Invited Oral Presentations
List of Invited Oral Presentations
Authors |
Title & Abstract |
Ken Elder |
Ordering of Ultrathin Metal and Graphene Films on Metallic Surfaces In this talk I would like to discuss the patterns formed by atomically thick films on metallic surfaces. The nature of the films is to a large extent determined by the differences between the film and substrate lattices (symmetry and lattice constant) and the strength of the coupling between the film and substrate. For concreteness the cases of two dimensional triangular and honeycomb lattices on a substrate with a surface that is triangular is examined. The former is relevant to systems such as Cu/Ru(0001) and Pd/Ru(0001) and the latter is relevant to graphene on various metallic substrates. Typically it is shown that for weak, intermediate and strong film/substrate coupling (small, medium and large film/substrate lattice mismatch) honeycomb superstructures, striped and ordered film patterns emerge respectively. A discussion of the dynamics and methods used to obtain the patterns on micron scales with atomistic resolution will also be discussed. |
Alain Karma, Yechuan Xu |
Phase-field Crystal Modeling of Nanocrystalline Grain Growth Nanocrystalline materials with a grain size in the range of ten to hundred nanometers exhibit unique mechanical, chemical and electrical properties. However, the large grain boundary content can drive rapid grain growth resulting in a loss of those properties. Grain growth can also be induced by external stresses leading to stress concentration and fracture around larger grains. This talk will discuss recent progress made to understand fundamental aspects of both curvature- and stress-driven grain growth in nanocrystalline materials using a combination of phase-field crystal and molecular dynamics simulations for fcc metals. For curvature driven grain growth, growth exponents and grain size distributions are characterized that depart markedly from those expected for normal grain growth. Some basic mechanisms that underlie those differences are identified. For stress-driven grain growth, shear-coupled grain boundary motion is found to play a dominant role in the coarsening behavior of the grain structure. Similarities and differences between mechanisms of grain boundary motion and grain growth behavior in phase field crystal and atomistic simulations are highlighted. |
Peter Voorhees |
The Dynamics of Dislocation-Mediated Grain Growth: Rotation and Translation The evolution of grains in nanocrystalline materials is examined using the phase field crystal (PFC) method. The strength of the PFC method lies in its ability to follow the atomic scale motion of a process that occurs on diffusive timescales. Due to the atomic scale resolution of the PFC method, the evolution of the dislocation structure of a grain boundary and the local atomic displacements of atoms near the boundary during grain growth can be determined. For example, we find that grain boundary dislocation interactions give rise to a repeating faceting-defaceting transition during grain shrinkage that leads to a different relationship between the grain area and time than seen in classical curvature-driven grain growth. The evolution of the dislocation structure during grain growth in nanocrystalline materials can also induce grain rotation and, in particular, grain translation. The mechanism for dislocation-mediated grain translation during grain growth will be discussed. |
Yanli Lu, Tingting Hu, Hong Mu, Zheng Chen |
Phase-field Crystal Study for the Effect of Heating Rate on the Premelting and Melting of the Grain Boundary The effect of heating rates on the premelting and melting of grain boundary is investigated by the phase-field crystal method. The microstructure evolution of premelting and melting under different heating rates is discussed, and the film width of grain boundary is quantitatively calculated in terms of the excess mass method. Results show that the liquid film appears as the melting point (Tm) is approached from below and that the film thickness is affected by heating rate and misorientation angle. In the heating process, at the same temperature (close to Tm), the liquid film width is larger when the temperature increases with lower heating rate. In the insulation process, the liquid film thickness increase until to its equilibrium value with holding time prolonging. For low-angle grain boundaries, the higher heating rate the system has during heating stage, the larger equilibrium value of liquid film will be reached during insulation stage. For high-angle grain boundaries, the liquid film thickness has the same equilibrium value during the insulation process although the heating rate is different during heating process. At the same temperature, the width of the liquid film increases with misorientation angle increasing. |
Jincheng Wang, Can Guo |
A Modified Phase-field Crystal Model for Solid-Liquid Phase Transition Phase field crystal (PFC) model is a powerful numerical method to simulate micro-structural evolution at the atomic length scale but the diffusive time scale. in recent years, a variety of PFC models have been developed, in which the two-mode PFC model proposed by Greenwood et al. has been constructed very skillful and considered as one of the most potential models. This model can give rise to realistic materials phase diagram, and is successful in stabilizing body-centered-cubic (BCC), face-centered-cubic (FCC), hexagonal-close-packed (HCP) lattices. However, this model can’t describe the solid-liquid interface and its anisotropic well. Here, we have proposed a modified PFC model which is quite suitable to model the solid-liquid phase transition, and the three-dimension dendritic growth has been chosen as an example to demonstrate the ability and to verify the validity of the modified model in the investigation of solid-liquid interface issues. |
L. Gránásy, F. Podmaniczky, G. I. Tóth, G. Tegze, T. Pusztai |
Recent Advances in Phase-Field Crystal Modeling of Heterogeneous Crystal Nucleation Freezing of the undercooled melts often starts by heterogeneous crystal nucleation assisted by container walls or foreign particles. The efficiency of these heterogeneities in promoting crystallization depends on the ordering of the liquid adjacent to the foreign surfaces: it may either help or hinder nucleation on the surface, depending on the size, structure, lattice constant, and the roughness of the foreign surface. In this work, we present recent results for heterogeneous crystal nucleation obtained using a phase-field crystal model, in which the substrate is represented by periodic external potentials and envelope functions. In treating the problem numerically, we rely on the Euler-Lagrange method outlined in Refs. [1, 2]. First, we investigate the effect of the mismatch between the lattice constants of the nucleating crystal on nucleation. We show that it plays a decisive role in determining the contact angle and nucleation barrier, which quantities were found to be non-monotonic functions of lattice mismatch [1, 2]. Next, we study the effect of surface roughness on the nucleation barrier. Finally, we explore the validity rage on the free growth limited model of particle-induced crystallization by Greer and co-workers. [1] G. I. Tóth, T. Pusztai, G. Tegze, G. Tóth, L. Gránásy: Amorphous nucleation precursor in highly nonequilibrium fluids. Phys. Rev. Lett. 107, 175702 (2011). [2] L. Gránásy, F. Podmaniczky, G.I. Tóth, G. Tegze, T. Pusztai: Heterogeneous nucleation of/on nanoparticles: a density functional study using the phase-field crystal model. Chem. Soc. Rev., Advance Article 2014. DOI: 10.1039/c3cs60225g. |
Thomas Werz, Carl E. Krill III, Michael Heinze, Stefan Odenbach, Nan Wang, Long-Qing Chen |
