Motion of a droplet for the mass-conserving stochastic Allen-Cahn equation

We study the stochastic mass-conserving Allen-Cahn equation posed on a bounded two-dimensional domain with additive spatially smooth space-time noise. This equation associated with a small positive parameter describes the stochastic motion of a small almost semicircular droplet attached to domain's boundary and moving towards a point of locally maximum curvature. We apply It\^o calculus to derive the stochastic dynamics of the droplet by utilizing the approximately invariant manifold introduced by Alikakos, Chen and Fusco for the deterministic problem. In the stochastic case depending on the scaling, the motion is driven by the change in the curvature of the boundary and the stochastic forcing. Moreover, under the assumption of a sufficiently small noise strength, we establish stochastic stability of a neighborhood of the manifold of droplets in $L^2$ and $H^1$, which means that with overwhelming probability the solution stays close to the manifold for very long time-scales

Advances in the theory of III-V Nanowire Growth Dynamics

Nanowire (NW) crystal growth via the vapour_liquid_solid mechanism is a complex dynamic process involving interactions between many atoms of various thermodynamic states. With increasing speed over the last few decades many works have reported on various aspects of the growth mechanisms, both experimentally and theoretically. We will here propose a general continuum formalism for growth kinetics based on thermodynamic parameters and transition state kinetics. We use the formalism together with key elements of recent research to present a more overall treatment of III_V NW growth, which can serve as a basis to model and understand the dynamical mechanisms in terms of the basic control parameters, temperature and pressures/beam fluxes. Self-catalysed GaAs NW growth on Si substrates by molecular beam epitaxy is used as a model system.Comment: 63 pages, 25 figures and 4 tables. Some details are explained more carefully in this version aswell as a new figure is added illustrating various facets of a WZ crysta

Computer-assisted proof of heteroclinic connections in the one-dimensional Ohta-Kawasaki model

We present a computer-assisted proof of heteroclinic connections in the one-dimensional Ohta-Kawasaki model of diblock copolymers. The model is a fourth-order parabolic partial differential equation subject to homogeneous Neumann boundary conditions, which contains as a special case the celebrated Cahn-Hilliard equation. While the attractor structure of the latter model is completely understood for one-dimensional domains, the diblock copolymer extension exhibits considerably richer long-term dynamical behavior, which includes a high level of multistability. In this paper, we establish the existence of certain heteroclinic connections between the homogeneous equilibrium state, which represents a perfect copolymer mixture, and all local and global energy minimizers. In this way, we show that not every solution originating near the homogeneous state will converge to the global energy minimizer, but rather is trapped by a stable state with higher energy. This phenomenon can not be observed in the one-dimensional Cahn-Hillard equation, where generic solutions are attracted by a global minimizer

A phase-field study of ternary multiphase microstructures

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Materials Science and Engineering, 2010.Cataloged from PDF version of thesis.Includes bibliographical references.A diffuse-interface model for microstructures with an arbitrary number of components and phases was developed from basic thermodynamic and kinetic principles and applied to the study of ternary eutectic phase transformations. Gradients in composition and phase were included in the free energy functional, and a generalized diffusion potential equal to the chemical potential at equilibrium was defined as the driving force for diffusion. Problematic pair-wise treatment of phases at interfaces and triple junctions was avoided, and a cutoff barrier was introduced to constrain phase fractions to physically meaningful values. Parameters in the model were connected to experimentally measurable quantities. Numerical methods for solving the phase-field equations were investigated. Explicit finite difference suffered from stability problems while a semi-implicit spectral method was orders of magnitude more stable but potentially inaccurate. The source of error was found to be the rich temporal dynamics of spinodal decomposition combined with large timesteps and a first-order time integrator. The error was addressed with a second-order semi-implicit Runge-Kutta time integrator and adaptive timestepping, resulting in two orders of magnitude improvement in efficiency. A diffusion-limited growth instability in multiphase thin-film systems was discovered, highlighting how ternary systems differ from binary systems, and intricate asymmetries in the processes of solidification and melting were simulated. A nucleation barrier for solidification was observed and prompted development of a Monte-Carlo-like procedure to trigger nucleation. However when solid was heated from below the melting point, premelting was observed first at phase triple junctions and then at phase boundaries with stable liquid films forming under certain conditions. Premelting was attributed to the shape and position of the metastable liquid curve, which was found to affect microstructure by creating low energy pathways through composition space. Slow diffusivity in solid relative to liquid was shown to produce solutal melting of solid below the melting point. Finally, the multiphase method was used to produce the first reported simulation of the entire transient liquid phase bonding process. The model shows promise for optimizing the bonding process and for simulating non-planar solidification interfaces.by Daniel A. Cogswell.Ph.D

