Conference-NC-AFM 2010-Wavelet Transforms to Probe the Torsional Modes of a Thermally Excited Cantilever across the Jump-to-Contact Transition : Preliminary Results

The response of the torsional modes of a thermally excited cantilever across the jump-to-contact transition shows a modification of the oscillation amplitude, frequency, and damping. The measurement of these parameters is important because their analysis provides nanoscale information on the physical, chemical, and topographic properties of the sample. The tip-surface interaction potential is usually reconstructed by Fourier analysis of the cantilever oscillations around its equilibrium position. However, Fourier analysis can be correctly interpreted only in the case of stationary systems. The wavelet transform analysis overcomes these limitations, revealing the temporal evolution of the spectral content of a temporal trace. The one-dimensional time signal from the photodiode is converted into a two-dimensional time-frequency topography, which simultaneously exhibits the time and frequency behavior of the cantilever thermal fluctuations. In the present study, we show preliminary data obtained using wavelet transforms to analyze thermally excited torsional cantilever modes during jump-to-contact transition on a highly oriented pyrolitic graphite surface in air. [DOI: 10.1380/ejssnt.2011.228]


I. INTRODUCTION
The thermal motion or Brownian motion of the tip of the cantilever is related to the local mechanical compliance via the fluctuation-dissipation theorem.Modification of the thermal motion of the cantilever due to the tip-surface interaction forces allows the interaction potential to be reconstructed, and information on various kinds of surface forces can be obtained [1][2][3][4].The influence of the local environment on the cantilever thermal fluctuations around the equilibrium position is usually analyzed in the frequency domain using the Fourier transform.By doing so, the oscillation eigenmodes of the cantilever appear as resonance peaks.However, Fourier analysis can be correctly interpreted only if the frequency spectrum is correlated with a temporally invariant physical system.If the physical state of the system changes due to a timedependent interaction, the Fourier spectrum shows only a time average of the spectral features of the interacting state.
There exists a powerful and well-developed mathematical tool that circumvents these limitations, namely, wavelet transform (WT) analysis [5,6].Recently, this method has been applied to analyze dynamic force spectroscopy data [7,8].The WT analysis allows to represent the temporal evolution of the spectral content of an oscillating cantilever.A wavelet transform converts one-dimensional time signals into two-dimensional timefrequency topography, in which time and frequency structures are displayed simultaneously in the time-frequency plane.Here, we analyze the evolution of the torsional mode resonances, as the cantilever approaches a HOPG surface, across the jump-to-contact (JTC) transition, using the wavelet transforms.

II. WAVELET TRANSFORMS
Wavelet transforms are computed by correlating the signal f (t) with families of time-frequency atoms called wavelets [6].The wavelets are functions with limited time support (in contrast to the Fourier basis) whose oscillation behavior sets the frequency resolution.The basis functions of WT are referred to as daughter wavelets, which are generated by dilatations and translations of a mother wavelet Ψ(t) as Ψ s, , where d is the delay and s the scale parameter.The WT is expressed as The WT coefficients are resemblance coefficients, which measure the similitude between the signal and the wavelet function at various delays and scales.The wavelet dilatations set by the scale parameter s are inversely proportional to the frequency.The square modulus of the wavelet coefficients |W f (s, d)| 2 represents the local timefrequency signal energy density, referred to as the scalogram of the signal.The time-frequency resolution of these transforms is limited by the time-frequency resolution of the corresponding atoms.
Unlike the FT, the basis of the WT is not unique.Here, we choose the complex Gabor wavelet, also referred to as the Gaussian wavelet, as the mother wavelet, because it has the least spread in both the frequency and time domains and, consequently, the best time-frequency resolu- tion.The mother wavelet is represented as where the parameters η and σ control the shape of the Gabor mother wavelet, σ controls the amplitude of the Gaussian envelope, and thus its time/frequency resolution, and η is the carrier frequency, i.e., the number of oscillations within the envelope width [6,9].
The time evolution of the cantilever thermal motion, sampled with a digitizing oscilloscope, forms a onedimensional string of units corresponding to the value of the signal at the specific sampling time and representing the discretized signal.Each unit is temporally connected to the next by a fixed sampling interval T .In this framework, the temporal parameter t is a discrete index and σ and η are adimensional wavelet parameters defining the wavelet shape over the discrete sampling string.The wavelet adimensional center frequency at scale s in sampling units is given by f s = η/(2πs).The pseudo frequency F s (in Hz) is associated with the scale s by considering that f s is sampled with a time interval T , so that F s = f s /T .The associated pulsation is ω s = 2πF s .
