Why is so important the thermal conduction in ceramics and polymers?

Thermal properties

Heat flows across temperature differences. There are three modes of heat transfer: conduction, radiation, and convection. Conduction and radiation are fundamental physical mechanisms, while convection is really conduction as affected by fluid flow.

• Conductionis an exchange of energy by direct interaction between molecules of a substance containing temperature differences. It occurs in gases, liquids, or solids and has a strong basis in the molecular kinetic theory of physics.
• Radiationis a transfer of thermal energy in the form of electromagnetic waves emitted by atomic and subatomic agitation at the surface of a body. Like all electromagnetic waves (light, X-rays, microwaves), thermal radiation travels at the speed of light, passing most easily through a vacuum or a nearly “transparent” gas such as oxygen or nitrogen. Liquids, “participating” gases such as carbon dioxide and water vapor, and glasses transmit only a portion of incident radiation. Most other solids are essentially opaque to radiation.
• Convection may be described as conduction in a fluid as enhanced by the motion of the fluid. It may not be a truly independent mode

Figure 1: Models of heat transfer

Thermal Conductivity is a term analogous to electrical conductivity with a difference that it concerns with the flow of heat unlike current in the case of the latter. It points to the ability of a material to transport heat from one point to another without movement of the material as a whole, the more is the thermal conductivity the better it conducts the heat.  Generally the heat transfer in solid (conduction) has two components:

1. Lattice thermal conduction
2. Electronic thermal conduction

Both types of heat conduction occur in solids but one is dominant over the other depending upon the type of material.

Figure 2:  Valence band vs Conduction band

In case of insulating materials, lattice conduction contributes to heat conduction. This is mainly due to the fact that in insulators the electrons are tightly held by their parent atoms and free electrons do not exist. Hence the heat is transferred from one end to another by vibration of atoms held in the lattice structure. Obviously insulators are bad heat conductors since they do not possess enough heat transfer capability due to lack of free electrons.

However in case of metals we have large number of free electrons, and hence heat conduction is primarily due to electronic conduction. The free electrons of the metals can freely move throughout the solid and transfer the thermal energy at a rate very high as compared to insulators. It is due to this that the metals posses high thermal conductivity. It is also observed that among the metals the best electrical conductors also exhibits best thermal conductivity. Since both electrical as well as thermal conductivity is dependent on the free electrons, factors such as alloying effect both the properties.

Figure 3: Thermal conduction in metals

On the other hand, metals crystallize in three fundamentally crystalline systems (BBC, FCC, HCF). These crystal systems each provide a different packing density of the atoms. The metals that present the FCC structure have the highest thermal conductivity, since they present the highest compaction

Figure 4: Crystalline structure

Figure 5: Atomic packing factor

Thermal conduction in ceramics and polymers

Theory

In these materials, electrons are not free, so their contribution to electrical conductivity, and also to thermal conductivity, is practically nil except at very high temperatures. In this case, the heat transfer is due not to the movement of the electrons, but to the vibration of the lattice itself. This vibration is transmitted through the propagation of waves or, in the dual conception of quantum mechanics, through elementary corpuscles called phonons.

Phonons are very complicated, but conceptually, phonons are waves of vibrations in the lattice of a solid. Vibrational energy in one part of a solid will be transmitted to other parts of the solid as phonons. What limits how quickly heat is transferred through this mechanism is phonon scattering.  There are several phonon scattering mechanisms that can limit the value of the phonon mean free path. The dispersal mechanisms are:

• Interaction between phonons “Umklapp-processes”.
• Scattering of phonons by point defects (see figures 7-9) such as impurities, isotopes; crystal atoms with the same number of protons but different numbers of neutrons, etc. Scattering of phonons along the boundaries of the specimen or crystallites.
• Phonon scattering by dislocations.

Figure 6: Different types of phonon scattering

Figure 7: Defects in a crystal

Figure 8: Typical point defects in a crystal

Figure 9: Typical lineal defects in a crystal

Each mechanism has a mean free path associated with it; lunk, limp, lfro, ldis…, the values ​​of the different mean free paths can be combined to achieve a global mean free path (l) (The average distance that the phonons travel without being scattered or without interacting with each other is called the mean free path).

