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Contact stiffness modulation in contact-mode atomic force microscopy

Ilham Kirrou, Mohamed Belhaq n

Laboratory of Mechanics, University Hassan II-Casablanca, Morocco

a r t i c l e i n f o

Article history:Received 27 January 2012Received in revised form17 April 2013Accepted 22 April 2013Available online 14 May 2013

Keywords:Atomic force microscopyContact-modeNon-linear dynamicsStiffness modulationFrequency shiftNanomechanics

a b s t r a c t

The effect of fast contact stiffness modulation on the frequency response in contact-mode atomic forcemicroscopy is studied analytically near primary resonance. Based on the Hertzian contact theory, alumped single degree of freedom oscillator is considered for modeling the contact-mode dynamicsbetween the tip of the microbeam and the sample. Averaging method and perturbation analysis areperformed to obtain the modulation equations of the slow dynamic. The influence of the contact stiffnessmodulation on the non-linear characteristic of the frequency response is examined. We find that theamplitude of the contact stiffness modulation influences significantly the amplitude of the tip oscillationas well as the shift direction of the frequency response indicating that such a modulation can be used tocharacterize the local elastic properties of the sample. Comparison between the analytical predictionsand the numerical simulations is given and application to a real atomic force microscope example isprovided.

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction

In atomic force microscopy (AFM) [1], a micro-scale cantileverbeam with a sharp tip is employed to scan the topography of aspecimen surface. Typically, the contact-mode AFM is used in suchapplications to confine the surface force to a Hertzian contactregime between the tip and the moving surface. The performanceof this contact-mode AFM in scanning requires the contact-moderegime to be maintained during the scan in order to obtainquantified results in terms of vibrational amplitude and amplituderesponse. It is known that in macro-scale mechanisms, contact-mode in a harmonically forced Hertzian contact regime has soft-ening characteristic and for a slight increase of the amplitude ofthe excitation, contact losses occur near resonances causingimpacts, thereby a possible deterioration of the device [2]. Thisloss of contact phenomenon has been observed for an idealizedpreloaded and non-sliding dry Hertzian contact modeled by asingle-degree-of-freedom (sdof) system [3]. Based on numericalsimulations, analytical approximation and experimental testing[3,4], it was concluded that the loss of contact is generally initiatedby jumps near resonances. In order to control the location of suchjumps, three strategies were developed [5,6]. The first strategyintroduced a fast harmonic excitation (added to the basic harmo-nic forcing) from above, the second strategy used a fast harmonicbase displacement, while the third one considered a fast harmonicparametric stiffness. It was concluded that fast harmonic base

displacement shifts the resonance curve left, whereas fast para-metric stiffness shifts the resonance curve right, suggestingmethods for controlling contact losses in systems evolving inHertzian contact regime.

Although the effect of fast excitation has been analyzedanalytically and numerically in various engineering applica-tions [5–8] showing various non-trivial effects, it has receivedlimited attention in micro-electromechanical devices [9,10] and inAFM [11].

In this paper, we report on the effect of fast contact stiffnessmodulation on the frequency response to primary resonance incontact-mode AFM considering a lumped sdof oscillator modelingthe contact-mode dynamics between the tip and the sample.In the case where the stiffness modulation is absence, Turner[12] investigated the non-linear vibrations of a linear beam withcantilever Hertzian contact boundary conditions assuming that thebeam remains in contact with the moving surface at all times. Heused the method of multiple scales (MMS) [13] to approximate theresponse of the probe-tip sample system to primary resonanceexcitation. The softening behavior of the response was obtainedfor the first four modes, and it was concluded that the response ofthe first mode is more willing to loose contact near the resonancebefore a significant change of parameters. Following Hajj et al.[14], Abdel-Rahman and Nayfeh [15] used the MMS to estimate thenon-linear coefficients of the contact stiffness using the subhar-monic resonance of the contact-mode AFM. Contact-mode AFMcan be used in various applications including, for instance, thedetermination of the viscoelastic properties of materials [16],identification of the interaction modes [17] and evaluation ofadhesion energy [18].

