Calibrating a nanoindenter for very shallow depth indentation using equivalent contact radius Damir R Tadjiev, Russell J Hand, Simon A Hayes
To cite this version:
Damir R Tadjiev, Russell J Hand, Simon A Hayes. Calibrating a nanoindenter for very shallow depth indentation using equivalent contact radius. Philosophical Magazine, Taylor & Francis, 2010, 90 (13), pp.1819-1832. 10.1080/14786430903571420. hal-00587293
HAL Id: hal-00587293 https://hal.archives-ouvertes.fr/hal-00587293 Submitted on 20 Apr 2011
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Calibrating a nanoindenter for very shallow depth indentation using equivalent contact radius
Journal: Philosophical Magazine & Philosophical Magazine Letters
Manuscript ID: TPHM-09-Jul-0315.R2
Journal Selection: Philosophical Magazine
Date Submitted by the 13-Nov-2009 Author:
Complete List of Authors: Tadjiev, Damir; University of Sheffield, Engineering Materials Hand, Russell; University of Sheffield, Engineering Materials Hayes, Simon; University of Sheffield, Engineering Materials
Keywords: nanoindentation, mechanical properties
Keywords (user supplied): tip area function, calibration, shallow depths
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1 2 3 4 Calibrating a nanoindenter for very shallow depth indentation using 5 equivalent contact radius 6 7 † 8 Damir R. Tadjiev, Russell J. Hand and Simon A. Hayes 9 10 Department of Engineering Materials, University of Sheffield, Sheffield, UK 11 12 Mappin Street, Sheffield S1 3JD, UK 13 14 15 (Received 21 July 2009; final version received 2009) 16 For Peer Review Only 17 Nanoindenter tips are usually modeled as axisymmetric cones, with calibration involving 18 finding a fitting function relating the contact area to the contact depth. For accurate 19 calibration of shallow depth indentation this is not ideal because it means that deeper 20 indents tend to dominate the fitting function. For an axisymmetric object it is always 21 possible to define an equivalent contact radius (which in the case of nanoindentation is 22 linearly related to the reduced modulus) and to obtain a fitting function relating this 23 equivalent contact radius to indentation depth. The equivalent contact radius approach is 24 used here to provide shallow depth calibration of a nanoindenter tip at three separate 25 times. The advantage of the equivalent contact radius methodology is that it provides a 26 clearer physical interpretation of the changes in tip shape than a conventional area based 27 fit. We also show that the minimum depth for a reliable hardness measurement is 28 obtainable increases as the tip blunts with age but that consistent measurements of very 29 near surface elastic moduli can be made if the blunting of the tip over time is fully 30 accounted for in the tip area function calibration. 31 32 Keywords: tip area function, calibration, nanoindentation, shallow depths 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 † Corresponding author. Tel.: +44 (0)114 2225465. Email address: [email protected] (R.J. Hand).
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1 2 3 4 1. Introduction 5 6 7 8 Nanoindentation differs from conventional indentation in that the results are based not 9 10 on imaging of the residual indent (which would be very challenging, and largely 11 12 13 impracticable, for very low load indents), but rather on a calibrated tip area function 14 15 (TAF) which describes the variation of the contact area ( Ac) of the tip as a function of 16 For Peer Review Only 17 contact depth ( h ). In the ideal case the indenter has a simple axisymmetric shape and, 18 c 19 20 hence, the TAF can be described by a simple mathematical function. Thus even the 21 22 commonly used Berkovich indenter, which is actually a triangular pyramid, is usually 23 24 25 modelled by an equivalent axisymmetric cone, as assumed in the conventional Oliver 26 27 and Pharr (O&P) approach. 28 29 In actuality any real tip shape will deviate from its ideal geometry to some 30 31 32 greater or lesser extent, even when new, and this will become more marked as the tip 33 34 becomes worn over time due to its use in indentation [1] and, commonly, secondary 35 36 use in imaging the residual indent [2]. Thus an equivalent cone model of a Berkovich 37 38 39 tip will break down at some point [3, 4] and one approach to this problem is to treat 40 41 the very near tip region using a Hertzian analysis (see, for example, [5]); clearly this 42 43 still treats the tip as an axisymmetric object. 