Session B23: Alloy Theory

Optimized basis-set representation for electronic-structure methods: Better Energetics

We derive an analytic expression for an optimal, and rapidly computed, representation for site-centered basis-set expansion (e.g., spherical harmonic). An optimal site-dependent radius are determined from the local saddle-points derived in terms of overlapping atomic charge densities, typically already used for L\"{o}wden construction of the starting potentials. These ``saddle-point adjusted'' sphere radii separate the ``spherical'' density and potential around an atom from the symmetry-induced, ``non-spherical'' part in the interstitial, and more properly accounts for charge and size of atoms. These radii also properly determine the weighted Voronoi cells (i.e., power diagram or generalized Wigner-Seitz cells) which are mathematically guaranteed to be convex and space filling. For full-potential and forces, exact integrations over Voronoi interstitials is accomplished easily by isoparametric integration. We implement these ideas in a general Korringa-Kohn-Rostoker coherent potential approximation (KKR-CPA) code within the atomic-sphere-approximation (ASA). For several large-atom/small-atom systems, we show that ASA using saddle-point-adjusted spheres now agrees with formation energies from full-potential calculations and experiments for both chemically ordered and disordered cases, and, hence, predict the stability of the correct phases and its temperature scale.   

Applications of the KKR-DCA: A Finite-Temperature Density Functional Theory to Predict Chemical Short-Range Order Effects in Disordered Metallic Alloys

Short-range order (SRO) is ubiquitous in metallic alloys, affecting changes in their electronic, thermodynamic, mechanical, magnetic, and structural properties. For example, SRO is responsible for the yield-strength anomalies observed in Cu-Al at high temperatures, i.e., the materials is more resistant to dislocation motion at high temperature than it is at room temperature. Within the Korringa-Kohn-Rostorker (KKR) electronic-structure method, we present results using the dynamical cluster approximations (DCA) to obtain the temperature-dependent SRO in disordered alloys. We obtain the KKR-DCA SRO energetics versus local neighbor SRO parameters and minimize it at fixed temperature to predict the SRO. We show that the calculated SRO at fixed temperature compares well with available experimental results, and then correlate the results to the electronic structure. We discuss how an accurate analytic estimate can be made for the SRO in most metals due to the dependence of the grand potential on SRO.   

Electronic and Magnetic properties of NbFe$_2$: An itinerant magnet near a quantum critical point

NMR studies show that pure C14 Laves phase NbFe$_2$ is a weak antiferromagnet below 13K with magnetic moment per Fe of no more than 0.1$\mu_B$. However, the Nb-rich samples do not show antiferromagnetism down to 1.8K, which suggests that they are close to antiferromagnetic QCP. Here we report density functional studies of the magnetic properties, band structure and Fermiology. We elucidate the nature of the ordering between the two distinct Fe sites and discuss the results in relation to the quantum criticality.   

First-principles calculations of free energies of unstable phases: The case of fcc W

Ab initio density-functional theory molecular dynamics simulations are used to solve the long-standing problem of calculating the free energies of harmonically unstable phases, such as fcc W. We find that fcc W is mechanically unstable with respect to long-wavelength shear at all temperatures considered (T$>$2500 K), while the short-wavelength phonon modes are anharmonically stabilized. The calculated fcc/bcc enthalpy and entropy differences at T=3500 K (308 meV and 0.74 kB per atom, respectively) agree well with the recent values derived from analysis of experimental data. The proposed method can be used in first-principles modeling of the thermodynamics of unstable phases and calculations of the thermodynamic driving forces for martensitic transformations in pure elements and alloys.   

Anomalies in the bulk properties of single crystalline Niobium

The thermodynamic properties of single crystal Niobium are presented. Anomalies in thermal expansion, specific heat, elastic constants, and electrical resistivity are observed. The linear coefficient of thermal expansion, $\alpha$, exhibits a large, broad peak in the range 200 K $<$ T $<$ 280 K, with a nearly two-fold increase in $\alpha$. The elastic constants show anomalies over a similar temperature range, while anomalies in heat capacity and resistivity are much narrower. This is surprising since crystalline Nb is a simple system, with only one naturally occurring isotope and a body centered cubic structure. Measurements on a second single crystal and on high purity polycrystalline Nb will also be presented.   

