Editing DLVO theory (section)

===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.