DLVO Theory

DLVO theory [1–3] describes the stabilization of colloidal dispersions by an interplay of van der Waals and electrostatic forces (as opposed to steric repulsions of colloids by polymeric solubilizers). The theory was developed in the 1940s by Derjaguin and Landau [4] and by Verwey and Overbeek [5]. In DLVO theory, the two determining interactions for the stability of a colloidal system are the attractive van der Waals interactions between the colloidal particles and the repulsive electrostatic Coulomb interactions. When salt is added, the alteration of the electrostatic interactions affects the stability.

The strength of the van der Waals interactions is determined by the size and the shape of the colloidal particles and by the chemical composition of the system, which is described by the Hamaker constant A. Between two similar particles, the van der Waals forces are always attractive and A is a positive constant. For spherical particles of radius R at separation d, the van der Waals energy is given as

$$ V(d)={-AR \left/ {12d } \right.}. $$

This relation is valid until the particles are in contact, where a steep repulsion prevents steric overlap. For other shapes, the dependence on the separation d is different, but for all cases, a 1/d n dependence is observed. The attractive van der Waals forces, which depend only weakly on the salt concentration, are strong at small separations and give the dominant energy contribution.

The second interaction that is relevant for the stability is of electrostatic nature: In the presence of inert salts (salts that are composed of ions that do not adsorb on the colloidal surface), an electrostatic double layer is formed around a charged colloidal particle. The simplest description, which holds for monovalent salts at low concentrations, is the Gouy-Chapman solution of the Poisson-Boltzmann equation, which yields that the resulting electrostatic potential decays exponentially with the distance from the colloid. How quick this decay is depends on the salt environment and is characterized by the screening length k−1. According to the Gouy-Chapman model, the extent of the repulsion becomes shorter with increasing salt concentration: k−1 ∼ χ−1/2. During the coagulation process, two identical particles approach each other, and the electrostatic interaction between the two spherical particles, including their double layers, is

$$ V(d)={RZ \left/ {2} \right.}\ \mathrm{ Exp}\left[ {-\mathrm{ k}d} \right] $$

In the following, we will consider particles with similar surface chemistry, for which Z is a positive constant that depends on the surface potential of the particles. For different shapes, rather similar expressions that depend on Exp[−kd] are obtained. The electrostatic forces are therefore repulsive and act against coagulation. The sign and magnitude of the surface potential can be influenced by so-called potential-determining ions, which react with or bind to the surface. In many cases, colloidal particles have acidic or basic surface groups, and the surface potential control is achieved by changing the pH.

Two different electrostatic boundary conditions are frequently combined with DLVO theory: In the constant charge boundary condition, the surface charge does not depend on the particle separation, in contrast to the constant potential boundary condition, which accounts for changes in the surface charge of one particle that are induced by the potential of the other one. Under constant charge, the predicted repulsive force is an upper limit for the experimental force, while a lower estimate is given under constant potential conditions. Charge-regulation models are closer to the real situation, but they require additional input.

The DLVO approach to colloidal stability identifies the interactions between particles as the sum of the van der Waals energy and the screened Coulomb energy. Depending on their relative strength, different behavior arises: For high surface potentials, there are two minima: a “primary” minimum at contact of the particles in the coagulated state, and a “secondary” minimum that describes flocculation, where a loose agglomerate of colloids is reversibly formed. Both minima are separated by an electrostatic barrier that prevents coagulation. This case is shown in Fig. 1a. For weaker surface potentials, this barrier gets smaller and eventually becomes zero. In this case, there is still a weak secondary minimum, but the colloidal dispersion is metastable, and rapid coagulation sets in. This case is shown in Fig. 1b. When the surface potential gets even smaller, the secondary minimum totally disappears. Particles always attract each other in this case, and coagulation from this unstable system will occur instantly. This case is shown in Fig. 1c.

DLVO Theory, Fig. 1
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Coulomb and vdW Energy between Colloidal Particles

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In summary: High surface potentials stabilize colloidal systems. The addition of inert salt leads to stronger screening and destabilizes the system. The point at which rapid coagulation (case b) sets in is defined as the critical coagulation concentration (ccc). One key result of DLVO theory is the explanation of the Schultze-Hardy rule, which states that the ccc depends on the counterion valency z like 1/z 6.

For the quantitative application of DLVO theory, it is important to notice that the origin, d = 0, of the van der Waals and Coulomb potential do not need to coincide. The screening layer starts at the outer Helmholtz plane (OHZ), which occurs at distances that are larger by a constant d than the origin of the van der Waals potential. This can drastically change the total potential: The primary minimum can be eliminated with values of d in the range of a few Ang. In this case, the system will be stable regardless of the extension of the screening layer.

At the times when DLVO theory was developed, the direct measurement of forces between colloidal particles and surfaces in solution was not possible, and the macroscopic observation of colloidal stability was the only experimental reference data. With increasing technological advancement, setups have been developed for the direct observation of such forces: The surface force apparatus (SFA) allows for the measurements of forces between surfaces in solution [6], and with an atomic force microscope (AFM), forces on a colloidal particle can be detected [7]. It is a major success that DLVO theory predicts forces that agree nicely with the measured forces for large particle separations (more than 3–10 nm), but at the same time, it is obvious that in the regime of short particle separations, not all effects are captured by DLVO. When the barrier for coagulation occurs at such low separations, the DLVO prediction for colloidal stability is not accurate (Fig. 2).

DLVO Theory, Fig. 2
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Experimental Results

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Despite its success, DLVO theory has various other limitations beside the failure at low particle separations. Intrinsically highly charged surfaces, where the ion’s electrostatic interactions with the surface dominate, are often well described by DLVO theory. However, in the case of surfaces with weak intrinsic charge, non-electrostatic forces between ions and the surface become important. DLVO theory does not account for these ion-specific effects [8], which have their origin in ion-type-dependant interactions of the ions with the colloidal particles and with the surrounding solvent. Consequently, DLVO is not able to predict the stability of proteins [9] or of simple air bubbles in salt solutions [10], where ion-specific effects cannot be neglected. Especially ions that are attracted to the particle surfaces are enriched in the particle solvation shells and induce deviations from DLVO theory: For surfaces with hydrophobic character, large ions like I− or SCN− show such deviations, at the surface of hydrophilic colloidal particles binding between specific ions and specific functional surface groups is important, an important example is ion pairing at ionized carboxyl groups on a protein surface.

Finally, another limitation of DLVO theory is that ion-ion correlations, which are important for multivalent ions or at high salt concentrations, are also not included.

Future Directions

The main shortcoming of DLVO theory is that it treats the interactions between the ions and the colloidal particle as purely electrostatic. Ions close to the particle are also subject to ion-type specific, often solvent-mediated attractive forces. Their inclusion on the level of an extended Poisson-Boltzmann equation leads to a more complex scenario, in which the salt concentration also changes the effective surface charge of the colloid [9]. This can induce resolubilization at high salt concentrations when a potential barrier reappears due to overcharging. The inclusion of ion-colloid dispersion interactions into the Poisson-Boltzmann equation can also induce similar effects.

Cross-References

Electrolytes, History

Specific Ion Effects, Evidences

Specific Ion Effects, Theory