The σ parameter determines the Heisenberg box associated with the square modulus of the mother wavelet, which corresponds to the time-frequency resolution with which the signal energy density can be represented.The Heisenberg box associated with the mother wavelet is given by a time resolution ∆ t = σ/ √ 2 and a frequency resolution ∆ ω = 1/( √ 2σ) [10].When the wavelet is subject to a scale dilatation s, the corresponding resolution is modified as ∆ s,t = s∆ t and ∆ s,ω = ∆ ω /s.As expected on the basis of the uncertainty principle, ∆ s,t ∆ s,ω = 1/2.The time-frequency Heisenberg box is ∆ s,t T × ∆ s,ω /T .

III. EXPERIMENTAL
The AFM [11] optical beam deflection system is based on a 600-nm laser diode coupled to a monomode fiber, which acts as a mode filter, providing a TEM00 beam output after recollimation.The fiber has a mode field diameter of 4 µm.Its collimated output is focused through an aspherical lens to a 10 µm spot on the cantilever free end.A four quadrant silicon diode monitors the cantilever flexural deflection by means of the up-down differential output and its torsion by means of the left-right differential output.The overall bandwidth of the optical lever deflection system exceeds 1 MHz.The flexural and torsional signals are recorded simultaneously by a digitizing oscilloscope with a vertical resolution of 8 bits, an analog bandwidth of 250 MHz, a maximum sampling rate of 1 Gsample/s, and a buffer memory of 128 Msample.The scanning system is based on a single scanner tube with a maximum vertical extension of 2 µm.
The nominal dimensions of the silicon cantilever are 40 µm × 460 µm × 2 µm.The nominal tip radius is R = 10 nm, and the tip height is 20-25 µm.The plan view dimensions are measured by an optical microscope.The Sader method gives a flexural spring constant of k = 0.13 N/m [12] and a torsional elastic constant of k ϑ = 9.9 nNm [13].
The sample consisted of a freshly cleaved highly oriented pyrolitic graphite (HOPG) surface.All the experiments have been conducted in air, with a relative humidity of less than 50%.Figure 1 shows a schematic diagram of the experimental apparatus.
The calibration of the force sensitivity of the cantilever for the static deflection was carried out before the experiment by taking the force spectroscopy curves.The deflection sensitivity calibration was performed on the hard HOPG surface, assuming negligible indentation and thus equal distances spanned by the cantilever tip and the piezotube.The obtained sensitivity is in the range of 50−200 nm/V, depending on the cantilever type, the laser spot position, and the laser beam power level.The contact point indicating the surface position is determined from the approach force curve as the distance at which no force acts on the cantilever (neutral position).This means that the surface corresponds to the position after the jump-to-contact transition at which the cantilever exhibits no deflection.
Volume 9 (2011) Malegori and Ferrini TABLE I: Comparison of measured and calculated [14,15] free cantilever resonant frequencies.The theoretical results are expressed as ratios with respect to the first flexural frequency, f1 = 10.908kHz.The Q factors are measured from the power density spectra.Here, t and l are the free torsional and lateral eigenmodes, respectively, and tc and lc are the contact torsional and lateral eigenmodes, respectively.In this case the contact measurements refer to a negative load of −0.5 nN on the tip.In order to determine the free cantilever resonance frequencies, the photodiode signal for the flexural and torsional modes is collected when the cantilever thermally oscillates far from the surface.Table I lists the resonance frequencies and quality factors Q of the two lower torsional modes and the first lateral mode that couples into the photodiode signal.The lateral modes are cantilever in-plane flexural modes, which is in contrast with the usual out-of-plane flexural modes.The measured resonance frequencies compare well with the calculated eigenfrequency of a rectangular cantilever beam [14,15], see Table I.The contact torsional resonance frequencies were measured when the tip was in contact with the surface, and the cantilever was deflected toward the surface by approximately 4 nm, with an applied negative load of −0.5 nN, Fig. 2b.Negative attractive or positive repulsive tip-sample forces are measured with respect to the cantilever neutral position.
