Calibrating a nanoindenter for very shallow depth indentation using equivalent contact radius

Transcription

1 Calibrating a nanoindenter for very shallow depth indentation using equivalent ontat radius Damir R Tadjiev, Russell J Hand, Simon A Hayes To ite this version: Damir R Tadjiev, Russell J Hand, Simon A Hayes. Calibrating a nanoindenter for very shallow depth indentation using equivalent ontat radius. Philosophial Magazine, Taylor Franis,, (), pp.-. <./>. <hal-> HAL Id: hal- Submitted on Apr HAL is a multi-disiplinary open aess arhive for the deposit and dissemination of sientifi researh douments, whether they are published or not. The douments may ome from teahing and researh institutions in Frane or abroad, or from publi or private researh enters. L arhive ouverte pluridisiplinaire HAL, est destinée au dépôt et à la diffusion de douments sientifiques de niveau reherhe, publiés ou non, émanant des établissements d enseignement et de reherhe français ou étrangers, des laboratoires publis ou privés.

2 Philosophial Magazine & Philosophial Magazine Letters Calibrating a nanoindenter for very shallow depth indentation using equivalent ontat radius Journal: Philosophial Magazine & Philosophial Magazine Letters Manusript ID: TPHM--Jul-.R Journal Seletion: Philosophial Magazine Date Submitted by the Author: -Nov- 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, mehanial properties Keywords (user supplied): tip area funtion, alibration, shallow depths

3 Page of Philosophial Magazine & Philosophial Magazine Letters Calibrating a nanoindenter for very shallow depth indentation using equivalent ontat radius Damir R. Tadjiev, Russell J. Hand and Simon A. Hayes Department of Engineering Materials, University of Sheffield, Sheffield, UK Mappin Street, Sheffield S JD, UK (Reeived July ; final version reeived ) Nanoindenter tips are usually modeled as axisymmetri ones, with alibration involving finding a fitting funtion relating the ontat area to the ontat depth. For aurate alibration of shallow depth indentation this is not ideal beause it means that deeper indents tend to dominate the fitting funtion. For an axisymmetri objet it is always possible to define an equivalent ontat radius (whih in the ase of nanoindentation is linearly related to the redued modulus) and to obtain a fitting funtion relating this equivalent ontat radius to indentation depth. The equivalent ontat radius approah is used here to provide shallow depth alibration of a nanoindenter tip at three separate times. The advantage of the equivalent ontat radius methodology is that it provides a learer physial interpretation of the hanges in tip shape than a onventional area based fit. We also show that the minimum depth for a reliable hardness measurement is obtainable inreases as the tip blunts with age but that onsistent measurements of very near surfae elasti moduli an be made if the blunting of the tip over time is fully aounted for in the tip area funtion alibration. Keywords: tip area funtion, alibration, nanoindentation, shallow depths Corresponding author. Tel.: + (). address: r.hand@shef.a.uk (R.J. Hand).

4 Philosophial Magazine & Philosophial Magazine Letters Page of. Introdution Nanoindentation differs from onventional indentation in that the results are based not on imaging of the residual indent (whih would be very hallenging, and largely impratiable, for very low load indents), but rather on a alibrated tip area funtion (TAF) whih desribes the variation of the ontat area (A ) of the tip as a funtion of ontat depth (h ). In the ideal ase the indenter has a simple axisymmetri shape and, hene, the TAF an be desribed by a simple mathematial funtion. Thus even the ommonly used Berkovih indenter, whih is atually a triangular pyramid, is usually modelled by an equivalent axisymmetri one, as assumed in the onventional Oliver and Pharr (O&P) approah. In atuality any real tip shape will deviate from its ideal geometry to some greater or lesser extent, even when new, and this will beome more marked as the tip beomes worn over time due to its use in indentation [] and, ommonly, seondary use in imaging the residual indent []. Thus an equivalent one model of a Berkovih tip will break down at some point [, ] and one approah to this problem is to treat the very near tip region using a Hertzian analysis (see, for example, []); learly this still treats the tip as an axisymmetri objet. In the ommonly adopted Oliver and Pharr approah TAF alibration is arried out by indenting a material of known modulus, whih is assumed to have i) an elasti modulus that is unvarying with depth and ii) a low E/H value whih means that there will be minimum pile-up and sink-in []. The stiffness, S, of the ombined indenter/ indented materials system is then assessed from the initial (elasti) part of the unloading urve and the area of the indentation is then obtained using