Topological Microstructure Analysis Using Persistence Landscapes

International audiencePhase separation mechanisms can produce a variety of complicated and intricate microstructures, which often can be difficult to characterize in a quantitative way. In recent years, a number of novel topological metrics for microstructures have been proposed, which measure essential connectivity information and are based on techniques from algebraic topology. Such metrics are inherently computable using computational homology, provided the microstructures are discretized using a thresholding process. However, while in many cases the thresholding is straightforward, noise and measurement errors can lead to misleading metric values. In such situations, persistence landscapes have been proposed as a natural topology metric. Common to all of these approaches is the enormous data reduction, which passes from complicated patterns to discrete information. It is therefore natural to wonder what type of information is actually retained by the topology. In the present paper, we demonstrate that averaged persistence landscapes can be used to recover central system information in the Cahn-Hilliard theory of phase separation. More precisely, we show that topological information of evolving microstructures alone suffices to accurately detect both concentration information and the actual decomposition stage of a data snapshot. Considering that persistent homology only measures discrete connectivity information, regardless of the size of the topological features, these results indicate that the system parameters in a phase separation process affect the topology considerably more than anticipated. We believe that the methods discussed in this paper could provide a valuable tool for relating experimental data to model simulations

Thermomechanical modelling of microstructure evolution in solder alloys

The reliability of soldered connections is a very important issue in electronics industry. This project aims to better understand the various factors influencing the lifetimeof solder joints through numerical modelling. Because of the high homologous temperatures at which they operate, the joints exhibit high temperature deformation mechanisms associated with creep and relaxation and are susceptible to lowcycle fatigue. Another issue is that due to the ongoingminiaturisation,microstructural sizes influence the material properties greatly. The project was initiated with the commercial eutectic tin–lead solder in mind. For this alloy the microstructure evolves, coarsens, significantly over time. To accurately capture the various, and sometimes very large, time scales that come into play, accelerated test methods are not suitable and real time testing is too time consuming. This is where numerical simulations can provide more insight and permit a significant reduction in cost and time of the design of highly reliable soldered connections in electronic packages and components. The first part of this thesis deals with the microstructure evolution of tin–lead due to diffusion. In the following two parts the proposed model is extended with respectively viscoplastic material behaviour and a nonlocal damage approach. In the final part attention is given to the tin–silver–copper system, which is one of the most likely replacement candidates of tin–lead in the strive towards a lead-free electronics industry. The evolving microstructure has been taken into account using a phase field model which is solved using the finite element method. The driving force for diffusion has been derived from a macroscopic free energy function, which describes phase segregation in microscopic model systems with long-range interactions evolving according to stochastic Kawasaki dynamics with nearest neighbour exchanges. Simulations of the static ageing process of eutectic tin–lead solder have been performed. The results predict break-up, coalescence, growth, and dissolution of phases, similar to experimental observations. It was also shown that external mechanical loading leads to faster coarsening rates. Quantitative comparison with experiments has been performed using the total interface length as the quantifying parameter and a good agreement was found. To capture the time dependent mechanical behaviour an elasto-viscoplastic material model law been used for the material model. The extensive information on the microstructure found with the phase field model was used to assign different parameter values to the individual phases and interfaces. Results from simulations of mechanical loading of eutectic tin–lead solder showed a strong dependency on the underlying microstructure. Aged microstructures exhibit more pronounced localisation of stresses and strains. To investigate the reliability of the solder, the model has been extended to include damage. The modelling of softening behaviour often leads to bad solutions using the finite element method. Although the viscous nature of the material model is known to regularise the solution, for practical purposes this effect is usually only sufficient for highly rate-sensitive materials or high loading rates and the numerical results still can show a mesh dependency. Therefore, a gradient enhanced nonlocal damage formulation has been implemented. The results of the phase field model are used to assign different damage parameter values to the phases and interfaces. For the tin–lead system the phase boundaries are known to be the crack initiation sites. The cracks next propagate preferably along tin–lead or tin–tin grain boundaries. The approach yields results that are qualitatively comparable with these experimental findings. Because, in the electronics industry, the tin–lead alloy needs to be replaced with a leadfree alternative in the near future, the final chapter deals with one of the most likely candidates, the near eutectic tin–silver–copper. Experiments performed on this ternary alloy revealed a disconcerting feature. Cyclic thermal ageing without any additional mechanical loading was already enough to lead to fracture along grain boundaries. In order to understand and explain this behaviour the experiments have been modelled using a three-dimensional finite element approach. The viscoplastic damage part of the model is extended to account for anisotropy of the material, both in the elastic as well as the thermal properties. Data obtained from Orientation ImageMicroscopy is used to take into account the microstructure at the grain level, which was found not to evolve over time. The results show a good qualitative agreement with the experiments, exhibiting stress concentrations leading to damage along the grain boundaries. The presented modelling approaches have been applied to simulate the complex behaviour of solder alloys. A multiphase alloy who’s mechanical behaviour is determined by its evolving microstructure is modelled and the results are compared with experimental data, showing satisfactory agreement. Furthermore, an industrially interesting material, near eutectic tin–silver–copper, has been successfully investigated, predicting damage in the same areas as seen experimentally, indicating that the elastic and thermal anisotropic properties play an important part in the fatigue life of this alloy

Group III-V Nanowire Growth and Characterization

Electronic and optical devices typically use bulk or quantum wells today, but nanowires are promising building blocks for future devices, due to their structural characterizations of larger aspect ratio and smaller volume. In situ growth of semiconductor devices is extremely attractive, as it doesn’t require expensive lithography treatment. Over the past ten years, a great deal of work has been done to explore NW, incorporation of group III-V materials and band engineering for the electronic and optoelectronic devices. Because pseudo one-dimensional heterostructures may be grown without involving lattice mismatch defects, NWs may give rise to superior electronic, photonic, and magnetic performances as compared to conventional bulk or planar structures