The crystallization takes place in two stages: in the beginning, nucleation center arise, and then nucleation center grow. There are two mechanisms for the nucleation of the nuclei: crystallization centers:

- Homogeneous (spontaneous, spontaneous without extraneous particles);
- Heterogeneous (not spontaneous, with the participation of foreign impurities at the ready interfaces.)

The growth of crystallization centers does not depend on the method of nucleation of crystallization centers and obeys general laws.

** **The homogeneous crystallization theory.

When homogeneous formation of nucleation centers, there are no foreign interfaces. The source of the nucleation center is the phase fluctuations temporarily creating clots with signs of long-range order. However, not every fetus can continue its growth. The grow only those whose size is more critical, and the rest are absorbed by the liquid.

What is this critical size and what does it depend on? To begin the crystallization, it is necessary to sub cool with respect to the temperature Т_{0}. The crystallization process is not a purely volumetric process, but is determined by the properties of the interface between the liquid melt and crystalline nucleation centers. The change in free energy associated with the formation of interfaces is positive and it counteracts the phase transition process, in addition, one should take into account the change in free energy, which is the result of elastic deformation of the nuclei during the phase transition. Thus, the total free energy of nucleation is the sum of three terms:

ΔG= -ΔGV+ΔGS+ΔG_{деф}

where ΔG_{V} – volumetric component, ΔG_{S} – surface component,

ΔG_{деф} – component, due to the energy of elastic deformation under structural change. The last term, in view of its smallness in the case of phase transitions, the pairs-the crystal and the melt-the crystal can be neglected. And in this case the equation has the form:

ΔG= -ΔG_{V}+ΔG_{S.}

If we take a particular case, when an nucleation center of the spherical shape is formed with a radius «r», Δq_{v }is the change in free energy per unit volume;

Δq_{s} – the change in free energy per unit area of the interphase boundary, i.e., this is in fact the surface tension.

So: ΔG_{V}=4/3πr^{3}Δq_{v}; ΔH_{пл}= ΔH_{пл} – TΔS_{пл}; ΔS= ΔH_{пл}/T.

Substituting into the formula, we get:

Δq_{v} = ΔH_{пл} – TΔS_{пл} = ΔH – T ΔH_{пл}/T_{пл} = ΔH_{пл}/T_{пл};

ΔG_{V} = 4/3πr^{3} H_{пл}/T_{пл}ΔТ;

ΔG_{S} = 4πr^{2}Δq_{s} = 4πr^{2}σ.

We substitute the values in the total energy balance, we get:

ΔG = -4/3πr^{3} ΔH_{пл}/T_{пл}ΔТ + 4πr^{2}σ.

Hence, ΔG is a function of the size of the nucleation center:

In Fig. 1 shows the volume and surface components of energy ΔG_{v} and ΔG_{s}, as well as their resulting ΔG – free energy of nucleation. Because of the opposite of the signs ΔG_{v} and ΔG_{s. }it follows that the curve ΔG first rises, passes through the maximum and at the end, is lowered. The critical radius of the crystallization nucleus r_{кр} corresponds to the maximum point on the curve ΔG. The embryo is smaller than this quantity and is unstable and dissolves again, because its increase is associated with an increase in free energy. Above the critical radius r_{кр} the embryo on the contrary is stable and can grow, since ΔG decreases. Particles – nuclei of crystallization with a value r<r_{кр} are called sub-embryos, and embryos with a value r>r_{кр} are called super-embryos. The maximum of the function given above is determined by taking the first derivative and equating it to zero.