Interpretation of latency time
In 1991, Dirksen et al. [DIR 91] proposed an expression for latency time:
Y: superficial crystal energy (J.m-2)
Q: molecular volume (m3.molecule-1) a: supersaturation
r *: critical radius (of the nucleus) (m) d: molecular diameter (m)
|l: solution viscosity (Pa.s).
This expression accounts for latency times in the order of microseconds in a liquid such as water at 20°C. In order to obtain latency of an hour for a highly saturated sucrose solution, which would be highly viscous, we cannot use the expression of D above.
Admitting that systematically we start from an initial temperature of 2-3°C above the saturation temperature, the latency time expression must account for the cooling rate. This is what we will consider next.
To begin, let us suppose that the supersaturation of the solution is obtained in a very short time relative to the latency time. This supersaturation then remains constant, which means that the crystals’ nucleation and growth are typically constant through latency; in other words, crystals appear with a mass low enough not to disturb the supersaturation.
If т is the instant at which a seed appears, the quantity of crystals that appear through time dT and per unit volume of suspension:
These crystals grow and, admitting that the nuclei have zero diameter, the radius of a nucleus at time t is:
This corresponds to volume:
The solid volume at instant t and per cubic meter of the solution is:
If we accept that the solid phase becomes visible for:
The corresponding instant characterizes latency, which is:
Now, supposing that we cool the solution at a constant rate dT/dt = cste, supersaturation increases proportionally to time:
We accept that the equilibrium curve dc*/dT remains constant.
The number of seeds formed per cubic meter and through time dT is:
At time t, the number of crystals is:
J depends on the instantaneous supersaturation:
The seeds that have appeared at time t grow over time t — T. The growth equation (for a seed supposed to be spherical with radius r) is:
For example, if we accept the mechanism of spiral dislocations:
If we accept the mechanism by simple diffusion:
The volume of a seed at time t is:
The volume of the solid phase per cubic meter of the solution is, at instant t:
As before, the crystals become detectable when:
Nielsen et al. [NIE 71] adopted this process, but by directly measuring J0. Furthermore, they supposed that the growth rate R(t) was determined by crossing the diffusion layer (see sections 2.3.2 and 2.3.4). They found a time in the order of seconds.
Thus, with reference to Figure 2.3,
- - development during cooling corresponds to tAB
- - development with a given supersaturation corresponds to time tBC
The “complete latency” is: