ENSC1004 - Lecture 6

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ENSC1004 - Lecture 5
Bonding & Properties of Materials - Slides

Interatomic Bonding Force

• Coulombic Attraction: the attraction force ($F_A$) between two oppositely charged species follows the relationship: $F_A=-A\frac{Q_1{Q}_2}{r^2}$
  • $Q_1,Q_2$ $-$ the charge of species 1 and 2, respectively
  • $r$ $-$ separation of the charge centres
  • $A$ $-$ a proportionality constant
• You have seen this equation in Y11+12 Physics!
  • However, in this context, we are applying it to charged attraction between ions at the sub-atomic level

There is always a repulsive force ($F_R$) from the overlapping of electron clouds and two positively charged nuclei when two charged species come too close. $F_R$ increases exponentially as $r$ decreases, which prevents the species from getting too close.

$F_R=-\frac{nB}{r^{n+1}}$

Our net force is therefore given by: $F=-A\frac{Q_1{Q}_2}{r^2}-\frac{nB}{r^{n+1}}$

• When $F$ is equal to $0$, (i.e. $F_A+F_R=0$), the equilibrium bond length $r_0$ occurs
  • Attraction and repulsion are perfectly balanced

Bonding Force and Equilibrium Separation

• Atoms will stay at their equilibrium separation ($r=r_0$)
  • Pulling atoms apart → resisted by attractive force $F_A$
  • Compressing atoms → resisted by repulsive force $F_R$
• Here is an $F-r$ curve:
  
  • The gradient $S$ represents the bond stiffness
    • Different materials have different bond stiffness
  • This is the origin of the elastic modulus of materials, $E\propto S$
• While $S$ is a function of $\Delta r=r-r_0$, in practice, $\Delta r$ is very small for elastic deformation and thus $S$ is regarded as a constant

The maximum force ($F_{max}$) a bond can endure defines the theoretical strength of the material. The stronger the bonding, the higher the maximum force $F_{max}$ required to break the bond. Thus, the stronger the bonding the higher the theoretical strength of the material.

Bonding Energy

• The need of a force to pull/compress atoms implies work done
  • Work is to counteract the bonding potential energy
• This potential energy ($U$) is expressed as: $U=A\frac{Q_1{Q}_2}{r}+\frac{B}{r^n}$
  

Thermal Expansion

• A solid expands on heating and contracts on cooling
• Coefficient of thermal expansion gives the rate of expansion/contraction
  • This is a material property

Given by:
${\alpha }_l=\frac{{ϵ}_T}{\Delta T}$

• ${ϵ}_T=\frac{\Delta l}{l}$

A higher value for ${\alpha }_l$ will mean it is easier for a material to expand when heat is applied to it

At temperatures above $0\text{K}$, atoms are energised above minimum energy state and vibrate about equilibrium position

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ENSC1004 - Lecture 7