Finite Element Modeling and Analysis of Multi-Channel Quartz Crystal Microbalance
(*)
F. Lu, (*)(**)H.P. Lee and (*)S.P. Lim
(*)
Department of Mechanical Engineering, National University of Singapore
(**)
Institute of High Performance Computation Singapore
Abstract:
Multi-channel Quartz Crystal Microbalance (MQCM)
devices have many promising potential applications as
sensors in biological engineering. In this work, a finite
element modeling of the Multi-channel QCM is
constructed using ANSYS. The coupling interference
between two adjacent QCMs will affect a MQCM in
measuring the surface loading quantitatively. For
reduction of the interference, grooves are created
between the QCMs. The thickness shearing mode
vibration is found to be trapped by the grooves. The
effects of the groove on the frequency interference are
studied.
equivalent circuit based on the Mason model has
been used to predict the performance of the QCM
with surface loading (mass, liquid et al.).
However, the equivalent circuit mode neglects
some mechanical important features, such as the
modes coupling, energy trapping of the electrode
and frequency interference of the elements of the
arrays of the MQCMs. The finite element method
is a numerical technique that can be used to model
the mechanical characteristics of the complex
structure accurately, as well as to optimize the
design parameters.
Keywords: Quartz Crystal Microbalance, Energy
Trapping, Frequency interference,
1. Introduction
AT-cut quartz crystal microbalance has been
applied to as mass sensitivity device to monitor
the mass changes on the surface of quartz crystal
over the past decades [1]. Recently, multi-channel
quartz crystal microbalance has been reported and
attracts the interest of researcher because of its
several potential applications [2-4]. Multi-channel
quartz crystal microbalance is a sensor array of
QCMs fabricated in one quartz chip. With
different coating recognition materials on each
channel, MQCM can be used to identify various
types of absorptions in the environment.
Frequency interference between the adjacent
resonators is one of the key problems for MQCM.
The surface mass absorbed on one channel results
in frequency decreases of not only its own but
also other channels. For purpose of the
miniaturization and reduction of cost, it is desired
to fabricate QCMs with narrow spaced, in which
the vibration overlap between two adjacent
resonators appears more severe. The energy
trapping effect of normal electrode design cannot
avoid this cross talk to an accepted range. This
frequency interference between the adjacent
resonators disables the MQCM in measuring
surface
loading
quantitatively.
Electrical
Figure 1 Schematic for MQCM on
one quartz chip
In this study, the finite element model of MQCM
is constructed with commercial software ANSYS.
The vibration shape of the thickness-shearing
mode shown that there is overlap between the two
adjacent QCMs, especially when the QCMs are
spaced with narrow distance. The self-mass
sensitivity and mutual mass sensitivity are defined
to investigate the frequency interference of
MQCM. The effects of groove designed between
the QCMs for reduction of the interference are
investigated.
2. Mass Sensitivity of Single Quartz Crystal
Microbalance
As schematic illustrated in figure 2, a QCM
consists of a thin disk of the AT-cut quartz crystal
wafer, with electrode patterned on the both sides.
The application of the period electrical voltage
between the electrodes on a certain frequency
induces the thickness shearing vibration of the
crystal. The mass absorbed on the quartz surface
induces the resonance frequency shift, which is
proportional to the mass absorbed.
left with f  22.62 KHz. The amount of mass
absorption is proportional to the frequency shift,
which is given by the equation [1]
f 
electrode
Network
analyzer
AT-cut
Quartz
0.4mm
70%
Figure 2, diagram of single QCM
Single QCM structure, with AT-cut quartz
thickness hq  0.4 mm, length L  12 mm, and
sliver electrode layer thickness he  20  m, is
modeled using ANSYS. The piezoelectric coupled
element plane-13 is used for finite element
modeling. Imposing a periodic electrical on the
electrode, the fundamental thickness shearing
mode is excited within the quartz. Due to the
finite size of the electrode layer on the quartz
wafer, the thickness shearing vibration is trapped
at the central portion.
2 f 02
f

m
(  q  q )1 / 2
(1)
where  f is the resonance frequency shift, f 0 is
the unperturbed resonance frequency of thickness
shearing mode,  q and  q are density and
thickness shearing modulus of the quartz,
m   m h is the surface mass density absorbed
on the quartz surface. The mass sensitivity is
proportion to square of the resonance frequency.
Equation (2) is derived for the infinite of the
quartz plate without consideration of the energy
trapping effect of the electrode.
20
15
Ux
10
5
0
-0.006
-0.004
-0.002
0
0.002
0.004
0.006
Figure 4 mass sensitivity and resonance
frequency as function of the
electrode covering area
-5
Figure 3 TS displacement profile of QCM
The vibration decays exponentially at the edge of
the electrode region. The amplitude of the
thickness shearing displacement is not unformed
distributed on the electrode area as shown in
figure 3. This non-uniform vibration amplitude
will do effect on the QCM measuring results],
especially when the mass is partially attached on
the QCM surface. In this study, the mass
absorption is assumed to be uniformly and rigid
attached on the surface.
