Phase Diagrams and Thermodynamic Modeling of Solutions

Part I – Thermodynamics and Phase Diagrams

Part I will follow quite closely the chapter entitled "Thermodynamics and Phase Diagrams" by A. Pelton in "Physical Metallurgy 5^th ed." to be published by Elsevier in 2013, but will be significantly expanded to approximately twice the length with a more detailed and less succinct presentation, more examples and some new topics.

1. Introduction:

As in Sec. 1 of Phys Met 5^th ed., examples of many different types of phase diagrams involving T, X(composition), P, V, H, µ(chemical potential), etc. will be presented by way of introduction.

2. Thermodynamic fundamentals:

(As in Sec. 2 of Phys Met 5^th ed. but in more detail)

- 1^st, 2^nd and 3^rd Laws, Fundamental equation of thermodynamics, H, G, chemical equilibria, predominance phase diagrams, measuring G, H and S, chemical potential, auxiliary functions, Gibbs-Duhem equation.

- Fully rigorous and complete, but directed at those who have already had a basic course in chemical thermodymamics

- Particular emphasis on concepts required to understand complex phase equilibria and to calculate and interpret all types of phase diagrams

- Several problem sets to help readers review the fundamentals.

3. Gibbs Phase Rule:

(As in Sec. 3 of Phys Met 5^th ed. but in more detail)

- The Phase Rule will be developed in a very general, rigorous and novel form suitable for understanding phase diagrams involving a wide range of variables (T,X, P, V, H, µ, etc.)

4. Fundamentals of the thermodynamics of solutions:

(As in Sec. 4.1 of Phys Met 5^th ed. but in more detail)

- H, S and G of solutions

- Relative partial properties.

- Tangent construction

- Chemical activity

- Ideal Raoultian solutions:

- Derivation of ideal entropy equation from Boltzmann equation

- Excess properties, activity coefficients

- Several problem sets for review purposes.

5. Binary T-composition phase diagrams:

(As in Secs. 4.2, 4.3 and 5 of Phys Met 5^th ed. but in more detail)

- The thermodynamic origin of isobaric binary T-X phase diagrams will be presented in the classical manner involving common tangents to curves of G or equality of activities.

- Several exercises for review purposes.

- A thorough discussion of all features of binary T-X diagrams with emphasis on the relationship between the phase diagram and the thermodynamic properties of the phases.

- Systems with minima and maxima in 2-phase regions, miscibility gaps, eutectics and other invariants, intermediate phases, etc.

- Regular Solution Theory (RST):

- The equations of RST will be developed from a model of the additivity of pair bonds.

- Thermodynamic origin of simple phase diagrams illustrated by RST (showing the effects of

various degrees of deviation from ideal solution behaviour on the phase diagram.)

- Ideal Henrian solutions:

- Derivation and origin of Henry’s Law in terms of "lattice stabilities."

- Limited mutual solubility in terminal solid solutions described by ideal Henrian behaviour

6. Ternary T-composition phase diagrams:

(As in Sec. 6 of Phys Met 5^th ed. but in more detail)

- Ternary composition triangle and space model

- Ternary isothermal sections – general topological rules

- Ternary isoplethal sections

- Polythermal projections

7. General phase diagram sections:

(As in Sec. 7 of Phys Met 5^th ed. but in more detail)

- The geometry of general phase diagram sections for multicomponent systems involving the variables T,X, P, V, H, µ, etc. will be developed thermodynamically in a completely rigorous way

- Many examples will be presented. It will be shown that all these diagrams, although seemingly quite different geometrically, actually obey one simple set of geometrical rules.

- "Corresponding" potential and extensive variables and "corresponding" phase diagrams

- Law of Adjoining Phase Regions and the general rules governing the geometry of all phase diagram sections

- Zero Phase Fraction lines

- Outline of a general algorithm to calculate any phase diagram section by minimization of G

- Choice of variables to ensure that phase diagram sections are single-valued

- Interpretation of phase diagrams involving different oxidation states, such as many phase diagrams of oxide systems.

- Phase diagrams of reciprocal systems

- Geometrical rules for polythermal projections

- Solidus projections and "first-melting projections."

8. Equilibrium and non-equilibrium cooling:

(As in Sec. 9 of Phys Met 5^th ed. but in more detail)

- Equilibrium cooling.

- General nomenclature for invariant and other reactions

- Scheil-Gulliver solidification and "Scheil-Gulliver phase diagrams."

- Paraequilibrium and paraequilibrium phase diagrams

9. Aqueous and concentrated aqueous systems:

- EpH diagrams

- Precipitation (evaporation) diagrams (equilibrium and non-equilibrium)

10. Phase diagrams involving second-order and higher-order transitions

Part II – Thermodynamic Modeling of Solutions

11. Introduction:

- In chapter 7 of Part I, a general algorithm was presented to calculate any general phase diagram section from the thermodynamic properties of the phases.

