Empirical estimation of thermal conductivity

Estimate the thermal conductivity using empirical models including the Clarke’s, Cahill–Pohl’s, and Slack's models.

The basic equations

Clarke model^1,2,3

κ_min = 0.87 k_B(N_A m ρ /M)^2/3 (E/ρ)^1/2,

κ_min = 0.87 k_B(Ω_avg)^-2/3 (E/ρ)^1/2,

κ_min = 0.87 k_B(m/Ω)^2/3 (E/ρ)^1/2,

where k_B is Boltzmann constant, M and m are the molecular mass and the number of the atoms per molecule, respectively, E is Young’s modulus, ρ is the density and N_A is Avogadro’s number.

Ω_avg = M/(mρN_A) = Ω/m,

Ω_avg is the average volume per atom.

The Eq.(3) in Ref. 3 has messed the definition of M and M_avg, and the correct way is to use M, rather the M_avg (the mean atomic mass).

Cahill–Pohl model^4,5,6

κ_min = k_B/2.48 (n/Ω)^2/3 (v_l+2v_t),

κ_min = 1/2 (π/6)^1/3k_B (n/Ω)^2/3 (v_l+2v_t),

where k_B is Boltzmann constant, Ω and n are the volume of unit cell and the number of the atoms in the unit cell, respectively. v_l and v_t and the longitudinal and transverse sound velocities, respectively, which are estimated from the bulk modulus B and shear modulus G as follows:

v_l= ((B+4G/3)/ρ)^1/2,

v_t=(G/ρ)^1/2.

Slack model^7,8

κ = A M_avg Θ_D³δ /(γ²n^2/3T),

where A is a constant parameter, M_avg is the mean atomic mass, Θ_D is the Debye temperature, n is the number of the atoms in the unit cell, δ³ is the volume per atom, and γ is the average Grüneisen parameter. The original equation in Ref. 7 (eq.1 therein) is independent on the parameter n^2/3. The derivation of this equation can be found in Ref. 8 (eq. 2.13 therein). The constant parameter A⁹ is approximated to be 3.1 ∗ 10^-6 when κ in W/m/K, M_avg in amu, and δ in Å.

The Debye temperature¹⁰ and Grüneisen parameter can be evaluated from the sound velocities, which can be measured experimentally, or can be obtained by the theoretically-calculated elastic modulus.

Θ_D = h/k_B (3n/(4πΩ))^1/3v_a,

where h and k_B are Planck and Boltzmann constants, respectively, n is the number of atoms in the unit cell, Ω is the cell volume, and v_a is the average sound wave velocity. The v_a is given in terms of v_l and v_t as

v_a = [(1/3)(1/v_l³+2/v_t³)]^-1/3.

The Grüneisen parameter γ is calculated from the relation proposed by Belomestnykh¹¹:

γ = [9-12(v_t/v_l)²]/[2+4(v_t/v_l)²],

which takes into account the contribution of acoustic sound velocities only.

Mixed model¹²

κ_a = (6π²)^2/3/(4π²) (M_avgv_a³)/(TΩ_avg^2/3γ²) (1/n^1/3),

κ_o = (3k_Bv_a)/(2Ω_avg^2/3) (π/6)^1/3 (1-1/n^2/3),

where M_avg is the mean atomic mass, v_a is the average sound wave velocity, T is the temperature, Ω_avg is the average volume per atom, i.e., Ω/n, Ω is the cell volume, n is the number of atoms in the unit cell, and γ is the average Grüneisen parameter, respectively. The v_a and γ are obtained by the theoretically-calculated elastic modulus, as shown above.

Usage

Input parameters

natoms_list: number of atoms for each species in the primitive unit cell. For example, Al₂Fe₃Si₃ in the primitive unit cell of its triclinic structure, the number of Fe, Al, and Si are 6, 4, and 6, respectively. So define: natoms_list = [6, 4, 6].
atomic_weight_list: the atomic weight (in amu) for each atom species. For example: the atomic weights of Fe, Al, and Si are 55.845, 26.982, and 28.086, respectively. So define: atomic_weight_list = [55.845, 26.982, 28.086].
vol: the volume of primitive unit cell (in in Å³). For example, the volume of the primitive unit cell of Al₂Fe₃Si₃ is 196.489731295 Å³. So define: vol = 196.489731295.
K: bulk modulus (in GPa). For example, K = 173.121
G: shear modulus (in GPa). For example, G = 173.121
E: Young's modulus (in GPa). For example, E= 286.239.
t: temperature (in K). t= 679.334852747

Call function

kappa_Clarke=thermal_cond_clarke(natoms_list, atomic_weight_list, vol, E)
kappa_Chill=thermal_cond_cahill(natoms_list, atomic_weight_list, vol, K, G)
kappa_Slack=thermal_cond_slack(natoms_list, atomic_weight_list, vol, K, G, t)
kappa_mixed=thermal_cond_latt_mixed(natoms_list, atomic_weight_list, vol, K)

print(kappa_Clarke)

print(kappa_Chill)

print(kappa_Slack)

print(kappa_mixed)

    #Fe, Al, Si
    natoms_list = [6, 4, 6]
    atomic_weight_list=[55.845, 26.982, 28.086]
    vol = 196.489731295
    E = 286.239
    K = 173.121
    G = 116.886
    T = 679.334852747

    print('Clarke model:')
    kappa_Clarke = thermal_cond_clarke(natoms_list, atomic_weight_list, vol, E)
    print('Thermal conductivity estimated by Clarke model [W/m/K]: {:15.5f}'.format(kappa_Clarke))
    print('\n')
    print('Cahil model')
    kappa_Cahill = thermal_cond_cahill(natoms_list, atomic_weight_list, vol, K, G)
    print('Thermal conductivity estimated by Cahill model [W/m/K]: {:15.5f}'.format(kappa_Cahill))
    print('\n')
    print('Mixed model')
    kappa_mixed = thermal_cond_latt_mixed(natoms_list, atomic_weight_list, vol, K, E,G, T)
    print('Thermal conductivity estimated by mixed model [W/m/K]: {:15.5f}'.format(kappa_mixed))
    print('\n')
    print('Slack model:')
    kappa_Slack = thermal_cond_slack(natoms_list, atomic_weight_list, vol, K, G, T)
    print('Thermal conductivity estimated by Slack model [W/m/K]: {:15.5f}'.format(kappa_Slack))

All the calculated thermal conductivities are given in the unit of W m^-1K^-1. Meanwhile, the other physics quantities such as Debye temperature, Grüneisen parameter, and sound velocities are also printed out.

We have used the Slack model to estimate of Al₂Fe₃Si₃. If you are interested in it, please refer to our paper:

Zhufeng Hou, Yoshiki Takagiwa, Yoshikazu Shinohara, Yibin Xu, and Koji Tsuda, Machine-learning-assisted development and theoretical consideration for the Al₂Fe₃Si₃ thermoelectric material, ACS Appl. Mater. Interfaces 11, 11545–11554(2019). DOI: 10.1021/acsami.9b02381.

References