DLVO theory

van der Waals force is actually the total name of dipole-dipole force, dipole-induced dipole force and dispersion forces,[14] 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/rⁿ, where C is a constant for interaction energy, decided by the molecule's property and n = 6 for van der Waals attraction.[15] 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.[14] So the net interaction energy for a molecule at a distance D away from the surface will therefore be

${\displaystyle w(D)=-2\pi \,C\rho _{1}\,\int _{z=D}^{z=\infty \,}dz\int _{x=0}^{x=\infty \,}{\frac {xdx}{(z^{2}+x^{2})^{3}}}={\frac {2\pi C\rho _{1}}{4}}\int _{D}^{\infty }{\frac {dz}{z^{4}}}=-{\frac {\pi C\rho _{1}}{6D^{3}}}}$

where

• • w(r) is the interaction energy between the molecule and the surface
  • ${\displaystyle \rho _{1}}$ is the number density of the surface.
  • z is the axis perpendicular with the surface and passes across the molecule. z = D at the point where the molecule is and z = 0 at the surface.
  • x is the axis perpendicular with z axis, where x = 0 at the intersection.

Then the interaction energy of a large sphere of radius R and a flat surface can be calculated as

${\displaystyle W(D)=-{\frac {2\pi C\rho _{1}\rho _{2}}{12}}\int _{z=0}^{z=2R}{\frac {(2R-z)zdz}{(D+z)^{3}}}\approx -{\frac {\pi ^{2}C\rho _{1}\rho _{2}R}{6D}}}$

where

• • W(D) is the interaction energy between the sphere and the surface.
  • ${\displaystyle \rho _{2}}$ is the number density of the sphere

For convenience, Hamaker constant A is given as

${\displaystyle A=\pi ^{2}C\rho _{1}\rho _{2}}$

and the equation will become

${\displaystyle W(D)=-{\frac {AR}{6D}}}$

With a similar method and according to Derjaguin approximation,[16] the van der Waals interaction energy between particles with different shapes can be calculated, such as energy between

two spheres: ${\displaystyle W(D)=-{\frac {A}{6D}}{\frac {R_{1}R_{2}}{(R_{1}+R_{2})}}}$

sphere-surface: ${\displaystyle W(D)=-{\frac {AR}{6D}}}$

Two surfaces: ${\displaystyle W(D)=-{\frac {A}{12\pi D^{2}}}}$ per unit area

A surface in a liquid may be charged by dissociation of surface groups (e.g. silanol groups for glass or silica surfaces[17]) 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 ${\displaystyle 1/\kappa }$. At a distance of two Debye screening lengths the electrical potential energy is reduced to 2 percent of the value at the surface wall.

${\displaystyle \kappa ={\sqrt {\sum _{i}{\frac {\rho _{\infty i}e^{2}z_{i}^{2}}{\epsilon _{r}\epsilon _{0}k_{\rm {B}}T}}}}}$

with unit of m⁻¹ where

• ${\displaystyle \rho _{\infty i}}$ is the number density of ion i in the bulk solution.
• z is the valency of the ion. For example, H⁺ has a valency of +1 and Ca²⁺ has a valency of +2.
• ${\displaystyle \varepsilon _{0}}$ is the vacuum permittivity, ${\displaystyle \epsilon _{r}}$ is the relative static permittivity.
• k_B is the Boltzmann constant.

The repulsive free energy per unit area between two planar surfaces is shown as

${\displaystyle W={\frac {64k_{\rm {B}}T\rho _{\infty }\gamma ^{2}}{\kappa }}e^{-\kappa D}}$

where

• ${\displaystyle \gamma }$ is the reduced surface potential

${\displaystyle \gamma =\tanh \left({\frac {ze\psi _{0}}{4k_{\rm {B}}T}}\right)}$

• ${\displaystyle \psi _{0}}$ is the potential on the surface.

The interaction free energy between two spheres of radius R is

${\displaystyle W={\frac {64\pi k_{\rm {B}}TR\rho _{\infty }\gamma ^{2}}{\kappa ^{2}}}e^{-\kappa D}}$[18]

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:

${\displaystyle W\left(D\right)=W(D)_{A}+W(D)_{R}\,}$

where W(D)_R is the repulsive interaction energy due to electric repulsion and W(D)_A is the attractive interaction energy due to van der Waals interaction.