The elastic energy of mixing for multi-component solid solutions is derived by generalizing Eshelby's sphere-in-hole model. By surveying the dependence of the elastic energy on the chemical composition and lattice misfit, we derive a lattice strain coefficient λ*. Studying several high-entropy alloys and superalloys, we propose that most solid solution multi-component alloys are stable when λ*<0.16, generalizing the Hume-Rothery atomic-size rule for binary alloys. We also reveal that the polydispersity index δ, frequently used for describing strain in multi-component alloys, directly represents the elastic energy (e) with e=qδ2, q being an elastic constant. Furthermore, the effects of (i) the number and (ii) the atomic-size distribution of constituting elements on the phase stability of high-entropy alloys were quantified. The present derivations and discussions open for richer considerations of elastic effects in high-entropy alloys, offering immediate support for quantitative assessments of their thermodynamic properties and studying related strengthening mechanisms. © 2021

Phase diagrams are the roadmaps for designing bulk phases. Similar to bulk, grain boundaries can possess various phases, but their phase diagrams remain largely unknown. Using a recently introduced density-based model, here we devise a strategy for computing multi-component grain boundary phase diagrams based on available bulk (CALPHAD) thermodynamic data. Fe-Mn-Cr, Fe-Mn-Ni, Fe-Mn-Co, Fe-Cr-Ni and Fe-Cr-Co alloy systems, as important ternary bases for several trending steels and high-entropy alloys, are studied. We found that despite its solute segregation enrichment, a grain boundary can have lower solubility limit than its corresponding bulk, promoting an interfacial chemical decomposition upon solute segregation. This is revealed here for the Fe-Mn-base alloy systems. The origins of this counter-intuitive feature are traced back to two effects, i.e., the magnetic ordering effect and the low cohesive energy of Mn solute element. Different aspects of interfacial phase stability and GB co-segregation in ternary alloys are investigated as well. We show that the concentration gradient energy contributions reduce segregation level but increase grain boundary solubility limit, stabilizing the GB against a chemical decomposition. Density-based grain boundary phase diagrams offer guidelines for systematic investigation of interfacial phase changes with applications to microstructure defects engineering. © 2021 Acta Materialia Inc.

The phase-like behavior of grain boundaries (GBs), recently evidenced in several materials, is opening up new possibilities in the design of alloy microstructures. In this context, GB phase diagrams are contributing to a predictive description of GB segregation and (interfacial) phase changes. The influence of chemo-mechanical solute-GB interactions on the GB phase diagram remains elusive so far. This is particularly important for multi-component alloys where the elastic interactions among solute atoms, of various sizes and bonding energies, can prevail, governing a complex co-segregation phenomenon. Recently, we developed a density-based model for GB thermodynamics that intrinsically accounts for GB elasticity in pure elements. In this work, we incorporate the homogeneous and heterogeneous elastic energies associated with the solutes into the density-based framework. We derive the multi-component homogeneous elastic energy by generalizing the continuum misfitting sphere model and extend it for GBs. The density-based free energy functional directly uses bulk CALPHAD thermodynamic data. The model is applied to binary and ternary Al alloys. We reveal that the elastic energy can profoundly affect the GB solubility and segregation behavior, leading to Cu segregation in otherwise Cu-depleted Al GBs. Consequently, GB segregation transition, i.e., a jump in the GB segregation as a function of alloy composition, is revealed in Al-Cu and Al-Cu-Mg alloy systems with implications for subsequent GB precipitation in these alloys. CALPHAD-informed elasticity-incorporated GB phase diagrams enable addressing a broader range of GB phenomena in engineering multi-component alloys. © 2021 Elsevier B.V.

The aluminum alloy 2618A is applied for engine components such as radial compressor wheels which operate for long time at elevated temperatures. This results in coarsening of the hardening precipitates and degradation in mechanical properties during the long-term operation, which is not taken into account in the current lifetime prediction models due to the lack of quantitative microstructural and mechanical data. To address this issue, a quantitative investigation on the evolution of precipitates during long-term aging at 190 °C for up to 25,000 h was conducted. Detailed transmission electron microscopy (TEM) was combined with Brinell hardness measurements and thorough differential scanning calorimetry (DSC) experiments. The results show that GPB zones and S-phase Al2CuMg grow up to < 1,000 h during which the GPB zones dissolve and S-phase precipitates form. For longer aging times, only S-phase precipitates coarsen, which can be well described using the Lifshitz–Slyozov–Wagner theory of ripening. A thorough understanding of the underlying microstructural processes is a prerequisite to enable the integration of aging behavior into the established lifetime models for components manufactured from alloy 2618A. © 2021, The Author(s).

