Observed in many physical systems, coarsening is an orderly decrease in the number of localized structures, such as particles, drops, shear bands, solitons, or point defects. Coarsening is a type of pattern formation in which the characteristic length scale between features grows while the total number of features decreases. These phenomena have been studied in many problems and several mathematical techniques for modeling these phenomena have been developed. This dissertation examines the aggregation of drops in the thin film equation, where drops may coarsen through two general mechanisms: collision and collapse. A series of simplifications to model this process is developed. Slender-body asymptotics is applied to the Navier-Stokes equations for fluid motion in order to derive the Reynolds lubrication equation. The lubrication equation is in turn simplified to a coarsening dynamical system (CDS) model for interacting drops through solvability conditions for a perturbation about a drop-type steady state. Lastly, the dynamical system is averaged into an ensemble model to describe the dynamics of the distribution of drop sizes. The ensemble model takes the form of an integro-differential equation for the distribution function, much like the model of Ostwald ripening proposed by Lifshitz and Slyozov. A convenient choice of scaling yields an intermediate asymptotic self-similar solution. This solution is compared to numerical simulations of the ensemble model and histograms of drop masses from the CDS model. The early-time dynamics before similarity are explored by varying the initial distribution of drop sizes. Interesting far-from-similarity ``stairstep'' behavior is observed in the coarsening rate when the initial distribution has a very small variance. A well-chosen initial condition with a fractal-like structure is shown to replicate the stairstep behavior. At very long times, the mean drop size grows large, requiring the inclusion of gravity in the model. The CDS model parameters are modified as a result of the dependence of drop shapes on both size and gravity. The new dynamical system predicts the coarsening rate slowing from a power law to an inverse logarithmic rate. The energy liberated by each coarsening event is shown to approach a gravity-dependent constant as the mean drop mass increases. This suggests a reason for the coarsening slow-down. / Dissertation

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Starting with the Mean Field Method (MFM) and Boundary Element Method (BEM), we investigate a mathematical model based on these two methods for studying particle-coarsening process in alloys. With MFM, second-phase particles are considered to be merged into bulk matrix, which greatly simplifies computation. However, the Mean-Field model itself is limited to a system with extremely small volume fractions of second phase. By combining BEM with MFM, this mathematical model shows the influence of second phase in particle-coarsening process. Our primary work demonstrates the robustness and capability of this model. This model is however limited to particle coarsening that is far away from grain boundaries. In this dissertation, we successfully extend the model to particle coarsening near grain boundaries. A major improvement made to the previous mathematical model is based on solute atoms conservation and diffusion theory. The capability and validity of the novel model is demonstrated by a binary alloy system. The simulation results are shown to quantitatively reproduce the essential features of particle coarsening near grain boundaries in certain alloys: a) precipitation Free Zones (PFZs) form near grain boundaries, b) the width of PFZs is proportional to square root of time, c) particles at the edge of PFZs are larger than those inside the grain. This novel model is shown to be well suited in describing particle coarsening near grain boundaries. On the other hand, it proves the credibility of the theories built in our mathematical model, i.e., the formation of PFZs near grain boundaries is caused by diffusion of solute atoms. / Thesis / Master of Applied Science (MASc)

In this thesis, we have used a phase field model for studying the effect of atomic mobility inside the precipitate phase on coarsening behaviour in two dimensional (2D) systems. In all the available coarsening theories, the diffusivity inside the precipitate phase is not explicitly taken into account; this would imply that there is no chemical potential gradient inside the precipitate. This assumption is valid if (a) the atomic mobility inside the precipitate is much higher than that in the matrix, or (b) the precipitate volume fraction is small (i.e. the interparticle spacing is far higher than the average particle size). We undertook this study to evaluate the potential effect of diffusivity in the precipitate on coarsening in situations where conditions (a) and (b), above, do not hold, by studying systems with moderate volume fractions (20% and 30%) and with low atomic mobilities in the precipitate. In our study, we have fixed the atomic mobility in the matrix at a constant value. We have used the well known Cahn-Hilliard model in which the microstructure is described in terms of a composition field variable. The evolution of microstructure is studied by numerically solving a non-classical diffusion equation known as the Cahn-Hilliard equation. We have used a semi-implicit Fourier spectral technique for solving the CH equation using periodic boundary conditions. The coarsening behaviour is tracked and analyzed using number density of particles, their average size and their size distribution. The main conclusion from this study is that, contrary to expectations, the atomic mobility in the precipitate phase has only a small effect on coarsening behavior. Specifically, with decreasing atomic mobility in the precipitate phase, we report a small increase in the number density, a slightly wider size distribution and a slightly smaller coarsening rate. We also add that these effects are too small to allow experimental verification. These results indicate that the need for chemical potential equilibration within each precipitate is not an important factor during coarsening.

