Crystallization in Monodisperse Emulsions with Particles in Size Range 20–200 nm - Food Colloids: Self-Assembly and Material Science - page 407

1 2

0.4 3

1

-5 0 5 10 15 20 25 30 35

4

-0.2

Temperature / degrees C

Figure 6 Solids content plotted against temperature for the 130-nm PGE1 sample.
Arrows indicate the direction of temperature change. Label 1 indicates begin-
ning of crystallization; 2, initiation of crystal melting; 3, the point where
information was lost due to increased attenuation of the sound; 4, the melted
material is different from that at the same temperature on the cooling cycle

theory using the methods described elsewhere. 17 Examination of Table 1 in
Ref. 17 indicates that all types of nucleation can be summarized on a single
graph in which ln[(1 f )/ f ]/T abs is plotted against time. Here f is the volume
fraction of the oil phase, which comprises crystals, and T abs is the absolute
temperature. In this case the slope of the curve is

exp DG
k B T abs

¼

Nk B
h

exp a DS
R

J
T abs

k x
L x T abs

¼ ð 1 Þ

where J is the nucleation rate for crystallization, N the number density of
catalytic impurities, k B Boltzmann's constant, h Planck's constant, exp( aDS/R)
the probability that a fraction a of the molecules is in the right conformation to
crystallize, and R the gas constant. The loss of entropy DS on incorporation of
material in a nucleus is given by DS ¼ DH/T m , where DH is the enthalpy of
fusion and T m is the melting temperature. The quantity DG* is the Gibbs
activation energy for the formation of a spherical nucleus. Here we have
introduced an arbitrary dimensionality x, which conventionally is 2 for surface
nucleation and 3 for volume nucleation, and L is a characteristic length corre-
sponding to the diameter for spheres. A correspondingly appropriate nucleation
rate constant k x is required.
The analysis is complicated by the fact that the temperature is also changing
slowly. For this reason the measured temperature is incorporated in Figure 9,
while in Figure 10 we plot ln[(1 f )/ f ]/T abs against time. The first thing to
notice is that, for the two Caflon emulsions, which were presumably identical in
all respects apart from particle size, the slopes whose measured ratio is 0.1001/
0.9544 ¼ 9.544 would have the theoretical ratio (330 10 9 ) x /(123 10 9 ) x ,
which for x ¼ 2 is 7.2, and for x ¼ 3 is 19.3. So we could conclude from this