Essential Physics of Low Temperature and Freezing - Freeze-drying of Pharmaceuticals and Biopharmaceuticals - page 26

a certain number of molecules, probably in a particular configuration,
that can promote the condensation of other molecules, to form a crystal
with its characteristically ordered structure. The actual structure of a
cluster, capable of initiating crystallisation, is still a subject of uncer-
tainty. 18 As regards the generation of ice in pure liquid water, nuclei
arise in the body of the liquid through random density fluctuations, and
nucleation kinetics can be reasonably well treated in terms of the growth
and decay rates of such molecular clusters and their ability to promote
crystal growth. The fluctuations in density originate from self-diffusion
(Brownian motion) of molecules, and the probability of nucleation
depends on the sizes and structures of high-density domains and their
lifetimes. In passing, we emphasise that nucleation of ice is of consid-
erable importance in ecological contexts, e.g. cloud physics at one
extreme, and freeze tolerance and avoidance mechanisms in living
species at the other. 18 It is beyond the scope of this topic to discuss
nucleation theory in detail, but it is instructive and relevant to industrial
freezing processes to summarise some of its quantitative aspects.
Most theoretical treatments of nucleation and crystal growth are based
on a model of stepwise addition of molecules to an embryo, up to a critical
size, at which the properties of the embryo (e.g. its surface free energy) are
equated with those of the known solid phase, i.e. the crystal. 19, 20
In the case of water and ice, such a model might be realistic in treating
the nucleation of liquid or solid from supersaturated vapour, as it occurs
in the upper atmosphere. It seems doubtful, on the other hand, whether
the model of stepwise molecular condensation can realistically be applied
to the liquid solid transition, because liquid water itself is already

extensively associated and exists as a three-dimensional distorted network
of hydrogen-bonded molecules, not too dissimilar from ice. Any mecha-
nistic model of ice nucleation, based on the condensation of individual
water molecules onto clusters of molecules with the properties of ice,
should therefore be treated with caution.
Ice nucleation is of enormous practical importance, and we therefore
summarise the basic features of the stepwise condensation model. We
take, as a starting point, a small spherical cluster of ''ice-like'' liquid
water of volume v, undercooled to a temperature T, corresponding to a
degree of undercooling DT ¼ (T m T), where T m is the equilibrium
freezing point of water (i.e. the melting point of ice). The free energy of
condensation is given by

-

DG c ¼ RT ln(p ice /p liquid ) (1)

where p ice and p liquid are the sublimation pressure of ice and the satu-
ration vapour pressure of undercooled water, respectively. For any