Heat transfer physics

Heat transfer physics describes the kinetics of energy storage, transport, and energy transformation by principal energy carriers: phonons (lattice vibration waves), electrons, fluid particles, and photons. Heat is energy stored in temperature-dependent motion of particles including electrons, atomic nuclei, individual atoms, and molecules. Heat is transferred to and from matter by the principal energy carriers. The state of energy stored within matter, or transported by the carriers, is described by a combination of classical and quantum statistical mechanics. The energy is different made (converted) among various carriers.
The heat transfer processes (or kinetics) are governed by the rates at which various related physical phenomena occur, such as (for example) the rate of particle collisions in classical mechanics. These various states and kinetics determine the heat transfer, i.e., the net rate of energy storage or transport. Governing these process from the atomic level (atom or molecule length scale) to macroscale are the laws of thermodynamics, including conservation of energy.

Introduction

Heat is thermal energy associated with temperature-dependent motion of particles. The macroscopic energy equation for infinitesimal volume used in heat transfer analysis is
$\nabla \cdot \mathbf = -\rho c_p \frac + \sum_ \dot s_,$
where is heat flux vector, is temporal change of internal energy ( is density, is specific heat capacity at constant pressure, is temperature and is time), and $\dot s$ is the energy conversion to and from thermal energy ( and are for principal energy carriers). So, the terms represent energy transport, storage and transformation. Heat flux vector is composed of three macroscopic fundamental modes, which are conduction (, : thermal conductivity), convection (, : velocity), and radiation ($\mathbf q_r = 2\pi \int_^ \int_^ \mathbf s I_ \sin(\theta) d\theta \, d\omega$, : angular frequency, : polar angle, : spectral, directional radiation intensity, : unit vector), i.e., .

Once states and kinetics of the energy conversion and thermophysical properties are known, the fate of heat transfer is described by the above equation. These atomic-level mechanisms and kinetics are addressed in heat transfer physics. The microscopic thermal energy is stored, transported, and transformed by the principal energy carriers: phonons (''p''), electrons (''e''), fluid particles (''f''), and photons (''ph'').

Length and time scales

Thermophysical properties of matter and the kinetics of interaction and energy exchange among the principal carriers are based on the atomic-level configuration and interaction. Transport properties such as thermal conductivity are calculated from these atomic-level properties using classical and quantum physics. Quantum states of principal carriers (e.g.. momentum, energy) are derived from the Schrödinger equation (called first principle or ''ab initio'') and the interaction rates (for kinetics) are calculated using the quantum states and the quantum perturbation theory (formulated as the Fermi golden rule). Variety of ''ab initio'' (Latin for from the beginning) solvers (software) exist (e.g., ABINIT, CASTEP, Gaussian, Q-Chem, Quantum ESPRESSO, SIESTA, VASP, WIEN2k). Electrons in the inner shells (core) are not involved in heat transfer, and calculations are greatly reduced by proper approximations about the inner-shells electrons.

The quantum treatments, including equilibrium and nonequilibrium ''ab initio'' molecular dynamics (MD), involving larger lengths and times are limited by the computation resources, so various alternate treatments with simplifying assumptions have been used and kinetics. In classical (Newtonian) MD, the motion of atom or molecule (particles) is based on the empirical or effective interaction potentials, which in turn can be based on curve-fit of ''ab initio'' calculations or curve-fit to thermophysical properties. From the ensembles of simulated particles, static or dynamics thermal properties or scattering rates are derived.

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. BTE considers particle states in terms of position and momentum vectors (x, p) and this is represented as the state occupation probability. 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). Central to the transport is the role of scattering which turn the distribution toward equilibrium. The scattering is presented by the relations time or the mean free path. The relaxation time (or its inverse which is the interaction rate) is found from other calculations (''ab initio'' or MD) or empirically. BTE can be numerically solved with Monte Carlo method, etc.

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.

