The thermal conductivity of a material is a measure of its ability to heat conduction. It is commonly denoted by $k$, $\backslash lambda$, or $\backslash kappa$.
Heat transfer occurs at a lower rate in materials of low thermal conductivity than in materials of high thermal conductivity. For instance, metals typically have high thermal conductivity and are very efficient at conducting heat, while the opposite is true for insulating materials like Styrofoam. Correspondingly, materials of high thermal conductivity are widely used in heat sink applications, and materials of low thermal conductivity are used as thermal insulation. The reciprocal of thermal conductivity is called thermal resistivity.
The defining equation for thermal conductivity is $\backslash mathbf\{q\}\; =\; \; k\; \backslash nabla\; T$, where $\backslash mathbf\{q\}$ is the heat flux, $k$ is the thermal conductivity, and $\backslash nabla\; T$ is the temperature gradient. This is known as Fourier's Law for heat conduction. Although commonly expressed as a scalar, the most general form of thermal conductivity is a secondrank tensor. However, the tensorial description only becomes necessary in materials which are anisotropic.
According to the second law of thermodynamics, heat will flow from the hot environment to the cold one in an attempt to equalize the temperature difference. This is quantified in terms of a heat flux $q$, which gives the rate, per unit area, at which heat flows in a given direction (in this case the xdirection). In many materials, $q$ is observed to be directly proportional to the temperature difference and inversely proportional to the separation:
The constant of proportionality $k$ is the thermal conductivity; it is a physical property of the material. In the present scenario, since $T\_2\; >\; T\_1$ heat flows in the minus xdirection and $q$ is negative, which in turn means that $k>0$. In general, $k$ is always defined to be positive. The same definition of $k$ can also be extended to gases and liquids, provided other modes of energy transport, such as convection and radiation, are eliminated.
For simplicity, we have assumed here that the $k$ does not vary significantly as temperature is varied from $T\_1$ to $T\_2$. Cases in which the temperature variation of $k$ is nonnegligible must be addressed using the more general definition of $k$ discussed below.
Energy flow due to thermal conduction is classified as heat and is quantified by the vector $\backslash mathbf\{q\}(\backslash mathbf\{r\},\; t)$, which gives the heat flux at position $\backslash mathbf\{r\}$ and time $t$. According to the second law of thermodynamics, heat flows from high to low temperature. Hence, it is reasonable to postulate that $\backslash mathbf\{q\}(\backslash mathbf\{r\},\; t)$ is proportional to the gradient of the temperature field $T(\backslash mathbf\{r\},\; t)$, i.e.
where the constant of proportionality, $k\; >\; 0$, is the thermal conductivity. This is called Fourier's law of heat conduction. In actuality, it is not a law but a definition of thermal conductivity in terms of the independent physical quantities $\backslash mathbf\{q\}(\backslash mathbf\{r\},\; t)$ and $T(\backslash mathbf\{r\},\; t)$.Bird, Stewart, and Lightfoot pp. 266267 As such, its usefulness depends on the ability to determine $k$ for a given material under given conditions. The constant $k$ itself usually depends on $T(\backslash mathbf\{r\},\; t)$ and thereby implicitly on space and time. An explicit space and time dependence could also occur if the material is inhomogeneous or changing with time.
In some solids, thermal conduction is anisotropic, i.e. the heat flux is not always parallel to the temperature gradient. To account for such behavior, a tensorial form of Fourier's law must be used:
where $\backslash boldsymbol\{\backslash kappa\}$ is symmetric, secondrank tensor called the thermal conductivity tensor.Bird, Stewart, & Lightfoot, p. 267
An implicit assumption in the above description is the presence of local thermodynamic equilibrium, which allows one to define a temperature field $T(\backslash mathbf\{r\},\; t)$.