Showdown! Pitting Phase Field Simulations Against Time-Resolved Experimental Measurements of Ostwald Ripening in 3D It’s not only moviegoers whose imagination has been captured by the 3D revolution in imaging technology—even materials scientists have been swept up in the hype! Their excitement is understandable, given the power of modern techniques like x-ray tomography to map out microstructural evolution in 3D with high spatial and temporal resolution. Not only can such measurements offer deeper insight into the mechanisms that underlie transformations in microstructure, but the resulting data sets may also constitute a stringent test for the validation of computational algorithms used to model the same phenomena. We offer an example for such validation via the study of Ostwald ripening in the model system Al-5 wt% Cu. First, we employed x-ray microtomography to obtain real-time 3D image sequences of particle growth and shrinkage, and then we used a phase field model to predict the coarsening occurring in the same sample region, starting from the measured initial state. Detailed comparison of simulation to experiment reveals a surprising degree of consistency between the two data sets but also telltale discrepancies. We discuss the origin of the latter and possible strategies for improving the predictive power of the computational simulations. |
Yong Du, Kaiming Cheng, Yingbiao Peng, Weibin Zhang, Weimin Chen, and Lijun Zhang |
Owing to high competition in the market and the trend to improve productivity while reducing costs of metalworking, cemented carbides have become more popular for a variety of tooling applications such as cutting and stamping tools. Cemented carbides usually consist of hard WC grains embedded in a more ductile binder phase (Co) and other hard-phase grains in the form of carbides or carbo-nitrides with cubic structure. The hard phase contains Cr, Ta, Ti, Nb, V and so on. It is well known that the alloy composition and the sintering conditions strongly influence the performance of cemented carbides. In order to find a quantitative way to control the microstructure evolution in cemented carbide, the reliable thermodynamic and kinetic (atomic mobility) databases for the 8-component W-C-Co-Cr-Ta-Ti-Nb-V cemented carbide were established firstly. Based on the databases, the gradient zone formation during sintering was then investigated experimentally (SEM, EPMA, and TEM) and theoretically (DICTRA and phase field simulation). The simulation of gradient layer growth as a function of temperature and time in one dimension (1D) was performed using the DICTRA software. The simulation on the effect of alloy compositions, atmosphere, sintering time and temperature, grain size on the microstructure evolution in cemented carbides was performed by using two-dimensional (2-D) phase field method. |
Yuichiro Koizumi, Toshihiro Yamazaki, Akihiko Chiba, Koji Hagihara, Takayoshi Nakano, Koretaka Yuge, Kyosuke Kishida, Haruyuki Inui |
Interfacial Segregation in MoSi2-Based Duplex Alloys: a Systematic Phase-field Study Combined with First-Principles Calculation MoSi2-based alloys are attracting attention as promising candidates for novel gas turbine materials to realize ultrahigh-efficiency power generation systems operated at high temperatures beyond the technological limit of Ni-based superalloys. However, further improvements in toughness and creep resistance are required for practical use, and the interfacial segregation is the key for the improvements via thermal stability of microstructures. In the present study, segregation of all the transition elements at C11b/C40-lamellar interface in (Mo,Nb)Si2 alloy has been investigated systematically by phase-field simulations combined with first principles calculation to find effective alloying elements. The experimentally known Cr-segregation at lamellar interface has been reproduced by a simulation using segregation energy evaluated by first principles calculation. Zr, Hf and Ta have been also suggested to segregate significantly. The relationship between the magnitude of segregation and the solute-interface interaction will be discussed with particular attention to the competition between the reduction of interfacial lattice mismatch and site preference of the alloying elements. |
Yu U. Wang |
As new applications of Phase Field Method, we present recent development of Diffuse Interface Field Approach (DIFA) to modeling and simulation of particle-based materials processes. The model employs diffuse interface fields to describe multiple moving particles of arbitrary shapes, sizes and configurations, as well as the charge and electric/magnetic dipole properties of the particles. Particle interactions of short-range (mechanical contact, steric repulsion, friction) and long-range (electrostatic, magnetostatic) natures, capillary forces of liquid interfaces (interfacial tension, Laplace pressure), and external fields (applied electrical and magnetic conditions, mechanical confinement) are taken into account. Microstructure evolution of particle ensemble and multi-phase fluids with/without applied external field (used to tune the long-range interactions and control configurations of the particles) are simulated without explicitly tracking the particle boundaries or liquid interfaces. Simulation examples are presented to illustrate the model’s capability and potential to explore particle-based processing routes to advanced materials with novel microstructures and improved properties, including random packing of particles with friction, in-situ firming of colloidal microstructures by capillary bridges, and self-assembly of colloid systems. |
Shibin Dai, Qiang Du |
Coarsening Mechanism for Cahn-Hilliard Equations with Degenerate Diffusion Mobility We study a binary phase transformation governed by a Cahn-Hilliard equation which adopts a continuously differentiable double-well potential and a diffusion mobility that is degenerate in one or both phases. Using the asymptotic matching method, we derive sharp interface models of the system under consideration to theoretically analyze the interfacial motion with respect to different scales of time. We describe in particular the regime that together with a surface diffusion process along the interface, there is a non-trivial porous medium diffusion process in the bulk which provides communications between disjoint components and accounts for the occurrence of coarsening. |
Hui-Chia Yu, Larry Aagesen, Ali Ramazani & Katsuyo Thornton |
Phase-Field Models with Smoothed Boundary Methods: Powerful Tool for Modeling Nano/Microstructure Evolution Simulations of morphological evolution often require boundary conditions set within the computational domain. Phase field models are typically solved on a regular grid, which makes it difficult to impose internal boundary conditions. In this talk, we briefly describe a numerical method, the Smoothed Boundary Method, that can be combined with phase field modeling to incorporate the effects of boundaries that may possess complex geometry. This method employs a domain parameter, similar to a phase-field order parameter, to differentiate domains where different governing equations apply and specify where the boundary conditions are applied. We will discuss its applications, including simulations of selective-area epitaxy and solidification within confined geometry, as time allows. |
G. Boussinot, E.A. Brener |
Kinetic cross coupling between the non-conserved and the conserved fields in phase field models We introduce a kinetic cross coupling between the non-conserved and the conserved fields in the phase field model. Here off diagonal terms are present in the Onsager formulation (relations between fluxes and driving forces) of the equations of motion. This brings a new kinetic parameter in the phase field model allowing for a correspondence between the number of kinetic coefficients in the phase field model and the number of kinetic coefficients in the macroscopic approach. We will present specific examples for which adding these new cross terms is crucial, focusing on the case of unequal finite diffusion coefficients in the two phases. |
Machiko Ode |
