Materials Science on CDROM User Guide
Nucleation in Metals and Alloys
Version 2.1
Andrew Green, MATTER
Phil Prangnell, UMIST/ Manchester Materials Science Centre
Assumed Preknowledge
Before using this package, it is assumed that the student is familiar with the
thermodynamic relationship DG=DHTDS. It would also be helpful if the concept of interfacial
energy had at least been encountered.
Module Structure
The module comprises 4 sections:
Homogeneous Nucleation
This section deals with the simplest nucleation event, namely the homogeneous
nucleation of solid crystals during the freezing of a pure metal. On completion, the
student should be able to:
 explain the term homogeneous, as applied to nucleation events,
 understand the concept of a critical size or radius, r* and a critical free
energy to nucleation, DG*,
 differentiate between unstable clusters (or embryos) and stable nuclei,
 derive expressions for r* and DG* in terms of
both volume free energy, DG_{v} and
undercooling, DT,
 recall that r* µ 1/DT
and DG* µ 1/DT^{ 2}.
The section starts by explaining that a driving force for solidification, DG_{v} exists below the equilibrium melting
temperature, T_{m} and that this is approximately proportional to the
degree of undercooling DT:

(1) 
where DH_{v} is the
change in enthalpy of solidification. (N.B. throughout this module, the convention that
(N.B. throughout this module, the convention that (N.B. throughout this module, the convention that
DG_{v} and DH_{v} are NEGATIVE below T_{m}
is used.)
H_{v} are NEGATIVE below T_{m}
is used.) H_{v} are NEGATIVE below T_{m}
is used.) H_{v} are NEGATIVE below T_{m}
is used.)
An animation is used to illustrate on an atomic scale the concept of small clusters of
crystallised solid forming from a liquid metal. These arise due to the random
motion of atoms within the liquid.
It is then shown that this driving force is opposed by the increase in energy due to
the creation of a new solidliquid interface.
By assuming that solid phase nucleates as spherical ‘clusters’ of radius, r,
it is shown that the net (excess) free energy change for a single nucleus, DG_{(r)} is given by:

(2) 
where g_{SL} is the
solid/liquid interfacial energy.
The student is shown stepbystep that from this expression, the critical radius r*
(defined as the radius at which DG_{(r)}
is maximum) is given by:

(3) 
The associated energy barrier to homogeneous nucleation, DG* is found by substituting r* into
equation (2):

(4) 
The temperaturedependence of these terms, i.e. that r* µ
1/DT and DG* µ 1/DT^{ 2} is emphasised.
An animation is provided to reinforce the concept of a critical radius. By clicking and
holding a button on screen, atoms are added to a subcritical cluster. If the button is
released before r = r*, the cluster is shown to redissolve, whereas
if r >r*, the stable nucleus continues to grow. The terms cluster
(unstable) and nucleus (stable) are thus differentiated.
The section is completed by an exercise to test whether the important concepts have
been understood and which allows the student to plot DG_{(r)}
as a function of r for different levels of undercooling.
This section continues to look at the liquidsold transformation, but now introduces
the idea that nuclei can form at preferential sites (e.g. mould wall, impurities or
catalysts, etc.). By so doing, the energy barrier to nucleation, DG*
can be substantially reduced. On completion the student should be able to:
 list some typical heterogeneous nucleation sites for solidification,
 identify all the relevant interfacial energy terms for a heterogeneous nucleus
forming as a spherical cap on a planar surface,
 understand the term wetting, or contact angle, q with respect to this geometry.
 prove that the critical nucleus size, r* is the same for both heterogeneous and
homogeneous nucleation,
 derive an equation for DG*
which takes into accounts nucleus geometry via the shape factor, S(q), and in so doing show that the energy barrier
to nucleation, DG* is lower for heterogeneous
nucleation than for homogeneous nucleation for all contact angles less than 180°,
 explain why the wetting angle is a measure of the ‘efficiency’ of a particular
nucleation site.
 write an expression relating the critical volumes of heterogeneous and homogeneous
nuclei.
The main example considers a solid cluster forming on a mould wall. An exercise is
provided in which the student is asked to identify on a diagram the newly created
interfaces (i.e. solidliquid and solidmould) and the destroyed interface (liquidmould).
These extra interfacial energy terms are incorporated into equation (2), such that:

(5) 
The student is then taken stepbystep through the manipulation of replacing the volume
and area terms for a spherical cap, radius r and wetting angle, q
until arriving at the final expressions:

(6) 
and

(7) 
where the shape factor S(q) is given by:

(8) 
(In an additional question, the student is asked to plot S(q)
as a function of q.)
It is stressed that the critical radius r* is the same for both homogeneous AND
heterogeneous nucleation. However, due to the shape factor, the volume of a critical
nucleus (and thus DG*) can be
significantly smaller for heterogeneous nucleation, depending on the wetting angle, q. The importance of
q in determining the ‘efficiency’ of a nucleation site is given special
attention.
In this section, the theories developed earlier in the module are developed to account
for nucleation during solidstate transformations. The most important difference is the
need to incorporate a misfit strain energy term, DG_{s} into the expression for the excess free energy of a
cluster/nucleus. As an illustration of misfit strain, the student is shown this diagram,
for which the relative lattice parameters of the a and b phases can be altered.
Using homogeneous nucleation as an example, the expressions for r* and DG* are shown to include the misfit strain
energy terms:

(9) 
and

(10) 
where g_{ab} is the ab
interfacial energy. Note that for solid state transformations, such as precipitation, the
volume free energy, DG_{v} is no longer simply
proportional to the undercooling, DT. There is no
attempt in this package to substitute DG_{v} by
the appropriate (and more complex) expression.
The importance of grain boundaries in acting as nucleation sites in solid state
transformations is considered. The geometry of a double spherical cap is used to derive
expressions for the critical values, r* and DG*
by incorporating the strain energy term, D G_{s}
into equations (6) and (7) as follows:

(11) 
and

(12) 
where S’(q) is the shape factor for a double
spherical cap (simply double that of a single spherical cap).
In the first 3 sections, we have been primarily concerned with energy balance equations
for the formation of a single cluster/nucleus. In this final section, we consider the
system as a whole and in so doing, obtain an insight into how the rate of nucleation, I
might vary with DT, the undercooling. This has real
practical significance as nucleation rate is a key factor in determining the shape of TTT
and CCT diagrams, etc.
The introductory page defines I=bn*, where n*
is the steadystate population of critical nuclei (m^{3}) and b is the rate at which atoms join critical nuclei (s^{1}),
thereby making them stable. Nucleation rate therefore has units of nuclei m^{3}s^{1}.
These two components are considered in turn.
Firstly, the steadystate population of critical nuclei is shown to be given by:

(13) 
where N is the number of potential nucleation sites per unit volume and
DG* is the critical free energy of nucleation.
The derivation of this equation is provided in the form of a ‘sidebranch’ for
students who need/want to know its background.
Since we are interested in the effects of undercooling on nucleation rate, equation (4)
is substituted for DG * in the above equation. n*
is thus dependent on both the absolute temperature, T and the undercooling, DT, giving rise to the type of graph shown here.
The rate at which atoms join critical nuclei is given by:

(14) 
where w is a relatively temperature
independent term incorporating vibrational frequency and the area to which atoms can join
the critical nucleus and Q is an activation energy for atomic migration (note that b is basically a diffusiondependent term).
The product of equations (13) and (14) are then presented graphically and algebraically
to give an expression for I.
It is stressed however, that in most practical situations, the nucleation process is
complete well before I reaches a maximum and that in reality, we are interested in
a much smaller range of values. When plotted on such a scale, the concept of a nucleation
‘explosion’ becomes apparent.
The section continues by looking at some of the extra considerations for solidstate
transformations, most notably the effect of misfit strain energy on the nucleation rate.
A final page presents an overview of the subject and allows the user to see how many of
the important variables interact to effect nucleation behaviour.
Bibliography
The student is referred to the following resources in this module:
Porter, D.A. and Easterling, K.E., Phase Transformations in Metals and Alloys,
Van Nostrand Reinhold, 1981
Ashby, M.F., and Jones, D.R.H., Engineering Materials 2, Pergamon, 1986
Brophy, J.H., Rose, R.M. and Wulff, J., Thermodynamics of structure, Wiley, 1964
Aaronson, H.I., Lectures on the theory of phase transformations, AIME, 1975
