Thermodynamic evaluation of the thixoformability of Al–Si–Cu alloys

The aim of this work was to evaluate the thixoformability of Al-(2 to 7 wt%) Si–Cu alloys by differential thermal analysis (DTA), differential scanning calorimetry (DSC) and CALPHAD simulation. Thermoanalytical data were generated for exothermic (rheocasting) and endothermic (thixoforming) cycles under different kinetic conditions (heating and cooling rates of 5, 10, 15, 20 and 25 °C/min). The findings indicate that the SSM critical temperatures and liquid fractions are directly affected by the kinetic conditions, chemical composition and heat-flow direction and that the measured values of these critical temperatures and liquid fractions vary according to the thermodynamic evaluation technique used (Calphad simulation, DSC or DTA). The SSM working window (a) became smaller as the heating/cooling rates and Si content increased; (b) was larger for rheocasting (solidification) than for thixoforming (melting) operations; (c) was on average approximately 12 °C wider and covered a range of mass fractions approximately 0.12 greater for DSC measurements than for DTA measurements; and (d) had a low sensitivity for all the conditions analyzed, indicating the thermodynamic stability of the Al–Si–Cu system. CALPHAD simulation successfully predicted several transformations and the thermodynamic behavior of the temperatures and liquid fractions analyzed. The DTA and DSC techniques yielded discrepant results for some of the critical points investigated, such as the limits of the SSM working window. The majority of the DSC cycles were more sensitive to variations in kinetic conditions, heat-flow direction and chemical composition than the corresponding DTA cycles. Furthermore, the tertiary Al₂Cu phase transformation could not be identified in many of the DTA cycles. For these reasons, DTA should be used with caution when predicting the thermodynamic behavior of potential raw materials for SSM processing.     

Computation of the Interval of Stability of Runge–Kutta–Nyström Methods

We present a Mathematica package to compute the interval of stability of Runge–Kutta–Nystrom methods fory">=f(t,y).     

Numerical investigation of homogenized Stokes–Nernst–Planck–Poisson systems

We consider charged transport within a porous medium, which at the pore scale can be described by the non-stationary Stokes–Nernst–Planck–Poisson (SNPP) system. We state three different homogenization results using the method of two-scale convergence. In addition to the averaged macroscopic equations, auxiliary cell problems are solved in order to provide closed-form expressions for effective coefficients. Our aim is to study numerically the convergence of the models for vanishing microstructure, i. e., the behavior for $\varepsilon \rightarrow 0$ ε → 0 , where $\varepsilon $ ε is the characteristic ratio between pore diameter and size of the porous medium. To this end, we propose a numerical scheme capable of solving the fully coupled microscopic SNPP system and also the corresponding averaged systems. The discretization is performed fully implicitly in time using mixed finite elements in two space dimensions. The averaged models are evaluated using simulation results and their approximation errors in terms of $\varepsilon $ ε are estimated numerically.     

Thermodynamic assessment of Sn–Cu–Ce system

Phase relations of Sn–Cu–Ce system are important in understanding metallurgical role of Ce in Sn–Cu based lead-free solder alloys. Thermodynamic assessment of Sn–Cu–Ce ternary system has been done based on experimental data about phase equilibria and thermodynamic properties by using the CALPHAD approach combined with first-principle calculations of formation enthalpy of key compounds. The solution phases (liquid, Fcc_A1, Bcc_A2 and Bct_A5) were treated as substitutional, of which the excess Gibbs energies were modeled by the Redlich–Kister polynomial. Considering its crystal structure and solid solubility range, intermetallic compound Ce₁₁Sn₁₀ was described with a three-sublattice model (Ce)_0.429(Sn)_0.429(Ce,Cu,Sn)_0.142. Other binary and ternary intermetallic compounds were described as stoichiometric phases because of their limited homogeneity ranges. During optimization, Ce–Sn binary system was first assessed; then phase relations in Sn–Cu–Ce ternary system were modeled by combining with the optimized Ce–Cu and Cu–Sn binary systems in literatures. A set of thermodynamic parameters for all known phases were obtained, which can reproduce most experimental data. The Scheil model was used to simulate the process of non-equilibrium solidification for a series of alloys.     

Non-uniqueness of solutions of the Hamilton–Jacobi–Bellman equation for time-average control

In control of diffusion processes a very useful instrument is the equation for optimal strategy and cost. For the version of infinite time horizon with time averaging this equation is much more complicated than for the version of finite time horizon, and even than for the version of infinite time horizon with discounting. In particular, the equation solution may be non-unique. This problem of non-uniqueness is researched in book of A. Arapostathis et al., 2012, for special models—near-monotone. The result received in the book is extended in the article to an important general case—models with restrictions in control which guarantee ergodicity of the process. Besides we correct the proofs from the book.     

