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In the current work we employ multi-phase-field simulations to understand the effect of remelting on microstructure evolution, especially on nucleation of new grains during selective electron beam melting (SEBM) of Ni-based super alloy. The phase-field model is coupled to both mass and heat transport phenomena including release of latent heat of solidification. We run remelting simulations in both as cast and homogenized conditions. Experimental observations show that remelting triggers the nucleation of new grains at the melt pool border. The simulation results shed more light on the local conditions at the melt pool border thereby enhancing our understanding of the mechanisms responsible for the nucleation. The simulation results are validated with experimental results obtained for the Ni–20.5 mol pct Al model binary alloy.

Computational Fluid Dynamics (CFD) simulations are used in this work to study the dynamic behavior of the melt pool and heat transfer during the single-track laser melting process of a nickel-based superalloy (CMSX-4). To include the effects of powder inhomogeneities and obtain a realistic distribution of the powder layer on the bed chamber, the CFD model is coupled with a Discrete Element Method (DEM) solver. The coupled model is implemented in the open-source software package OpenFOAM. In the CFD model’s governing equations, some key physical mechanisms, such as the Marangoni effect and recoil pressure, are considered. With the help of the coupled CFD-DEM model, we have investigated the effect of key process parameters, such as laser power, scanning speed of the laser, powder size, and powder shape, on the size and homogeneity of the melt pool. From the simulation results, it was discovered that high laser power and slow scanning speed create a deep and narrow keyhole that leads to porosity. In contrast, balling defects are found to be caused by a small melt pool obtained from fast scanning speeds and inadequate laser power.

The tool of phase-field modeling for the prediction of chemical as well as microstructural evolution during crystallization from a melt in a mineralogical system has been developed in this work. We provide a compact theoretical background and introduce new aspects such as the treatment of anisotropic surface energies that are essential for modeling mineralogical systems. These are then applied to two simple model systems - the binary olivine-melt and plagioclase-melt systems - to illustrate the application of the developed tools. In one case crystallization is modeled at a constant temperature and undercooling while in the other the process of crystallization is tracked for a constant cooling rate. These two examples serve to illustrate the capabilities of the modeling tool. The results are analyzed in terms of crystal size distributions (CSD) and with a view toward applications in diffusion chronometry; future possibilities are discussed. The modeling results demonstrate that growth at constant rates may be expected only for limited extents of crystallization, that breaks in slopes of CSD-plots should be common, and that the lifetime of a given crystal of a phase is different from the lifetime of this phase in a magmatic system. The last aspect imposes an inherent limit to timescales that may be accessed by diffusion chronometry. Most significantly, this tool provides a bridge between CSD analysis and diffusion chronometry - two common tools that are used to study timescales of magmatic processes.

For the purpose of numerically exploring different microstructure evolution processes, the phase-field approach has been developed. Large domains and adequate models that combine many physical fields in statistically relevant volume elements are needed to describe these complicated processes accurately. In order to facilitate such development for multi-physics simulations, OpenPhase was developed for application in material science and engineering. This chapter presents two examples simulated using the OpenPhase software, grain growth and dendritic solidification. The corresponding input files for OpenPhase are listed and explained. Also, some hints for postprocessing are given.

In this chapter a new paradigm for the understanding of the physical world is developed based on phase-field type non-linear wave solutions. Two 1-dimensional wave fronts, right- and left-moving are combined to define the doubloon, the basic element of mass and space. The interface energy of a standard phase field defines the mass of elementary particles. The non-zero values of the phase field, the core of the doubloon, defines space in an introverted view. The sum constraint of the multi-phase-field model defines a doubloon network which is closed in itself. The model can be applied to investigate scale formation in the universe.

The chapter reviews the basic analytic solutions of phase-field models at the mesoscopic scale. First the phase-field equation is derived by the Clausius-Duhem relation applied to a simple phase-field functional. The concept of variational derivative, related to the gradient contribution in the phase-field functional, is introduced. The traveling wave solution, or solitonian wave solution, for the phase-field equation in 1-dimension is derived for two relevant potential functions. Finally, the relations between model parameters and physically defined material parameters are derived.