Block copolymers have attracted interests for potential application ranging from dynamic photonic sensors to solid-state ion conductors. However, due to nucleation and growth mechanism, block copolymer inherently forms granular microstructure with defects such as grain boundaries. Understanding the microstructure of block copolymer is thus crucial in many applications because the microstructure determines the transport property of functional fillers such as ions in block copolymer template. Previous research has shown that athermal filler segregated to grain boundary of lamellae block copolymer and retards the grain coarsening. However, the kinetics of this grain boundary segregation during thermal annealing has not been revealed. Polystyrene-b-polyisoprene blended with deuterated polystyrene is used for neutron scattering study on studying the kinetics of grain boundary segregation. Deuterated polystyrene will segregate to grain boundaries, therefore, decorate grain boundary. The filler segregation behavior will be studied by comparing neutron scattering of polystyrene-b-polyisoprene/deuterated polystyrene with different annealing times (at T=130 deg C, duration of 0hr, 3hr, 1day, 3day and 7day, respectively). Invariant (Q) analysis along with grain mapping is conducted to quantitatively analyze the kinetics of grain boundary segregation. This kinetic was in good agreement with the McLean’s kinetic model for grain boundary segregation in metals. By applying Langmuir-Mclean’s segregation isotherm equation, we have predicted the equilibrium concentration of filler in grain boundary by calculating the strain energy stored in grain boundary.

H-matrix techniques use a data-sparse tree structure to represent a dense or a sparse matrix. The leaves of the tree store matrix sub-blocks that are represented in full-matrix format or Rk-matrix (low rank matrix) format. H-matrix arithmetic is defined over the H-matrix representation, which includes operations such as addition, multiplication, inversion, and LU factorization. These H-matrix operations approximate results with almost optimal computational complexity. Based on the properties of H-matrices, the H-matrix preconditioner technique has been introduced. It uses H-matrix operations to construct preconditioners, which are used in iterative methods to speed up the solution of large systems of linear equations (Ax = b). To apply the H-matrix preconditioner technique, the first step is to represent a problem in H-matrix format. The approaches to construct an H-matrix can be divided into two categories: geometric approaches and algebraic approaches. In this thesis, we present our contributions to algebraic H-matrix construction approaches and H-matrix preconditioner technique. We have developed a new algebraic H-matrix construction approach based on matrix graphs and multilevel graph clustering approaches. Based on the new construction approach, we have also developed a scheme to build algebraic H-matrix preconditioners for systems of saddle point type. To verify the effectiveness of our new construction approach and H-matrix preconditioner scheme, we have applied them to solve various systems of linear equations arising from finite element methods and meshfree methods. The experimental results show that our preconditioners are competitive to other H-matrix preconditioners based on domain decomposition and existing preconditioners such as JOR and AMG preconditioners. Our H-matrix construction approach and preconditioner technique provide an alternative effective way to solve large systems of linear equations.

The growth of Pb film on Si(111) is an unusual metal-semiconductor system. For a certain temperature range, Pb films have been found to grow in steep-edge and flat-top islands with uniform height on Si(111). This specific film morphology has been correlated to Quantum Size Effect (QSE) that the object size or film thickness affects the electronic structure of the films and results in certain thicknesses more stable than others. The X-ray diffraction technique has the advantages of long penetration length and high statistics, therefore it has been used to investigate the influence of QSE on the growth of Pb on the Si(111) 7x7 surface. It is demonstrated that the structure of Pb islands and the associated wetting layer are consistent with effects of quantum confinement. Specular reflectivity of 3 monolayer (ML) Pb films grown on the substrate at 227K has conclusively shown that the Pb islands do not reside on top of a Pb wetting layer, but directly on top of the Si substrate. The nucleating Pb nanocrystals transform the highly disordered Pb wetting layer beneath the islands into well-ordered fcc Pb. The surface then consists of fcc Pb islands directly on top of the Si surface with the disordered wetting layer between the islands. Moreover, it is found that QSE leads to novel behavior for the coarsening evolution of the Pb islands. The diffuse X-ray scattering experiments have been carried out as functions of temperature, deposition rate and coverage. A structural evolution of Pb islands was observed after deposition at very low coverages (0.2 -- 1.0 ML above the wetting layer coverage). Contrary to the classical scaling theory of nucleation and Ostwald ripening, a much lower island density is achieved with coarsening after deposition at high rather than low flux rates. The time constants of coarsening are found to be orders of magnitudes shorter than what is expected from the Gibbs-Thompson analysis. The rapid evaporation of unstable 3-layer islands shown in complementary STM suggests the role of QSE in the more efficient decay mechanism operating at low temperatures. These results have important applications for the controlled growth of nanostructures.

