Ron ReifenbergerBirck Nanotechnology Center

Purdue University

Lecture 6Interaction forces II –

Tip-sample interaction forces

Lecture 6Interaction forces II –

Tip-sample interaction forces

1

Summary of last lectureSummary of last lecture

r

Type of interaction

Ion-ion electrostatic

Dipole-charge electrostatic

Dipole-dipole electrostatic

Angle-averaged electrostatic (Keesom force)

Angle-averaged induced polarization force (Debye force)

Dispersion forces act between any two molecules or atoms (London force)

Q1 Q2r

p Q

r

p1

r

p2

1 2

0

( )4

Q QU r

r

p1

r

p2

2

0

cos( )( )

4

QpU r

r

1 2 1 2 1 23

0

2cos( )cos( ) sin( )sin( )cos( )( )

4

p pU r

r

2 21 2

2 6

0

1( )

3 4Keesom

B

p pU r

rk T

p1

r

p2

12 21 02 2 01

Debye 2 6

p +pU (r)=-

(4 ) r

01 02 1 22 6

0 1 2

( )( )3 1( )

2 (4 )London

I IU r

I I r

Adapted from J. Israelachvilli, “Intermolecular and surface forces”.2

From interatomic to tip-sample interactions-simple theory

From interatomic to tip-sample interactions-simple theory

First consider the net interaction between an isolated atom/molecule and a flat surface.Relevant separation distance will now be called “d”.Assume that the pair potential between the atom/molecule and an atom on the surface is given by U(r)=-C/rn.Assume additivity, that is the net interaction force will be the sum of its inter-actions with all molecules in the body – are surfaces atoms different than bulk atoms?No. of atoms/molecules in the infinitesimal ring is ρdV= ρ(2x dx d) where ρ is the number density of molecules/atoms in the surface.

=0√(

2+x2 )

d

x

=d

dx

2 20

3

3

( ) 2

2, 3

( 2)( 3)

, 66

x

nd x

n

VdW

dxU d C d

x

Cfor n

n n d

CU for n

d3

Number Densities of the ElementsNumber Densities of the Elements

Introduction to Solid State Physics, 5th Ed. C. Kittel, pg. 324

From interatomic to tip-sample interactions-simple theory

From interatomic to tip-sample interactions-simple theory

Next integrate atom-plane interaction over the volume of all atoms in the AFM tip. Number of atoms/molecules contained within the slice shown below is

x2 d= [R2-(R- d=(2Rtip-)d

Since all these are at the same equal distance d+ from the plane, the net interaction energy can be derived by using the result on the previous slide.

2

0

2 2

30

2 2

5

2

2 2

2

3

(2 )( )

,

2( ) ~

( 2)( 3) ( )

4

( 2)( 3)( 4)( 5)

, 6

( ) ~6 6

: ker

2

( 2)( 3) ( )

'

tipRtip

tip

tip

n

tip

n

tip

tip tipVdW

n

R dU d

If d R

R dCU d

n n d

C R

n n n n d

For d R n

R RU d

d dH H

C H

C

n n

ama

d

s con ( )Hstant also A or A

d

=0

+d

d

Rtip

2Rtip-

x

5

From interatomic to tip-sample interactions-some caveats

From interatomic to tip-sample interactions-some caveats

If tip and sample are made of different materials replace 2 by 12 etc.

Tip is assumed to be homogeneous sample also! Both made of “simple” atoms/molecules

Assumes atom-atom interactions are independent of presence of other surrounding atoms

Perfectly smooth interacting surfaces Tip-surface interactions obey very different power

laws compared to atom-atom laws

6

Measuring Macroscopic Surface Forces

An interface is the boundary region between two adjacent bulk phases

We call (S/G), (S/L), and (L/V) surfaces

We call (S/S) and (L/L) interfaces

L

L

V

G

S

S

L

L

L = LiquidG = GasS = SolidV = Vapor

S

S

7

Surface Energetics

Atoms (or molecules) in the bulk of a material have a low relative energy due to nearest neighbor interactions (e.g. bonding).

Performing work on the system to create an interface can disrupt this situation...

8

Atoms (or molecules) at an interface are in a state of higher free energy than those in the bulk due to the lack of nearest neighbor interactions.

(Excess) Surface Free Energy

interface

9

Surface Energy

The work (dW) to create a new surface of area dA is proportional to the number of atoms/molecules at the surface and must therefore be proportional to the surface area (dA):

is the proportionality constant defined as the specific surface free energy. It is a scalar quantity and has units of energy/unit area, mJ/m2.

acts as a restoring force to resist any increase in area. For liquids is numerically equal to the surface tension which is a vector and has units of force/unit length, mN/m.