Development of an Embedded-Atom Method Potential for Niobium

An embedded-atom method (EAM) potential [1,2] is developed for pure niobium as the first step in the construction of an EAM potential for titanium-niobium alloys. The potential is constructed using the force-matching method [3]: the functions comprising the potential are represented as cubic splines, and the spline knots are chosen such that the potential optimally reproduces a large database of forces, cohesive energies, and stresses computed via density functional theory. The code potfit [4] optimizes the splines using a combination of simulated annealing and conjugate gradient-like minimization algorithms. EAM results are compared to DFT and experimental results for the lattice constant, cohesive energy, single-vacancy formation energy, fcc-bcc and hcp-bcc structural energy differences, elastic constants, and phonon dispersions. \newline [1] M. S. Daw and M. I. Baskes, Phys. Rev. Lett. 50, 1285 (1983). \newline [2] M. S. Daw and M. I. Baskes, Phys. Rev. B 29 6443 (1984). \newline [3] F. Ercolessi and J. B. Adams, Europhys. Lett. 26, 583 (1994). \newline [4] P. Brommer and F. G\"{a}hler, Modelling Simul. Mater. Sci. Eng. 15, 295 (2007).   

Ab initio up to the melting point: Anharmonicity and vacancies in aluminum

At elevated temperatures, the heat capacity of metals strongly deviates from the harmonic prediction. This was pointed out long ago\footnote{M. Born and E. Brody, Zeitschrift f\"ur Physik 6, 132 (1921)} and various explanations have been considered. Ab initio calculations showed\footnote{B. Grabowski, T. Hickel, J. Neugebauer, Phys. Rev. B 76, 24309 (2007)} that a dominant part can be explained by quasiharmonic excitations. However, the {\it detailed} balance of further contributions, such as explicit anharmonicity and vacancies, is not clarified yet even for simple elementary metals. Aluminum is a prototypical example. Even though intensively studied, the ambiguous experimental situation has made a classification of the mechanisms impossible. To resolve the situation, we have calculated the full volume and temperature dependent {\it ab initio} free energy surface employing density-functional theory. In particular, we have included anharmonic and vacancy contributions using numerically highly efficient methods to coarse grain the configuration space. To obtain accurate vacancy energies, we have included the full spectrum of excitations: quasiharmonic, electronic, and explicitly anharmonic. The results are in contradiction to common belief, nevertheless the essential physics can be captured by a simple model.   

New Generation Structural Materials: Ab initio Based Modeling of High-Entropy Alloys

There is rapidly growing interest in a new generation of structural materials called high entropy alloys. This class of alloys is multi-component ($\sim $ five elements) with approximately equiatomic ratio, and thus have high entropy of mixing by which they are distinguished from conventional alloys. It has been reported experimentally that the single bcc-based AlCoCrFeNi, single fcc-based CoCrCuFeNi and FeCrMnNiCo high-entropy alloys exhibit promising mechanical properties with potential applications. In this work, we introduce ab initio based modeling for understanding structural, magnetic, and elastic properties based on relaxation of randomly generated supercells within the framework of density functional theory. We studied component-dependent phase stabilities, electronic structures, and magnetic properties with all solutes at fixed and relaxed positions. The properties are analysed in terms of the underlying electronic structure and suggestions are made for further experimental studies to further clarify the reasons for the unusual stability of these systems. Research sponsored by the Division of Materials Science and Engineering, Office of Basic Energy Sciences, U.S. DOE.   

High-throuput formalism and calculation of Ag, Au, Cd, Co, Cr, Ir, W, and Zn solubility in Ti from first-principles

Based on statistical-thermodynamic theory of a dilute lattice gas, we developed an approach for calculation of atomic solubility in alloys. The advantage of the approach consists in taking into account all known alloy ground states rather than just pure species. It is shown that the low-solubility obey the simple Arrhenius-type dependence on temperature determined by ``low-solubility formation energy.'' Such quantity is defined as the derivative of the compound formation energy, determined with respect to surrounding ground states, versus composition. ``Low-solubility formation energy'' coincides with the usual ``true'' defect formation energy only in the case of a phase-separating alloy having no intermediate ground states and vacancies. We present a high-throughput formalism where the ``low- solubility formation energy'' can be directly obtained through first-principles calculations. The developed approach is applied to solubility of transition metals in titanium. The obtained values and tendecies are in good qualitative correspondence with experiments.   

