## 2 Materials and Methods
### Electrical Conductivity Characterization of CFRPs
#### 2.1.2 Transverse Conductivity  
In the context of transverse conductivity, two primary mechanisms come into play. First, during the fabrication process, the likelihood of fiber-to-fiber contact can give rise to a conductive pathway within the sample. Second, when the gap between two fibers is within a threshold distance, quantum tunneling through the highly resistant epoxy barrier becomes a possibility. This section presents an in-depth explanation of a probabilistic conductivity model that encompasses both of these phenomena in conjunction with Monte Carlo simulations. Figure 4a illustrates the transverse microstructure of a CFRP lamina. Based on the morphology, a maximum fiber-fiber separation of 100 nm was selected for generating the RVE. A square RVE with 23 fibers was generated, accounting for the mean and standard deviation of fiber radius, range of fiber spacing, and fiber volume fraction. For a 60% fiber volume fraction, the resulting RVE size was \(\sim\)28.75 \(\upmu\)m. To gain insights into the spatial relationships between fibers and potential conductive pathways, a connectivity matrix  
Figure 4: (**a**) Transverse microstructure showing the fiber distribution used for estimating the overall inductance, (**b**) Interconnected RVE with straight line connections between pairs of fibers within the threshold distance. based on a threshold fiber pair distance (\(T\)) was established. It was observed that the tunneling probability was insignificant for fiber--fiber separation greater than 50 nm and was therefore chosen as the threshold for this study. This assumption is verified by the results presented in Section 3.2. Figure 4b illustrates the resulting connected RVE with doubly periodic boundaries, wherein all fiber pairs deemed potentially conductive are linked by straight lines. Next, the connected RVE is analyzed to identify the shortest conductive pathway across the RVE. This pathway holds significance since electrical charges follow the route of least resistance. Subsequently, this shortest conductive path is subjected to a more in-depth analysis using principles of quantum tunneling to compute the transverse conductivity. The details of this methodology are presented in Figure 5. Finally, the Monte Carlo method was employed, which involved conducting 1000 simulations of the process outlined in Figure 5. The results were used to calculate the mean transverse conductivity. To quantify the tunneling probability between each fiber pair within the conductive path, the one-dimensional, time-independent Schrodinger equation was solved, as given by Equation (8). \[-\frac{\hbar^{2}}{2m_{e}}\frac{\partial^{2}\Psi(x)}{\partial x^{2}}+U(x)\Psi( x)=E\Psi(x) \tag{8}\]  
where h is the reduced Planck's constant, \(m_{e}\) is the mass of the electron, and \(\Psi(x)\) denotes the wave function of the particle at a given location \(x\). The terms \(U\) and \(E\) represent the barrier potential energy and the particle energy, respectively. A schematic of fiber pairs with an enlarged view of the contact region is displayed in Figure 6. As illustrated in the figure, the fiber-fiber contact is divided into three regions. Regions A and C represent the fiber space, and region B represents the insulating matrix. By solving Equation (8) for each region individually, the unique wave functions \(\Psi_{A}\), \(\Psi_{B}\), and \(\Psi_{\text{C}}\) were computed. The solutions to these wave functions, along with the derivation of the tunneling probability, is detailed in Appendix A. The final expression for the tunneling coefficient \(C_{T}\) is expressed as follows:  
\[C_{T}=\left(1+\frac{\sinh^{2}(k_{b}a)}{4\eta(1-\eta)}\right)^{-1} \tag{9}\]  
where \(\eta\) represents the ratio of the particle's energy (\(E\)) to the barrier potential energy (\(U\)) and \(k_{b}\) is a constant. Equation (9) reveals that the likelihood of tunneling is directly proportional to the energy of the electrons and inversely proportional to the barrier size \(a\). The potential energy of the barrier is estimated by an equivalent circuit approach.