Warren-Cowley Short-range Order Parameters Research Articles

Atomistic simulations reveal strength reductions due to short-range order in alloys

A theory for strengthening for multicomponent non-dilute alloys possessing short-range order (SRO) has recently been developed. The theory predicts that, in addition to a well-known athermal strengthening, there is a notable effect of SRO on the solute–dislocation interactions that can decrease or increase the strength relative to a random alloy. Here, carefully designed atomistic simulations in a model binary NbW alloy are used to demonstrate that alloy strength due to solute–dislocation interactions can be increased or decreased, depending on the SRO and consistent with theoretical predictions. Specifically, SRO is introduced using fictitious solute–solute interactions in an alloy system with very small true solute–solute interactions, and the Nudged Elastic Band (NEB) method is then used to compute the energy barriers for edge dislocation motion for various levels of SRO. Energy barriers, and hence alloy strengths, can be decreased when the Warren-Cowley SRO parameters are negative (attraction of unlike solutes). The theoretical predictions for the same system are in reasonable quantitative agreement with the simulation results. These findings demonstrate the unexpected possibility of reduced strength due to SRO and also further validate the analytical theory as a tool to guide alloy design.

Modeling solute-grain boundary interactions in a bcc Ti-Mo alloy using density functional theory

Solute segregation in alloys is a key phenomenon which affects various material characteristics such as embrittlement, grain growth and precipitation kinetics. In this work, the segregation energies of Y, Zr, and Nb to a Σ5 grain boundary in a bcc Ti-25 at % Mo alloy are determined using density functional theory (DFT) calculations. A systematic approach is laid out by computing the solution energy distributions in the bulk alloy using Warren-Cowley short-range order parameters to find a representative bulk-solute reference energy. Additionally, different scenarios are considered when a solute atom replaces different sites in terms of their local Ti-Mo chemistry at the GB plane to calculate the distribution of segregation energies. The solute segregation to a Mo site at the GB plane is preferred rather than to a Ti site. Further analysis shows that these segregation energy trends can be rationalized based on a primarily elastic interaction. Thus the segregation energies scale with the solute size such that Y has the largest segregation energies followed by Zr and Nb.

Accurate prediction of chemical short-range order and its effect on thermodynamic, structural, and electronic properties of disordered alloys: Exemplified in Cu3Au

Electronic-structure methods based on density-functional theory (DFT) were used to directly quantify the effect of chemical short-range order (SRO) on thermodynamic, structural, and electronic properties of archetypal face-centered-cubic (fcc) Cu3Au alloy. We show that SRO can be tuned to alter bonding and lattice dynamics (i.e., phonons) and detail how these properties are changed with SRO. Thermodynamically favorable SRO improves phase stability of Cu3Au from − 0.0343 eV-atom−1 to −0.0682 eV-atom−1. We use DFT-based linear-response theory to predict SRO and its electronic origin, and accurately estimate the transition temperature, ordering instability (L12), and Warren-Cowley SRO parameters, observed in experiments. The accurate prediction of real-space SRO gives an edge over computationally and resource intensive approches such as Monte-Carlo methods or experiments, which will enable large-scale molecular dynamic simulations by providing supercells with optimized SRO. We also analyze phonon dispersion and estimate the vibrational entropy changes in Cu3Au (from 9kB at 300–6kB at 100 K). We establish from SRO analysis that exclusion of chemical interactions may lead to a skewed view of true properties in chemically complex alloys. The first-principles methods described here are applicable to any arbitrary complex solid-solution alloys, including multi-principal-element alloys, so hold promise for designing technologically useful materials.

