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Nonlinear Time-series Analysis Methods for the Evaluation & Study of Dynamical Phenomena in Atomic Force Microscopy Prediction of chaotic motion

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Nonlinear Time-series Analysis Methods for the Evaluation & Study of Dynamical Phenomena in Atomic Force Microscopy Prediction of chaotic motion
Georgakaki D.
a
, Mitsas Ch.
b
, Polatoglou H.M.
a
a
Physics Department, Solid State Physics Section, AUTH., 54124
b
Laboratory of Mechanical Measurements, ..., Thessaloniki, 57022
Abstract.
Atomic Force Microscopy (AFM) is a widely used tool for nanoscale surface analysis. Dynamic atomic force microscopy (tapping or non-contact mode) consists of a vibrating microcantilever with a nanoscale tip that interacts with a sample surface via short- and long-range intermolecular forces. Due to the need of high precision measurements, the interaction forces between the nanotip and the specimen must be carefully examined and analyzed in order to best evaluate the nonlinear dynamical response of the AFM system. Recent research on AFM systems has focused on detailed numerical analysis such that possible chaotic regions can be well defined. In this paper, nonlinear time-series analysis methods (NTSA) are applied to analyze signals generated by a numerical single-degree-of-freedom lumped-model of a dynamic-mode AFM. The signal characteristics will provide useful insight of the nonlinear system response leading in future work, to the evaluation of deterministic uncertainty in real-time AFM measurements that is being introduced because of these complex phenomena. The confirmation of chaos existence and the prediction of the regions where chaotic motion is possible remain crucial for the quality of AFM images. The methods used for the system analysis include Autocorrelation Function, Power Spectral Density Analysis, Phase-space analysis, Poincare Maps and Lyapunov Indicator. All the above computational techniques can then be applied to experimental AFM data in order to find system parameters and regimes, which ensure stable motions across the sample surface scanning process.
Keywords:
AFM, Dynamic mode, Dynamical system, Time series analysis
PACS:
05.45.Tp,
07.79.Lh
INTRODUCTION
Atomic Force Microscope (AFM), invented in 1986 by G. Binnig, C.F. Quate and C. Herber [1], is the most widely used tool of Scanning Probe Microscopy. AFM operational modes measure the topography of a sample as well as its electrical, mechanical, and chemical properties [2]. In a typical AFM system there exist three different open-loop operational modes, according to small changes in cantilever tip-sample surface separation: i) contact mode, ii) tapping or intermittent mode and iii) non-contact mode. The tapping mode of operation combines characteristics of both the contact and non-contact modes by oscillating the cantilever tip at or near its natural resonance frequency while allowing it to impact the target sample for a minimal amount of time [3,4]. This mode is the most potent non-destructive high-resolution technique for topographic imaging, ideal for the characterization of soft and fragile materials. It reduces adhesive and friction forces that are inevitable in contact mode and it provides repeatability and variety in measurements as it collects data from both attractive and repulsive force regions [5]. Though extremely promising, the dynamic AFM modes are often susceptible to unwanted dynamic phenomena in the cantilever tip displacement [6], thus leading to false topographical information and causing instabilities that reduce the quality of the AFM image. Due to these problems, research on AFM systems has focused on the 1177
CP1203,
7
th
International Conference of the Balkan Physical Union,
edited by A. Angelopoulos and T. Fildisis
© 2009 American Institute of Physics 978-0-7354-0740-4/09/$25.00
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application of various computational tools [12,14] in combination with the experimental techniques, in order to identify regions where chaotic trajectories of the system evolution are evident [4-8-9]. These tools will provide good insight into the complex system dynamics and will suggest cantilevers with proper characteristics and operational conditions suitable for precise noisy-free measurements.
FIGURE 1.
Atomic Force Microscope.
ATOMIC FORCE MICROSCOPY AS A DYNAMICAL SYSTEM
The cantilever tip - sample surface system is treated as a driven, nonlinear, dissipative dynamical system. By the term “dissipative” one can deduce that the system is not Hamiltonian and that is due to the damping effect of the environment as well as the nonlinear oscillations of the tip near the sample surface. The evolution of such dynamical system strongly depends on the choice of the free parameters and initial conditions. Every trajectory of a dissipative system is contained in a space R
d
and is attracted by a limit set of points, generally called an attractor. Attractors are divided in two categories, the chaotic and the non-chaotic ones. General characteristics of attractors are presented in the following table:
TABLE 1.
General characteristics of attractors
Features
Non chaotic Chaotic
Category
Limit point Limit circle Torus Strange attractor
Dimension
Integer Equal to topological dimension Non-integer or integer higher than topological dimension
Predictability
totally limited
Sensitivity to initial conditions
no yes
As the tip scans the surface the recorded deflection of the tip from the equilibrium position represents a unique trajectory evolution in the state-space with specified initial conditions. Another set of initial conditions will lead to a different trajectory. Any extreme sensitivity to initial conditions implies the existence of chaos. But again one cannot easily tell between chaos and colored noise. In this work, a variety of methods have been developed in order to distinguish between chaotic, periodic and quasi-periodic states. For this purpose, the system is treated as a single degree of freedom lumped-parameter system, in contrast to the distributed model approach [16], consisting of a sphere of radius R and effective mass m (tip), a plane (sample) and a vibrating bimorph connected with a spring constant k to the tip. Having placed the sample at a certain position z
0
-ξ and selecting the driving amplitude, a driving frequency near or at the excitation frequency of the tip is applied. The output-recorded signal involves changes in amplitude (AM-AFM) or phase (PM-AFM) of the system. 1178
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FIGURE 2.
