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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.

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.

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 investigate grain boundary motion and grain rotation in a body-centered cubic bicrystal composed of a spherical grain embedded in a single crystal matrix by three-dimensional phase-field crystal simulations. Structure and time evolution of dislocation networks formed on the grain boundary during the capillarity-driven grain shrinkage are examined. The results for initially spherical grains rotated about the [110] or [111] axes of the matrix grain reveal the formation of hexagonal dislocation networks (HDNs) on the grain boundary. We demonstrate that the anisotropic distribution of the HDNs is responsible for asymmetric shrinkage of the embedded grain. Through a detailed analysis of the HDNs, we clarify the mechanisms of dislocation reactions during the grain shrinkage in three dimensions, which include dissociation and recombination of a/2 and a dislocations. The configuration of the HDNs is strongly affected by the rotation axis of the embedded grain. For large misorientations, the high density of the HDNs accelerates dislocation reactions and leads to very small grain rotations. However, if the misorientation is small and the dislocations are further apart, the lack of dislocation reactions on grain shrinkage results in grain rotation. Even though the rotation axis and the misorientation strongly affect the grain shape and the grain rotation, the kinetics of the grain shrinkage associated with the grain rotation follow the classical theory for grain growth: area of the embedded grain shrinks linearly with time. We also show that the stagnation of the grain rotation slows the shrinkage of the embedded grain.

Coarsening plays a pivotal role in materials engineering, but our understanding of the dynamics of coarsening in morphologically complex systems is still limited. In this paper, we examine the correlations between the interfacial velocity and interfacial morphologies, and then predict the evolution of mean curvature based on the correlations. Three simulated structures with varying volume fractions, two bicontinuous and one nonbicontinuous, are generated using the Cahn-Hilliard equation. We find general correlations between interfacial velocity and mean curvature, as well as between interfacial velocity and the surface Laplacian of the mean curvature. Furthermore, we find that the probability of finding a patch of interface with a given normal velocity and the same local principal curvatures is described well by a Gaussian distribution, independent of the principal curvature values and the volume fractions of the structures. We also find that average interfacial velocity is described by a polynomial of the mean curvature and the net curvature. Based on this finding, we develop a semi-analytical approach to predicting the rate of change of the mean curvature, which determines the morphological evolution of complex microstructures.

We extend the existing framework [1–4] of two-point spatial correlations to allow for the quantification, analyses, interpretation and visualization of microstructure coarsening measured by time-resolved X-ray computed tomography. Specifically, extensions were made to facilitate (i) the incorporation of nonconventional local attributes such as solid-liquid interface, interface curvature, and interface velocity in the description of the local state, and (ii) the efficient computation of bulk spatial correlations when the local attributes are sparsely defined only at special locations in the three-dimensional volume (e.g., solid-liquid interfaces). We have explored multiple variants of spatial correlations, including Pearson correlation coefficients and two-point joint probabilities, and examined their relative merits in providing useful new insights into the coarsening process. Algorithmic enhancements needed to carry out these computations on the large datasets produced in the experiments are also described. The results demonstrate the remarkable ability of these new protocols in automated (unbiased) capture of the four-fold symmetry of the dendritic microstructure, and in providing quantitative and reliable estimates of the characteristic lengths associated with the dendritic microstructure (including the secondary and tertiary dendrite arm spacings, secondary dendrite arm diameter, and the solute diffusion length). These estimated quantities agree well with the direct measurements from the microstructure. The results also indicate that interfaces with high negative and near-zero mean curvature ( H ) have long range spatial auto-correlations, whereas all values of the interfacial normal velocity ( V ) are only auto-correlated in the short range in space. For mid-range (positive) values of H and non-extreme values of V , the spatial distributions are essentially random.

