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<p>A framework is described using the phase-field crystal model for the study of premelting in binary alloys through short-range interfacial interactions that arise from the structure of grain boundaries. A nonconserved model A formulation of PFC was used to model grain boundaries in two dimensions for several different angles of misorientation: 27.8, 21.8, 17.8, 13.2, and 5 degrees. The character of the premelting transition, whereby a liquid-like film develops at a defect at temperatures below the melting point, changed with misorientation angle. An excess mass over the grain boundary can be defined as an analog to the liquid layer thickness due to premelting. It is found that low-angle grain boundaries remain at a relatively constant value of excess mass, and indeed can remain solid above the melting point. High-angle grain boundaries have a logarithmically increasing width that diverges at the melting point. A width-dependent energy can be defined called the disjoining potential that takes into account structure, interfacial and bulk energies to describe the liquid-layer width. The form of this disjoinging potential was found to be exponential and monotonically decreased as width increased for high angles and produced an attractive minimum for low angles. The results of this work were compared to a pure material from a previous study.</p> / Master of Applied Science (MASc)

The continual research and development of nickel-based superalloys is driven by the global demand to improve efficiency and reduce emissions in the aerospace and power generation industries. Integrated Computational Material Engineering (ICME) is a valuable tool for reducing the cost, time, and resources necessary for the development and optimization of the mechanical properties of materials. In this work, an ICME approach for understanding the microstructure development and optimizing the mechanical properties in nickel-based superalloys is employed. Most nickel-based superalloys are precipitate strengthened by either the γ’ phase, γ” phase, or both. Therefore, understanding the precipitation kinetics and morphological evolution of these phases is critical for evaluating their hardening effects during heat treatment and degradation of the microstructure during high temperature service. To this end, a phase-field model has been developed to analyze the nucleation, growth and coarsening kinetics during isothermal and non-isothermal aging conditions. Utilizing the phase-field model, the γ” phase microstructure development and its coherency strengthening effect in Inconel 625 is studied. A novel multistage aging strategy to optimize the γ” phase strengthening effect and reduce aging times for Inconel 625 is proposed. Secondly, the coarsening kinetic and microstructure development of γ’ strengthening phase in nickel-based superalloys is studied, with the goal of understanding the effect of elastic inhomogeneity on the microstructure evolution at high volume fractions of the γ’ phase. The result shows deviation of the coarsening kinetics from the classical Lifshitz-Slyozov-Wagner (LSW) due to the effect of elastic inhomogeneity, highlighting the need for incorporating elastic energy into coarsening theories.

<p>Rapid solidification is a well established method to produce novel materials with improved mechanical or electrical properties. The sharp-interface kinetics of rapid solidification for a binary alloy is summarized. A Phase Field model mapping to this sharp interface model is summarized and solved by a new adaptive mesh refinement algorithm. Simulation results are consistent with experiments: The solidification velocity increases in power-law like fashion at low undercooling and approximately linearly at high undercooling; The solid/liquid interface undergoes a transition from four-fold dendritic to circular crystal structures; Solute trapping emerges and the solute partitioning approaches unity as the solidification velocity increases. Our Phase Field simulations are the first self -consistent predictions of velocity selection and morphological selection at both low and high undercoolings and also the first independent check of the solute trapping model in two dimensions.</p> / Thesis / Master of Applied Science (MASc)

This thesis develops a new quantitative multi-phase field model for polycrystalline solidification of binary alloys. We extend the thin interface formalism of Karma and co-workers to multiple order parameters. This makes it possible to model segregation and interface kinetics during equiaxed dendritic growth quantitatively, a feature presently lacking from polycrystalline or multi-phase solidification models. We study dendrite tip speed convergence as a function of interface width during free dendritic growth. We then analyze the steady state and grain coalescence properties of the model. It is shown that the model captures the correct physics of back diffusion and repulsive grain boundary coalescence as outlined by Rappaz and co-workers. Finally, the model is applied to simulate solidification and coarsening in delta-ferrite solidification. / Thesis / Master of Applied Science (MASc)

The workshop is composed of two main parts: the first part devoted to atomistic Monte Carlo simulations and the second part devoted to the Phase Field modelling. In each part a lecture will be accompanied by exercise activities.

