Interatomic Forces

Interatomic Forces
Overview
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Van der Walls (short range V~1/r6, weak ~0.01-0.1 eV)
Ionic (long range, V~1/r, strong ~5-10 eV)
Metallic (no simple dependence, ~0.1eV)
Covalent (no simple dependence, directional,~3 eV)
Hydrogen (partially ionic and covalent)
Interatomic Forces
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Van der Waals: related to dipole-dipole interactions
Induced charge separation
Fluctuation separates charge
- - +
- -l
-q
E
- - +
- -
r
l
-l’ l’
p1 +q
-q
+q
p2
Separate the molecules by some distance, a weak ATTRACTION is produced due to
molecular structure and Coulomb’s law. The internal motion of the nucleus and electron cloud
that can give attraction. This force is called the VAN DER WAALS force and is the attractive
part of the L-J potential
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From observation we know that:
• Molecules repel if too close together due to electrostatic repulsion
of electron clouds (r~1-2 Å)
• In solids and liquids the molecules tend to ‘stick’ together so at
some r molecules must attract each other
•At very large r there is no discernable force
Expressed graphically:
Strong repulsion due to electron overlap
F(r)
Vanishing force
Take some standard length
a as a unit of r
a
Attraction
2a
Interatomic Forces
3a
4a
r
3
Mathematical representation:
V(r)
E – electric field, p – dipole moment, q - charge
E ~ p1/r3 (follows from F~ qE ~ q2/r2), hence
p2 ~ αp1/r3, α - polarisability
V ~ pE ~ A/r6
V(r)= VRepulsive(r)+ VAttractive(r)
n
or
1
1
V (r ) = A   − B 
r
r
VRepulsive(r)
a0
r
ε
6
,n>9
VAttractive(r)
 σ  n  σ  6 
V ( r ) = 4ε   −   
 r  
  r 
Most commonly n=12 is used (“6 -12” potential mostly for convenience of calculations). In
general symbols p and q can be used (“p-q” potential) For repulsion ~e-r/a dependence is
also sometimes used (due to fall off in charge density in the outer edge of electron clouds of
each molecule as they overlap)
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• The ‘p-q’potential is just one example of an infinate set of
mathematical functions that might represent a molecular pairpotential
• For INERT GASES this is a good representation of potential
between molecules because these gases comprise single atoms
• Often use the LENNARD-JONES potential (often called the
‘6-12’ potential) to describe inert gases:
  σ 12  σ  6 
V ( r ) = 4ε    −   
 r  
  r 
This is sometimes written in terms of the distance a at which
V(r) is zero
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IONIC FORCES: in substances like Sodium Chloride or the alkali
halides generally, the effect of internal structure is different. Instead of
metal Na and Cl atoms, it is energetically favourable for an electron to
leave the Sodium cloud to joint the chlorine electron cloud leaving two
charged ions:
F (r ) = FRepulsion + FCoulomb
Na
r
V (r ) = VRepulsion + VCoulomb
V (r ) =
λ
r
p
−
e2
Constants
Cl
+
_
4πε o r
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- -+
- -
- -+
- -
q1 q 2
F (r ) =
4πε 0 r 2
The fundamental force here is the ELECTROSTATIC
COULOMB force between point charges.
