Transient viscoelasticity study of tobacco mosaic virus/Ba^{2+} superlattice
 Haoran Wang^{1},
 Xinnan Wang^{1}Email author,
 Tao Li^{2} and
 Byeongdu Lee^{2}
DOI: 10.1186/1556276X9300
© Wang et al.; licensee Springer. 2014
Received: 7 May 2014
Accepted: 6 June 2014
Published: 13 June 2014
Abstract
Recently, we reported a new method to synthesize the rodlike tobacco mosaic virus (TMV) superlattice. To explore its potentials in nanolattice templating and tissue scaffolding, this work focused the viscoelasticity of the superlattice with a novel transient method via atomic force microscopy (AFM). For measuring viscoelasticity, in contrast to previous methods that assessed the oscillating response, the method proposed in this work enabled us to determine the transient response (creep or relaxation) of micro/nanobiomaterials. The mathematical model and numerical process were elaborated to extract the viscoelastic properties from the indentation data. The adhesion between the AFM tip and the sample was included in the indentation model. Through the functional equation method, the elastic solution for the indentation model was extended to the viscoelastic solution so that the time dependent force vs. displacement relation could be attained. To simplify the solving of the differential equation, a standard solid model was modified to obtain the elastic and viscoelastic components of the sample. The viscoelastic responses with different mechanical stimuli and the dynamic properties were also investigated.
Keywords
Tobacco mosaic virus Viscoelasticity Atomic force microscopy NanoindentationBackground
The recognition of tobacco mosaic virus (TMV) since the end of nineteenth century [1] has sparked innumerable research towards its potential applications in biomedicine [2, 3] and biotemplates for novel nanomaterial syntheses [4, 5]. A TMV is composed of a singlestrand RNA that is coated with 2,130 protein molecules, forming a special tubular structure with a length of 300 nm, an inner diameter of 4 nm, and an outer diameter of 18 nm [6]. The TMVs observed under a microscope can reach several tens of microns in length due to its unique feature of headtotail selfassembly [7]. Practically useful properties of the TMVs include the ease of culture and broad range of thermal stability [8]. Biochemical studies have shown that the TMV mutant can function as extracellular matrix proteins, which guide the cell adhesion and spreading [8]. It has also been confirmed that stem cell differentiation can be enhanced by both native and chemically modified TMV through regulating the gene's expression [9–11]. Moreover, TMV can be electrospun with polyvinyl alcohol (PVA) into continuous TMV/PVA composite nanofiber to form a biodegradable nonwoven fibrous mat as an extracellular matrix mimetic [12].
A number of techniques for measuring the viscoelasticity of macroscale materials have been used. A comprehensive review of those methods can be found in the literature [35] that addresses the principles of viscoelasticity and experimental setup for time and frequencydomain measurements. When the sample under investigation is in micro or even nanometer scale, however, the viscoelastic measurements become much more complicated. In dynamic methods, shear modulation spectroscopy [36] and magnetic bead manipulation [37] are two common methodologies to obtain the micro/nanoviscoelastic properties. To improve the measurement accuracy, efforts have been made to assess the viscoelasticity of micro/nanomaterials using contactresonance AFM [38–41]. The adhesion between the AFM probe tip and sample, however, is usually neglected. Furthermore, in order for the dynamic method to obtain a sinusoidal stress response, the applied strain amplitude must be kept reasonably small to avoid chaotic stress response and transient changes in material properties [42]. In addition, the dynamic properties are frequency dependent, which is time consuming to map the viscoelasticity over a wide range of frequencies. An alternative way to measure the viscoelastic response of a material is the transient method. Transient indentation with an indenter was developed based on the functional equation methods [43], where the loading or traveling histories of the indenter need to be precisely programmed.