Surface tension acts to decrease the free energy of the system and leads to some well-known effects like liquid droplets forming spheres and meniscus effects in small capillaries.

dW dA

10

high surface energy ↔ strong cohesion ↔ high melting temperatures

Material Surface energy ( mJ/m2 )

mica 4500

gold 1000

mercury 487

water 73

benzene 29

methanol 23

Materials with high surface energies tend to rapidly adsorb

contaminants

Values depend on oxide contamination adsorbed contaminants surface roughness etc….

11

Work of Cohesion and Adhesion

For a single solid (work of cohesion):

For two different solids (work of adhesion):

11 1 12 2dW dA dA

12 1 2 12( )dW dA

1

1

2

1

1

1

2

1

12

interfacial

energy

1

2

12

Surface-surface interactionsSurface-surface interactionsFollowing the steps in previous slides it is possible calculate the inter- action energy of two planar surfaces a distance of ‘d ’ apart, specifically ρdV = ρ dA d for the unit area of one surface (dA=1) interacting with an infinite area of the other.

2

3

2

4

2

2

2 2

2

2( )

( 2)( 3) ( )

2 1

( 2)( 3)( 4)

6

( ) ( )

( )12

ker

1( )

12

nd

n

plane plane

H

Hplane plane

C dU d

n n

C

n n n d

For n

U d U d

Cper unit area

d

A Hama constant C

AU d per unit area

d

=d

=0

unit area

d

13

Typical Values for Hamaker ConstantTypical Values for Hamaker Constant

Typically, for solids interacting across a vacuum,

C ≈ 10-77 Jm6 and ρ ≈ 3x1028 m-3

22 77 6 28 3

19

10 3 10

10

HA J m m

J

Typically, most solids have

19(0.4 4.0) 10HA J

14

See Butt, Cappella, Kappl, Surf. Sci. Reps., 59, 50 (2005) for more complete list

15

Implications

Positive energyRepulsive force

Negative energyAttractive force

U or F

U (r)

Flocal max

Umin

r*=

r

( )

( )dU r

F rdr

r=σ

1612 13

1.246 7

rF=max=

1612

1.126

12 6

( ) 4 *oU r Ur r

atom-atom interaction

15

Equilibrium separation

** 2

11 1

1 * 2

( )12 ( )

2

24 ( )

HvdW

H

AU d r

r

dW

A

r

Estimating the interfacial energyEstimating the interfacial energy

When d≈r*, then per unit area, we should have:

This approach neglects the atomicity of both surfaces. Require an “effective” r*.

=d=r*

=0

unit area

d

16

In general, vdW interactions between macroscopic objects depends on geometry

In general, vdW interactions between macroscopic objects depends on geometry

Source J. Israelachvilli, “Intermolecular and surface forces”.17

The Derjaguin approximationThe Derjaguin approximation

Plane-plane interaction energies are fundamental quantities and it is important to correlate tip-sample force to known values of surface interaction energies. For a sphere-plane interaction we saw that

2 2

5

2 2

4

4 1( )

( 2)( 3)( 4)( 5)

4 1( )

( ) ( 2)( 3)( 4)

tip

n

tip

n

C RU d

n n n n d

C RdUF d

d d n n n d

( ) 2 ( )sphere plane tip plane planeF d R U d

1 2

1 2

( ) 2 ( )sphere sphere plane plane

R RF d U d

R R

It can be shown that for two interacting sphere of different radii

Comparing with previous slides we find that

18

Implications of Derjaguin’s approximationImplications of Derjaguin’s approximation

We showed this when U(r)=-C/rn - however it is valid for any force law - attractive or repulsive or oscillatory - for two rigid spheres.

As mentioned before, if two spheres are in contact (assuming no contamination), then d=r*.

The value of U(d=r*)plane-plane is basically dW11 the conventional surface energy per unit area to create a solid surface. Thus:

This approximation is useful because it converts measured Fadhesion in AFM experiments to surface energy dW11

1

1 2

1 1( *) 2 ( *)adhesion sphere sphere plane planeF F d r U d r

R R

dW11

19

How to Model the Repulsive Interaction at Contact?

20

tip ape

x

substrate

Source: Capella & Dietler

Maybe if the contact area involves tens or hundreds of atoms the description of net repulsive force is best captured by continuum elasticity models

Atom-Atom? Sphere-Plane?

20

Continuum description of contact - history

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

Sneddon’s analysis (1965) considers a rigid sphere (or other rigid shapes) on a linearly elastic half-space.

Neither Hertz or Sneddon considers surface forces.

