Materials Science on CDROM User Guide
Thermodynamics of Phase Diagrams
Version 2.1
Abigail Callanan, MATTER
Frank Noble, University of Liverpool
Trevor Myers, UMIST/University of Manchester
Assumed Preknowledge
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:
This is a large section which explains what free energy curves are, and how they
are calculated for any phase.
Introduction
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:
where k is Boltzmann’s constant and w is the
number of possible arrangements of atoms and is defined as:

(2) 
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, DG_{fusion} _{fusion }when a
solution solidifies/melts. It is shown that this can be calculated empirically using Richard’s
rule:
DG_{fusion} = 9.5(T_{m}T
)

(4) 
where T_{m }is the melting point and T is the current
temperature.
The free energy of solid phase, DG_{S} is
then defined as:
DG_{S} = DG_{L}+
DG_{fusion}

(5) 
This is followed by a simulation which enables the student to change temperature, and
plots the free energy of the solid phase DG_{S},
the free energy curve for the liquid phase DG_{L }and
the free energy of fusion DG_{fusion}.
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:
where N_{0 }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 DH_{mix }is
defined as:
DH_{mix} = WX_{A}X_{B}

(7) 
where W is the interaction parameter, X_{A}
and X_{B} are the molar fractions of atoms A and B respectively.
It is pointed out that when DH_{mix} 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:
DG_{S} = DH_{mix}
+ DG_{L} + DG_{fusion}

(8) 
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 AB. 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.
Bibliography
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
