nanohub.org resources 9820 download 2010.09.21-me597-l08-reifenberger
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7/29/2019 Nanohub.org Resources 9820 Download 2010.09.21-ME597-L08-Reifenberger
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Ron ReifenbergerBirck Nanotechnology Center
Purdue University
Lecture 8
Introduction to Contact Mechanics
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How to Model the Repulsive Interactionat Contact?
2
tip
apex
substrate
Source: Capella & Dietler
Maybe if the contact area involves tens or hundredsof atoms the description of net repulsive forceis best captured by continuum elasticity models
Atom-Atom? Sphere-Plane?
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What we want to know
Nature of the contact - reversible (elastic)?hysteretic?
Contact radius (contact area) as function ofapplied force
Any deformation?
Pull-off force (adhesion force)
What determines all these quantities?
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Continuum description of contact - history
Hertz (1881) takes into account neither surface forces nor adhesion,and assumes a linearly elastic sphere indenting an elastic surface
Sneddons analysis (1965) considers a rigid sphere (or other rigidshapes) on a linearly elastic half-space.
Neither Hertz or Sneddon consider surface forces discussed inlast lecture.
Bradleys analysis (1932) considers two rigid spheres interacting viatheLennard-Jones 6-12 potential
Derjaguin-Mller-Toporov (DMT, 1975) considers an elastic sphere withrigid surface but includes van der Waals forces outside the contactregion. Applicable to stiff samples with low adhesion.
Johnson-Kendall-Roberts (JKR, 1971) neglects long-range interactions outside contact area but includes short-range forces in the contact area.Applicable to soft samples with high adhesion.
Maugis (1992) theory is even more accurate shows that JKR and DMTare limits of same theory
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Tip-sample Interaction Models
From the Derjaguin approximation for rigid tip interacting with rigidsample we have
Real tips and samples are not rigid. Several theories are used to betteraccount for this fact (Hertz, DMT, JKR)
* These theories also apply to elastic samples, they are just shown onrigid sample to demonstrate key quantities clearly. For example D isthe combined tip-sample deformation in (b)
= = = + 13 23 2132 1( * ( *)) 2 2 2 ( )tip sample adhesion tip tip tipU r RWF r F R R
Rigid tip-rigid sample Deformable tip and rigid sample*
Rtip
sample
FaHertz
aJKR
F
as
Equilibrium Pull-off
1
2
3
D D
(a) (b) (c)
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Work of adhesion and cohesion: work done to separate unitareas of two media 1 and 2 from contact to infinity in
vacuum. If 1 and 2 are different then W12 is the work ofadhesion; if 1 and 2 are the same then W11 is the work ofcohesion. Think vdWs whenever you see work ofadhesion/cohesion.
Surface energy: This is the free energy change when thesurface area of a medium is increased by unit area. Thus
While separating dissimilar materials the free energy change
in producing an interfacial area by unit area is known astheir interfacial energy
Work of adhesion in a third medium
I. Surface energies - notation
11 12=
12 1 2 12W = +
12
132 13 23 12W = + 6
1 23
1 2
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10 MPa
five
ordersofmagnitude
1 Pa = 1 N/m2
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II. What is the Stiffness of the Tip/Substrate?
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FaHertz
aJKR
Equilibrium
D
(b)
=
13
tip
Hertz
tot
R Fa
E
+ =
13
132( 2 )
tip tip
DMT
tot
R F R Wa
E
+ + + =
132
132 132 132( 2 6 (3 ) )
tip tip tip tip
JKR
tot
R F R W R W F R Wa
E
Standard results
= =
2
1/32 2
Hertz
tip tip tot
a FD
R R E
( ) + = =
2
1/32
2132
2 tipDMT
tip tip tot
F R WaD
R R E
=
2
13262
3JKR
tip tot
W aaD
R E
= =
=
+
2 2
*
*
1 3 3 1
4 4
1
4
3
1s t
s ttot
tot
E
sometimes E
E E E
E
Contact radius a:
Deformation D:
Source: Butt, Cappella, Kappl 8
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F
as
Pull-off
D
(c)0HertzadhesionF =
tip 1322 R WDMT
adhesionF
=
tip 132
3R W
2
J KR
adhesionF
=
Standard results (cont.)
