thermal conductivity

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The thermal conductivity of a material is a measure of its ability to conduct heat. It is commonly denoted by $k$, $\lambda$, or $\kappa$. 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 Rockwool or Styrofoam. Correspondingly, materials of high thermal conductivity are widely used in
heat sink A heat sink (also commonly spelled heatsink) is a passive heat exchanger that transfers the heat generated by an electronic or a mechanical device to a fluid medium, often air or a liquid coolant, where it is thermal management (electronics), di ...
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 $\mathbf = - k \nabla T$, where $\mathbf$ is the
heat flux Heat flux or thermal flux, sometimes also referred to as ''heat flux density'', heat-flow density or ''heat flow rate intensity'' is a flow of energy per unit area per unit time (physics), time. In SI its units are watts per square metre (W/m2). ...
, $k$ is the thermal conductivity, and $\nabla T$ is the temperature gradient. This is known as
Fourier's Law Conduction is the process by which heat is Heat transfer, transferred from the hotter end to the colder end of an object. The ability of the object to conduct heat is known as its ''thermal conductivity'', and is denoted . Heat spontaneously flo ...
for heat conduction. Although commonly expressed as a scalar, the most general form of thermal conductivity is a second-rank
tensor In mathematics, a tensor is an mathematical object, algebraic object that describes a Multilinear map, multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as Vect ...
. However, the tensorial description only becomes necessary in materials which are anisotropic.

# Definition

## Simple definition

Consider a solid material placed between two environments of different temperatures. Let $T_1$ be the temperature at $x=0$ and $T_2$ be the temperature at $x=L$, and suppose $T_2 > T_1$. A possible realization of this scenario is a building on a cold winter day: the solid material in this case would be the building wall, separating the cold outdoor environment from the warm indoor environment. According to the
second law of thermodynamics The second law of thermodynamics is a physical law based on universal experience concerning heat and Energy transformation, energy interconversions. One simple statement of the law is that heat always moves from hotter objects to colder objects ( ...
, heat will flow from the hot environment to the cold one as the temperature difference is equalized by diffusion. This is quantified in terms of a
heat flux Heat flux or thermal flux, sometimes also referred to as ''heat flux density'', heat-flow density or ''heat flow rate intensity'' is a flow of energy per unit area per unit time (physics), time. In SI its units are watts per square metre (W/m2). ...
$q$, which gives the rate, per unit area, at which heat flows in a given direction (in this case minus x-direction). In many materials, $q$ is observed to be directly proportional to the temperature difference and inversely proportional to the separation distance $L$: :$q = -k \cdot \frac.$ 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 x-direction 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 Convection is single or Multiphase flow, multiphase fluid flow that occurs Spontaneous process, spontaneously due to the combined effects of material property heterogeneity and body forces on a fluid, most commonly density and gravity (see buoya ...
and
radiation In physics, radiation is the emission or transmission of energy in the form of waves or particles through space or through a material medium. This includes: * ''electromagnetic radiation'', such as radio waves, microwaves, infrared, visib ...
, are eliminated or accounted for. The preceding derivation assumes that the $k$ does not change significantly as temperature is varied from $T_1$ to $T_2$. Cases in which the temperature variation of $k$ is non-negligible must be addressed using the more general definition of $k$ discussed below.

## General definition

Thermal conduction is defined as the transport of energy due to random molecular motion across a temperature gradient. It is distinguished from energy transport by convection and molecular work in that it does not involve macroscopic flows or work-performing internal stresses. Energy flow due to thermal conduction is classified as heat and is quantified by the vector $\mathbf\left(\mathbf, t\right)$, which gives the heat flux at position $\mathbf$ and time $t$. According to the second law of thermodynamics, heat flows from high to low temperature. Hence, it is reasonable to postulate that $\mathbf\left(\mathbf, t\right)$ is proportional to the gradient of the temperature field $T\left(\mathbf, t\right)$, i.e. :$\mathbf\left(\mathbf, t\right) = -k \nabla T\left(\mathbf, t\right),$ where the constant of proportionality, $k > 0$, is the thermal conductivity. This is called Fourier's law of heat conduction. Despite its name, it is not a law but a definition of thermal conductivity in terms of the independent physical quantities $\mathbf\left(\mathbf, t\right)$ and $T\left(\mathbf, t\right)$. 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 $T\left(\mathbf, t\right)$ 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: :$\mathbf\left(\mathbf, t\right) = -\boldsymbol \cdot \nabla T\left(\mathbf, t\right)$ where $\boldsymbol$ is symmetric, second-rank
tensor In mathematics, a tensor is an mathematical object, algebraic object that describes a Multilinear map, multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as Vect ...
called the thermal conductivity tensor. An implicit assumption in the above description is the presence of local thermodynamic equilibrium, which allows one to define a temperature field $T\left(\mathbf, t\right)$. This assumption could be violated in systems that are unable to attain local equilibrium, as might happen in the presence of strong nonequilibrium driving or long-ranged interactions.

