Global minimization of the Gibbs energy of multicomponent systems Involving the presence of order/disorder phase transitions

We present in this paper a robust strategy to determine the equilibrium state, in the isobaric-isothermal (NPT) ensemble, of complex multicomponent systems in which solid solutions presenting order/disorder transitions are stable. The algorithm specifically designed to construct the first estimate of the phase assemblage describing the equilibrium state of the system is presented in detail in this work and tested on different binary and ternary systems in which solid solutions are modeled using i) the cluster site approximation or ii) the cluster variation method in the tetrahedron approximation for both the face-centered and body-centered cubic solutions. The performance of the sequential quadratic strategy using an exact Newton method and a linesearch method, implemented in this work for the specific resolution of Gibbs free energy minimization problems, is compared to the one of other large-scale optimization software packages: SNOPT, IPOPT, and KNITRO. Key subroutines implemented in the strategy to locate local minima, and specifically implemented to improve the convergence toward targeted local minima, are also presented in this work and highlight the robustness of our approach.

[1]  G. Inden,et al.  Phase equilibria in the Fe–Rh–Ti system II. CVM Calculations , 2007 .

[2]  Youjin Deng,et al.  Unified approach for cluster variation method calculations of phase diagrams in fcc substitutional alloys with interstitial species , 2006 .

[3]  K. I. M. McKinnon,et al.  A Generic Global Optimization Algorithm for the Chemical and Phase Equilibrium Problem , 1998, J. Glob. Optim..

[4]  W. A. Oates,et al.  Application of the cluster/site approximation to the calculation of multicomponent alloy phase diagrams , 2005 .

[5]  Jan-Olof Andersson,et al.  The Thermo-Calc databank system☆ , 1985 .

[6]  William C. Davidon,et al.  Variable Metric Method for Minimization , 1959, SIAM J. Optim..

[7]  Alan L. Myers,et al.  Numerical solution of chemical equilibria with simultaneous reactions , 1986 .

[8]  Zhang Jie,et al.  Improved cluster-site approximation for the entropy of mixing in multicomponent solid solutions , 2000 .

[9]  Selmer M. Johnson,et al.  Chemical Equilibrium in Complex Mixtures , 1958 .

[10]  P. Turchi,et al.  First Principles Calculations of Thermodynamic Quantities and Phase Diagrams of High Temperature BCC Ta-W and Mo-Ta Alloys , 2007 .

[11]  S. Gómez,et al.  Multiphase equilibria calculation by direct minimization of Gibbs free energy with a global optimization method , 2002 .

[12]  Y. Chang,et al.  Thermodynamic modeling of the Cr–Ir binary system using the cluster/site approximation (CSA) coupling with first-principles energetic calculation , 2009 .

[13]  Lorenz T. Biegler,et al.  On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming , 2006, Math. Program..

[14]  W. A. Oates,et al.  Cluster/site approximation calculation of the ordering phase diagram for Cd–Mg alloys , 2001 .

[15]  Gunnar Eriksson,et al.  A procedure to estimate equilibrium concentrations in multicomponent systems amd related applications , 1989 .

[16]  D. Morgan,et al.  Modeling of phase stability of the fcc phases in the Ni–Ir–Al system using the cluster/site approximation method coupling with first-principles calculations , 2008 .

[17]  Jenghan Wang,et al.  A General Method of Solution for the Cluster Variation Method in Ionic Solids, with Application to Diffusionless Transitions in Yttria-Stabilized Zirconia , 2010 .

[18]  H. Kageshima.  First-principles calculation I , 2006 .

[19]  Olaf Schenk,et al.  Matching-based preprocessing algorithms to the solution of saddle-point problems in large-scale nonconvex interior-point optimization , 2007, Comput. Optim. Appl..

[20]  E. Ressouche,et al.  Experimental study of phase equilibria in the Nb-Ti-Al system , 1999 .

[21]  John A. Trangenstein,et al.  Customized minimization techniques for phase equilibrium computations in reservoir simulation , 1987 .

[22]  V. Kress.  On the mathematics of associated solutions , 2003 .

[23]  J. Morral,et al.  Phase boundary, ZPF, and topological lines on phase diagrams , 1991 .

[24]  B. Sundman,et al.  Implementation of cluster variation method in the framework of a general thermodynamic databank , 1990 .

[25]  A. Böttger,et al.  Phase stabilities of pd-based alloys for membranes for hydrogen gas separation : A statistical thermodynamics approach , 2007 .

[26]  L. Armijo.  Minimization of functions having Lipschitz continuous first partial derivatives. , 1966 .

[27]  D K Smith,et al.  Numerical Optimization , 2001, J. Oper. Res. Soc..

[28]  Jorge Nocedal,et al.  A Reduced Hessian Method for Large-Scale Constrained Optimization , 1995, SIAM J. Optim..

[29]  Mark A. Stadtherr,et al.  Reliable prediction of phase stability using an interval Newton method , 1996 .

[30]  Marco Pretti,et al.  On the Convergence of Kikuchi’s Natural Iteration Method , 2004 .

[31]  Michael A. Saunders,et al.  SNOPT: An SQP Algorithm for Large-Scale Constrained Optimization , 2002, SIAM J. Optim..

[32]  Y. Chang,et al.  Study of the Ni-rich multi-phase equilibria in Ni-Al-Pt alloys using the cluster/site approximation for the face-centered cubic phases , 2010 .

[33]  Jorge Nocedal,et al.  Numerical Experience with a Reduced Hessian Method for Large Scale Constrained Optimization , 1995, SIAM J. Optim..

[34]  W. Bragg,et al.  The effect of thermal agitation on atomic arrangement in alloys , 1935 .

