Viscoelastic, mechanical, and dielectric measurements on ... · (GaPO 4) are in use, as well. These...
Transcript of Viscoelastic, mechanical, and dielectric measurements on ... · (GaPO 4) are in use, as well. These...
Viscoelastic, mechanical, and dielectric measurements on complex
samples with the quartz crystal microbalance
Diethelm Johannsmann*
Received 6th March 2008, Accepted 21st May 2008
First published as an Advance Article on the web 27th June 2008
DOI: 10.1039/b803960g
Piezoelectric resonators have been in use as mass-sensing devices for almost half a century. More
recently it was recognized that shifts in frequency and bandwidth can come about by a diverse
variety of interactions with the sample. The classical ‘‘load’’ consists of a thin film, which shifts
the resonance frequency due to its inertia. Other types of loading include semi-infinite viscoelastic
media, rough objects contacting the crystal via isolated asperities, mechanically nonlinear
contacts, and dielectric films. All these interactions can be analyzed within the small-load
approximation. The small-load approximation provides a unified frame for interpretation and
thereby opens the way to new applications of the QCM.
I. Introduction
The use of piezoelectric resonators for sensing (in addition to
frequency control1) is already described in the book by Mason
dating from 1948.2 Mason was concerned with torsional
resonators, which oscillate at frequencies in the kHz
range. These devices measured a viscosity without actually
inducing a macroscopic flow in the liquid.3,4 Although
torsional resonators are today much less common than the
thickness-shear resonators, Mason’s work anticipates much of
the conceptual basis for the current understanding of acoustic
resonators.
The field gained much momentum in 1959, when Sauerbrey
recognized that high-frequency thickness-shear mode (TSM)
resonators respond to the deposition of mass on their surface
with a rather extreme sensitivity.5,6 Molecular monolayers can
be easily detected. The quartz crystal microbalance (QCM)
exploits this effect.7–10 Many commercial instruments are
available nowadays. Until the mid-80s, the QCM was not
usually operated in liquids because the liquid overdamps the
oscillation unless suitable counter-measures are taken.11,12
One can either modify the oscillator circuit1,13 or ‘‘passively’’
determine the resonance frequency with a network analyzer.
The network analyzer measures the electrical impedance of the
device as a function of frequency without resorting to ampli-
fication and oscillation (Fig. 1a).14 The resonance frequency
and the bandwidth are derived by fitting these spectra with
resonance curves. The Chalmers group has commercialized a
different type of passive measurement which is based on ring-
down,15,16 rather than impedance analysis (Fig. 1b). This
instrument is called QCM-D. When the QCM was first
immersed into liquids, mass sensing at electrode surfaces was
the main application. This instrument is today called ‘‘electro-
chemical QCM (EQCM)’’.17,18 Later, many researchers inves-
Fig. 1 (a) Impedance analysis is based on the electrical conductance
curve. The central parameters of measurement are the resonance
frequency, fr, and the bandwidth, 2G. (b) Ring-down yields the
equivalent information in time-domain measurements.
Institute of Physical Chemistry, Clausthal University of Technology,Arnold-Sommerfeld-Str. 4, D-38678 Clausthal-Zellerfeld, Germany.E-mail: [email protected];Fax: +49(0) 5323-72-4835; Tel: +49 (0) 5323-72-3768
4516 | Phys. Chem. Chem. Phys., 2008, 10, 4516–4534 This journal is �c the Owner Societies 2008
PERSPECTIVE www.rsc.org/pccp | Physical Chemistry Chemical Physics
tigated the adsorption of biomolecules from buffer solutions to
functionalized resonator surfaces.19
Passively mapping out the resonance curve has many ad-
vantages. Firstly, one can compare data taken on different
overtones. Secondly, one immediately recognizes ‘‘bad’’ crys-
tals based on their irregular impedance spectrum. Finally and
importantly, one gets access to the resonance bandwidth in
addition to resonance frequency. The shift in bandwidth is
proportional to the amount of energy transferred from the
crystal to the sample per unit time. Below, the half-bandwidth
at half-maximum (‘‘bandwidth’’, for short) is used to quantify
such processes. Users of the QCM-D often display their results
in terms of frequency and ‘‘dissipation’’, D. D is the inverse
Q-factor: D = Q�1 = 2G/f.The bandwidth carries essential additional information,
which can, for instance, be used to determine viscoelastic
properties. For the Newtonian liquid, the relation connecting
the viscosity and the shifts of frequency and bandwidth20,21 are
rather similar to Mason’s original equations derived for tor-
sional resonators. There is a second, more practical benefit of
bandwidth measurements. It turns out that the bandwidth is
much less sensitive to fluctuations of temperature22 and me-
chanical stress23,24 than the frequency. Temperature and stress
and are sometimes difficult to control in complex environ-
ments. Some workers just show the shifts of bandwidth (or,
equivalently, dissipation) as a function of time, leaving out the
frequency shifts entirely. For qualitative comparisons, this
approach has proven rather useful. The bandwidth is the more
robust parameter.
Measurements of mass and viscosity are the most common
sensing applications of acoustic resonators to date. There are
more exotic, but still interesting kinds of loading, which have
not found widespread application, yet. For instance, a cou-
pling between the crystal and an external object touching it via
‘‘point contacts’’ increases the frequency, as was demonstrated
by Dybwad in 1985.25,26 Such point contacts typically occur
between rough surfaces.27,28 Dybwad derived a relation be-
tween the frequency shift and the spring constant of a point
contact. Monitoring the contact stiffness as a function of time,
vertical load, humidity, and other environmental parameters is
rather easy. Since the contact area depends on the load, the
force-displacement relations are often nonlinear.29 Even the
transition from stick to slip can be observed.
Another rather intriguing type of coupling is mediated by
piezoelectric stiffening.30,31 The sample’s electric and dielectric
properties affect the resonance properties if the front electrode
is not grounded well. Such electrical interferences are usually
avoided. However, they may be of interest. If the front
electrode leaves some part of the crystal surface open, the
electric field penetrates the sample and probes the sample’s
dielectric properties.
The name quartz crystal microbalance correctly suggests
that the main use of the QCM is microgravimetry. However,
many researchers who use quartz resonators for other pur-
poses, have continued to call the quartz crystal resonator
‘‘QCM’’. The current article follows this usage; all thickness
shear resonators are called ‘‘quartz crystal microbalance’’.
Actually, the term ‘‘balance’’ makes sense even for some
non-gravimetric applications if it is understood in the sense
of a force balance. At resonance, the force exerted onto the
crystal by the sample is balanced by a force originating from
the shear gradient inside the crystal. This is the essence of the
small-load-approximation.
When materials parameters are quoted below, they are
always related to AT-cut a-quartz. Thickness-shear resonatorsare also made of materials other than quartz. For example,
Langasite (La3Ga5SiO14, ‘‘LGS’’) or gallium-orthophosphate
(GaPO4) are in use, as well. These materials have better high-
temperature stability than quartz.32,33 Also, GaPO4 has a
higher coupling coefficient, which makes it attractive for
certain types of oscillating circuits.34 Silicon is also investi-
gated as a resonator material due its compatibility with the
existing semiconductor technology.35 The resonance can, for
instance, be excited by piezoelectric films sputtered onto the
crystal surface. The most serious drawback of silicon is the
absence of a temperature-compensated cut (like the AT-cut for
quartz).
The paper is organized as follows. In section II, the load
impedance and the small-load approximation (SLA) are in-
troduced. The SLA is the basis for the various modes of data
analysis. Section III reviews a classical set of applications,
which is the characterization of a planar viscoelastic layer
system. Such measurements are pursued by many groups. Still,
there are intricacies in the analysis, which deserve continued
attention. Section IV describes details of piezoelectric stiffen-
ing and the determination of a sample’s dielectric properties.
Section V discusses point contacts as they occur between
rough surfaces. The contact mechanics of rough surfaces
(including aging of contacts, yield, fracture, and slip) is a field
of immense practical importance.36 The macroscopic mechan-
ical properties of the contact zone are related to the micro-
scopic conditions at the load-bearing asperities in a highly
nontrivial way. Acoustic devices can probe these microscopic
contacts. Section VI describes the analysis of nonlinear
stress–speed relations, as they frequently occur in friction
and other contact-mechanics problems. Subjecting the sample
to high-frequency, low-amplitude oscillatory shear is a some-
what unorthodox way to investigate friction phenomena. Still,
the QCM has extreme sensitivity and can provide information
unavailable with other techniques. Section VII comments on
loads consisting of spatially separated adsorbed objects (such
as vesicles), which should not be described as a homogeneous
film. Such samples are often encountered in bio-related appli-
cations of the QCM. Section VIII concludes.
II. The load impedance and the small-load
approximation
In the following, the prevailing model of the quartz crystal
microbalance (QCM) is reviewed. The equations mostly apply
to thickness-shear resonators. Some other instruments (such as
shear-horizontal surface acoustic wave devices,37,38 Love-wave
devices,39 FBARs,40 and torsional resonators3,41) interact with
their environment in essentially the same way as the QCM. All
these devices measure the ratio of lateral stress to lateral speed
at the device–sample interface. This ratio is the load impe-
dance, ZL.
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The mathematical details of modeling have been provided in
a recent book chapter.42 The following remarks only cover the
outcome of these calculations. As a starting point, we adopt
the idealized view that the resonator can be modeled as a
laterally infinite plate undergoing a thickness-shear motion.
Modeling of this device can occur in three different frames,
which are called the ‘‘mathematical’’, the ‘‘optical’’, and the
‘‘equivalent circuit’’ approach. For a mathematician, the
search for the resonance frequencies amounts to the search
for the zeros of a determinant.43 When setting up the model,
one uses the wave equation inside all layers (crystal, electrodes,
film(s), and bulk) as well as continuity of stress and displace-
ment at the interfaces. A homogeneous set of linear equations
results. A non-trivial solution (with non-zero displacement)
only exists if the determinant of this equation system is zero.
The determinant depends on frequency. The zeros correspond
to the resonance frequencies. If losses are taken into account,
the resonance frequency turns into a complex parameter where
the imaginary part is the half bandwidth at half maximum, G.The mathematical approach is rigorous and—in a
way—aesthetic because it does not make use of analogies
and does not appeal to the reader’s intuition whatsoever. It
is straightforward in a mathematical sense, but is not easily
digested. The optical approach and the equivalent circuit
representation are less formal and provide intuitive insight.
The role, which the refractive index has in optics is taken by
the acoustic impedance, Zac, in acoustics.44 Zac = (Gr)1/2
(G the shear modulus and r the density) is a materials
constant, which governs the reflectivity at acoustic interfaces.