Contents lists available at SciVerse ScienceDirect

journal homepage: www.elsevier.com/locate/nlm

International Journal of Non-Linear Mechanics

0020-7462/$ - see front matter & 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.ijnonlinmec.2013.04.013

n Corresponding author. Tel.: +212 2522230674.E-mail address: [email protected] (M. Belhaq).

International Journal of Non-Linear Mechanics 55 (2013) 102–109


Owing to the fact that the first mode is predisposed to losecontact promptly with a slight change of parameters [12], atten-tion will be restricted to the analysis of the response to the firstmode of the microbeam. Thus, we consider a sdof system model-ing the cantilever dynamics of contact-mode AFM based onthe Hertzian contact theory. Such a system is often adopted tomodel the response of AFM cantilever neglecting the higher-orderflexural modes.

The next section presents the sdof model under a Hertziancontact condition in which the contact stiffness is assumed to bemodulated with high-frequency (HF) excitation. Then, the methodof direct partition of motion (DPM) is applied to obtain the mainequation describing the slow dynamic of the tip-sample system.In Section 3, the MMS is applied on the slow dynamic to obtain thecorresponding slow flow to primary resonance. This sectionincludes results of various parameters effect on the frequencyresponse and on jumps phenomena. An application to a real AFMcase is also provided in this section. Section 4 concludes the work.

2. Model and slow dynamic equation

A representative model of contact-mode AFM operation withcontact stiffness modulation is proposed. It consists of a lumpedsdof model, as shown in Fig. 1 [19], described by the equation ofmotion

m €x þ c1 _x þ kx¼−ðk0 þ k1 cos Ω2tÞðz0−xÞ3=2 þmg þ F cos Ω1t ð1Þ

where x denotes the effective displacement of the cantilever tip, mis the lumped cantilever mass, c1 (¼ c0 þ cn) is the effectivedamping constant, k is the free cantilever stiffness, k0 is theconstant given by the Hertz theory [20] which is associated withthe radius, AFM tip and substrate moduli and Poisson' s ratios, k1,Ω2 are the amplitude and the frequency of the contact stiffnessmodulation, respectively, z0 is the surface offset, g is the accelera-tion gravity, and F, Ω1 are, respectively, the amplitude and thefrequency of the excitation of the sample vibration, as consideredin atomic acoustic microscopy [21,22]. The displacement x isdefined by considering the static problem as x¼ xs þ X, and thequantity Δ¼ z0−xs as the static Hertz deformation, where xs is thestatic position and X is the displacement from the static position.From application view point, the contact stiffness modulation canbe introduced either by the excited cantilever using ultrasonictransducer attached to the cantilever and monitoring the ampli-tude of the z-piezo [23,24] or by monitoring the amplitude in the

transducer bounded underneath the sample as in the atomic forceacoustic microscopy [21,25].

Notice that the contact-mode AFM model under consideration,Eq. (1), takes into account only the Hertzian contact non-linearity.The non-linear attractive AFM mode opposing the behavior of theHertzian contact non-linearity is neglected, so that analyticalprediction is valid only for a limited amplitude and frequencyrange of the modulation.

Introducing the variable changes [12]: u¼ X=Δ, τ¼ ω0t, ω20 ¼

k=m, c¼ c1=mω0, β¼ 3k0Δ1=2=2k, β1 ¼ β=4, β2 ¼ β=24, f ¼ F=mω20Δ,

ω¼Ω1=ω0 and Ω¼Ω2=ω0, the dimensionless equation of motiontakes the form

€u þ c _u þ u−23 β þ 2

3βð1þ r cos ΩτÞð1−uÞ3=2 ¼ f cos ωτ ð2Þwhere _ðÞ ¼ d=dτ and r¼ k1=k0 is the dimensionless amplitude ofthe contact stiffness modulation given by the ratio between themodulated and the unmodulated contact stiffness coefficientswhich is assumed to be smaller than 1 ðk1ok0Þ or of order 1(k1≈k0). The coefficient β defines the stiffness of the unmodulatedcontact relative to the stiffness of the tip.

