The Use of a Vickers Indenter in Depth Sensing Indentation for Measuring Elastic Modulus and Vickers Hardness

Dpto. de Eng. Metalúrgica e de Materiais and Dpto. de Eng. Mecânica, Escola Politécnica, USP, Av. Prof. Mello Moraes, 2463, 05508-900 São Paulo SP, Brazil Dpto. Acadêmico de Mecânica, Centro Federal de Educação Tecnológica do Paraná Av. Sete de Setembro, 3165, 80230-901 Curitiba PR, Brazil c Núcleo de Pesquisas Tecnológicas, Universidade de Mogi das Cruzes Av. Dr. Cândido Xavier de A. Souza 200, 08780-210 Mogi das Cruzes SP, Brazil


Introduction
Load and displacement sensing indentation technique allows determining mechanical properties at penetration depths as low as 20 nm, avoiding the substrate effect on the measurements 1 .The possibility to carry out tests in so small scales makes this technique one of the tools chosen to characterize mechanical properties of thin films, coatings, second phase particles and magnetic hard disk recording media [2][3][4] .
A well defined indenter geometry is required to get well defined indentation impressions.A perfect tip shape is difficult to achieve.Berkovich is a three-sided pyramid, and provides a sharply pointed tip, compared to the Vickers indenter, which is a four-sided pyramid and has a slight offset (0.5-µm) 5,6 .This is the main reason why three-sided Berkovich indenters are used in depth sensing indentation machines.However, any indenter with a sharp tip suffers from a finite but an exceptionally difficult to measure tip bluntness.Experimental procedures have been developed to correct the tip shape, of both Vickers and Berkovich indenters 1,[7][8][9][10] .
When tests are carried out with Berkovich indenters, the registered data can be analyzed using the method pro-posed in 1992 by Oliver and Pharr 7 , which has its origins in an earlier treatment by Doerner and Nix 11 .Nowadays, both methods are accepted for the analysis of the indentation data by the ISO/FDIS 14577-1 standard 12 .The Oliver-Pharr method consists in a series of loading cycles to avoid thermal drift and plastic reversion, while the Doerner-Nix method uses only a single cycle to obtain the indentation data.The Fischerscope H100 -depth sensing indentation machine uses the latter method, which is less time consuming and simpler but takes into account only a few data points, leading to greater inaccuracy.Moreover, the Fischerscope H100 uses the constant hardness calibration method 8,13 to correct the indenter tip shape, which is not adequate since work-hardening may happen during the test.
The Oliver-Pharr method is widely used in depth sensing indentation machines with Berkovich indenter.However, it is equally applicable to the case depth sensing indentation using a Vickers indenter, with good results as mentioned in literature 8, 14,15 .The depth sensing indentation machine used in this work analyzes indentation data using a software based on the Doerner-Nix method, and uses an incorrect area function to describe the indenter tip shape, leading to overestimated elastic modulus and hardness values due to ISE effectindentation size effect 16 .Additionally, the Fischerscope apparatus software does not give Vickers numbers, which are useful to compare with wellknown data of phases and micro-constituents given in literature.
The present work aims to develop a computational routine based on the Oliver-Pharr method for measuring more precise values of elastic modulus and to obtain Vickers hardness numbers, using a Fischerscope H100 -depth sensing indentation apparatus, equipped with a Vickers diamond indenter.

