Tip-sample interaction, nanoindentation, nanolithography

Prof. E. Rabkin ,Department of Materials Engineering, TECHNION-Israel

Institute of Technology, Haifa, ISRAEL

Haifa, Bahai shrine and gardensMD simulations of nanoindentationof gold nanoparticles (D. Mordehai)

Moscow, MISiS, April 2009

Syllabus:

-Nanoindentation: main idea and instrumentation;

-Introducing load-displacement curve;

-What is measured? Olivier-Pharr method;

-Calibrating tip radius;

-Pile-ups and Joslin-Olivier method;

-Indentation size effect: theory of Nix and Gao;

-The role of grain boundaries and interfaces;

-Nanoindentation with ultralow forces; nanoindenting with AFM;

-Nanotribology and nanolithography;

-Atomistic theories of tip-sample interaction;

-Nanoindenting in the transmission electron microscope;

-Applications in biology and medicine .

Vickers microhardness testing

Hardness is the property of a material that enables it to resist plastic deformation, usually by penetration. The Vickers microhardness testing method consists of indenting the test material with a diamond indenter, in the form of a right pyramid with a square base and an angle of 136 degrees between opposite faces subjected to a load that does not exceed 1 kgf.

Contact mechanics

A contact of a hard sphere with elasticflat specimen:

*3

4

3

E

PRa

where E* is reduced modulus, orcombined modulus of a system

'

'111 22

* EEE

The deflection of original surface:

2

2

*2

42

31

a

r

a

P

Eh

The distance of mutual approach:

R

P

Eht

22

*3

4

3

Average pressure:

R

aEpm

34 *

Elastic-plastic contact

For hardness methods that employthe projected contact area, thehardness value,H, is given directly

by the mean pressure pm .Experiments show that

yCH where y is the yield stress of the

material and C is the constraintfactor .

For large values of E/y (metals)C3;

For low values of E/y (glasses)C1.5.

Johnson’s “expanding cavity” model:

216

214tan/ln1

3

2

y

y

Ep

Indentation response of a material:1. pm<1.1y-fully elastic;2. 1.1y<pm<Cy-plastic deformationbelow indenter, but constrained bysurrounding elastic material;3. pm>Cy-plastic region extends tothe surface.

Hardness test on the nanoscale

The idea of nanoindentation arose from the realization that an indentation test is an excellent way to measure very small volumes of materials. The only problem is determining the indentation area. For the indents below 1 m in size the resolution of optical microscope is insufficient.To solve this problem depth sensing indentation method was developed. In this method the load and displacement of the indenter are recorded during indentation. Mechanical properties of the indented volume can be determined without seeing the indent.

SEM micrograph of the indentmade in nc-Cu by Berkovichindenter under the load of 3mN.

Why nanoindentation?

Unique mechanics of small volumes(Krystin Van Vilet)

Are mechanical properties affected by

constraint?

Are E, y, n f(size)?

When structure dimensions approach

atomic dimensions, is

mechanical behavior quantized?

How the nanoindentation testing machine works.Force control

The simplest way to apply a controllablelow load on the sample is doing this withSFM. The elastic constant of the cantilever

beam should be properly characterized .Main disadvantages :

-The geometry is not strictly orthogonal;-maximal load is usually in sub-N range

Main advantage: the imaging of the indent iseasy.

Other ways of force control

Electromagnetic actuation (Micromaterials, NanoIndenter).Main advantages:

-Long displacement range (millimeters);-Good P(I) linearity;-High maximal load (few Newtons);-The indenter can be driven in and out.

Electrostatic actuation (Hysitron).Main advantages:

-Small size, high accuracy;-Good temperature stability;

Disadvantages: limited maximal load and displacement.

Extremely precise scheme (Burnham and Colton), butnot commercially realized .

Displacement control

Capacitance methodd

AC 0

C1-C2 is a linear function of small displacements

The primary coil is excited with an a.c. current (1 to 10kHz). The other two coils (the secondaries) are wound such that when a ferritic core (amature) is in the central linear position, an equal voltage is induced into each coil. Robust, cheap, no hysteresis, but lowersensitivity.

Linear Variable Differential Transducer (LVDT)

Displacement control (continued)

Interferometric. Fabry-Perot interferometer.The coherent laser light is channeled throughthe fiber 1 to fiber 2. It is partly reflected fromthe back end of fiber 2 and partly transmittedand reflected from the moving mirror. The

interference between the beams reflectedfrom the mirror and from the fiber 2 resultsin periodic variations of the intensity of light

at the detector .

Commercial instruments

NanoIndenter (MTS, US)

Large maximal load (500 mN);Positioning with the optical microscope .

http://www.mtsnano.com/f

Hysitron (US)

http://www.hysitron.com

The advantage of Hysitron is a combination of nanoindenting head with the SPM positioning stage.This allows a precise selection of theobject to be tested. The SPM is operating in the STM mode and the Control Unit is “cheated” byproviding the voltage from capacitancesensor instead of tunnel current.

SPM SPM + Nanoindenter

Micromaterials (UK)

http://www.micromaterials.co.uk

Original pendulum set-up. Makes high-temperature measurements easier .

Berkovich indenter

Berkovich Knoop Cube corner

Berkovich indenter was designed to givethe same projected area at a given depthas Vickers indenter. It is better suited for

small indents .

What is measured?

For calculation of nanohardness theknowledge of hc (contact depth at

maximal load) is necessary .

According to Oliver and Pharr (1992)

S

Phhc

maxmax

where =0.72 for conical indenter and =0.75 for paraboloid of revolution. S is a contact stiffness .

Loading curves for various materials

Oliver-Pharr method

cr AES2

1 .A fraction of unloading curve is fittedwith P=A)h-hf(m, where A and m are

fitting parameters .2 .The unloading curve fit is differentiated

analytically to find the stiffness at maximalload: S=)dP/dh(P=Pmax

Determining machinecompliance:

Canonical Sneddon stiffness equation:

cr

ftotAE

cc1

2

A series of LARGE indents is made in Al single crystal. The contact area forBerkovich indenter is determined according to Ac=24.5hc

2 .

machine compliance

Determining tip shape area function

After machine compliance was found, theempirical tip shape area function is determined

Using the set of indents in fused silica :Er=69.6 GPa for diamond indenters.

22

1

4 ftotr

cccE

A

The tip shape area function is determinedby fitting the obtained (Ac, hc) data to a poly-nomial of the form:

16/15

8/14

4/13

2/121

20 ccccccc hahahahahahaA

After tip shape area function is determined, the nanohardness is determined according to

cc hA

PH max where

S

Phhc

maxmax

The elastic modulus is determined by Sneddon stiffness equation.

Tip radius measurements

•Scanning of "calibration grid" (NT-MDT) with the Berkovich indenter: The

image of the indenter.•The tip radius: ~400nm.

0

500

1000

1500

2000

2500

3000

0 50 100 150 200

Depth [nm]

Load

[µN

]

Max shear stress estimation

• The loading response prior to the 1st pop-in can be described as purely elastic loading .

• Spherical approximation for the tip (shallow indents) - Hertz theory .

23

21

3

4tr hREP

GPaR

PEr 5.1926

31.031

23max

GPaa

Gbth 17

2

• Theoretical shear stress:

• homogeneous dislocation nucleation under the indenter.

Common sources of the artifacts

-Thermal drift (can be compensated);

-Surface preparation (defects and residual deformation);

-Tip-sample non-orthogonality and surface roughness;

- Incorrect extrapolation of tip area function;

-Pile-ups and sink-ins

Nanoindentation galleryMechanical properties of microstructure elements

T2=Mo5SiB2

Mo-22B-11Si

Nanoindentation of Au single crystals

Nanoindentation of thin Cu films

Pop-ins are especially pronouncedin the case of thin films. The stressof the first pop-in is close to the

theoretical shear strength of Cu .

Energetics of pop-ins in thin films

Elastic energy stored in the film:

3

3

0

2 b

h

e

ChdhChW

b

The interaction energy between twointeracting prismatic dislocation loopsof radius Rd>>:

1

8ln

1

2

d

di

RR

GbW

The summation over all loops in the film:

1

1

1

1

1

1

2

!ln8

ln1

N

j

N

j

N

j

dd

ti jj

RjR

GbW

For the first pop-in in Al film, Rd37 nm .

Bubble raft experiment

Measuring mechanical properties of thin films

0.1 0.2 0.3 0.4 0.5150

200

250

300

E

, G

Pa

Relative indentation depth

Sapphire substrate

SrTiO3 substrate

(a)

The indentation depth should be lessthan 1/3 of film thickness. Otherwise,

a combination of film and substratemechanical properties is measured.

For “soft” films on hard substrates theYoung’s modulus is more sensitive to

substrate than the hardness .

