# Quantitative Neutron Dark-Field Imaging of Milk: A Feasibility Study
## 3 Results and Discussion  
Figure 4 shows conventional neutron attenuation contrast images (AI) alongside dark-field contrast images (DFI) recorded at two different instrumental settings and probed lengths (130 nm ad 3.13 \(\upmu\)m) of the deuterated milk and milk gel samples. The conventional AI images, which, in terms of contrast, do not depend on the changing experimental settings, display only minor contrast variations with respect to fat content and gelation. This is clearly different for the DFI images, where the contrast increases obviously with increasing fat content, as well as with an increasing probed correlation length, \(\xi\). This is the first indication that the contribution of the scattering by FG dominates the scattering signal and that this scattering contrast is picked up for all samples, respectively. Figure 3: Schematic representation of the milk structure (fat globules and casein micelles) and the gelation process. In order to analyze the detected local scattering, i.e., the dark-field contrast over the probed correlation length, the DFI curves were extracted for different regions of interest (ROI). These curves can be modelled and fitted according to Section 2.4. In Figure 5a, the plotted DFI contrast values are median values of ROIs consisting of 60 \(\times\) 180 pixels as indicated exemplarily for two samples in Figure 4. Figure 5 displays the corresponding DFI curves of the deuterated milk samples. Initial visual inspection suggested the dominating structure sizes of up to approximately 1-2 \(\upmu\)m by observing where the curves became flat, as \(G(\xi)\) tended to zero (compare Equation (2)). The increasing contrast, i.e., the deeper levels of flat tails, indicates increased total scattering cross sections, \(\Sigma_{\text{s}}\), due to increasing volume fractions of the dominating scatterers, i.e., FG. The dashed lines represent initial fits with a simple isolated monodisperse hard-sphere model [19; 20; 26], and Table 2 presents the fit results in terms of fitted parameters. It is observed that such fits, as expected, do not fully account for the measured slopes, which is especially the case where these turn from a steep decay to flat tails. The more extended bending region in the measured curves shows the influence of polydispersity. The solid lines, therefore, represent subsequent nonlinear least square fits with the polydisperse hard-sphere model, where the projected real-space correlation function \(G(\xi)\) is calculated by a numerical Hankel transform, according to Ref. [32] (Equation (3)), of the analytic expression of the scattered intensity according to Equations (5)-(12). The model thus accommodates the form factor (FG) of the spheres, i.e., the hard-sphere structure factor, for their high concentration and thus correlation, as well as their polydispersity. These fits result in good agreement with the measured data of the DFI contrast for different probed correlation lengths in the ROIs (Figure 5a) and result in the fitted structural parameters for the FGs as shown in Table 3. The goodness of fits of FGs in the deuterated milk for the mono- and polydisperse hard-sphere models were considered using R\({}^{2}\) and the adjusted R\({}^{2}\) evaluation. The polydisperse hard-sphere model improves the quality of fit, in terms of both (R\({}^{2}\)) and adjusted R\({}^{2}\), compared to the monodisperse hard-sphere model, as shown in Table 3. Both reach values \(>\) 0.95. We also found that the use of a structure factor does not further improve these values, which might be expected due to the low actual fat volume fraction. Figure 4: (**a**) Attenuation contrast image (AI) and dark-field images (DFI) of deuterated milk samples with different volume fractions of fat at probed correlation lengths of 130 nm and 3.131 \(\upmu\)m; (**b**) Attenuation contrast image (AI) and dark-field images (DFI) of milk gel samples at probed correlation lengths of 130 nm and 3.131 \(\upmu\)m; scale bar is 10 mm.