Linear Viscoelasticity: Review of Theory and Applications in Atomic Force Microscopy

  • Marshall McCraw Department of Mechanical and Aerospace Engineering, The George Washington University, Washington, USA
  • Berkin Uluutku Department of Mechanical and Aerospace Engineering, The George Washington University, Washington, USA
  • Santiago Solares Department of Mechanical and Aerospace Engineering, The George Washington University, Washington, USA
Keywords: Atomic force microscopy, Linear viscoelasticity, Material property inversion


Recently, much research has been performed involving the mechanical analysis of biological and polymeric samples with the use of Atomic Force Microscopy (AFM).  Such materials require careful treatments which consider the rate-dependence of their viscoelastic response. Here, we review the fundamental theories of linear viscoelasticity, as well as their application to the analysis of AFM spectroscopy data.  An outline of general viscoelastic mechanical phenomena is initially given, followed by a brief outline of AFM techniques.  Then, an extensive outline of linear viscoelastic material models, as well as contact mechanics descriptions of AFM systems, are presented.


Aspden, R.M. (1991), “Aliasing effects in Fourier transforms of monotonically decaying functions and the calculation of viscoelastic moduli by combining transforms over different time periods”, Journal of Physics D: Applied Physics, Vol. 24 No. 6, pp. 803–808.

Binnig, G., Quate, C.F. and Gerber, C. (1986), “Atomic Force Microscope”, Physical Review Letters, Vol. 56 No. 9, pp. 930–933.

Binnig, G. and Rohrer, H. (1983), “Scanning tunneling microscopy”, Surface Science, Vol. 126 No. 1–3, pp. 236–244.

Bonfanti, A., Kaplan, J.L., Charras, G. and Kabla, A. (2020), “Fractional viscoelastic models for power-law materials”, Soft Matter, Vol. 16 No. 26, pp. 6002–6020.

Cheng, Y.-T. and Cheng, C.-M. (2005), “Relationships between initial unloading slope, contact depth, and mechanical properties for conical indentation in linear viscoelastic solids”, Journal of Materials Research, Vol. 20 No. 4, pp. 1046–1053.

Eaton, P. and West, P. (2010), Atomic Force Microscopy, Oxford University Press, available at:

Efremov, Y.M., Okajima, T. and Raman, A. (2020), “Measuring viscoelasticity of soft biological samples using atomic force microscopy”, Soft Matter, Vol. 16 No. 1, pp. 64–81.

Ferry, J.D. (1980), Viscoelastic Properties of Polymers, 3rd Edition, John Wiley & Sons, New York.

Findley, W.N., Lai, J.S. and Onaran, K. (1989), Creep and Relaxation of Nonlinear Viscoelastic Materials, 1st ed., Dover Publications, Inc, New York.

Gan, Y. (2009), “Atomic and subnanometer resolution in ambient conditions by atomic force microscopy”, Surface Science Reports, Vol. 64 No. 3, pp. 99–121.

Gannepalli, A., Yablon, D.G., Tsou, A.H. and Proksch, R. (2011), “Mapping nanoscale elasticity and dissipation using dual frequency contact resonance AFM”, Nanotechnology, Vol. 22 No. 35, p. 355705.

Garcia, P.D. and Garcia, R. (2018), “Determination of the viscoelastic properties of a single cell cultured on a rigid support by force microscopy”, Nanoscale, Vol. 10 No. 42, pp. 19799–19809.

Garcia, P.D., Guerrero, C.R. and Garcia, R. (2020), “Nanorheology of living cells measured by AFM-based force–distance curves”, Nanoscale, Vol. 12 No. 16, pp. 9133–9143.

Garcia, R. (2020), “Nanomechanical mapping of soft materials with the atomic force microscope: methods, theory and applications”, Chemical Society Reviews, Vol. 49 No. 16, pp. 5850–5884.

García, R. (2010), Amplitude Modulation Atomic Force Microscopy, Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, Germany, available at:

García, R., Martínez, N.F., Gómez, C.J. and García-Martín, A. (2007), “Energy Dissipation and Nanoscale Imaging in Tapping Mode AFM”, pp. 361–371.

García, R. and San Paulo, A. (1999), “Attractive and repulsive tip-sample interaction regimes in tapping-mode atomic force microscopy”, Physical Review B, Vol. 60 No. 7, pp. 4961–4967.

Greenwood, J.A. (2010), “Contact between an axisymmetric indenter and a viscoelastic half-space”, International Journal of Mechanical Sciences, Vol. 52 No. 6, pp. 829–835.