Introduction of thermodynamic-fluctuation-based nucleation to phase-field model Thermodynamic fluctuation-based nucleation model is introduced to phase-field simulations. The normally distributed random temperature fluctuation proposed by Landau and Lifshitz is imposed on each calculation grid, and if the grid temperature is lower than the average temperature, the undercooling is converted to the increase of solid fraction, where the conversion factor is given as a function of the latent heat, heat capacity, and the phase-field threshold for nucleation. For the numerical simulation, small three-dimensional cubes surrounded by a heat bath are prepared. The random temperature fluctuation and solid fraction are set as the initial conditions. We repeated the calculation more than 50 times under the same calculation conditions, starting with different random number seeds. The ratio of nucleation occurs to the total calculation trials is defined as the nucleation ratio. The nucleation ratio steeply increases at a certain range of undercooling and the transition temperature is regarded as the nucleation temperature. The obtained nucleation temperature is rather in good agreement with that by a classical nucleation theory. These nucleation criteria are successfully applied to the second-phase precipitation of peritectic solidification. |
Xiaofeng Yang |
Phase field methods, algorithms and simulations on the complex fluid system We present some efficient modeling approaches and algorithms for the two phase complex fluid system, that is the immiscible mixture of nematic liquid crystal fluid and viscous ambient fluids. The numerical algorithms are efficient with some advantages of decoupling, linear, and elliptic. Qualitative agreements with experimental results are observed. |
H. Neumann-Heyme, K. Eckert and C. Beckermann |
Coarsening and refinement phenomena in dendritic solidification Curvature-driven interface motion plays an important role in the formation of the final microstructure during dendritic solidification. Usually, such motion results in a coarser microstructure via coalescence or retraction of dendrite sidebranches. Under certain conditions, however, the microstructure can be refined due to curvature-driven pinching events that lead to dendrite fragmentation. Such pinching events are a strong function of the size and shape of the initial dendrite structure. In the present study, two- and three-dimensional phase-field simulations are performed to investigate coarsening and refinement phenomena during directional solidification of alloys. The phase-field model is solved using a finite element library that permits adaptive mesh refinement and exhibits excellent parallel scalability on supercomputing facilities. A semi-implicit time integration scheme is used to allow for adaptive time stepping, which is particularly important because curvature-driven interface motion occurs on much longer time scales than the initial growth. The model implementation and the automated analysis of the simulated dendrite morphologies are discussed in detail. Systematic parametric studies are conducted to obtain a comprehensive view of the various curvature-driven structural evolution processes. |
Munekazu Ohno, Keiji Haga, Tomohiro Takaki and Kiyotaka Matsuura |
Scaling of Tilt Angle of Directionally Solidified Dendrites in Alloy Systems Analyzed by Quantitative Phase-field Simulations The growth direction of columnar dendrites in directional solidification is one of the important factors determining microsegregation pattern and the grain orientation of as-cast materials. In this study, quantitative phase-field simulations of directional solidification in alloy systems were carried out to elucidate scaling behavior of tilt angle of columnar dendrites with the preferred growth direction differing from the heat flow direction. We found that the tilt angle of columnar dendrites can be uniquely described by a single curve with respect to a dimensionless quantity consisting of thermal diffusion length, solute diffusion length, capillary length and the primary arm spacing. |
Tomohiro Takaki, Munekazu Ohno, Takashi Shimokawabe, Takayuki Aoki |
3D Large-Scale Phase-field Simulations of Competitive Dendritic Growth During Directional Solidification During directional solidification of polycrystal with different preferred growth directions, it is generally recognized that favorably oriented (FO) dendrites, for which the inclination angle between the preferred growth direction and heat flow direction is small, can keep growing with stopping the growth of unfavorably oriented (UO) dendrites. In recent studies, new dendrite selection phenomenon where the UO dendrites can stop the growth of the FO dendrites is reported. In this study, by performing phase-field simulations of bi-crystal and polycrystal competitive growth during directional solidification of binary alloy, the new dendrite selection phenomenon is investigated in detail. Here, by using a graphics processing unit (GPU) supercomputer TSUBAME2.5 at the Tokyo Institute of Technology, large-scale simulations are performed in three-dimension using a quantitative phase-field model. We show the power and potential of GUP parallel phase-field computations. |
Peter Bollada |
In micro structure formation in alloy solidification, the phase field method is used to replace sharp interface models with the understanding that as the diffuse interface tends to zero the sharp interface model is recovered. From this point of view the phase field method is viewed as a computational device for solving, indirectly, the sharp interface model. The larger the diffuse interface width the computational problem becomes more tractable but at a cost to the physical results. To this end, the introduction of solute anti-trapping by Kharma enables the width to be much larger than would otherwise be practically feasible. A fresh approach is to view the phase field model as the model starting point with the motivation that the interface width is in fact finite. From this view point the sharp interface model is an approximation. There are further advantages in that more general models, e.g. Multi-phase, multi-species models are readily postulated simply by specifying the free energy functional. This is by generic non-equilibrium thermodynamic methods that enable dynamical equations to be derived from the free energy functional. One such method, introduced by Beris and Edwards (Thermodynamics of flowing systems 1994), can reproduce the single and multi phase field models in the literature to date: with the exception of the anti-trapping current. This is because this term is not readily derived from the variational procedure. However, the methods of Beris and Edwards predict new terms not previously presented in the phase field literature. Some of these terms do indeed appear to function as solute anti-trapping. We present the Beris and Edwards formulation as applied to phase field and give numerical results illustrating the effect of some of the new terms. |
Amer Malik, Gustav Amberg |
Phase-field Simulations of Martensitic Transformations The phase field method, or diffuse interface method is a way to derive governing equations for phase change problems from a specification of thermodynamic potentials. It is particularly well suited to modeling of phase change in materials science, such as solidification or solid state transformations. One particular case is martensitic transformations, which are governed by temperature and composition, but also strongly influenced by the stress state in the solid. I will discuss 3D simulations of martensitic transformations in polycrystalline materials. The effect of plastic accommodation is investigated by using a time dependent equation for evolution of plastic deformation. Elasto-plastic phase field simulations are performed in 2D and 3D for different boundary conditions to simulate FCC-BCT martensitic transformation in polycrystalline Fe- 0.3%C alloy. The simulated results show that the introduction of plastic accommodation reduces the stress intensity in the parent phase and hence causes an increase in volume fraction of the martensite. The simulations also show that an autocatalytic transformation may be initiated at the grain boundaries and grow into the parent phase. It has been concluded that stress distribution and the evolution of microstructure can be predicted with the current model in a polycrystal. |
Valery I. Levitas |
Phase-field Approach to Structural Changes: Advanced Mechanics and Surface-Induced Phenomena Our recent results on the following basic problems of phase field theory and finite element simulations will be presented: 1. Multivariant martensitic phase transformations and twinning: large strains, interface stresses, and quasi-static and dynamic problems on nanostructure evolution. 2. Interaction between phase transformations and dislocations with application to strain-induced phase transformations at high pressure and shear in rotational diamond anvil cell and temperature-induced martensitic plates. 3. Surface-induced phenomena: pre-transformations, multivariant martensitic transformations, barrierless nucleation, scale and kinetic effects, new phase diagrams, and morphological transitions. |
Matthias Militzer, Hamid Azizi-Alizamini, Morteza Toloui, Bengqiang Zhou |
Phase-field Modeling of Microstructure Evolution in Advanced Steels This paper provides an overview on the application of phase field models to simulate microstructure evolution during processing of advanced steels. In particular, model extensions will be discussed to describe the formation of complex microstructures including bainitic ferrite. The status of the model development will be illustrated with three examples: 1) Effect of microsegregation on banding in low carbon steels during continuous cooling transformation, 2) Microstructure evolution during rapid thermal cycles in the heat affected zone of linepipe steels, 3) Simulation of intercritical annealing of dual-phase steels. These examples demonstrate that the phase field model approach is a powerful tool for the development of next generation process models. |
H. Zapolsky, M. Certain, M. Lavsky and A.G. Khachaturyan |
In spite of importance of carbon steel and decades of intensive studies of its crystallography and morphology of the martensitic transformation, there are many fundamental questions that are still not answered. We revisit this problem because of recent development of instrumentation has opened new opportunities to obtain an information that was not available in the past. We employed recently developed atomic Density Field Approach, which allows atoms freely move in the isotropic space without any geometrical constraints. A modeling of the 3D bcc -> fcc martensitic transformation allowed us to predict the major structural characteristics of martensitic transformation at atomic scale including nucleation, growth and eventually formation of internally twinned and dislocated plates. We also modeled a behavior of interstitial C atoms in the tempered virgin martensite. The C atoms diffusion in the bcc host lattice of Fe atoms results in the spinodal decomposition followed by the formation of C -rich plate-like precipitates with the habit plane close to (210)bcc plane. The further evolution leads to an aggregation of the precipitates into plates with invariant plane habit. The plates consist of twin -related (110)bcc layers of orientation variants with alternating direction of the tetragonality axis between the <100>bcc directions. |
James Warren |
A Phase-field Model of Vacancy Formation and Creep We develop an irreversible thermodynamics framework for the description of creep deformation in crystalline solids by mechanisms that involve vacancy diffusion and lattice site generation and annihilation. The material undergoing the creep deformation is treated as a non-hydrostatically stressed multi-component solid medium with non-conserved lattice sites and inhomogeneities handled by employing gradient thermodynamics. Phase fields describe microstructure evolution which gives rise to redistribution of vacancy sinks and sources in the material during the creep process. We derive a general expression for the entropy production rate and use it to identify of the relevant fluxes and driving forces and to formulate phenomenological relations among them taking into account symmetry properties of the material. As a simple application, we analyze a one-dimensional model of a bicrystal in which the grain boundary acts as a sink and source of vacancies. The kinetic equations of the model describe a creep deformation process accompanied by grain boundary migration and relative rigid translations of the grains. They also demonstrate the effect of grain boundary migration induced by a vacancy concentration gradient across the boundary. |
Shenyang Hu |
Application of Phase-field Methods in Predicting and Monitoring Microstructure Evolution and Property Degradation in Irradiated Materials The displacement cascades of high energy neutrons generate defects and defect clusters such as vacancies and interstitials in structural materials of nuclear reactors, which affect both thermodynamic and kinetic properties of the materials. As a consequence, the neutron irradiation accelerates defect migration, phase separation, microstructure evolution, and material property degradation. Therefore, it is crucial to develop the modeling capability to simulate microstructure evolution kinetics and understand the effect of microstructures on material properties for predicting the degradation of nuclear reactor material properties, developing advanced non-destructive evaluation technologies, and reducing uncertainty in operational and safety margin. This talk will review the applications of phase-field methods in irradiated materials. A quantitative phase-field model is developed to describe generation, migration and reaction of defects, phase nucleation, and microstructure evolution in irradiated materials. A multi-scale modeling framework in simulating microstructure and property evolution, and the effect of microstructures on thermo-mechanical properties, and the effect of microstructures on magnetic and mechanical responses is proposed. The grand challenges will be discussed for the application of phase-field methods in irradiated materials. |
Anter El-Azab |
Diffuse Interface Modeling of Void Growth in Irradiated Materials. Mathematical, Thermodynamic and Atomistic Perspectives We present an assessment of the diffuse interface models of void growth in irradiated materials. Since the void surface is inherently sharp, diffuse interface models for void growth must be constructed in a way to make them consistent with the sharp-interface description of the problem. Therefore, we first present the sharp-interface description of the void growth problem and deduce the equation of motion for the void surface. We also compare two existing phase field models to determine which one corresponds to the sharp-interface analysis. It was shown that a phase field model of type C, which couples Cahn-Hilliard and Allen-Cahn equations, is the most adequate since this type of model can take into account the reaction of point defects at the void surface via an Allen-Cahn equation. Fixing the model parameters in the diffuse interface model is discussed from the points of view of asymptotic matching. Sample results for void growth in a single component metal based on sharp and diffuse interface models are presented. Finally, a perspective on the use of atomistic modeling in both constitutive and nucleation modeling within the phase field approach for void formation in irradiated materials is presented. |
R. Berger, B. Böttger, M. Apel |
