Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Analysis of the effect of mechanical properties of liquid and geometrical parameters of cantilever on the frequency response function of AFM


The dynamic behavior of atomic force microscopy (AFM) cantilevers in liquid is completely different from its behavior in air due to the applied hydrodynamic force. Exciting cantilever with frequencies close to resonance frequency and primary position alignment are two critical issues that should be considered in deriving frequency response function (FRF). In this paper, the hydrodynamic force has been modeled with string of spheres and the effect of the damping and the added mass on the model has been analyzed. Afterward, this force is applied to the dynamic equation so that the dynamic behavior of AFM cantilevers is studied in liquids by analyzing the effect of some important parameters such as added mass, internal, and fluid damping. By simulations of the dynamic equations for a silicon cantilever, FRF is determined in both air and liquid. In addition, the effects of two significant parameters of liquid mechanical properties (liquid viscosity and density) and geometrical parameters of cantilever on FRF are studied. The results for string of spheres model are compared with the other hydrodynamic model and the experimental data. When length/width ratio decreases, it is found that string of spheres model has a better agreement than the other hydrodynamic model with experimental data.

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


  1. 1.

    Hansma PK, Cleveland JP, Radmacher M, Walters DA, Hillner PE, Bezanilla M, Fritz M, Vie D, Hansma HG (1994) Tapping mode atomic force microscopy in liquid. Applied Physics Letters 64:1738–1740

  2. 2.

    Rankl C, Pastushenko V, Kienberger F, Stroh CM, Hinterdorfer P (2004) Hydrodynamic damping of a magnetically oscillated cantilever close to a surface. Ultramicroscopy 100:301–308

  3. 3.

    Song Y, Bhushan B (2007) Finite-element vibration analysis of tapping-mode atomic force microscopy in liquid. Ultramicroscopy 107:1095–1104

  4. 4.

    Putman CAJ, Van der Werf KO, De Grooth BG, Van Hulst NF, Greve J (1994) Tapping mode atomic force microscopy in liquid. Applied Physics Letters 64:2454–2456

  5. 5.

    Chen GY, Warmack RJ, Oden PI, Thundat T (1996) Transient response of tapping scanning force microscopy in liquids. American Vacuum Society B14:1313–1317

  6. 6.

    Burnham NA, Behrend OP, Oulevey F, Gremaudy G, Gallo PJ, Gourdon D, Dupas E, Kulik AJ, Pollock HM, Briggs GAD (1997) How does a tip tap? Nanotechnology 8:67–75

  7. 7.

    Lantz M, Liu YZ, Cui XD, Tokumoto H, Lindsay SM (1999) Dynamic force microscopy in fluid. Surface and Interface Analysis 27:354–360

  8. 8.

    Sader JE (1998) Frequency response of cantilever beams immersed in viscous fluids with applications to the atomic force microscope. American Institute of Physics 84:64–76

  9. 9.

    Ahmed N, Nino DF, Moy VT (2001) Measurement of solution viscosity by atomic force microscopy. American Institute of Physics 27:2731–2734

  10. 10.

    Vancura C, Dufour I, Heinrich SM, Josse F, Hierlemann A (2008) Analysis of resonating microcantilevers operating in a viscous liquid environment. Sensors and Actuators A141:43–51

  11. 11.

    Ghatkesar MK, Braun T, Barwich V, Ramseyer JP, Gerber C, Hegner M, Lang HP (2008) Resonating modes of vibrating microcantilevers in liquid. Applied Physics Letters 92:043106–3

  12. 12.

    Tsukada M, Watanabe N (2009) Theoretical analyses of cantilever oscillation for dynamic atomic force microscopy in liquids. Japanese Journal of Applied Physics 48:035001–6

  13. 13.

    Horng TL (2009) Analyses of vibration responses on nanoscale processing in a liquid using tapping-mode atomic force microscopy. Applied Surface Science 256:311–317

  14. 14.

    Basak S, Raman A (2006) Hydrodynamic loading of microcantilevers vibrating in viscous fluids. J Appl Phys 99:114906–10

  15. 15.

    Chu WH (1963) Vibration of fully submerged cantilever plates in water. Tech Rep 2:86396–2

  16. 16.

    Chon JWM, Mulvaney P, Sader JE (2000) Experimental validation of theoretical models for the frequency response of atomic force microscope cantilever beams immersed in fluids. J Appl Phys 87:3978–3988

  17. 17.

    Hosaka H, Itao K, Kuroda S (1995) Damping characteristics of beam-shaped micro-oscillators. Sensors and Actuators 49:87–95

  18. 18.

    Landau LD, Lifshitz EM (1959) Fluid mechanics. Pergamon Press, London

  19. 19.

    Song Y, Bhushan B (2006) Simulation of dynamic modes of atomic force microscopy using a 3D finite element model. Ultramicroscopy 106:847–873

  20. 20.

    Petyt M (1990) Introduction to finite element vibration analysis. Cambridge University Press, Cambridge

  21. 21.

    Derjaguin BV, Muller VM, Toporov YP (1975) Effect of contact deformations on the adhesion of particles. J Colloid Interface Sci 53:314–326

Download references

Author information

Correspondence to Moharam Habibnejad Korayem.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Korayem, M.H., Sharahi, H.J. Analysis of the effect of mechanical properties of liquid and geometrical parameters of cantilever on the frequency response function of AFM. Int J Adv Manuf Technol 57, 477–489 (2011).

Download citation


  • Tapping mode atomic force microscopy
  • Frequency response function
  • Hydrodynamic force