Double layer forces

Double layer forces occur between charged objects across liquids, typically water. This force acts over distances that are comparable to the Debye length, which is on the order of one to a few tenths of nanometers. The strength of these forces increases with the magnitude of the surface charge density (or the electrical surface potential). For two similarly charged objects, this force is repulsive and decays exponentially at larger distances, see figure. For unequally charged objects and eventually at shorted distances, these forces may also be attractive. The theory due to Derjaguin, Landau, Verwey, and Overbeek (DLVO) combines such double layer forces together with Van der Waals forces in order to estimate the actual interaction potential between colloidal particles. Double layer forces occur between charged objects across liquids, typically water. This force acts over distances that are comparable to the Debye length, which is on the order of one to a few tenths of nanometers. The strength of these forces increases with the magnitude of the surface charge density (or the electrical surface potential). For two similarly charged objects, this force is repulsive and decays exponentially at larger distances, see figure. For unequally charged objects and eventually at shorted distances, these forces may also be attractive. The theory due to Derjaguin, Landau, Verwey, and Overbeek (DLVO) combines such double layer forces together with Van der Waals forces in order to estimate the actual interaction potential between colloidal particles. An electrical double layer develops near charged surfaces (or another charged objects) in aqueous solutions. Within this double layer, the first layer corresponds to the charged surface. These charges may originate from tightly adsorbed ions, dissociated surface groups, or substituted ions within the crystal lattice. The second layer corresponds to the diffuse layer, which contains the neutralizing charge consisting of accumulated counterions and depleted coions. The resulting potential profile between these two objects leads to differences in the ionic concentrations within the gap between these objects with respect to the bulk solution. These differences generate an osmotic pressure, which generates a force between these objects. These forces are easily experienced when hands are washed with soap. Adsorbing soap molecules make the skin negatively charged, and the slippery feeling is caused by the strongly repulsive double layer forces. These forces are further relevant in many colloidal or biological systems, and may be responsible for their stability, formation of colloidal crystals, or their rheological properties. The most popular model to describe the electrical double layer is the Poisson-Boltzmann (PB) model. This model can be equally used to evaluate double layer forces. Let us discuss this model in the case of planar geometry as shown in the figure on the right. In this case, the electrical potential profile ψ(z) near a charged interface will only depend on the position z. The corresponding Poisson's equation reads in SI units where ρ is the charge density per unit volume, ε0 the dielectric permittivity of the vacuum, and ε the dielectric constant of the liquid. For a symmetric electrolyte consisting of cations and anions having a charge ±q, the charge density can be expressed as where c± = N±/V are the concentrations of the cations and anions, where N± are their numbers and V the sample volume. These profiles can be related to the electrical potential by considering the fact that the chemical potential of the ions is constant. For both ions, this relation can be written as where μ ± ( 0 ) {displaystyle mu _{pm }^{(0)}} is the reference chemical potential, T the absolute temperature, and k the Boltzmann constant. The reference chemical potential can be eliminated by applying the same equation far away from the surface where the potential is assumed to vanish and concentrations attain the bulk concentration cB. The concentration profiles thus become where β = 1/(kT). This relation reflects the Boltzmann distribution of the ions with the energy ±qψ. Inserting these relations into the Poisson equation one obtains the PB equation