Heat transfer physics

Variation of equilibrium particle distribution function with respect to energy for different energy carriers.

Kinetics of atomic-level energy transport and transition interaction[5]

Length-time scale regimes for ab initio, MD, Boltzmann transport, and macroscopic treatments of heat transfer.[5]

Heat is thermal energy associated with temperature-dependent motion of particles. The macroscopic energy equation for infinitesimal volume used in heat transfer analysis is[6]

$${\displaystyle \nabla \cdot \mathbf {q} =-\rho c_{p}{\frac {\partial T}{\partial t}}+\sum _{i,j}{\dot {s}}_{i-j},}$$

where q is heat flux vector, −ρc_p(∂T/∂t) is temporal change of internal energy (ρ is density, c_p is specific heat capacity at constant pressure, T is temperature and t is time), and ${\displaystyle {\dot {s}}}$ is the energy conversion to and from thermal energy (i and j are for principal energy carriers). So, the terms represent energy transport, storage and transformation. Heat flux vector q is composed of three macroscopic fundamental modes, which are conduction (q_k = −k∇T, k: thermal conductivity), convection (q_u = ρc_puT, u: velocity), and radiation (${\textstyle \mathbf {q} _{r}=2\pi \int _{0}^{\infty }\int _{0}^{\pi }\mathbf {s} I_{ph,\omega }\sin(\theta )d\theta \,d\omega }$, ω: angular frequency, θ: polar angle, I_ph,ω: spectral, directional radiation intensity, s: unit vector), i.e., q = q_k + q_u + q_r.

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).[7]

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.[1] Transport properties such as thermal conductivity are calculated from these atomic-level properties using classical and quantum physics.[5][8] 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).[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). 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.[10]

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.[11] 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.[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. 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.[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.[2]

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 [K/(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 ${\displaystyle {\dot {s}}_{i{\mbox{-}}j}}$. Heat transfer physics describes and predicts, c_v,p, K_p, R_p,b (or conductance G_p,b) and ${\displaystyle {\dot {s}}_{i{\mbox{-}}j}}$, 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

$${\displaystyle {\begin{aligned}\langle \varphi \rangle &=\langle \varphi \rangle _{\mathrm {o} }+\left.\sum _{i}\sum _{\alpha }{\frac {\partial \langle \varphi \rangle }{\partial d_{i\alpha }}}\right|_{\mathrm {o} }d_{i\alpha }+\left.{\frac {1}{2}}\sum _{i,j}\sum _{\alpha ,\beta }{\frac {\partial ^{2}\langle \varphi \rangle }{\partial d_{i\alpha }\partial d_{j\beta }}}\right|_{\mathrm {o} }d_{i\alpha }d_{j\beta }+\left.{\frac {1}{6}}\sum _{i,j,k}\sum _{\alpha ,\beta ,\gamma }{\frac {\partial ^{3}\langle \varphi \rangle }{\partial d_{i\alpha }\partial d_{j\beta }\partial d_{k\gamma }}}\right|_{\mathrm {o} }d_{i\alpha }d_{j\beta }d_{k\gamma }+\cdots \\&\approx \langle \varphi \rangle _{\mathrm {o} }+{\frac {1}{2}}\sum _{i,j}\sum _{\alpha ,\beta }\Gamma _{\alpha \beta }d_{i\alpha }d_{j\beta },\end{aligned}}}$$

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

$${\displaystyle m_{j}{\frac {d^{2}\mathbf {d} (jl,t)}{dt^{2}}}=-\sum _{j'l'}{\boldsymbol {\Gamma }}{\binom {j\ j^{\prime }}{l\ l'}}\cdot \mathbf {d} (j'l',T),}$$

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[15][16]

$${\displaystyle \mathbf {M} \omega _{p}^{2}({\boldsymbol {\kappa }}_{p},\alpha )\mathbf {s} _{\alpha }({\boldsymbol {\kappa }}_{p})=\mathbf {D} ({\boldsymbol {\kappa }}_{p})\mathbf {s} _{\alpha }({\boldsymbol {\kappa }}_{p}),}$$

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/dκ_p). The Hamiltonian of phonon system with the harmonic approximation is[15][17][18]

$${\displaystyle \mathrm {H} _{p}=\sum _{x}{\frac {1}{2m}}\mathbf {p} ^{2}(\mathbf {x} )+{\frac {1}{2}}\sum _{\mathbf {x} ,\mathbf {x} '}\mathbf {d} _{i}(\mathbf {x} )D_{ij}(\mathbf {x} -\mathbf {x} ')\mathbf {d} _{j}(\mathbf {x} '),}$$

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

$${\displaystyle b_{\kappa ,\alpha }={\frac {1}{N^{1/2}}}\sum _{\kappa _{p},\alpha }e^{-i({\boldsymbol {\kappa }}_{p}\cdot \mathbf {x} )}\mathbf {s} _{\alpha }({\boldsymbol {\kappa }}_{p})\cdot \left[\left({\frac {m\omega _{p,\alpha }}{2\hbar }}\right)^{1/2}\mathbf {d} (\mathbf {x} )+i\left({\frac {1}{2\hbar m\omega _{p,\alpha }}}\right)^{1/2}\mathbf {p} (\mathbf {x} )\right],}$$

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,

$${\displaystyle b_{\kappa ,\alpha }^{\dagger }={\frac {1}{N^{1/2}}}\sum _{\kappa _{p},\alpha }e^{i({\boldsymbol {\kappa }}_{p}\cdot \mathbf {x} )}\mathbf {s} _{\alpha }({\boldsymbol {\kappa }}_{p})\cdot \left[\left({\frac {m\omega _{p,\alpha }}{2\hbar }}\right)^{1/2}\mathbf {d} (\mathbf {x} )-i\left({\frac {1}{2\hbar m\omega _{p,\alpha }}}\right)^{1/2}\mathbf {p} (\mathbf {x} )\right].}$$

The Hamiltonian in terms of b_κ,α^† and b_κ,α is H_p = Σ_κ,αħω_p,α[b_κ,α^†b_κ,α + 1/2] and 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/dκ_p. Since phonon is a boson particle, its occupancy follows the Bose–Einstein distribution {f_p^o = [exp(ħω_p/k_BT)-1]⁻¹, k_B: Boltzmann constant}. 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[19]

$${\displaystyle c_{v,p}=\left.{\frac {dE_{p}}{dT}}\right|_{v}={\frac {9k_{\mathrm {B} }}{m}}\left({\frac {T}{T_{D}}}\right)^{3}n\int _{0}^{T_{D}/T}{\frac {x^{4}e^{x}}{\left(e^{x}-1\right)^{2}}}dx\qquad (x={\frac {\hbar \omega }{k_{\mathrm {B} }T}}),}$$

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 3n). This gives the Debye T³ law at low temperature and Dulong-Petit law at high temperatures.

From the kinetic theory of gases,[20] thermal conductivity of principal carrier i (p, e, f and ph) is

$${\displaystyle k_{i}={\frac {1}{3}}n_{i}c_{v,i}u_{i}\lambda _{i},}$$

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 > 0.2T_D. The interaction rates are reviewed in[21] and includes quantum perturbation theory and MD.

A number of conductivity models are available with approximations regarding the dispersion and λ_p.[17][19][21][22][23][24][25] 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[21][26]

$${\displaystyle k_{p,\mathbf {s} }={\frac {1}{8\pi ^{3}}}\sum _{\alpha }\int c_{v,p}\tau _{p}(\mathbf {u} _{p,g}\cdot \mathbf {s} )^{2}d\kappa \ \ \ \ \ {\text{ for component along }}\mathbf {s} ,}$$

$${\displaystyle k_{p}={\frac {1}{6\pi ^{3}}}\sum _{\alpha }\int c_{v,p}\tau _{p}{u}_{p,g}^{2}\kappa ^{2}d\kappa \ \ \ \ \ \ \ \ {\text{for isotropic conductivity}}.}$$

With the Debye model (a single group velocity u_p,g, and a specific heat capacity calculated above), this becomes

$${\displaystyle k_{p}=\left(48\pi ^{2}\right)^{1/3}{\frac {k_{\mathrm {B} }^{3}T^{3}}{ah_{\mathrm {P} }^{2}T_{\mathrm {D} }}}\int _{0}^{T/T_{\mathrm {D} }}\tau _{p}{\frac {x^{4}e^{x}}{\left(e^{x}-1\right)^{2}}}dx,}$$

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[27][28]

$${\displaystyle k_{p}=k_{p,S}={\frac {3.1\times 10^{12}\langle M\rangle V_{a}^{1/3}T_{D,\infty }^{3}}{T\langle \gamma _{G}^{2}\rangle N_{o}^{2/3}}}\qquad {\text{ high temperatures }}(T>0.2T_{D},{\text{ phonon-phonon scattering only)}},}$$

where ⟨M⟩ 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.[29] 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.

Quantum electron energy states for electron are found using the electron quantum Hamiltonian, which is generally composed of kinetic (-ħ²∇²/2m_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[19]

$${\displaystyle \mathrm {H} _{e}=-{\frac {\hbar ^{2}}{2m_{e}}}\nabla ^{2}+\varphi _{c}(\mathbf {x} ),}$$

where m_e is the electron mass, and the periodic potential is expressed as φ_c (x) = Σ_g φ_gexp[i(g∙x)] (g: reciprocal lattice vector). The time-independent Schrödinger equation with this Hamiltonian is given as (the eigenvalue equation)

$${\displaystyle \mathrm {H} _{e}\psi _{e,\mathbf {x} }(\mathbf {x} )=E_{e}({\boldsymbol {\kappa }}_{e})\psi _{e,\mathbf {x} }(\mathbf {x} ),}$$

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.

Electrons are affected by two thermodynamic forces [from the charge, ∇(E_F/e_c) where E_F is the Fermi level and e_c is the electron charge and temperature gradient, ∇(1/T)] because they carry both charge and thermal energy, and thus electric current j_e and heat flow q are described with the thermoelectric tensors (A_ee, A_et, A_te, and A_tt) from the Onsager reciprocal relations[30] as

$${\displaystyle \mathbf {j} _{e}=\mathbf {A} _{ee}\cdot \nabla {\frac {E_{\mathrm {F} }}{e_{c}}}+\mathbf {A} _{et}\cdot \nabla {\frac {1}{T}},\ \ {\text{and}}}$$

$${\displaystyle \mathbf {q} =\mathbf {A} _{te}\cdot \nabla {\frac {E_{\mathrm {F} }}{e_{c}}}+\mathbf {A} _{tt}\cdot \nabla {\frac {1}{T}}.}$$

Converting these equations to have j_e equation in terms of electric field e_e and ∇T and q equation with j_e and ∇T, (using scalar coefficients for isotropic transport, α_ee, α_et, α_te, and α_tt instead of A_ee, A_et, A_te, and A_tt)

$${\displaystyle \mathbf {j} _{e}=\alpha _{ee}\mathbf {e} _{e}-{\frac {\alpha _{et}}{T^{2}}}\nabla T\qquad (\mathbf {e} _{e}=\alpha _{ee}^{-1}\mathbf {j} _{e}+{\frac {\alpha _{ee}^{-1}\alpha _{et}}{T^{2}}}\nabla T),}$$

$${\displaystyle \mathbf {q} =\alpha _{te}\alpha _{ee}^{-1}\mathbf {j} _{e}-{\frac {\alpha _{tt}-\alpha _{te}\alpha _{ee}^{-1}\alpha _{et}}{T^{2}}}\nabla T.}$$

Electrical conductivity/resistivity σ_e (Ω⁻¹m⁻¹)/ ρ_e (Ω-m), electric thermal conductivity k_e (W/m-K) and the Seebeck/Peltier coefficients α_S (V/K)/α_P (V) are defined as,

$${\displaystyle \sigma _{e}={\frac {1}{\rho _{e}}}=\alpha _{ee},\ \ k_{e}={\frac {\alpha _{tt}-\alpha _{te}\alpha _{ee}^{-1}\alpha _{et}}{T^{2}}},\mathrm {and} \ \alpha _{\mathrm {S} }={\frac {\alpha _{et}\alpha _{ee}^{-1}}{T^{2}}}\ \ (\alpha _{\mathrm {S} }=\alpha _{\mathrm {P} }T).}$$

Various carriers (electrons, magnons, phonons, and polarons) and their interactions substantially affect the Seebeck coefficient.[31][32] The Seebeck coefficient can be decomposed with two contributions, α_S = α_S,pres + α_S,trans, where α_S,pres is the sum of contributions to the carrier-induced entropy change, i.e., α_S,pres = α_S,mix + α_S,spin + α_S,vib (α_S,mix: entropy-of-mixing, α_S,spin: spin entropy, and α_S,vib: vibrational entropy). The other contribution α_S,trans is the net energy transferred in moving a carrier divided by qT (q: carrier charge). The electron's contributions to the Seebeck coefficient are mostly in α_S,pres. The α_S,mix is usually dominant in lightly doped semiconductors. The change of the entropy-of-mixing upon adding an electron to a system is the so-called Heikes formula

$${\displaystyle \alpha _{\mathrm {S,mix} }={\frac {1}{q}}{\frac {\partial S_{\mathrm {mix} }}{\partial N}}={\frac {k_{\mathrm {B} }}{q}}\ln \left({\frac {1-f_{e}^{\mathrm {o} }}{f_{e}^{\mathrm {o} }}}\right),}$$

where f_e^o = N/N_a is the ratio of electrons to sites (carrier concentration). Using the chemical potential (μ), the thermal energy (k_BT) and the Fermi function, above equation can be expressed in an alternative form, α_S,mix = (k_B/q)[(E_e − μ)/(k_BT)]. Extending the Seebeck effect to spins, a ferromagnetic alloy can be a good example. The contribution to the Seebeck coefficient that results from electrons' presence altering the systems spin entropy is given by α_S,spin = ΔS_spin/q = (k_B/q)ln[(2s + 1)/(2s₀ +1)], where s₀ and s are net spins of the magnetic site in the absence and presence of the carrier, respectively. Many vibrational effects with electrons also contribute to the Seebeck coefficient. The softening of the vibrational frequencies produces a change of the vibrational entropy is one of examples. The vibrational entropy is the negative derivative of the free energy, i.e.,

$${\displaystyle S_{\mathrm {vib} }=-{\frac {\partial F_{\mathrm {mix} }}{\partial T}}=3Nk_{\mathrm {B} }T\int _{0}^{\omega }\left\{{\frac {\hbar \omega }{2k_{\mathrm {B} }T}}\coth \left({\frac {\hbar \omega }{2k_{\mathrm {B} }T}}\right)-\ln \left[2\sinh \left({\frac {\hbar \omega }{2k_{\mathrm {B} }T}}\right)\right]\right\}D_{p}(\omega )d\omega ,}$$

where D_p(ω) is the phonon density-of-states for the structure. For the high-temperature limit and series expansions of the hyperbolic functions, the above is simplified as α_S,vib = (ΔS_vib/q) = (k_B/q)Σ_i(-Δω_i/ω_i).

