interaction forces i

Ron ReifenbergerBirck Nanotechnology Center

Purdue University

Lecture 5Interaction forces I –

From interatomic and intermolecular forces to tip-sample interaction forces

1

Further reading

J. Israelachvilli, “Intermolecular and surface forces”. H. J.Butt,B. Cappella, M. Kappl, “Force measurements with the atomic

force microscope: technique, interpretation and applications” Surface Science Reports, Vol. 59 (1-6), 1-152, 2005.

K. L. Johnson “Contact Mechanics”.

2

3

What controls the force between tip and substrate?

tip apex

insulating substrate

scan

collective tip-substrate

interactions

+

atom-atom interactions

Frestore = -kΔzspring constant, k

ω ~ 1015 Hz

m ~ 10-30 kg

k~ ω2m ~ 1 N/m

Long range

+

Short range

Probing the Interaction Potential

4

The Electrostatic Force: Quick review

Coulomb’s Law

Point charges ONLY! The charges are stuck down!

1 2 1 22 2

12 2 2

9 2 2

1| |4

8.85 10 /

9 10 /

o

o

q q q qF kr r

C Nmk Nm C

πε

ε −

= =

= ×

= ×

In vacuum

5

Force is a vector!(same problem with four different choices for charge polarity)

q1 q2 q3

+ + + q1

q2 q3

F13 F12

q1 q2 q3

- + +

q1

q2 q3

F13F12

q3

- + -q1 q2 q3

q1

q2

F13

F12

q1 q2 q3+ + -

q1

q2 q3

F13

F12

- - -

6

The Electric Field produced by a Positive Point Charge

High field

Low field

The Electric Field produced by a Negative Point Charge

o

FEq

=

o

FEq

=

Bottom line: If there is an electric field at some point in space and you place a charge qo at that point, the charge qo will feel an electrostatic force.

7

Electric Fields Produced by Two Point Charges of Equal Magnitude but Opposite Polarity: the Electric Dipole

p qd

p qd

=

=

+-

d

p

q+q−

Historically the standard unit size of a dipole moment was defined by considering twocharges of opposite sign but with equal magnitude of 10-10 statcoulomb (also callede.s.u. - the electrostatic unit in older literature), which were separated by1 Angstrom = 1.0x10-8 cm.

This gives the following unit for molecular dipole moments:1 Debye =(1x10-10 statC)(1x10-8 cm )= 1x10-18 statC·cm

= 10-10 esu·Å

= 1⁄299,792,458 × 10−21 C·mm

≈ 3.33564 × 10−30 C·m

dipole moment

physical dipole

8

Material ε, dielectric constant

Dipole Moment (in Debye)

Acetone 21 2.9Isopropanol 18.3 1.7Ethanol 24 1.7Methanol 33 1.7Toluene 2.4 0.4Freon-11 (CCL3F) 2.0 0.45Water 80 1.8

A few common solvent molecules

9

Mechanics Electrostatics

Potential Energy, V V(r)=-W

Work W=∫F•dr W=∫E• dl

U(r)= - W

Potential Energy

If Force is conservative (not a function of time), then Potential Energy at point P (for linear motion) can be defined as:

( )( ) ( ) ( )P

PP

ref

V xV x F x d x F xx

∂≡ − ⇒ = −

∂∫If Electric field is conservative (not a function of time), then Potential

Energy at point P for electrostatic motion can be defined as:

( ) ( )P

P PU r E d F U r∞

≡ − ⋅ ⇒ = −∇∫

10

Electrostatic Potential Energy

+

-r12

2q

1q

1 2 1 2 1 2

12 12 12

( )( ) ( )1 1 14 4 4o o o

q q q q q qUr r rπε πε πε

−= = = −

r12

2q

1q

3qr23

r13

2 3 1 31 2

12 23 13

( )( ) ( )( )( )( )14 o

q q q qq qUr r rπε

= + +

Two charges:

Three charges:

11

Charge Distribution

Classical E&M Real World Examples Chemical

Formulapoint charges +Q or -Q ions Li+, Mg++ , Cl -

permanent dipoles p= Q d polar

molecules

acetone, acetonitrile,

ethanol, methanol,Freon-41,

water

(CH3)2COCH3CNC2H5OHCH3OHCH3FH2O

fluctuating dipoles p(t) = Q d

non-polar molecules,

neutral atoms

hexane, benzene, toluene

C6H14C6H6C7H8

(or C6H5CH3)

surface charge distribution σ charged

objectsmetal plate with voltage

volume charge distribution ρ charged

objectscharged non-conductors

Connecting charges and electric fields to the microscopic world

Interatomic and intermolecular forces

A wide variety of forces act between atoms and molecules can be loosely classified as follows (Israelachvilli):

Electrostatic forces: interactions between charged ions, perman-ent dipoles etc. fall into this category

Polarization forces: these forces occur due to dipoles induced in some molecules in response to electric fields from nearby charges or permanent dipoles

Dispersion forces: these are fundamentally electrodynamic in nature and occur between atoms and molecules even if they are charge neutral and without permanent dipoles

Covalent bond forces: when two or more atoms come together to form a molecule and share electrons, the forces that tightly bind the atoms together are called covalent forces.

