Double layer forces

For two similarly charged objects, this force is repulsive and decays exponentially at larger distances, see figure.

Adsorbing soap molecules make the skin negatively charged, and the slippery feeling is caused by the strongly repulsive double layer forces.

Once the potential profile is known, the force per unit area between the plates expressed as the disjoining pressure Π can be obtained as follows.

The starting point is the Gibbs–Duhem relation for a two component system at constant temperature[3] Introducing the concentrations c± and using the expressions of the chemical potentials μ± given above one finds The concentration difference can be eliminated with the Poisson equation and the resulting equation can be integrated from infinite separation of the plates to the actual separation h by realizing that Expressing the concentration profiles in terms of the potential profiles one obtains From a known electrical potential profile ψ(z) one can calculate the disjoining pressure from this equation at any suitable position z.

By expanding the exponential function in the PB equation into a Taylor series, one obtains The parameter κ−1 is referred to as the Debye length, and some representative values for a monovalent salt in water at 25°C with ε ≃ 80 are given in the table on the right.

In non-aqueous solutions, Debye length can be substantially larger than the ones given in the table due to smaller dielectric constants.

In practice, the DH approximation remains rather accurate up to surface potentials that are comparable to the limiting values given above.

The disjoining pressure can be obtained from the PB equation given above, which can also be simplified to the DH case by expanding into Taylor series.

This approximation thus suggests that one can simply add (superpose) the potentials profiles originating from each surface as illustrated the figure.

Since the potential profile passes through a minimum at the mid-plane, it is easiest to evaluate the disjoining pressure at the midplane.

This approximation turns out to be exact provided the plate-plate separation is large compared to the Debye length and the surface potentials are low.

Even if the potential is large close to the surface, it will be small at larger distances, and can be described by the DH equation.

Within the PB model, this effective potential can be evaluated analytically, and reads[4] The superposition approximation can be easily extended to asymmetric systems.

Analogous arguments lead to the expression for the disjoining pressure where the super-scripted quantities refer to properties of the respective surface.

While the superposition approximation is actually exact at larger distances, it is no longer accurate at smaller separations.

By solving the DH equation one can show that diffuse layer potential varies upon approach as while the surface charged density obey a similar relation The swelling pressure can be found by inserting the exact solution of the DH equation into the expressions above and one finds Repulsion is strongest for the CC conditions (p = 1) while it is weaker for the CP conditions (p = 0).

In this case, one must solve the PB equation together with an appropriate model of the surface charging process.

At lower salt levels, however, the range of this attraction is related to the characteristic size of the surface charge heterogeneities.

Three-body forces: The interactions between weakly charged objects are pair-wise additive due to the linear nature of the DH approximation.

These three-body contributions were found to be attractive on the PB level, meaning that three charged objects repel less strongly than what one would expect on the basis of pair-wise interactions alone.

[12][15][16][17] Explicit expressions for the forces are mostly not available, but they are accessible with computer simulations, integral equations, or density functional theories.

However, this description breaks down in the strong coupling regime, which may be encountered for multivalent electrolytes, highly charged systems, or non-aqueous solvents.

Around 1990, theoretical and experimental evidence has emerged that forces between charged particles suspended in dilute solutions of monovalent electrolytes might be attractive at larger distances.

[22] Despite the initial criticism, accumulative evidence suggest that the DLVO fails to account for essential physics necessary to describe the experimental observations.

While the community remains skeptical regarding the existence of effective attractions between like-charged species, recent computer molecular dynamics simulations with an explicit description of the solvent have demonstrated that the solvent plays an important role in the structure of charged species in solution, while PB and the primitive model do not account for most of these effects.

Based on this idea, simulations have explained experimental trends such as the disappearance of a scattering peak in salt-free polyelectrolyte solutions[25] and the structural inhomogeneities of charged colloidal particles/nanoparticles [24] observed experimentally that PB and primitive model approaches fail to explain.

Estimated force between two charged colloidal particles with radius of 1 μm and surface charge density 2 mC/m 2 suspended in monovalent electrolyte solutions of different molar concentrations as indicated. The scheme sketches the charged colloidal particles screened by the electrolyte ions.
Pictorial representation of two interacting charged plates across an electrolyte solution. The distance between the plates is abbreviated by h .
Electrostatic potential across an electrolyte solution within the superposition approximation. Dashed lines correspond to the contributions from individual plates.
Charge regulation within the DH model for ψ D = 20 mV and a monovalent salt of a concentration c B = 1 mM. From left to right: Dependence upon the separation distance of the surface change density, diffuse layer potential, and the disjoining pressure. Constant charge (CC, p = 1) and constant potential (CP, p = 0) boundary conditions, and superposition approximation ( p = 1/2).