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Van der Waals Forces
vs Casimir forces
Desireé Cotto Figueroa
Department of Physics & Astronomy
Ohio University
ABSTRACT
visible phenomena. He was the first to suggest
This paper is a summary of the basic concepts
how the dispersion forces might arise. The
and some applications of the Van der Waals
Dipole-dipole forces, are also called “Keesom
forces and the Casimir forces
forces”. Willem Hendrik Keesom (1876 – 1956)
was a Dutch physicist who, in 1926, invented a
method to solidify helium. In 1921 he produced
INTRODUCTION
the first mathematical description of the dipole-
The weak attractive forces between atoms or
molecules, Van der Waals forces, were named in
honor of Johannes van der Waals since he was
one of the first to postulate an intermolecular
force. Van der Waals (1837-1923), a Dutch
physicist, received the Nobel prize in physics in
1910 for his research on the gaseous and liquid
states of matter in which he developed a formula
for the continuity of all gases, the van der Waals
equation. There are two types of Van der Waals
forces : Dispersion forces and Dipole-Dipole
forces. The Dispersion forces are also known as
“London
Forces”.
Fritz
Wolfgang
London
(1900-1954), a German-born American physicist,
in the 1930s began to explore the ways in which
quantum principles could be used to explain
dipole interactions. The other main force that we
are interested in is the Casimir force. Have you
asked yourself what would happen if you arrange
two mirrors facing each other in empty space?
Your first thought would probably be "nothing at
all". But it results to be that the simple presence
of the vacuum made the two mirrors mutually
attracted to each other. This phenomenon was
first predicted in 1948 by the Dutch physicists
Hendrik B. G. Casimir (1909-2000) and Dirk
Polder (1919-2001) while participating in a
research
at
Philips
Research
Labs.
The
phenomenon is known as the Casimir effect,
while the force between the mirrors is known as
the Casimir-Polder force.
VAN DER WAALS FORCES
Dipole-Dipole Forces:
Dispersion Forces:
These exist between non-polar molecules or
atoms. They are the weakest of all the
intermolecular forces. 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 δ-, a temporary dipole moment. This
dipole moment will induce a temporary dipole in
Fig.2 The electron-rich region is colored red; the
electron-poor hydrogen atom is shown in blue. The
molecules align when they pass close to each other
because the positive end of one molecule is attracted
to the negative end of the other. The yellow glow
a neighboring atom by attracting or repelling its
indicates the formation of a weak intermolecular
electron charge cloud (Fig. 1). A fraction of a
attraction during a close encounter., the dipole-dipole
second later the electron distribution changes
force.
causing the temporary dipole-dipole attraction to
break.
Dipole Dipole forces occur in polar molecules,
that is, molecules that have an unequal sharing of
electrons. In a system composed of polar
molecules, the partial positive on one molecule
will be attracted to the partial negative on a
second molecule. For example, HCl comprised of
the atom Hydrogen and Chlorine is polar. The
Chlorine atom has an extra electron, which came
Fig.1 The electrons get piled up on one side of the
atom. This creates a temporary polarity in the atom.
A temporary polarity in the neighboring atom is also
then induced when its electrons are repelled by the
first The temporarily induced polarity allows the two
from the hydrogen atom. Because of this, the
chlorine part of the molecule is negatively
charged, and the hydrogen side of the molecule is
positively charged. So in a solution where there
atoms to be attracted to each other very weakly, when
are thousands of these molecules around that are
the partially negative end of the first atom is attracted
slightly charged on each side, the molecules
to the partially positive end of the neighboring atom.
naturally orient themselves. The positive part of
one molecule will move until it is next to the
0
The lowest energy solution of H is just the
negative part of a neighboring molecule. These
product of the ground states of the atoms A, B,
forces between molecules tend to make them
that is,
'stick' together.
0  100 A  100 B .
Van der Waals interaction:
Now for a large R ( R >> ao) we may expand the
We can calculate the long-range interaction, or
denominators in powers of rA / R, rB / R .
Van der Waals force, between two hydrogen
first non-vanishing terms are second order:
atoms in their ground states using pertubation
theory. The energy between the two hydrogen


V  e 2 rA  rB 
atoms is attractive and varies as r-6 for large
 R1  e
2

The

 r  r 3 rA  R rB  R
 A 3B 
R5
 R

At large separation the interaction between the
separation.
atoms is just a dipole-dipole interaction. Taking
now the z-axis in the direction R , this interaction
becomes:
e2
 xA xB  y A yB  2 z A zB 
R3
Now lets assume that the atom A an d the atom B
V
are in the states |nA> and |nB> respectively. Then
Fig 3. Two hydrogen atoms with their protons (+)
the unperturbed energy of the two atoms will be
separated by a fixed distance r and their electrons (-)
EnA +EnB. The first-order correction to the
at displacements ri (i=A,B) from them.
ground state energy of the two-atom system from
Consider the interaction between two hydrogen
the
atoms as in fig.3.
H 1  V and | n 0 > = |nA nB >. To the first order
Then the Hamiltonian is
H  H 0  V , where
H0  
2
2m
interaction
is
En1  n 0 H 1 n 0
the energy will be : E = EnA +EnB. +
2A  2B  
e2 e2

rA rB
where
Now,
if we look at the at the interaction term we have
and the electrostatic interaction between the two
atoms is :
e2
e2
e2
e2
V 


R R  rB  rA
R  rB
R  rA
,
.
 


