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The finite solid-liquid interface width in phase field models results in non-equilibrium effects, including solute trapping. Prior phase field modeling has shown that this extra degree of freedom, when compared to sharp-interface models, results in solute trapping that is well captured when realistic parameters, such as interface width, are employed. However, increasing the interface width, which is desirable for computational reasons, leads to artificially enhanced trapping thus making it difficult to model departure from equilibrium quantitatively. In the present work, we develop a variational phase field model with independent kinetic equations for the solid and liquid phases. Separate kinetic equations for the phase concentrations obviate the assumption of point wise equality of diffusion potentials, as is done in previous works. Non-equilibrium effects such as solute trapping, drag and interface kinetics can be introduced in a controlled manner in the present model. In addition, the model parameters can be tuned to obtain ``experimentally-relevant" trapping while using significantly larger interface widths than prior efforts. A comparison with these other phase field models suggests that interface width of about three to twenty-five times larger than current best-in-class models can be employed depending upon the material system at hand leading to a speed-up by a factor of $W^{(d+2)}$, where $W$ and $d$ denote the interface width and spatial dimension, respectively. Finally the capacity to model non-equilibrium phenomena is demonstrated by simulating oscillatory instability leading to the formation of solute bands., 51 pages, 9 figures, supplemental materials

Data - arguably the most important product of worldwide materials research investment - are rarely shared. The small and biased proportion of results published are buried in plots and text licensed by journals. This situation wastes resources, hinders innovation, and, in the current era of data-driven discovery, is no longer tenable. In this comment, we identify opportunities for synergistic, collaborative, and global actions to assemble large quantities of FAIR (Findable, Accessible, Interoperable, Reusable) (1) materials data.

In this work, we apply phase field simulations to examine the coarsening behavior of morphologically complex two-phase microstructures in which the phases have highly dissimilar mobilities, a condition approaching that found in experimental solid-liquid systems. Specifically, we consider a two-phase system at the critical composition ($50\%$ volume fraction) in which the mobilities of the two phases differ by a factor of 100. This system is simulated in two and three dimensions using the Cahn-Hilliard model with a concentration-dependent mobility, and results are compared to simulations with a constant mobility. A morphological transition occurs during coarsening of the two-dimensional system (corresponding to a thin film geometry) with dissimilar mobilities, resulting in a system of nearly-circular particles of high-mobility phase embedded in a low-mobility matrix. This morphological transition causes the coarsening rate constant to decrease over time, which explains why a previous study found lack of agreement with the theoretical $t^{1/3}$ power law. Three-dimensional systems with dissimilar mobilities resulted in bicontinuous microstructures that evolve self-similarly, as determined by quantitative analysis of the interfacial shape distribution. Coarsening kinetics in three dimensions agreed closely with the $t^{1/3}$ power law after the initial transient stage. A model is derived to explain a nearly-linear relationship between the coarsening rate constant and the variance of scaled mean curvature that is observed during this transient stage., 25 pages, 12 figures

International audience; Most metals, ceramics, semiconductors and rocks are composed of small crystals known as grains. When annealed, this polycrystalline structure coarsens, thus allowing the properties of a material to be tailored for a particular application. The mobility of grain boundaries is thought to be determined by the crystallography of the adjacent crystals, but experimental validation in bulk polycrystalline materials is lacking. Here we developed a novel fitting methodology by direct comparison of a time-resolved three-dimensional experimental data to simulations of the evolution of 1501 grains in iron. The comparison allows reduced mobilities of 1619 grain boundaries to be determined simultaneously. We find that the reduced mobilities vary by three orders of magnitude and in general exhibit no correlation with the boundary's five macroscopic degrees of freedom, implying that grain growth is governed by other factors.

Coarsening of bicontinuous microstructures is observed in a variety of systems, such as nanoporous metals and mixtures that have undergone spinodal decomposition. To better understand the morphological evolution of these structures during coarsening, we compare the morphologies resulting from two different coarsening mechanisms, surface and bulk diffusion. We perform phase-field simulations of coarsening via each mechanism in a two-phase mixture at nominal volume fractions of 50%-50% and 36%-64%, and the simulated structures are characterized in terms of topology (genus density), the interfacial shape distribution, structure factor, and autocorrelations of phase and mean curvature. We observe self-similar evolution of morphology and topology and agreement with the expected power laws for dynamic scaling, in which the characteristic length scale increases over time proportionally to $t^{1/4}$ for surface diffusion and $t^{1/3}$ for bulk diffusion. While we observe the expected difference in the coarsening kinetics, we find that differences in self-similar morphology due to coarsening mechanism are relatively small, although typically they are larger at 36% volume fraction than at 50% volume fraction. In particular, we find that bicontinuous structures coarsened via surface diffusion have lower scaled genus densities than structures coarsened via bulk diffusion. We also compare the self-similar morphologies to those in literature and to two model bicontinuous structures, namely, constant-mean-curvature surfaces based on the Schoen G minimal surface and random leveled-wave structures. The average scaled mean curvatures of these model structures agree reasonably with those of the coarsened structures at both 36% and 50%, but we find substantial disagreements in the scaled genus densities and the standard deviations of mean curvature., Main text: 20 pages, 7 figures. Supplemental Material: 9 pages, 6 figures

International audience; We present a novel self-assembly route to align SiGe quantum dots. By a combination of theoretical analyses and experimental investigation, we show that epitaxial SiGe quantum dots can cluster in ordered closely packed assemblies, revealing an attractive phenomenon. We compute nucleation energy barriers, accounting for elastic effects between quantum dots through both elastic energy and strain-dependent surface energy. If the former is mostly repulsive, we show that the decrease in the surface energy close to an existing island reduces the nucleation barrier. It subsequently increases the probability of nucleation close to an existing island, and turns out to be equivalent to an effective attraction between dots. We show by Monte-Carlo simulations that this effect describes well the experimental results, revealing a new mechanism ruling self-organisation of quantum dots. Such a generic process could be observed in various heterogeneous systems and could pave the way for a wide range of applications.

Highly anisotropic interfaces play an important role in the development of material microstructure. Using the diffusive atomistic phase-field crystal (PFC) formalism, we determine the capability of the model to quantitatively describe these interfaces. Specifically, we coarse grain the PFC model to attain both its complex amplitude formulation and its corresponding phase-field limit. Using this latter formulation, in one-dimensional calculations, we determine the surface energy and the properties of the Wulff shape. We find that the model can yield Wulff shapes with missing orientations, the transition to missing orientations, and facet formation. We show that the corresponding phase-field limit of the complex amplitude model yields a self-consistent description of highly anisotropic surface properties that are a function of the surface orientation with respect to the underlying crystal lattice. The phase-field model is also capable of describing missing orientations on equilibrium shapes of crystals and naturally includes a regularizing contribution. We demonstrate, in two dimensions, how the resultant model can be used to study growth of crystals with varying degrees of anisotropy in the phase-field limit., 40 pages, 9 Figures, Submitted to Physical Review Materials

International audience; Grain evolution in pure iron is determined in three dimensions using diffraction contrast tomography at a synchrotron source. During annealing for 75 minutes at 800 • C, the evolution of initially 1327 grains is quantified as a function of 15 time-steps. A comprehensive statistical analysis is provided based on the equivalent radius, the number of faces and the mean width parameters of the grains. We introduce analytical relations between these parameters, validate them, and discuss their physical meaning. While the sample is fully recrystallized, the growth is found not to be self-similar, as evidenced in changes in the distributions of normalized grain size and number of faces per grain. More importantly, a strong decrease in the slope of the growth rate over the mean width of grain faces is observed, indicating a slowdown of grain growth. The data is used to determine the applicability of the isotropic MacPherson-Srolovitz theory to an anisotropic material such as iron. Geometrical properties that are averaged over the entire grain ensemble are well described by the model, but the properties and evolution of the individual grains exhibit substantial scatter.

