# Atomistic Investigation of Plastic Deformation and Dislocation Motion in Uranium Mononitride
## 2 Computational Methods
### Deformation Behavior  
In this paper, we investigate the stress-strain behavior of UN single crystals at \(T=300\) K and strain rates of \(10^{8}\), \(5\times 10^{8}\), and \(10^{9}\) s\({}^{-1}\) using both the Tseplyaev and Kocevski potentials. Due to computational limitations, MD simulations of uniaxial tension are limited to nanometer-sized supercells and strain rates of at least \(10^{7}\) s\({}^{-1}\)[25], \(150\times 30\times 30\) UN supercells are first equilibrated at 300 K and zero pressure for 100 ps using a Nose-Hoover thermostat-barostat. After equilibration, the load is applied in the [100] direction at the target strain rate, and the material is allowed to deform in the isothermal-isobaric (\(NPT\)) ensemble up to fracture. Periodic boundary conditions (PBCs) are applied in all directions throughout the whole simulation. Relaxation is conducted such that the \(x\)-, \(y\)-, and \(z\)-directions are independent, but the supercell remains orthogonal. Note that during deformation, the \(x\)-direction is controlled by the strain rate. To estimate the nanoindentation hardness, we perform deformation simulations at strain rates of \(10^{7}\), \(10^{8}\), and \(10^{9}\) s\({}^{-1}\) but with PBCs applied only along the loading direction, i.e., the \(x\)-axis, whereas surfaces perpendicular to the \(y\)- and \(z\)-axes are modeled with fixed boundary conditions, yielding free surfaces along these planes. Before deformation, the supercells are equilibrated in the \(NPT\) ensemble for 50 ps (where the pressure is allowed to equilibrate only along the \(x\)-axis), and then in the canonical (\(NVT\)) ensemble for a further 50 ps. Uniaxial tension is then applied in the \(NVT\) ensemble until fracture. From the stress-strain curves of nanocrystalline samples with free surfaces, the nanoindentation hardness can be estimated by measuring the stress-strain curves at different strain rates, (\(\dot{e}\)), recording the maximum stress, \(\sigma_{\rm max}\), for each case, and then extrapolating \(\sigma_{\rm max}(\dot{e})\) to \(\dot{e}\to 0\)[6]. The linearly extrapolated value should correspond to the experimental nanoindentation hardness values. We define the maximum stress in the deformation process as the yield stress. [PERSON]'s modulus is obtained by fitting the linear part of the stress-strain curve excluding the transient at the origin and the nonlinearities near the yield point. The [PERSON]'s modulus along the [100] direction can also be calculated from the elastic constants according to the following formula [26]:  
\[E_{(100)}=\frac{(C_{11}-C_{12})\times(C_{11}+2C_{12})}{C_{11}+C_{12}} \tag{1}\]  
To monitor structural development during deformation, the centrosymmetry parameter (CSP) was calculated by OVITO for snapshots of the simulated supercells using the minimum-weight matching algorithm [27]. The CSP is a measure of the lattice disorder and is defined according to [28]  
\[\text{CSP}=\sum_{i=1}^{N/2}|\textbf{r}_{i}+\textbf{r}_{i+N/2}|^{2} \tag{2}\]  
where \(N\) is the number of nearest neighbors surrounding an atom (\(N=6\) for the NaCl crystal structure), and \(\textbf{r}_{i}\) and \(\textbf{r}_{i+N/2}\) are two vectors from the atom under consideration to a pair of opposite neighbor atoms. For an ideally centrosymmetric crystal, the contributions of all neighbor pairs cancel, and the resulting CSP is zero. The CSP algorithm is robust against thermal fluctuations; for a single crystal, it shows a single peak that starts at zero and whose width increases with increasing \(T\). A second peak in the CSP plot corresponds to a different local environment, e.g., dislocations and highly disordered grain boundaries (GBs) [27; 29]. The CSP has dimensions of squared distance and is usually given in units of \(a^{2}\), where \(a\) is the lattice parameter. The CSP algorithm is an indirect characterization technique. To directly visualize and analyze dislocations, OVITO's dislocation extraction algorithm (DXA) [30] is utilized and applied to the U sublattice.