Maximilian Forstenhäusler, Enrique A. López-Guerra, Santiago D. Solares

DOI Number
First page
Last page


We provide guidelines for modeling linear viscoelastic materials containing an arbitrary number of characteristic times, under atomic force microscopy (AFM) characterization. Instructions are provided to set up the governing equations that rule the deformation of the material by the AFM tip. Procedures are described in detail in the spirit of providing a simple handbook, which is accompanied by open-access code and workbook (Excel) sheets. These guidelines seek to complement the existing literature and reach out to a larger audience in the awareness of the interdisciplinary nature of science. Examples are given in the context of force-distance curves characterization within AFM, but they can be easily extrapolated to other types of contact characterization techniques at different length scales. Despite the simplified approach of this document, the algorithms described herein are built upon rigorous classical linear viscoelastic theory.


Modeling Viscoelasticity, Generalized Maxwell Model, Generalized Kelvin-Voigt Model, Multiple Characteristic Times, Numerical Simulation, Atomic Force Microscopy

Full Text:



Ferry, J.D., 1980, Viscoelastic Properties of Polymers, John Wiley & Sons.

Tschoegl, N.W., 2012, The Phenomenological Theory of Linear Viscoelastic Behaviour: an Introduction, Springer Science & Business Media.

Rouse Jr, P.E., 1953, A theory of the linear viscoelastic properties of dilute solutions of coiling polymers, The Journal of Chemical Physics, 21(7), pp. 1272-1280.

Plazek, D., 1993, Breakdown of the Rouse model for polymers near the glass transition temperature, The Journal of chemical physics, 98(8), pp. 6488-6491.

Roylance, D., 2001, Engineering viscoelasticity, Department of Materials Science and Engineering–Massachusetts Institute of Technology, Cambridge MA, 2139, pp. 1-37.

Gordon, V.D., 2017, Biofilms and mechanics: a review of experimental techniques and findings, Journal of Physics D: Applied Physics, 50(22), 223002.

Kovach, K., 2017, Evolutionary adaptations of biofilms infecting cystic fibrosis lungs promote mechanical toughness by adjusting polysaccharide production, npj Biofilms and Microbiomes, 3(1), 1.

López-Guerra, E.A., Shen H., Solares, S.D., Shuai, D., 2019, Acquisition of time–frequency localized mechanical properties of biofilms and single cells with high spatial resolution, Nanoscale, 11(18), pp. 8918-8929.

Shen, H., López-Guerra, E.A., Zhu, R., Diba, T., Zheng, Q., Solares, S.D., Zara, J.M., Shuai, D., Shen, Y., 2018, Visible-light-responsive photocatalyst of graphitic carbon nitride for pathogenic biofilm control, ACS applied materials & interfaces, 11(1), pp. 373-384.

Krisenko, M.O., Cartagena, A., Raman, A., Geahlen, R.L., 2015, Nanomechanical property maps of breast cancer cells as determined by multiharmonic atomic force microscopy reveal Syk-dependent changes in microtubule stability mediated by MAP1B, Biochemistry, 54(1), pp. 60-68.

Lekka, M., Laidler, P., 2016, Applicability of AFM in cancer detection, Nature nanotechnology, 4(2), pp. 72-72.

Plodinec, M., Loparic, M., Monnier, C.A., Obermann, E.C., Zanetti-Dallenbach, R., Oertle, P., Hyotyla, J.T., Aebi, U., Bentires-Alj, M., Lim, R.Y.,Schoenenberger, C.A., 2012, The nanomechanical signature of breast cancer, Nature nanotechnology,7(11), pp. 757-765.

Arrechea, S., Aljarilla, A., de la Cruz, P., Palomares, E., Sharma, G.D, Langa, F., 2016, Efficiency improvement using bis (trifluoromethane) sulfonamide lithium salt as a chemical additive in porphyrin based organic solar cells, Nanoscale, 8(41), pp.17953-17962.

Bruner, C., Dauskardt, R., 2014, Role of molecular weight on the mechanical device properties of organic polymer solar cells, Macromolecules, 47(3), pp. 1117-1121.

Noh, H., Diaz, A.J., Solares, S.D., 2017, Analysis and modification of defective surface aggregates on PCDTBT: PCBM solar cell blends using combined Kelvin probe, conductive and bimodal atomic force microscopy, Beilstein Journal of Nanotechnology, 8, pp. 579-589.

Efremov, Y.M., Wang, W.H., Hardy, S.D., Geahlen, R.L., Raman, A., 2017, Measuring nanoscale viscoelastic parameters of cells directly from AFM force-displacement curves, Scientific reports, 7(1), pp. 1-14.

Zhai, M., McKenna, G.B., Viscoelastic modeling of nanoindentation experiments: A multicurve method, Journal of Polymer Science Part B: Polymer Physics, 52(9), pp. 633-639.

López‐Guerra, E.A., Eslami, B., 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, 55(10), p. 804-813.

López‐Guerra, E.A., 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, 8(1), pp. 2230-2244.

Garcia, P.D., Guerrero, C.R., Garcia, R., 2017, Time-resolved nanomechanics of a single cell under the depolymerization of the cytoskeleton, Nanoscale, 9(33), pp. 12051-12059.

Garcia, P.D., Guerrero, C.R., Garcia, R., 2020, Nanorheology of living cells measured by AFM-based force–distance curves, Nanoscale, 12(16), pp. 9133-9143.

Parvini, C.H., Saadi, M.A.S.R., Solares, S.D., Extracting viscoelastic material parameters using an atomic force microscope and static force spectroscopy, Beilstein Journal of Nanotechnology, 11(1), pp. 922-937.

