I Dufour

E Lemaire

B Caillard

H Debéda

C Lucat

S M Heinrich

F Josse

O Brand

Effect of hydrodynamic force on microcantilever vibrations: applications to liquid-phase chemical sensing

Keywords: Hydrodynamic force, microcantilever vibration, rheology, chemical detection, in-plane vibration, MEMS

At the microscale, cantilever vibrations depend not only on the microstructure's properties and geometry but also on the properties of the surrounding medium. In fact, when a microcantilever vibrates in a fluid, the fluid offers resistance to the motion of the beam. The study of the influence of the hydrodynamic force on the microcantilever's vibrational spectrum can be used to either (1) optimize the use of microcantilevers for chemical detection in liquid media or (2) extract the mechanical properties of the fluid. The classical method for application (1) in gas is to operate the microcantilever in the dynamic transverse bending mode for chemical detection. However, the performance of microcantilevers excited in this standard out-of-plane dynamic mode drastically decreases in viscous liquid media. When immersed in liquids, in order to limit the decrease of both the resonant frequency and the quality factor, and improve sensitivity in sensing applications, alternative vibration modes that primarily shear the fluid (rather than involving motion normal to the fluid/beam interface) have been studied and tested: these include in-plane vibration modes (lateral bending mode and elongation mode). For application (2), the classical method to measure the rheological properties of fluids is to use a rheometer. However, such systems require sampling (no in-situ measurements) and a relatively large sample volume (a few milliliters). Moreover, the frequency range is limited to low frequencies (less than 200Hz). To overcome the limitations of this classical method, an alternative method based on the use of silicon microcantilevers is presented. The method, which is based on the use of analytical equations for the hydrodynamic force, permits the measurement of the complex shear modulus of viscoelastic fluids over a wide frequency range.

INTRODUCTION

In recent years, interest in microcantilever-based chemical and bio-chemical sensing systems has risen due to their projected high sensitivity [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][START_REF] Zhu | Microcantilever sensors in biological and chemical detections[END_REF]. The large ratio of surface area to volume makes the microcantilever extremely sensitive to surface processes. For (bio)chemical detection, the microcantilever is usually coated with a (bio)chemically sensitive layer whose purpose is to selectively sorb the analyte or molecule of interest. The sorbed substance can then be detected by monitoring either the resonant frequency shift (dynamic mode) or the quasi-static deflection (static mode). A simplified way to explain the basic principle of such sensors is to say that in the case of dynamic mode operation, the change in mass associated with the sorption of analyte onto/into the sensitive coating causes a shift in resonant frequency, which may be correlated to the ambient concentration of the target substance. For static-mode operation, the sorption of analyte causes a cantilever deflection that is induced by surface stress or by the tendency of the sensitive coating to expand or contract upon analyte sorption or desorption (modification of strain and stress in the coating). In this paper we will focus on the use of the dynamic mode.

At the microscale, cantilever vibrations depend not only on the microstructure's properties and geometry but also on the properties of the surrounding medium (density and viscoelastic properties). In fact, when a microcantilever vibrates in a fluid, the fluid offers resistance to the motion of the beam. The coupling between the structure and the surrounding viscous fluid influences the structure's mechanical resonance; in particular, the fluid causes a decrease of both the resonant frequency and the quality factor and, thus, negatively impacts the device's sensing capabilities. In gas, resonant frequencies of classical transverse flexural modes may be reduced by a few percent compared with the value in vacuum, whereas the quality factor may exhibit reductions of two orders of magnitude. This performance degradation for conventional out-of-plane flexural resonant modes is further exacerbated in liquids: the resonant frequency and quality factor values are approximately one order of magnitude smaller than their values in the gas phase (Fig. 1). Chemical detection using microcantilever-based sensors is based on the measurement of the resonant frequency shift r f  which is induced by the mass increase of the vibrating microstructure during sorption of the target species into the sensitive coating [START_REF] Fadel | Signal to noise ratio of resonant microcantilever type chemical sensors as a function of resonant frequency and quality factor[END_REF]:

2 rr m ff m    (1)
with f r and m being the resonant frequency and mass of the microstructure, respectively. According to the partition coefficient (K) of the analyte/sensitive coating pair [START_REF] Fadel | Signal to noise ratio of resonant microcantilever type chemical sensors as a function of resonant frequency and quality factor[END_REF], which is defined as the ratio of the concentration of the analyte in the coating and the ambient concentration, C A , of the analyte in the surrounding medium, the mass variation m  is proportional to the concentration C A . Thus, the sensitivity of the chemical sensor, S, is proportional to the resonant frequency of the microcantilever in the surrounding medium (gas or liquid) [START_REF] Fadel | Signal to noise ratio of resonant microcantilever type chemical sensors as a function of resonant frequency and quality factor[END_REF]:

2     rr L r A ff K V Sf Cm (2)
with V L defined as the volume of the sensitive layer.

The accuracy of the resonant frequency measurement depends on the sharpness of the resonant peak which is characterized by the quality factor, Q, associated with a particular resonant mode, while the noise N corresponding to the resonant frequency measurement can be considered to be inversely proportional to the quality factor [START_REF] Fadel | Signal to noise ratio of resonant microcantilever type chemical sensors as a function of resonant frequency and quality factor[END_REF]:

1 N Q  (3)
According to Eqs. 2-3, reduced values of the resonance characteristics (resonant frequency and quality factor) in liquid media (Fig. 1) adversely affect the sensitivity and the detection limit of these types of sensors. Therefore, for liquidphase chemical sensing applications it is imperative that the effect of the liquid on the resonant characteristics of the Preprint Sensors and Actuators B. doi: 10.1016/j.snb.2013.10.106 sensing device be understood and considered in the sensor design in order to achieve desired levels of sensing performance (sensitivity and limit of detection).

In the present paper, we propose alternative uses of microcantilevers, including the implementation of unconventional vibrational modes to overcome the aforementioned obstacles for liquid-phase operation. By understanding in detail the influence of the hydrodynamic force on the microcantilever's vibrational spectrum, the following two objectives may be achieved: (1) optimization of microcantilevers for chemical detection in liquid media and (2) extraction of the mechanical properties of complex fluids which is of particular importance for microfluidic systems and for chemical detection in these types of environments.

The paper is organized such that a review of hydrodynamic force expressions and their impact on the resonant frequency and quality factor is given in Section 2 for the cases of out-of-plane and in-plane vibrations. In Section 3 we present how one may use the knowledge of the hydrodynamic force to choose appropriate vibrational modes and microcantilever geometries for chemical detection in liquid media. A second example of how a firm grasp of fluid-structure interaction may be used to advantage is the implementation of a microcantilever to extract the rheological properties of a fluid, which is described in Section 4.

Notation (Fig. 2): The geometry of the cantilever is defined by the width b, thickness h, and length L. Coordinate x is measured along the beam length. The properties of the cantilever material are the Young's modulus E and the mass density . The fluid (gas or liquid) properties are defined as  f , the fluid's mass density, the fluid's dynamic viscosity and G* = G' + jG'', the complex shear modulus of the fluid.

INFLUENCE OF HYDRODYNAMIC FORCE ON MICROSTRUCTURE RESONANT CHARACTERISTICS

In this paper, we focus on three distinct vibration modes (Fig. 2): the transverse (out-of-plane) bending mode, the lateral (in-plane) bending mode and the elongation mode (also in-plane). A fourth mode that may also have interesting potential for both chemical detection and rheological sensing is the torsional mode, whose hydrodynamic force and resonant characteristics have been studied in [START_REF] Cai | Resonant characteristics of rectangular microcantilevers vibrating torsionally in viscous liquid media[END_REF][START_REF] Green | Torsional frequency response of cantilever beams immersed in viscous fluids with applications to the atomic force microscope[END_REF]; however, that mode is not included in the present study.

General modeling of the hydrodynamic force

When a microcantilever vibrates in a viscous fluid (gas or liquid), the fluid offers resistance to the motion. Depending on the vibration mode and on the cantilever surface there are hydrodynamic pressure forces (normal to the surface) and hydrodynamic viscous forces (tangential to the surface) (Fig. 3).

Figure 3.Schematic representation of the hydrodynamic force on a microcantilever cross section in the cases of transverse (outof-plane) bending, lateral (in-plane) bending and (in-plane) elongation modes.

The force per unit length, , which is the consequence of all normal and tangential stresses exerted by the fluid on all the surfaces of the cantilever, can be written in the frequency domain as [START_REF] Blevins | Formulas for natural frequency and mode shape[END_REF]:

(4)
where x is the longitudinal coordinate,  the radial frequency of vibration, w the microcantilever deflection in the case of bending modes or the microcantilever axial displacement in the case of the elongation mode, and and are functions depending on the fluid properties and the microcantilever cross-sectional geometry. They may be interpreted, respectively, as the distributed damping coefficient of the fluid and the distributed effective fluid mass (per unit length of beam). In general, these quantities are both frequency-dependent and x-dependent; however, in what follows g 1 and g 2 will be assumed uniform in x as this assumption has been shown to be sufficiently accurate in many cases of practical interest [START_REF] Chon | Experimental validation of theoretical models for the frequency response of atomic force microscope cantilever beams immersed in fluids[END_REF].

It is common to normalize into a dimensionless form called the hydrodynamic function,  r -j i , (where  r > 0 and  i > 0) [START_REF] Sader | Frequency response of cantilever beams immersed in viscous fluids with applications to the atomic force microscope[END_REF] (5)

The hydrodynamic function, , represents the total hydrodynamic force per unit length normalized by the force per unit length required to excite (at the same amplitude and frequency) a circular cylindrical volume of diameter b and density . The subscripts r and i in Eqs.5-6 denote the real and imaginary parts of the hydrodynamic function, respectively.

Resonant characteristics of the microstructure

The fluid effects (viscous term g 1 , and inertial term g 2 ) influence the dynamic response of the beam; in particular, the resonant frequency, f r , and the quality factor associated with viscous losses, , may be expressed as [START_REF] Chon | Experimental validation of theoretical models for the frequency response of atomic force microscope cantilever beams immersed in fluids[END_REF][START_REF] Sader | Frequency response of cantilever beams immersed in viscous fluids with applications to the atomic force microscope[END_REF] (7)

F       2 12     fluid F j g x, g x, w x,     1 g 2 g fluid F 2 2 4   r f g b 1 2 4    i f g b f  visc Q 0, 2 2 11 1 2 1 rv a c ff Q Lg m   2 0, 1 21 / / visc vac Lg m Qf Lg m    where
is the undamped natural frequency of the microcantilever in vacuum, Q is the total quality factor of the cantilever/fluid system (incorporating all losses), m is the total microcantilever mass and L is the microcantilever length.

The above equations illustrate, as stated in the introduction, the fact that in the general case of dynamic mode operation of a microcantilever, the resonant frequency and the quality factor depend on the fluid properties via the terms g 1 and g 2 .