Pauli repulsion forces: At very small interatomic distances the electron clouds of atoms overlap and a strong repulsive force determines how close two atoms or molecules can ultimately approach each other.

12

picometers up to few nanometers

Cou

lom

bic

in

orig

inQ

uant

um

in

orig

in

The Lennard-Jones interatomic potentialLet’s start with atom-atom interactions: Simple ad hoc model that tries to couple dispersion forces and Pauli

repulsion.

Uo is depth of potential, σ is value at which Uo(r=σ)=0 F = -dU(r)/dr While attractive part follows that from the general dispersion

relation, the repulsive part is adhoc.

13

σ σ = −

12 6

( ) 4 *oU r Ur r

-2.0

0.0

2.0

4.0

6.0

0.0 0.2 0.4 0.6 0.8

Pote

ntia

l (in

eV

)Fo

rce

(in

nN)

Separtion (in nm)

6-12 Potential; Uo=1.5 eV; σ=0.35 nm

potential

force

U

F

14

Implications

Positive energyRepulsive force

Negative energyAttractive force

U or F

U (r)

Flocal maxUmin

r*=

r

= −( )( ) dU rF r

dr

r=σ

1612 13 1.24

6 7σ σ⋅ = ⋅

rF=max=

1612 1.12

6σ σ =

6

6

6 77 6

4

( )

, 4 (0.5 ) (0.25 ) 8 10

ooften define U C so thatCU rr

typically C eV nm Jm

σ

−

=

= −

= ⋅ ⋅ ×

15

Electrostatic interatomic and intermolecular energies

1. ion-ion interactions2. ion-dipole interactions3. dipole-dipole interactions4. angle-averaged dipole-dipole interactions5. polarization forces (Keesom)6. dipole-induced dipole interactions (Debye)7. dispersion interactions (London)

= −( )( ) dU rF r

dr

Electrostatic interatomic and intermolecular forces

1. Charge-charge (ion-ion) interactions:

16

The Electrostatic Force: Coulomb’s Law

16

The Electrostatic Force: Coulomb’s Law

In dielectric

1 2 1 22 2

12 2 2

9 2 2

1| |4

8.85 10 /

9 10 /

o

o

q q q qF kr r

C Nmk Nm C

πε

ε −

= =

= ×

= ×

; 1o oε εε ε→ >

In vacuumπε

πε

=

= − =

1 2

0

1 22

0

( )4

( )| ( ) |4

Q QU rr

Q QdU rF rdr r

Electrostatic interatomic and intermolecular forces

17

Electrically neutral atom or molecule

(electrically neutral; no net charge but charge redistribution occurs)

equivalent electric dipole+ -

1 click

2. Permanent dipole-charge interactions:

The permanent dipole of a molecule is the separation of charges by d in the molecule times the magnitude of the charge separated

|p|=qd

18

θπεε

= − 20

cos( )( )4

QpU rr

Electrostatic interatomic and intermolecular forces

The interaction energy of a point charge Q with a dipole having a dipole moment p

+Q F1

F2

r1

r2Point Charge

-q

+q

1 click

=

| |p q dθ

r

d

19

3. Permanent dipole-permanent dipole interactions - fixed orientation:

[ ]θ θ θ θ φπεε

−= − 1 2 1 2 1 2

30

2cos( )cos( ) sin( )sin( )cos( ) 1( )4

p pU r

r

Electrostatic interatomic and intermolecular forces

already four parameters: separation distance (r) + 3 angles (Θ1,Θ2,φ)

+q2

-q1

+q1

-q2

Θ1 Θ2φ

r

20

diagram for s/d ≈ 1

Symbol Actual Charge

q1 + q

q2 - q

q3 + q

q4 - q

Electrostatic interaction between in-plane permanent dipoles

Fixed, NO rotation

q2

q3 q4q1

-x

+- x

y

Φ

s

+

d

-3.0

-1.5

0.0

1.5

3.0

-180 -120 -60 0 60 120 180

Uto

tal(

in10

-18

J)

Φ (degrees)

|q|=1x10-18 C; s=10 nm

d/s=0.75d/s=0.50d/s=0.30

Note: |p1|=|p2|

Strength of interactions

The strength of an interaction is measured by comparing the value of the potential energy at equilibrium with the typical thermal energy kBT (kB=1.38 × 10-23 J K-1).