Researchers familiar with optics often prefer to describe the
crystal as a cavity with an acoustic beam bouncing back and
forth. The frequency changes in response to shifts of the
acoustic reflectivity at the crystal–sample interface. For in-
stance, the reflectivity is given by (Zq � Zliq)/(Zq + Zliq) in
case the sample is a homogeneous, semi-infinite liquid. Viewed
it this way, the QCM is an acoustic reflectometer. Again, the
results are completely equivalent to the mathematical formu-
lation. It is a matter of language. The optical approach plays
out its strength when either the sample or the crystal is
laterally heterogeneous. For instance, the crystal may be
thicker in the center than at the rim. It will then form an
acoustic lens, focusing the acoustic beam to the center of the
crystal. This is the poor man’s explanation of energy trap-
ping.22 Should the sample be heterogeneous on a small scale,
the crystal surface will behave like a hazy mirror. Quantitative
calculations are demanding, but one may safely state that
acoustic scattering withdraws energy from the oscillation,
thereby increasing the bandwidth.45,46
A third, popular representation of the crystal is the equiva-
lent electrical circuit.47,48 Close to the resonance, the Butter-
worth–van Dyke (BvD) equivalent circuit applies. The
crystal’s mass, its elastic compliance, and its internal viscous
losses are pictured as an inductance, a capacitance, and a
resistance. Fig. 2a shows the BvD circuit in its simplest form.
C0 is an electrical capacitance, while L1*, C1*, and R1* are
‘‘motional’’ elements. The star is meant to indicate that these
parameters are effective parameters, comprising the crystal,
the sample, and piezoelectric stiffening. An electrical measure-
ment can, in principle, yield all four parameters C0, L1*, C1*,
and R1*. However, the extreme sensitivity (Df/f B 10�7) of
the QCM only pertains to the resonance frequency, which
is (4p2L1C1)�1/2. For the other parameters a typical accuracy
is 1%.
Central to equivalent circuits is the concept of the mechan-
ical impedance. The mechanical impedance is the ratio of
lateral force and lateral speed. It is the analogue of the
electrical impedance, which is the ratio of voltage to current.49
The term ‘‘impedance’’ is also in use for a stress–speed ratio
(rather than the force–speed ratio). This terminology is com-
mon in acoustics because the flux of momentum and energy is
usually normalized to area. Unfortunately, the stress–speed
ratio cannot be called ‘‘acoustic impedance’’ (which would be
a natural expression) because this term is already in use for a
materials constant. One has Zac = rc = (Gr)1/2 with r as the
density, c as the speed of sound, and G as the shear modulus.
Zac also has units of stress/speed, but it is a special
stress–speed ratio. The stress–speed ratios occurring in the
BvD circuit have a more general meaning.
In order to emphasize the difference between the electrical
and the motional branch of the BvD circuit, the circuit was
redrawn in Fig. 2b, such that it depicts the motional elements
Fig. 2 (a) Butterworth-van-Dyke (BvD) equivalent circuit. (b) BvD-
circuit, where the motional elements have been depicted as a spring, a
mass, and a dashpot. The sample and piezoelectric stiffening are
represented as separate elements. (c) Equivalent circuit for a crystal
with a hole in the front electrode. The electric impedance of the crystal
surface, U/I, contributes to the frequency shift via piezoelectric stiffen-
ing. See text for the definition of these variables.
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as a mass, mp, a spring, kp, and a dashpot, xp. These
parameters take the values
kp ¼AqGq
dq
ðnpÞ2
2¼ kq;stat
ðnpÞ2
2
mp ¼Aqrqdq
2¼ 1
2Amq
xp ¼AqZqnp2tanðdÞ
ð1Þ
Aq is the active area of the crystal, Gq is the shear modulus, dqis the thickness, n is the overtone order, kq,stat is the static
spring constant under thickness shear, rq is the density, mq is
the mass per unit area, Zq = 8.8 � 106 kg m�2 s�1 is the
acoustic impedance of AT-cut quartz, and tan(d) = Gq00/Gq
0 is
the loss tangent. The constants as defined in eqn (1) account
neither for piezoelectric stiffening50 nor for the presence of the
sample. Piezoelectric stiffening and the sample are represented
by the separate elements ZPES and AqZL. Coupling between
the motional and the electric branch is achieved via a trans-
former (an ‘‘impedance converter’’). It is characterized by a
‘‘ratio of the number of loops’’, 4f. While f is dimensionless
for usual transformers, it has the dimension of current/speed
here. For the origin of the factor of 4, see ref. 47. The following
equations hold47
I ¼2f _u
U ¼ 1
2fF ¼ 1
2fAqs
Zel ¼U
I¼ 1
4f2
Aqs_u¼ 1
4f2Zm
f ¼Aqe26
dq
ð2Þ
where I is an electrical current, u is the lateral displacement at
the crystal surface, _u is the lateral speed, U is an electrical
voltage, F is a force, s is the stress, Zel is an electrical
impedance, Zm is a mechanical impedance, and e26 =
9.54 � 10�2 C m�2 is the piezoelectric stress coefficient.51
The sample is usually depicted as the ‘‘load’’, AqZL. ZL (a
stress–speed ratio) is the ‘‘load impedance’’. If the sample
consists of a semi-infinite, viscoelastic medium, ZL is equal to
the acoustic impedance of this medium, Zac. Again, this is a
special case. The load impedance covers all kinds of samples,
however complicated they may be.
Equivalent circuits are a popular modeling tool for several
reasons. Firstly, anyone who knows how to add two resistors
in parallel or in series can calculate the spectrum of the total
(electrical) impedance across the electrodes. The spectrum
displays peaks at certain frequencies; finding these resonance
frequencies amounting to the solution of the problem at hand.
An equivalent circuit is not a cartoon, it is a recipe. Secondly,
equivalent circuits provide for an easy scheme for including
piezoelectric stiffening. Finally and importantly, the ‘‘small-
load approximation’’ (SLA) relates the load, ZL, to the shift of
the resonance properties in a very transparent way. In just
about all cases of practical relevance, the load impedance is
much smaller than the acoustic impedance of AT-cut quartz,
Zq. This condition defines the SLA. A small load implies
Df/f { 1. The author is not aware of a study where this
condition was not fulfilled. If the load is small, the frequency
shift can be approximated as52,53
Df �
fF� i
pZL
Zq¼ i
pZq
s_u
ð3Þ
Here fF is the frequency of the fundamental and Df* =
Df + iDG is the complex frequency shift.
Since Df* depends linearly on the load, the SLA can be
applied in an average sense.54,55 One may replace the stress, s,by an area-averaged stress, hsi. Assume that the sample is a
complex material, such as a cell culture, a sand pile, a froth, an
assembly of vesicles, or a droplet. If the average stress–speed
ratio at the bottom of the sample can be calculated in one way
or another, a quantitative analysis of the QCM experiment is
within reach.
The SLA may become inadequate when Df is large or whenthe dependence of Df and DG on the overtone order, n, is to be
analyzed in detail. Such detailed analysis might be needed to
derive the viscoelastic properties of the sample. A more
rigorous relation is
ZL ¼ �iZq tan pDf �fF
� �ð4Þ
Eqn (4) is implicit in Df* and must therefore be solved
numerically. There are approximate equations, which are
explicit but still closer to the exact numerical solution of
eqn (4) than the SLA.56
More generally, the limits of the equivalent circuit represen-
tation can be noticed quite severely when the three-dimensional
pattern of motion matters. This concerns, for instance, energy
trapping. Energy trapping may—somewhat artificially—
be introduced into the BvD circuit by replacing the area of
the plate, Aq, by an ‘‘active area’’. The active area is defined via
an integral over the amplitude distribution.
Aq ¼Z
crystal surface
u20ðrÞu20;max
d2r ð5Þ
Here u0 is the amplitude of the lateral displacement. When
probing the resonance properties via impedance analysis, the
active area is experimentally accessible via the relation57
Aq ¼np
32f 2FZqd226
1
R1Qð6Þ
The active area is approximately (but not strictly) the same as
the area of the back electrodes.58 In particular, it decreases
with overtone order because the efficiency of energy trapping
increases with overtone order.59
Energy trapping not only confines the active area to the
center of the resonator, but also entails a flexural contribution
to the mode of oscillation.60 Such distortions lead to the
emission of compressional waves.52,61,62 Equivalent circuits
can never capture the 3D pattern of motion. They are based on
plane waves. Importantly, deviations from pure thickness-
shear may be induced by the sample. For instance, the energy
trapping increases when the contact between the crystal and
the sample only occurs in the center.58 The sample then
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increases the lensing effect of the key-hole shaped back
electrode and thereby increases energy trapping.63,64
An analysis of the crystal’s 3D deformation will usually
require finite-element-method (FEM) simulations.65 FEM
software packages are now becoming increasingly available
and their use does not require extensive expertise. FEM
simulations on quartz crystals have been done in detail for
frequency control applications.66–69 In the context of sensing,
they are just emerging60,70 and it is worthwhile to put them
into the context of the SLA. If the sample is heterogeneous on
the scale of the crystal itself, there is no way around a full-
fledged simulation of the entire crystal in its environment.
Such simulations are demanding. If, however, the sample is
heterogeneous on a small scale, the problem can be simplified.
Assume that the sample contains small bubbles, a rough-
surface, colloidal spheres, or other objects with a size in the
micron range. In order to predict Df*, it is not necessary to
simulate the entire crystal. In fact, that would be difficult
because the mesh would have to be very fine and very large at
the same time. It suffices to simulate the crystal surface and its
immediate environment. Formally, the crystal is introduced
into the simulation as a tangentially moving wall. One first
computes the average stress onto the wall, hsi, and then
calculates Df* from the SLA. Thereby, the simulation itself
is limited to the microscopic scale.
III. Planar layer systems
Assumptions
For a number of configurations, there are explicit expressions,
relating Df* to the sample properties.71–74 The assumptions
underlying these equations are the following:
(a) The resonator and all cover layers are laterally homo-
geneous and infinite. For instance, roughness61,62 and air
bubbles75 at the surface are ignored. Nanoscopic air bubbles
at hydrophobic surfaces immersed in water have been claimed
to explain the slip.76
(b) The distortion of the crystal is given by a transverse plain
wave with the k-vector along the surface normal (thickness-
shear mode). There are neither compressional waves61,62 nor
flexural contributions to the displacement pattern.60 There are
no nodal lines in the plane of the resonator.77
(c) All stresses are proportional to strain. Linear visco-
elasticity holds.78
The last assumption is appropriate when the sample is a
planar layer system. The amplitude of oscillation, u0, and the
amplitude of stress, s0, are given by57
u0 ¼4
ðnpÞ2d26QUel
s0 ¼ioZLu0 ¼ 2p2nZqDfu0
ð7Þ
with d26 = 3.1 � 10�12 m V�1 the piezoelectric strain
coefficient, Q the quality factor, and Uel the amplitude of
electrical driving. Assume a driving voltage of 176 mV (�5dBm),79 a Q-factor of 100 000 (typical for experiments in air),
and a frequency shift of Df = �1 kHz. These values result in
u0 = 22 nm and s0 = 4 kPa. This stress is well below the yield
stress of most materials. Nonlinear stress–strain relations are
not usually observed (unless the stress is concentrated at point
contacts, see section VI).
In liquids, the oscillation amplitude is even lower than in air
because of the lower Q. While one might think that nonlinear
effects should be entirely irrelevant in liquids, there are scat-
tered reports of effects only occurring at high drive levels.
Particles have been ‘‘shaken off’’ by the oscillating crystal
surface.80 Also, the adsorption of vesicles was slowed down by
oscillating the crystal at a high amplitude.81 Finally, acoustic
second harmonic generation (scaling as the square of the
amplitude, u0) has been observed on rough surfaces.82
Semi-infinite viscoelastic medium
For a viscoelastic, semi-infinite medium, one has
Df �
fF¼ i
pZq
s_u¼ i
pZqZac ¼
i
pZq
ffiffiffiffiffiffiffiffiffiffiiroZ
p
¼ 1
pZq
�1þ iffiffiffi2p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiroðZ0 � iZ00Þ
p¼ i
pZq
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirðG0 þ iG00Þ
p ð8Þ
Z= Z0 � iZ00 is the viscosity, G= ioZ is the shear modulus, Zac
= rc = (Gr)1/2 is the acoustic impedance of the medium, r is
the density, and c is the speed of sound. Df and DG are equal
and opposite for Newtonian liquids (Z0= const, Z00= 0). They
both scale as n1/2. For non-Newtonian liquids (Z0 = Z(o), Z00
a 0), one can derive the complex viscosity from QCM data via
Z0 ¼G00
o¼ �
pZ2q
rliq fDfDGf 2F
Z00 ¼G0
o¼ 1
2
pZ2q
rliqfðDG2 � Df 2Þ
f 2F
ð9Þ
The QCM only probes the region close to the interface. The
shear wave decays into the liquid with a penetration depth of d= (2Z/or)1/2. In water, the penetration depth at f = 5 MHz is
about 250 nm. There are a number of effects interfering with
the measurement of the viscosity, which originate from nano-
scopic air bubbles, roughness, slip, and compressional waves.
Typically, the reproducibility of viscosity measurements be-
tween different crystals and mounts is in the range of �10%.
Also, the high-frequency viscosity is different from the visco-
sity in steady shear. Torsional resonators, which oscillate at
around 100 kHz, are closer to most technical applications in
this regard.41 In liquids, the amplitude of motion is small
enough to ensure linear behavior. The QCM cannot measure
the high-shear viscosity, unless one claims that the behavior at
high frequencies mimics the behavior at high shear according
to the Cox-Merz rule.83
Interfacial viscoelastic properties of polymers are very
interesting in the context of adhesion and friction of rub-
bers.84,85 Using the QCM for this purpose entails a difficulty,
because most polymers are so viscous that they just overdamp
the crystal. Such measurements are only feasible if the contact
area is confined to a small spot in the center of the plate. This
mode of measurement was first proposed by Flanigan et al.,
who combined the QCM with a JKR tester.58 The term
‘‘JKR’’ stands for Johnson, Kendall, and Roberts, who
formulated the contact mechanics model underlying this
4520 | Phys. Chem. Chem. Phys., 2008, 10, 4516–4534 This journal is �c the Owner Societies 2008
instrument.86 In a JKR tester, a hemisphere of the material
under test is pushed against a flat substrate. The JKR tester
determines the contact area and the compression as a function
of the vertical load. By applying the JKR model to these data,
one derives the elastic modulus of the material and the energy
of adhesion. Flanigan and coworkers have replaced the flat
substrate by a quartz resonator and used the frequency shift as
an additional source of information. Their working hypothesis
was that the frequency shift should be roughly proportional to
the area of contact, Ac. Such a scaling is expected if the
diameter of the contact is much larger than the wavelength
of sound. We term this model ‘‘sheet-contact model’’. The
sheet-contact model contrasts to the ‘‘point contact model’’
(section V), which applies in the opposing limit of small
contact radii.25,87 Combining the JKR tester with a QCM,
one obtains the near-surface moduli of the material (at
5 MHz) in addition to the bulk moduli.
Inertial loading (Sauerbrey equation)
The shift of frequency produced by a sample which is rigidly
coupled to the resonator (such as a thin film or an assembly of
nano-sized particles) is predicted by the Sauerbrey equation.
The stress, s, is governed by inertia. One has s = �o2u0mf,
where mf is the area-averaged mass per unit area. In the
following, the index ‘‘f’’ denotes the ‘‘film’’. Inserting this
stress into the SLA (eqn (3)), one finds
Df �
fF� i
pZq
�o2u0mf
iou0¼ � 2f
Zqmf ¼ �n
mf
mqð10Þ
n = f/fF is the overtone order and mq = Zq/(2fF) is the mass
per unit area of the crystal. If the density, rf, is known, one canderive the film thickness, df, from the mass per unit area of the
film, mf. The thus derived thickness is also called ‘‘Sauerbrey
thickness’’ in order to emphasize that it was derived via the
Sauerbrey equation.
Viscoelastic films
The frequency shift produced by a viscoelastic film in air is
Df �
fF¼ �Zf
pZqtanðkfdfÞ ð11Þ
Zf is the acoustic impedance of the film (Zf = (rfGf)1/2 =
rf cf), kf is the wave vector, and df is the thickness of the film.
The first pole of the tangent (at kfdf = p/2) defines the film
resonance.53,88–90 For thin films (kfdf { 1), perturbation
theory yields the following approximate result56
Df �fF¼ � 2f
Zqmf 1þ 1
3Jf
Z2q
rf� 1
!mf
mqpn
� �2" #
ð12Þ
The first term in square brackets is the Sauerbrey contribution,
while the second is the viscoelastic correction. Jf = 1/Gf is the
viscoelastic compliance of the film.
For thin films, a nonzero DG always results from viscous
dissipation. This is in contrast to the frequency shift, which has
mixed contributions from inertia and elastic softness. DG and
the viscous compliance, J00, are related by91
DG � 8
3rfZqf 4f m
3f n
3p2J 00 ð13Þ
For a viscoelastic film inside a liquid environment, the
frequency shift is92,93
Df �
fF¼ �Zf
pZq
Zf tanðkfdfÞ � iZliq
Zf þ iZliqtanðkfdfÞð14Þ
The reference state here is the crystal immersed in air. Some
researchers use the ‘‘Voigt model’’ to describe the same
situation. As shown in the Appendix, the Voigt model is
equivalent to eqn (14).
For thin films, eqn (14) can be Taylor-expanded to the first
order in df, yielding
Df �
fF� �omf
pZq1�
Z2liq
Z2f
!
� �orfdfpZq
1� Jfn2pifFrliqZliq
rf
� �ð15Þ
Apart from the term in brackets, eqn (15) is equivalent to the
Sauerbrey equation (eqn (10)). The term in brackets is a
viscoelastic correction. Softness reduces the Sauerbrey thick-
ness.
For films immersed in a liquid, the viscoelastic correction
scales asmf, whereas it scales as the 3rd power ofmf for films in
air (eqns (13) and (15)).94 In air, the film shears under its own
inertia, whereas in liquids, the film is ‘‘clamped’’ on its outer
side by the liquid. Since mf scales as �Df, the ratio DG/(�Df) isa materials parameter, independent of thickness. In the low-
thickness limit one has95
DG�Df � oZliqJ
0f ð16Þ
For two-layer systems91 and three-layer systems in va-
cuum,93 there are explicit equations predicting Df*, but theseare lengthy. A numerical calculation based on the multilayer
formalism92 may be easier. The multilayer formalism is itera-
tive and covers arbitrarily complex planar assemblies. Con-
tinuous viscoelastic profiles—typically described by the
function G(z)—can be approximated as multilayers with a
small slab width.92 There is an easier way to deal with
continuous profiles, which amounts to the solution of the
initial value problem
d
dzGðzÞ duðzÞ
dz
� �¼ �o2rðzÞuðzÞ
uðz ¼ 1Þ ¼ 1
du
dz
����z¼1¼ �
ffiffiffiffiffiffiffiffiffiffior1iZ1
ruðz ¼ 1Þ
ð17Þ
Toolboxes like Mathematica or Matlab readily solve such
ordinary differential equations. The load impedance is calcu-
lated from u(z = 0) and du/dz|z=0 as ZL = G(du/dz)/(iou).The initial condition is given by the amplitude, uN, and its
derivative, du/dz|z=N, at some location z far outside the
region of interest. The integration then occurs from ‘‘infinity’’
This journal is �c the Owner Societies 2008 Phys. Chem. Chem. Phys., 2008, 10, 4516–4534 | 4521
down to z = 0. The amplitude at infinity, uN, can be chosen
arbitrarily since it cancels in the calculation of ZL. The
derivative at infinity, du/dz|z=N, is inferred from the viscosity
of the bulk. At infinity, the bulk is homogeneous and the
solution u(z) will be of the form uNexp(�ikNz), with
kN = (orN/iZN)1/2. The derivative at infinity is therefore
du/dz|z=N = �ikNuN.
Determination of the shear modulus of thin films
Since Df* depends on the shear stiffness of the sample, one
can—at least in principle—derive the shear modulus of a film
from QCM data. However, there are certain caveats to be kept
in mind:
(a) It must never be forgotten that the derived moduli
pertain to MHz frequencies.96 For a typical polymer (assum-
ing time–temperature superposition) the QCM operates at an
equivalent temperature far below 0 1C.
(b) The viscoelastic parameters just about always depend on
frequency. The exact viscoelastic spectrum is usually not
known. Additional assumptions are needed when comparing
Df and DG from different overtones. The Chalmers group uses
a frequency dependence of the form G0 B G00(1 + xm(n � 1))
and Z0 B Z00(1 + xZ(n � 1)) with xm and xZ fit parameters.97
An alternative choice is G0 B G00(o/oref)a and G00 B
G000(o/oref)b, where the exponents a and b are fit parameters.
The exponents are the slopes in a log–log plot. Fitting QCM
data without viscoelastic dispersion is an exercise with limited
practical relevance.
(c) Electrode effects can be of importance. For films in air, in
particular, the SLA must be replaced by the corresponding
results from perturbation theory56 unless the films are
very soft.