With 70% electrode covering on the quartz plate,
the fundamental thickness–shear mode frequency
is 4.24866MHz. With the absorption mass volume
0.6% of the quartz substrate absorbed on the
electrode surface, the spectrum is shifted to the
In practical, the frequency of TS mode can be
modified with etching different size of electrode
on the surface. Figure 4 gives the mass sensitivity
of the QCM with different electrode size covering
on the quartz surface computed by FEM for a
finite size quartz substrate. It is shown that, the
mass sensitivity is almost same for different
electrode covering are, even the TS resonance
frequency is decreases with larger electrode area.
3. Analysis of the Frequency Interference of
MQCM
The MQCM consists of quartz crystal wafer of
0.4mm thickness and 21mm length. Silver
electrodes of thickness 20 m are etched on both
surfaces. The linear approximation of the
relationship between elastic absorption and TS-1
resonance frequency is about has to be found to
hold for mass fraction up to about 2% of the
resonator. This study only considers the
phenomena within the linear range. Based on
Mindlin’s studies, when the electrode size reduced
below Bechmann’s number [5], the inharmonic
overtones of the thickness-shearing mode are
vanished. Say, for AT-cut quartz crystal plate with
f TS 1 / fˆ  1.02 , yields a / h  23 . For vanished
the overtones within the linear range, the electrode
size is set to be less than 20 times of the thickness
quartz plate.
As shown in figure 5, two resonators with
electrodes covered are separated with a nonelectrode region. Two adjacent QCMs are
modeled by finite element method using ANSYS
QCM-A
QCM-B
interference is studied by calculating the
perturbation in frequency shift of QCM-A. The
mass sensitivity is defined as
 i j 
f i
,
f i m j
i, j  A, B
where m is mass added on surface and  f is
the frequency shift of TS mode, f i is the
resonance frequency of the thickness shear mode.
Defining the coupling factor of QCM-B to QCMA as the ratio
A 
 A B
 A A
(3)
On the Simulation, voltage is applied on one
QCM; another QCM is an open circuit.
I. Symmetrical Designed
Interval Distance 2mm
d
Figure 5 Analysis model for frequency
interference of MQCM
indicated as QCM-A and QCM-B.
(2)
Channels
with
Fabricating QCMs with a fixed size quartz wafer,
the coupling strength between QCMs is changed
with distance between adjacent QCMs. It is
intuition that the interference between the
channels can be varnished with enlargement of the
interval distance between the channels.
fraction of mass absorption
0.0% 0.3% 0.6% 0.9% 1.2% 1.5% 1.8% 2.1%
4.36
TF31
frequrncy (MHz)
4.32
TS2-B
TS2-A
4.28
SF
y = -2E-06x + 4.2555
TS1-B
4.24
TF31
4.2
y = -0.0025x + 4.2554
TS1-A
4.16
0
Figure 6 Finite element model for adjacent
QCMs with different interval design
The mass absorption on the one channel do effect
on itself, but also induces the frequency shift on
the adjacent channels. The frequency interference
coupling of the dual QCMs with changes of the
distance for fixed size of the wafer. By adding
mass on the surface of QCM-B, the frequency
5
10
15
20
mass absorption (kg/m2)
25
30
Figure 7 Frequency spectrum with mass absorbed
on channel A with 2mm interval
distance for symmetrical design
Figure 7 gives the frequency spectrum of the
MQCM with symmetrical design (QCM-A and
QCM-B have the same electrode size). The
distance between the two adjacent resonators are
Table 1 Interference factor for different resonator
pairs with interval d=2mm
Electrode length(mm)
5
6
7
 A A (KHz/kgm-2)
2.5
2.5
2.5
 A B ( KHz/kgm-2)
0.09
0.002
0.05
A
3.6%
0.08%
2%
The frequency interference between two QCMs is
every weak if the resonators operating with purely
thickness shear vibration because the thickness
shear displacement decays exponentially at the
edge of the electrode due to the energy trapping
effect. The disperse distance is less than 0.4mm as
shown in figure 3 for resonator with 4.2mm
electrode length. The frequency coupling is
mainly from other spurious mode vibration, for
example, thickness flexure, longitude compression
vibrations. For MQCM with interval distance is
larger than the TS displacement decay length, the
reduction of the frequency interference can be
achieved by reduce the displacement coupling
between the thickness shear and other spurious
modes, or cut the cross talk of those spurious
modes.
