## 3 Results
### Length and Sensitivity Correlation Based on Piezoresistive Effect  
To determine the correlation between sensor length and sensitivity, we proposed a unit gauge factor model of NCSS. The gauge factor (G. F.) is defined in Equation (1), and the strain (\(\varepsilon\)) of the cantilever is given in Equation (2). \[\text{G.F.}=\frac{R_{n}}{\varepsilon}=\frac{\frac{\Delta R}{R}}{\varepsilon} \tag{1}\]  
\[\varepsilon=\frac{3c(L-a)}{L^{3}}y(L) \tag{2}\]  
Here, \(c\) is the height from the center of the cantilever, \(L\) is the length of the cantilever, \(a\) is the distance from the fixed end of the cantilever to the center of the sensor, and \(y\) (\(L\)) is the displacement applied to the end of the cantilever. Substituting the above equation into the general gauge factor equation, we yield the following. \[\text{G.F.}=\frac{\frac{\Delta R}{R}}{\frac{3c(L-a)}{L^{3}}y(L)}=\frac{L^{3} \Delta R}{3c(L-a)y(L)R} \tag{3}\]  
Because the fillers dispersed in the matrix construct the electrical conductive paths with contact resistance, its conductivity is less than that of a metal. Therefore, we assumed that the NCSS is an electrical conductive series based on an electrical conductivity model of nanocomposites and tried to determine its piezoresistive sensitivity mechanism based on this assumption. The NCSS is considered as a chain series of piezoresistive units having the same geometric size (Figure 4a). The electrical resistance of the sensor can be expressed as a linear summation of each unit, as shown in Figure 4b. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Sensor** & \multicolumn{3}{c}{**0.35 wt.\%**} & \multicolumn{3}{c}{**0.5 wt.\%**} \\ \hline T = 1.2, W = 10 (mm) & L = 30 & L = 40 & L = 50 & L = 30 & L = 40 & L = 50 \\ \hline Sample 1 (k\(\Omega\)) & 9.47 & 11.37 & 14.33 & 1.53 & 1.87 & 2.33 \\ Sample 2 (k\(\Omega\)) & 8.63 & 10.89 & 14.10 & 1.44 & 1.87 & 2.50 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The resistance of the sensor according to the length and wt.% at no load. As strains happen across all of the units of the NCSS, the gauge factor with respect to the strain can be expressed as follows. \[\text{G.F.}=\frac{\frac{\Delta R_{S_{0}}}{\kappa_{S_{0}}}}{\epsilon_{S_{0}}}+ \frac{\frac{\Delta R_{S_{1}}}{R_{S_{1}}}}{\epsilon_{S_{1}}}+\frac{\frac{\Delta R _{S_{2}}}{R_{S_{2}}}}{\epsilon_{S_{2}}} \tag{4}\]  
In addition, the electrical resistance model of the nanocarbon composite, shown in Figure 4b, can be expressed as follows:  
\[R_{CNT-compsites}=R_{filler}+R_{tunnel}+R_{contact} \tag{5}\]  
where \(R_{filler}\) is the electrical resistance of the fillers in the composite, \(R_{tunnel}\) is the resistance between the fillers inside the composite, and \(R_{contact}\) is the contact resistance between fillers. Substituting the above equation into the gauge factor equation, the gauge factor of the nanocarbon composite sensor can be expressed as follows. \[\text{G.F.}_{CNT-compsites}=\frac{R_{N}}{\epsilon}=\frac{\frac{\Delta R}{R}}{ \epsilon}=\frac{\frac{\Delta R_{filler}+\Delta R_{tunnel}+\Delta R_{contact}}{ R_{filler}+R_{tunnel}+R_{contact}}}{\epsilon} \tag{6}\]  
Thus, the electrical resistance change of the composite is dominated by \(\Delta R_{tunnel}\)[12], and the resistance due to the tunneling effect can be expressed as follows in Equation (7) [31]:  
\[R_{tunnel}=\frac{h^{2}d}{Ae^{2}\sqrt{2m\lambda}}exp\bigg{(}\frac{4\pi d}{h} \sqrt{2m\lambda}\bigg{)} \tag{7}\]  
where \(h\) is Planck's constant, \(d\) is the spacing between fillers, \(A\) is the cross-sectional area of the tunnel, \(e\) is the quantum of electricity, \(m\) is the mass of electrons, and \(\lambda\) is the height of the matrix barrier. The amount of change in the tunneling resistance is generated exponentially by the applied strain of each single unit part.