Shifts of frequency and bandwidth of quartz crystal resonators ...

JOURNAL OF APPLIED PHYSICS 101, 114909 2007 

Shifts of frequency and bandwidth of quartz crystal resonators coated 

with samples of finite lateral size 

M. Herrscher and C. Ziegler 

Department of Physics, University of Kaiserslautern, Erwin-Schroedinger-Strasse 56, D-67663 

Kaiserslautern, Germany 

D. Johannsmann a 

Insitute of Physical Chemistry, Clausthal University of Technology, Arnold-Sommerfeld-Strasse 4, 

D-38678 Clausthal-Zellerfeld, Germany 

Received 17 November 2006; accepted 9 March 2007; published online 8 June 2007 

Recently, the viscoelastic properties of polymeric materials were probed by pushing a hemispherical 

cap of the respective material against the front surface of a quartz crystal resonator and measuring 

the induced shift of frequency and bandwidth, f and , as a function of the contact area, A c . 1 The 

shift of the resonance parameters was found to be proportional to the contact area, with the constant 

of proportionality containing the sample’s shear modulus. Confining the contact area to a small spot 

in the center of the plate is central to this approach, because the resonator would otherwise be 

overdamped. However, more detailed experiments have shown that there are small deviations from 

the proportionality of the frequency shift to the contact area. In particular, the ratio of and f 

the “D-f ratio”, which should reflect intrinsic material properties, was found to slightly depend on 

contact area. Employing a finite element method simulation, two hypotheses were tested for the 

nontrivial area dependence of f and , which are, first, scattering of the acoustic wave from the 

acoustically heterogeneous surface and, second, a change of the lateral amplitude distribution, 

“energy trapping” induced by loading the crystal in the center only. It can be concluded that 

scattering affects the D-f ratio only at very small contact radii 10 m, whereas in the range of 

millimeter-sized contacts, the change of energy trapping dominates. Employing a perturbation 

analysis, a relation of the form f A c 1+A c is found, where contains the viscoelastic 

parameters and the term in brackets is the nontrivial correction. Such a linear dependence agrees 

well with the results from simulation and experiment. An improved estimate of the material’s shear 

modulus is obtained by analyzing the ratios f /A c and /A c as a function of contact area A c , and 

extrapolating these values to the limit A c →0. © 2007 American Institute of Physics. 

DOI: 10.1063/1.2729451 

INTRODUCTION 

a Author to whom correspondence should be addressed; electronic mail: 

johannsmann@pc.tu-clausthal.de 

The quartz crystal microbalance QCM is a well-known 

tool for the determination of average film thicknesses in the 

nanometer range via a measurement of the added mass using 

the well known Sauerbrey relation. 2,3 As has been shown by 

numerous researchers in the past few years, 4 the use of the 

QCM is by no means limited to microgravimetry. For instance, 

there are now commercial viscosimeters around, 

which consist of an acoustic device in contact with a liquid 

or—more generally—a semi-infinite viscoelastic material. 5 

In this case, the shifts in frequency and bandwidth reflect the 

viscoelastic properties of the medium, as quantitatively expressed 

by the Borovikov-Kanazawa equation. 6–8 One 

prominent potential application of a microbalance using both 

shifts are biosensors. 9,10 Since the penetration depth of megahertz 

acoustic shear waves typically is of the order of a micron, 

the QCM gives access to the sample’s shear modulus in 

the immediate vicinity of the interface. The technique is surface 

specific. Clearly, the approach is highly attractive for the 

study of polymer surfaces, as well. Polymers display a diverse 

set of surface anomalies, mostly related to confinement 

and the surface energetics. More practical questions such as 

adhesion or sliding friction at a rubber-substrate interface are 

of much importance as well. 11,12 

Unfortunately, QCM-based studies of polymer viscoelasticity 

have proven to be difficult because the QCM does not 

work well when the viscosity of the load becomes too large. 

The resonance then is overdamped. Given that polymers are 

orders of magnitude more viscous than low-molecularweight 

liquids, the bandwidth of a crystal covered with a 

bulk polymer can easily approach tens of kilohertz. Such 

broad resonances cannot be distinguished from the background. 

The measurement of viscosities of polymer melts or 

elastomers is only feasible, if the contact area is confined to 

a small spot in the center of the plate. This mode of measurement 

was proposed by Flanigan et al., who combined the 

QCM with a JKR tester. 1 Here, the term “JKR” stands for 

Johnson, Kendall, and Roberts, who formulated the contact 

mechanics model underlying the instrument. 13 In a JKR 

tester, a hemisphere of the material under test is pushed 

against a flat substrate. The output parameters are the flattening 

of the contact indentation and the contact area as a 

function of the vertical load. Applying the JKR model to 

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114909-2 Herrscher Ziegler, and Johannsmann J. Appl. Phys. 101, 114909 2007 

these data, one derives the elastic modulus of the material 

and the energy of adhesion. Flanigan and co-workers 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 

about proportional to the area of contact. Such a scaling is 

expected if the diameter of the contact is much larger than 

the wavelength of sound, . We term this model “sheetcontact 

model.” The sheet-contact model contrasts to the 

“point-contact model,” which applies in the opposing limit of 

contact radii much smaller than . 14,15 In more quantitative 

terms, the sheet-contact model predicts 

f * 

= 

f f 

f + i 

f f 

= i 

Z q 

K A r c A c 

A Z ac 

= i K A r c A c 

G + iG, 

1 

Z q A 

where f * =f +i is the complex frequency shift, f f is the 

frequency of the fundamental, K A is a sensitivity factor see 

below, A c =r 2 c is the contact area, r c is the radius of contact, 

A is the active area of the crystal, Z ac =G 1/2 is the 

acoustic impedance of the sample, is the density of the 

material, and G=G+iG is the shear modulus. The imaginary 

part of the complex frequency, , is the half bandwidth 

at half maximum called “bandwidth,” for short, in the following. 