For instance, thermal conductance is defined as the quantity of heat that passes in unit time through a plate of particular area and thickness when its opposite faces differ in temperature by one kelvin. For a plate of thermal conductivity $k$, area $A$ and thickness $L$, the conductance is $kA/L$, measured in W⋅K^{−1}.Bejan, p. 34 The relationship between thermal conductivity and conductance is analogous to the relationship between electrical conductivity and electrical conductance.
Thermal resistance is the inverse of thermal conductance. It is a convenient measure to use in multicomponent design since thermal resistances are additive when occurring in series.Bird, Stewart, & Lightfoot, p. 305
There is also a measure known as the heat transfer coefficient: the quantity of heat that passes per unit time through a unit area of a plate of particular thickness when its opposite faces differ in temperature by one kelvin.
The heat transfer coefficient is also known as thermal admittance in the sense that the material may be seen as admitting heat to flow.
An additional term, thermal transmittance, quantifies the thermal conductance of a structure along with heat transfer due to convection and radiation. It is measured in the same units as thermal conductance and is sometimes known as the composite thermal conductance. The term Uvalue is also used.
Finally, thermal diffusivity $\backslash alpha$ combines thermal conductivity with density and specific heat:Bird, Stewart, & Lightfoot, p. 268
As such, it quantifies the thermal inertia of a material, i.e. the relative difficulty in heating a material to a given temperature using heat sources applied at the boundary.
In imperial units, thermal conductivity is measured in BTU/(hour⋅ft⋅Fahrenheit).1 Btu/(h⋅ft⋅°F) = 1.730735 W/(m⋅K)
The dimension of thermal conductivity is M^{1}L^{1}T^{−3}Θ^{−1}, expressed in terms of the dimensions mass (M), length (L), time (T), and temperature (Θ).
Other units which are closely related to the thermal conductivity are in common use in the construction and textile industries. The construction industry makes use of measures such as the Rvalue (resistance) and the Uvalue (transmittance or conductance). Although related to the thermal conductivity of a material used in an insulation product or assembly, R and Uvalues are measured per unit area, and depend on the specified thickness of the product or assembly.Rvalues and Uvalues quoted in the US (based on the inchpound units of measurement) do not correspond with and are not compatible with those used outside the US (based on the SI units of measurement).
Likewise the textile industry has several units including the tog and the clo which express thermal resistance of a material in a way analogous to the Rvalues used in the construction industry.
In comparison with solid materials, the thermal properties of fluids are more difficult to study experimentally. This is because in addition to thermal conduction, convective and radiative energy transport are usually present unless measures are taken to limit these processes. The formation of an insulating boundary layer can also result in an apparent reduction in the thermal conductivity.
Of all materials, allotropes of carbon, such as graphite and diamond, are usually credited with having the highest thermal conductivities at room temperature. An unlikely competitor for diamond as the best thermal conductor, Phys.org news (July 8, 2013). The thermal conductivity of natural diamond at room temperature is several times higher than that of a highly conductive metal such as copper (although the precise value varies depending on the diamond type)."Thermal Conductivity in W cm^{−1} K^{−1} of Metals and Semiconductors as a Function of Temperature", in CRC Handbook of Chemistry and Physics, 99th Edition (Internet Version 2018), John R. Rumble, ed., CRC Press/Taylor & Francis, Boca Raton, FL.
Thermal conductivities of selected substances are tabulated here; an expanded list can be found in the list of thermal conductivities. These values should be considered approximate due to the uncertainties related to material definitions.
Air  0.026  25 
Styrofoam  0.033  25 
WaterBird, Stewart, & Lightfoot, pp. 270271  0.6089  26.85 
Concrete  0.92  – 
Copper  384.1  18.05 
Natural diamond  895–1350  26.85 
On the other hand, heat conductivity in nonmetals is mainly due to lattice vibrations (). Except for highquality crystals at low temperatures, the phonon mean free path is not reduced significantly at higher temperatures. Thus, the thermal conductivity of nonmetals is approximately constant at high temperatures. At low temperatures well below the Debye temperature, thermal conductivity decreases, as does the heat capacity, due to carrier scattering from defects at very low temperatures.