5 Page of Philosophial Magazine & Philosophial Magazine Letters πs A= () β Er where β is a geometri orretion fator (in this work taken to be equal to.) [] and E r is the redued modulus, whih aounts for the fat that the measured displaement inludes ontributions from the sample and the indenter. E r is given by ν ν i = + () E E E r where E, E i and ν, ν i are the elasti moduli and Poisson s ratios of sample and the indenter respetively. For the diamond Berkovih indenter E i = GPa and ν i =., and for fused silia (whih largely obeys the two assumptions mentioned above) used as a standard in this work E = GPa and ν =., respetively; hene E r is equal to. GPa. The TAF is then obtained by fitting A as a funtion of the ontat depth, h. The work desribed here forms part of a larger projet to examine the mehanial properties of the surfae hydration layers formed on siliate glasses [, ]. This requires reliable alibration of TAFs for shallow (we have been fousing on less than nm indentation) depths. Calibration of the TAF for shallow depths is a partiular problem for a number of reasons: imaging is most diffiult in this region (and at the lowest loads impossible due to the ourrene of purely elasti behaviour); and although the data are more sensitive to random errors in this region (i.e. they are noisier) the data obtained at larger depths tend to have larger absolute (as distint from relative) residuals. The latter point means that any large depth data tend to dominate the urve fitting thereby reduing the auray of any fit at shallow depth. Conventionally this issue is addressed by providing different fits over different depth i

6 Philosophial Magazine & Philosophial Magazine Letters Page of ranges but an alternative method is to selet an appropriate data transform []. Here it has been therefore deided to fit r = A/π () as a funtion of h, where r is the ontat radius of the equivalent axisymmetri body. This has the advantage of giving an equal distribution of residuals due to the approximately linear saling between r and h that an be reasonably expeted for all tip geometries. Also ombining equations () and () gives S r = () βe r i.e. E / r meaning that the alibration is based entirely on a linear relationship r rather than a quadrati one.. Experimental proedure. Instrumentation and referene material A ommerial Hysitron Tribosope nanoindenter (Hysitron In., USA) mounted on a Dimension (Veeo Digital Instruments) nanosope equipped with a Berkovih tip (Hysitron, original tip radius nm) was used. The indenting system is housed in under a thermal and aousti isolation hood whih is mounted on a vibration resistant table. The nanoindenter has load and displaement resolution of µn and. nm, respetively. Calibration was undertaken by making indentations on standard fused silia samples to obtain data for ontat depth in the range of - nm. The first tip area funtion (TAF) was alibrated on mm fused silia sample that ame with nanoindenter and had surfae roughness, R a, of. nm, while the seond and the third (TAF and TAF) tip area funtions were alibrated on new mm fused silia slides (Heraeus, R a =. nm). data sets were olleted at