At large contact radii, there is a deviation from the scaling 

with contact area, which is related to the nonuniform 

amplitude distribution. When the size of the contact approaches 

the size of the back electrode where the latter 

roughly defines the active area via energy trapping 2 the 

outer portions of the contact experience a smaller stress than 

the center. These parts therefore only make a small contribution 

to the frequency shift. The effect is accounted for with a 

sensitivity factor K A r c . In this contribution, we limit the 

discussion to smaller contact radii, where such effects can 

safely be ignored. For small contact areas, K A r c is about 

constant. Its value depends on the definition of the active 

area. Flanigan et al. defined the active area as the area of the 

back electrode. 1 We take a different approach and calculate 

the active area from an integral over the displacement pattern. 

This definition automatically ensures that the sensitivity 

factor K A r c is equal to unity in the limit of small contact 

radius. In the limit of small contact area, the sheet-contact 

model states that 

f * 

i A c 

G + iG = 

i A c 

f f Z q A Z q A Z L. 

2 

In Eq. 2 the acoustic impedance of the polymer was 

replaced by the load impedance, Z L , in order to emphasize 

the generality of the model. Z L is the stress-speed ratio at the 

crystal surface. It is equal to the acoustic impedance for a 

thick viscoelastic material, but may take other values for 

other kinds of loads. For instance, Z L is equal to im f ,ifthe 

sample is a Sauerbrey film with areal mass density m f .A 

coating is called “Sauerbrey film” if it is a thin, rigid film 

with a thickness much smaller than the wavelength of the 

shear wave. Viscoelastic effects can be ignored in this case. 

FIG. 1. Shift of frequency and bandwidth induced by a water droplet panels 

a and b and a hemispherical cap of a polymer gel panels c and d. 

The gel was produced by swelling the triblock copolymer Kraton G 1650 in 

light mineral oil at polymer-solvent ratio of 1:3. In the first case, the area 

was varied by just letting the droplet evaporate, while in the second, the area 

is a function of the vertical pressure applied to the stamp. The contact area 

was determined with a microscope imaging the sample from above adapted 

from Ref. 18. 

Application of Eq. 1 to the experimental data entails a 

complication because the contact area, A c , has to be measured 

in one way or another. This measurement usually requires 

a JKR tester, which, in turn, requires a transparent 

sample. Importantly, the contact area can be eliminated from 

the data analysis by considering the ratio of and −f, 

termed “D-f ratio.” Since and f should, according to 

the sheet-contact model, scale with contact area in the same 

way, the ratio should reflect a material’s parameter, regardless 

of the value of A c . A similar approach to data analysis 

was previously proposed in the context of QCM 

microweighing, 16 where the ratio of and −f is independent 

of film thickness for very thin films and closely related 

to the film’s elastic compliance, J. 17 The sheet-contact 

model predicts 

 

− f = cot 2, 

where =arctanG/G is the loss angle. 

Experiments with small droplets of water as well as 

hemispherical caps of polymer gels have shown that the D-f 

ratio does in some cases slightly depend on the contact 

radius. 18 Figure 1 displays two examples. The data in panels 

a and b were obtained by letting a water droplet evaporate 

3 



and measuring f, , and A c as a function of time. Panel 

a displays f and vs A c . Both f and are about 

proportional to area, but there are systematic deviations. By 

plotting the ratio /−f vs A c panel b, the deviations 

become more evident. Note that an incorrect determination 

of the contact area would not have shifted the ratio of 

and −f. Since it is hard to see how the loss angle of water 

close to 90° should depend on the droplet size, one concludes 

that some artifact comes into play. Panels c and d 

display analogous data obtained on a polymer gel Kraton G 

1650, swollen in light mineral oil, polymer weight fraction 

25%. In this case various degrees of vertical pressure were 

applied in order to adjust the contact area. 

The first shortcoming of the sheet-contact model coming 

to mind is acoustic scattering of the wave from the contact. 

Acoustic scattering is not captured by Eq. 1, because it 

assumes plane waves. This explanation was elaborated on in 

Ref. 18. A second effect—investigated here—is a change of 

energy trapping induced by contacting the crystal in the center 

only. 19 Most crystals are shaped such that the acoustic 

thickness is higher in the center than at the rim. The crystal 

forms an acoustic lens which focuses the acoustic radiation 

to the center of the crystal. Acoustic focusing can be 

achieved by either employing convexly shaped crystals 

which is common in the frequency control community or 

by employing thick, keyhole shaped electrodes. The latter 

configuration is more common in sensing. Usually, the back 

electrode is thicker and smaller than the front electrode, ensuring 

that the edge of the front electrode is outside the active 

area of sensing. Importantly, the strength of energy trapping 

is changed, when a load is applied in the center of the 

crystal only. When touching the crystal in the center, one 

applies a phase shift to the central acoustic beam, thereby 

changing the effective curvature of the acoustic lens. A 

change in energy trapping, in turn, changes the resonance 

frequency. As is shown below, the effect is substantial and 

can account for area dependence of the D-f ratio. Strong 

energy trapping leads to strong shear gradients in the plane 

of the crystal, which increase the strain energy of the resonating 

structure. Even though the in-plane shear gradients are 

much weaker than the gradients along the surface normal, 

the effect, which energy trapping has on the resonance frequency, 

definitely is measurable. 20 

In order to discriminate between the effects of acoustic 

scattering, on the one hand, and of energy trapping, on the 

other, finite element method FEM simulations of partially 

loaded quartz crystals were performed. Within a finite element 

simulation, the device of interest is divided into a discrete 

mesh. The equations of continuum mechanics are 

solved for these elements. FEM simulations on quartz resonators 

are quite demanding because of the high Q factor. 21 

Most FEM simulations performed to date targeted the performance 

of resonators as acoustic clocks. 22,23 A prime concern 

is the separation of the main resonance from the anharmonic 

sidebands the “spurious modes”, which can be achieved by 

optimization of the geometry. Other topics of interest were 

the effects of gravity and stress. An FEM simulation in the 

context of sensing was reported by Friedt et al. 24 These authors 

looked into the flexural contribution to the displacement 

pattern by means of static simulations. 

FEM simulations allow us to distinguish between effects 

of energy trapping, on the one hand, and scattering, on the 

other, because they provide the amplitude distribution and 

the distribution of shear gradients. These are difficult to determine 

in experiment. Also, one can easily simulate samples 

with very small contact areas. In experiment, the frequency 

shift induced by very small contacts radius of contact 

100 m cannot be measured with sufficient accuracy. As 

we show below, both acoustic scattering and a change of 

energy trapping do play a role, but they do so in different 

ranges of contact size. 

The influence of energy trapping can also be calculated 

analytically based on perturbation theory. The small perturbation 

parameter is the contact area, A c , divided by the active 

area of the crystal, A. The first order perturbation result 

yields a term proportional to A c and is equivalent to the 

sheet-contact model. The second order result contains a term 

proportional to A c 2 and produces a deviation from the sheetcontact 

model. For an idealized geometry, the theory makes a 

quantitative prediction for size of the deviation. For realistic 

geometries, the result needs to be augmented with a numerical 

factor of order unity. 

FEM simulations 

For the sake of computational efficiency, the simulations 

were limited to a two-dimensional representation of the crystal, 

rather than a three-dimensional object. Leaving away the 

third dimension, we certainly sacrifice the claim to a truly 

realistic simulation. The simulation still elucidates the role of 

acoustic scattering and of energy trapping. In order to ensure 

that the simulation matches the experiment, a few checks 

were performed in selected cases. For instance, both the 

Sauerbrey equation 3 and the Borovikov-Kanazawa-Gordon 

result 7,8 were confirmed when bringing the resonator into 

contact with a thin film and a semi-infinite liquid, respectively. 

Detailed investigations to confirm the validity of the 

two-dimensional simulations are part of future research. 

The simulations were carried out with the software package 

FEMLAB 3.0 COMSOL AB, Stockholm, Sweden, which 

included the Structural Mechanics Extension module. In a 

first step, a static piezoelectric simulation was performed using 

the Multiphysics module. From this simulation, the relation 

between the external voltage and the stress distribution 

inside the resonator was obtained. In the following steps, this 

stress distribution multiplied by cost was used as the 

excitation field, rather than the electric voltage. In this way, 

the use of the piezoelectric tensor in the main simulation was 

avoided. The latter was based on conventional elasticity. 

The simulations were performed on a cross section of a 

quartz crystal oscillator as schematically shown in Fig. 2. 

The slab corresponds to a cut through an AT-cut, planar 

quartz disk along the x axis. The radius of the plate is 10 mm 

and its thickness is 166 m, resulting in a fundamental resonance 

frequency of 10 MHz. The crystal is clamped at the 

edge by imposing that the displacement be zero outside r 



FIG. 2. Discretization mesh used in the FEM simulation. The drawing is not 

to scale. 

=9.5 mm. The electrodes had a thickness of 100 nm and a 

radius of 7 mm. The material parameters were chosen to 

mimic gold. In order to analyze the influence of a partial 

loading of the quartz, a cylinder of the respective material 

was placed onto the center of the top electrode. The parameters 

of the applied materials are listed in Table I. The height 

of the sample was 10 m. The height must be larger than the 

penetration depth of the shear wave in order to ensure that 

the shear oscillation does not reach the outer interface. Reflections 

of acoustic waves from the back of the crystal can 

then be ignored. 

The mesh divides the crystal into finite elements, inside 

which displacement and stress are interpolated with shape 

functions. It is essential to adapt the mesh size distribution to 

the problem under investigation. Figure 2 shows the mesh 

employed in this work. The elements are small at the top, at 

the bottom, and in the center of the crystal. Even though the 

outcome of the simulation should not critically be affected 

by the anisotropy of the crystal, anisotropic elastic parameters 

were employed. Piezoelectrically stiffened shear 

moduli were applied. 25 The values inserted into the model 

were c¯11 =86.74 GPa, c¯12 =−8.27 GPa, c¯22 =129.8 GPa, and 

c¯66 =29.24 GPa. The bars indicate that these parameters were 

obtained after rotating the coordinate system about the y axis 

by 35° in order to describe the AT cut. 

Frequency and bandwidth of the crystal were derived by 

means of a simulated frequency sweep. Figure 3 shows the 

area-averaged amplitude at the top of the crystal, uf, asa 

function of driving frequency. Panels a and b show a 

slightly damped Q=160 000 and a heavily damped Q 

=730 crystal, respectively. In the latter case, the main resonance 

overlaps with an anharmonic sideband. The function 

uf was fitted with a resonance curve of the form 

TABLE I. Material parameters. 

g/cm 3 mPa s mPa s 

Water 1 1.0 0 

Elastomer 1 1.0 1.59 

Solid Film 1 0 159 000 G=10 GPa 

FIG. 3. Simulated resonance curves for a weakly loaded crystal Q 

=160 000, panel a and a heavily loaded crystal Q=730, panel b. Inthe 

latter case, the main resonance overlaps with an anharmonic sideband. The 

dotted line is the fit with a single resonance curve. In panel a, the simulation 

and the fit with a resonance curve cannot be distinguished. 

uf = 

C 

f0 2 − f 2 2 + 2f 2 , 

where C is a prefactor, f 0 is the resonance frequency, f is the 

excitation frequency, and describes the width of the curve. 

The parameter is equal to half the bandwidth at half maximum 

of the imaginary part of uf where the latter is proportional 

to the conductance of the crystal, G, and therefore 

is the basis of impedance analysis. The FEM analysis 

makes use of the modulus uf, in which case is the half 

bandwidth at 1/2 1/2 of the peak height. The simulated sweep 

is performed for both the unloaded and the loaded crystal in 

order to find the shifts, f and . The parameters f norm 

and norm are defined as 

r Q 

u 2 xdx 

f norm = f L − f u A Q 

= f L − f u −r 

r 

A 

C 

, 

c 

u 2 xdx 

−r C 

r Q 

u 2 xdx 

norm = L − u A Q 

= L − u −r 

r 

A 

C 

, 

c 

u 2 xdx 

−r C 

where the indices L and u denote the loaded and unloaded 

crystals, respectively, r Q is the radius of the crystal, r C 

is the 

radius of contact, and ux is the amplitude of oscillation at 

position x. The terms in the numerator and the denominator 

are the active area, A, and contact area, A c , respectively. If 

the sheet-contact model were rigorously correct, the normalized 

shifts of frequency and bandwidth as a function of contact 

radius would approach a horizontal line for A C 

A. The 

values of f norm and norm would obey the Sauerbrey and 

the Borovikov-Kanazawa equation for a film and a liquid, 

respectively. 

4 

5 



Energy trapping for convexly shaped crystals 

The change of energy trapping and the consequences of 

such changes can be calculated by means of a perturbation 

approach, which borrows from the quantum mechanical 

treatment of the harmonic oscillator. The mathematics described 

below applies to plane convexly shaped crystals. As 

in the FEM simulation, we treat the two-dimensional slab. 

The result can be transferred to other geometries by changing 

a numerical factor. 

Let the shape of the lower crystal surface be parabolic. 

Plane-convex crystals are, in fact, employed extensively in 

the frequency control community. These crystals have better 

energy trapping on the fundamental than the planar crystals 

usually employed in sensing. 26,27 Let d q be the thickness in 

the center of the crystal and let the radius of curvature, R, be 

much larger than the thickness. The crystal is assumed to be 

“locally flat,” meaning that all gradients in the surface plane 

are much smaller than the gradients along the vertical. The 

displacement pattern is given by a standing wave, where the 

vertical component of the wave number, k z x, at each location, 

x, is equal to n/dx, where n is the overtone order. 

The displacement pattern, Ux,z, is approximated as 

Ux,z = u 0 sin n 

dx z ux, 

where u 0 is an amplitude with dimension of a length. The 

function ux describing the amplitude distribution along x is 

dimensionless. The wave equation is 

2 U + 2 

c 2 U = − k 2 z x + 2 

x 2 + 2 

c 2u 0 uxsin 

n 

dx z 

6 

=0, 7 

where c is the speed of sound. The wave vector along the 

surface normal, k z , depends on location, x. For a planeconvex 

crystal, dx is given by d q 1−x 2 /2Rd q , resulting 

in 

k z x 

n 

d q 1−x 2 /2Rd q n 

d q1+ 

x2 

2Rd q. 

Further, for x 2 Rd q , the square of k z can be simplified to 

k 2 z x n 

d q 

21+ x2 

9 

Rd q, 

which yields 

− 2 u 

 

x 2 + n 

d q 

21+ x2 

Rd q − 2 

c 2u 

=− 2 u 

x 2 +x2 

 

4 

− u =0. 

10 

The eigenvalue, , is defined as 

8 

= 2 

c 2 

n 

− 

2 

d q 

. 11 

has the dimension of an inverse square length. The parameter 

is given by 

4 = Rd 3 

q 

n 2 . 

12 

Equation 10 is known from the quantum theory of the harmonic 

oscillator. 28 It resembles the Schrödinger equation, 

where 2 u/x 2 would be proportional to the kinetic energy 

and x 2 / 4 would be the potential. Its solutions are 

u 0 m x = a 0 m H m 

exp x − x2 

2 2. 

13 

The superscript 0 denotes the zeroth perturbation order. 

H m x are the Hermite polynomials and a 0 m 

are normalization 

constants. In the context of quartz crystal resonators, the 

index m labels the anharmonic sidebands as opposed to the 

overtone orders, which are labeled by n. The lowest Hermite 

polynomial is H 0 =1; the corresponding solution is the 

Gaussian. Higher Hermite polynomials contain nodes in the 

lateral amplitude distribution. These solutions correspond to 

the anharmonic sidebands. Here, only the main resonances 

are of interest, which have m=0. 

The solutions are normalized such that 

 

− 

 

u 0 m x 2 dx =− 

1. 

a m 0 2H m x 2 

exp− x2 

2dx 

14 

The parameter a m 0 2 has the dimension of an inverse 

length, termed “active area,” A. The active area here has 

dimension of a length, since a two-dimensional slab is considered, 

rather than a three-dimensional plate. The active 

area can be written as 

 

A = a 0 0 −2 =− 

exp− x2 

2dx = . 

15 

For modes with m0, the normalizing factors contain a numerical 

coefficient, m , given as 

a 0 m = m 1 1 

= . 

A 2 m m! A 

The eigenvalues 0 m 

are given by 

m 0 = 2m +1 1 2 . 

16 

17 

The resonance frequency of the hypothetical flat resonator 

R= is defined as ¯ =2f 0 =cn/d q . In the following, the 

resonance frequency is written as =¯ +˜ , where ˜ is a 

frequency shift. ˜ can be calculated by perturbation theory. 

The zeroth order term, ˜ 0 , only comprises the correction 

due to energy trapping. The first order contribution, ˜ 1 , will 

be the frequency shift induced by the sample as predicted by 

the sheet-contact model. The second order contribution, ˜ 2 , 

will be caused by the change of energy trapping due to the 

presence of the sample. The second order term will turn out 



to be the source of the area dependence of the D-f ratio. 

We first treat the zeroth order approximation. Assuming 

˜ 0 m 

¯ , 2 m can be approximated as 

2 m ¯ + ˜ 0 m 2 ¯ 2 +2¯ ˜ 0 m . 

18 

Using these variables and Eq. 11, Eq. 17 reads 

0 m = 2 

m 

c 2 − ¯ 2 

c 2 = 2 

2 

m n 

− 

c 2 d q = 2¯ ˜ 0 

m 

c 2 = 2m +1 1 2 . 

19 

The eigenvalues have dimension of an inverse square length, 

whereas the fundamental quantity in QCM measurements is 

a frequency. One translates between the two by multiplying 

with c 2 /2¯ , 

˜ 0 m = 1 

2¯ 2 m − ¯ 2 = 2m +1 

2¯ 2 = 2m +1 tr. 20 

The quantity tr denotes the frequency increase caused by 

energy trapping the “vibrational frequency” in the context of 

quantum mechanics, not be confused with the resonance frequency 

of the crystal. The contribution of the transverse 

shear gradients to the frequency is sizable, as can be seen 

from the ratio of tr /¯ for m=0, 

tr 

¯ = c 2 

2 

2¯ 2 2 = d q 

2n 2 2 . 

21 

Using d q =166 m, =1.66 mm, and n=1 yields tr /2 

=1/2 2 0.0110 MHz5 kHz. A change of energy trapping 

can easily produce a frequency shift of a few hertz. As a side 

result, we mention that the change of energy trapping is unessential 

in the simple Sauerbrey case. This can be seen by 

comparing the shift in tr to the shift in ¯ when the thickness 

of the crystal is increased by a small amount, d q . The detailed 

calculation shows that 

tr 

¯ = 1 tr 

2 ¯ . 

22 

The contribution of energy trapping to the Sauerbrey result is 

of the order of 10 −3 and may be neglected in view of the 

other uncertainties in microweighing. 

Touching the crystal in the center amounts to small perturbation. 

It is assumed that the contact area is larger than the 

thickness of the crystal, so that the vertical and the horizontal 

scale still separate well. The effect of the sample is introduced 

into Eq. 7 by modifying the parameter k z x. Dissipation 

is introduced by making k z x complex in the region 

of contact. The shift of the k vector induced by the load is 

inferred from the flat-plate analog: According to the small 

load approximation, 29 the complex fractional frequency shift 

is 

f + i 

= i Z L . 

23 

f f Z q 

The parameter Z L here is the load impedance, which is 

the ratio of stress and speed at the crystal surface. f f is the 

frequency of the fundamental. Using the frequency of the 

c 2 

unperturbed crystal at the given overtone order, f 0 =nf f , instead 

of the frequency of the fundamental, the small load 

approximation reads 

f + i i 

= Z L . 

24 

f 0 nZ q 

When the sample is a flat punch of a viscoelastic material, 

the load impedance is the same as the acoustic impedance of 

the sample, Z ac =G 1/2 . The perturbation calculation would 

also apply if the load were of some other nature such as a 

Sauerbrey film. 

A complex frequency shift as given by Eq. 23 implies 

a complex wave vector for the loaded crystal given by 

k z,L x = k z x + k z x = k z x + k z x iZ Lx 

, 25 

nZ q 

where the index L denotes the loaded crystal and k z is the 

2 

shift of k z induced by the load. For k z,L we find 

k 2 z,L x = k z x + k z x 2 k 2 z x +2k z xk z x 

= k 2 z x + k 2 z x 2iZ Lx 

. 26 

nZ q 

This yields the wave equation 

− 2 u 

x 2 + n2 2 

2 1+ x2 2iZ Lx 

d q 

Rd q1+ 

nZ q 

 

=− 2 u 

x 2 + x2 

+ n2 2 

d q 

2 

41+ 2iZ Lx 

nZ q 

 

− 2 

c 2u 

− 2 

c 2u 

+ n2 2 2iZ L x 

2 

d q 

nZ q 

− 2 u 

 

x 2 + x2 

4 + n2 2 2Z L x 

2 

− u =0. 27 

d q 

nZ q 

Equation 12 has been used when replacing n 2 2 /Rd 3 q by 

−4 . The relation Z L Z q was used in line 3 square brackets 

in line 2 replaced by unity. Below, it will be convenient to 

use eigenvalues with a dimension of a frequency termed ˜ . 

This is achieved by multiplying Eq. 27 with c 2 /2¯ cf. 

Eq. 20; 

− c2 

2¯ 

2 

c2 

x2u + 

2¯ 

1 

4x2 u + ¯ iZ Lx 

nZ q 

u − ˜ u =0. 

28 

In Eq. 28 ¯ =cn/d q was used. With regard to the algebraic 

details of the perturbation analysis, the approach from 

Ref. 30 is applied. Before proceeding, we cast Eq. 28 into 

a form matching the choices made there. Reference 30 uses 

the bracket notation 

H 0 + Vm = E m m, 

29 

where is the small perturbation parameter, H 0 is the Hamiltonian 

operator, V is a perturbing potential, m is a state, and 

E m is an eigenvalue. In quantum mechanics, the eigenvalues 

have the dimension of an energy, whereas they are frequencies 

here. Comparing Eqs. 29 and 28 the following correspondences 

are found: 



H 0 =− c2 

2¯ 

2 

x 2 + c2 

2¯ 

1 

4 , 

V = i¯ Z Lx 

nZ q 

, 

m = u m x, 

30 

E m = ˜ m. 

Importantly, V is not a Hermitian operator, and as a consequence 

kVmmVk * . 31 Because V is not Hermitian, the 

eigenvalues, ˜ m, are complex: they contain a frequency shift 

as well as an increase in bandwidth that is, an imaginary 

part. Borrowing from Ref. 30 we write 

m = m 0 + m 1 + ¯ , 

31 

m = E m − E 0 m = ˜ m − ˜ 0 m = 1 m + 2 2 m + ¯ , 

m 1 = m 0 Vm 0 , 

32 

33 

m 1 = k 0 k0 Vm 0 

km E 0 0 

, 34 

m − E k 

and 

2 m = m 0 Vm 1 m 

= 

0 Vk 0 k 0 Vm 0 

km E 0 0 

. 35 

m − E k 

Only the terms with m 0 =0 are of interest, because the main 

resonance is used for measurement. The anharmonic sidebands 

are essential in the calculation because the eigenstate 

in the presence of the sample m 1 contains contributions 

from k0 Eq. 34. This fact is illustrated in Fig. 4b. 

When superimposing the states 0 and 3 with relative 

weights of 0.9 and 0.1, respectively, an amplitude distribution 

is obtained, which looks different from the state 0 in 

the sense that it has a somewhat sharper peak in the center. 

