CHEE2940: Particle Processing

Lecture 16: van der Waals forces This Lecture Covers History of van der Waals forces Origin of van der Waals forces van der Waals interactions between particles and

surfaces Chee 2940: van der Waals forces

14.1 INTRODUCTION HISTORY OF VAN DER WAALS FORCES

• State equation for ideal gases: Relates the pressure, P, molar volume, V, of gases

PV RT=

T … absolute temperature R … universal gas constant (8.314 J mol-1 K-1) Ideal gases have no interaction between molecules.

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• State equation for real gases

( )( )2/P a V V b RT+ − = Famous van der Waals equation (1873)

- Parameter, a, depends on the attractive forces

between molecules.

- Parameter, b, depends on repulsive forces.

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Johannes van der Waals (1837-1923)

Obtained and published his famous equation in his doctorate in 1873 (at age 36).

- van der Waals forces between molecules - The specific volume of gas molecules.

Awarded Nobel Prize for Physics in 1910. Chee 3920: van der Waals forces 3

14.2 ORIGIN OF VAN DER WAALS FORCES WHAT ARE THE VAN DER WAALS FORCES?

They are the physical forces of attraction and repulsion existing between molecules & atoms.

•

•

They are not the chemical (covalent) forces that make up the molecule (chemical bonds).

For example: A water molecule is made up of hydrogen and oxygen, which are bonded together by the sharing of electrons. These

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electrostatic forces that keep a molecule intact are existent in covalent and ionic bonding but they are NOT the van der Waals forces. They come from the polarisation of molecules into dipoles and occurs due to the interactions between:

•

- Two freely rotating dipoles (Keesom), or - A freely rotating dipole and an induced dipole (Debye), or - Two induced dipoles (London).

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(Israelachvili, 1997)

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WHAT IS A (ELECTRIC) DIPOLE? Is a pair of electric charges (poles) of equal magnitude but opposite polarity, separated by a small distance.

Dipoles are characterised by their dipole moment, a vector quantity with a magnitude equal to the product of the charge and the distance from the positive to negative pole.

A half of electric field around a dipole

u q l= ⋅

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POLARISATION OF ATOMS AND MOLECULES: TEMPORARY FLUCTUATING DIPOLES AND INTERMOLECULAR ATTRACTION

On average, the negative charge of the electrons in an atom or molecule is spread evenly. For brief periods of time, the electrons are concentrated on one side of the atom or molecule more than the other. This gives the atom or molecule a temporary partial negative charge - a temporary dipole moment. Chee 3920: van der Waals forces 8

The dipole moment will induce a temporary dipole in a neighbouring atom/molecule by attracting its electron charge cloud (by the van der Waals attraction).

A fraction of a second later the electron distribution changes causing and the temporary dipole-dipole attraction (van der Waals attraction) to break. Chee 3920: van der Waals forces 9

VAN DER WAALS FORCE BETWEEN MOLECULES It’s due to the interaction between dipole moments of three (Keeson, Debye and London) polarisation effects.

It acts only over relatively short distances.

Its energy is proportional to the inverse of the sixth power of the intermolecular distances, r

( ) 6 6orient ind dispvdW

vdW

C C CCw rr r

+ += − = −

Chee 3920: van der Waals forces 10C … energy coefficients

Note the following alternative names: - The Keeson interaction = the orientation

interaction - The Debye interaction = the induction interaction - The London interaction = the dispersion

interaction

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(Israelachvili, 1997)

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In general, the dispersion interaction is the most significant component of the van der Waals interaction (compare the energy coefficients)

The van der Waals interaction is sometime called the dispersion interaction

The Lennard-Jones potential (referred to as the L-J potential or 6-12 potential)

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Unbounded atoms and molecules are subject to two distinct forces in the limit of large distance, and short distance: an attractive van der Waals force at long ranges and a repulsion force due to overlapping electron orbitals at short ranges.

( )12 6

4V rr rσ σε

= −

where ε is the well depth and σ is the hard sphere diameter. These parameters can be fitted to reproduce experimental data or deduced from results of accurate quantum chemistry calculations. Chee 3920: van der Waals forces 14

Interaction energy of argon dimer. The long-range part is due to London dispersion forces.

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14.3 VAN DER WAALS INTERACTIONS BETWEEN PARTICLES AND SURFACES

VAN DER WAALS ENERGY AND FORCE BETWEEN

PARTICLES AND SURFACES CAN BE OBTAINED BY

SUMMING (INTEGRATING) THE VAN DER WAALS

ENERGY BETWEEN ATOMS/MOLECULES COMPRISING

THE MACROSCOPIC BODIES.