The process of fragmentation of dendrite arms during coarsening remains poorly understood. We perform isothermal coarsening experiments of dendritic solid-liquid mixtures using PbSn alloys aboard the International Space Station (ISS), since arms that fission from the stem do not sediment and thus can be detected. The morphology of the structure and the number of fragments (fissioned arms) were determined using three-dimensional reconstructions. The evolution of the microstructure, change in length scale, interfacial shape distributions, number and distribution of fragments as well as the connectivity of the structures (handles) across coarsening time are discussed. We find that: the inverse of surface area per unit volume S V − 1 increases with time as t 1 / 3 in a manner that is almost identical to a sample coarsened on earth; the number of fragments per unit volume scaled by S V − 3 is independent of time. Thus, it is possible to predict the number of fragments during coarsening by a measurement of S V ; the connectivity of the structures as measured by the number of handles in the structure scaled by S V − 3 is also independent of coarsening time. We find that there is more coalescence of dendrite arms during coarsening than fragmentation.

Grain growth is generally driven to minimize the overall grain boundary energy. However, for low-angle grain boundaries the requirement that lattice planes be continuous across the boundary gives rise to a coupling between the normal motion of the grain boundary and the tangential motion of the lattice. We show through phase-field crystal simulations this coupling in polycrystalline systems can give rise to a rigid body translation of the lattice as a grain shrinks. The process is mediated by significant climb of the dislocations in the boundary and dislocation reactions at the trijunctions. Thus the grain growth process is coupled to vacancy diffusion processes as well as the dynamics of grain trijunctions. Moreover, grain shrinkage can cease because of dislocation behavior near the trijunction, illustrating that this coupling can have an influence on the grain growth process in polycrystals.

Phase separation in the γ − γ ’ alloy 5.2%Al–14.2%Cr-Ni at 873.15 K has been investigated by three-dimensional phase-field simulations, employing a model where both thermodynamic and kinetic parameters were experimentally verified. 510 individual nuclei corresponding to a density of 10 24 m −3 were introduced in accordance with classical nucleation theory, and the microstructural evolution was simulated up to a time of 32 h. The microstructural evolution was characterized, and regimes were identified according to the dominant mechanisms for microstructural evolution, where the final regime was found to be a coarsening regime obeying well-known power laws for the particle density and the average γ ’ particle radius. The evolution of the volume-averaged composition of the particles was found to follow a complicated trajectory, while the composition of individual particles were found to depart significantly from the average. The simulations were compared to experimental results from the literature based on atom probe tomography, and in general good qualitative correspondence was found, albeit with quantitative differences. These are discussed in terms of the assumptions inherent to the phase-field model, and by extension to most common continuum models of diffusive phase transformations.

This report describes a transmission X-ray microscopy observation of the high-temperature microstructural evolution of a solid oxide fuel cell Ni-Yttria-stabilized zirconia (Ni-YSZ) anode. Unlike prior studies that compared microstructural differences between different anodes, this three-dimensional measurement directly shows the changes occurring in the same region of an anode, enabling a new understanding of evolutionary processes. High-temperature ageing for 48 h at 1050 °C yielded substantial structural changes in the Ni, YSZ and pore networks, including coalescence of Ni particles, leading to a three-fold decrease in three phase boundary length. Implications for fuel cell long-term degradation are discussed.

Theoretical description of coarsening is of central importance to describing the kinetics of phase transformations in a wide range of materials. Through experiments on the International Space Station (ISS), the dynamics of coarsening was measured in a system that satisfies all assumptions of theory, in which the materials properties needed to compare experiment and theory are known, and has a rapid coarsening rate. The observed exponents for the temporal power laws match those predicted by theory. We find that the amplitudes of the power laws are slightly higher than predicted due to the presence of slow particle motion resulting from nonzero gravity present on the ISS. The measured particle size distributions are in agreement with those predicted by simulations. We conclude that interfacial energy driven coarsening is well described by theory and that simulations of coarsening can be used as a reliable tool for computational materials design.