In the following sections, the terms g 1 and g 2 will be analyzed for the different vibration modes.

Transverse (out-of-plane) bending vibrations of thin microcantilevers

The hydrodynamic function of a transversely vibrating beam of ribbon-like cross-section (i.e., width b much larger than the thickness h) was previously determined in [START_REF] Tuck | Calculation of unsteady flows due to small motions of cylinders in a viscous fluid[END_REF] by using the method of moments to solve for the velocity field in the fluid and in [START_REF] Sader | Frequency response of cantilever beams immersed in viscous fluids with applications to the atomic force microscope[END_REF] by performing a fit of the numerical results of [START_REF] Tuck | Calculation of unsteady flows due to small motions of cylinders in a viscous fluid[END_REF].

A more accurate approximation of the total hydrodynamic force acting on a transversely vibrating microcantilever in liquids should include both the effects of the pressure and shear stress exerted by the fluid on all faces of the beam of finite cross-sectional dimensions. Due to the symmetry of the problem, the hydrodynamic forces acting in the x or y direction on a beam with a rectangular cross-section vibrating transversely will cancel each other out. Only the hydrodynamic forces acting in the z direction will then affect the response of the beam. These forces (Fig. 3a) are the pressure forces acting on the large faces (of dimension b) and the shear forces acting on the small faces (of dimension h).

It can be assumed that the microcantilever is long enough so that the hydrodynamic force acting on the surface at the unsupported end of the beam is negligible. Recently, results for the hydrodynamic function in the case of transverse bending that accounted for both the beam's aspect ratio (h/b) and the Reynolds number (Re) of the fluid flow have been published [START_REF] Brumley | Oscillation of cylinders of rectangular cross section immersed in fluid[END_REF]. A Taylor series of the analytical function or a fitting of the numerical results can be made, and then the hydrodynamic function  can be written as

(9) (10)
witha 0 , a 1 , a 2 , b 0 , b 1 and b 2 being functions of aspect ratio b/h, and the Reynolds number, Re, defined as [START_REF] Sader | Frequency response of cantilever beams immersed in viscous fluids with applications to the atomic force microscope[END_REF] For the ribbon-like case, a 0 , a 1 , a 2 , b 0 , b 1 and b 2 have constant numerical values. In [START_REF] Maali | Hydrodynamics of oscillating atomic force microscopy cantilevers in viscous fluids[END_REF] a fit of the results of [START_REF] Sader | Frequency response of cantilever beams immersed in viscous fluids with applications to the atomic force microscope[END_REF]

In-plane vibrations of thin microcantilevers: lateral bending and elongation

In 1851, Stokes investigated the forces acting on an infinite rigid plane undergoing a sinusoidal, in-plane motion [START_REF] Stokes | On the effects of the internal friction of fluids on the motion of pendulums[END_REF]. This is commonly called Stokes's second problem. The fluid forces on a laterally vibrating beam of infinitely thin crosssection can be considered equivalent to those of Stokes's second problem if the beam under investigation is also quite wide relative to the boundary layer thickness of the fluid. In this case, the hydrodynamic force can be expressed as

4 2Re ri    (13)
This equation corresponds to the more general form of Eqs. 9-10 for the following values of coefficients:

(14) 0,vac f      01 2 11 Re Re r ab h ab h ab h         01 2 11 Re Re i bb h bb h bb h    2 Re 4 f b     01 2 01 2
00 . 9 0 0 3 0 00 . 9 0 0 3 0

aa a bb b    
However, Eq. 13 neglects edge effects, which could be significant for vibrations in which the boundary layer thickness is not negligible relative to width b. Furthermore, it also neglects the fluid resistance due to pressure on the thin faces of the beam (those of corresponding dimension h) in the lateral bending mode case and the fluid resistance due to shear on the same faces in the elongation mode case (Figs. 3b-3c). These neglected terms may be important for many practical beam geometries. As was the case for out-of-plane vibrations, it can be assumed that the microcantilever is long enough so that the hydrodynamic force acting on the surface at the unsupported end of the beam is negligible.

In the case of lateral bending, more accurate modeling has been studied in [START_REF] Brumley | Oscillation of cylinders of rectangular cross section immersed in fluid[END_REF][START_REF] Cox | Theoretical analysis of laterally vibrating microcantilever sensors in a viscous liquid medium[END_REF][START_REF] Cox | Characteristics of laterally vibrating resonant microcantilevers in viscous liquid media[END_REF]: the work takes into account both of the edge effects, which could be significant for beams whose boundary layer thickness is not much smaller than the width b, and the fluid resistance due to pressure on the thin faces of the beam. These results may also be fit to express them in the form of Eqs. 9-10.

In the case of the elongation mode, it is easy to modify Eq. 14 in order to take into account the hydrodynamic force on the thin faces of the beam (of dimension h):

(15)
In this mode, if we want to take into account the edge effects, numerical simulation may be performed and fitted to express them in the form of Eqs. 9-10.

CHOICE OF VIBRATIONAL MODES AND MICROCANTILEVER GEOMETRY FOR CHEMICAL DETECTION IN LIQUID MEDIA

From Eq. 7, it can be seen that the decrease of the resonant frequency when the microcantilever is immersed in a fluid is essentially due to the mass effect g 2 ; this effect can be minimized if the term g 2 is small in liquid. On the other hand, the strong decrease in the quality factor in liquid is due to the stronger influence of the viscous term g 1 compared to the (beneficial) influence of the displaced fluid mass (term due to g 2 ) in Eq. 8.

In order to limit the decrease of both the resonant frequency and the quality factor upon immersion in liquid media, alternative vibration modes that essentially shear the surrounding fluid rather than exerting normal stress on it have been studied (section 2.4); in particular, a comparison of values listed in Eqs. 12 and 14,within the context of the expressions for the hydrodynamic function (Eqs. 5, 6, 9 and 10), indicates quantitatively that the in-plane modes result in much smaller g 1 and g 2 values than the out-of-plane mode, meaning that the in-plane modes are less affected by immersion in liquid. Examples of analyses and experimental measurements of the resonant frequency and quality factor associated with the two unconventional modes of lateral bending and axial elongation are presented in what follows.

Case of lateral (in-plane) bending vibrations

Several cantilevers with widths of 45, 60, 75 and 90 µm, lengths of 200, 400, 600, 800, 1000 µm and nominal silicon thicknesses of 5, 8, 12 and 20 µm, corresponding to lateral bending resonant frequencies up to about 3 MHz, have been characterized in air and water. For operation in the lateral flexural vibration mode, thermal excitation and piezoresistive detection, both based on the use of diffused resistors, have been chosen as the actuating and sensing mechanisms, respectively (Fig. 4). The design of the actuation resistors and integrated Wheatstone bridge allows for a more efficient excitation of the in-plane flexural mode than the more classical out-of-plane flexural mode [START_REF] Beardslee | Thermal excitation and piezoresistive detection of cantilever in-plane resonance modes for sensing applications[END_REF]. The modeling of the fluid interaction terms, g 1 and g 2 , according to Eq. 13 for the lateral mode results in the resonant frequency being approximately proportional to b/L 2 (where b is the cantilever width and L the cantilever length), the relative resonant frequency shift from air to water being proportional to L/(hb 1/2 ) and the quality factor (in liquid) being proportional to hb 1/2 /L [START_REF] Heinrich | An analytical model for in-plane flexural vibrations of thin cantilever-based sensors in viscous fluids: applications to chemical sensing in liquids[END_REF][START_REF] Heinrich | An analytical model of a thermally excited microcantilever vibrating laterally in a viscous fluid[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]. These simple analytical results are based on elementary ("Euler-Bernoulli") beam theory, which is applicable to slender beams (i.e., small b/L for the lateral bending case), and a Stokes-type fluid resistance model. The simple expression for Q gives excellent quantitative agreement with the measured quality factor in water for aspect ratios up to h/b=0.2 [START_REF] Heinrich | An analytical model for in-plane flexural vibrations of thin cantilever-based sensors in viscous fluids: applications to chemical sensing in liquids[END_REF][START_REF] Heinrich | An analytical model of a thermally excited microcantilever vibrating laterally in a viscous fluid[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]. However, experimental results show that, for microcantilevers having larger widthto-length ratios, the behaviors predicted by this analytical model differ from measurements due to the violation of the slender beam assumption. To more accurately model microcantilever resonant behavior in viscous fluids and to improve the understanding of lateral-mode sensor performance, a new analytical model has been developed, incorporating both Stokes-type fluid resistance and "Timoshenko beam" effects (shear deformation and rotatory inertia) [START_REF] Schultz | Lateral-mode vibration of microcantilever-based sensors in viscous fluids using Timoshenko beam theory[END_REF][START_REF] Schultz | Timoshenko beam model for lateral vibration of liquid-phase microcantilever-based sensors[END_REF][START_REF] Schultz | Timoshenko beam effects in lateral-mode microcantilever-based sensors in liquid[END_REF]. The latter effects are neglected in Euler-Bernoulli beam theory but become more important at larger b/L ratios, which are the types of geometries that promise larger Q values in the lateral flexural mode according to the Euler-Bernoulli analytical formula for Q listed above. The results of the Timoshenko beam model [START_REF] Schultz | Lateral-mode vibration of microcantilever-based sensors in viscous fluids using Timoshenko beam theory[END_REF][START_REF] Schultz | Timoshenko beam model for lateral vibration of liquid-phase microcantilever-based sensors[END_REF][START_REF] Schultz | Timoshenko beam effects in lateral-mode microcantilever-based sensors in liquid[END_REF], presented in Fig. 5, exhibit a departure from linearity in both resonant frequency and Q for the shorter, "stubbier" specimens, trends that are also seen in the data. Results such as these may serve a useful purpose as guides for selecting microcantilever dimensions to maximize quality factors in liquids(in order to have small noise in the resonant frequency measurement) and to minimize the liquidinduced decrease in resonant frequency as compared to that in air (in order to yield higher sensitivity). Analyzing the experimental data in [START_REF] Beardslee | Geometrical considerations for the design of liquid-phase biochemical sensors using a cantilever's fundamental in-plane mode[END_REF] shows that (1) for the first lateral flexural mode the microcantilevers exhibit quality factors up to 86 in water (for out-of-plane modes, Q-factors in water usually do not exceed 10 [START_REF] Vancura | Liquid-phase chemical and biochemical detection using fully integrated magnetically actuated complementary metal oxide semiconductor resonant cantilever sensor systems[END_REF]); ( 2) the resonant frequencies of the first lateral flexural mode are only lowered by 3 % to 14 % in water compared to the values in air (for out-of-plane modes, this reduction is approximately 50 % [START_REF] Chon | Experimental validation of theoretical models for the frequency response of atomic force microscope cantilever beams immersed in fluids[END_REF]). These two characteristics of the lateral mode operation are promising for future sensing applications in liquid media. The first chemical detection data in liquid media generated with lateral-mode microcantilever devices are presented in [START_REF] Beardslee | Liquid-phase chemical sensing using lateral mode resonant cantilevers[END_REF].

