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A hybrid model to determine mechanical properties of soft polymers by nanoindentation
Mechanics of Materials 42 (2010) 1043–1047
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Mechanics of Materials journal homepage: www.elsevier.com/locate/mechmat
A hybrid model to determine mechanical properties of soft polymers by nanoindentation Qinzhuo Liao 1, Jianyong Huang 1, Tao Zhu, Chunyang Xiong ⇑, Jing Fang Department of Biomedical Engineering, Peking University, Beijing 100871, China Academy for Advanced Interdisciplinary Studies, Peking University, Beijing 100871, China
a r t i c l e
i n f o
Article history: Received 6 January 2010 Received in revised form 17 September 2010
Keywords: Nanoindentation Polymer Elasticity Adhesion
a b s t r a c t In nanoindentation tests for soft polymers, the elastic modulus estimated from the Hertz model varies with applied force, implying the effect of adhesion work needs to be considered in contact theory. In this article, a hybrid method of combining the Hertz model and the Johnson–Kendall–Roberts (JKR) model is presented, to analytically explain the descending phenomenon of the modulus estimation by considering adhesive effects. Thus both the force-independent elastic modulus and the adhesion work can be evaluated by fitting the experimental data, without need to know the adhesive force in advance. The successful application to the measurement of polydimethylsiloxane (PDMS) material’s elastic modulus demonstrates the method is applicable to the mechanical characterization of soft polymers by nanoindentation. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction Soft polymeric materials are finding greater use in microdevices and biomedical engineering applications (McDonald and Whitesides, 2002). For instance, polydimethylsiloxane (PDMS) is one of the most widely used materials in microfluidics (Whitesides, 2006), and polyacrylamide can be used as a soft substrate (Pelham and Wang, 1997; Qin et al., 2007; Huang et al., 2009a,b). With size reduction to micro- and nano-scales, suitable methods to characterize local mechanical properties of soft materials have become more important. Nanoindentation, also known as instrumented or depth-sensing indentation, is increasingly being used to measure mechanical parameters at small scales (Van Landingham, 2003; Schuh, 2006). Even though the technique has been successfully applied to estimate elastic and elastic–plastic properties of hard materials, it still requires further validation and suitable correction for soft materials ⇑ Corresponding author at: Department of Biomedical Engineering, Peking University, Beijing 100871, China. Tel.: +86 10 62757940; fax: +86 10 62752513. E-mail address: [email protected] (C. Xiong). 1 These authors contributed equally to this work. 0167-6636/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmat.2010.09.005
to obtain accurate and reproducible results (Zhao et al., 2003; Carrillo et al., 2005; Ebenstein and Pruitt, 2006; Gupta et al., 2007). In nanomechanics tests of hard materials, for instance, adhesion plays a negligible role in the sample and surface force can be neglected. Therefore, the classical Hertz model (Hertz, 1881) can be used to evaluate Young’s modulus of the elastic material. For soft materials such as polydimethylsiloxane (PDMS), many preliminary results have indicated that the adhesion energy at the tip-sample interface is a significant factor for a consistent modulus determination by nanoindentation (Johnson et al., 1971; Lim and Chaudhri, 2003; Cao et al., 2005; Carrillo et al., 2005; Ebenstein and Wahl, 2006; Gupta et al., 2007). For the Hertz model without adhesive effect considerations, the contact area and effective force experienced by the substrate might be underestimated, so that the sample modulus of soft matter would be overestimated (Johnson et al., 1971; Lim and Chaudhri, 2003). Some results also present obvious negative forces on the force–displacement curve, which is a direct evidence of adhesion behavior (Cao et al., 2005; Carrillo et al., 2005; Ebenstein and Wahl, 2006). Another big problem is that the elastic moduli evaluated by the Hertz model show significantly variation with the maximum indentation depth (or the peak indenter load) (Lim and
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Chaudhri, 2003; Gupta et al., 2007). These large variations are in disagreement with the actual properties of PDMS as a linearly elastic polymer. By considering adhesive interactions within the Johnson–Kendall–Roberts (JKR) adhesion contact model (Johnson et al., 1971), the difference in elastic moduli resulting from different peak loads can be reconciled (Gupta et al., 2007). This correction, however, needs to input a suitable adhesive force in advance, which is commonly difficult to be accurately measured and different authors provide various values for soft materials (Cao et al., 2005; Carrillo et al., 2005; Ebenstein and Wahl, 2006). In this study, a hybrid method of combining the Hertz model and the JKR model is presented to simultaneously determine both the Young’s modulus and adhesion work by nanoindentation. The Hertz theory is a classical method used in most nanoindentation tests of elastic materials, and the objective of data processing from the Hertz analysis is to obtain the modulus–force relation from multiple load– displacement curves. The effect of interfacial adhesion between probe tip and sample surface of the soft matter, on the other hand, is taken into account by introducing the JKR model to this approach. Specifically, by considering the Young’s modulus and the adhesion work as two inherent properties of the soft material and evaluating them as two constant parameters to fit the experimental data. In this way, the phenomenon of depth-dependence of Young’s modulus in the Hertz analysis can be described by the hybrid method and both the elastic modulus and the adhesion work can be obtained from the modulus–load curves, while the adhesive forces need not be known in advance. 