# Exploring Surface-Driven Mechanisms for Low-Temperature Sintering of Nanoscale Copper
## 3 Results and Discussion
### Mechanical Properties of Sintered Samples
#### 3.2.1 The Bulk Modulus and Shear Modulus  
The bulk modulus and shear modulus of sintered samples obtained at various sintering temperatures are illustrated in Figure 8a. The relevant measurement methods and techniques have been preliminarily described in the Method section. Figure 8b provides a schematic representation of the methods employed in this study to determine the elastic constants. The elastic constants characterize the stiffness of the material, defined by the linear relationship maintained between the stress and strain tensors under the limit of infinitesimal deformation. The approach used to measure the elastic constants has been reported in several articles and is known for its high accuracy. This method involves deforming one of the six directions of the proposed system at absolute zero and measuring the changes in the stress tensor during this process to estimate the derivative values of these linear relationships. It is important to note that the elastic tensor here is measured at absolute zero; however, this does not imply that the measured properties represent the material's attributes at absolute zero, as absolute zero is unattainable in practice. The samples used in this study were cooled to 300 K after sintering. The setting of absolute zero is employed solely to disregard the thermal vibrations of atoms in the simulation system, thereby minimizing stress fluctuations caused by thermal vibrations, reducing computational difficulty, enhancing convergence speed, and avoiding potential systematic and statistical errors. As shown in Figure 8a, the bulk modulus of the sintered samples monotonically increases from 300 K to 600 K, reaching approximately 130 GPa at a sintering temperature  
Figure 8: **(a)** The bulk modulus and shear modulus of samples obtained at different sintering temperatures. **(b)** Schematic diagram illustrating the morphological changes in the system during the determination of bulk modulus and shear modulus. of 600 K. When the temperature is from 600 K to 1000 K, the bulk modulus of the sintered samples exhibits a gradual increase. The bulk modulus of copper is around 140 GPa, and the samples obtained in this study have already approached the bulk modulus level of bulk copper at 600 K. In terms of the shear modulus, it continues to rise from 300 K to 600 K. At a sintering temperature of 600 K, the shear modulus of the sintered samples exhibits a step change, reaching a level nearly comparable to that of the samples at sintering temperatures of 600-1000 K. The shear modulus of copper is approximately 48 GPa, and the samples obtained in this study have reached around 43 GPa at 600 K, closely approaching the shear modulus level of bulk copper. The findings in the preceding section show that at a sintering temperature of 600 K, there is a significant reduction in the pore volume within the system, accompanied by a notable decrease in the energy levels before and after sintering. Both the bulk modulus and shear modulus indicate that at a sintering temperature of 600 K, the system can achieve mechanical properties that are close to those of bulk materials. At this temperature, the low-temperature sintering mechanism, dominated by rapid surface dynamics, plays a crucial controlling role. The atoms on the inner surfaces of the pores engage in rapid hopping movements within the interconnected regions of the pores, migrating from high-energy sites to low-energy sites. During this swift atomic transport process, the atoms that enter new sites act as nucleation points for local reconstruction, transforming originally high-energy heterogeneous regions (such as grain boundaries and slip planes) into low-energy homogeneous regions (FCC phase).