- - - -+
-+
-- - - Electron clouds overlap ⇒ DENSITY of –ve charge in the outer edge
of the molecule varies rapidly (∝exp[-r/a]) ⇒ strong repulsion
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• van der Waals binding energies in the range 0.01 – 0.1 eV
• Coulomb energy is typical of the energy it takes to pull an
electron out of an atom ≈ 5- 10 eV
⇒ IONIC FORCES are much stronger than van der Waals
|VvdW(r)|<<|VCoulomb(r)|
a0
• VvdW(a0)=- ε
a0
a0
6
 a0  − ε
 ≈
VvdW (2a0 ) ≈ −2ε 
≈ −0.03ε
32
 2 a0 
6
 a0  − 2ε
 ≈ 6 ≈ −0.003ε
VvdW (3a0 ) ≈ −2ε 
3
 3a0 
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− e2
VCoulomb (a0 ) =
4πε 0 a0
⇒ VCoulomb (2a0 ) = 1 VCoulomb (a0 )
2
1
⇒ VCoulomb (10a0 ) = VCoulomb (a0 )
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Van der Waals forces are SHORT RANGED
Coulomb forces are LONG RANGED
•A neutral molecule will exert significant force only on its nearest
neighbour
• A charged ion can exert a strong force on another ion many
neighbours away (causes problems in calculations)
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METALLIC BONDING: another form of bonding by electron
rearrangement. There is no simple L-J type expression for metallic
bonding (some textbooks do use a 6-12 potential but it has no
fundamental meaning as is the case for inert gases for example)
Electron
gas
…………..
+ + + + + + + + + + …………..
Metals:
• One or more loosely bound electrons
• Electrons detach (delocalisation) to form ‘electron gas’
surrounding the +ve ion
• Metal crystal behaves like a single giant molecule
• Repulsion on compression and attraction on extension
• Metallic bonding strong, ~ 0.1 eV
A FULL EXPLANATION OF METALLIC BONDING REQUIRES
QUANTUM PHYSICS
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COVALENT FORCES: related to electrostatics and the internal
structure of the molecules, e.g. H2 molecule:
- -+
- H
- -+
- H
H2 molecule
Binding energy~ 3 eV
Energetically favourable structure:
a0
- - - -+ -+
-- - - -
The electron cloud redistributes
The extra electron cloud between the nuclei ‘SCREANS’
the strong repulsion creating a strong COVALENT force
A FULL EXPLANATION OF COVALENT BONDING REQUIRES
QUANTUM PHYSICS
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E.g. Carbon: covalently bonded material
C
IA
IIA
IIIB
IVB
VB
VI B
VII B
VIII B
H
4 moveable
outer electrons
He
Li Be
B
C
N
O
Na Mg
Al Si
P
S Cl Ar
6 electrons
Triple bond
C
C
Double bond
F Ne
180º
C
4 electrons
C
C
C
120º
2 electrons
Single bond
C
C
Interatomic Forces
C
109.5º
12
Carbon allotropes
Fullerenes
Diamond
Graphite
covalent
d=0.7 nm
vdW
Carbon nanotubes
Carbyne
≡C – C ≡ C – C ≡ C –
d~1.4 nm
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Polymers
C
C
C
C C
C
C
C
The COVENT BOND is:
• Due to electron re-arrangement
• Strong
• Highly directional
• Cannot be represented by any simple mathematical force expression
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HYDROGEN BOND: H atoms form single covalent bond to form
H2, but fully or partially ionised can form essentially ionic bonds with
e.g. O (H2O), N (NH3) or F (HF). Bond is part (about 90%)
electrostatic and part (about 10%)
The water molecule
0.7e
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Summary
LENNARD-JONES (6-12) POTENTIAL
Origin in fluctuation
VAN DER WAALS bonding
of electron cloud
• Between neutral atoms and molecules
• Strength ~ 0.01 eV – 0.1 eV
  σ 12  σ  6 
• Short range
V ( r ) = 4ε    −   
 r  
• E.g. Ar, Kr
  r 
IONIC bonding
Origin in electron transfer
• Between charged atoms
• Strength ~ 5 eV
λ
e2
• Long range
V (r ) = p −
r
4πε o r
• E.g. Na+, ClInteratomic Forces
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Summary contd.
COVALENT bonding
• Origin in electron rearrangement (quantum mechanical)
• Strength ~ 3 eV
• Short range
• Directional
METALLIC bonding
• Origin in electron rearrangement (quantum mechanical)
• Strength ~ 0.1 eV
• Short range
Next Topic: Interatomic forces contd
Interatomic Forces
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