Nanoindentation Hardness Measurement in Piling up

Available online at www.sciencedirect.com

Physics Procedia 40 (2013) 100 – 112

2nd European Conference on Nano Films: ECNF-2012

Nanoindentation Hardness measurement in piling up SiO2 coating M. Cabibbo*, D. Ciccarelli, S. Spigarelli Dipartimento di Ingegneria Industriale e Scienze Matematiche (DIISM), Università Politecnica delle Marche, 60131-Ancona, Italy

Abstract In the last decades much attention has been focused on understanding the factors controlling the shape of the unloading curves obtained by the Oliver and Pharr nanoindentation analysis in order to estimate true contact area, and material parameters such as Young’s modulus and hardness. In fact, it is well known that the Oliver and Pharr analysis can overestimate the hardness of materials that plastically deform due to piling up around the indentation. In recent years, different visual and analytical methods have been proposed. The visual methods are based on direct measurements of the produced indentation by scanning probe microscopy (SPM) or by atomic force microscopy (AFM). In the present work, indentation hardness of a SiO2 coating was measured and analyzed by both visual and analytical methods. The SPM-based direct method showed a quite good qualitative and quantitative literature data agreement. This method was thus developed and improved to make it dependent on curve parameters, such as applied load and penetration depth, rather than on SPM measurements of the actual contact area. A correlation of the pile up phenomenon to the m exponent of the P = B(h-hf)m relationship was also discussed. © 2013 The Authors. Published by Elsevier B.V.

© 2012 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of VINF Selection and/or peer-review under responsibility of VINF. Keywords: Nanoindentation; pile up; Oliver and Pharr analysis; SiO2 coating.

1. Introduction In the sub-micron range, nanoindentation is an instrumented indentation method widely used to determine the mechanical properties of both bulk solids and thin coatings, which in principle does not need any indent imaging [1-3] (to cite but few). Nanoindentation provides accurate measurements of the continuous variation of indentation load, P, at loads as low as a few ȝN, as a function of the indentation

* Marcello Cabibbo. Tel.: +390712204728; fax: +390712204801 E-mail address: [email protected]

1875-3892 © 2013 The Authors. Published by Elsevier B.V. Selection and/or peer-review under responsibility of VINF doi:10.1016/j.phpro.2012.12.014

