### What is thermal conductivity?

Thermal conductivity is one of three methods of heat transfer; the others being convection and radiation. Thermal conductivity is the ability of a material to transport heat from areas of high temperature, and high molecular energy, to areas of low temperature and low molecular energy, and is quantified in the units of watts per meter-Kelvin (W/mK). A material with a high thermal conductivity can quickly transfer heat through a medium. Alternatively, a material with a low thermal conductivity transfers heat poorly. Materials are grouped into one of three categories when discussing thermal conductivity: gases, non-metallic solids, and metallic solids. Out of the three groups, metallic solids have the highest thermal conductivities.

Thermal conductivity can be thought of in the same terms as electrical conductivity, and in fact, the two are proportional to each other. A good thermal conductor will generally be an effective electrical conductor, and vice versa. Both properties measure the ease at which a material can transport energy. Just as there are electrical insulators, thermal insulators are materials with low thermal conductivities.

The coefficient of thermal conductivity is the value measured to define thermal conductivity in materials. Heat conduction is defined by Fourier’s law,  $$q = -K \frac{dT}{dx}$$ where $$q$$ is the heat flux at a point, $${K}$$ is the coefficient of thermal conductivity at that point, which is dependent on the material being tested, and $$\frac{dT}{dx}$$ is the change in temperature along a line perpendicular to the isothermal surface of the point.

Depending on the uniformity of an object, thermal conductivity can vary from point to point, such as in heterogeneous materials. Thermal conductivity also changes with temperature, though this change is usually miniscule for small temperature variations, and it also depends on the structure of a material, and how far heat must travel within the sample.

### Mathematical determination of thermal conductivity

Heat flow is governed by

$\nabla T- \frac{1}{\kappa} \frac{\partial T}{\partial t} = -\frac{A(x,y,z,t)}{K}$

where $$\kappa = K / \rho C$$ is the density, and $${A}$$ represents any heat put into the object. Solving this equation is necessary to find heat flow through a system, which leads to $$K$$, the thermal conductivity. It is assumed in this equation that $$K$$ is constant in position and time, which is always true for isotropic materials over small intervals. In the case where no heat is added, or the case of steady flow, the formula reduces to Laplace’s equation, $$\nabla T = 0$$.

### Mechanisms of thermal conduction

In solids, thermal energy can take the form of microscopic vibrations, or electron dissociation and movement. Each form can be described with its own thermal conductivity coefficient, and the entire material’s coefficient of thermal conductivity is simply the sum of the two forms. However, the two forms do not always work in concert. For example, raising the temperature of the sample will result in an increase in lattice vibrations and free electrons, but the larger vibrations may inhibit electron flow, depending on the structure of the material. Alternatively, in fluids and gases, thermal conduction happens through particle collisions.

### Measurement of thermal conductivity

Thermal conductivity is the most commonly measured thermal property, over thermal diffusivity and thermal effusivity. Thermal conductivity is a useful quantity when it is necessary to know the amount of heat that flows in a material, whereas thermal diffusivity determines how quickly heat flows within a material.

The measurement of thermal conductivity is easiest in isotropic, steady-state systems. Testing the thermal conductivity of anisotropic materials is slightly more difficult because heat moves more easily in either the axial or radial direction due to the arrangement of the structure. Simple apparatuses, such as Lee’s Disc, are based on measuring the amount of heat flow through a system of relatively low thermal conductivity, such as glass, or polymers. With few calculations, the thermal conductivity values can be easily found. Many modern pieces of equipment rely on simplistic methods, similar to Lee’s Disc.

Thermal conductivity measurements can also be performed using non-steady state systems, though measurements are usually made indirectly in these cases. Many of these common non-steady-state methods primarily measure thermal diffusivity, then use known relations to convert this property to a measurement of thermal conductivity. The laser flash method is an example of this. Other transient methods, like the transient plane source and transient hot wire, directly measure thermal conductivity.

### Internationally recognized standards

Commercialized methods that are used to test thermal conductivity must conform to national and international standards set in place by ASTM and ISO. These standards specify certain procedures and requirements during the production of an instrument, or method, to ensure methods are constructed safely and will develop reproducible results. Standardized methods and instruments are recognized globally and are therefore essential in facilitating international trade.

Many transient methods have standards set in place to accurately measure thermal conductivity, such as the ISO Standard 22007-4 and ISO 22007-2:2015, and the ASTM Standard D7896, ASTM D5334, and ASTM D5930. The ISO standards encompass the laser flash method and the transient plane source (hot disc) method for determining the thermal conductivity of plastics. As well, the previously mentioned ASTM standards detail the requirements when measuring the thermal conductivities of various materials by transient hot wire, transient-line source, and thermal needle probe procedures.

Although the techniques used to measure thermal conductivity have significantly evolved and improved over the years, the basic theories have remained the same. The following pieces of literature detail the fundamental principles for measuring the thermal conductivity of various materials.

1. Conduction of Heat in Solids. Authors: H.S. Carslaw and J.C. Jaeger (1959).
2. Basic Heat Transfer. Authors: Frank Kreith and William Zachary Black (1980).
3. Thermal Conductivity Volume 1. Author: Ronald P. Tye (1969).
4. Thermal Conductivity Volume 2. Author: Ronald P. Tye (1969).
5. Thermal Conductivity of Solids. Authors: John Edwin Parrott and Audrey D. Stuckes (1975).
6. Thermal Conductivity of Liquids. Authors: J.R. Woolf and W.L. Sibbitt (1954). Industrial and Engineering Chemistry 46(9).
7. Thermal Conductivity of Gases and Liquids. Authors: N.V. Tsederberg (1965).
8. Thermal Conductivity: Theory, Properties, and Applications. Author: Terry M. Tritt (2004).
9. Thermal Conductivity of Selected Materials. Authors: R.W. Powell, C.Y. Ho, and P.E. Liley (1966). National Standard Reference Data Series- National Bureau of Standards- 8 (Category 5-Thermodynamic and Transport Properties).
10. Thermal Conductivity of Selected Materials Part 2. Authors: C.Y. Ho, R.W. Powell, and P.E. Liley (1968). National Standard Reference Data Series- National Bureau of Standards- 16 (Category 5- Thermodynamic and Transport Properties).
11. Measurements of Thermal Conductivity and Related Properties: A Half Century of Radical Change. Author: Ronald P. Tye (1991). High Temperatures-High Pressures 23: 1-11.