On the use of nanoindentation for the determination of visco-elastic properties of polymers

//// Introduction

For several decades, spherical indentation has been used for the determination of materials characteristics such as elastic modulus, hardness and elastic- plastic properties [1-3]. One of the main reasons for the popularity of these measurements was that with a spherical indenter (see Fig. 1 for an SEM image) and sufficiently low loads essentially elastic deformations could be obtained. The contact problem can be solved relatively simply by solving well-known Hertz equations and the elastic constants of the material are easily calculated.

Such idea was very advantageous also for determination of visco-elastic properties of polymers, where low deformations would lead to suppression of irreversible flow of the material and the visco-elastic properties of the polymer could therefore be determined. This would be a great advantage over commonly performed indentation with Berkovich indenter, where immediately the limits of plastic flow are reached and the deformation is inevitably irreversible. Also, the stresses in many applications (e.g. damping components in civil and automotive engineering) susceptible to creep are relatively low and the viscoelastic deformations disappear some time after unloading. Therefore, the stresses in the indentation tests for the determination of the visco-elastic properties should also be low.

This Application note deals with several aspects of low load spherical indentation including long hold periods and demonstrates a way to calculated visco-elastic properties from these measurements Challenging aspects of low load spherical indentation on visco-elastic materials.

One of the main reason that hindered spread of low load spherical indentation was lack of suitable instruments providing sufficient force and displacement resolution. The situation has improved only recently with the introduction of the new generation of indentation systems such as the CSM Instruments Ultra Nanoindentation Tester (UNHT, see Fig. 2) with its patented active top referencing principle [1]. This instrument is capable to apply forces from the micronewton range and measure displacement from nanometer range with sufficient resolution. Another important issue in creep measurements was the phenomenon of thermal drift. This error in displacement signal, due to environment and the instrument itself, adds a false component in the displacement signal and yields erroneous results. This phenomenon becomes extremely important especially in the low load and low depth creep measurements, where hold periods are in the range of tens of minutes or even more. Furthermore, in order to remain in the visco-elastic regime, the deformations have to be reversible and the indenter displacement correspondingly small. This renders the problem of thermal drift more significant than in other types of measurement, where larger depths or much shorter indentation times are required.

//// Viscoelastic model

The procedure for obtaining of visco-elastic properties of the tested material was based on fitting the displacement evolution during a hold period at the peak load. The indentation was intentionally performed at high loading rates with load rise duration of 5 s. The hold period, following the load rise, was set to 30 minutes (1800 s). The length of the hold period was much longer than in many other cases thus better simulating real life conditions where the creep times are in the range of hours or even longer. The displacement signal was fitted using a formula proposed by Menčík et al [2] and the material was modeled by a series of springs, dashpots and Kelvin-Voigt bodies (Fig. 3).

The time depending behavior of this model in visco-elastic regime during a hold period at the peak indentation load can

be described using Menčík’s approach [3] as where h is penetration depth of the indenter, m and K are constanst depending on the geometry of the indenter (for a spherical indenter m = 3/2 and
K = 3/(4√R)), C0 is instantaneous compliance, Cj and tj are compliances and retardation times of Kelvin-Voigt bodies and the term is the so-called „ramp correction factor“, introduced by M. Oyen. This term takes into account definite loading time tR necessary to increase the load to its peak value.

This model can be easily adapted to various materials and measurement conditions, such as loading rate and hold period. Using this model for description of the results of creep measurements can also reveal information about the visco-elastic plastic materials that would be difficult to obtain by other methods such as dynamic mechanical analysis (DMA) in its indentation form. The DMA is based on relatively high frequency oscillations and determination of only two viscoelastic-plastic parameters, the storage and loss moduli. The creep measurements on the other hand, simulate more closely real applications where the component is subject to long term loading.

//// Experimental conditions

As a sample material, the widely used Polymethyl-methacrylate (PMMA) in form of a 4x4x3 mm large block, was used. The indentation experiments were performed using the CSM Instruments UNHT system equipped with spherical indenter made of ruby with nominal radius of 100 mm. The true radius of this indenter was verified on fused silica by a series of indentations at increasing peak loads. It was found that the radius was variable in the range of 0 nm to ~20 nm from the tip of the indenter while above this distance from the tip the radius reached its nominal value. The fitting formula (1) could therefore be used in its simple form with R = const.

The indentations were performed at several peak loads: 1 mN, 5 mN, 10 mN and 50 mN. The mean pressure in the PMMA at these peak loads calculated using Hertz relationships (the viscous component was neglect in the first approximation) is shown in Table 1. Compared to pressure conditions under Berkovich indenter, pressure under spherical indenter are significantly lower and one can therefore reasonably expect only elastic – and hence reversible – deformations. Therefore, the term visco-elastic rather than viscoelastic-plastic was used throughout this text.

In all cases, the peak load was reached in 5 seconds (except for the 50 mN load where the loading time was 15 seconds) and then maintained for 30 minutes to observe the increase in indentation depth due to time dependent behavior of the PMMA.

