Van der Waals Force

The van der Waals force represents the electromagnetic interaction of fluctuating dipoles in the atoms of the tip and surface. Basically, fluctuations of the electronic structure in one component induces dipoles in the other, the dipoles in both interact and a force between them is generated. This force is always attractive i.e. tip and surface are attracted to one another, and is present regardless of the tip/surface setup used or the environmental conditions of the experiment (excepting AFM experiments in liquids [64]). This is the same interaction mentioned in section 3.2, but in that case the number of atoms is small enough that the van der Waals force can be calculated explicitly by summing the interaction of all pairs of atoms. The full tip contains billions of atoms and it is impossible to sum all the interactions, therefore an approximation must be made based on the material and structure of the tip.

Assuming that the potential, , between two atoms separated by a distance is known, then the force between them is defined by the gradient of that potential:

$$f(r)=-\nabla V(r)$$ (3.32)

For the van der Waals interaction the potential is of the form:

$$V(r)=-\frac{C_{6}}{r^{6}}$$ (3.33)

where is the interaction constant as defined by London [65] and is specific to the identity of the interacting atoms. Hamaker [66] then performed the integration of the interaction potential to calculate the total interaction between two macroscopic bodies. Hamaker used the following hypotheses in his derivation:

• additivity - the total interaction can be obtained by the pairwise summation of the individual contributions.
• continuous medium - the summation can be replaced by an integration over the volumes of the interacting bodies assuming that each atom occupies a volume with a number density .
• uniform material properties - and are uniform over the volume of the bodies.

This then allows the total force between two arbitrarily shaped bodies to be given by:

$$F_{vdw}=\rho _{1}\rho _{2}\int _{v_{2}}\int _{v_{1}}f(r)dV_{1}dV_{2}$$ (3.34)

where and are the number densities and and are the volumes of bodies and respectively. The Hamaker constant for the interaction is then:

$$H=\pi ^{2}C_{6}\rho _{1}\rho _{2}$$ (3.35)

This study uses the derivation of Argento and French [67] for the total van der Waals force between a conical tip of angle and radius (see fig. 3.1) and a plane. The total force is given by:

$$F_{vdw}=\frac{HR^{2}(1-\sin \gamma )(R\sin \gamma -z_{0}\sin... ... +R\cos (2\gamma )]}{6\cos \gamma (z_{0}+R-R\sin \gamma )^{2}}$$ (3.36)

where is the tip-surface separation. Calculation of the van der Waals contribution to the total tip-surface force requires only knowledge of , and . and depend only on the tip-shape, and this can be estimated from experimental studies of NC-AFM tip properties (see section 4). The Hamaker constant, , depends on the geometry and materials of the tip and surface. However, the NC-AFM system is assumed to be well represented by a conical tip and a planar surface, so effectively depends only on the material of the tip and surface.