Parameters used in the calculations are given in the legend to Fig. 4. These have been selected to represent situations of interest in aquasols. The particle's radius, a, is set at 50 nm, in the lower part of the colloidal size range. The ionic strength (I) is set at 0.1, typical of estuarine waters and commonly considered to promote the coagulation of many colloids. A value of 10"20 J is chosen for the Hamaker constant (A); this is in the range of many aquasols and is also appropriate for the polystyrene latex-water-glass system often used in laboratory studies of particle deposition. The characteristic wavelength of the retarded van der Waals interaction (A) is assumed to be 100 nm. The surfaces of the suspended particles and the stationary flat plate are both given negative surface potentials of -30 mV, representative of surface properties of some aquasols at this ionic strength. The shapes of the curves presented in Fig. 4 are sensitive to these parameters.

Considering the interaction energy curves presented in Fig. 4, the attractive force dominates at very small separation distances and an infinitely deep well in potential energy is observed. This is termed the primary well. At still closer separations, a Born repulsion occurs. It is due to the interpénétration of ionic atmospheres, provides a strong repulsion at the surface and, with the DLVO forces, forms a primary energy minimum rather than a primary well. At larger separating distances, electrostatic repulsion between overlapping diffuse layers dominates van der Waals attraction and a repulsive energy barrier develops. The maximum energy of this barrier is denoted as VT max and, for the conditions in Fig. 4, has a value of about 7 kT. Since the average one-dimensional kinetic energy of a Brownian particle is only 0.5 kT, this barrier is a significant impediment to attachment of the aquasol to the plate.

At still larger separating distances, about 4 nm in Fig. 4, a second attractive region occurs. This is termed the secondary minimum and it is much shallower than the primary well or minimum. It is denoted as V2min. For the conditions represented in Fig. 4, the attractive secondary minimum has a depth of 2.8 kT, somewhat larger than the one-dimensional thermal energy of 0.5 kT. Association between the particle and the plate is possible for separating distances in this attractive region. For two surfaces to be attached, they must overcome the energy barrier and be held in the deep primary minimum or they can remain associated at larger separating distances in the secondary attractive region. Due in part to the differences in magnitude of these two attractive minima, contact in the primary minimum is thought to be irreversible while association in the secondary minimum can be reversible [37].

Aggregation and deposition rates, assuming attachment in the primary well, will depend on the height of the energy barrier, VTmax. When such a net repulsive interaction energy exists, the interaction is termed unfavorable and the kinetics of aggregation and deposition are termed slow. Conversely, when there is no energy barrier to attachment in the primary minimum, the interaction is termed favorable and the kinetics are termed fast.

Double Layer Theory Aquatic Chemistry
Fig. 4. Electrostatic, van der Waals, and net interaction energy curves for the sphere-plate case, constant potential double layer interaction, retarded van der Waals interaction: a = 50 nm, 1= 0.1, A = 10-20 J, X = 100 nm, y, = y/2 = -30 mV, T = 298 K. From [36]

Electrostatic repulsive forces and energies are very sensitive to the ionic strength of the solution. Ions in solution screen the coulombic interactions between the charged surfaces. This reduces the diffuse layer thickness («■"', Eq. (38)), thereby shortening the range of the repulsive interaction so that attractive van der Waals forces can be more effective. At low ionic strength (for example, the ionic strengths of fresh waters), a net repulsion can predominate over intermediate separating distances between aquasols. At high ionic strength (for example, estuarine and marine waters), this electrostatic energy barrier can disappear.

Effects of ionic strength on the total or net interaction energy between a sphere and a plate are illustrated in Fig. 5 from [36]. In these results, ionic strength is assumed to affect only the diffuse layer thickness; the surface potentials of the particle and the plate remain unchanged. Effects of ionic strength on the electrostatic terms in the Hamaker constant and the van der Waals interaction have been neglected. This is a good assumption for inorganic aquasols such as clays and metal oxides. Three ionic strengths are considered; other conditions are similar to those used in Fig. 4. Increasing the ionic strength from 10"3 to 10"' lowers the height of the maximum in the net interaction energy curve (VTmax) from 46 to 9 kT. At an even higher ionic strength this energy barrier would disappear, leading to fast deposition in the primary well. Negligible secondary minima {V2mjn < 0.2 kT) are formed at ionic strengths of 10"3 and 10"2 M, while a significant secondary minimum is formed at I = 10"'. This secondary minimum would disappear as the ionic strength was increased further.

Fig. 5. Net interaction energy curves as a function of ionic strength for the sphere-plate case, constant potential double layer interaction, retarded van der Waals interaction: a = 50 nm, A = 10"20 j, ¿=ioo nm, \f/1 = y/2 = -30 mV, T = 298 K. From [36]

Separation Distance (nm)

Fig. 5. Net interaction energy curves as a function of ionic strength for the sphere-plate case, constant potential double layer interaction, retarded van der Waals interaction: a = 50 nm, A = 10"20 j, ¿=ioo nm, \f/1 = y/2 = -30 mV, T = 298 K. From [36]

In addition to ionic strength, the electrostatic double layer interactions depend on the surface potentials of the particles (see Eq. (37) and also Table 3).

An important result in Eq. (36) is that the attractive and repulsive interaction energies and, consequently, the net interaction energy, all scale directly with the size of the particle, a. This result is true for any other force that may be considered between the two interacting solids. In addition to the sphere-plate interaction typical of particle deposition and described by Eq. (37), it also applies to sphere-sphere interactions in aggregation phenomena. Large colloidal particles have deeper primary minima, higher energy barriers, and deeper secondary attractive minima than smaller ones, and the differences are directly proportional to the sizes of the particles.

Results presented in Fig. 6 illustrate the effect of particle size on net interaction energy curves for the sphere-plate case. The effects of particle size on the height of the energy barrier VT max are observed best in Fig. 6b while particle size effects on the depth of the secondary minimum V2min are seen best in Figure 6a. The height of the energy barrier scales in direct proportion to the particle size, increasing from 43 to 114 to 290 kT as the particle radius increases from 23 to 60.5 to 154 nm. DLVO theory is seen to predict a very strong dependence of particle stability on particle size when stability is viewed as the difficulty in achieving contact in the primary well. Large colloids are predicted to be much more stable than small ones. The depth of the secondary well also scales in direct proportion to particle size. The depth of this minimum increases from 0.11 to 0.28 to 0.72 kT as particle radius is increased from 23 to 60.5 to 154 nm. DLVO theory predicts a dependence of reversible attachment in the secondary well on particle size with this attachment increasing significantly with increasing particle size. It is useful to note that the separating distances at which the energy barriers and the secondary minima are located (0.9 and 19 nm, respectively) are not affected by particle size.

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