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

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

$$\text{LLI}(\mathbf{x}, \mathbf{z}\_0) = \frac{A\_1 \mathbf{R}}{1260 \left(\mathbf{z}\_0 + \mathbf{x}\right)^7} - \frac{A\_2 \mathbf{R}}{6 \left(\mathbf{z}\_0 + \mathbf{x}\right)} \tag{8}$$

where 0 *Uxz* (, ) is the Lennard-Jones potential(LJ), <sup>2</sup> *<sup>A</sup>*1 1 21 *<sup>c</sup>* and <sup>2</sup> *<sup>A</sup>*2 1 22 *c* are the Hamaker constant for the attractive and repulsive potential, respectively, with 1 and 2 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

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

50 Nonlinearity, Bifurcation and Chaos – Theory and Applications

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

natural frequency, the system equations can be written as:

> 1 3

1 2

*x x*

1 2

3 2 2 *Z D <sup>s</sup>* and

*X X*

*<sup>Z</sup>* ,

Where 2 4 52

*s Z z*

following dimensionless form:

2

*<sup>Z</sup>* , *<sup>b</sup>*

*s s X*

2

*x*

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

*X XX*

2 23 1

3 2 11 21 2

*x ax ax c*

2

cos

sin

(7)

(6)

 

*wt* , the system (6) may be rewritten in the

(5)

*<sup>D</sup> A RH <sup>k</sup>* ,

1

*x*

*s X*

*<sup>Z</sup>* ,

<sup>1</sup> ( ) *<sup>b</sup>*

*Z X* 

*D*

2 23 1 2 11 21 2

 

where *Zb* the distance from the equilibrium position. The molecular diameter is 6

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

2 11 21 2 2

*<sup>D</sup> X XX f wt X Z X* 

 

*b*

2

1

*X a b aX* cos ' is the damping force. Considering the relations: <sup>1</sup>

*z x*

*b*

1

1 2

*X X*

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

force must be considered. These forces can be represented as the sum of the attractive and repulsive forces (Rutzel et al., 2003), expressed by:

$$F = -\frac{\partial \mathcal{U}}{\partial (\mathbf{x} + \mathbf{z}\_0)} = \frac{A\_1 R}{180 \left(z\_0 + \mathbf{x}\right)^8} - \frac{A\_2 R}{6 \left(z\_0 + \mathbf{x}\right)^2} \tag{9}$$

On an Overview of Nonlinear and Chaotic Behavior and

Their Controls of an Atomic Force Microscopy (AFM) Vibrating Problem 53

*ax ax a x*

(15)

1 1 1 cos

*ay ay a y*

 (14)

*d e <sup>p</sup> <sup>y</sup> ry by cy <sup>g</sup> <sup>y</sup>*

2 2 11 2 8 3 2 3

Considering the values of parameters: 1; r 0.1 ; b 1 ; c 0.35 ; d 4/27 ; e 0.0001 ; g 0.2 ; p 0.005 e a 1.6 (obtained by Zhang et al. (2009)). The displacement can be seen

According to Ashhab (1999) the chaotic behavior of AFM depends on the damping of the excitation and on the distance between the tip and the sample, suggesting that a feedback control of the states can be used to eliminate the possibility of chaotic behavior. According

> 3 2 11 21 2

numerically simulation results can be seen Figure 9. Additionally, The FFT and the

; 0.17602 *b* , 2.6364

*z x*

*b*

1

0.336 ) are shown in Figure 10.

sin

e 2.5 *z* , 1*a* 1 , 2*a* 14.5 ,

 

(16)

*d e <sup>p</sup> x rx bx cx g x <sup>x</sup>*

2 8 <sup>3</sup> cos

3

3

.

in Figure 8a and the phase portrait can be seen in Figure 8b.