Expanding the non-linear restoring force in Taylor series in thevicinity of the static load and keeping only terms up to order threein u, Eq. (2) reads

€u þϖ2uþ c _u þ β1u2 þ β2u

3

þ rð23 β−βuþ β1u2 þ β2u

3Þ cos Ωτ¼ f cos ωτ ð3Þwhere ϖ ¼

ffiffiffiffiffiffiffiffiffi1−β

pis the natural frequency of the system. Eq. (3)

contains a slow dynamic due to the external excitation of thesample and a fast dynamic produced by the frequency of thecontact stiffness modulation Ω. Assume that the natural frequency,ϖ, may be in resonance with the external excitation, ω, but not inresonance with Ω (supposed larger than ϖ). Further, in order tokeep ϖ small comparing to Ω, values of β have to be chosen asclose as possible to 1 with the condition βo1 to be satisfied.Taking these remarks into consideration, the effect of the contactstiffness modulation on the slow dynamic can be investigatedusing the method of DPM [26,27]. This method consists inintroducing two different time scales, a fast time T0 ¼Ωτ and aslow time T1 ¼ τ, and splitting up uðτÞ into a slow part zðT1Þ and afast part ψðT0; T1Þ asuðτÞ ¼ zðT1Þ þ ψðT0; T1Þ ð4Þwhere z contains a slow dynamic which describes the mainmotions at time-scale of the tip natural vibrations and ψ standsfor an overlay of the fast motions at time scale of the parametricexcitation. Performing the method of DPM, we obtain the mainequation governing the slow dynamic of the motion

€z þ ω21z þ c _z þ ρ1z

2 þ ρ2z3 þ H ¼ f cos ωτ ð5Þ

where the parameters ω21, ρ1, ρ2 and H are given, respectively, by

ω21 ¼ϖ2 þ 2β2r2

3Ω2 −5β3r2

36Ω4 −β4r4

48Ω6 ð6Þ

ρ1 ¼β

4−β2r2

3Ω2 þ β3r2

12Ω4 þ7β4r4

192Ω6 ð7Þ

ρ2 ¼β

24−

β2r2

48Ω2 þβ3r2

36Ω4 −35β4r4

1152Ω6 ð8Þ

H¼ −β2r2

3Ω2 þ β3r2

18Ω4 þβ4r4

216Ω6 ð9Þ

showing how the natural frequency and the non-linear compo-nents of the slow dynamic are related to the contact stiffnessmodulation parameters r, Ω and coefficient β. Details on thederivation of Eq. (5) are given in Appendix.Fig. 1. A schematic of sdof model of a tip-sample AFM.

I. Kirrou, M. Belhaq / International Journal of Non-Linear Mechanics 55 (2013) 102–109 103


3. Frequency response analysis

In this section we shall analyze the amplitude–frequencyresponse of the slow dynamic (5) in the absence and presenceof contact stiffness modulation. We apply the MMS to obtainthe slow flow system and we examine the effect of varioussystem parameters on the frequency response near the primaryresonance.

3.1. Case without contact stiffness modulation

In order that the cubic non-linearity balances the effect ofdamping and forcing, we scale parameters in Eq. (5) as c¼ ϵ2c,ρ2 ¼ ϵ2ρ2 and f ¼ ϵ2f (the other parameters being of order ϵ) sothat they appear together in the modulation equations. Thus,Eq. (5) reads

€z þ ω21z¼−ϵðρ1z2 þ HÞ−ϵ2ðc _z þ ρ2z

3−f cos ωτÞ ð10ÞA two-scale expansion of the solution to Eq. (10) is sought in theform

zðT0; T1; T2Þ ¼ z0ðT0; T1; T2Þ þ ϵz1ðT0; T1; T2Þ þ ϵ2z2ðT0; T1; T2Þ þ 0ðϵ3Þð11Þ

where T0 ¼ τ, T1 ¼ ϵτ and T2 ¼ ϵ2τ. In terms of the variables Ti(i¼ 0;1;2), the time derivatives become d=dτ¼D0 þ ϵD1 þ ϵ2D2 þOðϵ3Þ and d2=dτ2 ¼D2

