Physical Chemistry of Foods

longer distances than does double layer repulsion. For two spheres of equal
size it is given by

VEl;s¼

4 p[ 0 [R^2 c^20

2 Rþh

&

3 : 3? 10 
^10 R^2 c^20

2 Rþh

ð 12 : 9 Þ

where the part after the&sign applies to triglyceride oil as the medium.
Assuming, as an example, thatR¼ 2 mm andc 0 ¼25 mV, the repulsive free
energy ath¼0 amounts to 50kBT, which is a strong repulsion. However, at
h¼ 2 R¼ 4 mm, the repulsion is still quite strong at 25kBT. Since repulsion
acts over such large distances, the droplets sense about the same repulsive
force from all sides, unless the emulsion is very dilute (see Fig. 9.6). This
comes down to virtually no net repulsion. Equation (12.9) would also apply
to repulsion across a layer of (dry) air, withe¼1.

12.2.3 Total Interaction Energy

The basic idea of the DLVO theory is that the stability of lyophobic colloids
in aqueous systems is determined by the combination of van der Waals
attraction and electrostatic repulsion and that the two are exactly additive.
In other words, the total interaction free energyVTwould at any value ofh
be given by

VT¼VvdWþVEl ð 12 : 10 Þ

This turns out to be very nearly correct, provided that no other interaction
forces act.
Some calculated results are plotted in Figure 12.4. It may be noticed
that the range over which the interaction energy is of importance seems to
be restricted to about 10 nm; closer inspection shows that interparticle
distances up to about 20 nm may be relevant. Because the (negative) van der
Waals term is proportional to 1/h, and the (positive) electrostatic term
decreases about exponentially withh, attraction will always prevail at very
short and at large distances. At largeh, however, the attractive energy may
be negligibly small. It is generally assumed that aggregation in the primary
minimum (at y) is irreversible and that a sufficiently high maximum value of
V (at x) can prevent this aggregation. Aggregation in the secondary
minimum (at z), if it exists, is supposed to be reversible, since the minimum
is generally not very deep.

Important Variables. The results in Figure 12.4 are calculated for
a range of conditions as commonly experienced in food systems.