# Introduction to conduction

## Electrical conduction

It is important to not get confused by conduction, conductivity, resistance, and resistivity.

The materials properties are electrical conductivity, σ , and electrical resistivity, ρ

The electrical conductivity of a material is defined as the amount of electric charge transferred per unit time across unit area under the action of a unit potential gradient: J = σ E

where J is the current density (current per unit area) and E is the potential gradient. This is another way of expressing Ohm’s law, which is more commonly stated as \( V = I R \).

For an isotropic material:

\[ \sigma = \frac 1 \rho \]

The units of electrical resistivity are the ohm metre (**Ωm**), and for conductivity, the inverse (**Ω ^{-1} m^{-1} **). For an actual sample of length l, and cross sectional area A, the resistance, R, is calculated by :

\[ R = \rho \frac l A \]

Electrical signals propagate at close to the speed of light, though this does **not** mean the electrons themselves move this quickly. Instead, the typical electron *drift velocity* (their average velocity) is much lower: less than 1 mm s^{-1}. This is expanded upon in the Drude model section.

Another pertinent reminder is that of potential and current – current is the flow of electrons, and potential is the driving force that makes them flow. With sufficient potential, electrons may carry charge through any material, including a vacuum (see CRT), though they are powerless without any net current flow.

The best electrical conductors (apart from superconductors) are pure copper and pure silver, with resistivities of 16.78 and 15.87 nΩm respectively. For comparison, polystyrene has a resistivity of up to 10^{28} nΩm, 27 orders of magnitude different!

## Thermal conduction:

To understand thermal conductivity in materials, it is important to be familiar with the concept of heat transfer, which is the movement of thermal energy from a hotter to a colder body. It occurs in several circumstances:

- When an object is at a different temperature from its surroundings;
- When an object is at a different temperature to another object in contact with it;
- When a temperature gradient exists within the object.

The direction of heat transfer is set by the second law of thermodynamics, which states that the entropy of an isolated system which is not in thermal equilibrium will tend to increase over time, approaching a maximum value at equilibrium. This means heat transfer always occurs from a body at a higher temperature to a body at a lower temperature, and will continue until thermal equilibrium is reached.

A transfer of thermal energy occurs only through 3 modes: conduction, convection, and radiation. Each mode has a different mechanism and rate of heat transfer, and thus, in any particular situation, the rate of heat transfer depends on how much a certain mode is prevalent.

**Conduction** involves the transfer of thermal energy by a combination of diffusion of electrons and phonon vibrations – applicable to solids.

**Convection** involves the transfer of thermal energy in a moving medium – the hot gas/liquid moves through the cooler medium(normally due to density differences).

**Radiation** involves the transfer of thermal energy by electromagnetic radiation. The sun is a good example of energy transfer through a (near) vacuum.

This TLP focuses on conduction in crystalline solids.

Thermal conductivity, **Κ,** is the materials property that indicates the ability to conduct heat. Fourier’s first law gives the heat flux as proportional to the temperature difference, surface area, and length of the sample:

\[ H = \frac{\Delta Q}{\Delta t} = \kappa A\frac {\Delta T}{l}\]

where ΔQ / Δt is the rate of heat transfer, A is the surface area and l is the length.

The best metallic thermal conductors are pure copper and silver. At room temperature, commercially pure copper typically has a conductivity of about 360 Wm^{-1}K^{-1} (although the thermal conductivity of a single crystal of copper was measured at 12,200 Wm^{-1}K^{-1} at a temperature of 20.8 K). In metals, the movement of electrons dominates the conduction of heat.

The bulk material with the highest thermal conductivity (aside from the superfluid helium II) is, perhaps surprisingly, a non-metal: pure single crystal diamond, which has a thermal conductivity at room temperature of around 2200 Wm^{-1}K^{-1}. The high conductivity is even used to test the authenticity of a diamond. Strong covalent bonds within the molecule are responsible for the high conductivity even though there are no free electrons, heat is conducted by phonons. Most natural diamonds also contain boron atoms that replace carbon atoms in the crystal matrix, which also have high thermal conductance.