## Other quantities

In engineering practice, it is common to work in terms of quantities which are derivative to thermal conductivity and implicitly take into account design-specific features such as component dimensions. 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 Electrical resistivity (also called specific electrical resistance or volume resistivity) is a fundamental property of a material that measures how strongly it resists electric current. A low resistivity indicates a material that readily allows ...
and
electrical conductance The electrical resistance of an object is a measure of its opposition to the flow of electric current. Its Multiplicative inverse, reciprocal quantity is , measuring the ease with which an electric current passes. Electrical resistance shares s ...
. 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. There is also a measure known as the
heat transfer coefficient In thermodynamics, the heat transfer coefficient or film coefficient, or film effectiveness, is the Proportional (mathematics), proportionality constant between the heat flux and the thermodynamic driving force for the Heat transfer, flow of heat ...
: 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. In ASTM C168-15, this area-independent quantity is referred to as the "thermal conductance".ASTM C168 − 15a Standard Terminology Relating to Thermal Insulation. The reciprocal of the heat transfer coefficient is thermal insulance. In summary, for a plate of thermal conductivity $k$, area $A$ and thickness $L$, *thermal conductance = $kA/L$, measured in W⋅K−1. **thermal resistance = $L/\left(kA\right)$, measured in K⋅W−1. *heat transfer coefficient = $k/L$, measured in W⋅K−1⋅m−2. **thermal insulance = $L/k$, measured in K⋅m2⋅W−1. 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 Convection is single or Multiphase flow, multiphase fluid flow that occurs Spontaneous process, spontaneously due to the combined effects of material property heterogeneity and body forces on a fluid, most commonly density and gravity (see buoya ...
and
radiation In physics, radiation is the emission or transmission of energy in the form of waves or particles through space or through a material medium. This includes: * ''electromagnetic radiation'', such as radio waves, microwaves, infrared, visib ...
. It is measured in the same units as thermal conductance and is sometimes known as the ''composite thermal conductance''. The term '' U-value'' is also used. Finally, thermal diffusivity $\alpha$ combines thermal conductivity with
density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek language, Greek letter Rho (letter), rho), although the Latin letter ''D'' ca ...
and
specific heat In thermodynamics, the specific heat capacity (symbol ) of a substance is the heat capacity of a sample of the substance divided by the mass of the sample, also sometimes referred to as massic heat capacity. Informally, it is the amount of heat t ...
: :$\alpha = \frac$. 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.

# Units

In the
International System of Units The International System of Units, known by the international abbreviation SI in all languages and sometimes Pleonasm#Acronyms and initialisms, pleonastically as the SI system, is the modern form of the metric system and the world's most wid ...
(SI), thermal conductivity is measured in
watt The watt (symbol: W) is the unit of power or radiant flux in the International System of Units (SI), equal to 1 joule per second or 1 kg⋅m2⋅s−3. It is used to quantify the rate of energy transfer. The watt is named after James W ...
s per metre-kelvin ( W/( mK)). Some papers report in watts per centimetre-kelvin (W/(cm⋅K)). In
imperial units The imperial system of units, imperial system or imperial units (also known as British Imperial or Exchequer Standards of 1826) is the system of units first defined in the British Weights and Measures Act 1824 and continued to be developed thro ...
, thermal conductivity is measured in BTU/( hft°F).1 Btu/(h⋅ft⋅°F) = 1.730735 W/(m⋅K) The
dimension In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
of thermal conductivity is M1L1T−3Θ−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 R-value (resistance) and the U-value (transmittance or conductance). Although related to the thermal conductivity of a material used in an insulation product or assembly, R- and U-values are measured per unit area, and depend on the specified thickness of the product or assembly.R-values and U-values quoted in the US (based on the inch-pound 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 R-values used in the construction industry.