Phase-field Simulations of Reactive Air Brazing: Microstructure Evolution in Ag-Cu Brazing Fillers, DSC-Curves and Effective Thermal Expansion Reactive air brazing using Ag-Cu brazing fillers is a technology applied to join metallic and ceramic parts, e.g. in SOFC (solid oxide fuel cells) manufacturing. The multiphase field method in combination with thermodynamics databases is applied to simulate the microstructure evolution in the Ag-Cu brazing filler. Of particular importance is the interaction between the metallic alloy and oxygen from the ambient atmosphere. The effect of the ambient oxygen partial pressure and oxygen transport on the microstructure evolution in the filler metal will be discussed. The simulations show, e.g. that CuO formation and fcc-Ag solidification temperatures depend on the oxygen exchange between the ambient gas and the liquid brazing alloy. DSC (differential scanning calorimetry) curves are directly derived from the phase field simulations and confronted with experimental results which confirms the effect of oxygen. In addition, the effective thermal expansion of the two phase microstructure is calculated by means of mathematical homogenization and builds a bridge to a macroscopic mechanical FEM analysis of the brazing joint on the component level. |
Martin Z. Bazant |
Nonequilibrium Thermodynamics of Li-ion Batteries Li-ion batteries involve electrode materials, such as iron phosphate and graphite, which tend to separate into Li-rich and Li-poor phases upon intercalation of lithium. In nanoparticles, this bulk thermodynamic relaxation competes with surface electrochemistry, leading to the fundamental question: What is the reaction rate during a phase transformation? A consistent answer is provided by a phase-field theory of chemical thermodynamics that unifies and extends the Cahn-Hilliard and Allen-Cahn equations for chemical kinetics and charge transfer. The reaction rate depends on concentration gradients, elastic stress, and other non-idealities. Simulations based on the theory shed light on the complex nonlinear dynamics of ion intercalation and nucleation in battery nanoparticles and phase transformation dynamics in porous electrodes. |
Mikko Haataja |
Evolving Microstructures and Phase Transformations in Solid Oxide Fuel Cell Anode Materials Energy conversion processes in solid oxide fuel cell (SOFC) materials are strongly affected by a nonlinear coupling between mass/charge transport, heat transport, and morphology at nanometer and micrometer length scales in a multi-phase/multi-component system. Furthermore, under continuous operation, these complex morphologies and local compositions evolve over time in response to a multitude of physical, chemical, and mechanical cues at elevated temperatures. In order to understand and predict the stability of morphologies and their spatio-temporal evolution, a mesoscale approach, which accurately incorporates both atomic scale information and evolving microstructures, is required. In the first part of my talk, I will present our recent work on quantifying coarsening kinetics of Ni particles in Ni-YSZ SOFC anode materials based on phase-field simulations. In the second part of my talk, I will focus on the development of elastic stresses and resulting mechanical failure in SOFC anode materials driven by re-oxidation of Ni particles. |
Youhai Wen, Tianle Cheng |
Phase-field Modeling of Metal Oxidation Kinetics Oxidation of metals generally involves coupling between chemical reactions, mass transport and electrostatic interaction and oxidation kinetics is essentially a multi-scale problem. Existing theories mostly work for either a very thin oxide film or a thick one, leaving a length scale gap for oxidation kinetics. An electrochemistry based diffuse-interface model plus a multi-scale-relay scheme are developed to study oxidation kinetics in a gas/oxide/metal environment. The multi-scale-relay scheme allows the model to coherently cover a wide range of lengths and times and study the transition stage oxidation kinetics. The coupling between interfacial reactions and ionic transport with the moving boundary problem is solved, without using assumptions such as steady state, coupled-currents, local charge neutrality, or local chemical equilibrium. For the model oxidation system, in the thick-film limit perfect parabolic growth law is obtained with the rate constant in agreement with Wagner theory. Nevertheless, the Wagner-parabolic law is violated either when the oxide film thickness is on the order of the Debye length or when the interfacial reaction is rate-limiting. This work offers a unique opportunity for quantitatively studying oxidation kinetics at the intermediate length scale, thereby bridging the length scale gap in oxidation kinetics study. |
R. Spatschek, D. Korbmacher, C. Hüter, A. Chakrabarty, U. Aydin, J. Neugebauer |
Scale Bridging Modelling of Hydride Formation Hydrogen embrittlement is a serious issue for many high strength steels, and still a thorough understanding of this phenomenon is lacking. Among the different suggested models the hydrogen enhanced decohesion picture is based on a change of fracture energy due to hydrogen accumulation. We discuss the role of hydride formation, especially near free surfaces and in the vicinity of crack tips, from a combined ab initio, atomistic, mesoscale and thermodynamic perspective. Here, in particular the presence of elastic deformations can largely influence the hydride phase formation, and we demonstrate how this aspect can be covered in a scale bridging approach. |
Zhihua Xiao, Mingjun Hao and San-Qiang Shi |
Effect of Temperature on Hydride Morphology in Zirconium: A Phase-field Study Temperature plays an important role in hydride morphology in zirconium alloys, which in turn critically affect the mechanical properties of the alloys. This is mainly due to the fact that hydrogen solid solubility in zirconium is affected by temperature. We recently have developed a temperature dependent free energy functional for the Zr-H system. The newly developed free energy functional takes into account of crystallographic variants of hydrides, interfacial energy between hydride and matrix, interfacial energy between hydrides, elastoplastic deformation and interaction with externally applied stress. We have used this newly developed free energy functional in a phase-field scheme to study the effect of cooling rate and the effect of temperature gradients on the morphology of hydride precipitation in zirconium. The modeling results were compared with experimental data with a reasonable agreement. The model is fully quantitative in real time and real length scale. The work calls for experimental and/or theoretical investigations of some of the key material properties that are not yet available in the literature. |
Yichun Zhou, Limei Jiang, Jiyu Tang, Qiong Yang, Yi Zhang |
A Multi-Field Coupling Finite Element Model Considering Flexoelectric Effect for Ferroelectric Domain Evolution Flexoelectricity, the intrinsic coupling between strain gradient and polarization, opens up a way to switch polarization in ferroelectric thin films using mechanical force and may enable application in which memory bits are written mechanically and read electrically. In this work, a multi-field coupling finite element model considering flexoelectric effect for ferroelectric domain evolution is developed by combining finite-element and phase-field simulations. With this finite element model, we have studied the strain and polarization distribution inside the epitaxial film of archetypal ferroelectric PbTiO3, where the mismatch with the substrate is relaxed through the formation of domains (twins). The result shows a large different in out-of-plane strain, tensile in the acute corners and compressive in the obtuse ones, leading to a transverse gradient, which induces a horizontal rotation of ferroelectric polarization because of the flexoelectric effect. Agreement between modeling prediction and measurement evidences the validity of the proposed model. |
Yongmei M. Jin |
