Coarsening Model Research Articles

A multiscale procedure based on the stochastic kinetic mean field and the phase-field models for coarsening

We report a quantitative multiscale (MS) procedure based on the recently developed Stochastic Kinetic Mean Field approach (SKMF) [Erdélyi, M. Pasichnyy, V. Bezpalchuk, J. J. Tomán, B. Gajdics, and A. M. Gusak, Comput. Phys. Commun. 204, 31–37 (2016)], combined with the Phase Field model (PFM) and CALPHAD database, to study the nucleation-growth-coarsening process in alloys. The SKMF approach reproduces the nucleation and early growth of precipitates in the matrix, and the PFM then simulates the coarsening of the microstructure. To ensure the consistency of the procedure, the length and time scales of SKMF and PFM are explicitly connected. Moreover, both the effective interaction energies used in the SKMF and the free energy used in the PFM are taken from CALPHAD database. Two different implementations of the procedure are proposed. First, the postnucleation microstructure as provided by SKMF is used as the initial condition for subsequent PFM simulations. Second, only the particle size distribution and particle density are transferred to PFM, thereby giving access to bigger systems. The proposed procedure is tested in the specific case of the Ag-Cu model alloy.

Effect of cooling rate on the thermal and electrical conductivities of an A356 sand cast alloy

This study evaluates the effect of cooling rate on the microstructures and conduction properties of an A356 sand cast alloy. To evaluate effect of cooling rate, computer recordable sand mold with four step-thickness (5 mm– 20 mm) was used and microstructures of T6 heat treated specimens with different cooling rates were observed. It was found that with increasing cooling rate, eutectic reaction temperature shifted to lower temperature and the supercooling also increased. Based on the coarsening model via solute diffusion, when the cooling rate increase, secondary dendrite arm spacing decreases, providing the enhanced thermal and electrical conductivities.

A unified model for microstructural coarsening of deformed metals

In this work the kinetics of continuous coarsening of deformation microstructures are analysed. The coarsening examples span a large range in structural scale and driving force, including subgrain growth in a deformed subgrain structure, uniform coarsening in a deformed lamellar structure, and grain growth in a deformed nanocrystalline structure. The unified coarsening model for deformation microstructures considers the change of the apparent activation energy during coarsening and successfully describes the kinetics in each of the given coarsening examples. It is also found that the increase in the apparent activation energy dominates the coarsening kinetics and leads to a significant decrease in the coarsening rate as coarsening proceeds.

A new-curvature cellular automata model of austenite grain growth

A new-curvature three dimensional cellular automata model for grain coarsening is proposed. The model takes account of the effects of grain size on the local-curvature. The simulation results are analyzed from four aspects: grain growth kinetics, grain size distribution, dynamics of single grain, and topological properties. The grain growth kinetics of the present model is proved to agree well with the theoretical model of normal grain growth. The absolute values of average grain size are compared to the experimental results of a low alloy steel. The simulated grain size distribution can be satisfactorily described by the Werbull’s formula. The simulation results of single grain growth kinetics matches well with the Hillert’s theoretical model, and the topological properties are similar to Ding’s simulation results.

Reshaping, Intermixing, and Coarsening for Metallic Nanocrystals: Nonequilibrium Statistical Mechanical and Coarse-Grained Modeling.

Self-assembly of supported 2D or 3D nanocrystals (NCs) by vacuum deposition and of 3D NCs by solution-phase synthesis (with possible subsequent transfer to a support) produces intrinsically nonequilibrium systems. Individual NCs can have far-from-equilibrium shapes and composition profiles. The free energy of NC ensembles is lowered by coarsening which can involve Ostwald ripening or Smoluchowski ripening (NC diffusion and coalescence). Preservation of individual NC structure and inhibition of coarsening are key, e.g., for avoiding catalyst degradation. This review focuses on postsynthesis evolution of metallic NCs. Atomistic-level modeling typically utilizes stochastic lattice-gas models to access appropriate time and length scales. However, predictive modeling requires incorporation of realistic rates for relaxation mechanisms, e.g., periphery diffusion and intermixing, in numerous local environments (rather than the use of generic prescriptions). Alternative coarse-grained modeling must also incorporate appropriate mechanisms and kinetics. At the level of individual NCs, we present analyses of reshaping, including sintering and pinch-off, and of compositional evolution in a vacuum environment. We also discuss modeling of coarsening including diffusion and decay of individual NCs and unconventional coarsening processes. We describe high-level modeling integrated with scanning tunneling microscopy (STM) studies for supported 2D epitaxial nanoclusters and developments in modeling for 3D NCs motivated by in situ transmission electron microscopy (TEM) studies.

A Simple Model to Estimate Solidification Time–Dendrite Arm Spacing Relationships in Cast Aluminum Alloys with Two Major Alloying Additions: Application to the Al-Si-Cu System

Secondary dendrite arm coarsening model developed by Feurer and Wunderlin was used in this study. The parameters of the model were adjusted by following a simple approach to find an equivalent eutectic composition based on the alloying additions. Results showed the simple model proposed in this study provided good fits to data from literature on the 319 alloy as well as Al-Si-Cu alloys with various compositions.

Effects of Atomic Layer Deposition of ZrO2 on Infiltrated Mixed Ionic and Electronic Conducting Solid Oxide Cell Electrodes

Infiltration of mixed ionic and electronic conducting (MIEC) nanoparticles into ionic or MIEC scaffolds has been employed to improve the performance of solid oxide cells (SOCs) designed to operate at relatively low temperatures (<600°C). However, the high MIEC specific surface area also hastens sintering and coarsening processes that can lead to a high rate of performance degradation for the electrodes over time. Atomic layer deposited (ALD) coatings have been investigated as a means of curbing the degradation of nano-scale MIEC electrodes. ALD is desirable for coating porous electrodes because of its self-limiting layer-growth characteristic — fully conformal coating can ideally be achieved throughout the depth of a porous electrode, including the most electrochemically active region near the electrolyte. However, results reported to date have yielded mixed results: in some cases, the ALD coating suppresses degradation, whereas in others the coating actually degrades electrode performance. Successful performance improvement seems to be dependent on a combination of factors from choice of electrode substrate and its architecture, to the coating species, its thickness throughout an electrode, and the experimental procedures for coating. This presentation seeks to better elucidate the nature of these dependencies via a study of Sr0.5Sm0.5CoO3-δ (SSC) infiltrated Gd-doped ceria (GDC) electrodes with ALD-ZrO2 coating. Life tests were carried out with degradation accelerated by using a relatively high ageing temperature of 750°C, with impedance spectroscopy measured periodically at 600°C. Rises in polarization resistance with time were substantially mitigated by 30 ALD deposition cycles, and to a lesser extent by 60 cycles, relative to uncoated samples. The overall 0-hour performance is virtually the same across the coated and uncoated samples after a short break-in period, with the benefit from ALD being entirely in the increased sustained performance. The mechanistic benefit of ALD-ZrO2 on these electrodes is parsed with the aid of microstructural and chemical characterization. Scanning electron microscopy with energy dispersive spectroscopy of cross-sections of the coated samples reveals a substantial gradient in the ZrO2 coating thickness, with several nm thickness near the free surface decreasing to sub-monolayer thickness near the electrode/electrolyte interface. This is sensible because thicker ZrO2 coatings on SSC powders were found by powder x-ray diffraction to react extensively, forming Sr- and Sm-zirconates and cobalt oxide species. The results suggest that sub-monolayer ALD-ZrO2 coatings form a very thin layer of reactants that do not cover electrochemically-active surface sites but can help suppress particle coarsening. Based on this mechanism, a coarsening model previously developed for infiltrated SSC-GDC electrodes is shown to fit the polarization resistance versus time data by assuming that the ALD layer decreases the cation diffusion. The ideal thickness is that which balances the negative effect of the reaction of active SSC material with the positive effect of a reduced effective diffusion coefficient. This thickness is sensitive to the number of ALD cycles deposited as well as the aspect ratio of the pores the ALD precursors must traverse within the electrode. The results will be contrasted to those obtained from ALD-ZrO2 on conventional La0.6Sr0.4Co0.2Fe0.8O3-δ (LSCF) electrodes, which do not degrade by coarsening due to the larger feature sizes. In this case, there is no benefit from monolayer-thickness ALD-ZrO2, and thicker layers substantially degrade the performance. Figure 1

Long-term stability of precipitated phases in CLAM steel during thermal aging

China Low Activation Martensitic (CLAM) steel as one of 9%Cr ferritic-martensitic steels was designed as the structural materials for fusion reactors and lead-based Reactor in China. The long-term thermal stability of precipitated phases in CLAM steel at 550 °C was systematically studied in this paper, including coarsening kinetics, number density and shape stability. Based on large data statistics, the coarsening kinetics of M23C6 carbide, MX carbonitride and Laves phase were analyzed to develop a coarsening model of each precipitated phase, respectively. The results showed that the coarsening rate of Laves phase was about 100 times than that of M23C6 carbide after aging for 20,000 h at 550 °C. The number density of precipitated phases increased firstly and then decreased with increasing of aging time due to a dynamic equilibrium process of the precipitation and dissolution of precipitated phase. The results will help to deep understand degradation mechanisms after thermal aging.

On the origin of a remarkable increase in the strength and stability of an Al rich Al-Ni eutectic alloy by Zr addition

The effect of Zr addition to Al rich binary α-Al - Al3Ni eutectic cast alloy (Al-3.1 at.% Ni) in enhancing the microstructural stability and strength at high temperature is demonstrated. On subsequent heat treatment after casting, nanometric coherent L12 ordered Al3Zr precipitates form inside the α-Al that strengthen the alloy. Additionally, remarkable stability of eutectic microstructure was observed even after 100 h of annealing at 400 °C. The synergetic effect of the strengthening of the α-Al matrix by coherent Al3Zr precipitates and the low coarsening rate of the Al3Ni rods results in a significant increase in high temperature hardness and yield strength of the alloy. The tensile yield strength of the annealed Al-3.1Ni-0.15Zr alloy (400 °C, 10 h) tested at 250 °C is found to be 185 ± 10 MPa, which is 1.5 times higher than the corresponding binary Al-3.1Ni alloy. The experimentally determined average rod size (radius) during annealing at 400 °C follows the classical matrix diffusion controlled LSW-based coarsening model for both binary Al-3.1Ni and ternary Al-3.1Ni-0.15Zr alloys. The calculated coarsening rate constant values based on modified LSW coarsening model are 10.3 and 4.1 nm3/s for Al-3.1Ni and Al-3.1Ni-0.15Zr alloys, respectively. Atom probe tomographic (APT) investigations of the heat-treated ternary alloy unambiguously reveal segregation of Zr solute at the αAl/Al3Ni interface in addition to the presence of the strengthening Al3Zr ordered precipitates in the α-Al matrix. The segregation hinders the interdiffusion of Al and Ni in the eutectic and, thereby, increasing the stability of the eutectic phase at high temperature.

Reduced-order model for microstructure evolution prediction in the electrodes of solid oxide fuel cell with dynamic discrepancy reduced modeling

Microstructure evolution in the electrodes of solid oxide fuel cell is an important degradation mechanism which reduces active sites for redox reaction and the electric conductivity. Phase field models for microstructure evolution simulation are usually expensive for large scale simulations. In this work, a reduced-order coarsening model is developed using dynamic discrepancy reduced modeling, which reduces the model order by inserting Gaussian process stochastic functions into the dynamic equations of Ostwald ripening. The reduced order model has been calibrated on a dataset generated by a phase field model that has been well validated to experiments. A validating dataset has also been generated with which the model prediction show good agreement. This model is further applied to predict long term microstructure evolution in different SOFC electrodes. This work is the first attempt of building a degradation model of SOFC using data science techniques.

Investigation of LSM-YSZ Composite Cathode Performance Degradation with a Multistep Charge Transfer Model

The long-term performance degradation of LSM/YSZ composite cathodes is investigated via calibrated multiphysics simulation with a multistep oxygen reduction reaction (ORR) model and structural coarsening data from a phase field study. Multiphysics simulations, including a multistep ORR mechanism, are first developed for better understanding of the limiting processes in solid oxide fuel cells. The multistep ORR mechanism includes two parallel pathways, surface pathway and bulk pathway, which consist of elementary reaction steps. The multiphysics model is simultaneously calibrated with experimental polarization curves and impedance behavior for various air/fuel supply conditions. To our knowledge, this is the first time that a multistep ORR model is calibrated with such datasets. Next, the calibrated simulations are utilized to simulate a 2D half-cell constructed with measured microstructural data and random heterogeneity. Finally, the long-term performance degradation of the half-cell is predicted by the calibrated multiphysics model coupled with structural coarsening trends simulated using a phase field-based coarsening model. Degradation of both polarization curves and impedance behavior is investigated. Thorough analyses, including changes of contributions from different pathways, the resistance components, and overall reaction order, are performed to provide more insights into cathode performance degradation due to grain coarsening phenomena.

An unconditionally energy stable second order finite element method for solving the Allen–Cahn equation

In this paper, we design, analyze and numerically validate an unconditionally energy stable second order numerical method for solving the Allen–Cahn equation which represents a model for anti-phase domain coarsening in a binary alloy. The proposed scheme inherits the property of the decrease of the total energy from the Allen–Cahn equation. An error estimate for the fully discretized scheme is also established. Numerical examples are presented to confirm the accuracy, efficiency, and stability of the proposed method. In particular, the method is shown to be unconditionally energy stable and second order accurate in both time and space discretizations.

