Phase Separation In Alloys Research Articles

Optimal Control for Allen-Cahn Equations Enhanced by Model Predictive Control

The Allen-Cahn equation is a simple model of a nonlinear reaction-diffusion process. It is often used to model interface motion in time, e.g. phase separation in alloys. It has applications in many areas including material sciences, biology, geology, as well as image processing. We will consider a simple scalar Allen-Cahn equation subject to distributed control. Here, the nonlinear reaction term is obtained from using the standard double-well potential, leading to a cubic nonlinearity. We will describe a nonlinear feedback control strategy based on the concept of Model Predictive Control (MPC). We also show how to obtain the open-loop trajectory and control using numerical techniques for PDE-constrained optimization. The feedback control scheme is then applied to the spatially semi-discretized nonlinear optimal control problem. For the prediction and control step within the MPC scheme, we apply a linear-quadratic regulator/Gaussian design problem. The arising computational challenge consisting in solving the associated large-scale algebraic Riccati equations has already been shown in the literature to be feasible using reasonably fine discretizations.

Phase-field simulation of thermally induced spinodal decomposition in polymer blends

A phase-field simulation of thermally induced phase separation with various initial average concentrations under different quench depths is systematically carried out in this paper. The resultant morphology of the phase-separated materials is determined by the quench temperature and the average concentration. A detailed understanding of the effects of quench depth during the phase separation process of spinodal decomposition (SD) is presented. The coarsening mechanism with various average concentrations is investigated. In addition, hydrodynamic effects play an important role in phase separation in liquid immiscible alloys. Therefore, a systematic comparison between systems with and without hydrodynamic effects under different conditions is also considered. The model developed and the results obtained could shed light on using SD in immiscible polymer systems to obtain desired nanostructures for advanced applications.

First-principles modeling of temperature- and concentration-dependent solubility in the phase-separating alloy FexCu1−x

We present a novel cluster-expansion (CE) approach for the first-principles modeling of temperature and concentration dependent alloy properties. While the standard CE method includes temperature effects only via the configurational entropy in Monte Carlo simulations, our strategy also covers the first-principles free energies of lattice vibrations. To this end, the effective cluster interactions of the CE have been rendered genuinely temperature dependent, so that they can include the vibrational free energies of the input structures. As a model system we use the phase-separating alloy Fe-Cu with our focus on the Fe-rich side. There, the solubility is derived from Monte Carlo simulations, whose precision had to be increased by averaging multiple CEs. We show that including the vibrational free energy is absolutely vital for the correct first-principles prediction of Cu solubility in the bcc Fe matrix: The solubility tremendously increases and is now in quantitative agreement with experimental findings.

Phase separation in Beta Ti alloys through HRTEM characterization.

Extended abstract of a paper presented at Microscopy and Microanalysis 2012 in Phoenix, Arizona, USA, July 29 – August 2, 2012.

Phase separation in monotectic alloys as a route for liquid state fabrication of composite materials

The mechanism of liquid–liquid phase separation and factors determining the solid-state microstructure of monotectic alloys are discussed. The effect of the cooling rate on the phase-separated morphology is shown in examples of Al–In, Al–Pb, Ni–Nb–Y and Zr–Gd–Co–Al alloys solidified by different techniques. A remarkable improvement of the microstructure for the Al91Pb9 hypermonotectic alloy cast with TiB2 particles, which catalyze the phase separation, is demonstrated.

Thermoelectric Properties of Bi<SUB>2</SUB>Te<SUB>3</SUB>–PbTe Pseudo Binary Near the Eutectic Composition

(Bi2Te3)(1-x)(PbTe)(x) binary systems near eutectic composition were prepared by melting of elemental metals and a sequential water quenching process and their microstructures and thermoelectric properties were investigated. Multiple phases such as Bi2Te3, BiPbTe and PbTe were observed due to phase separation when the composition x was higher than the eutectic point. Also the electrical conduction type of the alloys converted from p-type to n-type in the phase separated alloys. The lattice thermal conductivities in the phase-separated alloys are lower than those in alloys without phase separation, attributable to increased boundary scattering.