Phase-field Modeling and Simulation of Domain Processes in Multiferroic Materials Phase Field Method is used to model and simulate the domain evolution and microstructure-property relationships in multiferroic materials. It explicitly describes the evolving domain microstructures of arbitrary complexity without tracking domain boundaries or imposing a priori constraints on evolution paths, and automatically treats multiple concurrent physical processes within the same modeling framework (ferroelastic strain, magnetization, polarization, electrical current). The model is employed to study the domain processes and mechanisms in ferromagnetic, magnetostrictive, magnetic shape memory, and magnetoelectric materials and composites. Domain wall behaviors in planar magnetic nanowires, magnetic field-induced strain response in magnetostrictive materials and magnetic shape memory alloys, and strain-mediated magnetization-polarization coupling in magnetostrictive-ferroelectric composites are simulated. The roles of long-range magnetostatic, electrostatic and elastostatic interactions, domain wall mobility, wall-wall interactions, and interactions between domain walls and electrical current are investigated. The resulting physical behaviors of multiferroic materials are discussed in relation to applications in device, sensor, actuator and transducer technologies. |
P. Chu, Y. Zhang, Y. L. Xie, and J. M. Liu |
Phase-field and Monte Carlo Simulations of Ferroelectric Domain Structures In the first part of this work, the two-dimensional ferroelectric (FE) domain structures in artificial square grid are investigated, based on the Landau phenomenological model. The aperture ratio dependent domain pattern evolution is carefully studied. In details, with increasing aperture ratio, the intact FE 90° domains split into multiple strip-like domain patterns. Given further increasing hollow area in the grid structure, the orientation of electric polarizations in each domain is completely subjected to the pattern of the grid. The underlying mechanism is understood by analyzing the energy evolution of each phenomenological term. We also investigate the electric field driven motion of 90°-domain walls and frequency-domain spectrum of the dielectric permittivity in normally strained FE lattice using the phase field simulations. Driven by the ac-electric field, spatial distribution of dielectric permittivity is the largest near the 90°-domain walls and the dielectric permittivity is dramatically reduced in tensile-strained lattice. This phenomenon is related to the shortened time for domain relaxation and the less-mobile 90°-domain walls. A tensile strain favors the parallel domains but suppresses the kinetics of the 90° domain wall motion, while the compressive strain results in the opposite behaviors. |
X. Q. Ma, H. B. Huang, J. J. Wang, Z. H. Liu, W. Q. He and L. Q. Chen |
A Combined Phase-field Model on Phase Transition in NiCoMnIn Metamagnetic Alloys We developed a computational model by combining thermodynamic modelling, the phase-field method and micromagnetic simulations to investigate the magnetic and structural phase transitions in metamagnetic shape memory alloys. The first principle study was also performed to investigate the mechanism of the phase transition. A thermodynamic potential including the ferromagnetic, antiferromagnetic, and martensite order parameters was developed and further applied to a phase field model to simulate the ferromagnetic, antiferromagnetic, and martensite domain structures. By this model, we studied properties of typical metamagnetic alloy NiMnInCo, especially the transition temperature from ferromagnetic austenite phase to antiferromagnetic martensite phase, the Curie temperature, as well as its response to an external magnetic field. The obtained magnetization hysteresis loops are well consistent with existing experimental measurements. Our simulations show that the walls of martensite twins are superimposed with the 900 magnetic domain walls in low temperature martensite phase due to the magneto-structural coupling between the structural and the magnetic transition. |
Jie Wang and Jianwei Zhang |
A Real-Space Phase-field Model for the Domain Evolution of Ferromagnetic Materials A real-space phase field model based on the time-dependent Ginzburg-Landau (TDGL) equation is developed to predict the domain evolution of ferromagnetic materials. The phase field model stems from a thermodynamic theory of ferromagnetic materials which employs the strain and magnetization as independent variables. The phase field equations are shown to reduce to the common micromagnetic model when the magnetostriction is absent and the magnitude of magnetization is constant. The strain and magnetization in the equilibrium state are obtained simultaneously by solving the phase field equations via a nonlinear finite element method. The finite-element based phase field model is applicable for the domain evolution of ferromagnetic materials with arbitrary geometries and boundary conditions. The evolution of magnetization domains in ferromagnetic thin film subjected to external stresses and magnetic fields are simulated and the magnetoelastic coupling behaviour is investigated. Phase field simulations show that the magnetization vectors form a single magnetic vortex in ferromagnetic disks and rings. The configuration and size of the simulated magnetization vortex are in agreement with the experimental observation, suggesting that the phase field model is a powerful tool for the domain evolution of ferromagnetic materials. |
D.C. Lv, D. McAllister, M. J. Mills, Y. Wang |
Phase-field Simulation of Phase Transformation and Plastic Deformation in IN718 Superalloy Microstructural evolution during co-precipitation of γ’ (L12) and γ” (D022) phases from a supersaturated γ (FCC) matrix during aging of superalloy Inconel 718 (IN718) is investigated by computer simulation using the phase-field method. The precipitation model is quantitative, using as model inputs ab initio calculations of elastic constants, experimental data on lattice parameters, precipitate–matrix orientation relationship, interfacial energy of each individual precipitate phase, interdiffusivities, and a Ni–Nb–Al pseudo-ternary thermodynamic database specifically developed for IN718. The various precipitate microstructures generated are used as inputs to a phase field model of precipitate-dislocation interactions to investigate deformation mechanisms and precipitation hardening effect. The model uses directly general stacking fault (GSF) energies of γ, γ’ and γ” phases as inputs. These GSF energies are obtained from ab initio calculations and then fitted to experimental data of key stacking fault energies to account for finite temperature and chemistry effects. These phase field models are able to predict effects of alloy chemistry and heat treatment on precipitate microstructure, operating deformation mechanisms and the corresponding strengthening. The simulation results are compared directly to experimental characterization. |
M. Cottura, Y. Le Bouar, A. Finel, B. Appolaire |
Coupling Phase-field Methods with Dislocation Density Based Plasticity Mechanical properties of metallic materials strongly depend on their microstructure, i.e. on the shape and spatial arrangement of the different phases in the materials. It is thus important, from both fundamental and industrial viewpoints, to understand and control the microstructure evolution. The phase field method as emerged as the most powerful method for tackling microstructure evolutions during phase transformations, especially when elastic coherency stresses are generated in solids. However, in many materials, the microstructure evolutions are coupled with a plastic activity, and there is currently a great research effort to extend the phase field method to take this coupling into account. Within the phase field approach, plasticity can be incorporated either at the scale of dislocations or in a continuous framework. In this work, a classical phase field model is coupled to a crystal plasticity model based on dislocation densities. This model includes the anisotropy as well as the size-dependence of the plastic activity, which is expected when plasticity is confined in region below few microns in size. The model uses a storage-recovery law for the dislocation density of each glide system and a hardening matrix to account for the short-range interactions between dislocations. The proposed coupled model will be first applied to study the growth of a hard precipitate inside an elasto-visco-plastic matrix. Then, the rafting of ordered precipitates observed in Ni-based superalloys will be addressed in the case of creep loadings along [001] and [011] directions. |
