Thermal conductivity

# Thermal conductivity

Overview
In physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

, thermal conductivity, , is the property of a material's ability to conduct
Heat conduction
In heat transfer, conduction is a mode of transfer of energy within and between bodies of matter, due to a temperature gradient. Conduction means collisional and diffusive transfer of kinetic energy of particles of ponderable matter . Conduction takes place in all forms of ponderable matter, viz....

heat
Heat
In physics and thermodynamics, heat is energy transferred from one body, region, or thermodynamic system to another due to thermal contact or thermal radiation when the systems are at different temperatures. It is often described as one of the fundamental processes of energy transfer between...

. It appears primarily in Fourier's Law for heat conduction
Heat conduction
In heat transfer, conduction is a mode of transfer of energy within and between bodies of matter, due to a temperature gradient. Conduction means collisional and diffusive transfer of kinetic energy of particles of ponderable matter . Conduction takes place in all forms of ponderable matter, viz....

.

Heat transfer across materials of high thermal conductivity occurs at a faster rate than across materials of low thermal conductivity.
Discussion

Encyclopedia
In physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

, thermal conductivity, , is the property of a material's ability to conduct
Heat conduction
In heat transfer, conduction is a mode of transfer of energy within and between bodies of matter, due to a temperature gradient. Conduction means collisional and diffusive transfer of kinetic energy of particles of ponderable matter . Conduction takes place in all forms of ponderable matter, viz....

heat
Heat
In physics and thermodynamics, heat is energy transferred from one body, region, or thermodynamic system to another due to thermal contact or thermal radiation when the systems are at different temperatures. It is often described as one of the fundamental processes of energy transfer between...

. It appears primarily in Fourier's Law for heat conduction
Heat conduction
In heat transfer, conduction is a mode of transfer of energy within and between bodies of matter, due to a temperature gradient. Conduction means collisional and diffusive transfer of kinetic energy of particles of ponderable matter . Conduction takes place in all forms of ponderable matter, viz....

.

Heat transfer across materials of high thermal conductivity occurs at a faster rate than across materials of low thermal conductivity. Correspondingly materials of high thermal conductivity are widely used in heat sink
Heat sink
A heat sink is a term for a component or assembly that transfers heat generated within a solid material to a fluid medium, such as air or a liquid. Examples of heat sinks are the heat exchangers used in refrigeration and air conditioning systems and the radiator in a car...

applications and materials of low thermal conductivity are used as thermal insulation
Thermal insulation
Thermal insulation is the reduction of the effects of the various processes of heat transfer between objects in thermal contact or in range of radiative influence. Heat transfer is the transfer of thermal energy between objects of differing temperature...

.

Thermal conductivity of materials is temperature dependent. In general, materials become more conductive to heat as the average temperature increases.

The reciprocal of thermal conductivity is thermal resistivity.

## Units of thermal conductivity

In the International System of Units
International System of Units
The International System of Units is the modern form of the metric system and is generally a system of units of measurement devised around seven base units and the convenience of the number ten. The older metric system included several groups of units...

(SI), thermal conductivity is measured in watts per meter kelvin (W/(m·K)).

In the imperial system of measurement thermal conductivity is measured in Btu/(hr·ft⋅F) where 1 Btu/(hr·ft⋅F) = 1.730735 W/(m·K). [Perry's Chemical Engineers' Handbook, 7th Edition, Table 1-4]

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 units such as the R-Value (resistance value) and the U-Value (thermal transmittance
Thermal transmittance
Thermal transmittance, also known as U-value, is the rate of transfer of heat through one square metre of a structure divided by the difference in temperature across the structure. It is expressed in watts per square metre per kelvin, or W/m²K...

). Although related to the thermal conductivity of a product R and U-values are dependent on the thickness of a product.

Likewise the textile industry has several units including the Tog
Tog (unit)
The tog is a measure of thermal resistance of a unit area, also known as thermal insulance, commonly used in the textile industry, and often seen quoted on, for example, duvets and carpet underlay....

and the Clo
Thermal comfort
Thermal comfort is a term used by the American Society of Heating, Refrigerating and Air-Conditioning Engineers, an international body. It is defined as the state of mind in humans that expresses satisfaction with the surrounding environment...

which express thermal resistance of a material in a way analogous to the R-values used in the construction industry.

