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Ron ReifenbergerBirck Nanotechnology Center
Purdue University
Lecture 5Interaction forces I –
From interatomic and intermolecular forces to tip-sample interaction forces
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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”.
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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
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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
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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
- - -
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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.
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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
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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
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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∞
≡ − ⋅ ⇒ = −∇∫
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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:
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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
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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
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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.
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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
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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
−
=
= −
= ⋅ ⋅ ×
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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
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Electrostatic interatomic and intermolecular forces
1. Charge-charge (ion-ion) interactions:
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The Electrostatic Force: Coulomb’s Law
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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
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Electrostatic interatomic and intermolecular forces
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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
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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
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3. Permanent dipole-permanent dipole interactions - fixed orientation:
[ ]θ θ θ θ φπεε
−= − 1 2 1 2 1 2
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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
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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|
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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
−
−
−
× ⋅ ×= = −
×
= − × ≈
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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
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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
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( )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
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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- +
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ϑ
pE
cosU p E pE ϑ= − ⋅ = −
Energetics of a static dipole in a uniform applied electric field
The interaction energy is:
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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
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Polarization forces
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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
= +
= +
= +
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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
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Electrostatic interatomic and intermolecular forces
7. Dispersion Forces
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• 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
= − = −+
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2 6 60
4( )2 (4 )
LLondon
I CU rr r
απε ε
= − = −
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31Source: http://en.wikipedia.org/wiki/Ionization_energy
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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.)
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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.
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( )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
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Source: http://www.ntmdt.com/spm-basics/view/intermolecular-vdv-force
CKeesom CLondonCDebye
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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?
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Interaction between electrically NEUTRAL objects!
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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
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Putting in some numbers
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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
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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)
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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
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