Toshiyuki Koyama |
Phase-field Modeling of Microstructures and Rapid Image-Based Calculation of Stress-Strain Curve The complex morphological changes of microstructure have been modeled realistically based on the phase-field method, and the various materials properties dependent on the internal microstructure have been estimated from image-based calculations. In this study, the image data of microstructure obtained from phase-field simulation is utilized as an input data for the image-based calculation of stress-strain (SS) curve. The modified secant method[1] is employed for the rapid calculation of SS curves on the two-phase composite having the microstructure with arbitrary morphology. The two-phase microstructures considered are ferrite-pearlite (F-P), ferrite-bainite (F-B) and the ferrite-martensite (F-M) in steels, and results obtained are as follows: (1) SS-curve of the F-P composite does not depend on the morphology of microstructure but the volume fraction of the constituent phase. (2) On the SS-curves of F-B and F-M composites, when the microstructure is changed from isotropic random structure to anisotropic structure (modulated structure or layered structure), the flow stress of SS-curve at initial low strain stage is increased and the elongation up to plastic instability is decreased. (3) The change of proof strength on F-M composite during spheroidization of lamella structure is reasonably characterized by using the topological parameter "genus". |
Jaber Rezaei Mianroodi, Bob Svendsen |
In this study, a 3D phase field (PF) approach is developed for dislocation processes (e.g., [1-2]) with two order parameters per glide plane. This is similar to the current Peierls-Nabaro (PN) models (e.g., [3]) which are based on two disregistry fields per glide plane. In both cases of PF and PN methods, the free energy model includes stacking fault (SF) energy of the material obtained from first-principle or atomic calculations. Assuming periodicity, the evolution equations and mechanical equilibrium are solved in Fourier space providing better scalability. Example simulations are performed for two fcc metals with low SF energy (Cu) and high SF energy (Al) with the corresponding microstructure evolution. In case of Cu, loading results in the formation of multiple SF layers and twinning. On the other hand, in Al, the deformation is governed almost solely by ideal dislocation glide. Analogous results from the PN model and additional examples will be discussed. |
P.-L. Valdenaire, A. Finel, Y. Le Bouar. B. Appolaire |
Continuum Theory of Dislocations: Coarse-Graining and Correlations Microstructures in crystalline solids evolve under the simultaneous action of various driving forces, commonly referred to as chemical, elastic and plastic. These driving forces originate from an ongoing phase transformation, coherency-induced elastic stresses and plastic relaxation. The Phase Field method has emerged as one of the most powerful tool to tackle this complex multi-physic problem. Whereas the phase transitions aspects and their interplay with elastic relaxation are now well understood, the incorporation of plasticity is still the subject of an intense activity. The major problem is to identify a physically-based mesoscopic continuum plasticity theory that incorporates properly the internal length scale effects, which are crucial in the context of microstructural evolutions. This communication focuses precisely on this aspect and elaborates on a key issue in the theory of crystal plasticity: the transition between the discrete, where plastic flow is resolved at the scale of individual dislocations, and the continuum, where dislocations are represented by densities. This transition requires the use of coarse-graining procedures, similar to the ones that are used in the statistical mechanic treatment of out-of-equilibrium many-particle systems. Several attempts along this route have been proposed in the recent past. Our aim here is to shed a new light and to clarify the coarse-graining procedure that enables the mathematical transition between the dynamics of discrete dislocations and the transport equations that control the flow of mesoscopic dislocation densities. We emphasize in particular the role of the coarse-graining length on the correlation-induced stresses that emerge as result of the closure of the coarse-grained transport equations. |
Charlotte Kuhn, Ralf Müller |
On Degradation Functions in Phase Field Fracture Models Unlike the description of cracks as sharp surfaces which requires different criteria to predict the onset of crack propagation, the direction of crack growth, possible crack branching, and the nucleation of new cracks, the phase field approach provides a unified description of all these fracture processes. Here, the entire crack evolution follows implicitly from the solution of a coupled system of equations formed by the mechanical field equations and the evolution equation of the phase field order parameter, that is used to differentiate between broken and undamaged material. A so called degradation function couples the order parameter to the elastic properties of the material in order to model the change in stiffness between broken and undamaged material. The nucleation of new crack in originally undamaged material is preceded by a localization of the crack field. Before the onset of this localization the material response of the phase field fracture model is mainly controlled by the degradation function. However, the degradation function frequently found in the literature yields a pronounced softening behavior before the onset of fracture which is not desirable when modeling brittle materials. In this work we discuss the potential of alternative degradation functions in the context of crack nucleation and propagation. |
Rajeev Ahluwalia, Murali Palla, Syed Khaderi |
Phase-field Simulations of Fracture in Biocomposites Biological materials such as bone are composites of stiff and brittle mineral phases as well as soft organic biopolymers. Such composite materials are refered to as biocomposites. A remarkable property of such composites is that the toughness of the composite is orders of magnitude higher than toughness of the constituent mineral and organic phases. Understanding the origin of this toughness enhancement is an important area of research which can help in design of artificial composites with high toughness. The underlying microstructure formed by mineral and organic phases is complex with several levels of hierarchies. How this complex microstructure influences the propagation of cracks is an important issue which could help in understanding the toughness enhancement. Phase field models of crack propagation are ideal to study crack propagation in such composites due to their ability to simulate crack paths in complex microstructures, without any apriori assumption of the crack trajectories. In this talk, I will present our recent efforts to predict crack paths in such mineral-organic composites, focusing on the role of modulus mismatch and the composite geometry. I will first show how the presence of organic layer can lead to crack arrest, crack branching close to interfaces and in some cases, delamination at the interface. I will also discuss how the microstructure and the modulus mismatch influences the toughness of such composites. In many biocomposites, the organic and mineral components are arranged in a “brick and mortar” microstructure. Finally, I will show the complex crack propagation that can be observed composites with such complex microstructures. |