44 45 46 In the commonly adopted Oliver and Pharr approach TAF calibration is 47 48 carried out by indenting a material of known modulus, which is assumed to have i) an 49 50 51 elastic modulus that is unvarying with depth and ii) a low E/H value which means that 52 53 there will be minimum pile-up and sink-in [6]. The stiffness, S, of the combined 54 55 indenter/ indented materials system is then assessed from the initial (elastic) part of 56 57 58 the unloading curve and the area of the indentation is then obtained using 59 60
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1 2 3 πS 2 4 A = (1) 5 2 2 4β Er 6 7 8 where β is a geometric correction factor (in this work taken to be equal to 1.05) [6] 9 10 and Er is the reduced modulus, which accounts for the fact that the measured 11 12 displacement includes contributions from the sample and the indenter. E is given by 13 r 14 15 16 For Peer Review2 2 Only 1 1−ν 1−ν i 17 = + (2) 18 Er E Ei 19 20 where E, E i and ν, νi are the elastic moduli and Poisson’s ratios of sample and the 21 22 indenter respectively. For the diamond Berkovich indenter Ei = 1141 GPa and νi = 23 24 25 0.07, and for fused silica (which largely obeys the two assumptions mentioned above) 26 27 used as a standard in this work E = 72 GPa and ν = 0.17, respectively; hence Er is 28 29 equal to 69.6 GPa. The TAF is then obtained by fitting A as a function of the contact 30 31 32 depth, hc. 33 34 The work described here forms part of a larger project to examine the 35 36 37 mechanical properties of the surface hydration layers formed on silicate glasses [7, 8]. 38 39 This requires reliable calibration of TAFs for shallow (we have been focusing on less 40 41 than 80 nm indentation) depths. Calibration of the TAF for shallow depths is a 42 43 44 particular problem for a number of reasons: imaging is most difficult in this region 45 46 (and at the lowest loads impossible due to the occurrence of purely elastic behaviour); 47 48 and although the data are more sensitive to random errors in this region ( i.e. they are 49 50 51 noisier) the data obtained at larger depths tend to have larger absolute (as distinct 52 53 from relative) residuals. The latter point means that any large depth data tend to 54 55 dominate the curve fitting thereby reducing the accuracy of any fit at shallow depth. 56 57 58 Conventionally this issue is addressed by providing different fits over different depth 59 60
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1 2 3 ranges but an alternative method is to select an appropriate data transform [9]. Here it 4 5 6 has been therefore decided to fit 7 8 r = A/π (3) 9 c 10 11 as a function of hc, where rc is the contact radius of the equivalent axisymmetric body. 12 13 This has the advantage of giving an equal distribution of residuals due to the 14 15 16 approximatelyFor linear scalingPeer between Review rc and hc that can be Only reasonably expected for all 17 18 tip geometries. Also combining equations (1) and (3) gives 19 20 21 S 22 r = (4) 23 c 2βEr 24 25 26 i.e. Er∝1/ r c meaning that the calibration is based entirely on a linear relationship 27 28 29 rather than a quadratic one. 30 31 32 2. Experimental procedure 33 34 35 2.1 Instrumentation and reference material 36 37 ® 38 A commercial Hysitron Triboscope nanoindenter (Hysitron Inc., USA) mounted on a 39 40 Dimension 3100 (Veeco Digital Instruments) nanoscope equipped with a Berkovich 41 42 43 tip (Hysitron, original tip radius 150 nm) was used. The indenting system is housed in 44 45 under a thermal and acoustic isolation hood which is mounted on a vibration resistant 46 47 table. The nanoindenter has load and displacement resolution of 1 µN and 0.1 nm, 48 49 50 respectively. Calibration was undertaken by making indentations on standard fused 51 52 silica samples to obtain data for contact depth in the range of 0-100 nm. The first tip 53 54 area function (TAF1) was calibrated on 10 × 10 × 3 mm fused silica sample that came 55 56 57 with nanoindenter and had surface roughness, Ra, of 0.69 nm, while the second and 58 59 the third (TAF2 and TAF3) tip area functions were calibrated on new 40 × 25 × 3 mm 60