(d) It is a wide-spread belief that the determination of
thickness is reliable even in situations where the results for
G0 and G00 may be doubted. That is not always correct. Fig. 3
shows a counter-example. These data were taken on a thick,
surface-adsorbed polymer gel. The details of the sample pre-
paration are unessential. Given that the sample presumably
has a gradient in its viscoelastic properties, one may take a
modest stance and say: let’s not attribute much significance to
the derived viscoelastic parameters, let’s just discuss the thick-
ness. However, for soft films with a thickness exceeding
100 nm, even the derived thicknesses have larger error bars
if one allows the viscoelastic constants (and their dependence
on overtone order!) to vary. The lines in Fig. 3 are both
reasonable fits, and the derived thicknesses differ by 25%.
(e) The calculation of the shear modulus from QCM data
becomes much easier if the film thickness is known indepen-
dently. As always, removing a free parameter from a fit
reduces the error bars on the remaining ones. In this particular
case, knowledge of the thickness indeed helps a lot. While one
might think that a combination of the optical and acoustic
techniques would always provide such a measure of thickness,
the ‘‘optical thickness’’ is not necessarily a geometric thick-
ness, either, because it entails an assumption on the film’s
refractive index. For films formed on the surface of an EQCM
via electropolymerization, the film thickness can be inferred
from the total charge passed through the electrode during film
growth.98 Note, however, that this procedure also needs
assumption, which is a value for the deposited mass per unit
charge (current efficiency).99
Physical interpretation of the Sauerbrey thickness
Since the Sauerbrey thickness is so easily obtained, it is
frequently discussed as a measure of film thickness. However,
the Sauerbrey thickness must not be naively identified with the
geometric thickness. This is particularly true in liquids where
viscoelastic effects and roughness can play a large role. The
following considerations can aid in the correct interpretation:
(a) On a fundamental level, the QCM is sensitive to the areal
mass density, mf, not to the geometric thickness, df. The
conversion from mf to df requires the density, rf, as an
independent input.
(b) If the viscoelastic contribution is significant, it will
usually induce a sizeable increase in DG and, also, lead to an
overtone-dependent Sauerbrey thickness. Therefore it is al-
ways advised to compare thicknesses derived on different
overtones. If the dissipation is small and if, further, Df/f is
the same for all harmonics, one may conclude that the
Sauerbrey equation applies. More specifically, the term
(1 � Zliq2/Zf
2) in eqn (15) is then close to unity.
(c) The Sauerbrey equation ignores roughness and hetero-
geneity.
(d) Samples with fuzzy outer interfaces should be modeled
within the multilayer formalism. A fuzzy interface between a
film and a bulk liquid will often lead to a viscoelastic correc-
tion and, as a consequence, to a non-zero DG as well as an
overtone-dependent Sauerbrey mass. In the absence of such
effects, one may conclude that the outer interface of the film
is sharp.
(e) If the viscoelasticity is of minor influence (as discussed in
b), the film may still be swollen in the solvent. A small
viscoelastic correction only implies that the film is more rigid
Fig. 3 Experimental data and fits with the acoustic multilayer model.
The sample was a thick (4100 nm), surface-adsorbed polymer gel. The
full line is a fit with df = 140 nm, rf = 1 g cm�3, G0f = 2.2 MPa �(f/fref)
1.5, and tan(d) = 1.2 � (f/fref)1.5. The dashed line is a fit with df
= 180 nm, rf = 1 g cm�3, G0f = 1.6 MPa � (f/fref)1.7, and tan(d) =
0.36 � (f/fref)1.63. The reference frequency is 35 MHz in both cases.
Both fits are reasonable and it is hard to tell from the QCM data which
of the two acoustic thicknesses (140 or 180 nm) is closer to the true
geometric thickness.
4522 | Phys. Chem. Chem. Phys., 2008, 10, 4516–4534 This journal is �c the Owner Societies 2008
than the bulk. The degree of swelling cannot be determined by
QCM measurements on the wet sample alone; one needs to
compare wet and dry thickness.
Some indicator of the degree of solvation is also obtained by
comparing QCM data to the results of surface plasmon
resonance (SPR) spectroscopy or ellipsometry.100,101 Since
the solvent takes part in the motion of the film, it does
contribute to the acoustic thickness. The solvent’s contribu-
tion to the optical thickness is much smaller because the
optical polarizability of a solvent molecule remains (almost)
unchanged when it enters the film. The integral polarizability
of the film is therefore weakly affected by swelling. In more
mathematical terms, the swollen film is an ‘‘effective medium’’.
The ways in which the properties of this medium are calculated
from the properties of the constituents are the same—with
regard to the mathematics—in optics and in acoustics, but
they yield much more different results after inserting the
numbers. For a more rigorous discussion of these differences
see ref. 102.
(f) As a final note, the competing techniques of thickness
determination are plagued by similar difficulties. This particu-
larly concerns optical reflectometry such as SPR spectro-
scopy,103 ellipsometry,104 or dual polarization inter-
ferometry.105 These techniques also work in the thin-film limit
(termed ‘‘large wavelength limit’’,106 as well), where the re-
fractive index (the analogue of Zf) and the geometric thickness
cannot be independently extracted from the data. Since there is
no easy way of predicting the refractive index of a dilute
adsorbed film, many researchers display the raw data (for
instance, in ‘‘reflectance units, RU’’) and compare data sets.
One would typically show a data set where ‘‘full coverage’’ has
been achieved in one way or another and normalize all other
results by the shift in RUs obtained with this reference sample.
The only technique to truly overcome the thin-film limit is
neutron reflectometry.107 X-Ray reflectometry also circum-
vents the thin-film limit, but is not easily applied in liquids.
IV. Dielectric and conductive loads
Many workers would view electrical interactions between the
crystal and its environment as a source of artifacts. In order to
stay away from such problems, grounding the front electrode
is always advised. p-Networks are employed for the same
purpose.108 A p-network is an arrangement of resistors, which
almost short-circuit the two electrodes. This makes the device
less susceptible to electric perturbations. Evidently, short-
circuiting must be incomplete, because the resonator would
otherwise be electrically separated from the testing equipment.
Poor grounding has two consequences. Firstly, there is the
danger of uncontrolled electrochemical processes at the front
electrode. The resonance frequency then drifts. There is a
second, more subtle consequence of electric contact between
the crystal and the sample, which has to do with piezoelectric
stiffening. Since the strain energy of a piezoelectric material
has an electrical contribution, the stiffness of the plate depends
on whether or not the shear-induced surface polarization is
compensated by an external charge. If the surface is fully
covered with electrodes, the external current—at least
partly—compensates for the surface polarization. The energy
of the electric field decreases accordingly. The contribution of
the electric strain energy to the total strain energy can amount
to up to 0.7% for AT-cut quartz. This translates to a fractional
frequency shift of 0.35% (16 kHz for a 5 MHz crystal).
Clearly, the effect is sizeable. Piezoelectric stiffening is well-
known in the frequency control community, because it can be
used to ‘‘pull’’ the resonance frequency with an external
capacitor.1 The additional capacitor partly short-circuits the
electrodes, thereby softening the crystal and lowering the
resonance frequency.
For the laterally homogeneous plate, piezoelectric stiffening
is accounted for by the element ZPES in the BvD circuit
(Fig. 2b). For the electrode-covered plate, ZPES is
ZPES ¼ �4f2 U
I¼ �4f2 1
ioC0¼ �4 Aqe26
dq
� �21
ioC0ð18Þ
where the parameter f translates between mechanical impe-
dances (in units of force/speed) and electrical impedances (in
units of voltage/current, cf. eqn (2)). C0 = Aqee0/dq is the
electrical (‘‘parallel’’) capacitance between the electrodes.
Importantly, the value of ZPES differs from eqn (18) if part
of the active region is not covered by the electrode. Consider
the geometry depicted in Fig. 4, where the front electrode has a
central hole of radius 1 mm.30 Inside the hole, the surface
polarization is uncompensated, which generates an excess
electric field both inside the crystal and in the adjacent sample.
The excess field creates a stress, which leads to an increase in
frequency. Should the sample, however, be a dielectric or
conductive medium, the polarization inside the sample will
partly compensate for the electric field, thereby decreasing the
field energy as well as the resonance frequency. A shear-
induced electric polarization of the sample can therefore be
described as a load.
An accordingly expanded equivalent circuit is shown in
Fig. 2c. The motion of the crystal couples to an electrical
impedance (U/I) at the crystal surface. The coupling
Fig. 4 Results from an FEM simulation of the electric field emanat-
ing from a small hole in the center of the resonator. The line on the
left-hand-side is the axis of rotational symmetry. The top and the
bottom electrodes have potentials of 176 and 0 mV, respectively.
Inside the hole (top left) there is a surface charge of 0.8 nC cm�2.
The lines denote constant electrostatic potential.
This journal is �c the Owner Societies 2008 Phys. Chem. Chem. Phys., 2008, 10, 4516–4534 | 4523
coefficient, fh, is different from f in eqn (1) because the
relevant area is the area of the hole, Ah, (rather than Aq).
The element ZPES is
ZPES � �4A2
he226
d2q
hUihI� � 4e226
d2q
AhhUihiosS
ð19Þ
The second relation makes use of I = ioAhsS with sS as thesurface polarization. In the spirit of the SLA, the (spatially
variable) voltage U was replaced by its average, hUih. Theaverage is taken over the hole, only. On resonance, the voltage
on the bare part of the front surface,U, is much larger than the
external voltage applied to the electrode, Uext, and one may
neglect the electrode in the calculation of hUi/I. We assume the
hole to be much smaller than the active area, Aq. The
amplitude of oscillation, u0, and the surface polarization, sS,are then constant inside the hole and averaging on sS is not
needed.
When hUih shifts due to the presence of the sample, the
corresponding frequency shift is calculated as:
Df �
fF¼ i
pZq
DZPES
Aq¼ � 1
pZq
4e226d2q
Ah
oAq
hDUihsS
¼ � i
pZqAq4f2
h
hDUihI
ð20Þ
Eqn (20) contains the term Aq in the denominator (cf.
eqn (3)) because ZPES is a force–speed ratio, whereas ZL is a
stress–speed ratio. The last transformation on the right-hand-
side was inserted in order to emphasize that the Df is propor-tional to the change of the electrical impedance of the surface
hDUih/I.This leaves the problem of calculating hUih from the surface
charge, sS. Since the geometry is non-planar, there are no
simple analytical solutions. However, finite-element method
(FEM) calculations readily yield a result. We used the software
package COMSOL (COMSOL Multiphysics, Gottingen, Ger-
many). Fig. 4 depicts the geometry. The vertical line on the left
is the axis of symmetry (r = 0). The model contains three
domains which are the ambient medium (vacuum or air, e= 1,
where e is the relative dielectric permittivity), the crystal (e =4.54), and a film situated on top of the central hole (df =
0.05. . .10 mm, e = 1. . .70, not discernible in Fig. 4). The
boundary conditions consist of a grounded bottom electrode,
a top electrode with U= Uext = 176 mV, and a surface charge
inside the hole. The external voltage of 176 mV corresponds to
a drive level of DL = �5 dBm according to U = 0.314 �10(DL[dBm]/20) V. The value of the surface charge results from:
sS ¼ e26u0
dq¼ e26
4
dqðnpÞ2Qd26Uext ¼
4e226
GqdqðnpÞ2QUext ð21Þ
The last identity makes use of the relation d26 = e26/Gq.