Results are shown in the following table. Chee 3920: van der Waals forces 16

1

2 3

van der Waals energy between atoms, molecules, particles and surfaces

Chee 3920: van der Waals forces 17(Israelachvili, 1997)

Hamaker constant

van der Waals energy between two particles

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1. Molecule-planar surface interaction - We sum the interaction between the molecule and the molecules contained in a circular ring with the cross-sectional area dxdz a d radius x

( ) ( ) 2z x

z

w D w r xdxdzπρ∞ =∞

=

= ⋅∫

m Chee 3920: van der Wa

n

=

∫

0D x=

# of molecules in the ring

olecule-molecule interaction

s forces 19

Density of solid molecules

al

Interaction energy between molecules: ( ) 6Cw rr

= −

Pythagoras’ theorem: z2 2 2r x= +

( )( )32 2

0

2z x

z D x

xdxdzw D Cx z

πρ=∞ =∞

= =

= −+

∫ ∫

( ) 36Cw DD

πρ= −

van der Waals energy between a molecule and a

planar surface Chee 3920: van der Waals forces 20

2. Interaction between planar surfaces - We sum the interaction, w(D), between a molecule and a p anar surface, giving

( ) 1236

z

z D

CW D dz

ρ zρ=∞

=

= − ⋅∫

msu e

Energy per

area

Chee 3920: van der Wa

l

π

ule-planar interaction

# of molecules pethe object 2 wit

surfac

r unit area in h the planar e

21

olecrfac

unit

als forces

( ) 1 2212CW D

Dπρ ρ

= −

Hamaker constant: 2

1 2A Cπ ρ ρ=

( ) 212AW DDπ

= −

van der Waals energy (per unit area) between

two planar parallel surfaces separated by a distance D

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3. Interaction between a sphere & a pla - We sum the interaction, w(D), between a molecule and a planar surface, giving

( )( )

221

230 6

z R

z

CW D x dzD zπρ ρ π

=

=

= − ⋅+∫

# of molecules i

molecule-planar surface interaction

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Molecule density in the sphere

nar surfaces

n the sphere

23

z

xR

R Pythagoras’ theorem: ( )22 2R x R z= + −

2 22x Rz z= −

( )( )

2 22

1 2 30

26

z R

z

Rz zC dD z

π ρ ρW D z=

=

−= −

+∫

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( ) 21 2

1 2ln3 2 6R D R D RW D CD D R D

π ρ ρ + + = − − +

In terms of the Hamaker constant

( ) 1 2ln3 2 6R D R D RW D AD D R D

+ + = − − +

For short separation distances: 0D→

( )6ARW DD

= −

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SUMMARY OF VAN DER WAALS FORCES AND

ENERGIES BETWEEN PARTICLES AND SURFACES Energy

Flat-flat surfaces ( ) 212AW DDπ

= − per unit area

Sphere-flat surface ( )6ARW DD

= −

Sphere-sphere ( ) 1 2

1 26A R RW DD R R

= −+

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( )EnergyForce

ddD

= −

Force

Flat-flat surfaces ( ) 36AF DDπ

= − per unit area

Sphere-flat surface ( ) 26ARF DD

= −

Sphere-sphere ( ) 1 22

1 26A R RF DD R R

= −+

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THE HAMAKER CONSTANT - Depends on the properties of the interacting

surfaces and the medium. - Weakly depends on the system geometry and

separation distance: The retardation effect • Can be examined using the Lifshitz quantum approach.

• Is beyond the scope of the Hamaker theory (and this course).

• Can be neglected at short distances. Chee 3920: van der Waals forces 28

- Is positive for (attractive) interactions between solid particles, air bubbles, and oil drops.

- Is negative for (repulsive) interaction between

an air bubble and a solid particle (in flotation).

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Typical Hamaker constants

Material 1 Medium Material 2 A (10-20J) Examples Alumina Alumina 5.0 Silica Silica 0.7Zirconia Zirconia 8.0 Titania

Water

Titania 5.5

Mineral suspensions

Metal Water Metal 30 to 40 Air Water Air 3.7 FoamsAlumina Alumina 15.0 Silica Silica 6.5Zirconia Zirconia 20.0 Titania

Air

Titania 15.0

Dry mineral powders

Water Octane Water 0.4 Water in oil emulsions Water Octane Air 0.5 Silica Water Air -0.9 Mineral flotation Octane Water Air -0.2 Oil flotation

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