fused silica slides (Heraeus, Ra = 0.54 nm). 3 d ata sets were collected at
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1 2 3 approximately 1 year time intervals (see table 1). All experiments were carried out in 4 5 6 a quasi-static mode and during each calibration run multiple arrays of 8 × 8 indents 7 8 spaced 4 µm apart on both the x and y axes were made, with each indent providing a 9 10 11 single measurement of stiffness and load. A standard loading scheme of 5 s up-load, 5 12 13 s dwell and 5 s unloading was used. All measurements were carried out at room 14 15 temperature (24±1°C) and a relative humidity of 60-80% with drift correction on. The 16 For Peer Review Only 17 18 indenter is continuously powered so that the electronics are maintained in a stable 19 20 state. Drift was measured by running 0.1 µN test with a 20 s hold at peak load and 21 22 measuring displacement drift during this hold. The remainder of the test was corrected 23 24 25 by the measured drift rate the typical value of which did not exceed 0.1 nm/s. 26 27 Approximately 1000 data points were collected for each calibration data set with an 28 29 emphasis on low load indentation values. 30 31 32 Prior to the measurements, the compliance of the indenter column (machine 33 34 compliance) was measured by making at least 25 high-load indentations (5000 to 35 36 37 10000 N) on fused silica and then plotting compliance (inverse stiffness) versus 1/P 38 39 (where P is applied load) for the obtained data. Reliability of the determined 40 41 compliance value was checked by plotting load over stiffness squared ( P/S 2) versus 42 43 44 contact depth and making sure that the slope of the curve is 0. This approach was 45 46 originally proposed by Oliver and Pharr [6] for the measurements at high-load 47 48 indentations, but it has been found in this work that it also works well at low-load 49 50 51 indentations. 52 53 Prior to each test the fused silica samples were cleaned by rinsing in pure 54 55 ethanol and drying using a warm air blower. The surface region to be indented was 56 57 58 then imaged using the instrument in the scanning probe microscope (SPM) mode to 59 60 ensure that the surface was free of dirt and contamination.
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1 2 3 2.2 Data analysis and TAF fitting 4 5 6 Following the standard O&P method [10] the initial unloading contact stiffness, S, 7 8 was obtained from the slope of a power law fit to the initial part of the unloading 9 10 11 curves using the installed Hysitron software. The power law used was 12 13 P = α h − h m (5) 14 ( f ) 15 16 For Peer Review Only where P is load, h is the measured indention depth, hf is the residual indentation depth 17 18 after removal of the indenter and α and m are fitting constants. The unloading stiffness 19 20 21 ( S = d P d h) was then obtained by differentiation and equations (1) were used to 22 23 24 obtain the equivalent axisymmetric contact radius. 25 26 Contact depth hc was calculated by the Hysitron indenter software using 27 28 S 29 hc = hmax − 75.0 (6) 30 Pmax 31 32 where hmax and Pmax are, respectively, the maximum depth and load during the 33 34 35 indentation [8]. 36 37 38 3. Results and discussion 39 40 41 Figure 1 shows a plot of rc versus hc for the tip when new. In this case a linear fit was 42 43 found to be applicable and a linear regression fit to the data shown gave 44 45 46 rc = .2 899 hc + 23 5. (7) 47 48 with an r2 value of 0.990. It is interesting to note that Thurn and Cook [11] proposed a 49 50 51 three term area function, which actually involves only two independent parameters, 52 53 to calibrate the TAF. They took a harmonic average of a spherical tip profile (with 54 55 radius R) and a perfect conical shape (with included angle 2 α) and obtained 56 57 π 58 A = h 2 + 4Rπh + 4R 2π cot 2 α (8) 59 cot 2 α c c 60 Combining equations (3) and (8) gives
