# Characterizing Vibrating Cantilevers for Liquid Viscosity and Density Sensing

Authors: Christian Riesch; Erwin K. Reichel; Franz Keplinger; Bernhard Jakoby
Journal: Journal of Sensors
 (2008)

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## Abstract

Miniaturized liquid sensors are essential devices in online process or condition monitoring. In case of viscosity and density sensing, microacoustic sensors such as quartz crystal resonators or SAW devices have proved particularly useful. However, these devices basically measure a thin-film viscosity, which is often not comparable to the macroscopic parameters probed by conventional viscometers. Miniaturized cantilever-based devices are interesting alternatives for such applications, but here the interaction between the liquid and the oscillating beam is more involved. In our contribution, we describe a measurement setup, which allows the investigation of this interaction for different beam cross-sections. We present an analytical model based on an approximation of the immersed cantilever as an oscillating sphere comprising the effective mass and the intrinsic damping of the cantilever and additional mass and damping due to the liquid loading. The model parameters are obtained from measurements with well-known sample liquids by a curve fitting procedure. Finally, we present the measurement of viscosity and density of an unknown sample liquid, demonstrating the feasibility of the model.


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## Body

## 1. Introduction

For many applications like online process or condition
monitoring, the liquid parameters viscosity and mass density are of high
relevance. The use of conventional laboratory equipment is often not applicable
due to its cost, space requirements, and other preconditions, for example,
vibration-free mounting. Furthermore, sample taking for such devices often
involves manual labor, tending to be time consuming and error-prone.Microacoustic sensors like quartz thickness shear mode
(TSM) resonators [1, 2]
and surface acoustic wave (SAW) devices, for example, [3] have proved particularly
useful alternatives to traditional viscometers [4]. However, these devices
measure viscosity at comparatively high-shear rates and small vibration
amplitudes. For non-Newtonian liquids, the results are, therefore, not directly
comparable to those obtained from conventional viscometers. For complex liquids
such as emulsions, it has also been shown that microacoustic devices may not be
sufficient to detect rheological effects which are present only on the
macroscopic scale [5].Micromachined vibrating structures usually feature
lower resonance frequencies and higher vibration amplitudes, making them more
suitable for non-Newtonian and complex liquids [6]. Microcantilevers commonly
used in atomic force microscopy [7–9] have been successfully used
as liquid property sensors. They allow for simultaneous measurement of the
liquid's viscosity and mass density, requiring sample volumes of less than 1 nL
[10]. However, a highly
sensitive optical readout is required to determine the beam's vibration
amplitudes. When immersed in liquid, the cantilevers face strong deterioration
of the quality factor due to high-dissipative effects [10]. Consequently, the vibration
amplitudes drops even more, limiting the sensor's measurement range to
low-viscous liquids. In other works, micromachined cantilevers and doubly
clamped beams driven by Lorentz forces [6, 11–13] or by the piezoelectric effect [14, 15] have been utilized as
liquid property sensors, and the feasibility of these sensors has been
demonstrated for viscosities in the range up to several Pa·s.In this contribution, we characterize resonating
cantilevers for the measurement of mass density and viscosity of liquids. They
measure viscosity in a rheological domain which is more comparable to that
probed by conventional laboratory instruments. The cantilevers feature
piezoelectric excitation as well as piezoelectric readout. The vibrating part
is about 55 mm long, but since only the cantilever tip is immersed in the
liquid, the induced damping of the cantilever vibration is kept low. The
sensors, therefore, exhibit high-quality factors, ranging from 20 to 60 even
for highly viscous liquids. Consequently, the detection of the resonances could
be accomplished by a simple readout electronics, and the measurement range is
greatly extended. On the other hand, the sensitivity of the sensor is
decreased. Furthermore, the sensor principle allows attaching different tips of
well-defined geometries to the cantilevers.When the cantilever tip is immersed in a liquid, the
resonance frequency and the damping of the cantilever are influenced by the
viscosity and density of the liquid. However, the cantilevers do not show a
simple relationship between the result of such a measurement and the liquid
parameters. Recently, several models have been devised to give a proper
description of the interaction of a vibrating cantilever and the surrounding
liquid, for example, [16–18] but most of these models assume fully immersed
cantilevers, which are long and thin, that is, the width w is much smaller than the length L. For the designs considered in this work, only the cantilever tip is immersed in
the liquid. For modeling the sensor-fluid interaction, the cantilever length L must therefore be replaced by the dipping
depth d, which is in the same range as w. A solution for cantilevers featuring w≈L, and accordingly w≈d is given in [19]. In [14] and other works, the
influence of the liquid loading on the cantilever's frequency characteristics
has been successfully modeled by approximating the forces acting on the
cantilever tip by those acting on a sphere oscillating in a fluid. The results
indicate that the cantilever tip is subject to an additional mass loading and
an additional damping caused by the surrounding liquid.In our work, we present a vibrating cantilever sensor
and a setup for the measurement of viscosity and density. The interaction of
the sensor and the liquid in which the tip is immersed is modeled by an
oscillating sphere. Possible simplifications of the model are discussed, and a
general model is devised. This model allows for simple calibration of the
sensor in a set of liquids with well-known properties. Therefore, the knowledge
of mechanical and electrical properties of the cantilever is not required, the
model parameters are instead obtained from the calibration procedure.


