## 2 Theoretical Model  
Generally, EE is fabricated by combining electrospinning and impregnation methods. The nanofibers are firstly prepared via electrospinning, followed by casting/printing on the electrolyte and co-sintering to generate the backbone. Subsequently, nanoparticles are coated on the surface of the backbone by impregnation. However, with increasing impregnation loading, the layers of the coated particles may vary from less than one layer to more than one layer. The multiple fabrication processes of EE lead to its distinctive microstructures. From a phenomenological viewpoint, the EE is a combination of the backbone and outer shell. The backbone closely resembles the packing of straight fibers because it is fabricated by sintering electrospinning fibers. Despite the sintering, the backbone still holds the straight fiber structure [33; 38]. The outer shell consists of impregnated nanoparticles, which adhere to the backbone surface. Based on the microscopic images [33; 34], the nanoparticles are spherical. To simulate the heat treatment, contact angles are allowed between the spheres and between fibers and spheres. Therefore, the backbone is considered to be a straight fiber structure in the theoretical model and the impregnated particles are considered to be spheres. In addition, all of the electrospun fibers and the impregnated particles in EE are of similar size. Thus, in order to simplify the model, the assumption that all of the electrospun fibers (the impregnated particles) have the same radii is adopted. The contact angle is widely used to describe the sintering process, which is generally set as \(15^{\circ}\) in CE and IE [29; 35]. As a result, it is reasonable to assume the contact angle between particles or between fiber and particle is \(15^{\circ}\). As we all know, in the impregnating process, the nanoparticles randomly stick to the surface of the fibers irregularly [18]. Therefore, we assume that the particles are randomly dispersed on the fiber face. In fact, fibers possess three surfaces, including the top, side, and bottom surface where the nanoparticles can stick. The intersection between the fibers covers part of the surface of the fibers. However, it is a formidable challenge for the quantitative calculation of the reduction of the fibers surface due to the intersection between the fibers. Thus, we assume that the total area of the top and bottom surface of fiber equals to the reduction of the fibers surface due to the intersection between the fibers. The nanoparticles can only stick to the side surface of the fibers in this theoretical model. In a word, the assumptions include the following:  
1. All electrospun fibers have constant radii. 2. All particles have constant radii. 3. The particles are coated on the fiber face; the contact degree is \(15^{\circ}\). 4. The contact degree between particles is \(15^{\circ}\). 5. If the outer shell is smaller than one layer, the particles will be randomly dispersed on the fiber face. 6. The intersecting region between the fibers is neglected. Figure 1 shows the schematic diagram of the developed theoretical model: Straight fibers (yellow cylinders) are randomly arranged in the electrode, acting as the backbone; nanoparticles (gray spheres) are coated on the fiber surfaces in three possible configurations. Depending on the impregnation loading, the shell may be smaller than one layer and one layer may be bigger (Figure 1, from left to right). For a shell, which is not smaller than one layer, the percolation ratio \(P^{shell}\) of the particles is 1. However, if the shell is smaller than one layer, the particles are randomly dispersed on the fiber face based on the fifth assumption (Figure 2a). In this case, the \(P^{shell}\) must be determined via percolation theory. Several particles cannot be connected to any of the percolation clusters (Figure 2a, red oval frame), and thus do not contribute to the TPB and need to be deleted, as shown in Figure 2b.