In condensed systems, like those where a new solid phase appears within another, variations in volume and deformations are produced whose energy should be taken into account.
Consider a system in which only one α phase is stable above the melting temperature Tf and that below Tf, two phases (α and β) are stable. The total variation in free energy associated with the formation – via homogeneous nucleation – of a β spherical precipitate in an α matrix, ∆G(r), is:
\[\triangle G(r)=V(\triangle G_{V}+\triangle G_{\epsilon})+S\gamma \tag{1.13} \]
where ∆GV is the variation in free energy by volume unit that results in the formation of a particle with radius r, ∆Gε is the variation of the strain energy of the matrix and γ the surface energy. Considering a spherical particle with radius r, we have V=(4/3)πr3 and S= 4πr2.
The values for the radius of a solid spherical nucleus of critical size (r*) and the energy barrier associated with the formation of critical sized nuclei ( ∆G*) can be found with the following expressions:
\[r^{*}=\frac{\text{2}\gamma}{(\triangle G_{\nu}+\triangle G_{\epsilon})} \tag{1.14} \]
\[\triangle G^{*} =\frac{\text{16}\pi\gamma^{3}}{3(\triangle G_{\nu}+\triangle G_{\epsilon})^{2}} \tag{1.15} \]
A coherent interface or a coherent particle is a situation when there is a perfect balance between the crystallographic planes and directions, through the interface that separates the nucleated particle and the matrix. As an example of this type of interface we can cite the low angle grain boundary. As the total energy of deformation associated with the particle tends to increase with size, the particle can lose coherence. An example of this is a high angle grain boundary. Figure 2.1.8 presents a diagram showing low and high angle grain boundary
In the case of a coherent interface, the energy of the deformation of the particle isn’t particularly dependent on the format but in the case of a noncoherent interface the format is quite important as seen in Figure 2.1.9 . In this figure the precipitate is considered to be an ellipsoid of revolution with semi-axes r1 and r2.
Figure 2.1.8 – Low and high angle grain boundaries in solids
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Figure 2.1.9 – The energy for deformation of a solid matrix is maximum for the spherical shape.
Because of the imparted restrictions, the velocity of nucleation I will be much lower, if compared to liquid-solid transformation. In fact, solid phase transformations nucleate extremely slowly to negligibly (for practical effects), close to melting temperature Tf, requiring supercooling (see table at the end of the previous post Homogeneous Nucleation: Nucleation speed in the Liquid-Solid transformation) to increase the transformation speed.
1Boundary of small angle: misalignment of atomic planes that separates two regions with different orientations in a crystal, with small disorientation angle (a few degrees).
1F. R. N Nabarro, Proc. Phys. Soc., 52, 90 (1940).
