WU2JING/empirical_thermal_conductivity

Estimate the thermal conductivity using empirical models

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,

or

κ_min = 0.87 k_B(Ω_avg)^-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_avg is the average volume per atom.

The Eq.(3) in Ref. 3 had 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),

or

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 model⁷

κ = 3.1 ∗ 10^-6 M_avg Θ_D³δ /(γ²n^2/3T),

where M_avg is the mean atomic mass (in amu), Θ_D is the Debye temperature (in K), n and the number of the atoms in the unit cell, δ³ is the volume per atom (in ų), and γ is the average Grüneisen parameter. The Debye temperature⁸ 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 Belomestnykh⁹:

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 model¹⁰

  κ_a = a₁ ∗ M

Usage

Input parameters

1. 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].
2. 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].
3. vol: the volume of primitive unit cell (in in ų). For example, the volume of the primitive unit cell of Al₂Fe₃Si₃ is 196.489731295 ų. So define: vol = 196.489731295.
4. K: bulk modulus (in GPa). For example, K = 173.121
5. G: shear modulus (in GPa). For example, G = 173.121
6. E: Young's modulus (in GPa). For example, E= 286.239.
7. 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)

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

1. D. R. Clarke, Materials selection guidelines for low thermal conductivity thermal barrier coatings, Surf. Coat. Technol. 163–164, 67–74(2003). DOI: 10.1016/S0257-8972(02)00593-5.
2. D. R. Clarke and S. R. Phillpot, Thermal barrier coating materials, Mater. Today 8, 22–29 (2005). DOI: 10.1016/S1369-7021(05)70934-2.
3. A. M. Limarga, S. Shian, R. M. Leckie, C. G. Levi, and D. R. Clarke, Thermal conductivity of single- and multi-phase compositions in the ZrO₂–Y₂O₃–Ta₂O₅ system, J. Eur. Ceram. Soc., 34, 3085-3094(2014). DOI: 10.1016/j.jeurceramsoc.2014.03.013.
4. D. G. Cahill and R. O. Pohl, Lattice vibrations and heat transport in crystals and glasses, Annu. Rev. Phys. Chem., 39, 93–121(1988). DOI: 10.1146/annurev.pc.39.100188.000521.
5. D. G. Cahill, P. V. Braun, G. Chen, D. R. Clarke, S. Fan, K. E. Goodson, P. Keblinski, W. P. King, G. D. Mahan, A. Majumdar, H. J. Maris, S. R. Phillpot, E. Pop and L. Shi, Appl. Phys. Rev., 1, 011305(2014). DOI: 10.1063/1.4832615.
6. D. G. Cahill, S. K. Watson, and R. O. Pohl, Lower limit to the thermal conductivity of disordered crystals, Phys. Rev. B 46, 6131(1992). DOI: 10.1103/PhysRevB.46.6131.
7. G. A. Slack, Nonmetallic crystals with high thermal conductivity, J. Phys. Chem. Solids 34, 321–335 (1973). DOI: 10.1016/0022-3697(73)90092-9.
8. P. Debye, Zur theorie der spezifischen wärmen, Annalen der Physik 344, 789–839 (1912). DOI: 10.1002/andp.19123441404.
9. V. N. Belomestnykh, The acoustical Grüneisen constants of solids, Tech. Phys. Lett. 30, 91–93 (2004). DOI: 10.1134/1.1666949.
10. Toberer, E. S.; Zevalkink, A.; Snyder, G. J. Phonon engineering through crystal chemistry. J. Mater. Chem. 21, 15843-15852(2011). DOI: 10.1039/c1jm11754h.