APPLIED PHYSICS LETTERS 93, 253106 共2008兲

Atomic force microscopy cantilever dynamics in liquid in the presenceof tip sample interaction
Sissi de Beer,a兲 Dirk van den Ende, and Frieder Mugele
Physics of Complex Fluids, Department of Science and Technology, University of Twente, P.O. Box 217,
7500 AE Enschede, The Netherlands

共Received 29 October 2008; accepted 27 November 2008; published online 23 December 2008兲
We analyze the dynamics of an atomic force microscopy 共AFM兲 cantilever oscillating in liquid atsubnanometer amplitude in the presence of tip-sample interaction. We present AFM measurementsof oscillatory solvation forces for octamethylcyclotetrasiloxane on highly oriented pyrolitic graphite
and compare them to a harmonic oscillator model that incorporates the effect of the finite driving
force for a typical AFM configuration with acoustic driving. In contrast to the general belief, we
find—in both experiments and modeling—that the tip-sample interaction gives rise to a pronounced
signature in the phase at driving frequencies well below resonance. © 2008 American Institute ofPhysics. 关DOI: 10.1063/1.3050532兴
Atomic force microscopy 共AFM兲 is more and more
evolving from a pure imaging technique to a tool for measuring quantitative tip-sample interaction forces, in particularalso in soft matter and biological systems. In this case ambient liquid damps the cantilever motion and reduces the quality factor. This poses a challenge, since established methods
for extracting tip-sample interaction forces from experimental data are based on low damping.1–3 It has been pointed outthat the motion of the base of acoustically driven AFM cantilevers 共which is negligible in air or vacuum兲 has to be taken
into account in ambient liquid.4–6 However, the consequences of this effect for the quantitative analysis of thecantilever dynamics in the presence of tip-sample interaction
forces have not been explored so far.
In this paper we present a harmonic oscillator model
which includes explicitly the finite amplitude of the base
movement and includes the effect of tip-sample interactions.
We compare the frequency-dependent amplitude and phase
response of the model to measurements of oscillatory solvation forces due to molecular layering.7 These forces weremeasured in octamethylcyclotetrasiloxane 共OMCTS兲, a nonpolar quasispherical model liquid, at various frequencies
close to and below resonance. For an AFM setup with acoustic driving and beam deflection detection, both the experiments and the analytical solution to the model display astrong phase response in the cantilever dynamics for low
frequencies.
The measurements were performed on a Veeco multimode 共with Nanoscope V controller and “A scanner”兲 using
rectangular gold coated cantilevers 共Mikromasch兲 with aspring constant of kc = 3 N / m and a resonance frequency off ⬇ 90 kHz in air. Prior to the measurements the cantilevers
were cleaned in a plasma cleaner for 30 min and after themeasurements the tip was characterized using high resolution
scanning electron microscopy imaging 共Rtip = 50 nm兲.
Acoustic driving was realized using an adapted cantilever
holder.8 The spring constant was determined in air using thethermal calibration method.9 The resonance frequency 共f 0
= 43.1 kHz兲 and quality factor 共Q = 3.1兲 in liquid were determined with the same method, 100 nm above the sample surface. The OMCTS used in the measurements was dried usinga兲Electronic mail: s.j.a.debeer@utwente.nl.

0003-6951/2008/93共25兲/253106/3/$23.00

4 Å molecular sieves. The highly oriented pyrolitic graphite
共HOPG兲 was freshly cleaved just before depositing theOMCTS on the surface.
Figure 1 shows the measured amplitude and phase distance curves for three different drive frequencies close to andwell below the cantilever resonance 共␻0 = 冑kc / m, with m being the effective mass兲 upon approaching the HOPG surface.
Far away from the surface the amplitude and phase responseare constant. 关The absolute value of the phase far away fromthe surface is shifted such in accordance with Fig. 3共b兲; seebelow.兴 At a distance of 5–6 nm the response changes due tothe tip-sample interaction. For a driving frequency ␻ close toresonance 共top panel兲 both amplitude and phase display clear
modulations due to the oscillating tip-sample interaction. The
periodicity of these oscillations 共⬇0.76 nm兲 reflects the molecular size of OMCTS 共0.9 nm兲, as reported before byothers.10–13 The curves shown in the middle panel were measured at an intermediate drive frequency on the wing of theresonance peak. Compared to the top panel, the oscillations
are more pronounced in the amplitude and less pronounced
in the phase. All these observations are expected from thestandard harmonic oscillator model: for ␻ ⬇ ␻0, the amplitude is close to its maximum and hence not very sensitive tosmall shifts of the resonance curve 共as induced by the tipsample interaction兲, whereas the variation in the phase ismaximum.14 If ␻ is chosen on the wing of the resonancepeak, the sensitivity of the amplitude becomes maximized
whereas the phase sensitivity continuously decreases 关seealso the thin gray lines in Figs. 3共a兲 and 3共b兲兴. However, for␻ Ⰶ ␻0共bottom panel兲 we find substantial deviations from thestandard picture: first, we observe an overall increase in theamplitude upon approaching the surface. Second, the periodicity in the amplitude oscillations doubles at separations below 2 nm. Third and most strikingly, the oscillations in thephase become more pronounced again, even more pronounced than at ␻ ⬇ ␻0. These observations have importantconsequences for the quantitative interpretation ofamplitude-force-distance curves. Figure 2共a兲 shows the schematic representation of the cantilever and its motion, which
we treat within the harmonic oscillator approximation. As
will become clear in the following, our observations arecaused by a combination of two—in principle well known—effects. 共i兲 In a highly damping environment 共i.e., for low Q兲