M. Cabibbo et al. / Physics Procedia 40 (2013) 100 – 112

depth, h, down to a few nanometers. The continuous load and displacement readings provide a material response to deformation from which Young’s modulus, E, and hardness, H, can be determined. Material hardness is generally obtained by dividing the maximum applied load by the projected residual contact area (classical micrometer-scale approach), and by dividing the maximum applied load by the contact area at that load [2-5]. To measure materials Young’s modulus and hardness, Doerner and Nix [3] were the first to propose the use of the unloading curve slope at the initial stage of unloading. Their method was later modified by Oliver and Pharr [4,6] to broadening to a larger scale the elastic recovery range during unloading. Thereafter, assumptions and limitations inherent in these methods were reviewed in [7]. The indenter contact area is the measurement key feature in the Oliver and Pharr's approach, which is directly evaluated from the load–displacement curves, thus making the imaging of the indent impression unnecessary [3,6-9]. Yet, elastic–plastic loading can generate a contact geometry where the material piles-up around the indenter [2,10,11]. The pile-up problem, as it pertains to indentation processes, has been extensively investigated in literature [1,4,8,12,13]. In instrumented depth-sensing nanoindentation, where the Oliver and Pharr analytical method is used for the contact area determination, the issue is critical since the contribution of the piled-up contact area is not included in the analysis, yielding a significant overestimation of the hardness and Young’s modulus [3,4,12,13]. In fact, according to Bolshakov and Pharr [12] hardness can be overestimated by up to 60% and Young’s modulus by up to 16% depending on the extent of pile-up. Hence, Oliver and Pharr, in their review paper on nanoindentation [14], specifically addressed the issue of accounting for pile-up without imaging the indentation and they concluded that it has to be considered yet unresolved. The contact depth to penetration depth ratio, hc/h, where hc is the actual contact depth, and h is the residual depth, can be either larger or lower than unity, depending on elastic deformation (hc/h < 1), or on the occurrence of pile-up or sink-in, respectively. Pile-up is often found in some ductile soft materials including bulks and thin films. Materials with large yield strength-to-Young's modulus ratio, Y/E, are likely to show sink-in upon nanoindentation loading, while materials with small Y/E can show either sinkin or pile-up, depending on the degree of work-hardening [15]. Comprehensive theoretical, computational and empirical approaches have been proposed to elucidate the contact mechanics and deformation mechanisms occurring on hardness and Young’s modulus evaluation of piling-up materials by nanoindentation (e.g., [3,8,10,16-18]). As a matter of fact, most of the Oliver and Pharr nanoindentation inaccuracies arise from the occurrence of pile-up or sink-in of the material around the indent, which is primarily affected by the plastic properties of the material [11]. In a low-strain-hardening alloy, plastically displaced material tends to flow up to, and pile-up against the faces of the indenter and consequently the geometrical result is a “barrel-shaped” impression. Methods able to solve the pile-up or sink-in issue are, thus, essential for the interpretation of the plastic properties of materials. A further important contribution of deviation from the actual hardness value in using the Oliver and Pharr approach and nanoindentation technique is due to the time-dependency effect (creep behaviour), which is recognized to play an important role since more than a decade [19-25] (to cite but few). In fact, the Oliver and Pharr unloading curve analysis essentially relies on the assumption of an elastic response of the material, even if the contact is elastic-plastic [19]. Many material shows a time-dependent behavior upon loading, which eventually leads to an inadequate estimation of hardness and Young's modulus [19]. A material showing time-dependent behavior is commonly called visco-elastic. A result of a visco-elastic material behavior is a delayed response to a given applied stress or strain (load), which is termed creep [20]. In particular, the occurrence of a visco-elastic phenomenon leads to an overestimation of the material rigidity, and, hence, to an overestimation of the material elastic modulus and an underestimation of its hardness. These mis-estimations are due to the presence of a significant relaxation deformation component in visco-elastic materials [21-23]. Thereafter, in nanoindentation measurements, the depth recorded at each load will be the result of a contribution coming from the elastic-plastic properties of the

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material and the contribution yield by the visco-plastic behavior, i.e. due to creep. This is due as nanoindentation tests involve complex mixtures of hydrostatic compression, tension, and shear into the material [19,24,25]. According to the standard procedure of which ISO14577:2002 [26], creep is simply expressed as a change in depth, or load, over time for fixed load, or fixed displacement depths. Following the results reported by Fischer-Cripps in [20], thermal drift (creep behavior) in fused silica, on thus in SiO2, is negligible, as creep times are comparatively short compared to the indentation test duration (holding segment in the trapezoid-shaped load-unload function). In fact, Chudoba and Richter [27] found that the holding period has to be long enough such that the creep behavior is kept at its minimum. Similar studies and results were reported by Feng and Ngan in [21]. Therefore, for crystalline materials, such metals, and semiconductor crystals, the viscous deformation component is negligible as compared to the entire plastic component in the unload curve. The viscous deformation, indeed, becomes significant only at testing temperatures close to the materials melting points. A different figure holds for non-crystalline materials, i.e. amorphous, where the visco-elastic component to the total mechanical response is significant [21-23]. With this repsect, different authors have reported and developed analytical approaches to directly extract visco-elastic properties from pyramidal indentation test functions and incorporating the time-dependence, in those cases where this behavior is present, to a visco-elastic constitutive loads [20,24-29]. Given these considerations, Atomic force microscopy (AFM) of the indents can be used to quantify the extent of pile-up and thus to correct hardness and Young’s modulus obtained by the Oliver and Pharr analysis [30]. In recent studies, Kese et al. [11,31], and Saha and Nix [1], have successfully developed a method to account for the presence of pile-up in the case of soda-lime glass. The method is based on AFM imaging of the piled-up imprint and by approximating the piled-up projected area as 3 semi-ellipses centered to the opposite corners, and thereafter inputting this pile-up area in the Oliver and Pharr analysis. According to Choi et al. [32], the actual contact depth, hc, can be defined as hc = hcOP +ǻhPU, where hcOP is the contact depth as determined by the Oliver and Pharr analysis, and ǻhPU is the induced pile-up depth increment. Thereafter, the actual contact area can be rewritten as Ac = AOP + ǻAPU. The plastically deformed pile-up morphology can be assumed to be invariant regardless of the loaded or unloaded states, thus assuring the validity of this approach [32,33]. This paper presents a comparative study of different ways to accounting for the hardness evaluation of a SiO2 coating on AISI316L stainless steel, in which pile up phenomena occur. The here presented true hardness measurement approaches are based on scanning probe microscopy (SPM) analyses of the nanoindentation prints after unloading. A interpretative model of true hardness evaluation without the need for a direct SPM direct analysis is also presented and developed.

2. Experimental details and Methods 2.1. Experimental details 300 nm-thick SiO2 coating was deposited by physical vapor deposition (PVD) on a 3 mm-thick 316L austenitic stainless steel sheet. Deposition was carried out by Arcelor-Mittal® who has also provided the material. SiO2 coating thickness was 250 nm. Nanoindentation measurements were carried out with a HysitronTM UBI®-1 nanoindenter. A three-sided Berkovich diamond tip, having a curvature radius of 70±5 nm, was used for all the indentation tests. The Berkovich tip was calibrated with a fused quartz reference sample. Unload curve analysis was performed to determine the coating hardness, according to the Oliver and Pharr analysis method and using a trapezoidal load function (loading for 10 s, followed by 100 s holding and then unloading in 10 s). The use of this trapezoidal function derives from an international round-robin series of measurements

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performed in different nanoindentation instruments and for different tips and coatings. Some of the results of these round-robin experiments were recently published by one of the present author (M Cabibbo) [34]. The length of the holding segment of the trapezoidal load function was motivated by the Chudoba and Richter [27] findings concerning the optimal holding time to be used for fused silica, and thence for the SiO2 as well. They showed that a holding time of 65 s is sufficient to avoid any time-dependent contribution to the analysis of the unloading curve. Thus, in the present case, where the holding time was fixed at 100 s, the possibly present time-dependent, i.e. creep, component is minimized to a irrelevant amount in the total plastic unloading curve analysis. For this reason, the hereafter proposed analysis methods (Section 2.2) does not includes any time-dependent contribution. In order to record the morphological images of the indents, the nanoindenter was also operated in scanning probe mode (SPM). Measurements were carried out with peak loads in the range 0.5 to 8Â103 μN. Indentations were spaced 100 μm apart and repeated in 3 different areas of the coating using a 8x8 indentation grid. The hardness of the 316L substrate was determined using a RemetTM HX-1000 tester. Mean substrate hardness was H316L = 4.0±0.2 GPa. 2.2. Methods The nanoindentation measurements were analyzed using the Oliver and Pharr method [35,36]. Therefore, hardness, H, and reduced Young’s modulus, Er, are obtained after a complete load-unloading cycle. The unloading curve is fitted with a power-law relationship (Eq. (1)): P = B(h-hf)m

(1)

where P is the tip load, h the displacement, hf is the displacement after unloading, B and m are fitting parameters. The contact depth, hc, can be estimated from the load-displacement data as (Eq. (2)): hc=hmax-ε(Pmax/S)

(2)

