A simple method to minimize displacement measurement uncertainties using dynamic nanoindentation testing

A new simple indentation method is presented, minimizing strongly the influence of uncertainties linked to the displacement measurement, in the context of homogeneous materials indented by a sharp tip. Based on the determination of the derivative of the contact depth with respect to the indentation depth, this method depends indirectly on the displacement measurement, making the technique less sensible to displacement uncertainties. After being validated on flat samples (fused silica, PMMA), the method was tested on a rough specimen (anodized aluminum, Ra = 1.7 μm) and on two heated flat samples (fused silica, PMMA, 60°C). The mechanical properties determined by this new technique are more precise than properties measured by classical nanoindentation measurement.


Introduction
Subsurface measurement of hardness and elastic modulus is of primary importance for tribological applications. For instance, most of wear laws are based on the hardness of contacting surfaces and elasto(plasto)hydrodynamic applications are strongly affected by both the hardness and the elastic modulus [1]. The nanoindentation technique is the most popular technique to determine the surface mechanical properties of materials [2]. The principle consists in applying a small normal load to an indenter of known geometry, and to record the applied load and the penetration of the indenter. The technique can be used to measure surface mechanical properties of homogeneous materials or layered materials [3,4]. The main mechanical properties measured by nanoindentation are the hardness H and the elastic modulus E [5][6][7][8]: Where P is the applied load, A c is the projected contact area, '* c E is the contact elastic modulus, '* i E is the reduced elastic modulus of the indenter, and Q and i Q are the Poisson ratio of the sample and the indenter respectively. S is the harmonic contact stiffness which can be measured at maximum load by a static method or continuously versus the indentation depth with a dynamic method (often called Continuous Stiffness Measurement [9]). The projected contact area A c is related to the contact depth h c with the following geometrical relation (for a perfect pyramidal indenter): 22 tan cc A h ST (4) Where ș is the angle of the equivalent conical indenter. The relation between the contact depth and the indentation depth is non-trivial. Pile-up or sink-in phenomena have to be taken into account, in relation with material behavior. Up to date, there is no general method that bypasses this problem.
There exists also a second source of uncertainties that needs to be considered to perform accurate nanoindentation measurements. They are linked to indentation displacement measurement. For instance, the thermal drift can induce uncertainties on the displacement measurement [10]. There exist some procedures to correct the thermal drift when indentations are performed at room temperature, but, at non-ambient temperature, the thermal drift is too important and perturbs greatly the displacement measurement and so, the determination of mechanical properties. In a recent paper, we have proposed a new method to reduce these uncertainties by using the measurement of second harmonic amplitude of the dynamic indentation signal [11]. Nevertheless, this technique requires to adapt the apparatus instrumentation, which is not accessible for most of standard users. Another uncertainty is linked to the surface roughness. Indenting a rough surface may cause difficulties to extract precisely the mechanical properties by nanoindentation [12,13]. Since the contact may occur on a peak or a valley, the determination of the contact depth at small penetration depth may vary strongly depending on the indentation location [14,15]. To avoid such a problem, it is recommended to indent at high depth in order to minimize the roughness effect, or to polish the sample, but mechanical polishing modifies the hardness near the surface [16,17]. Some authors have developed methods in order to minimize the roughness influence on measured properties but most of these techniques require a large number of experiments [18][19][20].
All of these uncertainties are encountered in the context of tribological applications. For instance, a lot of systems work at non-ambient temperature or friction can induce near-surface temperature elevation, thus it is necessary to determine accurately the nano-hardness as a function of temperature to predict the wear evolution. Measuring elastoplastic properties of rubbed surfaces often imposes to cope with rough surfaces too. Therefore the development of new instrumented indentation methods in these difficult experimental conditions will help to obtain more reliable and accurate values of near-surface mechanical properties for tribological studies.
In this paper, a simpler method is presented, usable without any additional instrumentation, to minimize these uncertainties. This new method is based on the calculation of the derivative of the contact depth with respect to the indentation depth. First, the new expressions used to measure the mechanical properties are presented. Then, an experimental validation of the method on well-known and prepared samples is given. In the last part, examples of applications on a rough sample and at high temperature are detailed and discussed.