Hurley, D.C., Shen, K., Jennett, N.M. and Turner, J.A. (2003), “Atomic force acoustic microscopy methods to determine thin-film elastic properties”, Journal of Applied Physics, Vol. 94 No. 4, pp. 2347–2354.

Igarashi, T., Fujinami, S., Nishi, T., Asao, N. and Nakajima, and K. (2013), “Nanorheological Mapping of Rubbers by Atomic Force Microscopy”, Macromolecules, Vol. 46 No. 5, pp. 1916–1922.

Jalili, N. and Laxminarayana, K. (2004), “A review of atomic force microscopy imaging systems: application to molecular metrology and biological sciences”, Mechatronics, Vol. 14 No. 8, pp. 907–945.

Jesse, S., Mirman, B. and Kalinin, S. V. (2006), “Resonance enhancement in piezoresponse force microscopy: Mapping electromechanical activity, contact stiffness, and Q factor”, Applied Physics Letters, Vol. 89 No. 2, p. 022906.

Jesse, S., Vasudevan, R.K., Collins, L., Strelcov, E., Okatan, M.B., Belianinov, A., Baddorf, A.P., et al. (2014), “Band Excitation in Scanning Probe Microscopy: Recognition and Functional Imaging”, Annual Review of Physical Chemistry, Vol. 65 No. 1, pp. 519–536.

Johnson, K.L. (1982), “One Hundred Years of Hertz Contact”, Proceedings of the Institution of Mechanical Engineers, Vol. 196 No. 1, pp. 363–378.

Johnson, K.L. (1985), Contact Mechanics, Cambridge University Press, available at:

Kareem, A.U. and Solares, S.D. (2012), “Characterization of surface stiffness and probe–sample dissipation using the band excitation method of atomic force microscopy: a numerical analysis”, Nanotechnology, Vol. 23 No. 1, p. 015706.

Killgore, J.P., Yablon, D.G., Tsou, A.H., Gannepalli, A., Yuya, P.A., Turner, J.A., Proksch, R., et al. (2011), “Viscoelastic Property Mapping with Contact Resonance Force Microscopy”, Langmuir, Vol. 27 No. 23, pp. 13983–13987.

Kim, Y., Liu, D., Lee, H., Liu, R., Sengupta, D. and Park, S. (2015), “Investigation of Stress in MEMS Sensor Device Due to Hygroscopic and Viscoelastic Behavior of Molding Compound”, IEEE Transactions on Components, Packaging and Manufacturing Technology, Vol. 5 No. 7, pp. 945–955.

Kocun, M., Labuda, A., Gannepalli, A. and Proksch, R. (2015), “Contact resonance atomic force microscopy imaging in air and water using photothermal excitation”, Review of Scientific Instruments, Vol. 86 No. 8, p. 083706.

Kolluru, P. V., Eaton, M.D., Collinson, D.W., Cheng, X., Delgado, D.E., Shull, K.R. and Brinson, L.C. (2018), “AFM-based Dynamic Scanning Indentation (DSI) Method for Fast, High-resolution Spatial Mapping of Local Viscoelastic Properties in Soft Materials”, Macromolecules, Vol. 51 No. 21, pp. 8964–8978.

Lee, E.H. and Radok, J.R.M. (1960), “The Contact Problem for Viscoelastic Bodies”, Journal of Applied Mechanics, Vol. 27 No. 3, pp. 438–444.

López-Guerra, E.A., Eslami, B., Solares, S.D., López‐Guerra, E.A., Eslami, B. and Solares, S.D. (2017), “Calculation of standard viscoelastic responses with multiple retardation times through analysis of static force spectroscopy AFM data”, Journal of Polymer Science Part B: Polymer Physics, Vol. 55 No. 10, pp. 804–813.

López-Guerra, E.A. and Solares, S.D. (2014), “Modeling viscoelasticity through spring–dashpot models in intermittent-contact atomic force microscopy”, Beilstein Journal of Nanotechnology, Vol. 5 No. 1, pp. 2149–2163.

López-Guerra, E.A. and Solares, S.D. (2017), “Material property analytical relations for the case of an AFM probe tapping a viscoelastic surface containing multiple characteristic times”, Beilstein Journal of Nanotechnology, Vol. 8, pp. 2230–2244.

Magonov, S. (2001), “VISUALIZATION OF POLYMERS AT SURFACES AND INTERFACES WITH ATOMIC FORCE MICROSCOPY”, Handbook of Surfaces and Interfaces of Materials, Elsevier, pp. 393–430.

McLean, M., Brown, W.L. and Vinci, R.P. (2010), “Temperature-Dependent Viscoelasticity in Thin Au Films and Consequences for MEMS Devices”, Journal of Microelectromechanical Systems, Vol. 19 No. 6, pp. 1299–1308.