2. Experimental procedure The PDMS samples were prepared by mixing the elastomer base with the curing agent (Sylgard 184, Dow Corning, Midland, MI) in two different ratios (10:1 and 20:1) respectively. The mixed solutions were poured into plastic containers to cure for 2 h at 85 °C. After cooling, the samples were cut from the bulk elastomer with dimensions 2 2 1 cm3. Nanoindentation tests were carried out using a Hysitron TriboIndenter (Hysitron Inc., Minneapolis, MN), with a displacement-controlled closed loop feedback mode. All indents were produced by a sapphire spherical probe with a nominal radius of 400 lm, which was suitable for soft materials with small indentation depths. For every specimen, three locations were chosen on the sample surface and each location had eight indents applied. Prior to indentation a small preload was applied for surface detection. The tip was then indented to the maximum displacement with a constant rate, held there for 5 s, and then drawn to unload with the same rate as loading. Indentations with eight different maximum indentation depths were implemented ranging from about 500 to 4000 nm and the rate of loading and unloading was kept at 100 nm/s. 3. Theory According to the Hertz model (Hertz, 1881), the reduced modulus EHertz of the sample material can be expressed as r
EHertz r
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi S3 ¼ 6RP max
ð1Þ
where R is the nominal radius of the indenter tip curvature, and Pmax is the maximum applied force corresponding to the maximum indentation depth hmax. The contact stiffness S (i.e., the slope of the initial portion of the unloading curve), is measured from the force–displacement curve of nanoindentation in the maximum depth, given by
S¼
dP dh h¼hmax
ð2Þ
As the sample is much softer than the indenter, the relation of the Young’s modulus EHertz and the reduced modulus is
EHertz ffi EHertz ð1 m2 Þ r
ð3Þ
where m is the Poisson’s ratio of the sample (m 0.5 for PDMS (Mark, 1999)). Basically, the Hertz analysis does not consider the behavior of adhesive force that exists in most interfaces between the soft material and the indenter. By the JKR model, on the other hand, the indentation depth h is evaluated by taking account of both the applied force P and the adhesive force Padh (Cao et al., 2005), expressed as
a2 1 þ h¼ 0 R
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!43 1 P=Padh 2
2a2 1 þ 0 3R
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!13 1 P=Padh a2 þ h0 0 2 3
ð4Þ
where h0 is the apparent displacement when the indentation force is zero (P = 0), (normally set h0 = 0 in the tests) and a0 is the radius of the contact area at zero load, related to the adhesive force Padh by
a30 ¼
3RP adh EJKR r
ð5Þ
From the adhesive force Padh, the adhesion work Dc (the interfacial energy per unit area) can be expressed as
Dc ¼
2 Padh 3 pR
ð6Þ
Substituting the depth h expressed by the JKR model [Eq. (4)] into that in the Hertz model [Eq. (2)], and then combining Eq. (5), the contact flexibility is derived as
a2 1 1 ~SðEJKR ; Padh ; Pmax Þ ¼ 1 ¼ dh pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 0 S dP P¼Pmax R Padh 1 P max =Padh 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!23 1 1 þ 1 Pmax =Padh 4 2 18 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!13 3 1 1 þ 1 Pmax =Padh 5 2 3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 3 1 3 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ RP adh EJKR 1 Pmax =Padh
Q. Liao et al. / Mechanics of Materials 42 (2010) 1043–1047
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2
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!23 1 Pmax =P adh 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!13 3 1 1 þ 1 Pmax =Padh 5 2 3
1 1þ 4 18
ð7Þ
Therefore, by substituting Eq. (7) into Eqs. (1) and (3), an analytical expression of combining the two models is obtained to relate the two types of elastic modulus given by
EHertz ¼ f ðEJKR ; Padh ; Pmax Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ¼ ð1 m2 Þ 6RP max ~SðEJKR ; Padh ; Pmax Þ3
ð8Þ
4. Results and discussion Fig. 1 shows the force–displacement curves of two PDMS specimens with different base-agent ratios, with maximum displacement of 500 nm in nanoindentation. Negative forces can be observed in both the initial loading and the unloading near the zero-displacement position, indicating the existence of adhesive force between the indenter and the sample in their contact zone. The modulus EHertz evaluated by the Hertz method, as given in Fig. 2, varies with the maximum applied force corresponding to the maximum indentation depth which ranged from 500 to 4000 nm, where the slopes of the initial unloading curve are used to obtain the elastic moduli. These force-dependent results of EHertz values are in accordance with those reported by Gupta et al. (2007), showing that the adhesion is an important factor which interferes with the estimation of sample modulus, especially when the indentation depth is shallow or the applied force is small. It should be pointed out that Figs. 1 and 2 plot data from eight indents of one location on each sample and that the results from the other two indent locations are similar and not shown.