The Seebeck coefficient derived in the above Onsager formulation is the mixing component α_S,mix, which dominates in most semiconductors. The vibrational component in high-band gap materials such as B₁₃C₂ is very important.
Considering the microscopic transport (transport is a results of nonequilibrium),

$${\displaystyle \mathbf {j} _{e}=-{\frac {e_{c}}{\hbar ^{3}}}\sum _{p}\mathbf {u} _{e}f_{e}^{\prime }=-{\frac {e_{c}}{\hbar ^{3}k_{\mathrm {B} }T}}\sum _{p}\mathbf {u} _{e}\tau _{e}\left(-{\frac {\partial f_{e}^{\mathrm {o} }}{\partial E_{e}}}\right)(\mathbf {u} _{e}\cdot \mathbf {F} _{te}),}$$

$${\displaystyle \mathbf {q} ={\frac {1}{\hbar ^{3}}}\sum _{p}(E_{e}-E_{\mathrm {F} })\mathbf {u} _{e}f_{e}^{\prime }={\frac {1}{\hbar ^{3}k_{\mathrm {B} }T}}\sum _{p}\mathbf {u} _{e}\tau _{e}\left(-{\frac {\partial f_{e}^{\mathrm {o} }}{\partial E_{e}}}\right)(E_{e}-E_{\mathrm {F} })(\mathbf {u} _{e}\cdot \mathbf {F} _{te}),}$$

where u_e is the electron velocity vector, f_e (f_e^o) is the electron nonequilibrium (equilibrium) distribution, τ_e is the electron scattering time, E_e is the electron energy, and F_te is the electric and thermal forces from ∇(E_F/e_c) and ∇(1/T). Relating the thermoelectric coefficients to the microscopic transport equations for j_e and q, the thermal, electric, and thermoelectric properties are calculated. Thus, k_e increases with the electrical conductivity σe and temperature T, as the Wiedemann–Franz law presents [k_e/(σ_eT_e) = (1/3)(πk_B/e_c)² = 2.44×10⁻⁸ W-Ω/K²]. Electron transport (represented as σ_e) is a function of carrier density n_e,c and electron mobility μ_e (σ_e = e_cn_e,cμ_e). μ_e is determined by electron scattering rates ${\displaystyle {\dot {\gamma }}_{e}}$ (or relaxation time, ${\displaystyle \tau _{e}=1/{\dot {\gamma }}_{e}}$) in various interaction mechanisms including interaction with other electrons, phonons, impurities and boundaries.

Electrons interact with other principal energy carriers. Electrons accelerated by an electric field are relaxed through the energy conversion to phonon (in semiconductors, mostly optical phonon), which is called Joule heating. Energy conversion between electric potential and phonon energy is considered in thermoelectrics such as Peltier cooling and thermoelectric generator. Also, study of interaction with photons is central in optoelectronic applications (i.e. light-emitting diode, solar photovoltaic cells, etc.). Interaction rates or energy conversion rates can be evaluated using the Fermi golden rule (from the perturbation theory) with ab initio approach.

Fluid particle is the smallest unit (atoms or molecules) in the fluid phase (gas, liquid or plasma) without breaking any chemical bond. Energy of fluid particle is divided into potential, electronic, translational, vibrational, and rotational energies. The heat (thermal) energy storage in fluid particle is through the temperature-dependent particle motion (translational, vibrational, and rotational energies). The electronic energy is included only if temperature is high enough to ionize or dissociate the fluid particles or to include other electronic transitions. These quantum energy states of the fluid particles are found using their respective quantum Hamiltonian. These are H_f,t = −(ħ²/2m)∇², H_f,v = −(ħ²/2m)∇² + Γx²/2 and H_f,r = −(ħ²/2I_f)∇² for translational, vibrational and rotational modes. (Γ: spring constant, I_f: the moment of inertia for the molecule). From the Hamiltonian, the quantized fluid particle energy state E_f and partition functions Z_f [with the Maxwell–Boltzmann (MB) occupancy distribution] are found as[33]

• translational
  $${\displaystyle E_{f,t,n}={\frac {\pi ^{2}\hbar ^{2}}{2m}}\left({\frac {n_{x}^{2}}{L^{2}}}+{\frac {n_{y}^{2}}{L^{2}}}+{\frac {n_{z}^{2}}{L^{2}}}\right)\ \ \ {\text{and}}\ \ \ Z_{f,t}\sum _{i=0}^{\infty }g_{f,t,i}\exp \left(-{\frac {E_{f,t,i}}{k_{\mathrm {B} }T}}\right)=V\left({\frac {mk_{\mathrm {B} }T}{2\pi \hbar ^{2}}}\right)^{3/2},}$$

