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” 2003 R.W. Carpick and D.S. Stone Tuesday 1
A Short Course on NanoscaleMechanical Characterization:
The Theory and Practice of ContactProbe Techniques
Tuesday, Part 1: Contact Mechanics
-Hertz & Boussinesq Problem-Adhesive elastic contact mechanics-Fracture approaches to the adhesive problem-Elastic-plastic problem-Viscoelastic problem-Rough surfaces
” 2003 R.W. Carpick and D.S. Stone Tuesday 2
Hertz & Boussinesq Problem
” 2003 R.W. Carpick and D.S. Stone Tuesday 3
• Assumptions:– contacting bodies are HILE
– contact radius is small compared to the size & curvature radius of the bodies
– frictionless surfaces
• Contact pressure, contact size, surface compression all depend on the load in anon-linear manner
• Consider two spheres in contact, radii d1/2=R1, d2/2=R2; moduli E1, E2; Poisson’sratios n1, n2. Applied load = P
Hertz Theory of Contact
question: what’swrong with thispicture? P P
” 2003 R.W. Carpick and D.S. Stone Tuesday 4
Hertz Equations
†
surface compression d = d1 +d 2 =a2
R =9P2
16RE *Ê
Ë Á
ˆ
¯ ˜
1/3
†
contact radius: a =3PR4E *
Ê Ë
ˆ ¯
1/3
†
Define: Reff = 1 R1 +1 R2( )-1, E * = 1- n12( ) E1 + 1-n1
2( ) E2( )-1
†
max. normal pressure at interface: p0 = 32 pave =
6PE*2
p 3R2
Ê
Ë Á
ˆ
¯ ˜
1/ 3
where pave = P pa2
d1
sphere 1 shown only
†
normal stress: s z(r) = -p0a2 - r2
a
†
at the contact zone edge:s r = 0.13p0, sq = -0.13p0, s z = 0
which is a state of pure shear!
” 2003 R.W. Carpick and D.S. Stone Tuesday 5
Hertz Theory of Contact• An analytic solution for all stress components at all points in space exists• Note: the largest shear stress tmax=0.31p0 occurs at z=0.48a below the
surface• The theory can also be applied to a sphere contacting a flat plane by letting
R1=Rsphere, R2=•
• Exercise: simplify the Hertz equations for this case, assuming the materialsare the same.
tmax=0.31p0
” 2003 R.W. Carpick and D.S. Stone Tuesday 6
Boussinesq Problem: Rigid flat punch• The stress distribution is very different for a rigid circular punch:
†
surface compression: d =P
2aE*
†
a = constant
†
max. normal pressure: p0 = 12 pave =
P2pa2
†
normal stress: s z(r) = -p0a
a2 - r2
” 2003 R.W. Carpick and D.S. Stone Tuesday 7
• Uniform pressure: stress is directly proportional to load, contact area is constant• Hertz pressure: stress depends nonlinearly on load, contact area is changing with
load
• Design considerations: bearing life is strongly influenced by contact stresses. Forexample, for ball bearings, fatigue life is related to p0
-9!! Thus, small reductions instress can substantially affect bearing performance.
Uniform pressure vs. Hertz pressure
r
stress
aa
sz
sr
sq
sz
sr
sq
sr
sq
sr
sqUniform Hertz
top view
sr
sq
” 2003 R.W. Carpick and D.S. Stone Tuesday 8
Adhesive elastic problem
” 2003 R.W. Carpick and D.S. Stone Tuesday 9
Rrecall: The JKR theory predicts the effect of adhesion on thecontact area
JKRHertz
R
L
aa A=π·a2
Hertz:
JKR:
A = p3R
4E* LÈ
Î Í
˘
˚ ˙
2 3
A = p3R
4E* L + 3pRg + 6pRgL + 3pRg( )2Ê Ë Á
ˆ ¯ ˜
È
Î Í
˘
˚ ˙ 2 3
A = contact area L = load
R = tip radius
E* = = reduced modulus
g = interfacial energy
1- n12
E1+
1 -n22
E2
Ê
Ë Á Á
ˆ
¯ ˜ ˜
-1
” 2003 R.W. Carpick and D.S. Stone Tuesday 10
The area of contact depends upon the range of theattractive forces
l>5 (JKR) l<0.1 (DMT)
ac ac
Johnson-Kendall-Roberts (JKR): short-range forces, high adhesion,compliant materials
Derjaguin-Müller-Toporov (DMT): long-range forces, low adhesion, stiffmaterials
Intermediate cases: Maugis, Colloid Interface Sci. 1992, 150, 243.