Finally, the force spectroscopy curves were collected while driving the piezotube motion at a constant velocity of 225 nm/s toward the tip.Since the sampling time is 240 ns, the signal string is composed of approximately 4 Msample per second.Flexural and torsional signals were collected simultaneously so that the flexural signal provides the cantilever deflection and the tip-sample distance, whereas the torsional signal is used for WT analysis.

IV. RESULTS AND DISCUSSION
In the torsional mode, the cantilever oscillates about its long axis and the tip moves approximately parallel to the surface.The probe tip is sensitive to in-plane forces, and, since the eigenfrequency of the torsional modes depends solely on the lateral stiffness of the sample, torsional modes are useful as shear stiffness sensors.An increasing shear stiffness increases the lateral spring constant and, consequently, the resonant frequency of the system [16].We investigate the spectra of thermally excited torsional modes of the cantilever as the tip approaches a graphite surface in air.Since we are interested in exploring what happens immediately after the JTC transition, the forces that predominate in this regime are the attractive adhesion forces.
The power density spectrum of the free cantilever first torsional mode is shown in Fig. 2a.The peak at 239.4 kHz with Q=310 is the first torsional mode (t 1 ), and the peak at 210.2 kHz with Q=590 is the first lateral bending mode (l 1 ) [15].In the spectrum, a minute contribution from the third flexural mode is also visible at 222 kHz.When the tip is brought close to the sample, the capillary forces attract the tip to the HOPG surface until the JTC transition occurs [17].Due to the modified mechanical boundary conditions, the cantilever end is no longer free.A clear shift of the torsional and lateral contact mode resonances is detected under a negative static load of −0.5 nN, as shown in Fig. 2b.The first contact torsional mode resonance frequency increases to 305.2 kHz with Q=14, and the contact lateral mode resonance frequency increases to 221.7 kHz with Q=200.In both cases, the dissipation increases for contact modes, particularly the first torsional eigenmode.
The torsional resonance variation of the thermally excited cantilever can be observed across the JTC transition with the wavelet transforms, as shown in Fig. 3.The JTC transition is located at time zero, separating the negative times of the free cantilever evolution, from the positive times of the clamped cantilever evolution.Note that the long-range forces and capillary phenomena, which usually interfere with the oscillations of the flexural modes [2,3,7,18], do not perturb the much stiffer torsional free modes until the jump-to-contact transition.The lateral mode frequency exhibits a very sharp frequency shift at JTC and remains fairly constant immediately thereafter.Instead, the torsional contact mode shows a detectable and continuous frequency increase after the JTC transition due to the tip interaction with the graphite surface.
Note the difference in appearance of the torsional mode frequency structure before and after JTC in Fig. 3 which can be qualitatively explained as a sudden increase in dissipation caused by the interaction with the surface.This demonstrates that a smooth transition did not occur during the JTC between the free and contact oscillations.
Taking into account the vertical velocity of the piezoscanner, it is possible to obtain a linear relation between time and cantilever deflection, enabling calculation of the contact loading force of the tip on the surface.The frequency evolution is provided by the wavelet ridges, which are the maxima of the normalized scalogram, as shown in Fig. ?? as black points.The ridges correspond to the instantaneous frequencies within the transform resolution limits [6].Using the wavelet ridges, the time-frequency representation is thus transformed into a contact-interaction-force vs. frequency shift representation after JTC transition.
Immediately after JTC transition, the force acting on the cantilever is attractive, providing a negative loading.In this case the tip is acted upon by adhesion forces that attract the tip towards the surface.The frequency shift of resonance frequencies with respect to the free cantilever oscillations is thus caused by the decrease in strength of adhesion forces, a transient that cannot be easily captured using standard or non-dynamic techniques.
Using a suitable model, this technique may enable detailed measurement of the properties of adhesion forces [15,19,20].Analytical and numerical models describing the free cantilever vibration as well as the contact resonances are well known and provide quantitative evaluation when complete contact-resonance spectra are measured.The contact-resonance frequencies of the cantilever are linked to the tip-sample contact stiffness, which depends on the elastic indentation moduli of the tip and the sample as well as on the shape of the contact.The spatial resolution depends on the tip-sample contact radius, which is usually in the range of 10-100 nm.Lateral stiffness determined by the contact-resonant noise spectra of the first torsional mode has been obtained in [20] using quasi-static force curve cycles.The improvement provided by WT analysis is related to the time required to detect the frequency shift vs. the load curve, which is on the order of a few milliseconds.This acquisition time is significantly shorter than that of the quasi-static techniques and is compatible with the development of realtime measurement.