In this study, the viscoelastic properties of the TMV/Ba^{2+} superlattice were investigated using AFMbased nanoindentation. AFM has the precision in both force sensing and displacement sensing, although it lacks the programing capability in continuous control of force and displacement. To realize the transient indentation in AFM, we introduced a novel experimental method. Viscoelastic nanoindentation theories were then developed based on the functional equation method [44]. The adhesion between the AFM tip and the sample, which significantly affected the determination of the viscoelastic properties [45], was included in the indentation model [20]. The viscoelastic responses of the sample with respect to different mechanical stimuli, including stress relaxation and strain creep, were further studied. The transition from transient properties to dynamic properties was also addressed.
Methods
The TMV/Ba^{2+} superlattice solution was obtained from the mixture of the TMV and BaCl_{2} solution (molar ratio of Ba^{2+}/TMV = 9.2 × 10^{4}:1) as stated in the reference [13]. It was further diluted with deionized water (volume ratio 1:1). A 10μL drop of the diluted solution on a silicon wafer was spun at 800 rpm for 10 s to form a monolayer dispersion of the sample. The sample was dried for 30 min under ambient conditions (40% R.H., 21°C) for AFM (Dimension 3100, Bruker, Santa Barbara, CA, USA) observation and subsequent indentation tests.
The sample was observed with FESEM and AFM. The indentation was performed using the AFM nanoindentation mode (AFM probe type: Tap150G, NanoAndMore USA, Lady's Island, SC, USA). The geometry of the cantilever was precisely measured using FESEM (S4700, Hitachi, Troy, MI, USA), with a length of 125 μm, width of 25 μm, and thickness of 2.1 μm. To accurately measure the tip radius, the tip was scanned on the standard AFM tip characterizer (SOCS/W2, Bruker) and the scanned data was curve fitted using PSIPlot (Poly Software International, Orangetown, NY, USA). The tip radius calculated to be 12 nm. For a typical indentation test, the tip was pressed onto the top surface of the sample until a predefined force of ~100 nN. The cantilever end remained unchanged in position during the controlled delay time. A series of indentations of the same predefined indentation force and different delay times were performed to track the viscoelastic responses. A 10min time interval of the two consecutive indentations was set for the sample to fully recover prior to the next indentation. The sample drift was minimized by turning off the light bulb in the AFM controller during scanning to keep the AFM chamber temperature constant and by shrinking the scan area gradually down to 1 nm × 1 nm on the top surface of the sample to rid the scanner piezo of the hysteresis effect.
Mathematical formulation
where F(t) is the contact force history, δ(t) is the indentation depth history, R is the nominal radius of the two contact spheres, w is the adhesive energy density, A_{ i } and B_{ i } (i = 0, 1, 2) are the parameters determined by the mechanical properties of two contact bodies, and the calculation of all these parameters can be found in the ‘Appendix.’
where H(t) is the Heaviside unit step function and D_{0} is the relative approach between the substrate and the end of the cantilever.
where a function with ‘^{∧}’ denotes Laplacetransformed function in s domain.
Solution to AFMbased indentation equation
It is observed from Figure 3 that the initial indentation force at t = 0 was measured to be 104.21 nN, then the force started to decrease and then remained constant at 38 nN after ~5,000 ms. The force decrease shown as red asterisks in Figure 3b fits qualatitatively well with the exponential function of Equation (5). E_{1}, E_{2}, and η, corresponding to the mechanical property parameters in Figure 2(a), can then be determined by fitting Equation (5) with the experimental data.
From the indentation data, D_{0} is obtained to be 78.457 nm. The pulloff force, 2πwR, calculated by averaging the pulloff forces of multiple indentations on the sample, is 16 nN. In comparison with the radius of the AFM tip, the surface of the sample can be treated as a flat plane. Hence, the nominal radius R = R_{tip} = 12 nm.
By invoking the force values at t = 0, t = ∞, and any intermediate point into Equation (5), the elasticity and viscosity components can be determined to be E_{1} = 32.0 MPa, E_{2} = 21.3 MPa, and η = 12.4 GPa ms. The coefficient of determination R^{2} of the viscoelastic equation and the experimental data is ~0.9639.
