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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

Copper (Cu) is a catalyst broadly used in industry for hydrogenation of carbon dioxide, which has broad implications for environmental sustainability. An accurate understanding of the degeneration behavior of Cu catalysts under operando conditions is critical for uncovering the failure mechanism of catalysts and designing novel ones with optimized performance. Despite the widespread use of these materials, their failure mechanisms are not well understood because conventional characterization techniques lack the necessary time and spatial resolution to capture these complex behaviors. In order to overcome these challenges, we carried out transmission electron microscopy (TEM) with a specialized in situ gas environmental holder, which allows us to unravel the dynamic behavior of the Cu nanowires (NWs) in operando. The failure process of these nanoscale Cu catalysts under CO2 atmosphere were tracked and further rationalized based on our numerical modeling using phase-field methods.

Multiscale and multimodal imaging of material structures and properties provides solid ground on which materials theory and design can flourish. Recently, KAIST announced 10 flagship research fields, which include KAIST Materials Revolution: Materials and Molecular Modeling, Imaging, Informatics and Integration (M3I3). The M3I3 initiative aims to reduce the time for the discovery, design and development of materials based on elucidating multiscale processing-structure-property relationship and materials hierarchy, which are to be quantified and understood through a combination of machine learning and scientific insights. In this review, we begin by introducing recent progress on related initiatives around the globe, such as the Materials Genome Initiative (U.S.), Materials Informatics (U.S.), the Materials Project (U.S.), the Open Quantum Materials Database (U.S.), Materials Research by Information Integration Initiative (Japan), Novel Materials Discovery (E.U.), the NOMAD repository (E.U.), Materials Scientific Data Sharing Network (China), Vom Materials Zur Innovation (Germany), and Creative Materials Discovery (Korea), and discuss the role of multiscale materials and molecular imaging combined with machine learning in realizing the vision of M3I3. Specifically, microscopies using photons, electrons, and physical probes will be revisited with a focus on the multiscale structural hierarchy, as well as structure-property relationships. Additionally, data mining from the literature combined with machine learning will be shown to be more efficient in finding the future direction of materials structures with improved properties than the classical approach. Examples of materials for applications in energy and information will be reviewed and discussed. A case study on the development of a Ni-Co-Mn cathode materials illustrates M3I3's approach to creating libraries of multiscale structure-property-processing relationships. We end with a future outlook toward recent developments in the field of M3I3.

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

The ability to watch the three-dimensional (3D) evolution of structural materials is a breakthrough in non-destructive characterization. In particular, X-ray tomographic imaging techniques have found success in revealing the underlying mechanisms of microstructural transformations in partially and fully solidified metals. Here we review the most important developments in four-dimensional X-ray microscopy, focusing on absorption- and diffraction-based techniques in the laboratory and the synchrotron. In light of recent progress in this area, we identify critical issues that point to directions for future research in imaging the evolution of heterogeneous microstructures at extreme space and time scales. IMPACT STATEMENT Four-dimensional X-ray tomography has opened a new paradigm in physical metallurgy, allowing us to characterize the various epochs of microstructural evolution in 3D and as a function of time.

Characterizing large four-dimensional materials datasets is difficult due to the presence of complex microstructures and time-varying length scales. We showcase the use of two-point statistics as an efficient and un-biased way of extracting materials parameters from an Al-Cu alloy during solidification. The evolution of dendrite primary arm thickness, average secondary arm spacing, and average tip-to-tip spacing were tracked using two-point Pearson auto-correlations of scaled mean curvatures. Insights into competitive side-branching are also reported. We show both visually and quantitatively that most length scales change rapidly during early stages of dendritic growth, but slow as diffusion fields of dendrites overlap.

Three-dimensional (3D) tomographic imaging of the structural, chemical, and physical properties of a material provides key knowledge that links the structure of a material to both its processing and structure that is central to studies across a broad spectrum of materials. For many decades, tomography using x-rays or electrons has proven to be an essential 3D characterization tool. In recent years, advances in technology have significantly pushed the envelope of these techniques in many respects, enabling new imaging capabilities at the nanometer and atomic scale. This article highlights several such developments in nanoscale x-ray and electron tomography. The five articles that appear in this issue of MRS Bulletin discuss research frontiers that include multimodal x-ray tomography at the nanoscale, x-ray spectroscopic tomography, dark-field x-ray microscopy, electron nanotomography for functional nanomaterials, and atomistic imaging by electron tomography. These articles give a holistic view of the status of these techniques and promising future directions, as well highlighting their applications for scientific problems.