Assuming Q = 125 000 and dq = 320 mm, one finds sS =
0.8 nC cm�2.
The boundary conditions together with the Poisson equa-
tion define a system of partial differential equations, which is
readily solved in a few seconds. Tests with a refined mesh
(increasing the computation time by a factor of 100) produced
the same results. Fig. 4 shows lines of equal potential for a
typical solution. The integral of the voltage over the area of
the hole is computed in the post-processing mode. For display
purposes, the integral was transformed to the average voltage
via AhhUih =RU(r) d2r. A typical value for hUih is 50 V.
Importantly, the potential inside the hole, hUih, slightly
decreases in the presence of a dielectric film because the latter
screens the electric field. Fig. 5 displays the difference in the
surface potential, hDUih, as a function of the hole radius, r
(panel a), the thickness of the film, df, (panel b), and the film’s
dielectric constant, e (panel c). The film’s polarizability there-
fore constitutes a load. There is a dependence of hDUih on the
size of the hole. hDUih is the largest when the diameter of the
hole matches the thickness of the crystal. Not surprisingly,
hDUih is proportional to the film thickness and to the film’s
polarizability, where the latter is proportional to e � 1:
hDUihUext
� 0:15
mmðe� 1Þdf
hDUihI� 32
ohm
mmðe� 1Þdf
ð22Þ
The numerical factors pertain to the particular electrode
geometry under consideration, which was a central hole with
a radius of 1 mm. The electrode shapes investigated in ref. 30
would each require a new run of the FEM simulation. Using
Fig. 5 Shift of the average voltage, hDUih, at the surface of the crystalinduced by a dielectric film which covers the hole and thereby partially
screens the field. (a) hDUih versus hole radius for a film with thickness
df = 10 mm and a dielectric permittivity of e = 5. (b) hDUih versus thefilm thickness for a hole radius of 1 mm and e = 5. (c) hDUih versus
e � 1 for a 10 mm film and a hole radius of 1 mm. The film constitutes a
dielectric load. The shifts in the surface potential induced by the film
translate to a frequency shift via eqn (20).
4524 | Phys. Chem. Chem. Phys., 2008, 10, 4516–4534 This journal is �c the Owner Societies 2008
eqn (20) with fF = 5MHz, Aq = 1 cm2, and dq = 320 mm, one
finds that a voltage of hDUih = 1V corresponds to Df =
�7 Hz. Such frequency shifts are well measurable.
With regard to sensing applications, the following remarks
come to mind.
(a) Since eqn (20) contains the frequency in the denomi-
nator, piezoelectric stiffening decreases with overtone order.
Dielectric measurements will therefore be typically carried out
on the fundamental. For the same reason, they should be
stronger when employing torsional resonators. Their fre-
quency is about a factor of 100 below the frequency of the
thickness-shear resonators. The 1/o–scaling can be used to
distinguish dielectric effects from inertial effects (scaling as o).(b) For dry polymer films, the dielectric properties of the
film will be of small influence on Df. As Fig. 5b shows, a 1 mm-
film with e = 5 induces a voltage of hDUi B 0.1 V, which
translates to Df B �0.7 Hz. This value is much smaller than
the inertial term and—depending on the softness—smaller
than the elastic correction (cf. eqn (12)), as well.
(c) Dielectric effects should be noticeable for conductive
films.109 Charge transport may come about by electronic or
ionic conductivity. The latter will be dominant in hydrophilic
films which have been soaked in salt solution. For instance, an
ionic conductivity of kel B 1 mS cm�1 will correspond to e00 E300 according to e00= kel/(oe0). The conductive film should be
insulated from the driving electrode, that is, there should be a
gap between the film located inside the hole and the ring-
shaped front electrode.
(d) Eqn (22) also holds for complex dielectric permittivities,
that is, for conductive films. A finite conductivity will increase
the bandwidth (rather than decreasing the frequency), because
e is imaginary.
(e) Dielectric effects are strong in bulk liquids.31 Measuring
the viscosity and the dielectric permittivity in parallel is
attractive when acoustic sensors are used to monitor the
viscosity of engine oil. Deviations from the target value usually
come about because either water or gasoline has leaked into
the system. The two failure scenarios can be distinguished
based on a measurement of the dielectric constant.
(f) There is a particularly interesting study where an EQCM
was immersed in an aqueous salt solution of varying ionic
strength.110 The authors find that Df and DG (XLf and Rf in the
language of ref. 110) shift with ionic strength in a peculiar way.
Plotting Rf versus XLf, the authors find a hemicycle, resem-
bling rather closely the hemicycles in Cole–Cole plots of
electrochemical impedance spectra.111 The authors attribute
the hemicycles to a change in the viscoelasticity of the electric
double layer with changing ionic strength. An alternative
explanation is based on the variable electric capacitance of
the double layer. As the ion concentration is increased, the RC
time of the double layer shifts across the inverse resonance
frequency, which straightforwardly explains the results
(assuming imperfect grounding of the front electrode).
(g) The hole in the front electrode also affects the
parallel capacitance, C0. Modified equivalent circuits
taking this change into account have been proposed.30,31 In
principle, one may even determine the sample’s
dielectric properties via C0. Note, however, that absolute
measurements of the impedance across the electrodes are
orders of magnitude less accurate than measurements of the
resonance frequency.
(h) Front electrodes with finite, variable conductivity have
been employed.109 Such electrodes will lead to a frequency
shift depending on the conductivity because of piezoelectric
stiffening, as well.
(i) Given that the surface potential inside the hole, hUih, canreach hundreds of volts, dielectric saturation and/or various
kinds of electrochemical processes are to be expected in
liquids. In order to avoid such nonlinearities, small drive levels
have to be employed. For instance, the FEM simulations yield
a voltage of hUih B 100 mV, if the Q-factor is 3000 (typical for
water) and the drive level is �35 dBm.
(j) It is emphasized below that nonlinear interactions are
covered by the SLA via eqn (31). One can extend eqn (31) to
the case of non-linear voltage–charge relations. Such nonli-
nearities may come about due to dielectric saturation or
electrochemical processes. Another source of nonlinearity is
triboelectricity.112 In the extension, one makes use of the fact
that a voltage across the crystal, hDUih, generates a stress
equal to e26hDUih/dq. One finds
DffF¼ 1
pZq
2
ou0
e26
dqhhDUðtÞih cosðotÞit
DGfF¼ 1
pZq
2
ou0
e26
dqhhDUðtÞih sinðotÞit
ð23Þ
V. Point contacts
Section III concerned samples, which—in essence—are planar.
More precisely, the acoustic wave entering the sample was
assumed to be a plane wave and the stress was calculated as
s= �G du/dz. For a heterogeneous sample, the calculation of
the average surface stress can be fairly complicated. There is a
second simple case, which are ‘‘point contacts’’. A point
contact is defined by the conditions rc { R and rc { l (rcthe radius of contact, R the local radius of curvature of the
contacting object, and l the wavelength of sound). A sphere-
plate contact would be a typical point contact in this sense.
Rough interfaces between sufficiently stiff materials also
behave in a similar way.
The mechanics of the contacts between rough surfaces is of
outstanding importance in such diverse disciplines as geophy-
sics,113 the food industry,114 mechanical engineering,115
MEMS technology,116 dry and wet granular media,117 tribo-
logy.36 As Boden and Tabor have emphasized, an understand-
ing of friction phenomena must include the microscopic
scale.118 What one typically encounters is a multi-asperity
contact, also termed multi-contact interface. The true contact
area is not easily determined. Macroscopically amenable
parameters related to the true contact area are electrical
conductivity,119 optical transmission,120 ultrasonic reflectivity
(employing conventional longitudinal waves),121 or lateral
contact stiffness under low frequency excitation.122 The mea-
surement of contact stiffness based on thickness-shear resona-
tors proposed below is part of this family of techniques. To
This journal is �c the Owner Societies 2008 Phys. Chem. Chem. Phys., 2008, 10, 4516–4534 | 4525
what extent the frequency shift correlates to the rupture force
or the coefficient of static friction will have to be seen.
The argument outlined below is simplistic in the sense that it
ignores the tensorial nature of the stress. It still illustrates the
essential argument. For point contacts, the acoustic wave is
spherical rather than planar:
uðrÞ � ucrc
re�ikr for r4rc
uðrÞ � uc for rorc
ð24Þ
where r is the radial coordinate originating at the point of
contact, k is the wavenumber, and uc is the displacement inside
the contact. Far away from the contact, the displacement
vanishes. The sphere (or, more generally, the contacting
object) rests in place in the laboratory frame due to inertia.
Via the contact, it exerts a restoring force onto the oscillating
crystal. Outside the contact area (r 4 rc) the stress can be
approximated as
s � Kru � Krc �1
r2� ik
r
� �e�ikru0 for r4rc ð25Þ
Here K is an effective modulus of the order of the shear
modulus. Again, K and s would have to be tensors in a more
rigorous description. For the sake of this simple estimate, we
approximate the stress inside the area of contact, sc, as
constant and equal to the stress at r = rc
sc � �K
rcu0ð1þ ikrcÞ ð26Þ
The force transmitted across the contact is about Krcu0. One
defines a spring constant as kS* = �prc2sc/u0 B pKrc(1 +
ikrc). The area-averaged impedance becomes ZL B kS*/(ioAq)
and the SLA yields25
Df �
fF¼ 1
pAqZq
k�So¼ 1
2p2AqZqfF
1
nk�S ð27Þ
kS has acquired a star in eqn (27) because it is complex. The
imaginary part may either be related to viscous dissipation in
the contact zone or to the term ikrc in eqn (26). The latter
accounts for the withdrawal of acoustic energy via sound
waves. Evidently, eqn (27) can be extended to layers of spheres
rather than single spheres. One then replaces 1/Aq by the
number density of contacts, NS. If the real part of the spring
constant, kS, is independent of frequency, eqn (28) predicts a
scaling of frequency shift with overtone order as Dfp n�1 (see
Fig. 7). For the bandwidth, it is more complicated. If the
bandwidth is mostly due to radiation of sound, eqn (28) and
(27) predict Df p n0, because the wavenumber (k in eqn (27))
scales as n. However, simple scaling relations are not usually
found in experiment, suggesting that friction processes at the
point of contact contribute.