# Measurement

There are several ways to measure thermal conductivity; each is suitable for a limited range of materials. Broadly speaking, there are two categories of measurement techniques: ''steady-state'' and ''transient''. Steady-state techniques infer the thermal conductivity from measurements on the state of a material once a steady-state temperature profile has been reached, whereas transient techniques operate on the instantaneous state of a system during the approach to steady state. Lacking an explicit time component, steady-state techniques do not require complicated signal analysis (steady state implies constant signals). The disadvantage is that a well-engineered experimental setup is usually needed, and the time required to reach steady state precludes rapid measurement. 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.

# Experimental values

The thermal conductivities of common substances span at least four orders of magnitude. Gases generally have low thermal conductivity, and pure metals have high thermal conductivity. For example, under standard conditions the thermal conductivity of
copper Copper is a chemical element with the Symbol (chemistry), symbol Cu (from la, cuprum) and atomic number 29. It is a soft, malleable, and ductility, ductile metal with very high thermal conductivity, thermal and electrical conductivity. A fre ...
is over times that of air. Of all materials, allotropes of carbon, such as
graphite Graphite () is a crystalline A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in al ...
and
diamond Diamond is a solid form of the element carbon with its atoms arranged in a crystal structure In crystallography, crystal structure is a description of the ordered arrangement of atoms, ions or molecules in a crystal, crystalline material ...
, are usually credited with having the highest thermal conductivities at room temperature. 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 below; an expanded list can be found in the list of thermal conductivities. These values are illustrative estimates only, as they do not account for measurement uncertainties or variability in material definitions.

# Influencing factors

## Temperature

The effect of temperature on thermal conductivity is different for metals and nonmetals. In metals, heat conductivity is primarily due to free electrons. Following the Wiedemann–Franz law, thermal conductivity of metals is approximately proportional to the absolute temperature (in
kelvin The kelvin, symbol K, is the primary unit of temperature in the International System of Units (SI), used alongside its metric prefix, prefixed forms and the degree Celsius. It is named after the Belfast-born and University of Glasgow-based eng ...
s) times electrical conductivity. In pure metals the electrical conductivity decreases with increasing temperature and thus the product of the two, the thermal conductivity, stays approximately constant. However, as temperatures approach absolute zero, the thermal conductivity decreases sharply. In alloys the change in electrical conductivity is usually smaller and thus thermal conductivity increases with temperature, often proportionally to temperature. Many pure metals have a peak thermal conductivity between 2 K and 10 K. On the other hand, heat conductivity in nonmetals is mainly due to lattice vibrations ( phonons). Except for high-quality 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.

## Chemical phase

When a material undergoes a phase change (e.g. from solid to liquid), the thermal conductivity may change abruptly. For instance, when ice melts to form liquid water at 0 °C, the thermal conductivity changes from 2.18 W/(m⋅K) to 0.56 W/(m⋅K). Even more dramatically, the thermal conductivity of a fluid diverges in the vicinity of the vapor-liquid critical point.

## Thermal anisotropy

Some substances, such as non- cubic
crystal A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macrosc ...
s, can exhibit different thermal conductivities along different crystal axes.
Sapphire Sapphire is a precious gemstone, a variety of the mineral corundum, consisting of aluminium oxide () with trace amounts of elements such as iron, titanium, chromium, vanadium, or magnesium. The name sapphire is derived via the Latin "sapphir ...
is a notable example of variable thermal conductivity based on orientation and temperature, with 35 W/(m⋅K) along the c axis and 32 W/(m⋅K) along the a axis.
Wood Wood is a porous and fibrous structural tissue found in the Plant stem, stems and roots of trees and other woody plants. It is an organic materiala natural composite material, composite of cellulose fibers that are strong in tension and emb ...
generally conducts better along the grain than across it. Other examples of materials where the thermal conductivity varies with direction are metals that have undergone heavy cold pressing, laminated materials, cables, the materials used for the
Space Shuttle thermal protection system The Space Shuttle thermal protection system (TPS) is the barrier that protected the Space Shuttle Orbiter during the searing heat of atmospheric reentry. A secondary goal was to protect from the heat and cold of space while in orbit. Materia ...
, and fiber-reinforced composite structures. When anisotropy is present, the direction of heat flow may differ from the direction of the thermal gradient.

