# Towards Transient Electronics through Heat Triggered Shattering of Off-the-Shelf Electronic Chips

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## Abstract

**:**

## 1. Introduction

_{2}), amorphous Si (a-Si), poly-crystalline Si (poly-Si), germanium (Ge), and silicon-germanium (Si-Ge) [32,33].

## 2. Transience Mechanism

_{Ic}is the critical stress intensity factor which is a measure of mode 1 fracture toughness of the brittle material and α is the geometric factor which depends on the crack geometry. Shattering will occur when the expanding material exerts enough stress for a given length of crack ‘a’ introduced in the backside of the silicon test chips such that the stress intensity factor equals or exceeds the critical stress intensity factor K

_{Ic}. The effect of geometry is qualitatively studied through finite element analysis (FEA) performed using COMSOL Multiphysics simulation software. The geometry used for the simulation mirrors the geometry of the grooves on the backside of the silicon chip with a crack of 250 µm and 450 µm. Using Equation (1), the critical stress σ

_{f}for a 250 mm deep groove was calculated as 24.2 MPa. For this study, we applied a stress just higher than the calculated σ

_{f}in our simulation model. Figure 1b,c shows the sites inside the groove where localized stresses exceed the yield strength of the silicon material for a 250 µm and 450 µm crack on application of a 25 MPa stress uniformly distributed on the groove boundary. For a uniform crack geometry without surface roughness stresses concentration is maximum at the edges and corners, which will act as sites for crack propagation. Hence, even though the silicon yield strength is close to 0.7 GPa, crack propagation can occur in silicon at much lower values of wedging stresses which follows mode 1 crack propagation predicted by Equation (1). This is only a qualitative analysis of the effect of groove depth on the stress concentrations. The complete analysis of crack propagation incorporating the effect of the geometry, several other extrinsic and intrinsic factors [43,44,45] that are not accounted for by the Griffith’s theory of crack propagation is beyond the scope of this article. Since satisfying the Griffith’s failure criteria alone does not guarantee the failure of backside grooves or cracks [46] we have attempted to explain this variability in the crack propagation without having to develop exact fracture models by using a form of weakest link theory given by Weibull’s failure probability (P

_{fo}) of a link in a chain under tensile stress as given in by Equation (2) [47,48,49].

_{o}is the characteristic stress of a material above which the failure probability is ~63% and m is the Weibull’s modulus which is the characteristic of strength distribution of the material and indicates the nature, severity, and distribution density of flaws. For crystalline silicon, the Weibull modulus could be between 1–10 depending on the defect density, intensity, and mode of testing [50]. A test chip with ‘N’ rows and columns of dicing street as shown in Figure 1d, is comparable to a chain with geometrically identical links. Where an individual link can be pictured as a single groove site as shown in Figure 1e. Under the application of stress from expanding material each of these individual “links” act as a potential site for crack propagation. If all such “links” crack completely, the total number of shattered pieces would be (N + 1)

^{2}. This situation accounts for 100% crack propagation probability. For all other cases in which the test chip shatters into ‘n < (N + 1)

^{2′}pieces, the failure probability P

_{fo}would then be n/(N + 1)

^{2}. Making σ

_{o}proportional to critical stress (σ

_{o}= C

_{i}σ

_{f}), Equation (2) can be rewritten using Equation (1) to correlate crack propagation probability with initial groove depth as is shown in Equation (3).

_{i}απ

^{1/2}/K

_{Ic}C

_{i}and n/(N + 1)

^{2}is the transience probability or transience efficiency. For this study, the transience threshold was defined such that a transience efficiency of at least 5% was achieved after shattering, resulting in ≥20 pieces. This transience efficiency is successfully modulated by varying the initial groove depth ‘a’ as shown in later sections of this study.