The effects of anisotropic interfacial properties and heterogeneous elasticity on the growth and ripening of plate-like θ’-phase (Al2Cu) in Al-1.69 at.% Cu alloy are studied. Multi-phase-field simulations are conducted and discussed in comparison with aging experiments. The precipi-tate/matrix interface is considered to be anisotropic in terms of its energy and mobility. We find that the additional incorporation of an anisotropic interfacial mobility in conjunction with the elastic anisotropy result in substantially larger aspect ratios of the precipitates closer to the experimental observations. The anisotropy of the interfacial energy shows comparably small effect on the precip-itate’s aspect ratio but changes the interface’s shape at the rim. The effect of the chemo-mechanical coupling, i.e., the composition dependence of the elastic constants, is studied as well. We show that the inverse ripening phenomenon, recently evidenced for δ’ precipitates in Al-Li alloys (Park et al. Sci. Rep. 2019, 9, 3981), does not establish for the θ’ precipitates. This is because of the anisotropic stress fields built around the θ’ precipitates, stemming from the precipitate’s shape and the interaction among different variants of the θ’ precipitate, that disturb the chemo-mechanical effects. These results show that the chemo-mechanical effects on the precipitation ripening strongly depend on the degree of sphericity and elastic isotropy of the precipitate and matrix phases. © 2021 by the authors. Licensee MDPI, Basel, Switzerland.

For more than half a century, spinodal decomposition has been a key phenomenon in considering the formation of secondary phases in alloys. The most prominent aspect of the spinodal phenomenon is the lack of an energy barrier on its transformation pathway, offering an alternative to the nucleation and growth mechanism. The classical description of spinodal decomposition often neglects the influence of defects, such as grain boundaries, on the transformation because the innate ability for like-atoms to cluster is assumed to lead the process. Nevertheless, in nanocrystalline alloys, with a high population of grain boundaries with diverse characters, the structurally heterogeneous landscape can greatly influence the chemical decomposition behavior. Combining atom-probe tomography, precession electron diffraction and density-based phase-field simulations, we address how grain boundaries contribute to the temporal evolution of chemical decomposition within the miscibility gap of a Pt-Au nanocrystalline system. We found that grain boundaries can actually have their own miscibility gaps profoundly altering the spinodal decomposition in nanocrystalline alloys. A complex realm of multiple interfacial states, ranging from competitive grain boundary segregation to barrier-free low-dimensional interfacial decomposition, occurs with a dependency upon the grain boundary character. © 2021

The effectiveness of the mechanism of precipitation strengthening in metallic alloys de-pends on the shapes of the precipitates. Two different material systems are considered: tetragonal γ′′ precipitates in Ni-based alloys and tetragonal θ′ precipitates in Al-Cu-alloys. The shape formation and evolution of the tetragonally misfitting precipitates was investigated by means of experiments and phase-field simulations. We employed the method of invariant moments for the consistent shape quantification of precipitates obtained from the simulation as well as those obtained from the experiment. Two well-defined shape-quantities are proposed: (i) a generalized measure for the particles aspect ratio and (ii) the normalized λ2, as a measure for shape deviations from an ideal ellipse of the given aspect ratio. Considering the size dependence of the aspect ratio of γ′′ precipitates, we find good agreement between the simulation results and the experiment. Further, the precipitates’ in-plane shape is defined as the central 2D cut through the 3D particle in a plane normal to the tetragonal c-axes of the precipitate. The experimentally observed in-plane shapes of γ′′-precipitates can be quantitatively reproduced by the phase-field model. © 2021 by the authors. Licensee MDPI, Basel, Switzerland.

Systematic microstructure design requires reliable thermodynamic descriptions of each and all microstructure elements. While such descriptions are well established for most bulk phases, thermodynamic assessment of microstructure defects is challenging because of their individualistic nature. In this paper, a model is devised for assessing grain boundary thermodynamics based on available bulk thermodynamic data. We propose a continuous relative atomic density field and its spatial gradients to describe the grain boundary region with reference to the homogeneous bulk and derive the grain boundary Gibbs free energy functional. The grain boundary segregation isotherm and phase diagram are computed for a regular binary solid solution, and qualitatively benchmarked for the Pt-Au system. The relationships between the grain boundary's atomic density, excess free volume, and misorientation angle are discussed. Combining the current density-based model with available bulk thermodynamic databases enables constructing databases, phase diagrams, and segregation isotherms for grain boundaries, opening possibilities for studying and designing heterogeneous microstructures. © The Royal Society of Chemistry.

