#
Grain Size Effects on Mechanical Properties of Nanocrystalline Cu_{6}Sn_{5} Investigated Using Molecular Dynamics Simulation

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## Abstract

**:**

_{6}Sn

_{5}is one of the main components of IMCs, and its mechanical properties considerably influence the reliability of solder joints. In this study, the effects of grain size (8–20 nm) on the mechanical properties (Young’s modulus, yield stress, ultimate tensile strength (UTS), and strain rate sensitivity) of polycrystalline Cu

_{6}Sn

_{5}were investigated using molecular dynamics simulations at 300 K and at a strain rate of 0.0001–10 ps

^{−1}. The results showed that at high strain rates, grain size only slightly influenced the mechanical properties. However, at low strain rates, Young’s modulus, yield stress, and UTS all increased with increasing grain size, which is the trend of an inverse Hall–Petch curve. This is largely attributed to the sliding and rotation of grain boundaries during the nanoscale stretching process, which weakens the interaction between grains. Strain rate sensitivity increased with a decrease in grain size.

## 1. Introduction

_{6}Sn

_{5}and Cu

_{3}Sn [2,4]. IMCs are inevitable byproducts of the soldering process, and they considerably influence the mechanical and electrical bonding.

_{6}Sn

_{5}were studied by molecular dynamics simulations. The previous studies suggested that the mechanical properties of IMCs could be affected by strain rate [18,20,27]. Strain rate effects on polycrystalline Cu6Sn5 have been studied in our previous work [20], and they cannot be ignored. Therefore, in this study, the influence of grain size on the mechanical properties of IMCs with different strain rates was also studied.

## 2. Methodology

#### 2.1. MEAM Potential

_{ij}is the distance between atoms i and j.

_{6}Sn

_{5}unit cell in this study is an η’ phase, which is a monoclinic crystal system with the C2/c space group [38], as shown in Figure 1 [16] (gray atoms represent Sn, and blue represents Cu). The parameters of the potential function selected refer to the literature values [20,39], and the validity of the potential functions has been verified in other studies [20].

#### 2.2. Establishment of Polycrystals

#### 2.3. Simulation Settings

^{−1}.

## 3. Results and Discussion

#### 3.1. Stress-Strain Responses at Different Strain Rates

_{6}Sn

_{5}(average grain size = 8–20 nm) were stretched to study their stress–strain response at different strain rates (Figure 4a–f). The tensile strain rates selected in this study were 0.0001, 0.001, 0.01, 0.1, 1, and 10 ps

^{−1}, respectively. The variations in stress–strain curves with grain size were almost identical; thus, only the stress-strain curves with grain sizes of 8, 12, 16, and 20 nm were selected for further analysis. In the case of higher strain rates (10, 1, 0.1 ps

^{−1}; Figure 4a–c), the differences of stress-strain curves with different grain sizes were small. As the strain rate decreased (0.01, 0.001, and 0.0001 ps

^{−1}; Figure 4d–f), the differences in the stress–strain curves under different grain sizes became more obvious. These differences are discernible via the slope at the linear stage (Young’s modulus) and via yield stress and the UTS when stretching. The influences of grain size on Young’s modulus, yield stress, and UTS are analyzed in detail in the following sections.

_{6}Sn

_{5}more prone to dislocation during plastic deformation. When grain size changes, the proportion of grain boundary in the whole polycrystal will be different, which will eventually affect the degree of plastic deformation. Therefore, grain size has a greater influence on the stress–strain curve at a lower strain rate.

#### 3.2. Elastic Properties of Polycrystalline Cu_{6}Sn_{5}

^{−1}. The results are shown in Figure 5a–f, which are the Young’s moduli calculated at strain rates of 10, 1, 0.1, 0.01, 0.001, and 0.0001 ps

^{−1}, respectively.

^{−1}fluctuated between 187 GPa and 193 GPa (Figure 5a,b). However, as the strain rate decreased, Young’s moduli showed an increasing trend with an increase in grain size (Figure 5c–f). In this study, the coefficient of variation (CV, the ratio of standard deviation to average value) was used to measure the dispersion degree of these Young’s moduli. The CVs of Young’s moduli are 0.01 at strain rates of both 10 and 1 ps

^{−1}. Therefore, it was concluded that when polycrystalline Cu

_{6}Sn

_{5}was drawn at higher strain rates (10 and 1 ps

^{−1}in this study), its Young’s modulus was almost unaffected by grain size, only fluctuating in a small range.

_{c}represents the Young’s modulus of the polycrystal, ${E}_{1}$ and ${E}_{2}$ represent the Young’s modulus of grain and grain boundary, respectively, and ${V}_{1}$ and ${V}_{2}$ represent the volume fraction of grain and grain boundary, respectively. With an increase in grain size, the volume proportion of the grain boundaries in the whole polycrystal is larger. The grain boundaries comprise disordered atoms with weaker interactions between these atoms compared to the grain, so the Young’s modulus of the grain boundaries is lower than the Young’s modulus of the grain itself. Therefore, with the increase in grain size, the proportion of the grain boundaries in the polycrystal becomes smaller, and thus, the total Young’s modulus of the polycrystal increases.

^{−1}is shown in Figure 6. The average Young’s modulus increased when the average grain size increased by 1 nm (Figure 5a–f: 0.33255, 0.65244, 0.76375, 1.50395 GPa at strain rates of 0.1, 0.01, 0.001, and 0.0001 ps

^{−1}, respectively). Therefore, Young’s modulus increases faster with increasing grain size at lower strain rates than that at higher strain rates. It was therefore concluded that the effect of grain size on Young’s modulus is greater at lower strain rates, which in turn means that it is more easily affected by grain size at a lower strain rate.

#### 3.3. Yield Stress of Polycrystalline Cu_{6}Sn_{5} with Different Grain Sizes

^{−1}, the yield stress fluctuates randomly with changes in grain size. When the strain rate decreases from 1 to 0.0001 ps

^{−1}, the yield stress increases proportionally with an increase in grain size. As the strain rate decreases, the stability of this trend improves. The trends of yield stress with changing grain size at different strain rates closely mirror those of Young’s modulus.

#### 3.4. Grain Size Effects on UTS of Polycrystalline Cu_{6}Sn_{5}

^{−1}: Figure 8a–e). The UTS fluctuated within the range of 16.0–16.5, 14.5 to 15.2, 12.4 to 13.2, 8.3 to 10.9, and 6.2 to 9.0 GPa with strain rates of 10,1,0.1, 0.01, and 0.001 ps

^{−1}, respectively. With a decrease in strain rate, the fluctuation range of UTS gradually increased. The CVs at the five strain rates mentioned above were found to be 0.008, 0.013, 0.018, 0.078, and 0.109, respectively. The corresponding mean UTS values are 16.16, 14.90, 12.83, 10.28, and 8.18 GPa, respectively. The UTS of the polycrystalline Cu

_{6}Sn

_{5}with different grain sizes only fluctuated randomly within a very small range at rapid stretching (10 ps

^{−1}). As the strain rate decreased, the fluctuation range of UTS increased gradually (determined from the CV), but there was still no obvious trend. A likely explanation for this effect is plastic deformation increasing in polycrystals during the stretching process with a decrease in strain rate. Therefore, the plastic deformation which leads to a random fluctuation of UTS in a large range is not completely dominant (Figure 4a–e). At a strain rate of 0.0001 ps

^{−1}, the UTS increases with the increasing grain size, which conforms to the inverse Hall–Petch curve [28]. This is largely due to the plastic deformation of polycrystalline Cu

_{6}Sn

_{5}, which is dominant before it reaches UTS when stretched (Figure 5f). Grain boundary sliding and rotation, which weaken the nanocrystal [28,43,44], are the main mechanisms for the mechanical properties with grains < 30 nm [28].

#### 3.5. Strain Rate Sensitivity of Polycrystalline Cu_{6}Sn_{5}

_{6}Sn

_{5}are affected by strain rate during tensile processes. Thus, strain rate sensitivity, m, is used to measure sensitivity, as shown in Equation (4) [45], where $\sigma $ and $\dot{\epsilon}$ represent flows stress and strain rate, respectively. Since the effects of grain size are more obvious at lower strain rates, the strain rates used in this calculation were 0.001 and 0.0001 ps

^{−1}, respectively. The strain range corresponding to flow stress is 0.02–0.08 (Figure 4e,f). Polycrystals with four grain sizes (20, 16, 14, and 8 nm) were selected for analysis. The results are shown in Figure 9. The results showed that the strain sensitivity of small grain size polycrystals is greater than that of larger grain size polycrystals. The strain rate sensitivity decreases with increasing grain size. At the same time, the strain rate sensitivity increases with increasing strain with the four-grain size polycrystals.

