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Dissipation and oscillatory solvation forces in confined liquids studied by small-amplitude
atomic force spectroscopy

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IOP PUBLISHINGNANOTECHNOLOGY

Nanotechnology 21 (2010) 325703 (11pp)

doi:10.1088/0957-4484/21/32/325703

Dissipation and oscillatory solvation forcesin confined liquids studied bysmall-amplitude atomic force spectroscopy
Sissi de Beer, Dirk van den Ende and Frieder Mugele
Physics of Complex Fluids, Department of Science and Technology, University of Twente,
PO Box 217, 7500 AE Enschede, Netherlands
E-mail: s.j.a.debeer@utwente.nl

Received 11 March 2010, in final form 6 June 2010
Published 19 July 2010
Online at stacks.iop.org/Nano/21/325703
Abstract
We determine conservative and dissipative tip–sample interaction forces from the amplitude andphase response of acoustically driven atomic force microscope (AFM) cantilevers using a
non-polar model fluid (octamethylcyclotetrasiloxane, which displays strong molecular layering)
and atomically flat surfaces of highly ordered pyrolytic graphite. Taking into account the basemotion and the frequency-dependent added mass and hydrodynamic damping on the AFM
cantilever, we develop a reliable force inversion procedure that allows for extracting tip–sample
interaction forces for a wide range of drive frequencies. We systematically eliminate the effectof finite drive amplitudes. Dissipative tip–sample forces are consistent with the bulk viscosity
down to a thickness of 2–3 nm. Dissipation measurements far below resonance, which we argue
to be the most reliable, indicate the presence of peaks in the damping, corresponding to an
enhanced ‘effective’ viscosity, upon expelling the last and second-last molecular layer.
S Online supplementary data available from stacks.iop.org/Nano/21/325703/mmedia(Some figures in this article are in colour only in the electronic version)layers [3, 4]. In contrast, other studies find essentially bulk-likeviscous dissipation, except for the molecular layers directly
adjacent to the solid, which behave more rigidly [5, 6].
According to molecular simulations both solidification andliquid-like behaviour can be achieved, depending on theinteraction potentials between the solid and the liquid. Whilemost simulations focused on equilibrium properties and theresulting conservative forces, Gao and Landman [7] also
reported that the diffusivity in confined liquids is maximum
for surface separations corresponding to a non-integer numberof molecular layers corresponding to states with a reduced
density. The fluctuation–dissipation theorem—if applicableto these systems—then suggests periodic variations in the(viscous) dissipation for strongly layered liquids.
Do such periodic variations exist? A number of recent
atomic force microscopy (AFM) experiments addressed thisquestion using various measurement techniques, including
conventional amplitude modulation (AM) AFM and frequency
modulation (FM) AFM and different driving schemes(acoustic/base drive versus magnetic drive). While studies

1. Introduction
The properties of liquids confined between solid surfaces are
crucial for understanding numerous technological problems
including lubrication and nanotribology, porous media andnanofluidic devices for biotechnological applications [1].
The key question is to what extent macroscopic continuum
physics—in particular hydrodynamics—can be applied todescribe the systems at such small scales and, if not,how deviations from the macroscopic behaviour manifest
themselves.
It is by now well established that the structure of the liquid
is altered close to the solid–liquid interface in such a waythat the liquid molecules assume a layer structure parallel tothe interface. This gives rise to oscillations in the averagedensity and to (conservative) oscillatory solvation forces upon
confining the liquid between two solid surfaces [2]. In
addition to this structuring, some studies also report a strongly
increased dissipation and, in some cases, even solid-like
friction for liquid films with a thickness of several molecular
0957-4484/10/325703+11$30.00

1© 2010 IOP Publishing Ltd Printed in the UK & the USA

Nanotechnology 21 (2010) 325703

S de Beer et al

diameter of 0.7–0.8 nm, which is known to display pronounced
layering when confined between two solid surfaces [2–11, 15].
To minimize the detrimental effect of residual water [16] wedried the liquid using 4 Å molecular sieves (Sigma Aldrich)for several days prior to the measurement. To test whether
water contamination influenced our results, we repeated themeasurements in a closed chamber with a dry nitrogen
atmosphere, which gave the same results (within experimental
error). As a substrate hydrophobic highly ordered pyrolytic
graphite (HOPG, Mikromasch grade ZYA) was used, which
was cleaved using adhesive tape just prior to the depositionof the liquid (OMCTS). Before each spectroscopy experiment,
the freshly cleaved HOPG substrate was imaged in the liquid
to ensure that the surface was clean and atomically smooth andthat the system was stable.

using the acoustic scheme [8, 9] reported oscillations in
the dissipation, a continuous increase by several ordersof magnitude was reported in the magnetically driven
systems [10, 11]. (Very recently a sharply peaked and periodic
dissipation was also reported in a very careful experimentusing the magnetic scheme [12].) Thus, even for simple(Lennard-Jones-like) model systems, such as the most widelyused octamethylcyclotetrasiloxane (OMCTS), the question
how confinement affects dissipation has yet to be answered.
From a technical perspective, the acoustic driving schemein AFM is frequently criticized for its known sensitivity toexperimental and modelling errors (see [13] in the present
context). Yet, AFM cantilevers driven at sufficiently small
amplitudes are known to behave as harmonic oscillators andare therefore in principle straightforward to model. This raises
the important—given the dominance of acoustically driven
AFMs—technical question under which conditions and to what
extent reliable interaction forces can be extracted from AMAFM measurements. Despite the fact that the relevance oftaking into account the motion of the cantilever base has been
known for a long time [14], it was noticed only recently [15]that the acoustic driving scheme gives rise to a particularly
strong phase response at low driving frequencies, which offers
the potential of an increased sensitivity.
In this paper we elaborate on our work which waspresented earlier in [15]. While in [15] we presented a model
to describe the cantilever dynamics, we now address both thetechnical question of obtaining reliable tip–sample interaction
forces from small-amplitude (AM)-AFM measurements andthe physical question regarding the effect of confinement on
the stiffness and dissipation in confined layers of OMCTS. To
do so, we measure and model the response curve of the AFM
cantilever as a function of the drive frequency incorporating (inaddition to the motion of the cantilever base which was alreadydiscussed in [15]) the frequency-dependent added mass anddissipation around the cantilever. Subsequently, we measure
the amplitude–phase-distance (APD) curves over a wide
range of driving frequencies and extract the corresponding
conservative and dissipative tip–sample interaction forces.
Eliminating the effect of finite drive amplitudes, we obtainconsistent conservative tip–sample interactions over the entire
range of drive frequencies. The corresponding dissipative
tip–sample forces display peaks at tip–surface distances
corresponding to the expulsion of the last two molecularlayers. The physical origin of (apparent) superimposed
oscillations observed at certain drive frequencies is discussed.
analysing the relative strength of various contributions to the
cantilever damping. We identify an upper limit for the possible
confinement-induced enhancement of the effective viscosity.