[35]  Long X. Nghiem,et al.  Computation of multiphase equilibrium phenomena with an equation of state , 1984 .

[36]  Gunnar Eriksson,et al.  The modified quasichemical model I—Binary solutions , 2000 .

[37]  W. A. Oates,et al.  Calculating phase diagrams using PANDAT and panengine , 2003 .

[38]  Mark S. Ghiorso,et al.  Gibbs energy minimization in gas + liquid + solid systems , 2000, J. Comput. Chem..

[39]  Rudolf A. Römer,et al.  On Large-Scale Diagonalization Techniques for the Anderson Model of Localization , 2006, SIAM J. Sci. Comput..

[40]  L. Höglund,et al.  Thermo-Calc & DICTRA, computational tools for materials science , 2002 .

[41]  Mark S. Ghiorso,et al.  Chemical mass transfer in magmatic processes IV. A revised and internally consistent thermodynamic model for the interpolation and extrapolation of liquid-solid equilibria in magmatic systems at elevated temperatures and pressures , 1995 .

[42]  M. Stadtherr,et al.  Validated Computing Approach for High-Pressure Chemical and Multiphase Equilibrium , 2004 .

[43]  Mark S. Ghiorso,et al.  Algorithms for the estimation of phase stability in heterogeneous thermodynamic systems , 1994 .

[44]  R. Kikuchi.  A Theory of Cooperative Phenomena , 1951 .

[45]  Brian W. Kernighan,et al.  AMPL: A Modeling Language for Mathematical Programming , 1993 .

[46]  R. Romero,et al.  The contribution of order to the phase stability of β-Cu–Zn–Al , 2011, Journal of physics. Condensed matter : an Institute of Physics journal.

[47]  M. Ghiorso.  Chemical mass transfer in magmatic processes , 1985 .

[48]  L. Onsager.  Crystal statistics. I. A two-dimensional model with an order-disorder transition , 1944 .

[49]  S. Lele,et al.  Computational thermodynamics of Sc–Zr and Sc–Ti alloys using cluster variation method , 2009 .

[50]  W. A. Oates,et al.  The cluster/site approximation for multicomponent solutions — A practical alternative to the cluster variation method , 1996 .

[51]  Hans Leo Lukas,et al.  Computational Thermodynamics: The Calphad Method , 2007 .

[52]  Lars Stixrude,et al.  Thermodynamics of mantle minerals - II. Phase equilibria , 2011 .

[53]  Y. Chang,et al.  Thermodynamic modeling of the Cr–Pt binary system using the cluster/site approximation coupling with first-principles energetics calculation , 2008 .

[54]  Jorge Nocedal,et al.  Knitro: An Integrated Package for Nonlinear Optimization , 2006 .

[55]  Robert J. Vanderbei,et al.  An Interior-Point Algorithm for Nonconvex Nonlinear Programming , 1999, Comput. Optim. Appl..

[56]  M. Enomoto,et al.  Modeling the microstructural evolution of Ni-base superalloys by phase field method combined with CALPHAD and CVM , 2007 .

[57]  C. Colinet.  Applications of the cluster variation method to empirical phase diagram calculations , 2001 .

[58]  W. A. Oates,et al.  Application of the cluster-site approximation (CSA) model to the f.c.c. phase in the Ni–Al system , 2003 .

[59]  Jianhua Wang,et al.  A thermodynamic assessment of the Au-Cu system☆ , 1998 .

[60]  Ryoichi Kikuchi,et al.  Superposition approximation and natural iteration calculation in cluster‐variation method , 1974 .

[61]  C. N. Yang.  A Generalization of the Quasi‐Chemical Method in the Statistical Theory of Superlattices , 1945 .

[62]  W. A. Oates,et al.  Application of the cluster/site approximation to fcc phases in the Ni–Al–Cr–Re system , 2005 .

[63]  R. Kikuchi,et al.  Phase diagrams of FCC and BCC ordered alloys , 1974 .

[64]  P. Balbuena,et al.  Combined ab initio and experimental study of A2 + L21 coherent equilibria in the Fe–Al–X (X = Ti, Nb, V) systems , 2011 .

[65]  Y. Kawazoe,et al.  Using first-principles results to calculate finite-temperature thermodynamic properties of the Nb–Ni μ phase in the Bragg–Williams approximation , 2006 .

[66]  K. Binder.  Ordering of the Face-Centered-Cubic Lattice with Nearest-Neighbor Interaction , 1980 .

[67]  Ying Chen,et al.  Combined ab initio/CALPHAD modeling of fcc-based phase-equilibria in the Ir–Nb system , 2008 .

[68]  R. Kikuchi,et al.  Cluster variation method in the computational materials science , 2002 .

[69]  E. A. Guggenheim,et al.  Statistical thermodynamics of super-lattices , 1940, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[70]  Lloyd S. Shapley,et al.  Mass Action Laws and the Gibbs Free Energy Function , 1965 .

[71]  Paul D. Asimow,et al.  Algorithmic modifications extending MELTS to calculate subsolidus phase relations , 1998 .

[72]  Journal of the Institute of Metals , 1918, Nature.

[73]  M. Stadtherr,et al.  Reliable Phase Stability Analysis for Excess Gibbs Energy Models , 2000 .

[74]  A. Gheribi,et al.  Accurate determination of the Gibbs energy of Cu-Zr melts using the thermodynamic integration method in Monte Carlo simulations. , 2011, The Journal of chemical physics.

[75]  Aage Fredenslund,et al.  Calculation of simultaneous chemical and phase equilibria in nonideal systems , 1989 .

[76]  Gunnar Eriksson,et al.  FactSage thermochemical software and databases - recent developments , 2009 .