## 4. Conclusions

_{6}Sn

_{5}were investigated. The results showed that the grain size affected both elastic and plastic deformation. Therefore, the grain size effects are different at different strain rates. The conclusions are as follows.

- The effect of grain size on the stress-strain curve increases with decreasing strain rate and is practically invisible at high strain. This conclusion can be particularised for Young modulus, yield stress, and UTS.
- Young’s modulus, yield stress, and UTS increase with the increasing grain size at a lower strain rate. Moreover, the growth rate of the Young’s modulus increases with a decrease in strain rate.
- Polycrystals with a small grain size are more sensitive to strain rate than those with a large grain size. The strain rate sensitivity of polycrystalline Cu
_{6}Sn_{5}increases with increasing strain rate.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Polycrystalline Cu

_{6}Sn

_{5}with average grain size of (

**a**) 17 nm, (

**b**) 14 nm, (

**c**) 12 nm, (

**d**) 11 nm, (

**e**) 9 nm, and (

**f**) 8 nm.

**Figure 3.**System states after relaxation of (

**a**) total energy; (

**b**) pressures in x, y, and z directions; and (

**c**) temperature.

**Figure 4.**Stress-strain response of polycrystalline Cu

_{6}Sn

_{5}at strain rates of (

**a**) 10 ps

^{−1}, (

**b**) 1 ps

^{−1}, (

**c**) 0.1 ps

^{−1}, (

**d**) 0.01 ps

^{−1}, (

**e**) 0.001 ps

^{−1}, and (

**f**) 0.0001 ps

^{−1}.

**Figure 5.**Young’s modulus of polycrystalline Cu

_{6}Sn

_{5}at strain rates of (

**a**) 10 ps

^{−1}, (

**b**) 1 ps

^{−1}, (

**c**) 0.1 ps

^{−1}, (

**d**) 0.01 ps

^{−1}, (

**e**) 0.001 ps

^{−1}, and (

**f**) 0.0001 ps

^{−1}.

**Figure 7.**Yield stress of polycrystalline Cu

_{6}Sn

_{5}at strain rates of (

**a**) 10 ps

^{−1}, (

**b**) 1 ps

^{−1}, (

**c**) 0.1 ps

^{−1}, (

**d**) 0.01 ps

^{−1}, (

**e**) 0.001 ps

^{−1}, and (

**f**) 0.0001 ps

^{−1}.

**Figure 8.**UTS of polycrystalline Cu

_{6}Sn

_{5}at strain rates of (

**a**) 10 ps

^{−1}, (

**b**) 1 ps

^{−1}, (

**c**) 0.1 ps

^{−1}, (

**d**) 0.01 ps

^{−1}, (

**e**) 0.001 ps

^{−1}, and (

**f**) 0.0001 ps

^{−1}.

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**MDPI and ACS Style**

Huang, W.; Pan, K.; Wang, B.; Gong, Y. Grain Size Effects on Mechanical Properties of Nanocrystalline Cu_{6}Sn_{5} Investigated Using Molecular Dynamics Simulation. *Materials* **2022**, *15*, 3889.
https://doi.org/10.3390/ma15113889

**AMA Style**

Huang W, Pan K, Wang B, Gong Y. Grain Size Effects on Mechanical Properties of Nanocrystalline Cu_{6}Sn_{5} Investigated Using Molecular Dynamics Simulation. *Materials*. 2022; 15(11):3889.
https://doi.org/10.3390/ma15113889

**Chicago/Turabian Style**

Huang, Wei, Kailin Pan, Bo Wang, and Yubing Gong. 2022. "Grain Size Effects on Mechanical Properties of Nanocrystalline Cu_{6}Sn_{5} Investigated Using Molecular Dynamics Simulation" *Materials* 15, no. 11: 3889.
https://doi.org/10.3390/ma15113889