# Thermal Induced Interface Mechanical Response Analysis of SMT Lead-Free Solder Joint and Its Adaptive Optimization

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## Abstract

**:**

## 1. Introduction

- The geometric model of SMT solder joints is greatly simplified. The reliability of SMT solder joints depends heavily on their geometry. SMT solder joints are often small in size. A tiny change in the solder joint morphology can lead to different simulation results.
- Most finite element analysis is assumed to be a plane stress or plane strain problem. The reliability analysis of SMT solder joints is inherently a three-dimensional problem. However, most current studies simplify the problem as a two-dimensional problem, which makes the analysis of the solder joint unreliable.
- An effective geometric shape optimization process is missing in the solder joint morphology design stage. Some existing studies use the Taguchi method to explore the optimal combination of geometric parameters [11,12,13]. However, the combination of optimization parameters is not necessarily globally optimal. Some other studies have introduced the numerical optimization algorithm, which has global optimization capability but is not suitable for high-cost problems, such as FEA, because of the large number of iterations [14,15].

## 2. Numerical Model

#### 2.1. CZM-Based Interfacial Delamination in Bi-Material Analysis Method

- An unknown crack propagation path requires the crack initiation force and crack propagation angle to be calculated by the crack propagation criterion.
- For a known crack propagation path, only relevant criteria are needed to determine the initiation force.

#### 2.2. Minimum Energy-Based SMT Solder Joint Geometry Modeling

#### 2.3. FEA Modeling Process

**Table 1.**Anand model parameters for SAC305 [22].

Parameter Symbol | Description | Value |
---|---|---|

${s}_{0}$ (MPa) | Initial value of deformation resistance | 2.15 |

$Q/k$ (K^{−}^{1}) | Activation energy | 9970 |

$A$ (s^{−}^{1}) | Pre-exponential factor | 17.994 |

$\zeta $ (dimensionless) | Stress multiplier | 0.35 |

$m$ (dimensionless) | Strain rate sensitivity of stress | 0.153 |

${h}_{0}$ (MPa) | Hardening/softening constant | 1525.98 |

$\widehat{s}$ (MPa) | Coefficient for saturation value of deformation resistance | 2.536 |

$n$ (dimensionless) | Strain rate sensitivity of the saturation value | 0.028 |

$a$ (dimensionless) | Strain rate sensitivity of the hardening/softening | 1.69 |

Materials | $\mathit{\rho}({\mathbf{g}/\mathbf{cm}}^{3})$ | $\mathit{E}(\mathbf{GPa})$ | $\mathit{\nu}$ | $\mathit{\alpha}(\times {10}^{-6}/\mathbf{K})$ |
---|---|---|---|---|

BaTiO3 | 6.02 | 76.5 | 0.32 | 8 |

SAC305 | 7.38 | Table 3 | 0.36 | Table 4 |

Cu | 8.92 | 117.0 | 0.34 | 16.6 |

Epoxy | 2.02 | 19.7 | 0.31 | 8.8 (x, y) 20 (z) |

Temperature (°C) | −65 | −55 | 0 | 25 | 65 | 105 | 130 |
---|---|---|---|---|---|---|---|

$E(\mathrm{GPa})$ | 59.0 | 56.7 | 43.8 | 37.4 | 28.6 | 20.5 | 13.5 |

Temperature (°C) | −55 | −35 | −15 | 5 | 22 | 50 | 75 | 100 | 125 |
---|---|---|---|---|---|---|---|---|---|

$\alpha (\times {10}^{-6}/\mathrm{K})$ | 21.1 | 21.6 | 22 | 22.2 | 22.4 | 23.1 | 23.7 | 24.3 | 24.9 |

