# Temperature Comparison of Looped and Vertical Carbon Nanotube Fibers during Field Emission
## 3 Results and Discussion
### Temperature Distribution Measurement and Analysis
It is clear that including the negative linear dependence of the thermal conductivity \(\kappa\prime\) gives a higher maximum temperature, yielding a closer fit to the measured temperature distribution data in Figure 5:2 A more accurate polynomial fit for \(\kappa(T)\) and a linear fit for \(\sigma(T)\) are also used in Equation (2) to numerically calculate the maximum temperature, giving \(T_{max}=300\)\({}^{\circ}\)C for the looped fiber and \(T_{max}=600\)\({}^{\circ}\)C at \(S_{max}=3.1\) mm for the vertical fiber, which are the closest fit to the experimental measurements in Figure 5:3 The results that were obtained from the experiments and from theory under various assumptions are summarized in Figure 5:4 Despite the agreement with experiments, we would like to emphasize again that the model in Equation (2) does not take into account the effects of the local emission site that is induced from fiber imperfection. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Carbon Nanotube (CNT)** & \multirow{2}{*}{**Measured**} & **Constant \(\sigma\)** & **Constant \(\sigma\) and** & **Linear \(\sigma(T)\) and** \\
**Fiber Emitter** & & & **and \(\kappa\)** & **Linear \(\kappa(T)\)** & **Polynomial Fit to \(\kappa(T)\)** \\ \hline \multirow{2}{*}{Vertical} & \(T_{max}\) [\({}^{\circ}\)C] & \(\sim\)600 & 580 & 595 & 600 \\  & \(S_{max}\) [mm] & \(\sim\)3.0 & 4 & 3.5 & 3.1 \\ \hline \multirow{2}{*}{Looped} & \(T_{max}\) [\({}^{\circ}\)C] & \(\sim\)300 & 254 & 286 & 300 \\  & \(S_{max}\) [mm] & \(\sim\)4 & 4 & 4 & 4 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of the measured maximum temperature and its location for both the vertical and looped fiber emitters, with the theory [27] of Equation (2) under various assumptions. In the above calculation, in order to give the best fit, the voltage drop across the emitters was adjusted to be \(V_{0}=0.79\) V and 0.61 V for the looped and the vertical fiber, respectively. These values are equivalent to a CNT fiber emitter resistance of \(R_{loop}=V_{0}/I_{0}=0.79\) V/3 mA \(=263\)\(\Omega\) and \(R_{vertical}=V_{0}/I_{0}=0.61\) V/3 mA \(=203\)\(\Omega\), which are much higher than the direct estimation of \(R=(1/\sigma_{0})L/\pi r^{2}=24\)\(\Omega\) for the single vertical fiber (and \(R/2=12\)\(\Omega\) for the looped fiber due to the parallel connection of two resistors, cf. Figure 2b). Several possible explanations are speculated as follows. First, even though the CNT fibers that were installed for the two emitters and for the \(\sigma\) and \(\kappa\) measurement are from the same fabrication batch, it was noticed that there were significant variations on the thermal conductivity values from sample to sample, as well as the number of measurements. We observed about 30% variation among the samples and 2 to 5% variation among the measurements of thermal conductivity. While the origin of this variation is yet to be identified (one possible origin is poor contact between the CNT fiber and the epoxy), the error during the conductivities measurement may lead to the difference in the calculations for the emitter resistance. It is also possible that the high current cycles during the FE measurement may have conditioned and damaged a fraction of the CNT fibers [6; 26; 34], resulting in a different \(\sigma\) and \(\kappa\) from those in the pristine samples that were used for the 3 Omega measurement. Another speculation is that during FE, only a small percentage of the CNT fibrils (sub-fiber units, \(\sim\)20-50 nm in diameter, formed by CNT tubes [5]) in the fiber were active to conduct current for field emission, which is postulated based on the fact that only a small fraction of the surface area of the emitter tip field emit [3; 6; 22]. It is also important to note that near the top of the looped fiber, only one side of the fiber faces the anode to emit current (Figure 2b).