While coarsening of spherical particles has been well documented, our understanding of coarsening of complex microstructures is still limited. The first step in developing a theory of coarsening of microstructures with complex morphologies is to study coarsening of microstructures that evolve self-similarly. In this paper, we examine the morphological evolution of a self-similar two-phase bicontinuous structure generated via nonconserved dynamics (i.e., motion by mean curvature) to elucidate the complex dynamics of coarsening. We find that the evolution proceeds with some interfaces evolving toward topological singularities (pinching) while the majority of interfaces flatten. These two processes were also illustrated through the evolution equation for the mean curvature, which has a term that depends solely on the local curvatures, as well as a term that is proportional to the surface Laplacian of the mean curvature. The first term causes an increase in the magnitude of the mean curvature, while the second term causes smoothing of the mean curvature in a manner similar to diffusion of chemical species on a surface. The second term causes a large dispersion in the values of the time derivative of mean curvature at various locations in the structure, characterized neither by the mean curvature nor the Gaussian curvature.

A method to determine material parameters by comparing the evolution of experimentally determined 3D microstructures to simulated 3D microstructures is proposed. The temporal evolution of a dendritic solid-liquid mixture is acquired in situ using x-ray tomography. Using a time step from these data as an initial condition in a phase-field simulation, the computed structure is compared to that measured experimentally at a later time. An optimization technique is used to find the material parameters that yield the best match of the simulated microstructure to the measured microstructure in a global manner. The proposed method is used to determine the liquid diffusion coefficient in an isothermal Al-Cu alloy. However, the method developed is broadly applicable to other experiments in which the evolution of the three-dimensional microstructure is determined in situ. We also discuss methods to describe the local variation of the best-fit parameters and the fidelity of the fitting. We find a liquid diffusion coefficient that is different from that measured using directional solidification.

A 3-D isotropic phase field simulation was used to predict the morphology of individual grains during grain growth. The simulation employed a polycrystalline array of titanium alloy Ti- β -21S experimentally characterized by X-ray tomography as an initial condition. The non-destructive nature of X-ray tomography allowed for a second characterization of the same sample following coarsening induced by a heat treatment. Thus, direct comparisons of individual grains between simulation and experiment could be made. Although the experimental system appeared isotropic from a statistical standpoint, direct examination of individual grains revealed very distinct anisotropy in the grain boundaries on the local scale. The comparison between experiment and phase-field simulations revealed regions with excellent agreement, despite the complex topological changes grains may undergo during grain growth. Thus, the sequence of topological transitions that occurred experimentally is correctly captured by the phase-field model. We therefore conclude that this phase-field model for isotropic systems has been verified experimentally.

The isothermal coarsening of dendritic Al–Cu microstructures is examined using time-resolved, in situ synchrotron-based X-ray tomographic microscopy. By obtaining the data at the coarsening temperature (in situ) with speeds that are on the order of or faster than the ongoing evolution of the microstructure (time-resolved, 4-D), we examine the dynamic morphological evolution of the solid–liquid interfaces in two solid volume fractions; as such, the relationship between the velocity of the evolving interface and characteristics that define the morphology of the interface is determined. We find that, while there is a correlation between velocity and mean curvature of the interface, there is a significant dispersion in the velocities for a given value of mean curvature. In addition, the Gaussian curvature plays a role in determining the interface velocity even though it has no effect on the chemical potential at the interface. We find that there are many interface patches of various morphologies and size scales that are not evolving during the coarsening process. At higher solid volume fractions and longer coarsening times, more of the structure is active in the coarsening process, and there is an increase in localized diffusional interactions.

A phase-field model has been developed to study the effect of triple junction (TJ) mobility on 2-D grain growth kinetics. The method captures the results of past work such as a linear increase in the average grain size with time, but can also follow the transition from TJ-limited to grain boundary energy-limited growth. The distribution of grain boundary curvature is examined. In the low TJ mobility simulations the distribution has a peak at zero curvature and approaches the grain boundary mobility-limited steady-state distribution at larger sizes. Even for extremely low TJ mobility, a small fraction of the grain boundary length has non-zero curvature and thus a lack of self-similarity is observed for all TJ-limited simulations, even when the average size is increasing linearly in time. We find that the topology of the grain structure is independent of the degree of TJ drag, within the range of parameters employed in the simulation. The effects of TJ mobility increase as the grain size decreases, suggesting that TJ mobility can play a significant role in nanocrystalline grain growth kinetics.

