the thermochemistry library thermochimica markus h.a. piro, april 2014

The Thermochemistry Library THERMOCHIMICA Markus H.A. Piro, April 2014

Outline Introduction Background Applications and capabilities Example problem Numerical methods and algorithms Accessing software Future plans Summary

Introduction THERMOCHIMICA is an open-source software library for computing thermodynamic equilibria with the primary purpose of direct integration into multi-physics codes. The software is written in Fortran and it can be called from a Fortran, C, or C++ Application Programming Interfaces (API) on a desktop workstation or high performance computing environment. Software development began during PhD at RMC*, it evolved during a Post- Doctoral fellowship at ORNL and it is currently being maintained by M.H.A. Piro. * M.H.A. Piro, Computation of Thermodynamic Equilibria Pertinent to Nuclear Materials in Multi-Physics Codes, PhD Thesis, Royal Military College of Canada (2011).

Brief Background Conditions for thermodynamic equilibrium: Gibbs Phase rule, Conservation of mass, and Gibbs energy of a closed system at constant T & P is a global minimum (derived from first and second laws of thermodynamics). Thermodynamic equilibrium is assumed (i.e., time dependency is not considered). The appropriateness of this assumption is problem specific. This is generally a good assumption when temperature is high and time scale is long. * M.H.A. Piro, Computation of Thermodynamic Equilibria Pertinent to Nuclear Materials in Multi-Physics Codes, PhD Thesis, Royal Military College of Canada (2011).

Applications The software is intended to provide input to material properties and boundary conditions for continuum mechanics and phase field simulations. THERMOCHIMICA can be used for various applications: Combustion Metallurgy Geochemistry Batteries Nuclear materials

Species mole fraction Chemical Potential Element Mass Database Gibbs energy Moles of Phases Enthalpy Heat capacity Pressure THERMOCHIMICA Temper- ature Input Output I/O

Applications of Thermochimica to Nuclear Engineering Applications Fuel performance and safety analysis: Fuel chemistry, Fuel melting, Fission gas retention (predicting fission product speciation), Iodine-induced stress corrosion cracking (I-SCC) / Pellet-cladding interaction (PCI), and Zirconium hydriding. Potential applications (more development needed): Aqueous chemistry: CRUD formation, fuel storage, fuel transportation.

Capabilities Parse ChemSage data-files as input. Data-files containing a maximum of 48 chemical elements, 1500 chemical species and 24 solution phases. Thermodynamic models: Pure condensed phases, Ideal solution phases, Substitutional Kohler-Toop model with regular polynomials, Substitutional Redlich-Kister-Muggiano model with Legendre polynomials, and Compound energy formalism with Legendre polynomials (up to 5 sublattices).

Compound Energy Formalism UO 2 Fluorite Crystal Structure M.H.A. Piro, PhD Thesis, Royal Military College, 2011. Reproduced from D.R. Olander, U.S. Dept. of Commerce, 1976.

Compound Energy Formalism UO 2 Fluorite Crystal Structure M.H.A. Piro, PhD Thesis, Royal Military College, 2011. Reproduced from D.R. Olander, U.S. Dept. of Commerce, 1976. Non-stoichiometric UO 2x (U 3+, U 4+, U 5+, O 2- ) Modelled with three sublattices by C. Gueneau et al, J. Nucl. Mater., 419 (2011) 145-167. This treatment is being expanded to represent irradiated fuel by T.M. Besmann et al, to be published.

Example Nuclear Fuel Thermochemistry Engineering motivation: Extend PWR fuel to very high burnup (i.e., ~ 100 GWd/t(U)). Maximize performance and safety. Experiments are extremely time-consuming (i.e., ~10 years in reactor + ~5 years in storage), expensive and dangerous. Simulations may help guide/minimize experiments. Description of problem: Fuel irradiated in European PWR to 100 GWd/t(U). Oxidation and compositional measurements performed at ITU (Germany). Numerical simulations predict fuel behaviour (chemistry, isotopic evolution and heat transfer). Coupled: AMP, Origen-S and Thermochimica. M.H.A. Piro, J. Banfield, K.T. Clarno, S. Simunovic, T.M. Besmann, B.J. Lewis and W.T. Thompson, J. Nucl. Mater., 441 (2013) 240-251.

Example Nuclear Fuel Thermochemistry Cont Oxygen partial pressure predictions with Thermochimica are in very good agreement with experimental measurements. Most codes that account for fuel chemistry assume fresh fuel. M.H.A. Piro, J. Banfield, K.T. Clarno, S. Simunovic, T.M. Besmann, B.J. Lewis and W.T. Thompson, J. Nucl. Mater., 441 (2013) 240-251.

Example Nuclear Fuel Thermochemistry Cont O/M cannot be measured directly. Experimentally inferred values for O/M were derived by ICP-MS, EPMA and assumptions regarding phase equilibria. M.H.A. Piro, J. Banfield, K.T. Clarno, S. Simunovic, T.M. Besmann, B.J. Lewis and W.T. Thompson, J. Nucl. Mater., 441 (2013) 240-251.

Example Nuclear Fuel Thermochemistry Cont SEM in high burnup structure [~75 GWd/t(U)] Noble metal HCP white phase Figure kindly provided by T. Wiss and V.V. Rondinella (ITU) M.H.A. Piro, J. Banfield, K.T. Clarno, S. Simunovic, T.M. Besmann, B.J. Lewis and W.T. Thompson, J. Nucl. Mater., 441 (2013) 240-251.