2.2. AFM measurements
The measurements were performed on a commercial Veeco
Multimode with Nanoscope V controller equipped with a lownoise head using deflection detection and a stable small piezo
scanner (‘Veeco A scanner’). We used rectangular cantilevers(Mikromasch NSC36, manufactured of bare silicon or goldcoated silicon) with various stiffnesses of kc = 3–7 N m−1
and resonance frequencies of f ≈ 80–120 kHz (in air), as
determined using the thermal calibration method [17]1 . The
resonance frequency ( f ≈ 30–60 kHz) and quality factor Q(∼3) in liquid were determined with the same method at areference distance of 100 nm above the sample surface (witha relative uncertainty of <5%). The specific cantilevers werechosen for their long tip height (h ∼ 25 μm) in order tominimize the change in hydrodynamic squeeze-out damping
between the cantilever and the solid surface. We verified
that possible variations of the added mass and hydrodynamic
damping around the cantilever were below the detection limit
between the reference distance and 10 nm above the surface.
Prior to the measurements the cantilevers were treated ina plasma cleaner (Harrick Plasma) for 1 min. After themeasurements the tip was imaged using high resolution SEM(HR-SEM Zeiss LEO 1550) yielding tip radii of Rtip = 15–30 nm. The cantilever oscillation was driven at the baseusing a modified tapping holder (Veeco MMMC), as describedin [18]. This yields stable drive amplitudes for several hours
and reduces the spurious resonance peaks that are typical ofcommercial liquid cells.
APD curves including the average (tapping mode,TM) deflection signal were recorded over 10 nm tip–
surface distances for variable free oscillation amplitudes anddrive frequencies from 0.07 to 0.5 nm at large distance(corresponding to peak-to-peak amplitudes of 0.14–1 nm)
and 5–60 kHz, respectively. For each setting a minimumof 25 measurement curves were recorded and analysed.
The deflection signal in contact with the substrate wasused to calibrate the deflection signal, which is appropriate
for both static and dynamic deflection since we use smallcantilevers compared the size of the laser spot [19] (see also

2. Materials and methods
2.1. Materials
Amplitude–phase-distance (APD) curves were measured
in octamethylcyclotetrasiloxane (OMCTS purum, 99.0%,
Sigma Aldrich), a non-polar liquid of slightly spheroid
molecules with a major diameter of 1.0–1.1 nm and a minor

1 We used the ‘Thermal Tune’ in the Nanoscope 7.20 software, which takes
all the relevant corrections (e.g. temperature, cantilever shape) into account.

2

Nanotechnology 21 (2010) 325703

S de Beer et al

figure 1 supplementary data available at stacks.iop.org/Nano/
21/325703/mmedia). Thanks to the small drive amplitudes, nohigher harmonics were generated within the (Brownian) noise
limit of our system, as verified using a spectrum analyser.
Typically, all APD curves in a set of measurements
displayed several oscillations in the amplitude and phase
response (see also section 4.1) over a period of several hours.
Cantilevers that did not display this stability (e.g. due to a bad
tip or contamination) were discarded. Test measurements did
not display any significant dependence of the results on theapproach speed within a range of 1–10 nm s−1 . Therefore aconvenient standard approach speed between 1 and 2 nm s−1
was chosen for all the experimental data presented here. The
drift normal to the surface was typically 0.08 ± 0.03 nm s−1 ,as determined from the change in z piezo-voltage at tip–
surface contact. All measurements were performed at a room
temperature of 22 ± 1 ◦ C. The temperature next to the sample
was measured to be 27 ◦ C, presumably due to local heatingfrom the electronics in the AFM head.

(a)

(b)

base driving

zb+d

zb

kc
cantilever tip
kint

γc

γint γtot
solid surface

Figure 1. (a) Illustration of the cantilever, where the total motion z ofthe cantilever consists of the measured deflection d plus the basemotion z b due to the acoustic driving mechanism. (b) Spring–dashpot
representation of the harmonic oscillator model for acoustic drivingwith deflection detection and linearized tip–sample forces.

cantilever. The usual deflection detection scheme measures thedeflection d(t) of the cantilever, which is related to the vertical
tip position by z(t) = d(t) + z b (t) (see also figure 1). Since
the quality factor Q is low in a liquid environment, both d(t)and z b (t) can be comparable (see, e.g., [14]). In particular for
driving frequencies off resonance the absolute motion of the tip
with respect to the sample surface can be significantly larger
than the measured deflection amplitude, which has importantconsequences for the amplitude and phase response of thecantilever (see also [15, 27]).
Since we use for our measurements very small amplitudes(much smaller than the characteristic length scale of thechanges in the interactions), we can linearize the tip–sample
force Fts around the (quasi-statically moved) average cantilever
position z c . A Taylor expansion of Fts yields

2.3. Numerical calculations
We simulated the dynamics of the AFM cantilever numericallyusing the approach of Garcia and Perez [20]. The programsfor the simulations were written and run in Matlab based on
the code used in [21]. The ordinary differential equation
for the cantilever dynamics including the tip–sample forces(see below) was solved using a fourth-order Runge–Kutta
scheme. Numerical APD curves for various drive amplitudes
were calculated starting 10 nm above the sample surface down
to 1 nm in 250 discrete steps. At each step, the steadystate amplitude and phase of the oscillation were extracted
from the oscillatory cantilever motion via a fast Fouriertransform. For fitting and validation purposes, the resulting
numerical amplitude and phase versus distance curves wereconverted into a distance-dependent stiffness using the sameforce inversion formulae (to be described in section 3) as forthe experimental data.

