Scanning Nanoindentation-

One example of a quantitative SPM technique

• Mechanical characterization on the nanoscale– The basic idea– Some models– Some issues

• Instrumentation

• Tribological characteriazation– Historical background– The difference between macro- and nano-tribology

Topics

SPM:

• Excellent surface imaging• Well-defined tip-positioning

(via imaging)• Quantitative mechanical

properties are not accessible

Nanoindentation:

• 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

+

Scanning nanoindentation

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

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

loading

unloading

Displacement, δ Lo

ad, P

AreaForcehardness =

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]]:

• Assumptions: Small pressures, R large compared to a, contact is frictionless. E*, the so called reduced modulus, is given by:

• Word of warning: Often ξ is mistaken to be δe , 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

2/32*

9ER16P ξ=

3/1

*E4RP3a

=

2

22

1

21

* E1

E1

E1 ν−

+ν−

=

Basics

• Many materials show an elastic-plastic type of behavior, that can be expressed in terms of an elastic and a plastic displacement (δ=δe+δp) of a series of two spring elements (conical indenter):

• Many theories are based on an approach by Loubet et al. [i], that incorporates the work of Tabor [ii] and Sneddon [iii]

[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

(a) ansatz: Loubet et al. (b) Sneddon

δe

δp

indenter flat-punch

surface

22

pe C1

C1P δ

+=

−

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 rotation-paraboloid, and a cone, respectively):

• Reduced modulus and hardness are given by:

[i]. W. C. Oliver und G. M. Pharr, J. Mater. Res., Vol. 7, No. 6, June 1992, 1564-1583

SPmax

PePc maxmaxθ−δ=δ−δ=δ

( )( )2

c

P 1*EA2

ddPS

max ν−πδ

=δ

≡

( )c

maxAPHδ

≡

S

loading

unloading

displacement, δ

load

, P

δc (θ = 1) δc (θ = 0,72)

possible range of δc

(Pmax, maxPδ )

δf

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 hfinto 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/Er2 may be expressed in terms of P and S2,

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, H0 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-of-mixture 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.

Film/substrate hardness models

• Korsunsky et al.:

• Bhattacharya und Nix:

0,1 1 1010

15

20

25

30

III

III

Measured values Korsunsky et al. Bhattacharya und Nix

mea

sure

d ha

rdne

ss [G

Pa]

δ / df

2

f

sfsm

dk1

HHHH

δ+

−+=

( ) fdsfsm eHHHH

δα−−+=

Example Modeling of film hardness: a-C:H / Si(100)

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:

• Surface imaging and tip-positioning

• Apply a load while measuring displacement of the tip

• Analyze the force vs. Displacement data

Additional options:

• Dynamic testing

• Tribological testing

x movement

springs

Center plate

Driving plates

Scanning probe microscope

Indenter sampleb

z movement

Time

Load

loading

unloading

holding: quasi-static dynamic

0

100

200

300

400

500

0 10 20 30 40 50 60

Load-Displacement Data(fused silica)

Load

[µN

]

Displacement [nm]

What is possible and what’s not:cube-corner / fused quartz (depth controlled indentation)

500 × 500 nm2 (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

226nm 14nm Si

x = 2 µmy = 12 µmz = 6 nm

DLC on Si

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

Some historical notes on tribology

• 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

Motivation

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

Tribology

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

Goal is to eliminate geometry effects in microscopic & mesoscopic tribological testing

Increasing contact pressure

phase IIb

phase IIa

0,01

0,1

1

10 100 1000 10000

Load [µN]

fric

tion

coef

ficie

nt

phase I

phase II

phase III

Example: 20 nm DLC on Si (100)

non-elastic processes(a)

non-elastic processes (b)

phase III

substrate contact

elastic

phase I

Different phases of a nanoscratch (film/substrate)

• 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)

3/2

*43

= Load

ER

A TipHertz π *0 E

pAA

ss κσπ

=

Hertz contact mechanics: Greenwood & Williams (mod. by Johnson):

True contact area

a-C:H-films on a Si(100) substrate / conical diamond indenter (6.5 µm tipradius)

0 1000 2000 3000 4000 50000,07

0,08

0,09

0,10

0,11

0,12

0,13 Luft, 50% RH Stickstoff, 0% RH Stickstoff, 30% RH Stickstoff, 70 % RH

Reib

ungs

koef

fizien

t µ

Last [µN]0 1000 2000 3000 4000 5000

0,07

0,08

0,09

0,10

0,11

0,12

0,13 Luft, 50% RH Stickstoff, 0% RH Stickstoff, 30% RH Stickstoff, 70 % RH

Reib

ungs

koef

fizien

t µ

Last [µN]

0 10 20 30 40 50 60 70 800,070

0,075

0,080

0,085

0,090

Coef

ficie

nt o

f fric

tion

µ (2

250

µN)

Relative humidity [%]0 1000 2000 3000 4000 5000

0,07

0,08

0,09

0,10

0,11

0,12

0,13 Air, 50% RH Nitrogen, 0% RH Nitrogen, 30% RH Nitrogen, 70 % RH

Coef

ficie

nt o

f fric

tion

µ

Load [µN]

Example: Influence of environment on friction