Electric potential energy between charges
We know from Coulomb's law that two charged objects attract or repel each other with a force proportional to 1/r2, i.e. inversely proportional to the distance squared. So when the charges get closer together, the force of attraction or repulsion gets stronger: a lot stronger. if you cut the distance between the charges in half, the force will be multiplied by four.
We can also describe this in terms of potential energy. Let's say two charges attract. Their electric potential energy falls as $1/r$ as shown at the right. As the charges get closer to each other, their potential energy gets more negative. This makes sense, because as they move closer together, they would accelerate, and therefore gain more kinetic energy, which means they're losing potential energy. It also makes sense because you would have to do work to split them apart. Not only that, but the slope of the potential energy gets steeper as the charges get closer together, indicating that the force is stronger.
Now let's say two charges repel. The functional form is the same — $1/r$ — but the sign of the product of the charge is now positive rather than negative, so the situation is reversed. As the charges are pushed closer together, their potential energy increases. (This time, go through the reasoning on your own in reverse, to convince yourself that this makes sense.) Again, the slope of the potential energy gets steeper as the charges get closer together, since the force is still stronger when the charges are closer, even though this force is now a repulsion rather than an attraction.
As you know, we can set the "zero" point of potential energy anywhere we want; all that really matters is the change in potential energy as something goes from point A to point B. So we'll set the potential energy to be zero when the two charges are "infinitely" far apart. ("Infinite" in physics doesn't have the same meaning as "infinite" in math; it just means "far enough away that they don't noticeably interact".)
Electric potential energy between neutral objects: Van der Waals forces
That explanation works for ions, which are charged, but doesn't explain how neutral atoms attract each other! Atoms have an equal number of protons and electrons, so the net charge is zero. So they don't experience electric forces! OR DO THEY?
Recall how the balloon stuck to the wall, even though the wall was neutral. What was going on there? Let's say the balloon had a net negative charge. Then the negative charges (electrons) in the wall are repelled by the negative charges in the balloon, and they move (slightly) farther away. Now the part of the wall closest to the balloon has a (slightly) positive net charge, and the negative charge in the balloon is attracted to this positive charge. (Try the PhET sim for fun.)
That's basically what happens with neutral atoms. Atoms are neutral overall, but as you know from chemistry, they are made up of a positive nucleus on the inside, and electrons on the outside. Imagine you're an atom, approaching another atom. You might find that, at a given time, the electrons in that other atom are not distributed completely symmetrically about the nucleus. As a result, when you get close enough, the part of the other atom that happens to be closest to you might look positive or negative. Let's say it looks negative (like the balloon) because the negative charges are closer to you. Then your electrons get repelled to the other side, and the side of you closest to the other atom becomes more positive, and you are attracted to the other atom. Or let's say the other atom (on the side closest to you) looks positive. Then your electrons get attracted to that side, and the side of you closest to the other atom becomes more negative, and once again, you are attracted to the other atom. Either way, the result is an attraction! This net attractive force is known as a Van der Waals force (or specifically a London dispersion force) which you may have heard about in your chemistry class. Here's an animation of this phenomenon, along with some questions to consider. (Many thanks to the CLUE (Chemistry, Life, the Universe, and Everything) project for these simulations.)
Like Coulomb's law, we expect the dispersion force to get stronger as the atoms get closer together, and weaker as they get farther apart. Except much more so! This attraction between atoms is only significant if the atoms are really close; otherwise they just look neutral.
How can we model this quantitatively? If we have two bare charges, we know the electric potential goes like $1/r$. If we have one bare charge and a dipole (neutral but with + and - charges not in the same place) the potential falls like $1/r^2$. If we have two dipoles, the potential falls like $1/r^3$. The fact that our dipoles are not fixed but fluctuating, sometimes looking like dipoles, sometimes not, makes the result fall off even faster — like $1/r^6$.
But there must be more to the story. If atoms just attracted each other, and this attraction continued to get stronger as they got closer together, then everything would eventually attract to everything, and all matter would collapse. So there has to be something that prevents atoms from getting too close.
In your chemistry classes, you've probably learned about the Pauli exclusion principle, which states basically that you can't have two electrons (of the same spin orientation) in the same place at the same time. This principle is the basis for all of chemistry, because it makes electrons end up in different orbitals, which gives different elements their chemical properties. The Pauli exclusion principle provides the answer to our puzzle: atoms can't get too close together, or they'll run into the problem of having two electrons in the same state.
This Pauli repulsion is the other half of the atom-atom potential. To model this repulsion quantitatively, let's think about what properties it needs to have: it needs to be even stronger than the attraction at very short distances (preventing all matter from collapsing), but even weaker at larger distances (so that there's still some attraction). But what could possibly be stronger at short distances (and weaker at long distances) than the sixth power? Oh, I don't know, how about THE TWELFTH POWER?
A model for the atom-atom potential was constructed by, John Lennard-Jones (1894-1954). It is therefore known as the Lennard-Jones potential and it is a reasonable approximation for the regions that atoms typically explore in biological systems. (Atoms don't get close enough to each other to see the details of the short ranged repulsion at biological temperatures.)
The Lennard-Jones potential includes two parts: an attraction proportional to 1/r6, and a repulsion proportional to 1/r12. What you get is shown in the figure at the right. See the follow-on for more discussion.
Ben Dreyfus 10/30/2011 and Joe Redish 11/15/11
Last Modified: February 27, 2019