  
• vibrational
  $${\displaystyle E_{f,v,l}=\hbar \omega _{f,v}\left(1+{\frac {1}{2}}\right)\ \ {\text{and}}\ \ Z_{f,v}\sum _{j=0}^{\infty }\exp \left[-\left(l+{\frac {1}{2}}\right){\frac {\hbar \omega _{f,v}}{k_{\mathrm {B} }T}}\right]={\frac {\exp(-T_{f,v}/2T)}{1-\exp(-T_{f,v}/T)}},}$$

  
• rotational
  $${\displaystyle E_{f,r,j}={\frac {\hbar ^{2}}{2I_{f}}}\ \ {\text{and}}\ \ Z_{f,r}\sum _{j=0}^{\infty }(2j+1)\exp \left[-{\frac {-\hbar ^{2}j(j+1)}{2I_{f}k_{\mathrm {B} }T}}\right]\approx {\frac {T}{T_{f,r}}}\left(1+{\frac {T_{f,r}}{3T}}+{\frac {T_{f,r}^{2}}{15T^{2}}}+\cdots \right),}$$

  
• total
  $${\displaystyle E_{f}=\sum _{i}E_{f,i}=E_{f,t}+E_{f,v}+E_{f,r}+\dots \ \ {\text{and}}\ \ Z_{f}=\prod _{i}Z_{f,i}=Z_{f,t}Z_{f,v}Z_{f,r}\dots .}$$

  

Here, g_f is the degeneracy, n, l, and j are the transitional, vibrational and rotational quantum numbers, T_f,v is the characteristic temperature for vibration (= ħω_f,v/k_B, : vibration frequency), and T_f,r is the rotational temperature [= ħ²/(2I_fk_B)]. The average specific internal energy is related to the partition function through Z_f, ${\displaystyle e_{f}=(k_{\mathrm {B} }T^{2}/m)(\partial \mathrm {ln} Z_{f}/\partial T)|_{N,V}.}$

With the energy states and the partition function, the fluid particle specific heat capacity c_v,f is the summation of contribution from various kinetic energies (for non-ideal gas the potential energy is also added). Because the total degrees of freedom in molecules is determined by the atomic configuration, c_v,f has different formulas depending on the configuration,[33]

• monatomic ideal gas
  $${\displaystyle c_{v,f}=\left.{\frac {\partial e_{f}}{\partial T}}\right|_{V}={\frac {3R_{g}}{2M}},}$$

  
• diatomic ideal gas
  $${\displaystyle c_{v,f}={\frac {R_{g}}{M}}\left\{{\frac {3}{2}}+\left({\frac {T_{f,v}}{T}}\right)^{2}{\frac {\exp(T_{f,v,i}/T)}{[\exp(T_{f,v,i}/T)-1]^{2}}}+1+{\frac {2}{15}}\left({\frac {T_{f,v}}{T}}\right)^{2}\right\},}$$

  
• nonlinear, polyatomic ideal gas
  $${\displaystyle c_{v,f}={\frac {R_{g}}{M}}\left\{3+\sum _{j=1}^{3N_{o}-6}\left({\frac {T_{f,v}}{T}}\right)^{2}{\frac {\exp(T_{f,v,i}/T)}{[\exp(T_{f,v,i}/T)-1]^{2}}}\right\}.}$$

  

where R_g is the gas constant (= N_Ak_B, N_A: the Avogadro constant) and M is the molecular mass (kg/kmol). (For the polyatomic ideal gas, N_o is the number of atoms in a molecule.) In gas, constant-pressure specific heat capacity c_p,f has a larger value and the difference depends on the temperature T, volumetric thermal expansion coefficient β and the isothermal compressibility κ [c_p,f – c_v,f = Tβ²/(ρ_fκ), ρ_f : the fluid density]. For dense fluids that the interactions between the particles (the van der Waals interaction) should be included, and c_v,f and c_p,f would change accordingly. The net motion of particles (under gravity or external pressure) gives rise to the convection heat flux q_u = ρ_fc_p,fu_fT. Conduction heat flux q_k for ideal gas is derived with the gas kinetic theory or the Boltzmann transport equations, and the thermal conductivity is

$${\displaystyle k_{f}={\tfrac {1}{3}}n_{f}c_{p,f}\langle u_{f}^{2}\rangle \tau _{f{\mbox{-}}f},}$$

where ⟨u_f²⟩^1/2 is the RMS (root mean square) thermal velocity (3k_BT/m from the MB distribution function, m: atomic mass) and τ_f-f is the relaxation time (or intercollision time period) [(2^1/2π d²n_f ⟨u_f⟩)⁻¹ from the gas kinetic theory, ⟨u_f⟩: average thermal speed (8k_BT/πm)^1/2, d: the collision diameter of fluid particle (atom or molecule), n_f: fluid number density].

k_f 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 k_f, 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.