For strong interactions, U ~ 100’s kBT

Example 1 (2 electron charges Q=1.602×10-19 C at a distance of 0.276 nm (equilibrium separation of Na+ and Cl-) in vacuum

21

−= × =

214.1 10 ( 300 ) 4.1Bk T J at K pN nm

( )229 192

1 29

19

9 10 1.6 1014 0.276 10

8.34 10 203o

B

Nm Cq q CUr m

J k T at room temperatureπε

−

−

−

× ⋅ ×= = −

×

= − × ≈

22

Example 2: What is the energy of interaction of two dipoles (p=1D) lined up and separated by 0.5 nm?

p1 p2

r

[ ]

[ ] ( )

θ θ θ θ φπεε

θ θ φ ε

πε

−

−−

−

−= −

= = = = = = × =

×= = × ⋅ = ×

×

1 2 1 2 1 23

030

1 2 1 2230

21 2 9 2123 9 3

0

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

4

0; 0; 0; 1 3.33 10 ; 1

2 3.33 102 1( ) 9 10 1.6 104 (0.5 10 )

( ) 0.4 B

p pU r

r

p p D Cm

Cmp p NmU r JCr

U r k T

about 100 times weaker than typical ion-ion interaction

23

Electrostatic interatomic and intermolecular forces

4. Angle averaged dipole-dipole interactions:

At large distances, kBT makes molecules fluctuate so that a weighted average using classical Boltzmann statistics is required.

rP

:

rB

rB

rB

rB

r

k T

rk T

r

r

k Tr

rr r

k Tr

r

Let probability of finding a state with energy

eP when in thermal equilibriume

Probability of finding any quantity y with values y

y ey y y P

e

ε

ε

ε

ε

ε−

−

−

−

=

=

= = =

∑

∑∑

∑

Partition function

24

( )πεε= −

2 21 2

2 60

1 1( )3 4

KeesomB

p pU rk T r

This gives (also known as the Keesom interaction):

cos, cos

B

B

kU

T

Uk T

e dfor example

e d

ϑϑ

−

Ω−

Ω

Ω≡

Ω

∫

∫

θ θ θ θ φπεε

ϑ ϑ ϕ

− = −

= Ω =

1 2 1 2 1 23

0

2 cos cos sin sin cos 1( )4

.... sin

p pU r

rwhere thermal average over d d d

……..when applied to dipole-dipole interactions

Electrostatic interatomic and intermolecular forces5. Polarization forces

All atoms and molecules not having an intrinsic dipole moment can be polarized by an external electric field.

E Highly magnified!

Electronic charge cloud Nucleus

pinduced

oinducedp Eα=

atom or molecule

α πε= × 304o R

α0 is called the electronic polarizability and has units of m3, it is proportional to molecule size

For water, α0 /4πεo = 1.48x10-30 m3

→ R=0.114 nm – compare to RH20 ≈ 0.135 nm

diam=2R- +

26

ϑ

pE

cosU p E pE ϑ= − ⋅ = −

Energetics of a static dipole in a uniform applied electric field

The interaction energy is:

27

2

1

2

1

cos

coscos

11 cos

1 133

3

cos

B

B

k T

B

k T B

B B

B

U

U

B

p E pE

e dk TpEcotanh

k T pEe d

pE pEwhenk

U

pT

k

k TE E

k TTpp p

ϑ

ϑϑ

ϑ

ϑ αα

−

Ω−

Ω

= − ⋅ = −

Ω

= = − Ω

≈

= = = ≡

∫

∫

If dipole fluctuates with thermal energy kT

ϑE

p

Langevin function (1905)

p

Atom or molecule

ϑ

cosp p

where thermal average

ϑ=

=

temperature dependent contribution to the

polarizability

Polarization forces

28

Usually, α depends on the frequency of the electric field.

α α= + −2

0 ( )3 B

p Debye Langevin equationk T

( )1

213

total induced

o

oB

p p p

E

p Ek T

α α

α

= +

= +

= +

29

6. Dipole-induced dipole interaction:

By taking the electric field generated by a fixed dipole and using the above equation and thermal angle averaging according to Boltzmann statistics, one finds the Debye interaction energy

where α01 and α02 are the electronic polarizabilities of the two molecules.

If one replaces α0 with the orientation polarizabilityα=p2/3kBT, we recover the Keesom force.