 2
Fts (z c + z, ż) = Fts (z c , 0) − kint z − γint ż − 12 kintz− γint z ż + H.O.T.

(2)
d Fts
|
dz z cwhere H.O.T. are the higher-order terms, kint =is the interaction stiffness, γint is the interaction damping(see figure 1) and Fts (z c , 0) is the equilibrium force on the
cantilever, which shows up in the average deflection. For small
static forces and small drive amplitudes, only the linear terms
need to be considered, Fts (z c + z, ż) = −kint (z c )z − γint (z c )ż ,leading to a simplified version of equation (1):

3. Cantilever dynamics and force inversion

m z̈ + γtot ż + ktot z = kc z bOver more than a decade many methods have been developed
to quantitatively map both the conservative and dissipativeinteraction forces in dynamic AFM based on different
methods and techniques, varying from small- [14] to largeamplitude [22] AFM as well as universal methods for both
amplitude modulation (AM) [23–25] as well as frequency
modulation (FM) AFM [26]. In this paper we used AM smallamplitude spectroscopy.
For typical base-driven atomic force microscopes thevertical motion z(t) of the cantilever tip can be modelled asa harmonic oscillator affected by the tip–sample interactions
Fts :m z̈ + γc ż + kc z = kc z b + Fts .
(1)

(3)

where ktot = kc + kint and γtot = γc + γint are the total stiffnessand damping, respectively.
We solve equation (3) using the ansatz z(t) =Atot exp(i(ωt + ϕtot )) = d(t) + z b (t) = A exp(i(ωt + ϕ)) +Ab exp(iωt), in which ω is the drive frequency, A and ϕ are
the measured deflection amplitude and phase of the deflection
and Ab is the amplitude of the base motion [15]:Ab (kc − ktot + mω2 )2 + (ωγtot )2

A=
(4a )
(ktot − mω2 )2 + (ωγtot )2and−kc ωγtot.
kc (−mω2 + ktot ) − (−mω2 + ktot )2 − (ωγtot )2(4b )As noted earlier, 
the total amplitude Atot of the tip motion,
given by Atot =(A sin ϕ)2 + (A cos ϕ + Ab )2 , can betan ϕ =

Here, kc is the cantilever’s spring constant, z b (t) describes thedisplacement of the cantilever base, m is the total effective
mass (including the added mass caused by the motion of thesurrounding liquid) and γc is the viscous damping around the3Nanotechnology 21 (2010) 325703

S de Beer et al

substantially larger than the measured deflection amplitude Ain equation (4a ).
In order to extract the physical interaction forces we needto solve these expressions for the interaction stiffness kint anddamping γint , yielding
(5a )

and

γint =−kc Ab A sin ϕ− γc
ω(A2b + A2 + 2 Ab A cos ϕ)(5b )A / Amax

kc Ab (Ab + A cos ϕ)
A2b + A2 + 2 Ab A cos ϕkint = −kc + mω2 +1.0

where Ab is calculated from the measured free amplitude A
far away from the surface (at 10 nm) using equation (4a )with the interactions set to zero2 . Note that these physicalparameters kint and γint are independent of the specificmeasurement technique and should look identical as equivalentdata extracted from, for example, frequency modulation AFM
measurements. Close to resonance, these formulae can beapplied using frequency-independent calibration constants (kc ,m and γc ) as determined from calibration measurements of theresonance frequency and the Q factor far away from the surface(at 100 nm). For drive frequencies varying over a substantial
range, however, one needs to take into account the frequency
dependence of the motion of the fluid around the oscillating
cantilever [29], which can be achieved by replacing m and γcby their frequency-dependent counterparts without changing
the structure of equations (4) and (5). Following Sader [30],
we describe the hydrodynamic loading on the cantilever by
a hydrodynamic function  =   + i  , which yields an
added mass and a damping given by m added = ρ π4 w2 L  and
γc = ρ π4 w2 Lω   [31]. Here, ρ , w and L denote resp. thedensity of liquid, and the width and the length of the cantilever.


0.5

0.0

0

1

2

ω / ω0ϕ[deg]

0

-90

-180

0

1

2

ω / ω0Figure 2. Amplitude and phase response of the cantilever versus
normalized drive frequency for ambient liquid ( Q = 3). Blue dashedlines: conventional harmonic oscillator (H.O.). Red solid lines:
deflection signal including base drive (constant damping and addedmass). Green dashed–dotted lines: deflection signal including basedrive with frequency-dependent damping and added mass. Blacklines: experimental response (Au-coated Si cantilever,ωres /2π = 38 kHz, the spurious peaks above ω/ω0 = 0.95 are due tothe piezo response). Grey line: thermal noise spectrum (Si cantilever,ωres /2π = 42 kHz). Coloured arrows indicate the measurement
frequencies of the data shown in figure 3.

2η
 depends on the viscous penetration depth δ =ρω (η :
viscosity of the liquid) and the cantilever geometry as   =a1 + a2 wδ and   = b1 wδ + b2 ( wδ )2 with a1 = 1.0533,
a2 = 3.7997, b1 = 3.8018 and b2 = 2.736. Using thecalibration constants (kc , Q air , Q liq , ωres,air and ωres,OMCTS )applied at resonance allows for calculating the frequencyindependent prefactor in both the frequency-dependent added
mass and damping.
The relevance of this frequency-dependent correction
becomes apparent from the amplitude and phase response of
the cantilever far away from the surface. Figure 2 shows
a comparison between the experimental frequency response(thick black solid lines) and various implementations of themodel. Above the resonance frequency, the experimental
data are distorted by various spikes, which are typical for
AFMs with acoustic driving. To avoid interference withthese features, all measurements described in this paper are
carried out at drive frequencies below the lowest spike. We
therefore also limit the comparison of the model curves tothe frequency range below resonance. The blue dashed lines
show the response curve of a simple harmonic oscillator(ignoring the base motion), which is shown for reference
and which is obviously not suitable in the present case of alow Q environment [14, 15, 27]. The red thin solid curves
represent the frequency response according to equations (4)(i.e. including the base motion) for constant m and γc . As
described in [15], this correction captures the most importantfeatures of the experimental curves, namely the vanishing
amplitude and the non-zero phase at low frequencies. Yet,the phase decreases much more quickly towards −90◦ thanobserved experimentally, which was left unnoticed previously.
This deviation leads to a substantial phase error if the model isused to extract tip–sample interaction forces from measured
APD curves at low frequencies. In contrast, the green
dashed–dotted model curve, which incorporates the frequency
dependence of m and γc , produces a much more satisfying
agreement also at the lowest frequencies and thus reproduces
the global shape of both amplitude and phase response overthe entire frequency range—notwithstanding some residual
deviations.