## 2. Sensor Fabrication

The cantilever sensors used in this work are based on commercially available lead zirconate
titanate (PZT) bimorph bending actuators (Argillon GmbH, Redwitz, Germany).
They feature a length of 49.95 mm, a width of 7.2 mm, and a total thickness of
0.8 mm [20]. The
cantilevers consist of two piezoelectric PZT layers on both sides of a carbon
fiber substrate (Figure 1(a)). The PZT layers are polarized in thickness
direction. Electrodes (1, 2, 3) allow for excitation of the actuator. Applying a voltage between the top electrode
(electrode 1) and the center electrode (2 in Figure 1(a)), which is used as
ground electrode (Figure 1(b)) causes a contraction or elongation in the upper
PZT layer but not in the substrate, and, therefore, deforms the cantilever. In
our setup (Figure 2(a)) a sinusoidal voltage is applied, which leads to bending
vibrations of the beam. A maximum voltage of 200 V can be applied to the
cantilever. The actual beam deflection is determined by measuring the voltage
at the sensing electrode.(a) PZT bending actuator with
attached tip. The bending actuator consists of a carbon fiber substrate and two
piezoelectric layers. The electrodes (1, 2, 3) allow for excitation and readout
of the sensor. (b) Applying a voltage to the upper layer leads to contraction
or elongation of the upper layer as indicated by the arrows.
(a)(b)Figure 2
Measurement
setup. The PZT bending actuator is rigidly clamped at one end. To the free end
of the cantilever, a tip of well-defined geometry (widthw) is attached and immersed in the sample
liquid (dipping depth d). The interaction of the cantilever tip and
the liquid changes the frequency characteristics, which is measured by the
lock-in amplifier (LIA).The bending actuator is clamped at one end, whereas
different tips of well-defined cross-sections are attached to the free end.
These tips are immersed in the sample liquid. The tip geometries and materials
are given in Table1. The clamping fixture is mounted on a rigid frame allowing
for vertical (x-direction) positioning of the sensor and
preventing vibrations of the entire setup. A lock-in amplifier (LIA) measures
the sensor voltage Vs,
resulting in the cantilever's frequency response. As voltage source Vd, the internal oscillator of the lock-in
amplifier is used.Table 1
Tip
geometries (rectangular cross-section) and tip materials.
TipWidthwThicknessMaterialTip A7 mm0.5 mmsiliconTip
B5 mm0.5 mmsiliconTip
C4 mm0.3 mmbrassTip
D2 mm0.5 mmsiliconThe cantilever exhibits several resonant vibration
modes. As an example, Figure3(a) shows the deflection (z-direction) for a cantilever with tip A (Table 1) vibrating in air. The corresponding sensing
electrode voltage is shown in the diagram below (Figure 3(b)). The resonance
frequencies are 100 Hz and 851 Hz for the first and second modes, respectively.
Further resonances were found at 2456 Hz (3rd mode) and 4870 Hz (4th mode). The
mode shapes for a uniform  cantilevered beam without a load at the tip are given
in Figure 4 [21].Frequency
characteristics of a PZT cantilever with tip A (Table1), Vd=400 mVrms driving voltage, vibrating in air. (a) The tip
deflection (peak-peak) was measured using a Polytec OFV-5000/OFV-505 laser
vibrometer. (b) The diagram below shows the corresponding sensing electrode rms
voltage.
(a)(b)Mode shapes of a uniform cantilevered
beam. The mode shapes have been obtained from solutions of the Euler-Bernoulli
beam equation and are also roughly valid for the piezoelectric bimorph.
(a)(b)(c)(d)


## 3. Theoretical Model

The vibration
behavior of the piezoelectric bending actuator is described by the
Euler-Bernoulli beam equation [21]:(1)EI∂4ψ(x,t)∂x4+μ∂2ψ(x,t)∂t2=∂2Mp(x,t)∂x2, where EI and μ are the effective bending stiffness and the
effective mass per unit length of the composite beam, ψ(x,t) is the beam deflection in z-direction (Figure 2), and Mp is the actuating moment due to the
piezoelectric effect. Since Mp is considered to be a constant moment along
the entire beam, ∂2Mp/∂x2=0.
Consequently, the boundary conditions for the clamped-free beam
are(2)ψ(0,t)=0,∂ψ(0,t)∂x=0,∂2ψ(L,t)∂x2=1EIMp,∂3ψ(L,t)∂x3=−1EIF,where L is the length of the beam and F is the force acting on the tip due to the
interaction with the surrounding liquid. The actuating moment Mp is given by [22](3)Mp=∫ApYpd31EzzdA=wbYpd31Vdzm,where Ap, Yp,
and d31 are the cross-sectional area, Young's
modulus, and the piezoelectric modulus of the actuating layer and Ez is the electric field in this layer in z-direction. wb is the width of the bending actuator, zm is the mean distance of the actuating layer
from the beam center, and Vd is the excitation voltage (Figure 2(a)).The measured voltageVs is calculated by applying(4)Dz=d31Txx+εEz,Txx=1K11(Sxx−d31Ez)to the sensing PZT layer, where Dz, Txx,
and Sxx are electric displacement, stress, and strain. ε and K11 are the permittivity and the compliance of the
PZT layer. Since Vs is measured by means of a voltage amplifier in
our setup, the current Is from the sensing electrode vanishes, and,
therefore, the charge:(5)Q=∫AeDzdA=0,where Ae is the surface area of the sensing electrode.
From (4), (5), and Sxx=−z∂2ψ(x,t)/∂x2,(6)Vs(t)=Kv∂ψ(L,t)∂xis obtained, where Kv is a constant depending on material and
geometry parameters [22]. The voltage at the sensing electrode is given by the
slope of the beam deflection ψ(x,t) at the free end, x=L,
of the cantilever. This fact is greatly confirmed by the measurements given by
Figure 3. Despite the lower tip deflection amplitude of the second mode the
resulting output voltage is higher than for the first mode of vibration.The interaction with the liquid surrounding, the cantilever
tip can be modeled by approximating the vibrating cantilever as an oscillating
sphere immersed in a liquid [14], exhibiting an effective sphere radius R and an effective sphere mass. The force F acting on such a sphere is given by [23](7)F=6πηR(1+Rδ)dudt+3πR22ηρω(1−2R9δ)d2udt2,where u is the sphere displacement, R is the sphere radius, ω is the angular oscillation frequency, and δ is the depth of penetration of the acoustic
wave, which is given by(8)δ=2ηωρ.The tips are attached to the
free ends of the cantilevers, therefore, the sphere displacement is u=ψ(L,t).In principle, solving the equations given above yields
the frequency characteristics of the vibrating cantilever with a tip immersed
in liquid for all modes of vibration. As the cantilever represents a composite
structure involving layers featuring different material properties, effective
parameters have to be used in the beam equation, for example, for Young's
modulus. As also the material parameters of the layers are not, or only partly,
available in the required accuracy and since the solution of the beam equation
would require the solution of a higher order system in order to determine the
coefficients by a suitable expansion (e.g., eigenmode expansion), we have
approximated the resonance behavior of the sensor in the vicinity of the first
mode resonance frequency as a second-order system, given by(9)(Me+Mi)d2udt2+(be+bi)dudt+Ku=F0ejωt,where u is the deflection of the cantilever tip in z-direction (Figure 2), Me and be are the effective mass and the intrinsic
damping of the cantilever and the tip, K is the spring constant, and F0 and ω are the driving force amplitude and angular
frequency. Mi and bi are the induced mass and damping due to the
liquid loading, given by (7)(10)Mi=3πR22ηρω(1−2R9δ),(11)bi=6πηR(1+Rδ).For sample liquids exhibiting high viscosities, and
low-vibration frequencies,R is negligible compared to δ,
leading to a simplification of (10) and (11) [14]. For the considered
cantilever tips and dipping depths (Table 1), the effective sphere radius R is in the range of a few millimeters. The
expected penetration depths δ for the sample liquids used in this work were
calculated for a vibration frequency of 100 Hz and are given in Table 2. The
results show that δ is in the same range as R,
and the prerequisites for said simplification are not fulfilled.Table 2
Reference values (dynamic
viscosityη and mass density ρ) of the sample liquids used in the
measurements and the expected depth of penetration δ at an angular frequency of ω=2π⋅100Hz,
see (8). The dynamic viscosity was measured using a Brookfield LVDV+II-CP
cone/plate rheometer at an ambient temperature of 23°C.
Sample liquidViscosityηDensityρδ (100Hz)AK150 silicone oil130mPa⋅s965kg/m30.65mmAK350
silicone oil440mPa⋅s979kg/m31.2mmSIL300
silicone oil211mPa⋅s1078kg/m30.79mmAlcatel
A120 oil270mPa⋅s881kg/m30.99mmConsequently, the consideration of the characteristic
penetration depthδ leads to a frequency dependence of the liquid
mass loading Mi,
(cf. (10)), and the liquid damping bi (cf. (11)). In the following, both Mi and bi are considered constant in the vicinity of the
resonance frequency, which is justified by the high-quality factors and the
accompanied narrow bandwidths of the resonances. Therefore, solving the
differential equation (9) using the Laplace transform yields(12)U(s)=11+(2Dn/ωn)s+(1/ωn2)s2F(s),where U(s) and F(s) are the Laplace transforms of the tip
deflection u and the driving force:(13)1ωn2=Me+MiK,2Dnωn=be+biK,where ωn is the resonance frequency and Dn the damping factor of the respective vibration
mode. From (7) and (13), one obtains(14)ωn2=ωn,air211+(2πR3/3Me)ρ+(6πR2/2Me)(1/ωn)ηρ,2Dnωn=2Dn,airωn,air(1+6πRbeη+6πR22beωnηρ),where ωn,air and Dn,air are the resonance frequency and damping factor
of the cantilever without liquid loading. The oscillating sphere model shows
that the cantilever's resonance frequency is affected by a term related to the
liquid density, and a second term containing the viscosity-density product,
whereas the time constant Tn=2Dn/ωn depends on the viscosity and the
viscosity-density product.For the interpretation of our measurement results
involving rectangular cross-sections, we devised a more generalized model.
Based on (14) and by introducing four independent coefficients c1, c2, c3,
and c4, we have
(15)ωn2=ωn,air211+c1ρ+c2(1/ωn)ηρ,(16)Tn=Tn,air(1+c3η+c4ωnηρ),where Tn,air is the in-air time constant 2Dn,air/ωn,air.
The values of the parameters ci are determined by the tip size and geometry,
the effective cantilever mass and damping, the dipping depth d (Figure 2(b)), and the respective mode of vibration n.Figure5 elucidates the model given by (15) and (16).
We consider four liquids LA, LB, LC,
and LD of which LA and LB are of the same density ρ1,
whereas LC and LD are of the same viscosity η3. The expected values of ωn and Tn are given by markers in the figure. The curves
in the figure represent the results expected for liquids of the same viscosity
and density, respectively.Time constantTn and resonance frequency ωn of a resonant cantilever dipping in 4
different sample liquids LA, LB, LC,
and LD as given by the generalized models (15) and (16).
(a)(b)For the determination of viscosity and density of an
unknown liquid, (15) and (16) have to be solved for η and ρ. The system of equations can be written as(17)A=Bρ+Cηρ,D=Eηρ+Fη, where A=ωn,air2/ωn2−1, B=c1, C=c2/ωn, D=Tn/Tn,air−1, E=c4ωn, and F=c3. Solving the system of equations yields
(18)ρ1,2=−12ACE−2ABF−C2DB(BF−CE)±14(ACE−2ABF−C2D)2B2(BF−CE)2−A2FB(BF−CE),η1,2=−12CDE−2BDF−AE2F(BF−CE)±14(CDE−2BDF−AE2)2F2(BF−CE)2−BD2F(BF−CE).For the cantilevers used in this
work, it turns out that BF−CE=c1c3−c2c4<0.
Furthermore, all the coefficients of (17) are positive, that is, A,B,C,D,E,F>0.
Therefore, in (18), only that ρ1,2 and η1,2,
respectively, are positive which exhibit a plus before the root sign, that
is,(19)ρ=−12(c2c4−2c1c3)(ωn,air2ωn2−1)−c22ωn(TnTn,air−1)c1(c1c3−c2c4)+{14[(c2c4−2c1c3)(ωn,air2ωn2−1)−c22ωn(TnTn,air−1)]2c12(c1c3−c2c4)2−c3(ωn,air2ωn2−1)2c1(c1c3−c2c4)}1/2,η=−12(c2c4−2c1c3)(TnTn,air−1)−c42ωn(ωn,air2ωn2−1)c3(c1c3−c2c4)+{14[(c2c4−2c1c3)(TnTn,air−1)−c42ωn(ωn,air2ωn2−1)]2c32(c1c3−c2c4)2−c1(TnTn,air−1)2c3(c1c3−c2c4)}1/2.


## 4. Measurements

With the setup depicted in Figure2(a), the frequency
response of the cantilevers is examined. The lock-in amplifier (Stanford
Research SR830) is used to drive the cantilever and to measure the sensor
voltage Vs and the phase shift ϕ(jω) between the driving voltage, that is, the
driving force and the sensor voltage, that is, the actual cantilever
deflection. The measurements are carried out with a sinusoidal driving voltage
of 200mVrms and within a frequency range from 70 to 110 Hz. At the resonance frequency of the first mode, a maximum tip deflection of 19μm (peak-peak) was measured in air by means of a
Polytec OFV-5000/OFV-505 laser vibrometer.A variety of oils are used as test liquids: AK150,
AK350 (Wacker Chemie), and SIL300 are silicone oils, and Alcatel 120 (A120)
oil. They exhibit liquid densities in the range from881 to 1078kg/m3 and viscosities from 145 to 440mPa⋅s.
These nominal liquid parameters are obtained from data sheets and measurements
by means of a Brookfield LVDV+II-CP cone/plate rheometer (Table 2). The
parameters of the liquids vary with temperature. As especially the
viscosity is highly temperature dependent, temperature control of the liquid
container has been established by means of a peltier heater/cooler system. All
results presented here have been obtained at 23°C.The dipping depth (Figure2(b)) was adjusted by
lowering the sensor until the tip touches the liquid surface, and then adding
the desired d=2mm, using a micrometer screw. For liquids
exhibiting low-surface tensions, like those considered in this work, a concave
meniscus is formed at the liquid-tip interface. We expect the resulting liquid
surface shape to increase the effective dipping depth, and, therefore, to
influence the sensor's behavior. Consequently, the sensor principle would be
limited to liquids with similar surface tensions, which can be assumed for the
considered oils.Figure6 summarizes the measurement results at mode 1
for tip A in air and immersed in the sample liquids. As expected from (15) and (16), the liquid
loading decreases the resonance frequency ω1 and increases the damping factor D1.Magnitude and phase response
of a cantilever with tip A vibrating in air and immersed in different liquids
(Table2). The dipping depth d is 2mm,
the driving voltage Vd=200mVrms.
(a)(b)From the cantilever's frequency response, we extract
the first mode resonance frequencyω1=2πf1 and the damping factor D1 by fitting a second-order transfer
function:(20)ϕ(jω)=arg{11+jω(2D1/ω1)−(ω2/ω12)}to the phase shift measurements
(Figure 6) with respect to the parameters ω1 and D1.Figure7 shows the results for the cantilevers with
tips A, B, C, and D (Table 1) and the sample liquids (Table 2). The
left column of Figure 7 shows the measured time constant T1=2D1/ω1 versus the dynamic viscosity η of the sample liquids. The results for each
cantilever appear to be nearly lying on a single trend curve, indicating that T1 is dominantly influenced by the liquid's
dynamic viscosity. According to model equation (16), the value of c4 thus must be small compared to c3.Resonance
frequencyω1 and time constant T1 obtained for the four different tips and a
variety of sample liquids. The crosses (+) in the diagrams represent the calculated
values obtained from the parameter fit and show good agreement between the
experimental results and the model.
(a)
Tip A(b)
Tip B(c)
Tip C(d)
Tip DFigure7, right column, depicts the resonance
frequency f1 versus the liquid density. The results are
widely spread in the ρ-f1 plane. Obviously, the resonance frequency of
the cantilever tip immersed in the liquid is strongly influenced by both the
liquid's density and viscosity. In our model, this relationship is described by
(15).Using the measurement results (Figure7),
the model parameters c1, c2, c3,
and c4 can be determined. Figure 8 illustrates the
model equations (15) and (16) and the parameter fit procedure. The values of ω1,air and T1,air are obtained from a single measurement of the