Computational modeling of phase separation and coarsening in solder alloys

Solders represent highly versatile and useful materials. They provide a broad range of technical applications such as soldering in automotive processing, microelectromechanical systems (MEMS) and solar panels. Due to the fascinating variety of microstructural changes solder materials underlie, their micromorphological dynamics have been extensively studied in the past decades by experimental, analytical and numerical approaches. The evolved microstructure exerts a significant effect, in particular, in very small components such as solder joints in microelectronic packages. In order to capture the essence of the microstructural evolution in solder alloys with a diffusion theory of heterogeneous solid mixtures we employ an extended Cahn–Hilliard phase-field model. In our contribution we introduce different numerical schemes to treat Cahn–Hilliard equation. Here we focus on the innovative isogeometric finite element approach and outline its considerable benefits in comparison to the other methods. To this end we present numerical simulations of phase decomposition and coarsening controlled by diffusion for eutectic binary solders Sn–Pb and Ag–Cu illustrating the versatility of this approach. A concluding computational study of a three-dimensional phase separation event within a solder ball geometry will corroborate the quality of our model.

Microstructure and phase selection in supercooled copper alloys exhibiting metastable liquid miscibility gaps

The influence of both bulk supercooling and cooling rate on the microstructure and phase selection during solidification of Cu–Co, Cu–Co–Fe, and Cu–Nb alloys exhibiting metastable liquid miscibility gaps were investigated using scanning electron microscopy, X-ray diffraction, and transmission electron microscopy. Containerless electromagnetic levitation was used to achieve large bulk supercoolings in the specimens. Supercooling of these alloys below a certain temperature resulted in metastable separation of the melt into two liquids, a Cu-lean (Co, Co + Fe, or Nb enriched) melt (L1) and a Cu-rich melt (L2). Usually, the microstructure of the phase-separated alloys consisted of spherulites corresponding to one of the phase-separated liquids embedded in a matrix corresponding to the other. The microstructure and phase selection are found to depend on factors such as: alloy composition, supercooling level, whether the material was dropped before or after recalescence, and the cooling rate during solidification. The following results were observed: (1) solidification of metastable e-Cu with enhanced Co (or Co + Fe, or Nb) solubility; (2) partitionless solidification of the L1 and L2 liquids; (3) spinodal decomposition of the supercooled liquid, and (4) secondary melt separation. The results are discussed and related to current solidification theories regarding solidification paths for the different conditions examined. The miscibility gap boundaries for the different alloys were determined and compared with those reported in the literature.

Magnetoresistance and Structural Study of Electrodeposited Ni-Cu/Cu Multilayers

Electrodeposition was used to produce Ni‐Cu/Cu multilayers by two-pulse plating (galvanostatic/potentiostatic control) from a single sulfate/sulfamate electrolyte at an optimized Cu deposition potential for the first time. Magnetoresistance measurements were carried out at room temperature for the Ni‐Cu/Cu multilayers as a function of the Ni‐Cu and Cu layer thicknesses and the electrolyte Cu2+ ion concentration. Multilayers with Cu layer thicknesses above 2 nm exhibited a giant magnetoresistance (GMR) effect with a dominating ferromagnetic contribution and with low saturation fields (below 1 kOe). A significant contribution from superparamagnetic (SPM) regions with high saturation fields occurred only for very small nominal magnetic layer thicknesses (around 1 nm). The presence of SPM regions was concluded from the GMR data also for thick magnetic layers with high Cu contents. This hints at a significant phase-separation in Ni-Cu alloys at low-temperature processing, in agreement with previous theoretical modeling and experiments. Low-temperature measurements performed on a selected multilayer down to 18 K indicated a strong increase of the GMR as compared to the room-temperature GMR. Structural studies of some multilayer deposits exhibiting GMR were performed by X-ray diffraction (XRD) and transmission electron microscopy (TEM). The XRD patterns of Ni‐Cu/Cu multilayers exhibited in most cases clear satellite peaks, indicating a superlattice structure which was confirmed also by cross-sectional TEM. The deterioration of the multilayer structure revealed by XRD for high Cu-contents in the magnetic layer confirmed the phase-separation concluded from the GMR data.

