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*Print version* ISSN 0327-0793

### Lat. Am. appl. res. vol.42 no.4 Bahía Blanca Oct. 2012

**ARTÍCULO**

**Analysis of the amplitude and phase response of the microcantilever in the tapping mode atomic force microscopy**

**M.H. Korayem and M.M. Eghbal**

^{?} *Robotic Research Laboratory, Center of Excellence in Experimental Solid Mechanics and Dynamics, School of Mechanical Engineering, Iran University of Science and Technology, Tehran, Iran. hkorayem@iust.ac.ir*

*Abstract* - In this paper, the amplitude and phase response of the microcantilever in the tapping mode atomic force microscopy is investigated as a function of excitation frequency, free oscillation amplitude, and tip-sample separation distance. For this purpose, the influence of the excitation frequency on the transition occurring in the amplitude-phase-separation curves is analyzed by taking into consideration, three vibration modes. The obtained results show that at the excitation frequency close to resonance, the participation of higher modes in the response is negligible, and accounting for only one mode in the response is sufficient. Moreover, increasing the excitation frequency a little over the resonance frequency causes the shifting of the transition location toward smaller separation distances and finally to no separation; while, lowering the excitation frequency around the resonance results in the displacement of the jump's location toward larger separation distances. To investigate the influence of the free oscillation amplitude on transition, the share of only one mode in the response is considered. The results indicate that, depending on the value of the free oscillation amplitude, the jump either has a steplike discontinuous shape or a smooth continuous one. Also, by increasing the free oscillation amplitude, the transition occurs at a larger separation.

*Keywords* - Tapping Mode Atomic Force Microscopy; Amplitude And Phase Response; Transitions; DMT Contact Mechanics Model.

**I. INTRODUCTION**

Since the invention of the Tapping Mode Atomic Force Microscope (TMAFM), it has attracted a great deal of attention in a vast range of applications. One of the reasons for the TMAFM's popularity is that it causes less damage to the soft tissues and biological samples during the scanning process (García and Pérez, 2002). The basis of the TMAFM's operation is the monitoring and control of the dynamics of the tip attached to the end of a flexible microcantilever, which interacts with the sample surface. Having a correct understanding of the tip-sample interaction regimes is very important in obtaining high-precision images and reliable data. Gleyces *et al.* (1991) were among the first researchers who demonstrated, experimentally and theoretically, the bistable behavior of the vibrating tip motion near the sample surface. Anczykowski *et al.* (1996), by using the MYD/ BHW model for the calculation of interaction forces, numerically simulated the high-amplitude dynamic response of the cantilever as a function of the tip-sample separation, and analyzed the hysteresis effects at different excitation frequencies. Tamayo and Garcia (1996) described the general characteristics of the tapping mode operations of the scanning force microscope and in this regard, they calculated factors such as force, deformation, and contact time as a function of tapping frequency, amplitude damping, and the sample's elastic and viscoelastic properties. Kühle* et al.* (1997) presented experimental and theoretical results that emphasized on the effective role of attractive forces in the behavior of the tapping mode atomic force microscope. Marth *et al.* (1999) investigated the experimental effects of the TMAFM (i.e., hysteresis and distortions of imaging), and by examining the phase space, they concluded that the common cause of all those effects is the bistable behavior of the cantilever motion under specific conditions. Garcia and San Paulo (1999) studied the attractive and repulsive regimes of the tip-sample interactions as a function of the tip-sample separation distance, free oscillation amplitude, and the properties of the sample, and proposed a definition for them in terms of the average forces. In another work, to get a better understanding of the TMAFM's operation, they investigated the experimental dependence of the amplitude on the tip-sample separation (García and San Paulo, 2000). They also provided a general scheme for the identification, classification, and the understanding of the features observed in the experimental curves, which is very useful for improving the imaging contrast in the dynamic AFM. In San Paulo and García (2002), they proposed a theoretical method for the study of the tapping mode atomic force microscopy, which is valid for any type of force interaction. They also showed that the combined participation of the attractive and repulsive interactions brings about the multi-value nature of the resonance curve, which creates the conditions for the coexistence of two stable oscillatory states for some excitation frequencies. The coexistence of the two oscillatory states depends on the properties of the sample (e.g., modulus of elasticity, adhesive force, and non-conservative interactions), and also on the operating parameters (e.g., free oscillation amplitude, and the cantilever's spring constant) (San Paulo and García, 2002; Chen *et al.*, 2000). Through theoretical, computational and experimental analysis, Lee *et al.* (2002) analyzed the nonlinear characteristics of the frequency response of a specific cantilever-tip system (such as the sudden jumps and the resulting hysteresis) in the tapping mode, and thoroughly investigated its relationship with the surface interaction potential. In another work (Lee *et al.*, 2003), they measured the nonlinear tip response of a TMAFM in two cases: first, a fixed static tip-sample separation and a variable driving frequency, and second, a fixed driving frequency and a variable separation; and they explained the connection between these two responses through the concept of bifurcation sets. In Nunes *et al.* (2003) a simple model is used for the simulation of the cantilever motion in the dynamic mode of the atomic force microscope. The obtained results indicate the dependency of the transition (from the non-contact to the tapping operating mode) on the free oscillation amplitude, frequency of the excitation force, height of the contamination layer, and on the sample's stiffness. Stark *et al.* (2003) investigated, through experiments and numerical simulation, the behavior of the TMAFM at different excitation frequencies. They showed that the excitation frequency plays a very effective role in the adjustment of the imaging regime. By using the double-stranded DNA (dsDNA) on mica as a sample, Santos *et al.* (2010) investigated the connection between theory and experiment in the tapping mode atomic force microscope. They demonstrated how the cantilever dynamics, especially the bi-stable behavior and the force regimes, can be controlled by changing the operating parameters. Korayem *et al.* (2011) investigated the tip-sample interaction regimes in the presence of hysteretic forces in the tapping mode atomic force microscopy. They also explored the role of hysteretic forces in producing the hysteresis of the amplitude-separation curves.