Models of configurationally-complex alloys made simple

We present a Python package for the efficient generation of special quasi-random structures (SQS) for atomic-scale calculations of disordered systems. Both, a Monte-Carlo approach or a systematic enumeration of structures can be used to carry out optimizations to ensure the best optimal configuration is found for given cell size and composition. We present a measure of randomness based on Warren-Cowley short-range order parameters allowing for fast analysis of atomic structures. Hence, optimal structures are found in a reasonable time for several dozens or even hundreds of atoms. Both SQS optimizations and analysis of structures can be carried out via a command-line interface or a Python API. Additional features, such as optimization towards partial ordering or independent sublattices allow the generation of atomistic models of modern complex materials. Moreover, hybrid parallelization, as well as distribution of vacancies, are supported. The output data format is compatible with ase, pymatgen and pyiron packages to be easily embeddable in complex simulation workflows. Program summaryProgram title: sqsgeneratorCPC Library link to program files:https://doi.org/10.17632/m2sb3wzcvc.1Developer's repository link:https://github.com/dgehringer/sqsgeneratorLicensing provisions: MITProgramming language: Python, C++Supplementary material:https://sqsgenerator.readthedocs.ioNature of problem: Many technological relevant materials, exhibit a crystalline disorder. Within atomistic modelling approaches such as Density Functional Theory (DFT) or Molecular Dynamics, disorder is modelled with a cell containing a (small) finite set of atoms. Such an atomic configuration is usually found by enumerating structures. However, since configurational space is growing exponentially efficient tools are needed to sample it properly.Solution method: Efficient quantification of disorder using a generalization of Warren-Cowley Short Range Order (WC-SRO) parameters [1,2]. By either a Monte-Carlo approach or systematic enumeration, optimal structures can be found. The software is distributed as a Python package offering a command line interface. Core parts are written in C++ and exhibit shared (OpenMP) and distributed (MPI) memory parallelism. For embedding into complex simulation workflows, the tool exposes a Python API to integrate into popular packages such as ase [3], pymatgen [4] or pyiron [5].

Theoretical assessment for the mixing properties of AlMg liquid alloys at 1073K

The alloying behavior of AlMg alloy in the liquid form at 1073 K has been theoretically investigated in the framework of four-parameter model which is based on Maclaurin series. The analytical expressions for thermodynamic functions such as excess free mixing energy, free mixing energy, enthalpy of mixing and entropy of mixing and microscopic functions such as concentration fluctuations at the long wavelength limit and Warren-Cowley chemical short range order parameter have been derived. These expressions have been used to compute the excess Gibbs free energy of mixing, Gibbs free energy of mixing, activity, enthalpy(heat) of mixing, excess entropy of mixing, entropy of mixing, concentration fluctuations in long wavelength limit and Warren-Cowley short range order parameters of AlMg liquid alloys at 1073 K. The investigation shows the excellent concurrence between the experimental and theoretical measurements of the mixing properties of AlMg liquid alloys at 1073 K. Interaction parameters of energy depends on temperature.

Statistical mechanical studies of Al rich Al–Cu melts

We evaluate the microscopic correlation functions in Al1−xCux melts (x= 0.10, 0.17, 0.25, 0.33 and 0.40) in the attractive and repulsive region of the square well (SW) potential under the mean spherical model (MSM) principle. We derive the temperature and concentration dependent transport coefficients through the computed structural functions and verify the Dzugutov’s scaling law in Al1−xCuxliquid alloys. The liquid Al-Cu alloys follow the Stokes–Einstein relation especially in Al-rich melts. The concentration–concentration fluctuations in the long-wavelength limit, i.e. SCC(0) and the Warren–Cowley short-range order parameters, α1 have been computed which explains the compound forming behavior in this melts. SCC(0) is also employed to determine the thermodynamic factor for calculating inter-diffusion coefficients from Darken’s approach in liquid alloys. The temperature variation of diffusion coefficients are being applied for the computation of activation energy in liquid Al-Cu alloys by following Arrhenius equation. The theoretically evaluated activation energies of the alloys are in fair agreement with reported values. The composition dependent surface tension is determined and compared with the available experimental results which are in satisfactory agreement. Thus we establish a closer relationship between structure, transport and surface properties in Al1−xCuxmelts without using any adjusting parameters.

Ordering effects in 2D hexagonal systems of binary and ternary C-B-N alloys

We present theoretical study of ordering phenomena in binary C1-xBx,C1-xNx, and ternary C1-2xBxNx alloys forming two-dimensional, graphene-like systems. In order to find equilibrium distribution of different species corresponding to minima of the energy, we use Monte Carlo approach in Metropolis regime. We employ empirical Tersoff potential to account for the interactions between atoms and to calculate the total energy. We model alloys considering large systems containing up to 20 000 atoms with various N and B concentrations. For quantitative description of ordering phenomena in the studied alloys, we determine Warren-Cowley Short Range Order (SRO) parameters for the first coordination shell. Our studies clearly demonstrate the presence of the SRO in the studied alloys. Binary alloys are neither random nor the clustering of N (or B) dopants takes place. In the ternary alloys, the h-BN domains are formed within the carbon areas.

