The Use of Computational Thermodynamics to Predict Properties of Multicomponent Materials for Nuclear Applications

Computational Thermodynamics is based on physically realistic models to describe metallic and oxide crystalline phases as well as the liquid and gas in a consistent manner. The models are used to a ...


Classical Thermodynamics
Thermodynamics is a phenomenological theory describing the relation between some observable properties like temperature, T , pressure, P, heat Q, etc. It is founded on two simple laws for macroscopic systems:

Classical Thermodynamics
Thermodynamics is a phenomenological theory describing the relation between some observable properties like temperature, T , pressure, P, heat Q, etc. It is founded on two simple laws for macroscopic systems:

Classical Thermodynamics
Thermodynamics is a phenomenological theory describing the relation between some observable properties like temperature, T , pressure, P, heat Q, etc. It is founded on two simple laws for macroscopic systems: 1. Energy cannot be destroyed or created, 2. Heat never flows spontaneously from a cold body to a hot body These laws, and some trivial mathematics, makes it possible to define a number of additional properties like internal energy, U, entropy, S, Gibbs energy, G etc. These are not observables but can be used to derive strict mathematical relations between many properties.
Bo Sundman and Christine Guéneau (INSTN and DEN/DANS/DPC/SCCME, CEA Saclay) The use of Computational Thermodynamics to predict properties of multicomponent mater December 5, 2012 4 / 26  Computational Thermodynamics: Models 1 The Gibbs energy has been selected for modelling materials properties using the Calphad method. The main reason is that most experimental data is known at constant T and P. T and P are intensive properties and simple polynomials can be used to describe the T and P dependence, but for very high pressures special models are needed. Computational Thermodynamics: Models 1 The Gibbs energy has been selected for modelling materials properties using the Calphad method. The main reason is that most experimental data is known at constant T and P. T and P are intensive properties and simple polynomials can be used to describe the T and P dependence, but for very high pressures special models are needed.
Modelling the composition dependence is the more complicated. There are two reasons for this: Computational Thermodynamics: Models 1 The Gibbs energy has been selected for modelling materials properties using the Calphad method. The main reason is that most experimental data is known at constant T and P. T and P are intensive properties and simple polynomials can be used to describe the T and P dependence, but for very high pressures special models are needed.
Modelling the composition dependence is the more complicated. There are two reasons for this: Computational Thermodynamics: Models 1 The Gibbs energy has been selected for modelling materials properties using the Calphad method. The main reason is that most experimental data is known at constant T and P. T and P are intensive properties and simple polynomials can be used to describe the T and P dependence, but for very high pressures special models are needed.
Modelling the composition dependence is the more complicated. There are two reasons for this: ◮ the amount of a component, N i is an extensive property, ◮ the configurational entropy is very important. For the configurational entropy one must take into account the formation of molecules in a gas phase, crystalline sites in solids, charge transfer between elements, clusters etc. In many cases it is necessary to introduce more constituents of the phases than just the components. Computational Thermodynamics: Models 2 Most materials consists of several crystalline phases with different structure and properties. The material often interact with other phases like gas and liquids. The Gibbs energy is an extensive property and it is possible to model each phase separately: where ℵ α is the number of moles and G α m is the molar Gibbs energy of of the phase α. Computational Thermodynamics: Models 2 Most materials consists of several crystalline phases with different structure and properties. The material often interact with other phases like gas and liquids. The Gibbs energy is an extensive property and it is possible to model each phase separately: where ℵ α is the number of moles and G α m is the molar Gibbs energy of of the phase α.The molar Gibbs energy is written as a function of the constituent fractions, y i , to model the configuration of the phase. In this way each phase can be modelled independently. Computational Thermodynamics: Models 2 Most materials consists of several crystalline phases with different structure and properties. The material often interact with other phases like gas and liquids. The Gibbs energy is an extensive property and it is possible to model each phase separately: where ℵ α is the number of moles and G α m is the molar Gibbs energy of of the phase α.The molar Gibbs energy is written as a function of the constituent fractions, y i , to model the configuration of the phase. In this way each phase can be modelled independently. The equilibrium is found by minimizing the total Gibbs energy for the given set of external conditions.
In a crystalline phase one may have several sublattices with are preferred by different elements.
where a s is the number of sites on sublattice s, b ij is the stoichiometric factor and y Computational Thermodynamics: Models 5 The end member is an important concept in CEF defining one specific constituent in each sublattice. This defines a compound and the surface of reference for the phase: where I has one constituent i in each subalttice s and • G I is the Gibbs energy of formation of this compound from the reference states of the elements, depending only on T and P. Computational Thermodynamics: Models 5 The end member is an important concept in CEF defining one specific constituent in each sublattice. This defines a compound and the surface of reference for the phase: where I has one constituent i in each subalttice s and • G I is the Gibbs energy of formation of this compound from the reference states of the elements, depending only on T and P.
In order to represent the interaction energy between the constituents in sublatticies there is an excess Gibbs energy: where J has one or more constituents in each sublattice and L J describe the properties of real phases. Computational Thermodynamics: Software 1 There are several commercial software for equilibrium calculations and they offer slightly different ways to control a system. The simplest way is to specify T , P and the amount of all components. Computational Thermodynamics: Software 1 There are several commercial software for equilibrium calculations and they offer slightly different ways to control a system. The simplest way is to specify T , P and the amount of all components. But a user may prefer to specify the chemical potential, µ i of a component i or its activity or one or more of the stable phases or maybe even the composition of a specific phase. Computational Thermodynamics: Software 1 There are several commercial software for equilibrium calculations and they offer slightly different ways to control a system. The simplest way is to specify T , P and the amount of all components. But a user may prefer to specify the chemical potential, µ i of a component i or its activity or one or more of the stable phases or maybe even the composition of a specific phase. By varying one of the conditions one can calculate how the system varies with this and that is known as a property diagram. Computational Thermodynamics: Software 1 There are several commercial software for equilibrium calculations and they offer slightly different ways to control a system. The simplest way is to specify T , P and the amount of all components. But a user may prefer to specify the chemical potential, µ i of a component i or its activity or one or more of the stable phases or maybe even the composition of a specific phase. By varying one of the conditions one can calculate how the system varies with this and that is known as a property diagram. Computational Thermodynamics: Software 1 There are several commercial software for equilibrium calculations and they offer slightly different ways to control a system. The simplest way is to specify T , P and the amount of all components. But a user may prefer to specify the chemical potential, µ i of a component i or its activity or one or more of the stable phases or maybe even the composition of a specific phase. By varying one of the conditions one can calculate how the system varies with this and that is known as a property diagram.  Computational Thermodynamics: Software 2 When two or more conditions are allowed to vary the software will calculate a phase diagram where the lines separate regions with different sets of stable phases. Computational Thermodynamics: Software 3 The diagrams below show the modelled Gibbs energy functions for two phases in a binary system at 3 different temperatures. Computational Thermodynamics: Software 3 The diagrams below show the modelled Gibbs energy functions for two phases in a binary system at 3 different temperatures. Computational Thermodynamics: Software 3 The diagrams below show the modelled Gibbs energy functions for two phases in a binary system at 3 different temperatures. The equilibrium state for any temperature and composition is the lowest Gibbs energy.The end points to a tangent to a Gibbs energy curve gives the chemical potential of the components, Computational Thermodynamics: Software 3 The diagrams below show the modelled Gibbs energy functions for two phases in a binary system at 3 different temperatures. The equilibrium state for any temperature and composition is the lowest Gibbs energy.The end points to a tangent to a Gibbs energy curve gives the chemical potential of the components, As the Gibbs energy curves are modelled outside the stable range of the phases it is possible to calculate metastable states. Computational Thermodynamics: Software 3 The diagrams below show the modelled Gibbs energy functions for two phases in a binary system at 3 different temperatures. Computational Thermodynamics: Software 3 The diagrams below show the modelled Gibbs energy functions for two phases in a binary system at 3 different temperatures. The solubility lines in the phase diagram are obtained by joining the points of the common tangents at varying temperatures. the system can, and must, be used to fit the model parameters.  All kinds of data that can be calculated from the Gibbs energy of the system can, and must, be used to fit the model parameters.   The difficulty with modelling oxides is the charge transfer. Normally each oxygen atom will take two electrons from the metallic atoms, some with multiple valencies, and a separate charge balance is needed for the equilibrium. In crystalline phases vacancies are often needed to describe defects or deviations from stoichiometry.
A simple case is wustite (periclas, halite) with a B1 structure modelled as (Fe +2 , Fe +3 , Va) 1 (O −2 ) 1 Modelling ionic systems 2 The C1 structure, CaF 2 , is the same structure as MO 2 in nuclear fuels modelled with several metallic valencies and defects on the oxygen sublattice and interstitial oxygen. The shaded plane is the neutral combination of defects.
The end members can be drawn in different ways, either varying occupancy of the oxygen sublattices at constant valency of U (top square prism) or varying U valencies at constant occupancy of the oxygen sublattices (bottom triangular prism). Modelling ionic systems 2 The C1 structure, CaF 2 , is the same structure as MO 2 in nuclear fuels modelled with several metallic valencies and defects on the oxygen sublattice and interstitial oxygen. The shaded plane is the neutral combination of defects.
The end members can be drawn in different ways, either varying occupancy of the oxygen sublattices at constant valency of U (top square prism) or varying U valencies at constant occupancy of the oxygen sublattices (bottom triangular prism).