# Exploring Surface-Driven Mechanisms for Low-Temperature Sintering of Nanoscale Copper
## 2 Methods  
In this work, a Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) [32] is used to conduct all simulations. The interactions between Cu atoms are described with embedded atom model potential, which is widely used in Cu systems [33; 34], with numerous studies applying it to the Cu-Cu bonding process [35; 36]. OVITO is used for subsequent analysis and visualization [37]. To obtain relatively random particle samples, several bulk copper structures are constructed, each with a lattice constant of 3.615 A at 300 K [38]. Gaussian distribution is applied to assign initial atomic velocities, with velocities distributed based on the average atomic velocity at 300 K. A predetermined random number seed is used to generate the velocity with consistent macroscopic properties yet slightly varied atomic structures. A total of 20 distinct random number seeds from a true random number generator were utilizedin this assignment process, producing 20 unique bulk copper structures. These structures are subsequently heated to 1500 K at a rate of 1 K/ps, surpassing their melting point, and then held for a 10 ps isothermal relaxation period. The molten bulk copper is subsequently divided into spherical nanoparticles with a diameter of 5 nm. These nanoparticles are cooled to 300 K at a rate of 1 K/ps and subjected to an additional 10 ps isothermal relaxation to achieve an equilibrium state. This process yielded 20 nanoparticles with varied shapes and internal structures, which were compiled into a random nanoparticle library for simulations. It should be noted that the particle size will change the intrinsic atomic mobility on the surface, but it does not significantly alter the overall trend of the conclusions based on single particle simulation [28]. Therefore, in this study, a diameter of 5 nm is selected for comprehensive analysis, considering the balance between model size and our computing ability. To correspond with the temperature range of the low-temperature process for nano-solders, seven discrete sintering temperatures are selected, ranging from 300 K to 1000 K in intervals of 100 K. During the modeling process, eight nanoparticles are randomly selected from a predefined nanoparticle library and positioned at the four corners of an 80 A \(\times\) 80 A \(\times\) 80 A cubic box. These nanoparticles are then displaced randomly around their center of mass along the three principal axes, with displacement values ranging from 0 A to a maximum of 10 A. Following displacement, each nanoparticle was rotated around its center of mass using the three axes as rotation axes, with angles determined by random values between 0\({}^{\circ}\) and 360\({}^{\circ}\). This configuration produced a structural file with eight randomly distributed nanoparticles. To expand the simulation system, the configuration is duplicated along the z-axis, resulting in a final model for low-temperature sintering simulation containing 16 nanoparticles with randomized orientations and spatial positions, avoiding the influence of particle shape and distribution. At the initiation of the sintering simulation, the system underwent a 10 ps isothermal relaxation at 300 K within the NVT ensemble. Subsequently, the temperature is raised to predetermined levels (300 K, 400 K, \(\ldots\), 1000 K) at a rate of 1 K/ps [39; 40], with the NVT ensemble still in use, followed by a 10 ps relaxation at each target temperature to ensure equilibrium. Following this stage, the ensemble is switched to NPT, maintaining a vacuum pressure of 10 ps to facilitate particle contact. Upon establishing contact, a 2 ns sintering process was carried out under the NPT ensemble, also at vacuum pressure. Post-sintering, the system was cooled to 300 K at a rate of 1 K/ps, concluding with a 10 ps relaxation to generate the post-sintering sample file for subsequent mechanical property evaluation. This sintering simulation protocol is repeated five times for each sintering temperature, obtaining a total of 35 distinct sintering samples.