# Quartz Enhanced Conductance Spectroscopy for Polymer Nano-Mechanical Thermal Analysis
## 3 Results  
The conductance spectra of the QTF in the vacuum sealed metal shell, in the air as well as modified by PMMA microwire, were measured at room temperature, respectively. The obtained conductance spectra as a function of frequencies are shown in Figure 3. The corresponding resonance frequencies, \(Q\) factors, and equivalent resistances of the QTFs calculated from Lorentz curve fitting are listed in Table 1. The resonance frequency of the QTF decreased from 32,764.3 Hz to 32,758.9 Hz and the \(Q\) factor reduced from 73,131 to 13,537 after the removal of the metal shell. This is due to the dampening effect caused by air. However, when the PMMA microwire with a diameter of \(\sim\)5 \(\upmu\)m was bridged across the QTF prongs, the resonance frequency of the MM-QTF increased to 33,362.2 Hz with a frequency shift \(\Delta f\) of \(\sim\)603 Hz. The resonance increasing, instead of decreasing, indicated that the bridged microwire had a comparable spring constant with the QTF and was able to resonate efficiently with the QTF [25,44]. The \(Q\) factor of a resonator represents the ratio of stored to dissipated energy per cycle. The \(Q\) factor of the QTF decreased from 13,537 to 920 resulting from the damping caused by the PMMA microwire, suggesting that the energy dissipating rate increases. In the measurement, the tri-prism cell was heated from room temperature (\(\sim\)28 \({}^{\circ}\)C) to over 180 \({}^{\circ}\)C. As shown in Figure 4a, the resonance frequency of MM-QTF decreased monotonically from 33,362.2 Hz to 32,825.2 Hz, yielding a frequency shift \(\Delta f\) of 463 Hz. This is attributed to the PMMA polymer chain segments' rearrangement when the temperature changes. The curve showed an abrupt plummet at \(T\)\(\sim\)180.5 \({}^{\circ}\)C, which corresponds to the sudden change in the QTF effective spring constant according to Equation (2). The inset of Figure 4a shows the resonance frequency of an unmodified QTF kept constant as a comparison, indicating that the large frequency shift was caused by the spring constant variation of the polymer microwire. Figure 4b shows the \(Q\) factor of the PMMA MM-QTF as a function of temperature. The \(Q\) factor showed a valley around 150 \({}^{\circ}\)C corresponding to the glass transition behavior. However, there was a steep jump of the \(Q\) factor from 346 to 11,440 around 180.5 \({}^{\circ}\)C, which revealed a sudden decrease in the energy dissipation rate. The inset of Figure 4b shows the \(Q\) factor of the QTF without modification by the PMMA microwire. The \(Q\) factor of bare QTF monotonously decreased slightly from 13,537 to 12,098. \begin{table}
\begin{tabular}{c c c c} \hline  & **Frequency (Hz)** & **Q Factor** & **Resistance (\(\Delta f\))** \\ \hline QTF in the vacuum metal shell & 32,764.3 & 73,131 & 12.07 \\ QTF in the atmosphere & 32,758.9 & 13,537 & 67.75 \\ PMMA microwire modified QTF (MM-QTF) & 33,362.2 & 920 & 390.83 \\ \hline \end{tabular}
\end{table} 
Table 1: The QTF parameters obtained by conductance spectroscopy. Figure 3: Conductance spectroscopy of QTF in the vacuum sealed metal shell (squares), QTF in the air (diamond), and QTF modified by polymethyl methacrylate (PMMA) microwire (triangle), respectively. The loss modulus \(E^{\prime\prime}\) is proportional to the ratio of the square of resonance frequency to the \(Q\) factor [25]. Figure 4c shows the \(E^{\prime\prime}\) variations of a PMMA MM-QTF with temperature. The \(E^{\prime\prime}\) increased to the maximum value \(E^{\prime\prime}\)max near the glass transition and then began to decrease due to the polymer chains' rearrangement. When the temperature of the material is lower than \(T\) (\(E^{\prime\prime}\)max), the polymer is in a glassy state and exhibits plasticity, and when the temperature is higher than \(T\) (\(E^{\prime\prime}\)max), the material is in a highly elastic state and exhibits elasticity, where \(T\) (\(E^{\prime\prime}\)max) is the temperature corresponding to the maximum loss modulus \(E^{\prime\prime}\)max.