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1 2 3 h 4 c rc = + 2Rcot α (9) 5 cot α 6 7 i.e. a straight line as seen in figure (1a). The original derivation of equation (8) 8 9 10 involved use of the binomial expansion which requires that the contact radius 11 12 a > 2R cot α , and implies that there is a flat end, the radius of which corresponds to 13 14 the constant term in equation (8), even though the original harmonic average does not 15 16 For Peer Review Only 17 assume this. Using equation (9) for the data shown in figure 1 gives an included 18 19 angle α = 70 9. o ± 3.0 o and R = 33 9. ± 2.1 nm indicating that the original tip 20 21 22 conformed reasonably closely to the expected equivalent cone angle of 70.32° (see, 23 24 for example, [12]). However, as noted above, equation (9) in fact indicates that there 25 26 r 27 is a flat right at the tip. In the current case the radius of this flat is c = 23 5. ± 9.0 nm . 28 29 For comparative purposes a 9 term Oliver and Pharr TAF fit was also 30 31 32 undertaken (see figure 1b). This gave 33 34 35 A = 23 .399 h2 + 604 .968 h + .8 694 h /1 2 + .6 444 ×10 −5 h /1 4 + .1 619 h 8/1 36 c c c c c (10) 16/1 −8 32/1 64/1 /1 128 37 + 30 .070 hc + .1 773 ×10 hc + 31 .478 hc + 41 .720 hc 38 39 with an r2 value of 0.988. 40 41 42 Figure 2a shows a later calibration on the same tip carried out after it had 43 44 received significant usage. The fact that the regression line from the first calibration 45 46 47 can be displaced upwards to overlay the data from the second calibration shows that 48 49 although the very near tip region has indeed become blunter away from this region the 50 51 tip is essentially unmodified. This fact cannot be simply inferred from a conventional 52 53 54 TAF plot (figure 2b). It would also appear that although the tip shape has been 55 56 modified that the effective radius of the flat right at the tip has remained essentially 57 58 unchanged as dataset 2 intercepts the y-axis at essentially the same point as dataset 1. 59 60
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1 2 3 The exact displacement between the two curves was evaluated by ensuring 4 5 6 that the residuals in the straight line portion were equally scattered about zero by 7 8 examining the moving average. The moving average was also used to identify an 9 10 11 appropriate break point below which an alternative curve fit was required. Empirically 12 13 it was found that using a function based on that proposed by Oliver and Pharr for the 14 15 tip area fit was most effective for this regime (see figure 3). The fitting was carried 16 For Peer Review Only 17 18 out using the non-linear regression routines supplied in SigmaPlot 10 (Systat). Thus 19 20 for the particular dataset shown in figures 2 and 3 the resultant fit was 21 22 r = .2 0115 h +12 .1929 h 2/1 +19 .7669 h < 48 nm 23 c c c c (11) 24 rc = .2 8969 hc + 61 67. hc ≥ 48 nm 25 26 2 2 27 In this case the overall r value for the fit was 0.999. The improved r value for this fit 28 29 compared to the original fit is due to a smaller spread in the initial data, which is due 30 31 to the use of a calibration piece of silica with lower surface roughness. Although a 32 33 34 split function approach could also be used for the TAF fit it is much less obvious as to 35 36 where to split the fitting function. Thus the equivalent radius approach potentially 37 38 offers some benefit in determining an appropriate fit. 39 40 41 Similar features were found with a third calibration carried out at a later date 42 43 (see figure 4). Again at the greater depths the calibration data appeared to be simply 44 45 46 displaced with respect to the original calibration but in the very near tip region the 47 48 detailed shape had further changed and once again an empirical fit was used in this 49 50 region. In this case the combined fit was 51 52 r = .2 5050 h + .6 1533 h 2/1 + 44 .9516 h < 55 nm 53 c c c c (12) 54 r = .2 8969 h + 68 94. h ≥ 55 nm 55 c c c 56 57 It can be seen that the fit is slightly less good right at the increasingly blunt indenter 58 59 tip but additional terms gave no benefit in terms of quality of fit. The overall r2 value 60 for the fit is 0.998. As well as showing how the tip geometry is being blunted over
