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The 'uncertainty principle' of quantitative AFM

Ok, it's not really Heisenberg's uncertainty principle we have to be concerned about (at least not yet), but a similar problem applies to AFM (as to any other measurement):

If we want to measure forces and stiffnesses we have to take the mechanics of the whole system into account, i.e. the sample and the instrument. This seems obvious. However, a quick look through the literature will show that this simple fact is often ignored. For example, we find the following statement in a paper where organic molecules are being studied by 'pulling' :

"In any case, the area under the force vs. distance curve, which is the energy required to break the material, is large."

Sounds reasonable, but when we start to look at the experimental conditions, we find that a 0.06 N/m lever was used. So, what ? Well, let's analyze the situation: Assume that the required energy to break the molecule was 1 eV, and that the stiffness of the molecule at this point was 10 N/m. These are reasonable values (10 N/m = 0.6 eV/Å²) for chemical bonds. We can model the situation as two springs connected in series: the cantilever (k₁, s₁) and the molecule (k₂, s₂):

What we measure is the displacement of the lever, s₁ (indicated by the arrow), which is then converted into a force, and the total displacement between tip and sample, s.

The following equations hold:

(eqn. 1)

The measured energy (the area under the force curve) is then the integral of the force over the total displacement and is thus given by:

(eqn. 2)

The elastic energy actually stored in the molecule, however, is given by:

(eqn. 3)

This means that at a critical energy of 1 eV, the critical extension length of the molecule is:

(eqn. 4)

The corresponding extension of the lever is (at k₁ = 0.06 N/m):

(eqn. 5)

Plugging this into equation 2, we find that the measured energy (area under the force curve) to break the molecule is Em = 167 eV (!!), while the energy stored in the molecule is only 1 eV. Why is this the case: If we have two springs in series, the weaker spring always ends up storing most of the energy, so in this case the cantilever stores lots of energy, while the molecule stores very little.

In general we find:

(eqn. 6a,b)

This means that if the stiffness of the lever is less than the stiffness of the sample (bottom, 6b), we measure absolutely nothing about the stiffness or elastic energy of the sample. Only if the lever is more stiff than the sample (top, 6a) can we say anything quantitative about the sample.

A simple analogy is trying to measure the stiffness of a nail in the wall. Typically, if we want to see how tight a nail is in the wall, we use pliers (high stiffness) and not a rubber band (low stiffness).

The moral of the story is: Adjust the stiffness of your cantilever to the sample you try to measure. Make sure the stiffness of the cantilever is at least as large as the maximum absolute value of the stiffness of the interaction you try to measure.