0 þ 2ϵD01 þ ϵ2D21 þ 2ϵ2D02 þ Oðϵ3Þ, where

Di ¼ ∂=∂Ti. Substituting (11) into (10) and equating the terms withthe same order of ϵ, yields

D20z0 þ ω2

1z0 ¼ 0 ð12Þ

D20z1 þ ω2

1z1 ¼−2D0D1z0−ρ1z20−H ð13Þ

D20z2 þ ω2

1z2 ¼−2D0D1z1−ðD21 þ 2D0D2Þz0

−cD0z0−2ρ1z0z1−ρ2z30 þ f cos ωτ ð14Þ

The solution of Eq. (12) can be written in the form

z0ðT0; T1Þ ¼ AðT1Þeiω1T0 þ cc ð15Þwhere AðT1Þ is a complex amplitude and cc stands for the complexconjugate of the preceding term. Also, the resonance conditionrequires that the frequency of excitation is assumed to remainnear the natural frequency according to

ω¼ ω1 þ ϵs ð16Þin which s is a detuning parameter representing the deviationfrom natural frequency. Substituting (15), (16) into (13), (14) andremoving secular terms, we obtain

ρnA2A þ ρlAþ ið−2ω1D2A−cω1AÞ þ

f2eisT2 ¼ 0 ð17Þ

where

ρn ¼10ρ213ω2

1

−3ρ2; ρl ¼2ρ1Hω21

ð18Þ

are, respectively, the effective non-linearity and the effectivelinearity induced by the contact stiffness vibration.

To better understand the dynamic of the oscillating tip, thevariation of these two quantities, ρn, ρl, as functions of theamplitude r will be examined below. Eq. (17) can be solved forthe complex amplitude by introducing its polar form as

A¼ 12ae

iθ ð19Þ

Hence, substituting (19) into (17) and separating real and imagin-ary parts, we obtain the modulation equations of amplitude and

phase as

dadt

¼ f2ω1

sin φ−c2a

dφdt

¼ f2ω1

cos φþ 5ρ2112ω3

1

−3ρ28ω2

1

!a3 þ sþ ρ1H

ω31

!a

8>>>><>>>>:

ð20Þ

in which φ¼ sT2−θ. Equilibria of this slow flow, correspondingto periodic solutions of Eq. (5), are determined by settingda=dt ¼ dφ=dt ¼ 0. This leads to the amplitude–frequency responseequation

AJ3 þ BJ2 þ CJ þ D¼ 0 ð21Þ

where A¼ ð34ρ2−ð5ρ21=6ω21ÞÞ2, B¼ 2cω1ð−2ω1s−ð2ρ1H=ω2

1ÞÞ, C ¼ðcω1Þ2 þ ð−2ω1s−ð2ρ1H=ω2

1ÞÞ2, D¼ −f 2 and J ¼ a2. In the rest ofthe paper we fix the parameter c¼0.02. Next, the effect ofexcitation amplitude, f, and contact stiffness, β, is analyzed. Fig. 2shows the variation of the amplitude–frequency response, as givenby Eq. (21), for different values of the amplitude f. The solid linesdenote stable solutions and the dashed lines denote unstable ones.Results obtained by direct numerical simulation of Eq. (5) (circles)using Runge–Kutta method are also plotted for validation. It can beseen from this figure an increase of the response amplitude andsoftening behavior when f is increased. In terms of the quantitiesdefined in the Hertzian contact case, values of r which are largerthan 1 mean loss of contact [12]. Fig. 3 depicts the variation of thefrequency response for different values of the contact stiffness β.One observes that an increase of β leads also to an increase insoftening behavior. This phenomenon (reported in [12] for a microcantilever) shows that the contact stiffness influences the non-linear characteristic of the system and then the analysis of the

−0.2 −0.1 0 0.1 0.20

0.2

0.4

0.6

0.8

1

σ

a

f=0.01

f=0.004

Fig. 2. Frequency response for r¼0, β¼ 0:8 and for different values of f. Analyticalprediction (solid lines for stable and dashed lines for unstable) and numericalsimulation (circles).

