# Crystal Formation and Growth

## Crystal formation

### Primary nucleation

All solutions contain agglomerates of solute that are increasingly numerous and significant on approaching the saturation temperature. These are known as crystal embryos.

As a result of a Ap supersaturation, the Gibbs energy variation of the embryonic solution (for crystal embryo formation) reaches its maximum. The crystal embryos with the corresponding size are the crystal seeds or nuclei. We should note that it is only by means of statistical fluctuations that the Gibbs energy can increase to its maximum. Next, the Gibbs energy of the seed-solution system naturally decreases as the nucleus increases in its size to form a crystal.

The free enthalpy variation corresponding to the crystal embryo formation of diameter d is the difference between the surface energy created and the over-potential Ap lost in the solution.

Q: molar volume (m^{3}.kmol^{-1})

y: superficial energy of the crystal embryo (J.m^{-2}).

*We refer to the publication of Nielsen et al. [NIE 71 ] for numerous values of the surface energy* y.

Дц: excess of chemical potential relative to saturation (J.kmol^{-1}).

By definition:

a: solute activity in solution

a*: solute activity at equilibrium

In crystallization operations:

*a*: relative supersaturation

Finally:

The maximum G is reached when: hence:

Coefficient v is added. This is the number of ions corresponding to the possible electrolyte dissociation.

In his work, Mutaftschief [MUT 01] gives the following expression for primary nucleation:

q_{re} is the partition function for the replacement of movement characteristics by their equivalents in the agglomerate. The corresponding parameters express transfers by a volume, vibrations by a surface and rotations by a length. In other words:

If, somewhat audaciously, we accept that V_{al} and l are similar for the agglomerate and the crystal, and if we also accept that V is in the order of molecular volume, we obtain:

The product a*n_{1} is the number of molecules that come into collision with the agglomerate per unit of time. Here again, and still in a somewhat risky manner, if we apply the Ranz formula for the exchange of material between the ambient environment (the solution) and a small molecule (the agglomerate), we obtain:

D: diffusivity (m^{2}.s^{-1})

c: concentration (molecules.m^{-3})

Hence:

We have written c rather than (c - c*) as here this is the molecular flow arriving in the agglomerate.

The number of molecules i present in the critical agglomerate is:

The first factor of the Mutaftschief formula is:

And finally:

which is equivalent to:

Primary nucleation can only occur for a relative supersaturation о greater than or equal to 0.5.

Example.-