Thermodynamic dataA tutorial course

Session 8: Calculation of phase equilibria from critically assessed thermodynamic data

Alan Dinsdale“Thermochemistry of Materials” SRC

Outline

• Describe how phase diagrams can be calculated from critically assessed thermodynamic data– What are critically assessed data ?

• How thermodynamic data are modelled– Temperature– Composition– Pressure

• Databases and software• Applications

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Why calculate phase diagrams ?

• Provides way to rationalise different but related sets of experimental properties

• ..... and to extrapolate thermodynamic data from small systems into higher order systems in such a way that they allow prediction of multicomponent phase equilibria

• Now software and data are sufficiently robust for – monitoring and controlling industrial plant– choosing materials used in everyday household appliances– understanding the way in which mankind affects the world and the environment in

which we live – devising how we can perhaps make the world a safer and better place for us and our

children to live in

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Principles of Phase Equilibrium Calculations

Experimental Data

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G(T,P,x) Model For Each Phase

Develop Parameters For SMALL Systems To Reproduce Experimental Data

DatabaseIndustrial Problem

Predictions For

LARGE Systems

Problem Solved

MTDATA

Validation

In a nutshell

Model Gibbs energy over a range of temperatures and compositions

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…. in order to calculate the phase diagram

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What are critically assessed data ?

What do we mean by critically assessed data ?

• Enthalpies of mixing of liquid Cu-Fe alloys

• Large scatter in experimental values

• Which data best represent reality ?

• .. and are these data consistent

• with …

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…. with the experimental phase diagram

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…… and measured activity data

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What is the aim of a critical assessment ?

• To generate a set of reliable data or diagrams which are self consistent and represent all the available experimental data

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• Involves a critical analysis of the experimental data

• ….. followed by a computer based optimisation process to reduce the experimental data into a small number of model parameters

• ….. using rigorous theoretical basis underlying thermodynamics

Critical analysis of experimental data

• Experimental data: Search and analysis– Search through standard compilations eg Hultgren, Massalski– Use a database of references to the literature eg Cheynet– Carry out a full literature search

• Which properties– Phase diagram information

• Liquidus / solidus temperatures• Solubilities

– Thermodynamic information • enthalpies of mixing• vapour pressure data• emf data• heat capacities• Enthalpies of transformation• Ab-initio calculations

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Obtaining model parameters

• Aim is to determine set of coefficients which gives best agreement with experimental data – by least-squares fitting of the thermodynamic functions to selected set

of experimental and ab-initio data

• It is usually carried out with the assistance of a computer– Using optimisation software

• MTDATA optimisation module• PARROT (inside ThermoCalc)• LUKAS program BINGSS• CHEMOPT

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Calculated enthalpies of mixing for liquid Fe-Cu alloys

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Calculated Fe-Cu phase diagram

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Calculated activities for Fe-Cu liquid alloys

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Variation of Gibbs energy of phases at fixed temperature

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How to model thermodynamic data

need to understand how G changes with T, P and x

Variation of Gibbs energy of phases at fixed temperature

Difference in Gibbs energy between fcc and liquid Fe

Difference in Gibbs energy between fcc and bcc Cu

Change in Gibbs energy with composition is complex

fcc phase is reference for both elements

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4th WRRS 25 June 2012 IoM3, London, UK

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Heat capacity of Sn for different phases

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Tfus = 505.078Ttrs

Enthalpy of Sn relative to 298.15 K

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ΔfusH

ΔtrsH

Entropy of phases of Sn

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Gibbs energy of phases of Sn relative to BCT

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Tfus

Ttrs

Mathematical description of Gibbs energy variation with temperature

• Heat capacity generally represented by 1 or more expressions of the form:– Cp = a + b T + c T2 + d T-2

(generally obtained from experiment)• With enthalpy of formation and entropies

at 298.15 K (or transition enthalpies and entropies) this leads to expressions for the Gibbs energy of the form:– G = A + B T + C T ln(T) + D T2 + ET3 + F T-1

– (relative to some defined reference state)

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Magnetic materials

• Magnetism has a big influence on the heat capacity and therefore on the Gibbs energy

Heat capacity of bcc Fe Gibbs energy of liquid and bcc phases of Fe relative to fcc

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Effect of pressure

G = H – T S + P V

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Effect of composition on Gibbs energy

Difference in Gibbs energy between fcc and liquid Fe

Difference in Gibbs energy between fcc and bcc Cu

Change in Gibbs energy with composition is complex

Basic approach is to use a simple theory to model what we measure and then

Fit any discrepancies (excess Gibbs energy) to a power series expression

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Simplest theory – ideal mixing• Assumes that the components (elements) mix randomly without

giving off or absorbing any heat and with no net volume change• The mixing does result in a change in the entropy and therefore

the Gibbs energy

Gideal = R T [xFe ln(xFe) + xCu ln(xCu)]

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Real materials – excess Gibbs energy

• In most cases mixing between components (elements) is accompanied by a significant heat effect

which may be simple or complex The deviation from ideal behaviour may also vary with

temperature leading to an “excess entropy of mixing”

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Overall Gibbs energy of mixing

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Gibbs energy of binary solutions

G = xFe GFe + xCu GCu

+ R T [ xFe ln(xFe) + xCu ln(xCu)]

+ xCu xFe [ a + b (xCu-xFe) + c (xCu-xFe)2

+ d (xCu-xFe)3 + …..]

