Scanning Nanoindentation - One example of a quantitative SPM technique

Scanning Nanoindentation - One example of a quantitative SPM technique

Topics Mechanical characterization on the nanoscale The basic idea Some models Some issues Instrumentation Tribological characteriazation Historical background The difference between macro- and nano-tribology

Scanning nanoindentation SPM: Nanoindentation: Excellent surface imaging Well-defined tip-positioning (via imaging) Quantitative mechanical properties are not accessible Quantitative mechanical characterization on the nano scale Blind to the surface + Scanning Nanoindentation: Good surface imaging Well-defined tip-positioning (via imaging) Quantitative mechanical characterization

Motivation Macroscopic (some basic examples) Brinell: Steel balls of various diameters are pressed into the material of interest Vickers: 4-sided pyramid (diamond) Rockwell: Steel ball or diamond cone is pressed into a material / indentation depth is used as a measure of hardness hardness = Force Area Microscopic In the ultra low load regime the remaining deformation starts to get too small to track it down optically Depth-sensing indentation instruments are used: A test yields a force-displacement curve Nanoindenter // AFM Load, P loading unloading Displacement, δ

Basics First closed solution of elastic contact (two elastic spheres) given by Hertz in 1881 [[i]] In case of a contact of a sphere with an elastic half-space the displacement change and the contact radius are given by [[ii]]: P = 16 R E 9 * 2 ξ 3 / 2 a = 3 P R * 4 E 1/ 3 Assumptions: Small pressures, R large compared to a, contact is frictionless. E*, the so called reduced modulus, is given by: E 2 2 1 ν1 1 2 * = + E1 E 2 1 ν Word of warning: Often ξ is mistaken to be δ e, the elastic penetration depth of the sphere into the half-space. This is only true for an infinitely rigid sphere (indenter). Misinterpretation can lead to wrong mechanical data as reported by Chaudhri [[iii]]. [i]. H. Hertz, Über die Berührung fester elastischer Körper, J. reine und angewandte Mathematik, 92, 1882, S. 156-171 [ii]. K. L. Johnson, Contact Mechanics, paperback edition, Cambridge University Press, Cambridge, 1987 [iii]. M. M. Chaudhri, A note on a common mistake in the analysis of nanoindentation data, J. Mat. Res., Vol. 16, No. 2, 2001, 336-339

Basics Many materials show an elastic-plastic type of behavior, that can be expressed in terms of an elastic and a plastic displacement (δ=δ e +δ p ) of a series of two spring elements (conical indenter): P = 1 C + 1 e C p 2 δ 2 Many theories are based on an approach by Loubet et al. [i], that incorporates the work of Tabor [ii] and Sneddon [iii] surface indenter flat-punch δ e δ p (a) ansatz: Loubet et al. (b) Sneddon [i]. J. L. Loubet, J. M. Georges und G. Meille, Microindentation techniques in material science and engineering, ASTM STP 889, P. J. Blau und B. R. Lawn, Eds., American Society for Testing Materials, 1986, 72-89 [ii]. D. Tabor, Review of physics in technology, Vol. 1, 1970, 145-179 [iii]. I. N. Sneddon, Int. J. Engng. Sci., Vol. 3, 1965, 47-57

Mechanical characterization (Oliver and Pharr model) Based on Loubet et al. and Sneddon they propose [i] (θ beeing 1, 0.75 and 0.72 for a punch, a rotationparaboloid, and a cone, respectively): δ c = δ P max δ e = δ P max θ P max S loading (P max, δ P max ) Reduced modulus and hardness are given by: load, P unloading S possible range of δ c S H dp dδ P A 2 A π ( δc ) E * ( 2 ν ) P 1 max max ( δ ) c = δ f δ c (θ = 1) δ c (θ = 0,72) displacement, δ [i]. W. C. Oliver und G. M. Pharr, J. Mater. Res., Vol. 7, No. 6, June 1992, 1564-1583 Nearly all of the elements of this analysis were first developed by workers at the Baikov Institute of Metallurgy in Moscow during the 1970's (for a review see Bulychev and Alekhin).