**Figure 8.** (a): Tip displacement (b): Phase Portrait

with the parameters: 0.14668

Lyapunov exponents ( <sup>1</sup>

**4. Chaos in mathematical model with cubic spring** 

to that, considering the system (7) in nondimensional form:

1 2

*x ax ax*

*x x*

0.336 , 2

Writing equation (14) into state space form results:

1 2

*x x*

where: 1 *x y* , 2 *x y* and 3 *x*

In the intermittent mode (TM-AFM) the probe touches the surface of the sample at the point of maximum amplitude of oscillation. During the scanning the microcantilever is driven to oscillate according to the force *f* cos*wt* , resulting that the tip-sample contact generates the force *F* . The contact between the tip and sample is delicate, and this mode of operation is suitable for fragile samples. Then, considering the Lagrangian *LTV* and the Euler-

$$\text{Lagrange equation } \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}\_i} \right) - \frac{\partial L}{\partial q\_i} = Q\_i, \text{ where}$$

$$T = \frac{1}{2}m\dot{\mathbf{x}}^2 \; \; \; V = \frac{1}{2}k\_l \mathbf{x}^2 + \frac{1}{4}k\_{nl} \mathbf{x}^4 \; \text{and} \; \; Q\_i = F + c\dot{\mathbf{x}} + c\_s \dot{\mathbf{x}} + f \cos \mathbf{w}t \tag{10}$$

are the kinetic energy, the gravitational potential energy, and nonconservative forces, respectively, the equation of motion for the microcantilever tip displacement *x* is given by:

$$m\ddot{\mathbf{x}} + c\dot{\mathbf{x}} + k\_l \mathbf{x} + k\_{nl} \mathbf{x}^3 = \frac{A\_1 \mathbf{R}}{180 \left(z\_0 + \mathbf{x}\right)^8} - \frac{A\_2 \mathbf{R}}{6 \left(z\_0 + \mathbf{x}\right)^2} + \frac{\mu\_{\text{eff}} B^3 L}{\left(\mathbf{x} + z\_0\right)^3} \dot{\mathbf{x}} + f \cos \mathbf{w} t \tag{11}$$

with:

$$\mathbf{c}\_s \dot{\mathbf{x}} = \frac{\mu\_{\rm eff} \mathbf{B}^3 \mathbf{L}}{\left(\mathbf{x} + \mathbf{z}\_0\right)^3} \dot{\mathbf{x}} \tag{12}$$

where *eff* is an effective coefficient of viscosity, *B* the width of the cantilever and *L* is the length of the cantilever (Zhang et al. (2009)).

Defining:

$$Z\_s = \left(\frac{2}{3}\right) \left(2D\right)^{\lambda\_3'} \tag{13}$$

where <sup>2</sup> 6 *A R <sup>D</sup> <sup>k</sup>* and considering the following relations in (11)

$$\text{If } \mathbf{r} = w\_1 \mathbf{t}', \quad \mathbf{y} = \frac{\mathbf{x}}{z\_s}, \quad \dot{\mathbf{y}} = \frac{\dot{\mathbf{x}}}{w\_1 z\_s}, \\
\mathbf{z} = \frac{z\_0}{z\_s}, \quad \mathbf{b} = \frac{k\_l}{k\_{at}}, \quad \mathbf{c} = \frac{k\_{nl}}{k\_{at}} z\_s^2, \\
\mathbf{d} = \frac{\mathbf{4}}{27}, \quad \mathbf{e} = \frac{\mathbf{2}}{405} \left(\frac{\mathbf{a}}{z\_s}\right)^6, \quad \mathbf{g} = \frac{f\_0}{k\_{at} z\_s}, \\
\mathbf{x} = \frac{\mathbf{1}}{z\_s}, \quad \mathbf{x} = \frac{\mathbf{1}}{z\_s}, \quad \mathbf{g} = \frac{f\_0}{k\_{at} z\_s}$$

3 3 1 *eff s B L p mw z* , 1 *w w* , <sup>1</sup> *r Q* . Equation (11) can be rewritten in the dimensionless form:

On an Overview of Nonlinear and Chaotic Behavior and Their Controls of an Atomic Force Microscopy (AFM) Vibrating Problem 53

$$\ddot{y} + r\dot{y} + by + cy^3 = -\frac{d}{\left(a+y\right)^2} + \frac{e}{\left(a+y\right)^8} + g\cos\Omega\,\tau - \frac{p}{\left(a+y\right)^3}\dot{y} \tag{14}$$

Writing equation (14) into state space form results:

$$\begin{aligned} \dot{\mathbf{x}}\_1 &= \mathbf{x}\_2\\ \dot{\mathbf{x}}\_2 &= -r\mathbf{x}\_2 - b\mathbf{x}\_1 - c\mathbf{x}\_1^3 - \frac{d}{\left(a + \mathbf{x}\_1\right)^2} + \frac{e}{\left(a + \mathbf{x}\_1\right)^8} + g\cos\Omega\mathbf{x}\_3 - \frac{p}{\left(a + \mathbf{x}\_1\right)^3}\mathbf{x}\_2 \end{aligned} \tag{15}$$

where: 1 *x y* , 2 *x y* and 3 *x* .

52 Nonlinearity, Bifurcation and Chaos – Theory and Applications

repulsive forces (Rutzel et al., 2003), expressed by:

Lagrange equation *<sup>i</sup>*

with:

where *eff* 

Defining:

where <sup>2</sup>

*w t*<sup>1</sup> ,

*p*

*B L*

*mw z* 

,

*s*

6 *A R <sup>D</sup>*

> *s x y <sup>z</sup>* ,

*y*

1 *w w* , <sup>1</sup> *r*

1 <sup>2</sup> 2

*l nl*

length of the cantilever (Zhang et al. (2009)).

<sup>1</sup>

1 *s x*

*w z* , <sup>0</sup>

*i i dL L <sup>Q</sup> dt q q* 

*T mx* , 1 1 2 4

, where

2 4

force must be considered. These forces can be represented as the sum of the attractive and

<sup>0</sup> 0 0 ( ) 180 6 *<sup>U</sup> AR AR <sup>F</sup>*

In the intermittent mode (TM-AFM) the probe touches the surface of the sample at the point of maximum amplitude of oscillation. During the scanning the microcantilever is driven to oscillate according to the force *f* cos*wt* , resulting that the tip-sample contact generates the force *F* . The contact between the tip and sample is delicate, and this mode of operation is suitable for fragile samples. Then, considering the Lagrangian *LTV* and the Euler-

are the kinetic energy, the gravitational potential energy, and nonconservative forces, respectively, the equation of motion for the microcantilever tip displacement *x* is given by:

*mx cx k x k x x f wt*

*eff*

3 3 0

*B L cx x x z* 

<sup>3</sup> <sup>2</sup> <sup>2</sup>

3 *Z D <sup>s</sup>* 

*at <sup>k</sup> <sup>b</sup>*

*<sup>k</sup>* , *nl* <sup>2</sup>

*s at k c z <sup>k</sup>* , <sup>4</sup>

*Q* . Equation (11) can be rewritten in the dimensionless form:

27 *d* ,

3 1 2

180 6

*s*

*<sup>k</sup>* and considering the following relations in (11)

*<sup>z</sup>* , *<sup>l</sup>*

*s z a*

 (11)

*z x z x xz*

is an effective coefficient of viscosity, *B* the width of the cantilever and *L* is the

*AR AR B L*

8 23 00 0

*x z zx zx* 

 1 2

8 2

*V kx k x l nl* and cos *Q F cx c x f wt i s* (10)

3

(12)

(13)

*e*

2 405 *<sup>s</sup> a*

6

, <sup>0</sup>

*at s <sup>f</sup> <sup>g</sup> k z* ,

*z* 

*eff*

cos

(9)

Considering the values of parameters: 1; r 0.1 ; b 1 ; c 0.35 ; d 4/27 ; e 0.0001 ; g 0.2 ; p 0.005 e a 1.6 (obtained by Zhang et al. (2009)). The displacement can be seen in Figure 8a and the phase portrait can be seen in Figure 8b.

**Figure 8.** (a): Tip displacement (b): Phase Portrait