Since the stress relaxation process is achieved by modeling a combination of the cantilever and the sample, the viscoelasticity of the sample can be obtained by subtracting the component of the cantilever from the results. The cantilever, acting as a spring, is in series with the sample, represented by a standard solid model. The schematic of the series organization is shown in Figure 2(b). Thus the component of E_{1} comprises of E_{1s} representing the elastic part from the sample and E_{1c} representing the elastic part from the cantilever. To clarify the sources of the components in the modified standard solid model, E_{2}, v_{2}, and η in Figure 2(a) are now respectively denoted by E_{2s}, v_{2s}, and η_{ s } in Figure 2(b), where the subscript ‘s’ denotes the sample.
where k is the spring constant of the cantilever, which is 5 nN/nm based on Sader's method [47] to calibrate k, δ_{cantilever} is the cantilever deflection, and δ is recorded directly as the Zpiezo displacement by AFM.
Results and discussion
The force decrease curve is shown in Figure 3b with the experimental data.
where E_{1s} = 3 GPa, E_{2s} = 21.3 MPa, and η_{ s } = 12.4GPa ms.
In the standard solid model, the initial experimental data point is determined by the instantaneous elastic modulus E_{1s}. For the indentation that is held for over 5,000 ms, the indentation force becomes steady at ~38 nN, when the force exerts on the two springs in series. In contrast to E_{1s}, E_{2s} is much smaller, as can be seen from the significant force decrease of from ~104 to ~38 nN. The tip traveled down 13.2 nm from the beginning of indentation. It is noted that for our indentation test, the ratio of the maximum indentation depth to the sample diameter is less than 10% [48, 49]; the substrate effect to the elastic modulus calculation is neglected.
From the determined viscoelastic model, the mechanical response of the superlattice under a variety of mechanical loads can be predicted. Several simulation results were included as follows.
where ϵ_{0} is the constantly applied strain.
where σ_{0} is the constantly applied stress.
The indentation force history has been obtained in Equation (5), where the elastic shear modulus G_{1} as a combined elastic response of two springs shown in Figure 2(b) should be replaced by G_{1s} of one spring only. Then, the simulated curves for the two situations can be found in Figures 6c,d. It is concluded that the creep depth variation under different forces gets larger through creep while the indentation force variation under different depths gets smaller through relaxation. Particularly, in Figure 6d, the force finally decreases to negative values, which represent attractive forces. The attraction cannot be found when G_{1s} and G_{2s} are very small. This phenomenon can be interpreted by the conformability of materials determined by the elastic modulus. When G_{1s} and G_{2s} get smaller, the materials are more conformable. Accordingly, in the final equilibrium state, the materials around the indenter tend to be more deformable to enclose the spherical indenter. This will result in a smaller attraction.
where G′ and G″ are storage and loss moduli, respectively, ω is the angular velocity which is related to the frequency of the dynamic system, and ${\mathit{G}}_{\mathit{s}}\left(\mathit{t}\right)={\mathit{G}}_{1\mathit{s}}+{\mathit{G}}_{2\mathit{s}}{\mathit{e}}^{{\mathit{G}}_{2\mathit{s}}\mathit{t}/\mathit{\eta}}$ is the shear stress relaxation modulus, determined by the ratio of shear stress and constant shear strain.
where G_{2s} = E_{2s} / 2(1 + v_{2s}).
Figure 7 shows the curves of storage and loss shear moduli vs. the angular velocity. The storage shear modulus, G′, increases with the increase of angular velocity, while the increasing rate of G′ decreases and the angular velocity of ~2 rad/s is where the increasing rate changes most drastically. However, the loss shear modulus, G″, first increases and then decreases reaching the maximum value, ~3.9 MPa, at the angular velocity of ~0.7 rad/s. The storage and loss moduli in other cases as uniform tensile, compressive, and indentation experiments can also be obtained.
Conclusions
This paper presented a novel method to characterize the viscoelasticity of TMV/Ba^{2+} superlattice with the AFMbased transient indentation. In comparison with previous AFMbased dynamic methods for viscoelasticity measurement, the proposed experimental protocol is able to extract the viscosity and elasticity of the sample. Furthermore, the adhesion effect between the AFM tip and the sample was included in the indentation model. The elastic moduli and viscosity of TMV superlattice were determined to be E_{1s} = 2.14 GPa, E_{2s} = 21.3 MPa, and η_{ s } = 12.4 GPa∙ms. From the characterized viscoelastic parameters, it can be concluded that the TMV/Ba^{2+} superlattice was quite rigid at the initial contact and then experienced a large deformation under a constant pressure. Finally, the simulation of the mechanical behavior of TMV/Ba^{2+} superlattice under various loading cases, including uniform tension/compression and nanoindentation, were conducted to predict the mechanical response of sample under different loadings. The storage and loss shear moduli were also demonstrated to extend the applicability of the proposed method. With the characterized viscoelastic properties of TMV superlattice, we are now able to predict the process of tissue regeneration around the superlattice where the timedependent mechanical properties of scaffold interact with the growth of tissue.
Appendix
Modeling of adhesive contact of viscoelastic bodies
The functional equation method was employed to develop a contact mechanics model for indenting a viscoelastic material with adhesion. A modified standard solid model was used to extract the viscous and elastic parameters of the sample.
Several adhesive contact models are available, such as JohnsonKendallRoberts (JKR) model [50], DerjaguinMullerToporov (DMT) model [46], etc. [51–53]. Detailed comparisons can be found in reference [54]. As the DMT model results in a simpler differential equation, it was used in this study for the simulation to solve the indentation on an elastic body with adhesion.
where R is the nominal radius of the two contact spheres of R_{1} and R_{2}, given by R = R_{1}R_{2}/(R_{1} + R_{2}); the adhesive energy density w is obtained from the pulloff force F_{ c }, where F_{ c } = 3πwR/2; and the reduced elastic modulus E^{ * } is obtained from the elastic modulus E_{ s } and Poisson's ratio ν_{ s } of the sample by ${\mathit{E}}^{*}=4{\mathit{E}}_{\mathit{s}}/\left[3\left(1{\mathit{v}}_{\mathit{s}}^{2}\right)\right]$ with the assumption that the elastic modulus of the tip is much larger than that of the sample.
In Equation (A.1), E^{ * }, which governs the contact deformation behavior, is decided by the sample's mechanical properties. In the functional equation method [43], E^{ * } needs to be replaced by its equivalence in the viscoelastic system, so that the contact deformation behavior can be governed by the viscoelastic properties. To achieve it, the elastic/viscoelastic constitutive equations are needed.
where i (i = 0, 1, 2,…) is determined by the viscoelastic model to be selected, t is time, and ${\mathit{p}}_{\mathit{i}}^{\mathit{d}}$, ${\mathit{q}}_{\mathit{i}}^{\mathit{d}}$, ${\mathit{p}}_{\mathit{i}}^{\mathit{m}}$, and ${\mathit{q}}_{\mathit{i}}^{\mathit{m}}$ are the components related to the materials property constants, such as elastic modulus and Poisson's ratio etc.
where G and K are the shear modulus and bulk modulus, respectively.
To evolve the elastic solution into a viscoelastic solution, the linear operators in the viscoelastic system need to be determined. To this end, the standard solid model, shown in Figure 2(a), was used to simulate the viscoelastic behavior of the sample, since both the instantaneous and retarded elastic responses can be reflected in this model, which well describes the mechanical response of most viscoelastic bodies.
