Essential Physics of Low Temperature and Freezing - Freeze-drying of Pharmaceuticals and Biopharmaceuticals - page 28

at a given temperature is given by

n(r*) ¼ n liquid exp( DG c /kT) (4)

where n liquid is the number of molecules per unit volume in the liquid
phase.
Apart from the dimensions of critical nuclei under varying conditions,
it is also important to consider the kinetics of nucleation, especially
relative to the kinetics of crystal growth. The nucleation rate (J)is
commonly estimated by means of transition state theory. With the aid of
the ''best'' available values for transport properties of undercooled
water, 21 it has been estimated that in the neighbourhood of 401C, J
increases rapidly with decreasing temperature, by about a factor 20 per
degree; nucleation is thus a well-defined event that is hardly affected by
the rate of cooling. 22
Figure 7 shows the relationship of the mass of liquid water, which is
likely to contain at least one active nucleus, as a function of temperature.
It is seen that in the range 233-243 K a temperature is reached at which 1
g of water is likely to contain at least one such nucleus, capable of

Figure 7 Mass of water (g) likely to contain one critical nucleus, as a function of
temperature, assuming a homogeneous nucleation mechanism. Reproduced
from Franks, 17 with permission from Cambridge University Press