Phonon

Phonon (quantized lattice vibration wave) is a central thermal energy carrier contributing to heat capacity (sensible heat storage) and conductive heat transfer in condensed phase, and plays a very important role in thermal energy conversion. Its transport properties are represented by the phonon conductivity tensor K_''p'' (W/m-K, from the Fourier law q_''k,p'' = -K_''p''⋅∇ ''T'') for bulk materials, and the phonon boundary resistance ''AR_p,b'' /(W/m²)for solid interfaces, where ''A'' is the interface area. The phonon specific heat capacity ''c_v,p'' (J/kg-K) includes the quantum effect. The thermal energy conversion rate involving phonon is included in $\dot_$. Heat transfer physics describes and predicts, ''c_v,p'', K_''p'', ''R_p,b'' (or conductance ''G_p,b'') and $\dot_$, based on atomic-level properties.

For an equilibrium potential ⟨''φ''⟩_o of a system with ''N'' atoms, the total potential ⟨''φ''⟩ is found by a Taylor series expansion at the equilibrium and this can be approximated by the second derivatives (the harmonic approximation) as
$\begin \langle\varphi\rangle &= \langle\varphi\rangle_\mathrm + \left.\sum_i\sum_\alpha\frac\_\mathrmd_ + \left.\frac\sum_\sum_\frac\_\mathrmd_d_ + \left.\frac\sum_\sum_\frac\_\mathrm d_d_d_+ \cdots \\ &\approx \langle\varphi\rangle_\mathrm + \frac\sum_\sum_\Gamma_d_d_, \end$

where d_''i'' is the displacement vector of atom ''i'', and Γ is the spring (or force) constant as the second-order derivatives of the potential. The equation of motion for the lattice vibration in terms of the displacement of atoms ''d(''jl'',''t''): displacement vector of the ''j''-th atom in the ''l''-th unit cell at time ''t''is
$m_j\frac = -\sum_ \boldsymbol \binom\cdot \mathbf (j' l', T),$
where ''m'' is the atomic mass and Γ is the force constant tensor. The atomic displacement is the summation over the normal modes ''s_''α'': unit vector of mode ''α'', ''ω_p'': angular frequency of wave, and κ_''p'': wave vector Using this plane-wave displacement, the equation of motion becomes the eigenvalue equation
$\mathbf \omega_p^2 (\boldsymbol_p,\alpha) \mathbf_\alpha(\boldsymbol_p) = \mathbf (\boldsymbol_p) \mathbf_\alpha(\boldsymbol_p),$
where M is the diagonal mass matrix and D is the harmonic dynamical matrix. Solving this eigenvalue equation gives the relation between the angular frequency ''ω_p'' and the wave vector κ_''p'', and this relation is called the phonon dispersion relation. Thus, the phonon dispersion relation is determined by matrices M and D, which depend on the atomic structure and the strength of interaction among constituent atoms (the stronger the interaction and the lighter the atoms, the higher is the phonon frequency and the larger is the slope ''dω_p''/''d''κ''_p''). The Hamiltonian of phonon system with the harmonic approximation is
$\mathrm_p = \sum_x \frac \mathbf^2(\mathbf) + \frac\sum_\mathbf_i(\mathbf)D_(\mathbf-\mathbf')\mathbf_j(\mathbf'),$
where ''D_ij'' is the dynamical matrix element between atoms ''i'' and ''j'', and d_''i'' (d_''j'') is the displacement of ''i'' (''j'') atom, and p is momentum. From this and the solution to dispersion relation, the phonon annihilation operator for the quantum treatment is defined as
b_ = \frac\sum_ e^\mathbf_\alpha(\boldsymbol_p)\cdot \left left(\frac\right)^\mathbf(\mathbf) + i\left(\frac\right)^\mathbf(\mathbf)\right
where ''N'' is the number of normal modes divided by ''α'' and ''ħ'' is the reduced Planck constant. The creation operator is the adjoint of the annihilation operator,
b_^\dagger = \frac\sum_ e^\mathbf_\alpha(\boldsymbol_p)\cdot \left left(\frac \right)^\mathbf(\mathbf)-i\left(\frac\right)^ \mathbf(\mathbf)\right
The Hamiltonian in terms of ''b_κ,α''^† and ''b_κ,α'' is H''_p'' = Σ_{''κ'',''α''}''ħω_p,α'' 'b_κ,α''^†''b_κ,α'' + 1/2and ''b_κ,α''^†''b_κ,α'' is the phonon number operator. The energy of quantum-harmonic oscillator is ''E_p'' = Σ_{''κ'',''α''} 'f_p''(''κ'',''α'') + 1/2'ħω_p,α''(κ_''p''), and thus the quantum of phonon energy ''ħω_p''.