7 Page of Philosophial Magazine & Philosophial Magazine Letters approximately year time intervals (see table ). All experiments were arried out in a quasi-stati mode and during eah alibration run multiple arrays of indents spaed µm apart on both the x and y axes were made, with eah indent providing a single measurement of stiffness and load. A standard loading sheme of s up-load, s dwell and s unloading was used. All measurements were arried out at room temperature (± C) and a relative humidity of -% with drift orretion on. The indenter is ontinuously powered so that the eletronis are maintained in a stable state. Drift was measured by running. µn test with a s hold at peak load and measuring displaement drift during this hold. The remainder of the test was orreted by the measured drift rate the typial value of whih did not exeed. nm/s. Approximately data points were olleted for eah alibration data set with an emphasis on low load indentation values. Prior to the measurements, the ompliane of the indenter olumn (mahine ompliane) was measured by making at least high-load indentations ( to µn) on fused silia and then plotting ompliane (inverse stiffness) versus /P (where P is applied load) for the obtained data. Reliability of the determined ompliane value was heked by plotting load over stiffness squared (P/S ) versus ontat depth and making sure that the slope of the urve is. This approah was originally proposed by Oliver and Pharr [] for the measurements at high-load indentations, but it has been found in this work that it also works well at low-load indentations. Prior to eah test the fused silia samples were leaned by rinsing in pure ethanol and drying using a warm air blower. The surfae region to be indented was then imaged using the instrument in the sanning probe mirosope (SPM) mode to ensure that the surfae was free of dirt and ontamination.

8 Philosophial Magazine & Philosophial Magazine Letters Page of. Data analysis and TAF fitting Following the standard O&P method [] the initial unloading ontat stiffness, S, was obtained from the slope of a power law fit to the initial part of the unloading urves using the installed Hysitron software. The power law used was ( h ) m P=α () h f where P is load, h is the measured indention depth, h f is the residual indentation depth after removal of the indenter and α and m are fitting onstants. The unloading stiffness ( S = d P d h ) was then obtained by differentiation and equations () were used to obtain the equivalent axisymmetri ontat radius. Contat depth h was alulated by the Hysitron indenter software using h S = hmax. () P where h max and P max are, respetively, the maximum depth and load during the indentation [].. Results and disussion Figure shows a plot of r versus h for the tip when new. In this ase a linear fit was found to be appliable and a linear regression fit to the data shown gave max r.h +. () = with an r value of.. It is interesting to note that Thurn and Cook [] proposed a three term area funtion, whih atually involves only two independent parameters, to alibrate the TAF. They took a harmoni average of a spherial tip profile (with radius R) and a perfet onial shape (with inluded angle α) and obtained Combining equations () and () gives π A= h Rπh R π ot α + + () ot α

9 Page of Philosophial Magazine & Philosophial Magazine Letters h r = + R otα () otα i.e. a straight line as seen in figure (a). The original derivation of equation () involved use of the binomial expansion whih requires that the ontat radius a> R otα, and implies that there is a flat end, the radius of whih orresponds to the onstant term in equation (), even though the original harmoni average does not assume this. Using equation () for the data shown in figure gives an inluded angle o o α =. ±. and R=.±. nm indiating that the original tip onformed reasonably losely to the expeted equivalent one angle of. (see, for example, []). However, as noted above, equation () in fat indiates that there is a flat right at the tip. In the urrent ase the radius of this flat is r =.±. nm. For omparative purposes a term Oliver and Pharr TAF fit was also undertaken (see figure b). This gave A=.h +.h +.h / with an r value of.. +.h +. / h / +. +.h h / / +.h +.h / / () Figure a shows a later alibration on the same tip arried out after it had reeived signifiant usage. The fat that the regression line from the first alibration an be displaed upwards to overlay the data from the seond alibration shows that although the very near tip region has indeed beome blunter away from this region the tip is essentially unmodified. This fat annot be simply inferred from a onventional TAF plot (figure b). It would also appear that although the tip shape has been modified that the effetive radius of the flat right at the tip has remained essentially unhanged as dataset interepts the y-axis at essentially the same point as dataset.