Even more dramatically, the thermal conductivity of a fluid diverges in the vicinity of the vaporliquid critical point.
When anisotropy is present, the direction of heat flow may not be exactly the same as the direction of the thermal gradient.
Low density gases, such as hydrogen and helium typically have high thermal conductivity. Dense gases such as xenon and dichlorodifluoromethane have low thermal conductivity. An exception, sulfur hexafluoride, a dense gas, has a relatively high thermal conductivity due to its high heat capacity. Argon and krypton, gases denser than air, are often used in insulated glazing (double paned windows) to improve their insulation characteristics.
The thermal conductivity through bulk materials in porous or granular form is governed by the type of gas in the gaseous phase, and its pressure. At lower pressures, the thermal conductivity of a gaseous phase is reduced, with this behaviour governed by the Knudsen number, defined as $K\_n=l/d$, where $l$ is the mean free path of gas molecules and $d$ is the typical gap size of the space filled by the gas. In a granular material $d$ corresponds to the characteristic size of the gaseous phase in the pores or intergranular spaces.
In a gas, thermal conduction is mediated by discrete molecular collisions. In a simplified picture of a solid, thermal conduction occurs by two mechanisms: 1) the migration of free electrons and 2) lattice vibrations (phonons). The first mechanism dominates in pure metals and the second in nonmetallic solids. In liquids, by contrast, the precise microscopic mechanisms of thermal conduction are poorly understood.
To incorporate more complex interparticle interactions, a systematic approach is necessary. One such approach is provided by Chapman–Enskog theory, which derives explicit expressions for thermal conductivity starting from the Boltzmann equation. The Boltzmann equation, in turn, provides a statistical description of a dilute gas for generic interparticle interactions. For a monatomic gas, expressions for $k$ derived in this way take the form
where $\backslash sigma$ is an effective particle diameter and $\backslash Omega(T)$ is a function of temperature whose explicit form depends on the interparticle interaction law.Chapman & Cowling, p. 167Bird, Stewart, & Lightfoot, p. 275 For rigid elastic spheres, $\backslash Omega(T)$ is independent of $T$ and very close to $1$. More complex interaction laws introduce a weak temperature dependence. The precise nature of the dependence is not always easy to discern, however, as $\backslash Omega(T)$ is defined as a multidimensional integral which may not be expressible in terms of elementary functions. An alternate, equivalent way to present the result is in terms of the gas viscosity $\backslash mu$, which can also be calculated in the Chapman–Enskog approach:
where $f$ is a numerical factor which in general depends on the molecular model. For smooth spherically symmetric molecules, however, $f$ is very close to $2.5$, not deviating by more than $1\%$ for a variety of interparticle force laws.Chapman & Cowling, p. 247 Since $k$, $\backslash mu$, and $c\_v$ are each welldefined physical quantities which can be measured independent of each other, this expression provides a convenient test of the theory. For monatomic gases, such as the noble gases, the agreement with experiment is fairly good.Chapman & Cowling, pp. 249251
For gases whose molecules are not spherically symmetric, the expression $k\; =\; f\; \backslash mu\; c\_v$ still holds. In contrast with spherically symmetric molecules, however, $f$ varies significantly depending on the particular form of the interparticle interactions: this is a result of the energy exchanges between the internal and translational degrees of freedom of the molecules. An explicit treatment of this effect is difficult in the Chapman–Enskog approach. Alternately, the approximate expression $f\; =\; (1/4)\{(9\; \backslash gamma\; \; 5)\}$ was suggested by Arnold Eucken, where $\backslash gamma$ is the heat capacity ratio of the gas.Bird, Stewart, & Lightfoot, p. 276
The entirety of this section assumes the mean free path $\backslash lambda$ is small compared with macroscopic (system) dimensions. In extremely dilute gases this assumption fails, and thermal conduction is described instead by an apparent thermal conductivity which decreases with density. Ultimately, as the density goes to $0$ the system approaches a vacuum, and thermal conduction ceases entirely. For this reason a vacuum is an effective insulator.
