# Quantitative Neutron Dark-Field Imaging of Milk: A Feasibility Study
## 4 Conclusions  
Dark-field imaging was applied to study the structure of deuterated milk and milk gel samples using a symmetric Talbot-Lau neutron-grating interferometer. In order to model the data, a numerical Hankel transform of the analytic expression of the small-angle scattering intensity in the case of polydisperse spheres was applied. This enabled us to fit the projected real-space correlation function resulting from the quantitative DFT scans, taking into account a polydisperse size distribution for the form factor. Moreover, a structure factor was considered, but showed little impact with regard to the relatively low concentration of fat globules. As a result, the mean size and log-normal size distribution of the fat globules, as well as their dependence on concentration, could be quantified. In addition, the mapping of the total scattering cross-section as a function of sample height revealed a subtle gradient in the concentration of fat globules towards the top surface of the samples, increasing with fat concentration. Moreover, the structure of the milk gel could be modelled as a fractal network with long-range correlations for skim milk gels related to the casein micelle aggregation enabled by acid and rennet enzymes, respectively. Slightly different structure parameters and fractal dimensions were modelled. For fat milk gels, the fat globules in the fat milk gels were analyzed by approximating their scattering contribution through normalization with the respective skim milk gel data. This approximation implied somewhat larger fat globules in the gels than in the milk samples with the same fat concentrations. In conclusion, this paper has demonstrated that dark-field imaging data enables the extraction of quantitative information for complex systems involving polydispersity and structure-factor contributions. In particular, quantitative dark-field contrast neutron imag \begin{table}
\begin{tabular}{c c c c c c c} \hline _Gel Sample_ & \(\mathbf{\Sigma_{s}t}\) & \(\mathbf{\alpha\) (\(\upmu\)m)}\) & \(\mathbf{H}\) & \(D_{f}\) & \(\mathbf{R^{2}}\) & **Adjusted \(\mathbf{R^{2}}\)** \\ \hline GA Skim & 0.3931 & 0.2170 & \(-\)0.1201 & 2.7598 & 0.9672 & 0.9425 \\ \hline GP Skim & 0.3214 & 0.3545 & \(-\)0.2287 & 2.5427 & 0.8677 & 0.7684 \\ \hline \end{tabular}
\end{table} 
Table 4: Structural parameters and goodness of fits for the fractal network of casein micelles (CM) in the deuterated skim milk gels. \begin{table}
\begin{tabular}{c c c c c c c} \hline _Gel Sample_ & \(\mathbf{\Sigma_{s}t}\) & \(\mathbf{\mu}\) & \(\mathbf{\sigma}\) & \(\mathbf{r_{0}\) (\(\upmu\)m)}\) & \(\langle\mathbf{r}\rangle\) (\(\upmu\)m) & \(\mathbf{R^{2}}\) & **Adjusted \(\mathbf{R^{2}}\)** \\ \hline GA Fat & 0.4141 & \(-\)14.5536 & 0.9906 & 0.4781 & 0.7808 & 0.9898 & 0.9822 \\ \hline GP Fat & 0.9128 & \(-\)14.9336 & 1.1449 & 0.3269 & 0.6300 & 0.9973 & 0.9953 \\ \hline \end{tabular}
\end{table} 
Table 5: The structural parameters and goodness of fits for fat globules (FG) in the fat milk gel. ing was found to be a well-suited quantitative tool for investigations of dairy system structures. The proven ability to analyze local variations through the imaging approach allows us to add not only the identification but also the assessment of macroscopic variations of microstructures resulting from external stimuli or processing and mixing of constituents to the body of research in this field. This approach holds the promise to be extended to other soft-matter and material science problems. Conceptualization: Y.K., F.B. and M.S. ; methodology; Y.K., J.V., J.K. and M.S. ; software: Y.K. and O.O. ; validation: Y.K., C.G. and M.S. ; formal analysis: Y.K. and M.S. ; investigation: Y.K., C.G., E.L., F.B. and J.V. ; resources: Y.K., J.K., S.W.L., F.B., E.L. and M.S. ; data curation: C.G., M.B. and M.S. ; writing--original draft preparation: Y.K. and M.S. ; writing--review and editing: J.V., O.O., J.K., C.G., S.W.L., F.B., E.L. and M.S. ; visualization: Y.K. ; supervision: M.S. ; project administration: M.S. ; funding acquisition: Y.K., S.W.L.