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1 2 3 time the results in figure (4a) once again emphasize how the stiffness of the 4 5 6 indentation system is affected by the tip geometry especially at low loads ( rc is 7 8 directly proportional to S; see equation 4). 9 10 11 The residuals for all 3 calibrations are shown in figure (5a). It can be clearly 12 13 seen that they are in general uniformly distributed about zero indicating both a good 14 15 fit and that, as desired, all points have essentially been equally weighted in obtaining 16 For Peer Review Only 17 18 the fit. As noted above it can be seen that the spread of data in the first fit seen in 19 20 figure (5) is wider (reflected in the poorer r2 value for this fit) and there is an 21 22 asymmetry in the residuals in the very near tip region for the third calibration 23 24 25 indicating a relatively poor fit in this region . Figure (5b) shows the corresponding 26 27 residuals for the Oliver and Pharr fit having re-expressed everything in terms of 28 29 equivalent radius for comparative purposes. It can be seen that the distribution of the 30 31 32 residuals is very similar, except in the region corresponding to the flat in the original 33 34 calibration run. Thus the two methodologies overall provide similar levels of accuracy 35 36 37 the real advantage of the methodology proposed here is that it directly provides a 38 39 clearer physical picture of progressive changes in indenter tip geometry. 40 41 As a further check equations (7), (10) and (11) were substituted in equation (4) 42 43 44 and the reduced modulus was recalculated for the three data sets and the results are 45 46 shown in figure (6a). In all cases the data are essentially scattered around the expected 47 48 value of 69.6 MPa. Figure (6a) also indicates that consistent measurements of very 49 50 51 near surface elastic moduli can be made if the blunting of the tip over time is fully 52 53 accounted for in the tip calibration. 54 55 It is obvious that the surface contamination and hydration may compromise 56 57 58 the accuracy of shallow depth indentations. As outlined above cleaning and SPM was 59 60 used to ensure that the indented region was clean prior to indentation. Also as figure
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1 2 3 (5a) shows that the residuals are essentially equally distributed across the entire data 4 5 6 range there is no evidence of any cleaning of contamination during successive 7 8 indents. Hydration is more of a problem. Hydration layer depths of 5 nm to 10 nm [7, 9 10 11 15] have previously been suggested for pure silica. Fundamentally this is an issue for 12 13 any shallow depth calibration of a nanoindenter in that all calibration techniques 14 15 assume that there is no variation of modulus with depth whereas a hydrated layer 16 For Peer Review Only 17 18 would be expected to result in a reduction in modulus as the surface is approached. In 19 20 this case one would expect to see an apparent decrease in equivalent contact radius 21 22 (i.e. an apparent sharpening of the tip) due to the over-estimation of the value of Er 23 24 25 used in equation (4). The fact that we do not observe this behaviour, rather the 26 27 opposite over time, suggests that for the silica sample examined here hydration is not 28 29 a significant problem although its presence cannot be absolutely ruled out. The fact 30 31 32 that the noise increases at the very shallowest of depths is, we believe, an inevitable 33 34 consequence of the fact that the measured quantities are particularly small in this 35 36 37 region and thus the signal to noise ratio is reduced. 38 39 For nanoindentation hardness is defined as 40 41 Pmax Pmax 42 H = = 2 (12) 43 A πrc 44 45 where P is the maximum load during the indentation. Hardnesses calculated in this 46 max 47 48 fashion are shown in figure (6b). In all cases it can be seen that the measured hardness 49 50 tends to decrease in the very near surface region. Unfortunately the greater amount of 51 52 noise present in the original calibration 1 data makes it difficult to identify a clear 53 54 55 trend in behaviour, although at very low depths there is an apparent decrease in 56 57 hardness with decreasing indentation depth; a clearer decrease in hardness with 58 59 60 decreasing indentation depth is apparent for the other two datasets and the minimum indentation depth required for meaningful hardness measurements appears to increase
