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

kTU thermal

In this lecture…

Chapter 3

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

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

dx

dUxF )(

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 ForcesYou 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

r

+6e+6e

Electrostatic potential

The potential energy of two molecules having charges q1 and q2 that are separated by a distance, x, is given by

x

qqxU

o4)( 21

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

221

4)(

x

qq

dx

dUxF

o

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 < 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?

We can estimate the range of an interaction by comparing the thermal energy to the size (magnitude) of the potential energy of two molecules

Small and large ranges are nebulous terms and depend upon the nature and strength of the interaction between two atoms/molecules

kTxU )(

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

kT

qqX

orange 4

21

kTX

qqxU

rangeo

4

)( 21

We can obtain an estimate of the range Xrange of the electrostatic interaction between two atoms/molecules such that

Rearranging gives

For x ≥ Xrange electrostatic interactions become unimportant

x < Xrange electrostatic interactions start to influence atoms/molecules

In water (=80) this has a special name- ‘Bjerrum length’

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

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

r

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

r

Me

r

eU

o

neighbournearest

nextnext

neighbournearestnextneighbour

nearestocoh 4

...3

8

2

126

4

22

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

a

ezU

oBorn 8

22

See OHP

Strong inter- atomic and intermolecular interactions II:

Covalent InteractionsCovalent 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 0.1-0.2 nm

The energies associated with the formation of these bonds are typically 100 – 300 kT per bond

You will learn more about these in the 3rd year Solid State module

p32

Strong inter- atomic and intermolecular interactions III:

Metallic bondingSharing 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

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!

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

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

Summary of key concepts

dx

dUxF )(

kTxU range )(

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 kTIonic U ~ kT

Metallic U ~100-300 kT