**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 *s <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 sample-sphere system is given by:

$$P = -\frac{A\_c R\_c}{6\left(Z\_b + X\right)} + \frac{1}{2}k\_{l\_s}X^2 + \frac{1}{4}k\_{nl\_s}X^4 \tag{2}$$

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

$$E = \frac{1}{2}\dot{X}^2 + \frac{1}{2}\alpha\_1^2 X^2 + \frac{1}{4}\alpha\_2^2 X^4 - \frac{Do\_1^2}{(Z\_b - X)}\tag{3}$$

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

$$\begin{aligned} X'\_1 &= \frac{\partial E}{\partial \mathbf{X}\_2} \\ X'\_2 &= -\frac{\partial E}{\partial \mathbf{X}\_1} \end{aligned} \tag{4}$$

**Figure 5.** Model of an AFM (Source: Wang, Father and Yau (2009))

The dynamic AFM system in Figure 5 is obtained replacing (2) and (3) into (4):

$$\begin{cases} \dot{X}\_1 = X\_2\\ \dot{X}\_2 = -\alpha\_1^2 X\_1 - \alpha\_2^2 X\_1^3 - \frac{D\alpha\_1^2}{(Z\_b + X\_1)^2} \end{cases} \tag{5}$$

On an Overview of Nonlinear and Chaotic Behavior and

(8)

*c* are the

 and 2 

 *<sup>c</sup>* and <sup>2</sup> *<sup>A</sup>*2 1 22 

Their Controls of an Atomic Force Microscopy (AFM) Vibrating Problem 51

**3.2. AFM Mathematical modeling: intermittent mode and hydrodynamic** 

sample interaction can be modelled as a sphere- flat surface interaction, given by:

where 0 *Uxz* (, ) is the Lennard-Jones potential(LJ), <sup>2</sup> *<sup>A</sup>*1 1 21

**Figure 6.** Microcantilver-tip-sample system

**Figure 7.** Physical model (Source: Zhang et al., 2009)

0 7

(, ) <sup>6</sup> <sup>1260</sup> *<sup>A</sup> R AR Uxz*

Hamaker constant for the attractive and repulsive potential, respectively, with 1

the densities of the interacting components, and 1*c* and 2*c* are constants from interaction. It should be noted that, when the "cantilever" is close to the sample, attractive van der Waals

The microcantilever schematic diagram of the AFM operating in intermittent mode can be seen in Figure 6. The base of the microcantilever is excited by a piezoelectric actuator generating a displacement *f wt* cos( ) . According to (Zhang et al., 2009), considering only the first vibration mode, the (AFM) can be modeled as a spring-mass-damper, as shown in Figure 7. The "tip" is considered as being a of radius R and *Z*0 is the distance from the equilibrium position of the cantilever to the sample. The position of the cantilever is given by *x*, measured from the equilibrium position. According to (Rutzel et al., 2003) the tip-

> 1 2

> >

*z x z x* 

<sup>0</sup> <sup>0</sup>

**damping** 

where *Zb* the distance from the equilibrium position. The molecular diameter is 6 *<sup>D</sup> A RH <sup>k</sup>* , where *Ah* is the Hamaker constant and *R* is the sphere radius. Considering only attractive Van der Waals force, and that the cantilever is being excited by *mf*cos(*wt*), where *w* is the natural frequency, the system equations can be written as:

$$\begin{aligned} \dot{X}\_1 &= X\_2\\ \dot{X}\_2 &= -\alpha\_1^2 X\_1 - \alpha\_2^2 X\_1^3 - \frac{D \alpha\_1^2}{\left(Z\_b - X\_1\right)^2} - f \cos wt - \phi X\_2 \end{aligned} \tag{6}$$

Where 2 4 52 *X a b aX* cos ' is the damping force. Considering the relations: <sup>1</sup> 1 *s X x <sup>Z</sup>* ,

2 2 *s s X x <sup>Z</sup>* , *<sup>b</sup> s Z z <sup>Z</sup>* , 1 3 3 2 2 *Z D <sup>s</sup>* and *wt* , the system (6) may be rewritten in the

following dimensionless form:

$$\begin{aligned} \dot{\mathbf{x}}\_1 &= \mathbf{x}\_2\\ \dot{\mathbf{x}}\_2 &= -a\_1 \mathbf{x}\_1 - a\_2 \mathbf{x}\_1^3 - \frac{b}{\left(\mathbf{z} + \mathbf{x}\_1\right)^2} + c \sin \tau \end{aligned} \tag{7}$$