Kinetic pathways for phase separation: An atomic-scale study in Ni–Al–Cr alloys

The kinetic pathways involved in the formation of γ′(L12 structure)-precipitates during aging of concentrated Ni–Al–Cr alloys at 873K, for three distinct alloy compositions, are studied experimentally by atom probe tomography, and computationally with lattice kinetic Monte Carlo (LKMC) simulations using parameters deduced from first-principles calculations of cohesive energies, and from experimental diffusion data. It is found that the compositional evolution of the γ′-precipitate phase does not follow the predictions of a classical mean-field model for coarsening of precipitates in ternary alloys. LKMC simulations reveal that long-range vacancy–solute binding plays a key role during the early stages of γ′-precipitation. With the aid of Monte Carlo techniques using the parameters employed in the LKMC simulations, we compute the diffusion matrix in the terminal solid-solutions and demonstrate that key features of the observed kinetic pathways are the result of kinetic couplings among the diffusional fluxes. The latter are controlled by the long-range vacancy–solute binding energies. It is concluded that, because it neglects flux couplings, the classical mean-field approach to phase separation for a ternary alloy, despite its many qualitatively correct predictions, fails to describe quantitatively the true kinetic pathways that lead to phase separation in concentrated metallic alloys.

Understanding atomic-scale phase separation of liquid Fe-Cu alloy

Using liquid Fe60Cu40 alloy as a model, the structure of liquid Fe-Cu alloy systems is investigated in the temperature range 1200-2200 K, covering a large metastable undercooled regime, to understand the phase separation of liquid Fe-Cu alloys on the atomic scale. The total pair distribution functions (PDFs) indicate that liquid Fe60Cu40 alloy is ordered in the short range and dis-ordered in the long range. If the atom types are ignored, the total atom number densities and PDFs demonstrate that the atoms are distributed homogenously in the liquid alloy. However, the segregation of Fe and Cu atoms is very obvious with decreasing tem-perature. The partial PDFs and coordination numbers show that the Cu and Fe atoms are not apt to get together on the atomic scale at low temperatures; this will lead to large fluctuations and phase separation in liquid Fe-Cu alloy.

Novel insight into microstructural evolution of phase-separated Cu–Co alloys under influence of forced convection

Cu–Co alloys of bulk compositions Cu75Co25 and Cu84Co16 were undercooled and solidified using electromagnetic levitation. Microstructure of the samples was characterized using optical microscopy, scanning electron microscopy, energy dispersive spectrometry, and micro X-ray diffraction analysis. It was found that besides a bimodal size distribution, droplets of the Co-rich L1 phase resulting from liquid phase separation have a broad composition distribution. The bimodal size distribution of the droplets is more pronounced for the Cu75Co25 sample than for the Cu84Co16 sample, whilst the composition distribution of the droplets is broader in the Cu75Co25 sample than in the Cu84Co16 sample. The droplets were found to also have different types of substructures in the two samples. The results provide novel insight into microstructural evolution of phase-separated alloys under influence of forced convection.

Quantum dot laser

Discovery of self-organized epitaxial quantum dots (QDs) resulted in multiple breakthroughs in the field of physics of zero-dimensional heterostructures and allowed the advancement of optoelectronic devices, most remarkably, lasers. The most advanced and well-understood results are obtained for lasers based on Stranski–Krastanow InGaAs–GaAs three-dimensional QDs; even significant progress in the understanding of basic lasing properties is also achieved for QDs made of II–VI materials and ‘native’ QDs formed by nanoscale alloy phase separation in the InGaN–AlGaN material system.

New Directions in Simulation, Control and Analysis for Interfaces and Free Boundaries

The field of mathematical and numerical analysis of systems of nonlinear partial differential equations involving interfaces and free boundaries is a flourishing area of research. Many such systems arise from mathematical models in material science, fluid dynamics and biology, for example phase separation in alloys, epitaxial growth, dynamics of multiphase fluids, evolution of cell membranes and in industrial processes such as crystal growth. The governing equations for the dynamics of the interfaces in many of these applications involve surface tension expressed in terms of the mean curvature and a driving force. Here the forcing terms depend on variables that are solutions of additional partial differential equations which hold either on the interface itself or in the surrounding bulk regions. Often in applications of these mathematical models, suitable performance indices and appropriate control actions have to be specified. Mathematically this leads to optimization problems with partial differential equation constraints including free boundaries. Because of the maturity of the field of computational free boundary problems it is now timely to consider such control problems. In order to carry out design, control and simulation of such problems interaction is required between distinct mathematical fields such as analysis, modeling, computation and optimization. By bringing together leading experts and young researchers from these separate fields we intended to develop novel research directions in applied and computational mathematics. The aim of the workshop here was to focus on emerging new themes and developments in these fields and to establish and extend links between them.

