# Metastable zone: supersaturation established slowly

Establishing the supersaturation slowly means over-cooling the solution beyond saturation at velocity v such that [NYV 85]:

Supersaturation accessible prior to nucleation can be represented by the interval separating the saturation curve and the nucleation curve as seen in the temperature-concentration system of coordinates.

**Figure 2.2. ***Metastable zone*

Massive nucleation occurs once the metastability threshold is crossed. However, often AT_{max} and Ac_{max} do not correspond to the same nucleation curve. In Figure 2.2, the hatched area is the metastable region (labile).

# Measurement of nucleation order n

The empirical laws of nucleation and growth adopted by Nyvlt *et al. *[NYV 85] are, respectively:

As supersaturation Ac increases consistently with time we will assume, like these authors, that:

Accordingly, if t is the duration of time that has passed since the start of the experiment, then:

The concentration of liquor is not sensitive to the precipitation of a few seeds, with supersaturation Ac(t) not dependent on c* (t). Initially, Ac(o) = 0 . Consequently:

Hence:

The size of a crystal formed at time т that has grown until time Tc at which the seeds become visible, is:

The total mass of the crystals precipitated at time t_{c} is:

Each crystal formed at time t and grew until time t_{c}. Writing out:

With:

Time t_{c} has passed while the difference in temperature from saturation reached value AT_{max}. Therefore, the moment at which the first crystals become visible is:

The precipitated mass is:

Writing out:

We obtain:

Moving to logarithms:

Writing out:

We obtain:

By varying v and measuring M and AT_{m}ax , we can deduce m since K and p are already known. Finally, the true order n of the empirical nucleation relation can be deduced. The measurement of M is made by passing the whole solution through a Coulter counter and not neglecting Ln M.

Unfortunately, since this is a delicate process, today we prefer to deduce nucleation according to the theory of a homogenous and continuous crystallizer.