Note: R-Values and U-Values quoted in the US (based on the imperial units of measurement) do not correspond with and are not compatible with those used in Europe (based on the SI units of measurement).

## Measurement

There are a number of ways to measure thermal conductivity. Each of these is suitable for a limited range of materials, depending on the thermal properties and the medium temperature. There is a distinction between steady-state and transient techniques.

In general, steady-state techniques are useful when the temperature of the material does not change with time. This makes the signal analysis straightforward (steady state implies constant signals). The disadvantage is that a well-engineered experimental setup is usually needed. The Divided Bar (various types) is the most common device used for consolidated rock samples.

The transient techniques perform a measurement during the process of heating up. Their advantage is quicker measurements. Transient methods are usually carried out by needle probes. A method described by Angstrom involves rapidly cycling the temperature from hot to cold and back and measuring the temperature change as the heat propagates along a thin strip of material in a vacuum.

## Definitions

The reciprocal of thermal conductivity is thermal resistivity, usually measured in kelvin-meters per watt (K·m·W−1). When dealing with a known amount of material, its thermal conductance and the reciprocal property, thermal resistance, can be described. Unfortunately, there are differing definitions for these terms.

### Conductance

For general scientific use, thermal conductance is 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 this is kA/L, measured in W·K−1 (equivalent to: W/°C). Thermal conductivity and conductance are analogous
Analogy
Analogy is a cognitive process of transferring information or meaning from a particular subject to another particular subject , and a linguistic expression corresponding to such a process...

to electrical conductivity (A·m−1·V−1) and electrical conductance (A·V−1).

There is also a measure known as heat transfer coefficient: the quantity of heat that passes in unit time through unit area of a plate of particular thickness when its opposite faces differ in temperature by one kelvin. The reciprocal is thermal insulance. In summary:
• thermal conductance = kA/L, measured in W·K−1
• thermal resistance = L /(kA), measured in K·W−1 (equivalent to: °C/W)
• 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

### Resistance

It is a thermal-property of a material to resist the flow of heat.

It is a resistance offered by a material(a metal in general and a heat sink material in particular) to the conduction or flow of heat through it.

Lesser the Thermal Resistance better will be the heat conduction and vice versa.

When thermal resistances occur in series
Series and parallel circuits
Components of an electrical circuit or electronic circuit can be connected in many different ways. The two simplest of these are called series and parallel and occur very frequently. Components connected in series are connected along a single path, so the same current flows through all of the...

, they are additive. So when heat flows through two components each with a resistance of 1 °C/W, the total resistance is 2 °C/W.

A common engineering design problem involves the selection of an appropriate sized heat sink
Heat sink
A heat sink is a term for a component or assembly that transfers heat generated within a solid material to a fluid medium, such as air or a liquid. Examples of heat sinks are the heat exchangers used in refrigeration and air conditioning systems and the radiator in a car...

for a given heat source. Working in units of thermal resistance greatly simplifies the design calculation. The following formula can be used to estimate the performance:
where:
• Rhs is the maximum thermal resistance of the heat sink to ambient, in °C/W (equivalent to K/W)
• is the temperature difference (temperature drop), in °C
• Pth is the thermal power (heat flow), in watts
• Rs is the thermal resistance of the heat source, in °C/W

For example, if a component produces 100 W of heat, and has a thermal resistance of 0.5 °C/W, what is the maximum thermal resistance of the heat sink? Suppose the maximum temperature is 125 °C, and the ambient temperature is 25 °C; then the is 100 °C. The heat sink's thermal resistance to ambient must then be 0.5 °C/W or less.

### Transmittance

A third term, thermal transmittance, incorporates the thermal conductance of a structure along with heat transfer due to convection
Convection
Convection is the movement of molecules within fluids and rheids. It cannot take place in solids, since neither bulk current flows nor significant diffusion can take place in solids....