Solid oxide cells (SOCs) have important applications as fuel cells and electrolyzers. The application for storage of renewable electricity is also becoming increasingly relevant; however, it is difficult to meet stringent area-specific resistance (ASR) and long-term stability targets needed to achieve required efficiency and cost. Here we show a new SOC that utilizes a very thin Gd-doped ceria (GDC)/yttria-stabilized zirconia (YSZ) bi-layer electrolyte, Ni–YSZ cell support with enhanced porosity, and electrode surface modification using PrOx and GDC nanocatalysts to achieve unprecedented low ASR values < 0.1 Ω cm2, fuel cell power density ∼3 W cm−2, and electrolysis current density ∼4 A cm−2 at 800 °C. Besides this exceptionally high performance, fuel cell and electrolysis life tests suggest very promising stability in fuel cell and steam electrolysis modes. Electrochemical impedance spectroscopy analysis done using a novel impedance subtraction method shows how rate-limiting electrode processes are impacted by the new SOC materials and design.

During the powder bed fusion additive manufacturing (AM) process, hundreds of grains may solidify simultaneously within the melt pool, depending upon the processing conditions and the grain size of the base plate and powder. Thus, a small variation in the shape of the melt pool or scan strategy may result in very different as-printed microstructures, since new grains may become more favourably aligned with the overall direction of growth. The microstructure affects the performance of the additively built component, and it is therefore critical to understand the interplay between the initial microstructure, material properties, heat source, and scan strategy on determining the as-printed microstructure. We examine columnar grain growth during AM using a phase-field model to perform simulations with single melt pools predicted by the cylindrically symmetric Rosenthal solution. Quantitative measures of the competitive grain growth are extracted from the simulation data. To aid in their understanding, we map these quantities onto visualizations of the microstructure and compare them to the inverse pole figure map, from which we find that the crystallographic directions of the grains that grow closest to the centre of the laser track may not be preferentially aligned with the scan direction.

In the classical theories of oxidation of metals it is assumed that the interface between the oxide and metal is in thermodynamic equilibrium. However, in many cases this is not true, the oxide grows too fast or the fluxes through the interface are too large for local interfacial equilibrium to exist, leading to nonequilibrium solute capture. We present a thermodynamic analysis using both an available database as well as density functional theory calculations of the thermodynamic conditions for this during the oxidation of Ni-Cr alloys. The analysis indicates that nickel atoms can be captured in the rocksalt or corundum crystallographies for a very wide range of compositions, consistent with recent experimental observations. The density functional theory analysis also provides information about the electronic structure of these oxides which is important to understand their properties, and also indicates that interpretation of spectroscopic data is not simple as mixed valence states as well as Cr4+ can occur under oxidizing conditions. We point out that across at least the first transition row of elements the thermodynamic conditions for nonequilibrium solute capture can easily be met.

Copper (Cu) is a catalyst broadly used in industry for hydrogenation of carbon dioxide, which has broad implications for environmental sustainability. An accurate understanding of the degeneration behavior of Cu catalysts under

Modeling and understanding dendrite growth on lithium metal anodes during charging is vital for improving battery safety. The phase field method has become an increasingly popular tool to achieve this due to its flexibility in handling complex morphologies and coupled physical processes. However, quantitative phase field models under realistic electrochemical conditions are rare. In this work, we derive a thermodynamically consistent phase field model in the grand potential framework. The design of this model emphasizes quantitative modeling of electrodeposition with experimental loading conditions and realistic length and time scales. First, electrostatic potentials in both the electrode and the electrolyte are considered to account for the interfacial potential difference. Second, the electrochemical driving force can be large even with an applied voltage of a few hundred millivolts; therefore, the grid size usually must be as small as a few nanometers to maintain a stable phase field profile. This hinders the modeling of system sizes of practical interest. To solve this problem, a mapping approach is introduced to maintain stability with moderate grid sizes and realistic applied voltages. Third, we thoroughly compare the phase field model with results from a sharp interface model for various cases that permit analytical solutions. Numerical studies demonstrate that the proposed model can quantitatively reproduce the sharp interface results. Moreover, lithium has highly anisotropic interfacial energy, which leads to corners on the equilibrium electrode-electrolyte interface shape. A convexification method is used to remove the ill-posedness due to this strong anisotropy. Finally, a parallel solver is developed to handle large systems. The overall framework is demonstrated by studying lithium dendrite morphologies under various conditions. In future work, the model can be used as a computational tool for the high-throughput screening process in battery design.