93, 253106-1© 2008 American Institute of Physics

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253106-2

Appl. Phys. Lett. 93, 253106 共2008兲

de Beer, van den Ende, and Mugele
0300

ϕ [°]

A [pm]

200-90100-180

300

0FIG. 1. 共Color兲 Amplitude and phaseof the AFM cantilever vs separation
between the solid HOPG surface andthe cantilever tip for different drive
frequencies 关top/yellow: 41 kHz
共␻ / ␻0 = 0.95兲, middle/green: 32 kHz
共␻ / ␻0 = 0.75兲, and bottom/red: 6 kHz
共␻ / ␻0 = 0.15兲兴 relative to resonance
共43 kHz兲. The periodicity of the oscillations reflects the size of the molecules 共OMCTS兲.

ϕ [°]A [pm]

200-90100-1803000

ϕ [°]A [pm]

200-90100-1800024680tip sample distance [nm]24

the dynamics of an acoustically driven cantilever can only beunderstood by properly including the base motion zd. 共This isin contrast to magnetically driven cantilevers, where themeasured motion is the only motion.10,13兲 共ii兲 Beam deflection systems measure the deflection x of the cantilever withrespect to the position zd of the base and not with respect tothe average position zc. 共This implies immediately that theamplitude measured via beam deflection goes to zero for ␻→ 0, in contrast, e.g., to an interferometric detection
system;11 see also Fig. 3.兲 Including the motion of the canti(a)zdz = zd + x
d8zc

lever base zd results in the following equation of motion forthe cantilever:mz̈ + ␥cż + kcz = kczd + Fts ,-Im

共1兲

where ␥c is the damping of the cantilever, kc is the spring
constant, and m is the effective mass of the cantilever including the added mass caused by the motion of the surrounding
liquid 共␻0 = 冑kc / m兲. Far away from the surface 共d ⱖ 6 nm, inthe present experiments兲 Fts is zero. For smaller d, Fts isfinite and changes the resonance behavior of the system. For
sufficiently small cantilever amplitude, Fts can be linearized
to Fts共z , ż兲 = Fts共zc , 0兲 − kint共zc兲z + −␥int共zc兲ż. Using the ansatz
that z is described by z = x + zd = Aei共␻t+␸兲 + Adei␻t, where A and␸ are the amplitude and phase measured in the experiments,Ad is the amplitude of the driving mechanism, and ␻ is thedrive frequency, Eq. 共1兲 can be solved for A and ␸:A=(b)6tip sample distance [nm]Ad冑共kc − kt + m␻2兲2 + 共␻␥t兲2

冑共kt − m␻2兲2 + 共␻␥t兲2⬇

兩kint兩
Adkt

共2a兲andkint<0kint>0zzdx

ϕReFIG. 2. 共Color兲 共a兲 Scheme of the cantilever dynamics, which can be accurately described by including the base motion. In 共b兲 the difference between
the drive signal zd 共blue arrow兲, the motion of the cantilever z 共green arrow兲,and the measured deflection x 共red arrow兲 is drawn. The solid lines show theresponse for a positive interaction stiffness and the dashed lines for a negative stiffness. The measured amplitude is the length of the vector x and thephase is the angle between the Re axis and the vector x.

tan ␸ =⬇

− k c␻ ␥ tkc共− m␻ + kt兲 − 共− m␻2 + kt兲2 − 共␻␥t兲22␻␥t,
kint共1 + kint/kc兲

共2b兲

where the approximations hold for ␻ Ⰶ ␻0. kt = kc + kint and␥t = ␥c + ␥int are the total stiffness and damping, respectively.
Figures 3共a兲 and 3共b兲 show the calculated amplitude andphase spectra for the cantilever used in the experiments. In
the absence of tip-sample interaction, the curves are similar