Alternative alloys for catalysts and platinum jewelry? New structures in Pt-Hf and Pt-Mo

The only known intermetallic structure with an 8:1 stoichiometry is that of Pt$_8$Ti. It is intriguing that an ordered phase would occur at such low concentrations of the minority atom, but this structure occurs in about a dozen binary intermetallic systems. The formation of an ordered phase in an alloy can significantly enhance the performance of the material, particularly the hardness. We have taken a broad look at possible systems where this phase forms. Using first-principles, we calculated the stability of this structure relative to experimentally known phases for more than 80 Pt/Pd binary systems. We find the Pt$_8$Ti structure is a possible ground state in more than 20 cases. Our experimental collaborators have verified our prediction in Pt-Mo and observed order-hardening in Pt-Hf. We discuss the discovery of new ground states that are likely to be verified experimentally and their impact on materials for Pt- and Pd-based catalysts and jewelry.   

New structures in Pd-rich ordered alloys

An intriguing intermetallic structure with 8:1 stoichiometry was discovered in the 1950s in the Pt-Ti system. Since then a handful of other Pt/Pd/Ni binary systems have been observed to exhibit this curious structure (Pt$_8$Zr, Pd$_8$Mo, Ni$_8$Nb, etc). This ordered structure can significantly increase the hardness of an alloy. For jewelry applications involving Pt and Pd, international hallmarking standards require that the alloys be at least 95\% pure by weight. However, Pt- and Pd-rich alloys are often soft when purity is high if the minority atoms are disordered. Because the 8:1 structure maintains a high weight percentage of Pt/Pd, it can satisfy purity standards while increasing performance. Recent calculations and experiments suggest that the 8:1 structure may form in about 20 previously unsuspected Pt/Pd binary systems. Using first-principles calculations and cluster expansion modeling, we have performed a ground state search to find the stable structures in Pd-Nb and Pd-Cu. In collaboration with Candace Lang's group at University of Capetown South Africa, we are working to experimentally validate the predicted ground states.   

Verification and refinement of the Al-Mg-Zn $\Phi$ phase crystal structure model

Density Functional Theory calculations were performed to validate the crystal structure proposed by L. Bourgeis et al.\ for the $\Phi$ phase of the Al-Mg-Zn system. Their model has ambiguous site occupancies for Zn and Al and definite locations for Mg. The model's simulated electron diffraction patterns agreed very well with experimental patterns. Using DFT calculations we are able to determine optimal Zn and Al aluminum locations. We will also show that the energetically optimal structure's element concentrations are within the experimentally observed range.   

An ab initio study of the crystal structure of the Tau-phase in Al-Mg-Zn alloys

Existing crystal structures for the intermetallic Tau-phase in Al-Mg-Zn alloy are studied by density functional theory calculations using projector augmented wave pseudopotentials. Favorable crystal structures are identified through volume optimization and formation energy calculations. Properties such as elastic constants and bulk modulus of the crystal structure are determined.   

Magnesium phase diagrams: Have you seen us?

Because of it's high strength-to-weight ratio, magnesium is seen as promising material for automotive applications. But magnesium alloys are far less understood that more common alloys such as steel or newer alloys such as aluminum. Even among simple binary magnesium systems, there is a great deal of missing information. There are binary magnesium systems for which no phase diagrams appear in the latest databases (the Pauling File, for example). Using a high-throughput approach, we have undertaken a broad search for ground states in 40 magnesium binary systems using more than 8000 fully-relaxed first-principles calculations. We find new, non-obvious ordering systems and many systems where there are unsuspected ground states. We discuss the results and their potential impact on magnesium alloys.   

First-principles thermodynamics of point defects and off-stoichiometry in \textit{$\beta $}-Mg$_{17}$Al$_{12}$

The mechanical strength of Mg-Al alloys may be enhanced by a fine spatial dispersion of \textit{$\beta $}-Mg$_{17}$Al$_{12}$ precipitates. Native point defects, i.e. vacancies and anti-sites, in Mg$_{17}$Al$_{12 }$are important for understanding the phase stability and unusually asymmetric observed off-stoichiometry in this precipitate phase. In an effort to provide a quantitative picture of the phase stability of this system, we have performed a series of first-principles density functional theory calculations of bulk and defect properties of Mg$_{17}$Al$_{12}$. We consider not only the T=0K static energetics, but also key entropic terms such as the configurational and vibrational entropies. The vibrational entropies are calculated from DFT via the direct force-constant approach using the quasiharmonic approximation. We investigate the effect of atomic vibrations on native point defect free energies of Mg$_{17}$Al$_{12 }$and combine the entropic contributions with the point defect formation energies to evaluate the thermodynamics of off-stoichiometry in this phase. We find there is a large vibrational entropy difference between Mg-rich and Mg-deficient defects in Mg$_{17}$Al$_{12}$, consistent with the strong asymmetry in the observed Mg-Al phase diagram.