The phonon dispersion relation gives all possible phonon modes within the Brillouin zone (zone within the primitive cell in reciprocal space), and the phonon density of states ''D_p'' (the number density of possible phonon modes). The phonon group velocity ''u_p,g'' is the slope of the dispersion curve, ''dω_p''/''d''κ_''p''. Since phonon is a boson particle, its occupancy follows the Bose–Einstein distribution . Using the phonon density of states and this occupancy distribution, the phonon energy is ''E_p''(''T'') = ∫''D_p''(''ω_p'')''f_p''(''ω_p,T'')''ħω_pdω_p'', and the phonon density is ''n_p''(''T'') = ∫''D_p''(''ω_p'')''f_p''(''ω_p,T'')''dω_p''. Phonon heat capacity ''c_v,p'' (in solid ''c_v,p'' = ''c_p,p'', ''c_v,p'' : constant-volume heat capacity, ''c_p,p'': constant-pressure heat capacity) is the temperature derivatives of phonon energy for the Debye model (linear dispersion model), is
$c_ = \left.\frac\_v = \frac \left(\frac \right)^3 n \int_0^ \frac dx \qquad (x = \frac),$
where ''T''_D is the Debye temperature, ''m'' is atomic mass, and ''n'' is the atomic number density (number density of phonon modes for the crystal 3''n''). This gives the Debye ''T''³ law at low temperature and Dulong-Petit law at high temperatures.

From the kinetic theory of gases, thermal conductivity of principal carrier ''i'' (''p'', ''e'', ''f'' and ''ph'') is
$k_i = \frac n_i c_u_i\lambda_i,$
where ''n_i'' is the carrier density and the heat capacity is per carrier, ''u_i'' is the carrier speed and ''λ_i'' is the mean free path (distance traveled by carrier before an scattering event). Thus, the larger the carrier density, heat capacity and speed, and the less significant the scattering, the higher is the conductivity. For phonon ''λ_p'' represents the interaction (scattering) kinetics of phonons and is related to the scattering relaxation time ''τ_p'' or rate (= 1/''τ_p'') through ''λ_p''= ''u_pτ_p''. Phonons interact with other phonons, and with electrons, boundaries, impurities, etc., and ''λ_p'' combines these interaction mechanisms through the Matthiessen rule. At low temperatures, scattering by boundaries is dominant and with increase in temperature the interaction rate with impurities, electron and other phonons become important, and finally the phonon-phonon scattering dominants for ''T'' > 0.2''T_D''. The interaction rates are reviewed in and includes quantum perturbation theory and MD.

A number of conductivity models are available with approximations regarding the dispersion and ''λ_p''. Using the single-mode relaxation time approximation (∂''f_p''^′/∂''t'', _''s'' = −''f_p''^′/''τ_p'') and the gas kinetic theory, Callaway phonon (lattice) conductivity model as
$k_ = \frac\sum_\int c_\tau_p(\mathbf_\cdot\mathbf)^2d\kappa \ \ \ \ \ \text \mathbf,$
$k_p = \frac\sum_\int c_\tau_p _^2\kappa^2d\kappa \ \ \ \ \ \ \ \ \text.$

With the Debye model (a single group velocity u_''p,g'', and a specific heat capacity calculated above), this becomes
$k_p = \left(48\pi^2\right)^ \frac \int_0^\tau_p \fracdx,$

where ''a'' is the lattice constant ''a'' = ''n''^−1/3 for a cubic lattice, and ''n'' is the atomic number density. Slack phonon conductivity model mainly considering acoustic phonon scattering (three-phonon interaction) is given as
$k_p = k_ = \frac\qquad \text ( T > 0.2 T_D,\text,$
where is the mean atomic weight of the atoms in the primitive cell, ''V_a''=1/''n'' is the average volume per atom, ''T_D,∞'' is the high-temperature Debye temperature, ''T'' is the temperature, ''N''_o is the number of atoms in the primitive cell, and ⟨γ²_G⟩ is the mode-averaged square of the Grüneisen constant or parameter at high temperatures. This model is widely tested with pure nonmetallic crystals, and the overall agreement is good, even for complex crystals.