Fig. 1. Force–displacement curves of two PDMS samples with different base-agent ratios (10:1, 20:1), showing negative forces near zerodisplacement indicating the existence of adhesive forces.
Fig. 2. The Young’s moduli estimated from the Hertz model, showing variation with the maximum applied indentation force.
Using the hybrid model to evaluate the modulus–force relation with Eq. (8), Fig. 3 presents the theoretical descriptions for the decreasing variation of the modulus EHertz with the maximum applied force Pmax. The adhesion work Dc, representing the influence of adhesive force Padh on the indentation contact, determines the descending rate of the modulus with the applied force, as shown in Fig. 3a. This effect is more obvious when the maximum applied force is relatively small in these Dc-dominated curves, where the various amounts of adhesive work produce different estimations of the elastic modulus as the indents are shallow. The other independent parameter, EJKR, mainly determines the modulus levels of EHertz evaluation, as shown in Fig. 3b where the adhesive work is kept at a certain level.
Fig. 3. Illustration of theoretical EHertz Pmax relation evaluated by Eq. (8) of the hybrid model, as dominated by (a) the adhesion work Dc, and (b) the modulus EJKR.
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This means these two parameters dominate a family of theoretical curves which cover the relation of EHertz Pmax as estimated by the Hertz model. Therefore, the solution of the elastic modulus EJKR for the soft materials becomes a process of finding an appropriate curve from Eq. (8), with two constant parameters to fit the data of EHertz Pmax from the nanoindentation test, in which the adhesion work Dc can also be evaluated without special measurement of adhesive force. Fig. 4 shows two analytical curves that fit well with our experimental data of EHertz Pmax relations, where the curve parameters EJKR and Dc are attached as the force-independent elastic modulus and the adhesion work, to represent the intrinsic mechanical behaviors of the PDMS material. The two estimated values of EJKR, as illustrated by the dashed lines of constants, show that EHertz tends to converge to EJKR as the maximum applied force becomes larger, indicating that the effect of surface adhesion becomes less in the cases of larger forces or deeper indents. In spite of this, EHertz will usually always be higher than EJKR for any indentation depths, which proves that the Hertz analysis without adhesion consideration normally overestimates the sample modulus (Johnson et al., 1971; Lim and Chaudhri, 2003). Through statistical analysis of indentation results of PDMS tests, Fig. 5 gives the mean values of the Young’s modulus EJKR with small standard deviations, for different base-agent ratios of the polymer. Taking the 10:1 ratio as an example, the estimated modulus EJKR = 1.74 ± 0.02 MPa is between the values of 1.9 ± 0.3 MPa (Cao et al., 2005) and 1.5 MPa (Gupta et al., 2007). Also for this case of base-agent ratio, the adhesion work Dc = 209 ± 9.1 mJ/m2 obtained from our PDMS-sapphire tip system, is within the range of the typical values (100–500 mJ/m2) reported by Bietsch and Michel (2000). The difference of results is due to several reasons: the curing/cooling time and temperature of materials, the sample-tip system, et al. For these two sorts of PDMS samples with different cross-linker, significant difference is observed in EJKR (p = 0.0005). However, the p-value of Dc (p = 0.0875) is slightly larger than 0.05, implying that the surface adhesive energy of PDMS material may be more consistent with the property
Fig. 4. The analytical curves determined by the appropriate parameters Dc and EJKR to fit the experimental data of EHertz Pmax.
