# 2.6.2 Cantilever deformations under the influence of lateral forces

To investigate friction, the Lateral Force Microscopy (LFM) is used. It is based on the probe lateral deflection recording during scanning. In the LFM, the cantilevers in the form of rectangular beam having length , width , and thickness ending with a tip having length (Fig. 1), are used. Fig. 1.  Rectangular cantilever with tip.

When the cantilever moves in the direction perpendicular to its axis, the tip twists due to the friction. As shown in chapter 2.1.4 the twisting angle is connected with horizontal lateral force as (1)

where – inverse stiffness coefficient which is defined in (12) chapter 2.1.2. The optical system registers this twist as a signal .

Another scan mode is available. Scanning in the direction produces frictional force directed longitudinally that results in the vertical beam deformation (see chapter 2.1.3). In this case the angle of bend is connected with longitudial force as (2)

Emphasize that the bend is due to not only lateral force but also vertical force component . Therefore, in general case, it is necessary to extend the expression (2) using formulas from chapter 2.1.3: (3)

In spite of the fact that first component of expression (3) includes a small factor , both term can be the same order, because the forces ratio is sufficiently great and ordinary varies from 1 to 100. The bend with angle results in a signal .

Thus using optical detection system it is impossible to isolate the value of lateral component . For its determination one should, for instance, find the force by others methods, and then subtract corresponding component from expression (3). Apparently, that this method is more difficult than first, in which the scanning is carried out in a transverse direction and component can be received immediately.

## Summary.

• Forces of friction can be measured both at transverse scanning – , and at longitudinal – .
• The transverse scan direction is more preferred, because the cantilever torsion registers as a separate signal .
• At longitudinal scan direction it will be difficult to extract from a signal the value of friction force because essentially of large strains due to normal load influence.
• The influencing of a surface topography on a signal is considered in chapter 2.6.4

## References.

1. Handbook of Micro/Nanotribology / Ed. by Bhushan Bharat. - 2d ed. - Boca Raton etc.: CRC press, 1999.