The influence of mixing on nucleation and crystal growth

In precipitations, just as in crystallization processes, there is a competition between nucleation and crystal growth. BoUi are controlled by the relative supersaturation, but nucleation is a higher order process. The rate of nucleation increases very sharply with increasing supersaturation. At lower values of the relative supersaturation Ac the nucleation rate appears to be proportional to Ac to a power of about 5 at higher values of Ac the nucleation rate increases exponentially with Ac. 

The rate of crystal growth is determined by two effects mass transfer toward the crystal surface, and in-building of ions in the crystal lattice. The first process is first order at low concentrations, the second process is proportional to Ac to a power between 1 and 2. Both processes are in principle proportional to the particle surface area, however, the rate of in-building may be different for different crystal planes. 

In principle, mass transfer and in-building can be considered as processes in series. If both processes could be approximated as first order processes with 

When mass transfer is not rate determining, a constant crystal growth rate may be expected. The crystal diameter would increase linearly wiUi time. 

However, when the mass transfer is governed by steady state diffusion, for which the relation Sh = 2 applies, the diffusion rate, and consequently the growth rate, are inversely proportional to the particle size  

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