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1 2 3 as the tip becomes increasingly blunt. At depths greater than the minimum indentation 4 5 6 depth for meaningful hardness measurements no indentation size effect is observed, 7 8 which is in agreement with the results obtained by other researchers [3, 10, 13, 14]. 9 10 11 The apparent decrease in hardness with decreasing depth very near the surface arises 12 13 because in fact fully elastic deformation is occurring in this region and thus hardness 14 15 is not actually a meaningful measurement in this region. The reduced elastic moduli 16 For Peer Review Only 17 18 (over the entire measured depth range) and hardness values (for indentation depths 19 20 >15nm) obtained with the three different calibrations are compared in Table 1. 21 22 Reduced moduli and hardness values were also calculated using the results of 23 24 25 9 term Oliver and Pharr TAF type fits (see equation (10) for the fit for calibration run 26 27 1) and the results are shown in figure 7. Although for calibration runs 2 and 3 the 28 29 differences between the two fitting techniques are relatively small there is significant 30 31 32 difference in the results from calibration run 1 where the Oliver and Pharr fit leads to 33 34 a significant over-estimation of both reduced modulus and hardness at low depths 35 36 37 (compare figures 6 & 7). One might argue that as a single 9 term Oliver and Pharr 38 39 type fit has been used for all 3 calibration runs whereas for the equivalent contact 40 41 radius technique for calibration runs 2 and 3 the data were split into 2 parts that we 42 43 44 are biasing things against the Oliver and Pharr fit. However the Oliver and Pharr fit 45 46 produced significant over-estimations of both reduced modulus and hardness at low 47 48 depths with calibration run 1 where the data were not split for either fitting process. 49 50 51 Thus the Oliver and Pharr type fit seems not to capture the initial tip geometry (which 52 53 is essentially ideal apart from a flat right at the tip) very well, even though once the tip 54 55 shape is changed through use the Oliver and Pharr fit works well. Overall we suggest 56 57 58 that the major advantage of the equivalent contact radius fitting over the Oliver and 59 60
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1 2 3 Pharr one is that it provides for a simple physical interpretation of the tip geometry 4 5 6 and how it is changing with time. 7 8 9 10 5. Conclusions 11 12 13 14 3 detailed shallow depth calibrations at approximately yearly intervals have been 15 16 For Peer Review Only 17 carried out on the same tip. As one would expect there is clear evidence of blunting 18 19 over time. When the tip was new the equivalent contact radius, which is related to the 20 21 contact area via equation (3 ), was linearly related to the contact depth apart from a flat 22 23 24 right at the tip thus demonstrating that the tip was geometrically good. The equivalent 25 26 contact radius approach therefore gives a clear physical interpretation of the tip 27 28 geometry and dealt with flat at the tip in a better fashion than a 9 term Oliver and 29 30 31 Pharr type fit. The equivalent radius approach means that the shallow depth 32 33 indentation data is automatically weighted equally in the fitting process. As the tip 34 35 became blunter with use the relationship between equivalent radius and the contact 36 37 38 depth starts to deviate from a linear relationship at very low depths but applies at 39 40 larger depths and still provides a clear physical interpretation of the changes at the tip. 41 42 43 Thus, although as good calibrations can be obtained using an Oliver and Pharr type 44 45 fit, the methodology proposed here provides a clearer physical picture of the tip 46 47 48 geometry. It is also worthy of note that Er∝1/ r c ∝ 1/ A so it arguably makes more 49 50 sense to calibrate directly using a linear relationship. Finally we note that c onsistent 51 52 53 measurements of very near surface elastic moduli can be made if the blunting of the 54 55 tip over time is fully accounted for in the tip calibration. 56 57 58 Acknowledgements 59 60 DT thanks the ORSAS, UK and the University of Sheffield for scholarships enabling him to undertake this work.
