Editing DLVO theory (section)

==Derivation==
DLVO theory is the combined effect of [[Van der Waals force|van der Waals]] and [[double layer (interfacial)|double layer]] force. For the derivation, different conditions must be taken into account and different equations can be obtained.<ref name="Elimelech">M. Elimelech, J. Gregory, X. Jia, R. A. Williams, ''Particle Deposition and Aggregation Measurement: Modelling and Simulation'' (Boston: 1995).</ref> But some useful assumptions can effectively simplify the process, which are suitable for ordinary conditions. The simplified way to derive it is to add the two parts together.

===van der Waals attraction===
{{main article|van der Waals force}}
van der Waals force is actually the total name of dipole-dipole force, dipole-induced dipole force and dispersion forces,<ref name="Jacob">Jacob N. Israelacvili, ''Intermolecular and Surface Forces'' (London 2007).</ref> in which dispersion forces are the most important part because they are always present.
Assume that the pair potential between two atoms or small molecules is purely attractive and of the form w = −C/r<sup>n</sup>, where C is a constant for interaction energy, decided by the molecule's property and n = 6 for van der Waals attraction.<ref name="London">London, F. (1937), ''Trans Faraday Soc'', '''33''', 8–26.</ref> With another assumption of additivity, the net interaction energy between a molecule and planar surface made up of like molecules will be the sum of the interaction energy between the molecule and every molecule in the surface body.<ref name="Jacob" /> So the net interaction energy for a molecule at a distance D away from the surface will therefore be
<math display="block">w(D) = -2 \pi \, C \rho _1\, \int_{z=D}^{z= \infty \,} dz \int_{x=0}^{x=\infty \,}\frac{x \, dx}{(z^2+x^2)^3} = \frac{2 \pi C \rho _1}{4} \int_D^\infty \frac{dz}{z^4} = - \frac{ \pi C \rho _1  }{ 6 D^3 }</math>

where 
* {{math|''w''(''r'')}} is the interaction energy between the molecule and the surface,
* <math> \rho_1 </math> is the number density of the surface,
* {{math|''z''}} is the axis perpendicular to the surface and passesding across the molecule, with {{math|1=''z'' = ''D''}} at the point where the molecule is, and {{math|1=''z'' = 0}} at the surface,
* {{math|''x''}} is the axis perpendicular to the {{math|''z''}} axis, with {{math|1=''x'' = 0}} at the intersection.

Then the interaction energy of a large sphere of radius ''R'' and a flat surface can be calculated as
<math display="block">W(D) = -\frac{2 \pi C \rho _1 \rho _2}{12} \int_{z=0}^{z=2R}\frac {(2R-z)zdz}{(D+z)^3} \approx  -\frac{ \pi ^2 C \rho _1 \rho _2 R}{6D}</math>

where
* ''W''(''D'') is the interaction energy between the sphere and the surface,
* <math>\rho_2</math> is the number density of the sphere.

For convenience, [[Hamaker constant]] ''A'' is given as
<math display="block"> A = \pi^2C\rho_1\rho_2, </math>
and the equation becomes
<math display="block">W(D) = -\frac{AR}{6D}. </math>

With a similar method and according to [[Derjaguin approximation]],<ref name="Derjaguin">Derjaguin B. V. (1934)''Kolloid Zeits'' '''69''', 155–164.</ref> the van der Waals interaction energy between particles with different shapes can be calculated, such as energy between

* two spheres: <math>W(D) = -\frac{A}{6D} \frac{R_1 R_2}{(R_1 +R_2 )},</math>
* sphere and surface: <math>W(D) = -\frac{AR}{6D},</math>
* two surfaces: <math>W(D) = -\frac{A}{12 \pi D^2}</math> per unit area.

===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.