where hmax is the maximum indenter penetration depth at the peak load, Pmax, ε is a tip dependant constant, S the sample stiffness. For a Berkovich tip, İ = 0.75. Hardness is determined as H = Pmax/A. The projected contact area, A, is a function of the contact depth, hc. For an ideal Berkovich tip, this value is given by A = 24.5hc2 [10]. For a full description of the Oliver and Pharr curve analysis method, the reader is referred to other literature works, such as [35,36]. Calibration was carried out on fused quartz (Er = 72GPa, Ȟ = 0.17), according to the specifications of which the ISO14577:2002 [26], the results reported in [36], and the findings of a specific international project, intended to overcome the difficulties of comparing nanoindentation hardness and Young's modulus measurement sets coming from different instruments [37]. When pile-up occurs, the actual contact depth, hc, is larger than hmax as the material plastically piles up around the indenter (hc is determined starting from the undeformed material surface). Thereafter the use of Eq. (2) generates an underestimation of the contact area, A. As a result, the hardness will be overestimated. Therefore, in a recent review of the subject [14], Oliver and Pharr commented that if pileup is large, accurate measurements of H and Er cannot be obtained using the contact area as deduced from the load–displacement curves; rather, the area measured from a SPM, or AFM image, should be used. Thence, correct H and Er values can be obtained by inputting the actual measured contact area in the Oliver and Pharr equations. Many researchers have already used this method and approach [1,17,38-40]. In the present study, different such approaches were followed and applied to a SiO2 coating on a 316L stainless steel. Fig. 1(a) is a schematic diagram of the triangular projection of the indent with one of the three pile-up lobes drawn in between the indent edge and the outer periphery of the piled-up zone of the material. The

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horizontal projection of the highest pile-up height to the undeformed material surface is radially displaced from the edge of the indentation by the distance ai (where i stands for the three potentially different pileup lobes at the three indent Berkovich edges). Let Atrue be the true contact area of the indentation. Then Atrue can be written as the sum of the contact area as calculated through the Oliver and Pharr analysis, AOP, and the contribution coming from the pile-up perimeter ridge, APU (Fig. 1(b)).

SPM image (c)

Fig. 1. a) Schematic diagram of the piled-up print showing the specific features used to determine the true contact area, Atrue, in the three SPM methods: Lobes (ai pile-up), Geometric (circle arc), and Corners (circumference of radius bx). A section view of the piled up indentation is shown in b). SPM mode image of one representative indent at 7 mN is shown in c).

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In order to determine the amount of APU, each pile-up contact perimeter ridge can be approximated to a semi-ellipse of major axis b (edge of the Berkovich indent) and minor axis ai. The three terms ai are directly measured on the indent profile image (SPM) as the horizontal distance of the pile-up contact point tip, T', from the edge, E, of the indentation (Fig. 1(a)). Fig. 1(c) reports a representative SPM image of a piled-up imprint. For an equilateral triangle of side b, the area is given by A eq = (b2/4)tan(60°) = 0.433b2. The projected contact area, Ac, determined at contact depth, hc, traces an equilateral triangle of side b. Thus, for a perfect Berkovich tip, Ac = 24.56hc2 = 0.433b2,from which b = 7.531hc. The area of each semi-elliptical pile-up projected contact area is ʌ(b/4)ai and the total pile-up contact area is APU = (ʌb/4)Σai, the summation being over the three semi-elliptical projected pile-up lobes. Thus, the true contact area, Atrue, is, Eq. (3): Atrue = AOP + 5.915hc ¦1 a i 3

(3)

At high loads, the pile-up forms a distinct ridge; yet, as the load decreases, the ridge smoothly reduces to a plateau. In such a case, the pile-up contact perimeter was taken to be the point of departure of a tangent parallel to the face of the pile-up [41]. This method to measuring the actual contact area, Atrue, has been called Lobes in the followings (AtrueLobes). A slightly different approximation for the pile up contact point ridge, respect to the one described above, is taking it as a projection of an arc around one edge of the indent and pointed to the opposite indent corner (point T in Fig. 1(a)), as proposed by Saha and Nix [1], Kese and Li et al. [29], and Zhou et al. [42]. In the present case, APU can be written as (Eq. (4)): § πR 2 3 2 · § 2π − 3 3 · 2 ¸b A PU = 3¨¨ b ¸¸ ≈ ¨¨ − ¸ 4 4 ¹ © 6 ¹ ©

