Microcantilevers with polymer coatings hold great promise as resonant chemical sensors. It is known that the coated cantilever sensitivity increases with coating thickness; however, the drawback of increasing the coating thickness is the increase of the frequency noise and thus the deterioration of the sensor\u27s limit of detection. In this paper, an analytical expression for the viscoelastic losses in the coating, hence the quality factor is established and is used to explain the observed increase of the frequency noise with the polymer thickness. This result is then used to demonstrate that an optimum coating thickness exists that minimise the limit of detectio

Dufour, Isabelle

Heinrich, Stephen M.

Josse, Fabien

Lochon, F.

Rebière, D.

Sampath, U.

English

epublications@Marquette

Marquette University e-Publications@Marquette Civil and Environmental Engineering Faculty Research and Publications Civil, Construction, and Environmental Engineering, Department of 10-2005 Effect of Viscoelasticity on Quality Factor of Microcantilever Chemical Sensors: Optimal Coating Thickness for Minimum Limit of Detection F. Lochon Université Bordeaux Isabelle Dufour Université de Bordeaux D. Rebière Université Bordeaux U. Sampath Marquette University Stephen M. Heinrich Marquette University, stephen.heinrich@marquette.edu See next page for additional authors Follow this and additional works at: https://epublications.marquette.edu/civengin_fac  Part of the Civil and Environmental Engineering Commons Recommended Citation Lochon, F.; Dufour, Isabelle; Rebière, D.; Sampath, U.; Heinrich, Stephen M.; and Josse, Fabien, "Effect of Viscoelasticity on Quality Factor of Microcantilever Chemical Sensors: Optimal Coating Thickness for Minimum Limit of Detection" (2005). Civil and Environmental Engineering Faculty Research and Publications. 61. https://epublications.marquette.edu/civengin_fac/61 Authors F. Lochon, Isabelle Dufour, D. Rebière, U. Sampath, Stephen M. Heinrich, and Fabien Josse This conference proceeding is available at e-Publications@Marquette: https://epublications.marquette.edu/civengin_fac/61  Marquette University e-Publications@Marquette  Civil and Environmental Engineering Faculty Research and Publications/College of Engineering  This paper is NOT THE PUBLISHED VERSION; but the author’s final, peer-reviewed manuscript. The published version may be accessed by following the link in the citation below.  SENSORS, 2005 IEEE, (October 30 – November 3, 2005). DOI. This article is © The Institute of Electrical and Electronics Engineers and permission has been granted for this version to appear in e-Publications@Marquette. The Institute of Electrical and Electronics Engineers does not grant permission for this article to be further copied/distributed or hosted elsewhere without the express permission from The Institute of Electrical and Electronics Engineers.   Effect of Viscoelasticity on Quality Factor of Microcantilever Chemical Sensors: Optimal Coating Thickness for Minimum Limit of Detection  F. Lochon IXL Laboratory, Talence Cedex, France L. Dufour IXL Laboratory, Talence Cedex, France D. Rebiere IXL Laboratory, Talence Cedex, France U. Sampath IXL Laboratory, Talence Cedex, France S.M. Heinrich Marquette University, Milwaukee, WI F. Josse Marquette University, Milwaukee, WI  Abstract: Microcantilevers with polymer coatings hold great promise as resonant chemical sensors. It is known that the coated cantilever sensitivity increases with coating thickness; however, the drawback of increasing the coating thickness is the increase of the frequency noise and thus the deterioration of the sensor's limit of detection. In this paper, an analytical expression for the viscoelastic losses in the coating, hence the quality factor is established and is used to explain the observed increase of the frequency noise with the polymer thickness. This result is then used to demonstrate that an optimum coating thickness exists that minimise the limit of detection SECTION I. Introduction Resonant microcantilever-based sensors have emerged as a new sensitive detection technique. In chemical sensing applications, the device consists of a microcantilever and a chemically-sensitive coating, which absorbs the molecule of interest (Fig. 1). The absorbed molecules can then be detected by monitoring the mechanical resonant frequency.  Figure 1. Geometry of the cantilever and its sensitive coating The choice of the coating thickness for minimum limit of detection (LOD) is known to be a difficult task because of its different effect on sensor characteristics. In fact, the sensor sensitivity increases with coating thickness whereas the frequency noise also increases. The aim of this paper is to demonstrate through rigorous analysis of a hybrid (elastic/viscoelastic) beam that there exists an optimal coating thickness leading to the minimal LOD. An analytical expression showing the effect of the viscoelasticity of the layer on the sensor quality factor, Q, is first established. Using this expression, the dependence of the LOD on the coating thickness is derived for a resonant frequency-based sensor for the cases of (1) intrinsic noise and (2) operation within an oscillator configuration. Theoretical results are then compared with preliminary measurements. SECTION II. Effect of Sensitive Layer Viscoelasticity on Sensor Quality Factor An expression for the sensitivity S of microcantilever-based chemical sensors [1], is given by (1) and is plotted in Fig. 2: 𝑆𝑆 = Δ𝑓𝑓𝐶𝐶𝑔𝑔= 𝐾𝐾ℎ2𝑓𝑓𝑟𝑟𝑟𝑟𝑟𝑟2(𝜌𝜌1ℎ1+𝜌𝜌2ℎ2) (1) where 𝑓𝑓𝑟𝑟𝑟𝑟𝑟𝑟 is the resonant frequency, Δ𝑓𝑓 is the frequency shift due to analyte sorption, K is the partition coefficient of the coating/analyte pair in the fluid environment, ℎ1 and ℎ2 are the microcantilever and sensitive coating thicknesses, and 𝜌𝜌1 and 𝜌𝜌2 are the respective mass densities. According to (1), the sensitivity may be improved by increasing the thickness, ℎ2, of the polymeric sensitive coating, provided that 𝜌𝜌2ℎ2 < 2𝜌𝜌1ℎ1 [1]. Although measurements (see section IV and [2]) have shown that increasing the sensitive coating thickness improves sensitivity, a corresponding decrease in Q is also observed. These effects adversely influences the sensor's frequency noise and, thus, its LOD. If only viscous losses [3] due to damping in the gas environment are considered, the quality factor increases with added mass. So, the observed decrease of the quality factor has to be explained by taking into account another loss phenomenon. It is hypothesized that this decrease is primarily due to internal losses in the viscoelastic sensitive coating.  Figure 2. Normalized sensitivity versus coating thickness (pib coating, silicon cantilever 4𝜇𝜇m × 50𝜇𝜇m × 300𝜇𝜇m). Normalization: 𝑆𝑆 = 1 for ℎ2 = 1𝜇𝜇m. A. Viscoelastic Losses The hybrid (elastic/viscoelastic) beam can be assumed to be replaced by an equivalent homogeneous viscoelastic beam [4] whose complex flexural rigidity, (EI)*, is given as: (𝐸𝐸𝐸𝐸)∗ = 𝐸𝐸1𝐸𝐸1 + 𝐸𝐸′ 2(𝜔𝜔)𝐸𝐸2 + 𝑗𝑗𝐸𝐸′′ 2(𝜔𝜔)𝐸𝐸2 (2) where 𝐸𝐸1 is the Young's modulus of the elastic material and 𝐸𝐸′ 2 + 𝑗𝑗𝐸𝐸′′ 2 the complex Young's modulus of the viscoelastic sensitive layer. 𝐸𝐸1 and 𝐸𝐸2 are the moments of inertia of the elastic and viscoelastic beam layers given respectively by 𝐸𝐸1 =𝑏𝑏ℎ1312+ 𝑏𝑏ℎ2 �ℎ𝑁𝑁 − ℎ2 −ℎ12�2𝐸𝐸2 =𝑏𝑏ℎ2312+ 𝑏𝑏ℎ2 �ℎ𝑁𝑁 −ℎ22�2  (3)(4) where b is the beam's width. The geometric properties (3) and (4) are with respect to the average position of the time-varying neutral axis of the hybrid beam, which is given by ℎ𝑁𝑁 =ℎ22+ ℎ1𝐸𝐸12(ℎ1+ℎ2)(ℎ1𝐸𝐸1+ℎ2𝐸𝐸′ 2)2(ℎ1𝐸𝐸1+ℎ2𝐸𝐸′ 2)2+ℎ2𝐸𝐸2′′2 (5) The equation of motion of an harmonically excited hybrid beam considering the only loss mechanism to be associated with the loss modulus of the sensitive layer, is (𝐸𝐸𝐸𝐸)∗ ∂4𝑤𝑤(𝑥𝑥,𝑡𝑡)∂𝑥𝑥4+ 𝑚𝑚𝐿𝐿∂2𝑤𝑤(𝑥𝑥,𝑡𝑡)∂𝑡𝑡2= 𝐹𝐹(𝑥𝑥)𝑒𝑒𝑗𝑗𝑗𝑗𝑡𝑡 (6) where 𝑤𝑤(𝑥𝑥, 𝑡𝑡) is the (complex and harmonically varying) transverse displacement, 𝐹𝐹(𝑥𝑥) is the arbitrary distribution of the force amplitude, mL is the mass per unit length of the beam and 𝜛𝜛 is the angular forcing frequency. An analysis of the solution of (6) leads to an expression for the resonant frequency, 𝑓𝑓𝑟𝑟𝑟𝑟𝑟𝑟, and quality factor associated with the viscoelastic losses in the sensitive layer, [4]: 𝑓𝑓𝑟𝑟𝑟𝑟𝑟𝑟 =1.87522𝜋𝜋𝐿𝐿2�𝐸𝐸1𝐼𝐼1+𝐸𝐸′ 2𝐼𝐼2𝑚𝑚𝐿𝐿𝑄𝑄𝑣𝑣𝑣𝑣𝑟𝑟𝑣𝑣𝑣𝑣𝑟𝑟𝑣𝑣 =12(1−�1− 𝐸𝐸′′ 2𝐼𝐼2𝐸𝐸1𝐼𝐼1+𝐸𝐸′ 2𝐼𝐼2) (7)(8) For a typical silicon cantilever and a polymeric sensitive coating, 𝐸𝐸′ 2 << 𝐸𝐸1 and 𝐸𝐸′′ 2 << 𝐸𝐸1. A first-order approximation to (8) yields 𝑄𝑄𝑣𝑣𝑣𝑣𝑟𝑟𝑣𝑣𝑣𝑣𝑟𝑟𝑣𝑣 ≈ℎ13ℎ2(ℎ22+3(ℎ1+ℎ2)2)𝐸𝐸1𝐸𝐸′′ 2 (9) This expression clearly shows that the quality factor due to coating losses decreases with increasing coating thickness. In the case of a small coating thickness, ℎ2 << ℎ1, the same expression as that of surface losses developed in [5] and [6] is found. An advantage of (8) is that it is valid for arbitrary values of thicknesses and moduli for the hybrid beam. B. Total Sensor Quality Factor When a microcantilever resonates in a gas or liquid medium, there are different mechanical loss mechanisms: viscoelastic and thermoelastic losses in the microcantilever, viscous and acoustic losses in the surrounding medium, losses due to radiation of elastic waves at the support, etc. Each of these loss mechanisms has an associated quality factor 𝑄𝑄𝑣𝑣  and the overall quality factor 𝑄𝑄𝑡𝑡𝑣𝑣𝑡𝑡𝑡𝑡𝑣𝑣 is obtained by 1𝑄𝑄𝑡𝑡𝑜𝑜𝑡𝑡𝑡𝑡𝑡𝑡= � 1𝑄𝑄𝑖𝑖𝑣𝑣 (10) In the case of chemical sensors, the surrounding medium is usually either a gas at atmospheric pressure or a liquid. Consequently, without the sensitive coating, the dominant losses are due to viscous damping, 𝑄𝑄𝑣𝑣𝑣𝑣𝑟𝑟𝑣𝑣𝑣𝑣𝑣𝑣𝑟𝑟, and the results of Sader [3] are then applicable. With the sensitive coating, both quality factors, 𝑄𝑄𝑣𝑣𝑣𝑣𝑟𝑟𝑣𝑣𝑣𝑣𝑣𝑣𝑟𝑟 and 𝑄𝑄𝑣𝑣𝑣𝑣𝑟𝑟𝑣𝑣𝑣𝑣𝑟𝑟𝑣𝑣, may need to be considered. In Fig. 3, the two quality factors are plotted as a function of the viscoelastic layer thickness. The total quality factor, 𝑄𝑄𝑡𝑡𝑣𝑣𝑡𝑡𝑡𝑡𝑣𝑣, is also plotted. These plots and other simulations in this paper are based on a poly(isobutylene) (PIB) coating with storage and loss moduli of 𝐸𝐸′ 2 =429𝑀𝑀𝑀𝑀𝑀𝑀 and 𝐸𝐸′′ 2 = 650𝑀𝑀𝑀𝑀𝑀𝑀 at the resonant frequency of 58kHz[7],[8]. If only viscous damping is considered, the quality factor increases with the added mass of the sensitive layer. The observed decrease of the total quality factor can be explained by the existence of viscoelastic losses in the sensitive coating.  Figure 3. Quality factors versus coating thickness (pib coating, silicon cantilever 4𝜇𝜇m × 50𝜇𝜇m × 300𝜇𝜇m). SECTION III. Dependence of Limit of Detection on Coating Thickness The choice of a coating thickness that minimizes limit of detection (LOD) is known to be a difficult task because of the different effects the thickness has on sensor characteristics. In fact, as shown in section II, the sensor sensitivity increases with coating thickness whereas the total quality factor decreases; thus, the frequency noise increases. The aim of this section is to demonstrate using (8) that there exists an optimal coating thickness that minimizes the LOD. Two cases are studied: (1) the case of intrinsic noise which is observed in direct spectrum analysis, and (2) the case of operation within an oscillator configuration. A. Intrinsic Noise Here the focus is on the intrinsic noise mechanisms since they determine the ultimate limits of the sensor's performance. When a microcantilever is in an ambient thermal environment, there is a continuous exchange of the mechanical energy accumulated in the microcantilever and the thermal energy of the environment. This exchange results in spontaneous microcantilever oscillation. Due to this energy exchange, the resonant frequency 𝑓𝑓𝑟𝑟𝑟𝑟𝑟𝑟 is subjected to frequency fluctuations Δ𝑓𝑓𝑛𝑛𝑣𝑣𝑣𝑣𝑟𝑟𝑟𝑟_𝑣𝑣𝑛𝑛𝑡𝑡𝑟𝑟𝑣𝑣𝑛𝑛𝑟𝑟 given by [9], Δ𝑓𝑓𝑛𝑛𝑣𝑣𝑣𝑣𝑟𝑟𝑟𝑟_𝑣𝑣𝑛𝑛𝑡𝑡𝑟𝑟𝑣𝑣𝑛𝑛𝑟𝑟 =1𝑧𝑧𝑚𝑚𝑡𝑡𝑚𝑚�2𝜋𝜋𝑓𝑓𝑟𝑟𝑟𝑟𝑟𝑟𝑘𝑘𝐵𝐵𝑇𝑇𝑇𝑇𝑘𝑘𝑄𝑄𝑡𝑡𝑜𝑜𝑡𝑡𝑡𝑡𝑡𝑡  (11) where 𝑘𝑘𝑇𝑇 = Boltzmann constant, T = absolute temperature, B = measurement bandwidth, k = microcantilever stiffness, and 𝑧𝑧𝑚𝑚𝑡𝑡𝑥𝑥 = microcantilever oscillation amplitude. The limit of detection is usually defined as the analyte concentration corresponding to a frequency shift equal to three times the frequency noise of the system measurement. Thus, (11) and (1) may be combined to yield the sensor's LOD when intrinsic noise is considered: 𝐿𝐿𝐿𝐿𝐷𝐷𝑛𝑛𝑣𝑣𝑣𝑣𝑟𝑟𝑟𝑟_𝑣𝑣𝑛𝑛𝑡𝑡𝑟𝑟𝑣𝑣𝑛𝑛𝑟𝑟 =3Δ𝑓𝑓𝑛𝑛𝑜𝑜𝑖𝑖𝑟𝑟𝑟𝑟_𝑖𝑖𝑛𝑛𝑡𝑡𝑟𝑟𝑖𝑖𝑛𝑛𝑟𝑟𝑆𝑆∝ 𝜌𝜌1ℎ1+𝜌𝜌2ℎ2ℎ2�𝑘𝑘𝑓𝑓𝑟𝑟𝑟𝑟𝑟𝑟𝑄𝑄𝑡𝑡𝑜𝑜𝑡𝑡𝑡𝑡𝑡𝑡 (12) In (12) only the terms depending on the coating thickness are kept in order to study the LOD dependence on the coating thickness. This expression allows one to determine the optimum sensitive coating thickness for minimum LOD. The case of a PIB coating on a silicon microcantilever 4𝜇𝜇m × 50𝜇𝜇m × 300𝜇𝜇m is presented in Fig. 4. Due to the viscoelastic effect, an optimum coating thickness exists, but if the coating thickness exceeds the optimal value, the resulting increase in LOD is not very significant.  Figure 4. Normalised limit of detection versus coating thickness in the case of intrinsic noise (PIB coating, silicon cantilever 4𝜇𝜇m × 50𝜇𝜇m × 300𝜇𝜇m). Normalization: 𝐿𝐿𝐿𝐿𝐷𝐷 = 1 for ℎ2 = 1𝜇𝜇m. B. Oscillator Configuration Usually, in order to accurately measure the resonant frequency change, the microcantilever is inserted in the feedback loop of oscillator. The output signal is then the oscillaton frequency that can be measured with a counter. For a high degree of accuracy, the oscillator must be as stable as possible. The oscillator stability may be characterized by the frequency noise, Δ𝑓𝑓𝑛𝑛𝑣𝑣𝑣𝑣𝑟𝑟𝑟𝑟_𝑣𝑣𝑟𝑟𝑣𝑣𝑣𝑣𝑣𝑣, which depends on the resonant frequency and the total quality factor as follows [10]: Δ𝑓𝑓𝑛𝑛𝑣𝑣𝑣𝑣𝑟𝑟𝑟𝑟_𝑣𝑣𝑟𝑟𝑣𝑣𝑣𝑣𝑣𝑣 ∝𝑓𝑓𝑟𝑟𝑟𝑟𝑟𝑟𝑄𝑄𝑡𝑡𝑜𝑜𝑡𝑡𝑡𝑡𝑡𝑡 (13) As in the case of intrinsic noise, the frequency fluctuation Δ𝑓𝑓𝑛𝑛𝑣𝑣𝑣𝑣𝑟𝑟𝑟𝑟_𝑣𝑣𝑟𝑟𝑣𝑣𝑣𝑣𝑣𝑣  can be used to relate the LOD to ℎ2: 𝐿𝐿𝐿𝐿𝐷𝐷𝑛𝑛𝑣𝑣𝑣𝑣𝑟𝑟𝑟𝑟_𝑣𝑣𝑟𝑟𝑣𝑣𝑣𝑣𝑣𝑣 =3Δ𝑓𝑓𝑛𝑛𝑜𝑜𝑖𝑖𝑟𝑟𝑟𝑟_𝑜𝑜𝑟𝑟𝑜𝑜𝑖𝑖𝑡𝑡𝑆𝑆∝ 𝜌𝜌1ℎ1+𝜌𝜌2ℎ2ℎ2𝑄𝑄𝑡𝑡𝑜𝑜𝑡𝑡𝑡𝑡𝑡𝑡 (14) Expression (14) may be used to determine the optimum sensitive coating thickness for minimum LOD. The case of a PIB coating on a silicon microcantilever (4𝜇𝜇m × 50𝜇𝜇m × 300𝜇𝜇m) is presented in Fig. 5. Clearly, when the viscoelastic losses are included, an optimum coating thickness exists. However, the oscillator configuration case differs significantly from the intrinsic noise case, in that an increase in coating thickness beyond the optimal value may seriously compromise the LOD. This phenomenon has been observed experimentally by Lange et al [2].  Figure 5. Normalised limit of detection versus coating thickness in the case of oscillator configuration (pib coating, silicon cantilever 4𝜇𝜇m × 50𝜇𝜇m × 300𝜇𝜇m). Normalization: 𝐿𝐿𝐿𝐿𝐷𝐷 = 1 for ℎ2 = 1𝜇𝜇m. SECTION IV. Experimental Results and Discussion In order to validate expression (8) for the quality factor associated with viscoelastic coating losses in the layer (which is the basis for determining the optimal coating thickness), some preliminary measurements have been made on a silicon microcantilever (𝐿𝐿 = 6000𝜇𝜇m,𝑏𝑏 = 200𝜇𝜇m,ℎ1 = 221𝜇𝜇m) for which 𝑓𝑓𝑟𝑟𝑟𝑟𝑟𝑟 =7.3kKz and 𝑄𝑄𝑡𝑡𝑣𝑣𝑡𝑡𝑡𝑡𝑣𝑣 = 1686. A PIB coating was sprayed onto the microcantilever and the quality factor was measured with a gain/phase analyzer (HP 4194A) for different coating thicknesses. The measurements are presented in Fig. 6. The modeling results with and without the viscoelastic losses (at 7.3 kHz, 𝐸𝐸′ 2 =11.4𝑀𝑀𝑀𝑀𝑀𝑀 and 𝐸𝐸′′ 2 = 19.6𝑀𝑀𝑀𝑀𝑀𝑀 [7], [8]) are also shown. Regarding Fig. 6, it clearly appears that, even if the viscoelastic quality factor (8) is very large (more than 27,500 here) compared to the total quality factor (less than 1686), the viscoelastic loss mechanism is the primary factor responsible for the observed decrease in the (total) quality factor. This being the case, one may use (12) or (14) and the relationship between quality factor and coating thickness (using (8) or measurements) to estimate the optimal coating thickness in order to achieve the minimum limit of detection.  Figure 6. Total quality factor versus coating thickness (pib coating, silicon cantilever 221𝜇𝜇m × 200𝜇𝜇m × 6mm). measurements and modeling (with and without viscoelastic losses) ACKNOWLEDGMENT This work was sponsored in part by the French Agency for Environment and Energy Management (ADEME) and the 'Région Aquitaine'. REFERENCES 1.F. Lochon, I. Dufour, D. Rebière, "An alternative solution to improve sensitivity of resonant microcantilever chemical sensors: comparison between using high-order modes and reducing dimensions", Sensors and Actuators B, vol 108, no1-2, pp. 979-985, 2005. 2.D. Lange, C. Hagleitner, A. Hierlemann, O. Brand and H. Baltes, "Complementary metal oxide semiconductor cantilever arrays on a single chip: mass-sensitive detection of volatile organic compounds", Analytical Chemistry, vol. 74, pp. 3084-3095, 2002. 3.J.E. Sader, "Frequency response of cantilever beams immersed in viscous fluids with applications to the atomic force microscope", Journal of Applied Physics, 84, pp.64-76, 1998. 4.U. Sampath, "Analytical Modeling of Microcantilever-Based Dynamic Microsensors," M.S. Thesis, Marquette University, Milwaukee, WI, 2005. 5.K.Y. Yasumura, T.D. Stowe, E.M. Chow, T. Pfafman, T.W. Kenny, B.C. Stipe and D. Rugar, "Quality factors in micron- and submicronthick cantilevers", Journal of microelectromechanical systems, 9, pp. 117-125, 2000. 6.P. Lu, F. Shen, S.J. O'Shea, K.H. Lee and T.Y. Ng, "Analysis of surface effects on mechanical properties of microcantilevers", Materials Physics and Mechanics, vol. 4, pp. 51-55, 2001. 7.Ferry, J.D.: Viscoelastic properties of polymers, John Wiley and Sons, 1961. 8.Polymer data handbook, Oxford University Press, 1999. 9.N.V. Lavrik, M.J. Sepaniak and P.G. Datskos, "Cantilever transducers as a platform for chemical and biological sensors", Review of Scientific Instruments, 75, pp. 2229-2253, 2004. 10.L. Fadel, I. Dufour, F. Lochon and O. Français, Signal-to-noise ratio of resonant microcantilever type chemical sensors as a function of resonant frequency and quality factor, Sensors and Actuators B, 99, pp.73-77, 2004  

Effect of Viscoelasticity on Quality Factor of Microcantilever Chemical Sensors: Optimal Coating Thickness for Minimum Limit of Detection

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Effect of Viscoelasticity on Quality Factor of Microcantilever Chemical Sensors: Optimal Coating Thickness for Minimum Limit of Detection

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