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Nanoscale multi-phase microstructures observed in multi-principal element alloys (MPEAs) such as $\rm AlMo_{0.5}NbTa_{0.5}TiZr$, $\rm Al_{0.5}NbTa_{0.8}Ti_{1.5}V_{0.2}Zr$, $\rm TiZrNbTa$, $\rm AlCoCrFeNi$ and $\rm Fe_{15}Co_{15}Ni_{20}Mn_{20}Cu_{30}$ that exhibit promising mechanical or functional properties may have evolved through spinodal-mediated phase transformation pathways (PTPs). The microstructures in such MPEA systems could be further engineered for targeted applications by appropriately designing the alloy composition and heat-treatment schedule. In this study, we investigate systematically how different heat treatment schedules such as single-step isothermal aging, two-step isothermal aging and continuous cooling alter the interplay among the various factors associated with alloy composition, such as volume fraction of individual phases, lattice misfit and modulus mismatch between the co-existing phases. We have determined the degree to which these factors influence significantly the spinodal-mediated PTPs and the corresponding microstructures by use of high-throughput phase-field simulations. In particular, we demonstrate that the microstructural topology (i.e., which phase forms the continuous matrix and which phase forms discrete precipitates) in the same MPEA having an asymmetric miscibility gap could be inverted simply by a continuous cooling heat treatment. Further, we reveal a rich variety of novel hierarchical microstructures that could be designed using two-step isothermal aging heat treatments in MPEA systems with symmetric or asymmetric miscibility gaps. These simulation results may shed light on novel microstructure design and engineering for the above-mentioned MPEA systems., Comment: Preprint submitted to Acta Materialia, 31 pages, 11 figures

Understanding the phase transformation pathways (PTPs) and microstructural evolution in multi-phase HEAs will aid alloy and process designs to tailor the microstructures for specific engineering applications. In this work, we study microstructural evolution in two-phase HEAs where a disordered parent phase separates into a mixture of two phases: an ordered phase ($\beta'$) + a disordered phase ($\beta$) upon cooling following two different PTPs: (i) congruent ordering followed by spinodal decomposition in the ordered phase and then disordering of one of the ordered phases, i.e., $\beta \rightarrow \beta' \rightarrow \beta_1' + \beta_2' \rightarrow \beta + \beta_2'$ and (ii) spinodal decomposition in the disordered phase followed by ordering of one of the disordered phases, i.e., $\beta \rightarrow \beta_1 + \beta_2 \rightarrow \beta_1+\beta'$. We systematically investigate the effects of equilibrium volume fractions of individual phases, free energy landscapes (in particular, the location of the critical point of the miscibility gap relative to the compositions of the final two equilibrium phases), and elastic modulus mismatch between the two equilibrium phases on the microstructural evolution of these HEAs. We focus on the following morphological characteristics: bi-continuous microstructures vs. precipitates + matrix microstructures, ordered matrix + disordered precipitates vs. disordered matrix + ordered precipitates, and the discreteness of the precipitate phase. This parametric study may aid in multi-phase HEA design for desired microstructures., Comment: Submitted to Acta Materialia

Researchers attributed the orderly arranged nanoscale phases observed in many multi-principal element alloys (MPEAs) to spinodal/spinodal-mediated phase transformation pathways. However, spinodal decomposition is not well understood in multicomponent systems. Although the theoretical background is available, CALPHAD databases were not used to explore the miscibility gap in MPEAs. In this work, we develop a CALPHAD framework utilizing the Hessian of free energy to study the stability of solid solutions in MPEAs. In particular, we utilize the geometry of higher dimensional Gibbs simplex in conjunction with the Hessian to calculate concentration modulations in early stages of spinodal decomposition. We apply this framework to a diverse set of multi-phase MPEAs that have been studied in the literature including TiZrNbTa (BCC/BCC), Fe15Co15Ni20Mn20Cu30 (FCC/FCC), Al0.5NbTa0.8Ti1.5V0.2Zr (BCC/B2), AlCo0.4Cr0.6FeNi (BCC/B2) and Al0.5Cr0.9FeNi2.5V0.2 (FCC/L12). We show that the MPEA systems are unstable only to certain concentration modulations which could be further explored to design microstructurally engineered alloys., Comment: Submitted to Scripta Materialia, 5 figures, 2 tables

The recently developed refractory multi-principle element alloy (MPEA), AlMo$_{0.5}$NbTa$_{0.5}$TiZr, shows an interesting microstructure with cuboidal precipitates of a disordered phase (${\beta}$, bcc) coherently embedded in an ordered phase (${\beta}'$, B2) matrix, unlike the conventional Ni-based superalloys where the ordered phase (${\gamma}'$, L12) is the precipitate phase and the disordered phase (${\gamma}$, fcc) is the matrix phase. It becomes critical to understand the phase transformation pathway (PTP) leading to this microstructure in order to tailor the microstructure for specific engineering applications. In this study, we first propose a possible PTP leading to the microstructure and employ the phase-field method to simulate microstructural evolution along the PTP. We then explore possible PTPs and materials parameters that lead to an inverted microstructure with the ordered phase being the precipitate phase and the disordered phase being the matrix phase, a microstructure similar to those observed in Ni-based superalloys. We find that in order to maintain the precipitates as highly discrete particles along these PTPs, the volume fraction of the precipitate phase needs to be smaller than that of the matrix phase and the elastic stiffness of the precipitate phase should be higher than that of the matrix phase., Comment: Preprint submitted to Applied Physics Letters (14 pages, 6 figures)