## 3. Materials and Methods

#### 3.1. Expansion Mechanism

_{start}–T

_{max}) in which they exhibit the best expansion properties [38]. The ability of the microspheres to expand depends on the ability of the shell to act as a good gas barrier for hydrocarbon vapors at elevated temperatures while deforming during expansion. These requirements are met by the thermoplastic shell’s copolymer composition [51]. On application of heat, the low boiling point hydrocarbon is the first to vaporize and exert a pressure on the shell. As the temperature is increased above the recommended T

_{start}the shell liquifies and expands rapidly under the exerted pressure from the vaporized hydrocarbon. The barrier properties of the shell functions if the temperature is kept below the recommended T

_{max}. On removal of external heat, the expanded polymer shell solidifies due to the thermoplastic nature of the shell material and keeps its expanded shape. This expansion of TEP particles then exerts a wedging pressure on the crack boundaries. The composite TEE material with the TEP particles dispersed inside a PDMS matrix will expand with the same mechanics as the volumetric thermal coefficient of PDMS is extremely low (~9.6 × 10

^{−4}°C

^{−1}) [52]. Therefore, the expansion is a result of the expansion of the TEP; however, due to the presence of a dense matrix, the diffusion barrier for the encapsulated hydrocarbon is expected to be much better which should allow the elastomer to be heated above the rated T

_{max}of the TEP particles used to make the elastomeric composite. The TEP particles, in this case, would expand against an elastomeric matrix which would exert an additional counter pressure on the expanding particle boundaries which would result in reduced volumetric expansion compared to the free expansion of TEP microspheres as is shown in later sections.

#### 3.2. Expansion Kinetics

_{3}and b

_{3}are parametric unknown constants, σ is the stress on the polymer shell, E is the elastic modulus of the thermoplastic shell material, η is the viscosity of the shell material, ε is the linear strain which is related to the material expansion as shown in Equation (5).

_{o}. At the critical point, the failure stress given by Equation (1) can be approximated as proportional to the strain of the expanding materials (σ

_{f}≈ C

_{i}ε

_{i}(t)). This relation can be used to find the parametric relation between the groove depth and transience time and is shown in Equation (6).

_{3}, B

_{4}, and C

_{4}are the unknown parametric constants. For this study, we are only interested in the TEP microspheres that demonstrate maximum strain and strain rate to achieve the highest transience efficiency in the least amount of time. To identify the best TEP particles we measured the maximum strain and strain rate for the expandable particles. This was achieved by heating the particles at their rated average maximum temp (T

_{end}) on a hot plate. The fully expanded particles were then imaged under a scanning electron microscope. To measure strain or expansion rate the TEP particles were placed in a stainless-steel scoop and heated on a hot plate for different lengths of time at the end of which they were immediately transferred onto a glass petri dish which was cooled using dry ice. In this way, the particles were frozen in the semi-expanded state. These particles were then imaged under an SEM microscope. The average size in the form of particle radius was calculated by measuring the radius of multiple particles in the SEM images and then calculating the average. Maximum strain at a given time was then calculated using Equation (5). Out of the four types of particles shown in Table 1, particle grades, P1 and P4 showed the highest maximum strain but polymer P4 showed a higher strain rate as shown in Figure 2b,c. However, due to a higher initial temperature for expansion (T

_{start}), the P4 particles were chosen as the filler actuator material to minimize the possibility of accidental transience of a functioning chip. A curve fitting of the experimental expansion data for the time-dependent linear strain of ‘P4′TEP particles was performed based on Equation (4) using the custom equation setting in Matlab’s curve fitting tool. The starting point for the parameters was chosen based on the y-axis intercept and the slope of the first two points in the experimental data. An excellent fit was obtained with R-squared value ~98.5% when a

_{3}= 0.8 and b

_{3}= 1.09 as shown in Figure 2d. This demonstrated a good agreement with the developed expansion theory.

_{1}and σ

_{2}are the stresses on the dashpot and spring of the Kelvin–Voigt element respectively, ε

_{1}is the strain in the dashpot representing the TEPs, ε

_{2}is the strain in the Kelvin–Voigt element of PDMS, ε

_{3}is the strain in the elastic component of the TEPs, η

_{1}is the viscosity of the TEPs, η

_{2}is the viscosity of the PDMS matrix, E

_{2}is the elastic modulus for the Kelvin–Voigt material (PDMS) and E

_{1}is the elastic modulus for the TEPs. Using the Laplace transforms from Table 2, the creep response of this 4-element model for a sudden application of load σ

_{o}[H(t)] can be easily derived using the constituent equations for each element and is shown in Equation (7) [54].