Bradley’s analysis (1932) considers two rigid spheres interacting via the Lennard-Jones 6-12 potential

Derjaguin-Müller-Toporov (DMT, 1975) considers an elastic sphere with rigid surface but includes van der Waals forces outside the contact region. 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 DMT are limits of same theory 21

Tip-sample Interaction ModelsTip-sample Interaction Models

From the Derjaguin approximation for rigid tip interacting with rigid sample we have

Real tips and samples are not rigid. Several theories are used to better account for this fact (Hertz, DMT, JKR)

* These theories also apply to elastic samples, they are just shown on rigid sample to demonstrate key quantities clearly. For example D is the combined tip-sample deformation in (b)

132 13 23 12( *) 2 ( *) 2 2 ( )tip sample adhesion tip tip tipF r F R U r R W R

Rigid tip-rigid sampleDeformable tip and rigid sample*

Rtip

sample

F

aHertz

aJKR

F

as

Equilibrium Pull-off

1

2

3D D

(a) (b) (c)

22

Work of adhesion and cohesion: work done to separate unit areas of two media 1 and 2 from contact to infinity in vacuum. If 1 and 2 are different then W12 is the work of adhesion; if 1 and 2 are the same then W11 is the work of cohesion.

Surface energy: This is the free energy change when the surface area of a medium is increased by unit area. Thus

While separating dissimilar materials the free energy change in expanding the “interfacial” area by unit area is known as their interfacial energy

Work of adhesion in a third medium

I. Surface energies Surface energies - notationI. Surface energies Surface energies - notation

11 12W

12 1 2 12W 12

132 13 23 12W

1 23

23

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

10 MPa

five o

rders

of

mag

nit

ud

e

1 Pa = 1 N/m2

24

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

F

aHertz

aJKR

Equilibrium

D

(b)

13

tipHertz

tot

R Fa

E

13

132( 2 )tip tipDMT

tot

R F R Wa

E

132

132 132 132( 2 6 (3 ) )tip tip tip tip

JKRtot

R F R W R W F R Wa

E

Standard results

2

1/32 2

Hertztip tip tot

a FD

R R E

2

1/322

1322 tip

DMTtip tip tot

F R WaD

R R E

213262

3JKRtip 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 25

F

as

Pull-off

D

(c)0Hertz

adhesionF

tip 1322 R WDMTadhesionF

tip 132

3R W

2JKRadhesionF

Standard results (cont.)

Pull-off Force F:

Source: Butt, Cappella, Kappl 26

Example

Hertz contact: Rtip = 30 nm; Fapp= 1 nN

Etip=Esub=200 Gpa; Poisson ratio = νtip=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

0.59tipHertz

tot

R Fa nm

E

2

1/32 2

12Hertztip tip tot

a FD pm

R R E

Fapp

aHertzD

60 nm

Contact radius:

Deformation:

Pull-off Force=0

Contact Pressure:

2

0.9 9000 .Hertz

FP GPa atmos

a 27

28

jump from contac

t

2 click

work of adhesion

29

Contact forces: Maugis’ TheoryContact forces: Maugis’ Theorya: normalized contact radiusδ: normalized penetrationP: normalized force

Penetration

Penetration

D. Maugis, J. Colloid Interface Sci.150, 243 (1992).

Non-contact Contact

Con

tact

Rad

ius

Load

ing

Forc

e

Non-contact

Att

racti

ve

Contact

Rep

uls

ive

1 click

tip

tot

R W

a E

1/3

132

20

2.06

2 21 11 3

4

* int

s t

tot s t

o

E E E

a r eratomic distance

λ 0: DMT (stiff materials)λ : JKR (soft materials)

30

adhesion

elasticity

tip

FF

W R

1/3

132

tip

tot

R W

a E

1/3

132

20

2.06

a equilibrium separation

typical atomic distance)0

(

2 21 11 3

4s t

tot s tE E E

31

applied force

work of adhesion

adhesion

elasticity

Validity of different models – converting measured adhesion force to work of

adhesion

Validity of different models – converting measured adhesion force to work of

adhesion

Comments on these theoriesComments 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 surfaces and found to apply extremely well

Most practical limitation for AFM is that no tip is a perfect smooth sphere, small asperities make a big difference.

Hertz, DMT describe conservative interaction forces, but in JKR, the interaction itself is non-conservative (why?) …for a force to be considered conservative it has to be describable as a gradient of potential energy.

32

Combining van der Waals force & DMT contactCombining 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) 33

Effect of capillary condensation – modifications to DMT model

Effect of capillary condensation – modifications to DMT model

Capillary force models range from simple to complex Strong dependence on humidity

34

capillary neck breaks

Next lectureNext lecture

Couple cantilever mechanics to tip sample interaction forces

F-Z vs. F-d curves

35