Etienne Lemaire
email: etienne.lemaire@u-bordeaux1.fr

Martin Heinisch

Benjamin Caillard

Bernhard Jakoby

Isabelle Dufour

Comparison and experimental validation of two potential resonant viscosity sensors in the kilohertz range

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Vibrating microstructures are known to interact with the surrounding medium they are immersed in [START_REF] Martin | Characterization of a quartz crystal microbalance with simultaneous mass and liquid loading[END_REF][START_REF] Jakoby | Miniaturized sensors for the viscosity and density of liquidsperformance and issues[END_REF]. The dynamic mechanical motion of such structures is used to sense the parameters of interest. The aim of the study is to compare two types of structures for a viscoelasticity sensing applications. The vibrating wires and microcantilevers described in this paper are electromagnetically actuated, which is compatible with most liquids. Vibration measurements are obtained electrically by a piezoresistive Wheatstone bridge set-up in the case of the microcantilever [START_REF] Dufour | The Microcantilever: A Versatile Tool for Measuring the Rheological Properties of Complex Fluids[END_REF] and by measuring the motion induced voltage in the case of the wire [START_REF] Heinisch | Tunable resonators in the low kHz range for viscosity sensing[END_REF]. These methods can be efficiently used to characterize the mechanical parameters in the vicinity of the resonant frequency. In order to cover a larger spectral range containing all modes of vibration, an optical measurement is also performed using a laser Doppler vibrometer (MSA-500 from Polytec). Complex viscous fluids feature frequency-dependent properties, e.g., viscoelastic phenomena can occur at specific shear rates or frequencies (in case of oscillating motions). In general, resonant microstructures can be used to measure hydrodynamic forces which depend on both, the fluid density and the complex viscosity [START_REF] Sader | Frequency response of cantilever beams immersed in viscous fluids with applications to the atomic force microscope[END_REF]. An analytical model, some parts of which are detailed in this paper, allows characterizing the complex fluids by recording the frequency dependent deflections of these microstructures being immersed in liquids. In order to obtain reliable results, a well-controlled experimental set-up as well as a complete physical characterization of the microstructures was performed. Transparent fluids (e.g., water or egg albumin) have been characterized using an optical characterization, whereas opaque fluids (e.g., yogurt) have been measured using an electrical measurement. The comparison between the results obtained with both structures allows for verifying the validity of the developed analytical model. The monitoring of the complex viscosity of yogurt during the fermentation process has been done in order to compare and validate the obtained rheological results with data from literature [START_REF] Soukoulis | Industrial Yogurt Manufacture: Monitoring of Fermentation Process and Improvement of Final Product Quality[END_REF][START_REF] Lee | Formation and Physical Properties of Yogurt 2010 Asian[END_REF].

Preprint Measurement Science and Technology, Vol. 24, 2013, 084005 (9 pages) (doi:10.1088/0957-0233/24/8/084005)

Materials and Methods

Description of the mechanical structures

The microcantilevers are designed and manufactured in collaboration with the LAAS in Toulouse (www.laas.fr.). The process uses SOI wafers enabling controlling the microcantilever's thickness (using the buried layer of silicon dioxide as etch-stop layer). Thus, the thickness is fixed between 19.7 and 20.3 µm. Three cantilevers with different geometries (see Table I) were designed, and are denoted as A, LL and LH. The cantilever's mass density is 2.45 10 3 kg.m -3 , (silicon, the main constituent of the cantilever, has a density of 2.33 10 3 kg.m -3 ). The conducting paths, deposited on the structure are constituted of titanium (100nm thick) and then gold (700 nm). An insulating layer of silicone dioxide, 600 nm thick, is deposited at the end of the process on the cantilever covering the metal. The Young modulus for the complete structure is found to be about 150 GPa. The cantilever design includes a piezoresistive strain gauge overlapping the cantilever's clamping region. A pair of the fabricated cantilevers is shown in figure 1. The cantilever on the right is released, allowing for out of plane oscillations. The second one on the left is more than ten times thicker and thus static. It is subjected to the same physical and electrical conditions and is used as reference in terms of noise and coupling for the strain gauge implanted on it. Both gauges are incorporated into a half-Wheatstone bridge in order to cancel both, the noise and the electrical coupling by differential measurements. The released structure is actuated to out of plane vibrations by means of Lorentz forces on sinusoidal currents (in the gold path, c.f. figure 1) in a magnetic field provided by a permanent magnet.

Preprint Measurement Science and Technology, Vol. 24, 2013, 084005 (9 pages) (doi:10.1088/0957-0233/24/8/084005) The second structure studied in this paper is named "U-shaped wire". The used wires are insulated nonmagnetic steel wires (the insulator is useful to avoid oxidation and corrosion problems). Their diameter is 79 µm and their mass density is 7.56 10 3 kg.m -3 . In previous studies, doubly clamped straight wires (or string) were investigated for characterizing liquids properties [START_REF] Heinisch | Tunable resonators in the low kHz range for viscosity sensing[END_REF]. However, these doubly clamped beams may be subjected to instabilities in their resonant characteristics due to (e.g., thermally induced) varying normal stresses. In this study, a "U-shaped" wire, simply magnetically clamped on a support (as shown in figure 2) is used to overcome these spurious instabilities. The "Ushape" has been proven to be one of the best geometrical configurations to probe the physical properties of surrounding media [START_REF] Agoston | Evaluation of a vibrating micromachined cantilever sensor for measuring the viscosity of complex organic liquids[END_REF][START_REF] Keplinger | Frequency and transient analysis of micromachined U-shaped cantilever devices for magnetic field measurement[END_REF]. These vibrating wires also use an electromagnetic actuation, as described above. In case of opaque fluids characterization, the steel wire has been clamped by soldering it onto a solid edge. This modification is necessary due to stability requirements and the lack of the ability visual control. In this case, the stability and the higher amplitude of the obtained electrical signal allow for accurate characterization of the mechanical resonance in opaque fluids.

Measurements techniques

Two methods have been used to characterize the mechanical resonance of both microstructures. For the first method, we used a laser vibrometer (MSA-500 Micro System Analyzer from Polytec). This technique is based on the interferences between a reference laser beam and another laser beam which Preprint Measurement Science and Technology, Vol. 24, 2013, 084005 (9 pages) (doi:10.1088/0957-0233/24/8/084005) is modulated by the structure's out-of-plane vibration. A picometer-range resolution is obtained when characterizing the investigated structures dynamic responses.

The second method utilizes a lock-in amplifier (Signal Recovery 7280 DSP) in order to measure the magnitude and phase of the motion-induced voltage in the electromechanical resonator. In the case of microcantilevers, the measured signal is on amplified Wheatstone bridge output voltage. For the wire, the input and the output signals (i.e. the actuation current and the motion induced voltage) are both carried by the conductive wire itself. Thus, the measured voltage consists of the ohmic voltage drop due to the wire's resistance and the motion-induced voltage. For both resonators, the first mechanical resonant mode has been measured electrically first in air and then in the examined liquids.

Test fluids and process

Three fluids have been characterized optically. Deionized water, a Newtonian fluid, was used for calibration. Then, two other complex viscous fluids have been investigated: hen egg albumin and a commercial shampoo (Garnier® FRUCTIS Shampoo) supposedly more viscous and exhibiting elastic properties. The measurements were performed in a cleanroom with temperature and humidity control. The temperature was varying between 18.8°C to 19.4 °C and the humidity rate was 45%. Milk mixture turning into yogurt during fermentation as well as water have been characterized electrically, using the lock-in amplifier.

The mass density of the probed fluids has been measured at one temperature using a microbalance specific setup (using the Archimedes force principle) and the viscosity values have been investigated using our methods. The viscosity of pure water at 25°C is known to be 1mPa.s (= 1cP) and is used as a reference value. These established data at 19 °C are summarized in Table 2.

Modeling and treatment

The differential equation describing the vibration of cantilevers (Euler-Bernoulli beam theory) is given as follows (this well-established equation is also presented e.g. in [START_REF] Nayfeh | Linear and Nonlinear Structural Mechanics[END_REF]):

4 2 4 (, ) (, ) (, ) (, ) L exc fluid wx EI m w x F x F x x        (1)
Here w(x,ω) is the amplitude of vibration at x coordinate and at angular frequency ω in the z-direction [m], E is the Young's modulus [Pa], I is the second moment of area of the cross section at coordinate x Preprint Measurement Science and Technology, Vol. 24, 2013, 084005 (9 pages) (doi:10.1088/0957-0233/24/8/084005) with respect to the y axis [m 4 ], F exc is the actuation force per unit length in x-direction [N.m -1 ]. F fluid is the force per unit length in x-direction exerted by the fluid on the wire or cantilever [N.m -1 ], m L is the mass per unit length of the microstructure [kg.m -1 ]. Equation ( 1) can be applied for both the cantilever and the U-shaped wire.

The hydrodynamic force per unit length exerted by the fluid on the microstructure can be written as:

2 12 (, ) ( ) (, ) ( ) (, ) fluid Fx j gw x g w x        (2)
The cross section of the U-shaped vibrating is circular whereas it is rectangular in case of the microcantilever. This difference has an impact on the hydrodynamic force modeling, but a common formalism can be used for the terms g 1 and g 2 [START_REF] Maali | Hydrodynamics of oscillating atomic force microscopy cantilevers in viscous fluids[END_REF]:

2 2 11 2 4 f b gb b bb            (3) 2 21 2 4 f b ga a b        ( 4 ) with 2 f      ( 5
)
where b is the cantilever width or the wire diameter, η is the fluid dynamic viscosity and ρ f is the mass density of the surrounding fluid. The coefficients a 1 , a 2 , b 1 and b 2 depend on the cross section geometry. For a circular cross section, a 1 =1, a 2 =4, b 1 =4, b 2 =123/32. This is obtained from a Taylor Series expansion when δ<<b of the expression given in [START_REF] Rosenhead | Laminar Boundary Layers ~Clarendon[END_REF]. For the rectangular cross section, these four coefficients have been determined by Maali et al. [START_REF] Maali | Hydrodynamics of oscillating atomic force microscopy cantilevers in viscous fluids[END_REF]. In our case they are, a 1 = 1 a 2 = b 1 = 3.7997, and b 2 = 2.7364.

The complex shear modulus of the liquid G is defined as:

Gj   ( 6 )
It can be split up into the real (elastic) part G' and an imaginary (viscous) part G". Thus, it follows that:

(' " ) jGj G j G      ( 7 )
The transfer function of the mechanical resonator near the resonant frequency can be approximated by:

0 2 00 H H( L, f ) f f 12 j f f       (8)
Where H 0 is the static deflection, f 0 is the eigenfrequency and ξ is the damping factor. The eigenfrequency f 0 is defined for a second order system as the free, undamped oscillation frequency. It is given by: Preprint Measurement Science and Technology, Vol. 24, 2013, 084005 (9 pages) (doi:10.1088/0957-0233/24/8/084005) 0, 0 20 ()

1 vacuum L f f g f m   (9)
Here f 0,vacuum is the eigenfrequency in vacuum.

The damping coefficient , which allows for estimating the decay of the transient oscillations is given by equation ( 10) which was already published in [START_REF] Youssry | A straightforward determination of fluid viscosity and density using microcantilevers: from experimental data to analytical expressions[END_REF] and then used in [START_REF] Mather | Liquid viscoelasticity probed by a mesoscale piezoelectric bimorph cantilever[END_REF]:

10 20 0, () () 41 vacuum L L gf g f fm m     (10)
Thus, if the fluid mass density is known, both, the real and imaginary parts of the shear modulus G (previously developed in [START_REF] Belmiloud | Vibrating microcantilever used as viscosimeter and microrheometer[END_REF]) can be given by:

    2 22 2 1 02 1 2 "( ) 2 4 2 2 g BB B Gf g C g g C DD D DD             (11) 2 02 1 1" '( ) " BG Gf g C Dg D G        ( 1 2 )
Where:

2 12 ;; 42 22 ff bb BbCb D         (13)
This model allows to calculate G'(f 0 ) and G"(f 0 ) from f 0 and ξ.

The damping coefficient  and the eigenfrequency f 0 are not extracted directly from the experimental spectra. Accuracy of the parameters f 0 and  is a key point. An up-to-date method to obtain these parameters from an experimental spectrum has been developed in [START_REF] Niedermayer | Methods for the robust of the resonant frequency and quality factor of significantly damped resonating devices[END_REF]. This method considers for second order system which are potentially affected by noise and electric coupling (cross talk). It allows subtracting noise and coupling for a significantly damped resonance. The method allows determing two parameters f r and Q which are obtained with a great robustness eliminating the incertitude due to noise or coupling. The relation between these parameters and the damping coefficient and eigenfrequency are given here:

1 2Q   ( 1 4 ) 0 r f f  ( 1 5 )
The presented model and parameter extraction are summarized in figure 3. This iterative method allows for fast determination of the complex viscosity or shear modulus. Furthermore, for calibration, only the eigenfrequency in vacuum is required, which is calculated from the measured value in air. This method has been applied to all of our (optical and electro-magnetical) measurements for both types of structures and for all of the investigated liquids. Eq. [START_REF] Mather | Liquid viscoelasticity probed by a mesoscale piezoelectric bimorph cantilever[END_REF][START_REF] Belmiloud | Vibrating microcantilever used as viscosimeter and microrheometer[END_REF] f 0 ξ

Eq. (8-10) In the case of cantilevers, an upgraded version of the presented model also exists. This version was initially presented by our team in [START_REF] Belmiloud | Rheological behavior probed by vibrating microcantilevers[END_REF] and improved in [START_REF] Youssry | On-chip characterization of the viscoelasticity of complex fluids using microcantilevers[END_REF]. It allows determining not only G'(f 0 ) and G"(f 0 ), but also G'(f) and G"(f) over a rather large frequency range (half a decade) around the resonant frequency when the experimental data is only negligibly affected by noise and coupling; good results are obtained in the case of optical characterization. The upgraded version includes improvements which mainly concern the strong hypothesis made in equation ( 8) which is not outlined in the present paper. In section 4.1, the values for G' and G" using cantilevers are obtained using this frequencydependent model.

g 1 (f 0 ) g 2 (f 0 ) Eq. (11-13) G'(f 0 ) G''(f 0 )

Experimental results

Optical characterizations (for transparent fluids)

The measurements of the amplitude and phase of the displacement of the test structures have been performed using the three cantilever geometries and the U-shaped wire, in air and in the three test liquids using optical measurements. Sample measurements of the amplitude are presented in figures 4 and 5. As expected from theoretical considerations, these optical measurements show that an increase in viscosity results in increasing in damping ratio and a decrease in the resonant frequency. In the case of shampoo, optical measurements for cantilever LL and for the U-shaped wire do not show a resonant peak. Since these cases do not meet the above-mentioned modeling requirements, results will not be exploitable. For both structures, the amplitude and phase data of the cantilever free-end deflection are directly provided by the laser vibrometer.

For the case of cantilevers it is possible to calculate the viscous modulus G" at each frequency in a large range of frequencies around the resonant frequency. The resulting rheograms are presented in Figure 6 for the three test liquids.

For U-shaped wires, the damping ratio ξ, the eigenfrequency in liquid f 0 , and the eigenfrequency in air (which is approximately the same as in vacuum, f 0,vacuum ) have been extracted from the spectrum obtained by vibrometry (when a resonant peak have been observed). The presented model has been used to calculate the viscous modulus at the eigenfrequency (G"(f 0 )) in the three examined liquids. In figure 6, the single values G"(f 0 ) obtained with U-shaped wires and the ranges of values G"(f) obtained with cantilevers A, LL and LH in the three different liquids are presented.

Preprint Measurement Science and Technology, Vol. 24, 2013, 084005 (9 pages) (doi:10.1088/0957-0233/24/8/084005) The behavior of the viscous modulus of water is well known (blue dotted curve in figure 6.a). The results obtained with cantilevers have been used to calibrate modeling parameters to obtain the Newtonian behavior of this reference fluid. The measurements performed with U-shaped wires only require calibrating f 0,air . For non-Newtonian fluids, the modeling parameters calibrated in water have been used. In both cases, for egg albumin and shampoo, the rheograms of cantilever LH and A follow the same trends (represented by the dashed lines in figure 6.b) and c)). The recorded spectra with cantilever LL did not exhibit a resonant behavior in the case of shampoo, and exhibited a strong damping in the case of egg albumin. In these cases, as noise and coupling are relatively large in respect to the low signal the upgraded model cannot be applied. Therefore, the viscous modulus corresponding to cantilever LL is inconsistent.

The values of G" calculated using the U-shaped wires are less accurate but they remain of the same order of magnitude without any calibration step. They were obtained with a magnetically clamped wire for which the accuracy of the final measurement is affected by stability issues as mentioned in section 2.1. More stability, reproducibility and accuracy on the viscosity measurement are achieved and detailed in the section 4.1 using a soldered clamping for the U-shaped structure.

In order to validate the calculating method detailed in this paper, the optical characterization of a microcantilever of geometry LH (see Table 1) has been performed in deionized water from 19 °C to 50 °C (using a Peltier element for temperature control). The measured magnitudes are shown in figure 7.a and the dependence of the quality factor with respect to temperature in figure 7 As expected, the quality factor of the LH cantilever increases with temperature due to the decrease of the viscosity of water. The viscosity of water measured between 19 °C and 50 °C follows the expected trend of theoretical dependence of the water's viscosity with respect to temperature (dashed blue line in figure 7.c). The great accuracy obtained with the optical measurements combined with the presented method allows calculating accurately the viscosity of water without any calibration. The same kind of measurements has to be tested for opaque viscoelastic liquids to confirm and validate the potential of the method.

Electrical characterizations (transparent and opaque liquids)

4.1.1. Preliminary results Electrical measurements are performed for both structures using a lock-in amplifier, as mentioned in section 2.2. Some results are shown in figures 8 and9.

For the case of cantilevers, the influence of the fluid on the signal, even if clearly observable, could not be determined due to a dominant electrical coupling. In air, a first approach of the electrical coupling can be modeled by a simple constant capacitive circuit and the cantilever resonance clearly overshoots the coupling base line (cf. figure 8). However, the modeling of coupling is much more complex in liquids. There, the coupling is affected by the surrounding liquid properties (e.g., electrical conductivity). Concretely, this means that the results from figure 8 are not reproducible. More generally speaking, in aqueous media, different cantilever spectrum acquisitions at different times give different results because of the variation of electrical coupling. This instability is a real limitation for continuous non-optical acquisition with respect to time. The reproducibility of electrical measurements Preprint Measurement Science and Technology, Vol. 24, 2013, 084005 (9 pages) (doi:10.1088/0957-0233/24/8/084005) of cantilevers in water-based solution has still to be improved in order to measure continuously an exploitable signal for viscosity monitoring measurements.

In the case of U-shaped wires, measurements in air and in water show more stability and reproducibility. As shown in figure 9, a reproducible and almost noiseless signal is obtained with the wire in both cases. The amplitude's resonance peak dynamic is the principal remaining problem: the motion-induced voltage is small compared to the driving voltage which can consume the main part of the digitalization scale. The numeric resolution has to be forced to the order of nV to be able to obtain a sufficient resolution for rheology purposes. A dedicated finely set differential compensation circuitry could solve this problem for future acquisitions (being careful about not restoring coupling issues). Preprint Measurement Science and Technology, Vol. 24, 2013, 084005 (9 pages) (doi:10.1088/0957-0233/24/8/084005)

Viscosity monitoring experiment with U-shaped wires

A viscosity monitoring experiment has been performed in order to completely validate the proposed method in the case of an opaque complex liquid mixture. For this purpose, the viscosity of yogurt was examined during fermentation. The experiment has been successfully reproduced three times using Ushaped structures. Some photographs and raw data of one experiment are presented in figure 10. The measurements have been done using an U-shaped structure packaged with a magnet. The U-shaped sensor has been immersed together with thermocouples (for temperature control) in the prepared milkmixture.

First, 168 mL of full-cream milk is mixed with 32 mL of yogurt. Then a first mass-density measurement was performed with a microbalance specific set-up; at 9 °C it was 1035.5 kg.m -3 . The mass-density was measured 15 hours later again, at 19.2°C, after the milk has turned into yogurt (see figure 10.c). At 19.2 °C, it was 1051.4 kg.m -3 (this last measurement is more complicated: due to the specific viscoelastic properties of yogurt, the measurement has to be stabilized during 2 hours to obtain a consistent value). In between, the mixture was heated to a temperature close to 37°C during 8 hours. The fermentation process occurs during this period. The deflection spectrum of the U-shaped structure used for this experiment has been measured in air before (cf. figure 10.c). The amplitude and phase of the motion induced voltage are continuously acquired by the lock-in amplifier. Both, the resonant frequency and the quality factor are calculated in real time after each spectrum acquisition using the correction developed in [START_REF] Niedermayer | Methods for the robust of the resonant frequency and quality factor of significantly damped resonating devices[END_REF]. The milk-mixture is simply heated by a hot plate and temperature is measured every minute during the experiment using the thermocouples. Once the temperature of the mixture is close to 37 °C the fermentation process starts efficiently and after 100 minutes the quality factor starts to decrease (cf. figure 10.d). The mixture becomes more viscous.

Once the yogurt is formed, the resonant frequency starts to decrease a lot (see figure 10.b). The elastic modulus, the viscous modulus and the complex viscosity have been calculated using the presented model. The measured mass density is used in the calculation. The complete monitoring of yogurt viscosity during the fermentation process is achieved and results are presented in figure 11. In this experiment, the viscosity is measured by the damping of a mechanical resonator and the associated resonant frequency shift.

In literature, the most widely used method for yogurt viscosity monitoring is performed by measuring the acidity of the mixture and calculating the viscosity with respect to the pH value [START_REF] Soukoulis | Industrial Yogurt Manufacture: Monitoring of Fermentation Process and Improvement of Final Product Quality[END_REF]. Viscosity

Preprint Measurement Science and Technology, Vol. 24, 2013, 084005 (9 pages) (doi:10.1088/0957-0233/24/8/084005) versus time curves is obtained, comparable to the one produced in figure 11.a. In [START_REF] Soukoulis | Industrial Yogurt Manufacture: Monitoring of Fermentation Process and Improvement of Final Product Quality[END_REF], authors have a starting plateau around 10 mPa.s and a final plateau around 200 mPa.s. Here, the starting plateau is around 4 mPa.s and the final one around 190 mPa.s. In between, results slightly differ: the dynamics of fermentation are not the same, due to different temperature profiles (up to 80 °C in [START_REF] Soukoulis | Industrial Yogurt Manufacture: Monitoring of Fermentation Process and Improvement of Final Product Quality[END_REF]) and of course due to the use of different types of ferments. Thus, the values of viscosity measured in this experiment are consistent and valid, close to literature's reference data (see discussion below). Furthermore, it is stated in literature, that the viscoelasticity of yogurt can be continuously measured with a temperature controlled rheometer. The elastic and the viscous modulus may be measured in oscillation mode at 1Hz [START_REF] Lee | Formation and Physical Properties of Yogurt 2010 Asian[END_REF], and compared to the rheograms presented in figure 11.b. After the 500 minutes, G' and G" have the same order of magnitude, which means that the viscoelastic character of yogurt is obtained. A quantitative comparison between results from literature [START_REF] Lee | Formation and Physical Properties of Yogurt 2010 Asian[END_REF][START_REF] Hassan | Microstructure and Rheology of Yogurt Made with Cultures Differing Only in Their Ability to Produce Exopolysaccharides[END_REF] and the results presented in figure 11.b is not possible due to the differences of operating frequencies (1 to 200 Hz using rheometers and 9.2 kHz using this sensor). However, the trends are remarkably comparable and, in both cases, the ratio of G' (after fermentation) and G' (before fermentation) are similar, very close to 200. The same verification was successfully achieved with the G" modulus. The presented rheograms are consistent in terms of variation of moduli.

The comparison of this investigation with literature shows that, the performed viscosity monitoring process during yogurt fermentation is valid.

Discussion

The current state-of-the-art high-frequency rheology shows deficits in terms of reproducibility of the measured values. Rheologists state that a margin of 20% of difference between supposedly Preprint Measurement Science and Technology, Vol. 24, 2013, 084005 (9 pages) (doi:10.1088/0957-0233/24/8/084005) identical experimental values obtained by different teams is acceptable [START_REF] Buchanan | High-frequency microrheology of wormlike micelles[END_REF]. The presented model is sufficient to accurately measure a valid complex viscosity compared to literature's data.

Due to their geometry, U-shaped structures allow minimizing the mechanical losses (as compared to cantilevers) which is an important issue when performing measurement in viscous media. Moreover, the electrical detection was sufficiently free of noise or electrical coupling for measuring accurately the desired resonance parameters (f r and Q) and therefore, for performing viscosity monitoring in significantly viscous opaque liquids. Nevertheless this method needs a resonant peak to extract f r and Q; this is a limitation which not allows to characterize fluids of high viscosity.

Compared to U-shaped structures, the microcantilevers used in this experiment were not suitable for opaque fluid characterization, mainly because of both, the electrical coupling and the mechanical damping. However, in the case of an accurate optical characterization, a good stability to characterize liquids of relatively small viscosity is demonstrated. Furthermore, an accurate frequency dependent model is available and allows calculating rheological parameters not only at the resonant frequency but also over a range of frequency around this value [START_REF] Youssry | On-chip characterization of the viscoelasticity of complex fluids using microcantilevers[END_REF]. Thus, the accuracy of this frequency dependent method remains insufficient in case of strongly damped spectra (cf. figure 6, 'commercial shampoo') due to some modeling limitations and to an excessive sensitivity to the existing incertitude about some parameters.

The detailed experiments associated to the proposed modeling have shown a great potential for viscosity monitoring of various viscoelastic products. The obtained results qualify the U-shaped sensor for viscosity monitoring. Improvements can be performed in order to reach better performances especially in term of quality factor. The clamping and the stiffness can be improved. The silicon cantilever has to be strongly modified in order to measure an electrical signal in opaque liquids with less electrical coupling. A U-shaped silicon based structure could be an option to demonstrate better performances and an increased integration. Nonetheless, the actual U-shaped structure is low-cost and low energy consuming; the change of both, the material and the clamping technique could be sufficient to obtain satisfying performances. Thus, a future sensor could cover a viscosity range from 0.5 to 1000 mPa.s. The actual system allows for measuring viscosities up to 250mPa.s.

Conclusion

As a conclusion, microcantilevers and vibrating wires are both potential viscosity sensors. The actual U-shaped design allows minimizing the damping losses and the electrical coupling; furthermore, it features easy fabrication and handling as well as low cost production. Results obtained with this structure allow considering it as a real microrheometer which could be integrated in an array of such structures. The simplicity of this design is its main asset, since a simple geometry can be modeled easily and because the model itself drives an important part of the accuracy of the viscoelasticity estimation. On the other hand, manufacturing of microcantilevers is more expensive; however a higher sensitivity can be achieved when using optical detection in a low viscous liquids. The validity and the accuracy of the viscosity calculated without calibration using the U-shaped structure is acceptable, reliable and integrated (i.e. no optical measurements required). The geometry associated with this robust analytical method appears really promising for future application. The accuracy of both measurement methods can still be improved. Thus, the presented results with these simple micro-mechanical-structures have proven their applicability as micro-sensors performing microrheological measurements in a frequency range which has hardly been investigated so far. Finally the integrated viscosity monitoring is now available using this method for many kinds of viscoelastic liquids.

Figure 1 .

 1 Figure 1. Example of microcantilevers fabricated at LAAS in Toulouse.Only the structure on the right is released

Figure 2 .

 2 Figure 2. a) "U-shaped" wire actuating set-up; b) Photograph of one of the examined wires immersed into a liquid.

Figure 3 .

 3 Figure 3. Synoptic of the presented model and parameter extraction.

Figure 4 .Figure 5 .

 45 Figure 4. Cantilever LL free-end deflection spectrum obtained by optical vibrometry in air (green curve), water (blue), shampoo (red).

Figure 6 .

 6 Figure 6. Rheograms: viscous part G" of the shear modulus calculated for cantilevers LL (green), LH (red), A (purple) and the U-shaped wires (o) in a) deionized water, b) hen egg albumin and c) a commercial shampoo. The theoretical viscous modulus of water is also depicted (--blue).

Figure 7 .

 7 Figure 7. a) Magnitudes measured optically with cantilever LH immersed into water from 19 °C (dark line) to 50 °C (red line). b) Quality factor with respect to temperature. c) Module of the complex viscosity calculated using the presented model (the theoretical trend is represented by the blue dashed line).

Figure 8 .

 8 Figure 8. Amplitudes containing the fundamental resonant frequency measured piezoresistively for cantilever LH in air (green line) and in water (dotted blue)

Figure 9 .

 9 Figure 9. Amplitudes of the motion induced voltage containing the fundamental resonant frequency for a U-shaped structure in air (green line) and in water (dotted blue line)

Figure 10 .

 10 Figure 10. a) Resultant yogurt. b) Measured resonant frequency with respect to time. c) Measured signal in air and yogurt. d) Temperature profile applied during the fermentation process (blue line), and measured quality factor (dotted green).

Figure 11 .

 11 Figure 11. Viscosity monitoring during yogurt fermentation process a) Module of complex viscosity (Pa.s) with respect to time (minutes). b) Elastic (G') and viscous modulus (G") in Pa with respect to time.

Table 1 .

 1 Dimensions of the fabricated microcantilevers.

	Cantilevers	A	LL	LH
	Length (µm)	500	2810	1440
	Width (µm)	100	100	285
	Thickness (µm)	20	20	20

Table 2 .

