Minimum secondary

The presence of the large repulsive potential barrier between the secondary minimum and contact prevents flocculation. One can thus see why increasing ionic strength of a solution promotes flocculation. The net potential per unit area between two planar surfaces is given approximately by the combination of Eqs. V-31 and VI-22 ... 

The charge on a droplet surface produces a repulsive barrier to coalescence into the London-van der Waals primary attractive minimum (see Section VI-4). If the droplet size is appropriate, a secondary minimum exists outside the repulsive barrier as illustrated by DLVO calculations shown in Fig. XIV-6 (see also Refs. 36-38). Here the influence of pH on the repulsive barrier between n-hexadecane drops is shown in Fig. XIV-6a, while the secondary minimum is enlarged in Fig. XIV-6b [39]. The inset to the figures contains t,. the coalescence time. Emulsion particles may flocculate into the secondary minimum without further coalescence. 

Fig. XrV-6. (a) The total interaction energy determined from DLVO theory for n-hexadecane drops for a constant ionic strength - 5.0 nm) at various emulsion pH (b) enlargement of the secondary minimum region of (a). (From Ref. 39.)...

At larger particle separation, a second minimum may occur in tire potential energy. In many cases, tliis minimum is too shallow to be of much significance. For larger particles, however, tire minimum may become of order kT. Aggregation in tliis minimum is referred to as secondary minimum flocculation. 

Flocculation a relatively reversible aggregation often associated with the secondary minimum of a potential energy diagram. Particles are held together loosely with considerable surface separations. 

As two particles approach in a liquid their charge fields may interact and form two minima as depicted in Figure 6.8. If the particles approach to a distance Li, known as the primary minimum they aggregate to form a configuration with minimum energy - and rapid coagulation is said to take place. On the other hand, if the particles remain separated at a distance L2, the secondary minimum, loose clusters form which do not touch. This is known as slow coagulation and is the more easily reversed. 

Hamiltonian in Eq. (39). The solid curve, for m = 0, indicates stable isomers at 0 = 0 and 0 = tt and a saddle point at 0 O.dStt. Note, however, that the secondary minimum disappears as the angular momentum increases. 

Figure 10. Divisions of the Em map for the quadratic Hamiltonian in Eq. (39). Va, Vb, and Vx are the energies of the two isomers and that of the saddle point between them. The solid lines are relative equilibria, and the cusp points, a , indicate points at which the secondary minimum and the saddle point in Fig. 9 merge at a point of inflection.

The second type of quantum monodromy occurs in the computed bending-vibrational bands of LiCN/LiNC, in which the role of the isolated critical point is replaced by that of a finite folded region of the spectrum, where the vibrational states of the secondary isomer LiNC interpenetrate those of LiCN [9, 10]. The folded region is finite in this case, because the secondary minimum on the potential surface merges with the transition state as the angular momentum increases. However, the shape of the potential energy surface in HCN prevents any such angular momentum cut-off, so monodromy is forbidden [10]. 

Verwey and Hamaker (10) have modified the Morse curve to take into account the approach of two charged bodies, as shown in Figure 4. Here, as one moves outward toward increasing distance of separation, an electrostatic repulsion is encountered because the charges are similar. A secondary minimum is then encountered as a result of the concentration of counter ions around each charged particle. The shallowness of the secondary minimum shows that the deflocculated system is metastable. The importance of the Verwey and Hamaker concept lies in its ability to show graphically the correlation between the secondary minimum and the metastable position. A, Figure 1. 

At a finite distance, where the surface does not come into molecular contact, equilibrium is reached between electrodynamic attractive and electrostatic repulsive forces (secondary minimum). At smaller distance there is a net energy barrier. Once overcome, the combination of strong short-range electrostatic repulsive forces and van der Waals attractive forces leads to a deep primary minimum. Both the height of the barrier and secondary minimum depend on the ionic strength and electrostatic charges. The energy barrier is decreased in the presence of electrolytes (monovalent < divalent [Pg.355]

The physicochemical forces between colloidal particles are described by the DLVO theory (DLVO refers to Deijaguin and Landau, and Verwey and Overbeek). This theory predicts the potential between spherical particles due to attractive London forces and repulsive forces due to electrical double layers. This potential can be attractive, or both repulsive and attractive. Two minima may be observed The primary minimum characterizes particles that are in close contact and are difficult to disperse, whereas the secondary minimum relates to looser dispersible particles. For more details, see Schowalter (1984). Undoubtedly, real cases may be far more complex Many particles may be present, particles are not always the same size, and particles are rarely spherical. However, the fundamental physics of the problem is similar. The incorporation of all these aspects into a simulation involving tens of thousands of aggregates is daunting and models have resorted to idealized descriptions. 

