**3. Mathematical modeling of the Atomic Force Microscope (AFM)**

The mathematical models governing the dynamics of the AFM micro-beams, usually result from the discretization of the classical beam equations, based on its mode of vibration, leading to one or more degrees of freedom. Several models of this kind are described in the literature, e.g., Wang et al. (2009), Garcia and San Paulo, (2000); (Laxminarayana and Jalili,

**Figure 4.** Force (F) versus Distance (*r*), (Source: Cidade et al., 2003).

"soft" (slightly stiff).

where

 and 

**2.1. Operation modes in AFM** 

of a pair of particles, and is given by:

force is represented in Figure 4.

tapping-mode technique is applied.

Typically, probes are made predominantly of silicon nitride (Si3N4); its upper surface is coated with a thin reflective surface, generally gold (Au) or aluminum (Al). The probe is brought to inside and out of contact with the sample surface, by using a piezo-crystal (Hilal and Bowen, 2009). In the illustration of a typical micro-manipulator, shown in Figure 3, it is possible to observe that it consists of a movable stage mounted under an optical microscope. The movement of the base can be controlled by an electronic control console. Large deflections are required to achieve high sensitivity. Therefore, the spring should be very

The probe mounted on the AFM performs the scan on the sample in a raster fashion. The movement of the microcantilever over the sample is carried out by the piezoelectric scanner, which comprises piezoelectric material that expands and contracts according to the applied voltage. There are several modes of operation for scanning and mapping surface. These modes include non-contact, contact and intermittent contact modes. These three modes of

The Lennard-Jones potential describes the relationship of the tip and sample interaction forces as depending on the tip and sample surface distance, considering the potential energy

are constants depending on the sample properties,

equal to the diameter of the particles involved. Deriving potential function (U) in relation to the distance (*r*) gives an expression for the force (F) versus distance (r) (Equation (1a)). This

<sup>2</sup> ( ) 24 *<sup>U</sup> F r*

**3. Mathematical modeling of the Atomic Force Microscope (AFM)** 

12 6

 

12 6 13 7

  (1)

is approximately

(1a)

*r r*

*r r r* 

The region above the r-axis corresponds to the region where the repulsive forces dominate (contact region). The region below the r-axis corresponds to the region where attractive forces dominate (non-contact region). Also in red, it can be seen the distance region that the

The mathematical models governing the dynamics of the AFM micro-beams, usually result from the discretization of the classical beam equations, based on its mode of vibration, leading to one or more degrees of freedom. Several models of this kind are described in the literature, e.g., Wang et al. (2009), Garcia and San Paulo, (2000); (Laxminarayana and Jalili,

 

operation differ from each other, basically, by the tip and sample distance.

*U r*() 4

2004), (Hu and Raman, 2007) (Raman et al. 2008), (Ashab et al. 1999), (Farrokh et al. 2009), (Lozano and Garcia, 2008) among others. Most of the mathematical models are linear massspring-damper systems incorporating nonlinear interaction between the tip and sample (Paulo & Garcia, 2002). Different AFM techniques provide a number of possibilities for topographical images of the samples, generating a wide range of information. In this chapter models for intermittent contact mode are discussed.
