Energetics Of Alloys Research Articles

Bayesian active machine learning for Cluster expansion construction

The Cluster expansion (CE) is a powerful method for representing the energetics of alloys from a fit to first principles energies. However, many common fitting methods are computationally demanding and do not provide the guarantee that the system’s ground states are preserved. This paper demonstrates the use of an efficient implementation of a Bayesian algorithm for cluster expansion construction that ensures all the input structural energies are fitted perfectly while reducing computational cost. The method incorporates an active learning scheme that searches for new optimal structures to include in the fit. As performance tests, we calculate the phase diagram of the Fe–Ir system and study the short range order in an equimolar MoNbTaVW system. The new method has been integrated into the Alloy Theoretic Automated Toolkit (ATAT).

Database-driven semigrand canonical Monte Carlo method: Application to segregation isotherm on defects in alloys.

The application of existing semigrand canonical ensemble Monte Carlo algorithms to alloys requires the chemical potential difference values between pairs of atomic species in the alloys as inputs. However, finding the appropriate values for a target system at a desired temperature and bulk composition is a time-consuming task consisting of multiple test runs to determine the chemical potential differences. This problem becomes more serious when dealing with systems containing three or more atomic species, such as medium- and high-entropy alloys, due to the increase of the number of chemical potential differences that need to be calculated. Here we propose a method for sampling from the semigrand canonical ensemble that relies on energy databases acting as an external atomic reservoir at the desired temperature and composition. Given these energy databases, the desired bulk composition and corresponding chemical potential differences can be satisfied in a "single" Monte Carlo simulation. Moreover, the energy databases shed light on the underlying energetics of alloys, reflecting their local chemical ordering. We demonstrate the validity of this method using analyses of segregation isotherms at grain boundaries and dislocations in two alloy systems: Fe-1-at.-%-Si and NiCoCr medium-entropy alloy. We also discuss the possibly relevant information contained in such energy databases.

Ab initio based modeling of interfacial segregation at Cu-rich precipitates in Fe–Cu–Ni alloys

Ni segregation at the interfaces of Cu-rich precipitates in Fe–Cu–Ni alloys is investigated using ab initio based models with Monte Carlo simulations and density functional theory calculations. It is demonstrated that Ni segregation decreases with increasing temperature and increases in proportion to global Ni concentration. The predicted Ni segregation profiles are in quantitative agreement with experiments. Analysis show that both the temperature dependent alloy energetics and the configurational entropy are important for predicting Ni segregation. The results also reveal that Ni segregation reduces the interfacial structure stability, especially at low temperatures and with high global Ni concentrations, which could be used to explain the Ni enhanced hardening in Fe–Cu–Ni alloys.

General method for incorporating CALPHAD free energies of mixing into phase field models: Application to the α-zirconium/δ-hydride system

This paper presents a general method of incorporating CALPHAD-based free energies into a phase field model. While CALPHAD-based free energy descriptions provide realistic energetics of alloys, their formulations may pose numerical difficulties for phase field simulations. Specifically, the free energies of mixing for different phases may not necessarily be defined over the same concentration ranges, and their derivatives may exhibit highly nonlinear behavior. The method presented approximates the free energies of mixing using piecewise functions to eliminate the aforementioned characteristics while largely retaining the free energy values and the first and second derivative information, which affect the thermodynamic and kinetic behavior of the system. The method is verified by planar interface simulations of the α-zirconium/δ-hydride system. The phase fractions and compositions obtained from the phase field simulations are compared to the values calculated from the common tangent construction on the original free energies. The results indicate a high degree of accuracy.

Short-range correlations in binary alloys: Spin model approach to AgcAu1−c and AgcPd1−c

Short-range correlations in Ag–Au and Ag–Pd alloys are investigated by analyzing the ab initio total energy of face centered cubic (fcc) based random AgcAu1−c and AgcPd1−c. Since the information on the atomic interactions is incorporated in the energetics of alloys it is possible with a suitable model, Bethe–Peierls–Weiss model is used in the present work, to invert the problem, i.e. to obtain information on the short-range correlation from the total energy of a random system. As an example we demonstrate how site correlations can be extracted from random alloy data. Bethe–Peierls–Weiss model predicts negative (positive) first neighbor correlator for substitutional fcc Ag–Au and (Ag–Pd) alloys at low temperature which can be related to the optimal structures of Ag0.5Au0.5 (and Ag0.5Pd0.5).