Fluid particles can interact with other principal particles. Vibrational or rotational modes, which have relatively high energy, are excited or decay through the interaction with photons. Gas lasers employ the interaction kinetics between fluid particles and photons, and laser cooling has been also considered in CO₂ gas laser.[34][35] Also, fluid particles can be adsorbed on solid surfaces (physisorption and chemisorption), and the frustrated vibrational modes in adsorbates (fluid particles) is decayed by creating e⁻-h⁺ pairs or phonons. These interaction rates are also calculated through ab initio calculation on fluid particle and the Fermi golden rule.[36]

Spectral photon absorption coefficient for typical gas, liquid, and solid phases. For the solid phase, examples of polymer, oxide, semiconductor, and metals are given.

Photon is the quanta of electromagnetic (EM) radiation and energy carrier for radiation heat transfer. The EM wave is governed by the classical Maxwell equations, and the quantization of EM wave is used for phenomena such as the blackbody radiation (in particular to explain the ultraviolet catastrophe). The quanta EM wave (photon) energy of angular frequency ω_ph is E_ph = ħω_ph, and follows the Bose–Einstein distribution function (f_ph). The photon Hamiltonian for the quantized radiation field (second quantization) is[37][38]

$${\displaystyle \mathrm {H} _{ph}={\frac {1}{2}}\int \left(\varepsilon _{\mathrm {o} }\mathbf {e} _{e}^{2}+\mu _{\mathrm {o} }^{-1}\mathbf {b} _{e}^{2}\right)dV=\sum _{\alpha }\hbar \omega _{ph,\alpha }\left(c_{\alpha }^{\dagger }c_{\alpha }+{\frac {1}{2}}\right),}$$

where e_e and b_e are the electric and magnetic fields of the EM radiation, ε_o and μ_o are the free-space permittivity and permeability, V is the interaction volume, ω_ph,α is the photon angular frequency for the α mode and c_α^† and c_α are the photon creation and annihilation operators. The vector potential a_e of EM fields (e_e = −∂a_e/∂t and b_e = ∇×a_e) is

$${\displaystyle \mathbf {a} _{e}(\mathbf {x} ,t)=\sum _{\alpha }\left({\frac {\hbar }{2\varepsilon _{\mathrm {o} }\omega _{ph,\alpha }V}}\right)^{1/2}\mathbf {s} _{ph,\alpha }\left(c_{\alpha }e^{i{\boldsymbol {\kappa }}_{\alpha }\cdot \mathbf {x} }+c_{\alpha }^{\dagger }e^{-i{\boldsymbol {\kappa }}_{\alpha }\cdot \mathbf {x} }\right),}$$

where s_ph,α is the unit polarization vector, κ_α is the wave vector.

Blackbody radiation among various types of photon emission employs the photon gas model with thermalized energy distribution without interphoton interaction. From the linear dispersion relation (i.e., dispersionless), phase and group speeds are equal (u_ph = d ω_ph/dκ = ω_ph/κ, u_ph: photon speed) and the Debye (used for dispersionless photon) density of states is D_ph,b,ωdω = ω_ph²dω_ph/π²u_ph³. With D_ph,b,ω and equilibrium distribution f_ph, photon energy spectral distribution dI_b,ω or dI_b,λ (λ_ph: wavelength) and total emissive power E_b are derived as

$${\displaystyle dI_{b,\omega }={\frac {D_{ph,b,\omega }f_{ph}u_{ph}d\omega _{ph}}{4\pi }}={\frac {\hbar \omega _{ph}^{3}}{4\pi ^{3}u_{ph}^{2}}}{\frac {1}{e^{\hbar \omega _{ph}/k_{\mathrm {B} }T}-1}}d\omega _{ph}\ {\text{or}}\ dI_{b,\lambda }={\frac {4\pi \hbar u_{ph}^{2}d\lambda _{ph}}{\lambda _{ph}^{5}(e^{2\pi \hbar u_{ph}/\lambda _{ph}k_{\mathrm {B} }T}-1)}}}$$

(Planck law),

$${\displaystyle E_{b}=\int _{0}^{\infty }dE_{b,\lambda }=\sigma _{\mathrm {SB} }T^{4}\ {\text{, where}}\ \sigma _{\mathrm {SB} }={\frac {\pi ^{2}k_{\mathrm {B} }^{4}}{60\hbar ^{3}u_{ph}^{2}}}}$$

(Stefan–Boltzmann law).

