F Lochon

I Dufour
email: dufour@ixl.fr

S Heinrich

F Josse

D Rebière

Effect of partial sensitive coating coverage on the limit of detection of microcantilever chemical gas sensors

The effect of partially coating the microcantilever's length in chemical sensor applications is studied. By only coating near the free end of the cantilever, energy loss associated with the coating's viscoelasticity can be reduced, which, in turns, improve the vibrating beam quality factor without substantially compromising sensitivity. As a result, combinations of coating thickness and beam coverage exist that minimizes the sensor limit of detection (LOD).

I.

INTRODUCTION

Microcantilever chemical sensors are typically fabricated by depositing a sensitive coating onto a microcantilever. Such sensors can be very sensitive depending on the coating's affinity for the target species and the coating thickness. Increasing the thickness usually improves sensitivity but reduces the quality factor, thereby increasing the measurement noise. Thus, any effort to improve the sensor's limit of detection (LOD) should take into account the influence of coating thickness on both the sensitivity and the measurement noise.

This study focuses on possible benefits of partially coating the microcantilever's length. In particular, the goal is a reduction in the LOD without any significant drop in sensitivity. To account for the energy losses in the coating, the sensitive material will be modeled as viscoelastic.

Because vibrating microcantilevers are most sensitive to mass loading near the free end, more efficiency may be achieved by depositing the sensitive coating only near the free end. Also, because beam deformations are largest near the support, moving the coating away from the support will help to reduce energy losses associated with the coating's viscoelasticity, thus improving the sensor's quality factor. These ideas provide the motivation to use an analytical model to explore the effects of partial coatings on the performance of microcantilever gas sensors. The sensor characteristics of interest include the sensitivity and the noise occurring in oscillator configuration.

A. Beam model

The equation of motion of an harmonically excited beam in a fluid is [START_REF] Sader | Frequency response of cantilever beams immersed in viscous fluids with applications to the atomic force microscope[END_REF]:

2 4 2 0 4 ( , ) ( ) ( , ) ( , ) 4 L b W x EI m W x F x x   πρ ∂ ω + ω + Γ ω ω = ω   ∂   ( 1
)
where EI is the flexural rigidity, W(x,ω) is the displacement, F(x,ω) is the force per unit length, m L is the beam mass per unit length, ω is the angular frequency, ρ 0 is the fluid mass density, b is the beam width and Γ(ω) is the hydrodynamic function [START_REF] Sader | Frequency response of cantilever beams immersed in viscous fluids with applications to the atomic force microscope[END_REF].

B. Simple beam model

When considering a beam without sensitive coating, the flexural rigidity of the beam is given by

3 1 1 12 EI E bh = ( 2 )
where E 1 is the beam's Young's modulus, h 1 is the beam thickness, and m L =ρ 1 bh 1 , with ρ 1 being the beam mass density.

C. Hybrid beam model

When the elastic beam is covered by a viscoelastic sensitive coating, the flexural rigidity of the beam is [START_REF] Sampath | Analytical Modeling of Microcantilever-Based Dynamic Microsensors[END_REF] ( )

* ' ' ' 11 2 2 2 2 ( ) ( ) EI EI E I E I jE I = = + ω + ω ( 3 )
where

' ' ' 2 2 ( ) ( ) E jE ω +
ω is the complex modulus of the coating, I 1 and I 2 are the moments of inertia of the beam and coating, respectively, which are given in [START_REF] Sampath | Analytical Modeling of Microcantilever-Based Dynamic Microsensors[END_REF], and

1 1 2 2 L m bh bh = ρ + ρ
, where 2 ρ and h 2 are the mass density and thickness of the sensitive coating.

D. Partially covered beam model

To study a beam partially covered with a sensitive coating, equation ( 1) must be separated into one part describing the bare portion of the beam ( 0 x L ≤ ≤ δ ) and another part describing the coated portion ( L x L δ < ≤ ) with δ, the uncoverage factor, satisfying 0 1 ≤ δ < . Continuity conditions must be imposed

at x L = δ on ( , ) W x ω , ( , ) W x x ∂ ω ∂ , 2 2 ( , ) W x EI x ∂ ω ∂ and 3 3 ( , ) W x EI x ∂ ω ∂ .
Then, by adding the boundary conditions for a clamped-free beam, an analytical expression for ( , ) W x ω is obtained.

E. Limit of detection in oscillator configuration

By using the sensitivity of the device and the frequency noise, it is possible to calculate the limit of detection (LOD), defined by fixed at optimal thickness corresponding to full coverage.)

3 g noise C f LOD S ∆ = , (4)
0 noise f f Q ∆ ∝ ( 5
)
where 0 f is the resonant frequency and Q is the quality factor.

III. DISCUSSION By calculating the resonant frequency, the quality factor and the sensitivity for a given beam, the LOD can be determined as a function of the coating thickness and the beam coverage.

From Fig. 2 it can be seen that, for the full coverage case (δ=0), an optimum coating thickness can be found that minimizes the limit of detection. For this thickness, if only the right half of the beam (near the free end) is coated, Fig. 3 shows that significant improvements in the quality factor and LOD may be achieved, while the corresponding drop in sensitivity may be negligible. These results suggest that, in the general case, using thicker coatings near the free end may be preferable to depositing thinner coatings along the entire beam as lower LOD values may be achieved without compromising the sensitivity in a significant manner.

Figure 1 .

 1 Figure 1. Geometry of the cantilever partially covered by a sensitive coating

Figure 2 .

 2 Figure 2. Influence of coating thicknesswith the beam fully covered.

SFigure 3 .

 3 Figure 3. Influence of partial coating coverage. (Coating thickness fixed at optimal thickness corresponding to full coverage.)

ACKNOWLEDGMENTS

This work was sponsored in part by the French Agency for Environment and Energy Management (ADEME) and the 'Région Aquitaine'.


Frédéric Lochon
email: lochon@ixl.fr

Isabelle Dufour

Dominique Rebière

INFLUENCE OF LOSSES ON MICROCANTILEVER-BASED CHEMICAL SENSORS DESIGN

Keywords: Microcantilever, quality factor, aspect ratio

The current trend in increasing microcantilever resonant frequencies leads to relatively low quality factors for microcantilevers resonating in a gas medium. In order to increase the quality factor of such high-frequency microcantilevers, an approach that takes into account the different types of losses is provided to specify appropriate values of microcantilever aspect ratios (length-to-thickness, width-to-thickness). Conventional values of aspect ratios result in low quality factors, while those proposed here lead to slightly higher values of quality factor.

INTRODUCTION

In order to design a very sensitive microcantilever-based chemical sensor, it is required to use high-frequency devices. Indeed, the sensitivity of such sensors is proportional to the resonant frequency. Moreover, to achieve a minimal limit of detection, high quality factors are preferable because they (a) increase the sharpness of the resonance peak and, thus, the frequency shift measurement accuracy, and (b) result in lower short-term noise in oscillator configurations. A theoretical approach is proposed to show the importance of taking into account different losses when designing a microcantileverbased sensor. The importance of increasing the resonant frequency is first presented and the four dominant losses that occur in a viscous medium are then detailed and analytical forms of equivalent quality factors are given. The relevance of taking into account each loss is detailed and illustrated. A conclusion is then made about aspect ratio values.

ASSUMPTIONS

This study is limited to uniform parallelepiped shaped microcantilevers with a negligible sensitive coating Young's modulus. The microcantilever vibration amplitude is considered far smaller than any dimension to neglect non linear effects.

SENSOR SENSITIVITY

The resonant frequency n f of a parallelepiped shaped microcantilever with a sensitive coating deposited on its top (Figure 1) is given by [START_REF] Blevins | Formulas for natural frequency and mode shape[END_REF]:

( ) λ π ρ ρ   =   +   2 3 1 1 1 1 1 22 1 4 3 n n E h f L h h (1)
Where n λ is a parameter depending on the mode order, 1 L is the microcantilever length, The sensitivity S of a sensor exposed to a gas concentration g C is given by:

L h , 1 1 L b , 2 1 h h ).
1 2 1 2 n n g f r S fK C r ρ ρ ∆ = = + (2)
where n f ∆ is the frequency shift due to gas sorption, K the partition coefficient of the sensitive coating -gas couple and r h h = .

Thus, high resonant frequency devices will have high sensitivity. But in order to have a low limit of detection, a high sensitivity is not sufficient: the noise has to be taken into account.

ABOUT LOSSES

The quality factor, a key parameter for the accuracy of the measurement, expresses the losses influence : low-loss microcantilevers will have a high quality factor.

For most microcantilevers resonating in vacuum, the quality factor is determined by thermoelastic losses and support losses. But when used in air, these cantilevers can have a much lower quality factor because of the viscous losses due to the surrounding fluid and because an acoustic radiation may occur for wide cantilevers.

To evaluate the global quality factor, each loss is first studied separately. Then the global quality factor is calculated using:

1 2 1 1 1 1 ... global n Q Q Q Q = + + + (3)
This approach is fully valid for low loss microcantilevers and thus it can be applied to determine how to minimize losses.

VISCOUS LOSSES

When vibrating in a fluid, a microcantilever is perturbed by viscosity and mass density of the environment. A general theoretical model describing viscous losses has been established by Sader [3]. For microcantilevers with a rectangular cross section, the equivalent quality factor associated with viscous losses can be analytically expressed:

1 1 0 1 4 ( ) ( ) r n viscous i n h b Q ρ ω πρ ω + Γ = Γ (4)
Where Sader [3] has also proposed an approximate analytical expression of the hydrodynamic function for a wide range of Reynolds number.

Using equation ( 4), it can be seen that viscous losses are important for high values of 1 1 L h or for very narrow microcantilevers.

SUPPORT LOSSES

To obtain equation [START_REF] Blevins | Formulas for natural frequency and mode shape[END_REF], it has been supposed an infinitely rigid support. In fact, the vibration of the microcantilever produces a mechanical work on the support. This mechanical work generates an elastic wave into the support. The generated wave power is lost by propagation.

Hao et al. [4] have proposed an analytical approach to quantify these losses. The associated equivalent quality factor for a silicon based microcantilever is given by: ( )

3 1 support 2 1 1 3.97 n n L Q h χ λ   =     (5)
With:

sin( ) sinh( ) cos( ) cosh( )

n n n n n λ λ χ λ λ - = + (6)
It can be seen from equation ( 5) that support losses are important for low values of 1 1 L h .

ACOUSTIC LOSSES

The vibration of the microcantilever in the fluid creates an acoustic pressure which propagates into the fluid. The evaluation of this acoustic radiation in the case of a microcantilever with a rectangular cross section is not trivial. By approximating the rectangular cross section with an elliptical cross section, Blake [5] has obtained an approximate expression for the radiated power: To calculate the microcantilever velocity we can approximate the microcantilever mode shape in presence of low losses to the microcantilever mode shape without losses. It then comes:

( )

, n n n U j w x ω ω ω = (9)
With:

( )

( , ) n v a c w x A x ω = Φ (10)
Where A is the oscillation amplitude and ( ) vac x Φ the mode shape of the microcantilever given by: ( )

1 1 1 1
cos cosh cos( ) cosh( ) sinh sin sin( ) sinh( )

n n vac n n n n n n x x x L L x x L L λ λ λ λ λ λ λ λ     Φ = -               + + -       +         (11)
Using ( 7) and (8) it then comes: The radiated energy during one period is given by:

2 n radiated n P W ω πω = (14)
The equivalent quality factor associated with acoustic radiation can be calculated using [4]:

2 cantilever rad radiated W Q W π = ( 15
)
With:

( )

1 2 2 12 0 1 , 2 L cantilever n n x W w x d x µ ω ω = = ∫ (16)
Where ( )

12 1 1 1 2 2 b h h µ ρ ρ = +
is the linear mass of the microcantilever with the sensitive coating.

Using equations ( 9) to ( 16), it then comes:

THERMOELASTIC LOSSES

Thermoelastic losses are caused by the microcantilever material heating due to vibration. Lifshitz et al. [6] have expressed analytically the equivalent quality factor associated with thermoelastic losses:

1 2 1 2 1 1 ˆ1 sinh( ) sin( ) 6 1 cosh( ) cos( ) p thermo C Q E T ρ ζ ζ ζ α ζ ζ ζ = + - + (18)
With:

1 1 1 2 n p C h ω ρ ζ κ = (19)
Where

1 p C , 1
α , T , κ , are the heat capacity, the thermal expansion coefficient, the temperature, and the thermal conductivity of the microcantilever material, respectively, and n ω is the microcantilever resonant pulsation.

From equation (18) it can be seen that thermoelastic losses are maximum for

. At this maximum loss, the thermoelastic quality factor for a silicon made microcantilever at room temperature is bigger than 10 4 .

TOTAL QUALITY FACTOR

The combination of all the losses shows that there is no particular ideal shape to minimize all losses. A compromise has to be made. In fact, when combining the losses, it comes that optimal aspect ratios can be found for a given length (Figure 2). Indeed, the thickness can be adjusted to equilibrate viscous losses and support losses (Figure 3), while the width can be adjusted to equilibrate viscous losses and acoustic losses (Figure 4). Because of the approximations in the evaluation of the quality factor associated with acoustic losses, no general equations can be given to choose the ideal aspect ratios ( 1 1 L h , 1 1

L b

). However, by determining the optimal width and thickness for a given microcantilever length, it is possible to observe that using A quick comparison with conventional aspect ratios ( 1 1 50

L h > , 1 1 10 L b > ) is made in table 1.
From figure 2, it can be seen that using these aspect ratios implies that viscous losses are dominant and thus sufficient to calculate the quality factor. But in that case, the quality factor can be much lower while the resonant frequency remains low because of a high 1 1

L h ratio.

CONCLUSION

The new aspect ratios that have been proposed ( 1 1 20 L h = , 1 1 3 L b = , in air) should be considered by designers to increase the sensor performance by increasing the resonant frequency while keeping high quality factors. This work has not taken into account losses that can occur in the sensitive coating (for example viscoelastic losses when using polymers). These aspect ratios can be significantly modified when using high loss sensitive coating. But if analytical expressions can be found for these losses, a similar work may be conducted to find new aspect ratios.

These aspect ratios are no longer valid in water, but similar work can be done since the equations can be applied on any fluid as long as viscosity and mass density are known.

Figure 1 :

 1 Figure 1: Geometry of the microcantilever and its sensitive coating

  n ω is the resonant pulsation, 0 ρ is the density of the fluid, and r Γ and i Γ

  at pulsation ω .

  calculations of (17) it can be seen that acoustic radiation becomes important for wide microcantilevers and for "acoustically long microcantilevers". The term "acoustically long microcantilevers" refers to cantilevers for which the structural wavelength (L 1 /λ n ) is comparable to the acoustic wavelength (

Figure 2 :

 2 Figure 2: Total quality factor and dominant losses for a 100 µm long microcantilever as a function of the microcantilever width and thickness

Figure 3 :Figure 4 :

 34 Figure 3: Quality factors (log scale) at the optimal width (42 µm) for a 100 µm long microcantilever as a function of the thickness (log scale)

  is a simple rule to obtain an almost optimal quality factor.

Table 1 :

 1 Comparison between different aspect ratios for a 100 µm long microcantilever

	Microcantilever aspect	Quality	Resonant
		ratios	factor	frequency
	1 L h = 1	20

ACKNOWLEDGEMENTS

This work was sponsored in part by the French Agency for Environment and Energy Management (ADEME) and the "Région Aquitaine".