Thermal Conductivity · aiwiki.page

Thermal conductivity is a material property describing how readily heat is transferred by conduction in response to a temperature difference. Usually denoted by $k$ , $\lambda$ , or $\kappa$ , it is the proportionality coefficient between conductive heat flux and the negative temperature gradient. Its SI unit is the watt per metre per kelvin, $\mathrm{W\,m^{-1}\,K^{-1}}$ . For the same temperature gradient, a material with higher thermal conductivity carries a larger heat flux. (web.mit.edu)

Definition and Fourier’s law

For an isotropic material—one whose conductive properties are independent of direction—Fourier’s law takes the form

\mathbf q=-k\nabla T,

where $\mathbf q$ is heat flux, measured in $\mathrm{W\,m^{-2}}$ , and $\nabla T$ is the temperature gradient. The minus sign indicates that conductive heat flows toward lower temperature. This is a constitutive relation: it describes a material’s response rather than stating a conservation law. (web.mit.edu)

For steady, one-dimensional conduction through a flat slab of thickness $L$ , area $A$ , and constant conductivity,

\dot Q=kA\frac{T_{\mathrm{hot}}-T_{\mathrm{cold}}}{L},

where $\dot Q$ is the heat-transfer rate in watts. The expression assumes negligible lateral heat losses and no internal heat generation. It separates the material property $k$ from the specimen’s geometry: a thicker slab transfers less heat, whereas a larger area transfers more. (web.mit.edu)

Conductivity, resistance, and diffusivity

Thermal conductivity must be distinguished from several related quantities.

Thermal conductance describes an entire object or heat-flow path. For the slab above, its conductance is $G=kA/L$ , in $\mathrm{W\,K^{-1}}$ . Thermal resistance is the reciprocal:

R_{\mathrm{th}}=\frac{L}{kA}, \qquad \dot Q=\frac{\Delta T}{R_{\mathrm{th}}}.

In building physics, resistance is often expressed per unit area, as $R_A=L/k$ , in $\mathrm{m^2\,K\,W^{-1}}$ . Thermal resistivity, by contrast, is a material property equal to $1/k$ for scalar conductivity. These quantities are related but are not interchangeable. (tsapps.nist.gov)

Thermal diffusivity describes the rate at which temperature variations spread through a material. In the usual constant-property model,

\alpha=\frac{k}{\rho c_p},

where $\rho$ is density and $c_p$ is specific heat capacity. Its unit is $\mathrm{m^2\,s^{-1}}$ . High conductivity does not by itself imply rapid temperature equalization: the material’s capacity to store thermal energy also matters. (ahtt.mit.edu)

Combining conductive heat transfer with energy conservation gives the heat equation. For a stationary, homogeneous material with constant properties and no internal heat source,

\frac{\partial T}{\partial t}=\alpha\nabla^2T.

Conductivity therefore governs heat transport, while volumetric heat capacity determines how much the temperature changes as energy is gained or lost. (ahtt.mit.edu)

Microscopic mechanisms

In solids, thermal transport commonly has both electronic and vibrational contributions:

k\approx k_{\mathrm e}+k_{\mathrm{ph}}.

Mobile electrons often dominate heat conduction in good metals. In many insulating solids and semiconductors, the principal carriers are phonons, the quantized modes of atomic vibration. Carrier scattering limits transport; impurities, defects, boundaries, and interactions between excitations influence conductivity. (web.mit.edu)

A simplified kinetic estimate for an isotropic phonon system is

k_{\mathrm{ph}}\approx \frac13 Cv\ell,

where $C$ is the carriers’ heat capacity per unit volume, $v$ a representative speed, and $\ell$ their mean free path. Actual materials contain many vibrational modes with different velocities and scattering rates, so this expression is an approximation. (web.mit.edu)

For many metals, the Wiedemann–Franz law relates electronic thermal conductivity to electrical conductivity $\sigma$ :

\frac{k_{\mathrm e}}{\sigma T}\approx L_0, \qquad L_0=\frac{\pi^2}{3}\left(\frac{k_{\mathrm B}}{e}\right)^2.

