**2.1. Operation modes in AFM**

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

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 of a pair of particles, and is given by:

$$\mathcal{U}I(r) = 4\varepsilon \left[ \left( \frac{\sigma}{r} \right)^{12} - \left( \frac{\sigma}{r} \right)^{6} \right] \tag{1}$$

On an Overview of Nonlinear and Chaotic Behavior and

Their Controls of an Atomic Force Microscopy (AFM) Vibrating Problem 49

sample-sphere system is given by:

*s*

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

models for intermittent contact mode are discussed.

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

**3.1. Mathematical modeling of AFM: with inclusion of the cubic (spring) term** 

The physical model of the AFM tip-sample interaction can be considered as shown in Figure 5 (Wang, Father and Yau, 2009). The microcantilever-tip-sample system is regarded as a sphere of radius *Rs* and mass *ms* , suspended by a spring of stiffness *s s l nl kk k* , where

*<sup>l</sup> <sup>k</sup>* and *<sup>s</sup> nl <sup>k</sup>* are the linear and nonlinear stiffness. The van der Waals potential for the

6 2 4 *s s*

2 22 24 1 1 2

 

2

1

*X*

22 4 ( ) *<sup>b</sup>*

*A R <sup>P</sup> kX k X*

1 1 2 4

(2)

(4)

*l nl*

2

(3)

*Z X* 

11 1

Replacing *X X* <sup>1</sup> and *X X* <sup>2</sup> , then, from equation (3) results:

The energy of the system scaled by the mass of the cantilever is given by *EXX Z* ( , ', ) :

*<sup>D</sup> EX X X*

 

1

*<sup>E</sup> <sup>X</sup> X*

*<sup>E</sup> <sup>X</sup>*

'

2

'

*Z X* 

*b*

*c c*

where and are constants depending on the sample properties, is approximately 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 force is represented in Figure 4.

$$F(r) = -\frac{\partial \mathcal{U}}{\partial r} = 24\varepsilon \left[ \frac{2\sigma^{12}}{r^{13}} - \frac{\sigma^6}{r^7} \right] \tag{1a}$$

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 tapping-mode technique is applied.