Phase stability of ternary fcc and bcc Fe-Cr-Ni alloys

The phase stability of fcc and bcc magnetic binary Fe-Cr, Fe-Ni, Cr-Ni alloys and ternary Fe-Cr-Ni alloys is investigated using a combination of density functional theory (DFT), Cluster Expansion (CE) and Magnetic Cluster Expansion (MCE). Energies, magnetic moments, and volumes of more than 500 alloy structures are evaluated using DFT, and the most stable magnetic configurations are compared with experimental data. Deviations from the Vegard law in fcc Fe-Cr-Ni alloys, associated with non-linear variation of atomic magnetic moments as functions of alloy composition, are observed. Accuracy of the CE model is assessed against the DFT data, where for ternary alloys the cross-validation error is smaller than 12 meV/atom. A set of cluster interaction parameters is defined for each alloy, where it is used for predicting new ordered alloy structures. Fcc Fe2CrNi phase with Cu2NiZn-like structure is predicted as the global ground state with the lowest chemical ordering temperature of 650K. DFT-based Monte Carlo (MC) simulations are used for assessing finite temperature fcc-bcc phase stability and order-disorder transitions in Fe-Cr-Ni alloys. Enthalpies of formation of ternary alloys calculated from MC simulations at 1600K combined with magnetic correction derived from MCE are in excellent agreement with experimental values measured at 1565K. Chemical order is analysed, as a function of temperature and composition, in terms of the Warren-Cowley short-range order (SRO) parameters and effective chemical pairwise interactions.

Change of Cr atoms distribution in Fe85Cr15 alloy caused by 250 keV He+ ion irradiation to different doses

Redistribution of Cr atoms in a Fe85Cr15 alloy caused by its irradiation with 250keV He+ ions to different doses, D=8⋅1016, 16⋅1016 and 48⋅1016ions/cm2 was investigated by means of conversion electrons Mössbauer spectroscopy. The redistribution was expressed in terms of the Warren–Cowley short-range order parameters α1, α2 and α12 pertaining to the first (1NN), second (2NN) and both i.e. 1NN+2NN shells, respectively. Clear evidence was found, both for non-irradiated and irradiated samples that the actual distribution of Cr atoms is characteristic of the shell, and for a given shell it depends on the irradiation dose. In particular, α1 is positive, hence indicates an under population of Cr atoms in 1NN with respect to the random case, α2 is negative, giving evidence thereby that 2NN is overpopulated by Cr atoms, and α12 is weakly positive. Under the applied irradiation the number of Cr atoms in both neighbor shells decreased signifying thereby a clustering of Cr atoms. The underlying decrease of Cr concentration within the 1NN–2NN volume around the probe Fe atoms was estimated at 1.5at.% ranging between 2.1 for the lowest and 0.8at.% for the highest dose.

Particle swarm optimization investigation of stable structures of Pt-Pd alloy nanoparticles

Pt-Pd alloy nanoparticles exhibit better catalytic activity and selectivity than pure Pt and Pd ones, and thus to explore their stable structures is crucial for understanding the catalytic performance of nanoparticles. In this paper, the particle swarm optimization algorithm and the quantum Sutton-Chen potentials are employed to investigate the stable structures of tetrahexahedral Pt-Pd alloy nanoparticles with different sizes and different composition. Results show that in Pt-Pd alloy nanoparticles, Pt atoms are preferential to locate of the core and Pd atoms to occupy the surface. Furthermore, the more symmetrical and ordered the structure, the lower the energy of the structure. The Warren-Cowley chemical short-range order parameters of three different sizes of nanoparticles increase accordingly with rising fraction Pt. The segregation degree of small sized nanoparticle is more remarkable than large sized one for the same content of Pt.

Ab initiothermodynamics of intrinsic oxygen vacancies in ceria

Nonstoichiometric ceria(CeO${}_{2\ensuremath{-}\ensuremath{\delta}}$) is a candidate reaction medium to facilitate two-step water splitting cycles and generate hydrogen. Improving upon its thermodynamic suitability through doping requires an understanding of its vacancy thermodynamics. Using density functional theory (DFT) calculations and cluster expansion-based Monte Carlo simulations, we have studied the high-temperature thermodynamics of intrinsic oxygen vacancies in ceria. The DFT$+U$ approach was used to get the ground state energies of various vacancy configurations in ceria, which were subsequently fit to a cluster expansion Hamiltonian to efficiently model the configurational dependence of energy. The effect of lattice vibrations was incorporated through a temperature-dependent cluster expansion. Lattice Monte Carlo simulations using the cluster expansion Hamiltonian were able to detect the miscibility gap in the phase diagram of ceria. The inclusion of vibrational and electronic entropy effects made the agreement with experiments quantitative. The deviation from an ideal solution model was quantified by calculating as a function of nonstoichiometry, (a) the solid state entropy from Monte Carlo simulations, and (b) Warren-Cowley short range order parameters of various pair clusters.

Size-dependent phase diagrams of metallic alloys: A Monte Carlo simulation study on order–disorder transitions in Pt–Rh nanoparticles

Nanoparticles of Pt–Rh were studied by means of lattice-based Monte Carlo simulations with respect to the stability of ordered D022- and 40-phases as a function of particle size and composition. By thermodynamic integration in the semi-grand canonical ensemble, phase diagrams for particles with a diameter of 7.8 nm, 4.3 nm and 3.1 nm were obtained. Size-dependent trends such as the lowering of the critical ordering temperature, the broadening of the compositional stability range of the ordered phases, and the narrowing of the two-phase regions were observed and discussed in the context of complete size-dependent nanoparticle phase diagrams. In addition, an ordered surface phase emerges at low temperatures and low platinum concentration. A decrease of platinum surface segregation with increasing global platinum concentration was observed, when a second, ordered phase is formed inside the core of the particle. The order–disorder transitions were analyzed in terms of the Warren–Cowley short-range order parameters. Concentration-averaged short-range order parameters were used to remove the surface segregation bias of the conventional short-range order parameters. Using this procedure, it was shown that the short-range order in the particles at high temperatures is bulk-like.

Near-surface microstructure ofNi-23 at. % Pt: Grazing incidence diffraction and first-principles calculations

Diffuse x-ray scattering under grazing incidence was measured of a $\text{Ni-}23.2\text{ }\text{at}\text{.}\text{ }\mathrm{%}\text{ }\text{Pt}(110)$ surface at 923 K. The modulation of in-plane and out-of-plane scattering is characterized by the maxima in short-range order scattering located at positions of the X type (as for the bulk microstructure). The Warren-Cowley short-range order parameters from diffuse scattering are in good agreement with theoretical simulations based on bulk effective pair interaction parameters from first-principles theory. The nearest-neighbor interaction parameter is dominant also when determined from short-range order scattering. Its value is larger than for the bulk microstructure of Pt-rich alloys, what might reflect the difference in composition and is not compellingly due to the near-surface microstructure. Allowing for the presence of a tetragonal site occupation, indications for a segregation profile are weak in diffuse scattering and hardly resolvable for the Warren-Cowley short-range order parameters. This is consistent with the findings for layer-resolved short-range order parameters from theory. Theoretical simulations also demonstrated that the strong Ni enrichment of the surface layer has the same origin as the segregation reversal for the (110) surface in the case of $\text{Ni-}50\text{ }\text{at}\text{.}\text{ }\mathrm{%}\text{ }\text{Pt}$.

Neutron polarization analysis study of the frustrated magnetic ground state ofβ-Mn1−xAlx

We have performed a neutron polarization analysis study of the short-range nuclear and magnetic correlations present in the dilute alloy, $\ensuremath{\beta}{\text{-Mn}}_{1\ensuremath{-}x}{\text{Al}}_{x}$ with $0.03\ensuremath{\le}x\ensuremath{\le}0.16$, in order to study the evolution of the magnetic ground state of this system as it achieves static spin-glass order at concentrations $x>0.09$. To this end we have developed a reverse--Monte Carlo algorithm which has enabled us to extract Warren-Cowley nuclear short-range order parameters and magnetic spin correlations. Using conventional neutron powder diffraction, we show that the nonmagnetic Al substituents preferentially occupy the magnetic site II Wyckoff positions in the $\ensuremath{\beta}\text{-Mn}$ structure---resulting in a reduction of the magnetic topological frustration of the Mn atoms. These Al impurities are found to display strong anticlustering behavior. The magnetic spin correlations are predominantly antiferromagnetic, persisting over a short range which is similar for all the samples studied---above and below the spin-liquid-spin-glass boundary---while the observed static (disordered) moment is shown to increase with increasing Al concentration.

