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.

This journal is �c the Owner Societies 2008 Phys. Chem. Chem. Phys., 2008, 10, 4516–4534 | 4517

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.


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


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


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’’


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.


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.


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).


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


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


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.


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.


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.


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)).


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


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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