A general theory of coarsening in a multicomponent alloy is developed, accounting for off-diagonal terms in the diffusion tensor. The analysis is valid for a non-ideal and non-dilute solution. The asymptotic analysis reveals that the temporal exponents for the average particle radius, number of particles per volume and both the precipitate and matrix compositions are identical to the binary limit. However, the amplitudes are different. It is found that the vector representing the matrix supersaturations coincides with the equilibrium tie-line, but in most alloys this is not the case with the precipitate compositions. It is also shown that considering only a low mobility species does not yield a description of the temporal evolution of the matrix and precipitate compositions, even though this can be the case for the average particle size and the number density of precipitates.

The dealloying process is directly imaged, for the first time, by using transmission X-ray microscopy for the case of an Ag–30 at.% Au wire dealloyed under free corrosion in nitric acid. The propagation of a sharp dealloying front separating the alloy from nanoporous Au was observed by two-dimensional real-time in situ imaging at 30 nm resolution and measured in detail in three dimensions by an ex situ nanotomography technique at fixed time intervals. The rate of the dealloying front propagation is independent of the dealloying time up to a 3 μm depth, indicating that the dealloying process to this depth is dominated by interfacial effects (i.e. gold surface diffusion and/or silver dissolution) rather than long-range transport effects (i.e. diffusion of acid and corrosion product in and out of the porous layer). The constant dealloying rate corresponds to a constant silver flux and a constant current density, even though the potential might be fluctuating under free corrosion conditions and the interfacial area is shrinking as a function of time. Free corrosion in this system generates a high current density, implying it is driven by a chemical potential difference that is much higher than the critical potential.

The microstructural evolution of a polycrystalline dual-phase material with a constant volume fraction of the phases was investigated using large-scale three-dimensional phase-field simulations. All materials parameters are taken to be isotropic, and microstructures with volume fractions of 50/50 and 40/60 were examined. After an initial transient, the number of grains decrease from ∼2600 to ∼500. It was found that the mean grain size of grains of both phases obeyed a power law with an exponent of 3, and the microstructural evolution was found to be controlled by diffusion. Steady-state distributions of grain sizes and topology were determined. It was found that the grain size distributions were in good agreement with experimentally characterized size distributions for solid particles coarsening in a liquid matrix, and that the distributions of the number of faces were in good agreement with the topology of single-phase grain structures as determined by experiment and simulation. The evolution of size and number of faces for the minority and majority phase grains in the 40/60 volume fraction simulation is presented and discussed. Non-constant curvature across some interphase boundaries was observed, even though the interfacial energies are isotropic.

The three-dimensional evolution of nanoligaments of nanoporous gold created by Ag–Au dealloying was studied during isothermal coarsening by X-ray nanotomography and microbeam Laue diffraction. The surface normal orientation, curvature and size of the gold nanoligaments were measured as a function of coarsening time (from 2 to 320 min). The following observations were made at 550, 600 and 650 °C. First, the distribution of orientations for the surfaces of the nanoligaments becomes more anisotropic with coarsening time, with an increasing area of the surfaces having a low surface energy, consistent with the growth of facets. Second, the curvature distribution of the nanoligaments (scaled by their size) also evolves during coarsening. The evolution of both surface orientation and scaled surface curvature indicates that coarsening does not occur in a self-similar manner, i.e. the interfacial shape distribution of the gold nanoligaments is not self-similar over time as they coarsen. This is consistent with the ligament size not being described by a classical temporal power law for coarsening systems. All three effects, and in particular the increased prevalence of surfaces with a low surface energy at long coarsening times, may affect the surface functionalities and properties of nanoporous gold in various applications, e.g. as catalysts, sensors and actuators.

We study two-dimensional grain growth at the nanoscale using the phase field crystal (PFC) model. Our results show that for circular grains with large misorientations the grain area decreases linearly with time, in good agreement with classical grain growth theory. For circular grains with small initial misorientations, grain rotation occurs as a result of the coupled motion between the normal motion of the grain boundary and the tangential motion of the adjacent grains. Despite this rotation and its effect on the grain boundary energy, the grain area decreases linearly with time. In addition, for intermediate initial grain misorientations, we find a repeating faceting–defaceting transition during grain shrinkage and a different relationship between the grain area and time, which suggests a different grain growth mechanism than that for small and large misorientations. For a circular grain embedded between a bicrystal with a symmetric tilt boundary, we find that the evolution of the embedded grain closely depends on dislocation reactions at triple junctions.

