## 3 TPB Length Calculation  
For the TPB length calculation, the fiber and particle are considered to be ionic and electronic conductors, respectively. Exchanging their roles will not influence the TPB calculation because they are pure (ionic or electronic) conductors. However, the influence on the TPB of the mixed ionic and electronic conductor (MIEC) is still under discussion [39]. Figure 1 shows that the impregnated fibers are randomly distributed in the electrode. They have different lengths, positions, and orientations. Because of the different volume fraction, the number of fibers also differs. For the TPB calculation, a repeating unit is first abstracted by tailoring the impregnated fiber, as shown in Figure 3. This way, the volume-specific effective TPB length \(\lambda_{\text{TPB}}^{\text{v}}\) can be given by:  
\[\lambda_{\text{TPB}}^{\text{v}}=2\pi r_{\text{el}}\sin\alpha_{0}\cdot Z_{\text {io}-\text{el}}n_{\text{v}}p_{\text{io}}p_{\text{el}} \tag{1}\]  
Figure 1: The schematic diagram of the theoretical model under different impregnation loadings. The shell of the impregnated particles may be smaller than one layer (left), one layer (middle), and bigger than one layer (right). Figure 2: The schematic diagram of the case in which the shell is smaller than one layer, (**a**) the particles are randomly dispersed on the fiber face and some particles do not percolate in the red oval frame and (**b**) shows the schematic after deleting the particles that do not percolate. where \(r_{\rm el}\) are the radii of the electron-conducting particles in the shell; \(\alpha_{0}\) is the contact angle between the fibers and particles, which is \(15^{\circ}\)[40, 41]; \(Z_{\rm io-el}\) is the coordination number between the fibers and particles in the repeating unit; \(n_{\rm v}\) is the number of repeating units per unit volume; \(p_{\rm io}\) is the percolation ratio of the ion-conducting fibers; and \(p_{\rm el}\) is the percolation ratio of the electron-conducting particles. Based on Equation (1), the primary task of determining \(\lambda_{\rm TPB}^{\rm v}\) is the determination of the parameters in the equation, that is, the coordination number between the fibers and particles in the repeating unit \(Z_{\rm io-el}\), number of repeating units per unit volume \(n_{\rm v}\), percolation ratio of the ion-conducting fibers \(p_{\rm io}\), and percolation ratio of the electron-conducting particles \(p_{\rm el}\). The coordination number between the fibers and particles in the repeating unit: the coordination number \(Z_{\rm io-el}\) between the fibers and particles in the repeating unit can be geometrically derived. Figure 4 illustrates the geometric relations in the repeating unit. The coordination number is:  
\[Z_{\rm io-el}=2\frac{360^{\circ}}{\theta}\eta \tag{2}\]  
where \(\theta\) is the central angle of the fiber occupied by one particle (see Figure 4a). Notably, two layers of particles surround the fiber in the repeating unit, as shown in Figure 3. The parameter \(\eta\) is the impregnation loading, which is defined as the ratio of the total impregnated particle number, to the particle number of the one-layer case. Therefore, \(\eta\) is 1 in the one-layer shell case. For shells with more than one layer, \(\eta\) is still 1 because the particles beyond the first layer are not in contact with the fibers, and thus do not influence the TPB length. However, \(\eta\) is smaller than 1 in the shells that are smaller than one layer. Figure 4: The schematic illustration of geometric relations in the repeating unit. (**a**) the define of crucial parameters. (**b**) the define of \(h_{3}\). Figure 3: The schematic diagram of the repeating unit.