Ag₃Sn Compounds Coarsening Behaviors in Micro-Joints.

As solder joints are being scaled down, intermetallic compounds (IMCs) are playing an increasingly critical role in the reliability of solder joints, and thereby an in-depth understanding of IMCs microstructure evolutions in micro-joints is of great significance. This study focused on coarsening behaviors of Ag3Sn compounds in Sn-3.0Ag-0.5Cu (SAC305) micro-joints of flip chip assemblies using thermal shock (TS) tests. The results showed that the Ag3Sn compounds grew and rapidly coarsened into larger ones as TS cycles increased. Compared with such coarsening behaviors during thermal aging, TS exhibited a significantly accelerating influence. This predominant contribution is quantitatively determined to be induced by strain-enhanced aging. Moreover, based on observations for Ag3Sn microstructure evolutions during TS cycling, one particular finding showed that there are two types of coarsening modes (i.e., Ostwald ripening and Necking coalescence) co-existing in the Ag3Sn coarsening process. The corresponding evolutions mechanism was elucidated in a combination of simulative analysis and experimental validation. Furthermore, a kinetic model of the Ag3Sn coarsening was established incorporating static aging and strain-enhanced aging constant, the growth exponent (n) was calculated to be 1.70, and the predominant coarsening mode was confirmed to be the necking coalescence.

Special Solutions to a Nonlinear Coarsening Model with Local Interactions

We consider a class of mass transfer models on a one-dimensional lattice with nearest-neighbour interactions. The evolution is given by the discrete backward fast diffusion equation, with exponent $\beta$ in the regime $(-\infty,0) \cup (0,1]$. Sites with mass zero are deleted from the system, which leads to a coarsening of the mass distribution. The rate of coarsening suggested by scaling is $t^\frac{1}{1-\beta}$ if $\beta \neq 1$ and exponential if $\beta = 1$. We prove that such solutions actually exist by an analysis of the time-reversed evolution. In particular we establish positivity estimates and long-time equililibrium properties for discrete parabolic equations with bounded initial data.

A thermodynamically consistent model for elastoplasticity, recovery, recrystallization and grain coarsening

We present a model for the thermomechanical behaviour and microstructure evolution of metallic materials during thermomechanical processing. Special emphasis is put on its derivation based on the Müller–Liu procedure which ensures consistency with fundamental physical principles. The model represents in principle the strong coupling between elastoplastic deformation, recovery, recrystallization, grain coarsening and related thermal effects. Furthermore, it takes the dragging effect of precipitates on grain boundaries and dislocations into account by which a higher strength can be achieved and the grain size is stabilised. For the microstructure description, we use a mean-field approach. In several numerical examples, we demonstrate the capability of the model to consistently predict the interplay between elastoplastic deformation, microstructure evolution, dynamic hardening and softening and the related temperature change.

Coarsening Model on $${\mathbb{Z}^{d}}$$ with Biased Zero-Energy Flips and an Exponential Large Deviation Bound for ASEP

We study the coarsening model (zero-temperature Ising Glauber dynamics) on $\mathbb{Z}^d$ (for $d \geq 2$) with an asymmetric tie-breaking rule. This is a Markov process on the state space $\{-1,+1\}^{\mathbb{Z}^d}$ of "spin configurations" in which each vertex updates its spin to agree with a majority of its neighbors at the arrival times of a Poisson process. If a vertex has equally many $+1$ and $-1$ neighbors, then it updates its spin value to $+1$ with probability $q \in [0,1]$ and to $-1$ with probability $1-q$. The initial state of this Markov chain is distributed according to a product measure with probability $p$ for a spin to be $+1$. In this paper, we show that for any given $p>0$, there exist $q$ close enough to 1 such that a.s. every spin has a limit of $+1$. This is of particular interest for small values of $p$, for which it is known that if $q=1/2$, a.s. all spins have a limit of $-1$. For dimension $d=2$, we also obtain near-exponential convergence rates for $q$ sufficiently large, and for general $d$, we obtain stretched exponential rates independent of $d$. Two important ingredients in our proofs are refinements of block arguments of Fontes-Schonmann-Sidoravicius and a novel exponential large deviation bound for the Asymmetric Simple Exclusion Process.

Windowless Observation of Evaporation-Induced Coarsening of Au-Pt Nanoparticles in Polymer Nanoreactors.