The morphology of a rod embedded in a matrix undergoing pinching by interfacial-energy-driven bulk diffusion is determined near the point of pinching. We find a self-similar solution that gives a unique temporal power law and interfacial shape prior to pinching and self-similar solutions after pinching. The theory is compared to experiments that employ in situ four-dimensional X-ray tomographic microscopy for rods of liquid or solid pinching by solute diffusion in the high-diffusivity liquid phase. The excellent agreement between experiment and theory confirms that the interfacial morphology near the singularity is universal both before and after pinching; the shape holds regardless of the material system and initial condition. This also implies that the predictions of the time-dependence of the process can be used to determine the time to pinching for a wide variety of physical systems, and thus provide estimates of the time required for capillarity-driven break-up of microstructures from the detachment of secondary dendrite arms to polymer blends.

Spatial correlations of interfacial curvature are compared for symmetric and asymmetric two-phase mixtures produced following spinodal decomposition as given by a numerical solution to the Cahn–Hilliard equation in three dimensions. By calculating radial distribution functions of the density of interfacial area as a function of the mean interfacial curvature of these bicontinuous microstructures, it is found that long-range diffusive interactions, in combination with the morphology of the system, yield a variety of correlations and anticorrelations over a range of length scales. The asymmetric mixtures show some similarities to the symmetric mixtures, as well as other unique features.

The morphological evolution of equiaxed Al– 20 wt. % Cu dendritic microstructures was studied in three dimensions. The microstructure evolved into a highly interconnected structure, where the inverse specific surface area scaled linearly with the cube root of time. As the size scale of the microstructure increased during coarsening, the scaled morphology of the interfaces changed only slightly. The distribution of interface normals indicated that the microstructure was approximately isotropic. These results are in contrast to those found using a directionally solidified Al–Cu alloy of a similar solid volume fraction, where the structure evolved into solid cylinders parallel to the growth direction used to create the sample prior to coarsening. Thus, we find that the initial morphology of a dendritic structure can have a major impact on its morphological evolution.

The three-dimensional microstructure of 78 and 52 vol.% Sn-rich particles coarsened within a liquid Pb–Sn matrix were determined by the reconstruction of serial sections. The three-dimensional particle size distribution (PSD) and the particle–particle contact distributions were determined. The three-dimensional PSDs do not match those predicted by particle coarsening theory, but there is reasonable agreement with the grain size distributions predicted by a grain growth simulation. In addition, when the particle–particle contact distribution is normalized to the average number of particle contacts, the distribution is statistically invariant with volume fraction. At 52 vol.% it is found that the number of contacts is proportional to the square of the average particle size, but this is not true for 78 vol.%. This is attributed to the increased shape distortions of the particles that are present in the higher volume fraction samples.

Quantitative characterization of coarsening is achieved through topological measurements such as the genus, the number of handles, and the number of independent bodies (liquid droplets). Topological analysis was performed on experimentally derived, three-dimensional reconstructions of dendritic microstructures. Measured topological quantities were reported on a per volume basis and scaled by the length scale of the system to remove effects of the changing length scale during coarsening. The scaled genus decreased with coarsening time due to the simplification of the topology of the microstructure, while the number of liquid droplets increased with coarsening time. These results were supplemented with calculations of the interfacial velocity determined using phase-field simulations that employ the experimental three-dimensional reconstructions as the initial condition. Through these calculations it is shown that liquid droplets form through capillary-driven instabilities of interconnected liquid channels, while liquid tubes are created through topological singularities occurring on large planar-like walls of liquid.

