13.6.2 Emulsions
LSW Theory. In most emulsions, the distance between drops is far
larger than in foams, which makes the de Vries theory less suitable. Often
the LSW theory is used (independently developed by Lifshitz and Slyozov
and by Wagner). It is presumed (a) that the concentration of the droplet
material in the continuous phase is higher thans?and attains a magnitude
corresponding tos(), whereis the number average droplet radius;
and (b) that concentration gradients only occur near the droplets. These
assumptions are illustrated in Figure 13.22b. It can be derived that Ostwald
ripening now leads to a size distribution of standard shape, which is thus
time invariant. After the distribution has attained this shape, the rate of the
process would be given by
d<a>^3
dt¼
4 x^0 Ds?
9 rD
ð 13 : 35 Þwheresis expressed in kg?m^3. The theory is well obeyed at very smallj,
but for higherjthe rate is faster than predicted (say by a factor of 2 or 3 for
j¼0.2). This is because the assumption about the concentration gradient
does not hold any more and because Brownian motion of the drops is
neglected. On the other hand, the rate can be much slower owing to the
surfactant layer producing a considerable surface dilational viscosity, as
discussed in the previous section.
Most foodoil-in-water emulsionsdo not show any Ostwald ripening
for the simple reason that the solubility of triglyceride oil in water is
negligible. Some plants, especially citrus fruits, contain essential oils, which
consist for a considerable part of various terpenes, which are soluble in
water. Emulsions of such oils, which are used as flavoring agents, can show
distinct Ostwald ripening.
Water is somewhat soluble in triglyceride oils, about 1.3 kg?m^3 at
room temperature, and this may cause significant Ostwald ripening inwater-
in-oil emulsions. For example, take pure water droplets of 2mm diameter in
pure triglyceride oil at 20 8 C. Then gOW & 0.03 N?m^1 and D &
10 ^11 m^2 ?s^1. The time needed for doubling the average droplet volume
is given by^3 over the right-hand side of Eq. (13.35), which would lead
to a value of about 4 days. This is not negligible in practice. (At higher
temperature, the rate will be markedly higher, by a factor of about 10 at
508 C, due to increased water solubility and diffusion coefficient.) In
margarine, Ostwald ripening tends to be far smaller for a number of reasons.
One is that surfactants are present and that the magnitude ofgOWwill thus
be lower (5–10 mN?m^1 ). A second reason is that solid particles, i.e., fat