Study of the size-dependant vibration behavior of an AFM microcantilever with a sidewall probe
محورهای موضوعی : فصلنامه شبیه سازی و تحلیل تکنولوژی های نوین در مهندسی مکانیک
محمد عباسی
1
(
مربی، دانشکده مکانیک، دانشگاه آزاد اسلامی واحد شاهرود
)
اردشیر کرمی محمدی
2
(
استادیار، دانشکده مکانیک، دانشگاه صنعتی شاهرود
)
کلید واژه:
چکیده مقاله :
In this paper, the resonant frequency and sensitivity of an atomic force microscope (AFM) with an assembled cantilever probe (ACP) are analyzed utilizing the modified couple stress theory. The proposed ACP comprises a horizontal microcantilever, an extension and a tip located at the free end of the extension, which make AFM capable of scanning the sample sidewall. First, the governing differential equation and boundary conditions for dynamic analysis are obtained by a combination of the basic equations of the modified couple stress theory and Hamilton principle. Then, a closed form expression for the resonant frequency are derived, and using this expression the sensitivity are also investigated. The results of the proposed model are compared with those of the classic beam theory. The comparison shows that the difference between the results predicted by these two theories becomes significant when the horizontal cantilever thickness comes approximately close to the material length scale parameter, in which for some values of contact stiffness the difference reaches its maximum. It can also be inferred that a decrease in the microcantilever thickness could have a knock on effect on the shifts of first frequency and first sensitivity caused by an increase in the extension length.
در این مقاله، فرکانس تشدید و حساسیت ارتعاشات یک نوع تیر مونتاژ شده میکروسکوپ نیرو اتمی با استفاده از روش تنش- کوپل اصلاح شده مورد بررسی قرار گرفته است. تیر مذکور شامل یک تیر یک سردرگیر افقی، یک رابط عمودی و یک نوک در انتهای سر آزاد رابط بوده که ساختار آن امکان روبش جدارههای نمونه نانو را برای میکروسکوپ نیرو اتمی فراهم میسازد. ابتدا با استفاده از تئوری تنش-کوپل اصلاح شده و بهرهگیری از اصل هامیلتون، معادله حرکت و شرایط مرزی، برای میکروتیر مذکور بدست آمده است. سپس رابطهای برای فرکانس میکروتیر مورد نظر استخراج شده که به کمک آن حساسیت ارتعاشات نیز مورد تحلیل قرار گرفته است. نتایج حاصل از تئوری تنش-کوپل اصلاح شده با نتایج بدست آمده از تئوری تیر کلاسیک مقایسه شده است. نتایج نشان میدهد که با نزدیک شدن ضخامت تیر به پارامتر طول مقیاس، اختلاف بین دو تئوری مذکور افزایش یافته و برای برخی از مقادیر سختی تماسی به حداکثر مقدار خود میرسد. همچنین معلوم شد که در مد اول ارتعاشات، کاهش ضخامت تیر و نزدیکتر شدن آن به پارامتر طول مقیاس، بر تغییرات فرکانس و تغییرات حساسیت که خود ناشی از تغییرات طول رابط نسبت به طول تیر میباشند، اثرگذار است
[1] Garcia R., Perez R., Dynamic atomic force microscopy methods, Surface Science Report, Vol. 47, 2002, pp. 197–301.
[2] Holmberg K., Matthews A., Coatings Tribology: Properties, Techniques and Applications in Surface Engineering, Second Ed., New York, Elsevier, 1994.
[3] Mahdavi M.H., Farshidianfar A., Tahani M., Mahdavi S., Dalir H., A more comprehensive modeling of atomic force microscope cantilever, Ultra microscopy, Vol. 109, 2008, pp. 54–60.
[4] Turner J.A., Hirsekorn S., Rabe U., Arnold W., High-frequency response of atomic-force microscope cantilevers, Journal of Applied Physics, Vol. 82(3), 1997, pp. 966-979.
[5] Wu T.S., Chang W.J., Hsu J.C., Effect of tip length and normal and lateral contact stiffness on the flexural vibration responses of atomic force microscope cantilevers, Micro electron Engineering, Vol. 71, 2004, pp. 15–20.
[6] Chang W.J., Fang T.H., Chou H.M., Effect of interactive damping on sensitivity of vibration modes of rectangular AFM cantilevers, Physics Letters A, Vol. 312, 2003, pp. 158–165.
[7] Shen K., Hurley D.C., Turner J.A., Dynamic behaviour of dagger-shaped cantilevers for atomic force microscopy, Nanotechnology, Vol. 15, 2004, pp. 1582-1589.
[8] Abbasi M., Mohammadi A.K., A new model for investigating the flexural vibration of an atomic force microscope cantilever, Ultramicroscopy, Vol. 110, 2010, pp. 1374–1379.
[9] Lee H.L., Chang W.J., Dynamic response of a cracked atomic force microscope cantilever used for nanomachining, Nanoscale Research Letters, Vol.7, 2012, pp. 131.
[10] Dai G., Wolff H., Pohlenz F., Danzebrink U.H., Wilkening G., Atomic force probe for sidewall scanning of nano- and microstructures, Applied Physics Letters, Vol. 88, 2006; pp. 171908.
[11] Chang W.J., Lee H.L., Chen T.Y.F., Study of the sensitivity of the first four flexural modes of an AFM cantilever with a sidewall probe, Ultra microscopy, Vol. 108, 2008, pp. 619-624
[12] Kahrobaiyan M.H., Ahmadian M.T., Haghighi P., Haghighi A., Sensitivity and resonant frequency of an AFM with sidewall and top-surface probes for both flexural and torsional modes, International Journal ofMechanical Science, Vol. 52, 2010, pp.1357-1365.
[13] Dai G., Wolff H., Weimann T., Xu M., Pohlenz F., Danzebrink H.U., Nanoscale surface measurements at sidewalls of nano- and micro-structures, Measurement Science and technology, Vol. 18, 2007, pp. 334.
[14] Fleck N.A., Muller G.M., Ashby M.F., Hutchinson J.W., Strain gradient plasticity: theory and experiment, Acta Metallurgical et Materialia, Vol. 42, No. 2, 1994, pp. 475–487.
[15] Stolken J.S., Evans A.G., Microbend test method for measuring the plasticity length scale, Acta Materialia, Vol. 46, No. 14, 1998, pp. 5109–5115.
[16] Lam D.C.C., Yang F., Chong A.C.M., Wang J., Tong P., Experiments and theory in strain gradient elasticity, Journal of the Mechanics and Physics of Solids, Vol. 51(8), 2003 pp. 1477 1508.
[17] Mindlin R.D., Micro-structure in linear elasticity, Archive for Rational Mechanics and Analysis, Vol. 16(1), 1964, pp. 51–78.
[18] Toupin R.A., Elastic materials with couple-stresses, Archive for Rational Mechanics and Analysis, Vol. 11(1), 1962, pp. 385–414.
[19] Fleck N.A., Hutchinson J.W., Strain gradient plasticity, Advances in Applied Mechanics, Vol. 33, 1997, pp. 296–358.
[20] Yang F., Chong A.C.M., Lam D.C.C., Tong, P., Couple stress based strain gradient theory for elasticity, International Journal of Solids and Structures, Vol. 39, 2002, pp. 2731.
[21] Kong S., Zhou S., Nie Z., Wang K., The size-dependent natural frequency of Bernoulli–Euler micro-beams, International Journal of Engineering science, Vol. 46, 2008, pp. 427.
[22] M.H. Kahrobaiyan, M. Asghari, M. Rahaeifard, M.T. Ahmadian, Investigation of the size-dependent dynamic characteristics of atomic force microscope microcantilevers based on the modified couple stress theory, International Journal of Engineering Science, Vol. 48, 2010, pp.1985–1994.
[23] Lee H.W., Chang W.J., Sensitivity of V-shaped atomic force microscope cantilevers based on a modified couple stress theory, Microelectronic Engineering, Vol. 88, 2011, pp. 3214.
[24] Sommerhalter C.h., Glatzel T.h., Mattes T.W., Waldau A.J., Steiner M.C., Applied Surface Science, Vol. 157, 2000, pp. 32.
[25] Lin S.M., Liauh C.T., Wang W.R., Ho S.H., Analytical solutions of the first three frequency shifts of AFM non-uniform probe subjected to the Lennard–Jones force, Ultra microscopy, Vol. 106, 2006, pp. 508–515.