An equation similar to eqn (27) was already derived by
Dybwad in 1985.25 Dybwad did not attempt continuum
modeling. He just assumed that the contact could be described
as a Hookean spring. In the Dybwad model, there is no
assumption about the mass being so large that it rests in place
in the laboratory frame. He assumes a sphere of mass mS being
coupled to the resonator via a spring with stiffness kS. Solving
the equation of motion leads to25,123
Df þ iDGfF
� 1
pAqZq
�mSo1� o2=o2
S
¼ 1
pAqZq
ok�So2 � o2
S
ð28Þ
Here oS = kS/mS is the characteristic frequency of the
sphere–plate contact. This derivation is interesting because it
entails both the Sauerbrey limit (small mass, oS c o) and the
point contact limit (large mass, oS { o). In the Sauerbrey
limit, the sphere is almost rigidly attached to the crystal and
takes part in its motion. The ‘‘sphere’’ might be a single
molecule. In the opposing limit of point contacts, the sphere
is ‘‘clamped’’ by inertia.
If the radius of contact, rc, is much smaller than the
wavelength of sound 2pk�1, the wavelength disappears from
the problem. The contact is then quasi-static. The Hertzian
sphere–plate contact under static lateral load has been ana-
lyzed by Mindlin.124,125 The lateral spring constant is found
to be
kS ¼ 4Keffrc
1
Keff¼ 1
2
2� n1G1
þ 2� n2G2
� � ð29Þ
Here Z is Poisson’s number and the indices 1 and 2 label the
contacting materials. With known viscoelastic properties, a
measurement of the frequency shift therefore allows to deter-
mine the radius of contact.
While one might think that an AFM would be a suitable
instrument to induce and study point contacts under MHz
shear, the numbers turn out to be too small. With fF B 5
MHz, Aq B 1 cm2, rc B 1 nm, and Keff B 10 GPa, eqn (29)
and (27) predict a frequency shift of 2 � 10�3 Hz. Colloidal
probe experiments,126,127 on the other hand, are feasible.
Consider a high-frequency crystal of the inverted mesa type128
with a fundamental frequency of 155 MHz, an effective area of
Aq B 1 mm2, and a contact radius of about 100 nm. These
values lead to Df B 20 Hz, which is measurable.
Fig. 6 shows two examples of colloidal probe measurements.
The crystal was of the inverted mesa type. The fundamental
frequency was fF = 155.9 MHz. The bandwidth in the
unloaded state was G0 = 2.2 kHz, corresponding to a Q-factor
ofQ= f/(2G0) = 35 000. The crystal was driven with a voltage
of 0.32 V, corresponding to an amplitude of 13 nm. The
central quartz membrane is 10 mm thick. The active area, as
determined by eqn (6), was about 1 mm2. The crystals were
mounted on the sample stage of a MultiMode scanning probe
microscope.126,127 Polystyrene spheres with a diameter of 10
mm were attached to the AFM tip (fres = 30 kHz, k B 40 N
m�1 according to the manufacturer) as described in ref. 127.
Force–distance curves were acquired with a ramp rate of 2 nm
s�1. Df and DG were acquired in parallel to the force–distance
curve. Fig. 6 shows two data sets, which were taken under dry
conditions (where the entire AFM was covered with a plastic
bag and a vessel of phosphorous oxide was placed into this
compartment) and under ambient conditions. In the latter
case, the humidity was about 30% rH (rH, relative humidity).
When contact is established, both the resonance frequency
and the bandwidth increase. For the experiments in humid air,
the transition is discontinuous, which is explained by capillary
4526 | Phys. Chem. Chem. Phys., 2008, 10, 4516–4534 This journal is �c the Owner Societies 2008
forces. Df is much larger than DG, suggesting that the contact
sticks. For a sliding contact, one would expect DG Z Df (seesection VI, eqn (37)). Stick is not necessarily expected. From
Df, one calculates a lateral spring constant in the range of kS B104 N m�1. With u0 = 13 nm, this translates to a peak lateral
force of about 100 mN. The vertical force can be estimated
from the deflection of about 0.5 mm and the spring constant of
the cantilever, which is around 40 N m�1 according to the
manufacturer. One finds a peak vertical force of about 20 mN,
which is lower than the lateral force by a factor of 5. If the
peak lateral force would have been applied as a steady lateral
load, sliding would have set in. The situation is different at
MHz frequencies. The peak lateral force only persists for a few
nanoseconds, which does not suffice to disrupt the contact.
Oscillation-induced rolling and sliding have been observed
with millimeter-sized spheres.129
For the dry case, Df is proportional to the vertical force, F>.
The external force is much stronger than the force of adhesion
as evidenced by the rather small jump into contact. For single-
asperity contacts, Df should scale as the radius of contact, rc,
which, according to the Hertz model, should scale as F>1/3. A
cubic-root dependence of Df on the vertical force was found in
experiments with macroscopic spheres,26 but is clearly not
observed here. A contact stiffness proportional to the vertical
load is indeed predicted for contacts between rough surfaces
within the Greenwood–Williamson formalism.130
Fig. 7 shows the results of a second contact mechanics
experiment, where the QCM was used to monitor capillary
aging.131 A monolayer of glass spheres (+ = 200 mm) was
deposited onto the crystal at t = 0 (‘‘I’’ in Fig. 7). These dry
contacts had virtually no effect on the frequency of resonance.
Even though the spheres did touch the crystal, the contacts
only transmitted a minute amount of stress. After about 10
min, the sample was exposed to humid air (85% rH), leading
to a substantial frequency increase (‘‘II’’). Capillary forces
strengthen the contacts, as is known from the sand-castle
effect.132,133 Granular media may solidify when they are
wetted with appropriate amounts of water. The menisci
around the points of contact turn a fragile material into an
elastic solid. Interestingly, a further, much stronger increase in
frequency was produced by bringing the humidity back down
(state ‘‘III’’). After having been exposed to water vapor, the
spheres form a cake. The latter transition is reversible: once
the assembly of spheres has been soaked in humid air, one can
go back and forth between the states II and III. Comparing the
frequency shifts on the different overtones, one confirms n�1
scaling. These experiments provide a new approach to the
study of capillary aging. Capillary aging was studied pre-
viously with a rotating drum.131 Note, however, that the
acoustic technique is non-destructive, while the critical angle
in the rotating drum is related to the forces needed to break
interparticle contacts.
VI. Nonlinear force-speed relations
The standard model for analyzing QCM data assumes linear
mechanics. All forces and stresses are proportional to displa-
cement or speed. Such linear behavior is a prerequisite for
equivalent circuits to apply. Contact mechanics, on the other
hand, just about always has non-linear stress–strain relations.
These have two sources, which are, firstly, the strong concen-
trations of stress at the points of contact and at the tips of
cracks and crevices and, secondly, the dependence of the true
contact area on the force. Stress concentrations are of out-
standing importance in fracture mechanics134 and the
stick–slip transition.36 The load-dependence of the contact
area has less dramatic consequences. It is widely observed,
even if the stresses are not particularly high. Consider, as an
example, the Hertzian sphere–plate contact under variable
vertical load, F>.135 F> is proportional to rc3. The compres-
sion, d, is equal to rc2/2R with R as the radius of curvature.
Combining these relations one finds F> B d3/2. The
Fig. 6 Experiments where the QCM was combined with a colloidal
force probe. Polystyrene spheres with a radius of 10 mm were glued to
the tip of an AFM. The shifts of frequency and bandwidth of the
oscillating crystal change when the sphere is pushed against the crystal
surface. Left: experiments in dry air. Right: experiments at 30% rH.
Fig. 7 Shifts of frequency (a) and bandwidth (b) experienced by a
quartz crystal covered with a monolayer of glass spheres (+ = 200
mm) exposed to humid air. States I, II, and III correspond to the initial
state right after deposition, to humid air, and to a dry state reached
after soaking the sample in humid air, respectively. Full line: 5 MHz,
dashed line: 15 MHz, dotted line: 25 MHz, dash-dotted line: 35 MHz
(reproduced with premission from ref. 123, copyright Springer
Verlag). The oblique lines indicate humidity.
This journal is �c the Owner Societies 2008 Phys. Chem. Chem. Phys., 2008, 10, 4516–4534 | 4527
sphere–plate contact forms a nonlinear spring with a differ-
ential spring constant k = dF/dd p d1/2. This increase of theeffective stiffness with load is characteristic of all granular
assemblies, where the exponent (1/2 in the Hertz contact)
varies. It is higher than 1/2 for multi-contact interfaces and
gravel.117
The Hertzian contact is also nonlinear under tangential
load.124 The phenomenon is termed ‘‘micro-slip’’ or ‘‘partial
slip’’. When exerting a lateral force, there is a stress singularity
at the rim of the contact area. The high local stress leads to a
ring-shaped area, inside which the two surfaces slide. The
induced damage pattern is called fretting wear.136 It is found
around contacts experiencing continued oscillatory forces, for
instance due to vibrations. As the stress increases, the sliding
portion of the contact zone increases in size, until it finally
covers the entire contact. At this point, partial slip turns into
gross slip and the contact ruptures.
The quantitative description of partial slip goes back to
Mindlin.125 The reader is referred to ref. 124 with regard to the
derivation. For oscillatory loading with a force F8(t), the
lateral displacement, u(t), is given by
uðtÞ ¼ � 3mSF?8rcKeff
2 1�Fjj;max � Fjj tð Þ
2mSF?
� �23
� 1� Fmax
mSF?
� �23
�1" #
¼� 3
2lS 2 1�
Fjj;max � Fjj tð Þ2mSF?
� �23
� 1� Fmax
mSF?
� �23
�1" #
ð30Þ
where mS is the static friction coefficient and Keff = 2((2 � Z1)/G1 + (2 � Z2)/G2)
�1 is an effective modulus. G is the shear
modulus and the indices 1 and 2 label the contacting materials.
u(t) and F8(t) in eqn (30) change sign at the turning points of
u(t). The quantity 4rcKeff is the lateral spring constant of the
contact in the low amplitude limit. For a proof, set F8(t) equal
to F||,max and Taylor-expand to first order in F||,max. The
characteristic length lS = mSF>/(4rcKeff) = mSF>/k0,M,
termed ‘‘partial slip length’’, was introduced for notational
convenience. lS is defined in analogy to the elastic length le =F>/k (k the lateral spring constant) used by the Paris group to
describe multi-asperity contacts.122 In macroscopic experi-
ments, le is a measure of roughness. lS differs from le in that
it contains the static friction coefficient, mS, as a prefactor.
Fig. 8 shows the force–displacement relation for oscillatory
tangential loading. The curve bends downward because the
part of the contact which sticks becomes smaller and smaller
as the lateral force increases. The energy dissipated per cycle
corresponds to the area inside the loop. The Mindlin model
describes a quasi-static motion in the sense that the dissipated
energy entirely originates from hysteresis. Viscous dissipation
is neglected.137 The central feature of the Mindlin model
(partial slip) occurs for non-Hertzian contacts, as well. For
instance, partial slip also occurs in multi-asperity contacts
because those micro-contacts, which are located at the rim
of the macroscopic contact area, rupture first.138 These con-
tacts experience the largest lateral stress and the smallest
vertical load.