The chapter introduces the meaning of Phase Field from the aspect of thermodynamics on the one hand and numerics of moving boundary solutions on the other hand. A moving boundary solution here means the evolution, motion, of grain boundaries, phase boundaries or surfaces in multicrystalline materials as described by a set of partial differential equations. The thermodynamic aspect relates to the concept of an order parameter, identifying a phase, in thermodynamics in general. Here the interfaces, grain- or phase boundaries and surfaces, are described by a gradient contribution in the free energy functional, the gradient of the phase field when the order changes between different grains. The history of both approaches is reviewed shortly considering their pros and cons.

In this chapter the coupling to solute concentration in an alloy is reviewed. During transformation the concentration between the phases must be redistributed, causing solute diffusion. First different approached to handle this problem are discussed comparatively. Special emphasis is given to the problem of scale in a mesoscopic phase field model. In detail the currently most advanced model, the so-called finite interface dissipation model is explained in detail. The chapter concludes with the discussion of the effect of multi-component diffusion and its handling in a phase-field model. As an example, solidification under additive manufacturing condition is presented with special emphasis of the microsegregation after solidification in a multi-component alloy.

In this chapter first a multi-phase-field model considering transformation strain and elastic energy is developed. It utilizes the expansion into multiple phases of the multi-phase-field model. In particular, the treatment of the diffuse interface region as an effective medium is discussed in the context of homogenization theory. The model is applied to martensitic transformation within finite strain framework. In the second part of the lecture coupling for solute transport by melt flow is discussed. The model is applied to equiaxed dendritic solidification of MgAl in a shear flow.

In this chapter the extension of a phase-field model for two phases to multiple phases is presented. This relates to the treatment of triple lines and junctions between several phases, or grains in a multicrystalline structure. The conservation constraint of the sum of all fields in one material point is realized using a Lagrange formalism. The free energy functional is expanded in pairs of phases, as well as the equation of motion of individual phase fields in dependence on all other fields. As example coarsening and texture evolution in a multi grain structure with anisotropic interface energy is presented.

A novel approach is developed to support retrieval of 3D information from 2D experimental micrographs. The approach utilizes 3D phase-field simulations to train an artificial intelligence machine. In a first step, the phase-field simulations have to be validated to reproduce microstructural features which characterize elementary processes which govern processing and high temperature service exposure. The qualified 3D simulation setup is then applied to produce a high number of 2D simulated micrographs by automated sectioning. These simulated micrographs are then used to train a gradient boosting regression model together with the 3D information from the simulations. In the final step, the model is applied to 2D experimental micrographs to retrieve the hidden 3D features. The approach is generally applicable to all kinds of metallic materials, minerals or ceramics which can be treated quantitatively by phase-field simulations. In this paper we concentrate on the process of directional coarsening, referred to as ‘rafting’, in the field of creep of single crystal Ni-base superalloys. The experimental and modeling aspects of the evolution of the volume fraction of the γ ′ phase during long term creep are discussed.

The role of inclination dependence of grain boundary energy on the microstructure evolution and the orientation distribution of grain boundary planes during grain growth in polycrystalline materials is investigated by three-dimensional phase-field simulations. The anisotropic grain boundary energy model uses the description of the faceted surface structure of the individual crystals and constructs an anisotropic energy of solid-solid interface. The energy minimization occurs by the faceting of the grain boundary due to inclination dependence of the grain boundary energy. The simulation results for a single grain show the development of equilibrium shapes (faceted grain morphologies) with different families of facets which agrees well with the theoretical predictions. The results of grain growth simulations with isotropic and anisotropic grain boundary energy for cubic symmetry show that inclination dependence of grain boundary energy has a significant influence on the grain boundary migration, grain growth kinetics and the grain boundary plane distribution. It has been shown that the model essentially reproduces the experimental studies reported for NaCl and MgO polycrystalline systems where the anisotropic distribution of grain boundary planes has a peak for the low-index {100} type boundaries.

