# Phonon Scattering and Thermal Conductivity of Actinide Oxides with Defects
## 2 Methods  
### Reverse Non-Equilibrium Molecular Dynamics (RNEMD)  
Various classical MD algorithms, such as direct non-equilibrium molecular dynamics (NEMD), reverse non-equilibrium molecular dynamics (RNEMD), equilibrium molecular dynamics (EMD), and homogeneous non-equilibrium molecular dynamics (HNEMD) have been used to explore thermal transport in solids. In the present study, RNEMD [PERSON] _et al._ (2018) is selected. As a cause and effect reversed algorithm, RNEMD imposes a heat flux in a simulation structure to induce a temperature gradient across the structure. A simulation box is divided into multiple imaginary bins, and the bin located at the center of the simulation structure is chosen to be a hot bath while the bins at the two ends are chosen to be cold baths. A heat flux is applied by exchanging energy between the hot bath and the cold baths, inducing two identical temperature gradients across the structure, as shown in Figure 1. Thermal conductivity, \(k\), is calculated from the induced temperature gradient, _dT/dx_, and the imposed heat flux, \(q\), by using the following Fourier's law.  
\[k\ =\ -\frac{\langle q\rangle}{\langle dT/dx\rangle} \tag{1}\]  
Here, \(<\!\!q\!\!>\) is the heat flux and \(<\!\!dT/dx\!\!\!>\) is the temperature gradient averaged over time and space. In the present study, the force field developed by [PERSON] et al. [PERSON] _et al._ (1989); [PERSON] _et al._ (1998); [PERSON] _et al._ (1998); [PERSON] _et al._ (1999) is used with LAMMPS simulator [PERSON] _et al._ (2016). In this force field, the interatomic energy in actinide oxides takes the following potential form.  
\[E_{i}\ =\ \frac{1}{2}\sum_{j}\phi_{\alpha\beta}\Big{(}r_{ij}\Big{)}-G_{\alpha} \sqrt{\sum_{j}\sigma_{\beta}\Big{(}r_{ij}\Big{)}} \tag{2}\]The potential energy of the _ith_ atom is determined from the combination of the pairwise interaction between the _ith_ atom and its surrounding atoms, \(\phi_{\alpha\beta}\) (\(r_{\bar{\eta}}\)) and many body perturbations, \(\sigma_{\beta}\)(\(r_{\bar{\eta}}\)). \(G_{\alpha}\) is the constant of proportionality. This potential has been used successfully to predict thermal properties of actinide oxides [12]. For example, [PERSON] et al. [17] demonstrated the effectiveness of this model, and the authors estimated the thermal transport properties of actinide oxides with defects using the same force field [3; 4]. For the interactions between noble gas-actinide and noble gas-oxygen, parameters developed by [PERSON] et al. [15] are used; during their parameter fitting procedure for the gas interactions with actinide oxides, they used the gas-gas interaction developed by [PERSON] [18].  
The following equilibration steps are taken to equilibrate the simulation structure before running RNEMD. The purpose of equilibration is to make the atoms in the simulation structure forget their initial velocities and reach a thermodynamically stable state in order to obtain thermodynamic properties that are statistically meaningful. First, velocities are randomly imposed onto each atom in the simulation structure, and the simulation structure is energy-minimized with a time step of 1 fs. Then, the simulation structure is heated up to 500 K using isenthalpic (NPH) ensemble with Langevin thermostat. Next, the simulation structure is cooled down to the target temperature (300 K) with an isothermal-isobaric (NPT) ensemble. Energy exchange using the RNEMD algorithm is performed for 2 ns with a time-step of 1 fs (femtosecond) once the equilibration is complete. The authors observed that the amount of energy exchanged between the hot bath and cold bath is maintained to be constant in all simulation structures, and temperature profiles in the left and right segments of the simulation structures reach a steady state quickly, after only 0.2 ns.