Pull-off Force F:
Source: Butt, Cappella, Kappl 9
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Example
Hertz contact: Rtip = 30 nm; Fapp= 1 nN
Etip=Esub=200 Gpa; Poisson ratio = tip=vsub=0.3=v
= + =
= = =
22 2111 3 3 1
4 2
3 0.91 1.365146.5
2 (200 ) 200
tipsub
tot sub tip
tot
E E E E
E GPaGPa GPa
= =
13
0.59tip
Hertz
tot
R Fa nm
E
= = =
2
1/32 2
12Hertz
tip tip tot
a FD pm
R R E
Fapp
aHertz
D
60 nm
Contact radius:
Deformation:
Pull-off Force=0
Contact Pressure:
= 2
0.9 9000 .Hertz
FP GPa atmos
a 10
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d
R
3* 2
12 2
*
0, 0
( ) 4( ) , 0
3
1 1
ts
tip sample
tip sample
d
F dE R d d
EE E
>
=
= +
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12
2
3* 2
2
12 2
*
,6
( ) 4( ) ,
6 3
1 1
o
ts
o o
o
tip sample
tip sample
HRd a
dF d HR
E R a d d aa
E
E E
>
= +
= +
d
R
ao
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jumpfrom
contact
2 click
WJKR
13
* 3* 3
2
*
12 2 3
* *
12 2
*
0,
( ) 48 ,
3
2
3
2
2;
8
1 1
ts
J KR
adhesionJ KR
J KR
J KR critical J KRcritical critical
tip sample
tip sample
nocontact
F d E aW E a contact
R
FW
R
W aad
R E
W a a R Wd a
E R E
EE E
=
=
= +
= =
= +
f h
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Contact forces: Maugis Theorya: normalized contact radius: normalized penetrationP: normalized force
Penetration
Penetration
D. Maugis, J. Colloid Interface Sci.150, 243 (1992).
Non-contact Contact
Contact
Rad
ius
Loadin
g
Force
Non-contact
Attractive
Contact
Repulsive
1 click
=
tip
tot
R Wa E
1/3
132
2
0
2.06
= +
= =
2 21 11 3
4
* int
s t
tot s t
o
E E E
a r eratomic distance
0: DMT (stiff materials) : JKR (soft materials)
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adhesionelasticity
i it i t m ti m
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=
tip
FF
W R
1/3
132
=
tip
tot
R W
a E
1/3
132
2
0
2.06
=a equilibrium separation
typical atomic distance)
0
(
2 21 11 3
4
s t
tot s tE E E
= +
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applied force
work of adhesion
adhesion
elasticity
a i ity o i erent mo e s converting measureadhesion force to work of adhesion
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Comments on these theories
JKR predicts infinite stress at edge of contact circle.
In the limit of small adhesion JKR -> DMT Most equations of JKR and Hertz and DMT have been
tested experimentally on molecularly smooth surfacesand found to apply extremely well
Most practical limitation for AFM is that no tip is aperfect smooth sphere, small asperities make a bigdifference.
Hertz, DMT describe conservative interaction forces,
but in JKR, the interaction itself is non-conservative(why?) for a force to be considered conservative ithas to be describable as a gradient of potential energy.
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Combining van der Waals force & DMT contact
A : Hamaker constant (Si-HOPG)
R : Tip radius
E*: Effective elastic modulus
a0: Intermolecular distance
Rtip=Si
a0 =r*
sample = HOPG
z
Raman et al, Phys Rev B (2002), Ultramicroscopy (2003) 17
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