Nickel-based superalloys have a wide range of applications in high temperature and stress domains due to their unique mechanical properties. Under mechanical loading at high temperatures, rafting occurs, which reduces the service life of these materials. Rafting is heavily affected by the loading conditions associated with plastic strain; therefore, understanding plastic strain evolution can help understand these material’s service life. This research classifies nickel-based superalloys with respect to creep strain with deep learning techniques, a technique that eliminates the need for manual feature extraction of complex microstructures. Phase-field simulation data that displayed similar results to experiments were used to build a model with pre-trained neural networks with several convolutional neural network architectures and hyper-parameters. The optimized hyper-parameters were transferred to scanning electron microscopy images of nickel-based superalloys to build a new model. This fine-tuning process helped mitigate the effect of a small experimental dataset. The built models achieved a classification accuracy of 97.74% on phase-field data and 100% accuracy on experimental data after fine-tuning.

The general mechanisms needed to understand thermodynamic and kinetic stability of the microstructure of superalloys are reviewed. Microstructural transformations during processing and service is described as a diffusion controlled transformation, where microstructural stability rests on slow diffusing elements and on a large partitioning of these elements between the phases. The theoretical tools for thermodynamic stability and diffusion are implemented in CALPHAD type data bases and softwares. The phase-field method, as the most advanced tools for simulation solidification, heat treatment and rafting under creep conditions, is shortly introduced.

Recent advances in the field of diffusion in multiprincipal element systems are critically reviewed, with an emphasis on experimental as well as theoretical approaches to determining atomic mobilities (tracer diffusion coefficients) in chemically complex multicomponent systems. The newly elaborated and augmented pseudobinary and pseudoternary methods provide a rigorous framework to access tracer, intrinsic, and interdiffusion coefficients in alloys with an arbitrary number of components. Utilization of the novel tracer-interdiffusion couple method allows for a high-throughput determination of composition-dependent tracer diffusion coefficients. A combination of these approaches provides a unique experimental toolbox to access diffusivities of elements that do not have suitable tracers. The pair-exchange diffusion model, which gives a consistent definition of diffusion matrices without specifying a reference element, is highlighted. Density-functional theory–informed calculations of basic diffusion properties—asrequired for the generation of extensive mobility databases for technological applications—are also discussed.

We combine a phase-field model with a crystal plasticity model to simulate the microstructural evolution during creep in the Co-based superalloy ERBOCo-2Ta. Three-dimensional simulations of tensile and compressive creep tests in [100] direction were performed to study the rafting behavior in Co-based superalloys. The loss of coherency between γ matrix and γ′ precipitate, which is essential for the understanding of rafted structures, is modeled in relation to the dislocation activity in the γ-channels. Special attention is given to the interplay between creep deformation and microstructure stability. Appropriate constitutive modeling is applied to simulate realistic microstructure evolution under creep conditions. Thus, with the removal of the misfit stress, γ′ precipitates lose their cuboidal shape and form rafts. During N-type rafting more γ′ precipitates coalesce than during P-type rafting. The γ′ volume fraction during rafting increases under tensile stress but decreases under compressive stress. The morphological evolution of γ′ precipitates under tensile and compressive stresses in Co-based superalloy is consistent with the rafting characteristics in experimental observations.

We present three-dimensional phase-field simulations of martensite microstructure formation in low-carbon steel. In this study, a full set of 24 Kurdjumov-Sachs symmetry variants of martensite is considered. Three different carbon compositions are investigated in order to reveal the effect of carbon content on the martensite microstructure formation. The simulations are performed using the finite strain framework which allows considering real martensite transformation strains. Using Neuber elasto-plastic approximation to the mechanical equilibrium solution, realistic stresses and strains can be obtained during martensite formation resulting in realistic mechanical driving forces for the transformation. The simulated microstructures are compared to experimental results for three carbon compositions. Good agreement between simulated and experimental results is achieved.