Theoretical Aspects
Depth sensing indentation technique consists of printing an impression on the material surface by applying a known load with an indenter of known geometry and subsequently analyzing the load vs. displacement data.Equations from the elastic punch theory can be used to determine the elastic modulus, E, and hardness, H, provided that the following conditions during the initial withdrawal of the indenter are ensured: • the material's recovery follows an elastic behavior; • the contact area between the indenter and the specimen remains constant.In this case, the Sneddon's solutions 17,18 for the case of the indentation of an elastic half-space for a cylindrical punch approach to the elastic behavior.One of the Sneddon's solutions leads to a simple relation between the load, P, and the penetration depth, h, of the form: (1)   where a is the radius of the cylinder; µ is the shear modulus; and ν is Poisson's ratio.Knowing that the area of the contact circle projected onto the surface, A c , is equal to πa 2 and that the shear modulus is related to the elastic modulus in the following way: (2) and substituting (2) in (1) and differentiating the obtained expression with respect to h: (3) one can obtain the contact stiffness S = dP/dh.The elastic modulus, E, can be taken directly from the initial unloading slope, S, when Poisson' ratio, ν, and contact area, A c , are given.The latter can be measured independently as a function of contact depth, h c .
As the elastic modulus of the indenter is not infinite, Eq. 3 should be written in terms of combined elastic modulus specimen/indenter (E r ), which is, according to Hertz Equation: (4) where E, E i , ν and ν i are the elastic moduli and Poisson's ratios of the specimen and indenter, respectively.
Therefore, for the indentation of a plane surface of a semi-infinite elastic solid by a rigid punch, Eq. 3 can be rewritten: or (5) The above Equation shows that, for axisymmetric indenters, the relationship between unloading stiffness, S, and contact area, A c , does not depend upon indenter geometry.Pharr, Oliver, and Brotzen 19 have shown experimentally that the analysis used for determining elastic moduli and contact areas from contact stiffness S is not limited to punch geometry.Using finite elements method, King 20 has introduced to Eq. 5 a correction factor for non-axisymmetric indenters: In Figs 1 and 2, h max corresponds to the maximum depth, a to the half-diagonal projected on the surface, h f to the residual depth, h c to the contact depth, and h s to the deflection depth.In Fig. 3, S corresponds to the unloading stiffness for h = h max .
As the unloading from h max to h f is elastic, one of the Sneddon's solutions 8 , for a conical punch (Figs. 1 and 2), shows that the deflection of the surface at the contact is: (7) Another Sneddon's solution shows that, for h = h max , the load is related to the elastic depth: (8) Substituting (7) in ( 8) and noting that the contact area of interest is that at peak load, P = P max , it follows: To determine the contact depth from experimental data, one can note that in Fig. 1:   Considering that usually the indenters are not conical, but square or triangular base pyramids (Vickers or Berkovich indenters) it must take into account that for any revolution paraboloyd (including Vickers indenters), ε is about 0.75 19 .
As one can see in Fig. 4a, the contact area, A c , can be expressed as a function of the diagonal d 21 : (12) Substituting this expression in Eq. 6, one obtains the diagonal as a function of the indentation parameters:

Equipment and indentation procedure
A Fischerscope H100 -depth sensing indentation machine equipped with a Vickers diamond indenter, manufactured by Helmut Fischer GmbH, was used.Such equipment allows applying loads from 1 mN to 1000 mN and registering penetration depths as a function of applied loads.Other important testing characteristics of the equipment are: • control of loading and unloading rates (dP/dt and d√ P/dt); • long or short delay times at maximum load; • minimum step among indentations equal to 10 mm; • definition of the number and position of the indentations (mapping).Table 1 shows the main testing conditions.In all cases the indentation depth did not surpass 10% of the specimen thickness (film or substrate), avoiding any contribution from the substrate 22 .

Materials
The materials used in this study were chosen in order to include a wide range of hardness and elastic modulus values: PVD TiN and HVOF WC-12%Co coatings, alumina, AISI D2 and AISI H13 tool steels, AISI 316 stainless steel, aluminum, gold, soda-lime glass, Au-12%Pt and Co-25%Cr odontological alloys.Table 2 presents elastic moduli for these materials reported in the literature.
Bulk specimens were mechanically polished and finished until 1 µm diamond paste.The AISI H13 tool steel specimens coated with TiN and with WC -12%Co were carefully cleaned using an ultra-sound apparatus in alcohol bath and subsequently dried in warm air.The WC-12%Co coating thickness is about 300 µm, while the TiN coating thickness is about 5 µm.

