continuous phase. Hence a significant solubility is a prerequisite for Ostwald
ripening to occur at a perceptible rate. This change in dispersity is especially
important in foams (Section 13.6.1), and it can also occur in emulsions
(13.6.2). Several kinds of suspensions show it as well, especially small
crystals in a saturated solution. Such suspensions are not very common in
foods; we will briefly come back to them in Chapter 15.
Ostwald ripening is almost the only change in dispersity that proceeds
faster for smaller particles: the excess solubility is roughly inversely
proportional to particle diameter, and there is generally no free energy
barrier. Since the average particle size increases, the process will proceed
ever slower. Except in foams, the process is generally not important if the
particle radii are over, say, 10mm.
13.6.1 Foams
De Vries Theory. Most foams have a high volume fraction, and a
small bubble will generally be rather close to several large ones: see Figure
11.1. De Vries derived an equation for the disappearance of such a small
bubble, assuming it to be separated by an average distancedfrom bubbles
with negligible curvature. Figure 13.22a illustrates the assumed
concentration gradient. Furthermore, de Vries used the Laplace pressure
(pL¼ 2 g/a) rather than the Kelvin equation; since the solubility of a gas is
proportional to its pressure, the same relation is obtained (for ideal
solubility behavior). Then the small particle will shrink with timetaccording
to
a^2 ðtÞ¼a^20 RTDs?g
pd
t ð 13 : 33 Þwhere D is the diffusion coefficient of the gas in water (about
1.5? 10 ^9 m^2 ?s^1 at room temperature), andpis atmospheric pressure
(about 10^5 Pa). When puttinga^2 (t)¼0,tequals the lifetime of a bubble of
initial radiusa 0.
Some sample calculations may be enlightening. Assume that the gas is
either N 2 ðs?& 7 mmol?m^3 ?Pa^1 Þor CO 2 ðs?& 0 :4 mmol?m^3 ?Pa^1 Þ,
that the surface tension is 0.05 N?m^1 , that the original bubble radius is
50 mm, and thatdis 20mm. We then calculate for a nitrogen bubble a
lifetime of 3800 s, or about one hour, and for a carbon dioxide bubble 66 s,
or about a minute. For a nitrogen bubble of 10mmandadvalue of 5mm, the
lifetime would be 38 s. Small bubbles can thus disappear very fast, especially
if the gas is highly soluble, as CO 2 is, which is common in several foods.
Experimental results on single bubbles agree reasonably well with the