Grain boundary (GB) segregation has a substantial effect on the microstructure evolution and properties of polycrystalline alloys. The mechanism of nanoscale segregation at the various GBs in multicomponent alloys is of great challenge to reveal and remains elusive so far. To address this issue, we studied the GB segregation in a representative equiatomic FeMnNiCoCr high-entropy alloy (HEA) aged at 450 °C. By combining transmission Kikuchi diffraction, atom probe tomography analysis and a density-based thermodynamics modeling, we uncover the nanoscale segregation behavior at a series of well-characterized GBs of different characters. No segregation occurs at coherent twin boundaries; only slight nanoscale segregation of Ni takes place at the low-angle GBs and vicinal ς29b coincidence site lattice GBs. Ni and Mn show cosegregation of high levels at the general high-angle GBs with a strong depletion in Fe, Cr, and Co. Our density-based thermodynamic model reveals that the highly negative energy of mixing Ni and Mn is the main driving force for nanoscale cosegregation to the GBs. This is further assisted by the opposite segregation of Ni and Cr atoms with a positive enthalpy of mixing. It is also found that GBs of higher interfacial energy, possessing lower atomic densities (higher disorder and free volume), show higher segregation levels. By clarifying the origins of GB segregations in the FeMnNiCoCr HEA, the current work provides fundamental ideas on nanoscale segregation at crystal defects in multicomponent alloys. © 2020 authors.

Segregation to grain boundaries affects their cohesion, corrosion, and embrittlement and plays a critical role in heterogeneous nucleation. In order to quantitatively study segregation and low-dimensional phase separation at grain boundaries, here, we apply a density-based phase-field model. The current model describes the grain-boundary thermodynamic properties based on available bulk thermodynamic data, while the grain-boundary-density profile is obtained using atomistic simulations. To benchmark the performance of the model, Mn grain-boundary segregation in the Fe–Mn system is studied. 3D simulation results are compared against atom probe tomography measurements conducted for three alloy compositions. We show that a continuous increase in the alloy composition results in a discontinuous jump in the segregation isotherm. The jump corresponds to a spinodal phase separation at grain boundary. For alloy compositions above the jump, we reveal an interfacial transient spinodal phase separation. The transient spinodal phenomenon opens opportunities for knowledge-based microstructure design through the chemical manipulation of grain boundaries. The proposed density-based model provides a powerful tool to study thermodynamics and kinetics of segregation and phase changes at grain boundaries. © 2020, The Author(s).

In-situ TEM investigation of aging response in an Al–7.8 at.% Li was performed at 200 °C up to 13 hours. Semi-spherical δ′ precipitates growing up to an average radius of 7.5 nm were observed. The size and number of individual precipitates were recorded over time and compared to large-scale phase-field simulations without and with a chemo-mechanical coupling effect, that is, concentration dependence of the elastic constants of the matrix solid solution phase. This type of coupling was recently reported in theoretical studies leading to an inverse ripening process where smaller precipitates grew at the expense of larger ones. Considering this chemo-mechanical coupling effect, the temporal evolution of number density, average radius, and size distribution of the precipitates observed in the in-situ experiment were explained. The results indicate that the mechanism of inverse ripening can be active in this case. Formation of dislocations and precipitate-free zones are discussed as possible disturbances to the chemo-mechanical coupling effect and consequent inverse ripening process. © 2019, The Author(s).

Grain boundary segregation, embrittlement and phase nucleation are interconnected phenomena that are often treated separately, which is partly due to limitations of the current models to predict grain boundary segregation in non-ideal solid solutions. Here, a simple model is introduced to predict grain boundary segregation in solid solutions by coupling available bulk thermodynamic data with a mean-field description of the grain boundary character. The model is confronted with experimental results obtained in Fe-Mn alloys analysed by atom probe tomography. This model successfully predicts a first order transition or a discontinuous jump in the composition of the grain boundary which kinetically implies the formation of spinodal Mn fluctuations that tend to grow further with time within the segregated region. The increase in solute concentration at the grain boundary leads to an increase of the enthalpy of the boundary and to its embrittlement at lower temperatures. Once austenite is formed, the amount of segregated solute Mn on the grain boundaries is drastically reduced and the toughness of the grain boundary is increased. © 2019 Acta Materialia Inc.

