# Controlled supersaturation

We must, from the beginning, dispose of a precise number N_{0} of seeds. Therefore, if L is the intended size of the crystals and M is the crystals’ intended mass, then, in these conditions:

The seeds can result from:

- 1) seeding with N
_{o}seeds of size L_{o}, so with mass N_{o}a L^{3}o p_{c}; - 2) an initial primary nucleation of duration t.

The supersaturation over this initial period is:

Q: thermal power (W) r: vaporization heat (J. kg^{-1}).

The crystal mass is:

At time t, the number of crystals JdT have appeared at time т and have grown over time (t - т).

J is a function of AX = X - X* with X = M_{s} / M_{e} and dM_{s} = -dM_{e}.

The logical sequence of the calculations is as follows:

In other words:

The Runge-Kutta method of order 4 is suitable to study nucleation variations according to time (see Appendix 1).

In fact, d0/dt results in a decrease when the temperature 0 decreases (see exchanger theory). Similarly, power Q also decreases.

**Figure 4.12. ***(1) “Natural” program; (2) controlled program*

Finally, AX decreases due to the rapid increase in dM_{c}/dt with t, so that J nucleation reaches a maximum before decreasing until AX, having a moderate value, is only of use to crystal growth.

After this instant, d(AX/dt) moves towards zero, which provides the desired values for dT/dt and Q. The т parameter, which is the instant of crystal formation, has reached limit T_{lim}, so that, for t > T_{lim}, the J nucleation is very low.

According to Figure 4.12 in the case of non-seeded nucleation, supersaturation control leads to a reduction in the spike amplitude of primary nucleation.

Similarly, crystallization after seeding leads to a variation in the neighboring supersaturation of curve (2), for which the supersaturation maximum has been weakened significantly.