It turns out that the consequences of nonlinearity for Df andDG can be quantitatively predicted as long as the nonlinearities
are small. Since the influence of the sample on the crystal must
be small anyway in order for the SLA to apply, the nonlinear
perturbations of the crystal are always small, as well. As long
as Df/f { 1, the QCM is only weakly perturbed and—in
consequence—only weakly nonlinear (if nonlinear at all).
Using the two-timing approximation, one finds139,140
DffF¼ 1
pZq
2
ou0hsðtÞ cosðotÞit ¼
1
pAqZq
2
ou0hFðtÞ cosðotÞit
DGfF¼ 1
pZq
2
ou0hsðtÞ sinðotÞit ¼
1
pAqZq
2
ou0hFðtÞ sinðotÞit
ð31Þ
Angular brackets denote a time average and s(t) is the
(small) stress exerted by the external surface. For a single
contact, one would replace s(t) by F8(t)/Aq. In essence, the
QCM operates like a lock-in amplifier. Df and DG pick out the
in-phase and the out-of-phase component of the stress.
The function s(t) may or may not be harmonic. For
instance, let s(t) be proportional to the speed (s = ZL _u(t)
= �ZLou0sin(ot)) and let the constant of proportionality be
ZL. This special case leads to eqn (3). One can always test for
linear behavior by checking for a dependence of the resonance
parameters on the driving voltage. If Df and DG are indepen-
dent of amplitude, linear acoustics holds. Note, however, that
quartz crystals have an intrinsic drive level-dependence,78
which must not be confused with nonlinear interactions
between the crystal and the sample.
A set of equations similar to eqn (31) is applied in non-
contact atomic force microscopy.141,142 Here, the tip–sample
interaction perturbs the oscillation of the cantilever. As long as
the tip–sample interaction is weak compared to the
energy contained in the oscillation of the cantilever, the
interaction potential can be reconstructed from the frequency
Fig. 8 Force–displacement relation as predicted by the Mindlin
model. Since the central area, where the contact sticks, decreases with
increasing tangential load, the force increases sub-linearly with dis-
placement. The area under the hysteresis loop is the energy dissipated
per cycle. Reproduced with permission from ref. 9. Copyright Springer
Verlag.
4528 | Phys. Chem. Chem. Phys., 2008, 10, 4516–4534 This journal is �c the Owner Societies 2008
of the cantilever as a function of amplitude and mean vertical
distance.
If the stress, s, does not explicitly depend on time (that is, in
the absence of memory effects and hysteresis) the stress–
displacement relation is quantified by a nonlinear spring
constant, kS(u), and a nonlinear dashpot xS(dot _u). These can
be explicitly reconstructed from the amplitude dependence of
the Df(u0) and DG(u0).29,140 A unique solution is obtained. If,
on the other hand, memory and hysteresis may not be
neglected (which is rather typical, see for instance Fig. 8),
s(t) cannot explicitly be recovered from Df and DG. Still, anyassumption on the time-dependent stress s(t) can be inserted
into eqn (31). The prediction can be compared to the experi-
ment and the model can be refined until the two match. By
means of eqn (31), any given hypothesis about the force F(t)
(including stick-slip and hysteresis) can be checked against the
experiment.
As an example, consider the lateral force as given by the
Mindlin model (eqn (30)). Since the force only weakly perturbs
the motion of the crystal surface, u(t) is almost time-harmonic.
The force F(t) is a function of the displacement u(t). Inversion
of eqn (30) leads to:
FjjðtÞmSF?
¼ �Fjj;max
mSF?� 2þ 1ffiffiffi
2p 1� 2uðtÞ
3lSþ 1�
Fjj;max
mSF?
� �23
!32
24
35
ð32Þ
Inserting eqn (32) into eqn (31) and performing the integra-
tion leads to a prediction for Df and DG. In the small
amplitude limit (u(t) { lS) one finds:
DffF� 1
pAqZq
4rcKeff
o1� u0
6lS
� �
¼ 1
pAqZq
k�
o1� u0
6lS
� �ð33Þ
DGfF� 1
pAqZq
k�
o2
9pumax
lSþ DGus
fFð34Þ
4rcKeff is the spring constant in the small-amplitude limit. The
Mindlin model has vanishing dissipation in the small-ampli-
tude limit. Remember, however, that there is a second source
of dissipation, which is the radiation of sound (cf. eqn (26)).
Radiation of ultrasound was heuristically added into eqn (36)
as a term DGus/fF.
Fig. 9 shows data which can be well interpreted within the
Mindlin model of partial slip. A quartz crystal was covered
with a monolayer of glass spheres (+= 200 mm) and exposed
to humid air. The sample is in state II from Fig. 7. In this
particular experiment, Df and DG were determined by ring-
down.15 The dependence of the frequency and bandwidth on
amplitude is substantial. From the slopes in the plots of Df andDG vs. u0, one infers a partial slip length lS of the order of a
nanometer. The slip lengths derived from Df and DG differ,
indicating that the Mindlin model is not quantitatively applic-
able. This does not come as a surprise because, firstly, one
presumably has a multi-asperity contact, and, secondly, capil-
lary bridges play some role. Rolling (simultaneous release of
contacts on one side and formation of new ones on the other)
may also be part of the picture.143
As a second example, consider a contact which sticks at the
turning points of the oscillation and slips, otherwise. Let the
coefficients of static and dynamic friction in the Coulomb
sense be mS and mD. The stress is then given by
sðtÞ ¼ kSðu� umaxÞAq
after the turning points with
jkSðu� umaxÞjomSF?
sðtÞ ¼ � mDF?Aq
otherwise
ð35Þ
Here umax = �u0 is the displacement at the turning point.
Inserting this stress into eqn (31) and performing the integra-
tion leads to
DffF¼ 1
pAqZq
2
ou02mSF? 1� mDF?
kSu0
� � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2kSu0mSF?
� 1
s
DGfF¼ 1
pAqZq
2
ou02mSF? 1þ 2
mDmS� mDF?
kSu0
� � ð36Þ
Now consider a large amplitude (2kSu0 44 mSF>, kSu0 cmDF>), leading to a contact which slips most of the time
and sticks only briefly around u E umax. These
Fig. 9 Shift of frequency (a) and bandwidth (b) as a function of
amplitude for a quartz plate covered with a monolayer of glass spheres
(+ = 200 mm) versus oscillation amplitude. The data were acquired
via ring-down. There is a strong amplitude dependence. Such an
amplitude dependence is predicted by the Mindlin model of partial
slip. From the slopes, one infers the partial slip length, lS. Reproduced
with permission from ref. 9. Copyright Springer Verlag.
This journal is �c the Owner Societies 2008 Phys. Chem. Chem. Phys., 2008, 10, 4516–4534 | 4529
approximations yield
DffF� 1
pAqZq
4mSF?ou0
1�
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi2mDF?kSu0
s !
DGfF� 1
pAqZq
4mSF?ou0
1þ 2mDmS
� � ð37Þ
In this limit, the DG should be larger than Df, which is at
variance with the findings shown in Fig. 6. Interestingly, one
finds Df B DG in the limit of very large amplitude and
mD { mS. Whether or not this limit exists in experiment needs
to be seen.
VII. Loads consisting of spatially separated
adsorbed objects
Today, applications of the QCM in the life sciences abound.
Recent reviews are to be found in ref. 144–148 Many of these
studies are concerned with intermolecular interactions and
biorecognition. The topic per se is of immense complexity
(and beyond the scope of this review). For such studies, the
QCM is usually operated as a gravimetric device. Its strength
is in the ease of operation. Detailed understanding is mostly
gained from the comparison of different (potentially very
many) samples and different types of surface preparation (as
opposed to a highly sophisticated data analysis). The follow-
ing remarks touch upon some specific aspects of modeling and
interpretation, deserving consideration when the size of inter-
est is in the mesoscopic range (a few nm). In this size range,
advanced data analysis is worthwhile.
There appears to be somewhat of a language barrier be-
tween the life-science community and more mathematically
inclined researches, for instance expressed by the statement ‘‘J0
doesn’t say anything to a bioengineer.’’ Admittedly, there is no
direct link between biological function, on the one hand, and a
film’s elastic compliance at 15 MHz, J0. If there is no genuine
biological interest in J0, then there is no point in deriving it.
Also, remember that J0 per se only makes sense in the context
of continuum mechanics. It is a material property. Bioengi-
neers, on the other hand, do not usually view their samples as
materials. In biophysics, the focus is often on the properties of
spatially separated objects, such as proteins,149 DNA
strands,150,162 or vesicles.151 These objects may interact later-
ally to the extent that they effectively form a continuous
medium (a ‘‘film’’), but that is not necessarily a desirable
situation. Much more attractive are studies on the isolated
objects, as such. As a side remark, the QCM does not have
single-molecule sensitivity, today. One might—in princi-
ple—conceive nanomechanical devices with this capability,
but these do not exist yet. At this point, all QCMs output
the average stress–speed ratio at a substrate covered with a
large number of adsorbed objects. Actually averaged quanti-
ties are often more meaningful than non-averaged ones.
Importantly, there is no need to describe these adsorbates as
effective continuous media in order to predict a frequency
shift. One may, as well, perform a simulation (or, more
generally, a calculation) of the stress at the crystal surface
induced by the presence of the adsorbate without resorting to
continuum mechanics. For adsorbates of molecular size, this
task will presumably come down to a molecular dynamics
simulation. Such simulations have been done for polymer
brushes.152 Meso-scale objects such as vesicles or proteins will
presumably be treated with a more coarse-grained formalism.
Fig. 10 and 11 show the results obtained with a very simple
model. With regard to its quantitative outcome, it is more of
an illustration than a realistic description. This model is two-
dimensional. The adsorbed material forms a periodic array,
and the shape of the adsorbed object (a rectangular brick) is
somewhat simplistic. Still, the simulation demonstrates some
essential points. It was performed using the 2D incompressible
Navier–Stokes model supplied with the COMSOL Multiphy-
sics package (Gottingen, Germany). The inset in Fig. 10
Fig. 10 Distribution of stress underneath a rigid rectangular object of
size 20 � 10 nm adsorbed to a resonator surface. The stress is the
outcome of a 2D simulation employing the finite element method. The
inset shows the geometry where the width of the diagram corresponds
to 200 nm and the contour lines indicate equal horizontal speed of
motion. The stress peaks at the edge of the adsorbed object.