For large-scale phase-field simulations, the trade-off between accuracy and computational cost as a function of the size and number of simulations was studied. For this purpose, a large reference representative volume element (RVE) was incrementally subdivided into smaller solitary samples. We have considered diffusion-controlled growth and early ripening of δ′ (Al3Li) precipitate in a model Al-Li system. The results of the simulations show that decomposition of reference RVE can be a valuable computational technique to accelerate simulations without a substantial loss of accuracy. In the current case study, the precipitate number density was found to be the key controlling parameter. For a pre-set accuracy, it turned out that large-scale simulations of the reference RVE can be replaced by simulating a combination of smaller solitary samples. This shortens the required simulation time and improves the memory usage of the simulation considerably, and thus substantially increases the efficiency of massive parallel computation for phase-field applications. © 2018 Elsevier B.V.

Phase-field studies of solid-state precipitation under strong chemo-mechanical coupling are performed and benchmarked against the existing analytical solutions. The open source software packages OpenPhase and DAMASK are used for the numerical studies. Solutions for chemical diffusion and static mechanical equilibrium are investigated individually followed by a chemo-mechanical coupling effect arising due to composition dependence of the elastic constants. The accuracy of the numerical solutions versus the analytical solutions is quantitatively discussed. For the chemical diffusion benchmark, an excellent match, with a deviation <0.1%, was obtained. For the static mechanical equilibrium benchmark Eshelby problem was considered where a deviation of 5% was observed in the normal component of the stress, while the results from the diffuse interface (OpenPhase) and sharp interface (DAMASK) models were slightly different. In the presence of the chemo-mechanical coupling, the concentration field around a static precipitate was benchmarked for different coupling coefficients. In this case, it is found that the deviation increases proportional to the coupling coefficient that represents the strength of coupling concentration and elastic constants. Finally, the interface kinetics in the presence of the considered chemo-mechanical coupling were studied using OpenPhase and a hybrid OpenPhase–DAMASK implementation, replacing the mechanical solver of OpenPhase with DAMASK's. The observed deviations in the benchmark studies are discussed to provide guidance for the use of these results in studying further phase transformation models and implementations involving diffusion, elasticity and chemo-mechanical coupling effect. © 2018 Elsevier B.V.

A coupling between diffusional and mechanical relaxation raised from composition-dependent elastic constants, and its effects on the evolution of precipitates with finite misfit strain are investigated. Inverse ripening has been observed where smaller precipitate grows at the expense of a larger one. This occurs due to fluxes generated under elastically-strained solute gradients around precipitates that scales with Rr6 where R and r are the precipitate radius and the radial coordinate, respectively. Both isotropic and anisotropic dependency of elastic constants on the composition were considered. The latter leads to the emergence of new patterns of elastic anisotropy and rearrangement of precipitates in the matrix. © 2017

The multi-phase-field approach is generalized to treat capillarity-driven diffusion parallel to the surfaces and phase boundaries, i.e., the boundaries between a condensed phase and its vapor and the boundaries between two or multiple condensed phases. The effect of capillarity is modeled via curvature dependence of the chemical potential whose gradient gives rise to diffusion. The model is used to study thermal grooving on the surface of a polycrystalline body. Decaying oscillations of the surface profile during thermal grooving, postulated by Hillert long ago but reported only in few studies so far, are observed and discussed. Furthermore, annealing of multi-nanoclusters on a deformable free surface is investigated using the proposed model. Results of these simulations suggest that the characteristic craterlike structure with an elevated perimeter, observed in recent experiments, is a transient nonequilibrium state during the annealing process. © 2017 American Physical Society.

The open-source software project OpenPhase allows the three-dimensional simulation of microstructural evolution using the multiphase field method. The core modules of OpenPhase and their implementation as well as their parallelization for a distributed-memory setting are presented. Especially communication and load-balancing strategies are discussed. Synchronization points are avoided by an increased halo-size, i.e. additional layers of ghost cells, which allow multiple stencil operations without data exchange. Load-balancing is considered via graph-partitioning and sub-domain decomposition. Results are presented for performance benchmarks as well as for a variety of applications, e.g. grain growth in polycrystalline materials, including a large number of phase fields as well as Mg–Al alloy solidification. Program summary Program Title: OpenPhase Program Files doi: http://dx.doi.org/10.17632/2mnv2fvkkk.1 Licensing provisions: GPLv3 Programming language: C++ Nature of problem: OpenPhase[1] allows the simulation of microstructure evolution during materials processing using the multiphase field method. In order to allow an arbitrary number of phase fields active parameter tracking is used, which can cause load imbalances in parallel computations. Solution method: OpenPhase solves the phase field equations using an explicit finite difference scheme. The parallel version of OpenPhase provides load-balancing using over-decomposition of the computational domain and graph-partitioning. Adaptive sub-domain sizes are used to minimize the computational overhead of the over-decomposition, while allowing appropriate load-balance. Additional comments including Restrictions and Unusual features: The distributed-memory parallelism in OpenPhase uses MPI. Shared-memory parallelism is implemented using OpenMP. The library uses C++11 features and therefore requires GCC version 4.7 or higher. [1] www.openphase.de © 2017 Elsevier B.V.