In this talk, I introduce topics, that I have been working on, in solid oxide electrolysis cells involving complicated interfacial structures and dynamics of interfaces. Then I will focus on my recent work on nickel (Ni) particle migration in electrodes consisting of Ni, yttria-stabilized zirconia (YSZ), and pores during the operation of oxygen-ion conducting solid oxide electrolysis cells (o-SOECs) and Faraday efficiency in proton-conducting solid oxide electrolysis cells (p-SOECs) under electrolysis operations. SOECs can have a significant impact on climate change over the next decade and beyond, in applications such as balancing renewable grid electricity via electrolytic fuel production. However, long-term performance degradation remains a key issue that may limit further implementation of O-SOECs, and the dependency of operation conditions on Faraday efficiency in P-SOECs has been under debate. In particular, in Ni/YSZ/pore electrode of O-SOEC, a phase-field model is proposed that employs the Ni-YSZ 3D microstructure as the initial condition and large-scale numerical simulation is implemented that predicts the directional Ni migration. The results are thus directly comparable to experimental observations. Quantitative predictions of the evolution of the Ni/YSZ/pore system's microstructures due to Ni particles' migration are studied through theoretical analysis and data analysis. In P-SOECs, an electrochemical model is proposed to study the dependency of Faraday efficiency on operation conditions for P-SOECs with yttrium-doped barium zirconates (BZY) and co-doping barium zirconate-cerate oxides with ytterbium and yttrium (BCZYYb) as electrolytes respectively. Our numerical predictions are verified by experimental results obtained in INL. An optimal structure of electrolyte is proposed to boost the Faraday efficiency in P-SOECs.

Grain boundary formation during coarsening of nanoporous gold (NPG) is investigated wherein a nanocrystalline structure can form by particles detaching and reattaching to the structure. MicroLaue and electron backscatter diffraction measurements demonstrate that an in-grain orientation spread develops as NPG is coarsened. The volume fraction of the NPG sample is near the limit of bicontinuity, at which simulations predict that a bicontinuous structure begins to fragment into independent particles during coarsening. Phase-field simulations of coarsening using a computationally generated structure with a volume fraction near the limit of bicontinuity are used to model particle detachment rates. This model is tested by using the measured NPG structure as an initial condition in the phase-field simulations. We predict that up to [Formula: see text] 5% of the NPG structure detaches as a dealloyed [Formula: see text] sample is annealed at 300 °C for 420 min. The quantity of volume detached is found to be highly dependent on the volume fraction and volume fraction homogeneity of the nanostructure. As the void phase in the experiments cannot support independent particles, they must fall and reattach to the structure, a process that results in the formation of new grain boundaries. This particle reattachment process, along with other classic processes, leads to the formation of grain boundaries during coarsening in nanoporous metals. The formation of grain boundaries can impact a variety of applications, including mechanical strengthening; thus, the consideration and understanding of particle detachment phenomena are essential when studying nanoporous metals.

The stability of supported metal nanoparticles determines the activity and lifetime of heterogeneous catalysts. Catalysts can destabilize through several thermodynamic and kinetic pathways, and the competition between these mechanisms complicates efforts to quantify and predict the overall evolution of supported nanoparticles in reactive environments. Pairing in situ transmission electron microscopy with unsupervised machine learning, we quantify the destabilization of hundreds of supported Au nanoparticles in real-time to develop a model describing the observed particle evolution as a competition between evaporation and surface diffusion. Data mining of particle evolution statistics allows us to determine physically reasonable values for the model parameters, quantify the particle size at which the Gibbs-Thomson pressure accelerates the evaporation process, and explore how individual particle interactions deviate from the mean-field model. This approach can be applied to a wide range of supported nanoparticle systems, allowing quantitative insight into the mechanisms that control their evolution in reactive environments.

