The nucleation process that occurs in a homogeneous phase is called homogeneous nucleation. In homogeneous nucleation a particle can nucleate in any given position in the system as there are no prefered nucleation sites present.
Reactions in Condensed Systems: consider the solidification of pure metal; also consider that the solid particles are spherical – a rigorous study would demand consideration of the shape of the particle, together with the associated problems of atom aggregation to a solid surface; as these factors have minimal influence over the speed of nucleation it can be said with reasonable certainty that the solid particles obtain a simple spherical form and that these atoms (or molecules) aggregate on these particles over its entire surface. On an atomic scale the precipitation process starts with the aggregation of adjacent atoms. Figure 2.1.4. represents a nucleation schematic of a solid particle in a liquid matrix.
Homogeneous Nucleation: Liquid-Solid Transformation Equations (I)
The following are the fundamental equations considered important for nucleation models and the growth of spherical nuclei in a liquid. Remember that this is homogeneous nucleation where no preferred nucleation sites exist for nucleus formation.
Here V is the volume of a single particle and S is its surface area. The total variation in free energy associated with the formation of a particle of radius r, ∆G(r), is:
\[ \triangle G(r)=V \triangle G + S \gamma \tag{1.2} \]
where ∆Gv is the variation in free energy per unit of solidified volume and γ is the surface energy or energy per unit of area. As the particle is spherical with a radius of r, we have the following \[V= \frac{4}{3} \pi r^{3}\] and \[S= 4 \pi r^{2}\] Thus: \[∆G(r)=\frac{4}{3}\pi r^{3} ∆G_{v} + 4\pi r^{2} \gamma \tag{1.3} \]
Each term in this equation is represented in Figure 2.1.4, that shows the variation of ∆G(r) with the radius of a solid particle.
Also in Figure 2.1.4, it can be seen that r* is the maximum (1)1 of Eq. (3) and that ∆G* is the correspondent ordinate. Once again substituting in Eq. (3) the maximum obtained gives ∆G*. The derived equations are:
\[r^*=\frac{2\gamma}{∆Gv} \tag{1.4} \]
and \[∆G^*= \frac{16\pi \gamma^{3}}{3∆Gv^{2}} \tag{1.5} \]
Particles with r=r* are called critical nuclei or critical size nuclei. Particles with r < r*, are called embryos; particles with r > r*, are called nuclei. Particles with a radius larger than r* lower their free energy by way of growth by accession of atoms (or molecules); particles with a radius smaller than r*, to lower free energy, tend to dissolve because atoms (or molecules) will migrate off of the particle, returning to the liquid matrix. Particles where r=r* can grow or shrink, because both processes will lower free energy. The quantity ∆G* represents the energy barrier associated with the formation of a critical sized nucleus.
1The maximum point can be found by making it equal to the first derivative of Equation 3, which is a function of r, to zero.
