## 4 Results and Discussion
### Size Effect  
Figure 7a-c show electron micrographs of the Al\({}_{2}\)O\({}_{3}\) powders used to analyze the effect of the active material size on the tortuosity in an electrode, together with particle size analysis data (Figure 7d). A total of three types were used, with average diameters of approximately 10, 70, and 100 \(\upmu\)m (hereafter, they are called S10, S70, and S100, respectively). As previously mentioned, because active material aggregation was determined to have a significant effect on the tortuosity, the active material size effect was analyzed by removing the powder with a relatively low particle diameter (<10 \(\upmu\)m), which had a relatively high possibility of particle aggregation.  
Figure 8a shows the change in tortuosity according to the particle size of the active material. It could be confirmed that the tortuosity slightly increased with the diameter of the active material particles. This was contrary to the previously reported tortuosity change in relation to the active material's size in an all-solid-state battery electrode. [PERSON] et al. reported that as the particle size of the active material increased, the interparticular interstices were better filled by the electrolyte, and the tortuosity decreased [36]. However, these results were obtained using a very low active material ratio or a high electrolyte ratio (electrolyte volume ratio in the electrode: 50-90%) compared to this study, suggesting that sufficient electrolyte effectively filled the large particle gaps, which in turn resulted in low tortuosity [36, 44, 45]. On the other hand, in this study, a high active material ratio (electrolyte volume ratio: 32%) was used for a high-energy-density all-solid-state battery, which is a recent industrial development direction. Thus, the electrolyte content may not have been sufficient to fill all the particle gaps.  
Figure 7: Scanning electron micrographs of spherical Al\({}_{2}\)O\({}_{3}\) powders with different average particle sizes. (**a**) 10 \(\upmu\)m, (**b**) 70 \(\upmu\)m, and (**c**) 100 \(\upmu\)m. (**d**) Particle size analysis data.  
A comparison of the porosity results according to the particle size (Figure 8b) showed that the porosity values of the electrodes using S10 and S70 were statistically similar. This suggested that the S70 electrode had the same volume of unfilled pores as the S10 electrode, but had a high degree of tortuosity because of the poor connectivity of the solid electrolyte in the large particle gaps. On the other hand, the electrode using S100 showed an increase in porosity compared to the other two electrodes. Therefore, it was conjectured that the overall connectivity of the ion transport path in the S100 electrode had further deteriorated as a result of the increase in the volume of unfilled pores, along with the low connectivity of the solid electrolyte in the large particle gaps. The difference in the ion-path connectivities of the S10, S70, and S100 electrodes was also confirmed by the change in the Bruggeman coefficient (Figure 8c). In conclusion, when the electrode had a high active material ratio, it should be noted that the connectivity of the electrolyte may have deteriorated because of the limited amount of solid electrolyte, even if the particle size of the active material and interparticle gaps increased.