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1 2 3 References 4 5 6 [1] Mencik J. Determination of mechanical properties by instrumented 7 indentation. Meccanica 2007; 42: 19. 8 [2] Hysitron. User's Manual for Tribiscope Nanomechanical Test System NRL- 9 10 M-015. Minneapolis, USA, 2003: 63. 11 [3] Doerner MF, Nix WD. A method for interpreting the data from depth-sensing 12 indentation instruments. Journal of Materials Research 1986; 1: 601. 13 [4] Pethica JB, Hutchings R, Oliver WC. Hardness measurement at penetration 14 depths as small as 20-nm. Philosophical Magazine a-Physics of Condensed 15 Matter Structure Defects and Mechanical Properties 1983; 48: 593. 16 For Peer Review Only 17 [5] Gerberich WW, Yu W, Kramer D, Strojny A, Bahr D, Lilleodden E, Nelson J. 18 Elastic loading and elastoplastic unloading from nanometer level indentations 19 for modulus determinations. Journal of Materials Research 1998; 13: 421. 20 [6] Oliver WC, Pharr GM. Measurement of hardness and elastic modulus by 21 instrumented indentation: Advances in understanding and refinements to 22 23 methodology. Journal of Materials Research 2004; 19: 3. 24 [7] Hand RJ, Tadjiev DR, Hayes SA. Nano-indentation and surface hydration of 25 silicate glasses. Journal of the Ceramic Society of Japan 2008; 116: 846. 26 [8] Tadjiev DR, Hand RJ. Inter-relationships between composition and near 27 surface mechanical properties of silicate glasses. Journal of Non-Crystalline 28 29 Solids 2008; 354: 5108. 30 [9] NIST. Engineering Statistics Handbook Section 4.4.5.2. 31 http://www.itl.nist.gov/div898/handbook/pmd/section4/pmd452.htm , accessed 32 21.7.09. 33 [10] Oliver WC, Pharr GM. An improved technique for determining hardness and 34 elastic-modulus using load and displacement sensing indentation experiments. 35 36 Journal of Materials Research 1992; 7: 1564. 37 [11] Thurn J, Cook RF. Simplified area function for sharp indenter tips in depth- 38 sensing indentation. Journal of Materials Research 2002; 17: 1143. 39 [12] Hay JL, Pharr GM. Instrumented Indentation Testing. Ohio, USA, 2000: 231. 40 [13] Gong JH, Miao HZ, Peng ZJ. Analysis of the nanoindentation data measured 41 42 with a Berkovich indenter for brittle materials: effect of the residual contact 43 stress. Acta Materialia 2004; 52: 785. 44 [14] Qian LM, Li M, Zhou ZR, Yang H, Shi XY. Comparison of nano-indentation 45 hardness to microhardness. Surface & Coatings Technology 2005;195:264. 46 [15] Hench L.L. and Clark D.E., Physical chemistry of glass surfaces. J. Non-Cryst. 47 Solids 1978; 28: 83. 48 49 50 51 52 53 54 55 56 57 58 59 60
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1 2 3 Table 1. Summary of mechanical properties (quoted errors are 3 standard deviations). 4 5 Data No. of H (>15nm) 6 Date E /GPa 7 set indents r /GPa 8 TAF1 Aug 2007 1540 69.1 ± 0.6 8.9 ± 0.2 9 TAF2 July 2008 960 69.6 ± 0.3 8.8 ± 0.1 10 11 TAF3 May 2009 982 69.6 ± 0.3 8.8 ± 0.1 12 13 14 15 16 For Peer Review Only 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
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1 2 3 Figure 1. Calibration data obtained when the tip was new. 4 5 6 7 250 Calibration run 1 data 8 Linear regression fit to data 9 200 10 11 12 150 13 14 /nm c
15 r 100 16 For Peer Review Only 17 18 50 19 20 21 0 22 0 20 40 60 80 23 24 hc /nm 25 26 200x10 3 27 Calibration run 1 data 28 O&P 9 term fit 29 160x10 3 30 31 32 2 120x10 3 33 34 35 /(nm) 80x10 3 A 36 37 3 38 40x10 39 40 41 0 42 0 20 40 60 80 43 h /nm 44 c 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
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1 2 3 Figure 2. Original calibration data plus data from the second (later) calibration a) 4 5 plotted as rc versus hc and b) plotted as A versus hc. 6 7 350 8 Calibration run 1 data 9 300 Linear regression fit to 1st dataset 10 Calibration run 2 data 11 250 Displaced regression line from 1 12 13 14 200
