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Materials Science on CD-ROM User Guide

Nucleation in Metals and Alloys

Version 2.1

Andrew Green, MATTER
Phil Prangnell, UMIST/ Manchester Materials Science Centre

Assumed Pre-knowledge

Before using this package, it is assumed that the student is familiar with the thermodynamic relationship DG=DH-TDS. 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, DGv and undercooling, DT,
  • recall that r* 1/DT and DG* 1/DT 2.

The section starts by explaining that a driving force for solidification, DGv exists below the equilibrium melting temperature, Tm and that this is approximately proportional to the degree of undercooling DT:


where DHv 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 DGv and DHv are NEGATIVE below Tm is used.)

Hv are NEGATIVE below Tm is used.) Hv are NEGATIVE below Tm is used.) Hv are NEGATIVE below Tm 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 solid-liquid 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:


where gSL is the solid/liquid interfacial energy.

The student is shown step-by-step that from this expression, the critical radius r* (defined as the radius at which DG(r) is maximum) is given by:


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


The temperature-dependence of these terms, i.e. that r* 1/DT and DG* 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 sub-critical cluster. If the button is released before r = r*, the cluster is shown to re-dissolve, 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.

Heterogeneous Nucleation

This section continues to look at the liquid-sold 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. solid-liquid and solid-mould) and the destroyed interface (liquid-mould). These extra interfacial energy terms are incorporated into equation (2), such that:


The student is then taken step-by-step 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:




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


(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.

Solid-State Transformations

In this section, the theories developed earlier in the module are developed to account for nucleation during solid-state transformations. The most important difference is the need to incorporate a misfit strain energy term, DGs 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:





where gab is the a-b interfacial energy. Note that for solid state transformations, such as precipitation, the volume free energy, DGv is no longer simply proportional to the undercooling, DT. There is no attempt in this package to substitute DGv 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 Gs into equations (6) and (7) as follows:




where S’(q) is the shape factor for a double spherical cap (simply double that of a single spherical cap).

Nucleation Rate

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 steady-state 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-3s-1. These two components are considered in turn.

Firstly, the steady-state population of critical nuclei is shown to be given by:


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 ‘side-branch’ 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:


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 diffusion-dependent 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 solid-state 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.


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


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Copyright of The University of Liverpool 2000