## Electrical conductivity

In metals, thermal conductivity is approximately correlated with electrical conductivity according to the Wiedemann–Franz law, as freely moving
valence electron In chemistry and physics, a valence electron is an electron in the outer electron shell, shell associated with an atom, and that can participate in the formation of a chemical bond if the outer shell is not closed. In a single covalent bond, a sh ...
s transfer not only electric current but also heat energy. However, the general correlation between electrical and thermal conductance does not hold for other materials, due to the increased importance of phonon carriers for heat in non-metals. Highly electrically conductive
silver Silver is a chemical element with the Symbol (chemistry), symbol Ag (from the Latin ', derived from the Proto-Indo-European wikt:Reconstruction:Proto-Indo-European/h₂erǵ-, ''h₂erǵ'': "shiny" or "white") and atomic number 47. A soft, whi ...
is less thermally conductive than
diamond Diamond is a solid form of the element carbon with its atoms arranged in a crystal structure In crystallography, crystal structure is a description of the ordered arrangement of atoms, ions or molecules in a crystal, crystalline material ...
, which is an electrical insulator but conducts heat via phonons due to its orderly array of atoms.

## Magnetic field

The influence of magnetic fields on thermal conductivity is known as the thermal Hall effect or Righi–Leduc effect.

## Gaseous phases

In the absence of convection, air and other gases are good insulators. Therefore, many insulating materials function simply by having a large number of gas-filled pockets which obstruct heat conduction pathways. Examples of these include expanded and extruded
polystyrene Polystyrene (PS) is a synthetic polymer made from monomers of the Aromatic hydrocarbon, aromatic hydrocarbon styrene. Polystyrene can be solid or foamed. General-purpose polystyrene is clear, hard, and brittle. It is an inexpensive resin pe ...
(popularly referred to as "styrofoam") and silica aerogel, as well as warm clothes. Natural, biological insulators such as fur and
feathers Feathers are epidermis (zoology), epidermal growths that form a distinctive outer covering, or plumage, on both Bird, avian (bird) and some non-avian dinosaurs and other archosaurs. They are the most complex integumentary structures found in ...
achieve similar effects by trapping air in pores, pockets, or voids. Low density gases, such as
hydrogen Hydrogen is the chemical element with the Symbol (chemistry), symbol H and atomic number 1. Hydrogen is the lightest element. At standard temperature and pressure, standard conditions hydrogen is a gas of diatomic molecules having the chemical ...
and
helium Helium (from el, ἥλιος, helios, lit=sun) is a chemical element with the symbol (chemistry), symbol He and atomic number 2. It is a colorless, odorless, tasteless, non-toxic, inert gas, inert, monatomic gas and the first in the noble gas gr ...
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 Argon is a chemical element with the Symbol (chemistry), symbol Ar and atomic number 18. It is in group 18 of the periodic table and is a noble gas. Argon is the third-most abundant gas in Earth's atmosphere, at 0.934% (9340 Parts-per notatio ...
and
krypton Krypton (from grc, κρυπτός, translit=kryptos 'the hidden one') is a chemical element with the symbol (chemistry), symbol Kr and atomic number 36. It is a colorless, odorless, tasteless noble gas that occurs in trace element, trace amount ...
, 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 low 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 In physics, mean free path is the average distance over which a moving particle (such as an atom, a molecule, or a photon) travels before substantially changing its direction or energy (or, in a specific context, other properties), typically as a ...
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.

## Isotopic purity

The thermal conductivity of a crystal can depend strongly on isotopic purity, assuming other lattice defects are negligible. A notable example is diamond: at a temperature of around 100 K the thermal conductivity increases from 10,000 W· m−1· K−1 for natural type IIa diamond (98.9% 12C), to 41,000 for 99.9% enriched synthetic diamond. A value of 200,000 is predicted for 99.999% 12C at 80 K, assuming an otherwise pure crystal. The thermal conductivity of 99% isotopically enriched cubic boron nitride is ~ 1400 W· m−1· K−1, which is 90% higher than that of natural boron nitride.