0.1 0.2 0.3 0.4 0.5

6

8

10

12

H, G

Pa

Relative indentation depth

Sapphire substrate

SrTiO3 substrate

(b)

Nanoindentation at elevated temperatures

C.A. Schuh et al., Nature mater. 4 (2005) 617

Cumulative distributions:

Nanoindentation at elevated temperature: data analysis

The model also predicts rate dependence of the pop-in load

Activation volume obtained:0.5b3

Not consistent with homogeneous nucleation

Role of vacancies?

Nanoindentation creep

Voigt (a), Maxwell (b) and combined (c)models of visco-elastic creep.

Voigt, spherical indenter:

/exp1

11

4

32

21

2/3 tEEER

Pth

Maxwell, spherical indenter:

t

ER

Pth

11

4

3

1

2/3

1 m thick Al film, spherical indenter

Voigt fit

Diffusion creep

Load-displacement curvefor In at room temperature

For loads lower than the firstpop-in stress exponent is 1.5at the end of holding period. Thisis close to the exponent 1 expectedfrom diffusion mechanism.

The pile-ups

Stress-strain relationship in theplastic region:

xK Ey

for

Piling-up is expected for non strain-hardening materials with large E/y

)heavily deformed metals.(Sinking-in is more likely to occur in

materials with small E/y (ceramics, glasses.(

In the case of extensive piling-up or sinking-in determing the correctvalue of nanohardness is difficult

Example of pile-ups (1)

)a)-(c :(Nanocrystalline Cuobtained by severe plastic deformation;

)d-(annealed Cu polycrystal.3 mN load .

0 1 2 3 4

100

200

300

400

500

z, n

m

x, m

as-indented

24 h

101 nm

74 nm

Joslin-Oliver method

Using Sneddon relationship allows eliminating projected area from theexpression for nanohardness:

22

4

S

P

E

H

r

The method is good when the elasticmodulus is known with high accuracy.Eliminates the effect of pile-ups ,

surface roughness and indenter shapearea function.

Indentation size effect

For low loads hardness is a strongfunction of the indentation depth .

Therefore, there is little sense indirect comparison of hardness values.

A good way to get the FULL H(d) dependence in one indentation run is the use of partial unloading technique.

0 50 100 150 200

0

1

2

3

For

ce, m

N

Time, s

loading schedule:

0 100 200 300

0

1

2

3

Forc

e, m

N

Displacement, nmPartial unloading scheme

Theory of Nix and Gao

Geometry :s

b

a

htan

Total length of dislocation loops in plastic hemisphere:

b

ha

Density of geometrically necessary dislocations:

22

tan2

3

2

3

bhba

hg

Taylor’ relation for shear strength:

sgGb where 0.5

With von Mises flow rule 3yand H=3y we get:

h

h

H

H *

0

1

sGbH 330 2

0

22* tan2

81

H

Gbh

What good theory can do

The effect of grain boundaries

Local hardening in thesub-um range

Grain boundary hardness in Ni3Al

Cube corner indenter, 300 mN max. load

Wo & Ngan, J. Mater. Res., 2003

It was concluded that no grain boundary hardening effect can be observed in nanoindentation experiment

Truncated hemisphere modelT. Eliash, E. Rabkin, Acta mater., 2008

Ultra-microhardness of Zn at different loadsMuktipavela et al, J. Mater. Sci., 2008

Grain boundaries stop the deformation twinning

Inverse size effectYang & Vehoff, Acta mater., 2007

AFM as a force measuring apparatus

Frictional (lateral) force microscopy

Twisting of cantilever during scanningcan be used to measure the frictionalforce Ff. Amontons’ law:

nf FF

is the friction coefficient .

Cantilever twist can be measured eitherusing capacitance method (a) or optically

with quadrant PSD (b) . FFM image of gold islands depositedon organic-coated mica. Au exhibits low friction.

Force spectroscopy

Experimentally measured:zt – cantilever deflection;zs – sample movement.

The force, F:

tczF

The relative displacement of tip andsample, s:

s ts z z

zt)zs( = F)s(

“jump to contact”

attr

actio

nre

puls

ion

1-maximal attractive force;3-maximal adhesive force

Humidity and contaminations in force spectroscopy

In vacuum

In humid air and withcontaminations

Example of adhesion curve

TaS2 vs. SiO2 cantilever (E. Meyer, 1990)

-Maps of attractive and adhesive forces;

-The slope of the curve after contact is established gives an information on local atomic forces;

-Hysteresis is a measure of inelasticity in the tip-sample system .