Elastic/plastic approach (Field and Swain) An analysis of quasi-static nanoindentation with spherical indenters based on a Herzian contact solution has been carried out by Field and Swain and Bell, Field and Swain. They were able to extend the Hertzian approach to incorporate plastic deformation. In their model they treat the indentation as a reloading of a preformed impression with depth h f into reconformation with the indenter. Using a loading and partially unloading technique they are able to determine hardness and reduced modulus from the appropriate nanoindentation data.

Other quasi static models Joslin and Oliver showed that the ratio H/E r2 may be expressed in terms of P and S 2, quantities which are easily and directly measured without need of a tip shape function or contact model. Another quantity which is easily measured is the area enclosed within the indentation hysteresis. This area represents the non-recoverable work done on the material during indentation. (See Gubicza et al.) A number of analyses of the loading data have been performed (see for example Hainsworth et al. or Zeng and Rowcliffe). These always require an assumption about how much of the displacement is due to elastic vs. plastic deformation. Recently Oliver proposed a method to avoid detailed knowledge about the actual tip shape by analyzing the slope of the loading and unloading segment of the test.

Comments / Things not included so far Plastic deformation often leads to so-called pile-up around the indent true contact area is changed. Some models exist based on the self-similarity of the indentation process [i] Work hardening: The indentation process itself introduces geometrically necessary dislocations (main work in this area by Nix and Gao [ii]) The assumption of an infinitely rigid indenter does not hold true in case of very hard and stiff samples (hardness > 60 GPa) here the indenter itself will elastically/plastically deform [iii] Effect of surface roughness or any deviation from the ideal half-space geometry there are only very few papers out there dealing with this topic / no model is established Time depended effects (visco-elastic behavior, creep) this is studied by dynamic indentation (not discussed here) some basic models exist Adhesion: Hardly ever discussed in the context of nanoindentation as its usually more relevant in the load regime of an AFM. Existing models include DMT [iv], JKR [v], and Maugis [vi] model [i]. K. W. McElhaney, J. J. Vlassak, and W. D. Nix, J. Mater. Res., Vol. 13, No. 5, 1998, 1300-1306 [ii]. W.D. Nix and H. Gao, Journal of the Mechanics and Physics of Solids, Vol. 46, No. 3, p. 411, 1998 [iii]. J. C. Hay, A. Bolshakov und G. M. Pharr, J. Mater. Res., Vol. 14, No. 6, 1999, 2296-2305 [iv]. B.V. Derjaguin, V.M. Muller, and Yu.P.Toropov, J. Colloid. Interface Sci. 53, 314 (1975). [v]. K. L. Johnson, K. Kendall, and A. D. Roberts, Proc. R. Soc. London 1971, A324, 301-313. [vi]. D.J. Maugis, J. Colloid. Interface Sci. 150, 243 (1992).

The indentation size effect (ISE) It is often observed that hardness increases as indentation size decreases, even for tests of homogeneous materials. This is known as the indentation size effect (ISE). The ISE has been known for a long time, but the length scale at which it is reported to appear has been decreasing. Many reports of ISE are actually due to artifacts: surface layers that where not accounted for, poor tip shape calibration, etc. Recent explanations invoke the need for geometrical necessary dislocations to explain hardness effects: Although nanoindentation with a pyramid or cone indenter may be self similar, dislocations have a length scale fixed by the Burgers vector. Nix and Gao show that such considerations lead to In this equation H is the hardness, H 0 the hardness at infinite depth, h the indentation depth and h* a characteristic length that depends on the shape of the indenter

Surface roughness As sample roughness does have a significant effect on the measured mechanical properties, one could either try to incorporate a model to account for the roughness or try to use large indentation depths at which the influence of the surface roughness is neglectable. A model to account for roughness effects on the measured hardness is proposed by Bobji and Biswas. Nevertheless it should be noticed that any model will only be able to account for surface roughnesses which are on lateral dimensions significantly smaller compared to the geometry of the indent good bad

