are closely packed; it is called theStern layer. In practice, however, one may
take forc 0 the potential at the Stern layer (see below).
The ‘‘double layer’’ discussed should not be interpreted as a static
layer, adhering to the particle. Molecules and ions diffuse in and out. When
a charged particle is subject to an electrokinetic experiment—which means
that it is moving relative to the salt solution in which it is dispersed due to an
externally applied electrostatic potential gradient—the slipping plane
between particle and liquid is close to the geometrical particle surface; for
hard and smooth particles it roughly coincides with the outside of the Stern
layer. Thezeta potential, which is the electrical potential determined in such
an electrokinetic experiment (usually electrophoresis), is the potential at the
slipping plane. It tends to agree with the value ofc 0 needed to calculate the
electrostatic repulsion.
Repulsive Free Energy. When the surfaces of two particles in an
aqueous phase come close to each other, their electric double layers start to
overlap, as illustrated in Figure 12.3. This will cause a local increase in
potential, which implies that work must be applied to bring the particles
closer together. From this increase in potential, the repulsive free energy can
be calculated. In another approach to the same problem, the increase in
osmotic pressure caused by the overlap of the double layers is calculated,
which also yields a repulsive free energy. The two methods give identical
results. The mathematics of the theory is quite involved, and we will merely
give some results for the case thatjc 0 jis not too high, say below 30 mV; this
is nearly always true in food systems.
For identicalspheres, the relation becomes
VEl;s¼ 2 pe 0 eRc^20 lnð 1 þekhÞ& 4 : 5? 10 ^9 Rc^20 lnð 1 þekhÞð 12 : 7 Þwhere the part after the&sign is valid for water at room temperature (in SI
units). If the spheres do not have the same radius, this can be accounted for
as in Eq. (12.1). There is one other condition for Eq. (12.7) to hold, which is
thatkR 4 1. In practice, this means that the theory is valid forh 5 R.
For the repulsion between infiniteflat plates, the result is
VEl;p¼ 2 e 0 ekc^20 ekh& 1 : 4? 10 ^9 kc^20 ekh ð 12 : 8 ÞAgain, the part after the&sign is valid for water as the medium at room
temperature. The results of Eq. (12.7) are in J, those of (12.8) in J?m^2.
A different situation arises if the particles are separated by adielectric
medium(i.e., not conducting electricity). A case in point is a W–O emulsion,
e.g., aqueous droplets in a triglyceride oil. The droplets contain, say,