The Ni-base superalloy 718 is the most widely used material for turbomachinery in the aerospace industry and land-based turbines. Although the relationship between processing and the resulting properties is well known, an understanding of the specific deformation mechanisms activated across its application temperature range is required to create more mechanistically accurate property models. Direct atomic-scale imaging observations with high angle annular dark-field scanning transmission electron microscopy, complemented by phase-field modeling informed by generalized stacking fault surface calculations using density functional theory, were employed to understand the shear process of ${\gamma}''$ and ${\gamma}'/{\gamma}''$ co-precipitates after 1 \% macroscopic strain at lower temperature (ambient and $427 {\deg}C$). Experimentally, intrinsic stacking faults were observed in the ${\gamma}''$, whereas the ${\gamma}'$ was found to exhibit anti-phase boundaries or superlattice intrinsic stacking faults. Additionally, the atomically flat ${\gamma}'/{\gamma}''$ interfaces in the co-precipitates were found to exhibit offsets after shearing, which can be used as tracers for the deformation events. Phase-field modeling shows that the developing fault-structure is dependent on the direction of the Burgers vector of the $a/2 \langle110\rangle$ matrix dislocation (or dislocation group) due to the lower crystal symmetry of the ${\gamma''}$ phase. The interplay between ${\gamma}'$ and ${\gamma}''$ phases results in unique deformation pathways of the co-precipitate and increases the shear resistance. Consistent with the experimental observations, the simulation results indicate that complex shearing mechanisms are active in the low-temperature deformation regime and that multiple $a/2 \langle110\rangle$ dislocations of non-parallel Burgers vectors may be active on the same slip plane., Comment: 32 pages, 17+6 figures, submitted to Acta Materiallia

One of the advantages of the phase-field method (PFM) is its ability to incorporate elastic interactions that dominate solid-state processes including phase transformations and plastic deformation. As mechanical equilibrium is attained much faster than chemical equilibrium, the former should be used as a constraint explicitly in deriving the governing equations of time-evolution of PFM order parameters. Current models for elastically anisotropic and inhomogeneous media in the literature do not impose such a constraint in their governing equations. In particular, they ignore the dependence of the total strain on the order parameters while evaluating the variational derivative of the elastic energy (VDEE). There is no mathematical proof to support this treatment and the fundamental thermodynamic consistency of such a models could be challenged. In this work, we present a rigorous and physically transparent formulation of PFM governing equations for elastically anisotropic and inhomogeneous media in which the mechanical equilibrium is explicitly used as a constraint. From our formulation, we show that VDEE computed in the Leo-Lowengrub-Jou (1998) (LLJ) model is thermodynamically consistent. We also show that the Wang-Jin-Khachaturyan (2003) (WJK) model has significant error in VDEE calculation. We proposed a first-order correction term to the WJK model which nullifies the error in VDEE and makes the model thermodynamically consistent. Also, we computed the VDEE by a numerical method with-out using any assumptions and compared the numerical VDEE with the LLJ, WJK and modified WJK models., Comment: 13 pages, 4 figures

The origin of cube recrystallization texture in medium to high stacking-fault energy fcc metals has been debated for almost 70 years. Despite numerous experimental and simulation studies, many issues regarding the nucleation and growth of cube grains remain unresolved. Here we apply a full field crystal plasticity model utilizing a dislocation density based constitutive theory to study the deformation and texture evolution in copper (Cu) under plane strain compression. Additionally, we use the phase field method, along with a stochastic nucleation model, for static recrystallization simulations. Simulation results show that the volume fraction of the cube component during deformation decreases with increasing strain. Although cube grains are not stable during plane strain compression, some of the non-cube grains rotate towards cube and develop narrow cube bands near the grain boundary region. With increasing deformation, the cube component accumulates dislocation density faster than other texture components. High stored energy in the cube regions leads to preferential nucleation of cube grains during static recrystallization. These cube nuclei originate from the intergranular cube bands. Although the cube component has a clear nucleation advantage, none of the texture component appears to have a growth advantage. Instead, simulation results show that heterogeneous distribution of nuclei has a profound influence on the resulting grain size distribution. During recrystallization, a significant increase in cube volume fraction is observed mainly due to high nucleation frequency of cube grains.

Stacking fault energy (SFE) plays an important role in deformation mechanisms and mechanical properties of face-centered cubic (fcc) metals and alloys. In metastable fcc alloys, the SFEs determined from density functional theory (DFT) calculations and experimental methods often have opposite signs. Here, we show that the negative SFE by DFT reflects the thermodynamic instability of the fcc phase relative to the hexagonal close-packed one; while the experimentally determined SFEs are restricted to be positive by the models behind the indirect measurements. We argue that the common models underlying the experimental measurements of SFE fail in metastable alloys. In various concentrated solid solutions, we demonstrate that the SFEs obtained by DFT calculations correlate well with the primary deformation mechanisms observed experimentally, showing a better resolution than the experimentally measured SFEs. Furthermore, we believe that the negative SFE is important for understanding the abnormal behaviors of partial dislocations in metastable alloys under deformation. The present work advances the fundamental understanding of SFE and its relation to plastic deformations, and sheds light on future alloy design by physical metallurgy.

We show the correlation between nanoscale structural heterogeneity and mechanical property and glass forming ability of Zr-based metallic glasses (MGs). Detailed parameters of medium range ordering (MRO) that constitutes the structural heterogeneity, including the type, size, and volume fraction of MRO domains determined using 4-dimensional scanning transmission electron microscopy, directly correlate with the ductility and glass forming ability of Zr-Cu-Co-Al MGs. Mesoscale deformation simulation incorporating the experimentally determined MRO confirms that the diverse types and sizes of MRO can significantly influence the MGs' mechanical behavior.