Results and Discussion

The nanomechanical mapping of IIR surface is shown in Figure 3.15 [Nagai et al., 2009]. As explained earlier, Young’s modulus and adhesive energy mappings were obtained together from an artifact-free true topographic image. It seems that the analysis result is acceptable, whereas, in reality, several problems exist. Figure 3.16 shows two typical δ–F plots selected from Figure 3.15. Both calculated values seem to be accurate. However, the experimental and JKR theoretical plots are almost identical in Figure 3.16a, whereas those in Figure 3.16b show a significant discrepancy. Thus, we could not believe that all the data analyzed were correct.

Thus, it is necessary to discuss why such a difference emerged even on a single specimen surface. It is an appropriate argument based on previous reports on NR [Watabe et al., 2005; Nakajima et al., 2006] that a crosslinked rubbery surface some-times has nanometer-scale inhomogeneity. However, we should not forget the fact that there is a clear discrepancy between the experimental and theoretical plots. Such a discrepancy was never observed in the case of NR.

FIGURE 3.15 Nanomechanical mappings of IIR analyzed by the JKR method, showing the images of (a) adhesive energy, (b) actual height, and (c) Young’s modulus at the same position.

The measurement was performed using a conical probe (R= 20 nm).

TOWARD NANORHEOLOGICAL MAPPING 151

FIGURE 3.16 Two different δ–F plots selected from the same FV results as in Figure 3.15.

Curve (a) shows good agreement between the JKR theoretical curve and the experimental plot, whereas curve (b) exhibits some deviation between theory and experiment.

To elucidate the origin of the deviation, further investigation was carried out as fol-lows. Figures 3.17 and 3.18 show a series of δ–F plots from withdrawing (unloading) processes for IIR and PDMS10, respectively. In both cases, δ–F plots were taken at a single point on each surface with different scanning velocities. Other experimental conditions were kept identical. It is clear from Figure 3.17 that δ–F plots for IIR varied remarkably depending on the velocity. The JKR theoretical curves were also superimposed on the slowest (200 nm/s), middle (6.0␮m/s), and fastest (60 ␮m/s) velocities. The slowest and fastest theoretical curves were in sufficiently good agree-ment with the respective experiagree-mental curves. On the other hand, the theoretical result for 6␮m/s velocity failed to reproduce the experimental curve.

A more detailed examination tells us that the experimental curves for the slower velocities indicate soft behavior, whereas those for the faster velocities indicate rigid behavior. This observation was confirmed by the Young’s modulus. The velocity of 200 nm/s gave 3.2 MPa, while that of 60␮m/s gave 10.2 MPa. We speculate that this behavior is related to the glass–rubber transition seen in the bulk mate-rial at room temperature as shown in Figure 3.14 [Nakajima et al., 1997]. The glass–rubber transition occurs at the nanometer scale. The group of slower veloc-ities from 200 to 600 nm/s exhibit a rubbery characteristics. Considering a slight

-30 -20 -10 0 10

force (nN)

-400 -300 -200 -100 0 100

sample deformation (nm) 60 µm/s

20 µm/s 6 µm/s 2 µm/s 600 nm/s 200 nm/s

JKR curves

FIGURE 3.17 Dramatic variation of δ–F plots obtained from IIR at different scanning veloc-ities using a conical probe (R= 20 nm). The JKR fitting curves for the 200-nm/s and 60-␮m/s experimental plots show good agreement with experiment; the JKR plot for the 6-␮m/s curve shows substantial disagreement with experiment.

increase in adhesive energy observed on increasing velocity, some type of velocity-dependent energy dissipation should be involved in the observed phenomena, which will be the subject of future discussion. Similarly, the velocities near 6.0␮m/s exhibit a transition state and the faster velocities (20 and 60␮m/s) exhibit glassy characteristics.

JKR curves -6

-4 -2 0 2

-150 -100 -50 0 50

sample deformation [nm]

30 µm/s 3 µm/s 100 nm/s

force (nN)

FIGURE 3.18 δ–F plots of PDMS10 at different scanning velocities using a conical probe (R= 20 nm) with JKR curves fitted to each experimental plot.

TOWARD NANORHEOLOGICAL MAPPING 153

It is possible to test the validity of JKR analysis using Tabor’s equation [Tabor, 1977; Johnson and Greenwoods, 1997]. The dimensionless parameter µ is expressed as

µ=

 Rw² E²D³₀

1/3

(3.14)

where D0is referred to as the equilibrium separation between surfaces and is in the range 0.3 to 0.5 nm. It is said that JKR analysis is appropriate if µ > 5. Assuming that D0= 0.3 nm, we estimated the Tabor parameters for velocities of 200 nm/s (rubber) and 60␮m/s (glass). Their values were 190 and 62, thus fulfilling the criterion. We can conclude that these regions are within the applicable limit of JKR analysis. However, as with Figure 3.16b, δ–F plots for 2.0 and 6.0␮m/s showed heavily curved features and therefore could not be fitted accurately with JKR theory. This is interpreted as due to larger energy loss or surface-specific energy dissipation.

In Figure 3.18 PDMS10 shows results that are less velocity dependent than those of IIR. Three curves coincide and JKR analysis reproduced the experimental results accurately. PDMS10 can be regarded as an almost perfect elastic body, as reflected by the small loss tangent at room temperature (see Figure 3.13). Thus, we may expect that practically elastic materials are well described within the framework of JKR theory, independent of the scanning velocities. The nanomechanical mapping of PDMS10 is shown in Figure 3.19. The measured E and w are almost homogeneous. JKR the-oretical curves were also in good agreement with the experimental curves. Thus, nanomechanical evaluation could be regarded an being more accurate in this case.

Hence, we conclude that JKR analysis is only applicable to sufficiently elastic mate-rials and that it falls beyond an applicable limit for viscoelastic matemate-rials such as IIR.

FIGURE 3.19 Nanomechanical mappings obtained from PDMS10 using a conical probe (R= 20 nm). Contrary to Figure 3.15, the mappings of (a) adhesive energy, (b) topography, and (c) Young’s modulus at the same position display homogeneous features.

10

FIGURE 3.20 (a) Young’s modulus and (b) adhesive energy of IIR (black), PDMS3 (dark gray), and PDMS10 (light gray) at different scan rates are plotted as a function of maximum sample deformation. All results were obtained using a conical probe with a 20-nm tip radius, R.

For a more detailed discussion concerning viscoelastic effects, we plotted the relationship between E and w determined by the two-point method and the maximum sample deformation as shown in Figure 3.20. In a practical experiment, we varied the scanning velocity and the maximum loading force. The larger the maximum loading force, the larger the maximum sample deformation attained. Because the two-point method failed to describe experimental curves accurately under some conditions, as discussed above, the E or w obtained must be treated as an apparent value. However, interestingly, Young’s modulus values changed monotonously on maximum sample deformation for IIR, whereas, in contrast, PDSM3 or PDMS10 gave almost constant modulus values.

TOWARD NANORHEOLOGICAL MAPPING 155

A similar observation could be made for the adhesive energy. The adhesive energy must be intrinsically constant, independent of any experimental parameters, as seen in the case of PDMS10. However, results for PDMS3 showed a small variation and those for IIR showed a large variation. This is, of course, due to the failure of JKR analysis, but we suspect that these variations contain fruitful information about nanometer-scale rheological phenomena.

It has been proven that adhesive energy can be a function of crack propagation speed according to Maugis’s theory [Barquins and Maugis, 1981; Barquins, 1983].

Here in the case of AFM experiments, crack propagation occurs at the interface between the probe and sample surfaces. According to this theory, the strain-energy release rate G is expressed as a dimensionless function of crack propagation speed v and Williams–Landel–Ferry (WLF) shift factor aT as follows:

G− w = wφ(aTv) (3.15)

Here G can be treated as an apparent adhesive energy, and the condition G= w is accomplished in the equilibrium state. The function φ(aTv) represents viscoelastic loss at the crack front and is indeed a logarithmic function of v. Detailed analysis indicating that this equation supports a small increase in adhesive energy at elevated scanning velocity for PDMS3 is given elsewhere. However, the theory does not suf-ficiently explain the phenomena observed in IIR. We should consider a probe shape effect for further discussion. Actually, we note that the experimental data in Figures 3.15 to 3.20 were taken using a conical probe. In this case a small change in radius of curvature R due to a change in contact area becomes trivial compared to a serious shape change. We treated the tip of the conical probe as spherical to a first approx-imation, but we must consider the true conical shape when the sample deformation becomes large enough.

To check more strictly the loading–force dependence shown in Figure 3.20, we used a spherical probe with R= 150 nm for PDMS3 (somewhat viscoelastic) and PDMS10 (elastic) as shown in Figure 3.21. The experimental parameter was the maximum loading rate and thus the maximum cantilever deflection. Because of the spherical probe, the value of R in the JKR theory was fixed. The result for PDMS10 in Figure 3.21a showed a good trace with changes only in the maximum loading force.

Here the JKR analysis worked very well. In addition, loading and unloading processes exhibit the same trace. On the other hand, the results for PDMS3 in Figure 3.21b show a hysteresis between the loading and the unloading processes. The hysteresis became larger when the loading force was increased. The degree of discrepancy between the experimental and theoretical curves is also larger. This observation brings us to the conclusion that viscoelastic materials are strongly affected by the maximum loading force due to mechanical loss. The values analyzed are summarized in Table 3.2.

The Young’s modulus decreases in both cases. The magnitude of the decrease is substantially smaller PDMS10 than for PDMS3, although viscoelastic character began to emerge even in the case of PDMS10 when the loading force became larger. The loading force had less effect on the estimation of adhesive energy.

FIGURE 3.21 Maximum loading force dependence of δ–F plots obtained from (a) PDMS10 and (b) PDMS3 using a spherical probe (R= 150 nm).

We have discussed the recent progress in nanomechanical property evaluation using AFM based on JKR analysis. Sufficiently elastic materials such as PDMS10 can be treated within the JKR framework, whereas viscoelastic character causes a large deviation from JKR theory, as observed in the case of PDMS3 or IIR.

TABLE 3.2 Young’s Modulus and Adhesive Energy Calculated from Two Sets of Force–Deformation Plots in Figure 3.21

PDMS10 PDMS3

Maximum deflection (nm) 3.0 5.0 10.0 3.0 5.0 10.0

E (MPa) 2.6 2.5 2.4 2.1 2.0 1.7

w (J/m²) 0.044 0.044 0.044 0.048 0.048 0.049

REFERENCES 157

Experiments in which scanning velocity and maximum loading force were varied confirmed the important role played by viscoelastic effects in the latter two materials.

We are continuing to develop a theory that can treat viscoelastic materials based on Greenwood’s idea [Greenwood and Johnson, 1981]. However, we feel the argument delineating the applicable limit of JKR theory reported in this chapter will help better understanding the future nanorheological analyses.

3.5 CONCLUSIONS

The nanopalpation technique, nanometer-scale mechanical and rheological mea-surement based on AFM, was introduced and shown to be useful in analyzing nanometer-scale materials properties for the surfaces and interfaces of polymer nanoalloys and polymer-based nanocomposites. It enables us to obtain not only struc-tural information but also mechanical information about a material at the same place and time.

Acknowledgments

The authors are grateful to all our collaborators, who provided many interesting specimens. We also thank all our laboratory students. Without their help we could not have achieved these results. Finally, we are grateful for financial support from the National Institute of Advanced Industrial Science and Technology, the Japan Chemical Innovation Institute, and the New Energy Development Organization as a project in the Nanotechnology Program of the Ministry of Economy, Trade, and Industry of Japan.

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