Understanding the kinetics of passive oxide formation and breakdown has been an ongoing problem for corrosion scientists for several decades. Here, we present a model for the formation of a passive oxide film on a metal that is an extension of the Point Defect Model (PDM) by Macdonald and coworkers. We replace the potential description of the PDM with a boundary value problem and ensure that Gauss's law is satisfied at the oxide-solution interface by considering the presence of the Helmholtz layer in solution. Our model predicts the observed linear variation of the steady-state film thickness with anode potential, and an increasing film thickness with increasing pH. We perform a linear stability analysis of the model and show that, depending on the parameters, the passive film may be unstable to morphological perturbations of the film interfaces, leading to nonplanar films and potentially the formation of a pit in the oxide. This also implies that one-dimensional models, which assume planar interfaces, can be inapplicable in broad classes of corrosion processes. The analysis shows that a morphological instability exists if the oxide dissolution mechanism is such that an increasing Helmholtz layer potential drop leads to an increasing dissolution rate. The instability behavior is consistent with the literature on breakdown of passivity in the presence of chloride ions. The theory provides insights on the initiation of passivity breakdown leading to pitting corrosion and the role that interfacial energy plays in determining stability.

Solid oxide electrolysis cells (SOECs) convert renewable electricity to fuels with efficiency substantially higher than other electrolysis technologies. However, questions remain regarding degradation mechanisms that limit SOEC long-term stability. One of the key degradation mechanisms is oxygen electrode delamination; although prior studies have improved the understanding of this mechanism, it is still difficult to predict how degradation depends on SOEC materials and operating conditions, i.e., temperature, voltage, and current density. Here we present a study aimed at developing a quantitative understanding of oxygen electrode delamination. Experimentally, a life test study of symmetric and full cells with yttria-stabilized zirconia (YSZ) electrolytes and Gd-doped ceria (GDC) barrier layers was done with three different perovskite oxygen electrode materials. Fracture was observed at the perovskite–GDC interface above a critical current density and below a critical operating temperature. A theory is presented that combines a calculation of the effective oxygen pressure across the electrolyte with an estimation of the pressure required for fracture. Fracture is correctly predicted for a critical oxygen partial pressure of ∼7200 atm and an associated electrode overpotential of ∼0.2 V, occurring at the electrode/GDC interface because of the relatively low perovskite fracture toughness. Damage at the GDC/YSZ interface was also observed in some cases and explained by a peak in the oxygen pressure at this interface.

Solid oxide cells (SOCs) can have a significant impact on climate change over the next decade and beyond, in applications such as balancing renewable grid electricity via electrolytic fuel production, and producing electricity from bio-fuels combined with CO2 product sequestration. However, long-term performance degradation remains a key variable that may limit further implementation of SOCs. This talk focuses on the Ni-YSZ fuel electrode that is widely used but is known to be an important contributor to SOC degradation. Various processes that cause Ni-YSZ degradation are discussed. Results on 3D tomography measurements of accelerated Ni coarsening are described and a quantitative model is developed to predict long-term degradation. Although the results indicate that coarsening effects can be minimized in well-designed Ni-YSZ microstructures, degradation can still occur, especially during high-current-density electrolysis operation. For operation under low H2O/H2 conditions, high electrolysis current density can yield reduction of zirconia to form Ni-Zr compounds, along with substantial microstructural damage. For operation under high H2O/H2conditions, high electrolysis current density can yield Ni migration away from the electrolyte. A phase-field simulation is described that predicts this Ni migration, using actual Ni-YSZ microstructures measured using 3D tomography as the starting point, and compared with experimental observations. The model assumes that Ni transport is driven by a spatial gradient in surface tensions, i.e., a decrease in the Ni/YSZ contact angle with increasing distance from the electrolyte. Recent results on electrolysis and reversibly operated SOCs, making use of Ceria-infiltrated Ni-YSZ to improve stability, are described.