10 Philosophial Magazine & Philosophial Magazine Letters Page of The exat displaement between the two urves was evaluated by ensuring that the residuals in the straight line portion were equally sattered about zero by examining the moving average. The moving average was also used to identify an appropriate break point below whih an alternative urve fit was required. Empirially it was found that using a funtion based on that proposed by Oliver and Pharr for the tip area fit was most effetive for this regime (see figure ). The fitting was arried out using the non-linear regression routines supplied in SigmaPlot (Systat). Thus for the partiular dataset shown in figures and the resultant fit was r =.h +.h r =.h / h < nm h nm In this ase the overall r value for the fit was.. The improved r value for this fit ompared to the original fit is due to a smaller spread in the initial data, whih is due to the use of a alibration piee of silia with lower surfae roughness. Although a split funtion approah ould also be used for the TAF fit it is muh less obvious as to where to split the fitting funtion. Thus the equivalent radius approah potentially offers some benefit in determining an appropriate fit. Similar features were found with a third alibration arried out at a later date (see figure ). Again at the greater depths the alibration data appeared to be simply displaed with respet to the original alibration but in the very near tip region the detailed shape had further hanged and one again an empirial fit was used in this region. In this ase the ombined fit was r =.h +.h r =.h / h < nm h nm It an be seen that the fit is slightly less good right at the inreasingly blunt indenter tip but additional terms gave no benefit in terms of quality of fit. The overall r value for the fit is.. As well as showing how the tip geometry is being blunted over () ()

11 Page of Philosophial Magazine & Philosophial Magazine Letters time the results in figure (a) one again emphasize how the stiffness of the indentation system is affeted by the tip geometry espeially at low loads (r is diretly proportional to S; see equation ). The residuals for all alibrations are shown in figure (a). It an be learly seen that they are in general uniformly distributed about zero indiating both a good fit and that, as desired, all points have essentially been equally weighted in obtaining the fit. As noted above it an be seen that the spread of data in the first fit seen in figure () is wider (refleted in the poorer r value for this fit) and there is an asymmetry in the residuals in the very near tip region for the third alibration indiating a relatively poor fit in this region. Figure (b) shows the orresponding residuals for the Oliver and Pharr fit having re-expressed everything in terms of equivalent radius for omparative purposes. It an be seen that the distribution of the residuals is very similar, exept in the region orresponding to the flat in the original alibration run. Thus the two methodologies overall provide similar levels of auray the real advantage of the methodology proposed here is that it diretly provides a learer physial piture of progressive hanges in indenter tip geometry. As a further hek equations (), () and () were substituted in equation () and the redued modulus was realulated for the three data sets and the results are shown in figure (a). In all ases the data are essentially sattered around the expeted value of. MPa. Figure (a) also indiates that onsistent measurements of very near surfae elasti moduli an be made if the blunting of the tip over time is fully aounted for in the tip alibration. It is obvious that the surfae ontamination and hydration may ompromise the auray of shallow depth indentations. As outlined above leaning and SPM was used to ensure that the indented region was lean prior to indentation. Also as figure

12 Philosophial Magazine & Philosophial Magazine Letters Page of (a) shows that the residuals are essentially equally distributed aross the entire data range there is no evidene of any leaning of ontamination during suessive indents. Hydration is more of a problem. Hydration layer depths of nm to nm [, ] have previously been suggested for pure silia. Fundamentally this is an issue for any shallow depth alibration of a nanoindenter in that all alibration tehniques assume that there is no variation of modulus with depth whereas a hydrated layer would be expeted to result in a redution in modulus as the surfae is approahed. In this ase one would expet to see an apparent derease in equivalent ontat radius (i.e. an apparent sharpening of the tip) due to the over-estimation of the value of E r used in equation (). The fat that we do not observe this behaviour, rather the opposite over time, suggests that for the silia sample examined here hydration is not a signifiant problem although its presene annot be absolutely ruled out. The fat that the noise inreases at the very shallowest of depths is, we believe, an inevitable onsequene of the fat that the measured quantities are partiularly small in this region and thus the signal to noise ratio is redued. For nanoindentation hardness is defined as H P P max max = = () A πr where P max is the maximum load during the indentation. Hardnesses alulated in this fashion are shown in figure (b). In all ases it an be seen that the measured hardness tends to derease in the very near surfae region. Unfortunately the greater amount of noise present in the original alibration data makes it diffiult to identify a lear trend in behaviour, although at very low depths there is an apparent derease in hardness with dereasing indentation depth; a learer derease in hardness with dereasing indentation depth is apparent for the other two datasets and the minimum indentation depth required for meaningful hardness measurements appears to inrease

