## 3 Fibrous Structures
### Collagen
The lateral integral width of the meridional reflections is represented by \(b_{12}\), which can be used to determine the fibril diameter. \(b_{12}\) increases with the diffraction order, \(n\), due to an increase in tilt and disorder; therefore, a zero-order extrapolation approach was used using the equation below:  
\[b_{12,n}=b_{12,0}+Cn \tag{5}\]  
where \(b_{12,0}\) is the width without any broadening effect, \(b_{12,n}\) is the width of a reflection at a given \(n\), and C is a proportional constant. The average fibril diameter is estimated from the reciprocal of \(b_{12,0}\). An average fibril diameter of 85 nm was calculated, which was corroborated by the TEM result. The width of meridional reflections along the axial direction corresponds to the coherence length along the fibril. The total peak broadening (\(b_{3,n}\)) is the combination of the finite fibril size effect (\(b_{size}\)), instrumental broadening (\(b_{inst}\)), and lattice disorder (\(b_{dis}\)) as given below:  
\[b_{3,n}=\sqrt{b_{size}^{2}+b_{inst}^{2}+b_{dist}^{2}} \tag{6}\]  
Broadening from disorder effects was quantified using the equation:  
\[b_{3,n}=b_{3,0}+B^{\prime}n+B^{\prime\prime}n^{2} \tag{7}\]  
where \(b_{3,0}\) is the broadening without the disordering effects, and B\({}^{\prime}\) and B\({}^{\prime\prime}\) are fitting parameters for strain and paracrystalline disorders, respectively. The longitudinal coherence length is the reciprocal of \(b_{3,0}\), which is \(\sim\)1010 nm. The azimuthal intensity distribution in the diffraction patterns is related to the orientation distribution of the molecules. This is usually quantified by Hermann's orientation function [3]. Alternatively, the narrowness of the azimuthal distribution can be empirically expressed by an orientation index (\(OI\)) calculated using the equation below:  
\[OI=\frac{90^{\circ}-OA}{90^{\circ}} \tag{8}\]  
where \(OA\) is the azimuthal angle range containing 50% of the fibrils. If \(OI\) = 1, the fibril is perfectly aligned; if \(OI\) = 0, the fibrils are entirely disordered. Higher \(OI\) was observed in stretched-tanned leather, meaning that fibrils are more aligned, making the leather stiffer. The lower \(OI\) in the control leather indicates a random fibril orientation, indicating greater flexibility [63]. Fibril diameter distribution was analyzed by [PERSON] et al. using the equatorial reflections. The intensity was modeled based on the form factor, \(F(q)\) of the cylindrical fibrils, as shown below:  
\[F(q)=\rho A\frac{2J_{1}(qr)}{qr} \tag{9}\]  
where \(J_{1}\) is the first-order Bessel function of the first kind, \(\rho\) is the number of excess electrons per unit area and unit length, and \(A\) is the cross-sectional area of the fibril. The form factor describes how the size and shape of fibrils influence the SAXS intensity. The relationship between \(I(q)\) and \(F(q)^{2}\) for polydisperse systems was derived as:  
\[I(q)=b+c\sum_{i=1}^{N}r_{i}^{4}w_{i}[\frac{2J_{1}(qr)}{qr}]^{2} \tag{10}\]  
where variables b and c describe the baseline corrections and scaling factor, respectively, N is the total number of bins, and \(w_{i}\) is the frequency of a fibril in the system. This model considers various fibril sizes and accounts for the frequency of different diameters (Figure 2c) [62]. The fibril size distribution dictates the overall shape of the scattering [64]. This approach provides statistical analysis of a larger scale compared to the small sample size of TEM [62]. A practical application of SAXS analysis of collagen is in the processing of leather. Figure 3 shows a diffraction pattern and structure of leather before and after tanning. The diffraction pattern was used to determine the D-period, fibril diameter and orientation, and intermolecular spacing [63]. Figure 3a shows that the diffraction pattern of normal leather is nearly isotropic from the random orientation of collagen fibrils compared to the diffraction pattern of tanned leather in Figure 3b, which has an elliptical shape, indicating anisotropy.