where $N\_\backslash text\{A\}$ is the Avogadro constant, $V$ is the volume of a mole of liquid, and $v\_\backslash text\{s\}$ is the speed of sound in the liquid. This is commonly called Bridgman's equation.Bird, Stewart, & Lightfoot, p. 279
with k_{0} a constant. For pure metals such as copper, silver, etc. k_{0} is large, so the thermal conductivity is high. At higher temperatures the mean free path is limited by the phonons, so the thermal conductivity tends to decrease with temperature. In alloys the density of the impurities is very high, so l and, consequently k, are small. Therefore, alloys, such as stainless steel, can be used for thermal insulation.
The phonon mean free path has been associated directly with the effective relaxation length for processes without directional correlation. If V_{g} is the group velocity of a phonon wave packet, then the relaxation length $l\backslash ;$ is defined as:
where t is the characteristic relaxation time. Since longitudinal waves have a much greater phase velocity than transverse waves,
V_{long} is much greater than V_{trans}, and the relaxation length or mean free path of longitudinal phonons will be much greater. Thus, thermal conductivity will be largely determined by the speed of longitudinal phonons.Regarding the dependence of wave velocity on wavelength or frequency (dispersion), lowfrequency phonons of long wavelength will be limited in relaxation length by elastic Rayleigh scattering. This type of light scattering from small particles is proportional to the fourth power of the frequency. For higher frequencies, the power of the frequency will decrease until at highest frequencies scattering is almost frequency independent. Similar arguments were subsequently generalized to many glass forming substances using Brillouin scattering.
Phonons in the acoustical branch dominate the phonon heat conduction as they have greater energy dispersion and therefore a greater distribution of phonon velocities. Additional optical modes could also be caused by the presence of internal structure (i.e., charge or mass) at a lattice point; it is implied that the group velocity of these modes is low and therefore their contribution to the lattice thermal conductivity λ_{L} ($\backslash kappa$_{L}) is small.
Each phonon mode can be split into one longitudinal and two transverse polarization branches. By extrapolating the phenomenology of lattice points to the unit cells it is seen that the total number of degrees of freedom is 3 pq when p is the number of primitive cells with q atoms/unit cell. From these only 3p are associated with the acoustic modes, the remaining 3 p( q − 1) are accommodated through the optical branches. This implies that structures with larger p and q contain a greater number of optical modes and a reduced λ_{L}.
From these ideas, it can be concluded that increasing crystal complexity, which is described by a complexity factor CF (defined as the number of atoms/primitive unit cell), decreases λ_{L}. This was done by assuming that the relaxation time τ decreases with increasing number of atoms in the unit cell and then scaling the parameters of the expression for thermal conductivity in high temperatures accordingly.
Describing anharmonic effects is complicated because an exact treatment as in the harmonic case is not possible, and phonons are no longer exact eigensolutions to the equations of motion. Even if the state of motion of the crystal could be described with a plane wave at a particular time, its accuracy would deteriorate progressively with time. Time development would have to be described by introducing a spectrum of other phonons, which is known as the phonon decay. The two most important anharmonic effects are the thermal expansion and the phonon thermal conductivity.
