thus expectation values of operators localized in the spatial core region that are calculated with POs, deviate significantly from the same expectation values calculated with all-electron (AE) MOs. This study describes the reconstruction of AE MOs from POs, with a focus on POs generated by energy consistent pseudo Hamiltonians. The method reintroduces the nodal structure into the POs, thus providing an inexpensive and easily implementable method that allows to use nonrelativistic, efficiently calculated POs for good estimates of expectation values of core-like properties. The discussion of the method proceeds in two parts: Firstly, the reconstruction scheme is developed for atomic cases. Secondly, the scheme is discussed in the context of MO reconstruction and successfully applied to numerous numerical examples. Starting from the equations of the state-averaged multi-configuration self- consistent field method, used for the generation of energy consistent pseudo potentials, the electronic spectrum of the many-electron Hamiltonian is linked to the spectrum of the effective one-electron Fock operator by means of various models systems. This relation and the Topp–Hopfield–Kramers theorem, are used to show the shape-consistency of energy-consistent POs for atomic systems. Shape-consistency describes POs that follow distinct AOs exactly outside a core-radius r_core . In the cases presented here, shape-consistency holds to a high degree and it follows that in atomic systems every PO has one distinct partner in the set of AOs. The overlap integral between these two orbitals is close to one, as it is determined mainly by the spatial orbital parts outside r_core . Expanding, e.g., a 5s PO in occupied AOs, the 5s AOs will have the highest contribution. The POs itself contains contributions from high-energy unoccupied AOs as well (e.g. 15s), which damp the nodal structure of the POs near the nucleus. Consequently, neglecting contributions from unoccupied orbitals in a projection of the POs reintroduces the nodal structure. This approach is not directly suitable for the reconstruction of MOs, as they often need to be expanded in a full set of AOs at each atomic center, including all unoccupied orbitals, to properly account for the electron density distribution in the molecule. However, it is shown that the occupied MOs are well described by occupied and low-energy unoccupied AOs only and a mapping of the POs onto a basis containing only these orbitals reconstructs the nodal structure of the MO. The approach uses only standard integrals available in most quantum chemistry programs. The computational cost of these integrals scales with N^2 , where N is the number of basis functions. The most time consuming step is a Gram-Schmidt orthogonalization, which scales in this implementation with MN^2 , M being the number of reconstructed orbitals. The reconstruction method is subsequently tested: Valence orbitals of atomic, closed-shell systems were reconstructed numerically exactly. The influence of numerical parameters is investigated using the molecule BaF . It is shown that the method is basis set dependent: One has to ensure that the PO basis can be expanded exactly in the basis of AOs. Violating this rule of thumb may degrade the quality of reconstructed orbitals. Additionally, the representation of MOs by a linear combination of occupied and unoccupied AOs is investigated. For the exemplary systems, the shells included in the fitting procedure of the PP were sufficient. Reconstruction of the alkaline earth monofluorides showed that periodic trends can be reconstructed as well. Scaling of hyperfine structure parameters with increasing atomic number is discussed. For hydrogenic atoms, the scaling should be linear, whereas small deviations from the linear behavior were observed for molecules. The scaling laws computed from reconstructed and reference orbitals were almost identical. In this context, the failure of commonly used relativistic enhancement factors beyond atomic number 100 is discussed. Applicability of the method is also tested on parity violating properties for which the main contribution is generated by the valence orbitals near the nucleus. Symmetry-independence of the method is shown by successful reconstruction of orbitals of the tetrahedral PbCl_4 and chiral NWHClF. The reliable reconstruction of chemical trends is shown with the help of the NWHClF derivatives NWHBrF and NWHFI. The study of chiral compounds as, e.g., NWHClF and its group 17 derivatives, which have been proposed as paradigm for the detection of parity-violation in chiral molecules, remains of great importance. Especially the direct determination of absolute configuration of chiral centers is still non-trivial. The author contributed to this field with a self-written molecular dynamics (MD) program to simulate Coulomb explosions and thus to provide an insight especially into the early explosion stages directly after an instantaneous multi-ionization of the molecule CHBrClF, comparable to experiments using the Cold Target Recoil Ion Momentum Spectroscopy (COLTRIMS) technique. An algorithm for the determination of the investigated molecule’s absolute configuration from time-of-flight data and detection locations of molecular fragments is included in the program. The program was used to generate experiment-equivalent data which allowed for the first time the investigation of non-racemic mixtures by the analysis routines of the experiment. The MD program includes harmonic and anharmonic bond potentials. A charge-exchange model can model partial charges in early phases of the Coulomb explosion. Furthermore, Born–Oppenheimer MD simulations and statistical models are used to explain the relative abundance of products belonging to competing reaction channels, as obtained by photoion coincidence measurements. Additionally, qualitative statements about reaction branching ratios are made by comparing the partition functions of involved degrees of freedom. Analytic equations for partition functions of simple models are used to provide a simple formula allowing fast estimates of reaction branching ratios