Thermal radiation is electromagnetic radiation generated by the thermal motion of charged particles in matter. All matter with a temperature greater than absolute zero emits thermal radiation....

. It is measured in the same units as thermal conductance and is sometimes known as the composite thermal conductance. The term U-value is another synonym.

### Temperature

The effect of temperature on thermal conductivity is different for metals and nonmetals. In metals conductivity is primarily due to lattice vibrations and free electron, however, free electrons play a dominant role. Therefore any increase in temperature increases the lattice vibrations but affects the movement of free electrons adversely thereby decreasing the conductivity. On the other hand conductivity in nonmetals is only due to lattice vibrations which increases with increasing temperature, and so the conductivity of nonmetals increases with increasing temperature.

### Material phase

When a material undergoes a phase change from solid to liquid or from liquid to gas the thermal conductivity may change. An example of this would be the change in thermal conductivity that occurs when ice (thermal conductivity of 2.18 W/(m·K) at 0 °C) melts into liquid water (thermal conductivity of 0.58 W/(m·K) at 0 °C).

### Material structure

Pure crystalline substances exhibit very different thermal conductivities along different crystal axes, due to differences in phonon
Phonon
In physics, a phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, such as solids and some liquids...

coupling along a given crystal axis. Sapphire
Sapphire
Sapphire is a gemstone variety of the mineral corundum, an aluminium oxide , when it is a color other than red or dark pink; in which case the gem would instead be called a ruby, considered to be a different gemstone. Trace amounts of other elements such as iron, titanium, or chromium can give...

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.

### Electrical conductivity

In metal
Metal
A metal , is an element, compound, or alloy that is a good conductor of both electricity and heat. Metals are usually malleable and shiny, that is they reflect most of incident light...

s, thermal conductivity approximately tracks electrical conductivity according to the Wiedemann-Franz law, as freely moving valence electron
Valence electron
In chemistry, valence electrons are the electrons of an atom that can participate in the formation of chemical bonds with other atoms. Valence electrons are the "own" electrons, present in the free neutral atom, that combine with valence electrons of other atoms to form chemical bonds. In a single...

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
Phonon
In physics, a phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, such as solids and some liquids...

carriers for heat in non-metals. As shown in the table below, highly electrically conductive silver
Silver
Silver is a metallic chemical element with the chemical symbol Ag and atomic number 47. A soft, white, lustrous transition metal, it has the highest electrical conductivity of any element and the highest thermal conductivity of any metal...

is less thermally conductive than diamond
Diamond
In mineralogy, diamond is an allotrope of carbon, where the carbon atoms are arranged in a variation of the face-centered cubic crystal structure called a diamond lattice. Diamond is less stable than graphite, but the conversion rate from diamond to graphite is negligible at ambient conditions...

, which is an electrical insulator.

### Convection

Air and other gases are generally good insulators, in the absence of convection. Therefore, many insulating materials function simply by having a large number of gas-filled pockets which prevent large-scale convection. Examples of these include expanded and extruded polystyrene
Polystyrene
Polystyrene ) also known as Thermocole, abbreviated following ISO Standard PS, is an aromatic polymer made from the monomer styrene, a liquid hydrocarbon that is manufactured from petroleum by the chemical industry...

(popularly referred to as "styrofoam") and silica aerogel
Aerogel
Aerogel is a synthetic porous material derived from a gel, in which the liquid component of the gel has been replaced with a gas. The result is a solid with extremely low density and thermal conductivity...

