## 3.4.1 Shear behavior

Figure 8 shows the two typical shear curves of the Sn5Bi*x*(Sb, In)/Cu joints. According to the fracture behavior after ultimate shear force (USF) point, a sharp drop in force, or resistance to crack propagation with more smooth dropping before fracture, namely fracture mode 1 and mode 2 respectively. For mode 1, the cutting area after shear is generally greater than 50%, while it is less than 50% for mode 2. The former has good plasticity, while the latter has weak resistance to deformation (Fig. 8a). As shown in Fig. 8b, Sn5Bi fractures in both mode, consisting of 60% mode 1 and 40% mode 2. After adding *x* (Sb, In) elements, the fracture ratio of mode 1 decreased to at least 80%. Among them, when the amount of Sb added is 2 wt.% and the addition amount of In is 1 wt.% and 2 wt.%, all the joints fracture in mode 1. The fracture behavior of alloys has a good correlation with the work of crack growth. The total shear work (TW) refers to the area under shear force vs displacement graph from initial shear force to the fracture. The TW (blue triangle in Fig. 8b) provided that solder joints with 100% fracture ratio of mode 1 exhibit the highest TW, while solder joints with 60% fracture ratio of mode 1 has the minimum TW. These results reveal that In or Sb addition can improve the ductility of Sn5Bi/Cu solder joints and both 2 wt.% In/Sb addition present the best TW.

Addition In or Sb of also can enhance their USF as the red circle shown in Fig. 8b. With the increase of Sb/In addition, the USF of the solder joint presents a monotonous increase trend, and 3 wt.% Sb doping reached the maximum USF value of 42.11 N which increase by 23.06%. Combining with the results of USF and TW, the Sn5Bi2Sb/Cu solder joints demonstrate optimal mechanical properties.

## 3.4.2 Mechanical performance corresponded to Sb/In additions

To better understand the relationship between the microstructures and shear properties of Sn5Bix(Sb, In)/Cu solder joints, Fig. 9 illustrates the calculated phase fraction corresponding to the shear results of *x*Sb/In doped joints. The proportion of particles in Sn5Bi*x*Sb/Cu joints shows that the phase fraction of SbSn, Cu6Sn5 and the total second phases containing Bi particles increase linearly with the increase in the Sb content, and the USF also shows a linear relationship with Sb content (Figs. 9a-d). The crack propagation work (CPW) is defined as the area under the force vs displacement from USF to the end of the shear displacement. The CPW increases parabola, which first rises with increasing Sb, and reaches the maximum when the content of Sb is 2 wt.% (Fig. 9f). The TW for all the ductile fracture joints shows a similar trend in Fig. 9g.

For Sn5Bi*x*In/Cu joints (see Fig. 9b1-g1), the phase fraction of Bi particles, Cu6Sn5, and the sum of the two of them decreased with the In content increasing from 1 to 3 wt.%, showing a linear decline in slope. This trend is consistent with the conclusion of Belyakov et al. [4]. Increasing the content of In, on the one hand, increases the solid solution strengthening part caused by the solution of Bi and In in the matrix, on the other hand, the addition of In reduces the precipitation of Bi particles and Cu6(Sn, In)5, making the precipitation strengthening part weak. However, when In continues to increase, the solution strengthening and precipitation strengthening combine together. This result can be verified by the USF curve versus In content, almost linear growth in USF from 0 ~ 2 wt.% In, with increasing content of In to 3 wt.%, the curve deviates from the straight line and grows in a parabola shape. The CPW for In doped solder joints in Fig. 9f1 exhibit the same trend as the Sb doped solder joints. The TW for the ductile fracture joints shows that 1 wt.% In doped Sn5Bi1In joints displays a poor performance (Fig. 9g1).

Masaki Moriuchi et al.[20] first theoretically investigated the solution strengthening mechanism of SAC doped with Bi and Sb. In the solution strengthening mechanism (we suppose that it is precipitate strengthening here), the solute atmospheres that are formed around solute atoms distort the crystalline structure and become obstacles to dislocations. When the atomic radius between the solute atom and the matrix phase atom is different, the size effect caused by them can be expressed by equations (1) and (2) [31].

$$\begin{array}{c}{F}_{\text{m}}=\mu b{r}_{\text{s}}\left|\epsilon \right|\text{ }\text{(}\text{1}\text{)}\end{array}$$

$$\begin{array}{c}\epsilon =\left({r}_{\text{s}}-{r}_{\text{m}}\right)/{r}_{\text{m}}\text{ }\text{(}\text{2}\text{)}\end{array}$$

Where \({F}_{\text{m}}\)is the maximum force of the obstacle acting on a dislocation. \(b\) is the Burgers vector, \({r}_{\text{m}}\) is the atomic radius in the matrix phase, and\(\epsilon\) is the misfit strain. In Eq. (2),\({r}_{\text{s}}\) and \({r}_{\text{m}}\) are the solute atomic radius and the atomic radius in the matrix phase, respectively. For In and Sb in Sn5Bi matrix, the \(\epsilon\) and the \({F}_{\text{m}}\) could be expressed in the following.

