Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-27T14:09:47.328Z Has data issue: false hasContentIssue false
  • English
  • Français

Resonant Frequency and Sensitivity of a Caliper Formed With Assembled Cantilever Probes Based on the Modified Strain Gradient Theory

Published online by Cambridge University Press:  10 September 2014

Mohammad Abbasi  and
Seyed E. Afkhami
Mohammad Abbasi*
Affiliation:
School of Mechanical Engineering, Shahrood Branch, Islamic Azad University, Daneshgah Blvd., 36199-43189 Shahrood, Iran
Seyed E. Afkhami
Affiliation:
School of Mechanical Engineering, Shahrood Branch, Islamic Azad University, Daneshgah Blvd., 36199-43189 Shahrood, Iran
*
*Corresponding author. m.abbasi28@yahoo.com
Get access

Abstract

The resonant frequency and sensitivity of an atomic force microscope (AFM) with an assembled cantilever probe (ACP) is analyzed utilizing strain gradient theory, and then the governing equation and boundary conditions are derived by a combination of the basic equations of strain gradient theory and Hamilton’s principle. The resonant frequency and sensitivity of the proposed AFM microcantilever are then obtained numerically. The proposed ACP includes a horizontal cantilever, two vertical extensions, and two tips located at the free ends of the extensions that form a caliper. As one of the extensions is located between the clamped and free ends of the AFM microcantilever, the cantilever is modeled as two beams. The results of the current model are compared with those evaluated by both modified couple stress and classical beam theories. The difference in results evaluated by the strain gradient theory and those predicted by the couple stress and classical beam theories is significant, especially when the microcantilever thickness is approximately the same as the material length-scale parameters. The results also indicate that at the low values of contact stiffness, scanning in the higher cantilever modes decrease the accuracy of the proposed AFM ACP.

Keywords

atomic force microscopeassembled cantilever probestrain gradient theoryresonant frequencysensitivity
Type
Technology and Software Development
Information
Microscopy and Microanalysis , Volume 20 , Issue 6 , December 2014 , pp. 1672 - 1681
Copyright
© Microscopy Society of America 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abbasi, M. & Mohammadi, A.K. (2009). Effect of contact position and tip properties on the flexural vibration responses of atomic force microscope cantilevers. Int Rev Mech Eng 3, 196–202.Google Scholar
Abbasi, M. & Mohammadi, A.K. (2010). A new model for investigating the flexural vibration of an atomic force microscope cantilever. Ultramicroscopy 110, 1374–1379.CrossRefGoogle ScholarPubMed
Abbasi, M. & Mohammadi, A.K. (2014). A detailed analysis of the resonant frequency and sensitivity of flexural modes of atomic force microscope cantilevers with a sidewall probe based on a nonlocal elasticity theory. Strojniški vestnik – J Mech Eng 60, 179–186.Google Scholar
Aifantis, E.C. (1999). Strain gradient interpretation of size effects. Int J Fracture 95, 299–314.CrossRefGoogle Scholar
Akgoz, B. & Civalek, O. (2011 a). Application of strain gradient elasticity theory for buckling analysis of protein microtubules. Curr Appl Phys 11, 1133–1138.CrossRefGoogle Scholar
Akgoz, B. & Civalek, O. (2011 b). Strain gradient elasticity and modified couple stress models for buckling analysis of axially loaded micro-scaled beams. Int J Eng Sci 49, 1268–1280.CrossRefGoogle Scholar
Akgoz, B. & Civalek, O. (2012). Analysis of micro-sized beams for various boundary conditions based on the strain gradient elasticity theory. Arch Appl Mech 82, 423–443.CrossRefGoogle Scholar
Akgoz, B. & Civalek, O. (2013). A size-dependent shear deformation beam model based on the strain gradient elasticity theory. Int J Eng Sci 70, 1–14.CrossRefGoogle Scholar
Bhushan, B. (1999). Handbook of Micro/Nanotribology. Boca Raton, FL: CRC.Google Scholar
Chang, W.J., Lee, H.L. & Chen, T.Y. (2008). Study of the sensitivity of the first four flexural modes of an AFM cantilever with a sidewall probe. Ultramicroscopy 108, 619–624.Google Scholar
Dai, G., Wolff, H., Pohlenz, F., Danzebrink, H.U. & Wilkening, G. (2006). Atomic force probe for sidewall scanning of nano- and microstructures. Appl Phys Lett 88, 171908.CrossRefGoogle Scholar
Dai, G., Wolff, H., Weimann, T., Xu, M., Pohlenz, F. & Danzebrink, H.U. (2007). Nanoscale surface measurements at sidewalls of nano- and micro-structures. Meas Sci Technol 18, 334–341.Google Scholar
Eringen, A.C. (1972). Nonlocal polar elastic continua. Int J Eng Sci 10, 1–16.Google Scholar
Eringen, A.C. (1983). On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J Appl Phys 54, 4703–4710.Google Scholar
Fleck, N.A. & Hutchinson, J.W. (1993). Phenomenological theory for strain gradient effects in plasticity. J Mech Phys Solids 41, 1825–1857.Google Scholar
Fleck, N.A. & Hutchinson, J.W. (1997). Strain Gradient Plasticity. New York, NY: Academic Press.Google Scholar
Fleck, N.A. & Hutchinson, J.W. (2001). A reformulation of strain gradient plasticity. J Mech Phys Solids 49, 2245–2271.Google Scholar
Garcia, R. (2010). Amplitude Modulation Atomic Force Microscopy. Germany: Wiley-VCH.Google Scholar
Garcia, R. & Herruzo, E.T. (2012). The emergence of multifrequency force microscopy. Nat Nanotechnol 7, 217–226.Google Scholar
Garcia, R. & Perez, R. (2002). Dynamic atomic force microscopy methods. Surf Sci Rep 47, 197–301.Google Scholar
Giessibl, F.J. (2003). Advances in atomic force microscopy. Rev Mod Phys 75, 949–983.Google Scholar
Guo, X.H., Fang, D.N. & Li, X.D. (2005). Measurement of deformation of pure Ni foils by speckle pattern interferometry. Mech Eng 27, 21–25.Google Scholar
Kahrobaiyan, M.H., Ahmadian, M.T., Haghighi, P. & Haghighi, A. (2010 a). Sensitivity and resonant frequency of an AFM with sidewall and top-surface probes for both flexural and torsional modes. Int J Mech Sci 52, 1357–1365.Google Scholar
Kahrobaiyan, M.H., Asghari, M., Rahaeifard, M. & Ahmadian, M.T. (2010 b). Investigation of the size-dependent dynamic characteristics of atomic force microscope microcantilevers based on the modified couple stress theory. Int J Eng Sci 48, 1985–1994.Google Scholar
Kahrobaiyan, M.H., Tajalli, S.A., Movahhedy, M.R., Akbari, J. & Ahmadian, M.T. (2011). Torsion of strain gradient bars. Int J Eng Sci 49, 856–866.Google Scholar
Kitahara, M. (1985). Boundary Integral Equation Methods in Eigenvalue Problems of Elastodynamics and Thin Plates. Amsterdam: Elsevier.Google Scholar
Kong, S., Zhou, S., Nie, Z. & Wang, K. (2009). Static and dynamic analysis of micro beams based on strain gradient elasticity theory. Int J Eng Sci 47, 487–498.Google Scholar
Lam, D.C.C., Yang, F., Chong, A.C.M., Wang, J. & Tong, P. (2003). Experiments and theory in strain gradient elasticity. J Mech Phys Solids 51, 1477–1508.CrossRefGoogle Scholar
Lee, H.L. & Chang, W.J. (2011). Sensitivity of V-shaped atomic force microscope cantilevers based on a modified couple stress theory. Microelectron Eng 88, 3214–3218.Google Scholar
Lee, H.L. & Chang, W.J. (2012). Dynamic response of a cracked atomic force microscope cantilever used for nanomachining. Nanoscale Res Lett 7, 131.CrossRefGoogle ScholarPubMed
Mazeran, P.E. & Loubet, J.L. (1999). Normal and lateral modulation with a scanning force microscope, an analysis: Implication in quantitative elastic and friction imaging. J Tribol Lett 7, 1573–2711.Google Scholar
McFarland, A.W. (2004). Production and Analysis of Polymer Microcantilever Parts. Atlanta, GA: Georgia Institute of Technology.Google Scholar
McFarland, A.W. & Colton, J.S. (2005). Role of material microstructure in plate stiffness with relevance to microcantilever sensors. J Micromech Microeng 15, 1060–1067.Google Scholar
Mindlin, R.D. (1964). Micro-structure in linear elasticity. Arch Rational Mech Anal 16, 51–78.Google Scholar
Mindlin, R.D. & Tiersten, H.F. (1962). Effects of couple-stresses in linear elasticity. Arch Rational Mech Anal 11, 415–448.Google Scholar
Narendar, S., Ravinder, S. & Gopalakrishnan, S. (2012). Strain gradient torsional vibration analysis of micro/nano rods. Int J Nano Dimens 3, 1–17.Google Scholar
Niwa, Y., Kobayashi, S. & Kitahara, M. (1979). Application of integral equation method to eigen-value problems of elasticity. Proc Jpn Soc Civil Eng 285, 17–28.CrossRefGoogle Scholar
Niwa, Y., Kobayashi, S. & Kitahara, M. (1982). Determination of Eigenvalues by Boundary Element Methods. London: Applied Science Pub.Google Scholar
Rao, C.R., Toutenburg, H., Fieger, A., Heumann, C., Nittner, T. & Scheid, S. (1999). Linear Models: Least Squares and Alternatives. New York, NY: Springer.Google Scholar
Reinstaedtler, M., Rabe, U., Scherer, V., Turner, J.A. & Arnold, W. (2003). Imaging of flexural and torsional resonance modes of atomic force microscopy cantilevers using optical interferometry. Surf Sci 532–535, 1152–1158.Google Scholar
Rugar, D. & Hansma, P. (1990). Atomic force microscopy. Phys Today 43, 23–30.Google Scholar
Song, Y. & Bhushan, B. (2006 a). Coupling of cantilever lateral bending and torsion in torsional resonance and lateral excitation modes of atomic force microscopy. J Appl Phys 99, 094911.Google Scholar
Song, Y. & Bhushan, B. (2006 b). Simulation of dynamic modes of atomic force microscopy using a 3D finite element model. Ultramicroscopy 106, 847–873.Google Scholar
Turner, J.A. & Wiehn, J.S. (2001). Sensitivity of flexural and torsional vibration modes of atomic force microscope cantilevers to surface stiffness variations. Nanotechnology 12, 322–330.CrossRefGoogle Scholar
Wang, B., Zhao, J. & Zhou, S. (2010). A micro scale Timoshenko beam model based on strain gradient elasticity theory. Eur J Mech A-Solid 29, 591–599.Google Scholar
Wu, T.-S., Chang, W.-J. & Hsu, J.-C. (2004). Effect of tip length and normal and lateral contact stiffness on the flexural vibration responses of atomic force microscope cantilevers. Microelectron Eng 71, 15–20.Google Scholar
Yang, F., Chong, A.C.M., Lam, D.C.C. & Tong, P. (2002). Couple stress based strain gradient theory for elasticity. Int J Solids Struct 39, 2731–2743.Google Scholar
Supplementary material: File

Abbasi and Afkhami Supplementary Material

Supplementary Material

Download Abbasi and Afkhami Supplementary Material(File)
File 130.8 KB