The final contribution to the force is the van der Waals interaction. It includes the following contributions: i) between the macroscopic Si tip of conical shape with the sphere of radius R at the end and semi-infinite substrate; ii) the dispersion forces between the atoms in the sample; and iii) the interaction between the macroscopic part of the tip and the sample atoms. The first contribution is calculated analytically as discussed in chapter 3. In fact, the macroscopic contribution to the van der Waals force is the same in each of the three systems described below, as it depends only on the tip-surface separation, macroscopic sphere radius, cone-angle and Hamaker constant of the system. All these quantities are identical in each system we look at, so that the van der Waals force acts as a background attractive force independent of the microscopic properties of the system. The Hamaker constant needed for the calculation of the macroscopic van der Waals force is estimated to be 0.5 eV [130]. The second contribution is calculated directly in the atomistic simulation of the surface.
To estimate the importance of the third contribution, the dispersion interaction
between one atom and a spherical block of atoms has been calculated explicitly.
This interaction converges to a constant value when the radius of the sphere
exceeds about 30 Å. The force exerted on one atom due to this interaction
at characteristic tip-sample distances is several orders of magnitude smaller
than the force between macroscopic tip and substrate. Also the force decays
with distance as r
. This effectively means that only the top layer
of the sample contributes to the van der Waals force, and there are not enough
ions in that layer for it to be significant. Therefore this interaction was
neglected in further calculations. However, for much larger clusters (such as
those discussed in chapter 7) this contribution could be significant.