Lecture 2: Forces and
Potentials
What did we cover in the last
lecture?
Microscopic and nanoscale forces
are important in a number of
areas of nanoscience,
nanotechnology and biology
These forces start to become
important on the sub micron
length scale
The relevant energy scale on
these small length scales is the
thermal energy scale
U thermal  kT
In this lecture…
1) The relation ship between force and
potential
2) Recap of electrostatic interactions
3) Ionic interactions
4) The range of interactions and solubility of
ions
5) Covalent interactions and bonding
6) Metallic bonding
Chapter 3
Relationship between force and
potential
The force, F, on an object can be related to the gradient in its
potential energy, U, with respect to distance, x, by the
formula
dU
F ( x)  
dx
This is an extremely important formula in physics and we
will use it a lot in this module!!
The ‘sign’ of the potential
When the potential energy is
negative the interaction between
two bodies is favourable
Potential energy U < 0
When the potential energy is
positive, interactions are
unfavourable
Potential energy U > 0
The sign of the force
Positive forces are repulsive
|F| > 0
Negative forces are attractive
|F| < 0
Strong inter- atomic and
intermolecular interactions I:
Coulomb Forces
You have already met Coulomb interactions in the first year
module ‘Newton to Einstein’
Some atoms and molecules may acquire a net charge and
will interact via electrostatic forces
+6e
+6e
r
Electrostatic potential
The potential energy of two molecules having charges q1
and q2 that are separated by a distance, x, is given by
U ( x) 
q1q2
4o x
o=8.85 x 10-12 C2N-1m-1
=relative permittivity
Inverse power law dependence (x-n) is typical of many
different types of potential
Electrostatic forces
The force acting between two point molecules q1 and q2
separated by a distance, x, is given by
dU
q1q2
F ( x)  

2
dx 4o x
The sign of the force depends upon the signs of the
charges
If q1q2 < 0 force is attractive (negative sign)
If q1q2 > 0 force is repulsive (positive sign)
Problem 1
Two protons (mass m=1.67x10-27kg and charge q=+1.6x
10-19C) are fixed in place at the coordinates (0, 1) nm and
(0, -1) nm. A third proton is fired in the negative x direction
from the position (10, 0) nm with a speed v
a) Calculate the closest distance of approach between the
particles if v =14,850 ms-1
b) Sketch the form of the potential energy of the particle as
a function of the distance along the x axis
The range of an interaction
Thermal motion of atoms and molecules tends to disrupt
the interactions between them
At ‘small’ distances
interactions between atoms
and molecules are strong
enough to overcome the
effects of thermal motion
At ‘large’ distances
interactions between atoms
and molecules become too
weak to overcome the effects
of thermal motion
What do we mean by small and large, strong
and weak? How do we define the range?
Can we be more quantitative?
Small and large ranges are nebulous terms and depend upon
the nature and strength of the interaction between two
atoms/molecules
We can estimate the range of
an interaction by comparing
the thermal energy to the size
(magnitude) of the potential
energy of two molecules
U ( x)  kT
When |U(x)|  kT thermal motion disrupts the interactions
When |U(x)| > kT the atoms/molecules still feel the
interactions between them
Range of electrostatic interactions
We can obtain an estimate of the range Xrange of the
electrostatic interaction between two atoms/molecules such
that
q1q2
U ( x) 
 kT
4o X range
Rearranging gives
q1q2
X range 
4o kT
In water (=80) this
has a special name‘Bjerrum length’
For x ≥ Xrange electrostatic interactions become unimportant
x < Xrange electrostatic interactions start to influence
atoms/molecules
Problem 2
Calculate the range of the
electrostatic interaction (at room
temperature) between two positively
charged ions having a charge of 1.6 x
10-19 C in a) vacuum (=1), b) water
(=80) and c) toluene (=2.38)
Ionic Crystals
Electrostatic interactions are responsible for the formation
of ionic crystals
For example, consider NaCl
r
Each Cl- has
6 nearest neighbour Na+ ions at a distance r
12 next-nearest neighbour Cl- ions at √2r
8 next-next-nearest neighbour Na+ ions at √3r
Cohesive energy of ionic crystals
We can calculate the (cohesive) energy per ion holding the
crystal together by summing up the contributions to the
potential energy from all neighbours
U coh






2
2
e
12
8
Me
 6 


 ...   


4o r  
4o r
2
3


nearest
 neighbour next

next  next
nearest
nearest


neighbour
neighbour


Where M is called the Madelung constant (M=1.748 for NaCl)
e is the electronic charge (1.6 x 10-19C)
Problem 3
Calculate the cohesive energy of a NaCl crystal in a)
vacuum, b) water and c) toluene if the nearest
neighbour spacing between Na+ and Cl- ions is r=0.276
nm.
Use your answers to parts b) and c) to explain why
ionic crystals dissolve in water, but not in organic
solvents.
The Born (or ‘self’) energy
When considering the solubility of ions and charged
particles in different media, it is important to consider how
much energy is associated with placing the ion/particle in
the medium (even when it is not interacting with other
particles)
This energy is the energy required to bring the constituents
of the ion/particle from infinity and construct an ion/particle
of radius a. For an ion/particle with a charge of +ze this
energy is given by
U Born 
2 2
z e
8o a
See OHP
Strong inter- atomic and
intermolecular interactions II:
Covalent Interactions
Covalent bonds are highly directional
and are chemical in origin (e.g. pi bond
created by overlap of d electron
orbitals)
These interactions originate from the
sharing of valence electrons to
facilitate the filling of electronic shells
within atoms → stable structures
The directionality of the bonds is caused by the mutual repulsion of the
electrons in different bond orbitals
Properties of covalent bonds
Covalent forces (bonds) operate over short distances 0.10.2 nm
The energies associated with the formation of these bonds
are typically 100 – 300 kT per bond
p32
You will learn more about these in the 3rd year Solid State
module
Strong inter- atomic and
intermolecular interactions III:
Metallic bonding
Sharing of electrons between metal atoms gives rise to another form of
bonding
However, in metallic bonds, the electrons become delocalised
throughout the material in such a way that they are shared between
many nuclei
The ‘sea’ of delocalised electrons helps to screen the
repulsion between neighbouring nuclei
Properties of metallic bonds
Metallic bonds are comparable in strength to covalent bonds
(U=100-300kT per bond)
The delocalisation of electrons can be used to explain many/all of
the properties of metals
Optical – The optical properties of metals are
determined by how the ‘sea’ of electrons
responds to the oscillating electric and magnetic
fields associated with incident light
Electrical and Thermal – Delocalised electrons
are easier to move, so metals are good
conductors of heat and electricity
Again you will learn more about these in future
modules!
Summary of key concepts
The force on an object is the gradient in
potential energy
dU
F ( x)  
dx
The range of an interaction is the
distance at which thermal effects start
to dominate
U ( xrange )  kT
Electrostatics can be used to explain the interactions in ionic solids.
The cohesive energy and Born energy of ions/particles can be used to
explain why they are more soluble in water than other solvents
The energies associated with the formation/breaking of strong
covalent, ionic and metallic bonds are
Covalent U ~100-300 kT
Ionic
U ~ kT
Metallic U ~100-300 kT