Based on the kinetics and atomic structure consideration, a material with high crystalline and strong interactions, composed of light atoms (such as diamond and graphene) is expected to have large phonon conductivity. Solids with more than one atom in the smallest unit cell representing the lattice have two types of phonons, i.e., acoustic and optical. (Acoustic phonons are in-phase movements of atoms about their equilibrium positions, while optical phonons are out-of-phase movement of adjacent atoms in the lattice.) Optical phonons have higher energies (frequencies), but make smaller contribution to conduction heat transfer, because of their smaller group velocity and occupancy.

Phonon transport across hetero-structure boundaries (represented with ''R_p,b'', phonon boundary resistance) according to the boundary scattering approximations are modeled as acoustic and diffuse mismatch models. Larger phonon transmission (small ''R_p,b'') occurs at boundaries where material pairs have similar phonon properties (''u_p'', ''D_p'', etc.), and in contract large ''R_p,b'' occurs when some material is softer (lower cut-off phonon frequency) than the other.

Electron

Quantum electron energy states for electron are found using the electron quantum Hamiltonian, which is generally composed of kinetic (-''ħ''²∇²/2''m_e'') and potential energy terms (''φ_e''). Atomic orbital, a mathematical function describing the wave-like behavior of either an electron or a pair of electrons in an atom, can be found from the Schrödinger equation with this electron Hamiltonian. Hydrogen-like atoms (a nucleus and an electron) allow for closed-form solution to Schrödinger equation with the electrostatic potential (the Coulomb law). The Schrödinger equation of atoms or atomic ions with more than one electron has not been solved analytically, because of the Coulomb interactions among electrons. Thus, numerical techniques are used, and an electron configuration is approximated as product of simpler hydrogen-like atomic orbitals (isolate electron orbitals). Molecules with multiple atoms (nuclei and their electrons) have molecular orbital (MO, a mathematical function for the wave-like behavior of an electron in a molecule), and are obtained from simplified solution techniques such as linear combination of atomic orbitals (LCAO). The molecular orbital is used to predict chemical and physical properties, and the difference between highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) is a measure of excitability of the molecules.

In a crystal structure of metallic solids, the free electron model (zero potential, ''φ_e'' = 0) for the behavior of valence electrons is used. However, in a periodic lattice (crystal), there is periodic crystal potential, so the electron Hamiltonian becomes
$\mathrm_e = - \frac\nabla^2 + \varphi_c(\mathbf),$
where ''m_e'' is the electron mass, and the periodic potential is expressed as ''φ_c'' (''x'') = Σ_''g'' ''φ_g''exp 'i''(g∙x)(g: reciprocal lattice vector). The time-independent Schrödinger equation with this Hamiltonian is given as (the eigenvalue equation)
$\mathrm_e \psi_(\mathbf) = E_e(\boldsymbol_e) \psi_(\mathbf),$
where the eigenfunction ''ψ_e,κ'' is the electron wave function, and eigenvalue ''E_e''(κ_''e''), is the electron energy (κ_''e'': electron wavevector). The relation between wavevector, κ_''e'' and energy ''E_e'' provides the electronic band structure. In practice, a lattice as many-body systems includes interactions between electrons and nuclei in potential, but this calculation can be too intricate. Thus, many approximate techniques have been suggested and one of them is density functional theory (DFT), uses functionals of the spatially dependent electron density instead of full interactions. DFT is widely used in ''ab initio'' software ( ABINIT, CASTEP, Quantum ESPRESSO, SIESTA, VASP, WIEN2k, etc.). The electron specific heat is based on the energy states and occupancy distribution (the Fermi–Dirac statistics). In general, the heat capacity of electron is small except at very high temperature when they are in thermal equilibrium with phonons (lattice). Electrons contribute to heat conduction (in addition to charge carrying) in solid, especially in metals. Thermal conductivity tensor in solid is the sum of electric and phonon thermal conductivity tensors K = K_''e'' + K_''p''.

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