Severe plastic deformation and phase transformations

Large densities of crystalline defects are created in severely deformed metallic alloys, pushing systems far from the thermodynamic equilibrium. Thus, strain induced phase transformations may occur and affect the mechanical properties like in the well known white etching layer on the surface of rail tracks. Here we briefly report about different phenomena reported in the literature such as solid state amorphization, super saturated solid solutions, precipitate dissolution and disordering. Phase separation and precipitation in nanostructured metallic alloys by severe plastic deformation prior to aging are also discussed.

Naturally-Formed Nanoscale Phase Separation in Epitaxially-Grown III-V Semiconductor Alloys

Extended abstract of a paper presented at Microscopy and Microanalysis 2010 in Portland, Oregon, USA, August 1 – August 5, 2010.

Evolution of phase separation in In-rich InGaN alloys

Evolution of phase separation in InxGa1−xN alloys (x∼0.65) grown on AlN/sapphire templates by metal organic chemical vapor deposition has been probed. It was found that growth rate, GR, is a key parameter and must be high enough (>0.5 μm/h) in order to grow homogeneous and single phase InGaN alloys. Our results implied that conditions far from thermodynamic equilibrium are needed to suppress phase separation. Both structural and electrical properties were found to improve significantly with increasing GR. The improvement in material quality is attributed to the suppression of phase separation with higher GR. The maximum thickness of the single phase epilayer tmax (i.e., maximum thickness that can be grown without phase separation) was determined via in situ interference pattern monitoring and found to be a function of GR. As GR increases, tmax also increases. The maximum value of tmax for In0.65Ga0.35N alloy was found to be ∼1.1 μm at GR>1.8 μm/h.

Growth and Characterization of Telecommunication-Wavelength Quantum Dots Using Bi as a Surfactant

Telecommunication-wavelength quantum dots (QDs) were successfully grown by metalorganic vapor phase epitaxy using a novel growth method in which trimethylbismuth (TMBi) was supplied during the growth. Supplying TMBi during the growth was confirmed to have a surfactant effect, but did not result in the formation of a bismuth-containing alloy. Using this growth method, the photoluminescence intensity and wavelength of the QDs were much improved. It was found that the QD size was increased during the growth of the InGaAs covering layer; this effect partly resembled activated alloy phase separation reported for molecular-beam-epitaxy-grown QDs. For the realization of high density and multilayer QDs, we confirmed that a much higher V/III ratio than that of usual growth conditions and a strain-compensation structure are effective, respectively.

Numerical simulation of diffusion induced phase separation and coarsening in binary alloys

Solder materials are solid mixtures and show a variety of micro-structural changes, for example, separation of phases and phase coarsening (Ostwald ripening). The micro-structural changes influence strength and life expectation of these materials. They may be of significant effect, in particular, in very small components such as solder joints in microelectronic packages.In order to analyze the micro-morphological evolution with a diffusion theory of heterogeneous solid mixtures we apply an extended Cahn–Hilliard phase field model. The underlying equations involve a bipotential operator and, accordingly, fourth-order spatial derivatives. Thus, the variational formulation of the Cahn–Hilliard problem requires approximation functions which are piecewise smooth and globally C1-continuous; standard finite element techniques cannot be applied here. In our contribution we fulfil the continuity requirement by means of rational B-spline finite element basis functions. To illustrate the versatility of this approach numerical simulations of phase decomposition and coarsening controlled by diffusion and by mechanical loading are discussed and compared with experimental results. Studies of error and convergence are provided.

Lattice Boltzmann modelling of liquid-liquid phase separation of monotectic alloys

本文建立了一个模拟在弥散相液滴的扩散长大、碰撞凝并和Ostwald熟化等因素的作用下偏晶合金液-液相分离过程的二维格子玻尔兹曼方法 (lattice Boltzmann method, LBM) 模型.该模型结合了Shan-Chen的两相流模型和Qin的介观粒子相互作用势模型的优点，并在LB演化方程中引入了反映相变的源项.应用该模型模拟研究了偏晶合金液-液相分离过程中单液滴的生长、两液滴的合并和多液滴的生长规律.结果表明在两液相区中第二相单个液滴的生长是一个通过扩散从非平衡态到平衡态过渡的过程.两液滴合并