PbSe is a promising thermoelectric that can be further improved by nanostructuring, band engineering, and carrier concentration tuning; therefore, a firm understanding of the defects in PbSe is necessary. The formation energies of point defects in PbSe are computed via first-principles calculations under the dilute-limit approximation. We find that under Pb-rich conditions, PbSe is an n-type semiconductor dominated by doubly-charged Se vacancies. Conversely, under Se-rich conditions, PbSe is a p-type semiconductor dominated by doubly-charged Pb vacancies. Both of these results agree with previously performed experiments. Temperature- and chemical potential-dependent Fermi levels and carrier concentrations are found by enforcing the condition of charge neutrality across all charged atomic and electronic states in the system. The first-principles-predicted charge-carrier concentration is in qualitative agreement with experiment, but slightly varies in the magnitude of carriers. To better describe the experimental data, a CALPHAD assessment of PbSe is performed. Parameters determined via first-principles calculations are used as inputs to a five-sublattice CALPHAD model that was developed explicitly for binary semiconductors. This five sublattice model is in contrast to previous work which treated PbSe as a stoichiometric compound. The current treatment allows for experimental carrier concentrations to be accurately described within the CALPHAD formalism. In addition to the five-sublattice model, a two-sublattice model is also developed for use in multicomponent databases. Both models show excellent agreement with the experimental data and close agreement with first-principles calculations. These CALPHAD models can be used to determine processing parameters that will result in an optimized carrier concentration and peak zT value.

Hydriding and dehydriding transitions in bulk and nanocrystalline binary metal hydrides were studied using the Pd-H model system by measuring pressure-composition isotherms with in situ x-ray diffractometry. Nanocrystalline Pd showed a smaller pressure hysteresis, solvus hysteresis, and hysteresis in lattice parameter, compared to bulk Pd. The time-dependence of pressure equilibration was measured after dosing with aliquots of hydrogen, giving equilibration times that were much faster in the single-phase regions than in the two-phase plateaus. In the broad two-phase plateaus, the pressure relaxations were exponential functions of time. An explanation of hysteresis is developed that is based on a dissipative potential barrier that impedes the motion of the interface due to interactions between lattice defects and the two-phase interface. The exponential pressure relaxations and hysteresis are consistent for this mechanism. For a simple model of the pinning potential, the potential barrier maximum is an order of magnitude less than typical grain boundary energies. These pinning effects are substantially different in the nanocrystalline Pd, suggesting differences in the hydriding mechanism.

We use irreversible thermodynamics to formulate a model for the formation of a cation-deficient, $p$-type oxide on the surface of a pure metal in which local equilibrium is not necessarily maintained at the oxide interfaces and no a priori assumption regarding a rate limiting step is made. The dissipation rate derived here accounts for most of the conceivable processes that can occur during the oxidation of a metal, including bulk diffusion of cations and surface diffusion along the metal-oxide and gas-oxide interfaces. By including capillary effects in the formulation, shape changes in the oxide, which are usually neglected in most theories, can be described. We examine the steady-state solution of the model in one dimension for planar interfaces, and show that the growth law naturally captures the transition from linear to parabolic kinetics as the film thickens. A characteristic lengthscale for the transition is derived and can be obtained from oxide growth curves as a fitting parameter. Wagner's classical oxidation model is obtained as a limiting case of our more general model. We obtain expressions for the interfacial defect concentrations as a function of the departure from local equilibrium at the interfaces. From these expressions, we show that the exponent for the oxygen pressure dependence of the cation vacancy concentration can deviate from the usual values obtained from equilibrium considerations, and thus measuring nonstandard exponents can indicate a nonequilibrium effect. Our analysis serves to clarify the meaning and validity of the local equilibrium assumption in the context of metal oxidation.

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

A phase-field crystal model that spontaneously undergoes displacive phase transitions is introduced by explicitly studying a two-component two-dimensional square crystal reminiscent of a perovskite. When the intercomponent free energy is a simple polynomial, the crystal undergoes displacive transitions in the $\ensuremath{\langle}10\ensuremath{\rangle}$ and $\ensuremath{\langle}11\ensuremath{\rangle}$ directions. When the interaction is a correlation function, displacements in any direction can occur. This displacive phase-field crystal (DPFC) model maps to Landau-Ginzburg-Devonshire (LGD) theories for ferroelectrics, and the DPFC and LGD models are compared in terms of phase transitions and domain walls. Dynamical simulations of quadrijunctions were also performed and found stable spiraling quadrijunctions and unstable nonspiraling quadrijunctions. Last, domain coarsening across a small-angle grain boundary demonstrates multiple forms of non-mean-curvature-driven growth.

An interfacial energy can be a function of a bulk field such as temperature or electric field. We find that in a system with a gradient in temperature or electric potential, the resulting variation in interfacial energy can induce a particle to migrate by either surface or bulk diffusion. For a circular particle under a constant unidirectional gradient in the bulk field, the field dependence of the interfacial energy induces a circular particle to move as a circle at a constant velocity along the direction from higher to lower interfacial energy. A linear stability analysis of this steady state migration suggests that perturbations will damp out as time evolves, and thus under these conditions a migrating circular particle is morphologically stable. Other spatial distributions of interface energy can lead to the distortion of an initially circular shaped particle during migration. A phase field model is developed that captures these distortions and verifies the theoretical results mentioned above.

Here, we design thermally stable microstructures of Al3Cu2Mg9Si7 (Q) precipitates based on commercial aluminum alloy (AS7GU) for automotive applications utilizing low-diffusivity solutes in aluminum. We find that the newly developed alloys show a ~ 20% enhancement in mechanical properties at 300 °C compared to the baseline AS7GU alloy.

We present the second set of benchmark problems for phase field models that are being jointly developed by the Center for Hierarchical Materials Design (CHiMaD) and the National Institute of Standards and Technology (NIST) along with input from other members in the phase field community. As the integrated computational materials engineering (ICME) approach to materials design has gained traction, there is an increasing need for quantitative phase field results. New algorithms and numerical implementations increase computational capabilities, necessitating standard problems to evaluate their impact on simulated microstructure evolution as well as their computational performance. We propose one benchmark problem for solidification and dendritic growth in a single-component system, and one problem for linear elasticity via the shape evolution of an elastically constrained precipitate. We demonstrate the utility and sensitivity of the benchmark problems by comparing the results of (1) dendritic growth simulations performed with different time integrators and (2) elastically constrained precipitate simulations with different precipitate sizes, initial conditions, and elastic moduli. These numerical benchmark problems will provide a consistent basis for evaluating different algorithms, both existing and those to be developed in the future, for accuracy and computational efficiency when applied to simulate physics often incorporated in phase field models.

We propose a general theory of Ostwald ripening for prolate spheroidal particles in a nonideal nondilute multicomponent alloy. The diffusion problem of a growing or shrinking particle is solved using prolate spheroidal coordinates under the assumption that the spheroidal particle has a constant Wulff shape. The result shows that the diffusional growth rate increases with an increasing particle aspect ratio due to the increased surface area per volume. The anisotropic interfacial energy necessary to guarantee that the particles are always prolate spheroids with a given aspect ratio is also determined. We find that the chemical potential decreases with an increasing particle aspect ratio under a constant volume-equivalent radius. Based on the two correction factors, asymptotic analysis reveals that the temporal exponents for the coarsening laws for spheroid particles are identical to that for spherical particles. However, as the aspect ratio increases the amplitudes of the temporal power laws of the average equivalent radius, the matrix supersaturations, and the particle composition decrease, whereas the amplitude of the number of particles per volume increases. It is also shown that the particle shape anisotropy affects the amplitudes, but not the direction of the vector representing the matrix supersaturation and particle composition.

The microstructure developed by two-phase lamellar eutectics $$(\alpha )$$ – $$(\theta {\text {-Al}}_{2}{\text {Cu}})$$ in Al-Cu and Al-Cu-Ag alloys is analyzed. A model of two-phase eutectic growth in multicomponent alloys is used to determine the scaling law of the eutectic microstructure using the alloy thermophysical properties. The application of the model to these alloys shows that the addition of Ag to Al-Cu alloys does not significantly change the length scale of the microstructure, which is in agreement with previous experimental studies. This is explained by the combined phenomena of the decrease in interface energies with the addition of Ag and the superheating of the $$(\alpha )$$ phase interface induced by the Ag composition profile.