−0.2 −0.1 0 0.1 0.20

0.2

0.4

0.6

0.8

1

σ

a

β=0.8

β=0.7β=0.9

Fig. 3. Frequency response for r¼0, f¼0.008 and for different values of β.

I. Kirrou, M. Belhaq / International Journal of Non-Linear Mechanics 55 (2013) 102–109104


non-linear behavior of the system can provide important informa-tion on the tip-sample interaction.

3.2. Case with contact stiffness modulation

Now we consider the case where the contact stiffness ismodulated and we fix the parameter β¼ 0:8. Periodic solutionsof the slow dynamic, Eq. (5), correspond to the roots of Eq. (21).This Eq. (21) gives one or three real solutions depending on thesign of its discriminant Δ¼Q2 þ ð4P3=27Þ in which P ¼ ðC=AÞ−ðB2=3A2Þ and Q ¼ ð2B3=27A3Þ−ðBC=3A2Þ þ ðD=CÞ. The bifurcationcurves separating the existence domain of periodic solutions givenby the condition Δ¼ 0 are plotted in Figs. 4 and 5. Fig. 4a showsthe bifurcation boundaries in the parameter plane (f ;s) for twodifferent values of the modulation amplitude r (r¼0 for solid linesand r¼0.4 for dashed lines, respectively). In the regions betweenthe boundaries (Δ40), three solutions exist, two stable and oneunstable, while only one stable solution exists outside the bound-aries (Δo0). The stability analysis has been done using theJacobian of the slow flow system. It can be seen from this figurethat as the amplitude r is increased, the domain of bistabilitydecreases. Fig. 4b presents the bifurcation curves in the parameterplane (r; s) showing the zone inside the boundaries where threesolutions exist (two stable and one unstable). Outside the bound-aries only one stable solution exists. One observes that beyonda certain value of the amplitude r, the domain of bistabilitydisappears.

Fig. 5 shows the bifurcation curves in the parametric excitationplane (r;Ω) for s¼ 0 (Fig. 5a) and for s¼ −0:08 (Fig. 5b). Between

the boundaries curves of Fig. 5a, two stable and one unstablesolutions exist, while one stable solution lies outside. In Fig. 5b,the three solutions exist above the line and the stable solutionexists below it.

Next the effect of the amplitude, r, and frequency, Ω, of thecontact stiffness modulation is examined. Fig. 6 shows the effect ofΩ on the frequency response indicating that increasing Ω softensthe contact stiffness between the tip and the sample resulting in adecrease of the bistability domain.

In Fig. 7 we show the frequency response for various values of r.By inspecting this figure, one observes that increasing r from 0.4 to0.7, the amplitude response shifts toward higher frequencies whilechanging from softening to linear behavior. As the amplitude rcontinues to increase, the linear frequency response keeps shifting

Fig. 4. Bifurcation curves of periodic solutions of (5) in (a) the plane ðf ; sÞ and (b) the plane ðr;sÞ for f¼0.008 (in subfigure (b)) and Ω¼ 1.

Fig. 5. Bifurcation curves of periodic solutions in the plane ðr;ΩÞ for f¼0.008 and Ω¼ 1.

−0.1 −0.05 0 0.05 0.10

0.2

0.4

0.6

0.8

1

σ

a

Ω=1

Ω=2 Ω=1.5

Fig. 6. Frequency response for f¼0.008 and r¼0.5.



right until reaching a maximum position for a certain critical valueof r, and then shifts back toward lower frequencies (see the curvefor r¼1). This result reveals that increasing the amplitude of thecontact stiffness modulation r causes a substantial decreasing inthe resonance peak, eliminates the jumps and produces a changein the shift direction of the frequency response. This change in theshift direction results in the variation of the effective contactstiffness. Namely, one observes that for a given value of detuning sthe amplitude of the tip vibration may increase or decreaseproviding estimation of the local elastic properties of the surface.