_{4}, b

_{4}, b

_{5}, and c

_{3}are the unknown parametric constants. A relation between the groove depth and transience time can be derived in the same way as was done for TEP materials and is shown in Equation (8).

_{4}, B

_{4}, C

_{5}, B

_{6}, and C

_{6}are the unknown parametric constants. Since TEP polymer P1 and P4 showed the best expansion characteristics, they were chosen as the TE material to be dispersed in the PDMS matrix for the TEE material. The preparation involved dispersing both P1 and P4 TEP particles in a PDMS matrix prepared with 1 part by weight of curing agent for every 10 parts by weight of the PDMS elastomer. The mixture was degassed for 45 min in a desiccator to remove any air bubbles. This was followed by carefully flowing the mixture into a glass petri dish making sure that no gas bubbles are formed in the process. The mixture was rested on a horizontal surface until it evenly covered the petri dish. The composite mixture was then cured at 60 °C for 12 h in a gravity oven. The cured elastomer was peeled off from the petri dish and test samples were cut as 10 × 10 × 1 mm

^{3}pieces. The mixing ratio for the TEP particle in the PDMS matrix was varied between 100–600 mg mL

^{−1}. The test samples from the elastomer composites made with TEP particles were heated to 160 °C to study the effect of the mixing ratio on volumetric expansion. The dimensions of the fully expanded test sample were measured and the volumetric expansion in percentage was calculated.

^{−1}mixing ratio for the two TEP polymers. At a mixing ratio higher than 600 mg mL

^{−1}phase separation was observed and TEP particles were seen separating from the PDMS matrix on expansion. The effect of temperature on the volumetric expansion was also measured by heating the test samples prepared with P1 and P4 microspheres at 600 mg mL

^{−1}mixing ratio and varying the temperatures between 120–190 °C and the result is shown in Figure 3c. It was observed that the volumetric strain for the P1-PDMS elastomer is higher compared to the P4-PDMS elastomer, for the entire temperature range. Between temperature 160–190 °C, the volumetric strain drops from ~600% to ~580% for P1-PDMS elastomer however for the P4-PDMS elastomer the volumetric strain keeps on increasing with temperature and maximizes at 470%. The decrease in the volumetric strain for P1-PDMS elastomer could be attributed to the diffusion of the encapsulated hydrocarbon out of particle shell at temperatures higher than the rated maximum temperature for the P1 TEP particles. It was observed that expandable elastomer prepared with polymer P1 showed ~130% higher volumetric strain compared to elastomer prepared with P4 TEP particles. This could be attributed to the fact that P1 particles are much smaller in size compared to the P4 particles and hence are integrated more evenly in the PDMS matrix which would result in a more uniform heat distribution. The time-dependent expansion behavior of the TEP elastomer was also studied. The P1-PDMS elastomer was heated to 150 °C and the P4-PDMS elastomer was heated to 190 °C. The dimension of the expanded sample was measured at different instances of time, and the linear strain corresponding to length, width, and thickness was plotted as shown in Figure 3d,e. The P1-PDMS samples show a higher initial rate of expansion compared to the P4-PDMS sample, which is consistent with the higher expansion rate observed for P1 particles. The experimental data for the linear strain across the length of the test samples were fitted using Equation (7). Figure 3f shows an excellent fit between the model and experimental data with R squared value ~99% when a

_{4}= 0.04, b

_{4}= 7.4 × 10

^{−5}, b

_{5}= 23.8, and c

_{3}= 0.05. The parameter values given by the curve fitting tool are also displayed on the plot. Since P1-PDMS elastomer showed a higher expansion rate as well as a higher maximum expansion, it was chosen as the actuator filler material in this study.

## 4. Device Fabrication

#### 4.1. Backside Grooves

^{−1}for all the test samples. This process produces a series of alternating pillars and grooves which act as partial surface cracks. Groove depth can be controlled by controlling the depth of the partial cuts during the dicing process. Test samples were prepared with several groove depths to study its effect on the transience time.

