concept

thermal conductivity

Thermal conductivity is the property of a material to conduct heat, determining how efficiently thermal energy transfers through substances like thin films or superconductors.
49 related articles · 5 formulas · 2 who worked here

Description

Thermal conductivity (k) is the proportionality coefficient in Fourier's law: heat flux density q⃗ = –k ∇T, where ∇T is the temperature gradient. It characterizes the ability of a substance to transfer the kinetic energy of microscopic particles without macroscopic movement of the substance. In metals, heat transfer is mainly by conduction electrons; in dielectrics, it is by phonons (quanta of crystal lattice vibrations). Silver has the highest thermal conductivity among pure metals (k ≈ 430 W/(m·K)).

History

The law of thermal conduction was formulated by Joseph Fourier in 1822 in his 'Analytical Theory of Heat'. For over a century, thermal conductivity remained a phenomenological constant. The kinetic theory of gases (James Clerk Maxwell, Ludwig Boltzmann) made it possible to relate it to molecular motion. In the 20th century, the development of quantum solid-state theory (Peter Debye, Rudolf Peierls) led to the phonon picture and the prediction of lattice thermal resistance. At the end of the century, materials with extreme thermal conductivity (diamond, graphene, nanotubes) were discovered.

How it works

In solids, heat transfer occurs via two main carriers: conduction electrons and phonons. Phonons — collective lattice vibrations — behave like a gas of quasiparticles and obey Bose–Einstein statistics. The distribution of phonons over frequencies and wave vectors is described by dispersion relations depending on crystal symmetry. Thermal resistance arises because the phonon mean free path is finite, limited by anharmonicity of interatomic potentials (three-phonon processes), isotopic disorder, defects, and grain sizes. At high temperatures, scattering on other phonons (so-called Umklapp processes) dominates. In nanostructures, the mean free path can be limited by geometric dimensions, leading to reduced thermal conductivity (phonon ballistic regime).

In practice

Copper pipes in a radiator quickly release heat into the room, while a wool sweater retains body heat due to low thermal conductivity of air in fluffy fibers.

Fun fact

Diamond has a thermal conductivity five times higher than copper, although diamond is an electrical insulator. The reason is a rigid crystal lattice and light carbon atoms, along which phonons travel quickly.

Concept graph →

Formulas

$$k = \frac{1}{3} C v l$$
Thermal conductivity depends on heat capacity, particle speed, and mean free path.
What is in the formula
kvariablethermal conductivity — thermal conductivity watt per metre per kelvin
Cvariableheat capacity per unit volume — volumetric heat capacity joule per cubic metre per kelvin
vvariableaverage particle speed — speed metre per second
lvariablemean free path — mean free path metre
k=\frac{1}{3}cvl · 2409.14328
$$S_a^{1/2} = \sqrt{\frac{4 k_B T k}{m^2 \omega Q}}$$
Spectral density of thermal acceleration noise depends on temperature and resonator properties.
What is in the formula
S_a^{1/2}variableSpectral density of acceleration noise (square root) — acceleration noise spectral density metre per second squared per root hertz
TvariableTemperature — temperature kelvin
kvariableSpring constant of the resonator — spring constant newton per metre
mvariableMass of the resonator — mass kilogram
ωvariableAngular resonance frequency — angular frequency radian per second
QvariableQuality factor of the resonator — quality factor
k_BconstantBoltzmann constant, relates thermal energy to temperature — boltzmann constant = 1.380649e-23 joule per kelvin
s^{1/2}_{a,th}=\sqrt{\frac{4k_btk_q}{m^2ω_{n,q}q_q}} · 2606.14738v1
$$\sigma = \sigma_0 \exp\left(-\frac{E_a}{k_B T}\right)$$
Conductivity depends exponentially on temperature and activation energy.
What is in the formula
\sigmavariableelectrical conductivity — electrical conductivity siemens per metre
\sigma_0variablepre-exponential factor (conductivity at infinite temperature) — electrical conductivity siemens per metre
E_avariableactivation energy for conduction — energy joule
Tvariableabsolute temperature — temperature kelvin
k_BconstantRelates temperature to energy in statistical mechanics. — boltzmann constant = 1.380649e-23 joule per kelvin
σ=σ_0\exp(-\frac{e_a}{k_bt}) · 2606.24480
p\propto\exp(-\frac{δg}{k_bt}) · 2606.24480
δf^*=k_bt\ln(τ/τ_0) · 2606.29055
k_s\approx\frac{δe_{{pv}}}{h}\exp(-\frac{g_{{rac}}}{rt}) · 2607.12813
$$\sigma_{\mathrm{total}} = \sigma_{\mathrm{cond}} + \sigma_{\mathrm{fluct}}$$
Total conductivity equals sum of coherent and fluctuating parts.
What is in the formula
σ_totalvariabletotal conductivity — electrical conductivity siemens per metre
σ_condvariableconductivity from condensed (coherent) carriers — electrical conductivity siemens per metre
σ_fluctvariableconductivity from fluctuating (incoherent) carriers — electrical conductivity siemens per metre
g_{\uparrow\downarrow}(t)=g_{{cond}}(t)+g_{{fluct}}(t) · 2606.28877
$$G = \frac{e^2}{h} \sum T(E)$$
Conductance is quantum of conductance times transmission sum.
What is in the formula
GvariableElectrical conductance of a mesoscopic conductor — conductance siemens
T(E)variableTransmission probability for an electron at energy E — transmission probability
EvariableEnergy of the electron — energy joule
econstantMagnitude of the charge of an electron — elementary charge = 1.602176634e-19 coulomb
hconstantPlanck constant, fundamental quantum of action — planck constant = 6.62607015e-34 joule second
\sumoperatorSum over all transmission channels or energy levels — summation
g^{σ}=\frac{e^2}{h}\frac{1}{n}\sum_{{k}}t^{σ}_{{k}}(e) · 2607.03908

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