Analytical procedure used by Fischerscope equipment
Fischerscope procedure for measuring hardness and elastic modulus consists in estimating the initial slope from unloading data based on the Doerner-Nix analytical treatment and subsequently calculating the correct depth through a calibration curve, previously established for tip-shape indenter correction, frequently called hardness constant method 8, 13 .
Figure 5 shows the load-displacement data for alumina.Considering the unloading curve, the initial unloading slope, defined as contact stiffness, dP/dh = S, is determined taking the upper 1/3 of the unloading data and fitting it by a linear relation of the form: (16)   where .
A straight line is fitted to the unloading curve down to its first 1/3 region and then extrapolated to zero, determining the depth h s and stiffness S.
Fitting the unloading data by a linear relation, yields S = 0.4719 mN/nm and h s = 84 nm.Because the Vickers indenter tip is not perfect, h max is replaced by a corrected depth, h corr , in a depth function of the form: (17)   where h corr corresponds to equivalent depths from an ideal Vickers indenter, and k and n are empirical parameters.The surface area of the indenter is determined through the expression: for Measuring Elastic Modulus and Vickers Hardness (18)   In the above expression, the factor 26.43 is used instead of 24.5 because the universal hardness H u , taken for tip shape calibration, is based on surface area of an ideal Vickers indenter.
The surface area, A, together with the initial stiffness, S, is then substituted in Eq. 6, yielding a value of 325.4 GPa for reduced modulus, E r .Using Eq. 4 leads to E = 440 GPa for alumina.

Oliver-Pharr Analytical Procedure
Referring to the unloading curve obtained experimentally for alumina (Fig. 5), one can note that it presents a non-linear behavior.Pharr, Oliver, and Brotzen 19 showed that even metals present unloading curves with non-linear behavior and that these curves are better described by nonlinear relationships.
In the Oliver-Pharr method 7 , the data taken from the upper portion of the unloading curve (Fig. 5) are fitted by a power-law relation of the form: (19)   where P is the indenter load, m and A are empirical constants determined after unloading data fitting, h f is the residual depth, and h is the elastic displacement.
Although fittings corresponding to 80% of the unloading curve are recommended in the Oliver-Pharr method, the fitting used in the present work was only 52%.When trying to fit more than 52% of the data of the unloading curve, it was verified that the contact stiffness, S, was underestimated.Probably, this effect is associated with the limitation of the Fischerscope equipment, which does not allow the inclusion of multiple cycles for minimizing the effects of thermal drift and plastic reversion.
Fitting the unloading data of Fig. 5 by a power law relation, it is obtained m equal to 1.386620 and A equal to 0.053065, as one can see in Fig. 6.Rearranging Eq. 19, the residual depth can be predicted: When P = P max = 50 mN, the maximum penetration depth, h =h max , is 270 nm.Substituting the values of m, A, P and h in Eq. 20, a value of 130 nm is obtained for h f .The contact stiffness, S, is obtained by means of Eq. 19 differentiating P with respect to h: (21)   Substituting the values of m, A and h f in Eq.21, the contact stiffness S = 0.4966 mN/nm is obtained.
The indenter-specimen contact depth, h c , can be obtained using one of Sneddon's solutions given in Eq. 11.Substituting the values of ε, P max and S in the referred expression, a value of 195 nm is obtained for h c .
The projected contact area, A c , can be determined through the indenter area function, shown in Fig. 7.
This calibration curve for tip shape correction was previously established by means of an iterative procedure, called the constant elastic modulus method, described by Oliver-Pharr 7 .Fine fittings of the empirical constants were performed using an alternative method 32 consisting of comparing the indentation diagonals measured by image analysis with that predicted through Eq. 13.Then, substituting h c in the area function of the Vickers indenter: Then, after substituting this value together with that for contact stiffness in Eq. 6, a value of 293.2 GPa for reduced modulus E r is determined.Hence alumina modulus can be determined by using the Hertz expression (Eq.4):