Many regions of the Earth’s mantle deform in grain size-sensitive creep regimes. The grain size right below the transition zone is believed to be very small, and the grain size should in subsequent depths be mainly controlled by normal grain growth. The grain size evolution is commonly predicted using either existing grain growth laws in combination with grain boundary diffusion coefficients or by extrapolating empirically determined grain growth laws. Effects of Zenner pinning and different ratios of second phases have been studied, while the role of anisotropic grain boundary properties is currently neglected1. The grain boundary energy varies with the orientation of the grain boundary plane, as expressed through the typical crystal habitae (Wulff-shapes). Individual crystals in a polycrystalline material maintain a grain boundary energy anisotropy during grain growth. Here we study how grain boundary anisotropy impacts grain boundary migration and normal grain growth rates by three-dimensional phase-field simulations2. We imply grain boundary energy minimization by faceting/varying the grain boundary plane to minimize the grain boundary energy. The ideal grain boundary energy anisotropy for the solid-solid interface is taken from experimentally investigated grain boundary plane distributions and grain boundary energy distributions on periclase (MgO). We compare the grain size evolution in simulations with isotropic and anisotropic grain boundary energy of cubic crystal symmetry. We found that the grain boundary energy anisotropy has a significant influence on grain boundary migration and grain growth kinetics2.2- 0>2= ktThe change in grain size is given as variation between the initial and final average grain radius, r. The time is t, and a material-specific parameter that accounts for the grain boundary energy anisotropy is extracted from the simulations ask = A·µgb·σgbWhere, the grain boundary energy, σgb varies with orientation, and the grain boundary mobility, µgb assumed to be isotropic. We found that the rate of grain growth for periclase, A is a factor of 3 smaller compared to an isotropic material.These results are of three-fold importance: The theory of grain growth is developed for purely isotropic materials1and clearly fail to predict grain growth for materials with anisotropic grain boundary energy correctly. Predicting grain growth using existing rate-laws and grain boundary diffusion studies can lead to an incomplete conclusion and a distorted picture of the real phenomenon. Because energy reduction associated with grain growth is small, already small energy variations, such as those associated with grain boundary energy anisotropy, can have a significant effect. A better prediction of grain size evolution will need to develop an anisotropic theory for grain growth, that address our lake in knowledge regarding pressure effects on both grain boundary energy anisotropy and diffusion.1. Rohrer, G. S. Annu. Rev. Mater. Res. 20052. Salama et al. Acta Materialia 2020

Modeling and simulation is transforming modern materials science, becoming an important tool for the discovery of new materials and material phenomena, for gaining insight into the processes that govern materials behavior, and, increasingly, for quantitative predictions that can be used as part of a design tool in full partnership with experimental synthesis and characterization. Modeling and simulation is the essential bridge from good science to good engineering, spanning from fundamental understanding of materials behavior to deliberate design of new materials technologies leveraging new properties and processes. This Roadmap presents a broad overview of the extensive impact computational modeling has had in materials science in the past few decades, and offers focused perspectives on where the path forward lies as this rapidly expanding field evolves to meet the challenges of the next few decades. The Roadmap offers perspectives on advances within disciplines as diverse as phase field methods to model mesoscale behavior and molecular dynamics methods to deduce the fundamental atomic-scale dynamical processes governing materials response, to the challenges involved in the interdisciplinary research that tackles complex materials problems where the governing phenomena span different scales of materials behavior requiring multiscale approaches. The shift from understanding fundamental materials behavior to development of quantitative approaches to explain and predict experimental observations requires advances in the methods and practice in simulations for reproducibility and reliability, and interacting with a computational ecosystem that integrates new theory development, innovative applications, and an increasingly integrated software and computational infrastructure that takes advantage of the increasingly powerful computational methods and computing hardware., Modelling and Simulation in Materials Science and Engineering, 28 (4), ISSN:0965-0393, ISSN:1361-651X

