# Thermal Conductivity

## What is Thermal Conductivity?

Thermal conductivity refers to the ability of a given material to conduct/transfer heat. It is generally denoted by the symbol ‘k’ but can also be denoted by ‘λ’ and ‘κ’. The reciprocal of this quantity is known as thermal resistivity. Materials with high thermal conductivity are used in heat sinks whereas materials with low values of λ are used as thermal insulators.

Fourier’s law of thermal conduction (also known as the law of heat conduction) states that the rate at which heat is transferred through a material is proportional to the negative of the temperature gradient and is also proportional to the area through which the heat flows. The differential form of this law can be expressed through the following equation:

q = -k.∇T

Where ∇T refers to the temperature gradient, q denotes the thermal flux or heat flux, and k refers to the thermal conductivity of the material in question.

An illustration describing the thermal conductivity of a material in terms of the flow of heat through it is provided above. In this example, Temperature1 is greater than Temperature2. Therefore, the thermal conductivity can be obtained via the following equation:

Heat Flux = -k * (Temperature2 – Temperature1)/Thickness

### Formula

Every substance has its own capacity to conduct heat. The thermal conductivity of a material is described by the following formula:

K = (QL)/(AΔT)

Where,

• K is the thermal conductivity in W/m.K
• Q is the amount of heat transferred through the material in Joules/second or Watts
• L is the distance between the two isothermal planes
• A is the area of the surface in square meters
• ΔT is the difference in temperature in Kelvin

## Measurement

There exist several methods of measuring the thermal conductivities of materials. These methods are broadly classified into two types of techniques – transient and steady-state techniques.

### SI Unit

• Thermal conductivity is expressed in terms of the following dimensions: Temperature, Length, Mass, and Time.
• The SI unit of this quantity is watts per meter-Kelvin or Wm-1K-1.
• It is generally expressed in terms of power/(length * temperature).
• These units describe the rate of conduction of heat through a material of unit thickness and for each Kelvin of temperature difference.

• These methods involve measurements where the temperature of the material in question does not change over a period of time.
• An advantage of these techniques is that the analysis is relatively straightforward since the temperature is constant.
• An important disadvantage of steady-state techniques is that they generally require a very well-engineered setup to perform the experiments.
• Examples of these techniques are the Searle’s bar method for measuring the thermal conductivity of a good conductor and Lee’s disc method.

### Transient Techniques

• In these methods, the measurements are taken during the heating-up process.
• An important advantage of these methods is that the measurements can be taken relatively fast.
• One of the disadvantages of transient techniques is the difficulty in mathematically analyzing the data from the measurements.
• Some examples of these techniques include the transient plane source method, the transient line source method, and the laser flash method.

Thus, there exist various methods of measuring the thermal conductivity of materials, each with their own advantages and disadvantages. It is important to note that it is easier to experimentally study the thermal properties of solids when compared to fluids.

## Effect of Temperature on Thermal Conductivity

Temperature affects the thermal conductivities of metals and non-metals in a different manner.

### Metals

• The heat conductivity of metals is attributed to the presence of free electrons. It is somewhat proportional to the product of the absolute temperature and the electrical conductivity, as per the Wiedemann-Franz law.
• With an increase in temperature, the electrical conductivity of a pure metal decreases.
• This implies that the thermal conductivity of the pure metal shows little variance with an increase in temperature. However, a sharp decrease is observed when temperatures approach 0K.
• Alloys of metals do not show significant changes in electrical conductivity when the temperature is increased, implying that their heat conductivities increase with the increase in temperature.
• The peak value of heat conductivity in many pure metals can be found at temperatures ranging from 2K to 10K.

### Non-Metals

• The thermal conductivities of nonmetals are primarily attributed to lattice vibrations.
• The mean free path of the phonons does not reduce significantly when the temperatures are high, implying that the thermal conductivity of nonmetals does not show significant change at higher temperatures.
• When the temperature is decreased to a point below the Debye temperature, the heat conductivity of a nonmetal decreases along with its heat capacity.

## Other Factors that Affect Thermal Conductivity

Temperature is not the only factor which causes a variance in thermal conductivity of a material. Some other important factors that influence the heat conductivity of substances are tabulated below.

 Factor Effect on Thermal Conductivity The chemical phase of the material When the phase of a material changes, an abrupt change in its heat conductivity may arise. For example, the thermal conductivity of ice changes from 2.18 Wm-1K-1 to 0.56 Wm-1K-1 when it melts into a liquid phase Thermal Anisotropy The differences in the coupling of phonons along a specific crystal axis causes some substances to exhibit different values of thermal conductivity along different crystal axes. The presence of thermal anisotropy implies that the direction in which the heat flows may not be the same as the temperature gradients direction. The electrical conductivity of the material The Wiedemann-Franz law that provides a relation between electrical conductivity and thermal conductivity is only applicable to metals. The heat conductivity of nonmetals is relatively unaffected by their electrical conductivities. Influence of magnetic fields The change in the thermal conductivity of a conductor when it is placed in a magnetic field is described by the Maggi-Righi-Leduc effect. The development of an orthogonal temperature gradient is observed when magnetic fields are applied. Isotopic purity of the crystal The effect of isotopic purity on heat conductivity can be observed in the following example: the thermal conductivity of type IIa diamond (98.9% concentration of carbon-12 isotope) is 10000 Wm-1K-1 whereas that of 99.9% enriched diamond is 41,000 Wm-1K-1