where ${\mathit{p}}_{1}^{\mathit{d}}=\frac{\mathit{\eta}}{{\mathit{G}}_{1}+{\mathit{G}}_{2}},\phantom{\rule{0.5em}{0ex}}{\mathit{q}}_{0}^{\mathit{d}}=\frac{2{\mathit{G}}_{1}{\mathit{G}}_{2}}{{\mathit{G}}_{1}+{\mathit{G}}_{2}},\phantom{\rule{0.5em}{0ex}}{\mathit{q}}_{1}^{\mathit{d}}=\frac{2{\mathit{G}}_{1}\mathit{\eta}}{{\mathit{G}}_{1}+{\mathit{G}}_{2}},\phantom{\rule{0.5em}{0ex}}{\mathit{G}}_{1}=\frac{{\mathit{E}}_{1}}{2\left(1+{\mathit{v}}_{1}\right)},{\mathit{G}}_{2}=\frac{{\mathit{E}}_{2}}{2\left(1+{\mathit{v}}_{2}\right)},\phantom{\rule{0.5em}{0ex}}{\mathit{K}}_{1}=\frac{{\mathit{E}}_{1}}{3\left(12{\mathit{v}}_{1}\right)}$, E_{1}, E_{2}, v_{1}, and v_{2} are the elastic modulus and Poisson's ratio of the two elastic components, respectively, shown in Figure 2.
where A_{0} = 2q_{0} + 3K_{1}, A_{1} = p_{1}(3K_{1} + 2q_{0}) + (3p_{1}K_{1} + 2q_{1}), A_{2} = p_{1}(3p_{1}K_{1} + 2q_{1}), B_{0} = q_{0}(1 + 6 K_{1}), B_{1} = q_{0}(p_{1} + 6K_{1}p_{1}) + q_{1}(6K_{1} + 1), and B_{2} = q_{1}(p_{1} + 6K_{1}p_{1}).
Abbreviations
 AFM:

atomic force microscopy
 DMT:

DerjaguinMullerToporov
 FESEM:

field emission scanning electron microscopy
 JKR:

JohnsonKendallRoberts
 PVA:

polyvinyl alcohol
 TMV:

tobacco mosaic virus.
Declarations
Acknowledgements
Funding support is provided by ND NASA EPSCoR FAR0017788. Use of the Advanced Photon Source, Electron Microscopy Center, and Center of Nanoscale Materials, an Office of Science User Facilities operated for the U. S. Department of Energy (DOE) Office of Science by Argonne National Laboratory, was supported by the U.S. DOE under Contract No. DEAC0206CH11357.
Authors’ Affiliations
References
 Zaitlin M: Discoveries in Plant Biology, ed S D K a S F Yang. HongKong: World Publishing Co., Ltd; 1998:105–110.View ArticleGoogle Scholar
 Hou CX, Luo Q, Liu JL, Miao L, Zhang CQ, Gao YZ, Zhang XY, Xu JY, Dong ZY, Liu JQ: Construction of GPx active centers on natural protein nanodisk/nanotube: a new way to develop artificial nanoenzyme. ACS Nano 2012, 6: 8692–8701. 10.1021/nn302270bView ArticleGoogle Scholar
 Hefferon KL: Plant virus expression vectors set the stage as production platforms for biopharmaceutical proteins. Virology 2012, 433: 1–6. 10.1016/j.virol.2012.06.012View ArticleGoogle Scholar
 Atanasova P, Rothenstein D, Schneider JJ, Hoffmann RC, Dilfer S, Eiben S, Wege C, Jeske H, Bill J: Virustemplated synthesis of ZnO nanostructures and formation of fieldeffect transistors. Adv Mater 2011, 23: 4918–4922. 10.1002/adma.201102900View ArticleGoogle Scholar
 Balci S, Bittner AM, Hahn K, Scheu C, Knez M, Kadri A, Wege C, Jeske H, Kern K: Copper nanowires within the central channel of tobacco mosaic virus particles. Electrochim Acta 2006, 51: 6251–6257. 10.1016/j.electacta.2006.04.007View ArticleGoogle Scholar
 Klug A: The tobacco mosaic virus particle: structure and assembly. Philos Trans Biol Sci 1999, 354: 531–535. 10.1098/rstb.1999.0404View ArticleGoogle Scholar
 Wang XN, Niu ZW, Li SQ, Wang Q, Li XD: Nanomechanical characterization of polyaniline coated tobacco mosaic virus nanotubes. J Biomed Mater Res A 2008, 87A: 8–14. 10.1002/jbm.a.31617View ArticleGoogle Scholar
