Thermodynamic dataA tutorial course

Session 4: Modelling of data for solutions (part 4)

Alan Dinsdale“Thermochemistry of Materials” SRC

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What happens when things mix ?

• Some elements mix together (eg in the liquid phase) without giving out or absorbing heat – said to mix ideally eg Co and Ni

• Normally there is a net heat effect which could be either negative (giving out heat) or positive (heat is absorbed) … or even more complex

Ideal solutions

• Mixing between metals or compounds having a similar size

• Same number of nearest neighbours• No enthalpy change associated with the

mixing• No volume change on mixing• Assume that the metals or compounds mix

together randomly

Entropy of mixing for random solution

• From the Boltzmann relationship

– k is the Boltzmann constant– is the number of ways of arranging the system

• For a mole of material ie with atoms, of A atoms and of B atoms

• Using Stirling’s approximation

• The contribution to the Gibbs energy from ideal mixing is therefore

Gibbs energy associated with ideal mixing

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Note that there could still be an offset in the Gibbs energy for each component associated with a change in the reference state for the Gibbs energy

Regular solutions

• Ideal solutions are rare• Most solutions show either positive or negative

enthalpies of mixing• Deviation from ideality is called excess Gibbs energy • Simplest model to represent non-ideality is the “so-

called” regular solution model• Components again approximately the same size, same

number of nearest neighbours and no change in volume on mixing

• Assumes enthalpy effect is from short range interaction between pairs of atoms

• The interaction has no effect on the order within the solution ie random mixing is maintained

• Total lattice energy given by

• where , and are the number of A-A, A-B and BB bonds respectively

• The probability of finding two chosen adjacent sites occupied by two A atoms, two B atoms or one atom and one B atom will be , and respectively

• If there are nearest neighbours, the total number of bonds will be

• The number of bonds of each type will be the total number of bonds multiplied by its probability ie.

, and

• This leads to:

• The change in energy on mixing will be this quantity minus the lattice energies for the two elements

• Or, since

More complicated descriptions• In practice few solution phases are regular

– a more complicated description is required• Redlich – Kister expression

• This is the one used most often• Others

The parameters here can be equated to the partial enthalpies

• All these descriptions are numerically identical and it is possible to convert one for to another

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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Overall Gibbs energy of mixing

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In practice the variation of the enthalpy of mixing can be complex

regular solution negative enthalpy of mixing

Positive enthalpy of mixing

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Entropy of mixing• Generally two contributions

– First from random mixing of the elements (ideal contribution)

– “So called” excess entropy of mixing - really to account for deviations from ideality

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Gibbs energy of mixing eg G = H – T S

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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Calculation of binary phase equilibria

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.... over a range of temperatures

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Magnetics

• In the second session we discussed the magnetic contribution to thermodynamic properties

• Where • T* the critical temperature

– TC, the Curie temperature for ferromagnetic materials

– TN, the Neel temperature for antiferromagnetic materials

• B0 is the average magnetic moment per atom

Variation of magnetic contribution with composition

• Usual approach is to represent the variation of and with composition using a Redlich-Kister series

and

• They can vary in a quite complicated way

In this the magnetic effect causes a miscibility gap between two fcc phases – a so-called “Nishizawa” horn

Binary systems – Cu-Ni

Mixing ferromagnetic and antiferromagnetic materials

• Model assumes that the Néel temperature for antiferromagnetic materials is equivalent to a negative Curie temperature eg: bcc Fe-Mn

• For fcc phases the situation is more complex eg fcc Mn-Ni. Here it is necessary to divide the critical temperature by 3 in the antiferromagnetic region

Intermetallic phases

• In intermetallic phases elements or species occupy may different sublattices

• The common terminology is distinguish the different sublattices by separating them with a colon

• eg. Silicon Carbide could be designated as(Si):(C)

• In this particular case there are the same number of sites on each sublattice

• Often the number of sites is different• eg. Cementite (Fe)3:(C) has three sites for the

metallic atoms and one for the carbon

• 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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• The expression for the Gibbs energy then becomes:

• etc can be represented using the Redlich Kister series

• , , and are the site fraction of the elements on each sublattice

Peritectic system – Sb-Sn

Peritectic system – Cu-Zn

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

Calculation of phase diagrams

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Phase Diagram Calculations

Experimental dataG(T,P,x) Model for each phase

Develop parameters for SMALL systems to reproduce experimental data

DatabaseIndustrial problem

Predictions for

LARGE systems

Problem solved

MTDATAor similar

Validation

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3D liquidus surfaces

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Early MTDATA 3D from NPL in the late 1940s

MTDATA 3D

Ag-Au-Pd

Tem

pera

ture

/øC

x(Au)

960

970

980

990

1000

1010

1020

1030

1040

1050

1060

1070

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

x(Au)

Te

mp

era

ture

/°C

Ag Au

LIQUID

FCC_A1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1960

970

980

990

1000

1010

1020

1030

1040

1050

1060

1070

Tem

pera

ture

/øC

x(Pd)

1000

1100

1200

1300

1400

1500

1600

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

x(Pd)

Te

mp

era

ture

/°C

Au Pd

LIQUID

FCC_A1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11000

1100

1200

1300

1400

1500

1600

Tem

pera

ture

/øC

x(Ag)

0

200

400

600

800

1000

1200

1400

1600

1800

2000

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

x(Ag)

Te

mp

era

ture

/°C

Pd Ag

LIQUID

FCC_A1

FCC_A1+FCC_A1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

200

400

600

800

1000

1200

1400

1600

1800

2000

x(A

u)

x(Pd)

0.0

0.2

0.3

0.5

0.7

0.9

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

x(Pd)

x(A

u)

Ag

Au

Pd

1000 1100 1200 1300 1400 1500

x(A

u)

x(Pd)

0.0

0.2

0.3

0.5

0.7

0.9

0.0 0.2 0.4 0.6 0.8 1.00 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

x(Pd)

x(A

u)

T = 1400°C

Ag

Au

Pd

LIQUID

FCC_A1

x(A

u)x(Pd)

0.0

0.2

0.3

0.5

0.7

0.9

0.0 0.2 0.4 0.6 0.8 1.00 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

x(Pd)

x(A

u)

T = 1200°C

Ag

Au

Pd

FCC_A1

LIQUID

Models of ternary phase diagrams

Critical assessment of data

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What do we mean by critical assessment ?

• Enthalpies of mixing of liquid Cu-Fe alloys

• Large scatter in experimental values

• Which data best represent reality ?

• .. and are these data consistent

• with …

…. with the experimental phase diagram

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Published phase

diagrams may be wrong

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…… and measured activity data

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What is the aim of a critical assessment ?

• Aim of critical assessment process is to generate a set of reliable data or diagrams which are self consistent and represent all the available experimental data for the system

• It involves a critical analysis of the experimental data (Hultgren, Massalski etc)

• ….. 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

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How

• 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

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How to carry out an assessment : 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 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

Calculated Fe-Cu phase diagram

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Calculated Fe-Cu phase diagram

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Calculated enthalpies of mixing for liquid Fe-Cu alloys

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Calculated activities for Fe-Cu liquid alloys

Next session

• Thermodynamic models for other sorts of phases– Chemical ordering– Reciprocal systems– Spinels, Halite– Liquids with short range ordering

• Oxides, slags, mattes• Molten salts