Interaction energy secondary minimum

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.)...

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. 

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. 

Fig. 31 Overall interaction energy between two DNA-coated colloids, (a) Sketch of the interacting surfaces of two spheres of radius R0 separated by d. The maximum length of hybridized strands is 2L. (b) Total interaction energy as a function of d. It is the sum of the attractive I/DNA from the binding of accessible DNA strands, the repulsive I/rep from electrostatics and/or polymer steric effect, and the van der Waals attraction t/vdw. (c) For weak, short-range I/rep, particles which are unbound at high temperatures are irreversibly trapped in the van der Waals well after DNA hybridization at low temperatures, (d) For strong, medium-range I/rep, DNA binding produces a secondary minimum of reversible aggregation. Reproduced with permission from [138]...

Figure 6.11 Gibbs free interaction energy (in units of ki>T) versus distance for two identical spherical particles of R = 100 nm radius in water, containing different concentrations of monovalent salt. The calculation is based on DLVO theory using Eqs. (6.57) and (6.32). The Hamaker constant was Ah = 7 x 10 21 J, the surface potential was set to )/>o = 30 mV. The insert shows the weak attractive interaction (secondary energy minimum) at very large distances.

Figure 5.6 shows an example of a total interaction energy curve for a thin liquid film stabilized by the presence of ionic surfactant. It can be seen that either the attractive van der Waals forces or the repulsive electric double-layer forces can predominate at different film thicknesses. In the example shown, attractive forces dominate at large film thicknesses. As the thickness decreases the attraction increases but eventually the repulsive forces become significant so that a minimum in the curve may occur, this is called the secondary minimum and may be thought of as a thickness in which a meta-stable state exists, that of the common black film. As the... 

By including Born repulsion in the calculation of the interaction energy profile, the primaty minimum is finite and depends on ionic strength. Allowing the primary to be above the secondary minimum one is... 

The values of the interaction energy at the maximum and at the secondary minimum are proportional with the radius a ofthe particle or droplet and depend strongly on the Hamaker constant AH and on the hydration repulsion. As shown in the previous section, small modifications in the ratio (pje ) (and hence in the hydration repulsion) can lead to a large increase in the potential barrier between the primary and secondary minima, thus affecting the stability of the system. 

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