This research has focused on understanding the mechanicPhysics and chemistry based constitutive framework for thermo-chemo-mechanical responses of polymeric materialsPhysics and Chemistry Based Constitutive Framework for Thermo-Chemo-Mechanical Responses of Polymeric Materialss and multi-physics of soft materials with rate-and temperature-dependent matrices. Such materials are oftentimes exposed to extreme environmental conditions such as Ultra-Violet (UV) light, elevated temperatures, and oxygen which degenerate their mechanical properties and contribute to their permanent failure. The irreversible changes in the mechanical response of polymers induced by such deleterious processes is commonly referred to as polymer aging. The ultimate goal of this work has been to identify the relevant damage processes affecting the lifetime of polymeric materials, and to develop predictive physics- and chemistry-based, thermodynamically consistent constitutive frameworks to evaluate their response under extreme environmental condition. A series of interconnected experimental, theoretical, and numerical studies were developed regarding the chemical, morphological, and mechanical changes that polymers and elastomers exhibit under thermo-photo-chemo-mechanical conditions. Emphasis was placed on linking the aggravation of macrostructural changes (such as cross-link breakage/formation and transformation of linkages) to the macromechanical response of aged polymers, and the development of a mathematically verifiable procedure for incorporating stored and dissipated energies – obtained through chemical experiments – into the thermodynamic formalism. Fracture was considered using the phase-field approach to brittle failure through development of robust and efficient numerical algorithms intended to solve the highly coupled and nonlinear displacement-damage problems. Results demonstrate that several chemical characterization tests such as equilibrium swelling, differential scanning calorimetry (DSC), quartz crystal microbalance with dissipation (QCMD-D), and dynamic mechanical analysis (DMA) can indeed reveal crucial information regarding the physio-chemical changes manifested within polymer networks. Information obtained from these tests can then be employed to propose accurate predictive evolution functions for the mechanical as well as the fracture properties towards a complete physics- and chemistry-based constitutive framework. Numerical analyses were performed within finite element software Abaqus via several user-element and user-material subroutines (UEL, VUMAT) to investigate the predictive capabilities of the developed frameworks in describing complex loading configurations including fracture. The developed constitutive frameworks are all thermodynamics-based and rely solely on the outputs obtained through appropriate chemical characterization techniques. Not only are the predicted results highly accurate, but also and most importantly, the developed constitutive equations are completely self-contained and bypass the need for extra fitting variables. / Doctor of Philosophy / Material science is a fundamental field of research. Understanding how materials behave under various operating conditions can help scientists and engineers propose efficient and economical designs with the aim of potentially establishing a robust foundation for our infrastructure. This work focused on the study and prediction of the deleterious effects of several environmental factors such as elevated temperature, Ultra-Violet (UV) light, and oxygen on the mechanical and failure responses of polymer systems. Several interconnected experimental, theoretical, and numerical studies were developed with the aim of characterizing the chemical, morphological, and mechanical changes that such material systems exhibit under coupled dissipation phenomena. In particular, this research aimed to investigate the aggravation of macrostructural changes that manifested themselves within polymer systems upon exposure to thermo-oxidation and photo-oxidation. Predictive constitutive frameworks were developed based on principles of thermodynamics and continuum damage theories to understand the effects that heterogeneous aging has on the mechanical and fracture responses of these materials. Results achieved in this work helped fill several gaps on both the theoretical as well as numerical sides towards a complete physics and chemistry-based constitutive framework for the analysis of multi-physics phenomena in soft materials. Overall, results shed light on our understanding of the aging process and the predictive capabilities of our proposed equations in describing such degenerative processes as thermo-chemo-mechanical aging. Findings from this work will contribute to the design of high-performance polymers in other applications such as implanted bio-medical devices. Ultimately, describing aging under extreme environmental conditions will contribute to the understanding and prediction of plastic fragmentation processes and therefore, microplastic pollution.

Phase field modeling and simulation is employed to study the underlying mechanism of enhancing electromechanical properties in single crystals and polycrystals of perovskite-type ferroelectrics around the morphotropic phase boundary (MPB). The findings include: (I) Coherent phase decomposition near MPB in PZT is investigated. It reveals characteristic multidomain microstructures, where nanoscale lamellar domains of tetragonal and rhombohedral phases coexist with well-defined crystallographic orientation relationships and produce coherent diffraction effects. (II) A bridging domain mechanism for explaining the phase coexistence observed around MPBs is presented. It shows that minor domains of metastable phase spontaneously coexist with and bridge major domains of stable phase to reduce total system free energy, which explains the enhanced piezoelectric response around MPBs. (III) We demonstrate a grain size- and composition-dependent behavior of phase coexistence around the MPBs in polycrystals of ferroelectric solid solutions. It shows that grain boundaries impose internal mechanical and electric boundary conditions, which give rise to the grain size effect of phase coexistence, that is, the width of phase coexistence composition range increases with decreasing grain sizes. (IV) The domain size effect is explained by the domain wall broadening mechanism. It shows that, under electric field applied along the nonpolar axis, without domain wall motion, the domain wall broadens and serves as embryo of field-induced new phase, producing large reversible strain free from hysteresis. (V) The control mechanisms of domain configurations and sizes in crystallographically engineered ferroelectric single crystals are investigated. It reveals that highest domain wall densities are obtained with intermediate magnitude of electric field applied along non-polar axis of ferroelectric crystals. (VI) The domain-dependent internal electric field associated with the short-range ordering of charged point defects is demonstrated to stabilize engineered domain microstructure. The internal electric field strength is estimated, which is in agreement with the magnitude evaluated from available experimental data. (VII) The poling-induced piezoelectric anisotropy in untextured ferroelectric ceramics is investigated. It is found that the maximum piezoelectric response in the poled ceramics is obtained along a macroscopic nonpolar direction; and extrinsic contributions from preferred domain wall motions play a dominant role in piezoelectric anisotropy and enhancement in macroscopic nonpolar direction. (VIII) Stress effects on domain microstructure are investigated for the MPB-based ferroelectric polycrystals. It shows that stress alone cannot pole the sample, but can be utilized to reduce the strength of poling electric field. (IX) The effects of compressions on hysteresis loops and domain microstructures of MPB-based ferroelectric polycrystals are investigated. It shows that longitudinal piezoelectric coefficient can be enhanced by compressions, with the best value found when compression is about to initiate the depolarization process. / Ph. D.