The discontinuous appearance of the traces in the timefrequency plane is due to the discreteness of the thermal excitation and can be rationalized using a simple model.The action of (quasi)uncorrelated impulsive forces from the thermal bath on the cantilever produce large oscillations at the resonance frequencies on a statistical basis.When the cantilever exhibits a single thermally activated fluctuation, it responds as a damped harmonic oscillator with a decay time constant of the associated energy expressed as τ = Q/ω 0 and a Fourier linewidth of ∆ω = 2π∆f = 1/τ [21,22].Since the cantilever is first thermally excited and then damped to a steady state by random forces that act on a time scale that is much smaller than the oscillation period of the cantilever, the characteristic response time for an isolated excitation/decay event is of the order of 2τ , with an associated Lorentzian width of ∆ω.The time-frequency box having dimensions of 2τ × ∆ω represents the damped oscillator box, which is characteristic of a thermally-excited damped oscillator response [7].
In order to study the interplay between the wavelet resolution and the dimensions of the single thermal excitation in the time-frequency plane, we consider two extreme cases: the free cantilever and the clamped cantilever with positive loading.
The wavelet transform of the free thermal oscillations of the cantilever detected by the left-right sections of the quadrant photodiode shows the time evolution of the first torsional mode and the first lateral mode (see Fig. 5a).When the Q factor of a mode is high, see Table I, the corresponding frequency linewidth is small.In this case, the frequency resolution of a wavelet may be not sufficient to resolve the intrinsic linewidth of the mechanical resonance.The Heisenberg box dimensions are 0.050 ms × 6.35 kHz for the first torsional mode (t 1 ) and 0.057 ms × 5.6 kHz for the first lateral mode (l 1 ).The damped oscillator boxes for the same modes are 0.41 ms × 0.77 kHz (t 1 ) and 0.90 ms × 0.35 kHz (l 1 ).Thus, the frequency width of the time-frequency distribution is limited by the wavelet resolution, i.e., by the frequency width of the Heisenberg box, which is much larger than the frequency width of the oscillator box, see Fig. 5b.On the other hand, a high Q implies a long decay time associated with the oscillator energy.In this case, the time associated with the damped oscillator box is larger than the temporal wavelet resolution, i.e., the time width of the oscillator box is larger than the time width of the Heisenberg box.In Fig. 5b, the oscillator boxes (red) and the Heisenberg boxes (red boxes with white borders) have been superposed onto the time-frequency representation of the wavelet coefficients.In this case, the Heisen-berg box, i.e., the wavelet resolution, limits the frequency width of the signal structures, whereas their temporal extension of the structures is similar to the oscillator box time width.Such structures can be interpreted as cantilever excitation and decay to a steady state after a single thermal fluctuation event [7].
Figure 5c shows the contact cantilever vibrations after JTC transition at a static positive load of the tip on the graphite surface of approximately 1.6 nN.The Q factor of the first torsional contact mode (t 1c ) decreases, and the oscillator box is reshaped accordingly, reducing the damping time and increasing the frequency width (Fig. 5d).We found the Q factors of the contact modes to be approximately independent from the tip loading in the investigated range and to be similar to the values reported in Table I for the case of negative loading.The dimensions of the Heisenberg boxes shown in Fig. 5d are 0.053 ms × 6.0 kHz for the first torsional contact mode (t c1 ) at 316.72 kHz and 0.075 ms × 4.2 kHz for the first lateral mode (l c1 ) at 221.6 kHz.The damped oscillator boxes for the same modes are 0.013 ms × 23.3 kHz (t c1 ) and 0.28 ms × 1.25 kHz (l c1 ).As indicated by the data described above, the frequency resolution of the wavelet for the mode t c1 is sufficient to reconstruct the linewidth profile of the time-frequency trace, i.e., the spectral width of the Heisenberg box is smaller than the frequency width of the oscillator box.In contrast to the other modes, the time resolution of the wavelet does not allow the temporal evolution of the single thermal excitation to be tracked, because the time width of the oscillator box is smaller than the corresponding Heisenberg box dimension.
