There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

We present a new algorithm to generate Special Quasirandom Structures (SQS), i.e., best periodic supercell approximations to the true disordered state for a given number of atoms per supercell. The method is based on a Monte Carlo simulated annealing loop with an objective function that seeks to perfectly match the maximum number of correlation functions (as opposed to merely minimizing the distance between the SQS correlation and the disordered state correlations for a pre-specified set of correlations). The proposed method optimizes the shape of the supercell jointly with the occupation of the atomic sites, thus ensuring that the configurational space searched is exhaustive and not biased by a pre-specified supercell shape. The method has been implemented in the “mcsqs” code of the Alloy Theoretic Automated Toolkit (ATAT) in the most general framework of multicomponent multisublattice systems and in a way that minimizes the amount of input information the user needs to specify and that allows for efficient parallelization.

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Efficient stochastic generation of special quasirandom structures

The relative effects of pressure, temperature and oxygen fugacity on the solubility of sulfide in mafic magmas

Solidification microstructures: recent developments, future directions

Sn-rich alloys in the Sn-Ag-Cu system are being studied for their potential as Pb-free solders. Thus, the location of the ternary eutectic involving L, (Sn), Ag3Sn and Cu6Sn5 phases is of critical interest. Phase diagram data in the Sn-rich corner of the Sn-Ag-Cu system are measured. The ternary eutectic is confirmed to be at a composition of 3.5 wt.% Ag, 0.9 wt.% Cu at a temperature of 217.2±0.2°C (2σ). A thermodynamic calculation of the Sn-rich part of the diagram from the three constituent binary systems and the available ternary data using the CALPHAD method is conducted. The best fit to the experimental data is 3.66 wt.% Ag and 0.91 wt.% Cu at a temperature of 216.3°C. Using the thermodynamic description to obtain the enthalpy- temperature relation, the DTA signal is simulated and used to explain the difficulty of liquidus measurements in these alloys.

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Experimental and thermodynamic assessment of Sn-Ag-Cu solder alloys

A generalized approach is proposed to calculate the magnetic contribution to the thermodynamic functions of alloys. This approach is applied successfully to the Fe-Ni binary system. The predicted magnetic specific heat of the fcc phase at 75 at. Pct Ni is in agreement with the experimental data within the accuracies of the data and the predicted values. The magnetic contributions to the Gibbs energies of the fcc and bcc phases for the Fe-Ni alloys obtained from this approach are added to the nonmagnetic portion of the Gibbs energies. The nonmagnetic portion of the Gibbs energy of the fcc phase is obtained from extensive thermochemical data at high temperatures as discussed in the paper immediately following this one. The total Gibbs energies of the fcc, bcc, and the orderedγ′-(FeNi3) phases are then used to calculate/predict phase equilibria of the Fe-Ni binary at temperatures lower than 1200 K. The calculated equilibria are in agreement with available experimental data. In addition, a irascibility gap of the fcc phase at low temperatures is predicted, resulting in the formation of a monotectoid equilibrium at 662 K as given below: {fx1361-02} The existence of the miscibility gap is due to the magnetic Gibbs energy term of the fcc phase which is composition dependent. Experimental results reported in the literature support the predicted miscibility gap.

Magnetic contributions to the thermodynamic functions of alloys and the phase equilibria of Fe-Ni system below 1200 K

An empirical mathematical equation is proposed for the magnetic contribution to the specific heat of pure metals The corresponding functions for enthalpy, entropy, and Gibbs energy are of simple form Two parameters used for each element are the critical temperature,Tc, and the total magnetic entropy The parameters have been determined from a careful separation of magnetic and nonmagnetic contributions to the specific heat Debye temperatures for Ni, Co, and Fe have been determined considering data to much higher temperatures than other studies The magnetic specific heats extracted from experimental data agree very well with the proposed equation over the entire temperature range and for all three elements Comparisons with different mathematical functions found in the literature give agreement only for the case of iron The total magnetic entropy given by a classical relation is found to be high, and a quantitative correction is given Various magnetic standard states are discussed The lattice stabilities of bcc- and fcc-iron are calculated assuming that the difference of the nonmagnetic specific heats is linear from 500 K to 1810 K A simple equation is obtained in which the anomalous temperature dependence is explained by the independently determined magnetic contribution The calculated values agree very well with Orr and Chipman’s assessment The stability of bcc iron at low temperatures is quantitatively rationalized