2 21 02 2 01

2 6 60

( )(4 )

DDebye

p p CU rr r

α απε ε

+= − = −

Electrostatic interatomic and intermolecular forces

30

Electrostatic interatomic and intermolecular forces

7. Dispersion Forces

30

• The London or Dispersion Force is the most important contributor to van der Waals forces and acts between all molecules irrespec-tive of their polarization • relevant distances range from 0.2 nm to 10 nm. • fundamentally quantum mechanical in nature and arises from induced dipole-induced dipole interactions. • quantum mechanical calculation

It can be shown that for two similar atoms/molecules with an ionization energy I

For dissimilar atoms/molecules with ionization energies I1 and I2

01 02 1 22 6 6

0 1 2

3( )2 (4 )

LLondon

I I CU rr I I r

α απε ε

= − = −+

20

2 6 60

4( )2 (4 )

LLondon

I CU rr r

απε ε

= − = −

31Source: http://en.wikipedia.org/wiki/Ionization_energy

32

Extensions to frequency dependent polarizations by McLachlan (1963)

As distances increase (>100nm), the time taken for fluctuating dipoles to reach other atom/molecule needs to be taken into account-retardation effects (power law dependence becomes 1/r7 )

Casimir-Polder force is generalization to include finite conductivity

Electrostatic interatomic and intermolecular forces

7. Dispersion Forces (cont.)

van der Waals Forces

The van der Waals forces is the sum of three different forces each of whose potential varies as 1/r6, where r is the separation Orientation or Keesom Force. The Keesom force is the averaged

dipole-dipole interaction between two atoms or molecules. Debye Force is the angle averaged dipole-induced dipole

interaction between two atoms or molecules The London or Dispersion force is the most important contributor

to van der Waals forces and acts between all molecules.

33

( )α α α α

πε ε πε επεε

= + +

+= − − − = −

+

2 22 21 02 2 01 01 021 2 1 2

2 6 2 6 2 6 60 0 1 20

( ) ( ) ( ) ( )

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

VdW Keesom Debye London

B

U r U r U r U r

pr r r

pp p I I CI I rk T

34

Source: http://www.ntmdt.com/spm-basics/view/intermolecular-vdv-force

CKeesom CLondonCDebye

35

Summary of Coulombic Intermolecular Forces

Coulomb Energy

London Dispersion

Dipoles rotating

Ions Involved?

YESNO

Do the atoms or molecules have a

permanent dipole?

YES NO

Ionic Bonding

Ion – dipole Interaction

YESNO

Induced dipole-induced dipole

only

Is H bonded to O, N or F

(permanent dipoles)?

Dipole rotating

Keesom

Are there any atoms or

molecules ?

YES

NO

PolarizationForces

Dipole – dipole Interactions

(hydrogen bonding)

Dipoles fixed

van der Waals Forces

Debye

System of interacting atoms and/or molecules

and/or ions

Dipole fixed

εo or εεo?

Interaction between electrically NEUTRAL objects!

36

Dispersive or van der Waals Force

zo

rD

Uncharged sphere, radius R

Uncharged Insulating plate3

2 2

2R( )3 (2 )

F D AD R D

= −+

an Electrodynamic effect!1 click

Instantaneous, fluctuating dipoles

+q-q

-q+q

+q-q

+q+q-q

-q+q

-q+q

A=Hamaker’s constant

Putting in some numbers

37

R=10 nm; Q= 8 electrons

0.000

0.010

0.020

0.0 10.0 20.0 30.0 40.0D Surface-to-Surface Separation (nm)

Att

ract

ive

Fo

rce

(nN

)

neutral sphere

chargedsphere metalUncharged plate

D

sphere, radius

R

Neutral sphere

Charged sphere

Metal plateDielectric plate

ao

Implications for AFM

In AFM the tip usually has native oxide on the tip within which small trapped charges or permanent dipoles can exist. As debris accumulates more permanent dipoles and charges accumulate on tip

Dispersion potential U(r) scales as r-6 while electrostatics scales as

r-1 (ion-ion), or r-2 (ion-dipole), or r-3 (dipole-dipole)

In reality attractive forces are due to combination of vdW and short range electrostatics (i.e. covalent forces)

38

Summary: Tip-sample interaction forces in AFM Long-range electrostatic and magnetic

forces (up to 100 nm) Short-range electrostatic forces Short-range polarization forces Dispersion forces (few nm) that are

fundamentally quantum mechanical in nature

van der Waals and Casimir forces Capillary forces (few nm) Short-range chemical forces

(fraction of nm) Pauli repulsion Contact forces Electrostastic double-layer forces Solvation forces Hydrophobic and hydrophillic forces Nonconservative forces (Dürig (2003))Tip-sample gap

Tip-sample interaction force

Attractive

Rep

ulsi

ve

Nanosensors Gmbh

39