Dipole-dipole forces: explanation and derivation

Van der Waals forces are usually taken to include all forces between molecules in molecular solids, liquids and gases that do not amount to actual chemical bonds. For monatomic substances such as noble gases they constitute all forces between atoms. However there is a slight distinction to be made: the force law is different, depending on whether both the molecules in question have a permanent electric dipole or not. vdW forces include both situations.

If both have permanent dipoles, they will tend to align directions so as to produce a relatively strong attraction: thus the main contribution to the force between the two will not depend on momentary induced fluctuations. This is called a permanent dipole-permanent dipole force; it is an electrostatic interaction and decreases with the 4th power of distance

Fpermanent dipole = - constant × R-4

where the constant of proportionality just depends on the identity of the molecules involved.

If one or both molecules has no permanent dipole moment the forces involve induced dipoles. This means that the presence of one dipole (whether permanent or due to random fluctuation of charge density) exerts a force on the electrons of the other molecule, giving it in turn a temporary dipole moment. These forces (which also in fact occur between permanent dipoles) are weaker than permanent dipole forces and decrease with the 7th power of distance

Finduced dipole = - constant × R-7.

The (different) constant of proportionality again depends on the molecules involved. These are the so-called London forces or dispersion forces.

What follows is a non-rigorous derivation of these force laws based on some simplifying (NB pipelink) assumptions. The derivation is a relatively short and easy one and ends up neatly with a physically reasonable result, despite the multiple things that are swept under the carpet.

I will model a molecular dipole of moment p = q × d by two opposite charges q and -q separated by a distance d. This doesn't sound like a good picture of what is actually a fuzzy blob of charge density, but turns out to be a good approximation at sufficiently large distances from the molecule.

Here are the two dipoles:

-q1        +q1                 -q2          +q2
<-- d1 -->                     <--- d2 --->
<------------- R -------------->
Now we need to compute the total force on the charges labelled 2, which are supposed to model the 2nd molecule, due to the charges labelled 1, using Coulomb's law. The force on the positive charge q2 is

q1q2 [ (R +d1/2 +d2/2)-2 - (R -d1/2 +d2/2)-2 ] /ε

which can be expanded using the binomial theorem or a Taylor series in the limit d1<<R to obtain

2 p1q2 (R +d2/2)-3 /ε.

(Please don't tell me I've left out some powers of 4π, I'm using a set of units where the Coulomb law is just q1q2R^2.) Similarly the force on the negative charge -q2 is

-2 p1q2 (R -d2/2)-3 /ε.

Then we add and expand in the limit d2<<R to get

-6 p1p2R4,

the inverse fourth power law for permanent dipoles.

Now how do we get from here to the inverse seventh power for dipole-induced dipole forces? In this case, we have to realise that a net force exists because the second dipole p2 arises in response to the first. The second molecule is polarized by the forces acting on its charge distribution due to the electric field of the first dipole. But this means that the farther the second molecule is away, the less polarization there will be and the smaller p2 will be, for a given value of p1. We need to work out the dependence of p2 on p1 and R.

To do this we need a model of the second molecule telling us how its constituents move around. This is where the simplification comes in. Imagine that the molecule consists of the two opposite charges attached by a spring with spring constant ω, such that when there are no external forces acting the charges sit on top of each other. Then the extension of the spring, which gives us the distance d2, is just

d2 = (force acting on q2 - force acting on -q2) /ω = 4 p1q2 /ω ε R3     ⇒
p2 = 4 p1q22 /ω ε R3,

where we are approximating the forces on ±q2 by ±2 p1q2R3. Of course the actual value of p2 is subject to fluctuations, just as p1 is, but this reasoning gives us a reasonably good average value.

To make this step a bit more concrete you could look up the numbers for the polarizability of various molecules, which can be related to the size of the putative spring constant and point charges. The polarizability k is defined to be the size of the induced dipole p2 divided by the electric field at distance R from molecule 1, which is 2 p1R3 (remember that the electric field is just the force on a point charge divided by the charge). So

k = 2 q22 /ω.

To return to the derivation, we simply substitute p2 = k × 2 p1R3 into the previous equation to get the force between a dipole and an induced dipole to be

F = -12 k p12 / ε2 R7.

In fact we made yet another assumption, that the charges of molecule 2 will react instantaneously to fluctuations in molecule 1; and we don't know the size of p1 in the first place. But, based on these simple models, we have got the correct dependence on the distance R. This can be compared with the Lennard-Jones potential which has the form

V = (repulsive term) - constant /R6.

(Again I am neglecting all the details apart from the dependence on the distance R. The constant of proportionality should be one-sixth of the constant I introduced into the equation for Finduced dipole above.) The attractive force is just minus the gradient of the potential V, which gives us the inverse seventh power again.

Closing thoughts

A real derivation would have to be based on consistent physical principles, i.e. quantum mechanics (since that is the only coherent theory which allows us to calculate what happens to the charge density of atoms and molecules). Curiously, the inverse seventh power van der Waals forces turn out to be very closely related to the Casimir effect, which tells us that there is a force between two conducting bodies in a vacuum, due to quantum fluctuations in the electromagnetic field.

In both cases, what one is calculating is the same: the change in energy due to the interaction between the zero point fluctuations of the electromagnetic field and the electrons making up the bodies. The difference, if there is one, comes in the way the electrons are bound: usually one considers conductors in the Casimir effect, simply because the effect is much weaker for insulators in the experimental setup that is used to measure it.