Fig. 5. Statistical results of the Young’s modulus and adhesion work for the PDMS with different base-agent ratios.
of the base component. But it is still too early to say there is no significant difference between Dc, before further investigation is carried out. In addition, tests with different preloads (from 0.1 to 2 lN) in nanoindentation show that the measured moduli vary less than 1%, which suggests that the results are basically independent of the preload forces. Furthermore, different loading/unloading rates from 50 to 200 nm/s have been implemented in our tests, showing that the force-dependent phenomenon of the Hertzian modulus keeps its appearance in the results (Fig. 6). The elastic modulus EJKR, however, changes less than 6% and the adhesion work Dc varies less than 2%, which shows that the viscoelastic property has not much influence on the results. Besides, the measured E was found to increase with the increasing loading/unloading rate, similar to the conclusion drawn by Lim and Chaudhri (2003). The loading/unloading rates used in our experiment were smaller than the rate of 250 nm/s that was used by Gillies and
Fig. 6. Results of the Young’s modulus and adhesion work for the 10:1 PDMS with different loading/unloading rates.
Q. Liao et al. / Mechanics of Materials 42 (2010) 1043–1047
Prestidge (2004), where only slight hysteresis of the force curve was observed, implying that the viscoelastic property of PDMS could be neglected in this case of low loading speed and the polymer could be considered as an ideal material with mechanical behavior of linear elasticity (Cao et al., 2005; Carrillo et al., 2005). 5. Conclusions This study provides a theoretical model combining Hertz procedure and JKR analysis, which not only describes the descending phenomenon of the elastic modulus estimated from the Hertz model changing with the maximum applied force, but also predicts the analytical curves to fit the experimental data with two dominating factors, the Young’s modulus and the adhesion work in the JKR model. Therefore, in data analysis of nanoindentation test for soft materials, the Hertz estimation can continue to be used as the common procedure to obtain the modulus–force relation from the multiple force–displacement curves. The force-independent elastic modulus of the soft matter, as well as the adhesion work on the sample surface, can be evaluated as two constant parameters by fitting the experimental data, without need to know the adhesive force in advance, because the analysis of sample-indenter interaction has been taken into account in this hybrid model. Acknowledgements The authors acknowledge Y.Y. Huang and H.W. Ma for helpful discussions and M. Qiang for technical assistance in nanoindentation tests. The research is supported by the National Basic Research Program of China through Grant Nos. 2007CB935602 and 2011CB809106 and the National Science Foundation of China under Grant Nos. 90607004, 10872008, 11002003 and 11072004. The authors are highly grateful to the anonymous reviewer for his/her careful proofreading and helpful suggestions, which have remarkably improved the manuscript. References Bietsch, A., Michel, B., 2000. Conformal contact and pattern stability of stamps used for soft lithography. Journal of Applied Physics 88 (7), 4310–4318.
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Cao, Y.F., Yang, D.H., Soboyejoy, W., 2005. Nanoindentation method for determining the initial contact and adhesion characteristics of soft polydimethylsiloxane. Journal of Materials Research 20 (8), 2004– 2011. Carrillo, F., Gupta, S., Balooch, M., Marshall, S.J., Marshall, G.W., Pruitt, L., Puttlitz, C.M., 2005. Nanoindentation of polydimethylsiloxane elastomers: effect of crosslinking, work of adhesion, and fluid environment on elastic modulus. Journal of Materials Research 20 (10), 2820–2830. Ebenstein, D.M., Pruitt, L.A., 2006. Nanoindentation of biological materials. Nano Today 1 (3), 26–33. Ebenstein, D.M., Wahl, K.J., 2006. A comparison of JKR-based methods to analyze quasi-static and dynamic indentation force curves. Journal of Colloid and Interface Science 298 (2), 652–662. Gillies, G., Prestidge, C.A., 2004. Interaction forces, deformation and nanorheology of emulsion droplets as determined by colloid probe AFM. Advances in Colloid and Interface Science 108, 197–205. Gupta, S., Carrillo, F., Li, C., Pruitt, L., Puttlitz, C., 2007. Adhesive forces significantly affect elastic modulus determination of soft polymeric materials in nanoindentation. Materials Letters 61 (2), 448–451. Hertz, H., 1881. On the contact of elastic solids. Journal fuer die reine und angewandte Mathematik 92, 156–171. Huang, J.Y., Peng, X.L., Qin, L., Zhu, T., Xiong, C.Y., Zhang, Y.Y., Fang, J., 2009a. Determination