2 Somewhat differently looking, but equivalent, force inversion formulae were
derived by Jai et al [28].

4Nanotechnology 21 (2010) 325703(a)

S de Beer et alω / ω = 0.95
0

ω / ω = 0.66

(b)

0

(c)

ω / ω = 0.16

0

2

0

A [nm]

0.10

0.05

ϕ[deg]

0.00
0

-90

-180 0

2

4

6 0

2

4

6

4

6

tip surface distance [nm]Figure 3. Amplitude A and phase ϕ versus distance curves upon approaching the surface (total free amplitude (a) 0.117 nm, (b) 0.129 nm and(c) 0.320 nm; resonance frequency ωres /2π = 38 kHz, Au-coated Si cantilever) for three different driving frequencies: (a) ω/ω0 = 0.95,(b) ω/ω0 = 0.66 and (c) ω/ω0 = 0.16. The different curves in (b) indicate the reproducibility of the measurements (a smoothing filter wasapplied to eliminate high frequency noise).

holds for the phase response. The strong phase response at low
frequencies is caused by the base driving scheme. As discussed
in [15], the phase response (red thin solid lines) shows agradual reduction in the phase to −90◦ for ω → 0. In this limit,
ωγt,equation (4b ) can be approximated by tan ϕω→0 ≈ kint (1+k
int /kc )which is very sensitive to changes in the interaction stiffness
kint . In particular, ϕ is expected to vary between zero and 180◦
for ω → 0 as kint changes sign as a function of distance, as is
the case for oscillatory solvation forces.
The data in figure 3 represent typical APD curves selectedfrom a large dataset. The overall behaviour of the curves is very
reproducible, as evidenced by the selection of amplitude curves
shown in figure 3(b). Yet, details such as the number of visible
oscillations cycles can vary from curve to curve. Figure 4
shows the distribution of the number of oscillations observedin a number of consecutive APD curves at several locations
on the sample surface under otherwise identical conditions.
Regarding the fact that the tip–sample interaction is mediatedby merely a few hundred molecules under the tip, we attributethe variations in the number of oscillations to the stochastic
nature of the molecular motion. This distribution does notaffect the validity of the qualitative trends as a function of
frequency shown in figure 3.

4. Results
4.1. Measurements
Figure 3 shows APD curves measured with the same cantileverat three different frequencies close to and below resonance.
We chose these frequencies with special care. At thesefrequencies the calculated cantilever response matches themeasured response very well and no spurious peaks wereobserved in the spectra (figure 2). For all driving frequencies,
both the amplitude and the phase display oscillations with aperiodicity varying from 0.7 to 0.9 nm from curve to curvereflecting the diameter of the OMCTS molecules. Close toresonance (figure 3(a), ω/ω0 = 0.95), the oscillations areclearly visible in both the amplitude and the phase signal. The
oscillations in the phase are approximately symmetric around−90◦ . The oscillations in the amplitude are superimposedon an overall decrease of the amplitude at small distances.
On the wing of the resonance peak (figure 3(b), ω/ω0 =0.66), the oscillations in the amplitude are stronger and more
peaked (the different curves indicate the reproducibility), while
they almost disappear in the phase response. At the lowest
frequency (figure 3(c), ω/ω0 = 0.16), a strongly oscillatory
response appears again in the phase, whereas the oscillations in
the amplitude become less pronounced and are superimposedon an overall increase towards small distances. The same
trends as a function of frequency are observed for all freeamplitudes. For the highest free amplitudes, the amplitude andphase modulations decrease systematically, as noted earlier byothers [8]. Using the results of figure 2 we can qualitatively
understand the frequency-dependent response in figure 3, as
already discussed in [15]. Briefly, at resonance (ω/ω0 = 0.95)
the slope in the amplitude response is smaller than on thewing (ω/ω0 = 0.66), leading to a less-pronounced amplitude
response in figure 3(a) than in figure 3(b), whereas the opposite

4.2. Analysis of the measurements
4.2.1. Tip–sample interaction forces. To extract the tip–
sample interaction forces from the measured data, we apply
equations (5) to the measured APD curves shown in figure 3.
Before doing so, we need to realize that the measured phase
is affected by a phase offset due to the measurement system.
The agreement between the experimental phase response andthe model curve in figure 2 shows that (except for the spurious
peaks at high frequencies) this phase offset is a frequencyindependent constant. To account for it, we measure the5Nanotechnology 21 (2010) 325703

S de Beer et al

beyond typical variations from curve to curve) is a slight
reduction of the amplitude of the force oscillations at the lowest
frequencies. Yet, as we will discuss below, this effect iscaused by the somewhat larger drive amplitude required in offresonance measurements.
The extracted total damping shows more variability
between the different drive frequencies. For ω/ω0 = 0.95 and
0.66, an oscillatory behaviour appears, similar to and exactly
out-of-phase with the interaction stiffness with superimposed
peaks. For the lowest drive frequency, the oscillatory behaviour
is absent, yet the peaks at d ≈ 0.9 and 1.7 nm remain visible.
Such sharp peaks in the total damping were consistently foundin many independent experiments with various cantilevers (Sior gold-coated Si and for various tip radii and spring constants).
Note that the total damping at large distance approaches a∞, which decreases with decreasing frequency
constant value γtot
as expected from the expressions given in section 3.
The degree of consistency shown in figure 5 can only beachieved using the complete mechanical model for acoustic
driving with deflection detection (see figure 1) including thefrequency-dependent damping and added mass presented in
section 4.1. Neglecting the frequency dependence of theadded mass and damping leads to substantial deviations of
the damping forces at low frequencies. If the base motion
is neglected (i.e. upon using a simple harmonic oscillator
model) both conservative and dissipative forces are completely
inconsistent between low and high frequencies, as expected(see [14]). Both of these findings are obviously consistent
with the deviations found for the modelled frequency response
curves (figure 2).

50

count

40

30

20

10

0

0

1

2

3

4

5

6

7

8

9

10

# oscillations

Figure 4. Frequency of the number of oscillations in
amplitude–distance curves from 174 measurements at six different
positions on different terraces of the HOPG (ω/ω0 = 0.75).

phase at a distance of 10 nm above the surface (i.e. outside
the range of the tip–sample interaction) and shift this value
manually to the value obtained from the model curve following
a procedure suggested by Sader and Jarvis [32] in the context
of frequency modulation AFM measurements. Failure to carryout this procedure correctly leads to cross-coupling between
the phase and amplitude signal and thereby induces two
main effects: (i) the interaction stiffness and the interaction
damping do not decay to zero at large distance and (ii) artificial
oscillations in the dissipation are created in the case of
oscillatory conservative tip–sample forces (see [8, 13]).
Figure 5 shows the interaction stiffness and damping
extracted by inverting the curves shown in figure 3 using
equations (5) using the full frequency-dependent m and γc . The
conservative forces display a strongly oscillatory behaviour
that decays to zero within a few molecular layers, independentof the applied frequency. The only significant trend (i.e.

kint [N/m]1(a)ω / ω0 = 0.95

0

2

(b)

4.2.2. Conservative forces and amplitude dependence. For
tip–surface distances beyond 1 nm, the conservative force
curves can be fitted rather well with an exponentially decaying
cosine profile (see figure 2, supplementary data available atω / ω0 = 0.66

ω / ω0 = 0.16(c)0γtot [10-5 kg/s]

-1
0.5

0.0

4

6 0

2

4

6 0

2

4

6

tip surface distance [nm]Figure 5. The interaction stiffness kint and total damping γtot versus tip–surface distance extracted from the amplitude and phase response of
the cantilever (curves in figure 3) using equations (5) with frequency-dependent damping and added mass. Left column: ω/ω0 = 0.95;
middle: ω/ω0 = 0.66; right: ω/ω0 = 0.16. The black dashed lines denote the damping of the cantilever γc .