” 2003 R.W. Carpick and D.S. Stone Tuesday 11
Recall: Ac depends on the range of the attractive forces
ForceArea
distance
Actual
z0
ForceArea
distance
Hertz
ForceArea
distanceg
ForceArea
distance
DMT
g
JKR..
DMT=Derjaguin-Müller-Toporov modelDerjaguin, B. V.; Muller, V. M.; Toporov, Y. P. J. Colloid Interface Sci. 1975, 53, 314.
” 2003 R.W. Carpick and D.S. Stone Tuesday 12
Recall: Ac depends on the range of the attractive forces
JKR.
-2 -1 0 1 2 3 4 50
1
2
3
4
5
6
Load
(Co
nta
ct A
rea
) /
π
Hertz
JKR
DMT
intermediate
DMT
” 2003 R.W. Carpick and D.S. Stone Tuesday 13
Recall: How to tell if you are in the JKR or DMTregime (or in between)?
• compare the elastic deformation that adhesionforces cause with the range of the adhesion forcesthemselves
l =1.16 ⋅g 2 ⋅ R
E*2⋅ z0
3
Ê
Ë Á Á
ˆ
¯ ˜ ˜
1 3
g=adhesion energyR=tip radius
E*=elastic modulusz0=equilibrium spacing
” 2003 R.W. Carpick and D.S. Stone Tuesday 14
Fracture mechanics approaches toadhesive contact and friction
” 2003 R.W. Carpick and D.S. Stone Tuesday 15
Contact -- Crack• Tip/sample contact is analogous to an
external circular crack• System is otherwise elastic
crack tip
” 2003 R.W. Carpick and D.S. Stone Tuesday 16
JKR Theory of ContactsJohnson, Kendall & Roberts,Proc. Roy. Soc. A (1971) 324, 301.
• Griffith’s concept of brittle fracture
• Total load:
• Mode I stress intensity factor KI, Irwin’s relationship:
• Final result:
†
P = PHertz + -Pa( ), PHertz =43 E * ⋅ a3
R
†
KI =12
⋅s a ⋅ 2pa =Pa
2a paG =
KI2
2E* = g
†
a3 =R
43 E * P + 3pgR + 6pgRP + 3pgR( )2Ê
Ë ˆ ¯
a = contact radiusP = load R = tip radiusg = adhesion energy E* = reduced Young’s modulus
” 2003 R.W. Carpick and D.S. Stone Tuesday 17
Maugis’ Dugdale Model
Dugdale
D. Maugis, J. Coll. Interf. Sci., 1992, 150, 243.
” 2003 R.W. Carpick and D.S. Stone Tuesday 18
Maugis’ Dugdale ModelD. Maugis, J. Coll. Interf. Sci., 1992, 150, 243.
” 2003 R.W. Carpick and D.S. Stone Tuesday 19
Assumptions of Maugis’ model1. Linear elastic fracture mechanics applies2. Link between equilibrium spacing and cohesive
zone distance:
†
s0 = s theoretical ,LJ
s theoretical ,LJ =1.03 gz0
G = s0 ⋅d t , G = g
\ z0 = 1.03 ⋅d t
” 2003 R.W. Carpick and D.S. Stone Tuesday 20
Shear strength measurements using the MD model are nowbeing made by several groups
• Lantz et al. (Phys. Rev. B 1997, 55, 10776): Si/NbSe2• Pietrement & Troyon (Langmuir 2001, 17, 6540): SiNx/carbon fiber, SiNx/mica,
SiNx/SiO, SiNx/epoxy• observations include: load-dependent shear strengths in some cases, more near-
ideal shear strengths, correlation between friction and adhesion
• E. Barthel (J. Colloid Interface Sci. 1998, 200, 7), has shown that the details of theinterfacial potential do not strongly affect the contact behavior
• K.L. Johnson (Proc.Roy.Soc.A, 1997, 453, 163) has considered mixed-modefracture approaches in frictional contacts.
” 2003 R.W. Carpick and D.S. Stone Tuesday 21
Kim & Hurtado’s modelJ.A. Hurtado, K.-S. Kim, Proc. Roy. Soc. London A 1999, 455, 3363 and 3385
• why is the shear strength so large for AFM results,but typically much smaller for larger?