Level-set method is used to simulate phase transformations with anisotropic kinetics where the transforming interface is faceted. The method overcomes previous limitation of this simulation methodology in tracking dynamic evolution of a large number of growing grains. The method is then used to simulate multi-grain phase transformations where the facets are three low-index ([1 0 0], [1 1 1], [1 1 0]) planes that yield morphologies including cube, octahedron and rhombic dodecahedron. The microstructure evolves under site-saturated nucleation and constant nucleation rate. The cube morphology undergoes fastest transformation followed by octahedron, rhombic dodecahedron and sphere. It is also shown that the Johnson-Mehl-Avrami-Kolmorgorov theory can be used to describe the kinetics of the faceted phase transformation. The resulting microstructure shows non-convex grain shapes with highly corrugated surfaces. The structures are also characterized using the average grain length along certain low index crystallographic directions, the coherent length. This measurement shows that for a cubic morphology, there is a significant difference in the coherent length for low index directions, while there is no meaningful difference in the coherent length for kinetic Wulff shapes of other anisotropies examined.

The structural characterization of dealloyed nanoporous metals is a fundamental and active area of research, needed for the optimization of these structures for catalytic, electrosensing, biomedical, and mechanical functions. The prediction of properties requires identifying and quantifying salient structural characteristics, while insights into the relevant mechanisms of dealloying and coarsening can be achieved through in situ observations of structural evolution. Three-dimensional structural characterization techniques have advanced such that nanoscale quantification of topology, morphology, and crystallographic parameters are achievable, yet the field is new enough that the assessment and comparison of such parameters of different nanoporous metals are just beginning. Here, we explore the state of the art in structural characterization, focusing on nanoporous gold to exemplify the challenges, the achievements, and the potential associated with establishing an appropriate set of structural parameters for this unique class of materials.

Cobalt-based alloys with γ / γ ′ microstructures have the potential to become the next generation of superalloys, but alloy compositions and processing steps must be optimized to improve coarsening, creep, and rafting behavior. While these behaviors are different than in nickel-based superalloys, alloy development can be accelerated by understanding the thermodynamic factors influencing microstructure evolution. In this work, we develop a phase field model informed by first-principles density functional theory and experimental data to predict the equilibrium shapes of Co-Al-W γ ′ precipitates. Three-dimensional simulations of single and multiple precipitates are performed to understand the effect of elastic and interfacial energy on coarsened and rafted microstructures; the elastic energy is dependent on the elastic stiffnesses, misfit strain, precipitate size, applied stress, and precipitate spatial distribution. We observe characteristic microstructures dependent on the type of applied stress that have the same γ ′ morphology and orientation seen in experiments, indicating that the elastic stresses arising from coherent γ / γ ′ interfaces are important for morphological evolution during creep. The results also indicate that the narrow γ channels between γ ′ precipitates are energetically favored, and provide an explanation for the experimentally observed directional coarsening that occurs without any applied stress.

We examine the equilibrium morphology of Al 2 Cu ( θ ′) precipitates in Al-Cu alloys using a phase field method with the parameters supplied by first-principles density functional theory (DFT) calculations. The phase field method employed allows for an interfacial energy that is highly anisotropic: there are missing high-energy orientations and corners on the Wulff shape. This high degree of anisotropic interfacial energy yields a plate-shaped equilibrium θ ′ precipitate in two-dimensions and a disk-like shape in three dimensions. Also, we consider the effects of a mismatch in elastic-moduli (elastic inhomogeneity) of Al and θ ′, elastic anisotropy, as well as tetragonal misfit strain anisotropy to gain a fuller picture of the elastic energy contributions to the morphology of θ ′ precipitates. Based on our phase-field modeling, the results show that the aspect ratio of the precipitate morphology with the anisotropy of interfacial and strain energies as given by DFT is significantly smaller than the aspect ratio observed in the experiment after long aging times (∼50 h). Specifically, the computed length (54 nm) is almost ten-times smaller than the length (∼580 nm) observed in the Al-Cu experiment with similar precipitate thickness (∼10 nm). Thus, we conclude that the experimental morphology after long aging times (∼50 h) is strongly influenced by the kinetics of precipitate growth.

