*3.3. Analytical Model and Experimental Validation*

The analytical model of the effective elastic constants of porous solids is based on numerical simulation and microstructure measurement, and it is considered to be a relatively accurate empirical equation for predicting elastic properties. In order to ensure the rationality of this simulation route and accurately quantify the microstructural damage on elastic properties of micron silver sintered joints, two analytical models were used to calculate the elastic modulus at different thermal shocks and these results were compared with simulation results. The first model corresponds to the Ramakrishnan and Arunachalam (R&A) method [25] and is conducted as Equations (15) and (16).

$$E\_p = E\_0 (1 - p)^2 / \left(1 + b\_p p\right) \tag{15}$$

*Appl. Sci.* **2020**, *10*, 6368

$$v = \frac{1}{4} \frac{(4v\_0 + 3p - 7v\_0p)}{(1 + 2p - 3vwp)},\tag{16}$$

where, *bp* = 2 *to* 3*v*, *p* is the pore volume fraction. *v*0, *E*<sup>0</sup> is the Poisson's ratio and elastic modulus for undamaged material, and *Ep* is the elastic modulus corresponding to a certain damage state.

Another model, the modified value of porous materials (M) [26] in Equation (17) was used to estimate the effect of microscopic damage on elastic modulus. The parameter definition is the same as above.

$$E\_p = E\_0 - pE\_0 \left(\frac{9 - 4v\_0 - 5v\_0}{7 - 5v\_0}\right) \tag{17}$$

Furthermore, nanoindentation test was used to measure the load and corresponding displacement of pressureless micron sintered joints. The geometric vertices and center point of the silver welding layer were selected as test positions, and the average value was taken as the test result. If there is a large deviation in the above positions, it is determined that the silver bonding layer of the sample is uneven during sintering, and needs to be rejected. Meanwhile, a new sample is re-selected for testing. The designed stress curve is shown in Figure 13.

**Figure 13.** Applied nanoindentation stress curve.

For geometries based on SCR method, normalized calculation results of FEA and theoretical models and nanoindentation tests are plotted in Figure 14. The error among simulation and test and R&A method is within 15%, which indicates that the proposed simulation route and the elastic properties of micron silver sintered joints obtained by FEA are reliable and precise.

**Figure 14.** Relative comparison of FEA, analytical models and nanoindentation results.

#### **4. Results and Discussion**

#### *4.1. Microstructural Evolution*

The simulation illustrates that the effective elastic modulus degrades with the accumulation of shock cycles. In order to study the cause of performance degradation, the microstructural evolution is characterized from autocorrelation functions.

The specific surface, marked as s, of two-phase media is defined as the two-phase interface area per unit total volume, which can be obtained from the slope of the two-point autocorrelation function [27]. In this study, *s* of original images and reconstructed models is conducted from Equation (18):

$$\frac{d}{dr}S\_2(r)\Big|\_{r=0} = \begin{cases} -s/\pi, D=2\\ -s/4, D=3 \end{cases}.\tag{18}$$

Specific surface can be used to estimate the hydraulic diameter (*DH*) of porous media [28], which is calculated in Equation (19). *V* is the volume fraction of the medium:

$$D\_H = \frac{4V}{s} \tag{19}$$

The captured autocorrelation functions of original SEM images from Figure 5 changes with thermal shocks are shown in Figure 15. There is an increase in the slope at the origin and a decrease in the correlation length. Changes of the mean particle size, hydraulic diameter and specific surface in SEM pictures and reconstructed models are calculated as shown in Table 4.

**Figure 15.** Autocorrelation functions under the accumulation of thermal shocks.

**Table 4.** The particle size, hydraulic diameter and specific surface of original images and reconstructed models.


It can be seen that the average micron-silver particle size decreases, while this is in contrast to a slight increase in the size of silver grains (−45 ◦C~250 ◦C) reported in [29]. It should be noted that the surface diffusivity of silver is relatively lower at 125 ◦C [30] than the melting point and sintering temperature. Meanwhile the soak of −170 ◦C seems to prevent particle recrystallization growth and even reverse-driven particles from dispersing. The hydraulic diameter of the medium declines with fluctuation. If the porous medium is regarded as an interconnected pipeline network, the above situation can be interpreted as the rise of pore density and pore nucleation growth induced by cyclic loading, which is consistent with the measured increase in average pore size and porosity [31]. The whole tendency of the specific surface goes up, which physically means that the total surface area of pores increases. The results demonstrate that damage gradually accumulates with thermal shock testing, resulting in the increase in porosity and the role of pores as grain growth inhibitors. The contact area between adjacent grains and the size of the medium decreases. The medium particles tend to disperse. In Figure 15 autocorrelation functions slightly oscillate after 50 thermal shocks, which reflects in the influence of the roundness of micron silver particles. Furthermore, another important finding from visual the view of pores (Figure 8) is that the pore distribution area becomes uneven and the local density changes, which will further promote the initiation of cracks.