Compared to blackbody radiation, laser emission has high directionality (small solid angle ΔΩ) and spectral purity (narrow bands Δω). 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.[40][41][42]

The BTE for photon particle momentum p_ph = ħω_phs/u_ph along direction s experiencing absorption/emission ${\displaystyle \textstyle {\dot {s}}_{f,ph-e}\ }$ (= u_phσ_ph,ω[f_ph(ω_ph,T) - f_ph(s)], σ_ph,ω: spectral absorption coefficient), and generation/removal ${\displaystyle \textstyle {\dot {s}}_{f,ph,i}}$, is[43][44]

$${\displaystyle {\frac {\partial f_{ph}}{\partial t}}+u_{ph}\mathbf {s} \cdot \nabla f_{ph}=\left.{\frac {\partial f_{ph}}{\partial t}}\right|_{s}+u_{ph}\sigma _{ph,\omega }[f_{ph}(\omega _{ph},T)-f_{ph}(\mathbf {s} )]+{\dot {s}}_{f,ph,i}.}$$

In terms of radiation intensity (I_ph,ω = u_phf_phħω_phD_ph,ω/4π, D_ph,ω: photon density of states), this is called the equation of radiative transfer (ERT)[44]

$${\displaystyle {\frac {\partial I_{ph,\omega }(\omega _{ph},\mathbf {s} )}{u_{ph}\partial t}}+\mathbf {s} \cdot \nabla I_{ph,\omega }(\omega _{ph},\mathbf {s} )=\left.{\frac {\partial I_{ph,\omega }(\omega _{ph},\mathbf {s} )}{u_{ph}\partial t}}\right|_{s}+\sigma _{ph,\omega }[I_{ph,\omega }(\omega _{ph},T)-I_{ph}(\omega _{ph},\mathbf {s} )]+{\dot {s}}_{ph,i}.}$$

The net radiative heat flux vector is ${\textstyle \mathbf {q} _{r}=\mathbf {q} _{ph}=\int _{0}^{\infty }\int _{4\pi }\mathbf {s} I_{ph,\omega }d\Omega d\omega .}$

From the Einstein population rate equation, spectral absorption coefficient σ_ph,ω in ERT is,[45]

$${\displaystyle \sigma _{ph,\omega }={\frac {\hbar \omega {\dot {\gamma }}_{ph,a}n_{e}}{u_{ph}}},}$$

where ${\displaystyle {\dot {\gamma }}_{ph,a}}$ is the interaction probability (absorption) rate or the Einstein coefficient B₁₂ (J⁻¹ m³ s⁻¹), which gives the probability per unit time per unit spectral energy density of the radiation field (1: ground state, 2: excited state), and n_e is electron density (in ground state). This can be obtained using the transition dipole moment μ_e with the FGR and relationship between Einstein coefficients. Averaging σ_ph,ω over ω gives the average photon absorption coefficient σ_ph.

For the case of optically thick medium of length L, i.e., σ_phL >> 1, and using the gas kinetic theory, the photon conductivity k_ph is 16σ_SBT³/3σ_ph (σ_SB: Stefan–Boltzmann constant, σ_ph: average photon absorption), and photon heat capacity n_phc_v,ph is 16σ_SBT³/u_ph.

Photons have the largest range of energy and central in a variety of energy conversions. Photons interact with electric and magnetic entities. For example, electric dipole which in turn are excited by optical phonons or fluid particle vibration, or transition dipole moments of electronic transitions. In heat transfer physics, the interaction kinetics of phonon is treated using the perturbation theory (the Fermi golden rule) and the interaction Hamiltonian. The photon-electron interaction is[46]

$${\displaystyle \mathrm {H} _{ph-e}=-{\frac {e_{c}}{m_{e}}}\left(a+a^{\dagger }\right)\mathbf {a} _{e}\cdot \mathbf {p} _{e}=-\left({\frac {\hbar \omega _{ph,\alpha }}{2\varepsilon _{o}V}}\right)^{1/2}(\mathbf {s} _{ph,\alpha }\cdot e_{c}\mathbf {x} _{e})\left(a+a^{\dagger }\right)\left(ce^{i\mathrm {\kappa } \cdot \mathrm {x} }+c^{\dagger }e^{-i\mathrm {\kappa } \cdot \mathrm {x} }\right),}$$

where p_e is the dipole moment vector and a^† and a are the creation and annihilation of internal motion of electron. Photons also participate in ternary interactions, e.g., phonon-assisted photon absorption/emission (transition of electron energy level).[47][48] The vibrational mode in fluid particles can decay or become excited by emitting or absorbing photons. Examples are solid and molecular gas laser cooling.[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, ε_e,ω), spectral absorption coefficient (σ_ph,ω), and the complex refraction index (m_ω), are calculated for various interactions between photons and electric/magnetic entities in matter.[52][53] For example, the imaginary part (ε_e,c,ω) of complex dielectric function (ε_e,ω = ε_e,r,ω + i ε_e,c,ω) for electronic transition across a bandgap is[3]

$${\displaystyle \varepsilon _{e,c,\omega }={\frac {4\pi ^{2}}{\omega ^{2}V}}\sum _{i\in \mathrm {VB} ,j\in \mathrm {CB} }\sum _{\kappa }w_{\kappa }|p_{ij}|^{2}\delta (E_{\kappa ,j}-E_{\kappa ,i}-\hbar \omega ),}$$

where V is the unit-cell volume, VB and CB denote the valence and conduction bands, w_κ is the weight associated with a κ-point, and p_ij is the transition momentum matrix element. The real part is ε_e,r,ω is obtained from ε_e,c,ω using the Kramers-Kronig relation[54]

$${\displaystyle \varepsilon _{e,r,\omega }=1+{\frac {4}{\pi }}\mathbb {P} \int _{0}^{\infty }\mathrm {d} \omega '{\frac {\omega '\varepsilon _{e,c,\omega '}}{\omega '^{2}-\omega ^{2}}}.}$$