Rajabifar, B., Jadhav, J.M., Kiracofe, D., Meyers, G.F. Raman, A., 2018, Dynamic AFM on viscoelastic polymer samples with surface forces, Macromolecules, 51(23), pp. 9649-9661.

Chyasnavichyus, M., Young, S.L., Tsukruk, V.V., 2015, Recent advances in micromechanical characterization of polymer, biomaterial, and cell surfaces with atomic force microscopy, Japanese Journal of Applied Physics, 54(8S2), 08LA02.

Radmacher, M.,Tillmann, R., Gaub, H., 1993, Imaging viscoelasticity by force modulation with the atomic force microscope, Biophysical journal, 64(3), pp. 735-742.

Garcia, R., 2020, Nanomechanical mapping of soft materials with the atomic force microscope: methods, theory and applications, Chemical Society Reviews, 49(16), pp. 5850-5884.

Lee, E., Radok, J.R.M., 1960, The contact problem for viscoelastic bodies, Journal of Applied Mechanics, 27(3), pp. 438-444.

Ting, T., 1966, The contact stresses between a rigid indenter and a viscoelastic half-space, Journal of Applied Mechanics, 33(4), pp. 845-854.

Graham, G.A., 1965, The contact problem in the linear theory of viscoelasticity, International Journal of Engineering Science, 3(1), pp. 27-46.

Garcia, R., Perez, R, 2002, Dynamic atomic force microscopy methods, Surface science reports, 47(6), pp. 197-301.

Melcher, J., Hu, S., Raman, A., 2008, Invited Article: VEDA: A web-based virtual environment for dynamic atomic force microscopy, Review of Scientific Instruments, 79(6), 061301.

Guzman, H.V., Garcia, P.D., Garcia,R., 2015, Dynamic force microscopy simulator (dForce): A tool for planning and understanding tapping and bimodal AFM experiments, Beilstein Journal of Nanotechnology, 6(1), pp. 369-379.

Attard, P., 2007, Measurement and interpretation of elastic and viscoelastic properties with the atomic force microscope, Journal of Physics: Condensed Matter, 19(47), pp. 473201.

Amo, C.A., Garcia, R., 2016, Fundamental high-speed limits in single-molecule, single-cell, and nanoscale force spectroscopies, ACS nano, 10(7), pp. 7117-7124.

Cartagena, A., Raman, A., 2014, Local viscoelastic properties of live cells investigated using dynamic and quasi-static atomic force microscopy methods, Biophys J, 106(5), pp. 1033-43.

Garcia, P.D., Garcia, R., 2018, Determination of the elastic moduli of a single cell cultured on a rigid support by force microscopy, Biophysical Journal, 114(12), pp. 2923-2932.

Garcia, R., 2006, Identification of nanoscale dissipation processes by dynamic atomic force microscopy, Physical review letters, 97(1), 016103.

Herruzo, E.T., Perrino, A.P., Garcia, R., 2014, Fast nanomechanical spectroscopy of soft matter, Nature Communications, 5(1), pp. 1-8.

Hu, S., Raman, A., 2008, Inverting amplitude and phase to reconstruct tip–sample interaction forces in tapping mode atomic force microscopy, Nanotechnology, 19(37), 375704.

Forstenhäusler, M., López‐Guerra, E.A., 2020, J.L. AFMviscoelastic Github Repository, Available from:

Gardner, M.F., Barnes, J.L., 1956, Transients in Linear Systems Studied by the Laplace Transformation, J. Wiley & Sons.

Kreyszig, E., 2007, Advanced Engineering Mathematics, John Wiley & Sons.

Stroud, K.A., Booth, D.J., 2011, Advanced Engineering Mathematics, Palgrave Macmillan.

Brinson, H.F., Brinson, L.C., 2008, Polymer Engineering Science and Viscoelasticity, Springer.

Simon, S.L., Mckenna, G.B, Sindt, O., 2000, Modeling the evolution of the dynamic mechanical properties of a commercial epoxy during cure after gelation, Journal of Applied Polymer Science, 76(4), pp. 495-508.

Lee, E., 1995, Stress analysis in visco-elastic bodies, Quarterly of Applied Mathematics, pp. 183-190.

Graham, G., 1968, The correspondence principle of linear viscoelasticity theory for mixed boundary value problems involving time-dependent boundary regions, Quarterly of Applied Mathematics, 26(2), pp. 167-174.

Popov, V.L., Willert, E., Heß, M., 2018, Method of dimensionality reduction in contact mechanics and friction: A user’s handbook. III, Viscoelastic contacts, Facta Universitatis-Series Mechanical Engineering, 16(2), pp. 99-113.

Meurer, A., Smith, C.P., Paprocki, M., Čertík, O., Kirpichev, S.B., Rocklin, M., Kumar, A., Ivanov, S., Moore, J.K., Singh, S. Rathnayake, T., 2017, SymPy: symbolic computing in Python, PeerJ Computer Science, 3, 103.

Argatov, I.I., Popov, V.L., 2016, Rebound indentation problem for a viscoelastic half‐space and axisymmetric indenter—Solution by the method of dimensionality reduction, ZAMM‐Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik, 96(8), pp. 956-967.

Khan, I.R., Ohba, R., 2003, Taylor series based finite difference approximations of higher-degree derivatives, Journal of Computational and Applied Mathematics, 154(1), pp. 115-124.

Grubmüller, H., 1991, Generalized Verlet algorithm for efficient molecular dynamics simulations with long-range interactions, Molecular Simulation, 6(1-3), pp. 121-142.


  • There are currently no refbacks.

ISSN: 0354-2025 (Print)

ISSN: 2335-0164 (Online)

COBISS.SR-ID 98732551

ZDB-ID: 2766459-4