## Xv

Counter-ions Co-ions

FIGURE 3.6 The organization of ions near a charged surface is governed by two opposing tendencies: (1) electrostatic interactions which favor accumulation of counterions near a surface and (2) thermal energy which favors a random distribution of the ions.

the surface and the surrounding liquid (Kitakara and Watanabe 1984; Hunter 1986, 1989; Hiemenz 1986).

It is important to establish the relationship between the characteristics of a charged surface, the properties of the solution in contact with it, and the distribution of ions in its immediate vicinity, because this information is needed to calculate the strength of electrostatic interactions between emulsion droplets. A surface is usually characterized by its surface charge density (a) and its surface potential (^0). The surface charge density is the amount of electrical charge per unit surface area, whereas the surface potential is the amount of energy required to increase the surface charge density from zero to a. These values depend on the type and concentration of emulsifier present at a surface, as well as the nature of the electrolyte solution (e.g., pH, ionic strength, and temperature). The important characteristics of an electrolyte solution are its dielectric constant and the concentration and valency of the ions it contains.

A mathematical relationship, known as the Poisson-Boltzmann equation, has been derived to relate the electrical potential in the vicinity of a charged surface to the concentration and type of ions present in the adjacent electrolyte solution (Evans and Wennerstrom 1994):

where n0j is the concentration of ionic species of type i in the bulk electrolyte solution (in molecules per cubic meter), z. is their valency, e is the electrical charge of a single proton, £_ is the dielectric constant of a vacuum, e„ is the relative dielectric constant of the solution,

and y(x) is the electrical potential at a distance x from the charged surface. This equation is of central importance to emulsion science because it is the basis for the calculation of electrostatic interactions between emulsion droplets. Nevertheless, its widespread application has been limited because it does not have an explicit analytical solution (Hunter 1986). When accurate calculations are required, it is necessary to solve Equation 3.7 numerically using a digital computer (Carnie et al. 1994). For certain systems, it is possible to derive much simpler analytic formulas which can be used to calculate the electrical potential near a surface

by making certain simplifying assumptions (Evans and Wennerstrom 1994, Sader et al. 1995).

If it is assumed that the electrostatic attraction between the charged surface and the counterions is relatively weak compared to the thermal energy (i.e., z.ey0 < kT, which means that must be less than about 25 mV at room temperature in water), then a simple expression, known as the Debye-Huckel approximation, can be used to calculate the dependence of the electrical potential on distance from the surface (Hunter 1986, Hiemenz 1986):

This equation indicates that the electrical potential decreases exponentially with distance from the surface at a decay rate which is determined by the parameter k-1, which is known as the Debye screening length. The Debye screening length is a measure of the "thickness" of the electrical double layer and it is related to the properties of the electrolyte solution by the following equation:

For aqueous solutions at room temperatures, k-1 « 0.304/VI nm, where I is the ionic strength expressed in moles per liter (Israelachvili 1992). For example, the Debye screening lengths for NaCl solutions with different ionic strengths are 0.3 nm for a 1 M solution, 0.96 nm for a 100 mM solution, 3 nm for a 10 mM solution, 9.6 nm for a 1 mM solution, and 30.4 nm for a 0.1 mM solution.

The Debye screening length is an extremely important characteristic of an electrolyte solution because it determines how rapidly the electrical potential decreases with distance from the surface. Physically, k-1 corresponds to the distance from the charged surface where the electrical potential has fallen to 1/e of its value at the surface. This distance is particularly sensitive to the concentration and valency of the ions in an electrolyte solution. As the ion concentration or valency increases, k-1 becomes smaller, and therefore the electrical potential decreases more rapidly with distance (Figure 3.7). The physical explanation for this phenomenon is that the neutralization of the surface charge occurs at shorter distances when the concentration of opposite charge in the surrounding solution increases.

A charged surface can be considered to be surrounded by a "cloud" of counterions with a thickness equal to the Debye screening length and which depends strongly on the ion concentration and valency (Hunter 1986). As will be seen in later chapters, the screening of electrostatic interactions by electrolytes has important consequences for the stability and rheology of many food emulsions (Chapters 7 and 8).

The Poisson-Boltzmann theory assumes an electrolyte solution is a continuum that contains ions which are infinitesimally small. It therefore allows ions to accumulate at an unphysically large concentration near to a charged surface (Evans and Wennerstrom 1994, Kitakara and Watanabe 1984). In reality, ions have a finite size and shape, and this limits the number of them which can be present in the first layer of molecules that are in direct contact with the surface (Derjaguin et al. 1987, Derjaguin 1989, Israelachvili 1992). This assumption is not particularly limiting for systems in which there is a weak interaction between the ions and the charged surface. Nevertheless, it becomes increasingly unrealistic as the strength of the electrostatic interactions between a charged surface and the surround- FIGURE 3.7 Influence of ionic strength on the electric field near a charged surface. The electrical double layer shrinks as the electrolyte concentration increases or the ion valency increases.

ing ions increases relative to the thermal energy (Evans and Wennerstrom 1994). In these cases, the Poisson-Boltzmann theory must be modified to take into account the finite size of the ions in the electrolyte solution. It has proved convenient to divide the counterion distribution near a highly charged surface into two regions: an inner and an outer region (Figure 3.8).

### 3.4.2.1. Inner Region

In the inner region, the attraction between the counterions and charged surface is strong and therefore they are relatively immobile, whereas in the outer region, the attraction is much weaker and therefore the counterions are more mobile (Evans and Wennerstrom 1994). The thickness of the inner region (5) is approximately equal to the radius of the hydrated counterions, rather than their diameter, because the effective charge of an ion is located at its center (Hiemenz 1986). The inner region is sometimes referred to as the Stern layer, while the boundary between the inner and outer regions is referred to as the Stern plane (Figure 3.8), after Otto Stern, the scientist who first proposed this concept (Hiemenz 1986, Derjaguin 1989). The electrical potential at the Stern plane (^5) is different from that at the surface (¥q) because of the presence of the counterions in the Stern layer. For monovalent indifferent electrolyte counterions, is less than ¥0, because the surface charge is partly neutralized by the charge on the counterions (Figure 3.9a). The extent of this decrease depends on the number and packing of the counterions within the Stern plane (Derjaguin 1989). The same behavior is observed for multivalent indifferent electrolyte counterions at low concentrations, but at higher concentrations the surface may adsorb such a large number of oppositely charged multivalent counterions that its charge is actually reversed, so that has an opposite sign to ¥0 (Figure 3.9b). If a charged surface adsorbs surface-active co-ions (e.g., ionic emulsifiers), it is even possible for to be larger than (Figure 3.9c). An increase in surface charge may occur when the hydrophobic attraction between the nonpolar tail of a surfactant and a surface is greater than any electrostatic repulsive interactions. Thus the magnitude of the electrical potential at the Stern plane depends on the precise nature and concentration of the ions present in the system (Derjaguin 1989). FIGURE 3.8 When the electrostatic attraction between a charged surface and the surrounding counterions is relatively strong compared to the thermal energy, it is convenient to divide the electrolyte solution into an inner and an outer region.

A number of theories have been developed to take into account the effect of the finite size and limited packing of ions in the Stern layer on the relationship between and One of the most widely used is the Stern isotherm, which assumes that there are only a finite number of binding sites at the surface and that once these are filled, the surface becomes saturated and cannot adsorb any more ions:

where 8 is the fraction of occupied surface sites, K is is the adsorption equilibrium constant, and % is the mole fraction of the ion in the bulk phase. The adsorption equilibrium constant depends on the strength of the interaction between the ion and the surface compared to the thermal energy (Hiemenz 1986): 