No description available.

Between two neighboring air bubbles in a froth (or foam), a thin liquid film (TLF) is formed. As the bubbles rise upwards, the TLFs thin initially due to the capillary pressure created by curvature changes. As the film thicknesses (H) reach approximately 200 nm, the disjoining pressure created by surface forces in the films also begins to control the film drainage rate and affect the waves motions at the air/water interfaces. If the disjoining pressure is negative, both the film drainage and the capillary wave motion accelerate. When the TLF thins to a critical film thickness (Hcr), the amplitude of the wave motion grows suddenly and the two air/water interfaces touch each other, causing the TLF to rupture and bubbles to coalesce. In the present work, a new model that can predict Hcr has been developed by considering the film drainage due to both viscous film thinning and capillary wave motion. Based on the Hcr model, bubble-coarsening in a dynamic foam has been predicted by deriving the geometric relation between the thickness of the lamella film, which controls bubble-coalescence rate, and the Plateau border area, which controls liquid drainage rate. Furthermore, a model for predicting bubble-coarsening in froth (3-phase foam) has been developed by developing a film drainage model quantifying the effect of particles on pc. The parameter pc is affected by the number of particles and the local capillary pressure around particles, which in turn vary with the hydrophobicity and size of the particles in the film. Assuming that films rupture at free films, the pc corrected for the particles in lamella films has been used to determine the critical rupture time (tcr), at which the film thickness reaches Hcr, using the Reynolds equation. Assuming that the number of bubbles decrease exponentially with froth height, and knowing that bubbles coalesce when film drains to a thickness Hcr, a bubble coarsening model has been developed. The model predictions are in agreement with the experimental data obtained using particle of varying hydrophobicity and size. / Ph. D.

A first principle flotation model has been derived from the basic mechanisms involved in the bubble-particle and bubble-bubble interactions occurring in flotation. It is a kinetic model based on the premise that the energy barrier (E1) for bubble-particle interaction can be reduced by increasing the kinetic energy (Ek) for bubble-particle interaction and by increasing the hydrophobic force in wetting films. The former is controlled by energy dissipation rate (𝜀), while the latter is controlled by collector additions. The model consists of a series of analytical equations to describe bubble generation, bubble-particle collision, attachment and detachment, froth recovery, and bubble coalescence in froth phase. Unlike other flotation models that do not consider role of hydrophobic force in flotation, the first principle model developed at Virginia Tech can predict flotation recoveries and grades from the chemistry parameters such as 𝜁-potentials, surface tension (𝛾), and contact angles (𝜃) that may represent the most critical parameters to control to achieve high degrees of separation efficiencies. The objectives of the present work are to i) validate the flotation model using the experimental data published in the literature, ii) incorporate a froth model that can predict bubble coarsening due to coalescence in the absence of particles, iii) develop a computer simulator for a froth model that can predict bubble coarsening in the presence of particles, and iv) study the effects of incorporating a regrinding mill and using a stronger collector in a large copper flotation circuit. The model validation has been made using the size-by-class flotation rate constants (kij) obtained from laboratory and pilot-scale flotation tests. Model predictions are in good agreement with the experimental data. It has been found that the flotation rate constants obtained for composite particles can be normalized by those for fully liberated particles (kmax), which opens the door for minimizing the number of flotation products that need to be analyzed using a costly and time-consuming liberation analyzer. A bubble coarsening froth model has been incorporated into the flotation model to predict flotation more accurately. The model has a limitation, however, in that it cannot predict bubble-coarsening in the presence of particles. Therefore, a new computer simulator has been developed to predict the effects of particle size and particle hydrophobicity on bubble coarsening in froth phase. In addition, the first principle flotation model has been used to simulate flotation circuits that are similar to the Escondida copper flotation plant to study the effects of incorporating a re-grinding mill and using a more powerful collector to improve copper recovery. The flotation model developed from first principles is useful for predicting and diagnosing the performance of flotation plants under different circuit arrangements and chemical conditions. / Master of Science