Graphical Abstract Figure
(a) The schematic illustration of the spherical oscillatory indentation and (b) the schematic representation of the imposed displacement and resultant force. The phase angle can be observed in the zoom-in view.
Graphical Abstract Figure
(a) The schematic illustration of the spherical oscillatory indentation and (b) the schematic representation of the imposed displacement and resultant force. The phase angle can be observed in the zoom-in view.
Abstract
The oscillatory indentation has become an attractive approach to characterizing the viscoelastic properties of soft biological solids. However, the influences of surface tension on oscillatory responses are ignored, which might lead to an inaccurate measurement of mechanical properties. In this work, the influences of surface tension on spherical oscillatory indentation for viscoelastic materials are investigated through the finite element method. The viscoelasticity of solids is characterized by the standard linear solid model and a sinusoidal displacement is applied as the excitation signal. During an entire cycle at the steady-state of oscillation, both the average value of contact radius and the dissipated energy decrease due to the presence of surface tension. For the oscillatory responses at various frequencies, the existence of surface tension results in an increase in average force but a decrease in phase angle. The force amplitude at low frequencies becomes higher when surface tension is considered. For the evaluation of the complex modulus, neglecting the surface tension would lead to a significant overestimation of storage modulus at low frequencies and an obvious underestimation of loss modulus when the normalized frequency approaches one. Our results provide a comprehensive understanding of the effects of surface tension on the mechanical responses of oscillation and the determination of viscoelastic properties through oscillatory indentation.
References
1.
Oliver
, W. C.
, and Pharr
, G. M.
, 1992
, “An Improved Technique for Determining Hardness and Elastic-Modulus Using Load and Displacement Sensing Indentation Experiments
,” J. Mater. Res.
, 7
(6
), pp. 1564
–1583
. 2.
Efremov
, Y. M.
, Okajima
, T.
, and Raman
, A.
, 2020
, “Measuring Viscoelasticity of Soft Biological Samples Using Atomic Force Microscopy
,” Soft Matter
, 16
(1
), pp. 64
–81
. 3.
Johnson
, K.
, 1985
, Contact Mechanics
, Cambridge University Press
, Cambridge, UK
.4.
Sneddon
, I. N.
, 1965
, “The Relation Between Load and Penetration in the Axisymmetric Boussinesq Problem for a Punch of Arbitrary Profile
,” Int. J. Eng. Sci.
, 3
(1
), pp. 47
–57
. 5.
Krieg
, M.
, Fläschner
, G.
, Alsteens
, D.
, Gaub
, B. M.
, Roos
, W. H.
, Wuite
, G. J. L.
, Gaub
, H. E.
, Gerber
, C.
, Dufrêne
, Y. F.
, and Müller
, D. J.
, 2019
, “Atomic Force Microscopy-Based Mechanobiology
,” Nat. Rev. Phys.
, 1
(1
), pp. 41
–57
. 6.
Papangelo
, A.
, and Ciavarella
, M.
, 2022
, “Viscoelastic Dissipation in Repeated Normal Indentation of an Hertzian Profile
,” Int. J. Solids Struct.
, 236–237
, p. 111362
. 7.
Magerle
, R.
, Zech
, P.
, Dehnert
, M.
, Bendixen
, A.
, and Otto
, A.
, 2024
, “Rate-Independent Hysteretic Energy Dissipation in Collagen Fibrils
,” Soft Matter
, 20
(12
), pp. 2831
–2839
. 8.
Weber
, A.
, Tyrakowski
, D.
, and Toca-Herrera
, J. L.
, 2022
, “Power Laws Describe Bacterial Viscoelasticity
,” Langmuir
, 38
(50
), pp. 15552
–15558
. 9.
Mierke
, C. T.
, 2021
, “Viscoelasticity Acts as a Marker for Tumor Extracellular Matrix Characteristics
,” Front. Cell Dev. Biol.
, 9
, p. 785138
. 10.
Sheng
, J.-Y.
, Mo
, C.
, Li
, G.-Y.
, Zhao
, H.-C.
, Cao
, Y.
, and Feng
, X.-Q.
, 2021
, “AFM-Based Indentation Method for Measuring the Relaxation Property of Living Cells
,” J. Biomech.
, 122
, p. 110444
. 11.
Jannati
, S.
, Maaref
, Y.
, Tibbits
, G. F.
, and Chiao
, M.
, 2023
, “Investigating Viscoelastic Properties of Myofibrils Isolated From hiPSC-CMs Using Atomic Force Microscopy and Quasi-Linear Viscoelastic Model
,” ASME J. Appl. Mech.
, 91
(1
), p. 011009
. 12.
Lim
, C. T.
, Zhou
, E. H.
, and Quek
, S. T.
, 2006
, “Mechanical Models for Living Cells—A Review
,” J. Biomech.
, 39
(2
), pp. 195
–216
. 13.
Smith
, B. A.
, Tolloczko
, B.
, Martin
, J. G.
, and Grütter
, P.
, 2005
, “Probing the Viscoelastic Behavior of Cultured Airway Smooth Muscle Cells With Atomic Force Microscopy: Stiffening Induced by Contractile Agonist
,” Biophys. J.
, 88
(4
), pp. 2994
–3007
. 14.
Nalam
, P. C.
, Gosvami
, N. N.
, Caporizzo
, M. A.
, Composto
, R. J.
, and Carpick
, R. W.
, 2015
, “Nano-Rheology of Hydrogels Using Direct Drive Force Modulation Atomic Force Microscopy
,” Soft Matter
, 11
(41
), pp. 8165
–8178
. 15.
VanLandingham
, M. R.
, 2003
, “Review of Instrumented Indentation
,” J. Res. Natl. Inst. Stand. Technol.
, 108
(4
), pp. 249
–265
. 16.
Cheng
, Y.-T.
, Ni
, W.
, and Cheng
, C.-M.
, 2006
, “Nonlinear Analysis of Oscillatory Indentation in Elastic and Viscoelastic Solids
,” Phys. Rev. Lett.
, 97
(7
), p. 075506
. 17.
Koruk
, H.
, Koc
, H. O.
, Yurdaer
, S. B.
, Besli
, A.
, and Pouliopoulos
, A. N.
, 2024
, “A New Approach for Measuring Viscoelastic Properties of Soft Materials Using the Dynamic Response of a Spherical Object Placed at the Sample Interface
,” Exp. Mech.
, 64
(1
), pp. 21
–32
. 18.
Chen
, S.-W.
, Liang
, X.-M.
, and Wang
, G.-F.
, 2024
, “Influence of Adhesion on Oscillatory Indentations of Viscoelastic Biomaterials by a Rigid Cone
,” J. Phys. D: Appl. Phys.
, 57
(31
), p. 315401
. 19.
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 Biol.
, 4
(5
), p. 140046
. 20.
Mahaffy
, R. E.
, Shih
, C. K.
, MacKintosh
, F. C.
, and Käs
, J.
, 2000
, “Scanning Probe-Based Frequency-Dependent Microrheology of Polymer Gels and Biological Cells
,” Phys. Rev. Lett.
, 85
(4
), pp. 880
–883
. 21.
Alcaraz
, J.
, Buscemi
, L.
, Grabulosa
, M.
, Trepat
, X.
, Fabry
, B.
, Farré
, R.
, and Navajas
, D.
, 2003
, “Microrheology of Human Lung Epithelial Cells Measured by Atomic Force Microscopy
,” Biophys. J.
, 84
(3
), pp. 2071
–2079
. 22.
Rigato
, A.
, Miyagi
, A.
, Scheuring
, S.
, and Rico
, F.
, 2017
, “High-Frequency Microrheology Reveals Cytoskeleton Dynamics in Living Cells
,” Nat. Phys.
, 13
(8
), pp. 771
–775
. 23.
