**2. Reliability analysis and experiments**

## **2.1 Reliability regarding viscoelasticity**

#### *2.1.1 Polymer viscoelasticity*

Viscoelasticity is a distinguishing characteristic of materials such as polymer. It exhibits both elastic and viscous behavior. The elasticity responding to stress is instantaneous, while the viscous response is time-dependent and varies with temperature. The viscoelastic behavior can be expressed with Hookean springs and Newtonian dashpot, which correspond to elastic and viscous properties, respectively. To measure the viscoelastic characteristics, stress relaxation or creep tests are often implemented. Stress relaxation of viscoelastic materials is commonly

described using a generalized Maxwell model, which is shown in **Figure 3**. It consists of a number of springs and dashpots connected in parallel, which represent elasticity and viscosity, respectively.

The Maxwell model can be described as a Prony series, which can be expressed with Eqs. (1) and (2) as below [47]:

$$E(t) = \frac{\sigma(t)}{\varepsilon\_0} = E\_\infty + \sum\_{i=1}^{N} E\_i \exp\left(-\frac{t}{\tau\_i}\right) \tag{1}$$

$$
\pi\_i = \frac{\eta\_i}{E\_i} \tag{2}
$$

*2.1.2 Viscoelasticity-induced stability problem of MEMS*

*Stress-relaxation test results of an epoxy die attach adhesive.*

*Reliability of Microelectromechanical Systems Devices DOI: http://dx.doi.org/10.5772/intechopen.86754*

storage [53].

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**Figure 5.**

**Figure 4.**

The viscoelasticity-related issue has become one of the most critical steps for assessing the packaging quality and output performance of highly precise MEMS sensors [50]. Applying the viscoelastic property to model the MEMS devices could yield a better agreement with the results observed in experiments than the previous elastic model [51]. The packaging stress in the MEMS was influenced not only by the temperature change but also by its change rate due to the time-dependent property of polymer adhesives [52]. Besides, the viscoelastic behavior influenced by moisture was recognized as the cause of the long-term stability of microsensors in

*Shift distance plot of individual relaxation curve with reference temperature of 25°C and polynomial fit.*

where *E*(*t*) is the relaxation modulus; *σ*(*t*) is the stress; *ε*<sup>0</sup> is the imposed constant strain; *E*<sup>∞</sup> is the fully relaxed modulus; *Ei* and *τ<sup>i</sup>* are referred to as a Prony pair; *Ei* and *η*<sup>i</sup> are the elastic modulus and viscosity of Prony pair; *τ<sup>i</sup>* is the relaxation time of *i*th Prony pair; *N* is the number of Prony pairs.

The relaxation data, for normalization, can be modeled by a master curve, which translates the curve segments at different temperatures to a reference temperature with logarithmic coordinates according to a time-temperature superposition [48]. The master curve can be fitted by a third-order polynomial function, such as

$$\log a\_T(T) = \mathbf{C\_1}(T - T\_0) + \mathbf{C\_2}(T - T\_0)^2 + \mathbf{C\_3}(T - T\_0)^3 \tag{3}$$

where *aT* is the offset values at different temperatures (*T*); *C*1, *C*2, and *C*<sup>3</sup> are constants; and *T*<sup>0</sup> is the reference temperature.

Taking an epoxy die adhesive used in a MEMS capacitive accelerometer as an example, a series of stress-relaxation tests were performed using dynamic mechanical analysis [49]. The experimental temperature range was set from 25 to 125°C with an increment of 10°C and an increase rate of 5°C/min, and at each test point, 5 min was allowed for temperature stabilization, and 0.1% strain was applied on the adhesive specimen for 20 min, followed by a 10 min recovery. The test results are shown in **Figure 4**.

The shift distance of individual relaxation curve with reference temperature of 25°C was shown in **Figure 5**. Then the three coefficients of the polynomial function (Eq. (3)) were determined to be *C*<sup>1</sup> = 0.223439, *C*<sup>2</sup> = �0.00211, and *<sup>C</sup>*<sup>3</sup> = 5.31163 � <sup>10</sup>�<sup>6</sup> .