Surface tension of Sn-Ag and Sn-Zn binary liquid alloy

Thermodynamic models are used to correlate the bulk and surface properties of the mixing of Sn-Ag and Sn-Zn binary liquid alloys. The energetics of alloys which are used to study the surface composition and surface tension of Sn-Ag and Sn-Zn binary liquid alloys. Our investigation indicates the surface is enormously rich with-Sn concentration throughout the composition of the alloy.

Alloy microstructures with atomic size effects: A Monte Carlo study under the lattice statics formalism

We present in this paper atomic scale Monte Carlo simulations of microstructure evolutions in the presence of atomic size mismatch, performed on a rigid lattice. The lattice statics formalism is used to obtain effective pair interactions (EPIs) from a continuous empirical description of alloy energetics that includes elastic relaxations in the harmonic approximation. These EPIs are long-ranged and are introduced in a Monte Carlo scheme to compute alloy properties. The influence of atomic size mismatch on EPIs as well as on microstructure evolutions is investigated in the case of a model binary fcc alloy. The microstructure evolution with atomic size mismatch operates mainly by shape changes and by the development of spatial correlations between precipitates. Our aim is also to provide a method to include elastic effects at the atomic scale starting from material macroscopic data, without any requirement of realistic continuous interatomic potential calibration. A commonly found approach in microscopic mean-field simulations is to use, at the atomic scale, effective elastic interactions inherited from continuum linear elasticity. We derive here a different and original method: the discrete lattice approach (DLA) that both converges to the continuum theory of elasticity (long-range aspect) and takes into account the discreteness of the lattice. It allows us to correctly reproduce the whole range of elastic interactions, down to the atomic scale, successfully improving the simple use of continuum elasticity, without the prior knowledge of any interatomic potential. Finally, the accuracy of the DLA is quantitatively verified through a detailed comparison between the microstructural evolutions obtained with this approach and the original lattice statics approach.

Energetics of quaternary III-V alloys described by incorporation and clustering of impurities

The energetics of alloy formation is generally modeled either by explicit sampling of the possible alloy configurations or by considering only noninteracting impurities in the dilute limit. We describe a model that bridges the two approaches by taking into account the thermodynamic probability to form small clusters by association of impurities, thereby extending the validity of the impurity model to higher concentrations. Since we express the alloy energetics in terms of pair and cluster binding energies there is no need for computationally intensive sampling over the full configuration space. The application to the ${\text{Ga}}_{1\ensuremath{-}x}{\text{In}}_{x}{\text{P}}_{1\ensuremath{-}y}{\text{N}}_{y}$ highlights the importance of short-range ordering due to ``small atom/large atom'' correlation in such quaternary III-V alloys.

First-principles calculations of structural energetics of Cu–TM (TM = Ti, Zr, Hf) intermetallics

A systematic and comprehensive study of cohesive properties of Cu–TM (TM = Ti, Zr, Hf) intermetallics is carried out using a first-principles method. Specifically, the total energies and equilibrium cohesive properties of 95 intermetallics in the Cu–TM (TM = Ti, Zr, Hf) systems are calculated employing electronic density-functional theory (DFT), ultrasoft pseudopotentials and the generalized gradient approximation for the exchange-correlation energy. The intermetallic phases considered in our first-principles investigation are classified as stable, metastable and virtual types. The concentration dependence of the heat of formation (Δ E f) in the Cu–Ti system is only slightly asymmetric, while in the Cu–Zr and Cu–Hf systems they are distinctly asymmetric, being skewed towards the Cu-rich side with a minimum in Δ E f at Cu 10TM 7. Based on the observed differences between ab initio and calorimetric heat of formation, we conclude that additional careful experiments are needed to validate ab initio alloy energetics. We also note that the calphad model parameters representing alloy energetics vary significantly from one assessment to another in these systems. For the stable intermetallics, the calculated zero-temperature lattice parameters agree to within ±1% of experimental data at ambient temperature. For the stable phases with unit cell-internal degree(s) of freedom, the results of ab initio calculations show a good agreement when such data are available from X-ray and other diffraction results. For intermetallic compounds where no such experimental data are available, we provide optimized unit cell geometries which may be verified in future experiments. For most structures we also provide zero-temperature bulk moduli and their pressure derivatives, as defined by the equation of state. The bonding between Cu and Zr is discussed based on the analyses of density of states and bonding charge densities in Cu 5Zr and CuZr.