Case of elongation vibrations

For the same reason as for the lateral bending mode, the longitudinal mode may be of potential interest for sensing applications in liquid media. To assess the characteristics of this mode, self-actuated resonant microcantilevers based on a PZT piezoelectric thick-film sandwiched between two gold electrodes have been fabricated using screen-printing technology associated to a sacrificial layer process [START_REF] Lucat | Microsystems elements based on free-standing thickfilms made with a new sacrificial layer process[END_REF]. The microcantilevers comprising Au/PZT/Au layers [START_REF] Castille | Longitudinal vibration mode of piezoelectric thick-film cantilever-based sensors in liquid media[END_REF] were partially released from an alumina substrate (Fig. 6). Using an impedance analyzer (HP4194A), frequency spectra have been measured in air and in various other fluids with viscosities ranging from 1.5 to 500 cP for a piezoelectric cantilever (8x2x0.1 mm 3 ) with 7 µm thick gold electrodes (Fig. 7). Even though the quality factor of 353 in air is relatively low (probably due to internal mechanical losses and piezoelectric losses), the quality factors in liquids are significantly higher compared to those of classical out-of-plane modes. The piezoelectric cantilever exhibits a decrease of the quality factor from 353 to 101 when the cantilever is brought from air to dodecane (with 1.5cP viscosity --on the order of that of water, 1cP). At the same time, a decrease of resonant frequency of the first longitudinal mode of only 1.8% (68.8 to 67.6 kHz) is observed when the cantilever is brought from air to dodedane (1.5cP). To demonstrate the potential advantages of the (in-plane) elongation mode in comparison with the (out-of-plane) transverse bending mode, the resonant characteristics of two cantilevers of comparable geometries in three different fluids are summarized in Table 1. The elongation-mode data corresponds to the PZT cantilever described in the previous paragraph, while the data for the transverse bending mode was obtained from measurements on a silicon cantilever of similar dimensions (6x0.6x0.11mm 3 ) [START_REF] Youssry | A straightforward determination of fluid viscosity and density using microcantilevers: from experimental data to analytical expressions[END_REF]. Despite the fact that the measurements were made on cantilevers of different dimensions and material properties, the data of Table 1 clearly suggests that the detrimental effects of the liquid on the resonant characteristics (i.e., lowering Q and f r ), which are quite prominent in the transverse (out-of-plane) bending case, may be significantly reduced when the elongation mode is employed. This result is consistent with the earlier observations concerning lateral bending vs. transverse bending and the superiority of in-plane modes over out-of-plane modes in liquid-phase sensing applications.

As was the case with lateral-mode silicon cantilevers having an integrated scheme for actuation and measurement, the thick-film piezoelectric cantilevers actuated in the longitudinal mode are promising in terms of sensitivity and limit of detection for (bio)chemical detection in liquid media.

USE OF HYDRODYNAMIC FORCE STUDY FOR THE MEASUREMENT OF FLUID RHEOLOGICAL PROPERTIES

Using the frequency spectra of microcantilevers vibrating in fluids and the equations of section 2, it is possible to the mechanical properties of fluids. As noted in the introduction and based on the results of the previous section, it is clear that successful chemical detection and analysis in liquids (Newtonian or non-Newtonian) requires that the liquid properties be accurately characterized. Different methods for fluid property determination have been developed using the transverse bending mode which, as noted earlier, is more influenced by the fluid properties than other modes [START_REF] Dufour | The microcantilever: a versatile tool for measuring the rheological properties of complex fluids[END_REF]. The first of these permits the determination of both the mass density and viscosity of the fluid at the resonant frequency of the microcantilever in the fluid [START_REF] Youssry | A straightforward determination of fluid viscosity and density using microcantilevers: from experimental data to analytical expressions[END_REF]; the second enables one to extract the same properties over a frequency range that includes the in-fluid resonant frequency [START_REF] Belmiloud | Rheological behaviour probed by vibrating microcantilevers[END_REF]; two other methods are based on the same principle, but with the assumption that the fluid's mass density is known. These latter methods allow for the determination of the real and imaginary parts of the fluid's shear modulus, which characterize both the elasticity and viscosity of the fluid [START_REF] Youssry | On-chip characterization of the viscoelasticity of complex fluids using microcantilevers[END_REF]. In this section we will focus on presenting a more detailed description of the fourth method (estimation of real and imaginary parts of the shear modulus over a range of frequencies).

Using the amplitude and phase frequency characteristics of the transverse-bending microcantilever, it is possible to determine the value of the hydrodynamic terms, g 1 and g 2 , at each measurement frequency.

Then, based on Eqs. 5-6 and 9-12, two analytical equations can be used to obtain the numerical values of both the real G' and the imaginary G'' parts of the fluid shear modulus at the frequency f:

    2 22 2 1 21 2 "2 4 2 2 g BB B Gg C g g C DD D DD             ( 16
) 2 2 1 1" ' " BG Gg C Dg D G        (17)
In order to measure the amplitude and phase transfer characteristic of microcantilevers in different fluids, chips with silicon-based microcantilevers of uniform rectangular cross section were provided by LAAS-CNRS of Toulouse, France (Fig. 8). A series of microcantilevers, termed LL, LH, and A, having the following lengths and widths were tested:

L LL =2810µm, L LH =1440µm, L A =500µm, b LL =100µm, b LH =285µm, b A =100µm.
Due to the use of SOI wafers, the silicon thicknesses of all microcantilevers are the same, h=20µm. The microcantilevers were designed to be electromagnetically actuated, i.e., excited by a Lorentz force [START_REF] Youssry | A straightforward determination of fluid viscosity and density using microcantilevers: from experimental data to analytical expressions[END_REF]. For this purpose, a conducting path was deposited on the top surface of the microcantilever and an external magnet was used. For this method of rheological measurement, vibration amplitude measurements were made using an optical vibrometer that detects the out-of-plane velocity using Doppler interferometry (MSA 500 Polytec), after which the deflection is deduced from the velocity.

CONCLUSION

In a liquid medium, the influence of the hydrodynamic force on the microcantilever's dynamic response is much more pronounced than in a gas medium. Appropriate modeling of this hydrodynamic force, taking into account the particular mode of vibration, is necessary in order to understand the complex interplay among the various system parameters and to achieve the promising potential of these types of devices in liquid-phase chemical sensing applications and beyond. In this paper, we have provided an overview of the relevant hydrodynamic force equations for three different modes of vibration (transverse bending, lateral bending and elongation) and presented two different applications for these modeling efforts: (a) the choice of optimal geometry and vibration mode for liquid-phase chemical detection and (b) the determination of the frequency-dependent rheological properties of liquids, which is also of paramount importance in achieving successfully detection of chemical species in more complex fluid environments.

Figure 1 .

 1 Figure 1. Measurements of three microcantilevers with different geometries (named in section 4 LL, LH and A respectively): 2810 x 100 x 20µm 3 (blue), 1440 x 285 x 20µm 3 (green) and 500 x 100 x 20µm 3 (red). a) spectrum in air. b) spectrum in water.

Figure 2 .

 2 Figure 2.Schematic representation of the transverse (out-of-plane) bending mode, the lateral (in-plane) bending mode and the (inplane) elongation mode.

Figure 4

 4 Figure 4.a) Schematic of the Wheatstone bridge connection, b) Schematic layout of heating resistors and piezoresistors at the microcantilever clamped-end, c) photograph of a lateral-mode silicon microcantilever with piezoresistors and heating resistors at the clamped-end.

Figure 5 .

 5 Figure 5. Resonant frequency and quality factor comparison (lateral mode in water, h=7.02 μm, E=151 GPa): Timoshenko-based model, Euler-Bernoulli-based model, and experimental data [21,24].

Figure 6 .

 6 Figure 6. Schematic of piezoelectric cantilever for elongation mode operation (top); photo of one screen-printed Au/PZT/Au microcantilever used for in-plane elongation vibrations (bottom). P represents the electrical polarization and E elec the electric field.

  : L=200 m; b=(60, 75) m Exp. Data: L=400 m; b=(45, 60, 75, 90) m Exp. Data: L=600 m; b=(45, 60, 75) m Exp. Data: L=800 m; b=(45, 60, 75, 90) m Exp. Data: L=1000 m; b=(45, 60, 75, 90) m Euler-

Figure 7 .

 7 Figure 7. Effects of liquid on the resonant frequency and quality factor associated with the first elongation mode of a 8x2x0.1 mm 3 PZT cantilever.

Figure 8 .Figure 9

 89 Figure 8. Schematic (left) and photo (right) of silicon microcantilever for operation in the tranverse (out-of-plane) bending mode.The rheograms presented in Fig.9illustrate the Newtonian behavior of water and of a silicone oil (20cP) obtained from spectrum measurements using three different cantilevers (LL, LH and A) covering different frequency ranges: LL between 1kHz and 5kHz, LH between 5kHz and 50kHz and A between 50kHz and 100kHz. The obtained viscous shear modulus, G'', over the whole frequency range is continuous and proportional to the frequency. This means that the

Figure 10 .

 10 Figure 10.Rheogram of CTAB measured with a classical cone-plate rheometer and three cantilever geometries (LL, LH and A) and covering the frequency range 10 3 -10 5 Hz using Eqs. 16-18. Comparison with literature DWS (Diffusing Wave Spectroscopy)measurements[START_REF] Galvan-Miyoshi | Diffusing wave spectroscopy in Maxwellian fluids[END_REF].

Table 1 .

 1 Comparison of experimental data for resonant characteristics of first elongation mode (EL) of PZT cantilever and first transverse bending mode (TB) of Si cantilever[START_REF] Youssry | A straightforward determination of fluid viscosity and density using microcantilevers: from experimental data to analytical expressions[END_REF]. (Beam dimensions are comparable.)

	Liquid Viscosity		1.5 cP	20 cP		100 cP
	Mode	EL	TB	EL	TB	EL	TB
	Q	101	19	54	4	31	2
	Relative
	Decrease in f r	1.8%	36%	2.7%	46%	3.6%	53%
	(air-to-liquid)

ACKNOWLEGMENTS

The authors would like to thank (1) Cédric Ayéla for the cantilever mask designs for the rheological measurements, (2) Liviu Nicu, Fabrice Mathieu and Daysuke Saya for the fabrication of these silicon cantilevers and (3) Luke Beardslee for the electrical measurements on silicon microcantilevers vibrating in lateral mode. This work was partially supported by the CPER Pôle 4N Nanosciences en Aquitaine (GP-206-action 216/1) with the contribution of the Conseil Régional d'Aquitaine, the FEDER and the Ministry of Education and Research, by the French National Agency (MicRheo project n°ANR-08-NANO-004), by the Conseil Régional d'Aquitaine (n°20091102001) and by NSF grant ECCS-0824017 and ECCS-1128992.


M T Boudjiet

J Bertrand

C Pellet

I Dufour

New characterization methods for monitoring small resonant frequency variation: Experimental tests in the case of hydrogen detection with uncoated silicon microcantilever-based sensors

Keywords: Resonant frequency, quality factor, uncoated silicon microcantilever, chemical detection, signal-to-noise ratio

The uncoated silicon microcantilever (USMC) operated in the dynamic mode is a new concept in the field of microcantilever-based chemical sensors. Due to the absence of a sensitive layer, this kind of microsensor can only be used for specific applications where it is known that only one chemical species may be varying in concentration, such as monitoring hydrogen release in radioactive waste disposal facilities. Usually, the relative variation of the USMC resonant frequency expected for low concentrations (≤2%) of hydrogen in nitrogen is below 50ppm. As a result, the measurement of both the resonant frequency, f r , and the quality factor, Q, by classical methods, based on the gain spectrum (resonant peak and -3dB bandwidth), is not sufficiently accurate. In this paper, new measurement methods for monitoring f r and Q variations are proposed: (1) variation of gain and phase at fixed frequencies and (2) polynomial approximations of gain and phase spectra. The performance study of these characterization methods shows that monitoring f r by using phase linearization yields the best signal-to-noise ratio (e.g., 100 at 0.6% of H 2 in N 2 ), with 0.02% as a limit of detection for hydrogen.

INTRODUCTION

Hydrogen release in radioactive waste disposal facility

After having concluded a feasibility study on deep geological disposal for high-level and Intermediate level long-lived radioactive waste in 2005, the French national radioactive waste management agency (Andra) was charged by the Planning Act n°2006-739 to study the design and the creation of an industrial center for geological disposal called Cigéo which must be reversible for at least a century-long period. Within the framework of this geological repository project, the observation and surveillance must contribute to acquire the knowledge required to run the disposal and its reversible management. Hydrogen release is expected in the radioactive waste disposal facility. It originates from (i) radioactive waste release (ii) and anoxic corrosion of metallic materials. In fact, some radioactive wastes (containing α, , radioactivity) resulting from the reprocessing of irradiated fuels are embedded in bitumen matrix. The self-irradiation of the bituminized waste leads mainly to the production of radiolytic hydrogen (75-95 vol. % of produced radiolytic gas). In the deep geological disposal environments steel components would corrode to more oxidized corrosion products and hydrogen gas. The kinetic of Anoxic corrosion of Fe components is not well known, it is actually considered an average rate about 10 μm/year. This assumption still awaits experimental verification in the Underground Research Laboratory sited at Bure in the Meuse district and aims at studying the feasibility of the reversible geological disposal of high-level and long-lived intermediate-level radioactive waste in the Callovo-Oxfordian clay formation (east of France). Despite the fact that hydrogen releases are expected to be small (in the order of 430 mmol/hour for each intermediate level nuclear waste), when ventilation stops with cell closure, concentrations would slowly and regularly increase.

The first calculation gives 4 % hydrogen content in the atmosphere of the radioactive waste cell in less than one year by the end of the oxic corrosion period.

Monitoring of repository structures contributes to security, safety and reversible management of the repository [START_REF] Buschaert | Developing the tools for geologic repository monitoring -Andra's monitoring R&D program[END_REF]. It is important as it relates to the guidance of the disposal process and to the corresponding decision-making process. In the upstream repositorydesign phases, the hydrogen monitoring system was planned to work under normal operating conditions and to withstand radiation exposure in case of an accidental event. There exist different kinds of hydrogen sensors, including those that are commercially available and others still under development [START_REF] Hübert | [END_REF]. In the present paper, the preliminary results concerning the development of microcantilever-based sensors without sensitive coating (Uncoated Silicon MicroCantilever, USMC) for hydrogen detection are presented.

Resonant frequency shifts of USMC-based sensors

In recent years, interest in microcantilever-based chemical sensing systems has risen due in part to their projected high sensitivity [START_REF] Goeders | Microcantilevers: sensing chemical interactions via mechanical motion[END_REF][START_REF] Zhu | Microcantilever sensors in biological and chemical detections[END_REF]. Because of the large ratio of surface area to volume, the microcantilevers are extremely sensitive to surface processes and can be used as micromechanical sensors [START_REF] Boisen | Cantilever-like micromechanical sensors[END_REF] for high-resolution sensing and probing [START_REF] Lee | Integrated microcantilevers for high-resolution sensing and probing[END_REF]. For chemical detection, the USMC-based sensors operated in the dynamic mode are used to measure mass density changes of one scalable analyte in gas mixture [START_REF] Tétin | Modeling and performance of uncoated microcantilever-based chemical sensors[END_REF]. The operating principle of this unconventional use of a microcantilever is based on the fact that the resonant frequency of the microresonator depends on both the density and the viscosity of the surrounding medium. In the case of the out-of-plane mode of a wide microcantilever, it has been shown that the predominant effect causing the change in the microcantilever resonant frequency is due to the gas mass density variation [START_REF] Tétin | Modeling and performance of uncoated microcantilever-based chemical sensors[END_REF]. In fact, when the surrounding fluid mass density increases (decreases), the equivalent effective mass of the microcantilever increases (decreases), thereby causing the resonant frequency to decrease (increase). The absence of a sensitive coating leads to a more reliable and reversible behavior because there is no significant absorption and desorption phenomena. However, the major drawbacks of such sensors is that there is no intrinsic selectivity; thus, these sensors can only be used for specific applications, such as when monitoring environments where it is known that only one gas concentration can vary [START_REF] Dufour | Unconventional uses of microcantilevers as chemical sensors in gas and liquid media[END_REF].

The conventional method for monitoring the resonant frequency shift consists of searching for the resonant peak on the amplitude spectrum and tracking the change in the corresponding frequency. The relative variation of the resonant frequency in the case of low concentration of hydrogen (H 2 ) in nitrogen (N 2 ) is very small. For instance, using an USMC (length=5mm, width=1mm and thickness=170µm, resonant frequency approximately 10 kHz), a concentration of 0.2% of H 2 in N 2 corresponds to a resonant frequency change of approximately 0.02 mHz. Therefore, the conventional method does not prove to be sufficiently accurate to measure such a small shift in resonant frequency. This is mainly due to the difficulty of identifying precisely the resonant peak location due to the measurement noise.

The quality factor is a key parameter in resonating sensors because it determines the stability of the resonator in terms of resonant frequency. It is defined as the ratio of the vibrational energy stored in the system to the total energy lost per cycle. This parameter is also commonly used to characterize resonating sensors [START_REF] Schmid | Electrostatically actuated nonconductive polymer microresonators in gaseous and aqueous environment[END_REF][START_REF] Lee | Direct inference of parameters for piezoresistive micromechanical resonators embedded in feedthrough[END_REF] that are typically used for viscosity measurements [START_REF] Abdallah | Measurement error estimation and quality factor improvement of an electrodynamicacoustic resonator sensor for viscosity measurement[END_REF][START_REF] Schilowitz | Measuring hydrocarbon viscosity with oscillating microcantilevers[END_REF]. In high quality factor systems, and when the resonant peak is sufficiently high to contain the -3dB bandwidth, the -3dB measurement method is often used to calculate the quality factor. However, when the magnitude of the resonant peak is less than 3dB, this method cannot be applied.

Other methods for monitoring the shift in resonant frequency consist of making measurements at a fixed frequency instead of using a frequency sweep. By fixing the measurement frequency at the value corresponding to the maximum slope of the gain [START_REF] Albrecht | Frequency modulation detection using high-Q cantilevers for enhanced force microscope sensitivity[END_REF], impedance [START_REF] Xu | A novel method for monitoring mass-change response of piezoelectric-excited millimeter-sized cantilever (PEMC) sensors[END_REF] or phase spectra, changes in the resonant frequency can be detected simply by monitoring variations of response magnitude.

In order to reduce the impact of the noise, polynomial approximations for the gain or phase spectrum can be used. These polynomial approximations can also be used to analytically extract various response characteristics such as natural frequency, resonant frequency and quality factor.

In this paper, the different characterization methods using gain and phase spectra have been tested on measurement data obtained with a USMC device with integrated actuation and read-out systems. To generate measurement data, the sensor has been subjected to various concentrations of hydrogen in nitrogen. The study is focused on the performance of the different characterization methods in terms of signal-to-noise ratio (SNR) and the limit of detection (LOD) corresponding to a signal-to-noise ratio of 3. Following a description of the different methods to measure small shifts in resonant characteristics in section 2, the experiments using an uncoated silicon microcantilever-based sensor (USMC) for hydrogen detection are presented in section 3. Finally, a comparison of the results of the different methods applied to these measurements are presented and discussed in section 4.

CHARACTERIZATION METHODS

In this section, two conventional methods (sections 2.1 and 2.2) and four novel characterization methods (sections 2.3-2.6) are presented.

Variation of resonant frequency: Δf r

The measurement of the resonant frequency shift, f r , can be simply performed by looking for the resonant peak in the gain spectrum, then associating it with the corresponding frequency as shown in Fig. 1.a. However, for small changes of the resonant frequency (USMC case) this technique is not very efficient because of the noise on the gain spectrum inducing significant measurement uncertainty. Nevertheless, eliminating the noise on the gain using appropriate fitting methods (section 2.5) can considerably improve the efficiency of this characterization method.

Variation of quality factor: ΔQ

The conventional measurement method for extracting the quality factor from experimental measurements is the -3dB bandwidth method, as shown in Fig. 1.b. This technique can only be applied to high quality factor systems and it consists of calculating the ratio between the resonant frequency and the -3dB bandwidth (Δf -3 dB ), measured on the gain spectrum:

3dB f r Q f    (1)
Just as was the case for the Δf r method of the previous section, measuring changes in the quality factor by the -3dB bandwidth method is not effective in the case of small resonant frequency changes and small quality factor changes, because this method requires not only the measurement of Δf r , but requires also the measurement of Δ(Δf -3 dB ), i.e., the change in the bandwidth.

Variation of gain at fixed frequency: ΔG FF

After setting the measurement (actuation) frequency, f mes , on the point approximately representing the highest slope in the gain spectrum, the characterization of the resonant frequency variation is possible, thanks to the gain variation obtained at f mes as shown in Fig. 1.c.

Moreover, to increase the measurement sensitivity and eliminate the noise on the gain, a linear fit can be applied to the portion of the gain spectrum that includes the maximum slope (inflection point). In practice, it is not efficient to use a general equation to determine the inflection point, because, the shape of the spectrum depends strongly on the coupling between the actuation and sensing circuits (crosstalk). However, a visual inspection of the gain spectrum is sufficient to determine approximately the inflection point, which allows determining the appropriate measurement frequency, f mes , thus improving the efficiency of this characterization method.

Another approach can also be used to eliminate the noise present on the gain spectrum around the resonant peak. In fact, around the resonant frequency, f r , the gain spectrum can be approximated by a parabolic curve by using the least-squares method. This approach allows not only the optimization of the ΔG FF and Δf r methods for small-span measurements (span ≤ 4Hz in our case), but also the determination of other characteristic parameters as presented in section 2.5.

Variation of phase at fixed frequency: ΔP FF

The linear appearance and high slope around the resonant frequency shown in the phase spectrum (Fig. 1.d) is a major asset for the measurement of small resonant frequency variation, as was seen in the ΔG FF method presented previously. The ΔP FF method is based on the same principle as the ΔG FF method; however, in this case the value of f mes is fixed at the inflection point of the phase spectrum (corresponding to a phase of -π/2) as shown in Fig. 1.d.

The fit of the phase spectrum around the resonant frequency, f r , by a straight line, using least-squares method, allows one to not only eliminate noise on the phase and, thus, optimize the performance of this method, but also to deduce other characteristic parameters as discussed in the section 2.6.

Parabolic approximation of the gain spectrum near the resonance peak

As mentioned in the Δf r method, to reduce the impact of noise-induced measurement uncertainty on the gain spectrum, a parabolic approximation can be performed around the resonance peak. Using the least-squares method, the a G , b G and c G coefficients in the following fitting equation can be calculated:

2 GG G H ab fc f   ( 2 )
The resonant frequency, f r , is the maximum of equation ( 2

); thus, 2 G r G b f c   (3)
The gain transfer function of the microcantilever operating in the linear bending domain is similar to that of a second-order system and is given by

0 2 2 2 2 0 0 12 H H f f f f            ( 4 )
where H 0 is the static gain, f 0 is the natural frequency and ξ the damping ratio.

To determine an analytical parabolic expression of the gain around the resonance peak in terms of system parameters f 0 and ξ, a second-order Taylor series expansion is performed on expression (4), yielding

H f f H f f                   ( 5 )
Matching coefficients on equations ( 2) and ( 5), we obtain

() 1 3 2 () 1 3 2 1 3 2 () 1 3 GG G G G G ac a b a H b a f c b                                           (6) 22
Equation 6.a may be rewritten to give an explicit expression for the quality factor in terms of the fitting coefficients:

 

1 31 1 3 1 2 24 1 A A Q A     (7)
with

GG G ac A b  2 .
Remarks:

• The parabolic approximation of the gain is only valid on the part of the gain spectrum very close to the resonance (frequency span< Δf -3dB ).

• The second root,  

1 31 1 3 1 2 24 1 A A A    
, of equation (6a) gives a false Q-factor value, so only the root given by equation ( 7) is considered.

.

Linear approximation of the phase spectrum near the resonance peak

In the linear bending regime the phase transfer function of the microcantilever can be written as

 Arctan f f f f f           0 2 2 0 2 1 (8)
A linear fit of the phase spectrum around f 0 using the least-squares method allows for the numerical determination of the coefficients a P and b P in the fitting equation below:

  PP f =a +b f  (9)
Furthermore, an analytical expression of the phase in the vicinity of the eigenfrequency f 0 may be used to determine f 0 and ξ of the microcantilever. To obtain this expression, a first-order Taylor series expansion is performed on the phase transfer function around f 0 (i.e., where the phase is to -π/2). The result of this calculation yields

0 1 () 1 2 f f f          (10)
from which one may obtain the following:

                     (11)

Remarks:

• The linear approximation of the phase is only valid in the part of the phase spectrum that is very close to resonance (span< Δf - 3dB ).

• Once f 0 and ξ are calculated, it is easy to deduce the resonant frequency:

EXPERIMENTS

In order to apply the different characterization methods presented in the section 2, measurements of hydrogen in nitrogen have been conducted using: (1) a USMC with integrated electromagnetic actuation and piezoresistive sensing (Fig. 2), and (2) a gas line for mixing gas with controlled flow and concentrations (Fig. 3).

Microcantilever design

The microcantilever has been fabricated by ESIEE group (Paris, France, www.esiee.fr) using classical silicon technologies [START_REF] Fadel | Chemical sensing: millimeter size resonant microcantilever performance[END_REF].

To generate vibrations, an AC current is passed through the conductive wire placed at the cantilever periphery. In the presence of a magnetic field collinear to the longitudinal axis of the beam, an AC Lorenz force is created at the microcantilever free-end and induces out-of-plane vibrations. In order to detect these vibrations, semiconductor strain gauges (boron-doped piezoresistors) are fabricated during the process. They are arranged in a half Wheatstone bridge configuration: a first gauge is located where the strains are maximum (at the clamped-end of the beam) and the other one is on the rigid substrate (Fig. 2). The USMC characteristics in air at 23C° are: length=5mm, width=1mm, thickness=170µm, f r ≈ 9662Hz and Q ≈ 1350.

Experimental setup

The microcantilever is placed in a hermetic gas chamber (total volume: 500 µl) under a controlled gas flow. The gas line (Fig. 3) contains mass-flowmeters allowing control of flow (100-1000 ml/min) and concentration of each gas. To perform chemical detection, a gas stream containing a H 2 -N 2 mixture flows into the gas chamber in which the USMC has been placed. The tested succession of the different concentrations of H 2 in N 2 is reported in different detection curves. In order to study the behavior of the USMC in the presence of different concentrations of H 2 in N 2 , a spectrum analyzer (HP4194A) is used to acquire gain and phase spectra with the following configuration: INTEG.TIME=Short, AVRG=8bits. The feature of this device is that it has only 400 measurement points. Therefore, for small f r shifts (≤1Hz), small spans must be applied to increase the measurement sensitivity. For example, to detect f r variations of at least 20mHz, a measurement span of no more than 8Hz (8Hz/400 points) can be used. In our case a span of 4Hz is used for the detection of different H 2 in N 2 concentrations. Each H 2 in N 2 concentration lasts 3 minutes with a constant flow of 100 ml/min.

After following the experimental protocol detailed previously, the gain and phase spectra obtained for the concentrations of H 2 in N 2 are presented in Fig. 4. The gain and phase translations to the right observed in Fig. 4.a and Fig. 4.b, respectively, are consistent with the decrease in the gas mass density when the H 2 concentration increases. This type of resonant frequency shift in the USMC due to density changes in the surrounding gas has been discussed in an earlier work [START_REF] Tétin | Modeling and performance of uncoated microcantilever-based chemical sensors[END_REF].

RESULTS AND DISCUSSION

In this section, the different characterization methods presented in this paper are tested. In order to compare performances of these methods in terms of signal-to-noise ratio, SNR, the relative variation of f r , f 0 , Q, G FF and P FF are presented.

The results of Fig. 5 demonstrate the validity of polynomial approximations of gain and phase spectra around resonance for a frequency span of 4Hz. These approximations are used to fit spectra, enabling one to increase the SNR (section 4.1) and to deduce f r , f 0 and Q (section 4.2) using equations ( 3), ( 7) and [START_REF] Abdallah | Measurement error estimation and quality factor improvement of an electrodynamicacoustic resonator sensor for viscosity measurement[END_REF]. The measurement frequencies (f mes ), which are different for the G FF method and P FF method, are also indicated in the gain (Fig. 5.a) and phase (Fig. 5.b) spectra; they are respectively placed at the maximum slope of the gain (f mes =9564Hz) for the G FF method and at the natural frequency (f mes =f 0 =9562Hz) for the P FF method.

Performance of the Δf r , ΔG FF and ΔP FF methods

The detection results of the Δf r , ΔG FF and ΔP FF characterization methods, obtained for concentrations of H 2 in N 2 ranging from 2% to 0.2% are presented in Fig. 6.

As expected, monitoring f r by the conventional method without fitting the gain spectrum is not efficient because of the noise on the gain (Fig. 6.a). However, fitting the gain with a parabolic approximation and using relation (3) to monitor f r drastically improves the SNR (blue line in Fig. 6.a). The ΔG FF (Fig. 6.b) and ΔP FF (Fig. 6.c) methods show good sensitivity; they are able to detect a concentration of 0.2% of H 2 in N 2 even without fitting. Moreover, the polynomial fits have greatly improved the SNR of both methods (blue lines). Table 1 presents the performance of each method, calculated at 0.6% of H 2 in N 2 . The noise is estimated by calculating the standard deviation (σ x ).

From Table 1, it can be seen that the ΔP FF method has the best SNR (S/N=100), the best relative variation (Δx/x= 0.86%) and the best limit of detection (hydrogen concentration corresponding to a SNR equal to 3, = 0.018% of H 2 in N 2 ).

The performance of the ΔG FF method can be optimized by fixing f mes at the point of maximum slope of the gain spectrum. Thus, another experiment is carried in the same conditions (the same concentrations of H in N 2 and the performances are calculated at 0.6% of H 2 in N 2 ), but using a frequency span of 10Hz in order to identify the maximum slopes of both the gain and phase spectra. In these conditions, the parabolic fitting cannot be applied to the whole spectrum (Fig. 7, red line with 'x' marks); it can only be applied closely around resonance (Fig. 7, green line with triangle marks). However, for large span measurements only the tangent to the curve at the inflection point (Fig. 7, black line with circle marks) is used to improve the SNR. The performance of each method is reported in Table 2. The comparison between the ΔG FF and ΔP FF methods in their optimal conditions shows that the ΔP FF method yields a SNR of 32 in comparison with 25 for the ΔG FF method, thereby confirming that ΔP FF is the best characterization method that we have examined to date. This is due to the fact that the phase spectrum presents a higher slope and a lower noise than the gain spectrum.

Remark:

Because of the large frequency span (10Hz) chosen in this measurement, the signal-to-noise ratios (Table 2) are relatively small compared to the values found in Table 1. This is due to the configuration of the gain/phase analyzer (HP5194A) that is maintained at the same level (INTEG.TIME=Short, AVRG=8bits) as for the measurements performed with the short span (4Hz). This results in a higher noise level associated with the larger frequency span.

Performance of polynomial approximation methods

In the previous part, polynomial approximations using the least-squares method were applied to fit the gain and phase spectra near resonance in order to decrease the impact of the noise. Using the polynomial coefficients of the fitting methods, equations ( 3), ( 7), (6.c) and ( 11) can be used to deduce f r , Q and f 0 . In this experiment, we compare only the performance of each fitting method in terms of the determination of f 0 and Q. The results of this processing and the corresponding performance values are presented in Fig. 8 and Table 3, respectively, for a parabolic approximation of the gain and a linear approximation of the phase.

It can be observed that variations of Q and f 0 in the presence of different concentrations of H 2 in N 2 are detected using the different polynomial methods. All performance metrics are reported in Table 3, where it can be seen that Q measurements (green lines with stars in Fig. 8) present a poor SNR (2.5 and 1.8 for gain and phase fits, respectively) and cannot detect H 2 concentrations in N 2 below 1%. Also, the actual Q value is around 1350, while the measured Q values based on the polynomial approximations are approximately 900-950. This estimation error is mostly caused by the span measurement (4 Hz) which is relatively large in comparison with the range of validity of these polynomial methods. However, the exact Q measurement is not necessary; only the Q variation is important in the present application, i.e., to sufficiently characterize USMC response to a change in H 2 concentration in N 2 . On the other hand, f 0 measurements utilizing both the gain and phase polynomial methods (blue lines in Fig. 8) yield a good SNR (46 and 82 for gain and phase spectra, respectively) and a good f 0 estimation (at 0% of H 2 in N 2 , temperature ≈ 23C° and f 0 ≈ 9562 Hz). The comparison between the f 0 determinations shows that the linear approximation of the phase near resonance results in the best SNR and consequently the best LOD. The analysis of the performance tables (Tables 123) shows that phase linearization around resonance optimizes the measurement sensitivity of the resonant frequency variation using the ΔP FF method (SNR=100). Therefore, it would be interesting to assess the corresponding limit of detection (LOD).

In theory, the LOD of the ΔP FF method is approximately 0.02% of H 2 in N 2 . To verify this value, another experiment is carried out using a concentration of 0.025% of H 2 in N 2 .Because of the accuracy limit of the flowmeters, the gas flow is increased in this experiment to 200ml/min in order to achieve this low concentration of H 2 in N 2 . The measurement result using the ΔP FF method is presented in Fig. 9, where it can be seen that a concentration of 200 ppm (0.025%) has been successfully detected.

CONCLUSION

In summary, the phase linearization using the least-squares method and the linear model around resonance to measure the phase variation at a fixed frequency (Table 1) or the natural frequency variation (Table 3) present the best SNRs among all characterization methods studied in this work. This is due to the fact that the phase spectrum results in lower noise and a higher compared to the gain spectrum. Using the FF method, 0.025% of H 2 in N 2 has been detected using an USMC with the characteristics mentioned in section 3.1.

Unlike the conventional resonant frequency variation method (Δf r ), the ΔP FF method allows accurate measurement of small resonant frequency variation of the USMC, thanks to its high phase slope around resonance (high Q). In addition, the linearized phase model enables one to extract an accurate value of natural frequency variation (Δf 0 ) and to monitor accurately both small and large (if the frequency span is large enough) resonant frequency variations (but with less sensitivity than the ΔP FF method for small resonant frequency variation).

Another advantage of the ΔP FF method is the simplicity involved in making an integrated measurement with only one circuit that performs the phase comparison between the actuation signal at a fixed frequency (f ≈ f r ) and the read-out signal. The major drawback of this method is that it can only be used for small resonant frequency variation (due to the smallness of phase linearity domain).

engineering at the University of Bordeaux and her research interests are in the areas of microcantilever-based sensors for chemical detection, rheological measurements, material characterization and energy harvesting.

Fig. 1 :

 1 Fig. 1 : Principle of the different characterization methods: (a) Δf r : measurement of the resonant frequency variation corresponding to the maximum of the gain. (b) ΔQ: measurement of the quality factor variation using the -3dB bandwidth method, (c) ΔG FF : measurement of the gain variation at a fixed frequency, f mes , corresponding approximately to the maximum slope of the gain, (d) ΔP FF : measurement of the phase variation at a fixed frequency, f mes , corresponding approximately to the maximum slope of the phase.

Fig. 2 :

 2 Fig. 2 : Uncoated silicon microcantilever-based sensor (USMC) design: a conductive wire is placed at the free-end of the cantilever for the electromagnetic actuation by Lorentz force; piezoresistors are placed on the clamped-end of the cantilever and on the substrat for cantilever vibration measurement.

Fig. 3 :

 3 Fig. 3 : Experimental setup. Gas line constituted of hydrogen (4% H 2 -N 2 ) and nitrogen (100% N 2 ) gas bottles and mass-flowmeters controlled by a computer. The uncoated silicon microcantilever-based sensor is put in a hermetic gas chamber connected to the H 2 -N 2 mixture gas flow (usually 100ml/min but can be increased up to 1000ml/min).

Fig. 4 :

 4 Fig. 4 : Evolution of spectra of the USMC (5mm x 1mm x 170µm) near the resonance as H 2 in N 2 concentration is changed from 0% to 2% with a gas flow of 100ml/min. (a) Gain spectrum. (b) Phase spectrum. Small modifications of both the gain and phase spectra are observed due to the increase of the resonant frequency with the hydrogen concentration.

Fig. 5 :

 5 Fig. 5 : Agreement between least-squares polynomial approximations and raw spectra. (a) Gain spectrum. (b) Phase spectrum. The measurement frequencies are shown for both the gain measurement (highest frequency of the measurements which corresponds to the maximum slope of the gain) and the phase measurement (frequency corresponding to a phase equal to -90° which corresponds to the maximum slope of the phase).

Fig. 6 :Fig. 7 :

 67 Fig. 6 : Detection curves of the USMC (5mm x 1mm x 170µm) as H 2 in N 2 concentration is changed from 2% to 0.2% with a gas flow of 100ml/min. The duration of each concentration step is 3min and between each concentration step pure nitrogen is injected in the gas chamber during 3min. The same measurements are used to test the three different methods: (a) Δf r method. (b) ΔG FF method. (c) ΔP FF method.

Fig. 8 :

 8 Fig. 8 : Detection curves of the USMC (5mm x 1mm x 170µm) as H 2 in N 2 concentration is changed from 2% to 0.2% with a gas flow of 100ml/min. The duration of each concentration step is 3min and between each concentration step pure nitrogen is injected in the gas chamber during 3min. It is the same data measurements as the one of Fig 6 and the methods presented in sections 2.5 and 2.6 are tested. (a) The parabolic approximation of the gain spectrum near the resonance peak is used and both the eigenfrequency f 0 and the quality factor Q are plotted using Eqs. 6c and 7; (b) the linear approximation of the phase spectrum near the resonance peak is used and both the aigenfrequency f 0 and the quality factor Q are plotted using Eqs. 11a and 11b.

Fig. 9 :

 9 Fig. 9 : Detection curve of the USMC (5mm x 1mm x 170µm) using the ΔP FF method as H 2 in N 2 concentration is changed from 0% to 0.025% and to 0% with a gas flow of 200ml/min..

Table 1 : Performance of ΔP FF and Δf r methods ( / ∶ relative variation. : noise estimation (standard deviation of the measured relative variation). LOD: limit of detection). Frequency span =4 Hz. Characterization methods

 1

			. %	%	%	. %	LOD
	Without fitting	Δf r ΔG FF ΔP FF	-------0.72 0.88		0.82 x10 -3 82 x10 -3 54x10 -3	------8.8 16	-------0.2% 0.11%
	With fitting	Δf r ΔG FF ΔP FF	0.72 x10 -3 0.60 0.86		15 x10 -6 16 x10 -3 8.5 x10 -3	48 38 100	0.037% 0.047% 0.018%

Table 2 : Performance of ΔP FF and ΔG FF methods ( / ∶ relative variation. : noise estimation (standard deviation of the measured relative variation). LOD: limit of detection). Frequency span =10 Hz. Characterization methods

 2

			. %	%	%	. %
		ΔG FF	0.71		140 x10 -3	5.1
	Without fitting With fitting	ΔP FF ΔG FF ΔP FF	0.77 0.52 0.72		94 x10 -3 20 x10 -3 22 x10 -3	8.2 26 33

Table 3 : Performance of polynomial gain and phase approximations ( / ∶ relative variation. : noise estimation (standard deviation of the measured relative variation). LOD: limit of detection). Frequency span= 4Hz. Characterization methods

 3

	. %	%	%	. %	LOD

Acknowledgement

This work is supported by Andra. The authors are grateful for interesting discussions with Stephen Heinrich on the paper content and for his suggestions for improving the paper.


Riadh Lakhmi

Hélène Debéda

Mario Maglione

Isabelle Dufour

Claude Lucat

Hélène Debeda
email: helene.debeda@ims-bordeaux.fr

Study of screen-printed PZT cantilevers both self-actuated and self-read-out

teaching and research institutions in France or abroad, or from public or private research centers.

Index Terms-cantilever, integrated actuation, integrated read-out, PZT, sintering, in-plane vibrations, chemical detection, liquid media, gaseous media.

I. INTRODUCTION

As other devices,microcantilever-based silicon ones are commonly used for chemical sensor applications, since their relatively simple structure allows high sensitivity to the environment [START_REF] Boisen | Cantilever-like micromechanical sensors[END_REF][START_REF] Sepaniak | Microcantilever Transducers: A new Approach in Sensor Technology[END_REF][START_REF] Goeders | Microcantilevers: Sensing Chemical Interactions via Mechanical Motion[END_REF][START_REF] Länge | Surface acoustic wave biosensors: a review[END_REF][START_REF] Gronewold | Surface acoustic wave sensors in the bioanalytical field: Recent trends and challenges[END_REF][START_REF] Arshak | A review of gas sensors employed in electronic nose applications[END_REF]. In the latter case, the actuation part (electrothermal, electromagnetic, piezoelectric or electrostatic) and the integrated read-out (piezoresistive, piezoelectric…) are included in most of the microcantilever devices developed for species detection in gaseous or liquid media. The integration of those actuation and read-out parts into microcantilever devices enables to get rid of most of the external equipment to perform the detection and follow the current tendency to head towards so-called labs-on-a-chip.

Measurement can be achieved with static or various dynamic modes [START_REF] Zhu | Microcantilever Sensors in Biological and Chemical Detections[END_REF]. Assuming that piezoelectric materials can be used for both actuation and detection, one may consider that they are suitable for simplified and low-cost devices [START_REF] Yan | Design and fabrication of thick-film PZTmetallic triple beam resonators[END_REF][START_REF] Lou-Moeller | Screenprinted piezoceramic thick films for miniaturised devices[END_REF][START_REF] Zhihongwang | Fabrication of piezoelectric MEMS devices-from thin film to bulk PZT wafer[END_REF]. The electromechanical properties of mass-sensors made by screen-printing pure PZT layers on silicon microcantilevers show the influence of the porosity of PZT structures fired at 800°C [START_REF] Hong Park | Effects of the material properties on piezoelectric PZT thick film micro cantilevers as sensors and self actuators[END_REF]. Higher sintering temperatures (1200°C-1300°C) leading to higher densification of pure PZT would confer high actuating force or sensitivity to sensor/actuator devices. To obtain equivalent electromechanical properties using the screen-printing process (T firing <1100°C), addition of sintering aids is recommended for PZT densification at lower firing temperature.

A new type of PZT-based cantilevers containing 7wt% borosilicate glass-frit, fired at low temperature (850°C) has been successfully processed [START_REF] Castille | Étude de MEMS piézoélectriques libérés et microstructurés par sérigraphie : application à la détection en milieu gazeux et en milieu liquide[END_REF][START_REF] Lucat | Microsystems elements based on free-standing thick-films made with a new sacrificial layer process[END_REF]. This fabrication combines screenprinting process and SrCO 3 sacrificial layer method set-up in our laboratory. From these results, it is expected that the optimization of powder composition (nature and concentration of sintering aid) and firing conditions would improve the piezoelectric performances of screen-printed PZT cantilevers. On this way, Corker et al. [START_REF] Corker | Liquid-phase sintering of PZT ceramics[END_REF] show that with low eutectic concentration (3wt% PbO-Cu 2 O), PZT ceramics fired at 900°C exhibit good piezoelectric properties similar to thoseobtained with 1.06wt% Li 2 CO 3 -Bi 2 O 3 -CuO (LBCu) eutectic for ceramics fired at 880°C [START_REF] Wang | Piezoelectric properties, densification behavior and microstructural evolution of low temperature sintered PZT ceramics with sintering aids[END_REF]. Other studies using 2-3wt%LBCu eutectic phase for the fabrication of ceramics and screen-printed thick films on Al 2 O 3 substrates, confirm that good piezoelectric behavior is observed for samples fired at 900°C-950°C [START_REF] Seveyrat | Elaboration et caractérisation de films épais piézoélectriques sérigraphiés sur alumine, aciers inoxydables et vitrocéramiques[END_REF]. These results show the potentialities of LBCu eutectic for the fabrication of PZT-based layer.

Moreover, it is noticed that thick-films directly screen-printed on the substrate do not favor the 31 inplane mode, conversely to PZT cantilever structure [START_REF] Castille | Free-standing piezoelectric thick-films for MEMS applications[END_REF].

In this work, we propose the realization of PZT cantilevers with improved piezoelectric properties, using the association of the sacrificial layer to the standard screen-printing technology. SEM characterizations and electromechanical properties of cantilevers prepared from a mixture of a PZT powder with 3wt% LBCu or reduced concentration of borosilicate glass-frit (5wt%), are investigated for different sintering temperatures (850°C, 900°C and 950°C). Correlations between experimental results and modeling of the different vibrations modes are studied.

II. DESIGN AND FABRICATION OF SCREEN PRINTED CANTILEVERS

The simple design of PZT cantilevers used for this study is shown in figure 1. Thanks to the combination of the standard screen-printing technology with a sacrificial layer process, free-standing piezoelectric cantilevers are fabricated and released from the alumina substrate. This multi-layered component is achieved by deposition of successive layers dried or polymerized 20min at 120°C after each printing step (figure 2): (a) PZT pad, (b) sacrificial layer, (c) bottom electrode, (d) PZT cantilever, (e) top electrode. Once all the layers are deposited and dried, the samples are isostatically pressed 1min at 1kbar to improve the densification [START_REF] Gouverneur | New densification process of thick films[END_REF]. The samples are then co-fired 2h at 850°C, 900°C and 950°C with an heating rate of 20°C/min and a cooling rate of 20°C/min. Finally, the removal of the sacrificial layer is performed in a diluted acidic solution (figure 2.f) and thus allows free movement of one extremity of the two-face electroded cantilever while the other one is clamped on the substrate via the pad. The decomposition of SrCO 3 into SrO under controlled air atmosphere, taking place at a temperature around 850°C, will not have any effect on the final device properties. Yet, it will affect the removal of the sacrificial layer since, contrary to SrCO 3 ,SrO is not soluble in acidic solutions. The introduction of CO 2 in the firing atmosphere would prevent the decomposition of SrCO 3 and thus the SrO formation. This cantilever is self-actuated thanks to the simple structure PZT beam sandwiched between symmetrical electrodes, also used at the same time to perform the signal transduction.

Two piezoelectric inks based on PZT powder (Pz26 from Ferroperm) are prepared by adding different metal oxide sintering aids. Each ink contains 3wt% LBCu (25wt% Li 2 CO 3 , 40wt% Bi 2 O 3 , 35wt% CuO) or 5wt% borosilicate glass-frit (PbO,B 2 O 3 , SiO 2 ). After drying, the piezoelectric powders are mixed with an organic binder (ESL400 from Electro-Science Laboratories) and homogenized between two rotating plates to form a PZT paste with a suitable viscosity and rheology for the subsequent screen-printing process. The sacrificial layer paste is composed of SrCO 3 dispersed in an epoxy binder CV59 from Electro-Science Laboratories. Concerning the bottom and top electrodes, commercial gold paste ESL8836 from Electro-Science Laboratories is used. For the two compositions of fired PZT cantilevers, the dimensions used in this study are for the length L p =8mm, width b p =2mm and thickness h p ~105µm (figure 3). The symmetrical electrodes (thickness h Au ~7µm) have a slightly smaller area than those of PZT beam in order to prevent short-circuits.

III. CHARACTERIZATION

A. Microstructural characterization

Porosity plays an important role in piezoelectric ceramic. Indeed, a high porosity will have a detrimental effect on the piezoelectric properties of the ceramic since the pores act like small capacitors decreasing the efficiency of the electric field. Therefore, according to the firing conditions and the sintering aid addition, a control of the samples' porosities is required. The porosity has been evaluated thanks to SEM images by grey level analysis on different parts of the cantilever. Then, the estimated porosity is a mean value of the porosities observed. SEM analyses of samples containing 5wt% glass-frit and fired 15min at 850°C show a porous microstructure with grain dimensions and agglomerates from a few hundreds of nm to about 10µm, corresponding roughly to the initial PZT grain sizes and/or agglomerates (figure 4). Moreover, the sample exhibits a very high porosity estimated at 45% with macroporosities reaching 10µm diameter.

The PZT layer is composed of grains having no cohesion giving rise to poor mechanical properties and stability. These preliminary results explain the choice of sintering time of 2h at higher temperature to improve the densification of the samples.

Figure 5 shows the SEM analyses of the samples fired at 850°C, 900°C and 950°C for (a-c) 5wt% borosilicate glass-frit and (a'-c') 3wt% LBCu respectively. Concerning the 5wt% glass-frit samples, it can be noticed that rising the sintering time from fifteen minutes to two hours improves the compacity of the PZT layer up to 70%. Increasing more the sintering temperature leads to lower porosity (~25%) for samples fired at 900°C and ~20% for those fired at 950°C. Concerning the 3wt%LBCu samples fired at 850°C for two hours high porosity (~45%) is observed.

As well, sintering at 900°C and 950°C decreases the porosity of the PZT layer to estimated values ~35% and ~25% respectively.

Castaing microprobe analyses of PZT cantilevers shows low diffusion, into PZT layer, of SrO (figure 6). This strontium oxide is formed during the decarbonation of sacrificial layer during the firing process, starting from 850°C under air atmosphere.

B. Electrical characterization

Prior to piezoelectric properties studies, the PZT cantilevers are poled under dry helium atmosphere to avoid any conduction phenomenon between the electrodes leading to sample's breakdown.

Therefore, a vacuum of 0.1 mbar is achieved before the introduction of helium, chosen for its good thermal conduction properties enabling better temperature homogenization inside the poling chamber. This poling is performed by applying an electric field of 5kV.mm -1 between the electrodes for 15min at 280°C, just below the measured Curie temperature (T C #284°C). It is worth noticing that the leakage current is controlled to make sure that it doesn't exceed 20µA. Then, the temperature is lowered to room temperature before turning off the electric field.

The ferroelectric hysteresis loop is obtained at room temperature thanks to a Sawyer-Tower circuit for samples fired at 900°C for 2h (figure 7). The measured coercive field E c =2.6kV/mm remains lower than the electric field E p =5kV.mm -1 used to pole the samples. Compared to the remanent polarization of PZT ceramics (30<P r (µC.cm -2 )<50), those observed for the screen-printed cantilevers P r =2.2µC.cm - 2 are relatively low. This may be explained by the high porosity remaining in PZT thick-films, the low diffusion of SrO at the electrode/PZT interface, etc.

The vibration spectrum of PZT cantilevers is recorded with an impedancemeter HP4194A. The first resonance peaks are observed for resonant frequencies around 75kHz, 220kHz and 370kHz (figure 8).

From those peaks, corresponding to in-plane 31-longitudinal vibration modes, information on the quality factor Q and piezoelectric characteristics like d 31 can be extracted. At higher frequencies (around 10MHz), another resonance peak corresponding to out-of-plane 33thickness vibration mode is observed (figure 9).

The fabrication and poling processes have been validated on PZT-cantilevers whose measured first in-plane 31-longitudinal and first out-of-plane 33-thickness modes are respectively within the range 65-75kHz and 8-10MHz. Indeed, quite reproducible results are obtained, the differences between the samples being mostly due to thickness variations.

For the in-plane modes and the out-of-plane ones, the quality factor has been calculated from the admittance spectra with (Δf taken at -3dB) (table 1). Piezoelectric coefficients d 31 , dielectric constant K T 33 and dielectric dissipation factor (tanδ) are respectively extracted from admittance spectra and capacitance (C p )/tanδ spectra (figure 10) of the PZT cantilevers containing 5wt% borosilicate glass-frit and 3wt% LBCu composition fired at 850°C, 900°C and 950°C (table 1) [START_REF] Lou-Moeller | Screen-printed piezoceramic thickfilms forminiaturised devices[END_REF]. K T is defined as the ratio of the material's permittivity to the permittivity of free space while tanδ is defined as the ratio of power dissipated by the ceramic on the active power. In a previous study [START_REF] Castille | Étude de MEMS piézoélectriques libérés et microstructurés par sérigraphie : application à la détection en milieu gazeux et en milieu liquide[END_REF], it has been pointed out that PZT layers containing 7wt% of the same borosilicate glass-frit but fired 15min at 850°C exhibited Q values around 300 and 20 respectively for the in-plane longitudinal modes and the out-of-plane thickness mode, with a coefficientd 31 =-20pC.N -1 .

For our samples, Q values ranging from 240 to 770 for in-plane longitudinal modes and from 20 to 40 for the out-of-plane thickness mode are obtained. The lower Q values of out-of-plane thickness mode could be explained by the fact that for this type of resonance, the waves move through the sample's thickness whose relative roughness (5%) is not negligible. Moreover, this resonance is close to the electrical resonance of the sample coupling to the electrical connection of the impedance-meter which also affects the resonance peak. Results related to Q values are comparable to those obtained for samples fired 15 min at 850°C and even better in some cases (Table 1). Actually, the quality factor depends not only on sintering but also on poling conditions and remanent polarization of the poled sample. Therefore, the obtained quality factors cannot be directly linked to the sintering temperature even if slightly better quality factors are obtained for samples fired 2h at 900°C.

The dielectric constant is an important property for a piezoelectric material since it is related to the capacity of the electroded piezoelectric material to store charge under a given electric field; so as the loss tangent which quantifies its inherent dissipation of electrical energy. A high porosity results in a poor effective dielectric constant K T 33 and a reduced efficiency of the electric field which is, then, mainly applied on the porosities. For the poled samples, dielectric constant goes from 250 to 400.

Conversely, the observed porosity has a positive impact on the dielectric dissipation values which remains quite low (0.5 to 0.9%) for all the used firing cycles.

The piezoelectric constant d 31 is the ratio between the mechanical strain produced by converse piezoelectric effect and the applied electric field. In case of actuation applications, where a high electromechanical coupling is needed, the d 31 value is of high interest. Compared to samples fired 15min at 850°C, improved values 80pC/N<-d 31 <100pC/N are obtained for 850<T(°C)<950. These best mechanical and piezoelectric properties observed for samples fired at 900°C confirm the results of D.L. Corker [START_REF] Corker | Liquid-phase sintering of PZT ceramics[END_REF] and X.X. Wang [START_REF] Wang | Piezoelectric properties, densification behavior and microstructural evolution of low temperature sintered PZT ceramics with sintering aids[END_REF] on PZT ceramics presenting improved electromechanical properties at an optimal low firing temperature, related to the nature and the concentration of sintering aid. Indeed, X.X. Wang et al [START_REF] Wang | Piezoelectric properties, densification behavior and microstructural evolution of low temperature sintered PZT ceramics with sintering aids[END_REF] observed on PZT ceramics a maximum value for Kp (and thus piezoelectric coefficient d31) at 880°C with the same LBCu sintering aids, though the dielectric constant is continuously decreasing with sintering temperature. It is worth noticing that in their case the compacity is well higher than 95%. In the same way, D.L. Corker et al [START_REF] Corker | Liquid-phase sintering of PZT ceramics[END_REF] have shown optimum piezoelectric properties at given sintering aid concentration 3<wt(%)<7 at temperatures 800<T(°C)<850. In their case, the variations of tangdelta and piezoelectric properties are not linear with density. Moreover, the slight decrease of piezoelectric properties and increase of dielectric losses may result from the low diffusion of SrO at the electrode/PZT interface at increasing sintering temperature.

For samples containing borosilicate glass-frit and fired at 850°C for two hours, tested samples electrically broke down. Therefore, a higher sintering time (13h) has been tried and the influence of sintering time has been evaluated for samples with borosilicate glass-frit and fired at 850°C and 900°C. Results are shown in table 2. For both sintering temperatures, the effect of the sintering time on the piezoelectric and dielectric properties is positive since the samples fired at 850°C for 13h can be poled and exhibit good properties. Dielectric properties of the sample fired at 900°C for 13h have also been improved. The piezoelectric constant d 31 is almost unchanged.

C. Mechanical characterization

Whatever the firing temperature and the sintering aid, electrical characterizations show different resonance peaks that we attributed to in-plane longitudinal vibrations and out-of-plane thickness vibrations. Mechanical analysis performed with a vibrometerPolytec MSA 500 confirms that the assumed resonance modes are the correct ones. Indeed, the vibrometer can be used in its "out-of-plane mode" to detect the cantilever movements along the z axis or in its "in-plane mode" to detect movements in the x and the y axis.Moreover, it enables to highlight the classical bending mode whose electrical signature is not sufficient to be electrically detected in the case of our cantilevers.

Electromechanical characterizations are performed on samples fired 2h at 900°C with LBCu sintering aid.

 Out-of-plane measurements

First, the vibrometerPolytec MSA 500 is used with its out-of-plane detection mode based on the Doppler Effect analysis of a laser beam reflection onto the cantilever's surface. The analysis is performed for an actuation voltage of a few mV. At low frequencies (figure 11), peaks corresponding to the bending transverse modes are detected.

Our cantilever structure is composed of one PZT layer between two gold electrodes as seen in figure 4. The well-known equation for the transverse bending resonant frequency in this case is [START_REF] Blevins | Flow induced vibration of bluff structures[END_REF]:

f - (1)
Where n,λ (1) bend = 1.875, λ (2) bend = 4.694, λ (3) bend = 7.855, L, b, h, E, ρ are respectively the mode's order, the cantilever's length, width, thickness, Young's modulus, density. The indexes p and Au refer respectively to the PZT layer and the gold electrodes.

The first measured bending resonant frequency (f (1) bend ) is located around 600Hz. At this frequency, the noise level is high and the peak resolution is not quite good. This is why we decided to focus on the second transverse bending mode whose related resonant frequency is f (2) bend = 3.757kHz.

The frequency limit of the optical vibrometer is 2.5MHz. Therefore, the thickness mode could not be mechanically detected. For this mode the equation of the resonant frequency is [START_REF] Blevins | Flow induced vibration of bluff structures[END_REF]:

f (2)
whereλ (n) 33 =nπ .

 In-plane measurements

To analyze the in-plane vibrations of the cantilever, the vibrometerPolytec MSA 500 is used with its in-plane detection mode. In this case, the frequency spectrum is obtained by stroboscopic effect.The analysis is performed for an actuation voltage of 9V. Peaks are detected around 75kHz and 220kHz

(figures 12 and 13) corresponding respectively to the first and the second in-plane longitudinal modes observed with the impedancemeter. In order to verify that the observed modes correspond to in-plane 31-longitudinal modes, that's to say vibrations along the x axis, the displacement magnitudes on the 3 axis directions were compared.. For the in-plane mode observed at 75kHz, peaks' amplitudes of 800 nm, 50 nm and a few hundreds of pm are measured respectively for x (figure 12 a), y (figure 12 b) and z displacements. Besides, the same observation of a vibration mainly along the x axis (figure 13) has been made for the resonance mode at 220kHz. Displacements along y axis cannot be clearly determined because of the signal's noise. Thus, those measurements confirm that both peaks at 75 and 220kHz are related to in-plane 31-longitudinal modes whose resonant frequency for this multilayer geometry can be expressed by the following equation [START_REF] Blevins | Flow induced vibration of bluff structures[END_REF]:

(

where  .

 Resonant frequency agreement

PZT thickness (105µm), Young's modulus (E Au =55GPa), density (ρ Au =18500kg/m 3 ), thickness (7µm) of gold layers have been determined with vibrometer and profilometer measurements [START_REF] Lakhmi | Determination of Young's Moduli for free-standing screen-printed thick film layers used in[END_REF]. For samples fired 2 hours at 900°C with LBCu sintering aid, PZT density of 5500kg/m 3 is estimated thanks to SEM photograph. Because of PZT's brittleness, Young's modulus cannot be measured directly with a monolayer PZT beam. Assuming that PZT is isotropic, the only unknown Young's modulus parameter is varied in order to obtain the best frequency fit with equation ( 3). The in-plane 31-longitudinal mode is the most reliable one for extraction of E p ~36±2GPa (mean value obtained on 20cantilevers), lower than the PZT ceramic's Young's modulus (~70GPa). Indeed, the out-of-plane thickness and bending modes are both very sensitive to the layers'thicknesses because of the layers' roughness. Therefore, extraction of Young's modulus value is done thanks to fitting with 31-longitudinal mode equation [START_REF] Goeders | Microcantilevers: Sensing Chemical Interactions via Mechanical Motion[END_REF]. Then, errors made on the calculated out-of-plane bending and 33thickness resonant frequencies are evaluated using the extracted E p value. In the worst cases, measurements and calculated results for out-of-plane bending and 33-thickness modes revealed errors of respectively 23% and 12%, as it can be seen on table 3.

IV. CONCLUSION

In this study, a self-actuated and self read-out PZT cantilever has been realized by screen-printing technique associated to a SrCO sacrificial layer. The microstructure revealed an evolution according to the sintering aid and the firing temperature used. Porosities of 20% and 25% are obtained respectively for samples with borosilicate glass-frit fired 2h at 950°C and samples with LBCu fired 2h at 950°C. Mechanical characterizations performed on PZT cantilevers confirm the assumed vibration modes.

Furthermore, analytical equations are used to predict the resonance behavior of the samples. 31longitudinal mode is used to extract Young's modulus value of PZT and the errors on the calculated out-of-plane resonant frequencies are evaluated with this extracted value. The errors on the out-ofplane bending and thickness modes reach respectively 20% and 12%.

Moreover, the cantilevers are very attractive for species detection in gas and liquid media for many reasons, the high quality factor of the in-plane 31-longitudinal mode, the good mass sensitivity S m = -2.3Hz.µg -1 or sensitivity to surfacic mass =-37Hz.mm 2 .µg -1 , the low actuation power (P<10µW), etc.

From

  the electrical characterization, two resonance modes are identified: an in-plane longitudinal mode around 75kHz and an out-of-plane thickness vibration mode around 10MHz. The quality factors (20 to 40), quite low for the out-of-plane thickness mode extracted from electrical measurement are much better for the in-plane vibration mode (300 to 400). Though the dielectric constant (250 to 400) is lower than those of ceramics, the piezoelectric coefficient d 31 (-75 to -100pCN -1 ) approaches the values generally obtained with ceramics.

Figure 1 .Figure 2 .Figure 3 .Figure 4 .

 1234 Figure 1. Design of the PZT cantilever

Figure 6 .Figure 7 .

 67 Figure 6. Microprobe analysis of the PZT/electrode interface close to the sacrificial layer

Figure 8 .Figure 9 .Figure 10 .Figure 11 .Figure 12 .

 89101112 Figure 8. Electrical signature of the in-plane longitudinal vibration modes

Figure 13 .

 13 Figure 13.Second in-plane longitudinal mode for vibrations along x axis.

TABLES Table 1 .

 TABLES1 Electro-mechanical properties of samples fired 2h

		T firing (°C)	Quality factor (Q) First 31 mode Second 33 31 mode mode	K T 33 (at 1 kHz)	tan(δ) (at 1 kHz)	-d 31 (pC/N)
	Glass-frit	850 900 950	-320 430	-770 270	-20 20	-340 390	-0.007 0.009	-90 80
		850	240	320	20	330	0.007	100
	LBCu	900	330	370	40	340	0.006	100
		950	330	340	30	270	0.009	80

Table 2 .

 2 Firing temperature time influence on the properties of samples containing borosilicate glass-

	frit

Table 3 .

 3 Comparison between experimental and calculated values for the flexural and thickness modes for the cantilevers 1, 2, 3 (L=8mm /w=2mm /t=120µm)

		f (2) (Hz) flex exp	f (2)	flex model (Hz)	Error (%)	f 33 exp (MHz)	f 33 model (MHz)	Error (%)
	Cantilever 1	4135		4516	9.2	8.702	9.599	10.3
	Cantilever 2	4073		4425	8.6	9.525	9.298	2.4
	Cantilever 3	3757		4625	23.1	8.923	9.947	11.5