Every
term in the expression above is zero
because the ground
widely separated hydrogen in their ground states.
states are spherically
This potential is known as a Van der Waals
symmetric. Therefore the first correction to the
interaction. Even though the hydrogen atom has
energy appears in second order. To second order,
no permanent electric dipole moment in its
the energy correction is given by:
ground state, a second atom near to the first one
will induce a dipole moment on the first one.
CASIMIR FORCES
In the classical everyday sense we think of a
where m is not equal to n.
vacuum as what is left after we have removed all
of the stuff, molecules atoms etc. But that still
We can notice from this interaction that if both
leaves photons, so if we remove those as well,
sums are in their ground states then the second
including all the thermal energy then surely we
order correction is negative. We can also notice
should now have an absolute vacuum which
that the interaction energy falls off as the inverse
contains precisely nothing and is featureless.
sixth power of the separations of the atoms. If
However, in quantum mechanics , the vacuum is
we define ζ as :
a much more complex entity. It is far from
featureless and far from empty. Every field have
fluctuations, specially the electromagnetic fields.
This means that
at any moment their values
varies around a constant, mean value. Even a
ζ is a dimensionless constant with a value of 6.5
perfect vacuum at absolute zero has fluctuating
from detailed calculations for the case of two
fields known as "vacuum fluctuations" which
hydrogen atoms in their ground state. Then the
have observable consequences. An example of a
total energy of the two atoms is given by:
consequence of the vacuum fluctuations is an
atom that is not in the ground state and return to
it by emitting a photon. If you try to hold a
.
pencil upright on the end of your finger it will
The second order energy acts as an effective
stay there if your hand is stable and nothing
potential energy for the interaction between two
pertubs the equilibrium.
Once there is a
pertubation the pencil will fall into a more stable
and they also exert pressure on surfaces which
position. Similarly, vacuum fluctuations cause an
increases with the energy of the electromagnetic
excited atom to fall into its ground state and the
field and hence its frequency.
Casimir force is the most famous mechanical
pressure inside the cavity is stronger than outside
effect of this fluctuations
at a cavity-resonance frequency, therefore the
mirrors are pushed apart.
The radiation
In the other hand the
mirrors are drawn towards each other when te
radiation pressure inside the cavity is smaller
than outside, which happens with a frequency out
of resonance. It results to be that the attractive
components, on balance, have a slightly stronger
impact that the repulsive components.
The
Casimir force is attractive for two perfect, plane,
parallel mirrors, therefore the mirrors are pulled
together. The Casimir force can be measured for
Fig.4
mirrors that are within microns of each other, for
Lets consider the gap between two plane mirrors
larger distance it is too small. For two plane
in
All
parallel metallic plates of area of 1cm2 separated
electromagnetic fields have a characteristic
by a distance of 1 µm, the value of the attractive
spectrum. And in a free vacuum there is not a
force is approximately 10-7N. This force is quite
frequency that is more important than the rest.
small but the Casimir force becomes the strongest
But the situation is different inside the cavity,
force between two neutral objects at distances
where the field is reflected back and forth
below a micrometre. The net attractive force per
between the mirrors. The field is amplified if
unit area between two plates is given by:
the
figure
above
as
a
cavity.
integer multiples of half a wavelength (which
corresponds to a "cavity resonance") can fit
exactly inside the cavity ad it is suppressed at
other wavelengths. Another important physical
quantity is the field radiation pressure.
The
vacuum field and every other field carries energy
All electromagnetic fields can propagate in space
where c is the speed of light,
is Planck
constant/2p, and d is the separation between the
plates. When the separation between the surfaces
decreases, the Casimir pressure increases rapidly,
reaching about 1 atmosphere at d~ 10 nm..
Measurements of the Casimir Force
Technology
in
Stockholm,
Sweden,.
He
Although Casimir proposed his effect in 1948 it
measured the force between two gold-coated
was very difficult to measure using the
cylinders and his results agreed to within 1% of
equipment of that time. One of the first
the theory values (see fig. 6).
experiments that showed evidence for the
Casimir force was carried out in 1958 by Marcus
Spaarnay
at
Philips
in
Eindhoven.
He
investigated the Casimir force between two flat,
metallic mirrors made from either aluminium,
chromium or steel, but experimental uncertainties
were too large for a quantitative verification of
the effect.
It was not until 1997 that S. K.
Lamoreaux measured the force using a torsion al
pendulum.
He found that his experimental
measurements agreed with theory to an accuracy
of 5%. The following year the casimir force was
confirmed by U Mohideen and Anushree Roy,
using an atomic force microscope. They were
able to measure the Casimir force to within 1% of
the expected theoretical value (see fig. 5).
In
both cases, one of the surfaces was chosen to be
Fig 5. (http://physicsweb.org/articles/world/15/9/6)
spherical to avoid the problem of keeping two flat
The experiment of U Mohideen and Anushree Roy
surfaces
huge
measures the Casimir force between a metallized plate
The
and a metallized sphere fixed to the tip of the
Casimir force when one plate is spherical is given
cantilever of an atomic force microscope. When the
parallel
which
cause
the
uncertainty in the experiment of 1958.
by :
sphere is brought near to the plate, an attractive
Casimir force causes the cantilever to bend. This
bending is monitored by bouncing a laser off the top
of the cantilever and using photodiodes to record the
where R is the radius of the spherical surface.
reflected light. The electron micrograph shows a
Another experiment using an atomic force
metallized sphere attached to the triangular cantilever
microscope to study the Casimir effect was made
tip of an atomic force microscope.
by Thomas Ederth at the Royal Institute of
Carugno, and Roberto Onofrio at the University
of Padova in Italy. They measured the force
between a rigid chromium-coated plate and the
flat surface of a cantilever made from the same
material.
Their result for the Casimir force
agreed to within 15% of the expected theoretical
value.
Fig.6
(http://physicsweb.org/articles/world/15/9/6)
This experiment of Thomas Ederth measures the
As we can see the percent increases
because of the difficulties involved in the
experiment.
Casimir force between two gold-coated cylinders
positioned at right angles to one another. The upper
cylinder can be lowered using the piezoelectric tube,
VAN DER WAALS FORCES & CASIMIR
which changes shape when a voltage is applied. The
FORCES
lower cylinder is mounted on a piezoelectric
deflection sensor (known as a bimorph spring) that
Initially the Casimir force was thought to be
generates a charge when it is bent. When the two
similar to the van der Waals force. The Van der
cylinders are close together, the Casimir force causes
Waals force results from the fluctuating dipole
the lower cylinder to be attracted to the upper one,
moment of the materials involved.
thereby deflecting the spring in the process. The
related to the Casimir effect for dielectric media.
linearly variable displacement transducer (LVDT)
monitors the nonlinear expansion of the piezotube.
However,
only a
few
number
of
recent
experiments have measured the Casimir force
using
the
original
configuration.
This
configuration is more difficult because the
They are
The first detailed calculations of this were done
in 1955 by E. M. Lifshitz. He generalized the
Van der Waals force between two extended
bodies as the force between fluctuating dipoles
induced by the zero point electromagnetic fields
The Casirmir force :
mirrors have to be kept perfectly parallel during
the experiment.
It is more easier to bring a
,
sphere close up to a mirror because the separation
is obtained by letting the dielectric constant € in
between the mirror and the sphere is just the
the Lifshitz theory approach infinity. It is also
distance
recent
important to notice that the Van der Waals force
experiment that tried to replicate Casimir's
is always attractive, whereas the sign of the
original set-up was carried out by Gianni
Casimir force depends on the geometry.
of
closest
approach.
One
For
example, if a thin spherical conducting shell is