First-principles calculation of structural energetics of Al–TM (TM = Ti, Zr, Hf) intermetallics

The total energies and equilibrium cohesive properties of 69 intermetallics in the Al–TM (TM = Ti, Zr and Hf) systems are calculated from first-principles employing electronic density-functional theory, ultrasoft pseudopotentials and the generalized gradient approximation. This work has been undertaken to investigate systematics in Al–TM alloying energetics, and to augment available calorimetric data for enthalpies of formation in support of the development of accurate multicomponent thermodynamic databases for these technologically interesting systems. The accuracy of our calculations is assessed through comparisons between theoretical results and experimental measurements (where available) for lattice parameters, elastic properties and formation energies. The concentration dependence of the heats of formation for all three binary systems are very similar, being skewed towards the Al-rich side with a minimum around Al 2TM. In all three binary Al–TM systems, the calculated zero-temperature intermetallic formation energies generally agree well, within a few kJ/mol, with calorimetric data obtained by direct reaction synthesis. This level of agreement suggests high accuracy for the calculated enthalpies of formation reported for structures where no such measured data are currently available. Several intermetallic phases which have previously been suggested to be stabilized by impurity effects are indeed found to be higher energy states compared to their stable counterparts. It is noted that the CALPHAD model parameters representing alloy energetics vary significantly from one assessment to another in these systems, demonstrating the clear need for additional enthalpic data for all competing phases to derive unique thermodynamic model parameters. For the stable intermetallics, the calculated zero-temperature lattice parameters agree to within ±1% of experimental data at ambient temperature. For the stable phases with unit cell-internal degree(s) of freedom, the results of ab initio calculations show excellent agreement when compared with data obtained by rigorous structural analysis of X-ray and other diffraction results. For intermetallic compounds where no such experimental data is available, we provide optimized unit cell geometries. For most structures we also provide zero-temperature bulk moduli and their pressure derivatives, as defined by the equation of state.

BFS Simulation and Experimental Analysis of the Effect of Ti Additions on the Structure of NiAl

The Bozzolo–Ferrante–Smith (BFS) method for alloy energetics is applied to the study of ternary additions to NiAl. A description of the method and its application to alloy design is given. Two different approaches are used in the analysis of the effect of Ti additions to NiAl. First, a thorough analytical study is performed, where the energy of formation, lattice parameter and bulk modulus are calculated for a large number of possible atomic distributions of Ni, Al and Ti. Substitutional site preference schemes and formation of precipitates are thus predicted and analyzed. The second approach used consists of the determination of temperature effects on the final results, as obtained by performing a number of large-scale numerical simulations using the Monte Carlo–Metropolis procedure and BFS for the calculation of the energy at every step in the simulation. The results indicate a sharp preference of Ti for Al sites in Ni-rich NiAl alloys and the formation of ternary Heusler precipitates beyond the predicted solubility limit of 5 at. % Ti. Experimental analysis of three Ni–Al–Ti alloys confirms the theoretical predictions.

First-principles calculations of bulk and interfacial thermodynamic properties for fcc-based Al-Sc alloys

The configurational thermodynamic properties of fcc-based Al-Sc alloys and coherent ${\mathrm{A}\mathrm{l}/\mathrm{A}\mathrm{l}}_{3}\mathrm{Sc}$ interphase-boundary interfaces have been calculated from first principles. The computational approach used in this study combines the results of pseudopotential total-energy calculations with a cluster-expansion description of the alloy energetics. Bulk and interface configurational-thermodynamic properties are computed using a low-temperature-expansion technique. Calculated values of the {100} and {111} ${\mathrm{A}\mathrm{l}/\mathrm{A}\mathrm{l}}_{3}\mathrm{Sc}$ interfacial energies at zero temperature are, respectively, 192 and $226{\mathrm{m}\mathrm{J}/\mathrm{m}}^{2}.$ The temperature dependence of the calculated interfacial free energies is found to be very weak for {100} and more appreciable for {111} orientations; the primary effect of configurational disordering at finite temperature is to reduce the degree of crystallographic anisotropy associated with calculated interfacial free energies. The first-principles-computed solid-solubility limits for Sc in bulk fcc Al are found to be underestimated significantly in comparison with experimental measurements. It is argued that this discrepancy can be largely attributed to nonconfigurational contributions to the entropy which have been neglected in the present thermodynamic calculations.

Embedded-atom-method effective-pair-interaction study of the structural and thermodynamic properties of Cu-Ni, Cu-Ag, and Au-Ni solid solutions.

The structural and thermodynamic properties of Cu-Ni, Cu-Ag, and Au-Ni solid solutions have been studied using a computational approach which combines an embedded-atom-method (EAM) description of alloy energetics with a second-order-expansion (SOE) treatment of compositional and displacive disorder. It is discussed in detail how the SOE approach allows the EAM expression for the energy of a substitutional alloy to be cast in the form of a generalized lattice-gas Hamiltonian containing effective pair interactions with arbitrary range. Furthermore, we show how the SOE-EAM method can be combined with either mean-field or Monte Carlo statistical mechanics techniques in order to calculate short-range-order (SRO) parameters, average nearest-neighbor bond lengths, and alloy thermodynamic properties which include contributions from static displacive relaxations and dynamic atomic vibrations. We demonstrate that the contributions to alloy heats of mixing arising from displacive relaxations can be sizeable, and that the neglect of these terms can lead to large overestimations of calculated phase-transition temperatures. The effects of vibrational free-energy contributions on the results of composition-temperature phase diagram calculations are estimated to be relatively small for the phase-separating alloy systems considered in this study.It is shown that within the SOE approach displacive effects can act only to displace the peak in the Fourier-transformed SRO parameter away from Brillouin-zone-boundary special points and towards the origin. Consistent with this result, we show that the unusual SRO observed in diffuse scattering experiments for Au-Ni solid solutions can be understood as arising from a competition between chemical and displacive driving forces which favor ordering and clustering, respectively. \textcopyright{} 1996 The American Physical Society.

Cluster expansions of alloy energetics in ternary intermetallics.

Cluster expansions of alloy energetics in ternary intermetallics.

Theoretical modelling of AFM for bimetallic tip-substrate interactions

Recently, a new technique for calculating the defect energetics of alloys based on equivalent crystal theory has been developed. This new technique successfully predicts the bulk properties for binary alloys as well as segregation energies in the dilute limit. We apply this method for the calculation of energy and force as a function of separation of an AFM tip and substrate. This study was done for different combinations of tip and sample materials. The validity of the universality discovered for same metal interfaces is examined for the case of different metal interactions.

Method for calculating alloy energetics.

A semiempirical method for the computation of alloy energies is introduced. It is based on the equivalent-crystal theory of defect-formation energies in elemental solids. The method is both simple and accurate. Heats of formation as a function of composition are computed for some binary alloys of Cu, Ni, Al, Ag, Pd, Pt, and Au using the heats of solution in the dilute limit as experimental input. The separation of heats into strain and chemical components helps in understanding the energetics. In addition, lattice-parameter contractions seen in solid solutions of Ag and Au are accurately predicted. Good agreement with experiment is obtained in all cases.

Extension of a New Semiempirical Method (BFS) and the Study of Ground State Properties of Binary Alloys

ABSTRACTWe extend the method of Bozzolo, Ferrante and Smith (BFS) for the study of alloy energetics to include a description of the local environment in specific ordered structures. The concept of bond-diagrams is introduced and applied to fcc binary compounds. A simple example of the parameterization of the bond-diagrams is done with reference to available first-principles calculations of Ni-Pt ordered alloys.