## 3. FEA Results and Discussion

#### 3.1. Interface Delamination between SMT Solder Joint and Termination under TC Load

#### 3.2. Interface Delamination between SMT Solder Joint and Land under TC Load

#### 3.3. Discussion and Analysis

## 4. Adaptive Surrogate Model-Based SMT Geometry Parameter Optimization

#### 4.1. A Brief Review: Kriging Surrogate Model and EGO Method

#### 4.2. IEGO: An Improved Adaptive Optimization Method

**Step 1**: Determining the upper search weight limits for local and global sample points, determining the stagnation convergence algebra and convergence threshold, and the correlation function $R({x}_{i},{x}_{j})$ of the Kriging surrogate model;

**Step 2**: Determine the initial sample points

**x**(using Optimal Latin Hypercube Sampling) and the corresponding outputs

**y**;

**Step 3**: Using the initial sample to generate the Kriging surrogate model, the parameters of the Kriging model, i.e., the weighting factor ${\beta}_{j}$, are calculated. The DACE (Design and Analysis of Computer Experiments) toolbox is used to generate the Kriging surrogate model in this paper;

**Step 4**: Based on the EI criterion and the numerical optimization algorithm (e.g., DE), the Kriging surrogate model is used to determine the EI function in the design space;

**Step 5**: Calculate the current and historical average optimal change rates, and determine the corresponding search weight. The corresponding relationship between the two can be set as linear; that is, when the change rate is greater than the set stagnation threshold, the search weight is reset to the maximum. When the change rate decreases, the search weight decreases synchronously. When the change rate is close to 0, the search weight is reset to the minimum value of 0; that is, the RMSE infill criterion is adopted, the optimum model of the RMSE infill criterion is:

**Step 6**: Determine the new sample point

**x**

**and calculate the corresponding true output**

_{a}**y**

**;**

_{a}**Step 7**: Determine if the result iteration is stalled. If convergence occurs, skip step (8), otherwise skip

**Step 10**;

**Step 8**: Determine if the convergence threshold is met. If satisfied, skip step (11), otherwise skip

**Step 9**;

**Step 9**: Add a new sample point

**x**

**according to the MAE criterion and calculate the corresponding real output**

_{b}**y**

**;**

_{b}**Step 10**: Update the sample set

**x**= [

**x**,

**x**

_{a}_{&b}],

**y**= [

**y**,

**y**

_{a}_{&b}], and then skip

**Step 3**;

**Step 11**: Output the optimal solution and the true value.

#### 4.3. Optimization of Interface Contact Pressure

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

#### Minimum Energy-Based Solder Joint Geometry Modeling

- There is no penetration between the land and the solder.
- There are no holes or other defects inside the solder materials.
- During the wetting process, the properties of the solder materials are stable, and there is no change in properties.
- The total solder volume is not affected by external factors and remains unchanged.

Component | Parameter | Variable | Initial Value (mm) |
---|---|---|---|

Ceramic dielectric | Length | ${L}_{1}$ | 2 |

Width | ${W}_{1}$ | 1.25 | |

Thickness | ${H}_{1}$ | 0.6 | |

Metal termination | Length | ${L}_{2}$ | 0.4 |

Solder | Length | ${H}_{3}$ | 0.32 |

Extension length | ${L}_{3}$ | 0.6 | |

Internal length | ${L}_{5}$ | 0.2 | |

Width | ${W}_{3}$ | 0.13 | |

Land | Length | ${L}_{4}$ | 1.4 |

Width | ${W}_{4}$ | 1.3 | |

Gap between land and termination | ${H}_{2}$ | 1.2 |

^{3}, the wetting angle with metal termination is 30 degrees, and the wetting angle with copper land is 36 degrees [33,34].

