DLVO theory

The DLVO theory is named after Derjaguin and Landau, Verwey and Overbeek.

The theory describes the force between charged surfaces interacting through a liquid medium. It combines the effects of the van der Waals attraction and the electrostatic repulsion due to the so called double layer of counterions. The electrostatic part of the DLVO interaction is computed in the mean field approximation in the limit of low surface potentials - that is when the potential energy of an elementary charge on the surface is much smaller than the thermal energy scale, $k_{B}T$. For two spheres of radius $a$ with constant surface charge $Z$ separated by a center-to-center distance $r$ in a fluid of dielectric constant $\epsilon$ containing a concentration $n$ of monovalent ions, the electrostatic potential takes the form of a screened-Coulomb or Yukawa repulsion,

$\beta U(r)=Z^{2}\lambda _{B}\,\left({\frac {\exp(\kappa a)}{1+\kappa a}}\right)^{2}\,{\frac {\exp(-\kappa r)}{r}},$

where $\lambda _{B}$ is the Bjerrum length, $\kappa ^{-1}$ is the Debye-Hückel screening length, which is given by $\kappa ^{2}=4\pi \lambda _{B}n$, and $\beta ^{-1}=k_{B}T$ is the thermal energy scale at absolute temperature $T$.

History

In 1923, the first successful theory for ionic solution was developed by Debye and Hückel. The framework of linearized DH theory was applied to describe colloidal dispersions. After that, Levine and Dube found that between colloidal particles there were both a medium-range strong repulsion and a long-range strong attraction, but they could not describe the stabiliy and instability of colloidal dispersion. In 1941, Derjaguin and Landau provided the initial theory of combination of attraction and repulsion forces. Seven years later, Verwey and Overbeek got the same answer. Both groups got the result independently. In their work they corrected the defect of the Levine-Dube theory for colloidal system and formulated the classical standard theory of colloidal dispersions which described successfully the irreversible process of coagulation of colloidal particles.

Derivation of DLVO theory

DLVO theory is the combined effect of van der Waals and double layer force. During the derivation, different conditions must be taken into account and different equations can be got.. But some useful assumption can effectively simplified the process, which is actually 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, diple-induced dipole force and dispersion forces, 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.. With another assumption of additivity, the net interaction energy betwween 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. So the net interaction energy for a molecule at a distance D away from the surface will therefore be

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

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

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

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

For conveience, Hamaker constant A is given as
$A=\left(\pi ^{2}C\rho _{1}\rho _{2}\right)$
and the equation will become
$W(D)=-{\frac {AR}{6D}}$

With similay method and according to Derjaguin approximation, the van der Waals interaction energy between particles with different shapes can be got, such as energy between
two spheres: $W(D)=-{\frac {A}{6D}}{\frac {R_{1}R_{2}}{(R_{1}+R_{2})}}$

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

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

Double layer force

Main article: Double Layer (interfacial)

A surface in a liquid may be charged by dissociation of surface groups (e.g. silanol groups for glass surfaces) or by adsorption of charged molecules such as protein ions from surrounding solution. This results in the development of a wall surface potential which will attract counterions from the surrounding solution and exclude co-ions. In equilibrium, the wall surface charge is balanced by an equal but oppositely charge of counterions. The region of counterions 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 and thus relatively mobile ions. The total electrical double layer due to the formation of the counterion layers results in electrostatic screening of the wall charge and minimizes the Gibb's free energy.

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

$\kappa =(\sum _{i}\rho _{\infty i}e^{2}z_{i}^{2}/\epsilon \epsilon _{0}kT)^{1/2}$
with unit of m^-1 where

• • $\rho _{\infty i}$ 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.
  • $\epsilon _{0}$ is the electric constant, $\epsilon$ is the relative static permittivity.
  • k is the boltzmann constant.

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

$W=(64kT\rho _{\infty }\gamma ^{2}/\kappa )e^{-\kappa D}$
where

• • $\gamma$ is the reduced surface potential
    

$\gamma =tanh\left({\frac {ze\psi _{0}}{4kT}}\right)$

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

The interaction free energy between two speres of radius R is

$W=(64\pi kTR\rho _{\infty }\gamma ^{2}/\kappa ^{2})e^{-\kappa D}$

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 got as:
$W\left(D\right)=W(D)_{A}+W(D)_{R}$

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

Application of DLVO theory

Since 1940s, the DLVO theory has been used to explan phenomena found in colloidal science, adsorption and many other fields. According to the appearance of nanoparticles, DLVO theory becomes even more popular. Because it can be used to explain both general nanoparticles such as fullerenes particles and microorganisms.

Shortcomings of the DLVO theory

The theory is not effective in describing ordering processes such as the evolution of colloidal crystals in dilute dispersions with low salt concentrations. It also can not explain the relation between the formation of colloidal crystals and salt concentrations.