Editing DLVO theory (section)

===Double layer force===
{{main article|Double layer forces}}
A surface in a liquid may be charged by dissociation of surface groups (e.g. silanol groups for glass or silica surfaces<ref>{{cite journal | url=https://doi.org/10.1063/1.1404988 | doi=10.1063/1.1404988 | title=The charge of glass and silica surfaces | date=2001 | last1=Behrens | first1=Sven H. | last2=Grier | first2=David G. | journal=The Journal of Chemical Physics | volume=115 | issue=14 | pages=6716–6721 | arxiv=cond-mat/0105149 | bibcode=2001JChPh.115.6716B | s2cid=19366668 }}</ref>) or by adsorption of charged molecules such as [[polyelectrolyte]] from the surrounding solution. This results in the development of a wall surface potential which will attract counterions from the surrounding solution and repel co-ions. In equilibrium, the surface charge is balanced by oppositely charged counterions in solution. The region near the surface of enhanced
counterion concentration is called the electrical double layer (EDL). The EDL can be approximated by a sub-division into two regions. Ions in the region closest to the charged wall surface are strongly bound to the surface. This immobile layer is called the Stern or Helmholtz layer. The region adjacent to the Stern layer is called the diffuse layer and contains loosely associated ions that are comparatively mobile. The total electrical double layer due to the formation of the counterion layers results in electrostatic screening of the wall charge and minimizes the [[Gibbs free energy]] of EDL formation.

The thickness of the diffuse electric double layer is known as the [[Debye screening length]] <math>1 / \kappa</math>. At a distance of two Debye screening lengths the electrical potential energy is reduced to 2 percent of the value at the surface wall.

<math display="block">\kappa = \sqrt{\sum_i \frac{\rho_{\infty i} e^2z^2_i}{\epsilon_r \epsilon_0 k_\text{B} T}}</math>

with unit of {{math|m<sup>&minus;1</sup>}}, where
* <math>\rho_{\infty i}</math> is the [[number density]] of ion i in the bulk solution,
* {{math|''z''}} is the valency of the ion (for example, H<sup>+</sup> has a valency of +1, and Ca<sup>2+</sup> has a valency of +2),
* <math>\varepsilon_0</math> is the [[vacuum permittivity]], <math>\epsilon_r</math> is the [[relative static permittivity]],
* {{math|''k''<sub>B</sub>}} is the [[Boltzmann constant]].

The repulsive free energy per unit area between two planar surfaces is shown as
<math display="block">W = \frac{64k_\text{B} T\rho_{\infty } \gamma ^2}{\kappa}e^{-\kappa D}</math>
where
* <math>\gamma</math> is the reduced surface potential, <math>\gamma = \tanh\left(\frac{ze\psi_0}{4k_\text{B}T}\right)</math>,
* <math>\psi_0</math> is the potential on the surface.

The interaction free energy between two spheres of radius ''R'' is<ref>
{{Citation
 | last1=Bhattacharjee
 | first1=S.
 | last2=Elimelech
 | first2=M.
 | last3=Borkovec
 | first3=Michal
 | year=1998
 | title=DLVO interaction between colloidal particles: Beyond Derjaguins approximation
 | journal=Croatica Chimca Acta
 | volume=71
 | pages=883–903
}}</ref>
<math display="block">W = \frac{64\pi k_\text{B} TR\rho_{\infty} \gamma ^2}{\kappa ^2}e^{-\kappa D}.</math>

Combining the van der Waals interaction energy and the double layer interaction energy, the interaction between two particles or two surfaces in a liquid can be expressed as
<math display="block">W(D) = W(D)_\text{A} + W(D)_\text{R},</math>
where ''W''(''D'')<sub>R</sub> is the repulsive interaction energy due to electric repulsion, and ''W''(''D'')<sub>A</sub> is the attractive interaction energy due to van der Waals interaction.