# Molecular origins

The molecular mechanisms of thermal conduction vary among different materials, and in general depend on details of the microscopic structure and molecular interactions. As such, thermal conductivity is difficult to predict from first-principles. Any expressions for thermal conductivity which are exact and general, e.g. the Green-Kubo relations, are difficult to apply in practice, typically consisting of averages over multiparticle correlation functions. A notable exception is a monatomic dilute gas, for which a well-developed theory exists expressing thermal conductivity accurately and explicitly in terms of molecular parameters. 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 non-metallic solids. In liquids, by contrast, the precise microscopic mechanisms of thermal conduction are poorly understood.

## Gases

In a simplified model of a dilute
monatomic In physics and chemistry, "monatomic" is a combination of the words "mono" and "atomic", and means "single atom". It is usually applied to gases: a monatomic gas is a gas in which atoms are not bound to each other. Examples at Standard temperature ...
gas, molecules are modeled as rigid spheres which are in constant motion, colliding elastically with each other and with the walls of their container. Consider such a gas at temperature $T$ and with density $\rho$,
specific heat In thermodynamics, the specific heat capacity (symbol ) of a substance is the heat capacity of a sample of the substance divided by the mass of the sample, also sometimes referred to as massic heat capacity. Informally, it is the amount of heat t ...
$c_v$ and molecular mass $m$. Under these assumptions, an elementary calculation yields for the thermal conductivity :$k = \beta \rho \lambda c_v \sqrt,$ where $\beta$ is a numerical constant of order $1$, $k_\text$ is the Boltzmann constant, and $\lambda$ is the
mean free path In physics, mean free path is the average distance over which a moving particle (such as an atom, a molecule, or a photon) travels before substantially changing its direction or energy (or, in a specific context, other properties), typically as a ...
, which measures the average distance a molecule travels between collisions. Since $\lambda$ is inversely proportional to density, this equation predicts that thermal conductivity is independent of density for fixed temperature. The explanation is that increasing density increases the number of molecules which carry energy but decreases the average distance $\lambda$ a molecule can travel before transferring its energy to a different molecule: these two effects cancel out. For most gases, this prediction agrees well with experiments at pressures up to about 10 atmospheres. On the other hand, experiments show a more rapid increase with temperature than $k \propto \sqrt$ (here, $\lambda$ is independent of $T$). This failure of the elementary theory can be traced to the oversimplified "elastic sphere" model, and in particular to the fact that the interparticle attractions, present in all real-world gases, are ignored. 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 :$k = \frac \frac c_v,$ where $\sigma$ is an effective particle diameter and $\Omega\left(T\right)$ is a function of temperature whose explicit form depends on the interparticle interaction law. For rigid elastic spheres, $\Omega\left(T\right)$ 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 $\Omega\left(T\right)$ is defined as a multi-dimensional 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 The viscosity of a fluid is a measure of its drag (physics), resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water. Viscosity quant ...
$\mu$, which can also be calculated in the Chapman–Enskog approach: :$k = f \mu c_v,$ 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 $k$, $\mu$, and $c_v$ are each well-defined 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 noble gases (historically also the inert gases; sometimes referred to as aerogens) make up a class of chemical elements with similar properties; under Standard conditions for temperature and pressure, standard conditions, they are all odorle ...
, the agreement with experiment is fairly good. For gases whose molecules are not spherically symmetric, the expression $k = f \mu c_v$ 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 $f = \left(1/4\right)$ was suggested by Eucken, where $\gamma$ is the heat capacity ratio of the gas. The entirety of this section assumes the mean free path $\lambda$ 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 A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or "Void (astronomy), void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Ph ...
, and thermal conduction ceases entirely.

