# Exploring Surface-Driven Mechanisms for Low-Temperature Sintering of Nanoscale Copper
## 3 Results and Discussion
### Sintering of Multiple Copper Nanoparticles
#### 3.1.1 Energy Changing During Sintering  
The energy changes during the sintering process provide insights into the structural quality of the system. A structure that closely approximates a perfect face-centered cubic (FCC) phase will exhibit lower potential energy at a given temperature, while deviations from this ideal structure will result in higher potential energy. In a sintering system, a lower number of pores and a morphology closer to a spherical shape are also associated with reduced potential energy. As shown in Figure 2, both the average atomic potential and kinetic energy throughout the simulation are monitored. For clarity, the figure displays only the changes in kinetic and potential energy at a sintering temperature of 600 K. It is clear that the average kinetic energy of the atoms varies with temperature, and the entire simulation system experienced a four-stage process: isothermal heating, heating, isothermal cooling, and  
Figure 1: Schematic diagram of the system modeling process and the sintering simulation process. cooling. Throughout the heating and cooling phases, the rate of temperature change remained stable, with no notable instances of overheating or overcooling. Regarding the potential energy curve, it similarly experiences the four-stage process of isothermal heating, heating, isothermal cooling, and cooling. At the beginning of the simulation, the potential energy is relatively high due to the redistribution of atomic velocities according to a Gaussian distribution at the corresponding temperature during initialization, resulting in an unstable initial state. However, as the system undergoes isothermal relaxation at room temperature, the excess potential energy present dissipates quickly, leading to a state of equilibrium. Subsequently, during the gradual heating process, the structure of the system does not undergo significant adjustments, and the potential energy remains relatively stable. Once the temperature exceeds a certain threshold, the potential energy begins to rise slightly with the increase in temperature, indicating that some structural adjustments occur within the system. After reaching 600 K, the potential energy begins to decrease slowly, signifying the progression of the sintering process. During this phase, the pores within the system are gradually eliminated, and the migration and merging of slip and dislocations occur, further stabilizing the system. By approximately 2.25 ns, the energy of the system stabilizes, indicating that the sintering process is nearly complete. Following this, the temperature of the system begins to decrease linearly, and the potential energy also exhibits a linear decline. This linear trend in potential energy relates to the thermal expansion coefficient of copper. Since the temperature is far below the melting point, the thermal expansion coefficient of copper exhibits minimal variation. The rate of change in the system's volume with temperature remains nearly constant, while the equilibrium distance between atoms gradually decreases with the lowering temperature, demonstrating the trend of linear decline in potential energy. From the analysis above, it can be concluded that the potential energy of the system significantly decreases after sintering. Furthermore, the difference in potential energy varies with different sintering temperatures. Figure 3 extracts the differences in potential energy for the sintering process at various temperatures. It is noteworthy that when the temperatures of the systems are equal, the average atomic kinetic energy within them remains consistent. The extracted differences in potential energy represent the overall energy change. This energy variation can significantly reflect the sintering conditions at different temperatures. Figure 2: The variations in average atomic kinetic energy and atomic potential energy throughout the entire sintering process at 600 K.  
As shown in Figure 3a, when the sintering temperature is below 500 K, the energy difference before and after sintering does not change significantly.