This is what is found in the simulation, as well Fig. 4a. 

Setting m 0 =0 and inserting Eq. 30 into Eq. 33 

yields the first order result for the frequency shift, 

1 0 =0 i¯ Z 

Lx 

0 

u i¯ Z Lx 

nZ 0 u 0 

q 

nZ 0 dx 

q 

= 

i¯ 

 

AnZ q 

− 

0 =− 

Z L xexp− x2 

2dx. 

36 

For a contact with a viscoelastic medium, Z L x is constant 

and equal to Z L =Z ac =G 1/2 between −r c and r c r c the 

contact radius. If the contact radius is much smaller than the 

radius of the active area, exp−x 2 / 2 can be approximated 

by unity. In the language of the sheet-contact model, this 

implies K A r c 1. This leads to 

f* 1 = f 1 + i 1 = 0 1 

2 = f 2r c iZ L 

0 . 37 

A nZ q 

Here, f 0 ¯ /2/2 is the frequency of the unperturbed 

resonator. Identifying 2r c with the contact area, A c , 

FIG. 4. a Amplitude distribution of a bare crystal full line and a crystal 

loaded with a water droplet r c =1 mm, dotted line. The load modifies the 

amplitude distribution and introduces additional lateral shear gradients. 

Panel b illustrates how a cusp in the center of the amplitude distribution 

can be produced by mixing the main resonance with a contribution from the 

third anharmonic sideband. 

and, also, substituting nf f for f 0 , Eq. 37 becomes equivalent 

to the sheet-contact model Eq. 2. 

Having rederived the sheet-contact model in this somewhat 

complicated way, we feel encouraged to move forward 

to the second order term. The first order displacement pattern 

is given by 

0 1 = k 0 k0 V0 0 

k0 E 0 0 

0 − E . 

k 

The functions k 0 are 

k 0 = k 

A 

H k x exp − x2 

2 2. 

The term in the denominator is Eq. 20 

1 

E 0 0 

0 − E =− 1 . 

k 

2k tr 

The scalar product k 0 V0 0 is 

k 0 V0 0 = i¯ Z L 

nZ q 

1 

A−r c 

exp− x2 

2dx. 

r c 

k H k x H 0 x 

In order to solve the integral, we substitute x by =x/, 

38 

39 

40 

41 



FIG. 5. Shift of frequency f norm and of bandwidth norm of a crystal 

loaded with a droplet of water. The lines are fits to Eq. 50 with B and B 

as a free parameters. 

r c / 

k 0 V0 0 = i¯ Z L 

nZ q A k H k H 0 exp− −r 2 d. 

c / 

42 

The second order correction 0 2 is given by 

2 0 = 0 0 V0 1 0 

= 

0 Vk 0 k 0 V0 0 

k0 E 0 0 

0 − E k 

=− ¯ Z L 

nZ q2 

 

k0 

r c / 

−r c / 

1 

− tr 

2 

A 2 1 

2k 

k H k H 0 exp− 2 d2. 43 

The integral can be evaluated with software packages such as 

MATHEMATICA. The outcome is 

r c / 

−r c / 

k H k H 0 exp− 2 d 

 

= 2k Fk/2,3/2,− rc / 2 

, 44 

2 k k!1−k/2 

where F is the generalized hypergeometric function and the 

Gamma-function x not to be confused with the bandwidth 

interpolates the factorial. Since only small contact 

areas are of interest, we expand the square of the integral to 

second order in r c /, 

r c / 

−r c / 

 

k H k H 0 exp− 2 d2 

2 

2 k r 2 

c 

. 45 

k!1−k/2 

Finally, the sum evaluates to 

FIG. 6. Shift of frequency f norm of half bandwidth at half maximum 

norm of a crystal loaded with a viscoelastic material. The lines are fits to 

Eq. 50 with B and B as a free parameters. 

 

 

k0 

1 

r c / 

2k−r c / 

k H k H 0 exp− 2 d2 

1.07 r 2 

c 

2 . 46 

Since the numerical factor of 1.07 depends on the geometry, 

we replace it with the variable B. Collecting the parameters 

yields 

2 0 = B ¯ Z L 

nZ q 

2 2 

1 r c 

tr A 2 . 

47 

Substituting r c =A c /2, ¯ =2nf f , 0 2 =2f* 2 , 

tr =c 2 /2¯ 2 , and c=2d q f f leads to 

f* 2 

f f 

= B Z L 

2 2 

A c 

Z q 

2 

A 2 2 n 

2d q 

2 . 

48 

Collecting the results from the first and the second order 

yields 

f* 

f f 

= f*1 + f* 2 

f f 

= A c 

A 

iZ L 

+ B Z L 

Z q 

2 2 

A c 

Z q 

2 

A 2 2 n 

2d q 

2 

= A c iZ Z L 

L A c 2 n 

A Z q1−iB 

Z q A 2d 

2. 

49 

q 

The equations simplify, if 2 is expressed in terms of the 

active area. We need to treat the two-dimensional slab and 

the three-dimensional plate separately. For the twodimensional 

slab, is replaced by A using Eq. 15. Further, 

we separate the real and the imaginary part, which yields 

f 

f f 

 

f f 

=− A c 

A 

= A c 

A 

Z L 1 A c An 

2 

Z q1+B 

Z q 2d q 

Z L 1 A c An 

2 

Z q1+B 

Z q d q 

Z L 2 − Z L 2 

Z L . 

Z L 

, 

50 

Equation 50 is to be compared with the simulation data 

see Figs. 5–7 and Table II. The data are fitted with a function 

of the form 



FIG. 7. Shift of frequency f of half bandwidth at half maximum of 

a crystal loaded with a Sauerbrey film. The line is a fit to Eq. 50 with B 

as a free parameter. 

f 

f f 

= 1+A c , 

 

f f 

= 1+A c . 

