# Quantitative Neutron Dark-Field Imaging of Milk: A Feasibility Study
## 3 Results and Discussion
The median radius, \(r_{0}\), and the mean radius, \(r\), of FG are calculated by extracting the structural parameters from the fitted log-normal polydispersity, chosen based on the literature about corresponding systems [33]. The radii are calculated from the mean \(\mu\) and the standard deviation \(\sigma\) of the log-normal distribution, respectively. We found that the median radius and the mean radius of FG both increased with an increase in volume fraction. The mean size of FG was found to increase from approximately 0.36 \(\upmu\)m to approximately 1.36 \(\upmu\)m in mean diameter (\(2r\)) when considering fat concentrations of 5 to 50% fat milk. When considering the homogeneity of the samples by choosing several smaller ROIs from top to bottom for each sample, as this imaging technique enables, a subtle increase of total scattering from bottom to top was found. Figure 6 shows an evaluation of the total neutron scattering with an increment of approximately 2.5 mm from the bottom to the top of the samples. The corresponding ROIs are indicated in the exemplary image on the right-hand side of Figure 6. The total neutron scattering increases clearly, in particular for the samples with higher fat concentrations. This indicates the presence of larger volume \begin{table}
\begin{tabular}{c c c c c} \hline _Milk Sample_ & \(\mathbf{\Sigma_{s}t}\) & \(\mathbf{r\) (\(\upmu\)m)} & \(\mathbf{R^{2}}\) & **Adjusted \(\mathbf{R^{2}}\)** \\ \hline
5 wt\% & 0.2805 & 0.0937 & 0.9975 & 0.9965 \\ \hline
10 wt\% & 0.4299 & 0.1393 & 0.9841 & 0.9778 \\ \hline
25 wt\% & 0.9454 & 0.1812 & 0.9156 & 0.8818 \\ \hline
50 wt\% & 1.4360 & 0.1647 & 0.5422 & 0.3591 \\ \hline \end{tabular}
\end{table} 
Table 2: Structural parameters and goodness of fits for fat globules (FG) in the deuterated milk samples with different fat contents derived using the simple isolated monodisperse hard-sphere model. Figure 5: (a) Dark-field image (DFI) contrast of the diluted milk over probed correlation length. The solid lines represent fits by projected real-space correlation functions of the polydisperse interacting hard-sphere model for fat globules (FG), and the dashed lines represent the results of simple isolated monodisperse hard-sphere model. (b) The log-normal size distribution of the polydisperse FG modelled for the different deuterated milk samples. (Color in online). \begin{table}
\begin{tabular}{c c c c c c} \hline _Milk Sample_ & \(\mathbf{\Sigma_{s}t}\) & \(\mathbf{\mu}\) & \(\mathbf{\sigma}\) (\(\upmu\)m) & \(\mathbf{\langle r\rangle}\) (\(\upmu\)m) & \(\mathbf{R^{2}}\) & **Adjusted \(\mathbf{R^{2}}\)** \\ \hline
5 wt\% & 0.2974 & \(-\)16.7194 & 1.4899 & 0.0550 & 0.1668 & 0.9996 & 0.9993 \\ \hline
10 wt\% & 0.4845 & \(-\)15.7739 & 1.4156 & 0.1411 & 0.3843 & 0.9999 & 0.9998 \\ \hline
25 wt\% & 1.0627 & \(-\)15.3192 & 1.0670 & 0.2223 & 0.3928 & 0.9980 & 0.9966 \\ \hline
50 wt\% & 1.5560 & \(-\)15.4524 & 0.8750 & 0.1946 & 0.2853 & 0.9731 & 0.9530 \\ \hline \end{tabular}
\end{table} 
Table 3: Structural parameters and goodness of fits for fat globules (FG) in the deuterated milk samples with different fat contents derived using a model for polydisperse interacting spheres (Equations (7)–(13)). fractions and/or larger sizes of FG towards the top, which is intuitive, considering the lower density of fat. The fits do not, however, provide a clear conclusion about increasing radii towards the top of the samples. It is, therefore, possible that FG concentrates at the top without coalescing, due to a persistent slight repulsion. The results of the deuterated milk gel samples are presented in Figure 7. Different analyses strategies and models were required to be chosen for the skim milk gels and the fat milk gels. This was due to the fact that once again the FG signal dominated in the case of the fat milk gels. The fractal network of CM in the skim milk gels was modelled using an exponential function substituting the Bessel function in Equation (13), as described in Section 2.4.