. Natural, biological insulators such as fur and feathers achieve similar effects by dramatically inhibiting convection of air or water near an animal's skin.
Light gases, such as hydrogen
Hydrogen
Hydrogen is the chemical element with atomic number 1. It is represented by the symbol H. With an average atomic weight of , hydrogen is the lightest and most abundant chemical element, constituting roughly 75% of the Universe's chemical elemental mass. Stars in the main sequence are mainly...

and helium
Helium
Helium is the chemical element with atomic number 2 and an atomic weight of 4.002602, which is represented by the symbol He. It is a colorless, odorless, tasteless, non-toxic, inert, monatomic gas that heads the noble gas group in the periodic table...

typically have high thermal conductivity. Dense gases such as xenon
Xenon
Xenon is a chemical element with the symbol Xe and atomic number 54. The element name is pronounced or . A colorless, heavy, odorless noble gas, xenon occurs in the Earth's atmosphere in trace amounts...

and dichlorodifluoromethane
Dichlorodifluoromethane
Dichlorodifluoromethane , is a colorless gas, and usually sold under the brand name Freon-12, is a chlorofluorocarbon halomethane , used as a refrigerant and aerosol spray propellant. Complying with the Montreal Protocol, its manufacture was banned in the United States along with many other...

have low thermal conductivity. An exception, sulfur hexafluoride
Sulfur hexafluoride
Sulfur hexafluoride is an inorganic, colorless, odorless, and non-flammable greenhouse gas. has an octahedral geometry, consisting of six fluorine atoms attached to a central sulfur atom. It is a hypervalent molecule. Typical for a nonpolar gas, it is poorly soluble in water but soluble in...

, a dense gas, has a relatively high thermal conductivity due to its high heat capacity
Heat capacity
Heat capacity , or thermal capacity, is the measurable physical quantity that characterizes the amount of heat required to change a substance's temperature by a given amount...

. Argon
Argon
Argon is a chemical element represented by the symbol Ar. Argon has atomic number 18 and is the third element in group 18 of the periodic table . Argon is the third most common gas in the Earth's atmosphere, at 0.93%, making it more common than carbon dioxide...

, a gas denser than air, is often used in insulated glazing
Insulated glazing
Insulated glazing also known as double glazing are double or triple glass window panes separated by an air or other gas filled space to reduce heat transfer across a part of the building envelope....

(double paned windows) to improve their insulation characteristics.

## Experimental values

Thermal conductivity is important in building insulation
Building insulation
building insulation refers broadly to any object in a building used as insulation for any purpose. While the majority of insulation in buildings is for thermal purposes, the term also applies to acoustic insulation, fire insulation, and impact insulation...

and related fields. However, materials used in such trades are rarely subjected to chemical purity standards. Several construction materials' k values are listed below. These should be considered approximate due to the uncertainties related to material definitions. In the opposite end of the spectrum, solutions for computer cooling
Computer cooling
Computer cooling is required to remove the waste heat produced by computer components, to keep components within their safe operating temperature limits.Various cooling methods help to improve processor performance or reduce the noise of cooling fans....

usually use high thermal capacity materials such as silver, copper and aluminium, to cool down specific components.

The following table is meant as a small sample of data to illustrate the thermal conductivity of various types of substances. For more complete listings of measured k-values, see the references.

This is a list of approximate values of thermal conductivity, k, for some common materials. Please consult the list of thermal conductivities for more accurate values, references and detailed information.
Material
Material
Material is anything made of matter, constituted of one or more substances. Wood, cement, hydrogen, air and water are all examples of materials. Sometimes the term "material" is used more narrowly to refer to substances or components with certain physical properties that are used as inputs to...

Thermal conductivity
[W/(m·K)]
Silica Aerogel
Aerogel
Aerogel is a synthetic porous material derived from a gel, in which the liquid component of the gel has been replaced with a gas. The result is a solid with extremely low density and thermal conductivity...

0.004 - 0.04
Air 0.025
Wood
Wood
Wood is a hard, fibrous tissue found in many trees. It has been used for hundreds of thousands of years for both fuel and as a construction material. It is an organic material, a natural composite of cellulose fibers embedded in a matrix of lignin which resists compression...