51 

For the sake of comparison, the parameter B was allowed to 

be different for f and , that is, two parameters B and B 

are derived from experiment and simulation. B and B are 

computed from the fit parameters and . If the above 

analysis were strictly correct, B and B would take the same 

value. 

For the three-dimensional plate, the area, A, and 2 are 

related as 

A =− 

 

 

 

− 

exp− x2 + y 2 

2 dxdy = 2 . 52 

FIG. 8. Data from Fig. 1, displayed in analogy to Figs. 5–7. The lines are 

fitstoEq.53. 

f 

f f 

 

f f 

=− A c 

A 

= A c 

A 

Z L 1 A c n 

2 

Z q1+B 

Z q 2d q 

Z L 1 A c n 

2 

Z q1+B 

Z q d Z L . 

q 

Z L 2 − Z L 2 

Z L 

, 

53 

Using this result in Eq. 49 and again separating real and 

imaginary parts lead to 

TABLE II. Normalized slopes and derived values for B. 

mm −1 B mm −1 B 

Simulation 

Water 0.011 0.099 0.51 

Elastomer −0.216 0.37 0.040 0.47 

Film 100 nm 0.055 0.50 

Expt. 

Water mm −2 B mm −2 B 

n=5 0.11 19.5 0.15 4.11 

n=7 0.10 7.72 0.15 2.19 

n=9 0.089 4.03 0.17 1.60 

n=11 0.050 1.64 0.13 0.84 

n=13 0.055 1.39 0.15 0.81 

Expt. 

Polymer gel 

n=3 0.10 −1.7 0.16 1.4 

n=5 −0.10 0.9 0.26 0.8 

n=7 −0.17 0.66 0.33 0.6 

Equation 53 is to be compared with experiment Fig. 8 and 

Table II in the same way as Eq. 50 is compared to the 

simulation. 

Before moving on to the comparison with simulation and 

experiment, we check that Eq. 53 produces an increase of 

the D-f ratio with increasing contact area for the case of a 

water droplet Fig. 1b. For water, the frequency shift is unaffected 

by energy trapping because water has Z L 

=i 1/2 and therefore Z L =Z L . The bandwidth does increase, 

and so does the D-f ratio, /−f. 

Such a result should be also obtained with anisotropic 

three-dimensional plates and keyhole shaped back electrodes. 

Importantly, this change of dimension and geometry 

only affects the numerical factor B. The functions u m x in 

Eq. 13 turn into more complicated functions of radius, r, 

and angle, . Also, the frequencies of the anharmonic sidebands 

will not be described by Eq. 20. This changes the 

elements of the sum in Eq. 43. The integral in Eq. 43 runs 

over an area, rather than a line, and its outcome must therefore 

be an area. Everything falls in place, if A and A c are 

reinterpreted as areas with dimensions of meters, 2 rather than 

lengths with dimensions of meters. Of course, the numerical 

factor B changes. But it should not change by orders of magnitude. 



FIG. 9. Normalized shift of frequency and bandwidth as function of contact 

area calculated for a crystal in contact with a droplet of water panels a and 

b. Panel c displays the D-f ratio. Regimes I, II, and III are dominated by 

acoustic scattering, variable energy trapping, and a variable sensitivity factor 

K A r c cf. Eq. 1, respectively. The focus here is on regime II. 

RESULTS AND DISCUSSION 

Three different kinds of samples were simulated, which 

are a droplet of water, a flat punch of an elastomer, and a thin 

100 nm film of rigid material a “Sauerbrey film”. The 

material parameters were varied to some extent, but these 

variations did not yield much interesting results. By and 

large, the Sauerbrey scaling and Borovikov-Kanazawa scaling 

were confirmed for the film and the semi-infinite medium, 

respectively. Below, the focus is on the dependence of 

f and on the contact area. The material parameters were 

maintained fixed as shown in Table I. 

In an initial step, the radius of contact was varied over a 

few orders of magnitude in order to obtain an overview of 

the situation. Figure 9 shows the results for water. Three 

different regimes can be identified labeled with roman capitals. 

Regime I contains the small contact radii up to about 

10 m. The D-f ratio varies strongly in this range, the reason 

being scattering. The D-f ratio increases with contact 

radius because f and scale differently with the size of 

the scatterer. f scales with the scattering amplitude, which 

in turn scales about linearly with the size of the scatterer. , 

on the other hand, scales with the square of the scattering 

amplitude and therefore with the square of the size. 18 In the 

limit of small size, the ratio of the two therefore scales as the 

size. 

Next to regime I, there is a “regime II” spanning the 

range from about r c 10 m tor c 500 m, inside which 

the dependence of the D-f ratio on size is rather small. The 

size dependence is about the same as observed in experiment. 

Also, the normalized shifts f /A c and /A c can be 

well fitted by straight lines of the form 1+A c . Therefore, 

we identify regime II with the regime covered by the experiments 

and by the perturbation calculation. We focus the discussion 

on this size range. At large contact radii, there is a 

regime III, where r c comes close to the radius of the crystal. 

In this range the details of the amplitude distribution play a 

role. Both the normalized frequency shift and the normalized 

bandwidth decrease, but the decrease does not go in parallel. 

In regime III, the outcome of the simulations varied considerably, 

depending on simulation details and the nature of the 

load. Regime III can be of considerable importance for experiments 

in liquid environments. 32 Here, the load is nonzero 

at the rim. Still, we focus on experiments in air in the following. 

In such experiments, one will usually avoid regime 

III because the load would be too large for reproducible operation 

of the QCM. 

In order to demonstrate that energy trapping does change 

when a load is applied to the center of the crystal, the amplitude 

distribution of an unloaded crystal and a crystal 

loaded with a water droplet r c =1 mm is shown in Fig. 4a. 

The load causes a sharpening of the maximum of the amplitude 

distribution. Figure 4b demonstrates that such a cusp 

in the amplitude distribution can be explained by mode mixing. 

The dotted line is the superposition of the pure thickness 

shear mode and the third anharmonic sideband at a ratio of 

9:1. 