15 /nm c
16 Forr 150 Peer Review Only 17 18 100 19 20 50 21 22 0 23 0 20 40 60 80 100 24 25 hc /nm 26 27 350x10 3 28 Calibration run 1 data 29 300x10 3 Calibration run 2 data 30 31 250x10 3 32 2 33 200x10 3 34 35 3 /(nm) 150x10
36 A 37 100x10 3 38 39 50x10 3 40 41 0 42 0 20 40 60 80 100 43 44 h /nm 45 c 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
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1 2 3 Figure 3. Data from the second calibration showing a) a fit of r = ah + bh 1/2 + c over 4 c c c 5 the lowest depth region b) the combined fit given by equation (10) over the entire data 6 range. 7 8 9 240 10 Calibration run 2 data 11 200 Fitted curve 12 13 160 14 15 16 For120 Peer Review Only /nm c
17 r 18 80 19 20 21 40 22 23 0 24 0 10 20 30 40 50 25 26 hc /nm 27 28 350 29 Calibration run 2 data 30 300 Fitted curve 31 32 250 33 34 200 35 /nm
36 c
r 150 37 38 100 39 40 50 41 42 0 43 0 20 40 60 80 100 44 45 hc /nm 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
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1 2 3 /1 2 4 Figure 4. a) Comparison of all 3 calibrations and b) a fit of rc = ah c + bh c + c over 5 the lowest depth region combined fit with a straight line fit at greater depths for 6 calibration run 3. 7 8 9 350 Calibration run 1 data 10 Linear regression fit to 1st dataset 11 300 Calibration run 2 data 12 Calibration run 3 data 13 250 14 15 200 16 For Peer Review Only /nm 17 c
r 150 18 19 100 20 21 50 22 23 0 24 0 20 40 60 80 100 25 26 hc /nm 27 28 350 29 Calibration run 3 data 30 300 Fitted curve 31 32 250 33 34 200 35 /nm 36 c
r 150 37 38 100 39 40 50 41 42 0 43 0 20 40 60 80 100 44 45 hc /nm 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
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1 2 3 4 5 Figure 5. Residuals from the fits to all 3 calibration runs a) obtained by fitting to rc 6 and b) obtained from the Oliver and Pharr fit but plotted in terms of rc for 7 comparative purposes 8 9 10 11 20
12 /nm c
13 r 14 10 15 16 For Peer Review Only 17 18 0 calculated
19 −
20 c r 21 -10 22 Calibration run 1 23 Calibration run 2 24 Calibration run 3 25 -20 Measured 26 0 20 40 60 80 100 27 28 hc /nm 29 30 31 32 33 20 34 35
36
37 1/2 10 ) /nm 38 π / 1/2 )
39 A π 40 / 0 41 A 42 43 44 -10 fitted ( Calibration run 1 Measured( 45 − Calibration run 2 46 Calibration run 3 47 48 -20 49 0 20 40 60 80 100 50 h /nm 51 c 52 53 54 55 56 57 58 59 60
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1 2 3 Figure 6. a) Reduced modulus and b) hardness plots obtained using calibration 4 5 equations (7), (11) and (12). 6 7 8 120 9 10 11 100 12 13 80 14 15 60
16 For/GPa Peer Review Only 17 E 40 18 19 Calibration 1 20 20 Calibration 2 21 Calibration 3 22 0 23 0 20 40 60 80 100 24 25 hc /nm 26 27 28 30 29 30 25 31 32 20 33 34 35 15 36 /GPa H 37 10 38 Calibration 1 39 5 40 Calibration 2 41 Calibration 3 42 0 43 0 20 40 60 80 100 44 h /nm 45 c 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
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1 2 3 Figure 7. a) Reduced modulus and b) hardness plots obtained using 9 term Oliver and 4 5 Pharr TAF type fits. 6 7 120 8 9 10 100 11 12 80 13 14 60 15 /GPa
16 ForE Peer Review Only 17 40 18 Calibration 1 O&P 9 term fit 19 20 Calibration 2 O&P 9 term fit 20 Calibration 3 O&P 9 term fit 21 22 0 23 0 20 40 60 80 100 24 h /nm 25 c 26 27 30 Calibration 1 O&P 9 term fit 28 Calibration 2 O&P 9 term fit 29 25 Calibration 3 O&P 9 term fit 30 31 20 32 33 34 15 35 /GPa H 36 10 37 38 39 5 40 41 0 42 0 20 40 60 80 100 43 44 hc /nm 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
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