 Lee LA, Nguyen QL, Wu LY, Horyath G, Nelson RS, Wang Q: Mutant plant viruses with cell binding motifs provide differential adhesion strengths and morphologies. Biomacromolecules 2012, 13: 422–431. 10.1021/bm2014558View ArticleGoogle Scholar
 Petrie TA, Raynor JE, Dumbauld DW, Lee TT, Jagtap S, Templeman KL, Collard DM, Garcia AJ: Multivalent integrinspecific ligands enhance tissue healing and biomaterial integration. Sci Transl Med 2010, 2: 1–6.View ArticleGoogle Scholar
 Kaur G, Wang C, Sun J, Wang Q: The synergistic effects of multivalent ligand display and nanotopography on osteogenic differentiation of rat bone marrow stem cells. Biomaterials 2010, 31: 5813–5824. 10.1016/j.biomaterials.2010.04.017View ArticleGoogle Scholar
 Kaur G, Valarmathi MT, Potts JD, Jabbari E, SaboAttwood T, Wang Q: Regulation of osteogenic differentiation of rat bone marrow stromal cells on 2D nanorod substrates. Biomaterials 2010, 31: 1732–1741. 10.1016/j.biomaterials.2009.11.041View ArticleGoogle Scholar
 Wu LY, Zang JF, Lee LA, Niu ZW, Horvatha GC, Braxtona V, Wibowo AC, Bruckman MA, Ghoshroy S, zur Loye HC, Li XD, Wang Q: Electrospinning fabrication, structural and mechanical characterization of rodlike virusbased composite nanofibers. J Mater Chem 2011, 21: 8550–8557. 10.1039/c1jm00078kView ArticleGoogle Scholar
 Li T, Winans RE, Lee B: Superlattice of rodlike virus particles formed in aqueous solution through likecharge attraction. Langmuir 2011, 27: 10929–10937. 10.1021/la202121sView ArticleGoogle Scholar
 Li T, Zan X, Winans RE, Wang Q, Lee B: Biomolecular assembly of thermoresponsive superlattices of the tobacco mosaic virus with large tunable interparticle distances. Angew Chem Int Ed 2013, 52: 6638–6642. 10.1002/anie.201209299View ArticleGoogle Scholar
 Agrawal BK, Pathak A: Oscillatory metallic behaviour of carbon nanotube superlattices  an ab initio study. Nanotechnology 2008, 19: 135706–135706. 10.1088/09574484/19/13/135706View ArticleGoogle Scholar
 Hultman L, Engstrom C, Oden M: Mechanical and thermal stability of TiN/NbN superlattice thin films. Surface Coatings Technol 2000, 133: 227–233.View ArticleGoogle Scholar
 Jaskolski W, Pelc M: Carbon nanotube superlattices in a magnetic field. Int J Quantum Chem 2008, 108: 2261–2266. 10.1002/qua.21750View ArticleGoogle Scholar
 Wu MJ, Wen HC, Wu SC, Yang PF, Lai YS, Hsu WK, Wu WF, Chou CP: Nanomechanical characteristics of annealed Si/SiGe superlattices. Appl Surf Sci 2011, 257: 8887–8893. 10.1016/j.apsusc.2011.05.015View ArticleGoogle Scholar
 Xu JH, Li GY, Gu MY: The microstructure and mechanical properties of TaN/TiN and TaWN/TiN superlattice films. Thin Solid Films 2000, 370: 45–49. 10.1016/S00406090(00)009408View ArticleGoogle Scholar
 Wang HR, Wang XN, Li T, Lee B: Nanomechanical characterization of rodlike superlattice assembled from tobacco mosaic viruses. J Appl Phys 2013, 113(024308):1–6.Google Scholar
 Belfiore LA, Floren ML, Paulino AT, Belfiore CJ: Stresssensitive tissue regeneration in viscoelastic biomaterials subjected to modulated tensile strain. Biophys Chem 2011, 158: 1–8. 10.1016/j.bpc.2011.04.008View ArticleGoogle Scholar