We investigate the evolution of large number of δ′ coherent precipitates from a supersaturated Al-8 at.% Li alloy using large-scale phase-field simulations. A chemomechanical cross-coupling between mechanical relaxation and diffusion is taken into account by considering the dependence of elastic constants of the matrix phase onto the local concentration of solute atoms. The elastic constants as a function of solute concentration have been obtained using density functional theory calculations. As a result of the coupling, inverse ripening has been observed where the smaller precipitates grow at the expense of the larger ones. This is due to size-dependent concentration gradients existing around the precipitates. At the same time, precipitates rearrange themselves as a consequence of minimization of the total elastic energy of the system. It is found that the anisotropy of the chemomechanical coupling leads to the formation of new patterns of elasticity in the matrix thereby resulting in new alignments of the precipitates. © 2017 American Physical Society.

Experimental and phase field studies of age hardening response of a high purity Al-4Cu-1Li-0.25Mn-alloy (mass %) during isothermal aging are conducted. In the experiments, two hardening phases are identified: the tetragonal θ' (Al2Cu) phase and the hexagonal T1 (Al2CuLi) phase. Both are plate shaped and of nm size. They are analyzed with respect to the development of their size, number density and volume fraction during aging by applying different analysis techniques in TEM in combination with quantitative microstructural analysis. 3D phase-field simulations of formation and growth of θ' phase are performed in which the full interfacial, chemical and elastic energy contributions are taken into account. 2D simulations of T1 phase are also investigated using multi-component diffusion without elasticity. This is a first step toward a complex phase-field study of T1 phase in the ternary alloy. The comparison between experimental and simulated data shows similar trends. The still unsaturated volume fraction indicates that the precipitates are in the growth stage and that the coarsening/ripening stage has not yet been reached. © 2017 by the authors.

Zener drag and pinning in composites reinforced with cylindrical particles is investigated using three-dimensional phase-field simulations. Detailed systematic studies clarify the effect of relative orientation of the particle and length/diameter ratio on the kinetics of drag. It is shown that a combination of local equilibrium at junctions in contact with the particles, initial driving force of the migrating grain boundaries, and configuration of the particles within the polycrystalline matrix determine the intensity and persistence of drag and pinning effects. © 2015 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

The classical mean field approach for normal grain growth in polycrystalline materials is revisited. We re-drive and study possible self-similar solutions and show that the grain size distribution can be determined only by the geometry of neighbouring grains for any given configuration. In three dimensions, it is shown that a single index 〈R〉2/〈R2〉 can represent the geometrical characteristic of grains and has a one-to-one relationship with the mean field parameter γ. We reinvestigate the results of our recent phase-field study [Darvishi Kamachali R, Steinbach I. Acta Mater 2012;60:2719] in the light of new analytical results and found a value γ≈3.5-3.2 for the stable regime. © 2015 Acta Materialia Inc. All rights reserved.

Experimental and phase-field studies are performed to investigate mechanisms of preferential growth which lead to improved formability in AZ31 Mg sheets. A compression/annealing treatment is specialized to modify the initial texture in thin sheets. The texture and stress states of materials are studied via electron back scattered diffraction (EBSD) technique before and after annealing. Using the EBSD data on microstructure and residual stresses, a phase-field model is constructed to simulate the texture evolution after initial compression. The results suggest that the residual-stresses induced by in-plane compression are the main driving force for recrystallization and grain growth. The inhomogeneous stress distribution leads to preferential growth of {21¯1¯0} texture along the normal to the sheet, which are at lowest stress state, at the expense of initial basal texture. Limited mobility of twin boundaries changes the mixture of textures but the non-basal textures are still preferred. The formability tests confirm a significant enhancement of the final product compared to as-received sheets. © 2015 Elsevier B.V. All rights reserved.

