Van der Waals Forces

In the near-field scanning microscopy configuration, several forces exist between the probe tip and the specimen, the most important of which are the van der Waals forces. The potential energy between the tip and the specimen can be expressed as a function of the distance between the lowest atom on the tip and the nearest specimen surface atom. The van der Waals contribution to the total tip-specimen force varies with the electronic properties of the particular atoms involved. This tutorial explores the dependence of these forces on the distance between the NSOM probe tip and the specimen, for a variety of atom pairs.

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The graph appearing in the tutorial window displays the interaction potential between various atom pairs that may be representative of the NSOM probe tip and specimen. The van der Waals interaction consists of two opposing forces: a long-range attraction and a short-range repulsion. The curve representing the van der Waals potential results from the combination of the attractive and repulsive forces. Each force is plotted as a function of the tip-to-specimen (inter-atom) separation. A pull-down menu labeled Atom Pairs may be utilized to select various combinations of atom species, for which new curves are calculated. After selecting an atom pair, the Tip-Specimen Separation slider can be adjusted to change the height of the illustrated NSOM probe tip above the specimen surface. The location of the black dot on the van der Waals curve reflects the interaction potential for the selected tip-to-specimen separation.

On the right-hand side of the tutorial graph (greater tip-to-specimen separation), the atoms are seen to exhibit a very weak attraction to one another. This attractive force increases slightly as the separation distance is reduced until the atoms are so close together that the electron clouds begin to repel each other electrostatically (note the rise in the blue curve). This repulsion opposes the attractive force as the interatomic separation decreases, and the combined force (van der Waals curve) goes to zero when the interatomic distance reaches approximately 0.2 nanometer, which is about the length of a typical chemical bond. At closer distances, the combined forces are repulsive in nature.

The attractive force component of the van der Waals interaction is proportional to C(n)/r(n), and the repulsive force is proportional to C(m)/r(m), where m and n are integers, and C(n) and C(m) are constants that depend on the equilibrium separation between the tip and specimen and the depth of the energy well between them. For the case where the end of the tip and the specimen are considered as two instantaneous dipoles, the Lennard-Jones parameters (n=12, m=6) can be used to model the van der Waals interaction, combining the two force components.

The equation utilized in the tutorial is given as follows:

V(r) = 4εts[(σ/r)12 - (σ/r)6]

where ε(ts) is the pairwise potential energy or well depth, and σ is related to the equilibrium separation, r(eqm), by the following equation:

σ = reqm/(21/6)

A range of values of equilibrium separation and well depth are represented by the different possible atom-pair combinations available in the Atom Pairs pull-down menu.

In addition to the van der Waals forces, two additional forces are exerted on the probe tip during ambient NSOM operation. The first is a capillary force produced by the thin layer of water that is usually present on a specimen surface under ambient conditions, depending upon the humidity level of the air. This hydro-attractive force arises as the water layer is drawn up and around the tip, producing a strong attractive force (approximately 10 piconewtons) that effectively binds the tip to the water layer. The magnitude of this force varies with the tip-to-specimen separation. An additional force is exerted by the feedback mechanism of the NSOM probe, which behaves similarly to a compressed spring, in accordance with Hooke's Law:

F = kΔs

where k is the spring constant of the cantilever and Δs is the cantilever displacement.

Contributing Authors

Jeremy R. Cummings, Matthew Parry-Hill, Thomas J. Fellers, and Michael W. Davidson - National High Magnetic Field Laboratory, 1800 East Paul Dirac Dr., The Florida State University, Tallahassee, Florida, 32310.