Fig. 11 Shifts of frequency and bandwidth induced by the adsorbate
shown in Fig. 10 as a function of coverage. Df saturates at a coverage
around 50% because the liquid entrained between the adsorbed object
takes part in the object’s motion. There is a dependence on overtone
order, caused by hydrodynamic effects (as opposed to a finite softness,
cf. eqn (15)).
4530 | Phys. Chem. Chem. Phys., 2008, 10, 4516–4534 This journal is �c the Owner Societies 2008
displays the geometry. A rectangular object with a size of 20 �10 nm is adsorbed to the crystal surface (straight horizontal
line on the bottom). The contour lines represent loci of equal
lateral speed of the solvent. The flow is deviated by the
obstacle. The adsorbed object is an incompressible rectangle
with a shear modulus of G = (30 + 5i) MPa. The bulk
medium is a Newtonian liquid with a viscosity of 1 mPa s. The
width of the simulation box varied between 11 and 100 nm, its
height was 2 mm. Periodic boundary conditions were applied
on both sides. The top surface was neutral. The bottom
surface performed an oscillatory motion with an amplitude
of 0.01 nm. The stress onto that surface was obtained via
boundary integration in the post-processing mode. In the
absence of the adsorbed brick, the model readily reproduces
eqn (8) within 0.1%. Tests with an adsorbed film (that is, with
an adsorbed object as wide as the simulation box) also
reproduced eqn (14).
Fig. 10 shows the stress distribution underneath the ad-
sorbed rectangle, whereas Fig. 11 shows the derived shifts of
frequency and bandwidth as a function of coverage. Coverage
here is the ratio of the width of the obstacle and the width of
the simulation box. These two figures illustrate some impor-
tant points:
(a) There is stress maximum at the rim of the contact. The
same is true for sphere-plate contacts in air (see the Mindlin
model above) or for liquid drops deposited on the crystal
surface.153,163 Continuum modeling does not apply right at the
three phase line because the stress goes to infinity. Still, the
stress distribution underneath an isolated adsorbed object is
strongly heterogeneous, even in liquids.
(b) The surface stress even changes sign for some locations
outside the contact, but close to the three-phase line. The sign
reversal is caused by vorticity of the flow of solvent. This
finding highlights the importance of rolling.
(c) At a distance from the object of about its lateral size
(where the latter was 20 nm in this particular case) the stress
distribution levels out to a value corresponding to the bare
surface. ‘‘Laterally dilute’’ objects should be placed at a
relative distance more than twice their size.
(d) The frequency shift versus coverage is linear below a
coverage of 20% (Fig. 11). At this coverage, lateral interac-
tions have a small effect. Considering the entire range of
0–100% coverage, frequency and coverage are by no means
proportional to each other. It is problematic to convert
frequency shift to coverage for layers of sub-monolayer
thickness.
(e) The shifts of frequency and bandwidth depend on over-
tone order, n, even for these rigid objects. On heterogeneous
surfaces, hydrodynamic effects induce differences between the
Sauerbrey masses on the different overtones. What would be
interpreted as softness within the frame of eqn (15) may be
caused by hydrodynamics.
The simulations described above were performed assuming
a flat resonator surface. Many of the bio-inspired QCM
experiments use gold surfaces. Part of the motivation for using
gold is the availability of a reliable linker chemistry for surface
functionalization.154 Importantly, evaporated gold surfaces
are not usually atomically flat. Neither is the underlying
resonator surface, which is mechanically polished. From a
physics point of view, it would be highly desirable to do
experiments on surfaces with a more well-defined geometry.
Spin-cast glassy polymer surfaces are smooth on the nano-
meter scale, although their roughness still exceeds the atomic
scale. Evidently, smoothness is paid for by a less reliable
surface chemistry. Experiments with mica glued to the crystal
surface have been reported,155–157 but the procedure is tedious
and the composite resonators often had poor Q-factors on the
high harmonics. Finding a manageable solution to the rough-
ness problem would greatly advance our understanding on
interactions between interfacial shear waves and bio-adsor-
bates.
VIII. Conclusions
This contribution has described applications of the QCM
which go beyond microweighing. The analysis has relied on
the small-load-approximation and has led to a comprehensive
picture covering a wide variety of configurations. We conclude
by pointing out in what way high-frequency characterization
differs from complementary low-frequency techniques.
(a) Firstly and most obviously, the high frequency precludes
relaxation processes with rates lower than the frequency. For
instance, the shear modulus determined via eqn (9) is a MHz
shear modulus. A similar, albeit less clear-cut, argument
applies to the stick-slip transition, which was not observed
in the colloidal probe experiments, even though the peak
lateral force exceeded the normal force by a factor of at least
5. Mechanical properties at high frequency are, for example,
highly relevant in tire friction158 and in fracture mechanics.
High-frequency properties are sometimes derived with con-
ventional rheometers by assuming time–temperature super-
position (TTS). TTS is often not valid for filled samples or
samples with internal structure.
(b) The high frequency often implies a highly confined
spatial range of interaction. A viscosity measurement with a
QCM always probes the part of the sample located inside the
evanescent tail of the acoustic wave. The depth of penetration
typically is below 1 mm.
(c) Inertia comes into play rather strongly at MHz fre-
quency. By balancing the force of interest (such as the force
transmitted through the sphere–plate contact) against the
force of inertia (holding the sphere in place), one gets around
the often cumbersome problem of calibrating springs. In low
frequency measurements, for example performed with the
atomic force microscope, determination of the spring constant
of the test device is a steady practical challenge. In a way, the
situation is reminiscent of astronomy a few hundred years ago,
where the force of gravity was determined based on the orbital
frequency of the planets (which, admittedly, is in a vastly
different range).
(d) Although the QCM is limited to oscillatory excitation, it
is not true that the QCM is blind to nonlinearities. The QCM
measures a peculiarly weighted average of the force, but it can
certainly access nonlinear phenomena via the amplitude de-
pendence of Df and DG.(e) Due to the small amplitude of excitation, the contacts
between a crystal and an external object are not usually
broken by the oscillation. The QCM is a device for
This journal is �c the Owner Societies 2008 Phys. Chem. Chem. Phys., 2008, 10, 4516–4534 | 4531
non-destructive testing of interfacial contacts. The evolution
of the contact strength with time, temperature, exposure to
solvent vapor, or vertical pressure can be monitored without
ever breaking a bond.
Recent years have seen continued evolution in acoustic
instrumentation and device fabrication. MEMS resonators
are currently being introduced for frequency control applica-
tions.159 The first applications for sensing have shown sensi-
tivity close to the single-molecule regime.160 Cantilever
resonators are used for similar purposes.161 Many of the
principles described above apply on the molecular scale. On
the other hand, the nano-scale certainly holds the potential for
new physics and new sensing principles, as well.
Appendix: on the equivalence of the Voigt model and
eqn (14)
The literature contains two derivations of the frequency shift
induced by a viscoelastic film in a liquid.92,93 The language
differs, but they are equivalent. The former model uses acous-
tic terminology, while the latter (the ‘‘Voigt model’’) uses the
wave equation and continuity of stress and strain at the
interfaces. The Voigt model contains two parameters m and
Z, which would be designated G0 and G00/o in the acoustic
formulation. These were originally considered to be indepen-
dent of frequency and additive. The total stress was written as
s = (m + ioZ) du/dz. This contrasts to the Maxwell model,
which has du/dz = (m�1 + (ioZ)�1) s. When m and Z depend
on frequency, the term Voigt model loses its original meaning.
One then by definition always has G(o) = G0(o) + iG00(o).The Voigt model and the acoustic model are equivalent.
Start from eqn (15) in ref. 93:
Df ¼ ImðbÞ2pr0h0
; DD ¼ � ReðbÞpf r0h0
b ¼ k1x11� Ae2x1h1
1þ Ae2x1h1
A ¼ k1x1ð1þ le2x2Dh1Þ � k2x2ð1� le2x2Dh1Þk1x1ð1þ le2x2Dh1Þ þ k2x2ð1� le2x2Dh1Þ
l ¼ k2x2 þ k3x3 tanhðx3Dh2Þk2x2 � k3x3 tanhðx3Dh2Þ
x ¼ ik; k ¼ G
io
ð38Þ
The parameters r0 and h0 are the density and the thickness of
the quartz crystal, r1 and h1 are the density and the thickness
of the first film, r2 and Dh1 are the density and the thickness of
the second film, r3 and Dh2 are the density and the thickness of
the third film, k is the wavenumber, and G is the shear
modulus. The shear modulus is termed m + ioZ in ref. 93.
ref. 93 considers a three layer geometry. The case of a single
viscoelastic layer immersed in a bulk liquid is derived from
eqn (38) by inserting h1 = 0, k1 = k2, x1 = x2 and Dh2 = N.
The term tanh(x3Dh2) then turns to one. Also, one may replace
the product kx by Z, where Z is the acoustic impedance. This
equality follows from Z = rc = r(G/r)1/2 = G/c = oGk. Inorder to match the notation of eqn (14), we replace the indices
‘‘2’’ and ‘‘3’’ by ‘‘f’’ and ‘‘liq’’ (for film and liquid). Further, we
replace r0h0 by Zq/(2fF). Finally, comparison with eqn (3)
shows that b = �ZL. With these replacements, eqn (38) reads
DffF¼ �1
pZqImðZLÞ
DDfF
f
2¼ 1
pZqReðZLÞ
ZL ¼ ZfA� 1
Aþ 1
A ¼ ð1þ le2ikfdf Þ � ð1� le2ikfdf Þð1þ le2ikfdf Þ þ ð1� le2ikfdf Þ ¼ le2ikfdf
l ¼ Zf þ Zliq
Zf � Zliq
ð39Þ
Inserting the expressions for l and A into the expression for
ZL yields
ZL ¼ Zf
ZfþZliq
Zf�Zliqe2ikfdf � 1
ZfþZliq
Zf�Zliqe2ikfdf þ 1
¼ ZfðZf þ ZliqÞe2ikfdf � ðZf � ZliqÞðZf þ ZliqÞe2ikfdf þ ðZf � ZliqÞ
ð40Þ
Inserting this expression into the SLA yields eqn (7) in ref. 92.
One may further simplify according to
ZL ¼ZfðZf þ ZliqÞe2ikfdf � ðZf � ZliqÞðZf þ ZliqÞe2ikfdf þ ðZf � ZliqÞ
¼ZfZfðeikfdf � e�ikfdf Þ þ Zliqðeikfdf þ e�ikfdf ÞZfðeikff df þ e�ikfdf Þ þ Zliqðeikfdf � e�ikfdf Þ
¼iZfZf tanðkfdfÞ � iZliq
Zf þ iZliq tanðkfdfÞ
ð41Þ
This result yields eqn (14) when inserted into eqn (3).
Acknowledgements
The author is indebted to B. Y. Du and I. Gubaidullin for
making data available prior to publication and to Arne
Langhoff for critical reading of the manuscript.
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