## 2 Materials and Methods
### Hysteresis and Sensitivity Mechanisms
#### 2.1.2 Mechanisms of Sensitivity  
Sensitivity is a crucial property of a pressure sensor, indicating its capacity to transduce pressure into an electrical output. The slope of the pressure-resistance response curve can be intuitively visualised [45]. Sensitivity pertains to several electronic conduction theories, including field emission theory, tunnelling effect theory, and conduction route theory. This study primarily focuses on tunnelling effect theory and classical percolation theory.  
The fundamental equation for the sensitivity of piezoresistive sensing materials is as follows:  
\[S=\frac{\Delta R/R_{0}}{P} \tag{12}\]  
where \(\Delta R\) represents the change in resistance, \(R_{0}\) is the initial resistance of the composite sensing material (\(\Omega\)), and \(P\) is the pressure exerted on the sensing material (Pa).  
The change in resistance of the sensing material presented in this paper, when subjected to pressure, is accompanied by a relatively minor and inconsequential alteration in its cross-sectional area. The relative change in resistance can be articulated using the following equation:  
\[\frac{\Delta R}{R_{0}}=\frac{R_{0}-R}{R_{0}}=1-\frac{\rho}{\rho_{0}}\lambda \tag{13}\]  
In this context, \(R\) denotes the resistance of the composite post-compression (\(\Omega\)), \(\rho_{0}\) represents the initial resistivity of the composite (\(\Omega\cdot\)m), \(\rho\) indicates the resistivity of the composite following compression (\(\Omega\cdot\)m), and \(\lambda\) signifies the thickness compression ratio, defined as \(d/d_{0}\).  
Classical seepage theory posits that a scaling law [46] can characterise conductivity. The initial resistance of the sensing material can be determined by the following equation:  
\[\rho_{0}=\frac{1}{\sigma}=\frac{1}{\sigma_{0}(\varphi_{0}-\varphi_{\mathrm{c }})^{t}} \tag{14}\]  
In this context, \(\sigma\) represents the composite's conductivity (S/m), \(\sigma_{0}\) denotes the scaling factor, \(\varphi_{0}\) signifies the filler content, \(\varphi_{\mathrm{c}}\) indicates the percolation threshold, and \(t\) is the critical exponent, which is solely dependent on the dimensionality of the conducting system.  
The resistivity \(\rho\) can be determined using Equations (5) and (6) along with the fundamental equation for resistance:  
\[\rho=R\frac{S}{M}=\bigg{(}\frac{SL}{MN}\bigg{)}\bigg{(}\frac{8\pi rhs}{3a^{2} \gamma e^{2}}\bigg{)}\mathrm{exp}(\gamma s) \tag{15}\]  
\(M\) represents the length of the sensing material.  
The sensitivity analytical model for composite materials can be derived by putting Equations (13)-(15) into Equation (12):  
\[S=\frac{1-\frac{SL}{MN}\left[\frac{8\pi\eta hs}{3a^{2}\
u\varepsilon^{2}}\text{ exp}(\gamma s)\right]\left[\sigma_{0}(\varphi_{0}-\varphi_{\text{c}})^{t} \right]}{E\lambda} \tag{16}\]  
\(E\) represents the modulus of elasticity of the sensing material.  
The sensitivity of the composite sensing material is intimately linked to Young's modulus of the rubber matrix, the nanofiller's dispersion state, and the composite's percolation threshold, as indicated by the equation above. The nature and composition of the added fillers, together with the production process, influence these characteristics. This work will examine the sensitivity of composite sensing materials by analysing the effects of three types of nanoparticles--Fe\({}_{3}\)O\({}_{4}\), CNT, and MLG--and the MT as the primary affecting factors.