#### *4.2. Mechanical Response Characteristics*

Extrema of mechanical response characteristics of nodal solutions in previous FEA simulation could be observed in Figures 11 and 12, which illustrate the displacement and stress of reconstructed micron silver sintered joints suffering different thermal shock cycles in the X, Y, and Z directions. It can be seen that the maximum and minimum values of the displacement and stress show a similar fluctuation in the X and Z directions with increasing thermal shocks. This is because the x–z plane represents the parallel interface of this sandwich structure, and the Y axis is the longitudinal constrained sintering direction, so there is a certain degree of anisotropy.

It can be found that the maximum displacement point appears near the junction of silver phase and larger pores, or the corner of pores, and the value shows an increasing trend, which demonstrates that the rise of porosity and particle dispersion caused by increased thermal shocks leads to the decrease in the size of the load-transmission area between particles, and can further result in a large displacement field across the local bearing surface. This situation also indicates that the elastic performance of micron silver sintered joints declines. On the other hand, the variation range of triaxial stress gradually increases as the thermal shock cycles increases. In addition, stress concentrations exist in all three directions. This situation could be explained by more likely stress concentrations resulting from the growing porosity.

Table 5 illustrates the triaxial elastic modulus and the effective elastic modulus by FEA, and the apparent density of samples. It can be seen that the *Y*-axis elastic modulus is a little greater than those of the X and Z axes. This phenomenon should be derived from the sample structure, and the constrained sintering state leads to a slight anisotropic microstructure [32]. Considering the slight degree of anisotropy, the effective elastic modulus is adopted to evaluate the degradation behavior of the micron sintered silver joints. As the thermal shock cycles increase, the apparent density of micron silver joints decreases, indicating that cyclic thermal stress can drop the material densification.


**Table 5.** The elastic modulus in three directions, the effective elastic modulus and apparent density.

#### *4.3. Elasticity Degradation*

Based on the data in Table 5, Figure 16 illustrates the change of effective elastic modulus and porosity of micron silver sintered joints with accumulated extreme thermal shocks. The porosity changes from 13.19% to 25.75% and the effective elastic modulus decreases by about 36%. Since the

elastic modulus of three axes is calculated with deviation among different directions, it can be concluded that the elastic properties of micron silver joints are also related to the microscopic topological structure. The closer the reconstruction is to the original, the less the simulation fluctuates.

**Figure 16.** Changes in effective elastic modulus and porosity.

Compared with analytical predictions which only consider porosity as the main factor, it is found that FEA is close to the R&A method for solving the problem of modulus uncertainty by using the special model of the change of the effective Poisson's ratio. However, the forecast is slightly larger. The large deviation of the M method may be related to the premise that the composite material consists of a continuous matrix phase with a high concentration of rigid spherical inclusion suspension. The comparison with the test results also indicates that FEA can easily realize the prediction of elastic properties in micron silver sintered joints. The calculation taking errors caused by structural randomness into account basically covers nanoindentation results (Figure 17).

**Figure 17.** Effective elastic modulus degradation with porosity.

To clarify the elasticity degradation, a mechanism analysis is taken and discussed below. Micron silver atoms are held together by interaction and the elastic properties are directly related to the relative movement between atoms. Due to the mismatching of the thermal expansion coefficient of the sandwich structure, the adhesive layer is mainly subjected to shear force in the deep space environment regarding approximately monotonic loading. This may lead to lattice shift and dislocation slip in the microstructure (Figure 18).

**Figure 18.** Dislocation from the perspective of a slip plane.

If the dislocation stops due to the concentration of micro-stress in sintered silver joints, a constrained region will be formed, where other dislocations may stop. At this time, the subsequent dislocation of the same dislocation increment occurs, piling up as shown in Figure 19. This defect can cause a decreasing number of atomic bonds; consequently, the effective elastic modulus shows degradation. Moreover, this defect can be observed as the growth of pores and dispersion of silver grains.

**Figure 19.** Ductile dislocation piling up.

#### **5. Conclusions**

This study is undertaken to provide a cost-effective modeling and simulation method for the elastic mechanical properties of pressureless micron silver sintered joints based on microstructure reconstruction and analyzing the cause of its elasticity degradation in a deep space environment. The following conclusions are drawn:


**Author Contributions:** Conceptualization, B.W.; methodology, W.G.; software, W.G.; validation, W.G. and B.W.; formal analysis, W.G.; investigation, G.F.; resources, G.F.; data curation, M.Z.; writing—original draft, W.G.; writing—review and editing, B.W.; visualization, W.G.; supervision, G.F.; funding acquisition, M.Z. All authors have read and agreed to the published version of the manuscript.

**Funding:** The research was funded by the Equipment Pre-research Fund Project in China, grant number 61400020105.

**Acknowledgments:** The authors would thank the Institute of Microelectronics, Chinese Academy of Sciences for its assistance in making samples.

**Conflicts of Interest:** The authors declare no conflict of interest.
