page_title: Double layer forces
section_title: Poisson–Boltzmann model
hierarchical_section_title: Double layer forces / Poisson–Boltzmann model
caption_attribution_description: English: Double Layer Forces Scheme Plates
context_page_description: 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. For unequally charged objects and eventually at shorted distances, these forces may also be attractive. The theory due to Derjaguin, Landau, Verwey, and Overbeek 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 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.
context_section_description: 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  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
  The potential profile between two plates is normally obtained by solving this equation numerically.
  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
  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. Alternative derivation of the same relation for disjoining pressure involves the stress tensor.
caption: Pictorial representation of two interacting charged plates across an electrolyte solution. The distance between the plates is abbreviated by h.
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