Numerical Methods From a mathematical point of view, this is a numerical optimization problem of a non-convex function with linear and non-linear equality and inequality constraints. Also, the active set of constraints change throughout the iteration process. The overall objective is to minimize the integral Gibbs energy of the system subject to the mass balance constraints and Gibbs Phase Rule. Numerical methods employed by THERMOCHIMICA are described in the literature (1-4). 1. M.H.A. Piro and S. Simunovic, CALPHAD, 39 (2012) 104-110. 2. M.H.A. Piro, S. Simunovic, T.M. Besmann, B.J. Lewis and W.T. Thompson, Comp. Mater. Sci., 67 (2013) 266-272. 3. M.H.A. Piro, T.M. Besmann, S. Simunovic, B.J. Lewis and W.T. Thompson, J. Nucl. Mater., 414 (2011) 399-407. 4. M.H.A. Piro and B. Sundman, to be published.

Numerical Methods Overview Initialization Local Minimization Global Minimization Leveling & Post-Leveling algorithms Gibbs energy minimization algorithm Modified branch and bound algorithm

Numerical Methods Initialization (Leveling) A procedure is required to initiate the non-linear solver. The Leveling algorithm of Eriksson & Thompson is first used. The premise is to temporarily drop the non-linear terms(i.e., mixing) from the chemical potentials, converting this to a linear minimization problem. G. Eriksson and W.T. Thompson, CALPHAD, 13(4) 1989 389-400. M.H.A. Piro and S. Simunovic, CALPHAD, 39 (2012) 104-110. Initialization Local Minimization Global Minimization One can then compute the chemical potentials of the system components directly. An iterative process is required to determine a unique assemblage of stable phases (i.e., species are treated as pure phases). 00

Numerical Methods Initialization (Post-Leveling) The Post-Leveling algorithm of Piro and Simunovic is then used to improve upon the estimates from Leveling. The premise is to include the ideal mixing terms of only the dominant species, which are treated numerically as phases. M.H.A. Piro and S. Simunovic, CALPHAD, 39 (2012) 104-110. Initialization Local Minimization Global Minimization 0 Performance is enhanced with this algorithm. 0 ~300%! ~40%

Numerical Methods Local Minimization Minimize the integral Gibbs energy of the system (i.e., 1 st and 2 nd law of thermodynamics). Minimize a system of Lagrangian multipliers: Subject to the following linear equality constraints (i.e., conservation of mass): and inequality constraints (i.e., non-negative mass and Gibbs Phase Rule): M.H.A. Piro, S. Simunovic, T.M. Besmann, B.J. Lewis and W.T. Thompson, Comp. Mater. Sci., 67 (2013) 266-272. Initialization Local Minimization Global Minimization

Numerical Methods Global Minimization At equilibrium, the following must be satisfied for all stables phases: M.H.A. Piro and B. Sundman, to be published. M.H.A. Piro, T.M. Besmann, S. Simunovic, B.J. Lewis and W.T. Thompson, J. Nucl. Mater., 414 (2011) 399-407. Initialization Local Minimization Global Minimization and the following must be satisfied for meta-stable phases:

Numerical Methods Global Minimization A modified branch and bound approach has been adopted for the non-linear inequality constraints. This is tested when a local minima has been reached. Minimize the following function (for meta-stable phases): M.H.A. Piro and B. Sundman, to be published. Initialization Local Minimization Global Minimization Which is subject to the following linear equality and inequality constraints: By exploiting the fact that the variables (i.e., x) are linearly constrained (i.e., bounded), the domain is decomposed into multiple sub-domains (i.e., branches) to search for a global minimum.

Numerical Methods Updating the Phase Assemblage Throughout all of the foregoing processes, provisions must be made to allow for the predicted assemblage of stable phases to change. This is the most challenging component of the entire programming: Singularities, Cyclical sets of constraints, Inefficiencies resulting from poor choices, Initialization Local Minimization Global Minimization M.H.A. Piro and S. Simunovic, CALPHAD, 39 (2012) 104-110. The Euclidean Norm Method of Piro & Simunovic greatly accelerates the process. Compute the Euclidean norm in multi- dimensional space between the composition of the phase to be added to the system relative to all other existing phases. Very simple and inexpensive. ~30% ~120%

Accessing the code The software is maintained on a Subversion (SVN) repository and can be accessed online: https://sites.google.com/site/thermochimica/ https://sites.google.com/site/thermochimica/ Prerequisites: BLAS / LAPACK linear algebra libraries Fortran compiler (gfortran/Intel) Operating system: Intended for Linux/Mac OS-X

Current and Future plans (Piro) A database conversion tool is under development to convert between various established formats (i.e., TDB, DAT). A thermodynamic model optimization tool is being developed to facilitate model development.

Summary THERMOCHIMICA is an open-source thermodynamic equilibrium solver for integration into multi-physics codes to provide material properties and boundary conditions. THERMOCHIMICA can be used for a multitude of applications, including combustion, metallurgy, geochemistry, nuclear materials and batteries. Please feel free to contact me ([email protected]) should you have any [email protected]

the thermochemistry library thermochimica markus h.a. piro, april 2014

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