J. Clayton, J. Knap |
Phase-field Modeling of Fracture and Twinning in Nonlinear Anisotropic Solids Phase field theory for coupled twinning and fracture in single crystal domains is developed. Distinct order parameters denote twinned and fractured domains, finite strains are addressed, and elastic nonlinearity is included. An incremental energy minimization approach is implemented for prediction of equilibrium morphologies for quasi-statics. Aspects of the theory are analyzed in detail for a material element undergoing simple shear deformation. Predicted criteria for shear stress-driven fracture or twinning are often found to be in closer agreement with test data for several types of real crystals than those based on the notion of theoretical strength. Numerical 3D solutions are presented for a notched solid, demonstrating validity of the model for capturing crack propagation across a misoriented interface and crack deflection around a spherical elastic inclusion. |
Yong Ni, Linghui He, Ai Kah Soh |
Phase-field Modeling of Combined Cracking, Delamination and Buckling of Films The structure of a thin film with internal stress on an elastic substrate has been widely used in thin film technology. The compressed film often leads to buckling instability with formation of wrinkle, buckle-delamination blister, ridge cracking or their combinations. Understanding the film morphology driven by the buckling instability is important in various applications, i.e., flexible microelectronics, graphene nanotechnology, thin film metrology, mechanics-based nanofabrication. In this talk, I will discuss the phase field modeling of combined cracking, delamination and buckling in film-substrate systems. The model couples substrate elasticity, nonlinear film deformation and interface adhesion by integrating Green function method, Föppl-von Kármán plate theory and cohesive zone model. The buckling, cracking and delamination processes are formulated using the time-dependent Ginzburg-Landau kinetic equations, driven by minimizing the film-substrate total free energy. This model is able to capture formation of various buckling morphologies including wrinkling, buckle-delamination as well as combined buckling and cracking. The mechanism and conditions for their formations are identified, and good agreements with the experimental observations are obtained. The model may provide a computational tool for exploring the effects of pre-strain, substrate compliance and adhesion on the formation of complex nonlinear buckling and cracking patterns in film-substrate systems. |
H. Zapolsky, A. Khachaturyan |
Self-Organization of Complex Structures at Different Length Scales A general approach to the evolution of complex atomic structures in terms of the Phase Field Field Theory is proposed. It is shown that this approach can be extended to atomic scale and can be used to get an insight on the atomic mechanisms of solid-solid phase transformation. This new approach is illustrated by examples of systems of different physical nature, scales and complexities. |
John Lowengrub |
Phase-field Modeling of Biomembranes and Endocytosis |
W. Craig Carter |
Making Materials from Simulations This will not be a talk about materials science and simulations. I will not present any new results or new techniques. However, I will present something that derives materials simulations which has have given me deep satisfaction and renewed enthusiasm for the craft of developing simulations. I will present the results of a collaboration that uses materials science simulation techniques to make physical objects. As computational materials scientists, we can simulate of a large variety of materials processes and structures. Our phase field and level set methods produce instructive images and videos which are appreciated as beautiful to others and deeply satisfying to the creators of the images and simulations. These are art and we are artisans. In the last several years, I’ve been collaborating with the artist Neri Oxman of MIT’s Media Lab. We’ve been using existing techniques of materials simulation and developing new algorithms to produce material objects created by additive manufacturing and numerically controlled machining. The results have been exhibited at MoMa, the Modern Art Museum in Paris, the Smithsonian, the Paris Haute Couture Fashion show, and many other venues. I will describe a few of these objects and describe two examples where Cahn-Hilliard and level set methods were used in our creative process. |
A.Umantsev |
A number of very different recent experiments with nanoparticles (NPs) produced very similar results: in NPs of sizes above critical the sequence of transformations is similar to that of the bulk while in NPs of sizes below the critical a novel, amorphous (disordered) phase appears and remains stable in a significant domain of variation of the control parameters. A natural question arises: What is the origin of this phase? In a series of recent publications the author has developed a field theory of the nanophase stability, which claims that the phase that appears in NPs of sizes below the critical is a transition state between the stable bulk phases in the space of the order parameter that distinguishes between the symmetries of the bulk phases. Such change of stability of the transition state from unstable to stable can occur only in conditions of conservation: energy, matter, or volume. As known, in large systems with the conservation constraint the transformation will result in a two-phase state, which is a mixture of the two stable bulk phases. The theory claims that in a system of the size below the critical such state is energetically impossible due to high ‘energy cost’ of the phase separating interface. Then, the two-phase state is replaced by the homogeneous transition state. |
Michael Tonks, Derek Gaston, Cody Permann, Yongfeng Zhang, Xianming Bai |
Multiscale Modeling of Nuclear Fuel Performance using the Phase Field Method with the MOOSE Framework The phase field method is a powerful tool for investigating the impact of microstructure evolution on material performance. In this presentation, we demonstrate how the phase field method, implemented using INL’s finite element based multiphysics object-oriented simulation environment (MOOSE), is being used to develop improved materials models for modeling nuclear reactor fuel performance. The MOOSE framework provides a powerful tool for coupling the phase field model to other physics, such as heat conduction and deformation, and for determining bulk material properties as the microstructure evolves. The simulation results are then used to develop and refine mechanistic materials models for macroscale modeling. We demonstrate our multiscale approach for various aspects of nuclear fuel behavior. |
O.Shchyglo, E.Borukhovich, P.Engels, A.Monas, R.D.Kamachali, M.Stratmann, D.Medvedev, I.Steinbach |
OpenPhase - The Open Source Phase-field Simulation Library We present an open source phase field simulation library “OpenPhase” which is based on a multi-phase field multi-component model of Steinbach et al. [Steinbach et al., 1996; Steinbach, 2009]. The library is intended for high quality quantitative simulations of processes that involve structural and phase transformations. It contains modules that allow simultaneous solving the multi-phase field equations, equations for advection and diffusion of different components, Navier-Stokes equation for a liquid phase flow including the interaction with solid structures and elastic problem. The modular structure of the library allows for easy extension and modification by adding new modules. Several simulation examples related to the metallurgical processes will be presented. The library is distributed in the form of an open and free of charge source code which is available at www.OpenPhase.de. |