Topology optimization of cables, cloaks, and embedded lattices

Materials play a critical role in the behavior and functionality of natural and engineered systems. For example, the use of cast-iron and steel led to dramatically increased bridge spans per material volume with the move from compression-dominant arch bridges to tensile-capable truss, suspension, and cable-stayed bridges; materials underlie many of the major technological advancements in the auto and aerospace industries that have made cars and airplanes increasingly light, strong, and damage tolerant; and the great diversity of biological materials and bio-composites enable complex biological and mechanical functions in nature. Topology optimization is a computational design method that simultaneously enhances efficiency and design freedom of engineered parts, but is often limited to a single, solid, isotropic, linear-elastic material. To understand how the material space can be tailored to enhance design freedom and/or promote desired mechanical behavior, several topology optimization problems are explored in this dissertation in which the space of available materials is either relaxed or restricted. Specifically, in a discrete topology optimization setting defined by 1D (truss) elements, tension-only systems are targeted by restricting the material space to that of a tension-only material and tailoring a formulation to handle the associated nonlinear mechanics. The discrete setting is then enhanced to handle 2D (beam) elements in pursuit of cloaking devices that hide the effect of a hole or defect on the elastostatic response of lattice systems. In this case the material space is relaxed to allow for a continuous range of stiffness and the objective is formulated as a weighted least squares function in which the physically-motivated weights promote global stiffness matching between the cloaked and undisturbed systems. Continuous 2D and 3D structures are also explored in a density-based topology optimization setting in which the material space is relaxed to accommodate an arbitrary number of candidate materials in a general continuum mechanics framework that can handle material anisotropy. The theoretical and physical relevance of such framework is highlighted via a continuous embedding scheme that enables manufacturing in the relaxed (or restricted) design space of lattice-based microstructural-materials. Implications of varying the material design space on the mechanics, mathematics, and computations needed for topology optimization are discussed in detail.Ph.D

Metal Citrate Cubanes: Synthesis, Characterization and Properties

La tesis, titulada "Cubanos Metálicos con el Ligando Citrato: Obtención, Caracterización y Propiedades," se basa en el estudio de compuestos conocidos comúnmente como cubanos, formados por el ligando citrato (tetra anión del ácido cítrico) y cobalto o manganeso. El trabajo se ha dividido en tres partes para una exposición más clara de los resultados obtenidos. En la primera parte se ha llevado a cabo un completo estudio de la unidad cubano. En él, aspectos importantes de dicha unidad tales como su estructura, quiralidad y topología de su periferia han sido descritos y comparados. Así mismo, se ha propuesto un sistema de nomenclatura para el sistema cubano con el objetivo de simplificar futuras descripciones de su estructura y coordinación y de unificar los resultados. Finalmente, se ha revisado y clasificado los modos de coordinación de las diferentes unidades cubano obtenidas así como los de los ejemplos previamente publicados. La segunda parte de la tesis se centra en el estudio de los cubanos de citrato y cobalto. Se ha comprobado que estos compuestos son capaces de comportarse como ¿Single Molecule Magnets¿ (SMMs), exhibiendo interesantes propiedades magnéticas. Se ha logrado la síntesis de una familia de compuestos relacionados entre sí que presentan diferentes dimensionalidades. El primer capítulo de esta parte es una introducción al magnetismo y a los SMMs desde una aproximación química. El capítulo 3 está dedicado al estudio de una pareja de compuestos relacionados a través de una trasformación reversible de monocristal a monocristal (SC-SC, por sus siglas en inglés). Ambas especies son moléculas discretas. Una de ellas presenta una estructura cristalina modulada. Los capítulos 4 y 5 constan de un conjunto de citrato-cubanos de cobalto con estructuras 2-D y 3-D respectivamente. Finalmente, la tercera y última parte de la tesis se centra en los compuestos de citrato y manganeso cuya unidad básica es el cubano. Estos compuestos, a pesar del evidente parecido estructural con los compuestos de cobalto, no presentan comportamiento de SMM. Sin embargo, se ha observado que son capaces de dar lugar a reacciones de pérdida y ganancia de agua de forma reversible en condiciones ambiente a pesar de tratarse de materiales no porosos. Además, este proceso tiene lugar manteniendo la cristalinidad de las muestras lo que nos ha permitido su estudio mediante difracción de Rayos X. Esto ha aportado valiosa información acerca del mecanismo que opera en cada caso. Así pues, el primer capítulo de esta parte es una introducción a la química y reactividad en estado sólido así como al transporte de protones en diferentes materiales. Los siguientes capítulos, números 7, 8 y 9 se centran en el estudio y propiedades de compuestos 0-D, 1-D y 2-D respectivamente basados en cubanos de citrato y manganeso