Standard methods
When a homogeneous material is indented by a perfectly sharp tip, the dynamic contact stiffness S is expected to be proportional to the indentation depth h, and the hardness and the elastic modulus are independent of the applied load [3,21]: In this case, the ratio of load divided by squared contact stiffness is also independent of the applied load and is given by [21]: Because this ratio is obtained directly from measured values, it is not affected by displacement measurement uncertainties. But, to go further, a second equation is required to determine the hardness and the elastic modulus. Generally, the chosen equation relates the contact depth to the indentation depth as proposed for example by Loubet et al. [22][23][24]: Where Į is a constant related to pile-up or sink-in which is equal to 1.2 for a Berkovich indenter, and h 0 is the equivalent height of the tip defect. This coefficient can be easily calculated by plotting the harmonic contact stiffness S as a function of the plastic depth h r' as shown in Fig. 1, with ' / r hh P S . With Eq. (7), it is clear that any errors related to the indentation depth measurement will strongly affect the calculation of hardness and Young modulus values.

An alternative method for homogeneous materials
Taking advantage of Eq. (7), the derivative of the contact depth with respect to the indentation depth appears to be independent of the indentation depth and the tip defect h 0 . From Eqs. (5), (6) and (7), we obtain: In this expression, the coefficients Į and ș are known. The term dh c /dh can be determined by a simple linear fit. Consequently, the ratio '* / c HE can be computed with the following equation: Combining with Eq. (6), the hardness and the elastic modulus are now given by: Where '* c E is calculated with Eq. 11. Similarly to the second harmonic method, thermal drifts effects should thus be minimized, because the calculations are not directly related to the absolute measurement of the indentation depth. Moreover, it can be noticed that the indenter tip defect estimation is no more necessary and the identification of the initial contact point has less influence on results. Nevertheless, this method can be used only if the indented material is homogeneous at the indentation scale to take advantage of the linear dependence of the contact depth with respect to the indentation depth. For instance, it is not possible to use such an approach when considering the indentation of thin films on substrates. Nevertheless, standard contact models also fail in this case as shown by Perriot et al [25].
In this paper the expressions (11) and (12)  Where İ is a constant parameter. In this case, the hardness and elastic modulus are expressed by the following equations: Where '* c E is calculated with Eq. 14.

A new simple indentation procedure
The test procedure is similar to standard nanoindentation experiments based on the standard dynamic mode (CSM

Application to smooth samples: fused silica and PMMA
In order to validate the method described above, experiments were performed on two well-known smooth and homogeneous materials at ambient temperature: fused silica and PMMA. A Nano Indenter SA2 ® equipped with a DCM head system (Agilent Technologies) was used. This load-controlled apparatus has high resolution in displacement and force measurements (0.2 pm and 1 nN respectively).

Rough sample
It is well known that sample surface has to be as smooth as possible to determine the mechanical properties with the best accuracy. In some cases, polishing the surface by mechanical or electrochemical methods may induce changes in the hardness value near the surface [26], which is prejudicial for the interpretation of nanoindentation measurements. To avoid such problems, it is  Moreover, as shown in Fig. 8, like for flat samples, the value of dh c /dh is easily calculable with a linear fit. As shown in Fig. 9, a larger scattering is observed when the standard method is used. With the new method proposed in this paper, the scattering is strongly reduced, and a mean hardness value can be extracted with more accuracy. Our interpretation is that the effects of roughness on the indentation depth are minimized thanks to the linear fit of the h c -h curve. This result is of primary importance because it highlights that it is possible to estimate the hardness of rough samples without any specific surface preparation.