• use a dislocation mechanics approach• predicts a scale dependence of shear strength
” 2003 R.W. Carpick and D.S. Stone Tuesday 22
Kim & Hurtado’s model
” 2003 R.W. Carpick and D.S. Stone Tuesday 23
Kim & Hurtado’s model
” 2003 R.W. Carpick and D.S. Stone Tuesday 24
Elastic-plastic problem
” 2003 R.W. Carpick and D.S. Stone Tuesday 25
Elastic-plastic indentation
from Johnson, Contact Mechanics
” 2003 R.W. Carpick and D.S. Stone Tuesday 26
Contact of Rough Surfaces
” 2003 R.W. Carpick and D.S. Stone Tuesday 27
Mechanics Issues in MEMS: Friction and wear
• Currently, there are no commercial MEMS devicesthat involve contacting surfaces with extendedsliding
10 mm
” 2003 R.W. Carpick and D.S. Stone Tuesday 28
10 x 10 µm scan, 100 nm scale
RMS roughness~11.5 nm 500 x 500 nm scan, 30 nm height
scale
AFM images of polycrystalline MEMS surface(E. Flater & R. Carpick)
” 2003 R.W. Carpick and D.S. Stone Tuesday 29
A multi-scale understanding of friction could bedeveloped through experiment and modeling
•
micromachine interfaceprediction,
design,understanding
topographicstudies;
constitutive lawsfor friction & wear
mechanics ofmulti-asperity
contacts
Single asperity:Ff=tAc (?)
Multiple asperities:Ff=µL (?)
” 2003 R.W. Carpick and D.S. Stone Tuesday 30
00.20.40.60.811.2-60
-40
-20
0
20
40
60
00.20.40.60.811.2
-60
-40
-20
0
20
40
60 d
Contact mechanics analysis:the Greenwood-Williamson Theory
rigid plane, lowered
rough elastic surfacedistribution ofsurface heights
distribution ofsummit heights
Greenwood & Williamson, Proc. Roy. Soc. A 295 300 (1966)
” 2003 R.W. Carpick and D.S. Stone Tuesday 31
GW Theory: Assumptions• linear, elastic, homogeneous, isotropic• distribution of peak heights is Gaussian• radius of peaks is constant (or narrowly distributed
about a mean)• Hertzian contact (no adhesion, no tangential
stresses)• key parameters:
– number of peaks/area– peak curvature– std. dev. of peak height distribution– elastic modulus
” 2003 R.W. Carpick and D.S. Stone Tuesday 32
The GW Theory: Derivation
• Greenwood & Williamson, Proc. Roy. Soc. A 295 300 (1966)
†
Li = KR1/2d i3/ 2
†
Ai = pRd i
z
d
zi
di
rigid surface
ith summit out of N total summits
†
di = zi - d
Hertz:
†
Li = KR1/2(zi - d)3/ 2
†
Ai = pR(zi - d)
†
di = zi - d
” 2003 R.W. Carpick and D.S. Stone Tuesday 33
The GW Theory
• Greenwood & Williamson, Proc. Roy. Soc. A 295 300 (1966)
†
Li = KR1/2(zi - d)3/ 2
†
Ai = pR(zi - d)
†
Let f(z) = asperity height distributioni.e. there are Nf(z)dz asperities between z and z + dz
†
toatal contact area A = NpR (z - d) ⋅f(z)dzd
•
Ú
†
total load L = NKR1/2 (z - d)3/ 2 ⋅f(z)dzd
•
Ú
” 2003 R.W. Carpick and D.S. Stone Tuesday 34
The GW Theory
• Greenwood & Williamson, Proc. Roy. Soc. A 295 300 (1966)
†
toatal contact area A = NpR (z - d) ⋅f(z)dzd
•
Ú
†
total load L = NKR1/2 (z - d)3/ 2 ⋅f(z)dzd
•
Ú
†
Assume Gaussian distr. of summit heights: f(z) = 1s 2p e-z 2 2s 2
†
then AL =
pR1/2
Ks1/2 ⋅(x - a)e- x 2 dx
a
•
Ú(x - a)3/ 2e-x 2 dx
a
•
Ú=
pR1/2
Ks 1/ 2 ⋅ I
” 2003 R.W. Carpick and D.S. Stone Tuesday 35
Area is directly proportional to load!!!•
10-5 10-4 10-3 0.01 0.1 10
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
I, in
tegr
al ra
tio
LNKR 1/2 s3/2
The effect of increasing the load is not just to increase thearea of contact between asperities already in contact, butalso to bring new asperities into contactNote: real contact area may only be a fraction of a percent of theapparent contact area!