To clarify this change phenomenon in the shift direction, weplot in Fig. 8 the detuning of the pick response as a function of theamplitude r. It can be clearly seen that as r increases from 0 to 1,the values of the pick response increase toward higher frequen-cies, reach a maximum for a certain critical value of r, and thendecrease toward lower frequencies (which is coherent with theresult shown in Fig. 7). The analytical prediction (solid line) andresults obtained by numerical simulations (circles) using Runge–Kutta method are compared showing a good match.

To better understand this interesting phenomenon of thechange in the frequency shift direction when increasing r, we plotin Fig. 9 the variation of the effective non-linearity, ρn, and theeffective linearity, ρl, given by (18), as functions of r. This figuredepicts two critical values for r; the first one, rl ¼ 0:665, corre-sponds to the condition

ρn þ ρl ¼ 0 ð22Þat which the frequency response meets a linear behavior, and thesecond one, rs ¼ 0:770, given by the condition

ρn ¼ 0 ð23Þcorresponding to the location where the frequency shift changesits direction. Fig. 9 reveals that increasing r from 0 to rl, thefrequency response shifts right while the softening characteristic

decreases until meeting a linear behavior at rl. Increasing r bet-ween rl and rs, the frequency response keeps shifting right (seeFig. 7 for r¼0.7) until reaching its maximum position at r¼ rs.By increasing r beyond rs, the frequency response undergoes ashift back to the left (see Fig. 7 for r¼1).

Fig. 10 shows in the parameter plane ðr;ΩÞ the boundary givenby the condition (22) separating the regions where the frequencyresponse is softening or has a linear behavior. The curves shown inthe small boxes inset Fig. 10 are obtained for values of r and Ω asgiven in the legend.

In Fig. 11 we show the curve given by the condition (23) in theplane ðr;ΩÞ. When r increases along the line labeled L1, the

−0.1 −0.05 0 0.05 0.1 0.150

0.2

0.4

0.6

0.8

1

σ

a

r=0.4

r=0.7

r =1

Fig. 7. Frequency response for f¼0.008, Ω¼ 1 and different values of r (pickedfrom Fig. 11 on the line L2 (dots)).

0 0.2 0.4 0.6 0.8 1−0.15

−0.1

−0.05

0

0.05

r

Det

unin

g (σ

) of t

he p

eak

resp

onse

Fig. 8. Detuning of the pick response versus r. Analytical approximation (solid line)and numerical simulation (circles) for f¼0.008 and Ω¼ 1.

0 0.2 0.4 0.6 1

−0.1

−0.05

0

0.05

0.1

0.15

r

ρ i

rl rs

ρn

ρl

Fig. 9. The variation of the effective non-linearity and the effective linearity asfunctions of r for f¼0.008 and Ω¼ 1.

Fig. 10. Curve separating softening and linear domains of the response in the planðr;ΩÞ for f¼0.008. In box (a) Ω¼ 0:6; r¼ 1, (b) Ω¼ 1; r¼ 1 and (c) Ω¼ 1:5; r¼ 0:6.

Fig. 11. Curve corresponding to the change in the frequency shift direction in theplan ðΩ; rÞ for f¼0.008.



frequency response shifts toward higher frequencies while theresonance amplitude decreases (as shown in Fig. 13 for values of rpicked on the line L1 (squares)). In this region no change in thefrequency shift direction is observed. When r increases on the linelabeled L2, the frequency response first shifts to the right untilreaching a maximum position for a value of r lying on the curve,and then shifts back to the left while increasing r above the curve(as shown in Fig. 7 for values of r picked on the line L2 (dots)).In other words, three regions can be distinguished in this figure.When increasing r, region I corresponds to softening behavior anda shift to the right, region II corresponds to linear response shiftingright, whereas in region III, the response has a linear behavior butshifting left.

