diameter 4mm has about that volume. Upon cooling an emulsion of the
same milk fat, one would thus predict that droplets of about 4mm contain
one crystal. Actually it would, because of statistical variation, be zero, one,
or a few crystals. The proportion of droplets containing no crystals is about
as predicted. However, the other droplets do not contain one or two
crystals, as predicted, but many, at least 100. This must be due to secondary
nucleation. In other fats something similar is observed, though the number
of crystals often is smaller. In emulsions of paraffin mixtures, or of a
solution of hexadecane in oil, one does observe droplets with one or two
(large) crystals and no droplets with many crystals, in accordance with the
presence of a limited number of catalytic impurities, hence no secondary
nucleation would have occurred.
The phenomenon is of considerable practical importance for the size
of the crystals obtained. There is no generally accepted theory for secondary
nucleation. It appears to occur at high supersaturation in systems where
(nevertheless) crystal growth is slow.
Spinodal Decomposition. Liquid–liquid phase separation has
been briefly discussed for polymer solutions in Section 6.5; see especially
Figure 6.17. The theory applied also yields an alternative treatment for
nucleation of the new phase. This will be briefly discussed for the simplest
case, i.e., a binary mixture (solution).
The change in free energy upon phase separation only involves a
mixing term, since a transition in the physical state (crystallization or
evaporation) does not occur. In other words,DtrG¼DGmix. For an ideal
solution (see Section 2.2), phase separation cannot occur, since it implies
thatDHmixequals zero, andDSmixis always positive. For nonideal solutions,
it depends on the shape of the free energy curve whether phase separation
can occur.
Figure 14.8 gives a hypothetical example. In (a) a phase diagram is
shown (as in Figure 6.17, but the variable is now temperature rather than
solvent quality). The coexistence line or binodal, whereDGmix¼0, bounds
the region where phase separation will occur. Also the spinodal is given. For
the conditionT¼T 1 , the free energy curve is given in (b). Part of this curve
is concave towards them-axis. If now the composition of the mixture is
between A and F, phase separation into two liquids of composition A and F
can occur, since this will lead to a lowerGvalue. The total free energy then
is given by the broken line, which is tangent to the curve at two points.
These points, A and F, correspond to points atT 1 on the binodal.
The question is now whether the phase separation will occur
spontaneously. Consider the second derivative of the free energy
G^00 :ðq^2 G=qm^2 ÞT;p. The points whereG^00 ¼0 denote inflection points, in