The equilibrium shapes of misfitting precipitates in elastically anisotropic systems are obtained in both two and three dimensions, and the corresponding Gibbs–Thomson equation is derived as a function of the characteristic ratio between elastic and interfacial energies, L 0 . The effect of elastic inhomogeneity is investigated systematically. For soft or moderately hard particles, the stable equilibrium shape bifurcates from a fourfold symmetric shape to a twofold symmetric one in 2D and from a cubic symmetric shape to a plate-like one in 3D. For a very hard particle, the shape bifurcation is not observed in 2D for the range of L 0 investigated, but both plate-like and rod-like shapes are found in 3D. The computed Gibbs–Thomson equation is well approximated by a piecewise linear function of L 0 . Predictions are made for coarsening of many-particle systems based on an established mean-field theory. The results predict that the elastic stress has no effect on coarsening kinetics where most particles are highly symmetric (fourfold in 2D and cubic in 3D), and the exponent remains 1/3 but the rate constant increases if stress is sufficient to induce symmetry-breaking bifurcation on most particles.

We present the results from large-scale simulations of Ostwald ripening of misfitting second-phase particles in an elastically anisotropic system. We employ the sharp interface description and perform simulations in two dimensions using boundary integral methods combined with state-of-the-art numerical methods such as the fast multipole method. We find that particle shapes are perturbed by elastic interactions at sufficiently large area fractions and particle sizes, and thus the shapes are not given by the equilibrium morphology of an isolated particle. Unlike isolated particles, the morphology transitions from fourfold to twofold shapes are not sharp, but are smeared due to interparticle interactions. However, for a very low area fraction system, the particles remain fourfold symmetric well beyond the bifurcation point, indicating that elastic interactions are essential in inducing a particle shape bifurcation. We find that the evolution of the microstructure is not scale invariant. However, the microstructure is unique, in a statistically averaged sense, for a given ratio of the elastic and interfacial energies.

Ostwald ripening of misfitting second-phase particles in an elastically anisotropic solid is studied by large-scale simulations. The coarsening kinetics for the average particle size are described by a t 1/3 power law with a rate constant equal to its stress-free value when the particles are fourfold symmetric. However, the rate constant increases when the elastic stress is sufficient to induce a large number of twofold-symmetric particles. We find that interparticle elastic interactions at a 10% area fraction of particles do not affect the overall coarsening kinetics. A mean-field approach was used to develop a theory of Ostwald ripening in the presence of elastic stress. The simulation results on the coarsening kinetics agree well with the theoretical predictions. The particle size distribution scaled by the average particle size is not time invariant, but widens slightly with an increasing ratio of elastic to interfacial energies. No time-independent steady state under scaling is found, but a unique time-dependent state exists that is characterized by the ratio of elastic energy to interfacial energy.

Phase boundaries play an important role in setting the properties of multicomponent materials at both the meso- and nano-scales. In this review, we provide an overview of the modelling methods utilized in state-of-the-art research and engineering applications. We review the current physical understanding of how phase boundaries evolve, focusing on multicomponent systems. The recent advances in numerical modelling, fueled by powerful computers, have provided accurate and robust results that allow problems that are beyond the reach of analytic methods to be addressed. While the approaches used in engineering-oriented applications employ simplified microstructures, it is found that such models are quite useful in many problems. The ability to simulate realistic microstructures will further increase the power of materials modelling. We also highlight the differences between the sharp interface and diffuse interface approaches for modelling microstructural evolution. In addition, we identify future research topics in this area.

We investigate the three-dimensional structure of a model g–g alloy using serial sectioning and digital reconstruction. The particles align in parallel two-dimensional sheets along the 100 crystallographic directions. The sheets in different regions of the microstructure are aligned along different 100 crystallographic directions. A variety of particle morphologies are observed. These morphologies are relatively independent of particle size, indicating a significant departure from single-particle equilibrium morphologies. The primary determinant of particle morphology appears to be the interparticle elastic interactions, both within sheets of particles and at the intersections of sheets. There is no evidence of particle coalescence despite the small interparticle separation.  2002 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved.

