# On the Measurement of Particle Contact Curvature and Young's Modulus Using X-ray \(\mu\)CT
## 5 Results and Discussion
### Discussion  
1. The [PERSON]'s modulus reduces with the increase of the number of fitting sections when the 3 rd polynomial fitting method is used. The [PERSON]'s modulus obtained in 10 fitting sections with a semi-particle is larger than that by the traditional averaging method. The majority of the [PERSON]'s modulus obtained in 20 and 50 fitting sections of semi-particle is larger than that by the traditional averaging method 3963 MPa, which is calculated by Equation (10), substituting the parameter values in Table 1. The [PERSON]'s modulus obtained using the circular and spherical fitting is constantly lower than 3963 MPa for the three fitting sections within a semi-particle. The summation of the principal curvatures changes dramatically under higher fitting sections. The [PERSON]'s modulus from the circular and spherical fitting methods is constantly lower than 3963 MPa. The value of the [PERSON]'s modulus reduces significantly in higher fitting sections, indicating that the [PERSON]'s modulus determined by principal curvature is more sensitive to local fitting regions.  
2. The largest value of the Young's modulus calculated from polynomial and spherical fitting is respectively 4855 MPa and 3773 MPa within Section 1 of the semi-particle scope whereas the smallest value of the corresponding Young's modulus is 2165 MPa and 3108 MPa in fitting section number 50. This indicates that a drastic change of the Young's modulus when fitting from globally to locally, which results from a drastic change of contact curvature.  
3. Interestingly, a large scatter of Young's modulus is also observed for the zeolite particles with similar sizes in the literature. For example, the Young's moduli for zeolite 4 AK with mean granule size \(d_{50}\) at 1.75, 2.05 and 3.5 mm are shown as 5400 \(\pm\) 1570 MPa, 2450 \(\pm\) 640 MPa, and 3340 \(\pm\) 1050 MPa, respectively [21]. Another study of Young's modulus for zeolite 4A shows the value of 4160 \(\pm\) 950 MPa [38]. This can be attributed to individual variability of sphericity, porosity and surface roughness [39]. Moreover, the structural inhomogeneities such as pores, microcracks, solid bridge bonds and non-uniform surface curvature. These factors can play collectively significant roles in differentiating the macroscopic behaviour, ultimately leading to dramatic scatter in Young's modulus and statistically distributed breakage probability.  
Figure 26: The Young’s modulus determined by different fitting methods in semi-particle scope of 50 sections.  
4. The preceding study has shown that the calculated Young's modulus of a particle exhibits a non-negligible variability, and is highly associated with its profile. The contact curvature becomes significant in determining the Young's modulus when taking particle shape into account. The significance of particle shape can provide plausible reason to account for a large scatter of [PERSON]'s modulus for zeolite particles by the compression test [21]. An accurate and repeatable topological characterization measurement is still an enigma [40]. However, a criterion to determine [PERSON]'s modulus should consider the effect of contact curvature where the particle is irregularly shaped.