Lecture 4
Interactions between neutral
atoms and molecules (dispersion
forces)
What did we cover in the last
lecture?
• Polar interactions and
dipoles
• E field due to a dipole
• Energy of a dipole in
an external field
• Polarisability
In this lecture…
• Dispersion interactions
• Dispersion interaction potential and force
• Range of dispersion interactions
• Repulsive potentials
• The total interaction potential and force
(Lennard-Jones)
• Sketching interaction potentials
Dispersion forces
Dispersion interactions are
always present and act between
all atoms and molecules (in
contrast to other types of
interaction e.g. ionic, covalent etc
which depend upon the type of
molecule)
These interactions arise as the
result of temporary local
fluctuations in charge density
on atoms/molecules which
result in an attractive (usually!)
force between them and their
neighbours
Properties of dispersion forces
• Can be attractive or repulsive (usually former)
• Tend to have a weak orientational effect on neighbouring
atoms/molecules
• Can be relatively long ranged between macroscopic
bodies (0.2nm to >10nm)
• As we will see, they play a role in a many important
phenomena such as adhesion, surface tension, wetting,
flocculation and aggregation of particles and in
determining structures of polymers and proteins.
Physical origin of dispersion forces
Consider two neutral atom or molecules…
Small fluctuations in the
position of the electron clouds
on one atom/molecule result in
the production of an
instantaneous dipole
This dipole induces a
separation of charge
in a neighbouring
atom/molecule
The dipoles on the atoms/molecules interact (quasi-)
electrostatically
Dispersion interaction energy
What is the energy of interaction due to dispersion forces?
Ans: We need to
consider the energy
of a dipole in the
field due to another
dipoel (see OHP)
Recall (from last lecture) that
Field due a dipole at origin
along x axis
qd
1
E
i
3 ~
~
2o x
Potential energy of a dipole
in an external field
U dipole   p . E
~
~
Dispersion interaction potential
The potential energy, U, of an atom/molecule with
polarisability, , at a distance, x, from a dipole (p = qd i) is
given by
 (qd ) 2 1
C
U ( x)   2 2 2 6   6
4   x
x
 (qd ) 2
C 2 2 2
4  
o
o
London derived a similar dependence on x in 1937. He
used quantum mechanical perturbation theory!
 h Bohr 1
U London ( x)  
2
6
x
4o 
2
London equation
See p84-85 Surface
and intermolecular
forces
Dispersion forces
Attractive dispersion force between two atoms/molecules
is given by
dU
3 (qd ) 1
6C
F ( x)  
 2 2 2 7  7
dx
2   o x
x
2
How do we determine the range
of dispersion interactions?
Thermal motion of atoms
and molecules tends to
disrupt the effects of
dispersion interactions
When U becomes comparable to thermal energies dispersion
interactions become less important. We can define the range
of interaction xrange by setting the two energies to be equal
 (qd ) 2 1
U ( x)  2 2 2
 kT
6
4   xrange
o
Estimate of interaction range
So
  (qd ) 2 
xrange   2 2 2 
 4   kT 
o


1
6
Range of dispersion interactions
Inserting typical values of =9 x 10-40 C2m2J-1 and
p=qd=5 x 10-30 Cm for individual atoms at T=300 K gives
xrange ~ 0.3 nm
Compare this with the radius of an atom ~ 0.1 nm
Dispersion interaction between atoms is short range, but
strong enough to hold them in close contact against
thermal agitation! (simple liquids and organic solids)
So why don’t atoms and
molecules collapse into one
another?
The form of the dispersion
potential predicts that
interactions get stronger as
atoms and molecules get
closer together
F ( x)  
So why don’t they continue to attract and collapse into
each other?
Ans: there has to be a repulsive part of the potential
that acts at even shorter ranges
6C
x7
Origin of repulsive potential
When atoms and
molecules are in contact,
their electron clouds
overlap and interact
directly
The atoms/molecules behave like hard spheres at very
short range
Different forms have been used for the dependence of the
hard sphere potential on separation. Many of these are
convenient empirical equations and have no physical basis
(see p111-113, Israelachvili)
The ‘6-12’ potential
A convenient choice for the repulsive potential that works
quite well has a power law dependence of x-12
The total interatomic/molecular potential is given by summing
the attractive and repulsive potentials
The use of this form gives rise to the ‘Lennard-Jones’ or ‘6-12’


12
6

A C
 
  

U ( x)   12  6  4     
 x 
x
x
x

 





repulsive
attractive


Sketching potential vs. distance
   12   6 
U ( x)  4       
 x 
 x  

Total Interaction force
The total interaction force is given by
  12  6 
dU
F ( x)  
 24  2 13  7 
dx
x 
 x
Summary of key results
Dispersion interactions arise due to
instantaneous dipole fluctuations
The potential energy due to attractive
dispersion interactions between atoms
and molecules has the form
C
U ( x)   6
x
Range of dispersion forces is ~ 0.3 nm
The total interaction (dispersion + hard
sphere repulsion) between neutral
atoms and molecules can be described in
terms of a ‘6-12’ potential
   12   6 
U ( x)  4       
 x 
 x  
