It should be noted that Figure 14.8 gives just one example and that the
relations can be very different. The dependence of solubility on temperature
may be very weak (in many polymer solutions) or even opposite. Other
variables may be important, notably solvent quality. For many systems, the
curve ofGversusmis convex towards them-axis over the whole range,
which implies that phase separation cannot occur. For demixing of solutions
of two polymers, the same considerations hold, but the treatment is more
complicated. The treatment is not restricted to polymer solutions. For
instance, mixtures of a higher alcohol and water may give relations very
similar to those in Figure 14.8.
In a sense, the spinodal gives the composition and hence the
supersaturation needed for homogeneous nucleation. People have tried to
develop more general nucleation theories from the thermodynamics of
mixtures along these lines. It is, however, difficult, if not impossible, to
predict free energy curves with sufficient accuracy. When working with
polymer solutions or aqueous polymer mixtures, one often has experimen-
tally to estimate the spinodal, to find the conditions needed for spontaneous
demixing.
Question
Sweetened condensed milk is supersaturated with lactose, and the lactose crystals
formed tend to settle and also give the product a sandy mouth feel. To prevent this,
crystal size should be at most 8mm, and one generally tries to achieve this by adding
crystalline seed lactose, about 0.3 g per kg product. Calculate what the maximum size
of the seed particles may be. One kg of sweetened condensed milk contains about
110 g lactose, 440 g sucrose, and 260 g water. Consult also Table 2.2.
Answer
The product contains 440/260¼1.69 kg sucrose per kg water and, according to Table
2.2, the solubility of lactose then is close to 200 g per kg water. This implies that
about 58 g of lactose will eventually crystallize. Assuming that the number of crystals
formed equals the number of seed crystals added, the volume of a seed crystal must
be at most 0.3/58 times the volume of a crystal eventually formed. Assuming the
crystals to be of the same shape, the maximum diameter of a seed crystal should then
be (0.3/58)1/3 68 mm¼1.38mm.