 Coulombe PA, Wong P: Cytoplasmic intermediate filaments revealed as dynamic and multipurpose scaffolds. Nat Cell Biol 2004, 6: 699–706. 10.1038/ncb0804699View ArticleGoogle Scholar
 Drozdov AD: Viscoelastic Structures: Mechanics of Growth and Aging. San Diego, CA, the United States: Academic Press; 1998.Google Scholar
 Tan SCW, Pan WX, Ma G, Cai N, Leong KW, Liao K: Viscoelastic behaviour of human mesenchymal stem cells. BMC Cell Biol 2008, 9: 40–40. 10.1186/14712121940View ArticleGoogle Scholar
 Rico F, Picas L, Colom A, Buzhynskyy N, Scheuring S: The mechanics of membrane proteins is a signature of biological function. Soft: Matter; 2013.Google Scholar
 Rayaprolu V, Manning BM, Douglas T, Bothner B: Virus particles as active nanomaterials that can rapidly change their viscoelastic properties in response to dilute solutions. Soft Matter 2010, 6: 5286–5288. 10.1039/c0sm00459fView ArticleGoogle Scholar
 Jang D, Meza LR, Greer F, Greer JR: Fabrication and deformation of threedimensional hollow ceramic nanostructures. Nat Mater 2013, 12: 893–898. 10.1038/nmat3738View ArticleGoogle Scholar
 Schaedler TA, Jacobsen AJ, Torrents A, Sorensen AE, Lian J, Greer JR, Valdevit L, Carter WB: Ultralight metallic microlattices. Science 2011, 334: 962–965. 10.1126/science.1211649View ArticleGoogle Scholar
 Bawolin NK, Chen XB, Zhang WJ: A method for modeling timedependant mechanical properties of tissue scaffolds. 2007 IEEE International Conference on Mechatronics and Automation, Vols IV, IEEE Conference Proceedings, Harbin, Heilongjiang, China 2007, 1423–1427.View ArticleGoogle Scholar
 Leung LH, Naguib HE: Characterization of the viscoelastic properties of poly(epsiloncaprolactone)hydroxyapatite microcomposite and nanocomposite scaffolds. Polym Eng Sci 2012, 52: 1649–1660. 10.1002/pen.23108View ArticleGoogle Scholar
 Nemoto N, Schrag JL, Ferry JD, Fulton RW: Infinitedilution viscoelastic properties of tobacco mosaicvirus. Biopolymers 1975, 14: 409–417. 10.1002/bip.1975.360140213View ArticleGoogle Scholar
 Graf C, Kramer H, Deggelmann M, Hagenbuchle M, Johner C, Martin C, Weber R: Rheological properties of suspensions of interacting rodlike Fdvirus particles. J Chem Phys 1993, 98: 4920–4928. 10.1063/1.464947View ArticleGoogle Scholar
 Huang F, Rotstein R, Fraden S, Kasza KE, Flynn NT: Phase behavior and rheology of attractive rodlike particles. Soft Matter 2009, 5: 2766–2771. 10.1039/b823522hView ArticleGoogle Scholar
 Schmidt FG, Hinner B, Sackmann E, Tang JX: Viscoelastic properties of semiflexible filamentous bacteriophage fd. Phys Rev E 2000, 62: 5509–5517. 10.1103/PhysRevE.62.5509View ArticleGoogle Scholar
 Lakes RS: Viscoelastic measurement techniques. Rev Sci Instrum 2004, 75: 797–810. 10.1063/1.1651639View ArticleGoogle Scholar
 Wahl KJ, Stepnowski SV, Unertl WN: Viscoelastic effects in nanometerscale contacts under shear. Tribol Lett 1998, 5: 103–107. 10.1023/A:1019169019617View ArticleGoogle Scholar
 MacKintosh FC, Schmidt CF: Microrheology. Curr Opin Colloid Interface Sci 1999, 4: 300–307. 10.1016/S13590294(99)900109View ArticleGoogle Scholar
 Mahaffy RE, Shih CK, MacKintosh FC, Kas J: Scanning probebased frequencydependent microrheology of polymer gels and biological cells. Phys Rev Lett 2000, 85: 880–883. 10.1103/PhysRevLett.85.880View ArticleGoogle Scholar