Eq 2

Thermal conductivity is strongly influenced by the different phonon scattering mechanisms that can manifest themselves in the transfer of thermal energy in solids. The frequency of occurrence of phonon scattering events largely determines thermal conductivity. This is why crystalline substances (like diamond) tend to have higher thermal conductivity than amorphous ones (like glass or polymers), as phonons are more dispersed when they don’t have a nice ordered crystalline structure through which to travel.

As with electrical conductivity, increases in temperature cause vibrations of greater amplitude and energy and also facilitates the excitation of some electrons towards the conduction band, which causes conductivity to increase with temperature. The following figure represents the previous hypothesis, in which the atoms vibrate around their equilibrium position, according to a harmonic oscillator model.

Figure 10: Vibration model

Atoms vibrate around their equilibrium position, given by the average distance of separation from their neighbors, with a specific frequency and amplitude. The characteristic vibration frequency of the harmonic oscillator obeys the expression:

Eq 3

where k is a constant related to the elastic deformations around the equilibrium point in the bond potential energy-interatomic distance graph, different in each material. The maximum amplitude of vibration A0 is a function of the temperature T:

Eq 4

Increases in temperature cause vibrations of greater amplitude and energy and also the excitation of some electrons towards the conduction band is facilitated, which causes the conductivity to increase with temperature. Conductivity increases with increasing temperature, E-modulus, and material density. Thermal conductivity in non-metals can be estimated by the relationship:

where:

E  is the modulus of elasticity
d  is density
Ce is the specific heat
d or l, is the average path of thermal vibration.

The value d, of the vibration average path, increases with the temperature of the material, which accounts for the observed increase in thermal conductivity when the material is heated. The above equation justifies several experimentally proven facts:

• Materials with more compact structures and higher modulus of elasticity have higher conductivities.
• Crystalline solids have higher conductivity than if they are amorphous.

Bonding Forces and Energies

Considering the interaction between two isolated atoms as they are brought into close proximity from an infinite separation.

Figure 11: Bonding Forces vs distance

Figure 12: Atomic mechanism of elastic deformation

Figure 13: Types of Bonds

Examples

Metal vs Metal oxides

Slack (1979) carried out a systematic comparison of calculated and measured thermal conductivities for face-centered cubic lattices. Lower thermal conductivities are observed when the difference in atomic masses between anions and cations increases or the number of atoms that make up the unit cell increases. Basically, a large mass difference improves the anharmonicity of the lattice vibrations, where the conductivities of the oxides decrease in the sequence BeO → MgO → Al2O3. Such arguments also apply to other classes of ceramics, such as nitrides or carbides.

Figure 14 : Ionic Bond/Electronegativity

The table shows some examples showing that thermal conductivity drops drastically in metal oxides

Table 1: Thermal conductivity of metals and their respective metal oxides

Figure 15: Thermal conductivity of metals at room temperature

Figure 16: Covalent bond

The metallic oxides have lower thermal conductivity than the starting metals.  That is due to the following factors:

• Electronegativity difference between atoms (ionic character)
• Crystal structure of crystallization (packing density)
• Defects in the crystal structure
• Mass difference between atoms

Figure 17: Comparison of Ionic and Covalent Bonding

The thermal conductivity of non-metallic crystals, including the majority of oxides, exhibits four distinct regimes when plotted against temperature. This is shown in the next Figure. The key to understanding the temperature dependence of thermal conductivity is to realize that the phonons responsible for carrying heat undergo scattering events proportional to the number of scattering sites present including defects and other phonons.

• At low temperatures, in the A or I regime, there is very little thermal energy to excite phonons. Therefore, the phonon mean-free-path is limited by the physical dimensions of the material, the grain size, and the spacing between dislocations. The conductivity in this region increases rapidly (∝ T³), thus reaching a region where thermal conductivity has its maximum value and which we define as region II. The T³ temperature dependence of thermal conductivity must then be explained by the heat capacity having a T³ dependence at very low temperatures.

• The B or II regime, characterized by a maximum in thermal conductivity occurs when the phonon mean-free-path of phonon-phonon collisions is equal to phonon defect collisions. Although this temperature will certainly vary with defect type and concentration, this value to be approximately 0.1θD-0.2 θ In region II, contributions are reduced due to the effect of grain size, dislocations and lack of harmony (anharmony) in the network.

• At temperatures above this peak, in the C or III regime, the thermal conductivity exhibits a 1/T dependence. This is because anharmonic phonon scattering, due to phonon-phonon interactions dominate. If more phonons are present then the likelihood of scattering should increase.

