Nucleation

From classical theory, nucleation occurs in locations where there are fluctuations in density of chemical concentrations or defects, e.g. interface and inclusions. In a pure system, the overall driving force for nucleation is provided by the free energy differences between the newly formed phase and the initial phase (and sometimes intermediate metastable states should be considered). However, this energy is also used for the creation of the new surface between the phases. If the nucleus is constrained by the surrounding initial phase, additional strain energy should also be

considered. For homogeneous nucleation of α ferrite in a γ austenite matrix, the overall free energy change can be calculated from [4, 20]:

where the nucleus is assumed to be a sphere shape with a radius of , is the driving force for the formation of ferrite from austenite, is the strain energy per unit volume, and is the interfacial energy per unit area between ferrite and austenite.

The critical free energy barrier occurs at a critical radius when the free energy change is maximum, which is shown in Figure 2.18 and can be calculated by the integral of Equation 2.14:

Figure 2.18: The free energy change during the homogeneous nucleation of a spherical nucleus with a radius of r. G* and r* are the critical free energy barrier and

the critical radius for the nucleation [4].

Therefore, the critical radius and the critical free energy barrier can be calculated, respectively:

If a nucleus is smaller than the critical radius, it will be dissolved for the free energy reduction. If a nucleus is larger than the critical radius, the growth will happen to reduce the free energy. Therefore, the number of nuclei which are energetically favoured to grow, can be expressed as a probability multiplied by the total number of atoms in an initial vapour phase [21]:

where is the total number of atoms in the initial phase, is the statistical distribution function for nuclei of critical nuclei, and is the Boltzmann constant.

This equation is based on a vapour phase. After many modifications, the homogeneous nucleation rate in a solid phase can be expressed as [4, 20]:

(

)

where is the number of atoms per unit volume, h is the Planck constant, is the vibration frequency, is the activation energy of the transfer of atoms across the nucleus/matrix interface, and is the Zeldovich factor, which is [22]:

However, in reality, heterogeneous nucleation which happens at high energy sites e.g. defects, interface, or impurities, is much more common. In the case of low carbon steels, prior austenite grain boundaries are favoured sites for the heterogeneous nucleation of proeutectoid ferrite. From Figure 2.19, there are three types of austenite grain boundary sites for ferrite nucleation: grain faces, grain edges, and grain corners. Nucleation at each type of site results in a different interfacial

energy change, and thus the probability of nucleation is also varied. However, the densities of the grain boundary sites are also different. Therefore, the nucleation rate at each grain boundary site should be calculated separately. The allotriomorphic ferrite grain boundary nucleation rate is [4, 23]:

( ( )

)

where j denotes grain faces, grain edges or grain corners (f, e, or c), is a site factor about density of nucleation sites per unit area of grain boundary, is a shape factor about austenite/ferrite interfacial energy per unit area, is the activation energy for self-diffusion of iron, with a typical value of 240 kJ/mol, and is the critical energy for nucleation. If the strain energy caused by nucleation is ignored, the critical energy for nucleation can be expressed as [22, 23, 24]:

where is the free energy per unit volume for ferrite nucleation from supersaturated austenite, and is the austenite/ferrite nucleus interfacial energy per unit area, which is assumed not to vary with interfacial orientation or alloy chemistry.

The total nucleation rate is the summation of the nucleation rates at the three types of sites:

The number of grain faces nucleation sites per unit area of boundary can be calculated by assuming that each atom can act as a nucleation site, but this number should be halved because there are two sides of a boundary [4, 23]:

where is a factor to describe active fraction of the total number of grain face sites, and is the atomic spacing, which is typically taken as m. The ratio of face to edge sites and the ratio of edge to corner sites are both (where is the average austenite grain size), and thus the number of grain edge sites and the number of grain corners sites can be expressed as [4]:

Figure 2.19: Schematic diagram of grain faces, grain edges, and grain corners in a typical austenite grain [25].

From Parker [4], the contribution of the three types of nucleation sites to the overall nucleation rate vary with temperatures and austenite grain size. Generally, at a temperature slightly below , the small undercooling makes corner nucleation, which requires a low activation energy barrier, to dominate the rate. With decreasing temperatures, edge nucleation becomes more important, and then face nucleation takes over. A large austenite grain size reduces the fraction of grain edges and grain corners, and thus face nucleation becomes more significant with increasing austenite grain size. The nucleation rate calculation is also largely dependent on the chosen values for and . The values suggested by Reed and Bhadeshia [23] are listed

in Table 2.1. Parker [4] made further development based on the work of Reed and Bhadeshia, and the values of site and shape factors are listed in Table 2.2.

Table 2.1: Values of site and shape factors for nucleation rate calculation [23]

Factor Value

K1

Table 2.2: Values of site and shape factors for nucleation rate calculation [4]

Factor Value