Model for tip-sample interaction.
The equations used to solve numerically the equation of motion are given below:
( )
( )
( )
( )
( )
( )
σ
= −− −
62 8
6 180
h h
R A R AP x t z x t z x t
(1)
( ) ( ) ( )
( )
( ) ( )
( )
( )
( )
′′ ′ ′+ − + − =
e
m x t c x t d t k x t d t P x t
(2) By changing the parameter z
0
different displacement x(t) and velocity v(t) diagrams are obtained. It is useful to observe the behavior of two different regimes, one close to the sample, where the influence of Lenard-Jones potential is stronger, and far away from it at a distance sufficient enough to be compared to the free vibrations of the tip [14]. Note that the simulation runs starting from a minimum tip-sample distance to a maximum tip-sample separation. Such an example is given at the image below:
FIGURE 3a.
Time series of displacement for z
0
=4nm.
FIGURE 3b.
Time series of displacement for z
0
=60nm.
3a.
For a very close tip-sample separation the oscillations of the tip are chaotic.
3b.
As we move the tip away from the sample, the amplitude of the oscillation increases until a certain time where the tip can sense the sample and the amplitude decreases.
3c.
Finally, we compare the previous situations with the free vibrations of the tip without an interaction with the sample.
FIGURE 3c
. Time series of displacement for free vibration.
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TIME SERIES ANALYSIS METHODS Autocorrelation Function
The autocorrelation function given by equation (3), defines the relation of each measurement x
i
with x
i+τ
where τ denotes the time interval or lag between the two measurements. For a system with a strong periodic component (driving frequency near the excitation frequency) theory claims that the data exhibit short-range correlations, and the ACF function bounces around zero without decreasing (as for example in an AR(1) procedure with φ<0 [10]). Unfortunately ACF cannot tell between a chaotic area and a harmonic one so we have to proceed to different methods in order to obtain this kind of information.
121
(())(())()(())
N iii N ii
xtxxtxr xtx
τ
τ τ
−==
− + −=−
∑∑
(3)
Power Spectral Density
The Power Spectrum is a very popular method used in dynamical systems. By applying Fast Fourier Transform in the time series data we get the intensity of Fourier coefficients in the frequency domain. If the tip is placed at a distance very close to the sample surface where van der Waals forces are very strong, a continuous spectrum with a variety of peaks (besides the peak at the excitation frequency) is observed. Otherwise if the tip is placed far away from the sample we expect only one peak, at the excitation frequency of the system [10]. The following diagrams prove the theoretical assumptions:
FIGURE 4a.
Harmonic regime with a peak at the excitation frequency of the cantilever.
FIGURE 4b.
Chaotic regime with various peaks at different frequencies.
Phase Space
By plotting the displacement versus the velocity of the tip, the phase space diagram of the AFM system is generated. As the tip is moved from a close distance to the sample surface far away from it, the observed chaotic attractor becomes a limit circle. The chaotic behavior is traced at a distance where van der Waals forces are very strong and at such small tip-sample separations, the physical properties of the sample influence the cantilever response at a significant level. The limit circle is observed at z
0
= 80-90nm, and it should be noted that the same image is obtained when we simulate the free vibration response of the tip [14]. In the following pictures [18] the transition from the chaotic regime to the quasiperiodic regime is fully recorded.
FIGURE 5a
. For
z
0
=3nm, a chaotic attractor is observed.
FIGURE 5b.
For an intermediary z
0
, the attractor expands to the entire space.
FIGURE 5c.
For z
0
=80nm, a limit circle is observed.
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Poincare Maps
Poincare maps are another convenient method to recognize chaotic vibrations. For a distance z
0
=90nm a closed quasiperiodic curve is observed [14] but as we minimize z
0
chaos is evident. By increasing the driving amplitude and lowering z
0
even more, it seems as the trajectory of the system evolution covers the entire phase space and is not contained to a specific topological subspace.
FIGURE 6a
. For
z
0
=90nm, a closed quasiperiodic trajectory is observed.
FIGURE 6b.
For z
0
=5nm chaotic motions of the tip are observed.
FIGURE 6c.
For z
0
=3nm and increasing driving amplitude, chaos is inevitable.
Fast Lyapunov Indicator
The most precise way to calculate chaoticity of a given trajectory is the maximum Lyapunov Exponent, method claimed to be computationally slow and laborious. Instead, various chaoticity indices have been proposed, amongst them is the Lyapunov Indicator [13]. So, given a flow dx/dt=f(x), we examine the evolution of a vector v which is given by the tangent flow
( )
[ ]
∂= =∂
,
i ij j
f dv Df x v Df dt x
(4) where Df is the matrix of the variations of the flow. Integrating the above system of equations with specific initial conditions x
0
, v
0
we get the FLI(t):
( ) ( )
φ
=
0 0
, log ,
t x v t x
(5) So, the FLI(t) is a monotonically increasing function and according to its slope it indicates chaos, weak chaos, quasiperiodic and periodic behavior [13]. Another solid proof of chaos is given in figure (7b) where two trajectories with slightly different initial conditions, start in accordance but at a specific time they begin to diverge from each other very quickly.
FIGURE 7a.
A chaotic, z
0
=5nm (light blue) and a weakly chaotic, z
0
=35nm, trajectory (dark blue).
FIGURE 7b.
Two trajectories, at z
0
=35nm, with very close initial conditions begin to diverge from each other.
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