# Attraction - Dispersion energies

## Introduction

"The first quantitative theories of molecular forces, however, could be developed only after the structure of atoms and molecules was elucidated. The notion of molecular dipoles gave rise to Debye's theory of orientational forces and Keesom's theory of inductional forces. But the molecular interaction between nondipolar molecules remained unexplained, and especially the interaction between the molecules of noble gases with their spherically symmetrical structure of electron shells. Only be using the newly born quantum mechanics was London able to explain the existence of these forces and to develop, to a first approximation, the general qantitative theory of molecular forces." Derjaguin, Surface Forces, p. 86, 1987.

## Dispersion forces between molecules

Size and shape considerations:

"How molecular size affects the strength of the dispersion forces

The boiling points of the noble gases are

helium -269°C neon -246°C argon -186°C krypton -152°C xenon -108°C radon -62°C

All of these elements have monatomic molecules.

The reason that the boiling points increase as you go down the group is that the number of electrons increases, and so also does the radius of the atom. The more electrons you have, and the more distance over which they can move, the bigger the possible temporary dipoles and therefore the bigger the dispersion forces.

Because of the greater temporary dipoles, xenon molecules are "stickier" than neon molecules. Neon molecules will break away from each other at much lower temperatures than xenon molecules - hence neon has the lower boiling point.

This is the reason that (all other things being equal) bigger molecules have higher boiling points than small ones. Bigger molecules have more electrons and more distance over which temporary dipoles can develop - and so the bigger molecules are "stickier"."

"How molecular shape affects the strength of the dispersion forces

The shapes of the molecules also matter. Long thin molecules can develop bigger temporary dipoles due to electron movement than short fat ones containing the same numbers of electrons.

Long thin molecules can also lie closer together - these attractions are at their most effective if the molecules are really close. "

Hmmm... would elliptical colloids then have stronger Van Der Waals forces?

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From:

F. London, The general theory of molecular forces, Trans. Faraday Soc., 33, 8, 1937. Casimir and Polder, Phys. Rev., 73, 360, 1948.

 Using a perturbation theory to solve the Schroedinger equation for two atoms, London found: \begin{align}  & U\left( r \right)=-\frac{\Lambda _{ab}}{r^{6}}\text{ } \\ & \text{where }\Lambda _{ab}=\left( \frac{h\nu _{a}h\nu _{b}}{h\nu _{a}+h\nu _{b}} \right)\alpha _{a}\alpha _{b} \\ & \nu \text{ is a characteristic frequency} \\ & \text{and }\alpha \text{ is polarizability} \\  \end{align} Casimir and Polder refined the theory to account for the finite speed of light, c. This “retardation” diminishes the attraction. $U\left( r \right)=-\frac{23hc\alpha ^{2}}{4\pi ^{2}r^{7}}$

## Hamaker model for dispersion-force attraction between particles

 The intermolecular attraction is due to London (dispersion) energies: $U_{11}=-\frac{3}{2}\Lambda _{11}r^{-6}$ Summing (algebratically or by integration) over all the interactions between molecules in particle 1 and all the molecules in particle 2. For two spheres (per pair): $\Delta G_{11}=\frac{-A_{11}d}{24H}$ For two flat plates (per unit area): $\Delta G_{11}=\frac{-A_{11}}{12\pi H^{2}}$

## What is Hamaker constant?

The Hamaker constant A can be defined for a van der Waals (vdW) body-body interaction:

$A=\pi^2\times C \times \rho_1 \times \rho_2$

where $\rho_1$ and $\rho_2$ are the number of atoms per unit volume in two interacting bodies and C is the coefficient in the particle-particle pair interaction.

The Hamaker constant provides the means to determine the interaction parameter C from the van der Waals (vdW) pair potential, $w(r)=-C/r^6$.

Hamaker's method and the associated Hamaker constant ignores the influence of an intervening medium between the two particles of interaction. In the 1950s Lifshitz developed a description of the vdW energy but with consideration of the dielectric properties of this intervening medium (often a continuous phase).

The van der Waals forces are effective only up to several hundred angstroms. When the interactions are too far apart the dispersion potential decays faster than $1/r^6$; this is called the retarded regime.

## Hamaker constants

 Typical Hamaker constants

## Effect of intervening liquid

 The effect of an intervening medium calculated by the principle of Archimedean buoyancy: $A_{121}=A_{11}+A_{22}-2A_{12}\,\!$ Introducing the approximation: $A_{12}\simeq \left[ A_{11}A_{22} \right]^{1/2}\,\!$ Which leads to: \begin{align}  & A_{121}=\left( A_{11}^{1/2}-A_{22}^{1/2} \right)^{2} \\ & and \\ & A_{123}=\left( A_{11}^{1/2}-A_{22}^{1/2} \right)\left( A_{33}^{1/2}-A_{22}^{1/2} \right) \\  \end{align}\,\!

## Lifshitz theory

 Limitation of Hamaker theory: all molecules act independently Lifshitz theory: the attractions between particles are a result of the electronic fluctuations in the particle. What describes the electronic fluctuations in the particle? the absorption spectra: uv-vis-ir Result: The Lifshitz constant depends on the absorption spectra of the particles. The absorption spectra is measured. Often a single peak in the UV and an average IR is sufficient. That is two amplitudes and two wavelengths. The dielectric spectrum is calculated from the absorption spectrum. The only additional information needed is the static dielectric constant. The Lifshitz constant is a double summation of products of dielectric functions: $A_{123}=\frac{3kT}{2}\sum\limits_{n=0}^{\infty }{\text{ }\!\!\acute{\ }\!\!\text{ }\sum\limits_{m=1}^{\infty }{\frac{\left( \Delta _{12}\Delta _{32} \right)^{m}}{m^{3}}}}$ The dielectric functions are differences in dielectric constants over a series of frequencies: $\Delta _{12}=\frac{\varepsilon _{1}\left( i\xi _{n} \right)-\varepsilon _{2}\left( i\xi _{n} \right)}{\varepsilon _{1}\left( i\xi _{n} \right)+\varepsilon _{2}\left( i\xi _{n} \right)}\text{ and }\Delta _{32}=\frac{\varepsilon _{3}\left( i\xi _{n} \right)-\varepsilon _{2}\left( i\xi _{n} \right)}{\varepsilon _{3}\left( i\xi _{n} \right)+\varepsilon _{2}\left( i\xi _{n} \right)}$ The frequencies are: $\xi _{n}=n\frac{4\pi ^{2}kT}{h}$ where k is the Boltzmann constant, T is the absolute temperature, h is Planck's constant, and the prime on the summation indicates that the n = 0 term is given half weight. At 21°C, $\xi _{1}$ is 2.4 × 10^14 rad/s, a frequency corresponding to a wavelength of light of about 1.2 µm. As n increases, the value of $\xi$ increases and the corresponding wavelength decreases, hence $\xi$ takes on more values in the ultraviolet than in the infrared or visible. Lifshitz calculation vs measurement. Titania has an angle-dependent refractive index; both values were used in this calculation. Larson, I.; et al JACS, 1993, 115,11885-11890.