Under the generic term ``Stranski-Krastanov'' are grouped two different growth mechanisms of SiGe quantum dots. They result from the self-organized Asaro-Tiller-Grinfel'd (ATG) instability at low strain, while at high strain, from a stochastic nucleation. While these regimes are well known, we elucidate here the origin of the transition between these two pathways thanks to a joint theoretical and experimental work. Nucleation is described within the master equation framework. By comparing the time scales for ATG instability development and three-dimensional (3D) nucleation onset, we demonstrate that the transition between these two regimes is simply explained by the crossover between their divergent evolutions. Nucleation exhibits a strong exponential deviation at low strain while ATG behaves only algebraically. The associated time scale varies with $exp(1/{x}^{4})$ for nucleation, while it only behaves as $1/{x}^{8}$ for the ATG instability. Consequently, at high (low) strain, nucleation (instability) occurs faster and inhibits the alternate evolution. It is then this different kinetic evolution which explains the transition from one regime to the other. Such a kinetic view of the transition between these two 3D growth regimes was not provided before. The crossover between nucleation and ATG instability is found to occur both experimentally and theoretically at a Ge composition around 50% in the experimental conditions used here. Varying the experimental conditions and/or the system parameters does not allow us to suppress the transition. This means that the SiGe quantum dots always grow via ATG instability at low strain and nucleation at high strain. This result is important for the self-organization of quantum dots.

International audience; During casting, often a dendritic microstructure is formed, resulting in a columnar or an equiaxed grain structure, or leading to a transition from columnar to equiaxed growth (CET). The detailed knowledge of the critical parameters for the CET is important because the microstructure affects materials properties. To provide unique data for testing of fundamental theories of grain and microstructure formation, solidification experiments in microgravity environment were performed within the European Space Agency Microgravity Application Promotion (ESA MAP) project Columnar-to-Equiaxed Transition in SOLidification Processing (CETSOL). Reduced gravity allows for purely diffusive solidification conditions, i.e., suppressing melt flow and sedimentation and floatation effects. On-board the International Space Station, Al-7 wt.% Si alloys with and without grain refiners were solidified in different temperature gradients and with different cooling conditions. Detailed analysis of the microstructure and the grain structure showed purely columnar growth for nonrefined alloys. The CET was detected only for refined alloys, either as a sharp CET in the case of a sudden increase in the solidification velocity or as a progressive CET in the case of a continuous decrease of the temperature gradient. The present experimental data were used for numerical modeling of the CET with three different approaches: (1) a front tracking model using an equiaxed growth model, (2) a three-dimensional (3D) cellular automaton–finite element model, and (3) a 3D dendrite needle network method. Each model allows for predicting the columnar dendrite tip undercooling and the growth rate with respect to time. Furthermore, the positions of CET and the spatial extent of the CET, being sharp or progressive, are in reasonably good quantitative agreement with experimental measurements.

We present the first set of benchmark problems for phase field models that are being developed by the Center for Hierarchical Materials Design (CHiMaD) and the National Institute of Standards and Technology (NIST). While many scientific research areas use a limited set of well-established software, the growing phase field community continues to develop a wide variety of codes and lacks benchmark problems to consistently evaluate the numerical performance of new implementations. Phase field modeling has become significantly more popular as computational power has increased and is now becoming mainstream, driving the need for benchmark problems to validate and verify new implementations. We follow the example set by the micromagnetics community to develop an evolving set of benchmark problems that test the usability, computational resources, numerical capabilities and physical scope of phase field simulation codes. In this paper, we propose two benchmark problems that cover the physics of solute diffusion and growth and coarsening of a second phase via a simple spinodal decomposition model and a more complex Ostwald ripening model. We demonstrate the utility of benchmark problems by comparing the results of simulations performed with two different adaptive time stepping techniques, and we discuss the needs of future benchmark problems. The development of benchmark problems will enable the results of quantitative phase field models to be confidently incorporated into integrated computational materials science and engineering (ICME), an important goal of the Materials Genome Initiative.