The goal of this paper is to analyze the amplitude and phase response of the microcantilever as a function of excitation frequency, free oscillation amplitude, and tip-sample rest separation. For this purpose, based on the Euler-Bernoulli elastic beam theory, the equation governing the motion of the microcantilever is derived and by numerically solving this equation, the changes of the dynamic parameters (such as the oscillation amplitude, phase difference, average force, and the contact time) with respect to the tip-sample rest separation are investigated. The tip-sample interactions are assumed to include the attractive components (Van der Waals forces) and the adhesive-repulsive components (DMT contact mechanics model). Here, the excitation caused by the piezoelectric actuator has been accounted in the overall displacement of the cantilever as well as the generation of the external force term.

**II. MODELING**

**A. Tip-Sample Interaction**

For the analysis of the tip-sample interactions, the Van der Waals force and the DMT contact mechanics model between a sphere (tip apex) and a flat surface (sample) are used. These forces consist of (García and San Paulo, 1999):

(1) |

In the above relations, *H* is the Hamaker constant, *?* is the surface energy, *R* is the tip radius, *d* is the instantaneous tip-sample separation distance, *a _{0}* is the intermolecular distance (contact begins at this distance),

*E*is the modulus of elasticity,

*E*is the effective modulus of elasticity, and

^{*}*ν*is the Poisson coefficient. The subscripts "

*t*" and "

*s*" denote "tip" and "sample", respectively. Figure 2 shows the tip-sample interaction for the polystyrene sample. The interaction can be divided into the Van der Waals force regime and the DMT contact regime. A negative interaction indicates an attractive force, while a positive interaction indicates a repulsive one (the elastic restoring force).

Fig. 2: Interaction force associated with the polystyrene sample, including the Van der Walls force and the DMT contact mechanics model

**B. Equation of Microcantilever Motion**

The dynamic equation of motion of a microcantilever exposed to base harmonic excitation can be derived by applying the Euler-Bernoulli theory and by considering the non-inertial coordinate system with its origin at the microcantilever's base, in the following way:

Fig. 1: Schematic diagram of vibrating cantilever near the sample surface

(2) |

where q (x, t)is the transverse displacement of the cantilever relative to the considered coordinate system, E_{c} is the Young's modulus, *I _{c}* is the bending moment of inertia of the cross section,

*?*is the mass density,

_{c}*S*is the cross-sectional area, and

_{c}*L*is the cantilever length. Here, the excitation by the piezoelectric actuator is modeled as the harmonic motion of the base

*y*(

*t*) with frequency O i.e.,

*y*(

*t*)=

*Y*sin(O

*t*), which leads to an external force term in the equation of motion, and also to time-dependent terms in the interaction force through the overall dynamic displacement of the cantilever,

*w*(

*x,t*). In such a case, the

*w*(

*x, t)*is calculated via the following relation:

(3) |

Equation (2) is strongly nonlinear and non-autonomous, and its discretization can be appropriately obtained by projecting the motion dynamics onto the system's linear modes. By using the separation of variables method, the general answer of the equation of motion can be written in the following form:

