The MD formalism is a powerful tool in studying the mechanical performance of alloys. It is a computational approach that simulates the behavior of atoms and molecules in a system over time, allowing for the prediction of mechanical outputs (parameters) such as strength, ductility, and fracture toughness [20]. At its core, MD formalism is based on classical mechanics and statistical thermodynamics. It involves solving the equations of motion for each atom in a system, taking into account the interatomic forces that govern their behavior [21]. This allows for the simulation of the system's dynamics over time, providing insight into its mechanical properties as below equation [22],

$${F_i}=\sum\limits_{{i \ne j}} {{F_{ij}}={m_i}\frac{{{d^2}{r_i}}}{{d{t^2}}}={m_i}\frac{{d{v_i}}}{{dt}}}$$

1

To solve this equation, force field of modeled system is important parameter [23]. In current research, to define atomic interactions, the modified embedded-atom method (MEAM) implemented [24]. The MEAM force field defined as Eq. (2),

$$E=\sum _{i}\{{F}_{i}\left({\rho }_{i}\right)+\frac{1}{2}\sum _{i\ne j}{\phi }_{ij}\left({r}_{ij}\right)\}$$

2

Here, F is the embedding energy parameter and describes the electron density ρ parameter in modeled sample, and φ introduced the pair interaction inside sample. The last parameter is calculated for various neighbors’ parties which located inside cutoff radius of center particle. Force field implementing inside box, Newton’s evolution equation fulfilled and various particles displacement in defined condition predicted. Technically, integrate of the Newton’s formalism is calculated by the velocity Varlet method as equations (3) and (4) [25, 26],

$$r\left( {t+\Delta t} \right)=r\left( t \right)+v\left( t \right)\Delta t+\frac{1}{2}a\left( t \right)\Delta {t^2}+O\left( {\Delta {t^4}} \right)$$

3

$$v\left( {t+\Delta t} \right)=v\left( t \right)+\frac{{a\left( t \right)+a\left( {t+\Delta t} \right)}}{2}\Delta t+O\left( {\Delta {t^2}} \right)$$

4

Where, r(t + Δt)/v(t + Δt) parameters are the coordinate and velocity of various atoms in t + Δt time of simulations, respectively. Currently, described computational method used to estimate the hardening process effects on mechanical performance of Al-Zn-Mg alloy. For this, MD simulations divided into 2 main phases as below,

**Equilibrium Phase –** In the first step, the extra stress of model Al-Zn-Mg alloy eliminated by using the NVE ensemble for 1 ns. Next, the temperature of system changed from target value (773 K) to 298 K by implementing NVT ensemble (with 1 fs time step and 0.1 fs for temperature damping ratio). MD simulation done for 10 ns in this step. Computationally, the MD box size set to 120×340×340 Å3 by using periodic boundary condition [27]. Our atomic representation of modeled Al-Zn-Mg alloy with 31850 atoms depicted in Fig. 1. In this sample, modeled alloy consists of 92%Al- 5.5%|Zn- 2.5% Mg.

**Mechanical Evolution Phase –** Secondary, the mechanical test settings implemented to equilibrated alloys. This setting caused the structural expansion of models with 0.1 fs− 1 ratio. In this step, the Nose-Hoover thermostat controlled the temperature of allies in expansion procedure [28, 29]. Our used MD settings in our numerical research presented in Table 1.

Table 1

The MD method parameters value in atomic/mechanical description of Al-Zn-Mg alloy.

MD Simulation Parameter | Parameter Setting |

Computational Box Length | 120×340×340 Å3 |

Number of Atoms | 31850 |

Boundary Setting | PBC |

Barostat/Thermostat | Nose-Hoover |

Temperature of Initial Condition | 773 K |

Time Step | 1 fs |

Temperature Damping Ratio | 0.1 fs |

Simulation Total Time | 20 ns |