13 Page of Philosophial Magazine & Philosophial Magazine Letters as the tip beomes inreasingly blunt. At depths greater than the minimum indentation depth for meaningful hardness measurements no indentation size effet is observed, whih is in agreement with the results obtained by other researhers [,,, ]. The apparent derease in hardness with dereasing depth very near the surfae arises beause in fat fully elasti deformation is ourring in this region and thus hardness is not atually a meaningful measurement in this region. The redued elasti moduli (over the entire measured depth range) and hardness values (for indentation depths >nm) obtained with the three different alibrations are ompared in Table. Redued moduli and hardness values were also alulated using the results of term Oliver and Pharr TAF type fits (see equation () for the fit for alibration run ) and the results are shown in figure. Although for alibration runs and the differenes between the two fitting tehniques are relatively small there is signifiant differene in the results from alibration run where the Oliver and Pharr fit leads to a signifiant over-estimation of both redued modulus and hardness at low depths (ompare figures & ). One might argue that as a single term Oliver and Pharr type fit has been used for all alibration runs whereas for the equivalent ontat radius tehnique for alibration runs and the data were split into parts that we are biasing things against the Oliver and Pharr fit. However the Oliver and Pharr fit produed signifiant over-estimations of both redued modulus and hardness at low depths with alibration run where the data were not split for either fitting proess. Thus the Oliver and Pharr type fit seems not to apture the initial tip geometry (whih is essentially ideal apart from a flat right at the tip) very well, even though one the tip shape is hanged through use the Oliver and Pharr fit works well. Overall we suggest that the major advantage of the equivalent ontat radius fitting over the Oliver and

14 Philosophial Magazine & Philosophial Magazine Letters Page of Pharr one is that it provides for a simple physial interpretation of the tip geometry and how it is hanging with time.. Conlusions detailed shallow depth alibrations at approximately yearly intervals have been arried out on the same tip. As one would expet there is lear evidene of blunting over time. When the tip was new the equivalent ontat radius, whih is related to the ontat area via equation (), was linearly related to the ontat depth apart from a flat right at the tip thus demonstrating that the tip was geometrially good. The equivalent ontat radius approah therefore gives a lear physial interpretation of the tip geometry and dealt with flat at the tip in a better fashion than a term Oliver and Pharr type fit. The equivalent radius approah means that the shallow depth indentation data is automatially weighted equally in the fitting proess. As the tip beame blunter with use the relationship between equivalent radius and the ontat depth starts to deviate from a linear relationship at very low depths but applies at larger depths and still provides a lear physial interpretation of the hanges at the tip. Thus, although as good alibrations an be obtained using an Oliver and Pharr type fit, the methodology proposed here provides a learer physial piture of the tip geometry. It is also worthy of note that E / r / A so it arguably makes more r sense to alibrate diretly using a linear relationship. Finally we note that onsistent measurements of very near surfae elasti moduli an be made if the blunting of the tip over time is fully aounted for in the tip alibration. Aknowledgements DT thanks the ORSAS, UK and the University of Sheffield for sholarships enabling him to undertake this work.