• Finally, at very high temperatures the D or IV regime tends towards having no temperature dependence. The physical explanation for this is that phonons are waves of displaced atoms and the mean-free-path cannot be decreased smaller than the distance between two neighboring atoms.

Figure 18: Thermal conductivity versus temperature for non metallic crystal. The four characteristics regimes correspond to different scattering processes

 Diamond vs graphite

Diamond is one of the best known thermal conductors, in fact, diamond is a better thermal conductor than many metals. The carbon atoms in diamond are attached to 4 other carbon atoms located at the vertices of a tetrahedron. Therefore, the bond in diamond is a continuous and uniform three-dimensional network of single C-C (sigma) bonds. Graphite, on the other hand, is formed from carbon atoms that form a continuous two-dimensional network of sigma and pi bonds. This two-dimensional lattice forms graphite sheets, but there is little connection between the sheets, in fact the sheet-to-sheet separation is a whopping ~ 3.4 angstroms.

This could lead us to suspect that the heat conduction in the two-dimensional graphite sheet would be higher than that of the diamond, but that the heat conduction between the graphite sheets would be very low. This is, in fact, an accurate description of thermal conduction in graphite. Thermal conductivity parallel to the graphite sheets = 1950 W/mK, but thermal conduction perpendicular to the sheet = 5.7 W/mK. Therefore, when we consider thermal conduction in all possible directions (anisotropic), diamond would be superior to graphite.

Figure 19 : Crystal structure of diamond and graphite

The reason why diamond, in particular, is an especially good thermal conductor, even compared to other well-ordered crystals, comes down to two factors: the mass of the carbon atoms and the strength of the bonds that connect them.

As much as a harmonic oscillator has a higher frequency with a stronger spring (the force of the C-C bonds) and a lighter mass (the C atoms), diamond can support higher-energy phonons. This means that at a given temperature, there will be fewer phonons near the material boundary in diamond than in a material with a smaller boundary. The details are not important, but when phonons near the boundary interact, they spread out in such a way that the phonons do not travel through the structure. Because the diamond boundary is so high, the probability of this type of scatter is less.

Phonons and electrons travel very well along the graphene sheets of graphite, but poorly between them, due to the weak interactions between layers and the large distance between the layers, which explains the anisotropy of their thermal and electronic conductivities.

Figure 20 : Allotropes of carbon

The same happens to crystallized boron nitride in the hexagonal system (similar to graphite), along the sheets there is a very good conduction but between layers the conductivity is low.

Figure 21: Structure of NB and Graphene

In the case of silicon carbide, despite having a diamond-like structure (covalent bonds and a crystalline structure throughout the network), its conductivity drops to a value of 490 W / mK

Silicium carbide                                               Diamond

Figure 22: Crystal structure of silicium carbide vs diamond

A material with an ambient temperature value of K greater than 100W / m K is considered a material with high thermal conductivity. At the high temperature limit, the conductivity expression as:

where B is a constant and ΘD is the Debye temperature. This result suggests four criteria for choosing materials with high thermal conductivity:

• low atomic mass,
• strong interatomic bond,
• simple crystal structure and
• low anharmonicity.

Conditions (i) and (ii) help to increase the quantity M¯ Θ3 D in the equation condition (iii) means a low number of atoms per unit cell, which results in fewer optical branches and therefore fewer anharmonic interactions, and condition (iv) means a reduction in the strength of anharmonic interactions. According to this suggestion, at least 12 semiconductors and insulators can be classified as high thermal conductivity materials. In order of decreasing conductivity at room temperature, these are: C (diamond), BN, SiC, BeO, BP, AlN, BeS, BAs, GaN, Si, AlP, and GaP. The following table shows the thermal conductivity of the following materials:

Table 2: K values ​​at room temperature for non-metallic single crystals with high thermal conductivity. Units are W / m K

From all of the above, the thermal conductivity depends on:

• Non-metallic materials thermal conductivity is due to phonons (vibrations of the crystal lattice)
• Metallic materials the fundamental mechanism of heat transmission is through electrons
• Type of crystal lattice. Dispersion of phonons in different directions of the crystal.
• Orderly structures of crystals
• Impurities
• Types of atoms (types of bonds: covalent, ionic … etc)
• Difference in masses between the different atoms.
• Single-element structures or elements of similar atomic weight (no distorted lattice)