Formation of the Ni4Ti3 precipitate has a strong effect on the shape memory properties of NiTi alloys. In this work, growth of this precipitate is studied using phase-field modelling and density functional theory (DFT) calculations. Using first-principles calculations, the composition-dependent stability and elastic properties of the B2 phase are obtained. Composition-dependent elastic constants are incorporated into our phase-field model to investigate the interplay between stress and concentration fields around the precipitate. The model introduces a source of diffusion due to mechanical relaxation which is accompanied by local softening/hardening of the B2 phase. The results are discussed in light of previous experimental and simulation studies. © 2014 IOP Publishing Ltd.

We demonstrate that the distortion of a crystal, caused by secondary phase precipitates, can stabilize a solutal gradient around the precipitate. The gradient persists in the quasi-static state stabilized by the gradient of the elastic energy around the precipitate. The peak concentration at the interface between precipitate and matrix hereby is independent of the radius of the precipitate and no mechanism of ripening is active in an arrangement of precipitates of different size. The model offers an explanation of experimental observations of the anomalous stability of nano-precipitates in Al-Cu. © 2013 Taylor & Francis.

The characteristics of 3-D grain growth are investigated by a topological analysis of phase-field simulation results compared with theoretical mean-field theories. We found that the size distribution of the grains starting from an arbitrary narrow distribution crosses the self-similar Hillert distribution, and ends in a distribution with relatively longer tails of large grains in which the central peak shifted towards smaller grain size. The distribution of topological classes, as characterized by the number of facets per grain, is found to be time-invariant for the process as a whole. The obtained shape function is in good agreement with the analytical distribution function derived based on the average N-hedron model [Rios PR, Glicksman ME. Act Mater 2008;56:1165]. The volumetric growth rate per topological class also correlates well with the analytical approach obtained by Mullins [Mullins WW. Acta Mater 1956;3:900]. The relationship between grain size and its shape, however, deviates from theoretical predictions. © 2011 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Nanocrystalline tricalcium phosphate powder was synthesized via the solution- precipitation method followed by heat treatment in order to achieve phase evolution, which was then studied by XRD and TEM techniques. The crystallites sizes were estimated by the Scherrer method and results were confirmed by TEM micrographs. The experimental observations showed that nanocrystalline tricalcium phosphate can be successfully prepared from raw materials by the precipitation technique. This technique is a competitive method for nanocrystalline tricalcium phosphate synthesis compared to other techniques. Moreover, a simple kinetic growth investigation was performed on the nanocrystalline growth process during heat treatment. Results have shown growth rate to increase exponentially with temperature and the growth rate constants to increase with time. The average activation energies of tricalcium phosphate grain growth obtained by this method were 84.78 and 134.38 KJ/mol. © Wroclaw University of Technology.

In this study, the authors first review the previously developed, thermodynamics-based theory for size dependency of the cohesion energy of free-standing spherically shaped Al nanoparticles. Then, this model is extrapolated to the cubic and truncated octahedron Al nanoparticle shapes. A series of computations for Al nanoparticles with these two new shapes are presented for particles in the range of 1-100 nm. The thermodynamics computational results reveal that there is a second critical size around 1.62 and 1 nm for cubes and truncated octahedrons, respectively. Below this critical size, particles behave as if they consisted only of surface-energy-state atoms. A molecular dynamics simulation is used to verify this second critical size for Al nanoparticles in the range of 1-5 nm. MD simulation for cube and truncated octahedron shapes shows the second critical point to be around 1.63 and 1.14 nm, respectively. According to the modeling and simulation results, this second critical size seems to be a material property characteristic rather than a shape-dependent feature.

In this work, multi-phase field and molecular dynamics simulations have been used to investigate nanoscale grain growth mechanisms. Based on experimental observations, the combination of grain boundary expansion and vacancy diffusion has been considered in the multi-phase field model. The atomistic mechanism of boundary movement and the free volume redistribution during the growth process have been investigated using molecular dynamics simulations. According to the multi-phase field results, linear grain growth in nanostructured materials at low temperature can be explained by vacancy diffusion in the stress field around the grain boundaries. Molecular dynamics simulations confirm the observation of linear grain growth for nanometresized grains. The activation energy of grain boundary motion in this regime has been determined to be of the order of onetenth of the self-diffusion activation energy, which is consistent with experimental data. Based on the simulation results, the transition from linear to normal grain growth is discussed in detail and a criterion for this transition is proposed. © Carl Hanser Verlag GmbH & Co. KG.