# Molecular Dynamics Simulation of Bulk Cu Material under Various Factors
## 2 Method of Calculation  
To study the structural characteristics of bulk Cu material, we applied the following procedure:  
Originally, Cu atoms were randomly sown into a cube with the size l, determined by Formula (1):  
\[1=\sqrt[3]{\frac{\text{N}}{\rho}} \tag{1}\]  
and the molecular dynamics (MD) simulation method with the embedded interaction potential of Sutton-Chen (SC) (2) [61; 74; 75], and periodic boundary conditions were applied. As emphasized in [73], this choice of potential is the most appropriate one.  
\[\text{F}_{\text{i}}(\rho)=A_{\text{i}}E_{\text{i}}^{0}\rho\ln\rho; \tag{2}\] \[\rho_{\text{i}}^{\text{a(l)}}(\text{R})=e^{-b^{*}},\text{ b}^{*}=\beta_{\text{i}}^{\text{(l)}}\left(\frac{\text{R}}{\text{R}^{\text{l} }-1}\right),\left(\overline{\rho}_{\text{i}}\right)^{2}=\sum\limits_{\text{l}= 0}^{3}t_{\text{i}}^{\text{(l)}}\left(\rho_{\text{i}}^{\text{(l)}}\right)^{2}\]  
The corresponding parameters of bulk Cu [61; 74; 75] are given in Table 1.  
The program code LAMMPS [74; 75] based on Embedded Atom Method (EAM) potential [49] was applied to describe the interaction between Cu atoms.  
Initially, to make bulk Cu material, the material was left to run 5 \(\times\) 10\({}^{4}\) MD steps of recovery statistics with a heating rate of 4 \(\times\) 10\({}^{11}\) K/s and a heating step of 1 fs at temperature (T) = 4500 K. Then, the temperature decreased from T = 4500 K to T = 1000, 900, 800, 700, 600, 500, 400, and 300 K to increase the crystallization state of the material. To determine the effect number of atoms, models with the corresponding number of atoms (N) were applied: N = 2916 atoms Cu (Cu\({}_{2916}\)), 4000 atoms Cu (Cu\({}_{4000}\)), 5324 atoms Cu (Cu\({}_{5324}\)), and 6912 atoms Cu (Cu\({}_{6912}\)). Then, the material Cu\({}_{5324}\) was selected as the standard to study the effect of temperature, and the glass transition temperature of the material and the annealing time (t) = 200 ps at T = 652 K were determined to increase the crystallization of the material.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline
**Parameters** & **E\({}_{\text{i}}^{0}\) (eV)** & **R\({}_{\text{i}}^{0}\) (Å)** & **\(\alpha_{\text{i}}\)** & **A\({}_{\text{i}}\)** & **\(\beta_{\text{i}}\)(0)** & **\(\beta_{\text{i}}\)(1)** \\ \hline Value & 3.54 & 2.48 & 5.11 & 1.07 & 3.63 & 2.2 \\ Parameters & \(\beta_{\text{i}}\)(2) &To study the structural features, radial distribution function (RDF), size (I), and the total energy of the system (E\({}_{\text{tot}}\)) [77] were determined through the values of the shape quantities. All samples are conducted through the phase transition with the laws of [PERSON] [78], [PERSON] [79], and the Common Neighbors Analysis (CNA) [80; 81] was applied on the basis of the glass transition temperature (T\({}_{\text{g}}\)) by the E\({}_{\text{tot}}\) and T. For the number of atoms distribution in the material structure, the radial distribution function (RDF) was applied [82] to determine g(r):  
\[\text{g(r)}=\frac{\text{n(r)}}{4\pi\text{r}^{2}\rho_{0}\text{dr}} \tag{3}\]  
where g(r), n(r), \(\rho_{0}\), and r are functions used to determine the probability of finding an atom i in the space, the atom number, the atomic density, and the distance from atom i to another atom.  
To determine the diffusion mechanism of the atoms, mean squared distance (MSD) was applied using the following expression [83]:  
\[\text{r(t)}=\sqrt{\frac{1}{\text{N}}\left\{\sum_{\text{i=1}}^{\text{N}}\left| \overrightarrow{\text{r}_{\text{i}}}(\text{t})-\overrightarrow{\text{r}_{ \text{i}}}(0)\right|^{2}\right\}} \tag{4}\]  
where N is the number of atoms in the system, and r\({}_{\text{i}}\)(0) \(\rightarrow\) r\({}_{\text{i}}\)(t) is the position of atom i at time t. For visualization, the software OVITO [55; 56] was used. Finally, to check the accuracy of the results, Dual Energy X-ray Absorptiometry (DXA) was used [84]. These programs were edited and applied on the computer system of the Institute of Physics, University of Zielona Gora, Poland.