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253106-3

Appl. Phys. Lett. 93, 253106 共2008兲

de Beer, van den Ende, and Mugele

FIG. 3. 共Color兲 Amplitude and phase vs frequency. 关共a兲 and 共b兲兴 Thick lines:
calculated model curves following Eqs. 共2a兲 and 共2b兲 for variable interaction
stiffness kint = 0, ⫾0.1kc 共at ␥int = 0兲. Thin dotted lines: standard harmonic
oscillator model. Colored arrows indicate the drive frequencies for the datain Fig. 1. 共c兲 Measured amplitude and phase response as well as thermal
response vs frequency. 共kc = 3.7 N / m and the resonance frequency is f= 49 kHz兲. The spurious peaks above resonance are due to the response of
the piezodrive.

We thank A. Maali for his assistance during the earlystages of the experiment and we thank K. Smit and P.
Markus 共Veeco兲 for the technical support. This work has
been supported by the Foundation for Fundamental Researchon Matter 共FOM兲, which is financially supported by theNetherlands Organization for Scientific Research 共NWO兲.

to those calculated by others, displaying, in particular, a decrease in the amplitude to zero5,6 and a reduction in the phaseto −90° for ␻ → 0.4 It is particularly interesting to analyze
the behavior of the curves in the presence of a finite tipsample interaction, as shown here for two examples withpositive and negative interaction stiffnesses of +0.1kc and−0.1kc, respectively. In line with the asymptotic expressions
in Eq. 共2兲, the phase becomes increasingly sensitive to variations in kint for ␻ → 0. This explains the experimental behavior of the phase shown in Fig. 1: for the oscillatory tipsample interaction due to the confined OMCTS, theinteraction stiffness varies between positive and negativevalues and thereby gives rise to dramatic oscillations of thephase. The physical origin of this behavior becomes clear
from Fig. 2共b兲, where we indicate the position of the cantilever base zd and the tip z in the complex plane. As explained
above, the quantity measured by beam deflection in an

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E. T. Herruzo and R. Garcia, Appl. Phys. Lett. 91, 143113 共2007兲.
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共2007兲.
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1.0

A/Amaxdrive frequency [kHz]acoustically driven AFM is the cantilever deflection, i.e., thedifference vector x = z − zd. For low frequencies, the tip displacement z 关which is correctly described by the standard
harmonic oscillator; dotted lines in Figs. 3共a兲 and 3共b兲兴 displays very little variation in the phase but a finite amplitude
variation 共as a function of Fts兲. As a consequence, the phaseof the difference vector varies a lot, as found in the experiments. Figure 2共b兲 also shows why the periodicity in the low
frequency amplitude response doubles in Fig. 1: when theinteraction stiffness varies back and forth between a positive
and a negative value, z moves along the trajectory in thecomplex plane that is indicated by the dotted line. The measured amplitude of the difference vector x, however, displays
twice as many maxima and minima in agreement with theasymptotic expression in Eq. 共2兲. Finally, Fig. 3共c兲 shows anexperimental frequency response curve measured far awayfrom the surface together with a thermal noise spectrum.
While the response curves display several spurious resonances at ␻ ⬎ ␻0 共which are related to the usual resonancesin the driving piezo8兲, the low frequency behavior corresponds nicely to the model curves shown in Figs. 3共a兲 and3共b兲. In particular, the phase displays the marked decreasethat the model predicts for ␻ → 0.
In summary we have shown that the combination ofbeam deflection detection and acoustic driving gives rise to avery strong sensitivity of the cantilever’s phase to the tipsample interaction for low driving frequencies. The effects
described here are relevant for any experiment attempting tomeasure quantitative tip-sample interaction forces, including
nonoscillatory ones, in low Q environments with an AFM
that makes use of this 共by far most widely spread兲 design.
The consequences of the present observations for quantitative force inversion procedures will be reported shortly in a
separate communication.15

(a)

0.5

0.00.0(b)

0.5

1.0

1.5

2.0

ω / ω00magnetic
kint = 0
ϕ [°]kint > 0kint < 0-90-180
0.0(c)

0.5

1.0

1.5

2.0

ω / ω01.5
0
1.0

A/Amax-240

ϕ [°]

-120
brownian
active acoustic
phase

0.5
-360
0.0
0

20

40

60

-480
80

12

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