Nanohub.org Resources 9820 Download 2010.09.21-ME597-L08-Reifenberger

7/29/2019 Nanohub.org Resources 9820 Download 2010.09.21-ME597-L08-Reifenberger

1/17

Ron ReifenbergerBirck Nanotechnology Center

Purdue University

Lecture 8

Introduction to Contact Mechanics

1


2/17

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?

2


3/17

3

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?


4/17

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

4


5/17

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)

5


6/17

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


7/17http://www-materials.eng.cam.ac.uk/mpsite/interactive_charts/stiffness-density/NS6Chart.html

10 MPa

five

ordersofmagnitude

1 Pa = 1 N/m2

7

II. What is the Stiffness of the Tip/Substrate?


8/17

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


9/17

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


10/17

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


11/17

11

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

>

=

= +


12/17

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


13/17

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


14/17

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)

14

adhesionelasticity

i it i t m ti m


15/17

=

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

= +

15

applied force

work of adhesion

adhesion

elasticity

a i ity o i erent mo e s converting measureadhesion force to work of adhesion


16/17

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.

16


17/17

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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