Fig. 19 Typical free energy profile for the deposition of a colloid. Note the logarithmic distance scale and the divided energy scale for the relatively shallow secondary minimum and the much deeper primary minimum.

So far we have derived an expression for the overall rate constant k. As with the copper system, we should also like an expression for the first-order rate constant k, describing the passage of particles from the secondary minimum over the barrier. The equilibrium constant describing the population of the secondary minimum is given by the same expression (49) as (36), where, as in Fig. 14, xR describes the width... 

PVA and TaM -for the 88%-hydrolyzed PVA. The same dependence was found for the adsorbed layer thickness measured by viscosity and photon correlation spectroscopy. Extension of the adsorption isotherms to higher concentrations gave a second rise in surface concentration, which was attributed to multilayer adsorption and incipient phase separation at the interface. The latex particle size had no effect on the adsorption density however, the thickness of the adsorbed layer increased with increasing particle size, which was attributed to changes in the configuration of the adsorbed polymer molecules. The electrolyte stability of the bare and PVA-covered particles showed that the bare particles coagulated in the primary minimum and the PVA-covered particles flocculated in the secondary minimum and the larger particles were less stable than the smaller particles. 

Combined Electrostatic and Steric Stabilization. The combination of the two mechanisms is illustrated in Figure 4, taken from Shaw s textbook, (13) where the repulsion of the steric barrier during a collision falls off so rapidly as the colliding particles bounce apart that the dispersion force attractions hold the particles together in the "secondary minimum". This is exactly what happens in the system investigated in this paper. 

In such systems the requirement of the electrostatic contribution to colloidal stability is quite different than when no steric barrier is present. In the latter case an energy barrier of about 30 kT is desirable, with a Debye length 1/k of not more than 1000 X. This is attainable in non-aqueous systems (5), but not by most dispersants. However when the steric barrier is present, the only requirement for the electrostatic repulsion is to eliminate the secondary minimum and this is easily achieved with zeta-potentials far below those required to operate entirely by the electrostatic mechanism. 

While the system conforms to coagulation in a secondary minimum, the redispersion region is best accounted for in terms of gel formation originating from the rod-like shape of the particles and hydrated surface. During the final coagulation process additional attractive forces such as dipolar and hydrogen bonding form floes which are irreversible and denser than those formed at a lower salt concentration. 

DLVO Theory Applied to MCC Sols. In order to interpret the stability of MCC sols, several stability calculations were made. First, by using a spherical particle radius of 706A, A = 2.6 kT, ijto = 14 mV and T = 296°K at a salt concentration of 0.9 x 10 3M (region II), a secondary minimum was obtained at 500 A, V x =... 

Schematic forms of the curves of interaction energies (electrostatic repulsion Vr, van der Waals attraction Va, and total (net) interaction Vj) as a function of the distance of surface separation. Summing up repulsive (conventionally considered positive) and attractive energies (considered negative) gives the total energy of interaction. Electrolyte concentration cs is smaller than cj. At very small distances a repulsion between the electronic clouds (Born repulsion) becomes effective. Thus, at the distance of closest approach, a deep potential energy minimum reflecting particle aggregation occurs. A shallow so-called secondary minimum may cause a kind of aggregation that is easily counteracted by stirring.

The presence of polymers or polyelectrolytes have important effects on the Van der Waal interaction and on the electrostatic interaction. Bacterial adhesion, as discussed in Chapter 7.9 may be interpreted in terms of DLVO theory. Since the interaction in bacterial adhesion occurs at larger distances, this interaction may be looked at as occurring in the secondary minimum of the net interaction energy (Fig. 7.4). Particle Size. The DLVO theory predicts an increase of the total interaction energy with an increase in particle size. This effect cannot be verified in coagulation studies. 

Figure 2.16 The pair potential calculated for rutile particles with a radius of 100 nm, background electrolyte concentration of 10 4moldm 3 and a -potential of —50 mV. Curve a was calculated for an isolated pair of particles and curve b corresponds to the potential for a pair ofparticles in a dispersion at a volume fraction of 0.45. Note how the increased electrolyte content due to the counterions introduced with the particles shorten the range of the repulsion enough for a small secondary minimum to be found at h 70 nm...

---