Interaction forces III

8/8/2019 Interaction forces III

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Ron ReifenbergerBirck Nanotechnology Center

Purdue University

Lecture 7Interaction forces III

Tip-sample interaction forces

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Summary 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 Q2rp Q

r

p1r1

p2 2

1 2

0

( )

4

Q QU r

r

=

p1r

p2

=

2

0

cos( )( )

4

QpU r

r

[ ]

=

1 2 1 2 1 2

30

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

( ) 4

p pU r

r

( )=

2 2

1 22 6

0

1( )

3 4Keesom

B

p pU r

rk T

p1r

p2

0

12 21 02 2 01Debye 2 6

p +pU (r)= -

(4 ) r

01 02 1 2

2 6

0 1 2

( )( )3 1( )

2 (4 )London

I IU r

I I r

=

+

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


3/20

From interatomic to tip-sample interactions-simple theoryFirst consider the net interaction between an isolated atom/molecule and a flatsurface.

Relevant separation distance will now be called d.Assume that the pair potential between the atom/molecule and an atom on thesurface 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 bulkatoms?

No. of atoms/molecules in the infinitesimal ring is dV= (2x dx d) where isthe number density of molecules/atoms in the surface.

=0

d

x

=d

dx

( )

= =

= =

=

+

= >

= =

2 20

3

3

( ) 2

2, 3

( 2)( 3)

, 66

x

n

d x

n

VdW

dxU d C d

x

Cfor n

n n d

CU for n d

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4/20

Number Densities of the Elements

Introduction to Solid State Physics, 5th Ed. C. Kittel, pg. 32 4


5/20

From interatomic to tip-sample interactions-simple theory

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

x2 d = [R2-(R-)2] 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.

=

=

=

=

=


6/20

From interatomic to tip-sample interactions-some caveats

If tip and sample are made of different materials replace2 by 12 etc.

Tip is assumed to be homogeneous sample also! Bothmade of simple atoms/molecules

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

Perfectly smooth interacting surfaces

Tip-surface interactions obey very different power laws

compared to atom-atom laws

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7/20

Measuring Macroscopic Surface Forces

An interface is the boundary region between two adjacent bulk phases

We call (S/G), (S/L), and (L/V) surfacesWe call (S/S) and (L/L) interfaces

LL

V

GS

S

L

L

L = LiquidG = GasS = SolidV = Vapor

SS

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Surface Energetics

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

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

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9/20

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

(Excess) Surface Free Energy

interface

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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 thereforebe proportional to the surface area (dA):

is the proportionality constant defined as the specific surface freeenergy. 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 andleads to some well-known effects like liquid droplets forming spheresand meniscus effects in small capillaries.

dW dA=

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11/20

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 energiestend to rapidly adsorb contaminants

Values depend on oxide contamination adsorbed contaminants surface roughness etc.

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Surface-surface interactionsFollowing the steps in previous slides it is possible calculate the inter-action energy of two planar surfaces a distance of d apart, specificallydV = 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

n

d

n

plane plane

H

Hplane plane

C dU d

n n

C

n n n d

For n

U d U d

Cper unit aread

A Hama constant C

AU d per unit area

d

=d

=0

unit area

d

13


14/20

Typical Values for Hamaker Constant

Typically, for solids interacting across a vacuum,C 10-77 Jm6 and 3x1028 m-3

( )2

2 77 6 28 3

19

10 3 10

10

H A 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


15/20

15

Implications

Positive energy

Repulsive force

Negative energyAttractive force

U or F

U (r)

Flocal maxUmin

r*=

r=

( )( )

dU rF r

dr

r=

1612 13

1.246 7

= rF=max=

1612

1.126

=

=

12 6

( ) 4 *oU r Ur r

atom-atominteraction

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Equilibrium

separation


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I l dW i i b


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In general, vdW interactions betweenmacroscopic objects depends on geometry

Source J. Israelachvilli, Intermolecular and surface forces. 17


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The Derjaguin approximation

Plane-plane interaction energies are fundamental quantities andit is important to correlate tip-sample force to known values of

surface interaction energies. For a sphere-plane interaction wesaw that

=

= =

2 2

5

2 2

4

4 1( )

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

4 1( )( ) ( 2)( 3)( 4)

tip

n

tipn

C RU d

n n n n d

C RdUF dd d n n n d

=( ) 2 ( )sphere plane tip plane plane F d R U d

= + 1 2

1 2

( ) 2 ( )sphere sphere plane plane R R

F d U d R R

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

Comparing with previous slides we find that

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19/20

Implications of Derjaguins approximation

We showed this when U(r)=-C/rn - however it is valid forany force law - attractive or repulsive or oscillatory - fortwo 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

theconventional surface energy per unit area to create asolid surface. Thus:

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

= = = +

1

1 2

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

R R

dW11

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How to Model the Repulsive Interactionat Contact?

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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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Interaction forces III

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