6Nanotechnology 21 (2010) 325703

S de Beer et al
2

Kint [N/m]Atot / Ainf

tot

1.2

1.0

0.8

2

4

1

6
0
0.0

tip surface distance [nm]

0.2

0.4

Ainf
tot [nm]ϕtot [deg]80Figure 7. Apparent amplitude K int of the fitted interaction stiffness
versus total cantilever amplitude A ∞tot extracted from the amplitude
and phase response (ω/ω0 = 0.92, resonance frequency
ω0 /2π = 36 kHz). Symbols: experimental results (Si cantilever);error bars denote the standard error with 95% confidence interval.
Solid line: numerical calculation (see text for details).

60

40

2

4

figure 6 (see also [8]). Converting APD curves obtained forvariable A∞
tot between 0.09 and 0.4 nm at a fixed frequency
into interaction forces and fitting the conservative forces using
equation (6), we find experimentally that the amplitude of theinteraction stiffness gradually decreases approximately from
1.2 N m−1 for the smallest amplitude to 0.3 N m−1 for thelargest amplitude (see symbols in figure 7)3 .
In the context of the force inversion method used here,
the linearization of the tip–sample interaction forces, as
described in equation (2), provides a criterion for the maximum
acceptable total amplitude A∞
tot . For typical numbers for theperiodicity σ = 0.8 nm and for the decay length ξ =
1.1 nm, we find the requirement A∞
tot  2k int /k int , so∞
Atot  0.29 nm. To quantify the expected reduction, we
calculated APD curves inserting the full nonlinear tip–sample
interaction force according to equation (6) into (1) for variabledriving amplitude at a fixed drive frequency close to resonance.
(For simplicity the interaction damping was left out in thesecalculations.)
As shown in figure 6, the amplitude of the oscillationsin Atot and ϕtot decreases strongly with increasing A∞
tot , inqualitative agreement with the experimental data. Note that
appreciable smoothing effects can already be observed forA∞
tot = 0.16 nm.
Inserting the numerically calculated cantilever response
into equations (5) leads to the apparent interaction stiffness
as a function of the free cantilever amplitude. The solid line
in figure 7 shows the corresponding stiffness amplitude K int
decreases as a function of A∞
tot , in excellent agreement withexperimental measurements.

6

tip surface distance [nm]Figure 6. Numerically calculated amplitude and phase response of
the cantilever (ω/ω0 = 0.93, kc = 1.9 N m−1 and Q = 2.7) versus
tip–surface distance in the presence of oscillatory tip–sample forces( K int = 0.3 N m−1 , σ = 0.8 nm and ξ = 1.1 nm) for variable freeamplitude (80 pm, 160 pm, . . . 400 pm) increasing along the arrowfrom dark blue to turquoise.

stacks.iop.org/Nano/21/325703/mmedia), which is known toapproximate the shape of oscillatory solvation forces rather
well [2]:kint (z c ) = K int cos(2π z c /σ ) exp(−z c /ξ ).

(6)

Fitting equation (6) to the experimental data at variablefrequency, we find an average periodicity σ = 0.78 ± 0.1 nm
and a decay length ξ = 1.2 ± 0.2 nm. Equation (6) provides agood fit to the experimental data for distances beyond the firstmolecular layer, as illustrated in supplementary data (figure 2
available at stacks.iop.org/Nano/21/325703/mmedia). (Theregion d <∼ 0.75 nm was excluded in the fitting process.)
The fitting parameter K int yields a measure for the strength ofthe oscillatory solvation forces.
Before comparing the data at variable frequencies, we notethat the (apparent) strength of the oscillatory solvation forces
depends on the amplitude of the cantilever oscillation. If thefree amplitude of the cantilever becomes comparable to thecharacteristic length scale of the interactions the amplitudeof the oscillations in both amplitude and phase as well as
in the resulting interaction forces decreases, as shown in

3 To exclude the possibility of systematic changes in the experimental
conditions with time, e.g. due to contamination or temperature variation,
the free amplitude was increased and decreased in random order during theexperiments.

7

Nanotechnology 21 (2010) 325703

S de Beer et al

f /R is related to the amplitude K int of the distance-dependent
oscillatory stiffness kint (fitted using equation (6)) by2

Kint [N/m]

K int
F

=
R
2 π R ( 1 )2 + ( 1 )2
2π ξ

0.5

σ

where R is the tip radius. For small drive amplitudes
and using the fitted interaction stiffness, we find a force
amplitude F/R = 9 ± 2 mN m−1 , in agreement with earlier
measurements in SFA and AFM [2, 33]. Based on the factthat tip–sample distances of about one molecular layer andbelow need to be excluded from the fitting procedure, one
may speculate that the molecules immediately adjacent to thesubstrate surface are in a different, perhaps solidified, state ascompared to the other molecular layers.
The fact that the strength of the solvation forces extracted
from distances beyond the first molecular layer is independentof the drive frequency indicates that the system does not have
any relevant relaxation times within the corresponding timeinterval of 0.02–0.2 ms. For a fluid this is not surprising since
typical molecular relaxation times are substantially faster.

