## R

electrical double layer which accumulates and contains both positive and negative ions.

Electrostatic forces arise from the interaction of the electrical double layers surrounding particles in suspension (see Fig. 7.3). This interaction leads to repulsion if the particles have surface charges and surface potentials of the same sign and magnitude. When the surface charge is produced by the adsorption of potential-determining ions the surface potential, is determined by the activity of these ions and remains constant during interaction with other particles, if the extent of adsorption does not change. The interaction therefore takes place at constant surface potential. In emulsion systems where the adsorbed layers can desorb, or in conditions of low availability of potential-determining ions, the interaction takes place not at constant surface potential but at constant surface charge (or at some intermediate state). The electrostatic repulsive force decays as an exponential function of the distance and has a range of the order of the thickness of the electrical double layer, equal to the Debye-Huckel length, 1/k:

For small values of simplifies to

The equations do not take into account the finite size of the ions; the potential to be used is yd, the potential at the Stern plane (the plane of closest approach of ions to the surface), which is difficult to measure. The nearest experimental approximation to yjs is often the zeta potential (£) measured by electrophoresis.

In the DLVO theory the combination of the electrostatic repulsive energy VR with the attractive potential energy VA gives the total potential energy of interaction

Vtotal plotted against the distance of separation H gives a potential energy curve showing certain characteristic features, illustrated in Fig. 7.4. The maximum and minimum energy states are shown. At small and at large distances the van der Waals energy (proportional to H where x varies from 1 to 7) is greater than the repulsion, which is proportional to exp(-KH). If the maximum is too small, two interacting particles may reach the primary

where e0 is the permittivity of the vacuum, e is the dielectric constant (or relative permittivity) of the dispersion medium, R is the gas constant, T is temperature, F is the Faraday constant, and ci and zi are the concentration and the charge number of the ions of type i in the dispersion medium. For monovalent ions in water, c = 10 -15k2 (with c in mol dm 3 and k in cm-1). No simple equations can be given for the repulsive interactions. However, for small surface potentials and low values of k (that is, when the double layer extends beyond the particle radius) and at constant the repulsive energy is

Distance of surface separation, H

Figure 7.4 Schematic form of the curve of total potential energy (VtotaJ against distance of surface separation (H) for interaction between two particles, with Vtotal = VA + VR.

and exp(-KH) this

Distance of surface separation, H

Figure 7.4 Schematic form of the curve of total potential energy (VtotaJ against distance of surface separation (H) for interaction between two particles, with Vtotal = VA + VR.

minimum and in this state of close approach the depth of the energy minimum can mean that escape is improbable. Subsequent irreversible changes in the system may then occur, such as sintering and recrystallisation in suspensions or coalescence in emulsions forming irreversible structures. When the maximum in Vtotal is sufficiently high, the two particles do not reach the stage of being in close contact. The depth of the secondary minimum is important in determining events in a hydrophobic dispersion. If the secondary minimum is smaller than the thermal energy, kT (where k is the Boltzmann constant), the particles will always repel each other, but when the particles are large enough the secondary minimum can trap particles for some time as there is no energy barrier to overcome. At intermediate distances the energy of repulsion may be the larger of the two.

### Effect of electrolytes on stability

Pharmaceutical colloids are rarely simple systems. The influence of additives including simple and complex electrolytes has to be considered. Electrolyte concentration and valence (z) are accounted for in the term (X ctz2) in equation (7.3) and thus in equations (7.4) and (7.5). Figure 7.5 gives an example of the influence of electrolyte concentration on the electrostatic repulsive force. In this example, a # 10 5 cm, A # 10019 J, and y 0 = RT/F q 26.5 mV. As the electrolyte concentration is increased, k increases due to compression of the double layer with consequent decrease in 1/k.

At low electrolyte concentrations (low k) the range of the double layer is high and VR extends to large distances around the particles. Summation of VR and VA gives a total energy curve having a high primary maximum but no secondary minimum. The decrease of the double layer when more electrolyte is added produces a more rapid decay in VR and the resultant total energy curve now has a small primary maximum but, more importantly, a secondary minimum. This concentration of electrolyte would produce a stable suspension, since flocculation could occur in the

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