Loubet
, J. L.
, Oliver
, W. C.
, and Lucas
, B. N.
, 2000
, “Measurement of the Loss Tangent of Low-Density Polyethylene With a Nanoindentation Technique
,” J. Mater. Res.
, 15
(5
), pp. 1195
–1198
. 24.
Odegard
, G. M.
, Gates
, T. S.
, and Herring
, H. M.
, 2005
, “Characterization of Viscoelastic Properties of Polymeric Materials Through Nanoindentation
,” Exp. Mech.
, 45
(2
), pp. 130
–136
. 25.
Lee
, E. H.
, and Radok
, J. R. M.
, 1960
, “The Contact Problem for Viscoelastic Bodies
,” ASME J. Appl. Mech.
, 27
(3
), pp. 438
–444
. 26.
Ting
, T. C. T.
, 1966
, “The Contact Stresses Between a Rigid Indenter and a Viscoelastic Half-Space
,” ASME J. Appl. Mech.
, 33
(4
), pp. 845
–854
. 27.
Huang
, G.
, Wang
, B.
, and Lu
, H.
, 2004
, “Measurements of Viscoelastic Functions of Polymers in the Frequency-Domain Using Nanoindentation
,” Mech. Time-Depend. Mater.
, 8
(4
), pp. 345
–364
. 28.
Cao
, Y.
, Ma
, D.
, and Raabe
, D.
, 2009
, “The Use of Flat Punch Indentation to Determine the Viscoelastic Properties in the Time and Frequency Domains of a Soft Layer Bonded to a Rigid Substrate
,” Acta Biomater.
, 5
(1
), pp. 240
–248
. 29.
Cao
, Y.
, Ji
, X.
, and Feng
, X.
, 2011
, “On Determination of the Damping Factor of Linear Viscoelastic Materials Using Dynamic Indentation: A Theoretical Study
,” Sci. China Phys. Mech.
, 54
(4
), pp. 598
–605
. 30.
Style
, R. W.
, Boltyanskiy
, R.
, Che
, Y.
, Wettlaufer
, J. S.
, Wilen
, L. A.
, and Dufresne
, E. R.
, 2013
, “Universal Deformation of Soft Substrates Near a Contact Line and the Direct Measurement of Solid Surface Stresses
,” Phys. Rev. Lett.
, 110
(6
), p. 066103
. 31.
Jagota
, A.
, Paretkar
, D.
, and Ghatak
, A.
, 2012
, “Surface-Tension-Induced Flattening of a Nearly Plane Elastic Solid
,” Phys. Rev. E
, 85
(5
), p. 051602
. 32.
Ding
, Y.
, Wang
, J.
, Xu
, G. K.
, and Wang
, G. F.
, 2018
, “Are Elastic Moduli of Biological Cells Depth Dependent or Not? Another Explanation Using a Contact Mechanics Model With Surface Tension
,” Soft Matter
, 14
(36
), pp. 7534
–7541
. 33.
Cui
, Y.
, Leong
, W.-H.
, Liu
, C.-F.
, Xia
, K.
, Feng
, X.
, Gergely
, C.
, Liu
, R.-B.
, and Li
, Q.
, 2022
, “Revealing Capillarity in AFM Indentation of Cells by Nanodiamond-Based Nonlocal Deformation Sensing
,” Nano Lett.
, 22
(10
), pp. 3889
–3896
. 34.
Hajji
, M.
, 1978
, “Indentation of a Membrane on an Elastic Half Space
,” ASME J. Appl. Mech.
, 45
(2
), pp. 320
–324
. 35.
He
, L. H.
, and Lim
, C. W.
, 2006
, “Surface Green Function for a Soft Elastic Half-Space: Influence of Surface Stress
,” Int. J. Solids Struct.
, 43
(1
), pp. 132
–143
. 36.
Koguchi
, H.
, 2008
, “Surface Green Function With Surface Stresses and Surface Elasticity Using Stroh’s Formalism
,” ASME J. Appl. Mech.
