The electrostatic double-layer force

If two charged surfaces approach each other and the electric double layers overlap, an electrostatic double-layer force arises. This electrostatic double-layer force is essential for the stabilization of dispersion in aqueous media. 

Please note that the electrostatic double-layer force is fundamentally different from the Coulomb force. The difference is the presence of free charges (ions) in solution. They screen the electrostatic field emanating from the surfaces. 

The interaction between two charged surfaces in liquid depends on the surface charge. Here, we only consider the linear case and assume that the surface potentials are low. If we had to use the nonlinear Poisson-Boltzmann theory, the calculations would become substantially more complex. In addition, only monovalent salts are considered. An extension to other salts can easily be made. 

---

Please note that the electrostatic double-layer force is fundamentally different from the Coulomb force. For example, if we consider two identical spherical particles of radius R you cannot take Eq. (6.1), insert the total surface charge as Qi and Q2, use the dielectric permittivity of water and expect to get a reasonable result. The main differences are the free charges (ions) in solution. They screen the electrostatic field emanating from the surfaces. 

In an aqueous medium, the electrostatic double-layer force is present. For distances x larger then the Debye length A it decays roughly exponentially F oc exp (—x/A >). 

For the case of purely attractive forces (such as Lon-don-van der Waals forces) the length Sjr over which they act is a useful characteristic. An attractive force which acts over a distance which is much less than Sc will not contribute substantially to the overall rate. When repulsive forces (such as the electrostatic double-layer forces) are also present, they may effectively prevent particles from arriving at the collector, even when they act only over a very short distance. For this reason the decay length alone cannot characterize the relative importance of the joint effect of attractive and repulsive farces. Useful characteristics of their combined effect may be obtained by considering the total potential energy of interaction between the particle and the collector. 

Even though the components of the total interaction potential between such complex adsorbents as solid carbons and a wide range of adsorbates can be grouped in many different ways 315,316], it is convenient and meaningful to consider only the London dispersion (induced dipole) forces and the electrostatic (double-layer) forces [620,621,76,77]. 

First measurements of the electrostatic double-layer force with the AFM were done in 1991 [9, 10]. The electrostatic double layer depends on the surface charge density (or the surface potential) and the ionic strength. A brief introduction to the theory of the electrostatic force is given in Chap. 4. The electrostatic double-layer force is in many cases responsible for the stabilization of dispersions. An AFM experiment can be regarded as directly probing the interaction between a sample surface and a colloidal particle (or the AFM tip). Since the AFM tip is relatively small, this interaction can be probed locally. The lateral spacial resolution can be of the order of few nanometers. 

With the SFA, the main predictions of the DLVO theory were verified. In particular the electrostatic double-layer force was analyzed for difierent salts and under... 

Characteristic decay lengths Xo determined with the SFA, the osmotic stress method, or the AFM range from 0.2 to 1.4 nm. Typical amphtudes are A = 10 to 10 J m . In contrast to the electrostatic double-layer force, hydration forces tend to become stronger and longer ranged with increasing salt concentration, especially for divalent cations. 

Since the first measurements of the electrostatic double-layer force with the AFM not even 10 years ago, the instrument has become a versatile tool to measure surface forces in aqueous electrolyte. Force measurements with the AFM confirmed that with continuum theory based on the Poisson-Boltzmann equation and appKed by Debye, Hiickel, Gouy, and Chapman, the electrostatic double layer can be adequately described for distances larger than 1 to 5 nm. It is valid for all materials investigated so far without exception. It also holds for deformable interfaces such as the air-water interface and the interface between two immiscible liquids. Even the behavior at high surface potentials seems to be described by continuum theory, although some questions still have to be clarified. For close distances, often the hydration force between hydrophilic surfaces influences the interaction. Between hydrophobic surfaces with contact angles above 80°, often the hydrophobic attraction dominates the total force. 

The electrostatic double-layer force can be calculated using the continuum theory, which is based on the theory of Gouy, Chapman, Debye, and Hiickel for an electrical double layer. The Debye length relates the surface charge density of a surface to the electrostatic surface potential /o via the Grahame equation, which for 1 1 electrolytes can be expressed as... 

Assuming constant potentials of the sample % and the tip titj-, for constant charge conditions the electrostatic double-layer force is... 

In contrast to the electrostatic double-layer force, hydration forces tend to become stronger and longer ranged with increasing salt concentration, especially for divalent cations [13]. 

The electrostatic double-layer forces are obtained by solving the Poisson-Boltzmann equation under a variety of different boundary conditions. There exists an extensive literature concerning the calculation of the electrostatic repulsion between interfaees, and therefore only two of the classic results will be given here as illustrative examples. 

The electrostatic double-layer force dominates at relatively medium-large separations. However, when very far away from the surfaces (very large separations) and when the surfaces are brought very close to each other, the attractive van der Waals forces (may) overcome the repulsive forces and dominate the interactions. If vdW forces dominate the surfaces will be pulled into a strong adhesive contact (attraction -instability) whereas stability is obtained in the region where the repulsion forces dominate. 

The major effect in the interaction between identical protein particles comes from the electrostatic double-layer forces (vdW forces are weak). Interestingly, these forces are indeed repulsive for monovalent cations and agree with the simulation results (points in the figures shown above). However, when the medium contains divalent salts. 

As one example, the electrostatic double-layer force was calculated for a sphere of Ri =3 j,m radius interacting with a flat surface R2 = 00). The surface charge was adjusted by O1/2 = EEoXtj)i 2 so that at large distances both lead to the same potential. 

Here, Ilvdw is the van der Waals force. He is the electrostatic double-layer force, and list contains steric and short-range structural forces. In addition, other forces might contribute. 

---