Heat Transfer in Bulk Matter

The heat in materials is transferred from high to low temperature areas. The heat flow can be described by Fourier’s law of heat conduction:

= − ∙ (2.2)

where κ is the thermal conductivity, q the heat flux density and ∇T the temperature gradient. The law of heat conduction states that the time rate of heat transfer through a material is proportional to the negative temperature gradient (δT/δx), which is the driving force.

Three different mechanisms exist for thermal transport: radiative (mediated by photons), convective (mediated by the physical movement of heat carriers), and conductive.⁷⁴ For many materials, like colloidal assemblies, the dominant mechanism is thermal conduction.

Crystalline materials

In dielectric, crystalline materials, the thermal energy is conducted by propagating lattice vibrations with a set of characteristic frequencies. The energies of the lattice vibrations are quantized by

= +1

2 ℎ (2.3)

where n is an integer, which is zero at 0 °C and h is the Planck’s constant. If the solid is heated/cooled, E will increase/decrease in integer steps of hv. The similarity between this process and the absorption/emission of light (photons) is evident. Therefore, the quantum unit of lattice vibrational energy is called a phonon.

By treating phonons as quasi-particles, the thermal conductivity can be described by the Debye equation:

= 1

3 (2.4)

where Cν is the heat capacity, νg the phonon velocity, and Λ the average phonon mean free path (MFP), which is the average traveling distance between two successive collisions.

Table 2.1. Typical heat transfer properties at 293 K: Lattice spacing l, Debye temperature θD, phonon MFP Λ, and thermal conductivity κ.^75-76

θD [K] Λ [nm] κ [W/mK]

Diamond^c 1860 315 2300

Sapphire^c 600 4.0 46

Silicon^c 645 43 153

Silica^a 290 0.6 1.4

c crystalline structure, ^a amorphous structure.

The phonon MFP can range from 1 nm to more than 1 µm, depending on the material and its temperature and as a result on the existing scattering mechanisms (see Table 2.1). Conceivable interactions are (i) phonon-phonon scattering, (ii) boundary scattering (iii) impurity scattering, (iv) phonon-imperfection scattering, and (v) phonon-grain boundary scattering. The different scattering mechanisms are depicted in Figure 2.4a.

Figure 2.4. (a) Phonon scattering mechanism in dielectric materials: Phonon-phonon scattering, phonon-boundary scattering, phonon-impurity scattering, phonon-imperfection scattering, and phonon-grain boundary scattering. Adapted from Ashegi⁷⁷ with permission fromAIP Publishing. (b) Typical temperature-dependence of dielectric materials. Adapted from Kaviany⁷⁸ with permission from Cambridge University Press.

Phonon-phonon collisions result from the anharmonicity of the lattice potential and dominate in pure crystals. The process can be divided into the normal process (N-process) and the Umklapp process (U-process).⁷⁹ The U-process is necessary to obtain thermal equilibrium, due to the formation of a thermal resistance (i.e., finite thermal conductivity). This process is almost entirely dominating at high temperatures. The probability of these processes decreases with decreasing temperatures. As a result, the phonon MFP and the thermal conductivity increase with T^-1 until reaching a maximum at about 10 % of the Debye temperature θD

(compare Table 2.1).⁸⁰ Further temperature reduction leads to a decrease of the thermal conductivity, following the T³ dependence of the heat capacity.⁸¹ In this temperature regime, boundary scattering dominates. The location of the maximum depends on the sample size, defects, and crystal size. In nanostructured materials, the maximum is lower in magnitude and shifted to higher temperatures.^{74, 81}

At very high temperatures, the thermal conductivity is independent of the temperature leading to a minimum thermal conductivity value due to the MFP being comparable to the interatomic spacing. However, most materials melt below this temperature. A typical temperature dependency of a dielectric crystalline material in comparison to an amorphous one is depicted in Figure 2.4b. The

thermal conductivity in amorphous materials is significantly reduced compared to crystalline materials. Furthermore, the temperature-dependency is different. This is discussed in the following section, covering heat transport in amorphous materials.

Amorphous materials

Amorphous materials exhibit no translational symmetry or periodicity. The structural defects/disorder leads to significant wave scatterings and even localization of the wave propagation.⁸² Thus, the thermal conductivity is strongly reduced (e.g., amorphous silicon has a thermal conductivity of ~1 – 4 W m^-1 K^-1, while its crystalline counterpart has a thermal conductivity of 148 W m^-1 K^-1).⁸¹ Therefore, the vibrational modes cannot be longer named extended phonon waves.

A new concept arose by Allen and Feldman to describe disordered vibrational modes (vibrons).^83-84 They classify these vibrons in propagons, diffusons, and locons. Propagons are propagating and delocalized (i.e., phonon-like), and are typically found in the low-frequency range of the vibrational spectrum (see the spectrum of vibrons for a model of amorphous silicon, Figure 2.5). Diffusons are non-propagating and delocalized modes, located in the middle-frequency range.

Propagons and diffusons are also called extendons due to their delocalization. The mobility edge marks the transition from delocalized to localized modes (locons).

Locons are non-propagating and spatially localized modes and are typically found in the high frequency range of the vibrational spectrum.

The fundamental differences in the vibrating modes between amorphous and crystalline materials lead to completely different temperature dependencies of the thermal conductivity.⁸⁵ While the thermal conductivity of crystalline materials decreases at high temperatures, the thermal conductivity of amorphous materials increases monotonically over the whole temperature range.⁸⁶ Moreover, the dependency at low temperatures is different for disordered materials. They show a T²-dependence, instead of a T³-dependence.⁸⁷ The increase at high temperatures

results not only from the increased heat capacity but also from stronger anharmonic coupling between the modes.^{85, 88}

Figure 2.5. Vibrational spectrum of a model of amorphous silicon. Adapted from Allen et al.⁸⁴ with permission from Taylor & Francis Group.