Chapter 3

Van der Waals forces

3.1 Introduction

Microscopic observations of colloidal particles in the nineteenth century have showntheir tendency to form persistent aggregates, even for uncharged particles with-out specific reactive site. This behaviour indicates attractive interparticle force.This attraction arises from dipolar interaction: local fluctuations in the polarizationwithin one particle induces correlated response in the others via the propagation ofelectromagnetic waves. A general description, with many-body interaction, is verycomplicated. Thus, de Boer [1] and Hamaker [2] assumed the intermolecular forcesto be strictly pairwise additive. Later, Lifshitz et al [3] proposed a continuum theorywhere many-body effects are taken into account by treating the particles and thesubphase as individual macroscopic phases characterized by their dielectric prop-erties. The basis of this treatment lies in quantum electrodynamic theory but aneasier description was reformulated by van Kampen et al [4] and the first quantita-tive implementation of the theory to real systems began with Parsegian and Ninham[5]. In vacuum, the solution of Laplace’s equation for the electric potential due to apoint dipole is given by

If a second dipole of moment is placed in this field at position r, an interactionenergy U results which reads

Molecules that do not possess permanent dipole posses a non-zero instantaneousdipole moment because of fluctuations caused for instance by electromagnetic ra-diation fields. Then instantaneous values of dipolar moment must be taken into

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48 CHAPTER 3. VAN DER WAALS FORCES

account: This dipole can then induce a dipolar moment in a secondmolecule which is a function of its polarisability Also, this kind of interactionexists even for unpolar molecules.Let us go back to the definition of the electric dipolar moment in a molecule.The instantaneous configuration of a molecule consisting of a set of nuclei and elec-trons with charges at positions The net charge is given by

and the electric dipole moment is defined as

When two molecules A and B are brought from an infinite separation to a distanceR, the charges on each particle will interact. The interaction energy isdefined by

with and the ground state energy of each particle and the energyof the new system. The interaction energy operator, in the total Hamiltonianof the system can be written as

in a vacuum, with R the position of the center of mass of the molecule B relativeto the center of mass of the molecule A. We suppose here that there is no spin-spininteraction and that the electromagnetic interaction is an electrostatic one.If the intermolecular distance R is greater than and , the last equation canbe expanded in powers ofFor neutral particles,

and the terms with only or are equals to zero and

If A and B are polar molecules,

3.1. INTRODUCTION 49

with the permanent dielectric moment.If A and B are non-polar molecules, it is necessary to calculate to thesecond order i.e. to consider how causes the internal state of each molecule tochange.At this approximation order,

where

If A and B are symmetrical, we can rewrite eq. 3.10 as

where

with

and

is the pulsation of the electromagnetic radiation that would cause the transitionfrom ground state to the excited state of the isolated molecule.

is the z co-ordinate of the dipole moment operator with a spherical assumptionThis expression of is the first term of the

infinite series

Concerning colloidal science, it has been shown that the term is theleading term [6]. A description of the unsymmetric case has been given in the samereference.

The next picture gives the main interaction formulas for charged and unchargedparticles.

50 CHAPTER 3. VAN DER WAALS FORCES

3.2. INTERACTION BETWEEN POLAR MOLECULES (SMALL PARTICLES) 51

3.2 Interaction between polar molecules (smal-l particles)

Wo can rewrite the interaction free energy as :

where the coefficients due to the induction the orientation and thedispersion are given by:

and

In most cases, the dispersion forces are dominant except for small and highly polarmolecules like water. These expressions are in good agreement with experimental

data and when the studied interaction arises between dissimilar molecules 1 and 2,is close to the geometric mean of and

The important case of water does not obey this empiric law because it is an highlypolarised molecule.The London theory assumes that molecules have only one single ionization potentialand therefore one absorption frequency and it does not handle the effect of thesolvent to the particle interaction potential.In 1963, McLachlan [7, 8] gave a new expression for the Van der Waals free energyof two small particles (1 and 2) in a medium (3):

where is the polarisability of the molecule j at the imaginary frequencyand is the relative dielectric permitivity of the medium at the same frequencyand

All these quantities can be measured independently.

52 CHAPTER 3. VAN DER WAALS FORCES

If we consider only the zero frequency and a particle j with a permanent dipolemoment is reduced to the Debye-Langevin equation:

In vacuum equation 3.22 becomes:

And we find again the equation 3.18.

Van der Waals forces have peculiar properties. As the polarisability is generallyanisotropic (except for ideal spherical particles), Van der Waals forces are anisotropictoo. However, the orienting effects of the anisotropic dispersion forces are usuallyless important than other forces like dipole-dipole interactions. Another problemin the description of these forces for a system is that Van der Waals forces are notgenerally pairwise additive. This property is very important for large particles andsurfaces in a medium. The last specificity is the retardation effect. Concerning thedispersion energy: as the speed of the interaction between particles is limited bythe speed of light in a vacuum), the states of the fluctuating dipolesare not the same when the fisrt one sends the information as when the second onereceives it. The consequence is that the power law could be closer to than

This latter effect is more important in a medium where the speed of light isslower than in the vacuum and can become very important for macroscopic bodies.When the size of the body becomes greater than a distance where this decay inducesa change in the power law, retardation effect must be taken into account.

For a more detailed description on this topic the interested reader can be referredto refs [9, 10]

3.3 Interaction between surfaces (big particles)

The three most important forces for the long range interaction between macroscopicparticles and a surface are steric-polymer forces, electrostatic interactions and Vander Waals forces. If we assume than the Van der Waals interactions between twoatoms in a vaccuum are non-retarded and additive, we saw in the previous chapterthat the form of the Van der Waals pair potential is: where C is thecoefficient in the atom-atom pair potential and D is the distance between the two

3.3. INTERACTION BETWEEN SURFACES (BIG PARTICLES) 53

atoms. We can then integrate the energy of all the atoms wich form the surface andthe studied atom. In the same way, we can calculate interactions between surfaceswith different geometries making the integration for all the atoms on each surface.The resulting interaction law is given for different geometries in the next figureswhere the Hamaker constant is introduced:

where is the atom density (number of atom per unit volum) of each body.

54 CHAPTER 3. VAN DER WAALS FORCES

3.3. INTERACTION BETWEEN SURFACES (BIG PARTICLES) 55

Bibliography

[1] J.H. de Boer Trans. Faraday Soc. 32 (1936) 10.

[2] H.C. Hamaker Physica 4 (1937) 1058.

[3] E.M. Lifshitz et al. Soviet Physics JETP 2 (1956) 73.; I.E. Dzaloshinskii, E.M.Lifsitz and L.P. Pitaevskii Adv. Phys. 10 (1961) 165.

[4] N.G. van Kampen, D.R.A. Nijboer and K. Schram Phys. Lett. 26A (1968) 307.

[5] V.A. Parsegian and B.W. Ninham Nature (London) (1969) 1197.

[6] J.O. Hirschfelder, C.F. Curtis and R.B. Bird Molecular theory of gases andliquids Wiley N.Y., Chapman and Hall, London, 1954.

[7] A. D. McLachlan Proc. Roy. Soc. Lond. Ser. A202 (1963) 224.

[8] A. D. McLachlan Mol. Phys. 6 (1963) 423.

[9] J. Israelachvili Intermolecular and surface forces, Academic Press, San Diego,1992.

[10] W.B. Russel, D.A. Saville and W.R. Schowalter Collidal dispersions, G.K.Batchelor, Cambridge University Press, Cambridge, 1989.

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