The effects of elastic stresses on the kinetics of coarsening were investigated. The coarsening of g particles in NiAl alloys was quantified by measuring the g-g interfacial area per volume, instead of a characteristic average particle size. This allowed the kinetics of coarsening to be measured meaningfully at very long times when most particles do not have an equiaxed shape. It was found that the interfacial area per volume follows a t 1/3 rate law, with no change in the exponent or the rate constant even at long coarsening times when elastic energy is a significant contribution to the total energy of the system, and the microstructure is not self-similar. The coarsening rate constant was found to vary with volume fraction as predicted by theory in the absence of elastic stress, in contrast to previous experimental results which quantify coarsening kinetics using an average particle radius.  2002 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved.

The microstructural evolution and kinetics of Ostwald ripening were studied in the high volume fraction regime by numerically solving the time-dependent Ginzburg–Landau (TDGL) and Cahn–Hilliard equations. It is shown that the growth exponent m is equal to 3, independent of the volume fraction, and the kinetic coefficient k increases as the volume fraction increases. The shape of size distributions changes significantly with increasing volume fraction of the coarsening phase; the skewness changes continuously from negative to positive while the kurtosis decreases in the low fraction regime and increases in the high volume fraction regime.  2002 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved.

The first complete characterization of a dendritic microstructure is presented. The dendrite morphology is obtained via a novel serial-sectioning process that allows the three-dimensional microstructure of opaque materials to be determined in a routine manner. Using the reconstructed dendrite we determine the spatial distribution of the mean and Gaussian curvature as well as the probability density distributions of these curvatures. These measurements yield many insights into the local processes that shape these topologically complex structures that are vital in setting many material properties.

The coarsening of solid-Sn particles in a Pb–Sn liquid has been studied under microgravity conditions. These experiments permit an unambiguous comparison between theory and experiment to be made. In contrast to steady-state theories, such as those due to Lifshitz and Slyozov and Wagner, the scaled particle size distributions evolve in samples containing 0.1 and 0.2 volume fractions of solid. Steady state was not reached even though the average particle radius increased by a factor of three during the experiment. In addition, the scaled spatial correlation functions were also found to be time dependent in samples containing 0.1, 0.2, and 0.3 volume fractions of solid. The size distributions and correlation functions for all coarsening times at the fractions ≤0.3 agree with the predictions of a theory for transient coarsening. We show that the microstructures have not reached the steady-state regime for all volume fractions, are thus not self-similar, and that given our initial experimental conditions the time required to reach steady-state coarsening increases with increasing volume fraction. In these experiments, and we suspect in others as well, the transients are sufficiently long that steady-state theories cannot adequately describe the evolution of the microstructure.

The coarsening of solid-Sn particles in a Pb–Sn liquid was studied under microgravity conditions. Spatial correlation functions were measured on plane sections in a low-volume fraction system undergoing Ostwald ripening. The correlation functions changed with time in a way that indicated that the microstructure initially consisted of clusters of particles and evolved into one which was more dispersed. The model by Akaiwa and Voorhees (AV) was used to study the effect of spatial correlations on the ripening process. We found that the initially highly correlated structure had no observable effect on the evolution of particle size distributions, but did have an effect on the coarsening rate of the system. Specifically, we determined that a structure consisting of clusters of particles coarsened faster than a system with a random, spatial arrangement of non-overlapping particles. We also found that the approach of the microstructure towards the steady-state regime could be monitored more sensitively using spatial correlations rather than using particle size distributions. The spacial correlations and the particle size distributions measured from the experiment agreed well with those calculated from the AV simulations using the initial experimental correlations and size distribution.

A semi-infinite, uniform film on a substrate tends to contract from the edge to reduce the surface energy of the system. This work studies the two-dimensional retraction of such a film step, assuming that the film evolves by capillarity-driven surface diffusion. It is found that the retracting film edge forms a thickened ridge followed by a valley. The valley sinks with time and eventually touches the substrate. The ridge then detaches from the film. The new film edge retracts to form another ridge accompanied again by a valley, and the mass shedding cycle is repeated. This periodic mass shedding is simulated numerically for contact angle α between 30 and 180°. For smaller α, a small-slope late-time solution is found that agrees with the numerical solution for α=30°. Thus, the complete range of α is covered. The long-time retraction speed and the distance traveled per cycle agree quantitatively with experiments.

