As such, thermal conductivity is difficult to predict from first-principles. Any expressions for thermal conductivity which are exact and general, e. 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 mechanism dominates in non-metallic solids. In liquids, by contrast, the precise microscopic mechanisms of thermal conduction are poorly understood. In a simplified model of a dilute monatomic gas, molecules are modeled as rigid spheres which are in constant motion, colliding elastically with each other and with the walls of their container.

Under these assumptions, an elementary calculation yields for the thermal conductivity. For most gases, this prediction agrees well with experiments at pressures up to about 10 atmospheres. 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. More complex interaction laws introduce a weak temperature dependence. For monatomic gases, such as the noble gases , the agreement with experiment is fairly good.

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An explicit treatment of this effect is difficult in the Chapman-Enskog approach. In extremely dilute gases this assumption fails, and thermal conduction is described instead by an apparent thermal conductivity which decreases with density. For this reason a vacuum is an effective insulator. 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 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.

Elementary calculations then lead to the expression. This is commonly called Bridgman's equation. 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.

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So the only temperature-dependent quantity is the heat capacity c , which, in this case, is proportional to T. For pure metals such as copper, silver, etc. 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. Heat transport in both amorphous and crystalline dielectric solids is by way of elastic vibrations of the lattice phonons.

This transport mode is limited by the elastic scattering of acoustic phonons at lattice defects. The phonon mean free path has been associated directly with the effective relaxation length for processes without directional correlation.

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Since longitudinal waves have a much greater phase velocity than transverse waves, [39] 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. 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.

## 5. Thermal insulation materials, technical characteristics and selection criteria

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. Each phonon mode can be split into one longitudinal and two transverse polarization branches. 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. 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. 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.

At steady state conditions and local thermal equilibrium are assumed we get the following equation. 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.

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 q 2 and q 3 points outside of the Brillouin zone the momentum is conserved and the process is normal scattering N-process.

To U-process 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. Only momentum non-conserving processes can cause thermal resistance. 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.

### Supplementary files

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.

Specific thermal conductivity is a material property used to compare the heat-transfer ability of different materials to each other. Absolute thermal conductivity , however, is a component property used to compare the heat-transfer ability of different components to each other. Components, as opposed to materials, take into account size and shape, including basic properties such as thickness and area, instead of just material type. In this way, thermal-transfer ability of components of the same physical dimensions, but made of different materials, may be compared and contrasted, or components of the same material, but with different physical dimensions, may be compared and contrasted.

It is therefore often-times 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 material thermal conductivity into absolute thermal resistance values for use in thermal resistance calculations. Heat conduction or thermal conduction is the movement of heat from one object to another, that has a different temperature, through physical contact.

Heat can be transferred in three ways: conduction, convection and radiation. Heat flows from the object with the higher temperature to the colder one. Thermal transfer takes place at the molecular level, when heat energy is absorbed by a surface and causes microscopic collisions of particles and movement of electrons within that body. The process of heat conduction mainly depends on the temperature gradient the temperature difference between the bodies , the path length and the properties of the materials involved.

Not all substances are good heat conductors - metals, for example, are considered good conductors as they quickly transfer heat, but materials like wood or paper are viewed as poor conductors of heat. Materials that are poor conductors of heat are referred to as insulators. Some of the potential applications for graphene-enabled thermal management include electronics, which could greatly benefit from graphene's ability to dissipate heat and optimize electronic function.

In micro- and nano-electronics, heat is often a limiting factor for smaller and more efficient components. Therefore, graphene and similar materials with exceptional thermal conductivity may hold an enormous potential for this kind of applications. Applied Graphene Materials recently added new adhesive materials to their portfolio, aimed at the Space and Defense sectors.

Read the full story Posted: Sep 10, Researchers gain a better understanding of heat distribution processes Understanding atomic level processes can open a wide range of prospects in nanoelectronics and material engineering. A team of scientists from Peter the Great St.