http://antoine.frostburg.edu/chem/senese/
cut in half, the two hemispheres will experience a
101/liquids/faq/h-bonding-vs-london-
mutual repulsive force. The Casimir force and
forces.shtml
the Van der Waals forces are quite different.

http://www.ithacasciencezone.com/chemz
one/lessons/03bonding/mleebonding/van_
der_waals_forces.htm
REFERENCES

Modern Quantum Mechanics, J.J. Sakurai
(323-324).

Lectures on Quantum Mechanics, Baym,
(234-237).

The Casimir effect: a force from nothing.
Available at :
http://physicsweb.org/articles/world/15/9/6

Van der Waals forces between atoms
Available at:
http://galileo.phys.virginia.edu/classes/752.mf
1i.spring03/vanderWaals.htm

H. B. Chan et al. 1941 (2001); 291
Science

U. Mohideen and Anushree Roy; Phys.
Rev. Lett. 81, 4549 - 4552 (1998)

T. H. Boyer; Phys. Rev. 174, 1764
(1968).

G. L. Klimchitskaya , U. Mohideen and
V. M. Mostepanenko;Phys. Rev. A 61,
062107 (2000)

S. K. Lamoreaux; Phys. Rev. Lett. 78, 5 8 (1997)

http://www.wikipedia.org