# Phonon Scattering and Thermal Conductivity of Actinide Oxides with Defects
## 3 Results and Discussion
### The Effect of Point Defects on Thermal Transport in Actinide Oxides
According to the popular analogy of phonon transport to the kinetic theory of gas, the effective phonon mean free path, \(l_{eff}\), is expressed using Mathiessen rule [23; 24] as  
\[\frac{1}{l_{eff}}\ =\ \frac{1}{l_{ph-ph}}+\frac{1}{l_{b}} \tag{3}\]  
where \(l_{ph-ph}\) is phonon-phonon scattering length, and \(l_{b}\) is the phonon-boundary scattering length which is the length of the simulation box. As discussed in the simulation method section, the cross-sectional area (i.e., sample lengths in \(y\) and \(z\) directions) does not affect the 1D heat conduction in the simulation structure. Therefore, it is concluded that the dominant factor that limits thermal transport in actinide oxide is the sample length in the direction of 1D heat conduction (\(x\)-direction in the present study). Since \(l_{b}\) represents the phonon scattering that arises from the simulation boundaries, it can be considered to be the sample length in \(x\)-direction. If thermal conductivity is assumed to be proportional to the effective phonon mean free path based on the kinetic theory of gas, the thermal conductivity of the sample can be expressed as  
\[\frac{1}{k}\propto\frac{1}{l_{eff}}\ =\ \frac{1}{l_{ph-ph}}+\frac{1}{l_{b}} \tag{4}\]  
Figure 5: (a) Thermal conductivity of stoichiometric \(\mathrm{U}\mathrm{O}_{2}\) and \(\mathrm{U}\mathrm{O}_{2}\) with various defects. (b) Thermal conductivity of stoichiometric \(\mathrm{Pu}\mathrm{O}_{2}\) and \(\mathrm{Pu}\mathrm{O}_{2}\) with various defects. This relationship indicates that the inverse of thermal conductivity, \(k\), is a linear curve when it is plotted as a function of the inverse of a simulation structure size. Therefore, thermal conductivity for a bulk size sample can be obtained by the extrapolation of the plot; i.e., in the limit when \(1/l_{b}\to 0\). However, the bulk thermal conductivity for the simulation structures with high defect concentrations cannot be estimated using this traditional Matthiessen rule [23], because the thermal conductivity (or effective phonon mean free path) is independent on the sample length (or \(l_{b}\)) (Figure 5). The thermal conductivity calculation in the present study is supported by the literature. [PERSON] et al. [25] estimated the thermal conductivity of PuO\({}_{2}\) to be 15.2 W/m-K. [PERSON] et al. [26] reported 14.6 W/m-K for the thermal conductivity of UO\({}_{2}\) using MD. [PERSON] et al. [21] estimated the thermal conductivities of UO\({}_{2}\) and PuO\({}_{2}\) to be \(\approx\)17.5 W/m-K and \(\approx\)22.5 W/m-K respectively by using MD. [PERSON] [27] calculated the thermal conductivity of UO\({}_{2}\) to be \(\approx\)13.5 W/m-K. [PERSON] et al. [28] used first-principles simulations to calculate the thermal conductivities of UO\({}_{2}\) and PuO\({}_{2}\) to be \(\approx\)12.5 W/m-K and 7.5 W/m-K.  
As can be seen in Figure 5, the effect of oxygen interstitials on the thermal conductivities of UO\({}_{2}\) and PuO\({}_{2}\) is similar to that of oxygen vacancies. Several researchers have obtained correlations for the thermal conductivity actinide oxide fuels as a function of oxygen-to-metal ratio using experimental results. For example, [PERSON] et al. [29] proposed \(\lambda=1/(\text{A}+\text{BT})\) for the thermal conductivity (\(\lambda\)) of PuO\({}_{2}\). Here, \(\text{A}=2.85x+0.035\) m-K/W and \(\text{B}=(-7.15x+2.86)\times 10^{-4}\) m/W, where \(x\) is the deviation from stoichiometry (\(x=2-\text{O/M}\)). If this correlation is used, the thermal conductivities of PuO\({}_{2}\) with 1%, 2%, and 5% oxygen vacancies are calculated to be 5.0, 3.58, and 1.93 W/m-K respectively. Our results exhibit thermal conductivities that are consistent with these experimental results; for the case of PuO\({}_{2}\) with \(N_{x}=100\), thermal conductivities are calculated to be 5.82, 3.93, and 2.16 W/m-K for 1%, 2%, and 5% oxygen vacancy defects, respectively. However, it should be noted that the direct comparison of the results obtained in the present study to the previous experimental results is difficult.