## 3 DMA Analysis and Thermal Activation Energy Calculation  
Dynamic mechanical analysis (DMA 8000, PerkinElmer, OH, USA) allows temperature and frequency sweeps to measure dynamic viscoelastic properties at constant frequency and temperature, respectively. It gives the storage modulus, which is proportional to the energy stored, the loss modulus, which is proportional to the energy dissipated, and the ratio of loss modulus to storage modulus is defined as \(tan\delta\), which is a measure of the damping capacity of the material. The DMA 8000 allows temperature scanning up to 400 \({}^{\circ}\)C and frequency range up to 600 Hz \(\pm\) 0.001 Hz. The force and displacement ranges are \(\pm\)10 N and \(\pm\)1000 \(\upmu\)m with resolutions 0.002 N and 1 nm, respectively. The resolutions of the measured moduli are 0.0001 Pa, and for tan delta the resolution is 0.00001. Besides the viscoelastic and damping properties, the DMA can also identify the glass transition temperature, \(T_{g}\). Typically, differential scanning calorimetry (DSC) is used to measure \(Tg\). However, according to the ASTM D4065 standard [51], the peak of loss modulus vs. temperature curve can be considered as an alternative to identify \(T_{g}\) for practical purposes. The tan delta vs. temperature peak can also be used for the same purpose. However, there is usually a discrepancy in the data when the comparison is made between both results. Thus, an error margin of around 25 \({}^{\circ}\)C is usually expected when deciding the \(T_{g}\)[42]. While samples can be mounted using various fixtures, three-point bending fixture was utilized in this study, as it can eliminate errors resulting from clamping the sample. A maximum displacement of 0.05 mm was used for all experiments. For the temperature scan, a temperature range from 30 \({}^{\circ}\)C to 170 \({}^{\circ}\)C was considered to provide a reliable fit to the Arrhenius equation used for the calculation of thermal activation energy. In the most basic form, this equation is given by:  
\[k=A\ exp\left(\frac{-E}{RT}\right) \tag{1}\]  
where \(k\) is the rate constant, \(A\) is the pre-exponential factor, \(E\) is the activation energy, \(R\) is the gas constant and \(T\) is the absolute temperature [44]. For viscoelastic analysis, this equation is modified to calculate activation enthalpy, \(\Delta H\), by the time-temperature superposition principle. The \(tan\delta\) or loss modulus peaks at different frequencies are superimposed on a logarithmic time axis, where they are shifted by a factor of \(\log(a_{T})\)[43]. Thus, the original equation is modified as:  
\[f=f_{0}exp\left(\frac{-\Delta H}{RT}\right) \tag{2}\]  
where \(f\) and \(f_{0}\) are analogous to \(k\) and \(A\), respectively. Due to the change in frequency, \(\Delta H\) is calculated as follows:  
\[\frac{f_{1}}{f_{2}}=\frac{exp\left(\frac{-\Delta H}{RT_{g}\pm}\right)}{exp\left( \frac{-\Delta H}{RT_{g}\pm}\right)} \tag{3}\]Applying logarithmic of either sides of the above equation,  
\[\log(a_{T})=\left(\frac{\Delta H}{R}\right)\left[\frac{1}{T_{g2}}-\frac{1}{T_{g1 }}\right]log(e) \tag{4}\]  
where \(a_{T}=\left(\frac{f_{1}}{f_{2}}\right)\). The final equation for \(\Delta H\) can be rearranging the terms in the above equation as:  
\[\Delta H=-R\left[\frac{d(lnf)}{d\left(1/T_{g}\right)}\right] \tag{5}\]  
This equation infers that the slope of the curve \(\ln(f)\) vs. \(1/T_{g}\) gives \(\Delta H\) for polymers using dynamic mechanical analysis. To reduce errors due to increasing temperature in this equation, constant heating rates of 2 \({}^{\circ}\)C min\({}^{-1}\) were used [44]. To compare the damping characteristics of the different composites' configurations, temperature and frequency sweeps were performed for each configuration. The dimensions of the samples were in the range of 2.0 in \(\times\) 0.20 in \(\times\) 0.024 in.