Pure component Gibbs energies

Ideal contribution to Gibbs energy

Excess Gibbs energy – in this case “Redlich-Kister expression”

where a, b, c, d …. could be temperature dependent(in practice for Fe-Cu we may need only one or possibly two parameters)

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Calculated phase diagram involving two phases

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From binary to multicomponent• Multicomponent Gibbs energy given by

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Kohler Muggianu Toop

G = Σ xi Gi + R T Σ xi ln(xi) + Gexcess

Various models used to extrapolate excess Gibbs energy into ternary and higher order systems from data for binary systems. Extra ternary terms used if required

Compound phases• Stoichiometric phases: variation of Gibbs energy

with T similar to that for phases of elements• Many important compound phases are stable over

ranges of homogeneity. Crystal structure indicates sublattices with preferred occupancy.– eg: sigma, mu, gamma brass

• Use compound energy formalism to allow mixing on different sites– Laves phases: (Cu,Mg)2 (Cu,Mg)1

– Interstitial solution of carbon: (Cr,Fe)1 (C,Va)1

– Spinels: (Fe2+,Fe3+)1 (Fe2+,Fe3+)2 (O2-)4

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Gibbs energy using compound energy formalism eg (Cu,Mg)2 (Cu,Mg)1

• Gibbs energy again has 3 contributions• Pure compounds with element from each sublattice

Cu:Cu, Cu:Mg, Mg:Cu, Mg:Mg• Ideal mixing of elements on each sublattice

• Cu and Mg on first and on second sublattices

• Non-ideal interaction between the elements on each sublattice but with a specific element on the other sublattice

– Cu,Mg:Cu Cu,Mg:Mg Cu:Cu,Mg Mg:Cu,Mg

•

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Thermodynamic databases and software

Thermodynamic databases• Two types of databases

– Collections of datasets suitable for potentially diverse materials eg SGSUB, SGSOL

– Application specific databases: comprehensive set of data for particular material types

• All databases need to be based upon a set of standards– Consistent data for the elements– Consistent models to represent thermodynamic data

based (where possible) on crystallographic information

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Database providers• Software developers

– MTDATA (NPL)– ThermoCalc (TCSAB)– FactSage (GTT, Ecole Polytechnique Montreal– PANDAT (CompuTherm)– JMatPro (ThermoTech)

• Specialist centres eg SGTE, CEA, Tohoku, Claustal• International Collaborative Projects

– COST507, COST531, COST MP0602– Superdata, III-V semiconductor project

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Software packages

General phase diagram calculations• ThermoCalc: TCSAB, Stockholm, Sweden• FactSage: GTT, Herzogenrath, Germany• Pandat: CompuTherm, Madison, USA• MTDATA: NPL, Teddington, UK

Calculation of materials properties• JMatPro: Sente Software Ltd, Guildford, UK

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General facilities available• Single calculation of phase equilibria for defined temperature,

pressure and composition• Calculations over a range of temperatures for fixed composition

• Mass of phases in equilibrium (and compositions)• Partial pressures• Enthalpy changes and heat capacity• Transformation temperatures

• Phase diagram calculations (binary, ternary, isopleths, liquidus projections)

• Database management• Assessment and optimisation capability• Simple simulations (eg Scheil solidification)

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Applications

Liquidus projection for solders

x(C

u)

x(Sn)

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x(Sn)

x(C

u)

Ag

Cu

Sn

Cu3Sn()

(Cu)

(Ag)

Ag3Sn()

(Sn)

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Te

mp

era

ture

/ C

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x(Ag)

Te

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ture

/ C U 1

U 2

U 3

U 4

U 5

E 1

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Isopleths: Mixing electrician’s solder with lead free solder

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Plot of mass of phases with variation of temperature

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Calculated heat capacity and volume change

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Use of thermodynamic data to predict thermophysical properties

Calculation of surface tension:equilibrium between bulk and surface

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Oxide liquid modelled as non-ideal mixture of species

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Calculated viscosity

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Engineering toolkits

Virtual Measurement Systems• Use MTDATA to calculate properties such as liquidus and solidus

temperatures, enthalpy, heat capacity and density• Develop and use thermodynamics as basis for modelling other

thermophysical properties• Simple interface - user is shielded from complexity of models• Easy to export data to Excel and other software

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Amalgam Toolkit for compact fluorescent lamp design

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• Provide an easy to use tool for lamp design engineers

Amalgam thermodynamic database + MTDATA api

Automatic selection of elements and compositions for amalgams to optimise light output

Light output calculations derived from calculated materials chemistry

Relative Light Output results

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Final thoughts

• Now possible to calculate phase equilibria for a wide range of materials

• Rely on good quality critically assessed data compiled into comprehensive databases

• A number of organisations are involved in developing database

• Other sorts of properties can be modelled from a thermodynamic basis

• Engineering toolkits offer potential to bring phase diagram calculations to the non specialist

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