1

0
0.0

(7)

1.0

ω / ω0Figure 8. Extrapolated amplitude K int of the fitted interaction
stiffness versus measurement frequency for various cantilevers. (Redcircles: Au-coated Si, ωres /2π = 51 kHz; green triangles: Si,ωres /2π = 37 kHz; yellow diamonds: Si, ωres /2π = 59 kHz; blue
squares: Au-coated Si, ωres /2π = 38 kHz; error bars denote thestandard error with 95% confidence interval.)This result explains the apparent reduction of theoscillatory interaction forces in figure 5(c) (recorded far below
resonance) compared to figures 5(a) and (b) (recorded onresonance): the off-resonance measurements were performed
using a relatively large amplitude A∞
tot ≈ 0.3 nm (see figure 3)
in order to achieve a similar signal-to-noise ratio as forthe on-resonance measurements. For such large amplitudes,
linearization of the tip–sample interaction forces is no longer
justified, causing the observed apparent reduction of K int .
This observation may explain some of the discrepancies
between different dynamic AFM measurements carried out
with different free amplitudes.

5.2. Dissipative tip–sample interaction
Compared to the conservative interactions, the picture
emerging from the dissipation extracted for various frequencies
is somewhat less consistent. For all frequencies the dissipation
decays to a constant frequency-dependent asymptotic value∞
γtot(ω), as expected. Yet, the data recorded close to resonance(figures 5(a) and (b)) display oscillations involving bothenhancements and reductions of γtot by several tens of per cent∞of γtot, while the low frequency data (figure 5(c)) only display
positive peaks at distances corresponding to the minima in kint .
To elucidate whether oscillations in the damping are
realistic and to what extent confinement may enhance the local
dissipation, we consider the dynamics of the cantilever in theframework of hydrodynamics. The experimental observationthat A and ϕ (as well as kint and γtot ) are constant withinseveral tens of nanometres from the surface (see section 2
and figure 3 of supplementary data available at stacks.iop.org/
Nano/21/325703/mmedia) suggests that it is justified to split
the dissipation into a background contribution due to the global
motion of the cantilever, which is constant within the distance
range covered by the APD curves, and a distance-dependent
contribution arising from the local dynamics in the vicinity ofthe tip. This idea is consistent with an approach by O’Shea
and Welland [11], who decomposed the total damping into
separate contributions arising from the motion of different partsof the cantilever, namely the cantilever beam, the cone of
the tip (modelled as an effective sphere) and the nanoscale
tip itself. For each of these components separate solutionsof the time-dependent Stokes equations for the respective
idealized geometry are known and allow for estimating theirrelative importance. The analysis, which we describe for our
experimental situation in the appendix, shows that the total
damping is dominated by the damping of the cantilever beamin the fluid rather than the squeeze-out damping at the tip.
This is due to the much larger characteristic size of the former

4.2.3. Conservative tip–sample interaction at variable drive
frequency. Using the result of figure 7, we can correct theextracted apparent values of the amplitude of the interaction
stiffness K int and extrapolate them to their intrinsic value
for small drive amplitudes. Figure 8 shows the result for adataset consisting of various different cantilevers and types ofcantilevers for a range of drive frequencies from 5 to 58 kHz.
Within the experimental error, we find that the strength of theoscillatory solvation forces is identical, irrespective of the drive
frequency and of the material of the cantilever.

5. Discussion
5.1. Conservative tip–sample interaction
To compare the present results to previous SFA and
AFM measurements, we convert the interaction stiffness
kint into the normalized force f /R using the Derjaguin
approximation [19, 33, 34]. f /R is proportional to theinteraction energy per unit area of two parallel surfaces at
separation z c . Approximating the local oscillatory pressureby P(z c ) = P0 cos(2π z c /σ ) exp(−z c /ξ ), the resulting
amplitude F/R of the distance-dependent oscillatory force
8Nanotechnology 21 (2010) 325703

S de Beer et al(typically 100 μm versus 20 nm). Inserting numerical values,
one finds indeed that this background contribution is expected
to be constant, within ∼0.25%, over the range of the APD
curves. The most interesting distance-dependent contribution
is obviously the squeeze-out damping due to the confined fluid
the between tip and sample. It can be described by Reynolds’
classical expression:
2
Rtip

d

.

η= 1000 ηbulk

γint [10-5 kg/s]

γR = 6πηloc

1.0

0.5

(8)

100 ηbulk

For distances d of a few nanometres and for a local
viscosity ηloc under the tip equal to the bulk viscosity, this tip
damping is several orders smaller than the beam damping [11].
Only for d → 0 (i.e. 1 nm) and/or if the local viscosity isdramatically enhanced, e.g. due to confinement effects, thiscontribution can lead to a substantial and measurable excess
damping.
The above considerations have important consequences
regarding the nature of variations in the damping, which—to our knowledge—have not been expressed explicitly sofar: since the background damping is constant within theexperimental error and a few orders of magnitude larger than
the tip damping, γtot can only increase but never significantly
decrease within the uncertainty of AFM measurements(typically a few per cent). This conclusion results directlyfrom the relative order of magnitude of the global and the local
dissipation and therefore applies to all AFM measurements
independent of the specific measurement technique (AMAFM versus FM-AFM; base drive versus magnetic drive).
As a consequence, we conclude that any reduction of the
global damping, such as the ones shown in figures 5(a)and (b), must be artificial and caused by either uncertainties
in calibration constants and/or imperfections in the model ofthe cantilever dynamics. (For the present case of AM-AFM,
the experimental uncertainties include in particular errors in
the absolute phase [8, 13], in addition to Q and ω0 , which arealso important in FM-AFM.)
Since the background damping of the cantilever is constantwithin the experimental errors, it is justified to subtract that
contribution and consider the remaining interaction damping∞
γint = γtot − γtot
. Using hydrodynamics as a reference
framework, we compare the resulting γint to the expectations
based on equation (8) (see figure 9).
We plot the low frequency data from figure 5(c) in figure 9
because they do not display artificial negative values of γint ,
caused by excessive reductions of γtot as discussed above.
Moreover, the low frequency measurements are less sensitive
to calibration errors, while the amplitude and phase response(as mentioned above) is equally sensitive to variations in the
stiffness and damping (see also supplementary data available
at stacks.iop.org/Nano/21/325703/mmedia and partially [15]).
Figure 9 also shows the calculated Reynolds squeeze-out
damping under the tip (equation (8)), for increasing valuesof the viscosity. For ηloc = ηbulk (figure 9), the Reynolds
damping under the tip γR is negligible for all realistic values ofd , in agreement with the discussion above. Yet, the Reynolds
damping resulting from an already 100 times enhanced local
viscosity lies above the experimental data at all distances.

10 ηbulk

0.0
0

2

4

6

tip surface distance [nm]Figure 9. The extracted interaction damping for ω/ω0 = 0.16 (green
curve). The black curves show the calculated Reynolds damping γR
for increasing viscosity ( Rtip = 20 nm).