(4)

which holds by approximating the pile-up arc, R, with the indent edge, b (Fig. 1(a)). AOP is geometrically determined as the actual area of the impression, AOP = (√3/4)b2, which, considering the Berkovich tip geometry, can be written as AOP = 3√3hc2tan2(65.3°) and therefore b = 2√3tan(65.3°)hc. Atrue is thus given by the sum of APU and AOP. This latter relationship implies that Atrue can be obtained indirectly from the contact depth, hc, without the need to physically measure the indent edge, b. This second method has been called Geometric in the followings (AtrueGeometric). In a third method, to measure the total contact area, Atot, the three corners of the indent were taken into account (C1, C2, C3 in Fig. 1(a)). The Atrue was then determined by considering the three corner points belonging to a circumference, and setting it equals to the corresponding circle area. The hardness was measured by dividing the applied load to the corresponding Atrue (=πb2/3). This method has been called Corner in the followings (AtrueCorners). Fig. 1(c) revealed that significant pile-up occurs only along the edges of triangle indent and not at the corners, thus justifying and validating the third SPM-based method (Corners). The indent projected equilateral triangular area of side b yield virtually same values of AOP than that resulted from the Oliver and Pharr analysis. The experimental accuracy of the SPM measurements was within 10 %, and the experimental estimation of the AOP was thus within the value derived by the Oliver and Pharr curve analysis. Indeed, the slight disagreement between the load-on contact area (i.e., the one determined by the Oliver and Pharr analysis) and the load-off contact area (i.e., the projected area measured by the SPM inspections) was within the experimental error and, for this reason, they were considered coincident. In all the three cases, the hardness is measured by dividing the maximum load, Pmax, by the calculated true area, Atrue: HtrueLobes = Pmax/AtrueLobes, HtrueGeometric = Pmax/AtrueGeometric, HtrueCorners = Pmax/AtrueCorners.

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3. Results and Discussion 3.1. SPM-based methods of hardness measurements Fig. 2 reports the mean hardness values obtained by the Oliver and Pharr analysis and by direct SPM measurements of the pile up ridges and indentation projected areas according to the different methods presented in the Method section: Lobes, Geometric, Corners. For the whole load range, the Lobes and the Corners hardness results were quite close, and they were some 25-30 % lower with respect to the Oliver and Pharr hardness. The hardness values obtained using the Geometric method stayed constantly much lower by some 35-50 % with respect to the Oliver and Pharr results, and lower by some 20 % respect to the Lobes and Corners method obtained values, within a normalized depth, hc/t, of 0.4 (where t is the coating thickness). It is interesting to note that the hardness determined using all the three methods decreased steadily with the load from hc/t > 0.4. At the maximum load of 8mN, corresponding to a normalized penetration dept, hc/t = 0.8, the true hardness, as determined by the three methods, were quite close, and indeed essentially equal to the stainless steel substrate hardness (i.e. 2 GPa). In all the cases, the Oliver and Pharr curve analysis overestimated the hardness estimation of the SiO2 coating, irrespectively of the applied load. Lobes and Corners methods estimated a hardness ranging 5.8 to 5 GPa, while the Geometric method estimation lies in the lower range, being of 4.9-5 GPa. All three methods revealed that for a reliable measurement of the hardness in this SiO2 /316L SS, the penetration depth of the Berkovich tip must not exceed the normalized value, hc/t, of 0.4. Moreover, the hardness obtained using the Lobes method showed quite good agreement with other literature results [43,44].

Fig. 2. H vs. normalized penetration depth, hc/t. HOP is the hardness calculated directly from the load–depth curve using the Oliver and Pharr method. HLobes, HGeom, and HCorners are the hardness values calculated by dividing the maximum load by the projected contact area as measured directly from the corresponding SPM image (according to the Lobes, Geometrical, and Corners method, respectively).

