*2.1. Structure of Cu6Sn5 Unit Cell*

We performed MD simulations using a large-scale atomic/molecular massively parallel simulator (LAMMPS) [26] with the modified embedded atom method (MEAM) potential proposed by Baskes [27].

The Cu6Sn5 unit cell in this study is an η' phase belonging to a monoclinic crystal system with the C2/c space group [28]. In Figure 1, the gray and blue balls represent Sn and Cu atoms, respectively [12,28].

**Figure 1.** Cu6Sn5 (η' phase) cell structure [12].

#### *2.2. Modified Embedded Atom Method (MEAM) Potential*

In the MEAM theory, the total energy of the system consists of two parts: the energy embedding an atom in the background electron density and a pair interaction, which is described by Equations (1) and (2) [27,29].

$$E\_{\text{tot}} = \sum\_{i} \left[ F\_{i}(\overline{p}\_{i}) + \frac{1}{2} \sum\_{j \left( \neq i \right)} \phi\_{ij}(R\_{ij}) \right] \tag{1}$$

$$F\_i(\overline{\rho}\_i) = A E\_c \frac{\overline{\rho}\_i}{\rho\_{i0}} \ln(\frac{\overline{\rho}\_i}{\rho\_{i0}}) \tag{2}$$

where *Fi* is the embedding energy of the atom *i*, *Φij* is the interaction potential between atoms *i* and *j*, and *Rij* is the distance between atoms *i* and *j*. The parameters in the MEAM potential and their implications can be obtained from the literature [27,29]. For pure metals, the parameters in the MEAM potential function mainly include the cohesive energy *E*c, equilibrium nearest-neighbor distance *r*0, exponential decay factor for the universal energy function *α*, scaling factor for the embedding energy, the exponential decay factors for the atomic densities *β* (*i*, *i* = 0, 1, 2, 3), the weighting factors for the atomic densities *t* (*i*, *i* = 1, 2, 3), and the density scaling parameter *ρ*. For alloys, in addition to the above parameters, interaction parameters between the two pure metals could also be included, such as the binding energy between two different metal atoms, equilibrium nearest-neighbor distance between two types of atoms, and exponential decay factor for the universal energy function of the two atoms. Table 1 list the primary parameters in the potential according to references [29]. There is a many-body screening function in the MEAM, and the screening parameter *C* is adopted to determine the screening extent. *Cmin* and *Cmax* represent the minimum and maximum values of the screening boundaries, respectively [29]. Values of the *Cmin* and *Cmax* in this study are shown in Table 2 according to reference [29].

**Table 1.** Parameters of Cu6Sn5 MEAM potential.




## **3. Results and Discussion**

*3.1. Details of the MD Simulation*

3.1.1. Monocrystalline and Polycrystalline Structures of Cu6Sn5

Based on a Cu6Sn5 unit cell, polycrystalline Cu6Sn5 can be created by Atomsk through Voronoi tessellation [30]. The box size and number of the grain seeds can be determined when creating polycrystals. For example, a 100 Å × 100 Å × 100 Å polycrystal with 10 grains is shown in Figure 2. Different colors represent different grain identifications (IDs).

**Figure 2.** Polycrystalline Cu6Sn5 of 100 Å × 100 Å × 100 Å with 10 grains.

3.1.2. Validation of the MEAM Potential

Table 3 lists the stiffness constants calculated via the large-scale atomic/molecular massively parallel simulator (LAMMPS) with the MEAM potential in our study and by the first-principles method in reference [31]. Table 4 lists the elastic moduli of polycrystals, including bulk modulus (*B*), shear modulus (*G*), and Young's modulus (*E*), obtained via the

Voigt–Reuss–Hill (VRH) methods. Though some of the stiffness constants obtained from both methods differ, the value of the elastic moduli of the polycrystalline Cu6Sn5 is within the calculated result by Ghosh [32] and Lee [31]. Our MD simulations were performed based on this MEAM potential.


**Table 3.** Stiffness constants of monocrystalline Cu6Sn5.



#### 3.1.3. Simulation Setting

By using the proposed MD simulation, the stress–strain relations of the polycrystals were studied by stretching the polycrystals at different temperatures and strain rates. The polycrystal was fully relaxed under NVE, NVT, and NPT ensembles alternately to make the system reach target setting temperatures and pressures and was stretched under the NPT ensemble. To access the effects of the strain rate, the polycrystals were stretched at stain rates of 0.00001 to 100 ps<sup>−</sup>1. In addition, the temperature effect over a temperature range of 250–500 K with intervals of 50 K was considered in our study.

## *3.2. Isotropic Analysis of Tensile Properties of Polycrystals*

To determine the isotropic characteristics of the polycrystalline Cu6Sn5, it was stretched along the x-, y-, and z-axis, respectively, at 300 K with a strain rate of 1 ps<sup>−</sup>1. Figure 3 shows the stress–strain response. Stress\_x, Stress\_y, and Stress\_z represent stretching along the x-, y-, and z-axis, respectively. The three curves almost overlapped before the polycrystalline Cu6Sn5 cracked, specifically in the linear elastic deformation phase. This indicates that the tensile properties of the polycrystalline Cu6Sn5 along the x, y, and z directions are very similar to each other. Therefore, the polycrystalline Cu6Sn5 in our study is isotropic, which is consistent with the real polycrystal materials [33].

**Figure 3.** Stress–strains curves when stretching along x-, y-, and z-axis.