Creep measurements by nanoindentation Although most mechanical property measurements, including elastic and elastic/plastic nanoindentation analyses, assume a single monotonic relationship between stress and strain, in reality plastic deformation in all materials is time and temperature dependent on some scale. In practical terms, time-dependent deformation is usually thought to be important when the temperature is greater than 0.4-0.5Tm, where Tm is the absolute melting temperature. If one plots log(σ) vs. log(dε/dt), the data fall on a straight line with slope n = 1/m, where n is the strain rate sensitivity In an indentation experiment, there is a distribution of stress and strain. Nonetheless, Mayo and Nix presented a method whereby the strain rate sensitivity can be obtained from nanoindentation measurements. The stress is simply obtained as the average pressure under the indenter, which is just the hardness. At each point under the indenter, the strain rate must scale with the indenter descent rate divided by the current contact depth. They thus consider the average strain rate to be

Thin film models (Young s Modulus) The majority of models proposed in the context of determination of mechanical properties of thin films are based on a phenomenological or semi-phenomenological approaches. A "rule of thumb" for hardness measurements which is still popular and well-known is the 1/10- rule of Bückle Models that describe the behavior of the measured Young's Modulus of a film substrate system: Doerner und Nix propose a fundamental approach. They treat Indenter, film and substrate as a series of springs. Bhattacharya and Nix pick up the model of Doerner and Nix and introduce some smaller changes. They obtain a good agreement between their model and some FEM calculations. Gao, Chiu and Lee draft one of the few analytical models utilizing an approach similar to the image of Loubet, Georges and Meille of the possibility to ascribe the elastic part of the unloading to an unloading of a flat punsh indenter with identical contact area. Swain and Weppelmann are able to show that it is possible to apply the approach of Gao et al. to spherical indenters too. A complete analytical solution of the stress- and strain-field for a spherical indenter in case of a Hertzian contact is given by Schwarzer, Richter and Hecht.

Thin film models (Hardness) Models that describe the behavior of the measured hardness of a film substrate system: Bhattacharya and Nix propose a simple model based on FEM observations. Fabes et al. present a volume fraction model - an advancement of the area-law-ofmixture approach from Joensson and Hogmark - and compare this model with the one from Bhattacharya and Nix. In this case their model showed a better performance in reproducing their experimental data; in generell the model from Fabes et al. is yet a bit more demanding with respect to knowledge of parameters that should be (or have to be) known prior to modelling. Korsunsky et al., whose work bases on the work of McGurk et al. and McGurk and Page, propose a good model for the case of hard films on soft substrates which was originally derived in order to understand cracking in those systems.

Example Modeling of film hardness: a-c:h / Si(100) Film/substrate hardness models Korsunsky et al.: Hf H H s m = Hs + 2 δ 1+ k df Bhattacharya und Nix: Hm = Hs + δ α d ( H H ) e f f s measured hardness [GPa] 30 25 20 15 10 I II Measured values Korsunsky et al. Bhattacharya und Nix III 0,1 1 10 δ / d f

Dynamic indentation (nanodma) Models used by (a) Pethica and Oliver, and (b) Syed Asif and Pethica. M is the mass of the indenter, C and K are the damping and spring constants. The indices i, m and s represent the indentation transducer, machine load frame, and sample, respectively. With Km = Asif and Pethica conclude: with

How to generate a force pro long displacement range (on the order of mm) approximately linear I-P behavior over the entire displacement range wide load range (up to several N) pick and push possible con large (10 cm) and heavy (on the order of kg) current in the load coil generates heat that leads to thermal drift pro small size of the system good temperature stability con limited load range limited displacement range (in the order of tenths of microns) usually only one possible direction of tip movement - its possible to push but not to pull.