Mijailovic, A.S., Qing, B., Fortunato, D. and Van Vliet, K.J. (2018), “Characterizing viscoelastic mechanical properties of highly compliant polymers and biological tissues using impact indentation”, Acta Biomaterialia, Vol. 71, pp. 388–397.

Morris, V.J., Kirby, A.R. and Gunning, A.P. (2009), Atomic Force Microscopy for Biologists, IMPERIAL COLLEGE PRESS, available at:

Oppenheim, A. V. and Schafer, R.W. (1975), Digital Signal Processing, Pretince-Hall, New Jersey.

Parvini, C.H., Saadi, M.A.S.R. and Solares, S.D. (2020), “Extracting viscoelastic material parameters using an atomic force microscope and static force spectroscopy”, Beilstein Journal of Nanotechnology, Vol. 11, pp. 922–937.

Payam, A.F., Martin-Jimenez, D. and Garcia, R. (2015), “Force reconstruction from tapping mode force microscopy experiments”, Nanotechnology, Vol. 26 No. 18, p. 185706.

Proksch, R., Kocun, M., Hurley, D., Viani, M., Labuda, A., Meinhold, W. and Bemis, J. (2016), “Practical loss tangent imaging with amplitude-modulated atomic force microscopy”, Journal of Applied Physics, Vol. 119 No. 13, p. 134901.

Rabe, U. and Arnold, W. (1994), “Acoustic microscopy by atomic force microscopy”, Applied Physics Letters, Vol. 64 No. 12, pp. 1493–1495.

Radmacher, M., Cleveland, J.P., Fritz, M., Hansma, H.G. and Hansma, P.K. (1994), “Mapping interaction forces with the atomic force microscope”, Biophysical Journal, Vol. 66 No. 6, pp. 2159–2165.

Radmacher, M., Fritz, M., Kacher, C.M., Cleveland, J.P. and Hansma, P.K. (1996), “Measuring the viscoelastic properties of human platelets with the atomic force microscope”, Biophysical Journal, Vol. 70 No. 1, pp. 556–567.

Rother, J., Nöding, H., Mey, I. and Janshoff, A. (2014), “Atomic force microscopy-based microrheology reveals significant differences in the viscoelastic response between malign and benign cell lines”, Open Biology, Vol. 4 No. 5, p. 140046.

Roylance, D. (2001), “Engineering Viscoelasticity”, Massachusetts Institute of Technology: MIT OpenCouseWare.

Saadi, M.A.S.R., Uluutku, B., Parvini, C.H. and Solares, S.D. (2020), “Soft sample deformation, damage and induced electromechanical property changes in contact- and tapping-mode atomic force microscopy”, Surface Topography: Metrology and Properties, Vol. 8 No. 4, p. 045004.

Takino, H., Nakayama, R., Yamada, Y., Kohjiya, S. and Matsuo, T. (1997), “Viscoelastic Properties of Elastomers and Tire Wet Skid Resistance”, Rubber Chemistry and Technology, Vol. 70 No. 4, pp. 584–594.

Ting, T.C.T. (1966), “The Contact Stresses Between a Rigid Indenter and a Viscoelastic Half-Space”, Journal of Applied Mechanics, Vol. 33 No. 4, pp. 845–854.

Ting, T.C.T. (1968), “Contact Problems in the Linear Theory of Viscoelasticity”, Journal of Applied Mechanics, Vol. 35 No. 2, pp. 248–254.

Tschoegl, N.W. (1989), The Phenomenological Theory of Linear Viscoelastic Behavior, Springer Berlin Heidelberg, Berlin, Heidelberg, available at:

Uluutku, B., López-Guerra, E.A. and Solares, S.D. (2021), “A New Method for Obtaining Model-Free Viscoelastic Material Properties from Atomic Force Microscopy Experiments Using Discrete Integral Transform Techniques”, Beilstein Arch., Vol. 202142, available at:

WILLIAMS, M.L. (1964), “Structural analysis of viscoelastic materials”, AIAA Journal, Vol. 2 No. 5, pp. 785–808.

Yuya, P.A., Hurley, D.C. and Turner, J.A. (2008), “Contact-resonance atomic force microscopy for viscoelasticity”, Journal of Applied Physics, Vol. 104 No. 7, p. 074916.

How to Cite
McCraw, M., Uluutku, B., & Solares, S. (2021). Linear Viscoelasticity: Review of Theory and Applications in Atomic Force Microscopy. Reports in Mechanical Engineering, 2(1), 156-179.