Magnetic contributions to the thermodynamic functions of pure Ni, Co, and Fe

Liquid alloys with a rapid increase of component activities over a narrow composition range may be modelled by an associated solution model. Associated species with a fixed stoichiometry in that composition range are postulated in equilibrium with elemental species. Palrwise interactions among all species are described by Margules-type equations. The success of this description is, of course, not a verification of the physical existence of associated species. The thermodynamic background of this model is investigated in this study. For binary liquid alloys, the key equations governing the calculation of activities in internal equilibrium are presented graphically. Formulae for the calculation of the Gibbs energy, enthalpy and entropy of mixing in the postulated species-system are developed and related to the corresponding effects in the binary liquid alloys. The formation of very asymmetric and twin miscibility gaps is discussed. The difference between the Gibbs energy of species and real alloys at stoichiometry is pointed out. Limiting cases of weak and strong association are discussed and formulae for terminal values of activity coefficient and partial enthalpy of solution are developed. Structural related properties of Sn-Te liquid alloys are calculated and compared to experimental data. A quantitative prediction of data in ternary and multicomponent alloys is one major motivation for the precise description of binary alloys and a suitable extension of the associated solution model is presented. The model is compared to other associate models and to sublattice models. Despite the different physical picture, close phenomenological and mathematical resemblance is discovered among the associated solution and the sublattice model.

A thermodynamic study on an associated solution model for liquid alloys

A quasi-subregular solution model is used to describe the thermodynamic properties of the liquid phase; values of the solution parameters are obtained from extensive and consistent thermochemical data reported in the literature. For the fcc and bcc phases, the same model is used to account for the nonmagnetic part of the Gibbs energy and the magnetic contribution is taken from the previous paper. Again, the values for the quasi-subregular solution parameters for the fcc phase are obtained from extensive and consistent thermochemical data reported in the literature at high temperatures. The values of the solution parameters for the bcc phase are obtained from the thermodynamic values of the liquid and fcc phases and the known phase boundary data. The calculated phase equilibria are in good agreement with the available data. Based on the thermodynamic data, the metastablel + γ andl + δ phase boundaries as well as theT
 0
 (γ + l) andT
 0(δ +l) curves are calculated.

A thermodynamic analysis of the phase equilibria of the Fe-Ni system above 1200 K

The relevant thermodynamic and phase equilibrium data for the Fe-S binary system have been reevaluated in light of more recent data. An associated solution model is used to describe the thermodynamic properties of the liquid phase as a function of composition and temperature. For the pyrrhotite phase, a statistical thermodynamic model based on the formation of Frenkel defects in the lattice is used. For the austenite and ferrite phases, the solute is assumed to follow Henry’s law, and pyrite is taken to be a stoichiometric compound. The model parameters for the various phases are obtained by evaluating all relevant experimental data reported in the literature. The calculated phase diagram is in good agreement with the experimental data. The calculated sulfur activity values for the liquid phase agree well with experimental values including those at higher sulfur concentrations. This is an improvement of an earlier evaluation by Sharma and Chang using the same model with the same number of parameters.

Thermodynamics and Phase Relationships of Transition Metal-Sulfur Systems: Part V. A Reevaluation of the Fe-S System Using an Associated Solution Model for the Liquid Phase

Y. Austin Chang

Papers

Magnetic contributions to the thermodynamic functions of alloys and the phase equilibria of Fe-Ni system below 1200 K

Magnetic contributions to the thermodynamic functions of pure Ni, Co, and Fe

A thermodynamic study on an associated solution model for liquid alloys

A thermodynamic analysis of the phase equilibria of the Fe-Ni system above 1200 K

Thermodynamics and Phase Relationships of Transition Metal-Sulfur Systems: Part V. A Reevaluation of the Fe-S System Using an Associated Solution Model for the Liquid Phase