Energetics driving the short-range order in CuxPd1–x/Ru(0001) monolayer surface alloys

The energetics determining the distinct short-range order in two-dimensional (2D) monolayer Cu(x)Pd(1-x) surface alloys on a Ru(0001) substrate were investigated by Monte Carlo simulations and density functional theory calculations. Using a 2D lattice gas Hamiltonian based on effective pair interaction (EPI) parameters, the EPIs were derived for different Cu concentrations with Monte Carlo (MC) simulations by comparing with the atomic distributions obtained from atomic resolution STM images and the related Warren-Cowley short-range order parameters (Hoster et al., Phys. Rev. B, 2006, 73 165413). The ground state structures and mixing energies at 0 K derived from these EPIs agree well with mixing energies determined from DFT calculations of different ordered surface alloys. Additional MC simulations yield rather low transition temperatures which explain the absence of ordered 2D phases in the experiments. The consequences of our findings for the use of alloy surfaces and surface alloys as model systems for adsorption and catalytic reaction studies are discussed.

ORDERING BEHAVIOR OF FCC AND HCP ALLOYS IN Ni-Hf AND Re-Ti SYSTEMS

The concentration dependence of atomic short range order (SRO) has been studied in Ni 100-x Hf x and Re 100-x Ti x alloys (where x = 2, 5, 10, 15, 20 at.%) using the framework of transition metal model pseudopotential (TMMP). The ordering energies and Warren–Cowley SRO parameters were calculated for the Ni-Hf and Re-Ti alloys. It was found that this model predicted the correct sign of SRO for Ni 100-x Hf x system but failed in the case of Re 100-x Ti x alloys.

Inverse Monte Carlo calculation of the effective pair interactions in FePd

Warren–Cowley short-range order parameters deduced from diffuse neutron scattering in anFePd single crystal have been used to calculate the effective pair interaction energies usingan inverse Monte Carlo method. The pair interaction energies agree well with thoseobtained recently using an inverse cluster variation method and show a strong attractionbetween second-nearest-neighbour like atoms. The inverse Monte Carlo calculations show,in contrast to the direct cluster variation method, that the pair interactions inFePd are vanishing beyond the sixth atomic shell. The relevance of the calculatedpair interactions is tested in terms of the order–disorder transition temperaturesimulation. A satisfactory agreement with the experimental value is observed.

X-ray study of the short-range order structure in Cu–24.3 at.% Mn alloy

The diffuse X-ray scattering from a single crystal of Cu–24.3 at.% Mn aged within the fcc phase at 770 K was measured at room temperature. Diffuse peaks were found at 2 n 1+0.2, 2 n 2+0.2, 0 and equivalent positions ( n 1 and n 2 are integers) in reciprocal space. The Warren–Cowley short-range order parameters were derived from the separated short-range order scattering and used to generate the corresponding local atomic arrangements on a computer. The generated arrangements indicated that Mn atoms form clusters. A set of effective pair interaction energies obtained using an inverse Monte Carlo method showed that the interaction between the first nearest neighbors mainly affects the clustering of like atoms.

Short-range order in Ni-8.4 at.% Au above the miscibility gap

The elastic diffuse neutron scattering was measured for a Ni-8.4 at.% Au single crystal above the miscibility gap at 1083 K. The Ni-58 isotope was used to reduce the incoherent scattering. In contrast to previous x-ray investigations, short-range order scattering reflects local atomic arrangements characteristic of decomposition. This can be uniequivocally concluded from the diffuse scattering within the first Brillouin zone. For larger scattering vectors, the diffuse scattering is dominated by the displacement scattering due to the large difference in atomic sizes. The magnitudes of the extracted Warren-Cowley short-range order parameters show a smooth decrease with increasing distance. Crystals modeled on the basis of these parameters reveal basic arrangements of small Au clusters consisting of a tetrahedron and an attached fifth Au atom. The effective pair interaction parameters obtained from an inverse Monte Carlo analysis give no hint for a metastable state with ${L1}_{2}$ structure.

Concentration Fluctuations and Chemical Complexes in the Zn-Au Liquid Alloy

The concentration-concentration fluctuations in the long wavelength limit S cc(0) and the Warren-Cowley short range order parameters (SRO) α1 have been calculated for the Zn-Au liquid alloy at 1173 K by applying the Quasi Chemical Approximation (QCA). The results of the calculations show that Zn-Au liquid alloy manifest hetero-coordination throughout its whole concentration range. There exist evidence for the presence of two complexes Zn4AU3 and Zn4Au6. The Zn4Au3 complex which manifests dominance in the thermodynamic properties of the liquid alloy yielded a minimum short range order value of about αmin 1 ≈ −0.26 while Zn4Au6 has its minimum as αmin 1 ≈ −0.34. Studies from the S cc(0) indicate that these complexes are weak and hence cannot encourage glass formation in the liquid alloy.