vibrating cantilever in air. Furthermore, each measurement in a sample liquid
yields a pair of (ω1,T1).
The values of viscosity and density, η and ρ,
of each sample liquid are also known (Table 2). Finally, a fit algorithm
applied to the model equations results in the model parameters c1, c2, c3,
and c4 for a particular cantilever. The results of
this parameter fit are depicted in Figure 9. The diagram shows
the model parameters c1, c2, c3, c4 for the cantilevers with different tips. It is
important to note that the parameters do not only depend on the width of the
attached tip, but also on the mass and the intrinsic damping of the entire
cantilever. Therefore, the parameter values do not necessarily decrease with
the tip width, as shown by the location of the tip D marker in the c3-c4 plane (Figure 9). The PZT bending actuator,
with tip D has being attached, was not sealed by a protective coating, and thus
features a lower intrinsic damping. However, the parameter fitting process
described above yields parameter values considering such different behaviors of
the cantilevers.Figure 8
Obtaining the model parametersc1, c2, c3,
and c4 from measurements (Figure 7) in air and in
reference liquids with known viscosities and densities (Table 2).Fitted parameter valuesc1, c2, c3,
and c4 obtained by fitting the model equations to the
measurement results.
(a)(b)For a validation of the model, the parameters
determined above were used to calculate the values ofω1 and T1 for each tip-liquid combination. These
calculated values are indicated by crosses (+) in Figure 7 and are in good agreement with
the experimental data.At last, we use the model parameters obtained above to
determine the viscosity and density of an olive oil sample. The measurement
results, that is, the resonance frequenciesω1 and the time constants T1 of the cantilevers immersed in the sample
liquid are given in Table 3. From these results, the density and the viscosity
of the liquid are calculated using (19),
respectively.Table 3
Determination of viscosity and density of an olive oil sample by means of vibrating cantilevers. The densityρ and the viscosity η have been calculated from the measurement
results ω1=2πf1 and T1 using (19). The density and viscosity
obtained by means of weighting and a cone-plate rotational viscometer,
respectively, are ηref=66.4mPa⋅s and ρref=913kgm−3.
Cantileverf1T1ViscosityηDensityρTip A95.07Hz88.24μs66.4mPa⋅s919.7kgm−3Tip B103.15Hz58.80μs62.9mPa⋅s921.4kgm−3Tip
C94.44Hz50.43μs64.6mPa⋅s920.1kgm−3Tip
D95.83Hz40.88μs67.6mPa⋅s908.6kgm−3


## 5. Conclusion

The change of
the dynamic behavior of a vibrating cantilever allows to investigate the
physical properties of liquids. Various types of small tips of different
geometries are attached to the cantilever and immersed into the sample
solutions. The liquid surrounding the cantilever tip changes both the resonance
frequency and the damping of the entire cantilever structure. To be able to
conclude, from the measured frequency response to the liquid's parameters, an
analytical model is needed. The developed model is based on the forces acting
on an oscillating sphere in liquid, but generalized model parameters are used
to consider the actual geometries of the applied cantilever tips. These
parameters, furthermore, include the electrical and mechanical characteristics
of the beam, which, therefore, must not be known. The model proved to be well
suited for the characterization of various cantilevers and tip geometries by
measuring in liquids with known density and viscosity. To extract the model
parameters from the measured data, a curve fitting procedure was performed. The
obtained parameters are specific for each cantilever tip and allow the
subsequent simultaneous determination of density and viscosity of unknown
liquids. Compared to other works, the sensor features high-quality factors of
the considered resonance mode even for highly viscous liquids, greatly
extending the measurement range.



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*Source: 697062-2008-10-09.xml*