The recently proposed pair-exchange concept for multicomponent diffusion is analyzed in detail. The concept defines the differences of chemical potential gradients of two elements as general driving forces for interdiffusion and defines the corresponding proportionality coefficients as pair-mobilities for atomic exchange fluxes of a pair of elements at the mesoscopic scale. The total fluxes of alloying elements are given as the sum over corresponding pair-contributions, which rely on an interdependent set of forces to maintain a meaningful symmetric form. Onsager's reciprocal relations are satisfied as well as the consistency between description of chemical- and tracer-diffusion. It is confirmed that a consistent definition of interdiffusion coefficients requires the fulfillment of the Gibbs-Duhem relation by the applied thermodynamic Gibbs energy description.

The development of new Ni-base superalloys with a complex composition consisting of eight or more alloying elements is a challenging task. The experimental state-of-the-art development cycle is based on the adaption of already existing compositions. Although new alloy compositions with potentially improved material properties are expected to be similar to already known superalloys, this procedure impedes efficiently finding these compositions in the large multi-dimensional design-space of all alloying elements. Modern alloy development combines numerical optimization methods with experimental validation to guide the development towards promising compositions. In this work, an improved numerical multi-criteria optimization tool using CALPHAD calculations and semi-empirical models for alloy development is presented. The model improvements to its predecessor are described and the successful application for the development of rhenium-free single-crystal Ni-base superalloys ERBO/13 and ERBO/15 is revisited. The optimization tool is described and the designed alloys are discussed regarding phase stability. Finally, a possible phase stability model extending the optimization tool and improving the alloy composition predictions is presented.

Rejuvenation of the initially hot isostatic pressing (HIP) heat-treated single-crystal Ni-base superalloy (SX) ERBO/1 was examined experimentally and via phase field simulation to establish rejuvenation treatments as a cost-effective alternative for another interval of service life. Creep was performed at 950 °C and 350 MPa, and the specimens were crept to 0.6 pct (creep rate minimum) or 2 pct strain, respectively. A slight coarsening of the γ/γ′ microstructure was observed experimentally and via simulation at 0.6 pct and rafting at 2 pct strain. The damaged microstructures were rejuvenated in a novel hot isostatic press that provides fast quenching rates before the same specimens were recrept under the same initial creep conditions. High-resolution microscopy proves that the rejuvenation re-establishes the original γ/γ′ microstructure in the dendrite core of the precrept specimens (0.6 and 2 pct). However, the interdendritic areas of the 2 pct precrept and rejuvenated specimen still contain elongated γ′ particles enwrapped by interfacial dislocation networks that survived the applied rejuvenation. The subsequent experimental and simulated creep tests after rejuvenation demonstrated that the creep behavior is only reproducible by the proposed rejuvenation for specimens that had crept until the end of the primary creep regime.

For large-scale phase-field simulations, the trade-off between accuracy and computational cost as a function of the size and number of simulations was studied. For this purpose, a large reference representative volume element (RVE) was incrementally subdivided into smaller solitary samples. We have considered diffusion-controlled growth and early ripening of δ ′ (Al3Li) precipitate in a model Al-Li system. The results of the simulations show that decomposition of reference RVE can be a valuable computational technique to accelerate simulations without a substantial loss of accuracy. In the current case study, the precipitate number density was found to be the key controlling parameter. For a pre-set accuracy, it turned out that large-scale simulations of the reference RVE can be replaced by simulating a combination of smaller solitary samples. This shortens the required simulation time and improves the memory usage of the simulation considerably, and thus substantially increases the efficiency of massive parallel computation for phase-field applications.