Our experimental setup Indentation and scratch testing: 500 Load-Displacement Data (fused silica) Surface imaging and tip-positioning Apply a load while measuring displacement of the tip Analyze the force vs. Displacement data Load [µn] 400 300 200 100 holding: quasi-static dynamic x movement 0 0 10 20 30 40 50 60 Displacement [nm] springs Load unloading z movement loading Time Center plate Additional options: Driving plates sample b Indenter Dynamic testing Tribological testing Scanning probe microscope

What is possible and what s not: cube-corner / fused quartz (depth controlled indentation) 500 500 nm 2 (5 nm z-scale)

Available tip shapes Berkovich Indenter (standard tip) Three sided pyramid (same depth to area relation as a Vickers indenter but easier to fabricate with small tip radii) Typical tip radius of 100-150 nm Cube-corner indenter Usually used in the context of ultra thin films (< 10 nm) Tip radius about 50 nm Conical indenters Usually used in tribological applications as well as to study any crystallographic effects on mechanical testing Tip radii vary from 0.5 to 100 µm (commercially available) Flat-punch indenters Custom made tips from CAS (Prof. Gu)

Tribological properties of a surface on the nanoscale some considerations DLC on Si x = 2 µm y = 12 µm z = 6 nm 226nm 14nm Si

Some historical notes on tribology 1452-1519 Leonardo da Vinci Basic documentation of frictional forces 1663-1705 Guillaume Amontons Frictional forces depend on normal force present Roughness is used to explain friction 1707-1783 Leonhard Euler Detailed studies of friction phenomena Introduction of the friction coefficient µ 1736-1806 Charles Augustin Coulomb Fundamental description based on the work of Amontons

Motivation Macroscopic approaches Application based test methods Pin-on-disk tester Carlo-Tester Scratch tester Micro/nanoscopic Micro-Scratch-Tester AFM-based scratch- and area wear tests Combination of AFM and Nanoindenter Scanning Nanoindenter

Tribology Parameters of interest Test environment Test method Humidity, ambient, temperature Contact geometry Macroscopic (sphere-plane, plane-plane, others) Microscopic (roughness, asperity radii, etc.) Materials properties Mechanical properties Chemical properties Egyptians using lubricant to aid movement of Colossus, El-Bersheh, about 1800 BC

If all environmental conditions are kept constant Focus on geometry and materials effects on friction and wear Geometry Macroscopic friction is often described by Amontons law and is independent of real contact area Microscopic contacts: real and apparent contact area are similar Possible to observe contact area dependence of friction (in microscopic case) Materials parameters These are the ones one is usually interested in Increasing contact pressure Goal is to eliminate geometry effects in microscopic & mesoscopic tribological testing

Different phases of a nanoscratch (film/substrate) elastic non-elastic processes (b) phase I 1 phase IIb non-elastic processes (a) phase IIa friction coefficient phase I phase III 0,1 phase II 0,01 10 100 1000 10000 Load [µn] Example: 20 nm DLC on Si (100) substrate contact phase III

True contact area Hertz contact mechanics: Greenwood & Williams (mod. by Johnson): A Hertz 3 RTip = π Load 4 * E 2/3 A π p = A0 σ s κ s E * Iteration process as one has to plug an effective reduced modulus into the Hertz equation (this one assumes an ideal sphere half-space contact) Once the nominal contact area is identified it is possible to calculate the true area of contact for any surface that shows a Gaussian height distribution by using the model of Johnson (based on Greenwood and Williams)

Example: Influence of environment on friction a-c:h-films on a Si(100) substrate / conical diamond indenter (6.5 µm tipradius) Coefficient of friction µ Reibungskoeffizient µ Air, 50% RH 0,13 0,13 Nitrogen, Luft, 50% RH 0,12 Nitrogen, Stickstoff, 30% 0% RH RH 0,12 Nitrogen, Stickstoff, 70 % 30% RH RH 0,11 Stickstoff, 70 % RH 0,11 0,10 0,10 0,09 0,09 0,08 0,08 0,07 0,07 0 1000 2000 3000 4000 5000 0 1000 2000 3000 4000 5000 Load [µn] Last [µn] Coefficient of friction µ (2250 µn) 0,090 0,085 0,080 0,075 0,070 0 10 20 30 40 50 60 70 80 Relative humidity [%]