Here $k_{\mathrm B}$ is the Boltzmann constant and $e$ the elementary charge. The relation depends on the applicable scattering regime and concerns the electronic contribution, not necessarily the total conductivity. (web.mit.edu)

Material values and temperature dependence

Thermal conductivity spans several orders of magnitude. The following approximate room-temperature values illustrate the contrast between familiar metals and nonmetals; they are not universal specifications for every grade or specimen. (web.mit.edu)

Material	Approximate conductivity $\mathrm{W\,m^{-1}\,K^{-1}}$
Silver	420
Copper	390
Aluminium	200
Iron	70
[[water	Liquid water]]
Wood	0.2
Cork	0.04
Air	0.026

The figures show, for example, why substituting a metal for an insulating material can greatly increase conductive heat transfer without changing geometry. Conductivity also varies with temperature, so calculations over a substantial temperature interval may require $k(T)$ rather than a single tabulated value. (web.mit.edu)

There is no universal temperature dependence. In crystalline solids, changing phonon populations and scattering rates can produce a conductivity maximum followed by a decrease. In metals, the balance between electronic and lattice transport and the relevant scattering mechanisms determines the trend. (web.mit.edu)

Directional and effective conductivity

In an anisotropic material, thermal conductivity is a second-rank tensor rather than a single scalar. Fourier’s law becomes

q_i=-\sum_j k_{ij}\frac{\partial T}{\partial x_j}.

Consequently, heat flux need not point exactly opposite the temperature gradient. Directional measurements are necessary when a crystal, film, or structured material conducts differently along different axes. (mit.edu)

For heterogeneous insulation and porous materials, measurements often report an effective or apparent thermal conductivity. This quantity characterizes the specimen as a whole rather than an individual constituent. In insulation tests, the measured heat transfer may include radiation and, under some conditions, convection as well as conduction. An apparent conductivity therefore depends on the measurement conditions and should not automatically be interpreted as a purely microscopic conduction coefficient. (tsapps.nist.gov)

Measurement methods

Conductivity measurements either establish a steady heat flow or infer transport properties from a time-dependent thermal response.

Guarded hot plate. A specimen is placed between controlled hot and cold surfaces. Guard heaters suppress lateral heat flow so that conductivity can be obtained from the heat-transfer rate, thickness, area, and temperature difference. This method is especially important for insulation. (nist.gov)
Transient hot wire. A thin electrically heated wire serves as a heat source and thermometer. Conductivity is inferred from its temperature rise over time, with corrections for wire geometry and the surrounding boundaries. The method is used for fluids, including gases and liquids. (tsapps.nist.gov)
Laser flash. A short pulse heats one face of a specimen, and the temperature response at the opposite face is recorded. The measurement primarily determines diffusivity; conductivity is then calculated using density and specific heat capacity. (arxiv.org)
Thermoreflectance. Optical heating and changes in reflected light provide a thermal response that is fitted to a transport model. Time-domain and frequency-domain techniques can investigate thin films, directional conductivity, and conductance across interfaces. (nist.gov)

A reported conductivity should identify the temperature, specimen characteristics, measurement method, and measurement uncertainty. Heat leakage, specimen dimensions, temperature measurement, and departure from the assumed heat-flow geometry can affect the result. (tsapps.nist.gov)

Applications and limits

Thermal conductivity is central to calculating heat leakage through insulation and to selecting materials for controlled heat transfer. Its measurement was an early priority in refrigeration and building-material research. Conductivity measurements also inform manufacturing processes in which the thermal response of solids and powders affects heating and cooling. (nist.gov)

In thermoelectric materials, reducing lattice heat conduction can help preserve the temperature difference needed for energy conversion. Nanostructuring is one approach: interfaces can scatter phonons and lower their contribution to conductivity. The resulting performance depends on electrical transport as well as thermal transport. (web.mit.edu)

The interpretation of conductivity as a local bulk property has limits. Fourier’s law assumes diffusive transport arising from sufficiently frequent internal scattering. When relevant dimensions become comparable to carrier mean free paths, boundaries and interfaces may strongly affect the response, and a single size-independent bulk conductivity may not describe the experiment. Nanoscale thermal transport can also exhibit coherent or wave-like behavior beyond the ordinary diffusion model. (web.mit.edu)

References

2 Introduction to Conductionweb.mit.edu
NIST Technical Note 1606tsapps.nist.gov
Solid State Physicsweb.mit.edu
NanoEngineering: Thermal Transportweb.mit.edu
43 Advanced Thermodynamics—Comprehensive Video Textbookmit.edu
Instrumentationnist.gov
Transient Thermal Response of a Guarded-Hot-Plate Apparatus for Operation Over an Extended Temperature Rangenist.gov
Transient Hot-Wire Thermal Conductivity Studytsapps.nist.gov
Rear-Surface Integral Method for Calculating Thermal Diffusivity from Laser Flash Experimentsarxiv.org
Thermal Property Measurement Methods and Analysis for AM Solids and Powdersnist.gov
Early Guarded-Hot-Plate Apparatusnist.gov