#### 4.2. TE Transience Test Chip

^{−1}were filled in the backside grooves of the OTS Si-chip. Any excess material was scraped off using a scalpel and then the chip was cured at 55 °C for 12 h. Figure 4a,b shows optical images of backside grooves of a Silicon chip tightly filled with TEP particles and TE elastomer. The test chips were then transferred on a 500 µm thick silicon die with a thermal grease used as an encapsulant for TEP particles. The excess grease flows to the side of the microchip and prevents the microparticles from falling out of the grooves and dispersing in the surrounding. There was no thermal grease used for TE elastomer. The front side of the test chip was then encased with an acrylic window sealed with a 0.14 mm thin transparent polymer sheet. Figure 4c shows the step-by-step fabrication procedure for the test chips with both kinds of actuator material used in the study. The heat for triggering the actuator material was provided through a flexible heating pad ensuring that the acrylic encasing was never in contact with the heating pad. The transience event was imaged through a digital camera mounted on a stand.

#### 4.3. Thermal Transience Test Set-Up

## 5. Results and Discussions

#### 5.1. Heat Triggered Transience via TEP Microspheres

^{2}value of 0.98 when constant D

_{3}= 0.04.

#### 5.2. Heat Triggered Transience via TE Elastomer

^{−1}and heating at 160 °C. The post-transience images of the test chip with different groove depths are shown in Figure 6a,b. Transience time was calculated and plotted against ‘√a’ using Equation (8) for curve fitting. Transience time varied from 0.8 s to 12 s as the groove depth was varied from 450 µm to 120 µm. As seen in Figure 6c an excellent fit was achieved between the developed theory and the experimental results. A higher overall transience efficiency was observed compared to the transience experiment performed with the TEP material. This can be attributed to the even distribution of actuator material in the cracks which expands uniformly under heat, leading to a higher number of crack propagation. The transience efficiency has been shown to increase from ~6% to ~60% with increasing groove depth for the Weibull modulus (m) of 4. Using Equation (3) and Matlab’s curve fitting tool transience efficiency data was fitted against the square root of initial groove depth √a. A good fit was achieved with an R

^{2}value of 0.99 when the constant D

_{4}= 0.04 as is shown in Figure 6d.

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Transience mechanism through Griffith’s mode 1 crack propagation. (

**b**,

**c**) COMSOL simulation results of a groove under stress. Simulated stress maps of a geometrically perfect crack in the form of partial grooves on the backside of a silicon wafer, showing regions of stresses higher than the yield strength of silicon on application of stress applied by expanding actuator material. Stress concentration on the edges and corners can be seen for (

**b**) 250 µm initial groove depth and (

**c**) 450 µm initial groove depth. Longer cracks have a larger area of high stress concentration which would result in more efficient transience, (

**d**) 3D representation of the backside of a test chip with orthogonal grooved created by partial dicing, (

**e**) the enlarged inset shows a typical crack site with characteristic dimensions where ‘a’ is the groove depth. (

**f**) TE microsphere expansion under heat, resulting in >100 times increase in radius. The microspheres before expansion show thick walls compared to the expanded particles.

**Figure 2.**(

**a**) A lumped parameter model of a Maxwell’s material representing a TEP material with the constitutive stresses and strain. (

**b**) Maximum radial strain as observed for 4 different grades of TEP microspheres listed in Table 1. Polymer P1 and P4 show the maximum expanded capacity. (

**c**) Experimental plot of time-dependent radial strain for TEP particles P1 and P4. (

**d**) Experimental data for time-dependent radial strain for P4-TEP particles fitted with the strain equation for a Maxwell’s model showing excellent fit which means that TEP expansion can be modeled as a Maxwell’s material.