0.04 - 0.4
Hollow Fill Fibre Insulation 0.042
Alcohol
Alcohol
In chemistry, an alcohol is an organic compound in which the hydroxy functional group is bound to a carbon atom. In particular, this carbon center should be saturated, having single bonds to three other atoms....

s and oil
Oil
An oil is any substance that is liquid at ambient temperatures and does not mix with water but may mix with other oils and organic solvents. This general definition includes vegetable oils, volatile essential oils, petrochemical oils, and synthetic oils....

s
0.1 - 0.21
Polypropylene
Polypropylene
Polypropylene , also known as polypropene, is a thermoplastic polymer used in a wide variety of applications including packaging, textiles , stationery, plastic parts and reusable containers of various types, laboratory equipment, loudspeakers, automotive components, and polymer banknotes...

0.25
Mineral oil
Mineral oil
A mineral oil is any of various colorless, odorless, light mixtures of alkanes in the C15 to C40 range from a non-vegetable source, particularly a distillate of petroleum....

0.138
Rubber
Rubber
Natural rubber, also called India rubber or caoutchouc, is an elastomer that was originally derived from latex, a milky colloid produced by some plants. The plants would be ‘tapped’, that is, an incision made into the bark of the tree and the sticky, milk colored latex sap collected and refined...

0.16
LPG 0.23 - 0.26
Cement
Cement
In the most general sense of the word, a cement is a binder, a substance that sets and hardens independently, and can bind other materials together. The word "cement" traces to the Romans, who used the term opus caementicium to describe masonry resembling modern concrete that was made from crushed...

, Portland
0.29
Epoxy
Epoxy
Epoxy, also known as polyepoxide, is a thermosetting polymer formed from reaction of an epoxide "resin" with polyamine "hardener". Epoxy has a wide range of applications, including fiber-reinforced plastic materials and general purpose adhesives....

(silica-filled)
0.30
Epoxy (unfilled) 0.12 - 0.177
Water
Water
Water is a chemical substance with the chemical formula H2O. A water molecule contains one oxygen and two hydrogen atoms connected by covalent bonds. Water is a liquid at ambient conditions, but it often co-exists on Earth with its solid state, ice, and gaseous state . Water also exists in a...

(liquid)
0.6
Thermal grease
Thermal grease
Thermal grease is a viscous fluid substance, originally with properties akin to grease, which increases the thermal conductivity of a thermal interface by filling...

0.7 - 3
Thermal epoxy 1 - 7
Glass
Glass
Glass is an amorphous solid material. Glasses are typically brittle and optically transparent.The most familiar type of glass, used for centuries in windows and drinking vessels, is soda-lime glass, composed of about 75% silica plus Na2O, CaO, and several minor additives...

1.1
Soil
Soil
Soil is a natural body consisting of layers of mineral constituents of variable thicknesses, which differ from the parent materials in their morphological, physical, chemical, and mineralogical characteristics...

1.5
Concrete
Concrete
Concrete is a composite construction material, composed of cement and other cementitious materials such as fly ash and slag cement, aggregate , water and chemical admixtures.The word concrete comes from the Latin word...

, stone
1.7
Ice
Ice
Ice is water frozen into the solid state. Usually ice is the phase known as ice Ih, which is the most abundant of the varying solid phases on the Earth's surface. It can appear transparent or opaque bluish-white color, depending on the presence of impurities or air inclusions...

2
Sandstone
Sandstone
Sandstone is a sedimentary rock composed mainly of sand-sized minerals or rock grains.Most sandstone is composed of quartz and/or feldspar because these are the most common minerals in the Earth's crust. Like sand, sandstone may be any colour, but the most common colours are tan, brown, yellow,...

2.4
Mercury
Mercury (element)
Mercury is a chemical element with the symbol Hg and atomic number 80. It is also known as quicksilver or hydrargyrum...

8.3
Stainless steel
Stainless steel
In metallurgy, stainless steel, also known as inox steel or inox from French "inoxydable", is defined as a steel alloy with a minimum of 10.5 or 11% chromium content by mass....

12.11 ~ 45.0
Lead is a main-group element in the carbon group with the symbol Pb and atomic number 82. Lead is a soft, malleable poor metal. It is also counted as one of the heavy metals. Metallic lead has a bluish-white color after being freshly cut, but it soon tarnishes to a dull grayish color when exposed...