Subsequently the master curve can be acquired, and a Prony series having nine Prony pairs (Eq. (1)) was used to fit to the master curve, as shown in **Figure 6**. The coefficients of the Prony pairs are listed in **Table 1**, where *E*<sup>0</sup> is the instantaneous modulus when time is zero.

**Figure 3.** *Generalized Maxwell model to describe the viscoelastic behavior.*

*Reliability of Microelectromechanical Systems Devices DOI: http://dx.doi.org/10.5772/intechopen.86754*

described using a generalized Maxwell model, which is shown in **Figure 3**. It consists of a number of springs and dashpots connected in parallel, which represent

The Maxwell model can be described as a Prony series, which can be expressed

*N i*¼1 *Ei* exp � *<sup>t</sup>*

*τi* 

<sup>2</sup> <sup>þ</sup> *<sup>C</sup>*3ð Þ *<sup>T</sup>* � *<sup>T</sup>*<sup>0</sup>

(1)

(2)

<sup>3</sup> (3)

¼ *E*<sup>∞</sup> þ ∑

*<sup>τ</sup><sup>i</sup>* <sup>¼</sup> *<sup>η</sup><sup>i</sup> Ei*

where *E*(*t*) is the relaxation modulus; *σ*(*t*) is the stress; *ε*<sup>0</sup> is the imposed constant strain; *E*<sup>∞</sup> is the fully relaxed modulus; *Ei* and *τ<sup>i</sup>* are referred to as a Prony pair; *Ei* and *η*<sup>i</sup> are the elastic modulus and viscosity of Prony pair; *τ<sup>i</sup>* is the relaxation time of

The relaxation data, for normalization, can be modeled by a master curve, which translates the curve segments at different temperatures to a reference temperature with logarithmic coordinates according to a time-temperature superposition [48]. The master curve can be fitted by a third-order polynomial function, such as

where *aT* is the offset values at different temperatures (*T*); *C*1, *C*2, and *C*<sup>3</sup> are

Taking an epoxy die adhesive used in a MEMS capacitive accelerometer as an example, a series of stress-relaxation tests were performed using dynamic mechanical analysis [49]. The experimental temperature range was set from 25 to 125°C with an increment of 10°C and an increase rate of 5°C/min, and at each test point, 5 min was allowed for temperature stabilization, and 0.1% strain was applied on the adhesive specimen for 20 min, followed by a 10 min recovery. The test results are

The shift distance of individual relaxation curve with reference temperature of 25°C was shown in **Figure 5**. Then the three coefficients of the polynomial function (Eq. (3)) were determined to be *C*<sup>1</sup> = 0.223439, *C*<sup>2</sup> = �0.00211, and

Subsequently the master curve can be acquired, and a Prony series having nine Prony pairs (Eq. (1)) was used to fit to the master curve, as shown in **Figure 6**. The coefficients of the Prony pairs are listed in **Table 1**, where *E*<sup>0</sup> is the instantaneous

elasticity and viscosity, respectively.

*Reliability and Maintenance - An Overview of Cases*

with Eqs. (1) and (2) as below [47]:

*E t*ðÞ¼ *<sup>σ</sup>*ð Þ*<sup>t</sup> ε*0

log*aT*ð Þ¼ *T C*1ð Þþ *T* � *T*<sup>0</sup> *C*2ð Þ *T* � *T*<sup>0</sup>

*i*th Prony pair; *N* is the number of Prony pairs.

constants; and *T*<sup>0</sup> is the reference temperature.

.

*Generalized Maxwell model to describe the viscoelastic behavior.*

shown in **Figure 4**.

*<sup>C</sup>*<sup>3</sup> = 5.31163 � <sup>10</sup>�<sup>6</sup>

**Figure 3.**

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modulus when time is zero.

**Figure 4.** *Stress-relaxation test results of an epoxy die attach adhesive.*

**Figure 5.** *Shift distance plot of individual relaxation curve with reference temperature of 25°C and polynomial fit.*