$${\epsilon }_{\text{I}\text{n}}=\left({r}_{\text{s}}-{r}_{\text{m}}\right)/{r}_{\text{m}}=\left(167-140\right)/140=0.19$$

$${F}_{\text{m},\text{I}\text{n}}=\mu b167\text{*}0.19=31.73 {\mu }_{\text{I}\text{n}}{b}_{\text{I}\text{n}}$$

$${\epsilon }_{\text{S}\text{b}}=\left({r}_{\text{s}}-{r}_{\text{m}}\right)/{r}_{\text{m}}=\left(128-140\right)/140=0.08$$

$${F}_{\text{m},\text{S}\text{b}}=\mu b128\text{*}0.08=10.24{\mu }_{\text{S}\text{b}}{b}_{\text{S}\text{b}}$$

Thus, the atomic-induced lattice distortion caused by the In solid solution is greater and that of Sb is smaller, assumed that \({\mu }_{\text{S}\text{b}}{b}_{\text{S}\text{b}} \text{a}\text{n}\text{d} {\mu }_{\text{I}\text{n}}{b}_{\text{I}\text{n}}\) are generally identical. According to the above results from microstructure and the calculated phase diagram, Sb addition will form fine SbSn IMC, which is no longer solute in Sn matrix. Therefore, the Sb addition made the atomic misfit no longer computed further for the critical resolved shear stress. The critical resolved shear stress dependent on \({F}_{\text{m}}\) and concentration of solute atoms in the solid solution alloy *C* can be expressed by Labusch limit [31] as shown in Eq. (3). It is considered that the multiple solute atoms are involved in the elementary step of a one-time dislocation motion, when the solute atomic concentration is high.

$$\begin{array}{c}{\tau }_{\text{a}}={\left[\left({F}_{\text{m}}^{4}{C}^{2}w\right)/\left(4\mu {b}^{9}\right)\right]}^{1/3} \text{(}\text{3}\text{)}\end{array}$$

where, \({\tau }_{\text{a}}\) is the critical shear stress, *C* is the concentration of the solute atoms, *w* is a parameter that describes the range to which the interaction between the dislocations and the solute atoms extends, which is approximately 5*b* [20]. Thus, the critical resolved shear stress caused by In addition can be further expressed by Eq. (4).

$$\begin{array}{c}{\tau }_{\text{a}，\text{I}\text{n}}=108.21{{\mu }_{\text{I}\text{n}}{b}_{\text{I}\text{n}}}^{-4/3}{C}^{2/3} \text{(}\text{4}\text{)}\end{array}$$

In this paper, the proportion of elements in Sn5Bi*x*In (*x* = 1, 2, 3) is calculated by the weight percentage, while \(C\) is the atomic percentage of solute atoms, which can be obtained by calculation:

$${C}_{1\text{I}\text{n} }=1.06\%$$

$${C}_{2\text{I}\text{n} }=2.11\%$$

$${C}_{3\text{I}\text{n} }=3.17\%$$

Take the value of \(C\) into Eq. (4) to get \({\tau }_{\text{a}，\text{I}\text{n}}\):

$${\tau }_{\text{a}，1\text{I}\text{n}}=\text{5.22}{{\mu }_{\text{I}\text{n}}{b}_{\text{I}\text{n}}}^{-4/3}$$

$${\tau }_{\text{a}，2\text{I}\text{n}}=\text{8.26}{{\mu }_{\text{I}\text{n}}{b}_{\text{I}\text{n}}}^{-4/3}$$

$${\tau }_{\text{a}，3\text{I}\text{n}}=\text{10.84}{{\mu }_{\text{I}\text{n}}{b}_{\text{I}\text{n}}}^{-4/3}$$

As for Sb, it is actually an IMC particle existent in tin matrix, the critical shear stress should be related to Eq. (5) related to the precipitation strengthening caused by IMC particles [32].

$$\begin{array}{c}\varDelta {\tau }_{p}=a{f}^{1/2}{p}^{-1} \text{(}\text{5}\text{)}\end{array}$$

where *a* is a material-related constant, *f* is the volume percentage of IMC particles, and *p* is the average radius of particles. The greater the particle density and the smaller the particle size, the more obvious the effect of this strengthening mechanism. This ensures that the Sb added Sn5Bi*x*Sb alloy is stronger than the In doped one. For \(\varDelta {\tau }_{p}\propto\) \({f}^{1/2}\), the In/Sb doped joints reinforced via IMCs can be shown in Fig. 10a, which shows an opposite trend, differently from the USF results shown in Fig. 9e1. Further plot the solution induced strengthening curve shown in Fig. 10b, it is increasing with In addition which may compensate the weaker precipitate strengthening compared to Sb induced precipitate strengthening. Making a parabolic increase not linear curve resulted by Sb addition, see Fig. 9e1. This kind of USF change may result from changes in the strengthening mechanism by the inner characteristics of the element.