” 2003 R.W. Carpick and D.S. Stone Tuesday 36
An explanation for macroscopic friction?
•
†
then AL =
pR1/2
Ks1/2 ⋅(x - a)e- x 2 dx
a
•
Ú(x - a)3/ 2e-x 2 dx
a
•
Ú=
pR1/2
Ks 1/ 2 ⋅ I
†
Ff = tA = tpR1/ 2
Ks1/ 2 L = mL
m = tpR1/ 2
Ks1/ 2
Assumes only elastic deformation!
” 2003 R.W. Carpick and D.S. Stone Tuesday 37
Mistakes in the analysis1. “Peaks” found on a line are not true“summits” within a given area
• How to correct for this?• The answer comes from oceanography!
– developed by Longuet-Higgins, Phil. Trans. R. Soc. Lond. (1957)– adapted to contact problems by Nayak, J. Lubr. Technol. (1971)– see work by J. I. McCool
2. Roughness can’t be representedby a single length scale
summit
line peak
” 2003 R.W. Carpick and D.S. Stone Tuesday 38
Rough surfaces with adhesion• Fuller & Tabor: JKR case• The effect of surface roughness on the adhesion of elastic solids
• Proceedings of the Royal Society of London A. 1975; 345(1642): 327-42
• Maugis: DMT case• On the contact and adhesion of rough surfaces
• Journal of Adhesion Science and Technology. 1996; 10(2): 161-75
1© 2003 R.W. Carpick and D.S. Stone
Scale Effects in Plasticity
39
2© 2003 R.W. Carpick and D.S. Stone
Hall-Petch Effect
2/10
−+= kdσσd = grain size
Yield stress increases with decreasing grain size Hardness of molybdenum thin films
Yoder et al, 2003N.J. Petch, Fracture, Proceedings of Swampscott Conference 1959, p. 54.
40
3© 2003 R.W. Carpick and D.S. Stone
Indentation Size Effect & Reverse Indentation Size Effect
41
4© 2003 R.W. Carpick and D.S. Stone
Indentation Size Effect & Reverse Indentation Size Effect
Ma and Clark J Mater. Res. 10(4)
41a
5© 2003 R.W. Carpick and D.S. Stone
Indentation Size Effect & Reverse Indentation Size Effect
Tambwe et al, J Mat. Res. 1999
41b
42” 2003 R.W. Carpick and D.S. Stone Tuesday
Surfaces
• Surface crystallography
• Relaxation and reconstruction
• Surface states
• Surface energy
• Characterization methods
43” 2003 R.W. Carpick and D.S. Stone Tuesday
Surface crystallography
• A 2-D lattice is referred to as a net• The area unit is referred to as a mesh• 5 Bravais nets in 2-D
– Square– Hexagonal– Rectangular– Centered rectangular– Oblique
• Mesh defined by mesh basis vectors c1, c2
• For a reconstructed surface, (c1, c2)≠(a1,a2)
44” 2003 R.W. Carpick and D.S. Stone Tuesday
Wood’s notation:-used for reconstructed surfaces or overlayer (adsorbate) structures
• General relation between c’s and a’s:
• Wood’s notation:
– a omitted if it is 0°.– p or c is included at the beginning to indicate primitive or
centered, e.g. c(4x2), p(2x2)• The p is optional
– The surface is listed first, e.g. Si(111) (7x7)– If an adsorbate structure is being described, it is included
afterward, e.g. Si(111) (1x1)H
†
c1
c2
Ê
Ë Á
ˆ
¯ ˜ = P
a1
a2
Ê
Ë Á
ˆ
¯ ˜ =
P11 P12
P21 P22
Ê
Ë Á
ˆ
¯ ˜
a1
a2
Ê
Ë Á
ˆ
¯ ˜
†
c1
a1
¥c2
a2
Ê
Ë Á
ˆ
¯ ˜ Ra, e.g. 3 ¥ 3( )R30°, (2 ¥1)
45” 2003 R.W. Carpick and D.S. Stone Tuesday
46” 2003 R.W. Carpick and D.S. Stone Tuesday
Examples:• See: http://www.chem.qmw.ac.uk/surfaces/scc/ (Prof. R. Nix, U. London)
Substrate : fcc(100)c( 2 x 2 )(√2 x √2)R45
Substrate : fcc(110)c( 2 x 2 )
Substrate : fcc(111)(√3 x √3)R30
47” 2003 R.W. Carpick and D.S. Stone Tuesday
Surface Relaxation and Reconstruction
• Surfaces are defects: bonds are broken (all of them!),creating dangling bonds
• Atoms will re-arrange to make up for this lack ofcoordination
• Surface relaxations: motion of atoms inward oroutward from surface plane– More frequently observed with metallic and ionic materials
(much weaker in ionic materials)– See table– Compensation for “electron overspill” (sketch)
• Surface reconstruction: rebonding of surface and sub-surface atoms– Surface lattice is not in registry with the bulk– More frequently observed with covalent materials– Can be removed by terminating the surface with other atoms– e.g. Diamond(111)
48” 2003 R.W. Carpick and D.S. Stone Tuesday
Surface relaxations
back
49” 2003 R.W. Carpick and D.S. Stone Tuesday
Relaxation
• Relaxation: motion perpendicular to surface– change in the interlayer spacing.
