Essential Physics of Low Temperature and Freezing - Freeze-drying of Pharmaceuticals and Biopharmaceuticals - page 37

water-soluble glasses, which forms an important feature of pharmaceu-
tical freeze-drying, is explored in detail in several later chapters.
An analysis of crystallisation rates is conveniently performed in terms
of the so-called time-temperature-transformation (TTT) curves, which
relate the time taken to crystallise a given fraction of the undercooled
liquid or the supersaturated solution to the temperature. Experimen-
tally, the crystallisation rates are measured by quenching the liquid
phase to some predetermined temperature T and measuring the time
taken for the solid to crystallise at that temperature, either by monitor-
ing the latent heat of crystallisation or by microscopic observation. The
volume fraction f(T) that crystallises out in time t is given by one form
of the Avrami equation:

f(T) ¼ (pJu 3 t 4 )/3

(6)

where J is the rate of nucleation and u the rate of crystal growth.
Examples of TTT curves for glycerol from its melt and of ice from
undercooled water are shown in Figure 13. The so-called nose of the
TTT curve indicates the maximum crystallisation rate that can be
attained. Any cooling rate high enough not to intersect the TTT curve
will, therefore, lead directly to vitrification. In the case of water, the
maximum rate of ice growth occurs near 131Catca. o 1nsnm 1 ,
indicating that cooling rates in excess of 50001/s need to be applied in
order to prevent freezing. Cooling rates ( dT/dt) are proportional to
(k/C p d), where k is the thermal conductivity, C p the heat capacity and d
the density. Both water and ice have low thermal conductivities and
large heat capacities. It therefore becomes difficult to achieve high
cooling rates, especially for bulk samples, but vitrification has been
achieved with specialised quenching techniques applied to microdroplets
and thin films. 31, 165
Inspection of simple binary solid-liquid phase diagrams, as shown in
Figure 1, indicates that, if a solution, the composition of which lies to
the right-hand side of the eutectic point, is cooled, the salt, and not
water, will be the component that crystallises first. A TTT curve for salt
precipitation could thus be constructed, but the rates would be much
lower than those shown in Figure 13 for ice crystallisation. The impor-
tance of salt crystallisation during freeze-drying is further elaborated in
Chapter 5.
Yet two further aspects associated with ice growth in a freezing solution
deserve mention. For the sake of simplicity, we consider the advance of a
planar ice front during controlled cooling, shown in Figure 14 for a 2%
w/w solution of NaMnO 4. A thin film is held between two thin glass plates
set 10 mm apart and is seeded with an ice crystal and cooled inwards from