Nanoindentation with AFM

Non-Hertzian behavior because penetration depth is comparable with the tip radius:

221

s

s

d

dkFIndented with Si tip, R<10 nm

)J. Fraxedas et al., PNAS 2002(

Spatial maps of surface elasticityInput: periodic modulation of zs. Measured: caused by it periodic changes of zt

ts

t

tss zz

cz

zz

Fc

cs and c-effective spring constant of the surface and of cantilever, respectively.

st

s

c

c

z

z1

Cantilever should be selected properly!

32 m2 topography (a) and modulus map (b) images of epoxy-carbon fiber composite (Maivald et al., 1991)

Nanolithography with AFMSundararajan & Bhushan, J. Mater. Res., 2001

Most commercial AFMs are equipped with the nanolithography software which enables controlled “plowing” of the surface

Scratch test of Si (001) surface, with monotonously increasing normal force and friction force continuously monitored.

Rapid increase in friction force corresponds to the wear groove

formation .

Nanolithography with AFM (contd.)

The morphology of wear debries indicate that debries were removed by nano-cutting

Nanofabrication with AFMBhushan, Wear, 2005

Advantages by “scratching” nanofabrication:

-Good control of normal load, speed, etc;.-Can be performed on any surface;

-Dry process, no wet chemicals needed;

But:-Debris formed during process may be

a problem;

Atomistics of tip-sample contact

Interatomic potential allows calculating the energy of any atomic configuration. Embedded atom method (EAM) semi-empiric potentials provide a good description of the fcc metals.

i ij

ijii rFU 2

1 F Embedding energy function;

r Pairwise interaction energy

An example: Interatomic EAM potential for gold by Grochola et al. (2005):

Top

Bottom[001]z

xy

A

B

Example 1: asperity contact (MEMS switch)Jun & Srolovitz, Acta mater. 2007

Interface energy AB

Work of adhesion

AABBAB (measured at 0

Kelvin)

1366 erg/cm2 1157.94 erg/cm2 998.82 erg/cm2 881.81 erg/cm2

735 erg/cm2 694.23 erg/cm2 610.90 erg/cm2 532.64 erg/cm2

Morphologies of the contact after one approach-separation cycle

Force-displacement curves

Comparison with experimentRubio et al., PRL, 1996

Abrupt drops in the load-displacement curves correspond to nucleation of Shockley partials;

Between the nucleation events the area of the contact remains constant.

Transfer criterion

About stacking faults and Shockley partials…http://www.tf.uni-kiel.de/matwis/amat/def_en/kap_5/backbone/r5_4_1.html

1126

ab

fcc stacking: ABCABCAB…Intrinsic stacking fault:

ABCACABCA…

Nanoindenter as a microcompression tester

Uniaxial compression of FIB-machined pillars became popular in recent years.

Metallic pillars are found to exhibit strong size effect, known as “smaller is stronger”.

Greer et al., Acta Mat. 53, 1821 (2005) Dislocations are found to starve inside pillars.Pros: Avoids strain gradients plasticity.Cons: Contaminated by Ga ions.‘Soft’ substrate.

Difficult to produce pillars.Limited possible structures.

Chemically-etched Mo Pillars – produced by etching the matrix around Mo pillars in NiAl-Mo composite grown by directional solidification eutectic.

Bei et al., Scripta Mat. 57, 397 (2007)

Pros: Avoids strain gradients plasticity, Relatively dislocations-free.Cons: Pillars solid solution with Al (10%) and Ni(4%). Limited to certain shapes and materials. ‘soft’ substrate.

Pillars compression test showed no size effect.

“Natural” nano- and micropillars

Size effect in microcompression tests is still a subject of controversy!

Nanoindentation in TEM

Hysitron “Picoindenter”

In-situ nanoindentation in TEM

“Local” recrystallization

M. Jin et al., Acta mater. 2004

“Mechanical annealing” by nanoindentationShan et al., Nature mater. 2008

Contact mechanics and nanoindentation in biology and medicine

Head-neck contacts in bugs:

(a) and (b) – head part;(c) and (d) – neck partChitin fibers in the protein matrix.

Hard head joined with a softer neck.Good combination for fighting anddigging

Wing locking mechanism in beetle

)c-(wing part; (d)-body part.

Thicker and shorter hairs of the wingpart result in a higher hardness.

Cicada’s wings

The Young’ modulus of cicada wing membrane was very close to themodulus of some synthetic amorphous protein polymers (gosline).Materials selection for micro-drones .

Why it is dangerous to drink a lot of orange juice

SFM topography images of the treated tooth enamel after exposure towater (a), blackcurrant drink (b), and orange juice (c).

Exposure to water Exposure to orange juice