# Quartz Enhanced Conductance Spectroscopy for Polymer Nano-Mechanical Thermal Analysis
## 2 Experimental Section  
The spring constant of a QTF can be given by  
\[k_{\text{QTF}}=\text{E}_{\text{QTF}}t^{3}w/4t^{3} \tag{1}\]  
where \(\text{E}_{\text{QTF}}\) is the Young's modulus of the QTF prongs, \(t\) is the QTF thickness, \(w\) is the QTF width, and \(l\) is the QTF length [28]. The length, width, thickness, and prong spacing of the QTF are measured as 3.65 mm, 0.57 mm, 0.33 mm, and 0.29 mm, respectively. The resonance frequency of the QTF can be expressed in terms of fundamental material properties  
\[f_{\text{QTF}}=\sqrt{k_{\text{QTF}}/m_{\text{QTF}}}/2\pi \tag{2}\]  
where \(f_{\text{QTF}}\), \(m_{\text{QTF}}\). are the resonance frequency and effective mass of the QTF, respectively. Therefore, the effective \(k_{\text{QTF}}\). which was related to the QTF spring constant and the polymer microwire thermal dynamics can be calculated from the \(f_{\text{QTF}}\) of polymer microwire modified QTF (MM-QTF). Loss modulus \(\text{E}^{\prime\prime}\) is the metric of energy loss when material is deformed. \(\text{E}^{\prime\prime}\) is proportional to the square of resonance frequency \(f\) and related to the inverse of the \(Q\) factor [25].  
\[\text{E}^{\prime\prime}\propto\frac{f_{\text{QTF}}^{2}}{Q} \tag{3}\]  
Resistance film heaters were used to melt the polymer powder samples quickly. A tungsten microwire with a diameter of \(\sim\)10 \(\upmu\)m was used to pull and form polymer microwires from polymer melt. The polymer microwire solidified quickly in the air as a conic shape with a diameter from \(\sim\)15 \(\upmu\)m (head) to \(\sim\)4 \(\upmu\)m (tail). The polymer microwires were transferred to the microscope and cut into pieceswith different diameters. The polymer microwire was transferred across the prongs of a QTF by use of a three-dimensional translation stage. The photograph of MM-QTF under an optical microscope is shown in Figure 1.  
A schematic diagram of the experimental setup is depicted in Figure 2. The MM-QTF was placed in a custom-made tri-prism heat cell made of stainless steel. Three side walls of the heat cell were heated by three resistance heating films, respectively.  
Based on the piezoelectric effect of the quartz crystal, the conductance spectroscopy of the MM-QTF in the cell was measured [25]. A function generator (Agilent 33210A) was used to generate a sinusoidal signal to excite the MM-QTF. The frequency of the sinusoidal signal applied to the QTF pin was scanned across the resonance frequency of the QTF, from 33,200 Hz to 33,550 Hz with a step increment of 1 Hz. The output current of the MM-QTF was firstly converted into voltage by a custom-made preamplifier with a feedback resistance of 10 M\(\Omega\) (not shown in Figure 2), and then demodulated by a lock-in amplifier (Stanford Research System model SR830) working in 1\(f\) mode [31,43]. The time constant and the slope of the lock-in amplifier was 1 s and 12 dB/octave, respectively, resulting in a detection bandwidth of 0.25 Hz. The output signal of the lock-in amplifier was recorded by a personal computer (PC). The conductance spectroscopy of the MM-QTF in the frequency scanning range was acquired. The resonance frequency and \(Q\) factor of the MM-QTF can be obtained by fitting the square of the conductance spectra to a Lorenz curve.  
Figure 1: Photograph of a quartz tuning fork (QTF) with polymer microwire bridged across its prongs under a 45x stereomicroscope: (**a**) front view, (**b**) top view.  
Figure 2: Schematic of the experimental setup.