 2 Density, viscosity and elasticity of probed fluids at 19°C.

Properties/Fluids Water Albumin Shampoo Yogurt Density (kg.m -3 ) 998 1027 1018 1051 Viscosity (mPa.s) 1

		.03	NA	NA	NA
	Elasticity (Pa)	0	NA	NA	NA
	NA : Not Available

Acknowledgments

The authors would like to thank Cédric Ayéla for the cantilevers mask design, Liviu Nicu, Fabrice Mathieu and Daisuke Saya for the fabrication of the silicon cantilever, Alexander Niedermayer for the resonant frequency and quality factor extraction procedure. This work was partially supported by the French National Agency (MicRheo project n°ANR-08-NANO-004) and by the Conseil Régional d'Aquitaine (n°20091102001). The contributions of Martin Heinisch and Bernhard Jakoby were supported by the Austrian Competence Center in Mechatronics (ACCM). The collaboration between Austria and France has been founded by a project PHC-AMADÉUS.


Keywords: PZT-cantilever, thick-layer, longitudinal vibration mode, toluene, PEUT, in-plane vibrations

Self-actuated and self-read-out piezoelectric cantilevers made of Au/PZT/Au (8x2x0.095mm 3) beam layers on an alumina substrate are fabricated thanks to the association of screen-printing and sacrificial layer techniques. Then, PZT cantilevers used as piezoelectric resonant type sensors are functionalized with an organic PEUT layer (30µm) for toluene detection at room temperature. The first and second unusual in-plane 31-longitudinal modes presenting high resonant frequencies (respectively 70kHz and 214kHz) and good quality factors (200) are used for toluene detection. Sensitivities and LOD are compared for both modes. Due to their higher resonant frequency, PZT cantilevers using these longitudinal vibration modes have higher sensitivities (15.0mHz.ppm-1 at 70kHz) than those actuated with the classical flexural mode of silicon cantilever (0.9mHz.ppm-1 at 3kHz). These interesting results demonstrate the potentialities of printed piezoelectric cantilevers for gas detection. Their performances are compared with those of silicon cantilevers actuated with transverse classical flexural modes.

Introduction

For a few decades, microcantilever-based chemical sensors have been widely investigated because of their high sensitivity at room temperature [START_REF] Goeders | Microcantilevers: Sensing Chemical Interactions via Mechanical Motion[END_REF][START_REF] Boisen | Cantilever-like micromechanical sensors[END_REF][START_REF] Li | Integrated microcantilevers for high-resolution sensing and probing[END_REF][START_REF] Sepaniak | Microcantilever Transducers: A new Approach in Sensor Technology[END_REF]. To perform detection of chemical species, in gas or liquid media, the microcantilever is usually coated with a chemically sensitive layer that aims to selectively sorb the target analyte. The sorbed species modifies the mechanical properties of the sensitive coating and therefore the mass, the rigidity and the surface stress of the microstructure. The sorbed species can consequently be detected either by measuring the microcantilever's resonant frequency shift (dynamic mode) or the microcantilever's quasi-static deflection (static mode). Polymers are often used as sensitive layers for gas detection (e.g. VOCs) because they allow reversible sorption. A wide range of polymers are employed for this purpose: PEUT, PDMS, PIB, PECH, PVA, PEI, PVP, PA, PMMA… [START_REF] Lange | Complementary metal oxide semiconductor cantilever arrays on a single chip: mass-sensitive detection of volatile organic compounds[END_REF][START_REF] Donga | Characterization of the gas sensors based on polymer-coated resonant microcantilevers for the detection of volatile organic compounds[END_REF]. More recently, silicon cantilevers have been functionalized with inorganic coatings like zeolite [START_REF] Urbitzondo | Zeolite-modified cantilevers for the sensing of nitrotoluene vapors[END_REF][START_REF] Urbitzondo | Detection of organic vapours with Si cantilevers coated with inorganic (zeolites) or organic (polymer) layers[END_REF]. The major drawback of both polymer and zeolite layers is the partial selectivity to different species, and consequently the lack of sensor selectivity if only one cantilever with one sensitive layer is used. Usually, to address this issue, a multisensor approach using different sensitive layers with different partition coefficients for various species is used [START_REF] Lang | Cantilever array sensors[END_REF][START_REF] Archibald | Independent component analysis of nanomechanical responses of cantilever arrays[END_REF]. In this paper, we will not focus on the sensitive layer performance but on the advantage of an unusual mode of vibration for cantilever-based sensor in dynamic mode. Usually, the resonating cantilevers used for chemical sensing are vibrating in their first transverse flexural mode (out-of-plane vibrations, also called transverse bending mode), which is the resonant mode having the smallest resonant frequency. Due to the fact that the sensor sensitivity increases with the resonant frequency, alternative operational modes have already been used: higher transverse flexural modes [START_REF] Lochon | An alternative solution to improve sensitivity of resonant microcantilever chemical sensors: comparison between using high-order modes and reducing dimensions[END_REF][START_REF] Maraldo | Mass-change sensitivity of high-order mode of piezoelectricexcited millimeter-sized cantilever (PEMC) sensors: Theory and experiments[END_REF], lateral flexural modes [START_REF] Beardslee | Liquid-Phase Chemical Sensing Using Lateral Mode Resonant Cantilevers[END_REF][START_REF] Beardslee | Geometrical considerations for the design of liquid-phase biochemical sensors using a cantilever's fundamental in-plane mode[END_REF][START_REF] Gil | Selective modal excitation in coupled piezoelectric microcantilevers[END_REF][START_REF] Ruiz | Modelling and characterization of AlN-actuated microcantilevers vibrating in the first in-plane mode[END_REF] or torsional modes [START_REF] Green | Torsional frequency response of cantilever beams immersed in viscous fluids with applications to the atomic force microscope[END_REF][START_REF] Cai | Resonant Characteristics of Rectangular Microcantilevers Vibrating Torsionally in Viscous Liquid Media[END_REF]. We propose here to use the axial mode, which is an inplane mode also called longitudinal or extensional mode [START_REF] Fu | Dual-Mode Piezo-on-Silicon Resonant Temperature and Humidity Sensor for Portable Air Quality Monitoring Systems[END_REF][START_REF] Ruiz | Optical and electrical characterization of in-plane modes of AlN-based piezoelectric vibrating MEMS[END_REF]. To actuate and have an integrated read-out of this particular vibration mode, symmetric screen-printed piezoelectric cantilevers are designed, fabricated and tested. With the screen-printing thick film process, the thicknesses of the cantilevers are in the range of few tens of micrometers. The cantilever's width and length are in the millimeter range. Comparison to classical silicon cantilever having comparable dimensions used in transverse bending mode is done. For this purpose, toluene detection is performed with the same PEUT (PolyEtherUreThane) polymer sensitive coating in both cases.

In this paper, the fabrication of the screen-printed PZT cantilevers and their functionalization is first detailed. Electromechanical characterizations of the piezoelectric transducers, with and without sensitive coating, are then presented as well as correlations with analytical models. Finally, toluene detections are performed with the first and the second longitudinal modes. In this last part, a comparison between PZT and silicon microcantilevers allows focusing on advantages and drawbacks of the proposed microsensors.

Fabrication and functionalization of screen-printed PZT cantilevers

PZT cantilever

The self-actuated 8x2x0.095 mm 3 piezoelectric cantilever is realized through screen-printing deposition associated to a sacrificial layer [START_REF] Lucat | Microsystems elements based on freestanding thick-films made with a new sacrificial layer process[END_REF]. It is composed of a PZT (Pb(Zr 0,52 Ti 0,48 )O 3 ) layer between two gold electrodes (Fig. 1).The gold ink is commercially available (ESL8836 from ElectroScience Laboratories), whereas PZT and sacrificial layer inks are prepared at the IMS laboratory. The sacrificial layer ink is made of an epoxy-strontium carbonate (SrCO 3 ) mixture. The piezoelectric paste is prepared from a commercial piezoelectric PZT powder (PZ26 from Ferroperm) and 3wt% LBCu (25wt% Li 2 CO 3 , 40wt% Bi 2 O 3 , 35%wt CuO) sintering aid blended with ESL400 organic vehicle (ElectroScience Laboratories) [START_REF] Lakhmi | Study of screen-printed PZT cantilevers both self-actuated and self-read-out[END_REF]. The different fabrication steps are given in figure 2. The anchor pad, based on PZT is first screen-printed on the alumina substrate and dried 20min at 120°C. Then, sacrificial layer is deposited on the substrate and polymerized 20min at 120°C. Finally, bottom electrode, PZT layer and top electrode are printed successively after drying at 120°C during 20min between each deposition. To improve green sample's densification before firing, the dried layers are isostatically pressed 1min at 1kbar. The samples are then fired 2 hours at 900°C in air atmosphere. Finally, dissolution of sacrificial layer is performed in the 0.9mole.l -1 H 3 PO 4 aqueous solution. Au/PZT/Au cantilevers shown on Fig. 3 are then poled with an electric field of 50kVcm -1 , at a temperature of 280°C, just below the PZT's Curie temperature.

Functionalization

For the deposition of a sensitive layer (polymer or inorganic) on a microcantilever, spray coating [START_REF] Fadel | Chemical sensing: millimeter size resonant microcantilever performance[END_REF], drop coating [START_REF] Amirola | Micromachined silicon microcantilevers for gas sensing applications with capacitive read-out[END_REF] or spin coating [START_REF] Amirola | Micromachined silicon microcantilevers for gas sensing applications with capacitive read-out[END_REF] are classically used. The polymer PEUT (PolyEtherUreThane) is chosen because of its good partition coefficient to toluene (K=1610) [START_REF] Kurzawski | Detection and discrimination capabilities of a multitransducer single-chip gas sensor system[END_REF]. However, selectivity towards other BTEX compounds (benzene, ethylbenzene or xylene) and alcohols vapors is not achieved. Prior to its deposition by drop-coating on the top surface of the cantilevers, PEUT is dissolved in the dichloromethane solvent . A manual dispenser is used for this purpose. A thickness of approximately 30µm is measured with an optical profilometer (VeecoNT9080).

Electro-mechanical characterization of screen-printed PZT cantilevers

Measurements

Electrical measurements of poled samples are carried out with an impedancemeter (HP 4194A). An example of the conductance curve obtained on the 8x2x0.095mm 3 uncoated cantilever confirms the piezoelectric behavior of the PZT cantilevers (Fig. 4). The resonance peaks are attributed to the inplane 31-longitudinal vibration modes (31 refers to the piezoelectric effect: the electrical polarization is perpendicular to the mechanical displacement). Without coating, the three first resonant frequencies values are respectively 70, 214 and 362kHz. The quality factor Q~200 of the first peak, slightly decreases for the second and third modes. Mechanical analysis performed with an optical vibrometer allowing both out-of-plane (Doppler interferometry) and in-plane (stroboscopy) vibration measurements (Polytec MSA 500) confirm that the resonant modes seen on the impedance measurements are in-plane longitudinal resonant modes. Moreover, the optical measurement enables to estimate the deflection of the out-of-plane transverse bending modes (first and second modes) which are not visible on the impedance measurement. Few hundreds of picometers are measured with the optical vibrometer (Fig. 5). The small bending deflection value is due to the fact that the microcantilever is designed with a symmetrical geometry in order to minimize the bending-modes deflection. Thanks to the design, the minimization of the unwanted bending modes' amplitude is confirmed by the electrical signature where these modes are not detected, while detecting the desired in-plane 31-longitudinal modes (Fig. 4). Thus, in future work this symmetrical microcantilever design will facilitate operation of the microresonator in closed-loop operation, where the microcantilever is the frequency-determining element in an amplifying feedback loop. Besides, high resonant frequencies and good quality factors justify their choice for gas sensing. These modes, easily recordable by impedance analysis, show the PZT cantilever's ability for actuation/sensing with low power consumption.

After deposition of the PEUT sensitive layer, electrical measurement shows a decrease of both resonant frequency and quality factor (Fig. 6) (Tab.1).

Correlations with analytical models

For the multilayer piezoelectric cantilever structure, the analytical expression of the 31-longitudinal modes' resonant frequency can be expressed by [START_REF] Blevins | Formulas for Natural Frequency and Mode Shape[END_REF]:

() () 31 31 2 22 n n PZT PZT Au Au PZT PZT Au Au h E h E f L h h        (1)
where

  () 31 21 2 n n     (2)
with n, L, h, E and ρ the mode's order, the length, the thickness, the Young's modulus and the mass density respectively. The indexes PZT and Au refer to the piezoelectric layer and the gold electrodes, respectively. To take into account of the sensitive layer the equation ( 1) of the resonant frequency has to be modified:

() () 31 31 2' 22 n n PZT PZT Au Au PEUT PEUT PZT PZT Au Au PEUT PEUT h E h E h E f L h h h         (3)
where the index PEUT is for the sensitive layer and the exponent ' on the Young's modulus is for the real part when it is a complex value due to viscoleasticity.

Due to its viscoelastic properties, the PEUT layer's Young's modulus depends on the frequency. For the considered resonant frequencies (less than 500kHz), polymers such as PIB, PEUT,.. have Young's modulus lower than 0.5GPa. According to estimated Young's moduli (E PZT =36.5GPa and E Au = 55GPa) and thicknesses (h PZT = 80µm and h Au = 7.5µm) of screen-printed PZT and gold layers, the modification of the rigidity (numerator in Eq.3) due to the PEUT coating is negligible. The decrease of the resonant frequency can then be fully attributed to the cantilever's mass increase (denominator of Eq.3). Then, using the mass density of the different materials ( AUT = 18500kg/m 3,  PZT = 6000kg/m 3,  PEUT = 1040kg/m 3 ), the sensitive layer's thickness can be estimated at 21µm using Eq.3. The fact that the value differs from the measured PEUT thickness with the optical profilometer (30m) is attributed to the incertitude on the estimated values for both the Young's moduli and thicknesses of the gold and PZT layers. The quality factor is not very high in air even before the polymer deposition because of the internal losses in the piezoelectric layer. The cantilever's mass increase with the PEUT layer deposition is supposed to increase the quality factor if no additional losses are added. Then, the decrease of the quality factor is due to losses in the layer which are viscoelastic losses as in the case of transverse bending vibration [START_REF] Dufour | Effect of coating viscoelasticity on quality factor and limit of detection of microcantilever chemical sensors[END_REF]. Using the numerical values of Tab.1, the quality factor Q PEUT , due to the viscoelastic losses in the layer, is estimated to 610. In this multilayer configuration and for the longitudinal modes, the quality factor associated to viscoelastic losses in the PEUT layer can be expressed as follow:

2'

''

PZT PZT Au Au PEUT PEUT PEUT PEUT PEUT h E h E h E Q hE   (4)
where the exponent '' is related the Young's modulus's imaginary part.

As in the case of the resonant frequency, the term due to the elastic part of the PEUT Young's modulus is negligible compared to the other ones (numerator of Eq.4). Then, using numerical values at f r,31 =70kHz (Tab. 1), it is possible to estimate the imaginary part of PEUT's Young's modulus E" PEUT ~0.3GPa, whose value is close to that of PIB [START_REF] Ferry | Viscoelastic Properties of Polymers[END_REF].

Toluene detection

Experiments

 Screen-printed PZT cantilevers coated with PEUT The functionalized PZT cantilevers are placed in a gas chamber (500mm 3 ) under a controlled gas flow (100ml/min) of nitrogen. Different concentrations of toluene are introduced into the chamber using a vapor generator. During the experiments, the impedance of the functionalized cantilevers is monitored using the HP4194A impedancemeter. The whole system is controlled under a LabVIEW environment. During the tests, first and second 31-longitudinal modes are measured. Influence of oscillating voltage and bias voltage is first studied to have the best sensitivity and lowest noise. For a 0.5V oscillating voltage, a bias voltage of 40V doesn't lead to better sensitivities than without bias. It indicates that the polarization of the piezoelectric material is already maximized and stable. On the other hand, without bias, oscillating voltage allowing actuation of the cantilever has an influence on the detection performance of the cantilevers, more precisely on the measurement noise. By varying the oscillating voltage from 0.3V to 1V, the sensitivity to toluene remains the same whereas the signal noise changes. Indeed, the amplitude's noise decreases with the oscillating voltage amplitude, of more than a factor 10 for a change between 1V and 0.5V or 0.5V and 0.3V.This is of importance in order to reduce the Limit Of Detection LOD = 3f noise /S CA which is proportional to the noise. Note that with a lower tested oscillating voltage (amplitude 0.1V), no actuation of the PZT cantilever is observed with the impedance measurement. For the comparison of first and second longitudinal modes, a 0.3V oscillating voltage is thus chosen. The resonant frequency evolution under different toluene concentrations (from 260ppm to 1050ppm) is shown in Fig. 7. Using a single 260ppm concentration, a good reproducibility of the PZT cantilever's response is observed (Fig. 8). The relative resonant frequency shifts under the different toluene concentrations (C A ) are given in Fig. 9. The measured sensitivities

A n CA dC df S ) ( 31

  Silicon PEUT-coated cantilevers

Microfabricated silicon cantilevers of uniform rectangular section, coated with sprayed-PEUT, have been also used in previous works for toluene detection (Fig. 10). The process developed by ESIEE (http:/www.esiee.fr), calls upon 5 levels of masks [START_REF] Fadel | Chemical sensing: millimeter size resonant microcantilever performance[END_REF]. The three first steps ensure the development of the strain gauge and the electrical contact. The two last steps define the geometry of the millimeter sized microcantilever. The actuation is electromagnetic and allows the out-of-plane transverse bending resonance mode. The performances of the silicon cantilevers also tested at IMS laboratory are compared to those of PZT cantilevers (Tab. 2). For this purpose, because of PEUT deposit of different thicknesses, a normalized relative sensitivity is defined (S CA /f res /h PEUT ).

Discussion

As expected, sensitivity to gas S CA is higher for the second longitudinal mode whereas the relative sensitivities (S CA /f 31 ) are equivalent for both longitudinal modes in accordance to equation ( 5)

     ( ) ( ) 31 31 22 A nn PEUT C A PZT PZT AU Au PEUT PEUT df f Kh S dC h h h (5)
where C A and K are respectively the gas concentration and partition coefficient of the PEUT with respect to the analyte (Fig. 9). A signal noise of 5.0Hz for the second longitudinal mode is nevertheless higher than those of the first longitudinal mode of 0.4Hz. Consequently, the first longitudinal mode appears to be more interesting for detection at low gas concentrations. For the first longitudinal mode, calculations of theoretical sensitivities are made for L = 8mm, h PZT = 80µm, h Au = 7.5µm, h PEUT = 30µm,  PZT = 6000kg/m 3 , ρ Au = 18500kg/m 3 ,  PEUT = 1040kg/m 3 . Discrepancy observed between theoretical and experimental sensitivities (S CAexp /S CAtheor = 1.8) may be attributed to inhomogeneous PEUT layers. In this case, the Limit of Detection (LOD = 3f noise /S CA ) is estimated to 80ppm. Sensitivities of low cost screen-printed PZT cantilevers under toluene are close to those of silicon functionalized cantilevers. Nevertheless, the quality factor of the silicon cantilever is still higher and the LOD of silicon cantilever is lower assuming the lower noise value.

Conclusion

Tables

E elec b) c) d) f) e) V g)

Biographies of Authors

Hélène Debéda engineer in Physics of materials received the PhD degree from Bordeaux University, France, in 1996, for her work on the selectivity and reliability of thick film methane gas sensors. From 1996-1998 she obtained an European postdoc grant at the IMT Laboratory of Karlsruhe, Germany, to work on "Integrated piezoactuators made by the LIGA technique". Thanks to both the PhD and the post-doctoral position Hélène Debéda acquired a microsystem culture. She is now associate professor at the University of Bordeaux, IMS Laboratory where she is currently working on thick film technology applied to Microsystems and passive components.

Riadh Lakhmi graduated from Ecole Nationale Supérieure de Chimie et Physique de Bordeaux in 2008. He obtained, in the same year, his master in inorganic chemistry from the University of Bordeaux. He acquired PhD thesis at the University of Bordeaux in 2011. His research works concerns the realization of a physical force sensor and a chemical fluid sensor based respectively on a piezoresistive screen-printed layer upon a cantilever and a piezoelectric self-actuated and self-read out cantilever. Both cantilevers have been realized by combination of screen-printing technology and the use of a sacrificial layer. He's currently working, as a research engineer, on the application of printed piezoelectric devices.

Claude Lucat received his PhD in inorganic Chemistry from Bordeaux University in 1980. He joined the Centre National de la Recherche Scientifique in 1979. Currently, he is Director of Research at IMS laboratory, University of Bordeaux, France. His main research interests were in thick-film materials for hybrid components and chemical sensors. Now, his research efforts are on the fabrication and characterization of free-standing structures for Microsystems applications, using the association of screen-printing technology and sacrificial layer method specifically developed at IMS. Microcantilevers or self-supporting devices for gas sensing, force sensors, energy harvesting, etc. are studied.

Isabelle

Fig. 1 .Fig. 2 .Fig. 3 .Fig. 4 .Fig. 5 .Fig. 6 .Fig. 7 .Fig. 8 .Fig. 9 .Fig. 10 .Fig. 1 :

 123456789101 Fig.1. Scheme of the Au/PZT/Au screen-printed cantilever Fig.2. Fabrication steps of the PZT cantilever (a) screen-printing of the PZT pad (b) screen-printing of the sacrificial layer (c) screen-printing of the bottom gold electrode (d) screen-printing of the PZT cantilever (e) screen-printing of the top gold electrode (f) sacrificial layer elimination with H 3 PO 4 solution after co-firing (g) poling of the PZT cantilever (cross-view) Fig.3. Photograph of the 8x2x0.095mm 3 piezoelectric cantilever Fig.4. Conductance measurements revealing the three in-plane 31 longitudinal modes for the 2x8x0.095mm 3 PZT cantilever (oscillating voltage 0.5V) Fig.5. Mechanical spectrum observed for the 2 nd bending mode of the 2x8x0.095mm 3 PZT cantilever (oscillating voltage 0.25mV) Fig.6. Spectrum of the conductance for the first longitudinal mode before and after deposition of a PEUT layer (0.3V oscillating voltage) Fig.7. Resonant frequency evolution under different toluene concentrations measured for the first and second longitudinal 31 modes of the 8x2x0.095mm 3 PZT cantilever (0.3V oscillating voltage) Fig.8. Repeatability of the cantilever's response for a toluene concentration of 260ppm (0.3V oscillating voltage) Fig.9. Relative resonant frequency shifts as a function of toluene concentrations measured for the first and second longitudinal 31 modes of the 8x2x0.095mm 3 PZT cantilever (0.3V oscillating voltage) Fig.10. Silicon cantilever (5x0.4x0.059mm 3 ) with electromagnetic actuation and piezoresistive measurement (a) photograph (b) electrical connection diagram

Fig. 2 .Fig. 3 .Fig. 4 :Fig. 5 :Fig. 6 .

 23456 Fig.2. Fabrication steps of the PZT cantilever (a) screen-printing of the PZT pad (b) screen-printing of the sacrificial layer (c) screen-printing of the bottom gold electrode (d) screen-printing of the PZT cantilever (e) screen-printing of the top gold electrode (f) sacrificial layer elimination with H 3 PO 4 ; after co-firing (g) poling of the PZT cantilever (cross-view)

Fig. 7 .Fig. 8 .Fig. 9 .Fig. 10 .

 78910 Fig.7. Resonant frequency evolution under different toluene concentrations measured for the first and second longitudinal 31modes of the 8x2x0.095mm3 PZT cantilever (0.3V oscillating voltage)

  Dufour graduated from Ecole Normale Supérieure de Cachan in 1990 and received the Ph.D. and H.D.R. degrees in engineering science from the University of Paris-Sud, Orsay, France, in 1993 and 2000, respectively. She was a CNRS research fellow from 1994 to 2007, first in Cachan working on the modelling of electrostatic actuators (micromotors, micropumps) and then after 2000 in Bordeaux working on microcantilever-based chemical sensors. She is currently Professor of electrical engineering at the University of Bordeaux and her research interests are in the areas of microcantilever-based sensors for chemical detection, rheological measurements and materials characterisation.


Martin Heinisch

Erwin K Reichel

Isabelle Dufour

Bernhard Jakoby

Tunable resonators in the low kHz range for viscosity sensing

Keywords: viscosity, mass density, sensor, frequency, Lorentz forces

come   L'archive ouverte pluridisciplinaire

Introduction

Miniaturized viscosity sensors are attractive devices for condition monitoring applications involving fluid media. Most recently introduced devices utilize vibrating resonant mechanical structures interacting with the fluid where the resonant behavior (resonance frequency and quality factor) is affected by the fluid's mass density and viscosity [START_REF] Jakoby | Miniaturized sensors for the viscosity and density of liquids -performance and issues[END_REF][START_REF] Riesch | A suspended plate viscosity sensor featuring in-plane vibration and piezoresistive readout[END_REF]. So far, the ability of tuning the resonance frequency of such devices in a larger frequency range has hardly been reported. However, such a feature is highly desirable particularly when investigating complex liquids showing viscoelastic behavior. We devised two different types of resonating viscosity sensors with tunable resonance frequencies in a range of over two octaves for a fixed geometry. Analytical models relating measured data to the viscosity of the examined liquid have been presented earlier together with first measurement data where we briefly discussed the benefits of different geometries in terms of high tunability and their sensitivity to viscosity [START_REF] Heinisch | A suspended plate in-plane resonator for rheological measurements at tunable frequencies[END_REF][START_REF] Heinisch | A study on tunable resonators for rheological measurements[END_REF][START_REF] Heinisch | Miniaturized resonating viscometers facilitating measurements at tunable frequencies in the low khz-range[END_REF].

Complex fluids often show viscoelastic behavior which means that they exhibit a significant frequency dependence in their shear modulus (whose imaginary part is related to the shear viscosity) when using oscillatory rheometers. Laboratory rheometers work in a limited frequency range not exceeding typically 100 Hz and commonly, they are not suitable for online monitoring applications as they are expensive and maintenance intensive (e.g., due to macroscopically moving parts involving bearings etc.). Miniaturized devices sensing viscosity in the MHz range have been discussed extensively in the past, e.g. thickness shear mode resonators [START_REF] Martin | Characterization of a quartz crystal microbalance with simultaneous mass and liquid loading[END_REF]. Between these devices, providing a "high-frequency viscosity" and the aforementioned laboratory viscometers is a gap, which we are aiming to bridge with devices as presented in this contribution.

Recently, we proposed a design for a tunable in-plane resonating sensor for viscosity [START_REF] Heinisch | A suspended plate in-plane resonator for rheological measurements at tunable frequencies[END_REF]. As an alternative, a single wire device has been investigated [START_REF] Heinisch | A study on tunable resonators for rheological measurements[END_REF]. Both types of these resonating sensors provide a large range of achievable resonance frequencies (over two octaves up to ∼ 4 kHz for our first devices with a fixed vibrating length of ∼ 3 cm). The tuning of the resonance frequency is achieved by the variation of tensile stresses in the wires by an appropriate tensioning mechanism.

These and previously reported sensors (e.g., [START_REF] Heinisch | A resonating rheometer using two polymer membranes for measuring liquid viscosity and mass density[END_REF]) underlie the same principles of actuation and read out, namely actuation by means of Lorentz forces F L and read out via the (motion) induced voltage V M , where

dF L = I in B dl a (1)
(I in : sinusoidal input (driving) current, B: magnetic flux density provided by a permanent magnet (assembly), l a : length of the electrical conductive path used for actuation) and

dV M = v B dl ro (2)
(v: velocity of the electrical conductive path used for read out, l ro : length of the readout path) Here, orthogonality of the direction of the current flow, B and v was assumed.

The motion induced voltage is the quantity representing the resonator's movement in the fluid-structure interaction (FSI) which is thus related to the fluid's dynamic viscosity for a known mass density (and vice versa).

This contribution explains the principles and the physical relations (by means of closed form models) of the Suspended Plate and the Wire Viscometer in Sec. 2 and 3, respectively. Section 4 briefly shows experimental results revealing both sensors' response (and sensitivity) to viscosity. The following section focuses on the tunability of the resonance frequencies of both resonators, see Sec. 5. Based on these first (feasibility) concepts discussed in the aforementioned sections, this work is extended by several (indispensable) improvement strategies effecting higher readout signals and more reliable setups allowing for more accurate sensing, see Sec. 6.

Suspended Plate Viscometer

Setup

The principle of the Suspended Plate Viscometer (which is fully surrounded by the sample liquid) is depicted in Fig. 1(a). 1: In (a) a schematic drawing of the suspended plate rheometer is depicted. The sensor, surrounded by an experimental well, consists of two parallel wires (placed in external field B) and a small platelet. In this figure, the left wire is used for exciting lateral vibrations by means of Lorentz forces. The second wire on the right used for pick-up, is coupled to the excitation wire with a rigid plate, thus following the movement of the left wire which effects an induced voltage which is used as read-out. By changing the tensile stresses in the wires, the resonance frequency can be changed. In (b) the scheme of the modeling approach is depicted. The upper figure shows the electrical equivalent circuit consisting of two galvanically isolated circuits representing the excitation and read-out circuits. (V g : Voltage of the function generator, 50 Ω: Internal resistance of the function generator, R 1 : resistance of the excitation wire, R 2 : resistance of the pick-up wire, V M : Motion induced voltage, V ct : Induced voltage trough electrical cross-talk, 10 MΩ: Input resistance of the lock-in amplifier, I in , I out , V in , V out : Input and output currents / voltages). The lower figure shows the mechanical lumped-element model. (k 0 : intrinsic spring constant, r 0 : intrinsic damping parameter, F F : Fluid forces, F L : Lorentz forces, u x : displacement in x-direction, B: external magnetic field). In (c) a comparison between measured data (in this case for isopropanol) and the model obtained by fitting the parameters from Eq. 6 is shown.

A small platelet which is suspended by two wires is forced to lateral vibrations by means of Lorentz forces on AC-currents in the excitation wire. This in-plane fluid-structure interaction of the platelet with the surrounding liquid is recorded via the motion induced voltage in the second wire which follows the movement of the platelet (and the excitation wire). Varying the tensile stresses within both wires (by an appropriate tensioning mechanism) allows for setting the desired resonance frequency (in a given bandwith, see Sec. 5).

In first setups, 100 µm thick tungsten wires were used (because of tungsten's high ultimate strength exceeding 1.5 GPa). For the platelet's material glass or PET was used which yielded both good results and mainly differed in the fabrication technique, see [START_REF] Heinisch | A suspended plate in-plane resonator for rheological measurements at tunable frequencies[END_REF]. Similar designs have also been fabricated using one material only (nickel brass) and their tunability was briefly investigated but has not been published yet. Resonating platelets made of one material only and fabricated with micromachining techniques are reported e.g., in [START_REF] Reichel | Analysis and experimental verification of a metallic suspended plate resonator for viscosity sensing[END_REF][START_REF] Reichel | Shear wave sensors for viscoelastic properties[END_REF].

Semi lumped element model

As the platelet's lateral dimensions are much larger than its thickness and as its surface A is much larger than the wire's surfaces, boundary effects emerging from displacement of the liquid at the platelets front sides as well as the flow around the wires are not taken into account. The fluid forces F F acting on the platelet are modeled by one-dimensional shear wave propagation,

F F = -2 T (z = 0) A (3)
where the shear stress is [START_REF] Heinisch | A suspended plate in-plane resonator for rheological measurements at tunable frequencies[END_REF][START_REF] Martin | Characterization of a quartz crystal microbalance with simultaneous mass and liquid loading[END_REF] (using complex notation assuming time dependence e jωt )

T (z = 0) = u x (1 -j) µ ρ ω 3 2 (4)
With this, the transfer function of the mechanical oscillator, which is the quotient of displacement in x-direction u x and Lorentz forces Eq. 1 can be written as (see also Fig. 1(b) and Eq. 1)

G m = u x F L (5) = 1 -ω 2 m + A µ ρ 2 ω + j ω r 0 + A µ ρ ω 2 + k 0
Finally, the motion-induced voltage (see Eq. 2) in the pickup wire can be written as follows

V M = j ω G m B 2 l * a l * ro R 1 V in (6)
here, l * a and l * ro are the wire's effective lengths (which depend on the mode shape but are in the order of the actual length).

It may be necessary to consider the effect of electrical crosstalk in the measured voltage as well, see [START_REF] Heinisch | A feasibility study on tunable resonators for rheological measurements[END_REF], which can be easily taken into account with

V ct = j ω M R 1 V in ( 7
)
where M is the mutual inductance describing the inductive coupling between excitation and read-out circuit. As the input resistance of the lock-in amplifier is 10 MΩ, it is assumed that I out ≈ 0 and thus the total amount of output voltage is given by

V out = V M + V ct (8)
A comparison between this simple (fitted) model and measured data for isopropanol is shown in Fig. 1(c). A 100 µm thick tungsten wire carrying sinusoidal currents is placed in an external magnetic field and thus oscillating due to Lorentz forces. For excitation, a function generator is used for the input signal. A series resistors is used to limit the excitation current. The motion induced voltage (i.e., the read-out signal) is measured with lock-in amplifier. The resonance frequency can be changed by (mechanically) changing the tensile stresses in the wire. In (b) the electrical equivalent circuit of the wire viscometer is illustrated. (R s : series resistor to limit the excitation current, R tw : resistance of the tungsten wire). In (c) a comparison between measured data (in this case for isopropanol) and the model obtained by fitting the parameters from Eq. 14 is shown.

Wire Viscometer

Resonating viscometers using a single wire only, are known as wire viscometers in literature [START_REF] Retsina | The theory of a vibrating-rod viscometer[END_REF][START_REF] Seibt | Schwingdrahtviskosimeter mit integriertem Einsenkkorperdichtemessverfahren fur Untersuchungen an Gasen in groseren Temperatur-und Druckbereichen[END_REF][START_REF] Sullivan | On the nonlinear interpretation of a vibrating wire viscometer operated at large amplitude[END_REF]. These viscometers were designed to be operated at one fixed resonance frequency only. In our work, we focus on the investigation of tuning the resonance frequencies of the vibrating sensors within a preferentially large range and focus on miniaturized devices.

Setup

Figure 2(a) shows a schematic sketch of the wire viscometer. A tungsten wire is stretched over two electrical conductive blades. For first (test) setups these blades, determining the vibrating length and assuring for electrical contact, were made of PCB (copper coated fiberglass) which were later replaced by machined metal blades, see Sec. 6. The series resistor limits the excitation current to avoid non linear effects such as e.g., Duffing behavior, see [START_REF] Sullivan | On the nonlinear interpretation of a vibrating wire viscometer operated at large amplitude[END_REF]. For measurements in air about 10 kΩ and in liquids, resistors between 50 Ω and 1 kΩ were used. The external magnetic field was generated with different Nd-FeB magnet assemblies yielding magnetic flux densities from 0.3 T up to 1.4 T, see [START_REF] Heinisch | Tunableminiaturizedvis-cosity sensors operating in the khz-range[END_REF].

Modeling

For the modeling of the wire viscometer, the transversal movements w(x, t) of the wire subjected to an axial force N, driving Lorentz forces F L and the loading of the fluid forces F F can be described by the following linear, inhomogenous, partial differential equation in time domain [START_REF] Weaver | Vibaration Problems inEngineering[END_REF][START_REF] Elwenspoek | Mechanical Microsensors[END_REF]:

E I ∂ 4 w(x, t) ∂x 4 -N ∂ 2 w(x, t) ∂x 2 + m ∂ 2 w(x, t) ∂t 2 (9) = F L (x, t) + F F (x, t)
(E: Young's modulus, I: second moment of inertia, m: wire's mass, : quantity in respect of unit length) Eq. 9 is transformed to the frequency domain again assuming a time dependence e jωt and the forces per unit length on the right hand side of Eq. 9 acting on the wire are substituted by the following expressions

F L = I in (ω) B (10)
and from [START_REF] Landau | Fluid Mechanics[END_REF]:

F F (x, ω) = (11) -j ω g 1 (x, ω) w(x, ω) + ω 2 g 2 (x, ω) w(x, ω)
( I in : input current, B: flux density of the external magnetic field which is provided by a permanent magnet, ω: angular frequency of the harmonic oscillation) g 1 and g 2 can be calculated by using the analytical relation of the fluid forces acting on a circular cylinder [START_REF] Rosenhead | LaminarBoundaryLayers[END_REF], [START_REF] Sader | Frequency response of cantilever beams immersed in viscous fluids with applications to the atomic force microscope[END_REF]:

F F (x, ω) = (12) π ρ f ω 2 R 2         1 - 4 j K 1 j √ -j Re √ -j Re K 0 j √ -j Re         w(x, ω) where Re = ρ ω R 2 µ ( 13
)
(R: wire radius, K 0 , K 1 : modified Bessel functions, µ: dynamic viscosity) After the analytical calculation of the deflection of the wire w(x, ω), the induced voltage is calculated as follows:

V out = j ω B l 0 w(x, ω) dx + R tw I in (ω) (14)
(l: wire length, R tw : wire ohmic resistance) Fig. 2(c) shows a comparison of measurement data and a fit of this model.

Response to viscosity

From the resonant behavior (which is monitored by the motion induced voltage in the wire) the viscosity of a sample liquid is determined (knowing the liquid's mass density). Methanol: 0.47 mPa s, 0.79 g/cm 3 Di Water: 0.86 mPa s, 1.0 g/cm 3 2-Propanol: 2.1 mPa s, 0.78 g/cm 3 Measurement Model (b) Wire Viscometer Figure 3: In both figures the sensors measured and theoretical frequency responses including the fundamental harmonic in four different liquids are illustrated (Excitation Voltage V g =0.5V). For both sensors, clearly, it is visible that higher viscosities yield higher damping. The effect of the the viscosity and the mass density on the resonance frequency shift can be observed in the according theoretical frequency responses. The deviation in the resonance frequencies between model and experiment are mainly caused by undesired de-tuning of the sensor.

The effect of different viscosities on the resonant behavior was examined by recording the frequency response including the fundamental mode in four different liquids. The selected liquids only have small differences in their viscosity, ranging from 0.21 mPa s to 2.1 mPa s to investigate the sensitivity to viscosity. The used liquids were acetone, methanol, isopropanol and DI-water, see Fig. For the measurements in this study, we observed shifts in the resonance frequencies, see Fig. 3, which are mainly not caused by the different physical properties of the liquids but by de-tuning of the sensor (e.g., from thermal stresses, slacking of the entire setup, etc.). However the liquid's physical properties (theoretically) have a significant effect on the shift of the resonance frequency which can nicely be observed in the heoretical frequency responses in Fig. 3. From these theoretical frequency responses it can also bee seen, that the wire viscometer is more sensitive to mass density. (Compare the theoretical frequency responses of water, which has a higher mass density than the examined solvents.)

For obtaining these theoretical frequency responses, the models described in Sec. 2.2 and 3.2 were compared to the four measured frequency responses. In the theoretical evaluation of the wire viscometer and the suspended plate sensor the normal stresses and the intrinsic spring constant, respectively, were kept constant to show the effect of the physical properties on the frequency shift.

Based on the insights gained with these measurements, we aim at setups yielding stable resonance frequencies, see discussion in Sec. 6, for accurate and reliable sensing.

Discussion and comparison

For a rough estimate, to compare the sensors' sensitivity to viscosity, we evaluated the Q-factors Q = f r /( f +f -) when the sensors were immersed in the different sample liquids detecting the maximum and -3 dB values in the magnitude of the frequency response for both, the measured and the theoretical frequency responses. Both sensors show a similar sensitivity to viscosity when evaluating their quality factors in the examined range of viscosity, see Fig. 4. We note that with the wire viscometer it is in principle possible to measure both, mass density and viscosity with a single measurement only, however, with restricted accuracy [START_REF] Seibt | Schwingdrahtviskosimeter mit integriertem Einsenkkorperdichtemessverfahren fur Untersuchungen an Gasen in groseren Temperatur-und Druckbereichen[END_REF]. The sensitivity of the wire viscometer to mass density can be observed in the theoretic frequency response in Fig. 3(b). There, the shift of the resonance frequency is mainly affected by the mass density which is nicely visible for the case of water which has a higher mass density than the other examined liquids.