Here, ${\displaystyle \mathbb {P} }$ denotes the principal value of the integral.

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

$${\displaystyle {\frac {\varepsilon _{e,\omega }}{\varepsilon _{e,\infty }}}=1+\sum _{j}{\frac {\omega _{\mathrm {LO} ,j}^{2}-\omega _{\mathrm {TO} ,j}^{2}}{\omega _{\mathrm {TO} ,j}^{2}-\omega ^{2}-i\gamma \omega }},}$$

where LO and TO denote the longitudinal and transverse optical phonon modes, j is all the IR-active modes, and γ is the temperature-dependent damping term in the oscillator model. ε_e,∞ is high frequency dielectric permittivity, which can be calculated DFT calculation when ions are treated as external potential.

From these dielectric function (ε_e,ω) calculations (e.g., Abinit, VASP, etc.), the complex refractive index m_ω(= n_ω + i κ_ω, n_ω: refraction index and κ_ω: extinction index) is found, i.e., m_ω² = ε_e,ω = ε_e,r,ω + i ε_e,c,ω). The surface reflectance R of an ideal surface with normal incident from vacuum or air is given as[55] R = [(n_ω - 1)² + κ_ω²]/[(n_ω + 1)² + κ_ω²]. The spectral absorption coefficient is then found from σ_ph,ω = 2ω κ_ω/u_ph. The spectral absorption coefficient for various electric entities are listed in the below table.[56]

Mechanism	Relation (σph,ω)
Electronic absorption transition (atom, ion or molecule)	${\displaystyle {\frac {n_{e,A}\pi ^{2}u_{ph}^{2}{\dot {\gamma }}_{ph,e,sp}}{\omega _{e,g}^{2}\int _{\omega }\mathrm {d} \omega }}={\frac {n_{e,A}\pi \omega _{e,g}|{\boldsymbol {\mu }}_{e}|^{2}}{3\varepsilon _{\mathrm {o} }\hbar u_{ph}\int _{\omega }\mathrm {d} \omega }}}$, [ne,A: number density of ground state, ωe,g: transition angular frequency, ${\displaystyle {\dot {\gamma }}_{ph,e,sp}}$: spontaneous emission rate (s−1), μe: transition dipole moment, ${\displaystyle \textstyle \int _{\omega }\mathrm {d} \omega }$: bandwidth]
Free carrier absorption (metal)	${\displaystyle {\frac {2\omega \kappa _{\omega }}{u_{ph}}}=\left({\frac {2n_{e,c}e_{c}^{2}\langle \langle \tau _{e}\rangle \rangle \omega }{\varepsilon _{\mathrm {o} }\varepsilon _{e}m_{e}u_{ph}^{2}}}\right)^{1/2}}$ (ne,c: number density of conduction electrons, ${\displaystyle \langle \langle \tau _{e}\rangle \rangle }$: average momentum electron relaxation time, εo: free space electrical permittivity)
Direct-band absorption (semiconductor)	${\displaystyle {\frac {e_{c}^{2}|\langle \varphi _{v}|e_{c}\mathbf {x} |\varphi _{c}\rangle |^{2}D_{ph-e}[f_{e}^{\mathrm {o} }(E_{e,v})-f_{e}^{\mathrm {o} }(E_{e,c})]}{\varepsilon _{\mathrm {o} }^{2}m_{e,e}^{2}u_{ph}n_{\omega }\omega }}}$ (nω: index of refraction, Dph-e: joint density of states)
Indirect-band absorption (semiconductor)	with phonon absorption: ${\displaystyle {\frac {a_{ph\mathrm {-} e\mathrm {-} p,a}(\hbar \omega -\Delta E_{e,g}+\hbar \omega _{p})^{2}}{\mathrm {exp} (\hbar \omega _{p}/k_{\mathrm {B} }T)-1}}}$ (aph-e-p,a phonon absorption coupling coefficient, ΔEe,g: bandgap, ωp: phonon energy ) with phonon emission: ${\displaystyle {\frac {a_{ph-e-p,e}(\hbar \omega -\Delta E_{e,g}-\hbar \omega _{p})^{2}}{1-\exp(-\hbar \omega _{p}/k_{\mathrm {B} }T)}}}$ (aph-e-p,e phonon emission coupling coefficient)

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