- Several large integrated thermodynamic databases have been developed in recent years. These databases have been prepared by the following procedure. For every compound and solution phase of a system, an appropriate model is first developed giving the thermodynamic properties as functions of T, P and composition. Next, all available thermodynamic and phase equilibrium data from the literature for the entire system are simultaneously optimized to obtain one set of critically-evaluated, self-consistent parameters of the models for all phases in 2-component, 3-component and, if data are available, higher-order subsystems. Finally, the models are used to estimate the properties of multicomponent solutions from the databases of parameters of the lower-order subsystems. The Gibbs-energy minimization software then accesses the databases and, for given sets of conditions (T, P, X, H,V, µ, ...), calculates the compositions and amounts of all phases at equilibrium. By calculating the equilibrium state as the variables are varied systematically, the software generates the phase diagram.

- In Part II, the models used for the solution phases will be discussed along with procedures of optimization.

12. Single-sublattice random-mixing (Bragg- Williams (BW)) models:

- Binary systems:

- Ideal, Regular and Henrian solutions were discussed in Part I, chapters 4 and 5.

- Extension of RST to polynomial expansions of excess Gibbs energy g^E. – Redlich-Kister

expansions

- Systems with two sublattices but with one sublattice occupied by only one species (such

as common-ion salt solutions)

- Darken’s Quadratic Formalism – Introduction of concept of lattice stabilities and

end-member Gibbs energies.

- Introduction of coupled thermodynamic/phase diagram optimization:

- General principles. (However, the numerical techniques employed will not be discussed

in detail.)

- Examples of optimizations of some simple binary systems using single-sublattice BW

models

- Use of "virtual data" from first principles and molecular dynamics calculations.

- Multicomponent systems:

- "Geometrical models" (Kohler, Toop, Muggianu,...) and their physical interpretation

- Examples of predictions of properties and phase diagrams of ternary and higher-order

systems based only on optimized binary parameters

- Ternary parameters and ternary optimizations

- Wagner’s formalism and the Unified Interaction Parameter Formalism for dilute solutions

13. Single-sublattice models with short-range-ordering (SRO):

- The single-sublattice Modified Quasichemical Model (MQM) in the pair approximation for binary systems

- Examples of optimizations of binary systems with the MQM (metals, salts,....)

- Effect of SRO on binary miscibility gaps

- MQM with composition-variable coordination numbers. – Use of "equivalent fractions" rather than mole fractions

- MQM for ternary and higher-order systems. – Predictions from optimized binary parameters

- "Associate" (or "molecular") models of SRO

14. Two-sublattice models – Molten salts

Although molten salt solutions exhibit no long-range-order and therefore cannot strictly-speaking be said to consist of two sublattices, it is nevertheless very fruitful to treat them as such, with cations occupying one sublattice and anions the other. Multiple-sublattice models will be introduced via molten salt solutions partially for historical reasons, but mainly because there exists a wealth of experimental data for molten salt systems over wide composition regions, thereby permitting the models to be easily tested. The Compound Energy Formalism discussed in chapter 15 will be based upon the model equations developed in this chapter.

- Two-sublattice random-mixing (BW) model for molten salts; calculation of phase diagrams of reciprocal ternary systems and comparison to data.

- Introduction of first-nearest-neighbour (FNN) short-range-ordering (SRO) to the model; calculation of phase diagrams of reciprocal ternary systems and comparison to data.

- Charge-asymmetric systems with composition-variable first coordination numbers (variable ratio of the number of sites on the two sublattices.) – Use of equivalent fractions

- Coupled FNN SRO and second-nearest-neighbour (SNN) SRO. – Simplified treatment. Detailed treatment will be developed in chapter 16.

15. Multiple sublattice models – The Compound Energy Formalism (CEF):

- Long-range-ordering (LRO) versus SRO

- The CEF will be developed based on the random-mixing (BW) molten salt model of chapter 14 with fixed ratios of the numbers of sites on the sublattices since it applies to solid solutions.

- Concept of hypothetical end-members.

- Examples of application of CEF to models of:

- intermetallic solutions

- interstitial solutions

- ceramic solutions (such as spinels, perovskites, etc.)

- solutions with point defects

stressing the importance of separating the formalism parameters (such as the Gibbs energies of hypothetical end-members) from the model parameters which are functions of the formalism parameters which have physical meaning.

- The Ionic Liquid Model.

16. The Modified Quasichemical Model in the quadruplet approximation:

- Coupling of FNN SRO and SNN FRO in a two-sublattice model

- Examples of applications and optimizations

17. Order-disorder transitions

- Treatment using the Compound Energy Formalism

- The Cluster Variation Method and the Cluster Site Approximation – coupling of SRO and LRO

- The thermodynamics of magnetism

18. Applications of models to various classes of solutions:

- The equation for the limiting slope of phase boundaries in infinitely dilute solutions and its importance in choosing correct models

- Molten salts:

- Temkin model

- "complex ions" – are they real

- concentrated aqueous solutions – hydrated species

- Slags, magmas, glasses:

- silicates, borates, .....

- dissolution of non-oxide species (sulfide capacity,.....)

- the charge compensation effect

- Ceramic solutions:

- spinels, olivines, perovskites, etc.

- non-oxide ceramics

- Steel

- Solutions with more than one composition of maximum SRO

- Other classes of solutions