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DYNAMICAL STUDY OF A PIECEWISE-SMOOTH MODEL IN TAPPING MODE ATOMIC FORCE MICROSCOPE

Rodrigues, K. S.1 , Balthazar, J. M.

2 , Tusset, A. M.

3, Pontes Jr, B. R.4

1

DYNAMICAL STUDY OF A PIECEWISE-SMOOTH MODEL IN TAPPING MODE ATOMICFORCE MICROSCOPE

Kleber dos Santos Rodrigues1

, José Manoel Balthazar 2 , Angelo Marcelo Tusset

3 , Bento Rodrigues de Pontes Jr

4

1 Unesp - Universidade Estadual Paulista, Bauru, Brasil, [email protected] Unesp - Universidade Estadual Paulista, Rio Claro, Brasil, [email protected]

3 UTFPR, Ponta Grossa, Brasil, [email protected] Unesp – Universidade Estadual Paulista, Bauru, [email protected]

Abstract: This paper deals with the dynamics of apiecewise-smooth model used to represents the TappingMode Atomic Force Microscope behavior. Created byBinning et al (1986), Atomic Force Microscopy became apowerful tool in scanning probe process. A micro-cantilevervibrates, excited by a piezo electric transducer and itsvibration is detected by a photo detector. Working close

from its resonance, the micro-cantilever has a nonlinearbehavior, due to this, chaotic behavior may occur.Represented by a piecewise SDOF system, the dynamics of the model are investigate via numerical simulations and thestudy of figures like, phase portrait, FFT, time history andLyapunov exponent show evidences of chaotic oscillationsof micro-cantilever tips in dynamic atomic force microscopy(AFM). This irregular behavior appears in a defined intervalof time.

Palavras-Chave: Atomic force microscopy, chaos.

1. INTRODUCTION

In last decades, the Atomic Force Microscope (AFM)

became a powerful tool in scanning probe microscopy

(SPM). Atomic Force Microscope (AFM) is a powerful tool,

its application includes manipulation of carbon nanotubes,

DNA, imaging and actuation in nano-electronics, etc.

(Rützel et al., 2003). In AFM, a micro-cantilever with a tip

at its free end vibrates and sends a signal to a photo detector,

the acquired images come from this movement and the

scanning process starts.

Invented by Binning et al (1986) [3] the AFM has three

modes of operation, non contact, tapping mode andintermittent mode. the non contact mode is characterized bythe presence of the attractive forces, tapping mode andintermittent mode are characterized by the presence of attractive and repulsive forces. The most used operationmode is the tapping mode and in this case, strongnonlinearities may occurs [1].

Figure 1 – Tip sample interaction scheme [2].

Deterministic chaos underpins the dynamics of manynonlinear systems that display a very high degree of sensitivity to initial conditions, in AFM, chaotic behavior isvery common and when this type of behavior occurs, theimages are affected, what is not a desirable outcome.

Mathematical models are commonly used to understand

the behavior of an AFM micro-cantilever. Numericalsimulation and analysis of the obtained images are essentialto understand the behavior of the system and to determine if there is or not chaos in the system. In other works, like [7],[6], [10], and [11], the study of the system was essential tofind chaotic behavior and to project control methods tostabilize the system.

In AFM, chaos is characterized by an undesirable motionof the micro cantilever that vibrates close from itsresonance. By [5], the presence of chaos can lead to errors,while the imaging of samples are obtained, therebyintroducing an element of deterministic uncertainty innanometrology.

In this paper, a model proposed by [2] that uses a one-degree-of-freedom model with linear coefficients to describeattractive and repulsive forces is used. Numericalsimulations using software Matlab® obtained phaseportraits, time history and Lyapunov exponents.

2. MATHEMATICAL MODEL

The tip-sample interaction is characterized by a short-range of the repulsive forces and by a long-range of theattractive forces.

