Essential Physics of Low Temperature and Freezing - Freeze-drying of Pharmaceuticals and Biopharmaceuticals - page 29

initiating freezing. Although estimates vary considerably, some dimen-
sions and concentrations of such nuclei in water are summarised in
Table 1. 17 Thus, if a practical aim is to stabilise undercooled water, then
the total mass should be subdivided into small droplets, say of 5-10 nm
radius; this would yield ca. 10 18 drops g 1 . Referring to the results in
Table 1, the proportion of drops susceptible to freezing would then
increase from 10 450 at 263 K to 10 18 at 233 K. 175
It will now be apparent that the size of an active nucleus decreases
with decreasing temperature, thus increasing the probability of their
existence at low subzero temperatures. At the same time, the self-
diffusion rate also decreases with decreasing temperature, thus increas-
ing the lifetime of the nucleus once it has been formed. Taken together,
these results indicate that nucleation rates increase sharply with decreas-
ing temperature in deeply undercooled solutions. In fact, for water
droplets of nm radii, there appears to be a limiting temperature, ca.
401C, referred to as the ''homogeneous nucleation temperature''
(T hom ), at which the vast majority of drops will contain at least one
active nucleus, capable of initiating ice crystal growth.
Nucleation, although it can be treated as a first-order kinetic process,
does not display the typical Arrhenius behaviour, since the rate increases
with decreasing temperature. The rate J of nucleus generation can be
adequately expressed in terms of a reduced temperature and degree of
undercooling by

J ¼ A exp(Bt y ) (5)

where t y ¼ [y 3 (Dy) 2 ] 1 , y ¼ T/T m and Dy ¼ (T m T)/T m .
Nucleation rates, when plotted according to Equation (5), confirm
that the relationship can adequately account for the process over several
orders of magnitude in J, see Figure 8. 23 The experimental window for
nucleation measurements of ice in real time is narrow, because of the
extremely high sensitivity of J to temperature and the degree of under-
cooling. In the temperature range 35 4 T 4 451C, ln J changes from
approximately 20 to 3, with a peak value of 34 at 401C. 24
Reference was made earlier to the use of undercooled water in
enhancing the long-term stability of enzymes in solution and the viabil-
ity of isolated cells. The data in Table 1 and Figure 8 confirm that clean,
undercooled water at 201C, dispersed in the form of small droplets, is
most unlikely to freeze, and can be safely used as storage medium for
many years.
First-order phase transitions can be of several types, e.g. vapour -

liquid, vapour solid. In practice, they

are brought about by changes in pressure and/or temperature. Figure 9

- solid, liquid - solid, solid -