The ability to quickly analyze large imaging datasets is vital to the widespread adoption of modern materials characterization tools, and thus the development of new materials. Image segmentation can be the most subjective and time-consuming step in the data analysis workflow. A promising approach to segmentation of large materials datasets is the use of convolutional neural networks (CNNs). However, a major challenge is to obtain the images and segmentations needed for CNN training, since this requires segmentations performed by humans. We show that it is possible to segment experimental materials science data using a SegNet-based CNN that was trained only using simple phase field simulations. A test image from an in-situ solidification experiment of an Al-Zn alloy was used to parameterize the phase field simulations. The most important microstructural features required for the best CNN to “understand” the contents of the image are ranked as: (1) having training images with diffuse particle-background interfaces, (2) modifying the images by adding noise, (3) removing particles at the image edges, and (4) adding sub-images to the particles to account for the feint bands present on some dendrites. The CNN trained on phase field images segmented the experimental test image with 99.3% accuracy, comparable to CNNs trained on experimental data. This approach of using computationally generated images to train CNNs capable of segmenting experiments will accelerate the rate of materials design and discovery.

A number of degradation mechanisms have been observed during the long-term operation of solid oxide electrolysis cells (SOEC). Using an electrolyte charge carrier transport model and a diffuse interface treatment for a multilayer electrolytes, we quantify the oxygen potentials across the electrolyte and thereby provide insights into these degradation mechanisms. Our model describes the transport of charge carriers in the electrolyte when the oxygen partial pressure is extremely low by accounting for the spatial variation of the concentration of oxygen vacancies in the electrolyte which is closely related to the degradation of the SOEC near the interface of hydrogen electrode and electrolyte. Moreover, we identify four quantities that characterize the distribution of oxygen partial pressure in the electrolyte, which are directly related to the degradation mechanisms in the electrolyte as well, and give analytical estimates for them. These analytical expressions provide guidance on the parameters that need to be controlled to suppress the degradation observed in the electrolyte.

Additive manufacturing of structural alloys results in the formation of complex microstructures, often with long, columnar grains that grow epitaxially from existing grains. A phase-field modeling framework is presented that considers solidification along a single track of 316L stainless steel in a regime where the solid-liquid interface is moving sufficiently fast that there is absolute interfacial stability with negligible composition variations. By coupling to a thermal field of a moving laser source, the model captures the trajectory of grains solidifying around a melt pool. The effect of interfacial kinetic anisotropy on the predicted as-solidified microstructures is examined through three-dimensional simulations. Through a combination of qualitative and quantitative analyses, we find that kinetic anisotropy has the largest impact along the center of the laser track and that the grain morphology in the early stages of solidification is predominantly due to the shape of the melt pool.

We present nucleation phase field model benchmark problems, expanding on our previous benchmark problems on diffusion, precipitation, dendritic growth, linear elasticity, fluid flow and electrochemistry. Nucleation is the process in which either a new thermodynamic phase or a new structure is created, such as solidification from the melt, or self-assembly of particulates. Based on where the nucleation occurs, it can be divided into two main categories: homogeneous nucleation and heterogeneous nucleation. In the first nucleation benchmark problem, we focus on homogeneous nucleation for both single seed under different initial conditions and multiple seeds. The second nucleation benchmark problem focuses on athermal heterogeneous nucleation and nucleation behavior near the free growth limit with different undercooling driving forces.

We investigate a phase transformation in a Ni-Cr-Fe-based high-temperature alloy during the technologically important carburization process, with multi-length scale experimental techniques. The study focuses on the formation and stability of novel austenitic islands formed within carbide structures during the M23C6 to M7C3 phase-transformation. We demonstrate that the austenitic islands nucleate near the M23C6/M7C3 transformation front, from a supersaturation of metal atoms in the M23C6-carbide as M7C3 grows. After formation, the austenitic islands equilibrate their composition with the matrix temporally but remain relatively stable inside the carbides due to their large sizes (a few hundred nanometers in diameter) and a weak Gibbs-Thomson effect.