In the damping measured for d > 2 nm, the data areconsistent with the bulk viscosity and approximately 10× thebulk viscosity (within the experimental error), corresponding
to 22 mPa s. Larger enhancement factors are incompatible with
our experimental data. Note that the same conclusion can also
be drawn on the basis of all our data recorded, irrespective ofthe drive frequency (including the artificial oscillations visible
close to resonance).
A direct conversion of the measured damping into aneffective viscosity seems to provide a reasonable descriptionof the system down to distances of 2–3 nm. At smaller
separations, the hydrodynamic picture becomes progressively
more questionable given the small number of molecules
involved (a few hundred) and the discrete layer structure of
the confined liquid. The position of the maxima at d ≈ 0.9
and 1.7 nm in the damping in figure 9 coincides with theminima in the interaction stiffness. It also coincides with aregion of (small) negative slope of the average deflection of thecantilever (data not shown). The maxima in the damping thus
occur at distances corresponding to attractive forces, where
according to a standard hard sphere-like model as well as
numerical simulations, the average density of the fluid inside
the gap is reduced compared to the bulk value. Such a
reduced density implies more free volume for the molecules
and thus more freedom to exchange sites, which may enhancethe opportunities to dissipate energy compared to the morecompact configuration of completely filled layers that provide
a more elastic response. Hofbauer et al [12] recently proposeda similar mechanism to explain similar (yet more pronounced)
peaks in the dissipation for confined dodecanol.
Finally, we note that for all distances, except for theminima in the interaction stiffness (corresponding to theattractive part of the conservative forces), the damping always
assumes values that are compatible with the bulk viscosity,
indicating that the ‘effective viscosity’ in complete layersof OMCTS is similar to the value found in the bulk.
Notwithstanding the differences in the confinement geometry
between the surface force apparatus and the AFM, it is9Nanotechnology 21 (2010) 325703

S de Beer et al

remarkable how this result agrees with some of the earlier SFA
measurements [5, 6].
The fact that the conservative tip–sample interaction
forces follow the behaviour of hard sphere fluids (equation (6))
and the fact that the effective viscosity is compatible with thebulk viscosity down to a thickness of three monolayers clearlysuggest that the system behaves liquid-like. The consistency ofthis behaviour (including in particular the absence of artificial
negative interaction damping) for the various experimental
conditions investigated support this conclusion. We notethat this finding is at variance with the conclusion of Patilet al [9] who reported signs of a jamming-like solidification
for confined OMCTS for the range of approach rates used in
the present study and liquid-like behaviour only for approach
rates below 0.6 nm s−1 . Their interpretation is based on a
Maxwellian model of a viscoelastic liquid, yet they do notprovide any details regarding the modelling of the cantileverdynamics. We currently have no explanation for the deviationfrom our data.

Appendix
Consider an AFM cantilever [11] consisting of a rectangularbeam with length L = 120 μm, width w = 35 μm and at
the end a tip with cone radius Rcone = 5 μm and a tip end
with radius Rtip = 10 nm. The cantilever is in the vicinity of asolid surface and completely immersed in a liquid with density
ρ = 2330 kg m−3 and viscosity η = 2.2 mPa s. The distance
between the beam and the surface is h = d + 25 μm, between
the tip cone and the surface is D = d + 8 μm, with the distance
between the tip and the surface d (∼1–2 nm).
For each part of the cantilever, there are contributions
due to the free oscillation far away from the surface andcontributions due to squeeze-out damping close to the surface.
The former contain a constant and a frequency-dependent
component due to the varying added mass, which is responsible
for the reduction of the total damping for large distances at low
frequencies (see figure 5).
The total damping of the cantilever is built up by:(1) Hydrodynamic damping due to viscous drag and the added
mass over the beam:3γcb = 0.24 · 3πηL + 4 wL 2ρηω .

6. Conclusions
In summary, the results presented in this paper show that it
is indeed possible to extract consistent tip–sample interaction
forces over a wide range of driving frequencies from acoustic
drive amplitude modulation AFM measurements provided thatthree elements are correctly taken into account, namely (i) the
motion of the cantilever base, (ii) the frequency dependence of
the added mass and the viscous damping around the cantileverand (iii) the finite oscillation amplitude of the cantilever. Whilethe consistency with respect to the conservative forces is verysatisfying, known residual discrepancies due to cross-couplingwith conservative forces limit the accuracy of dissipative forces
measured at AFM drive frequencies close to resonance. This
problem can be circumvented by choosing drive frequencies
substantially below resonance, where the measurement system
is less sensitive to errors in the phase and/or uncertainties in
the calibration constants.
The dissipative forces extracted in this way show that the‘effective’ viscosity of the liquid confined between tip andsample is bulk-like down to approximately three molecularlayers. Only for the last two layers are local maxima inthe dissipation found at tip–sample distances corresponding tominima in the interaction stiffness where the average densityof the confined fluid is expected to be minimal from molecularsimulations.

(2) Hydrodynamic damping due to viscous drag and the added
mass at the tip cone:2γcc = 12 (6πη Rcone + 3π Rcone2ρηω).
(3) Hydrodynamic damping due to viscous drag and the added
mass at the tip:2γct = 12 (6πη Rtip + 3π Rtip
2ρηω).
(4) Reynolds squeeze-out damping under the beam, γRb =3
0.24η wh 3L .
(5) Reynolds squeeze-out damping under the cone, γRc =
2
Rcone
6πη D
.
(6) The interaction damping under the tip, γint .
(a) Reynolds squeeze-out damping under the tip, γint R =R2

6πηlocal dtip .
(b) Changes in the hydrodynamic damping due tolocal changes in the added mass and viscosity
1
under the√ tip, γint c √= 2 (6π(ηlocal − η)Rtip +2
3π Rtip ( 2ρηlocal ω − 2ρηω)).
We define damping 1–5 as the background damping γc anddamping 6 as the interaction damping, γint .
First of all, using these equations, it is easy to show that,
due to the change in Reynolds damping between 100 and 1 nm
tip–surface distance, the cantilever’s damping γc increases by
0.25%. This is far within our experimental error and therefore
our assumption that the extracted changes in damping are
due to tip–sample interactions is valid. (Remember that theassumption of a constant γc is supported by the experimental
observation that the APD curves are constant for tip–sample
distances varying between 100 and 10 nm.)Acknowledgments
We thank J E Sader for his comments at SPM-UK 2009.
We thank P Markus (Veeco) and K Smit for the technicalsupport and M Smithers for his help on the SEM images. This
work has been supported by the Foundation for Fundamental
research on Matter (FOM), which is financially supported by
the Netherlands Organization for Scientific Research (NWO).
10

Nanotechnology 21 (2010) 325703

S de Beer et al

Second, note that the damping due to the Reynolds force
under the tip always results in an increase of the total damping
and therefore a positive interaction damping. Consequently
the negative interaction damping can only be a result of achange in hydrodynamic damping due to viscous drag and theadded mass at the tip. Since the local density (and accordingly
the viscosity) under the tip varies around its bulk value, thiscontribution can be both positive and negative. However, if, ina worst case scenario, the local viscosity under the tip would go
to zero, the maximum decrease in the calculated total damping
is 0.013%. Our experimentally found decrease in the total
damping is much larger and therefore unrealistic.

