# Temperature Comparison of Looped and Vertical Carbon Nanotube Fibers during Field Emission
## 3 Results and Discussion
### Temperature Distribution Measurement and Analysis
The boundary conditions for Equation (2) are \(T(s=0)=T(s=2L)=T_{0}=320\) K for the looped fiber (i.e., both ends are attached to the cathode base at \(\sim\)room temperature); and \(T(s=0)=T_{1}=T_{0}=320\) K and \(T(s=L)=T_{2}\) for the single vertical fiber (i.e., one end is attached to the cathode based at room temperature and the other end is at temperature \(T_{2}=580\)\({}^{\circ}\)C (853 K), which is estimated from Figure 6d). Equation (2) is solved with these boundary conditions to give the temperature distribution along the CNT fibers. Figure 8: Temperature dependent thermal and electrical conductivities of the CNT fiber, measured from the 3 Omega method. Dashed lines are for a linear fit to the electrical conductivity, \(\sigma[\text{S/m}\ ]=-366.09T[\text{K}]+582,096\) and to the thermal conductivity, \(\kappa[\text{W/mK}]=-0.1298T[\text{K}]+218.75\). Dotted lines are for a polynomial fit to the thermal conductivity, \(\kappa[\text{W/mK}]=2.4\times 10^{7}/T[\text{K}]^{2}-70,369/T[\text{K}]+161.11\). These curve fittings are used in Equation (2) to estimate the maximum temperature in the fiber emitters. By assuming constant electrical conductivity \(\sigma=\sigma_{0}\) and linear temperature dependence of thermal conductivity \(\kappa=\kappa_{0}+\kappa\prime(T-T_{1})\), Equation (2) can be solved analytically [27]. The maximum temperature for the looped fiber and for the single vertical fiber is found to be [27],  
\[T_{max}=T_{0}+\frac{\kappa_{0}}{\kappa^{\prime}}\Bigg{(}\sqrt{1+\frac{\kappa^{ \prime}V_{0}^{2}\sigma_{0}}{4\kappa_{0}^{2}}-1}\Bigg{)},\text{ Looped fiber} \tag{3}\]  
\[T_{max}=T_{1}+\frac{\kappa_{0}}{\kappa^{\prime}}\Bigg{(}\sqrt{1+\frac{\kappa^{ \prime}T_{\Delta}^{2}}{4V_{0}^{2}\sigma_{0}}\Bigg{(}\frac{V_{0}^{2}\sigma_{0}} {\kappa_{0}T_{\Delta}}+\frac{\kappa^{\prime}T_{\Delta}}{\kappa_{0}}+2\Bigg{)} ^{2}}-1\Bigg{)},\text{ Vertical fiber} \tag{4}\]  
where, for the looped fiber, the location of \(T_{max}\) always occurs at the top of the fiber loop (i.e., half the length of the total loop); for the vertical fiber, it occurs (measured from the cathode base) at \(s_{max}=L/2+L\big{(}\kappa^{\prime}T_{0}^{2}/2\sigma_{0}V_{0}^{2}+\kappa_{0} T_{0}/\sigma_{0}V_{0}^{2}\big{)}\) if \(s_{max}<L\), otherwise at \(s_{max}=L\) with \(T_{max}=T_{2}\). In Equations (3) and (4), \(T_{0}\) is the temperature at the cathode base, \(T_{\Delta}=T_{2}-T_{1}\) is the temperature difference between the two ends of the vertical fiber, \(V_{0}=I_{0}R\) is the voltage drop from the top to the bottom of the emitter during field emission, \(I_{0}\) is the emission current, and \(R\) is the total resistance of the emitter. When the temperature dependence coefficient \(\kappa^{\prime}\to 0\), Equations (3) and (4) become,  
Figure 5:0  
Figure 5:1  
Note that Equation (5) is identical to the V-T relation that is used in electrical contacts and ohmic heated nanowire [38; 39]. By using \(\sigma_{0}(320\text{ K})=4.64\times 10^{5}\) S/m, \(\kappa_{0}(320\text{ K})=175.28\) W/(mK), and \(\kappa^{\prime}=-0.1298\) W/(mK\({}^{2}\)) (cf. linear fit in Figure 8), the maximum temperature for the looped fiber is estimated to be \(T_{max}=254\)\({}^{\circ}\)C (527 K) and \(286\)\({}^{\circ}\)C (559 K), from Equations (5) and (3), respectively; and the maximum temperature for the vertical fiber is estimated to be \(T_{max}=580\)\({}^{\circ}\)C (853 K) at \(s_{max}=4\) mm and \(595\)\({}^{\circ}\)C (868 K) at \(s_{max}=3.5\) mm, from Equations (6) and (4), respectively.