## Liquids

The exact mechanisms of thermal conduction are poorly understood in liquids: there is no molecular picture which is both simple and accurate. An example of a simple but very rough theory is that of Bridgman, in which a liquid is ascribed a local molecular structure similar to that of a solid, i.e. with molecules located approximately on a lattice. Elementary calculations then lead to the expression :$k = 3\left(N_\text / V\right)^ k_\text v_\text,$ where $N_\text$ is the Avogadro constant, $V$ is the volume of a mole of liquid, and $v_\text$ is the
speed of sound The speed of sound is the distance travelled per unit of time by a sound wave as it propagates through an elasticity (solid mechanics), elastic medium. At , the speed of sound in air is about , or one kilometre in or one mile in . It depends ...
in the liquid. This is commonly called ''Bridgman's equation''.

## Metals

For metals at low temperatures the heat is carried mainly by the free electrons. In this case the mean velocity is the Fermi velocity which is temperature independent. The mean free path is determined by the impurities and the crystal imperfections which are temperature independent as well. So the only temperature-dependent quantity is the heat capacity ''c'', which, in this case, is proportional to ''T''. So :$k=k_0\,T \text$ with ''k''0 a constant. For pure metals, ''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.

## Lattice waves

Heat transport in both amorphous and crystalline
dielectric In electromagnetism, a dielectric (or dielectric medium) is an Insulator (electricity), electrical insulator that can be Polarisability, polarised by an applied electric field. When a dielectric material is placed in an electric field, electr ...
solids is by way of elastic vibrations of the lattice (i.e., phonons). This transport mechanism is theorized to be limited by the elastic scattering of acoustic phonons at lattice defects. This has been confirmed by the experiments of Chang and Jones on commercial glasses and glass ceramics, where the mean free paths were found to be limited by "internal boundary scattering" to length scales of 10−2 cm to 10−3 cm. The phonon mean free path has been associated directly with the effective relaxation length for processes without directional correlation. If Vg is the group velocity of a phonon wave packet, then the relaxation length $l\;$ is defined as: :$l\;=V_\text t$ 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), low-frequency phonons of long wavelength will be limited in relaxation length by elastic
Rayleigh scattering Rayleigh scattering ( ), named after the 19th-century British physicist Lord Rayleigh (John William Strutt), is the predominantly elastic scattering of light or other electromagnetic radiation by particles much smaller than the wavelength of the ...
. 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 ($\kappa$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 λL. 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 :$Q_x=\frac \sum_ \text$ 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 :$\frac=_+_\text$ 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 :$_\text=-\text\frac,$ 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 :$_\text=-_\frac\frac\text$ Using the relaxation time approximation for the Boltzmann equation and assuming steady-state 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 :$_=\frac\sum _v\left\left(q,j\right\right)\Lambda \left\left(q,j\right\right)\frac\epsilon \left\left(\omega \left\left(q,j\right\right),T\right\right),$ where Λ is the mean free path for phonon and $\frac\epsilon$ denotes the heat capacity. This equation is a result of combining the four previous equations with each other and knowing that $\left \langle v_x^2\right \rangle=\fracv^2$ for cubic or isotropic systems and $\Lambda =v\tau$. 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 q-conservation 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 λL is determined by the specific heat and is therefore proportional to T3. 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 $\hslash _=\hslash _+\hslash _$ and quasimomentum $\mathbf_=\mathbf_+\mathbf_+\mathbf$, where q1 is wave vector of the incident phonon and q2, q3 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 ''q''-vectors are excited, because unless the sum of q2 and q3 points outside of the Brillouin zone the momentum is conserved and the process is normal scattering (N-process). The probability of a phonon to have energy ''E'' is given by the Boltzmann distribution $P\propto ^$. To U-process to occur the decaying phonon to have a wave vector q1 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 $\sim k\Theta /2$, which is a significant fraction of Debye energy that is needed to generate new phonons. The probability for this is proportional to $^$, with $b=2$. Temperature dependence of the mean free path has an exponential form $^$. 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 non-conserving 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 $^$ by making the following approximation $^\propto x\text,\text\left\left(x\right\right) < 1$ and writing $x=\Theta /bT$. This dependency is known as Eucken's law and originates from the temperature dependency of the probability for the U-process 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.

# Prediction

Because thermal conductivity depends continuously on quantities like temperature and material composition, it cannot be fully characterized by a finite number of experimental measurements. Predictive formulas become necessary if experimental values are not available under the physical conditions of interest. This capability is important in thermophysical simulations, where quantities like temperature and pressure vary continuously with space and time, and may encompass extreme conditions inaccessible to direct measurement.

