Heat transfer physics

The state of energy stored within matter, or transported by the carriers, is described by a combination of classical and quantum statistical mechanics.

Once states and kinetics of the energy conversion and thermophysical properties are known, the fate of heat transfer is described by the above equation.

[9] Variety of ab initio (Latin for from the beginning) solvers (software) exist (e.g., ABINIT, CASTEP, Gaussian, Q-Chem, Quantum ESPRESSO, SIESTA, VASP, WIEN2k).

[12][13] At yet larger length scales (mesoscale, involving many mean free paths), the Boltzmann transport equation (BTE) which is based on the classical Hamiltonian-statistical mechanics is applied.

The occupation has equilibrium distributions (the known boson, fermion, and Maxwell–Boltzmann particles) and transport of energy (heat) is due to nonequilibrium (cause by a driving force or potential).

The relaxation time (or its inverse which is the interaction rate) is found from other calculations (ab initio or MD) or empirically.

[14] Depending on the length and time scale, the proper level of treatment (ab initio, MD, or BTE) is selected.

Heat transfer physics analyses may involve multiple scales (e.g., BTE using interaction rate from ab initio or classical MD) with states and kinetic related to thermal energy storage, transport and transformation.

So, heat transfer physics covers the four principal energy carries and their kinetics from classical and quantum mechanical perspectives.

This enables multiscale (ab initio, MD, BTE and macroscale) analyses, including low-dimensionality and size effects.

where di is the displacement vector of atom i, and Γ is the spring (or force) constant as the second-order derivatives of the potential.

[17][19][21][22][23][24][25] Using the single-mode relaxation time approximation (∂fp′/∂t|s = −fp′/τp) and the gas kinetic theory, Callaway phonon (lattice) conductivity model as[21][26]

Solids with more than one atom in the smallest unit cell representing the lattice have two types of phonons, i.e., acoustic and optical.

Hydrogen-like atoms (a nucleus and an electron) allow for closed-form solution to Schrödinger equation with the electrostatic potential (the Coulomb law).

where me is the electron mass, and the periodic potential is expressed as φc (x) = Σg φgexp[i(g∙x)] (g: reciprocal lattice vector).

In practice, a lattice as many-body systems includes interactions between electrons and nuclei in potential, but this calculation can be too intricate.

DFT is widely used in ab initio software (ABINIT, CASTEP, Quantum ESPRESSO, SIESTA, VASP, WIEN2k, etc.).

In general, the heat capacity of electron is small except at very high temperature when they are in thermal equilibrium with phonons (lattice).

Also, study of interaction with photons is central in optoelectronic applications (i.e. light-emitting diode, solar photovoltaic cells, etc.).

The net motion of particles (under gravity or external pressure) gives rise to the convection heat flux qu = ρfcp,fufT.

where ⟨uf2⟩1/2 is the RMS (root mean square) thermal velocity (3kBT/m from the MB distribution function, m: atomic mass) and τf-f is the relaxation time (or intercollision time period) [(21/2π d2nf ⟨uf⟩)−1 from the gas kinetic theory, ⟨uf⟩: average thermal speed (8kBT/πm)1/2, d: the collision diameter of fluid particle (atom or molecule), nf: fluid number density].

kf is also calculated using molecular dynamics (MD), which simulates physical movements of the fluid particles with the Newton equations of motion (classical) and force field (from ab initio or empirical properties).

For calculation of kf, the equilibrium MD with Green–Kubo relations, which express the transport coefficients in terms of integrals of time correlation functions (considering fluctuation), or nonequilibrium MD (prescribing heat flux or temperature difference in simulated system) are generally employed.

The quanta EM wave (photon) energy of angular frequency ωph is Eph = ħωph, and follows the Bose–Einstein distribution function (fph).

where ee and be are the electric and magnetic fields of the EM radiation, εo and μo are the free-space permittivity and permeability, V is the interaction volume, ωph,α is the photon angular frequency for the α mode and cα† and cα are the photon creation and annihilation operators.

Lasers range far-infrared to X-rays/γ-rays regimes based on the resonant transition (stimulated emission) between electronic energy states.

[39] Near-field radiation from thermally excited dipoles and other electric/magnetic transitions is very effective within a short distance (order of wavelength) from emission sites.

[49][50][51] Using ab initio calculations based on the first principles along with EM theory, various radiative properties such as dielectric function (electrical permittivity, εe,ω), spectral absorption coefficient (σph,ω), and the complex refraction index (mω), are calculated for various interactions between photons and electric/magnetic entities in matter.

where V is the unit-cell volume, VB and CB denote the valence and conduction bands, wκ is the weight associated with a κ-point, and pij is the transition momentum matrix element.

In another example, for the far IR regions where the optical phonons are involved, the dielectric function (εe,ω) are calculated as