Only when the phonon number ‹n› deviates from the equilibrium value ‹n›^{0}, can a thermal current arise as stated in the following expression
where v is the energy transport velocity of phonons. Only two mechanisms exist that can cause time variation of ‹ n› in a particular region. The number of phonons that diffuse into the region from neighboring regions differs from those that diffuse out, or phonons decay inside the same region into other phonons. A special form of the Boltzmann equation
states this. When steady state conditions are assumed the total time derivate of phonon number is zero, because the temperature is constant in time and therefore the phonon number stays also constant. Time variation due to phonon decay is described with a relaxation time ( τ) approximation
which states that the more the phonon number deviates from its equilibrium value, the more its time variation increases. At steady state conditions and local thermal equilibrium are assumed we get the following equation
Using the relaxation time approximation for the Boltzmann equation and assuming steadystate conditions, the phonon thermal conductivity λ_{L} can be determined. The temperature dependence for λ_{L} originates from the variety of processes, whose significance for λ_{L} depends on the temperature range of interest. Mean free path is one factor that determines the temperature dependence for λ_{L}, as stated in the following equation
where Λ is the mean free path for phonon and $\backslash frac\{\backslash partial\}\{\backslash partial\; T\}\backslash epsilon$ denotes the heat capacity. This equation is a result of combining the four previous equations with each other and knowing that $\backslash left\; \backslash langle\; v\_x^2\backslash right\; \backslash rangle=\backslash frac\{1\}\{3\}v^2$ for cubic or isotropic systems and $\backslash Lambda\; =v\backslash tau$.
At low temperatures (< 10 K) the anharmonic interaction does not influence the mean free path and therefore, the thermal resistivity is determined only from processes for which qconservation does not hold. These processes include the scattering of phonons by crystal defects, or the scattering from the surface of the crystal in case of high quality single crystal. Therefore, thermal conductance depends on the external dimensions of the crystal and the quality of the surface. Thus, temperature dependence of λ_{L} is determined by the specific heat and is therefore proportional to T^{3}.
Phonon quasimomentum is defined as ℏq and differs from normal momentum because it is only defined within an arbitrary reciprocal lattice vector. At higher temperatures (10 K < T < Θ), the conservation of energy $\backslash hslash\; \{\backslash omega\}\_\{1\}=\backslash hslash\; \{\backslash omega\}\_\{2\}+\backslash hslash\; \{\backslash omega\}\_\{3\}$ and quasimomentum $\backslash mathbf\{q\}\_\{1\}=\backslash mathbf\{q\}\_\{2\}+\backslash mathbf\{q\}\_\{3\}+\backslash mathbf\{G\}$, where q_{1} is wave vector of the incident phonon and q_{2}, q_{3} are wave vectors of the resultant phonons, may also involve a reciprocal lattice vector G complicating the energy transport process. These processes can also reverse the direction of energy transport.
Therefore, these processes are also known as Umklapp (U) processes and can only occur when phonons with sufficiently large qvectors are excited, because unless the sum of q_{2} and q_{3} points outside of the Brillouin zone the momentum is conserved and the process is normal scattering (Nprocess). The probability of a phonon to have energy E is given by the Boltzmann distribution $P\backslash propto\; \{e\}^\{E/kT\}$. To Uprocess to occur the decaying phonon to have a wave vector q_{1} that is roughly half of the diameter of the Brillouin zone, because otherwise quasimomentum would not be conserved.
Therefore, these phonons have to possess energy of $\backslash sim\; k\backslash Theta\; /2$, which is a significant fraction of Debye energy that is needed to generate new phonons. The probability for this is proportional to $\{e\}^\{\backslash Theta\; /bT\}$, with $b=2$. Temperature dependence of the mean free path has an exponential form $\{e\}^\{\backslash Theta\; /bT\}$. The presence of the reciprocal lattice wave vector implies a net phonon backscattering and a resistance to phonon and thermal transport resulting finite λ_{L}, as it means that momentum is not conserved. Only momentum nonconserving processes can cause thermal resistance.
At high temperatures ( T > Θ), the mean free path and therefore λ_{L} has a temperature dependence T^{−1}, to which one arrives from formula $\{e\}^\{\backslash Theta\; /bT\}$ by making the following approximation $\{e\}^\{x\}\backslash propto\; x\backslash text\{\; \},\backslash text\{\; \}\backslash left(x\backslash right)\; <\; 1$ and writing $x=\backslash Theta\; /bT$. This dependency is known as Arnold Eucken law and originates from the temperature dependency of the probability for the Uprocess to occur.