– Usually affects the first 1-3 layers, with the strongest effectat the surface
50” 2003 R.W. Carpick and D.S. Stone Tuesday
Example: Bulatov et al, JOM-e, 49 (4) (1997)
Figure 1. The ions in a 552 ion cluster of LaF3 areallowed to move, subject to the forces from all other ionsin the nanocluster, until the target temperature of 10 K isreached.
Figure 2. Cross-section view of the ionmovement depicted in Figure 1.
http://www.tms.org/pubs/journals/JOM/9704/Bulatov/Bulatov-9704.html
51” 2003 R.W. Carpick and D.S. Stone Tuesday
Relaxation: NaCl(001)
A sodium chloride (NaCl) surface. There is a slight rearrangement of the surface: sodium ions moveinto the surface and chloride ions relax away from the surface.
http://www.molecularuniverse.com/Bound/bound1.htm
52” 2003 R.W. Carpick and D.S. Stone Tuesday
Surface Reconstruction
• Re-bonding by re-arrangement ofsurface and sub-surface atoms(parallel and perpendicular to thesurface plane)
• Si(100)
• Si(111)
53” 2003 R.W. Carpick and D.S. Stone Tuesday
Diamond (111), with andwithout H
diamond(111)-(1x1)H
Bulk termination
diamond(111)-(2x1)
Reconstructed
54” 2003 R.W. Carpick and D.S. Stone Tuesday
Surface energy
• All surfaces are energetically unfavorable becausethey have a positive free energy of formation withrespect to the bulk crystal– Bonds must be broken to form the surface
• Surface energy is minimized by:– Reducing the amount of exposed surface area– Exposing surface planes with low surface tension– Altering local atomic geometry (relaxation and
reconstruction)
• Low surface energy=high surface stability
55” 2003 R.W. Carpick and D.S. Stone Tuesday
Rules of thumb for lowsurface energies (in vaccum)• High surface atom density
• High coordination number
fcc(100) fcc(110) fcc(111)
Arrange these surfaces from lowest to highest stability.
56” 2003 R.W. Carpick and D.S. Stone Tuesday
Surface tension
• Total energy: U=TS-PV+µN+gA• g = surface tension
= the reversible work done in forming a unit area of surface
≠ surface energy, but lots of people call it that!
units: energy/area or force/distance (e.g. soap film)
†
g =∂U∂A T ,V ,N
†
Ss = -A∂g∂T e
†
s ij = gdij +∂g∂eij T
†
Ss = surface entropyT = temperatures ij = surface stress (force/distance)eij = surface strain
We can derive the following relations (Zangwill):
57” 2003 R.W. Carpick and D.S. Stone Tuesday
Surface tension is anisotropic in a crystal• To minimize the excess surface free energy,
†
gÚ dA = minimum. But g = g(hkl)
\ gÚ (hkl)dA(hkl) = minimum.
http://www.fysik.dtu.dk/CAMP/hot0009_img3.html
PALLADIUM NANOCRYSTALS ON AL2O3
Wulff construction: easy method forfinding the low-temperatureequilibrium shape
Pb
58” 2003 R.W. Carpick and D.S. Stone Tuesday
Surface tensions of variousmaterials
Material Surface energy (mJ/m2)
W 4400Cu 2000Ag 1500Al 1100
Water 73Ethanol 22.8Teflon 18.3
from Israelachvili, “Intermolecular and Surface Forces”
59” 2003 R.W. Carpick and D.S. Stone Tuesday
Summary