Energy trapping does influence the resonance frequency 

because in-plane shear gradients increase the strain energy in 

about the same way as the vertical shear gradients associated 

with the standing wave which is not a plane wave. Only if 

the lateral shear gradients are infinitely small, can they be 

neglected but they are not 20 . In order to make the argument 

more quantitative, the contribution of lateral shear gradients 

to the elastic strain energy, , was calculated according to 

d q /2 

 

−r Q 

q /2 

=−d 

d q /2 

−d q /2 

 

−r Q 

r Q 

c¯xx ux,z/x 2 

r Q 

c¯zz ux,z/z 2 . 

54 

A plot of and the fractional frequency shift f / f versus 

contact radius is shown in Fig. 10. The two correlate well. 

Being confident that a change of energy trapping is at 

least one source of the variability in the D-f ratio, we quantitatively 

compare the simulation results and the experimental 

data to Eqs. 50 and 53, respectively. When doing this, 

B was allowed to take different values for f and , 

termed B and B, although the numbers should be same 

according to Eqs. 50 and 53. The fit parameters are collected 

in Table II. The result of the comparison can be stated 

as follows: 

a 

b 

There is a range of contact areas, where the f /A c 

and /A c are well described by fit functions of the 

form 1+A c . The size range, where this fit works, 

is about the same as the size range of interest to the 

experimentalist. 

Calculating the parameters B and B from the normalized 

slopes, and , values of order unity are 

found. There is an exception for the simulated result 

for water, which produces an infinite B. This occurs 

because should be zero according to Eq. 50, 



because the total energy consumed by surface waves 

is small. Still, they deserve a mention. 

FIG. 10. Normalized frequency shift a and fractional contribution of lateral 

shear gradients to the total strain energy according to Eq. 54 b as a 

function of the radius of contact. There is a clear correlation. The lateral 

shear gradients contribute to the overall elastic energy and thereby increase 

the resonance frequency. 

c 

d 

while it is slightly nonzero in the fit. However, the 

slope is much smaller for f than for , , 

which is in line with the predictions. 

The parameter B takes different values for f and . 

For the experimental results, B decreases with overtone 

order. Also, B is larger than expected on the low 

overtones. 

The facts that fits with straight lines work well and, also, 

that the parameters B and B come out to be of order unity, 

give support to the perturbation approach and to the hypothesis 

that variable energy trapping is part of the explanation. 

On the other hand, the perturbation prediction is not quantitatively 

correct. Shortcomings of the perturbation analysis 

are listed below. 

a 

b 

c 

The analysis assumes the resonator is “locally flat,” 

meaning that the width of the amplitude distribution is 

much larger than the wavelength of shear sound and, 

also, that the radius of contact is larger than the crystal 

thickness. Both assumptions have their limitations. 

Interestingly the former assumption is rather well fulfilled 

on high harmonics, whereas it works less well 

on low ones. Comparing the results for the different 

overtone orders, one finds the agreement between perturbation 

result and experiment to be much better on 

high overtone orders. Generally speaking, theory and 

experiments on the QCM often agree better on higher 

harmonics as long as the anharmonic sidebands do 

not interfere. In this regard, the present calculation 

confirms a time-honored rule of thumb. 

When it comes to inadequacies of the viscoelastic 

model, there is—as always—the issue of compressional 

waves. The FEM simulation does pick up flexural 

modes, but is entirely unclear how these influence 

frequency and bandwidth at different contact radii. 

The crystal may launch surface waves at the outer 

edge of the sample. Generally speaking, their influence 

onto frequency and bandwidth should be small, 

The analysis above has an important implication for data 

analysis. The fact that fitting with straight lines works well in 

all cases indicates that the perturbation ansatz is correct. 

There only is a problem with the value of the second order 

coefficient. Possibly, other factors of influence such as compressional 

waves can also be treated within a perturbation 

ansatz and scale as r c 2 in leading order. 

With regard to the analysis of QCM data, the perturbation 

approach suggests a procedure to eliminate the second 

order effects from the data analysis. Regardless of what the 

values of B and B actually are, their influence can be removed 

from the data analysis by extrapolating to zero contact 

area. An improved value for the shear modulus of the 

material can be gained by measuring f and at different 

contact areas, fitting f /A c and /A c with a straight line as 

in Figs. 5–8, and inferring the shear modules from the intercept 

with the y axis. 

CONCLUSIONS 

The QCM can be used to determine the near-surface 

viscoelastic parameters of materials touching its surface, 

even if these are rather rigid. Since highly viscous materials 

would overdamp the sensor if they would cover the entire 

crystal, these can only be analyzed if the contact area is 

confined to small spot in the center of the crystal. The analysis 

then has to be based on the normalized shifts of frequency 

and bandwidth, f /A c and /A c . 

Experiments as well as FEM simulations show that these 

normalized shifts do slightly change as a function of contact 

size. However, the variability can be described by a linear 

relation. Such a linear relation can be rationalized by a perturbation 

approach. Assuming that a change of energy trapping 

is the dominating source of the variability of f /A c and 

/A c , one can make a prediction for the slope of these 

lines. Perturbation analysis, FEM simulations, and experiment 

all produce slopes of the same order of magnitude, but 

the values differ by up to a factor of 10. While a change of 

energy trapping certainly is part of the picture, it is not the 

entire explanation, either. 

Regardless of whether or not energy trapping is the only 

explanation, the perturbation ansatz itself implies that one 

may fit f /A c A c and /A c A c with straight lines and 

extrapolate to A c =0 in order to remove the size dependence 

from the analysis. The limiting values of f /A c and /A c 

for A c →0 should then be used for further analysis. In this 

way, data obtained on finite-sized contacts can be mapped 

onto the theory derived for samples of infinite lateral extension, 

be this the Sauerbrey relation, the Borovikov- 

Kanazawa relation, or any one of the more complicated models 

of viscoelastic layer systems. 

ACKNOWLEDGMENT 

Partial support by the State Program “Materials for 

Micro- and Nanosystems” MINAS at the University of 

Kaiserslautern is acknowledged. 



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