Kolloidpartikel die von zwei nicht mischbaren Fluiden benetzt werden, tendieren dazu sich an der fluiden Grenzfläche aufzuhalten um die Oberflächenspannung zu minimieren. Bei genügender Anzahl solcher Kolloide werden diese zusammengedrückt und lassen die fluide Grenzfläche erstarren. Das gesamte System aus Fluiden und Kolloiden bildet dann eine spezielle Emulsion mit interessanten Eigenschaften. In dieser Arbeit wird ein kontinuum Model für solche Systeme entwickelt, basierend auf den Prinzipien der Massenerhaltung und der themodynamischen Konsistenz. Dabei wird die makroskopische Zwei-Phasen-Strömung durch eine Navier-Stokes Cahn-Hilliard Gleichung modelliert und die mikroskopischen Partikel an der fluiden Grenzfläche durch einen Phase-Field-Crystal Ansatz beschrieben. Zur Evaluation des verwendeten Strömungsmodells wird ein Test verschiedener Navier-Stokes Cahn-Hilliard Modelle anhand eines bekannten Benchmark Szenarios durchgeführt. Die Ergebnisse werden mit denen von anderen Methoden zur Simulation von Zwei-Phasen-Strömungen verglichen. Desweiteren wird eine neue Methode zur Simulation von Zwei-Phasen-Strömungen in komplexen Gebieten vorgestellt. Dabei wird die komplexe Geometrie implizit durch eine Phasenfeldvariable beschrieben, welche die charakteristische Funktion des Gebietes approximiert. Die Strömungsgleichungen werden dementsprechend so umformuliert, dass sie in einem größeren und einfacheren Gebiet gelten, wobei die Randbedingungen implizit durch zusätzliche Quellterme eingebracht werden. Zur Einarbeitung der Oberflächenkolloide in das Strömungsmodell wird schließlich die Variation der freien Energie des Gesamtsystems betrachtet. Dabei wird die Energie der Partikel durch die Phase-Field-Crystal Energie approximiert und die Energie der Oberfläche durch die Ginzburg-Landau Energie. Eine Variation der Gesamtenergie liefert dann die Phase-Field-Crystal Gleichung und die Navier-Stokes Cahn-Hilliard Gleichungen mit zusätzlichen elastischen Spannunngen. Zur Validierung des Ansatzes wird auch eine sharp interface Version der Gleichungen hergeleitet und mit der zuvor hergeleiteten diffuse interface Version abgeglichen. Die Diskretisierung der erhaltenen Gleichungen erfolgt durch Finiten Elemente in Kombination mit einem semi-impliziten Euler Verfahren. Durch numerische Simulationen wird die Anwendbarkeit des Modells gezeigt und bestätigt, dass die oberflächenaktiven Kolloide die fluide Grenzfläche hinreichend steif machen können um externen Kräften entgegenzuwirken und das gesamte System zu stabilisieren. / Colloid particles that are partially wetted by two immiscible fluids can become confined to fluidfluid interfaces. At sufficiently high volume fractions, the colloids may jam and the interface may crystallize. The fluids together with the interfacial colloids compose an emulsion with interesting new properties and offer an important route to new soft materials. Based on the principles of mass conservation and thermodynamic consistency, we develop a continuum model for such systems which combines a Cahn-Hilliard-Navier-Stokes model for the macroscopic two-phase fluid system with a surface Phase-Field-Crystal model for the microscopic colloidal particles along the interface. We begin with validating the used flow model by testing different diffuse interface models on a benchmark configuration for a two-dimensional rising bubble and compare the results with reference solutions obtained by other two-phase flow models. Furthermore, we present a new method for simulating two-phase flows in complex geometries, taking into account contact lines separating immiscible incompressible components. In this approach, the complex geometry is described implicitly by introducing a new phase-field variable, which is a smooth approximation of the characteristic function of the complex domain. The fluid and component concentration equations are reformulated and solved in larger regular domain with the boundary conditions being implicitly modeled using source terms. Finally, we derive the thermodynamically consistent diffuse interface model for two-phase flow with interfacial particles by taking into account the surface energy and the energy associated with surface colloids from the surface PFC model. The resulting governing equations are the phase field crystal equations and Navier-Stokes Cahn-Hilliard equations with an additional elastic stress. To validate our approach, we derive a sharp interface model and show agreement with the diffuse interface model. We demonstrate the feasibility of the model and present numerical simulations that confirm the ability of the colloids to make the interface sufficiently rigid to resist external forces and to stabilize interfaces for long times.

A preconditioner to improve the convergence properties of Krylov subspace solvers is derived and analyzed in this work. This method is adapted to linear systems arising from a finite-element discretization of a phase-field crystal equation.