In conclusion, we note that WT analysis is better suited to the measurement of the oscillator energy dissipation (damping) in the time domain in high-Q environments and in the frequency domain in low-Q environments.In both cases, WT analysis provides the instantaneous characterization of the energy fluctuation-dissipation effects.

V. CONCLUSION
Dynamic force spectroscopy using torsional modes is a useful tool for imaging in-plane forces, which cannot be observed by means of other AFM techniques.Moreover, torsional modes are interesting from the perspective of their use in liquids, where their lower hydrodynamic damping allows excitation of clean torsional resonances [23].The wavelet analysis is especially useful in capturing the temporal evolution of the spectral response of the thermally driven cantilever interacting with the surface forces rapidly and continuously across the jumpto contact transition.Traditional AFM techniques enable the construction of the spectral response by varying the cantilever interaction step by step, however, in this way, it is not possible to analyze transients.Instead, the wavelet analysis allows to detect transient spectral features that are not accessible through steady state techniques, as shown in Fig. 3.Moreover, the ability to capture the relevant spectral evolution in a time of the order of tens of milliseconds enables surface chemical kinetics or surface force modification to be tracked in real time with dynamic force spectroscopy.More fundamentally, the wavelet transforms highlight the thermodynamic characteristics of the cantilever Brownian motion, enabling the tip-sample fluctuation-dissipation interactions to be investigated.
In conclusion, although the results of the present study are only preliminary, the proposed technique is interesting in view of its simplicity and connection with fundamental thermodynamic quantities.
FIG. 1: (Color online) Block diagram of the optical beam detection system.The power spectral density is provided by the thermal left-right photodiode signal of the free cantilever.
FIG. 2: a) Thermal power spectral density of the free cantilever torsional fluctuations showing the first torsional (t1) and first lateral (l1) resonance peaks.The arrow indicates the small contribution from the third flexural mode (f3) at 223 kHz.b) Same as a) but with the tip in contact with the sample at constant negative load (−0.5 nN)

FIG. 3 :
FIG. 3: Wavelet transform of the cantilever thermal torsional oscillation across the jump-to-contact transition, showing the evolution of the first free torsional mode t1 into the contact torsional mode tc1 and the evolution of the first free lateral mode l1 into the contact lateral mode lc1.The wavelet coefficients |W f (s, d)| are coded in color scale.The origin of the time axis is at the onset of the jump-to-contact transition.Both modes exhibit a shift as the tip is attracted to the surface.

FIG. 4
FIG. 4: a) Wavelet transform (in gray scale) showing the freshift of the first contact torsional mode, tc1, versus tip load.The frequency shift is calculated from the resonant frequency of the first free torsional mode.b) Wavelet transform (in gray scale) showing the frequency shift of the first contact lateral mode, lc1, versus tip load.The frequency shift is calculated from the resonant frequency of the first free lateral mode.In both figures, the black points are the wavelet ridges, showing the instantaneous frequency within the limit of the scalogram resolution.The continuous vertical lines at time zero correspond to the onset of the jump-to-contact transition.The dotted lines are provided to guide the eye.
FIG. 5: a) Wavelet transform of the cantilever thermal fluctuations far from the surface of the first lateral (l1) and first torsional (t1) modes.The wavelet coefficients |W f (s, d)| are coded in color scale.The dotted lines are centered on the resonant frequencies of the modes.b) Same image as in a) but coded in saturated gray scale in order to clarify the shape of the discontinuous structures.c) Wavelet transform of the cantilever thermal fluctuations in contact mode of the first lateral (lc1) and first torsional (tc1) modes at a constant positive load of approximately 1.6 nN.d) Same image as in c) but coded in saturated gray scale in order to clarify the shape of the discontinuous structures.In b) and d), the red rectangles with a white border represent the Heisenberg boxes.The red rectangles represent the damped oscillator boxes.The Heisenberg boxes and the oscillator boxes overlap in order to compare the structures in the time-frequency representation.Note that the oscillator boxes for the modes t1, l1, and lc1 (high Q) have a frequency width much smaller than that of the mode tc1 (low Q).