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:

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 temperature-dependence 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 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:

and

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

and

where g_ab is the a-b 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:

and

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:

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.

Bibliography

The student is referred to the following resources in this module: