Skip to main content
SpringerLink
Log in

Modal interactions in contact-mode atomic force microscopes

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

Atomic force microscopes (AFM) are used to estimate material and surface properties. When using contact-mode AFM, the sample or the probe is excited near a natural frequency of the system to estimate the linear coefficient of the contact stiffness. Because higher modes offer lower thermal noise, higher quality factors, and higher sensitivity to stiff samples, their use in this procedure is more desirable. However, these modes are candidates for internal resonances, where the energy being fed into one mode may be channeled to another mode. Ignoring such interactions could distort or affect the accuracy of measurements. The method of multiple scales is used to derive an approximate analytical expression to the probe response in the presence of two-to-one autoparametric resonance between the second and third modes. We examine characteristics of this solution in relation to a single-mode response and consider its implications in AFM measurements. We find that the influence of this interaction extends over a considerable range of the tip-sample contact stiffness.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Binnig, G., Quate, C.F., Gerber, C.: Atomic force microscope. Phys. Rev. Lett. 56, 930–933 (1986)

    Article  Google Scholar 

  2. Lennard-Jones, J.E.: Cohesion. Proc. Phys. Soc. 43, 461–482 (1931)

    Article  MATH  Google Scholar 

  3. Hertz, H.R.: Ueber die Beruehrung Elastischer Koerper (On contact between elastic bodies). In: Gesammelte Werke (Collected Works), Leipzig, Germany, vol. 1, p. 155 (1895)

  4. Timoshenko, S., Goodier, J.N.: Theory of Elasticity. McGraw-Hill, New York (1951)

    MATH  Google Scholar 

  5. Johnson, K.L., Kendall, K., Roberts, A.D.: Surface energy and the contact of elastic solids. Proc. R. Soc. Lond., Ser. A 324, 301–313 (1971)

    Article  Google Scholar 

  6. Derjaguin, B.V., Muller, V.M., Toporov, Y.P.: Effect of contact deformations on the adhesion of particles. J. Colloid Interface Sci. 53, 314–326 (1975)

    Article  Google Scholar 

  7. Lin, D.C., Dimitriadis, E.K., Horkay, F.: Robust strategies for automated AFM force curve analysis, I: non-adhesive indentation of soft, inhomogeneous materials. J. Biomech. Eng. 129, 430–440 (2007)

    Article  Google Scholar 

  8. Lin, D.C., Dimitriadis, E.K., Horkay, F.: Robust strategies for automated AFM force curve analysis, II: adhesion-influenced indentation of soft, elastic materials. J. Biomech. Eng. 129, 904–912 (2007)

    Article  Google Scholar 

  9. Cappella, B., Dietler, G.: Force-distance curves by atomic force microscopy. Surf. Sci. Rep. 34, 1–104 (1999)

    Article  Google Scholar 

  10. Rabe, U., Janser, K., Arnold, W.: Vibrations of free and surface-coupled atomic force microscope cantilevers: theory and experiment. Rev. Sci. Instrum. 67, 3281–3293 (1996)

    Article  Google Scholar 

  11. Mazeran, P.E., Loubet, J.L.: Force modulation with a scanning force microscope: analysis. Tribol. Lett. 3, 125–132 (1997)

    Article  Google Scholar 

  12. Vairac, P., Cretin, B.: Frequency shift of a resonating cantilever in A.C. force microscopy: towards a realistic model. Appl. Phys. A 66, S227–S230 (1998)

    Article  Google Scholar 

  13. Rabe, U., Kester, E., Arnold, W.: Probing linear and non-linear tip-sample interaction forces by atomic force acoustic microscopy. Surf. Interface Anal. 27, 386–391 (1999)

    Article  Google Scholar 

  14. Crozier, K.B., Yaralioglu, G.G., Degertekin, F.L., Adams, J.D., Minne, S.C., Quate, C.F.: Thin film characterization by atomic force microscopy at ultrasonic frequencies. Appl. Phys. Lett. 76, 1950–1952 (2000)

    Article  Google Scholar 

  15. Yamanaka, K., Nakano, S.: Quantitative elasticity evaluation by contact resonance in an atomic force microscope. Appl. Phys. A 66, S313–S317 (1998)

    Article  Google Scholar 

  16. Yamanaka, K., Noguchi, A., Tsuji, T., Koike, T., Goto, T.: Quantitative material characterization by ultrasonic AFM. Surf. Interface Anal. 27, 600–606 (1999)

    Article  Google Scholar 

  17. Rabe, U., Amelio, S., Kester, E., Scherer, V., Hirsekorn, S., Arnold, W.: Quantitative determination of contact stiffness using atomic force acoustic microscopy. Ultrasonics 38, 430–437 (2000)

    Article  Google Scholar 

  18. Kester, E., Rabe, U., Presmanes, L., Tailhades, Ph., Arnold, W.: Measurement of Young’s modulus of nanocrystalline ferrites with spinal structures by atomic force acoustic microscopy. J. Phys. Chem. Solids 61, 1275–1284 (2000)

    Article  Google Scholar 

  19. Yamanaka, K., Tsuji, T., Noguchi, A., Koike, T., Mihara, T.: Nanoscale elasticity measurement with in situ tip shape estimation in atomic force microscopy. Rev. Sci. Instrum. 71, 3281–3293 (2000)

    Article  Google Scholar 

  20. Yamanaka, K., Maruyama, Y., Tsuji, T., Nakamoto, K.: Resonance frequency and Q factor mapping by ultrasonic atomic force microscopy. Appl. Phys. Lett. 78, 1939–1941 (2001)

    Article  Google Scholar 

  21. Muraoka, M., Arnold, W.: A method of evaluating local elasticity and adhesion energy from the nonlinear response of AFM cantilever vibrations. JSME Int. J. Ser. A, Solid Mech. Mater. Eng. 44, 396–405 (2001)

    Google Scholar 

  22. Tsuji, T., Irihama, H., Yamanaka, K.: Observation of anomalous dislocation behavior in graphite using ultrasonic atomic force microscopy. Jpn. J. Appl. Phys. 41, 832–835 (2002)

    Article  Google Scholar 

  23. Rabe, U., Amelio, S., Kopycinska, M., Hirsekorn, S., Kempf, M., Göken, M., Arnold, W.: Imaging and measurement of local mechanical material properties by atomic force acoustic microscopy. Surf. Interface Anal. 33, 65–70 (2002)

    Article  Google Scholar 

  24. Hurley, D.C., Shen, K., Jennett, N.M., Turner, J.A.: Atomic force acoustic microscopy methods to determine thin-film elastic properties. J. Appl. Phys. 94, 2347–2354 (2003)

    Article  Google Scholar 

  25. Tsuji, T., Ogiso, H., Akedo, J., Saito, S., Fukuda, K., Yamanaka, K.: Evaluation of domain boundary of a piezo/ferroelectric material by ultrasonic atomic force microscopy. Jpn. J. Appl. Phys. 43, 2907–2913 (2004)

    Article  Google Scholar 

  26. Jalili, N., Laxminarayana, K.: A review of atomic force microscopy imaging systems: Application to molecular metrology and biological sciences. Mechatronics 14, 907–945 (2004)

    Article  Google Scholar 

  27. Carpick, R.W., Agraït, N., Ogletree, D.F., Salmeron, M.: Measurement of interfacial shear (friction) with an ultrahigh vacuum atomic force acoustic microscope. J. Vac. Sci. Technol. B 14, 1289–1295 (1996)

    Google Scholar 

  28. Abdel-Rahman, E.M., Nayfeh, A.H.: Contact force identification using the subharmonic resonance of a contact-mode atomic force microscopy. Nanotechnology 16, 199–207 (2005)

    Article  Google Scholar 

  29. Aahhab, M., Salapaka, M.V., Dahleh, M., Mezic, I.: Melnikov-based dynamical analysis of microcantilevers in scanning probe microscopy. Nonlinear Dyn. 20, 197–220 (1999)

    Article  Google Scholar 

  30. Turner, J.A., Wiehn, J.S.: Sensitivity of flexural and torsional vibration modes of atomic force microscope cantilevers to surface stiffness variations. Nanotechnology 12, 322–330 (2001)

    Article  Google Scholar 

  31. Turner, J.A.: Nonlinear vibrations in contact atomic force microscopy. In: Proceedings of the ASME 2003 Design Engineering Technical Conferences and Computers and Information in Engineering Conference, DEC2003/BIB-48524, Chicago, IL (2003)

  32. Turner, J.A.: Non-linear vibrations of a beam with cantilever-Hertzian contact boundary conditions. J. Sound Vib. 275, 177–191 (2004)

    Article  Google Scholar 

  33. Nayfeh, A.H.: Perturbation Methods. Wiley, New York (1973)

    MATH  Google Scholar 

  34. Nayfeh, A.H.: Introduction to Perturbation Techniques. Wiley, New York (1981)

    MATH  Google Scholar 

  35. Nayfeh, A.H.: Nonlinear Interactions. Wiley, New York (2000)

    MATH  Google Scholar 

  36. Wu, W., Pragai, S., Gottlieb, O.: Nonlinear multi-mode dynamics of a microbeam for noncontact atomic force microscopy in ultra-high vacuum. In: Proceedings of the ASME 2005 International Design Engineering Technical Conferences, DEC2005-85742, Long Beach, CA, September 24–28 (2005)

  37. Rabe, U., Turner, J., Arnold, W.: Analysis of the high-frequency response of atomic force microscope cantilevers. Appl. Phys. A 66, S277–S282 (1998)

    Article  Google Scholar 

  38. Nayfeh, A.H.: Resolving controversies in the application of the method of multiple scales to higher orders. Nonlinear Dyn. 40, 61–102 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  39. Nayfeh, A.H., Balachandran, B.: Applied Nonlinear Dynamics. Wiley, New York (1995)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dynamic Loads Group, Cessna Aircraft Company, Wichita, KS, 67215, USA

    Haider N. Arafat

  2. Department of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University, Blacksburg, VA, 24061, USA

    Ali H. Nayfeh

  3. Department of Systems Design Engineering, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada

    Eihab M. Abdel-Rahman

Authors
  1. Haider N. Arafat
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ali H. Nayfeh
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Eihab M. Abdel-Rahman
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Haider N. Arafat.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Arafat, H.N., Nayfeh, A.H. & Abdel-Rahman, E.M. Modal interactions in contact-mode atomic force microscopes. Nonlinear Dyn 54, 151–166 (2008). https://doi.org/10.1007/s11071-008-9388-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-008-9388-5

Keywords

Search

Navigation