15 Page of Philosophial Magazine & Philosophial Magazine Letters Referenes [] Menik J. Determination of mehanial properties by instrumented indentation. Meania ; :. [] Hysitron. User's Manual for Tribisope Nanomehanial Test System NRL- M-. Minneapolis, USA, :. [] Doerner MF, Nix WD. A method for interpreting the data from depth-sensing indentation instruments. Journal of Materials Researh ; :. [] Pethia JB, Huthings R, Oliver WC. Hardness measurement at penetration depths as small as -nm. Philosophial Magazine a-physis of Condensed Matter Struture Defets and Mehanial Properties ; :. [] Gerberih WW, Yu W, Kramer D, Strojny A, Bahr D, Lilleodden E, Nelson J. Elasti loading and elastoplasti unloading from nanometer level indentations for modulus determinations. Journal of Materials Researh ; :. [] Oliver WC, Pharr GM. Measurement of hardness and elasti modulus by instrumented indentation: Advanes in understanding and refinements to methodology. Journal of Materials Researh ; :. [] Hand RJ, Tadjiev DR, Hayes SA. Nano-indentation and surfae hydration of siliate glasses. Journal of the Cerami Soiety of Japan ; :. [] Tadjiev DR, Hand RJ. Inter-relationships between omposition and near surfae mehanial properties of siliate glasses. Journal of Non-Crystalline Solids ; :. [] NIST. Engineering Statistis Handbook Setion... aessed... [] Oliver WC, Pharr GM. An improved tehnique for determining hardness and elasti-modulus using load and displaement sensing indentation experiments. Journal of Materials Researh ; :. [] Thurn J, Cook RF. Simplified area funtion for sharp indenter tips in depthsensing indentation. Journal of Materials Researh ; :. [] Hay JL, Pharr GM. Instrumented Indentation Testing. Ohio, USA, :. [] Gong JH, Miao HZ, Peng ZJ. Analysis of the nanoindentation data measured with a Berkovih indenter for brittle materials: effet of the residual ontat stress. Ata Materialia ; :. [] Qian LM, Li M, Zhou ZR, Yang H, Shi XY. Comparison of nano-indentation hardness to mirohardness. Surfae & Coatings Tehnology ;:. [] Henh L.L. and Clark D.E., Physial hemistry of glass surfaes. J. Non-Cryst. Solids ; :.

16 Philosophial Magazine & Philosophial Magazine Letters Page of Table. Summary of mehanial properties (quoted errors are standard deviations). Data No. of H (>nm) Date E set indents r /GPa /GPa TAF Aug. ±.. ±. TAF July. ±.. ±. TAF May. ±.. ±.

17 Page of Philosophial Magazine & Philosophial Magazine Letters r Figure. Calibration data obtained when the tip was new. A /(nm) Calibration run data Linear regression fit to data x x x x x h Calibration run data O&P term fit h

18 Philosophial Magazine & Philosophial Magazine Letters Page of Figure. Original alibration data plus data from the seond (later) alibration a) plotted as r versus h and b) plotted as A versus h. r A /(nm) x x x x x x Calibration run data Linear regression fit to st dataset Calibration run data Displaed regression line from x h Calibration run data Calibration run data h

19 Page of Philosophial Magazine & Philosophial Magazine Letters Figure. Data from the seond alibration showing a) a fit of r = ah + bh / + over the lowest depth region b) the ombined fit given by equation () over the entire data range. r r Calibration run data Fitted urve Calibration run data Fitted urve h h

20 Philosophial Magazine & Philosophial Magazine Letters Page of / Figure. a) Comparison of all alibrations and b) a fit of r = ah + bh + over the lowest depth region ombined fit with a straight line fit at greater depths for alibration run. r r Calibration run data Linear regression fit to st dataset Calibration run data Calibration run data Calibration run data Fitted urve h h

21 Page of Philosophial Magazine & Philosophial Magazine Letters Figure. Residuals from the fits to all alibration runs a) obtained by fitting to r and b) obtained from the Oliver and Pharr fit but plotted in terms of r for omparative purposes Measured r alulated r Measured (A/π) / fitted (A/π) / - Calibration run Calibration run Calibration run - - h - h Calibration run Calibration run Calibration run

22 Philosophial Magazine & Philosophial Magazine Letters Page of Figure. a) Redued modulus and b) hardness plots obtained using alibration equations (), () and (). E /GPa H /GPa Calibration Calibration Calibration h Calibration Calibration Calibration h

23 Page of Philosophial Magazine & Philosophial Magazine Letters Figure. a) Redued modulus and b) hardness plots obtained using term Oliver and Pharr TAF type fits. E /GPa H /GPa h h Calibration O&P term fit Calibration O&P term fit Calibration O&P term fit Calibration O&P term fit Calibration O&P term fit Calibration O&P term fit