DLVO force-distance curves

It is clear from this chapter that the coulombic attraction theory potential is much better adapted to explain the experimental phenomena described in Chapter 1 than the DLVO theory potential (Equation 1.2). Of course, if you predict an interaction potential, you predict force-distance curves along the swelling axis. There have been a lot of arguments about how direct measurements of forces between spherical colloidal particles refute the coulombic attraction theory. Let us get the facts first. We now examine the experimental curves for the n-butylammonium vermiculite system. 

FIGURE 3.9 DLVO prediction for the force-distance curves. DLVO theory predicts collapse into the primary minimum at y = 2.5 or y = OTSy. The straight line is for a pure repulsion. 

DLVO [2,3] theory estimates the repulsive and attractive force due to the overlap of electric double layers and London-van der Waals force in terms of inter particle distance, respectively. The summation of them gives the total interaction force and can be used for the interpretation of colloid stability in terms of the nature of interaction force-distance curve. If a small interparticle separation (H) is assumed, van der Waals forces for a sphere and substrate can be expressed to... 

Figure 5. (a) Force—distance curves recorded upon compression of PLL(20)-g-PEG(2) polymer adlayers on niobiaby aSi02 microsphere in 10 mM HEPES buffer solution (pH 7.4), with PEG chain densities from 0 to 0.54 PEG nm". (b) Surface potential from DLVO fits (not shown) as a function of the grafting ratio, g, of the PLL-g-PEG polymer, highlighting charge reversal at g = 4. 

The combination of van der Waals attraction with steric repulsion (combination of mixing and elastic interaction) forms the basis of the theory of steric stabilization [20]. Figure 1.3 (b) gives a schematic representation of the force-distance curve of sterically stabilized systems. This force-distance curve shows a shallow minimum at separation distance h comparable to twice the adsorbed layer thickness (28) and when h < 28, very strong repulsion occurs. Unlike the V-h curve predicted by the DLVO theory (which shows two minima), the V-h curve of sterically stabilized systems shows only one minimum whose depth depends on the particle or droplet radius R, the Hamaker constant A and the adsorbed layer thickness 8. At a given R and A, the depth of the minimum decreases with increasing the adsorbed layer thickness 8. When the latter exceeds a certain value (particularly with small particles or droplets) the minimum depth can become < kT and the dispersion approaches thermodynamic stability. This forms the basis of stability of ntmodispersions. 

A schematic representation of the force (energy)-distance curve, according to DLVO theory, is shown in Figure 10.10. 

We finish this section with an important conclusion, which is that considerable hysteresis can occur in the force-distance or the free energy distance curves between approach and moving apart of two particles. Classical DLVO theory proceeds from the assumption that the interaction force is at equilibrium under all conditions, but this is mostly not the case if polymers are involved. The shape of a curve as depicted in Figure 12.9 will also depend on the rate at which the particles are moved. 

The stability of dispersions in aqueous media can often be described by the DLVO theory, which contains the double-layer repulsion and the van der Waals attraction. Specific force versus distance curves depend on the Hamaker constant, the surface charges, and the salt concentration. In general, DLVO theory predicts an electrostatic repulsion at intermediate distance and a van der Waals dominated attraction at short distance. 

The pair potential of colloidal particles, i.e. the potential energy of interaction between a pair of colloidal particles as a function of separation distance, is calculated from the linear superposition of the individual energy curves. When this was done using the attractive potential calculated from London dispersion forces, Fa, and electrostatic repulsion, Ve, the theory was called the DLVO Theory (from Derjaguin, Landau, Verwey and Overbeek). Here we will use the term to include other potentials, such as those arising from depletion interactions, Kd, and steric repulsion, Vs, and so we may write the total potential energy of interaction as... 

Figure 2.14 Measured electrostatic double-layer and van der Waals forces between two surfaces of curved mica of radius 1 cm in (a) water and (b) dilute KNO3 and Ca(N03)2 solutions. The lines are the predictions of the DLVO theory with a Hamaker constant of 2.2 x 10 J in the limits of constant surface charge and constant surface potential here xfrQ = -(j/s, the particle surface potential. (The lines for constant surface charge are slightly higher than those for constant surface potential at small D.) The inset in (b) is the measured force in 0.1 M KNO3, which shows a force minimum at a distance of around 7 nm. Since this minimum in force occurs away from the deep minimum at the surface, it is called a secondary minimum. (From Israelachvili and Adams 1978 and Israelachvili 1992, reprinted with permission from Academic Press.)...

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