# When do atomic bonds break?

Fracture is the separation of atoms. We normally do this by applying a force to the body. How big a force should we need?

We want to know how the energy, U, changes with distance, r, between two atoms. A commonly used expression (known as the Lennard-Jones potential is

$$U(r) = 4\varepsilon \left[ {\left( {{{{\beta ^{12}}} \over {{r^{12}}}}} \right) - \left( {{{{\beta ^6}} \over {{r^6}}}} \right)} \right]$$

Other potentials exist, but the argument is essentially the same.

This contains a short range repulsive term, and a long range attractive term. The parameter ε is a measure of the depth of the potential well, and β is the non-infinite distance where the interparticle potential equals zero.

Use the animation to see how the energy changes as the distance between two atoms is varied.

The force between a pair of atoms is calculated by taking the derivative of the energy function, giving:

$$F = {{{\rm{d}}U{\rm{(r)}}} \over {{\rm{d}}r}} = 24\varepsilon \left[ { - 2\left( {{{{\beta ^{12}}} \over {{r^{13}}}}} \right) + \left( {{{{\beta ^6}} \over {{r^7}}}} \right)} \right]$$

Now see how the force between the atoms changes with distance. What is the force needed to separate the atoms in a crystal?

Because we are interested in the material’s properties we normally think of failure *stresses*, so in a unit area of crystal we need to know how many bonds in an area of crystal normal to the applied force are carrying the load, say 6.25 × 10^{−22} m^{2}.

From the animation it is clear that materials should have breaking stresses of the order of 10 GPa, but a piece of glass normally breaks at a stress of 70 MPa.

## Where have we gone wrong?

What have we just estimated? It is the stress required to break the bonds on a given plane simultaneously, what becomes the fracture surface. So do all the bonds on a given plane break at once?

Anybody who has opened a packet of crisps or torn a sheet of paper knows the answer is ‘no’ – the bonds break a few at a time – at the tip of a growing crack.

So the analysis above does not describe what actually happens when a material breaks, which is a relief as the predictions were no good anyway.