The Gibbs free energy of grain boundaries (GBs) in Al bicrystals has been investigated by Molecular Dynamics (MD) simulations. In our novel approach, one grain is fully embedded in a large matrix grain with fixed misorientation. Hence all inclinations are considered simultaneously since the boundary covers the full orientation subspace. A synthetical driving force is employed to counteract the shrinkage of the embedded grain by the capillary forces. Hence, the number of atoms of the embedded grain is kept constant, but its shape adjusts itself at elevated temperatures in order to minimize the total GB energy. The quasi-equilibrium shapes are used to derive the GB energy γ ( n ) as functions of the GB plane normal n . For GBs with the misorientations Σ 5 〈 001 〉 and Σ 7 〈 111 〉 , analytical functions were derived and validated in a mesoscopic front-tracking simulation: the latter simulations recovered the grain shapes observed in MD simulations. For the Σ 5 〈 001 〉 misorientation it is shown that the anisotropy of γ ( n ) varies quite strongly with temperature. For a Σ 9 〈 110 〉 misorientation, the derived numerical energy function was found to be rather complex, showing pronounced energy minima for mixed tilt/twist GBs parallel to 111 crystal planes.

The present work takes a new look at a modified Bridgman process (Bridgman seed technique, BST) for the production of laboratory Ni-base single crystal (SX) superalloy cylinders of 12/120mm diameter/length. This type of specimen is needed to perform inexpensive parametric studies for the development of new SX and for understanding the evolution of microstructures during SX casting. During melting, the seed partially melts back. The elementary segregation processes cause a certain type of constitutional heating/cooling. Competitive growth eventually establishes a constant average dendrite spacing. In the present work it is documented how this dendrite spacing varies in one cylindrical ingot, and how it scatters when a series of SX ingots is produced. This type of information is scarce. The calculated temperature gradient across the solid/liquid interface (calculated by FEM) is in excellent agreement with predictions from the Kurz-Fisher equation which yields a dendrite spacing based on the experimental withdrawal rate and the microstructurally determined average dendrite spacing. The presence of small angle grain boundaries on cross sections which were taken perpendicular to the solidification direction can be rationalized on the basis of small deviations from the ideal growth directions of individual primary dendrites. Keywords: Crystal growth, Superalloys, Directional solidification, Microstructural evolution, Solidification defects

The open-source software project OpenPhase allows the three-dimensional simulation of microstructural evolution using the multiphase field method. The core modules of OpenPhase and their implementation as well as their parallelization for a distributed-memory setting are presented. Especially communication and load-balancing strategies are discussed. Synchronization points are avoided by an increased halo-size, i.e. additional layers of ghost cells, which allow multiple stencil operations without data exchange. Load-balancing is considered via graph-partitioning and sub-domain decomposition. Results are presented for performance benchmarks as well as for a variety of applications, e.g. grain growth in polycrystalline materials , including a large number of phase fields as well as Mg–Al alloy solidification. Program summary Program Title: OpenPhase Program Files doi: http://dx.doi.org/10.17632/2mnv2fvkkk.1 Licensing provisions: GPLv3 Programming language: C++ Nature of problem: OpenPhase[1] allows the simulation of microstructure evolution during materials processing using the multiphase field method. In order to allow an arbitrary number of phase fields active parameter tracking is used, which can cause load imbalances in parallel computations. Solution method: OpenPhase solves the phase field equations using an explicit finite difference scheme. The parallel version of OpenPhase provides load-balancing using over-decomposition of the computational domain and graph-partitioning. Adaptive sub-domain sizes are used to minimize the computational overhead of the over-decomposition, while allowing appropriate load-balance. Additional comments including Restrictions and Unusual features: The distributed-memory parallelism in OpenPhase uses MPI. Shared-memory parallelism is implemented using OpenMP. The library uses C++11 features and therefore requires GCC version 4.7 or higher. [1] www.openphase.de