35.3
Aluminium
Aluminium
Aluminium or aluminum is a silvery white member of the boron group of chemical elements. It has the symbol Al, and its atomic number is 13. It is not soluble in water under normal circumstances....

237 (pure)
120—180 (alloys)
Gold
Gold
Gold is a chemical element with the symbol Au and an atomic number of 79. Gold is a dense, soft, shiny, malleable and ductile metal. Pure gold has a bright yellow color and luster traditionally considered attractive, which it maintains without oxidizing in air or water. Chemically, gold is a...

318
Copper
Copper
Copper is a chemical element with the symbol Cu and atomic number 29. It is a ductile metal with very high thermal and electrical conductivity. Pure copper is soft and malleable; an exposed surface has a reddish-orange tarnish...

401
Silver
Silver
Silver is a metallic chemical element with the chemical symbol Ag and atomic number 47. A soft, white, lustrous transition metal, it has the highest electrical conductivity of any element and the highest thermal conductivity of any metal...

429
Diamond
Diamond
In mineralogy, diamond is an allotrope of carbon, where the carbon atoms are arranged in a variation of the face-centered cubic crystal structure called a diamond lattice. Diamond is less stable than graphite, but the conversion rate from diamond to graphite is negligible at ambient conditions...

900 - 2320
Graphene
Graphene
Graphene is an allotrope of carbon, whose structure is one-atom-thick planar sheets of sp2-bonded carbon atoms that are densely packed in a honeycomb crystal lattice. The term graphene was coined as a combination of graphite and the suffix -ene by Hanns-Peter Boehm, who described single-layer...

(4840±440) - (5300±480)

## Physical origins

Heat flux is exceedingly difficult to control and isolate in a laboratory setting. Thus at the atomic level, there are no simple, correct expressions for thermal conductivity. Atomically, the thermal conductivity of a system is determined by how atoms composing the system interact. There are two different approaches for calculating the thermal conductivity of a system.
• The first approach employs the Green-Kubo relations
Green-Kubo relations
The Green–Kubo relations give the exact mathematical expression for transport coefficients in terms of integrals of time correlation functions.-Thermal and mechanical transport processes:...

. Although this employs analytic expressions which in principle can be solved, calculating the thermal conductivity of a dense fluid or solid using this relation requires the use of molecular dynamics computer simulation.

• The second approach is based upon the relaxation time approach. Due to the anharmonicity within the crystal potential, the phonons in the system are known to scatter. There are three main mechanisms for scattering:
• Boundary scattering, a phonon hitting the boundary of a system;
• Mass defect scattering, a phonon hitting an impurity within the system and scattering;
• Phonon-phonon scattering, a phonon breaking into two lower energy phonons or a phonon colliding with another phonon and merging into one higher energy phonon.

### Lattice waves

Heat transport in both glassy and crystalline dielectric
Dielectric
A dielectric is an electrical insulator that can be polarized by an applied electric field. When a dielectric is placed in an electric field, electric charges do not flow through the material, as in a conductor, but only slightly shift from their average equilibrium positions causing dielectric...

solids occurs through elastic vibrations of the lattice (phonon
Phonon
In physics, a phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, such as solids and some liquids...

s). This transport is limited by elastic scattering of acoustic phonons by lattice defects. These predictions were confirmed by the experiments of Chang and Jones on commercial glasses and glass ceramics, where mean free paths were 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. Thus, if Vg  is the group velocity of a phonon wave packet, then the relaxation length is defined as:

where t is the characteristic relaxation time. Since longitudinal waves have a much greater phase velocity than transverse waves, Vlong is much greater than Vtrans, 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
Dispersion
Dispersion may refer to:In physics:*The dependence of wave velocity on frequency or wavelength:**Dispersion , for light waves**Dispersion **Acoustic dispersion, for sound waves...

), low-frequency phonons of long wavelength will be limited in relaxation length by elastic Rayleigh scattering
Rayleigh scattering
Rayleigh scattering, named after the British physicist Lord Rayleigh, is the elastic scattering of light or other electromagnetic radiation by particles much smaller than the wavelength of the light. The particles may be individual atoms or molecules. It can occur when light travels through...