of cellular tractions on elastic substrate based on an integral Boussinesq solution. Journal of Biomechanical Engineering – Transactions of the ASME 131 (6), 061009. Huang, J.Y., Qin, L., Peng, X.L., Zhu, T., Xiong, C.Y., Zhang, Y.Y., Fang, J., 2009b. Cellular traction force recovery: an optimal filtering approach in two-dimensional Fourier space. Journal of Theoretical Biology 259 (4), 811–819. Johnson, K.L., Kendall, K., Roberts, A.D., 1971. Surface energy and contact of elastic solids. Proceedings of the Royal Society of London Series A 324 (1558), 301–313. Lim, Y.Y., Chaudhri, M.M., 2003. Experimental investigations of the normal loading of elastic spherical and conical indenters on to elastic flats. Philosophical Magazine 83 (30), 3427–3462. Mark, J.E., 1999. Polymer Data Handbook. Oxford University Press, USA. McDonald, J.C., Whitesides, G.M., 2002. Poly(dimethylsiloxane) as a material for fabricating microfluidic devices. Accounts of Chemical Research 35 (7), 491–499. Pelham, R.J., Wang, Y.L., 1997. Cell locomotion and focal adhesions are regulated by substrate flexibility. Proceedings of the National Academy of Sciences of the United States of America 94 (25), 13661–13665. Qin, L., Huang, J.Y., Xiong, C.Y., Zhang, Y.Y., Fang, J., 2007. Dynamical stress characterization and energy evaluation of single cardiac myocyte actuating on flexible substrate. Biochemical and Biophysical Research Communications 360 (2), 352–356. Schuh, C.A., 2006. Nanoindentation studies of materials. Materials Today 9 (5), 32–40. Van Landingham, M.R., 2003. Review of instrumented indentation. Journal of Research of the National Institute of Standards and Technology 108 (4), 249–265. Whitesides, G.M., 2006. The origins and the future of microfluidics. Nature 442 (7101), 368–373. Zhao, Y.P., Shi, X.H., Li, W.J., 2003. Effect of work of adhesion on nanoindentation. Reviews on Advanced Materials Science 5 (4), 348–353.
Contents lists available at ScienceDirect
Mechanics of Materials journal homepage: www.elsevier.com/locate/mechmat
A hybrid model to determine mechanical properties of soft polymers by nanoindentation Qinzhuo Liao 1, Jianyong Huang 1, Tao Zhu, Chunyang Xiong ⇑, Jing Fang Department of Biomedical Engineering, Peking University, Beijing 100871, China Academy for Advanced Interdisciplinary Studies, Peking University, Beijing 100871, China
a r t i c l e
i n f o
Article history: Received 6 January 2010 Received in revised form 17 September 2010
Keywords: Nanoindentation Polymer Elasticity Adhesion
a b s t r a c t In nanoindentation tests for soft polymers, the elastic modulus estimated from the Hertz model varies with applied force, implying the effect of adhesion work needs to be considered in contact theory. In this article, a hybrid method of combining the Hertz model and the Johnson–Kendall–Roberts (JKR) model is presented, to analytically explain the descending phenomenon of the modulus estimation by considering adhesive effects. Thus both the force-independent elastic modulus and the adhesion work can be evaluated by fitting the experimental data, without need to know the adhesive force in advance. The successful application to the measurement of polydimethylsiloxane (PDMS) material’s elastic modulus demonstrates the method is applicable to the mechanical characterization of soft polymers by nanoindentation. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction Soft polymeric materials are finding greater use in microdevices and biomedical engineering applications (McDonald and Whitesides, 2002). For instance, polydimethylsiloxane (PDMS) is one of the most widely used materials in microfluidics (Whitesides, 2006), and polyacrylamide can be used as a soft substrate (Pelham and Wang, 1997; Qin et al., 2007; Huang et al., 2009a,b). With size reduction to micro- and nano-scales, suitable methods to characterize local mechanical properties of soft materials have become more important. Nanoindentation, also known as instrumented or depth-sensing indentation, is increasingly being used to measure mechanical parameters at small scales (Van Landingham, 2003; Schuh, 2006). Even though the technique has been successfully applied to estimate elastic and elastic–plastic properties of hard materials, it still requires further validation and suitable correction for soft materials ⇑ Corresponding author at: Department of Biomedical Engineering, Peking University, Beijing 100871, China. Tel.: +86 10 62757940; fax: +86 10 62752513. E-mail address: [email protected] (C. Xiong). 1 These authors contributed equally to this work. 0167-6636/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmat.2010.09.005
to obtain accurate and reproducible results (Zhao et al., 2003; Carrillo et al., 2005; Ebenstein and Pruitt, 2006; Gupta et al., 2007). In nanomechanics tests of hard materials, for instance, adhesion plays a negligible role in the sample and surface force can be neglected. Therefore, the classical Hertz model (Hertz, 1881) can be used to evaluate Young’s modulus of the elastic material. For soft materials such as polydimethylsiloxane (PDMS), many preliminary results have indicated that the adhesion energy at the tip-sample interface is a significant factor for a consistent modulus determination by nanoindentation (Johnson et al., 1971; Lim and Chaudhri, 2003; Cao et al., 2005; Carrillo et al., 2005; Ebenstein and Wahl, 2006; Gupta et al., 2007). For the Hertz model without adhesive effect considerations, the contact area and effective force experienced by the substrate might be underestimated, so that the sample modulus of soft matter would be overestimated (Johnson et al., 1971; Lim and Chaudhri, 2003). Some results also present obvious negative forces on the force–displacement curve, which is a direct evidence of adhesion behavior (Cao et al., 2005; Carrillo et al., 2005; Ebenstein and Wahl, 2006). Another big problem is that the elastic moduli evaluated by the Hertz model show significantly variation with the maximum indentation depth (or the peak indenter load) (Lim and
1044
Q. Liao et al. / Mechanics of Materials 42 (2010) 1043–1047
Chaudhri, 2003; Gupta et al., 2007). These large variations are in disagreement with the actual properties of PDMS as a linearly elastic polymer. By considering adhesive interactions within the Johnson–Kendall–Roberts (JKR) adhesion contact model (Johnson et al., 1971), the difference in elastic moduli resulting from different peak loads can be reconciled (Gupta et al., 2007). This correction, however, needs to input a suitable adhesive force in advance, which is commonly difficult to be accurately measured and different authors provide various values for soft materials (Cao et al., 2005; Carrillo et al., 2005; Ebenstein and Wahl, 2006). In this study, a hybrid method of combining the Hertz model and the JKR model is presented to simultaneously determine both the Young’s modulus and adhesion work by nanoindentation. The Hertz theory is a classical method used in most nanoindentation tests of elastic materials, and the objective of data processing from the Hertz analysis is to obtain the modulus–force relation from multiple load– displacement curves. The effect of interfacial adhesion between probe tip and sample surface of the soft matter, on the other hand, is taken into account by introducing the JKR model to this approach. Specifically, by considering the Young’s modulus and the adhesion work as two inherent properties of the soft material and evaluating them as two constant parameters to fit the experimental data. In this way, the phenomenon of depth-dependence of Young’s modulus in the Hertz analysis can be described by the hybrid method and both the elastic modulus and the adhesion work can be obtained from the modulus–load curves, while the adhesive forces need not be known in advance. 2. Experimental procedure The PDMS samples were prepared by mixing the elastomer base with the curing agent (Sylgard 184, Dow Corning, Midland, MI) in two different ratios (10:1 and 20:1) respectively. The mixed solutions were poured into plastic containers to cure for 2 h at 85 °C. After cooling, the samples were cut from the bulk elastomer with dimensions 2 2 1 cm3. Nanoindentation tests were carried out using a Hysitron TriboIndenter (Hysitron Inc., Minneapolis, MN), with a displacement-controlled closed loop feedback mode. All indents were produced by a sapphire spherical probe with a nominal radius of 400 lm, which was suitable for soft materials with small indentation depths. For every specimen, three locations were chosen on the sample surface and each location had eight indents applied. Prior to indentation a small preload was applied for surface detection. The tip was then indented to the maximum displacement with a constant rate, held there for 5 s, and then drawn to unload with the same rate as loading. Indentations with eight different maximum indentation depths were implemented ranging from about 500 to 4000 nm and the rate of loading and unloading was kept at 100 nm/s. 3. Theory According to the Hertz model (Hertz, 1881), the reduced modulus EHertz of the sample material can be expressed as r
EHertz r
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi S3 ¼ 6RP max
ð1Þ
where R is the nominal radius of the indenter tip curvature, and Pmax is the maximum applied force corresponding to the maximum indentation depth hmax. The contact stiffness S (i.e., the slope of the initial portion of the unloading curve), is measured from the force–displacement curve of nanoindentation in the maximum depth, given by
S¼
dP dh h¼hmax
ð2Þ
As the sample is much softer than the indenter, the relation of the Young’s modulus EHertz and the reduced modulus is
EHertz ffi EHertz ð1 m2 Þ r
ð3Þ
where m is the Poisson’s ratio of the sample (m 0.5 for PDMS (Mark, 1999)). Basically, the Hertz analysis does not consider the behavior of adhesive force that exists in most interfaces between the soft material and the indenter. By the JKR model, on the other hand, the indentation depth h is evaluated by taking account of both the applied force P and the adhesive force Padh (Cao et al., 2005), expressed as
a2 1 þ h¼ 0 R
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!43 1 P=Padh 2
2a2 1 þ 0 3R
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!13 1 P=Padh a2 þ h0 0 2 3
ð4Þ
where h0 is the apparent displacement when the indentation force is zero (P = 0), (normally set h0 = 0 in the tests) and a0 is the radius of the contact area at zero load, related to the adhesive force Padh by
a30 ¼
3RP adh EJKR r
ð5Þ
From the adhesive force Padh, the adhesion work Dc (the interfacial energy per unit area) can be expressed as
Dc ¼
2 Padh 3 pR
ð6Þ
Substituting the depth h expressed by the JKR model [Eq. (4)] into that in the Hertz model [Eq. (2)], and then combining Eq. (5), the contact flexibility is derived as
a2 1 1 ~SðEJKR ; Padh ; Pmax Þ ¼ 1 ¼ dh pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 0 S dP P¼Pmax R Padh 1 P max =Padh 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!23 1 1 þ 1 Pmax =Padh 4 2 18 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!13 3 1 1 þ 1 Pmax =Padh 5 2 3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 3 1 3 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ RP adh EJKR 1 Pmax =Padh
Q. Liao et al. / Mechanics of Materials 42 (2010) 1043–1047
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2
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!23 1 Pmax =P adh 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!13 3 1 1 þ 1 Pmax =Padh 5 2 3
1 1þ 4 18
ð7Þ
Therefore, by substituting Eq. (7) into Eqs. (1) and (3), an analytical expression of combining the two models is obtained to relate the two types of elastic modulus given by
EHertz ¼ f ðEJKR ; Padh ; Pmax Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ¼ ð1 m2 Þ 6RP max ~SðEJKR ; Padh ; Pmax Þ3
ð8Þ
4. Results and discussion Fig. 1 shows the force–displacement curves of two PDMS specimens with different base-agent ratios, with maximum displacement of 500 nm in nanoindentation. Negative forces can be observed in both the initial loading and the unloading near the zero-displacement position, indicating the existence of adhesive force between the indenter and the sample in their contact zone. The modulus EHertz evaluated by the Hertz method, as given in Fig. 2, varies with the maximum applied force corresponding to the maximum indentation depth which ranged from 500 to 4000 nm, where the slopes of the initial unloading curve are used to obtain the elastic moduli. These force-dependent results of EHertz values are in accordance with those reported by Gupta et al. (2007), showing that the adhesion is an important factor which interferes with the estimation of sample modulus, especially when the indentation depth is shallow or the applied force is small. It should be pointed out that Figs. 1 and 2 plot data from eight indents of one location on each sample and that the results from the other two indent locations are similar and not shown.
Fig. 1. Force–displacement curves of two PDMS samples with different base-agent ratios (10:1, 20:1), showing negative forces near zerodisplacement indicating the existence of adhesive forces.
Fig. 2. The Young’s moduli estimated from the Hertz model, showing variation with the maximum applied indentation force.
Using the hybrid model to evaluate the modulus–force relation with Eq. (8), Fig. 3 presents the theoretical descriptions for the decreasing variation of the modulus EHertz with the maximum applied force Pmax. The adhesion work Dc, representing the influence of adhesive force Padh on the indentation contact, determines the descending rate of the modulus with the applied force, as shown in Fig. 3a. This effect is more obvious when the maximum applied force is relatively small in these Dc-dominated curves, where the various amounts of adhesive work produce different estimations of the elastic modulus as the indents are shallow. The other independent parameter, EJKR, mainly determines the modulus levels of EHertz evaluation, as shown in Fig. 3b where the adhesive work is kept at a certain level.
Fig. 3. Illustration of theoretical EHertz Pmax relation evaluated by Eq. (8) of the hybrid model, as dominated by (a) the adhesion work Dc, and (b) the modulus EJKR.
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This means these two parameters dominate a family of theoretical curves which cover the relation of EHertz Pmax as estimated by the Hertz model. Therefore, the solution of the elastic modulus EJKR for the soft materials becomes a process of finding an appropriate curve from Eq. (8), with two constant parameters to fit the data of EHertz Pmax from the nanoindentation test, in which the adhesion work Dc can also be evaluated without special measurement of adhesive force. Fig. 4 shows two analytical curves that fit well with our experimental data of EHertz Pmax relations, where the curve parameters EJKR and Dc are attached as the force-independent elastic modulus and the adhesion work, to represent the intrinsic mechanical behaviors of the PDMS material. The two estimated values of EJKR, as illustrated by the dashed lines of constants, show that EHertz tends to converge to EJKR as the maximum applied force becomes larger, indicating that the effect of surface adhesion becomes less in the cases of larger forces or deeper indents. In spite of this, EHertz will usually always be higher than EJKR for any indentation depths, which proves that the Hertz analysis without adhesion consideration normally overestimates the sample modulus (Johnson et al., 1971; Lim and Chaudhri, 2003). Through statistical analysis of indentation results of PDMS tests, Fig. 5 gives the mean values of the Young’s modulus EJKR with small standard deviations, for different base-agent ratios of the polymer. Taking the 10:1 ratio as an example, the estimated modulus EJKR = 1.74 ± 0.02 MPa is between the values of 1.9 ± 0.3 MPa (Cao et al., 2005) and 1.5 MPa (Gupta et al., 2007). Also for this case of base-agent ratio, the adhesion work Dc = 209 ± 9.1 mJ/m2 obtained from our PDMS-sapphire tip system, is within the range of the typical values (100–500 mJ/m2) reported by Bietsch and Michel (2000). The difference of results is due to several reasons: the curing/cooling time and temperature of materials, the sample-tip system, et al. For these two sorts of PDMS samples with different cross-linker, significant difference is observed in EJKR (p = 0.0005). However, the p-value of Dc (p = 0.0875) is slightly larger than 0.05, implying that the surface adhesive energy of PDMS material may be more consistent with the property
Fig. 4. The analytical curves determined by the appropriate parameters Dc and EJKR to fit the experimental data of EHertz Pmax.
Fig. 5. Statistical results of the Young’s modulus and adhesion work for the PDMS with different base-agent ratios.
of the base component. But it is still too early to say there is no significant difference between Dc, before further investigation is carried out. In addition, tests with different preloads (from 0.1 to 2 lN) in nanoindentation show that the measured moduli vary less than 1%, which suggests that the results are basically independent of the preload forces. Furthermore, different loading/unloading rates from 50 to 200 nm/s have been implemented in our tests, showing that the force-dependent phenomenon of the Hertzian modulus keeps its appearance in the results (Fig. 6). The elastic modulus EJKR, however, changes less than 6% and the adhesion work Dc varies less than 2%, which shows that the viscoelastic property has not much influence on the results. Besides, the measured E was found to increase with the increasing loading/unloading rate, similar to the conclusion drawn by Lim and Chaudhri (2003). The loading/unloading rates used in our experiment were smaller than the rate of 250 nm/s that was used by Gillies and
Fig. 6. Results of the Young’s modulus and adhesion work for the 10:1 PDMS with different loading/unloading rates.
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Prestidge (2004), where only slight hysteresis of the force curve was observed, implying that the viscoelastic property of PDMS could be neglected in this case of low loading speed and the polymer could be considered as an ideal material with mechanical behavior of linear elasticity (Cao et al., 2005; Carrillo et al., 2005). 5. Conclusions This study provides a theoretical model combining Hertz procedure and JKR analysis, which not only describes the descending phenomenon of the elastic modulus estimated from the Hertz model changing with the maximum applied force, but also predicts the analytical curves to fit the experimental data with two dominating factors, the Young’s modulus and the adhesion work in the JKR model. Therefore, in data analysis of nanoindentation test for soft materials, the Hertz estimation can continue to be used as the common procedure to obtain the modulus–force relation from the multiple force–displacement curves. The force-independent elastic modulus of the soft matter, as well as the adhesion work on the sample surface, can be evaluated as two constant parameters by fitting the experimental data, without need to know the adhesive force in advance, because the analysis of sample-indenter interaction has been taken into account in this hybrid model. Acknowledgements The authors acknowledge Y.Y. Huang and H.W. Ma for helpful discussions and M. Qiang for technical assistance in nanoindentation tests. The research is supported by the National Basic Research Program of China through Grant Nos. 2007CB935602 and 2011CB809106 and the National Science Foundation of China under Grant Nos. 90607004, 10872008, 11002003 and 11072004. The authors are highly grateful to the anonymous reviewer for his/her careful proofreading and helpful suggestions, which have remarkably improved the manuscript. References Bietsch, A., Michel, B., 2000. Conformal contact and pattern stability of stamps used for soft lithography. Journal of Applied Physics 88 (7), 4310–4318.
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