Ni-base superalloys are materials which are designed to resist extreme thermal and mechanical conditions. In this regard, an essential factor is their microstructure consisting of γ ′ precipitates embedded in a γ matrix. The application of superalloys at high temperatures can however induce the topological phase inversion, where the γ ′-phase topologically becomes the matrix phase, resulting in subpar material properties. In this work, the topological inversion is analyzed via experiment and phase-field simulation. The evolution of the microstructure has been quantified in the second generation single crystal Ni-base superalloy ERBO/1, which belongs to the family of CMSX-4, submitted to long-term aging at 1100 ° C for up to 250 h. Phase-field simulations carried out using a multi phase-field approach deliver insight into the microstructure evolution driven by the loss of coherency of the γ ′ precipitates, which is induced by the accumulation of dislocations at the γ / γ′ interfaces. The obtained simulation results are in good agreement with the experimental results, and indicate that the mechanisms causing the topological inversion are linked to the accommodation of the lattice misfit, which enables coalescence and ripening of γ′ precipitates.

Within diffuse interface models for multiphase problems, the curvature of the phase boundary can be expressed as the difference of two terms, a Laplacian and a second, gradient, term of the diffuse interface variable, o. In phase field simulations of microstructure evolution, the second term is often replaced by f’(o) = df/do, where f(o) is the potential function in the free energy functional of the underlying physical model. We show here that this replacement systematically deteriorates the accuracy in local curvature evaluation as compared to a discretized evaluation of the second term. Analytic estimates reveal that the discretization errors in the Laplacian and in the second term have roughly the same spatial dependence across the interface, thus leading to a cancellation of errors in k. This is confirmed in a test case, where the discretization error can be determined via comparison to the exact solution. If, however, the second term is replaced by a quasi exact expression, the error in ∆ o enters k without being compensated and can obscure the behavior of the local curvature. Due to the antisymmetric variations of this error term, approaches using the average curvature, as obtained from an integral along the interface normal, are immune to this problem.

A monistic framework is set up where energy is the only fundamental substance. Different states of energy are ordered by a set of scalar fields. The dual elements of matter, mass and space, are described as volume- and gradient-energy contributions of the set of fields, respectively. Time and space are formulated as background-independent dynamic variables. The evolution equations of the body of the universe are derived from the first principles of thermodynamics. Gravitational interaction emerges from quantum fluctuations in finite space. Application to a large number of fields predicts scale separation in space and repulsive action of masses distant beyond a marginal distance. The predicted marginal distance is compared to the size of the voids in the observable universe.

A novel thermodynamic modeling strategy of stable solid alloy phases is proposed based on segmented regression approach. The model considers several physical effects (e.g. electronic, vibrational, etc.) and is valid from 0 K up to the melting temperature. The preceding approach has been applied for several pure elements. Results show good agreement with experimental data at low and high temperatures. Since it is not a first attempt to propose a “universal” physical-based model down to 0 K for the pure elements as an alternative to current SGTE description, we also compare the results to existing models. Analysis of the obtained results shows that the newly proposed model delivers more accurate description down to 0 K for all studied pure elements according to several statistical tests.

The phase-field method is used as a basis to develop a strictly mass conserving, yet simple, model for simulation of two-phase flow. The model is aimed to be applied for the study of structure evolution in metallic foams. In this regard, the critical issue is to control the rate of bubble coalescence compared to concurrent processes such as their rearrangement due to fluid motion. In the present model, this is achieved by tuning the interface energy as a free parameter. The model is validated by a number of benchmark tests. First, stability of a two dimensional bubble is investigated by the Young-Laplace law for different values of the interface energy. Then, the coalescence of two bubbles is simulated until the system reaches equilibrium with a circular shape. To address the major capability of the present model for the formation of foam structure, the bubble coalescence is simulated for various values of interface energy in order to slow down the merging process. These simulations are repeated in the presence of a rotational flow to highlight the fact that the model allows to suppress the coalescence process compared to the motion of bubbles relative to each other., 15 pages and 9 figures