Elastic Modulus
Table 3 compares the results of elastic moduli measured using the developed computational routine with that yielded by the software of the equipment, together with values reported in the literature.
With respect to the values determined using the proposed routine, a good agreement is observed when compared with that reported in literature.
On the other hand, Table 3 shows that elastic moduli measured by the equipment are overestimated for materials with high elastic moduli.One can see in Table 3 that elastic moduli determined for alumina, through the Fischerscope routine, are greater overestimated the less is the applied load.
This effect is as indentation size effect -"ISE effect" 16 , indicating that the analytical procedure used by the equipment introduces inaccuracies when low loads are applied.On the other hand, the proposed routine based on    the Oliver and Pharr method gives elastic moduli practical independent of the applied load, as one can see in Table 3.

Vickers Hardness
Figure 8 shows a set of measurements undertaken 5, 15, 25 and 35 µm beneath the nitrided surface of a H13 tool steel specimen and Table 4 compares the diagonals measured by image analysis with those determined by means of Eq. 13, based on the calculated S and E r values.It can be seen that the values given by both methods are very close.
Even in the case of large indentations (Fig. 9), the measured hardness values are in good agreement with those determined using Eq. 13, as can be seen in Table 5.
Finally, Table 6 shows the Vickers hardness numbers determined through the proposed routine for different materials presenting different elastic-plastic behaviors and in some cases obtained with different applied loads.The hardness values shown in Table 6 agree fairly well with the well-known Vickers hardness numbers of the different listed materials.
Therefore, artifacts in the measurements of the Vickers hardness owing to ISE effect alone can be ruled out.The new routine adapted to Fischerscope -depth sensing indentation apparatus allows measuring Vickers hardness of tribological coatings using very low loads without the necessity to derive expressions that relate coating hardness to substrate hardness, and to deal with the composite response of film and substrate.

Conclusion
• The proposed routine gives a better estimate of elastic moduli of materials, when compared to the values given by the equipment software.• The lesser the load, the greater are the differences between the two methods.• Very low loads can be used for determining elastic moduli of very thin layers, as the ISE effect was minimized.• The elastic contact theory equations and the Oliver-Pharr approach offer an optimum prediction of the indentation diagonals and, consequently, of the Vickers hardness numbers of the tested materials.

Figures 1
Figures 1 to 3 show the main parameters used in analyzing indentation data.In Figs1 and 2, h max corresponds to the maximum depth, a to the half-diagonal projected on the surface, h f to the residual depth, h c to the contact depth, and h s to the deflection depth.In Fig.3, S corresponds to the unloading stiffness for h = h max .As the unloading from h max to h f is elastic, one of the Sneddon's solutions 8 , for a conical punch (Figs.1 and 2), shows that the deflection of the surface at the contact is:

Figure 1 .
Figure 1.Profile of the surface before and after indentation.

Figure 2 .
Figure 2. Main parameters used in analyzing unloading vs. indenter depth curves.

( 13 )Figure 3 .
Figure 3. Schematic representation of load-displacement data for a depth sensing indentation experiment.

Figure 8 .
Figure 8. Vickers impressions taken beneath the nitrided layer of a H13 tool steel specimen.Load = 50 mN.

Figure 7 .
Figure 7. Area function, which takes into account roundness of the Vickers indenter tip, determined previously by the constant elastic modulus method and image analysis.

Figure 6 .
Figure 6.Unloading curve fitted by a power law relation of the form P = A (h -h f ) m .

Table 2 .
Elastic moduli reported in literature for the used materials.

Table 3 .
Comparison of elastic moduli measured by means of the analysis procedure based on Oliver Pharr method with those given by Fischerscope machine.Each measurement corresponds to the average of twenty curves.

Table 4 .
Indentation diagonals beneath the surface in a nitrided specimen, determined by means of Eq. 13 and measured by image analysis.Maximum load: 50 mN.

Table 5 .
Comparison of Vickers hardness number determined using Eq.11 with that measured by image analysis for an aluminum specimen.Load = 255 mN.