[16] Christenson H K and Blom C E 1987 Solvation forces andphase separation of water in a thin film of nonpolar liquid
between mica surfaces J. Chem. Phys. 86 419[17] Hutter J L and Bechhoefer J 1993 Calibration of atomic force
microscopy tips Rev. Sci. Instrum. 64 1868
[18] Maali A, Hurth C and Cohen-Bouhacina T 2006 Improved
acoustic excitation of atomic force microscope cantilevers inliquids Appl. Phys. Lett. 88 163504[19] Proksch R, Schaeffer T E, Cleveland J P, Calahan R C andViani M B 2004 Finite optical spot size and position
corrections in thermal spring constant calibration
Nanotechnology 15 1344
[20] Garcia R and Perez R 2002 Dynamic atomic force microscopy
methods Surf. Sci. Rep. 47 197–301[21] Zitzler L, Herminghaus S and Mugele F 2002 Capillary forces
in tapping mode atomic force microscopy Phys. Rev. B
66 155436[22] Hoelscher H 2006 Quantitative measurement of tip–sample
interactions in amplitude modulation atomic force
microscopy Appl. Phys. Lett. 89 123109[23] Lee M H and Jhe W H 2006 General theory of
amplitude-modulation atomic force microscopy Phys. Rev.
Lett. 97 036104[24] Hu S Q and Raman A 2008 Inverting amplitude and phase toreconstruct tip–sample interaction forces in tapping mode
atomic force microscopy Nanotechnology 19 375704[25] Katan A J, van Es M H and Oosterkamp T H 2009 Quantitative
force versus distance measurements in amplitude modulation
AFM: a novel force inversion technique Nanotechnology
20 165703[26] Sader J E and Jarvis S P 2004 Accurate formulas for interaction
force and energy in frequency modulation force
spectroscopy Appl. Phys. Lett. 84 1801[27] Xu X and Raman A 2007 Accurate force spectroscopy intapping mode atomic force microscopy in liquids J. Appl.
Phys. 102 034303[28] Jai C, Cohen-Bouhacina T and Maali A 2007 Analyticaldescription of the motion of an acoustic-driven atomic force
microscope cantilever in liquid Appl. Phys. Lett. 90 113512[29] Landau L and Lifshitz F 1971 Fluid Mechanics, Theoretical
Physics vol 6 (Moscow: Mir)[30] Sader J E 1998 Frequency response of cantilever beams
immersed in viscous fluids with applications to the atomic
force microscope J. Appl. Phys. 84 64[31] Maali A, Hurth C, Boisgard R, Jai C, Cohen-Bouhacina T and
Aime J-P 2005 Hydrodynamics of oscillating atomic force
microscopy cantilevers in viscous fluids J. Appl. Phys.
97 074907[32] Sader J E and Jarvis S P 2006 Coupling of conservative anddissipative forces in frequency-modulation atomic force
microscopy Phys. Rev. B 74 195424[33] Lim L T W, Wee A T S and O’Shea S J 2008 Effect of tip size
on force measurement in atomic force microscopy Langmuir
24 2271[34] O’Shea S J, Welland M E and Pethica J B 1994 Atomic force
microscopy of local compliance at solid—liquid interfaces
Chem. Phys. Lett. 223 336

References[1] Bhushan B, Israelachvili J N and Landman U 1995
Nanotribology: friction, wear and lubrication at the atomic
scale Nature 374 607[2] Israelachvili J N 1992 Intermolecular and Surface Forces
2nd edn (London: Academic)[3] Klein J and Kumacheva E 1995 Confinement-induced phasetransitions in simple liquids Science 269 816[4] Gee M L, McGuiggan P M, Israelachvili J N and
Homola A M 1990 Liquid to solid-like transitions ofmolecularly thin films under shear J. Chem. Phys. 93 1895
[5] Becker T and Mugele F 2003 Nanofluidics: viscous dissipationin layered liquid films Phys. Rev. Lett. 91 166104[6] Bureau L and Arvengas A 2008 Drainage of a nanoconfined
simple fluid: rate effects on squeeze-out dynamics Phys. Rev.
E 78 061501[7] Gao J, Luedtke W D and Landman U 1997 Layering transitions
and dynamics of confined liquid films Phys. Rev. Lett.
79 705[8] Maali A, Cohen-Bouhacina T, Couturier G and Aimé J-P 2006
Oscillatory dissipation of a simple confined liquid Phys. Rev.
Lett. 96 086105[9] Patil S, Matei G, Oral A and Hoffmann P M 2006 Solid or
liquid? solidification of a nanoconfined liquid under
nonequilibrium conditions Langmuir 22 6485
[10] Kaggwa G B, Kilpatrick J I, Sader J E and Jarvis S P 2008
Artifact-free dynamic atomic force microscopy reveals
monotonic dissipation for a simple confined liquid Appl.
Phys. Lett. 93 011909[11] O’Shea S J and Welland M E 1998 Atomic force microscopy atsolid–liquid interfaces Langmuir 14 4186
[12] Hofbauer W, Ho R J, Hairulnizam R, Gosvami N N andO’Shea S J 2009 Crystalline structure and squeeze-out
dissipation of liquid solvation layers observed bysmall-amplitude dynamic AFM Phys. Rev. B 80 134104[13] O’Shea S J 2006 Comment on ‘oscillatory dissipation of asimple confined liquid’ Phys. Rev. Lett. 97 179601[14] Burnham N A, Gremaud G, Kulik A J, Gallo P-J andOulevey F 1996 Materials’ properties measurements:choosing the optimal scanning probe microscope
configuration J. Vac. Sci. Technol. 14 1308
[15] de Beer S, van den Ende D and Mugele F 2008 Atomic force
microscopy cantilever dynamics in liquid in the presence of
tip sample interaction Appl. Phys. Lett. 93 253106

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