In a piecewise-smooth model of an AFM micro -cantilever studied by Sebastian et al., the attractive forcesare represented by a linear spring with negative stiffness, therepulsive forces are represented by a linear spring withpositive stiffness, and linear dampers are introduced to



ANALYSIS AND CONTROL OF A PIECEWISE-SMOOTH MODEL IN TAPPING MODE ATOMIC FORCE MICROSCOPE


2 , Tusset, A. M.

3

2

account for the energy dissipation during the tip-sampleinteraction.

The dynamics of the micro-cantilever are represented bya piecewise-smooth dynamical system.

Figure 2 – Model of the tapping mode atomic microscopy [2].

The equation of motion can be used to describe the tipmovement represented in fig 2:

2

0 02 cos( ) ( , ) x x x t h x x

(1)

Where ( , )h x x is the piecewise-linear function that

describes the tip-sample interactions:

0

( )

( ) ( )

a a

b b a a

k c x s d x

m m

k c k c x s x x s d x

m m m m

0

0

x s d

x s d

x s

Here, x measures the movement of the micro-cantilever

tip,0

k

m

is the resonance frequency,

02

c

m

is the

coefficient of intrinsic and ambient damping, and m and represents the excitation amplitudes and frequency

due to the oscillation of the dither piezo, respectively.

Moreover, s characterizes the equilibrium separationbetween the micro-cantilever tip and the onset of therepulsive tip-sample interactions. Finally, d represents thedistance from the onset of the attractive forces to the onsetof the repulsive forces and reflects the material properties of the sample.

The dimensionless system is:

cos * ( , )u Bu u t H u u (2)

0

0 * *

( , ) ( * *) 0 * *

( *) ( * *) * 0

a

b a

u s d

H u u A u s d B u u s d

A u s B u A u s d B u u s

With ( , ) ( , *) x t u t

0*t t ,

0

xu

u , 2

0

k

m , 2 b

b

k

m , 2 a

a

k

m ,

2

0

,

0

c B

m ,

0

a

a

c B

m ,

0

b

b

c B

m ,

2

2

0

a A

,

2

0 2

0

b A

,

0

*s

su

,

0

*d

d u

3. NUMERICAL SIMULATIONS

By using software Matlab®, ode 45, step length 310

, and

parameters,0

1 A ; 0a

B ; 0.0000001b

B ; * 0.01s ;

* 0.01d ; 0.000001 A ; 0.01 , the following figures

are obtained:

-0.02 -0.01 0 0.01 0.02 0.03 0.04-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

x

d x / d t

Figure 3

– Phase portrait with nonlinear behavior.



DYNAMICAL STUDY OF A PIECEWISE-SMOOTH MODEL IN TAPPING MODE ATOMIC FORCE MICROSCOPE


2 , Tusset, A. M.

3, Pontes Jr, B. R.4

3

0 50 100 150 200 250 300-0.02

-0.01

0

0.01

0.02

0.03

0.04

time

d i s p l a c e m e n t

Figure 4a – Time history with t = 300.

0 10 20 30 40 50 60 70 80 90 100

-0.02

-0.01

0

0.01

0.02

0.03

0.04

time

d i s p l a c e m e n t

Figure 4b – Time history with t=100.

0 50 100 150 200 2500

0.02

0.04

0.06

0.08

0.1

0.12

0.14

frequência (Hz)

Figure 5 – FFT with irregular spectrum and resonance peaks.

-0.02 -0.01 0 0.01 0.02 0.03 0.04-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

x

d x / d t

Figure 6 – Poincare maps with irregular behavior.

0 1 2 3 4 5 6 7 8 9 10

x 104

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

t

L y a p u n o v

E x p o n e n t s

Figure 7a – Lyapunov exponents with positive .

2000 4000 6000 8000 10000 12000 14000

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

Figure 7b – Lyapunov Exponents with zoom.

In figure 6a and 6b, with 410t , Lyapunov exponents

obtained the follow results:

1 2 30.0099, 0.0099, 0 .



ANALYSIS AND CONTROL OF A PIECEWISE-SMOOTH MODEL IN TAPPING MODE ATOMIC FORCE MICROSCOPE


2 , Tusset, A. M.

3

4

4. CONCLUSIONS AND DISCUSSIONS

In atomic force microscopy, the micro cantilever chaoticbehavior can results in an unsatisfactory obtaining of images. In the model presented in this paper, chaoticbehavior is obtained after a study of the motion of thesystem via numerical simulations.

In figure 3, is possible to observe the presence of strangeattractors, in figures 4a and 4b there is no presence of anonly period. In figure 5, the Poincare map shows astochastic motion [12], that commonly appears in undampedor lightly damped systems. The FFT shows picks of resonance and irregular spectrum, and finally, in figures 7aand 7b, the presence of a positive Lyapunov exponent implyin a chaotic behavior [12]. By this fact, is safe to affirm thatchaotic motion is detected in the system. The chaotic motionoccurs in a system without nonlinear term, by this, thesystem can be characterized as a strong nonlinear system

[13].In future work, the objective is to study the bifurcation

map and basins of attraction to better understand the system,and to find a periodic orbit to develop a control system tostabilize it.

5. AKNOWLEDGEMENTS

Authors thank CNPq and Fapesp for financial support. K.S.Rodrigues thanks CAPES for fellowship support.

6. REFERENCES

[1] Rutzel, S., Lee, S.I. and Raman, A., 2003, “Nonlinear dynamics of atomic-force-microscope probes driven inLennard-Jones”, Fractals Vol 33, pp. 711 – 715,

[2] Sebastian, A., Salapaka, M. V., Chen, D. J., andCleveland J. P., 2001, “Harmonic and power balance toolsfor tapping mode atomic force microscope”, Journal of Applied Physics Vol 89, pp. 6473 – 6480

[3] G. Binnig, C. Gerber, C. Quate, 1986, Atomic ForceMicroscope, Phys. Rev. Lett. 56, 930-933.

[4] Dankowitz H. and Zhao, X., 2006, “Characterization of intermittent contact in tapping mode force microscopy”,J.Comput. Nonlinear Dynamics – Vol 1, pp 109.

[5] Hu, S., Haman A., (2006), “Chaos in atomic forcemicroscopy”, Birckand NCN Publications. Paper 39.http://docs.lib.Purdue. edu/nanopub/39.

[6] Aime´, J. P., Boisgard, R., Nony, L. and Couturier, G.,1999, “Nonlinear dynamic behavior of an oscillating tip– micro lever system and contrast at the atomic scale”, Phys. Rev. Lett. Vol 82, pp. 3388 – 3391

[7] Nozaki, R., Pontes Jr, B.R, Tusset, A. M. and Balthazar,J.M., 2010, “Optimal linear control to an atomic forcemicroscope: (AFM) problem with chaotic

behavior”,Dincon’10- 9th Brazilian conference on dynamics,

control and their Applications (http://www.sbmac.org.br/dincon/ )

[8] Pyragas, K., 1992, “Continuous control of chaos by self -controlling feedback”, Phys. Lett. A, 170, pp.421-428

[9] Boisgard, R., Michel, D. and Aimé, J. P., 1999,“Hysteresis generated by attractive interaction: oscillatingbehavior of a vibrating tip – micro-cantilever system near asurface”, Surf. Sci. Vol 401, pp. 199 – 205

[10] Couturier, G., Nony, L., Boisgrad, R. and Aimé, J.-P.,2002, “Stability of an oscillating tip in noncontact atomicforce microscopy: theoretical and numerical investigations”, J. Appl. Phys. Vol 91, pp. 2537 – 2543

[11] Nony, L., Boisgrad, R. and Aimé, J.-P., 2001,“Stability criterions of an oscillating tip-cantilever system in

dynamic force Microscopy”, Eur. Phys. J. B Vol 24, pp.221 – 229.

[12] MOON, C. F., Chaotic and Fractal Dynamics, anintroduction for applied scientists and engineers, WileyVCH, New York, 1992.

[13] GURAN, A., Nonlinear Dynamics: The Richard Rand50th anniversary volume, World scientific publishing, Co PteLtda, Series on stability, vibration and control of systems.Series B: Vol 2 -24, 25, 1997.

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