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To verify the predictive ability of the Lobes method, and to extend it to situations where the ai’s are or cannot be measured, the measured ai ’s, for loads ranging 0.5 to 8 mN, were fitted using a linear equation of the form [11] (Fig. 3a, and Eq. (5a)): aav = ȖP + c

(5a)

where P is the peak indentation load at which the average pile-up width, aav, is sought, while Ȗ and c are curve fitting constants. In the present case Ȗ = 20.2 nm/mN, and c = 29 nm, and correlation factor R was 0.992. The term c has not a physical meaning, since as the load reduce to zero no pile up can occur. Therefore, in an analogy to Eq. (1), which describes the dependency of the penetration depth to the applied load thorough a power-law, a similar power-law interpolation approach (Fig. 3b) is here proposed. This is believed to better describe the relationship between a and P. Eq. (5b) is the fitting power law relationship: aav = APn

(5b)

where A = 50.1 nm/mN, and n = 0.608. This solution also gave a better correlation factor R = 0.998, with respect to a linear data interpolation (Eq. (5a)). Eq. (5b) is expected to hold for thicker SiO2 coatings, although is believed to be limited to this specific coating.

a)

b)

Fig. 3. Linear interpolation, a=γ P+c, a), and power law interpolation, a=APn, b), of the pile-up lobes average heights, aav, vs. the maximum applied load, Pmax.

3.2. Models for determining the true area function To avoid the need to directly measuring the indent pile-up ridges, a further improving step of the pile-up contact area corrective methods would consist in the correlation of this area to other analytically determined parameters, such as the contact stiffness or the contact depth (Eq. 6): P = (πH/4ȕ2Er2)S2 = kS2

(6)

Where β is constant depending on the indenter geometry, for triangular diamond Berkovich tip ȕ was reported to be 1.05 [14]. Er is the reduced Young’s modulus. Assuming constant H and Er (since, in the present case, the H and Er values obtained by the Lobes, Geometric, and Corners methods, support this assumption for P > 4mN), k then becomes the coefficient of S2 in the quadratic function between the load and the contact stiffness. The average hardness obtained in the 300 nm-thick SiO2 coating was H = 3.76, 3.57, and 2.88, in the Lobes, Geometric, and Corners methods, respectively. The average reduced

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Young’s modulus, for P > 4mN, was Er = 48, 51, and 47, in the Lobes, Geometric, and Corners methods, respectively. Therefore, inputting these values into Eq. (6), k = 1160±60, 1100±50, and 820±80 nm2/mN, for the Lobes, Geometric, and Corners methods, respectively. In the case of the Lobes method, the relationship between the pile-up contact width and the contact stiffness, a, can be expressed as, (Eq. (7)): a(S) = k’S2n

(7)

where k' has the same meaning of k in Eq. (6), i.e. a fitting parameter containing experimentally measured values. Using Eq. (7), the pile-up contact area can automatically be derived and incorporated in the analysis without the need for any physical direct measurement. Since piling-up is a flow process, in a general elastic-plastic case, the constants in Eq. (7) is expected to depend on the coating yield stress, strain hardening exponent, Poisson’s ratio, and indenter geometry (see, for example, Refs. [9,16]). With such an empirical function of a(S), Atrue can be written as follows: Atrue = AOP + 3(5.915)a(S)hc (8a) or in a more straightforward form, as: Atrue = AOP + (k”S2n)hc

(8b)

where, in the present case, the SiO2 coating yield a value of the fitting parameter k” = 1090±50 (Lobes method). Using this new relationship for the Atrue, the true hardness, Htrue (=Pmax/Atrue) results same as determined by direct SPM measurements. It must be here emphasized that the use of Eq. (8b) is limited to the considered SiO2 coating and to load above 4mN (for which the values of hardness and reduced Young’s modulus are load-independent). 3.3. Load versus penetration depth in piling up coatings The relationships between P and h of Eq. (1) is a power-law with an exponent m that is related to the geometry of the indenter, and a constant B containing geometric constants, such the indenter elastic modulus and the indenter Poisson’s ratio. For a flat-ended cylindrical punch, m = 1, for a paraboloid of revolution, m = 1.5 and for a cone and cone-like symmetry punch geometry, m = 2 [39]. According to the elastic assumption, the fitted power law exponent m at the unloading part of the load–displacement curve, for the Berkovich indenter should be 2, while experimental evidences showed m mainly distributed in the range 1.1–1.8, depending on the material [10,12,45,46]. Recently, Pharr and Bolshakov [35] reported m to be strongly associated with the material E/H ratio; high E/H values correspond to large values of m. The reason why the m index is not precisely equals to 2 is related to the material behavior on loading, which elastically recovers so that the sides of the residual impression remain straight. Indeed, in piled up materials the sides of the residual impression are curved upwards. The actual indent shape is thus embodied in the power law index m which is expected to be consistently lower than 2 [18,47-49]. Fig. 4 reports the experimental m mean value obtained by the load-unload curve. This fitting value was m = 1.62 ± 0.05. The relationship between the pile up width, a, and the corresponding actual penetration depth, hfpileup, can thus be expressed as power-law (Eq. (9a)):

