#### Title

Computational Modelling of Thermal Transport using Spectral Phonon Boltzmann Transport Equation

#### Date of Award

12-2017

#### Degree Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Nuclear Engineering

#### Committee Chair

Anter El-Azab

#### Committee Member 1

Ahmed Hassanein

#### Committee Member 2

Shripad Revankar

#### Committee Member 3

Xiulin Ruan

#### Abstract

Lattice vibration is the main microscopic mechanism for thermal transport in dielectric materials. The convenience of the analysis of atomic vibrations in the reciprocal space, motivated by the pioneering work of Debye and Peirels, made phonon transport theory is one of the standard paradigms adequate to study the microstructure and stoichiometry effects on thermal transport phenomena at mesoscale. UO2 is of theoretical as well as technological importance. The characteristic thermal transport phenomena at short length-scale (~ nanometer) and time-scale (~ picosecond) associated with radiation dictate close examination of available theoretical models and solution methods for thermal conductivity prediction, in addition to the validity of introduced approximations. By the virtue of INS experimental technique with powerful resolution, a direct benchmarking of simulated phonon properties results has been made possible. This provides by far a more accurate assessment criteria than thermal conductivity, and pave the way for founding sophisticated models of radiation effects on thermal transport with theoretical supports, beyond the currently available empirical or phenomenological models that succeed to reproduce the right macroscopic behavior in many cases just because of error cancellations and/or the use of adjustable parameters. Within time dependent perturbation theory (Fermi golden rule) framework, to represent the collision term of the semi-classical phonon Boltzmann Transport Equation, the bottleneck of the employed approach is to calculate intrinsic and extrinsic scattering rates of phonon modes. Being a highly correlated system with 5f electrons and magnetic phase transition at very low temperature, there are several challenges facing first-principle methods to leverage accurate phonon properties at finite temperature and imperfect structure that still need to be overcome. Moreover, it is common that using 3-phonon processes alone in other dielectrics overestimates lattice thermal conductivity at high temperatures (due to ignoring higher order phonon-phonon interactions), however, previous computational studies predicted values for UO2 conductivity lower than experiment by a factor of two about one third of the melting temperature. These observations assert the necessity of firstly investigating the impact of different introduced approximations for the calculation of intrinsic lattice thermal conductivity, to analyze the crucial parameters and to better understand this anomalous prediction. In this investigation, we present a critical assessment of several common approximations for the calculations of lattice thermal conductivity using spectral phonon Boltzmann Transport Equations. These approximations pertain to dispersion anisotropy and relations, Brillouin zone structure, and the coupling between the scattering rates of phonon normal modes. By employing harmonic approximation—perturbation theory to describe the scattering rates of a model system, FCC argon, our calculations show that widely spread approximations such as isotropic continuum and Single Mode Relaxation Time (SMRT) are not reliable, even for the case of cubic systems with their high symmetry properties. The success of these approximations is demonstrated to be a direct result of error cancellations. In addition, we show the essential importance of considering coupling terms at phonon mode level, and not in a statistical average sense as, for example, Callaway’s model does. By taking into account the coupling terms, the results evidence the crossover between the heat diffusion mediated by particle-like phonons (incoherent scattering) and the wave-like heat propagation due to phonon coherent scattering. Furthermore, this made possible revealing thermal conductivity anisotropy in cubic crystals. Finally, sensitivity of conductivity prediction to phonon spectrum is found to change over temperature. On the other hand, we challenge the widespread consensus that phonon-phonon interactions are inactive in the low temperature regime, which, in past investigations, led to the belief that the peak in lattice thermal conductivity (versus temperature) occurs because of two competing scattering mechanisms, umklapp and defect scattering mechanisms, dominant above and below the peak temperature, respectively. To the contrary, our study demonstrates that peak thermal conductivity, versus temperature, can still be obtained solely based upon phonon-phonon processes. This finding has been aided by considering the inelastic nature of 3-phonon scattering through applying energy conservation rule in a statistical average sense. Among the different statistical distributions examined to represent the regularized Dirac delta function appearing in Fermi Golden Rule, adopting Lorentz distribution, in analogy with phonon normal mode eigenenergy broadening due to the leading term of crystal anharmonicity, can uniquely reproduce the attained behavior in the low temperature limit. Simulation results, based on our adjustable parameter-free model, evidence that the heavy tail of the Lorentz distribution is the key. Unlike other models that similarly employ harmonic approximation—perturbation theory to describe the 3-phonon scattering rates, a maximum in the intrinsic thermal conductivity at finite temperature was strikingly obtained in our investigation, without the need to consider multi-step or higher order phonon interactions. By applying our approach to solid argon, considering only three-phonon scattering, good agreement with experiment was achieved for the first time in both the low (T2) and high (T-1) temperature regimes simultaneously, in addition to the peak temperature (~ 8 K). This indicates that phonon-phonon interactions can solely be used to interpret the shape of the conductivity versus temperature curve for the case of Ar, without the need to invoke defect or boundary scattering in the low temperature regime. At the same time, by employing the same computational model for UO2, the results show that phonon-phonon interactions are not predominant in the sub-peak regime, which suggest that phonon-magnon interactions should be considered at low temperature to reproduce experimental results.

#### Recommended Citation

Hamed, Ahmed Osama Ibrahim, "Computational Modelling of Thermal Transport using Spectral Phonon Boltzmann Transport Equation" (2017). *Open Access Dissertations*. 1557.

https://docs.lib.purdue.edu/open_access_dissertations/1557