(4) |

where _{i}(*x*) denote the eigenfunctions (modal shapes) of the cantilever and *?** _{i}*(

*t*) are the time-dependent generalized coordinates. By substituting Eq. (4) into Eq. (2), multiplying the obtained relation by the expression

_{i}(

*x*)

_{i}(

*L*), and integrating both sides in the range of 0 ≤ x ≤L, and also by adding the damping term, the following ordinary differential equations (for each of the modes of the vibrating cantilever tip) are obtained:

(5) |

In the above equation, *d _{0}* is the tip-sample rest separation distance,

*?*are the natural frequencies,

_{i }*Q*are the quality factors, and

_{i }*ζ*are the modal damping coefficients.

_{i}**III. SIMULATION**

In this section, by using the models proposed in the previous section for the interaction forces and the dynamics governing the microcantilever, the effects of the excitation frequency and free oscillation amplitude on the transitions appearing in the dynamic response curves are examined and analyzed. The forward-time simulation approach, introduced in Hashemi (2008), has been applied using the ode integrators of the MATLAB and engaging the software's event-managing capability. The numerical simulations have been performed for the polystyrene material (PS) and silicone cantilever with a rectangular cross-section. All the parameters required for the simulations and relevant calculations have been listed in Table 1.

Table 1: Constants and properties (of microcantilever and sample) used in numerical calculations

**A. Effect of Excitation Frequency on Transition**

In this subsection, the influence of the excitation frequency *f _{e}* on the transition occurring in the amplitude and phase curves is studied. For this purpose, for the free oscillation amplitude

*A*= 11 nm and for the excitation frequencies equal to fundamental resonance frequency (the first mode's natural frequency,

_{0}*f*) or its vicinity (

_{1}*f*- 400,

_{1}*f*+ 150,

_{1}*f*+ 400Hz), the amplitude and phase are calculated versus the tip-sample separation distance (Fig. 3).

_{1}

Fig. 3: Amplitude and phase curves for different separation distances and excitation frequencies with free oscillation amplitude of 11 nm, and the involvement of three vibration modes

In the estimation of the amplitude and phase, the effects of three vibration modes of cantilever have been accounted and for this purpose, the equations of motion related to the first three modes have been simultaneously solved. The amplitude-separation curves of the second and third modes at each of the excitation frequencies have been shown separately in Fig. 4. As the figure demonstrates, in these conditions, the contributions made by the higher modes to the response reach 0.17% at the most, and therefore, it would be acceptable to consider only one mode in the response. Naturally, if the excitation frequency moves away considerably from the fundamental frequency, the participation of higher modes in the response will not be negligible anymore. The solid curve in Fig. 3 shows the response when the cantilever is excited at a frequency which is 400 Hz less than *f _{1}* (

*f*- 400 Hz). At distances far from the surface, the amplitude is less than the free oscillation amplitude, and equal to 7.98 nm. By moving close to the surface, a minor increase is observed in the amplitude value. Then, at the point where the tip starts interacting with the surface, the amplitude rises suddenly and after that, it decreases almost linearly with distance. In fact, the amplitude initially increases because of the attractive interaction and then, as a result of repulsive interaction, it damps out.

_{1}

Fig. 4: Amplitude-separation curves related to the second and third modes at each excitation frequency

The curve formed by the hollow circles represents the case in which the excitation frequency is the same as the fundamental resonance frequency. In this case, after the amplitude decreases linearly, it goes through a jump at a specific separation distance. The solid line curve through the small squares shows the amplitude versus separation distance for the excitation frequency equivalent to *f*_{1}+150 Hz. Qualitatively, this curve has identical characteristics with the *f _{e}*=

*f*

_{1}case, except that here, the jump occurs at a closer distance to the sample surface. The dashed-line curve in Fig. 3 is related to the excitation frequency of

*f*

_{1}+400 Hz. In this case, no jump occurs in the curve and at large separations, the amplitude conforms to the solid-line curve and as the separation distance decreases, the amplitude diminishes with a mild slope. Getting closer to the sample surface causes the amplitude to decrease with a sharper slope until it finally plunges almost linearly with distance. Qualitatively, the phase-separation curves associated with different excitation frequencies display a trend similar to their corresponding amplitude-separation curves. As it can be seen from Fig. 4, the second and third modes also undergo a sudden change at the jump position, and since mode 2 has a higher share in the response, its changes in the diagram is more pronounced.

