諧振器的應用隨處可見，遍布在各個不同的領域當中，例如在交通產業上，如航空產業，亦或是自動駕駛系統不斷的進步，在精密儀器中，微機電系統感測器是不可或缺的一項重要元件，然而，空氣中可能存在水氣，外在環境中並不單單只有乾燥空氣，不同的含水量可能影響到最終結果的準確度，因此，探討在不同相對濕度下之外在環境，變成了一個重要的議題。
　　對於潮濕空氣而言，乾燥空氣與水蒸氣的組成會影響平均自由徑，由於其不同的分子動態直徑 (Kinetic diameter)、分子量、與水蒸氣及乾燥空氣之莫耳分率。在過去的研究中，對於平均自由徑的計算僅考量到水蒸氣與乾燥空氣之間的分壓變化，這會使得評估品質因子 (Quality factor)、頻率位移(Frequency shift)上出現低估的情形，進而導致相對濕度之量測誤差。因此，為了解決此問題，我們再次仔細地討論平均自由徑。並且本研究將討論反努森數 (Inverse Knudsen number)、壓力流率 (Poiseuille flow rate)，最後討論品質因子、頻率位移，以及模態 (Mode)。
　　本研究會分為以下部分探討，包含是平均自由徑、反努森數 (Inverse Knudsen number)、壓力流率、品質因子 (Quality factor)以及頻率位移 (Frequency shift)等。影響平均自由徑的主要因素有：(1)分子的大小，不同的分子大小意味著不同的分子直徑，或以平均自由徑中的動態半徑 (Kinetic diameter)表示； (2)分子的重量，不同的分子重量會影響到其移動速率，較重的分子移動速度較慢，較輕的分子移動速率較快；(3)系統的組成，系統中可能存在著單一分子，或是不同種類的分子。單一分子之情形較為簡單，然而，以A、B分子混合情形為例，若要計算A分子之平均自由徑，我們必須考慮A分子撞擊A分子的情形，以及A分子撞擊B分子時之情形，混合系統之情形在過去未曾被探討過，本研究將以此作為本篇核心重點，探討後續混合氣體後，溫度、壓力，以及相對濕度是如何影響平均自由徑的變化，進而影響到其他性質如反努森數、壓力流率因子、品質因子等。壓力變因主要分為兩種情況，第一種是固定總壓力 (p_total=p_a+p_v=constant)，隨著相對濕度的增加，水蒸氣增加，空氣壓力降低以維持總壓固定；以及第二種，固定乾燥空氣之壓力 (p_a=100,1000,10000,100000 Pa)，而隨著相對濕度的增加，水蒸氣壓上升，且總壓也隨之上升。

In moist air, the composition of dry air and water vapor affects the mean free path due to differences in molecular weight, kinetic diameter, and the molar fraction of water vapor and dry air. Previous research only accounted for changes in partial pressure, which led to underestimations of the quality factor and frequency shift, resulting in errors in humidity sensing. This study aims to address this issue by thoroughly considering the composition of moist air, including the aforementioned factors. Additionally, this study discusses the inverse Knudsen number and the Poiseuille flow rate, examining the quality factor, frequency shift, and mode. The results indicate that when the total pressure is fixed, the mean free path increases with rising relative humidity. Conversely, when the dry air pressure is fixed, the mean free path decreases as relative humidity increases. Therefore, considering the mean free path in mixed gases is crucial for evaluating the properties of resonators.

[1] M. H. Hasan, "Influence of environmental conditions on the response of MEMS resonators," Mechanical Engineering and Applied, MechanicsUniversity of Nebraska, 2018.
[2] W. G. Vincenti and C. H. Kruger, Introduction to physical gas dynamics. Robert E.Krieger Publishing Co. New York, 1965.
[3] C. R. Wilke, "A viscosity equation for gas mixtures," The journal of chemical physics, vol. 18, no. 4, pp. 517-519, 1950.
[4] P. Tsilingiris, "Thermophysical and transport properties of humid air at temperature range between 0 and 100 C," Energy Conversion and Management, vol. 49, no. 5, pp. 1098-1110, 2008.
[5] M. Mehregany and S. Roy, "Introduction to MEMS," Microengineering Aerospace Systems, pp. 1-38, 1999.
[6] C. E. Akbaba and Y. Tanrıkulu, "MEMS capacitive accelerometer: A review," Artıbilim: Adana Alparslan Türkeş Bilim ve Teknoloji Üniversitesi Fen Bilimleri Dergisi, vol. 6, no. 2, pp. 41-58, 2023.
[7] I. Stachiv, Z. Machů, O. Ševeček, Y.-R. Jeng, W.-L. Li, M. Kotoul, and J. Prásěk, "Achievable accuracy of resonating nanomechanical systems for mass sensing of larger analytes in GDa range," International Journal of Mechanical Sciences, vol. 224, p. 107353, 2022.
[8] L. Wei, X. Kuai, Y. Bao, J. Wei, L. Yang, P. Song, M. Zhang, F. Yang, and X. Wang, "The recent progress of MEMS/NEMS resonators," Micromachines, vol. 12, no. 6, p. 724, 2021.
[9] A. Uranga, J. Verd, and N. Barniol, "CMOS–MEMS resonators: From devices to applications," Microelectronic Engineering, vol. 132, pp. 58-73, 2015.
[10] C. Liu, J. Zhang, Q. Li, L. Su, X. Fu, W. Jin, W. Bi, and G. Fu, "Ultrasound detection based on optical tapered-knot resonator sensor," Sensors and Actuators A: Physical, vol. 369, p. 115214, 2024.
[11] H. Woo, H.-J. Lee, and J.-G. Yook, "Non-Invasive Detection of Glucose and NaCl Solutions With Environment Correction Using a Dual IDC-Based Microwave Sensor," IEEE Sensors Journal, p. 3390544, 2024.
[12] Z. Li, M. Li, H. Jia, H. Zhang, P. Xu, and X. Li, "High-Sensitive Hydrogen Detection in Oxygen-Free Environment with MEMS Differential Thermopiles," in 2024 IEEE 37th International Conference on Micro Electro Mechanical Systems (MEMS): IEEE, pp. 859-862, 2024.
[13] Z. Wei, J. Liu, X. Zheng, Y. Sun, and R. Wei, "Influence of squeeze film damping on quality factor in tapping mode atomic force microscope," Journal of Sound and Vibration, vol. 491, p. 115720, 2021.
[14] Q. C. Le, M. T. Phan, X. T. Trinh, H. L. Truong, V. K. T. Ngo, and C. C. Nguyen, "The combined effects of temperature and relative humidity on resonant frequency and quality factor of MEMS beam resonators in atmospheric pressure and gas rarefaction," Microsystem Technologies, vol. 29, no. 9, pp. 1357-1373, 2023.
[15] I. Zine-El-Abidine and P. Yang, "A tunable mechanical resonator," Journal of Micromechanics and Microengineering, vol. 19, no. 12, p. 125004, 2009.
[16] R. A. Barton, B. Ilic, A. M. Van Der Zande, W. S. Whitney, P. L. McEuen, J. M. Parpia, and H. G. Craighead, "High, size-dependent quality factor in an array of graphene mechanical resonators," Nano letters, vol. 11, no. 3, pp. 1232-1236, 2011.
[17] M. Imboden and P. Mohanty, "Dissipation in nanoelectromechanical systems," Physics Reports, vol. 534, no. 3, pp. 89-146, 2014.
[18] R. Lifshitz and M. L. Roukes, "Thermoelastic damping in micro-and nanomechanical systems," Physical review B, vol. 61, no. 8, p. 5600, 2000.
[19] 李文榮, "微/奈米機電系統的能量耗散機制探討," 國立成功大學奈米科技暨微系統工程研究所, 2010.
[20] 梅晉熙, "橫向等向性材料對MEMS諧振器品質因子的影響," 國立成功大學材料科學及工程學系, 2019.
[21] H. Campanella, Acoustic wave and electromechanical resonators: concept to key applications. Artech House, pp. 6-10, 2010.
[22] I. Crandall, "The air-damped vibrating system: Theoretical calibration of the condenser transmitter," Physical Review, vol. 11, no. 6, p. 449, 1918.
[23] W. S. Griffin, H. H. Richardson, and S. Yamanami, "A study of fluid squeeze-film damping," Journal of Basic Engineering, 1966.
[24] J. J. Blech, "On isothermal squeeze films," Journal of Lubrication Technology, pp. 615-620, 1983.
[25] R. B. Darling, C. Hivick, and J. Xu, "Compact analytical modeling of squeeze film damping with arbitrary venting conditions using a Green's function approach," Sensors and Actuators A: Physical, vol. 70, no. 1-2, pp. 32-41, 1998.
[26] C. Zhang, G. Xu, and Q. Jiang, "Characterization of the squeeze film damping effect on the quality factor of a microbeam resonator," Journal of Micromechanics and Microengineering, vol. 14, no. 10, p. 1302, 2004.
[27] J. W. Lee, "Analysis of fluid-structure interaction for predicting resonant frequencies and quality factors of a microcantilever on a squeeze-film," Journal of mechanical science and technology, vol. 25, pp. 3005-3013, 2011.
[28] M. Bao and H. Yang, "Squeeze film air damping in MEMS," Sensors and Actuators A: Physical, vol. 136, no. 1, pp. 3-27, 2007.
[29] B. A. Bircher, R. Krenger, and T. Braun, "Influence of squeeze-film damping on higher-mode microcantilever vibrations in liquid," EPJ Techniques and Instrumentation, vol. 1, pp. 1-13, 2014.
[30] S. J. Kim, R. Dean, R. L. Jackson, and G. T. Flowers, "An investigation of the damping effects of various gas environments on a vibratory MEMS device," Tribology international, vol. 44, no. 2, pp. 125-133, 2011.
[31] T. P. Burg and S. R. Manalis, "Suspended microchannel resonators for biomolecular detection," Applied Physics Letters, vol. 83, no. 13, pp. 2698-2700, 2003.
[32] M. T. Jan, F. Ahmad, N. H. B. Hamid, M. H. B. M. Khir, M. Shoaib, and K. Ashraf, "Experimental investigation of temperature and relative humidity effects on resonance frequency and quality factor of CMOS-MEMS paddle resonator," Microelectronics Reliability, vol. 63, pp. 82-89, 2016.
[33] M. Bao, H. Yang, H. Yin, and Y. Sun, "Energy transfer model for squeeze-film air damping in low vacuum," Journal of Micromechanics and Microengineering, vol. 12, no. 3, p. 341, 2002.
[34] R. Christian, "The theory of oscillating-vane vacuum gauges," Vacuum, vol. 16, no. 4, pp. 175-178, 1966.
[35] 陳宗煥, "以薄膜擠壓阻尼效應為基礎之壓電驅動電容感測共振型壓力感測器," 國立清華大學電子工程研究所, 2017.
[36] J. C. Maxwell, "III. On stresses in rarefied gases arising from inequalities of temperature," Proceedings of the Royal Society of London, vol. 27, no. 185-189, pp. 304-308, 1878.
[37] C. C. Nguyen and W. L. Li, "Effect of gas rarefaction on the quality factors of micro-beam resonators," Microsystem Technologies, vol. 23, pp. 3185-3199, 2017.
[38] C. C. Nguyen and W. L. Li, "Effects of surface roughness and gas rarefaction on the quality factor of micro-beam resonators," Microsystem Technologies, vol. 23, pp. 3489-3504, 2017.
[39] C. C. Nguyen and W. L. Li, "Influences of temperature on the quality factors of micro-beam resonators in gas rarefaction," Sensors and Actuators A: Physical, vol. 261, pp. 151-165, 2017.
[40] H. Yagubizade, M. I. Younis, and G. Rezazadeh, "The effect of squeeze-film damping on suppressing the shock response of MEMS," in ASME International Mechanical Engineering Congress and Exposition, vol. 43857, pp. 303-308, 2009.
[41] M. I. Younis and A. H. Nayfeh, "Modeling squeeze-film damping of electrostatically actuated microplates undergoing large deflections," in International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, vol. 47438, pp. 1969-1978, 2005.
[42] H. M. Ouakad, H. M. Al-Qahtani, and M. A. Hawwa, "Influence of squeeze-film damping on the dynamic behavior of a curved micro-beam," Advances in mechanical engineering, vol. 8, no. 6, p. 1687814016650120, 2016.
[43] A. H. Nayfeh and M. I. Younis, "A new approach to the modeling and simulation of flexible microstructures under the effect of squeeze-film damping," Journal of Micromechanics and Microengineering, vol. 14, no. 2, p. 170, 2003.
[44] S. Ghaffari, E. J. Ng, C. H. Ahn, Y. Yang, S. Wang, V. A. Hong, and T. W. Kenny, "Accurate modeling of quality factor behavior of complex silicon MEMS resonators," Journal of Microelectromechanical Systems, vol. 24, no. 2, pp. 276-288, 2014.
[45] H. Hosseinzadegan, O. Pierron, and E. Hosseinian, "Accurate modeling of air shear damping of a silicon lateral rotary micro-resonator for MEMS environmental monitoring applications," Sensors and Actuators A: Physical, vol. 216, pp. 342-348, 2014.
[46] W. Liang, M. Lv, and X. Yang, "The combined effects of temperature and humidity on initial emittable formaldehyde concentration of a medium-density fiberboard," Building and Environment, vol. 98, pp. 80-88, 2016.
[47] C. C. Nguyen, V. K. T. Ngo, H. Q. Le, and W. L. Li, "Influences of relative humidity on the quality factors of MEMS cantilever resonators in gas rarefaction," Microsystem Technologies, vol. 25, pp. 2767-2782, 2019.
[48] J. Lovell-Smith and H. Pearson, "On the concept of relative humidity," metrologia, vol. 43, no. 1, p. 129, 2005.
[49] J. W. Bencloski, "Air temperature and relative humidity: A simulation," The Journal of geography, vol. 81, no. 2, p. 64, 1982.
[50] S. Wang, Z. Pan, J. Zhang, Z. Yang, Y. Wang, Y.-S. Wu, X. Li, and A. Lukyanov, "On the Klinkenberg effect of multicomponent gases," in SPE Annual Technical Conference and Exhibition?: SPE, p. D021S019R009, 2017.
[51] A. Öchsner and A. Öchsner, " Euler–Bernoulli Beam Theory," Classical Beam theories of structural mechanics, pp. 7-66, 2021.
[52] 陳詩潔, "壓電複合型懸臂樑熱彈性阻尼分析," 國立成功大學材料科學及工程學系, 2020.
[53] S. Karim and L. Rosenhead, "The second coefficient of viscosity of liquids and gases," Reviews of Modern Physics, vol. 24, no. 2, p. 108, 1952.
[54] H.-S. Tsien, "Superaerodynamics, mechanics of rarefied gases," Journal of the Aeronautical Sciences, vol. 13, no. 12, pp. 653-664, 1946.
[55] W. L. Li, "A database for interpolation of Poiseuille flow rate for arbitrary Knudsen number lubrication problems," Journal of the Chinese Institute of Engineers, vol. 26, no. 4, pp. 455-466, 2003.
[56] H. Yamaguchi, Y. Matsuda, and T. Niimi, "Tangential momentum accommodation coefficient measurements for various materials and gas species," in Journal of Physics: Conference Series, vol. 362, no. 1: IOP Publishing, p. 012035, 2012.
[57] T. Veijola, H. Kuisma, and J. Lahdenpera, "Model for gas film damping in a silicon accelerometer," in Proceedings of International Solid State Sensors and Actuators Conference (Transducers' 97), vol. 2: IEEE, pp. 1097-1100, 1997.
[58] 盧沛恆, "高熵合金諧振器系統阻尼分析," 國立成功大學材料科學及工程學系, 2023.