Interfaces of A–B–C ternary alloys decomposed into two and three phases are studied. The effect of the gradient energy coefficients κ II , I=A, B, C, on the interface composition profiles of ternary alloys is examined. The adsorption of component C in ternary alloys is obtained numerically by finding steady-state solutions of the nonlinear Cahn–Hilliard equations and by solving the two Euler–Lagrange equations resulting from minimizing the interfacial energy, and analytically near the critical point. It is found that the solutions from both numerical methods are identical for a two-phase system. In symmetric ternary systems (equal interaction energy between each pair of components) with a minority component C, the gradient energy coefficient of C, κ CC , can have a very strong influence on the degree of adsorption. In the α and β two-phase regions, where α and β are the phases rich in the majority components A and B, respectively, as κ CC increases, the adsorption of the minority component C in the α and β interfaces decreases. Near a critical point, however, the degree of adsorption of minority component C is independent of the gradient energy coefficient.

We determine the three-dimensional equilibrium shapes of particles with a purely dilatational misfit in an elastically anisotropic medium with cubic symmetry. We have identified a succession of cuboidal shapes with four-fold rotational symmetry that minimize the total energy of the system. In the process of determining these equilibrium morphologies, we have also developed a computationally efficient approach to determine the equilibrium shape which is many orders of magnitude faster than a standard implementation of Newton's method. For small elastic stress a (100) cross-section of the three-dimensional equilibrium shape agrees well with the two-dimensional calculation. However, for larger values of the elastic stress, the agreement is not as good. Elastic-stress-induced configurational forces are identified as the reason for the non-spherical equilibrium shapes.

In this paper we develop conditions that govern the evolution of a fully faceted interface separating elastic phases. To focus attention on the effects of elastic stress, we restrict attention to interface-controlled kinetics, neglecting bulk transport; and to avoid geometric complications, we limit our discussion to two space-dimensions. We consider a theory in which the orientations present on the evolving particle are not necessarily those given by the Wulff shape: we allow for metastable crystallographic orientations as well as stable orientations. We find that elastic stress affects the velocity of a facet through the average value of the normal component of the jump in configurational stress (Eshelby stress) over the facet. Within our theory singularities in stress induced by the presence of corners do not influence the velocity of the facet. We discuss the nucleation of facets from corners; the resulting nucleation condition is shown to be independent of elastic stress. We also develop equations governing the equilibrium shape of a faceted particle in the presence of elastic stress.

A solid film freshly deposited on a substrate may form a non-equilibrium contact angle with the substrate, and will evolve. This morphological evolution near the contact line is investigated by studying the motion of a solid wedge on a substrate. The contact angle of the wedge changes at time t = 0 from the wedge angle α to the equilibrium contact angle β, and its effects spread into the wedge via capillarity-driven surface diffusion. The film profiles at different times are found to be self-similar, with the length scale increasing at t 1 4 . The self-similar film profile is determined numerically by a shooting method for α and β between 0 and 180°. In general, we find that the film remains a wedge when α = β. For α β, the film extends. For α = 90°, the results describe the growth of grain-boundary grooves for arbitrary dihedral angles. For β = 90°, the solution also applies to a free-standing wedge, and the thin-wedge profiles agree qualitatively with those observed in transmission electron microscope specimens.

Three-dimensional (3-D) images of the skeletal morphology that forms at a high volume fraction of solid during Ostwald Ripening are reconstructed from two-dimensional (2-D) sections. These images illustrate the topology of the skeleton, including the circuitous and multiple paths through which particles in the skeleton connect. Particles that connect by grain boundaries form chains that can cross-link and intersect. These chains can also give rise to multiple indirect contacts between two particles. Direct interparticle contacts can occur via grain boundaries or thin liquid films. All particles connect to the skeletal network by such “contacts”; there are no isolated particles or clusters within our 3-D reconstruction. Shape accommodation is prevalent. The particle morphology is a strong function of the spatial arrangement of neighboring particles. We find that isolated 2-D sections are thus limited in characterizing the microstructure. The relationship between the 3-D microstructure of the skeleton and its structural stability is discussed.