### What is thermal diffusivity?

Thermal diffusivity, as the name indicates, is a measure of how quickly heat diffuses through a material, and is quantified in the units of length squared over time, usually in meter squared or millimeter squared per second (m^{2} or mm^{2}/s). An object with high thermal diffusivity will readily disperse any heat added to it, while a poor thermal diffuser will take longer to disperse the same heat throughout the material. Like thermal conductivity, a thermally diffusive material will reach thermal equilibrium quickly, whereas a less diffusive material will not.

It is easy to confuse the properties of thermal diffusivity and thermal conductivity. Thermal diffusivity determines temperature distribution within a material as time goes on, whereas thermal conductivity represents exactly how much heat is flowing through a material. The amount of heat required to increase the temperature of an object depends on heat capacity and density. A material with high diffusivity will have a low specific heat capacity and density. Consider solid nickel and air, which have similar thermal diffusivities, ~23 mm^{2}/s, but very different conductivities (~87 W/mK and ~0.025 W/mK, respectively). When nickel and air are in contact with a heat reservoir, it will take them the same amount of time to match the temperature of the heat supply. However, nickel will have pulled much more heat from the reservoir compared to air.

### Mathematical determination of thermal diffusivity

This is not to say that thermal conductivity and thermal diffusivity are independent quantities, which causes confusion. They are both important when predicting the transfer of thermal energy through a material, but thermal diffusivity, \(D\), is related to thermal conductivity by \( D = \frac{K}{\rho c} \). This means that a conductive material will be a good diffuser and an insulative material will be a poor diffuser, barring factors like comparing solids to gases, as in the previous example. Quantifying thermal diffusivity is important when calculating thermal stress after a rapid temperature change, when testing small samples, or for measurements taking place at temperatures higher than 1000 °C. When there is no internal heat generation within a system, but the temperature inside the object changes as time goes on (eg. before the system reaches equilibrium), thermal diffusivity can be found by solving the equation

#### \[ \nabla ^{2} T- \frac{1}{D} \frac{\partial T}{\partial t} = 0 \]

Where *T *is temperature, *D *is diffusivity, and *t *is time. It is important to note that once equilibrium is established within the system and and \( \frac{\partial T}{\partial t} \) becomes 0, thermal diffusivity is no longer relevant, and thermal conductivity becomes increasingly important. As such, thermal diffusivity must be measured indirectly by finding thermal conductivity and heat capacity, or directly using transient methods.

### Mechanisms of thermal diffusivity

Given the close relationship between thermal conductivity and thermal diffusivity, it is no surprise that the factors which influence conductivity also affect diffusivity. These factors are electron motion and atomic vibrations for solids, and energy transfer through collisions for liquids and gases. An extra factor must be considered when finding the thermal diffusivity of gases, because unlike thermal conductivity, diffusivity has an added dependence on density, \( D = \frac{K}{\rho c} \). For near-ideal gases at low pressure, this turns into a dependence on pressure, that conductivity does not share; \( D = \frac{KRT}{PcM} \).

### Measurement of thermal diffusivity

In liquid or gas systems, the coefficient of thermal diffusivity may not be entirely accurate for real world applications. The bulk transfer of mass, or convection, plays an important role in thermal transport, but is often not accounted for theoretically. This can be resolved by maintaining the system in a steady state with respect to its mass. Porous media, however, can pose an additional problem, because conduction may not be the dominant method of thermal energy transfer in these types of materials, leaving diffusivity to be replaced by effective diffusivity, which is a theoretical value that cannot be realistically measured.

The laser flash apparatus is the most effective method to measure thermal diffusivity over a wide range of temperatures. The laser flash analysis directly measures the thermal diffusivity and specific heat capacity of the samples, which then can be used to calculate thermal conductivity as well. Other transient methods, like the transient plane source and the transient hot wire, involve the direct measurement of thermal diffusivity, though they are less accurate than the modern flash apparatus. The transient plane source directly measures thermal diffusivity and thermal conductivity, which are then used to calculate specific heat capacity and thermal effusivity.

### Internationally recognized standards

Methods developed to test the thermal diffusivity comply to various international and nationally recognized standards, such as the ASTM standard E1461 and D7896, which relate to the measurement of thermal diffusivity with the laser flash method, and using the transient hot wire method to measure the thermal conductivity of engine coolants and other fluids, respectively. The ISO Standard 22007-4, is the standard test for the laser flash method when determining the thermal conductivity and diffusivity of plastics, and 22007-2:2015 details the measurement of thermal conductivity and diffusivity with the transient plane source (hot disc) method. As well, the DIN EN Standard 821-2, overviews the laser flash analysis and heat pulse methods for determining the thermal diffusivity of ceramics.

### Additional literature

As described above, it is important to consider thermal diffusivity along with thermal conductivity and effusivity. The quantification and comprehension of all these properties will lead to a complete understanding of transient systems. For further information relating to the theory of thermal diffusivity, as well as experimental measurements, the following pieces of literature provide excellent insight.

- Flash Method of Determining Thermal Diffusivity, Heat Capacity, and Thermal Conductivity. Authors: W.J. Parker, R.J. Jenkins, C.P. Butler, and G.L. Abbott (1961).
*Journal of Applied Physics*32:1679. - A Laser Flash Apparatus for Thermal Diffusivity and Specific Heat Capacity Measurements. Authors: K. Shinzato and T. Baba (2001).
*Journal of Thermal Analysis and Calorimetry*64(1): 413-422. - Transient Plane Source Techniques for Thermal Conductivity and Thermal Diffusivity Measurements of Solid Materials. Author: Silas E. Gustafsson (1990).
*Review of Scientific Instruments*62(3). - On Thermal Diffusivity. Author: Agustin Salazar (2003).
*European Journal of Physics*24(4).