II Non-symmetrical
interval distance
channels
with
Two QCMs with different electrodes covering
area (QCM-A with 3.5mm and QCM-B with
4mm) are fabricated in one quartz wafer. By
changing the interval distance between e QCMs,
the mutual mass sensitivity of the nonsymmetrical QCM-A and QCM-B are studied
using finite element method. Figure 8 gives
coupling factor  A B as function of the interval
distance between the QCM-A and QCM-B. It is
shown that the coupling mass sensitivity is
reduced with increased the interval distance. And
after the interval distance exceed the 1mm, the
coupling due to the thickness shearing vibration is
decreased, but the coupling due to the thickness
flexure vibration is more significant, as shown in
figure 9, which gives the vibration profile of the
MQCM in QCM-A’s operation frequency for
different interval distances.
coupling factor (no unit)
Keep the wafer size constant; same computations
are done for the symmetrical resonators with
different size of electrodes. The corresponding
interference factors  are listed in table 1. It is
shown that the resonator with electrode size 6mm
gives the minimum interference.
modes only by design electrode length/thickness
ratio. The modification of the structure between
channels can be employed to reduce the frequency
interference between channels.
5.00%
4.00%
3.00%
2.00%
1.00%
0.00%
0
0.5
1.5
2
2.5
Figure 8 Coupling factor value  A with
different interval distance for QCMs
with 4.289MHz and 4.279MHz
40
36
32
28
24
2.0mm
20
16
small
For reduction of cost and miniaturization of the
MQCM structure, it is desired to reduce the
interval distance to fabricate more channels on
one quartz wafer. With different frequencies
resonators on one quartz wafer, it is not easy to
design all channels well away from spurious
1
interval distance between channels (mm)
U_X
2mm, electrode length is 6mm, which is 15 times
of the quartz thickness. With mass absorption
added on the channel-A, channel-A exhibits the
same mass sensitivity as the single QCM, i.e.
2.5KHz/(kg/m2). And there is tiny effect on the
resonance frequency of the thickness shear mode
of channel-B, i.e. 2Hz/(kg/m2). The frequency
interference between two channels is very small
even channel A and channel be designed with
same size of the electrodes. The coupling factor
 A is only 0.08%.
12
1.0mm
8
4
0
0.4mm
-4
QCM-A-8
-0.01
-0.005
QCM-B
0
0.005
0.01
postion on qaurtz wafer
Figure 9, displacement profile for different
interval distance
A groove mesa between the QCMs is designed to
decrease the vibration coupling due to the
thickness shearing vibration for case with 0.4mm
interval distance as shown in figure 6. The groove
enhances the energy trapping effect of thickness
vibration. The displacement coupled between two
channels is reduces as shown in figure 10. The
depth of the groove is set to be 40 m , which is
10% of the thickness of quartz wafer. The
vibration amplitude in non-operation channels is
reduced, so as to reduce the mutual frequency
interference of adjacent channels. Do mass
sensitivity computation with mass deposited on
the each QCM, the resonance frequencies of
QCM-A and QCM-B is changed to 4.29440MHz
and 4.28305MHz respectively. And the coupling
factor  A is reduced from 4.74% to 2.4%.
Reference
[1] D. S Ballantine Jr. R. M. White, S. J. Martin, A.J.
Ricco, G. C. Frye, E. T. Zellers and H. Wohltjen
Acoustic Wave Sensors: Theory, Design and
Physico-Chemical Application
ACADEMIC
PRESS 1997
[2] Takashi Abe and Masayoshi Esashi
One-Chip
Multichannel Quartz Crystal Microbalance (QCM)
Fabricated by Deep RIE
Sensors and Actuators
Vol.82 pp:139-143 2000
[3]
Steffen Berg and Diethelm Johannsmann
Laterally Coupled Quartz Resonators Anal. Chem.
Vol.73 No.6 pp:1140-1145 2001
[4] K H Lee, F. Shen et al. Frequency Interference
between two Quartz Crystal Microbalance. IEEE
Sensor 2002 pp:1148—1153.
[5] R. D Mindlin and P. C. Y. Lee Thickness-Shear
30
for normal
design
for groove
design
25
20
amplitude of U_X
15
10
5
QCM-A
QCM-B
0
-5
-0.01
-0.006
-10
-0.002
0.002
0.006
0.01
position alone the quartz
Figure 10 vibration profile of normal interval
design and that of grooves interval
Conclusion
With the FEM modeling, it is applicable to
optimal design complex structures of MQCM to
achieve the minimal coupling interference
between adjacent QCMs.
For MQCM structure with sufficient interval
distance, the adjacent coupling is mainly due to
the flexure wave, which can propagate along the
length direction of the MQCM without decay.
With convex wall designed between adjacent
channels, this interference can be reduced. For
cases with size consideration, can the interval
distance is, the non-symmetrical design is require.
The mutual mass-frequency interference is mainly
due to the thickness shearing vibration. The
dented grooves structure can be used to increase
the energy trapping effect of the QCM, so as to
reduce the interference between channels.
and Flexural Vibrations of Partially Plated, Crystal
Plates International Journal of Solids Structure Vol.
2 pp:125-139 1966