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aav = D(hfpile up)r

(9a)

where the two fitting parameters, D and r, are D = 0.75 ± 0.05, and r = 0.97 ± 0.02, with a correlation factor of R = 0.992. Eq. (9a) can be rewritten as, (Eq. (9b)): hfpile up = D-1/raav1/r = D’aavr’

(9b)

Combining Eqs. (5b) and (9b), hfpile up can expressed as (Eq. (9c)): hfpile up = D’Ar’Pnr’ = β’Pm*

(9c)

Reversing Eq. (9c), being hf(true)=hfO&P+Δhfpileup = hfpileup, Eq. (1), in presence of pile up, becomes (Eq. (10)): Ppile up = χ-1/m*(hfpile up)1/m* = β’(hfpile up)m’= Pmax(A/D)-1/n(hfpile up)r/n

(10)

where, β’ = Pmax(A/D)-1/n = 9.90± 0.01, and m* = r/n = 1.60 ± 0.05.

Figure 4. m fitting parameter of Eq. (1) vs. the maximum applied load, Pmax. The average m value, as determined by the Oliver and Pharr curve analysis, is shown as horizontal line and the corresponding value is directly reported in the plot.

That is, it seem that most of the discrepancy of the m value to the expected value of 2 is associated to the pile up phenomenon and to the fact that the Oliver and Pharr analysis does not take into account the extra piling up ridge above the measured hf. In other words, the evaluation of the exact value of m is likely to yield, with a reasonable approximation, the extent of the pile up above the Oliver and Pharr measured penetration depth. In the present case, the simple determination of the difference m’=2-m gives the opportunity to know the extent of the actual penetration depth (Eqs. (9b,c), and (10)), and thereafter the extent of the pile up width (Eq. (9a)). This value, through Eqs. (1) and (3), and the relationship Htrue = Pmax/Atrue, gives the actual hardness without the need to experimentally measuring the pile up ridges and widths. It must be pointed out that the Eq. (10) holds when the pile up phenomenon has saturated. In the present case, saturation occurred at P > 4mN. Same experimental approaches and same interpretative models were successfully used in a work by one of these authors (M. Cabibbo) on bulk copper [50]. With this respect, it must here emphasized that the half space model on which the present approach and entire formulation are based strictly holds assuming only small material surface curvatures.

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4. Summary The phenomenon of pile up on a SiO2 coating on AISI316 substrate was investigated. Three different SPM methods were used to measure the pile up ridges and thus accounting for the corrective coefficient of the actual contact area with respect to the contact area determined by the Oliver and Pharr analysis. The three SPM direct pile up measurement methods yield hardness considerably lower (20 %, and up to 30 %, depending on the model used) than the Oliver and Pharr analysis results. Starting form load of 4 mN, these method results were neither dependant on the penetration depth, nor dependant on the applied load. A relationship between Atrue and the material stiffness was found. This allowed to determine the actual hardness, without the need of measuring the pile-up width. The contribution of pile up to the m exponent of the P = B(h-hf)m relationship was determined and discussed. This contribution accounted for most part of the difference between the expect value (that is a value of 2) and the experimentally determined m value. The value difference between the expected value of 2 and the one found by the Oliver and Pharr analysis has been shown to yield a reliable estimation of the pile up ridge height and width, and ultimately a reliable estimation of the actual hardness. This latter approach made the direct measurement of the pile up ridges unnecessary.

Acknowledgments Authors acknowledge the EU-FP7 for having funded the project NANOINDENT (NMP3-CA-2008218659) entitled Creating and disseminating novel nanomechanical characterisation techniques and standards from which this work derives.

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