The interactions between nanoparticles and solvents play a critical role in the formation of complex, metastable nanostructures. However, direct observation of such interactions with high spatial and temporal resolution is challenging with conventional liquid-cell transmission electron microscopy (TEM) experiments. Here, a windowless system consisting of polymer nanoreactors deposited via scanning probe block copolymer lithography (SPBCL) on an amorphous carbon film is used to investigate the coarsening of ultrafine (1-3 nm) Au-Pt bimetallic nanoparticles as a function of solvent evaporation. In such reactors, homogeneous Au-Pt nanoparticles are synthesized from metal-ion precursors in situ under electron irradiation. The nonuniform evaporation of the thin polymer film not only concentrates the nanoparticles but also accelerates the coalescence kinetics at the receding polymer edges. Qualitative analysis of the particle forces influencing coalescence suggests that capillary dragging by the polymer edges plays a significant role in accelerating this process. Taken together, this work (1) provides fundamental insight into the role of solvents in the chemistry and coarsening behavior of nanoparticles during the synthesis of polyelemental nanostructures, (2) provides insight into how particles form via the SPBCL process, and (3) shows how SPBCL-generated domes, instead of liquid cells, can be used to study nanoparticle formation. More generally, it shows why conventional models of particle coarsening, which do not take into account solvent evaporation, cannot be used to describe what is occurring in thin film, liquid-based syntheses of nanostructures.

Influence of Cationic Surfactant and Temperature on the Growth of ZnO Nanoparticles

AbstractIn this study, ZnO nanoparticles were synthesized from Zn(OAc)2 · 2H2O and NaOH in 2-propanol. Nucleation and growth are fast, and hence at longer times the particle size is controlled by coarsening. The optical properties were investigated as a function of time. X-ray diffraction (XRD) also has been used to characterize ZnO nanoparticles. The wavelength onset (λons) of the absorption spectra was used to estimate the particle size. The effect of the temperature and the surfactant on the growth process of ZnO nanoparticles was followed by UV-Vis spectrophotometer using the effective mass model. Particle growth was fitted using the commonly used linear coarsening model. The linear coarsening model is applied to calculate the rate constant of growth of ZnO nanoparticles. In all cases the linear coarsening produced good fits of the data. The results reveal that the temperature and the CTAB concentration play an important role in growth process of ZnO nanoparticles.

A simplified approach to predict performance degradation of a solid oxide fuel cell anode

The agglomeration of nickel (Ni) particles in a Ni-cermet anode is a significant degradation phenomenon for solid oxide fuel cells (SOFCs). This work aims to predict the performance degradation of SOFCs due to Ni grain growth by using a simplified approach. Accelerated aging of Ni-scandia stabilized zirconia (SSZ) as an SOFC anode is carried out at 900 °C and subsequent microstructural evolution is investigated every 100 h up to 1000 h using scanning electron microscopy (SEM). The resulting morphological changes are quantified using a two-dimensional image analysis technique that yields the particle size, phase proportion, and triple phase boundary (TPB) point distribution. The electrochemical properties of an anode-supported SOFC are characterized using electrochemical impedance spectroscopy (EIS). The changes of particle size and TPB length in the anode as a function of time are in excellent agreement with the power-law coarsening model. This model is further combined with an electrochemical model to predict the changes in the anode polarization resistance. The predicted polarization resistances are in good agreement with the experimentally obtained values. This model for prediction of anode lifetime provides deep insight into the time-dependent Ni agglomeration behavior and its impact on the electrochemical performance degradation of the SOFC anode.

Logarithmic speed-up of relaxation in A-B annihilation with exclusion.

We show that the decay of the density of active particles in the reaction A+B→0 in one dimension, with exclusion interaction, results in logarithmic corrections to the expected power law decay, when the starting initial condition (i.c.) is periodic. It is well known that the late-time density of surviving particles goes as t^{-1/4} with random initial conditions, and as t^{-1/2} with alternating initial conditions (ABABAB⋯). We show that the decay for periodic i.c.'s made of longer blocks (A^{n}B^{n}A^{n}B^{n}⋯) do not show a pure power-law decay when n is even. By means of first-passage Monte Carlo simulations, and a mapping to a q-state coarsening model which can be solved in the independent interval approximation (IIA), we show that the late-time decay of the density of surviving particles goes as t^{-1/2}[ln(t)]^{-1} for n even, but as t^{-1/2} when n is odd. We relate this kinetic symmetry breaking in the Glauber Ising model. We also see a very slow crossover from a t^{-1/2}[ln(t)]^{-1} regime to eventual t^{-1/2} behavior for i.c.'s made of mixtures of odd- and even-length blocks.