Dynamic properties of the Fisher–Kolmogorov–Petrovskii–Piscounov equation with the deviation of the spatial variable

We consider the problem of the density wave propagation of a logistic equation with the deviation of the spatial variable and diffusion (the Fisher–Kolmogorov equation with the deviation of the spatial variable). The Ginzburg–Landau equation was constructed in order to study the qualitative behavior of the solution near the equilibrium state. We analyzed the profile of the wave equation and found conditions for the appearance of oscillatory regimes. The numerical analysis of the wave propagation shows that, for a fairly small spatial deviation, this equation has a solution similar to that the classical Fisher–Kolmogorov equation. An increase in this spatial deviation leads to the existence of the oscillatory component in the spatial distribution of solutions. A further increase in the spatial deviation leads to the destruction of the traveling wave. This is expressed in the fact that undamped spatiotemporal fluctuations exist in a neighborhood of the initial perturbation. These fluctuations are close to the solution of the corresponding boundary value problem with periodic boundary conditions. Finally, when the spatial deviation is large enough we observe intensive spatiotemporal fluctuations in the whole area of wave propagation.     

Experimental investigation of Ag–Pd–Sn ternary system

Isothermal sections of the Ag–Pd–Sn system at 500 and 800 °C were plotted based on the XRD and EDX results. The solubility of Sn in palladium and silver-based fcc solid solution has nearly zero minimum around the Ag corner. The Pd₃Sn and γ-Pd_2–xSn phases show significant dissolution of Ag and extend towards the Ag corner. The other Pd–Sn and Ag–Sn binary phases dissolve under 5 at.% of the third component. A new τ₁ ternary phase was discovered in the Pd-rich region. Its XRD pattern corresponds formally to In structure type, but the actual arrangement of the atoms in the τ₁ phase most probably corresponds to tetragonal Al₃Ti type structure.     

Thermodynamic assessment of the Al–Cu–Er system

The Cu–Er binary system had been thermodynamically assessed with the CALPHAD approach. The solution phases including Liquid, Fcc and Hcp were treated as substitutional solution phases, of which the excess Gibbs energies were formulated with the Redlich–Kister polynomial function. All the binary intermetallic compounds were treated as stoichiometric phases. Combining with the thermodynamic parameters of the Al–Cu and Al–Er binary systems cited from the literature, the Al–Cu–Er ternary system was thermodynamically assessed. The calculated phase equilibria were in good agreement with the experimental data.     

Thermodynamic modeling of the Co–Fe–O system

As a part of the research project aimed at developing a thermodynamic database of the La–Sr–Co–Fe–O system for applications in Solid Oxide Fuel Cells (SOFCs), the Co–Fe–O subsystem was thermodynamically re-modeled in the present work using the CALPHAD methodology. The solid phases were described using the Compound Energy Formalism (CEF) and the ionized liquid was modeled with the ionic two-sublattice model based on CEF. A set of self-consistent thermodynamic parameters was obtained eventually. Calculated phase diagrams and thermodynamic properties are presented and compared with experimental data. The modeling covers a temperature range from 298 K to 3000 K and oxygen partial pressure from 10⁻¹⁶ to 10² bar. A good agreement with the experimental data was shown. Improvements were made as compared to previous modeling results.     

Thermodynamic description of the Al–Fe–Zr system

The Fe–Zr and Al–Fe–Zr systems were critically assessed by means of the CALPHAD technique. The solution phases, liquid, face-centered cubic, body-centered cubic and hexagonal close-packed, were described by the substitutional solution model. The compounds with homogeneity ranges, hex.- Fe₂Zr, Fe₂Zr, FeZr₂ and FeZr₃ in the Fe–Zr system, were described by the two-sublattice model in formulas such as hex.- Fe₂(Fe,Zr), (Fe,Zr)₂(Fe,Zr), (Fe,Zr)Zr₂ and (Fe,Zr)(Fe,Zr)₃ respectively. The compounds Al_mZr_n except Al₂Zr in the Al–Zr system were treated as line compounds (Al,Fe)_mZr_n in the Al–Fe–Zr system. The compounds FeZr₂ and FeZr₃ in the Fe–Zr system were treated as (Al,Fe,Zr)Zr₂ and (Al,Fe,Zr)(Fe,Zr)₃ in the Al–Fe–Zr system, respectively. All compounds in the Al–Fe system and hex.- Fe₂Zr in the Fe–Zr system have no solubilities of the third components Zr or Al, respectively, in the Al–Fe–Zr system. The ternary compounds _λ1${\lambda }_1$ with C14 structure and _λ2${\lambda }_2$ with C15 structure in the Al–Fe–Zr system were treated as _λ1${\lambda }_1$- (Al,Fe,Zr)₂(Fe,Zr) with Al₂Zr in the Al–Zr system and _λ2${\lambda }_2$- (Al,Fe,Zr)₂(Fe,Zr) with Fe₂Zr in the Fe–Zr system, respectively. And the ternary compounds _τ1${\tau }_1$, _τ2${\tau }_2$ and _τ3${\tau }_3$ in the Al–Fe–Zr system were treated as (Al,Fe)₁₂Zr, Fe(Al,Zr)₂Zr₆ and Fe₇Al₆₇Zr₂₆, respectively. A set of self-consistent thermodynamic parameters of the Al–Fe–Zr system was obtained.     

Optimality proof of the Kise–Ibaraki–Mine algorithm

Kise, Ibaraki and Mine (Operations Research 26:121–126, 1978) give an O(n ²) time algorithm to find an optimal schedule for the single-machine number of late jobs problem with agreeable job release dates and due dates. Li, Chen and Tang (Operations Research 58:508–509, 2010) point out that their proof of optimality for their algorithm is incorrect by giving a counter-example. In this paper we give a correct proof of optimality for their algorithm.