, 75
(6
), p. 061014
. 37.
Chen
, W. Q.
, and Zhang
, C.
, 2010
, “Anti-Plane Shear Green’s Functions for an Isotropic Elastic Half-Space With a Material Surface
,” Int. J. Solids Struct.
, 47
(11–12
), pp. 1641
–1650
. 38.
Wang
, G. F.
, and Feng
, X. Q.
, 2007
, “Effects of Surface Stresses on Contact Problems at Nanoscale
,” J. Appl. Phys.
, 101
(1
), p. 013510
. 39.
Long
, J.
, Ding
, Y.
, Yuan
, W.
, Chen
, W.
, and Wang
, G.
, 2017
, “General Relations of Indentations on Solids With Surface Tension
,” ASME J. Appl. Mech.
, 84
(5
), pp. 051007
–051008
. 40.
Gao
, X.
, Hao
, F.
, Fang
, D.
, and Huang
, Z.
, 2013
, “Boussinesq Problem With the Surface Effect and Its Application to Contact Mechanics at the Nanoscale
,” Int. J. Solids Struct.
, 50
(16
), pp. 2620
–2630
. 41.
Xu
, X.
, Jagota
, A.
, and Hui
, C. Y.
, 2014
, “Effects of Surface Tension on the Adhesive Contact of a Rigid Sphere to a Compliant Substrate
,” Soft Matter
, 10
(26
), pp. 4625
–4632
. 42.
Zhao
, X. J.
, and Rajapakse
, R. K. N. D.
, 2009
, “Analytical Solutions for a Surface-Loaded Isotropic Elastic Layer With Surface Energy Effects
,” Int. J. Eng. Sci.
, 47
(11–12
), pp. 1433
–1444
. 43.
Hui
, C.-Y.
, and Jagota
, A.
, 2016
, “Effect of Surface Tension on the Relaxation of a Viscoelastic Half-Space Perturbed by a Point Load
,” J. Polym. Sci., Part B: Polym. Phys.
, 54
(2
), pp. 274
–280
. 44.
Rauker
, J.
, Moshtagh
, P. R.
, Weinans
, H.
, and Zadpoor
, A. A.
, 2014
, “Analytical Relationships for Nanoindentation-Based Estimation of Mechanical Properties of Biomaterials
,” J. Mech. Med. Biol.
, 14
(03
), p. 1430004
. 45.
Efremov
, Y. M.
, Wang
, W.-H.
, Hardy
, S. D.
, Geahlen
, R. L.
, and Raman
, A.
, 2017
, “Measuring Nanoscale Viscoelastic Parameters of Cells Directly From AFM Force-Displacement Curves
,” Sci. Rep.
, 7
(1
), p. 1541
. 46.
Papangelo
, A.
, and Ciavarella
, M.
, 2021
, “Viscoelastic Normal Indentation of Nominally Flat Randomly Rough Contacts
,” Int. J. Mech. Sci.
, 211
, p. 106783
. 47.
Niu
, T. X.
, and Cao
, G. X.
, 2014
, “Power-Law Rheology Characterization of Biological Cell Properties Under AFM Indentation Measurement
,” RSC Adv.
, 4
(55
), pp. 29291
–29299
. 48.
Cammarata
, R. C.
, 1994
, “Surface and Interface Stress Effects in Thin-Films
,” Prog. Surf. Sci.
, 46
(1
), pp. 1
–38
. 49.
Feller
, S. E.
, and Pastor
, R. W.
, 1999
, “Constant Surface Tension Simulations of Lipid Bilayers: The Sensitivity of Surface Areas and Compressibilities
,” J. Chem. Phys.
, 111
(3
), pp. 1281
–1287
. 50.
Cao
, G.
, and Chandra
, N.
, 2010
, “Evaluation of Biological Cell Properties Using Dynamic Indentation Measurement
,” Phys. Rev. E
, 81
(2
), p. 021924
. 
