# DLVO theory

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, kBT. 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 λB is the Bjerrum length, κ − 1 is the Debye-Hückel screening length, which is given by κ2 = 4πλBn, and β − 1 = kBT is the thermal energy scale at absolute temperature T.

## History

In 1923, Debye and Hückel reported the first successful theory for the distribution of charges in ionic solutions.  The framework of linearized Debye-Hückel theory subsequently was applied to colloidal dispersions by Levine and Dube   who found that charged colloidal particles should experience a strong medium-range repulsion and a weaker long-range attraction. This theory did not explain the observed instability of colloidal dispersions against irreversible aggregation in solutions of high ionic strength. In 1941, Derjaguin and Landau introduced a theory for the stability of colloidal dispersions that invoked a fundamental instability driven by strong but short-ranged van der Waals attractions countered by the stabilizing influence of electrostatic repulsions.  Seven years later, Verwey and Overbeek independently arrived at the same result.  This so-called DLVO theory resolved the failure of the Levine-Dube theory to account for the dependence of colloidal dispersions' stability on the ionic strength of the electrolyte. 

## Derivation of DLVO theory

DLVO theory is the combined effect of van der Waals and double layer force. For the derivation, different conditions must be taken into account and different equations can be obtained. 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

van der Waals force is actually the total name of dipole-dipole force, dipole-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 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. So the net interaction energy for a molecule at a distance D away from the surface will therefore be $w(r) = -2 \pi \, C \rho _1\, \int_{z=D}^{z= \infty \,}dz \int_{x=0}^{x=\infty \,}\frac{xdx}{(z^2+x^2)^3} = \frac{2 \pi C \rho _1}{4}\int_D^{\infty }\frac{dz}{z^4} = -\pi C \rho _1/ 6 D^3$

where

• w(r) is the interaction energy between the molecule and the surface
• ρ1 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.
• ρ2 is the number density of the sphere

For convenience, Hamaker constant A is given as $A=\left(\pi^2C\rho_1\rho_2\right)$

and the equation will become $W(D)=-\frac{AR}{6D}$

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

A surface in a liquid may be charged by dissociation of surface groups (e.g. silanol groups for glass or silica surfaces  ) 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 1 / κ. At a distance of two Debye screening lengths the electrical potential energy is reduced to 2 percent of the value at the surface wall. $\kappa = (\sum_i \rho_{\infty i} e^2z^2_i/\epsilon \epsilon_0 k_B T)^{1/2}$

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.
• ε0 is the electric constant, $\epsilon$ is the relative static permittivity.
• kB is the Boltzmann constant.

The repulsive free energy per unit area between two planar surfaces is shown as $W = (64k_B T\rho_{\infty } \gamma ^2 /\kappa )e^{-\kappa D}$

where

• γ is the reduced surface potential $\gamma =\tanh\left(\frac{ze\psi_0}{4kT}\right)$

• ψ0 is the potential on the surface.

The interaction free energy between two spheres of radius R is $W=(64\pi k_B TR\rho_{\infty} \gamma ^2 /\kappa ^2 )e^{-\kappa D}$

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 expressed 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 explain phenomena found in colloidal science, adsorption and many other fields. Due 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. 

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