DLVO Theory (and other stuffs) The system we want to study. Colloids in an electrolytic

solution (many ions agglomerate around them). Dealing with non-ideal systems (interactive particles).

Necessary approximations to deal with them. How these colloids interact each other.

Van der Waals attraction versus Double layer repulsive force. How to calculate these 2 terms.

An example of a system with strongly interacting particles, methods to model it. Phase transitions. Simulation of ferromagnetic nano-structures.

Colloids in a solution.*These charge particles may consist of many atoms and they tend to attract ions from the electrolyte which will form an agglomerate around them.* Colloids interact one another: VdW attraction and EDL repulsion. The balance between these 2 forces give stability to the system and avoids the formation of clusters.

Colloids in a solution.*These charge particles may consist of many atoms and they tend to attract ions from the electrolyte which will form an agglomerate around them.* Colloids interact one another: VdW attraction and EDL repulsion. The balance between these 2 forces give stability to the system and avoids the formation of clusters.

This system is cleary non-ideal: Interaction among its particles (the colloids).

Example of non-ideal systems: Real gas (Van der Waals attraction between molecules), Classical plasma (Coulomb interaction), Ferromagnets (Exchange-interaction: Coulomb + exclusion principle), Quantum ideal gas (exchange interaction).

These interactions lead to Phase Transitions (of 1st or 2nd order, at the critical point).

The problem is that these correlations make impossible to calculate the partition function exactly, that's why we need useful approximations like: Cluster expansion, Correlation corrections, Mean-Field-Theory, also B-P, BWA, RGT, Monte Carlo-Metropolis.

(Black-Board 1)

Black Board 1 (Non-ideal systems)

DLVO theory The DLVO theory is named after: Derjaguin and Landau (1941), Verwey and

Overbeek (1948). The theory explains the aggregation of aqueous dispersions quantitatively

and describes the force between charged surfaces interacting through a liquid medium.

Combines the effects of the van der Waals attraction (quite easy to compute) and the electrostatic repulsion due to the so-called double layer of counter-ions. The electrostatic part of the DLVO interaction is computed in the mean field approximation in the limit of low surface potentials (requires a high-T approximation to linearize the Poisson-Boltzmann equation).

History and Derivation 1923: Debye and Hckel reported the first successful theory for the

distribution of charges in ionic solutions. The framework of linearized DebyeHckel 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 LevineDube theory to account for the dependence of colloidal dispersions stability on the ionic strength of the electrolyte.

Derivation: Superposition Principle (VdW + EDL separately).

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.

Pair potential between 2 molecules:

We can calculate: The net interaction energy between a molecule and planar surface made up of like molecules (superposition).

And also: the interaction energy between 2 planes or 2 spheres (each consisting of many atoms together).

With the aid of the previous result we can calculate:

The case of 2 surfaces is most widely used. (Black-Board 2)

Electric Double layer repulsion 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.

*We seek the electric potential in order to compute the interaction energy between 2 colloids near each other.

We have a charge particle (the colloid) which creates an electric potential in a region where many ions are agglomerated, so we have a region with a charge density:

To find the potential we solve the Poisson equation, taking into account that the concentration of ions follows a Boltzmann-distribution. In this way we are led to a Boltzmann-Poisson equation for the potential, which can be solved considering the temperature to be high enough (in this way we linearize this P.D.E.):

The last equations were taken from Landau's Book Vol 5 and are used to treat a classical plasma (similar system).

For the system we are actually considering the last solution is a bit different:

The range is not infinite. It's actually short.

Now, like in the VdW case: U(r) ------> W(D) We can pass to the S.I system, so:

Finally we can compute the net interaction between eg. 2 surfaces:

Finally we can compute the net interaction between eg. 2 surfaces:

From a theoretical viewpoint, the zeta potential is the electric potential in the interfacial double layer (DL) at the location of the slipping plane relative to a point in the bulk fluid away from the interface.

In other words, zeta potential is the potential difference between the dispersion medium and the stationary layer of fluid attached to the dispersed particle.

The zeta potential is a key indicator of the stability of colloidal dispersions. The magnitude of the zeta potential indicates the degree of electrostatic repulsion between adjacent, similarly charged particles in a dispersion.

When the potential is small,attractive forces may exceed this repulsion and the dispersion may break and flocculate. So, colloids with high zeta potential (negative or positive) are electrically stabilized while colloids with low zeta potentials tend to coagulate or flocculate.

The most known and widely used theory for calculating zeta potential from experimental data is that developed by Marian Smoluchowski in 1903.

This theory was originally developed for electrophoresis; however, an extension to electroacoustics is now also available.

Smoluchowskis theory is powerful because it is valid for dispersed particles of any shape and any concentration (although it has its limitations).

SUMMARY:

OTHER SYSTEM OF HIGHLY INTERACTING PARTICLES

Ferromagnets----> Models of Ising and Heisenberg. Short Range and Strong Interaction. The partition function cannot be found analytically. Approximations are necessary: Cluster expansion fails. Useful Metods: MFA, BWA, BPA, Landau's Theory, RGT. All of them have an uncertainty that cannot be controlled. The best choice is to rely in the Monte Carlo method for this

strongly interacting system (error follow the LLN). So by increasing the number of Steps in the MC-Trajectory we reduce the uncertainty.

The system exhibits 2nd order phase transition in the Thermodynamic Limit (Yang-Lee Theorem).

I was able to simulate a system which can be made of large size by applying the BvK periodic B.C for different kinds of lattices-------> I studied the BULK of the material.

What would happen if I change the B.C and let the system to have a definite size (consisting of a definite number of atoms)?

There is possibility to simulate ferromagnetic nano-structures by making some variants in the codes I wrote.

In the case of the system Heisenberg-3D the total C++ program consisted in more than 1000 lines of codel. The functions which constructed the energy of the system occupied most of the space because of the needed BvK PBC.

In invoked Landau's Theory (and also some experiments) to justify some noise near the critical point.

Thanks for you attention

Lev Landau

Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 21Slide 22Slide 23Slide 24Slide 25Slide 26Slide 27Slide 28Slide 29Slide 30Slide 31Slide 32Slide 33Slide 34