Tests at non ambient temperature
During the past decade, important efforts have been devoted to the development of nanoindentation experiments under high temperature environment [28]. One of the most important challenges is to minimize the thermal drift induced by the device thermal regulation, which affects more specifically the indentation depth during the test. For instance, Fig. 10.a) shows load-displacement curves corresponding to ten nanoindentation experiments performed on a fused silica sample heated at 60°C.
These bad results are mainly due to the thermal regulation during the test which affects the displacement measurement. Nevertheless, it can be observed on Fig. 10.b) that the harmonic contact stiffness and the load are not perturbed by thermal regulation, which implies that the method proposed in this paper will be less affected by the thermal regulation.
In this section, experiments performed on fused silica and PMMA samples at non-ambient temperature are presented. The Nano Indenter XP® (Agilent Technologies) was used with a Berkovich diamond tip. The equivalent height of the tip defect h 0 was approximately 15 nm. This apparatus is equipped with a sample heater. Electrical components are protected by a thermal shield. One important drawbacks of this device is that the indenter can only be heated by contact with the sample, which results in an increase of the drift due to thermal dilation [29]. To minimize this effect, a test procedure proposed in the literature has been used. It consists in contacting the tip with the sample at very low load (100 μN) during approximately 3 hours, before making the indentation without losing the contact between the tip and the sample [30]. Further information about the apparatus are given in the documentation [27]. The maximum load applied to the specimens was 450 mN for fused silica sample and 300 mN for PMMA sample. The oscillation amplitude was set to 1 nm. The oscillation frequency was 32 Hz. The strain rate was constant with 1 03 . 0 s P P . Ten indentation tests were performed on each sample. The test temperature was 60°C. The tip heating procedure was used for each indentation test.
As shown on Figs. 11 and 12, the harmonic contact stiffness is linear and the P/S² ratio is constant versus the indentation depth from h j = 250 nm, in spite of the thermal drift which affects the displacement measurement. Consequently, the hardness can be calculated using Eqs. (11) and (12). In and 14.b)). The explanation of these better results is illustrated on Fig. 15 showing h c -h curves for fused silica and PMMA samples (mean value of 10 indents). As it can be seen, the contact depth is linear versus indentation depth. Consequently, the value of dh c /dh is easily calculable, and the mechanical properties are more precisely measured with this method.

Conclusion
In this paper, a new method is proposed to extract mechanical properties from nanoindentation testing.
It is based on the calculation of the derivative of the contact depth with respect to the indentation depth, which is simply obtained from the slope of a linear fit of the h c -h curve. As the method is no longer dependent of the absolute indentation depth measurement, it is less sensible to thermal drift, surface roughness, and tip defect. Nevertheless, it is important to keep in mind that the equations have been obtained with the strong hypothesis that the indented material is homogeneous, i.e. that the harmonic contact stiffness is proportional to the indentation depth and that P/S² ratio is constant versus depth.
The application of the method on two smooth samples (fused silica and PMMA) at ambient temperature has provided results in good agreement with classical measurement. Nanoindentation tests performed on a rough anodized aluminum sample has shown that this new method gives more precise results. Finally, it was shown that hardness can be accurately measured with this new method in the context of nanoindentation at elevated temperature whereas the standard method leads to strongly scattered results. The main advantage of this method is that the global slope of the h c -h curve is less affected by the sample roughness or by the thermal regulation than the standard one based on the absolute indentation depth measurement.
Thanks to these developments, it makes it possible for users of standard nanoindentation device to measure accurate mechanical properties in difficult experimental conditions such as not ambient temperature and rough samples. Fig. 1. Harmonic contact stiffness versus plastic depth for fused silica sample. The height of the tip defect is determined by extrapolating the stiffness curve on the plastic depth axis.

Research highlights
The new method is less sensible to nanoindentation thermal drift.
The new method is less sensible to contact point detection and tip defect.