## In fluids

For the simplest fluids, such as dilute monatomic gases and their mixtures, ''
ab initio ''Ab initio'' ( ) is a Latin term meaning "from the beginning" and is derived from the Latin ''ab'' ("from") + ''initio'', ablative case, ablative grammatical number, singular of ''initium'' ("beginning"). Etymology Circa 1600, from Latin, liter ...
'' quantum mechanical computations can accurately predict thermal conductivity in terms of fundamental atomic properties—that is, without reference to existing measurements of thermal conductivity or other transport properties. This method uses Chapman-Enskog theory to evaluate a low-density expansion of thermal conductivity. Chapman-Enskog theory, in turn, takes fundamental intermolecular potentials as input, which are computed ''ab initio'' from a quantum mechanical description. For most fluids, such high-accuracy, first-principles computations are not feasible. Rather, theoretical or empirical expressions must be fit to existing thermal conductivity measurements. If such an expression is fit to high-fidelity data over a large range of temperatures and pressures, then it is called a "reference correlation" for that material. Reference correlations have been published for many pure materials; examples are
carbon dioxide Carbon dioxide ( chemical formula ) is a chemical compound made up of molecules that each have one carbon Carbon () is a chemical element with the chemical symbol, symbol C and atomic number 6. It is nonmetallic and tetravalence, tetraval ...
,
ammonia Ammonia is an inorganic chemical compound, compound of nitrogen and hydrogen with the chemical formula, formula . A Binary compounds of hydrogen, stable binary hydride, and the simplest pnictogen hydride, ammonia is a colourless gas with a dis ...
, and
benzene Benzene is an Organic compound, organic chemical compound with the Chemical formula#Molecular formula, molecular formula C6H6. The benzene molecule is composed of six carbon atoms joined in a planar Ring (chemistry), ring with one hydrogen atom ...
. Many of these cover temperature and pressure ranges that encompass gas, liquid, and supercritical phases. Thermophysical modeling software often relies on reference correlations for predicting thermal conductivity at user-specified temperature and pressure. These correlations may be proprietary. Examples are REFPROP (proprietary) and CoolProp (open-source). Thermal conductivity can also be computed using the Green-Kubo relations, which express transport coefficients in terms of the statistics of molecular trajectories. The advantage of these expressions is that they are formally exact and valid for general systems. The disadvantage is that they require detailed knowledge of particle trajectories, available only in computationally expensive simulations such as molecular dynamics. An accurate model for interparticle interactions is also required, which may be difficult to obtain for complex molecules.

## In solids

* Copper in heat exchangers *
Heat pump A heat pump is a device that can heat a building (or part of a building) by transferring thermal energy from the outside using a Heat pump and refrigeration cycle, refrigeration cycle. Many heat pumps can also operate in the opposite direction, ...
*
Heat transfer Heat transfer is a discipline of thermal engineering that concerns the generation, use, conversion, and exchange of thermal energy (heat In thermodynamics, heat is defined as the form of energy crossing the boundary of a thermodynamic s ...
* Heat transfer mechanisms * Insulated pipes * Interfacial thermal resistance * Laser flash analysis * List of thermal conductivities * Phase-change material * R-value (insulation) *
Specific heat In thermodynamics, the specific heat capacity (symbol ) of a substance is the heat capacity of a sample of the substance divided by the mass of the sample, also sometimes referred to as massic heat capacity. Informally, it is the amount of heat t ...
* Thermal bridge * Thermal conductance quantum * Thermal contact conductance * Thermal diffusivity * Thermal effusivity * Thermal entrance length * Thermal interface material * Thermal rectifier * Thermal resistance in electronics * Thermistor * Thermocouple *
Thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed b ...
* Thermal conductivity measurement * Refractory metals

# References

## Sources

*

*. A standard, modern reference. * * * *

*Halliday, David; Resnick, Robert; & Walker, Jearl (1997). ''Fundamentals of Physics'' (5th ed.). John Wiley and Sons, New York . An elementary treatment. *. A brief, intermediate-level treatment. *. An advanced treatment.