**Figure 3.**(

**a**) A 4-element lumped parameter model with a Kelvin–Voigt and a Maxwell element connected in series representing a TEP-PDMS composite material with the constitutive stresses and strain. (

**b**) Volumetric expansion of TEP-PDMS composite prepared with P1 and P4 TEP microspheres with varying mixing ratios heated at 160 °C. Highest volumetric strain was observed for the mixing ratio of 500 mg mL

^{−1}for both the microspheres with the TEP-PDMS composite of P1 microsphere showing higher expansion. (

**c**) Volumetric strain of TEP-PDMS composite of P1 and P4 microsphere against different temperatures. P1-PDMS composite showed highest strain at 160 °C whereas P4-PDMS composite had the highest strain at 190 °C. For all temperature values, P1-PDMS composite showed a higher expansion compared to P4-PDMS composite. Time-dependent linear strain for the length, width, and thickness of TEP-PDMS composite test samples of (

**d**) P1-PDMS with 500 mg mL

^{−1}mixing ratio and heated at 160 °C and (

**e**) P4-PDMS with 500 mg mL

^{−1}mixing ratio and heated at 190 °C. (

**f**) Experimental data for linear expansion across the length of TEP-PDMS composite fitted with the time-dependent strain equation for the 4 element model with a Maxwell and K–V element in series for P1-PDMS composite.

**Figure 4.**(

**a**) SEM images of backside grooves filled with TEP microspheres as actuator material. (

**b**) Optical image of backside grooves filled with TEP-elastomeric actuator material. (

**c**) Fabrication schematic of heat-triggered transience test chip with TEP and TEE material. (

**d**) Schematic of test set-up for heat-triggered transience experiments.

**Figure 5.**Post transience optical images of OTS silicon test chips for heat-triggered transience via. P4 TEP microspheres with different groove depths (

**a**) 150 µm, (

**b**) 450 µm. (

**c**) Observed transience time vs. square root of initial groove depths for heat-triggered transience via. P4 TEP microspheres as actuator material fitted with using Equation (6). (

**d**) Transience efficiency for a Weibull’s modulus m = 4 showing an increase from 5 to 47% with increasing initial groove depths and fitted with the failure probability equation showing a good approximation of the relation between the chosen initial groove depths to achieve the desired transience efficiency.

**Figure 6.**Post-transience optical images of OTS silicon test chips for heat-triggered transience via P1 TEP-PDMS composite material with different groove depths (

**a**) 150 µm, (

**b**) 450 µm (

**c**) Observed transience time vs. square root of initial groove depths for heat-triggered transience via. P1-PDMS composite as actuator material fitted with Equation (8) shows a good approximation of the relation between the desired transience time on the chosen initial groove depth. (

**d**) Experimental transience efficiency for a Weibull’s modulus m = 4 showing an increase from 10 to 60% with increasing initial groove depths and fitted with the failure probability equation showing a good approximation of the relation between the chosen initial groove depths to achieve the desired transience efficiency.

Commercial Name | Size (μm) | T_{start}(°C) | T_{end}(°C) | Grade |
---|---|---|---|---|

461-DU-20 | 6–9 | 100–106 | 143–151 | P1 |

461-DU-40 | 9–15 | 98–104 | 144–152 | P2 |

920-DU-80 | 18–24 | 123–133 | 180–195 | P3 |

920-DU-120 | 28–38 | 133–143 | 192–207 | P4 |

f(t) | L{f(t)} |

α | α/s |

H(t) | 1/s |

δ(t − τ) | e^{−τs} |

dδ(t)/dt | s |

e^{−αt} | 1/(α+s) |

(1 − e^{−αt})/α | 1/s(α+s) |

t/α − (1 − e^{−αt})/α^{2} | 1/s^{2}(α + s) |

t^{n} | n!/s^{1+n}, n ≥ 0 |

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## Share and Cite

**MDPI and ACS Style**

Pandey, S.; Mastrangelo, C.
Towards Transient Electronics through Heat Triggered Shattering of Off-the-Shelf Electronic Chips. *Micromachines* **2022**, *13*, 242.
https://doi.org/10.3390/mi13020242

**AMA Style**

Pandey S, Mastrangelo C.
Towards Transient Electronics through Heat Triggered Shattering of Off-the-Shelf Electronic Chips. *Micromachines*. 2022; 13(2):242.
https://doi.org/10.3390/mi13020242

**Chicago/Turabian Style**

Pandey, Shashank, and Carlos Mastrangelo.
2022. "Towards Transient Electronics through Heat Triggered Shattering of Off-the-Shelf Electronic Chips" *Micromachines* 13, no. 2: 242.
https://doi.org/10.3390/mi13020242