Heterogeneous Nucleation

All equations remain the same for the calculation of heterogeneous nucleation rate within the framework of classic nucleation theory (CNT), but the nucleation energy and available nucleation site are different.

To a first approximation, the nucleation energy may be calculated by assuming an effective interfacial energy for each heterogeneous nucleation site.

For a rigorous treatment of heterogeneous nucleation the effect of wetting angles need to be considered.

Non-Spherical Particles and the Effect of Wetting Angle

Precipitation Module (TC-PRISMA) References

Non-spherical particles have been considered for grain boundary (GB) precipitation. Three morphologies are implemented for grain boundary, grain edge and grain corner precipitation, respectively, as shown in Figure 1. The parameter that defines the deviation from spherical shape is wetting angle (or dihedral angle), wetting angle, as described in Figure 2 and taken from Clemm and Fisher [1955Cle].

Non-spherical particles example

         (a)                                 (b)                                                          (c)

Figure 1: Non-spherical particles (β) that precipitate at grain boundaries of matrix phase(alpha) (a) grain boundary (two-grain junction) (b) grain edge (three-grain junction) (c) grain corners (four-grain junction). Images from [2004Zan].

The nucleus at the grain boundary.

Figure 2: The nucleus at the grain boundary. Image from [1955Clem].

Some physical parameters, mainly in nucleation models, have to be modified for the effect of the wetting angle. The following discussions contribute to this effect.

Shape Factors

Following Clemm and Fisher, the eliminated grain boundary area between eliminated grain boundary area grains due to the formation of nucleus of a new phase β is:

[Eq. 1]          eliminated grain boundary area between grains due to the formation of nucleus of a new phase

Where radius of curvature of the nucleus is the radius of curvature of the nucleus. The surface area of the nucleus is:

[Eq. 2]          surface area of the nucleus

and the volume of the nucleus is

[Eq. 3]          volume of the nucleus

The expression of a, b, c in Eq. 1 and Eq. 3 for grain boundary (two-grain junction), grain edge (three-grain junction) and grain corner (four-grain junction) can be found in the paper by Clemm and Fisher [1955Cle].

Critical Radius and Activation Energy

The energy change due to the formation of the new particle is thus

[Eq. 4]          energy change due to the formation of the new particle

where interfacial energy and grain boundary energy are the interfacial energy and grain boundary energy, respectively.

Then the critical radius should be

[Eq. 5]          Critical Radius and Activation Energy

And the activation energy barrier for nucleation is

[Eq. 6]          activation energy barrier for nucleation

The interfacial energy, grain boundary energy and wetting angle can be related as

[Eq. 7]          interfacial energy, grain boundary energy and wetting angle relationship

i.e.

[Eq. 8]          grain boundary energy equation example

which can be replaced into Eq. 5 and Eq. 6.

[Eq. 9]          Critical Radius and Activation Energy

[Eq. 10]          activation energy barrier for nucleation

The bulk, spherical precipitation equation is recovered by:

[Eq. 11]          bulk, spherical precipitation equation is recovered

so that

[Eq. 12]          Critical Radius and Activation Energy

[Eq. 13]          activation energy barrier for nucleation

And also the grain boundary precipitation with spherical shape (with weighting angle 90 degrees) follows Eq. 12 and Eq. 13 by k equals cosine theta equals zero.

Zeldovich Factor

The Zeldovich factor is modified as

[Eq. 14]          Zeldovich factor modification

with Zeldovich factor original value the original value, and volume factor the volume factor that is the ratio of the c in Eq. 3 to the spherical shape factor spherical shape factor .

[Eq. 15]          volume factor equation

Impingement Rate

The surface area factor in impingement rate calculation is changed from surface area factor in impingement rate calculation to surface area factor in impingement rate calculation.

Nucleation Site Density

In each time step, the occupied grain boundary area

[Eq. 16]          occupied grain boundary area

is deducted when calculating available nucleation site density.

Growth Rate

The radius radius defined in previous equations is the curvature of the precipitate surface (the Big radius) as shown in Figure 2. It is exactly the definition in the growth rate whose coarsening consideration relates to the curvature effect. Therefore, the growth rate equation can be directly used with radius being defined here. Again, for bulk, spherical precipitation it automatically corresponds to the particle radius.

Output

The volume of the precipitate should be calculated appropriately as volume of the precipitatewith curvature of the particle surface being the curvature of the particle surface as defined above. For output of the particle size (mean radius, critical radius and particle size distribution) in the user interface, it seems better to use the radius of the eliminated grain boundary area, i.e.

[Eq. 17]          radius of the eliminated grain boundary area

With alpha defined in Eq. 1.

For example, the size of the precipitate at grain boundary (two-grain junction) is the radius of the lens shown in Figure 2.

radius of the lens

The Number of Available Heterogeneous Nucleation Sites

An example of tetrakaidecahedron approximation of grains

Figure 3: An example of tetrakaidecahedron approximation of grains.

The number of different nucleation sites is dependent on the shape and size of grains in the matrix. Assuming all grains are equally large tetrakaidecahedra with the distance between one pair of square faces as H, and that between the other two pairs as D we obtain the densities density of grain boundary area, density of edge length, and density of grain corner number for the grain boundary area, edge length, and grain corner number, respectively in

[Eq. 18]          density of grain boundary area

[Eq. 19]          density of edge length

[Eq. 20]          density of grain corner number

where aspect ratio, defining the degree of elongation of the grains is the aspect ratio, defining the degree of elongation of the grains.

By default, the aspect ratio is set to 1 and the densities are then the same as obtained by Cahn [1956Cah]. Once the densities are known, if taking boundary thickness as one atomic layer, the available nucleation sites can be calculated by

[Eq. 21]          available nucleation sites equation

where molar volume of the matrix phase is the molar volume of the matrix phase and Avogadro number is the Avogadro number.

For a crystalline material, given a dislocation density dislocation density, the number of nucleation sites at the dislocations number of nucleation sites at the dislocations can be calculated with the same form as in

[Eq. 22]          number of nucleation sites at the dislocations

References

[1955Cle] P. J. Clemm, J. C. Fisher, The influence of grain boundaries on the nucleation of secondary phases. Acta Metall. 3, 70–73 (1955).

[1956Cah] J. W. Cahn, Transformation kinetics during continuous cooling. Acta Metall. 4, 572–575 (1956).

[2004Zan] L. Zang, Lecture 13: Heterogeneous Nucleation: Effects of Grain Boundaries and Surface Defects, The University of Utah, (2004), (available at http://www.eng.utah.edu/%7B$~$%7Dlzang/images/lecture-13.pdf).