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Materials Science on CD-ROM User Guide

Thermodynamics of Phase Diagrams

Version 2.1

Abigail Callanan, MATTER
Frank Noble, University of Liverpool
Trevor Myers, UMIST/University of Manchester

Assumed Pre-knowledge

In order for the student to gain maximum benefit from this module, a familiarity with basic thermodynamic concepts and the general concept of phase diagrams is required. For example it is assumed that the student will be familiar with the following concepts:

  • Enthalpy
  • Phase Enthalpy
  • Gibbs’ Free Energy
  • Internal Energy
  • Entropy

When these terms occur in the text, they are hyperlinked to definitions in the glossary.

It is suggested that the MATTER module ‘An Introduction to Phase Diagrams’ is used before this module to gain a familiarity with the general concepts of phase diagrams.

Module Structure

The module comprises 2 main sections:

Thermodynamics of Phase Diagrams

This is a large section which explains what free energy curves are, and how they are calculated for any phase.


This section begins with a simple animation that shows how the free energy for each phase in a pure component changes with temperature. This is followed by a screen that introduces the concept that for a binary alloy, the free energy of any phase is related to its composition. At this point the assumption is made that most alloys are totally miscible in the liquid phase i.e. they form ideal solutions. An animation shows how atoms behave in a liquid solution.

The main part of this section considers the free energy changes in the solid state. It begins by explaining that depending on how the different atoms interact with each other, the liquid solution will solidify to form one of four structures:

  • Single solid solution
  • Separated components
  • Two solid solutions
  • Compound and solid solution

Several ancillary screens show each structure and describe the ways that different atoms may interact with each other.

Ideal solutions

Over several screens the effect of changing composition and temperature on the free energy of an ideal solution is dealt with. First the concept of an ideal solution is defined, using the copper nickel alloy as an example. A simulation enables the user to change the molar fraction of nickel and see how the atomic structure and microstructure changes. After this the student is shown how the entropy changes on mixing for an ideal system. The change in entropy on mixing is defined as:

DSmix = kln(w)


where k is Boltzmann’s constant and w is the number of possible arrangements of atoms and is defined as:


A simulation enables the student to change composition and see how this affects the atomic structure and w. The free energy of an ideal liquid is then defined as:



The student is then introduced to ideal solid solutions and the concept that there is a change in free energy, DGfusion fusion when a solution solidifies/melts. It is shown that this can be calculated empirically using Richard’s rule:

DGfusion = -9.5(Tm-T )


where Tm is the melting point and T is the current temperature.

The free energy of solid phase, DGS is then defined as:

DGS = DGL+ DGfusion


This is followed by a simulation which enables the student to change temperature, and plots the free energy of the solid phase DGS, the free energy curve for the liquid phase DGL and the free energy of fusion DGfusion.

Regular solutions

The point is now made that few systems show ideal behaviour in the solid state, usually different atoms will interact with each other to cause a change in internal energy. A simple animation shows that the change in internal energy is due to the difference between the bond energies of like atoms and the average of the bond energies of unlike atoms, e.

The ideal model is now extended to allow for this change in internal energy by introducing the interaction parameter, W. This is defined as:

W = N0 ze


where N0 is Avogadro’s constant, z is the number of bonds per atom and e is the bond energy difference.

The change in internal energy on mixing DHmix is defined as:



where W is the interaction parameter, XA and XB are the molar fractions of atoms A and B respectively.

It is pointed out that when DHmix is negative, mixing will be endothermic, and when it is positive mixing will be exothermic.

Finally the free energy for a regular solid solution is defined as:

DGS = DHmix + DGL + DGfusion



Construction of a Phase Diagram

Users should try to complete the previous section first, as this section uses concepts that were introduced in ‘Thermodynamics of Phase Diagrams’.

This section starts by reminding the user of what a free energy curve is. This is followed by two screens which show how free energy curves are used to determine the phases which are in equilibrium. This is done by first considering a solid phase, which might exist as a mixture of components, as a mixture of two solid solutions or as a homogenous solution. An animation is used to show that the homogenous solution is the most stable. The same method is used to show how the common tangent is constructed and how to determine the composition of the equilibrium phases.

The next three screens introduce the Lever rule. A simple animation is used to show how the proportions of each phase change as composition changes. The user is also taken through a sample calculation step by step. This is followed by a simulation on the common tangent, where the user can change temperature and interaction parameter and see the plot of the resultant free energy curves and common tangents.

A series of screens show how a binary eutectic phase diagram can be constructed using free energy curves. Free energy curves of the silver copper alloy at five different temperatures are shown with the relevant portion of the phase diagram. In order to avoid cluttering the diagrams the phase fields and common tangents can be displayed separately.

Finally there is a complex simulation on phase diagrams, which plots the free energy curves and phase diagram for a hypothetical binary alloy A-B. The user can change the temperature and interaction parameter, and see how the phase diagram changes. Three different types of phase diagram can be generated; a simple phase diagram where A and B are completely miscible in both liquid and solid state, a diagram showing a miscibility gap and a simple eutectic diagram.


The student is referred to the following resources in this module

Ashby, M.F., and Jones, D.R.H., Engineering Materials 2, Pergamon, 1986

Cottrell, A.H., An Introduction to Metallurgy, Edward Arnold, 1975

Porter,D.A. and Easterling, K.E., Phase Transformations in Metals and Alloys, Van Nostrand Reinhold (UK), 1981


The University of Liverpool
Copyright of The University of Liverpool 2000