International audience; We compare the morphological and structural features of SiGe membranes fabricated by three different processes: direct deposition of Si0.5Ge0.5 on. Si(001) nominal substrate, direct deposition Of Si0.5Ge0.5 on silicon on insulator, and deposition of SiGe with low Ge concentration on silicon on insulator followed by Ge enrichment by condensation. We show that the formation of fully strained Ge-rich layers free of defects with a flat surface is possible only by the two-step epitaxy-condensation process. We demonstrate that the condensation based process enables the total inhibition of the morphological instability, together with the hindering of dislocations for critical thickness much greater than those commonly obtained by direct deposition. Those behaviors could be explained by the injection of self-interstitials in the Ge-rich layers during condensation. Such remarkable properties systems using a similar condensation process. could be generalized to many other systems using a similar condensation process.

International audience; A theory of two-phase eutectic growth for a multicomponent alloy is presented. Using the same hypotheses as the Jackson-Hunt theory, we find that the growth law of the microstructure given by Jackson and Hunt for binary alloys can be generalized to systems with N elements. Thermodynamic parameters involved in this theory are linked to the Gibbs free energies of the phases which makes it possible to compute these parameters with CALPHAD tools. A model is derived from this general theory for ternary alloys which does not contain any assumptions on the alloy thermodynamic properties, contrary to previous models. We find that a small addition of a ternary alloying element with a small diffusivity to a binary alloy can significantly alter the spacing of the eutectic.

The coarsening process of an Al–29.9wt%Si alloy is studied using four-dimensional phase contrast X-ray tomography. This alloy is composed of highly anisotropic, primary Si particles in an eutectic matrix. We analyze the morphology of the primary Si particles during coarsening by determining the interface normal distribution and the interface shape distribution. The inverse surface area per unit volume increases with the cube root of time despite the lack of microstructural self-similarity and highly anisotropic particle morphology. More specifically, over the time frame of the experiments, the Si particles evolve from mostly faceted domains to a more isotropic structure that is not given by the Wulff shape of the crystal. These trends can be rationalized by the presence of twin defects that intersect particle edges and that may provide the kink sites necessary for interfacial propagation, thus leading to a more isotropic structure. While in many cases the interfacial velocity of Si solid–liquid interfaces is highly anisotropic, the presence of many defects leads to a highly mobile interface and diffusion-limited coarsening.

International audience; The properties of a critical nucleus are derived using the capillarity theory in the framework of classical nucleation. An analytical solution for the composition of a critical nucleus is given for low supersaturation. The theory is valid for any multicomponent systems. It is found that the deviation in nucleus composition from the equilibrium tie-line is mainly due to the difference in the Hessian of the Gibbs energy of the phases and the magnitude of the deviation in composition from equilibrium is order of the supersaturation. Despite our analysis strictly holds for low supersaturation, this suggests strong deviations near the spinodal line.

We investigate the possibility to control the symmetry of ordered states in phase-field crystal models by tuning nonlinear resonances. In two dimensions, we find that a state of square symmetry as well as coexistence between squares and hexagons can be easily obtained. In contrast, it is delicate to obtain coexistence of squares and liquid. We develop a general method for constructing free energy functionals that exhibit solid-liquid coexistence with desired crystal symmetries. As an example, we develop a free energy functional for square-liquid coexistence in two dimensions. A systematic analysis for determining the parameters of the necessary nonlinear terms is provided. The implications of our findings for simulations of materials with simple cubic symmetry are discussed., 19 pages, 6 figures

First-principles calculations are used to calculate the strain dependencies of the binding and diffusion-activation energies for Ge adatoms on both Si(001) and Ge(001) surfaces. Our calculations reveal that the binding and activation energies on a strained Ge(001) surface increase and decrease, respectively, by 0.21 eV and 0.12 eV per percent compressive strain. For a growth temperature of 600 degrees C, these strain-dependencies give rise to a 16-fold increase in adatom density and a 5-fold decrease in adatom diffusivity in the region of compressive strain surrounding a Ge island with a characteristic size of 10 nm., 4 pages, 4 figures