Although the forces used in the experiments might seem relatively high, the corresponding depth growth during the 30 min hold period varied between only 7 nm and 180 nm. Hence excellent resolution in displacement measurement and especially long term stability of this signal was a crucial factor in this research. Although the UNHT has an unmatched thermal stability, its thermal drift was nevertheless verified by a series of measurements at the same conditions as the intended creep experiments on PMMA: 5 seconds load rise followed by at least 30 minutes hold period with the same R =100 mm spherical indenter.

The results proved that the thermal stability is excellent and the thermal drift never exceeds 0.2 nm/min. Figure 4 shows a typical evolution of displa-cement signal during 30 minutes long creep period on fused silica at 10 mN load. This value is by far negligible during conventional measurements and therefore the thermal drift of the UNHT does not have to be corrected.

However, while most of thermal drift corrections in other indentation instruments are based on a supposed linear increase (or decrease) of the displacement signal, the thermal drift of the UNHT is directly measured. There are no suppositions about its evolution made – simply because the error is so small, that in a vast majority of cases the experimental error is much larger.

//// Creep measurements and fitting

All the creep measurements were performed with the same loading time 5 seconds (rise time tR in equation (2)) except for the 50 mN load. The loading rate was thus varying according to the peak load from 12 mN/min to 200 mN/min. The hold period, on the other hand, remained constant for all measurements (30 minutes). The visco-elastic properties could therefore be determined for a larger set of loading conditions. An average evolution of indentation depth growth during the hold period is shown in Fig. 5, together with thermal drift measured at the same conditions. The PMMA creep blue solid line is an average of three measurements and the gray area around indicates the scatter of the results. The displacement signal in this graph, similarly as in all other measurements present in this paper, is neither corrected nor processed by any other means. In this case, the error in displacement measured on fused silica is comparable to the scatter of the data (grey area around the PMMA creep solid line) and therefore does not need to be corrected.

Typical force-displacement curves from indentation experiments on the three decorative gels together with the PS-6C elastomers are compared in Fig. 6. Clearly there were differences between all four tested materials with the PS-6C elastomer being much stiffer than the three decorative gels. In the first approximation, this difference was also confirmed by the values of elastic modulus calculated by the O&P model.

For higher peak loads, the ratio between the displacement error and the depth growth during the hold period was even smaller thus rejecting the necessity of thermal drift correction.

The fitting of the creep data was done during the hold period but taking into account also the depth increase during the loading. The data were fitted according to formula (1) and due to the duration of the whole experiment (~30 minutes) a model including three Kelvin-Voigt bodies was used. Such model allows for proper description of all viscous components acting in such long creep measurement. A typical creep indentation measurement of a 1800 seconds (30 minutes) long hold period at 5 mN is shown in Fig. 6. Part a) of the figure shows complete creep data while part b) shows a detail of the first 200 seconds of the hold period. Other datasets (1 mN, 10 mN and 50 mN) were also fitted with similarly good results. In all cases, the relative error between the measured displacement and the fit was well below 0.5 % for the whole duration of the measurement. It means that the selected model (equations 1 and 2) can well describe the visco-elastic behavior of the PMMA at low load conditions.

All the indentation experiments were intentionally performed at low loads in order to remain in the elastic regime, where the flowing of the PMMA would cease. This was somewhat limiting for the creep measurement since its values for the lowest loads were in the range of less than 10 nm in 30 minutes. For such small depth increase, the displacement error cannot be predicted and hence the displacement not corrected – any attempts to correct such low displacement signal should be critically observed. However, due to unmatched thermal stability of the UNHT system, reliable long term creep measurements can be performed already for depth increase of ~20 nm. The graphs in Fig. 6 clearly show that equations (1) and (2) can be successfully used for fitting the displacement data, at least in the elastic regime. The fit of the measured data was very accurate both at the beginning of the creep, where short-term components in Eq. 1 are dominant, and at after the first 200 seconds where the longer term (longer relaxation times) components played the major role. The parameters of the fit, summarized in Table 2, can therefore be successfully used for further modeling by finite element analysis or other similar programs for prediction of mechanical response of materials.

//// Conclusions

Instrumented indentation has proved to be a powerful method for the measurement of extremely soft gels with properties and structure similar to biological tissues. The utility of the method stems from the ability to nondestructively measure small samples while yielding indentation data that can be analyzed to extract the mechanical properties of the sample. A great advantage of the CSM Instruments Ultra Nanoindentation Tester is that the system can be used for such measurements without any hardware modifications. Thanks to its unique design and features such as absence of thermal drift and excellent force stability, the UNHT system is a fast and easy way for testing of
extremely soft materials. All this makes the indentation method very attractive for testing in biomedical research facilities and hospitals.

//// Acknowledgements

The authors would like to express his gratitude to Prof. Jaroslav Menčík of University of Pardubice (CZ) for his invaluable help.

//// References

1. Nohava J, Randall, N. X, Conté N.: Novel ultra nanoindentation method with extremely low
thermal drift: Principle and experimental results, J. Mater. Res., Vol. 24, No. 3, Mar 2009, p.
873-882.

2. J. Mencik, L. H. He, M. V. Swain: Determination of visco-elastic material parameters of
biomaterials by instrumented indentation. J. Mech. Behavior Biomed. Mater. 2, 318 (2009).

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