**B. Effect of Free Oscillation Amplitude on Transition**

In this subsection, we estimate the amplitude, average force and contact time as a function of free oscillation amplitude (*A*_{0}) and tip-sample separation distance (*d*_{0}). Here, the simulations have been carried out for *W*=*w*_{1}, and therefore, the contribution of only one mode has been considered in the response.

Figure 5(a) shows the oscillation amplitude versus separation distance for three different free oscillation amplitudes (*A*_{0}=11, 20, 45 nm). In all the diagrams, a flat section can be observed at large *d*_{0} distances. In this section, the attractive forces have a negligible influence on the oscillation. As the separation distance decreases, the cantilever starts feeling the influence of the long-range attractive forces. This usually results in the diminishing of the amplitude. Then, a transition occurs in the amplitude curve which causes the increase of the amplitude. After that, the amplitude decreases linearly with a uniform slope. Considering the increase of amplitude in the transition region, the amplitude curve could be divided into two separate oscillation branches, i.e., the high amplitude branch (*H*) and the low amplitude branch (*L*). In fact, here, transition means a transfer from the low amplitude branch to the high amplitude one. The shape of the transition curve depends on the free oscillation amplitude. For free oscillation amplitudes of 11 and 20 nm, the transition displays a steplike discontinuous shape and for *A*_{0} = 45 nm, it has a continuous form. Also, with the increase of the *A*_{0}, the transition occurs at a larger separation distance.

In Fig. 5(b), the average force of the tip-sample interaction in one stable oscillation cycle has been plotted. For *A*_{0} = 11 nm, at large separation distances, the average force is negative (attractive). With the decrease of d_{0}, the absolute value of the average force increases until the point where a sudden change of sign occurs. After that, the average force increases slightly until it reaches a maximum magnitude. Then, the average force decreases and finally, at very small separation distances, it goes through a sign change again, which this time has a continuous form. At *A*_{0} = 20 nm, a similar behavior is observed. However, at *A*_{0} = 45 nm, the average force of interaction varies continually with *d*_{0}, such that at small separations, no change of sign occurs and only the magnitude of the average force decreases.

In Fig. 5(c), the contact time normalized with the period of the oscillation *T* has been illustrated. The contact time, *t _{c}*, denotes the time duration in a cycle, in which the cantilever tip is in physical contact with the sample surface. For

*A*

_{0}=11 nm, the contact time is zero at separation distances prior to the transition. Therefore, in this situation, the steplike discontinuous transition can represent the transfer from the non-contact mode (

*t*=0) to the tapping mode (

_{c}*t*≠0). Also, for the free oscillation amplitude of 20 nm, the

_{c}*t*shows a steplike transition; although this time, a small continuous curve can be seen before the transition, from 0.0

_{c}*T*to 0.04

*T*. This means that, before the jump, the cantilever tip is in a periodic contact with the sample, and it is affected by the short-range repulsive forces. Therefore, the existence of steplike transition in the amplitude curves is not exclusively confined to the transition from non-contact to tapping modes, but is also always dictated by the existence of two stable oscillatory states in the cantilever response. For

*A*

_{0}=45 nm, the contact time shows a continuous transition with a sharp slope. In all of the diagrams, when the transition occurs, the tc increases uniformly with decreasing d

_{0}; however, at very small separation distances, its rate of increase goes up.

Fig. 5: Dependency of (a) amplitude, (b) average force, and (c) contact time on the tip-sample separation distance for three different free oscillation amplitudes

**IV. CONCLUSIONS**

In this paper, for the polystyrene material, the dependency of the transition jumps in the dynamic response curves of the TMAFM on the excitation frequency and free oscillation amplitude was investigated. The results indicated that, depending on the magnitude of the free oscillation amplitude, the jump either has a steplike discontinuous shape or a smooth continuous one. Also, with the increase of the free oscillation amplitude, the jump occurs at a larger separation distance. Moreover, the increase or decrease of the excitation frequency causes the jump's position to shift toward smaller or larger tip-sample separations, respectively.

Under practical conditions, because the transition jump occurring in the amplitude-separation curve can disrupt the operation of the feedback loop, the selection of proper set point amplitude for imaging purposes becomes very important. Since changing the excitation frequency and free oscillation amplitude causes the displacement of the transition location and consequently the change of the range of high and low amplitude oscillation branches, it is expected of these two parameters to play a significant role in creating appropriate imaging conditions.

**REFERENCES**

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**Received: July 16, 2011. Accepted: February 17, 2012 Recommended by subject editor: Jorge Solsona**