. This type of light scattering form 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
Brillouin scattering
Brillouin scattering, named after Léon Brillouin, occurs when light in a medium interacts with time dependent optical density variations and changes its energy and path. The density variations may be due to acoustic modes, such as phonons, magnetic modes, such as magnons, or temperature gradients...

.
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 λL (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 3pq 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 3p(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 λL.

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. Micheline Roufosse and P.G. Klemens derived the exact proportionality in their article Thermal Conductivity of Complex Dielectric Crystals at Phys. Rev. B 7, 5379–5386 (1973). This was done by assuming that the relaxation time τ decreases with increasing number of atoms in the unit cell and ten scaling the parameters of the expression for thermal conductivity in high temperatures accordingly.

Describing of anharmonic effects is complicated because 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 following expression

where 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
Boltzmann equation
The Boltzmann equation, also often known as the Boltzmann transport equation, devised by Ludwig Boltzmann, describes the statistical distribution of one particle in rarefied gas...

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 steady-state conditions, the phonon thermal conductivity λL can be determined. The temperature dependence for λL originates from the variety of processes, whose significance for λL depends on the temperature range of interest. Mean free path is one factor that determines the temperature dependence for λL, as stated in the following equation

where Λ is the mean free path for phonon. This equation is a result of combining the four previous equations with each other and knowing that for cubic or isotropic systems and .
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 due to the fact that it is only defined within an arbitrary reciprocal lattice vector. At higher temperatures (10 K and quasimomentum , 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 . 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 , which is a significant fraction of Debye energy that is needed to generate new phonons. The probability for this is proportional to , with . 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 λL, 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 λL has a temperature dependence T-1, to which one arrives from formula by making the following approximation
and writing . 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/structures.

### Electronic thermal conductivity

Hot electrons from higher energy states carry more thermal energy than cold electrons, while electrical conductivity is rather insensitive to the energy distribution of carriers because the amount of charge that electrons carry, does not depend on their energy. This is a physical reason for the greater sensitivity of electronic thermal conductivity to energy dependence of density of states and relaxation time, respectively.

Mahan and Sofo have showed in their article The best thermoelectric (PNAS 1996 93 (15) 7436-7439) that materials with a certain electron structure have reduced electron thermal conductivity. Based on their analysis one can demonstrate that if the electron density of states in the material is close to the delta-function, the electronic thermal conductivity drops to zero. By taking the following equation , where λ0 is the electronic thermal conductivity when the electrochemical potential gradient inside the sample is zero, as a starting point. As next step the transport coefficients are written as following , , , where , with a0 as the Bohr’s radius. The dimensionless integrals In are defined as , where s(x) is the dimensionless transport distribution function. The integrals In are the moments of the function , x is the energy of carriers. By substituting the previous formulas for the transport coefficient to the equation for λE we get the following equation . From the previous equation we see that λE to be zero the bracketed term containing In terms have to be zero. Now if we assume that , where δ is the Dirac delta function, In terms get the following expressions ,
, . By substituting these expressions to the equation for λE, we see that it goes to zero. Therefore, P(x) has to be delta function.

## Equations

First, we define heat conduction, H:

where is the rate of heat flow, k is the thermal conductivity, A is the total cross sectional area of conducting surface, ΔT is temperature difference, and x is the thickness of conducting surface separating the two temperatures. Dimension of thermal conductivity = M1L1T−3K−1

Rearranging the equation gives thermal conductivity:

I.E. It is defined as the quantity of heat, ΔQ, transmitted during time Δt through a thickness x, in a direction normal to a surface of area A, per unit area of A, due to a temperature difference ΔT, under steady state conditions and when the heat transfer is dependent only on the temperature gradient.

Alternatively, it can be thought of as a flux
Flux
In the various subfields of physics, there exist two common usages of the term flux, both with rigorous mathematical frameworks.* In the study of transport phenomena , flux is defined as flow per unit area, where flow is the movement of some quantity per time...

of heat (energy per unit area per unit time) divided by a temperature gradient (temperature difference per unit length)