Tunability of the resonance frequency

To show the capability of the tuning of the resonance frequency the normal stresses in the wires were changed by tensioning the wires with micro-stages with micrometer screws. (c) Theoretic Achievable Range Figure 6: In (a) a setup of the wire viscometer allowing for changing both, the normal stresses in the wire as well as its length is depicted. A tungsten wire is placed in an external magnetic field (not depicted; perpendicular to the wire). The conductive blades allow for electrical contact and for changing the wire's vibrating length. One end of the wire is rigidly affixed. The other is attached to a weight, which is varried for changing the normal stresses in the wire but keeping the normal stresses constant when changing the vibrating length. (b) Comparison of measured and theoretic resonance frequencies for different normal stresses N. The solid line shows the theoretic results obtained by using the given material parameters, whereas for the dashed line, these parameters were fitted. In (c) the results from a theoretical estimation of achievable resonance frequencies, when changing the normal stresses as well as the wire's vibrating length for a 100µm thick tungsten wire with L = 10 . . . 30 mm and T N = 0 . . . 1.4 GPa is depicted. (Tungsten has a yield stress of 1.5 GPa, approximately)

In first experiments in air resonances between 1000 Hz and 4250Hz in case of the wire viscometer and 820 Hz and 4040Hz in case of the suspended plate rheometer were achieved, see Fig. 5, [START_REF] Heinisch | A study on tunable resonators for rheological measurements[END_REF]. The intention of these experiments was to monitor which frequency range can be excited.

With a second experiment, see Fig. 6(a), we wanted to further investigate the achievable tunable range for the case of an oscillating wire when varying both, normal stresses as well as the vibrating length of the wire. Figure 6(b) shows the results for the experiment for a constant length l = 30 mm for different normal stresses in the wire, which shows the same range of resonance frequencies as in the first experiment, c.f., Fig. 5

(b).

A relatively large deviation of measured resonance frequencies to the theoretical values can be observed. This deviation results from frictional forces between the blades and the wire. That is to say that not the total amount of normal forces (provided by the weight) is transferred to the vibrating part of the wire, but that instead they are partially balanced at the blades due to frictional forces. This assumption was confirmed by a further experiment, see Sec. 6 below.

Figure 6(c) shows the (theoretically) total achievable range of resonance frequencies when changing the normal stresses as well as the wire's vibrating length for a 100 µm thick tungsten wire with the limiting factors L = 10 . . . 30mm and T N = 0 . . . 1.4 GPa. This range has been verified experimentally and good accordance was observed. 6. Improved setups using the Wire Viscometer as example

Reducing cross sensitivities

The previous sections showed a proof of concept of both types of sensors. They are well applicable for viscosity sensing and can be detuned in a range of over one decade. However, from the first experiments, it became clear that several improvements have to be achieved, which we want to discuss for the example of the wire viscometer. In this contribution we focus on the discussion on improving the stability of the resonance frequency (once it is set to a desired value). In [START_REF] Heinisch | Tunableminiaturizedvis-cosity sensors operating in the khz-range[END_REF] a discrete permanent magnet assembly is presented yielding a flux density of 1.2 T in a 4 mm airgap, which drastically improves the magnitude of the read-out signal.

We investigated three different types of wire clamping, see Fig. 7, in the laboratory during 25 hours recording the ambient temperature to study the range of resonance frequency changes under (randomly) varying conditions in a non-air-conditioned room. For the first case (Fig. 7(a)) we expected high shifts of the resonance frequency due to thermal stresses resulting from the different thermal expansion coefficient of the used materials. The problematic characteristic of setup (a) is not (only) this high temperature dependency but that for a given temperature the variation of resonance frequencies is still about 20 Hz. Thus the idea was to keep the normal stresses constant by pinning the wire instead of clamping, see Fig. 7(b). However, in this case, there is still a relatively large temperature dependency which can only be explained with friction forces between the blades and the wire, c.f. discussion in Sec. 5. Here, also a significant hysteresis can be observed within a temperature cycle. In the third setup, see Fig. 7(c) an almost frictionless linear stage was used to keep the normal stresses in the clamped wire as constant as possible. With this setup the variation of the resonance frequency is less than 3 Hz and mainly not dependent on the ambient temperature.

Conclusion

This work shows the feasibility of measuring viscosity with both presented sensors at tunable resonance frequencies. For the case of the wire viscometer an improved setup has been found allowing for stable resonance frequencies. Furthermore, first closed-form models have been developed which can be fitted into the measured data with high accuracy. Regarding future work, the stability of both sensors has to be investigated and most likely, improved with more sophisticated setups. For this investigation large measurement series with multiple test liquids over a long period of time have to be carried out.

Acknowledgement

We are indebted to the Austrian Competence Centre of Mechatronics (ACCM) for the financial support.

Figure

  Figure1: In (a) a schematic drawing of the suspended plate rheometer is depicted. The sensor, surrounded by an experimental well, consists of two parallel wires (placed in external field B) and a small platelet. In this figure, the left wire is used for exciting lateral vibrations by means of Lorentz forces. The second wire on the right used for pick-up, is coupled to the excitation wire with a rigid plate, thus following the movement of the left wire which effects an induced voltage which is used as read-out. By changing the tensile stresses in the wires, the resonance frequency can be changed. In (b) the scheme of the modeling approach is depicted. The upper figure shows the electrical equivalent circuit consisting of two galvanically isolated circuits representing the excitation and read-out circuits. (V g : Voltage of the function generator, 50 Ω: Internal resistance of the function generator, R 1 : resistance of the excitation wire, R 2 : resistance of the pick-up wire, V M : Motion induced voltage, V ct : Induced voltage trough electrical cross-talk, 10 MΩ: Input resistance of the lock-in amplifier, I in , I out , V in , V out : Input and output currents / voltages). The lower figure shows the mechanical lumped-element model. (k 0 : intrinsic spring constant, r 0 : intrinsic damping parameter, F F : Fluid forces, F L : Lorentz forces, u x : displacement in x-direction, B: external magnetic field). In (c) a comparison between measured data (in this case for isopropanol) and the model obtained by fitting the parameters from Eq. 6 is shown.

Figure 2 :

 2 Figure2: In (a) the schematic drawing of the wire viscometer is depicted. The experimental well for investigating sample liquids is not shown in this sketch. A 100 µm thick tungsten wire carrying sinusoidal currents is placed in an external magnetic field and thus oscillating due to Lorentz forces. For excitation, a function generator is used for the input signal. A series resistors is used to limit the excitation current. The motion induced voltage (i.e., the read-out signal) is measured with lock-in amplifier. The resonance frequency can be changed by (mechanically) changing the tensile stresses in the wire. In (b) the electrical equivalent circuit of the wire viscometer is illustrated. (R s : series resistor to limit the excitation current, R tw : resistance of the tungsten wire). In (c) a comparison between measured data (in this case for isopropanol) and the model obtained by fitting the parameters from Eq. 14 is shown.

3 .

 3 The indicated reference values for viscosity and mass-density were measured with a Stabinger Viscometer (SVM 3000) at 25 • C.

Figure 4 :

 4 Figure 4: Comparison of the Q-factor. For both sensors, the relative and the absolute values are indicated. In the investigated range, both sensors show a similar sensitivity to viscosity. The two dashed lines in both figures show the theoretical results for the Q-factor for the indicated mass densities.

Figure 5 :

 5 Figure 5: In both figures the band width of achievable resonance frequencies (in air) using 3 cm long and 100 µm thick tungsten wires are illustrated. In each case a detailed view of one characteristic resonance is depicted as well. In case (b) (wire viscometer) higher quality factors are obtained. (Note that the bandwidths in the detailed views are 40 Hz for the wire viscometer and 200 Hz for the suspended plate rheometer.) The obtained resonances in first measurements go from 1000 Hz to 4250 Hz in case of the wire viscometer and from 820 Hz to 4040 Hz for the suspended plate rheometer. In both cases, for changing the resonance frequencies for these measurements, the normal stresses in the wires were varied by tensioning the wires with micro-stages with micrometer screws.

Figure 7 :

 7 Figure 7: Effect of three different types of wire clamping on the resonance frequency. For each case the resonance frequency over temperature as well as resonance frequency and temperature over time is plotted.

Isabelle

  Dufour graduated from Ecole Normale Supérieure de Cachan in 1990 and received the Ph.D. and H.D.R. degrees in engineering science from the University of Paris-Sud, Orsay, France, in 1993 and 2000, respectively. She was a CNRS research fellow from 1994 to 2007, first in Cachan working on the modelling of electrostatic actuators (micromotors, micropumps) and then after 2000 in Bordeaux working on microcantilever-based chemical sensors. She is currently Professor of electrical engineering at the University of Bordeaux and her research interests are in the areas of microcantileverbased sensors for chemical detection, rheological measurements and materials characterisation. Bernhard Jakoby obtained his Dipl.-Ing. (M.Sc.) in Communication Engineering and his doctoral (Ph.D.) degree in electrical engineering from the Vienna University of Technology (VUT), Austria, in 1991 and 1994, respectively. In 2001 he obtained a venia legendi for Theoretical Electrical Engineering from the VUT. From 1991 to 1994 he worked as a Research Assistant at the Institute of General Electrical Engineering and Electronics of the VUT. Subsequently he stayed as an Erwin Schrödinger Fellow at the University of Ghent, Belgium, performing research on the electrodynamics of complex media. From 1996 to 1999 he held the position of a Research Associate and later Assistant Professor at the Delft University of Technology, The Netherlands, working in the field of microacoustic sensors. From 1999 to 2001 he was with the Automotive Electronics Division of the Robert Bosch GmbH, Germany, where he conducted development projects in the field of automotive liquid sensors. In 2001 he joined the newly formed Industrial Sensor Systems group of the VUT as an Associate Professor. In 2005 he was appointed Full Professor of Microelectronics at the Johannes Kepler University Linz, Austria. He is currently working in the field of liquid sensors and monitoring systems.