To have a complete view of the precedent observations,curves of Figs. 10 and 11 are plotted together in Fig. 12 showingdifferent situations of the frequency response in different regions.The arrows above the small figures (in boxes) indicate thedirection of the shift. In Fig. 13 we plot the frequency responsefor Ω¼ 0:6 and for values of r picked on the line L1 in Fig. 11(squares) showing a shift toward higher frequencies accompaniedby a substantial reduction of the resonance amplitude.

Finally, Fig. 14 shows the change of the shift direction phenom-enon for values of r picked on the line L2 (Fig. 11) but for a differentvalue of β¼ 0:95.

3.3. Application to a real AFM example

The mathematical model studied in the previous sections iscompared with a real AFM example. Those comparisons are madeusing the parameters typical to those found in AFM [28]. The

elastic modulus and density for ⟨100⟩ silicon, E¼169 GPa andρ¼ 2330 kg=m3, respectively, were used. The cantilever has widtha¼ 51 μm, thickness b¼ 1:5 μm, length L¼ 262 μm, the lumpedmass m¼ 1:13� 10−11 kg and the free stiffness k¼0.404 N/m.Also, the following parameters, corresponding to a single crystalsilicon tip interacting with a chromium surface were used in thenumerical results: R¼20 nm, Δ¼ 0:26� 10−6 μm, Et ¼ 130 GPa,νt ¼ 0:181, Es ¼ 204 GPa and νs ¼ 0:26, where R is the tip radius,Et, Es are the elastic modulus of the tip and surface, respectively,and νt , νs are Poisson's ratio of the tip and surface, respectively.

For comparison, Fig. 15 shows in the amplitude and frequencymodulation parameter plane ðr;ΩÞ the analytical curve given by(22) (solid line) and the curve obtained from the real AFM example(dashed line). Inside the boundary the frequency response is linearand outside the boundary it has a softening behavior.

The chosen value of free stiffness k¼0.404 N/m is rather small interms of what we usually use for quantitative measurements [28].However, using the values given above provides the same value for thecantilever stiffness. Probably, it entails that soft cantilevers exhibitmuch easier non-linear contact resonances.

4. Summary

The effect of contact stiffness modulation on the frequencyresponse of a contact-mode AFM was studied. A lumped sdofsystem modeling the cantilever dynamics of contact-mode AFMwas considered and emphasis was placed on the case when theAFM is driven near the primary resonance. The contact force isproduced by the Hertzian contact regime and the external harmonicexcitation is induced by the sample vibration. The technique of DPM

Fig. 12. Curves of Figs. 10 and 11 as given by Eqs. (22) and (23).

−0.1 −0.05 0 0.05 0.1 0.15 0.20

0.2

0.4

0.6

0.8

1

σ

a

r =0.4

r =0.7

r =0.9

Fig. 13. Frequency response for f¼0.008, Ω¼ 0:6 and for different values of r pickedfrom Fig. 11 on the line L1 (squares).

−0.2 −0.1 0 0.1 0.20

0.2

0.4

0.6

0.8

1

σ

a r =0

r =0.7

r =1

r =0.4

Fig. 14. Frequency response for f¼0.008, Ω¼ 1 and β¼ 0:95.

Fig. 15. Curve separating softening and linear domains of the response in the planðr;ΩÞ. Analytical prediction (solid line, picked from Fig. 10) and result for real AFMexample (dashed line).



as well as the MMS was used to determine the non-linear frequencyresponse of the slow dynamic. It was shown that increasing theamplitude of the contact stiffness modulation, the frequencyresponse shifts toward higher-frequencies, becomes linear for acritical value of the amplitude and then shifts back toward lowerfrequencies while the peak of the amplitude decreases substantially.This reveals that by monitoring the amplitude of the contactstiffness modulation, the frequency response can be shifted towardhigher or lower values of the frequencies leading the amplitude ofvibration to increase and decrease for given operating frequencies.This change of the amplitude oscillation is correlated with the localelastic properties of the surface and thus with the imaging of thesample.

