where AGads is the free energy associated with the adsorption of an ion to the interface: AGads = ze^s + The ze¥s term is due to the electrostatic attraction between the ion and the surface, while the ^ term accounts for any specific binding effects. These specific binding effects could be due to hydrophobic interactions (e.g., when an emulsifier adsorbs) or chemical interactions (e.g., -COO- + Na+ ^ -COO-Na+). The fraction of surface sites which are occupied increases as the free energy of adsorption of an ion increases. The electrical potential at the Stern layer can be related to the electrical potential at the charged surface using the following equation (Hiemenz 1986):
where a* is the surface charge density when the surface is completely saturated with ions and £s is the relative dielectric constant of the Stern layer. Equation 3.12 indicates that the difference between the potential at the surface and that at the Stern plane depends on the fraction of surface sites that are occupied. In principle, this equation can be used to calculate the change in the electrical potential of a surface due to ion adsorption. In practice, this equation is difficult to use because of a lack of knowledge about the values of S, and es in the Stern layer (Hiemenz 1986, Derjaguin 1989). These parameters are unique for every ion-surface combination and are difficult to measure experimentally. For this reason, it is usually more convenient to experimentally measure the electrical potential at the Stern plane, rather than attempting to predict it theoretically (Hunter 1986).
Experiments have shown that is closely related to the electrical potential at the shear plane (Hunter 1986). When a liquid flows past a charged surface, it "pulls" those counterions which are only weakly attached to the surface along with it, but leaves those ions that are strongly attached in place (i.e., those ions in the Stern plane). The shear plane is defined as the distance from the charged surface below which the counterions remain strongly attached and is approximately equal to the diameter of the hydrated ions (Figure 3.8). The electrical potential at the shear plane is referred to as the zeta potential (Z) and can be measured using various types of electrokinetic techniques (Chapter 10).
In the outer region, the electrostatic interaction between the surface and the counterions is usually fairly weak (because the ions in the Stern layer partially screen the surface charge), and so the variation in electrical potential with distance can be described by Equation 3.8, by replacing with ¥s:
where x is now taken to be the distance from the shear plane, rather than from the charged surface. The dependence of the electrical potential on distance from the shear plane can then be calculated once is known.
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