Lecture 2: Forces and Potentials

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Lecture 2 Forces and Potentials
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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
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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
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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
This is an extremely important formula in physics
and we will use it a lot in this module!!
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The sign of the potential
When the potential energy is negative the
interaction between two bodies is
favourable Potential energy U lt 0
When the potential energy is positive,
interactions are unfavourable Potential energy
U gt 0
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The sign of the force
Positive forces are repulsive F gt 0
Negative forces are attractive
F lt 0
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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
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Electrostatic potential
The potential energy of two molecules having
charges q1 and q2 that are separated by a
distance, x, is given by
eo8.85 x 10-12 C2N-1m-1 erelative permittivity
Inverse power law dependence (x-n) is typical of
many different types of potential
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Electrostatic forces
The force acting between two point molecules q1
and q2 separated by a distance, x, is given by
The sign of the force depends upon the signs of
the charges If q1q2 lt 0 force is attractive
(negative sign) If q1q2 gt 0 force is repulsive
(positive sign)
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Problem 1

• Two protons (mass m1.67x10-27kg and charge
  q1.6x 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
• Calculate the closest distance of approach
  between the particles if v 14,850 ms-1
• Sketch the form of the potential energy of the
  particle as a function of the distance along the
  x axis

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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?
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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
When U(x) ? kT thermal motion disrupts the
interactions When U(x) gt kT the atoms/molecules
still feel the interactions between them
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Range of electrostatic interactions
We can obtain an estimate of the range Xrange of
the electrostatic interaction between two
atoms/molecules such that
Rearranging gives
In water (e80) this has a special name- Bjerrum
length
For x Xrange electrostatic interactions become
unimportant x lt Xrange electrostatic
interactions start to influence
atoms/molecules
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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 (e1), b) water (e80) and
  c) toluene (e2.38)

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Ionic Crystals
Electrostatic interactions are responsible for
the formation of ionic crystals For example,
consider NaCl
r
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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
Where M is called the Madelung constant (M1.748
for NaCl) e is the electronic charge
(1.6 x 10-19C)
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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 r0.276 nm.
• Use your answers to parts b) and c) to explain
  why ionic crystals dissolve in water, but not in
  organic solvents.

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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
See OHP
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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
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Properties of covalent bonds
Covalent forces (bonds) operate over short
distances 0.1-0.2 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
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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
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Properties of metallic bonds
Metallic bonds are comparable in strength to
covalent bonds (U100-300kT 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!
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Summary of key concepts
The force on an object is the gradient in
potential energy The range of an interaction is
the distance at which thermal effects start to
dominate
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