We conclude from this work that for small values of dampingand external forcing, modulation of the contact stiffness at largefrequency may increase or decrease the Hertzian contact forcedynamically as well as the contact area. As a result, the contact-mode AFM can be monitored near the resonance such that a goodperformance of the AFM operation can be achieved in term ofscanning or measuring local elastic proprieties of the specimen.

Appendix

In the averaging procedure, the fast part ψ and its derivativesare assumed to be 2π�periodic functions of fast time T0 with zeromean value with respect to this time, so that ⟨uðtÞ⟩¼ zðT1Þ where⟨⟩≡1=2π

R 2π0 ðÞdT0 defines time-averaging operator over one period

of the fast excitation with the slow time T1 fixed. IntroducingDji≡∂

j=∂jT i yields d=dt ¼ΩD0 þ D1, d2=dt2 ¼Ω2D20 þ 2ΩD0D1 þ D2

1and substituting Eq. (4) into Eq. (3) gives

€z þ €ψ þ cð_z þ _ψ Þ þϖ2ðz þ ψÞ

þβ1ðz þ ψÞ2 þ β2ðz þ ψÞ3 þ 32rβ cos Ωτ

−rβðzþ ψÞ cos Ωτ þ rβ1ðzþ ψÞ2 cos Ωτ

þrβ2ðzþ ψÞ3 cos Ωτ¼ f cos ωτ ð24ÞAveraging (24) leads to

€z þ c _z þϖ2zþ β1z2 þ β1⟨ψ

2⟩þ β2z3 þ 3β2z⟨ψ

2⟩þ β2⟨ψ3⟩þ r½−β⟨ψ⟩

þ2β1z⟨ψ⟩þ β1⟨ψ2⟩þ 3β2z

2⟨ψ⟩þ 3β2z⟨ψ2⟩

þβ2⟨ψ3⟩� cos Ωτ¼ f cos ωτ ð25Þ

Subtracting (25) from (24) yields

€ψ þ ψ þ 2β1zψ þ β1ψ2−β1⟨ψ2⟩þ 3β2z

2ψ þ 3β2zψ2−3β2z⟨ψ2⟩

þβ2ψ3−β2⟨ψ

3⟩

þr½−βψ þ β⟨ψ⟩þ 2β1zψ−2β1z⟨ψ⟩þ β1ψ2−β1⟨ψ

2⟩

þ3β2z2ψ−3β2z

2⟨ψ⟩þ 3β2zψ2−3β2z⟨ψ

2⟩

þβ2ψ3−β2⟨ψ3⟩� cos Ωτ¼ −r½32β−βz þ β1z

2 þ β2z3� cos Ωτ ð26Þ

Using the inertial approximation [26], i.e. all terms in the left-hand side of Eq. (26), except the first, are ignored, the fast dynamicψ is written as

ψ ¼ rΩ2 ð23β−βz þ β1z

2 þ β2z3Þ cos Ωτ ð27Þ

This simplification when solving Eq. (26) consists in finding ψ inthe form of a sum of a small number of harmonics of the fast timeT0 taking into account that ψ is small compared to z and it ispossible to consider only the linear dominant terms. For moredetails on this approximation, the reader can refer to [26,Chapter 2]. Inserting ψ from Eq. (27) into Eq. (25), using that⟨cos2 T0⟩¼ 1=2, and neglecting terms of orders greater than threein z, give the main equation governing the slow dynamic of themotion (5).

Fig. 16 shows a comparison between the full motion uðτÞ, (3),and the slow dynamic zðτÞ, (5), for the given parameters c¼0.05,f¼0.008, β¼ 0:8, Ω¼ 3, s¼ 0:01 and r¼0.2. The agreementbetween the full motion and the slow dynamic validates theaveraging procedure.

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2000 2004 2008 2012 2016 2020−0.4

−0.2

0

0.2

0.4

τ

z(τ)

u(τ)

Fig. 16. Comparison between the full motion u(t), ð3Þ, and the slow dynamic z(t), (5).



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