 Yuya PA, Hurley DC, Turner JA: Contactresonance atomic force microscopy for viscoelasticity. J Appl Phys 2008, 104: 074916–17.View ArticleGoogle Scholar
 Yablon DG, Gannepalli A, Proksch R, Killgore J, Hurley DC, Grabowski J, Tsou AH: Quantitative viscoelastic mapping of polyolefin blends with contact resonance atomic force microscopy. Macromolecules 2012, 45: 4363–4370. 10.1021/ma2028038View ArticleGoogle Scholar
 Herbert EG, Oliver WC, Pharr GM: Nanoindentation and the dynamic characterization of viscoelastic solids. J Phys D Appl Phys 2008, 41: 074021–19.View ArticleGoogle Scholar
 Shaw MT, MacKnight WJ: Introduction to polymer viscoelasticity. Hoboken, New Jersey: John Wiley & Sons, Inc.; 2005.View ArticleGoogle Scholar
 Radok JRM: Viscoelastic stress analysis. Quart Appl Math 1957, 15: 198–202.Google Scholar
 Lee EH: Stress analysis in viscoelastic bodies. Quart Appl Math 1955, 13: 183–190.Google Scholar
 Gupta S, Carrillo F, Li C, Pruitt L, Puttlitz C: Adhesive forces significantly affect elastic modulus determination of soft polymeric materials in nanoindentation. Mater Lett 2007, 61: 448–451. 10.1016/j.matlet.2006.04.078View ArticleGoogle Scholar
 Derjaguin BV, Muller VM, Toporov YP: Effect of contact deformations on adhesion of particles. J Colloid Interface Sci 1975, 53: 314–326. 10.1016/00219797(75)900181View ArticleGoogle Scholar
 Sader JE, Larson I, Mulvaney P, White LR: Method for the calibration of atomic force microscope cantilevers. Rev Sci Instrum 1995, 66: 3789–3798. 10.1063/1.1145439View ArticleGoogle Scholar
 Gamonpilas C, Busso EP: On the effect of substrate properties on the indentation behaviour of coated systems. Mater Sci Eng A Struct Mater Properties Microstruct Process 2004, 380: 52–61. 10.1016/j.msea.2004.04.038View ArticleGoogle Scholar
 Tsui TY, Pharr GM: Substrate effects on nanoindentation mechanical property measurement of soft films on hard substrates. J Mater Res 1999, 14: 292–301. 10.1557/JMR.1999.0042View ArticleGoogle Scholar
 Johnson KL, Kendall K, Roberts AD: Surface energy and contact of elastic solids. Proc Royal Soc Lond A Math Phys Sci 1971, 324: 301–313. 10.1098/rspa.1971.0141View ArticleGoogle Scholar
 Maugis D: Extension of the JohnsonKendallRoberts theory of the elastic contact of spheres to large contact radii. Langmuir 1995, 11: 679–682. 10.1021/la00002a055View ArticleGoogle Scholar
 Maugis D: Adhesion of spheres  the jkrdmt transition using a Dugdale model. J Colloid Interface Sci 1992, 150: 243–269. 10.1016/00219797(92)90285TView ArticleGoogle Scholar
 Sneddon IN: The relation between load and penetration in the axisymmetric Boussinesq problem for a punch of arbitrary profile. Int J Engng Sci 1965, 3: 47–57. 10.1016/00207225(65)900194View ArticleGoogle Scholar
 Johnson KL, Greenwood JA: An adhesion map for the contact of elastic spheres. J Colloid Interface Sci 1997, 192: 326–333. 10.1006/jcis.1997.4984View ArticleGoogle Scholar
 Malvern LE: Introduction to the mechanics of a continuous medium. Englewood Cliffs, New Jersey: PrenticeHall, Inc; 1969.Google Scholar
Copyright
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.