## Appendix B

#### Benchmark Functions

- Michalewicz function

- 2.
- Ackley function

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**Figure 1.**Pin forms of common SMT packages. (

**a**) The gull-wing pin; (

**b**) the I pin; (

**c**) the J pin; (

**d**) the pin-less form (using metal termination).

**Figure 5.**SMT solder joint geometry model based on minimum energy. (

**a**) Solder joint geometry modeling evolved in surface evolver; (

**b**) reconstructed in ANSYS.

**Figure 9.**Contact pressure (unit: MPa) distribution at interface 1 and interface 2. (

**a**) at 125 °C; (

**b**) at −45 °C.

**Figure 10.**Critical points at interface 1 and interface 2 and contact pressure evolution in these critical points. (

**a**) Points a, b and c are the location of the critical point; (

**b**) contact pressure evolution in critical point a; (

**c**) contact pressure evolution in critical point b; (

**d**) contact pressure evolution in critical point c.

**Figure 13.**Contact pressure evolution in critical points. (

**a**) Contact pressure evolution in critical point d; (

**b**) contact pressure evolution in critical point e.

**Figure 15.**Benchmark function image and optimization convergence profile: (

**a**) 2-D Michalewicz function; (

**b**) local schematic of Michalewicz function; (

**c**) convergence profile of traditional EGO algorithm; (

**d**) convergence profile of IEGO algorithm.

**Figure 16.**Benchmark function image and optimization convergence profile. (

**a**) The 2-D Ackley function; (

**b**) local schematic of Ackley function; (

**c**) convergence profile of traditional EGO algorithm; (

**d**) convergence profile of IEGO algorithm.

**Figure 17.**Contact pressure optimization region. (

**a**) Optimized region of interfaces 1 and 2 (i.e., the interface between termination and solder joint); (

**b**) optimized region of interface 3 (i.e., the interface between land and solder joint).

**Table 5.**CZM parameters for SAC305 [27].

Parameter | ${\mathit{\sigma}}_{\mathbf{max}}\left(\mathbf{MPa}\right)$ | ${\mathit{G}}_{\mathit{n}}^{\mathit{c}}{(\mathbf{J}/\mathbf{m}}^{2})$ | ${\mathit{\tau}}_{\mathbf{max}}\left(\mathbf{MPa}\right)$ | ${\mathit{G}}_{\mathit{t}}^{\mathit{c}}{(\mathbf{J}/\mathbf{m}}^{2})$ |
---|---|---|---|---|

value | 47.5 | 480 | 56 | 600 |

Geometric Parameter | ${\mathit{h}}_{\mathit{s}}\left(\mathbf{mm}\right)$ | ${\mathit{l}}_{\mathit{w}}\left(\mathbf{mm}\right)$ | ${\mathit{h}}_{\mathit{g}}\left(\mathbf{mm}\right)$ |
---|---|---|---|

Optimal value | 0.703 | 1.288 | 0.125 |

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**MDPI and ACS Style**

Liu, S.; Yan, Y.; Zhou, Y.; Han, B.; Wang, B.; Zhang, D.; Xue, S.; Wang, Z.; Yu, K.; Shi, Y.; Wang, C. Thermal Induced Interface Mechanical Response Analysis of SMT Lead-Free Solder Joint and Its Adaptive Optimization. *Micromachines* **2022**, *13*, 908.
https://doi.org/10.3390/mi13060908

**AMA Style**

Liu S, Yan Y, Zhou Y, Han B, Wang B, Zhang D, Xue S, Wang Z, Yu K, Shi Y, Wang C. Thermal Induced Interface Mechanical Response Analysis of SMT Lead-Free Solder Joint and Its Adaptive Optimization. *Micromachines*. 2022; 13(6):908.
https://doi.org/10.3390/mi13060908

**Chicago/Turabian Style**

Liu, Shaoyi, Yuefei Yan, Yijiang Zhou, Baoqing Han, Benben Wang, Daxing Zhang, Song Xue, Zhihai Wang, Kunpeng Yu, Yu Shi, and Congsi Wang. 2022. "Thermal Induced Interface Mechanical Response Analysis of SMT Lead-Free Solder Joint and Its Adaptive Optimization" *Micromachines* 13, no. 6: 908.
https://doi.org/10.3390/mi13060908