DLVO approximation

It has long been demonstrated that the traditional double layer theory does not provide an accurate description for the repulsion between colloidal particles at small separations and high ionic strengths. Recent experiments on black films revealed that the traditional theory seems to be inadequate even for large separation distances and moderate ionic strengths. These results raise doubts about the validity of the DLVO approximations. It is argued here that the results can be, however, understood within the framework of the traditional theory if one accounts for the thermal undulation of the film interfaces, and a new treatment to account for the thermal undulations is suggested. 

Water plays an important role in the chemistry and physics of bulk solutions and interfaces, including electrochemistry and macromolecules in solution. Usually the water is treated as a structureless, dielectric continuum, such as in the Debye-Hiickle approximation for electrolytes, the Gony-Chapman-Stern " (GCS) approximation for the electrical double layer and the DLVO approximation for colloids. Properties sensitive to the molecular nature of water cannot be determined by these theories. 

Figure 2. Relative effective charge Z versus grain radius determined as Z = p/pD at r a. Other parameters are Z 25 I 0.1(h) 0.05(B). The curve (C) corresponds to the linear DLVO approximation the dashed line is the DH theory.

The DLVO theory, with the addition of hydration forces, may be used as a first approximation to explain the preceding experimental results. The potential energy of interaction between spherical particles and a plane surface may be plotted as a function of particle-surface separation distance. The total potential energy, Vt, includes contributions from Van der Waals energy of interaction, the Born repulsion, the electrostatic potential, and the hydration force potential. [Israelachvili (13)]. 

In filtration, the particle-collector interaction is taken as the sum of the London-van der Waals and double layer interactions, i.e. the Deijagin-Landau-Verwey-Overbeek (DLVO) theory. In most cases, the London-van der Waals force is attractive. The double layer interaction, on the other hand, may be repulsive or attractive depending on whether the surface of the particle and the collector bear like or opposite charges. The range and distance dependence is also different. The DLVO theory was later extended with contributions from the Born repulsion, hydration (structural) forces, hydrophobic interactions and steric hindrance originating from adsorbed macromolecules or polymers. Because no analytical solutions exist for the full convective diffusion equation, a number of approximations were devised (e.g., Smoluchowski-Levich approximation, and the surface force boundary layer approximation) to solve the equations in an approximate way, using analytical methods. 

The total DLVO interaction energy (Vs) between two spherical colloids (each of radius a and separated by distance H) is given by the following approximate equation ... 

Roughly 60 years ago Derjaguin, Landau, Verwey, and Overbeek developed a theory to explain the aggregation of aqueous dispersions quantitatively [66,157,158], This theory is called DLVO theory. In DLVO theory, coagulation of dispersed particles is explained by the interplay between two forces the attractive van der Waals force and the repulsive electrostatic double-layer force. These forces are sometimes referred to as DLVO forces. Van der Waals forces promote coagulation while the double layer-force stabilizes dispersions. Taking into account both components we can approximate the energy per unit area between two infinitely extended solids which are separated by a gap x ... 

Figure 5.15 shows an example of a disjoining pressure isotherm in which the steric force contributions have been superimposed on the classical DLVO force contributions. It can be seen that this creates two regions for meta-stable foam films. One region is the thick, common black film region, with film thicknesses of approximately 50 nm or so. The other region is the thin, Newton black film region, with film thicknesses of approximately 4 nm. While the common black films are mostly stabilized by electrostatic forces, the Newton black films are at least partly stabilized by the steric forces. 

The maximum repulsion generated by the asymmetric distribution of ions is however, small when the surface charge is zero (approximately one order of magnitude smaller than a typical van der Waals attraction) and therefore the neutral physiological colloids will coagulate, in agreement with the DLVO theory. 

For the sake of simplicity, in what follows it will be considered that the double layer potential is sufficiently small to allow the linearization of the Poisson—Boltzmann equation (the Debye—Hiickel approximation). The extension to the nonlinear cases is (relatively) straightforward however, it will turn out that the differences from the DLVO theory are particularly important at high electrolyte concentrations, when the potentials are small. In this approximation, the distribution of charge inside the double layer is given by... 

Let us first investigate the effect of the new equations alone, by using for both the DLVO theory and the present equations the same boundary conditions. For the surface charge density the constant value o = 5 x 10 4 C/m2 was employed (the value selected is low enough for the linear approximation to be accurate for all the electrolyte concentrations investigated here), while the polarization... 

At low electrolyte concentrations, the experimental results can be explained within the DLVO theory. A well-known approximation for the double layer interaction between weakly charged spheres, at constant surface charge, is1... 

The effect of electrolyte concentration on the transition from common to Newton black films and the stability of both types of films are explained using a model in which the interaction energy for films with planar interfaces is obtained by adding to the classical DLVO forces the hydration force. The theory takes into account the reassociation of the charges of the interface with the counterions as the electrolyte concentration increases and their replacements by ion pairs. This affects both the double layer repulsion, because the charge on the interface is decreased, and the hydration repulsion, because the ion pair density is increased by increasing the ionic strength. The theory also accounts for the thermal fluctuations of the two interfaces. Each of the two interfaces is considered as formed of small planar surfaces with a Boltzmannian distribution of the interdistances across the liquid film. The area of the small planar surfaces is calculated on the basis of a harmonic approximation of the interaction potential. It is shown that the fluctuations decrease the stability of both kinds of black films. 

Most of the water-mediated interactions between surfaces are described in terms of the DLVO theory [1,2]. When a surface is immersed in water containing an electrolyte, a cloud of ions can be formed around it, and if two such surfaces approach each other, the overlap of the ionic clouds generates repulsive interactions. In the traditional Poisson-Boltzmann approach, the ions are assumed to obey Boltzmannian distributions in a mean field potential. In spite of these rather drastic approximations, the Poisson-Boltzmann theory of the double layer interaction, coupled with the van der Waals attractions (the DLVO theory), could explain in most cases, at least qualitatively, and often quantitatively, the colloidal interactions [1,2]. 

As discussed earlier (see Secs. II and VI), for polystyrene spheres in water the DLVO pair potential provides an expression for the effective interparticle interaction that, with an appropriate renormalization of the charge, accounts for the main features of the structure of 3D homogeneous suspensions. One might think that the DLVO potential should be a good assumption under most circumstances. This, however, turns out to be the case at least for the systems being considered here. Then the question is, how to measure the effective pair potential One way to do it is described here in some detail. For sufficiently dilute suspensions, one can resort to the low concentration approximation to obtain the pair potential directly from the measured radial distribution functions, i.e.,... 

Thus, we can obtain the DLVO theory either from Eq. (71) or from Eq. (72). In the absence of any approximations, the two routes would yield the same results. In the PB approximation, we must use the HNC approximation (or some similar approximation involving In g(R), in order to obtain the result,... 

Because this result has been obtained by solving a generalized Poisson-Boltzmann equation with the linearization approximation, it is necessary to compare it with the DLVO theory in the limit where the Debye approximation holds. In this case, Verwey and Overbeek [2], working in cgs (centimeter-gram-second) units, derived the following approximate equation for the repulsive potential ... 

SI formalism and the calculations based on it above. First, there is ample evidence that the medium-range osmotic repulsion of DLVO theory is basically correct [11], A theory does not become textbook without being able to explain some phenomena well. Therefore, there must be something correct, at least to a leading approximation, about the repulsive potential... 

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