Thermodynamic data A tutorial course Session 4: Modelling of data for solutions (part 4) Alan Dinsdale “Thermochemistry of Materials” SRC

2 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

Gibbs energy associated with ideal mixing 5 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

More complicated descriptions

Gibbs energy of binary solutions G = x Fe G Fe + x Cu G Cu + R T [ x Fe ln(x Fe ) + x Cu ln(x Cu )] + x Cu x Fe [ a + b (x Cu -x Fe ) + c (x Cu -x Fe ) 2 + d (x Cu -x Fe ) 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) 10

Overall Gibbs energy of mixing 11

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

regular solution negative enthalpy of mixing Positive enthalpy of mixing

14 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

15 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 16

17 Calculation of binary phase equilibria

over a range of temperatures

19

Magnetics

Variation of magnetic contribution with composition

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: (Fe 2+,Fe 3+ ) 1 (Fe 2+,Fe 3+ ) 2 (O 2- ) 4 27

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 28

Peritectic system – Sb-Sn

Peritectic system – Cu-Zn

From binary to multicomponent Multicomponent Gibbs energy given by 33 KohlerMuggianu Toop G = Σ x i G i + R T Σ x i ln(x i ) + G excess 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

35 Phase Diagram Calculations Experimental data G(T,P,x) Model for each phase Develop parameters for SMALL systems to reproduce experimental data Database Industrial problem Predictions for LARGE systems Problem solved MTDATA or similar Validation

36 3D liquidus surfaces

37 Early MTDATA 3D from NPL in the late 1940s MTDATA 3D

Ag-Au-Pd

Models of ternary phase diagrams

Critical assessment of data

41 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 42

43 Published phase diagrams may be wrong

44 …… and measured activity data

45 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

46 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

47 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 48

49 Calculated Fe-Cu phase diagram

50 Calculated enthalpies of mixing for liquid Fe- Cu alloys

51 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