Thermal conductivity is usually described by the Boltzmann equation with the relaxation time approximation in which phonon scattering is a limiting factor. Another approach is to use analytic models or molecular dynamics or Monte Carlo based methods to describe thermal conductivity in solids.
Short wavelength phonons are strongly scattered by impurity atoms if an alloyed phase is present, but mid and long wavelength phonons are less affected. Mid and long wavelength phonons carry significant fraction of heat, so to further reduce lattice thermal conductivity one has to introduce structures to scatter these phonons. This is achieved by introducing interface scattering mechanism, which requires structures whose characteristic length is longer than that of impurity atom. Some possible ways to realize these interfaces are nanocomposites and embedded nanoparticles or structures.
In component datasheets and tables, since actual, physical components with distinct physical dimensions and characteristics are under consideration, thermal resistance is frequently given in absolute units of $\backslash rm\; K/W$ or $\backslash rm\; ^\{\backslash circ\}\; C/W$, since the two are equivalent. However, thermal conductivity, which is its reciprocal, is frequently given in specific units of $\backslash rm\; W/(K\backslash cdot\; m)$. It is therefore often necessary to convert between absolute and specific units, by also taking a component's physical dimensions into consideration, in order to correlate the two using information provided, or to convert tabulated values of specific thermal conductivity into absolute thermal resistance values for use in thermal resistance calculations. This is particularly useful, for example, when calculating the maximum power a component can dissipate as heat, as demonstrated in the example calculation here.
"Thermal conductivity λ is defined as ability of material to transmit heat and it is measured in watts per square metre of surface area for a temperature gradient of 1 K per unit thickness of 1 m".http://tpm.fsv.cvut.cz/student/documents/files/BUM1/Chapter16.pdf Therefore, specific thermal conductivity is calculated as:
where:
$\backslash Delta\; T$ = temperature difference (K, or °C) = 1 K during measurement
Absolute thermal conductivity, on the other hand, has units of $\backslash rm\; W/K$ or $\backslash rm\; W/^\{\backslash circ\}\; C$, and can be expressed as
where $\backslash lambda\_A$ = absolute thermal conductivity (W/K, or W/°C).
Substituting $\backslash lambda\_A$ for $\backslash frac\{P\}\{\backslash Delta\; T\}$ into the first equation yields the equation which converts from absolute thermal conductivity to specific thermal conductivity:
Solving for $\backslash lambda\_A$, we get the equation which converts from specific thermal conductivity to absolute thermal conductivity:
Again, since thermal conductivity and resistivity are reciprocals of each other, it follows that the equation to convert specific thermal conductivity to absolute thermal resistance is:
This value fits within the normal values for thermal resistance between a device case and a heat sink: "the contact between the device case and heat sink may have a thermal resistance of between 0.5 up to 1.7 °C/W, depending on the case size, and use of grease or insulating mica washer".
where $\backslash vec\; q$ is the heat flux (amount of heat flowing per second and per unit area) and $\backslash vec\; \backslash nabla\; T$ the temperature gradient. The sign in the expression is chosen so that always k > 0 as heat always flows from a high temperature to a low temperature. This is a direct consequence of the second law of thermodynamics.
In the onedimensional case, q = H/ A with H the amount of heat flowing per second through a surface with area A and the temperature gradient is d T/d x so
In case of a thermally insulated bar (except at the ends) in the steady state, H is constant. If A is constant as well the expression can be integrated with the result
where T_{H} and T_{L} are the temperatures at the hot end and the cold end respectively, and L is the length of the bar. It is convenient to introduce the thermalconductivity integral
The heat flow rate is then given by
If the temperature difference is small, k can be taken as constant. In that case

