# Quantitative Neutron Dark-Field Imaging of Milk: A Feasibility Study
## 1 Introduction  
Dairy milk has a very long history of use and forms a basic part of global nutrition, which is reflected in global production in 2019 of 881 hundred million tons [1]. Thus, the structure of milk during processing is of considerable commercial and consumer interest in the sense of creating a demand for new dairy products; optimizing processing; human nutrition; and oral drug delivery. The most basic and main constituents of milk, apart from water, salts, and whey proteins, are casein micelles (CM) and fat in the form of fat globules (FG) with sizes ranging between 10-10\({}^{2}\) nm for CM and approximately 10\({}^{3}\) nm for FG, respectively. Dairy products contain a hierarchy of length scales in the arrangements of these structures. Milk coagulation in the formation of dairy gels, by, for example, microbiological action, pH variation, or enzymatic action, results in the formation of continuous networks. In milk, the CM is a colloidal and electrostatically stabilized sol, where glycomacropeptide strands, \(\kappa\)-casein, decorate the CM surface in the form of brushes, providing electrostatic or steric repulsion. To enable gelation, the repulsion can either be neutralized or, in the case of rennet gels, the brush strands can be removed by enzymes. Conventionally, animal rennet was predominantly used, while modern trends have turned to rennet sourced from plants, bacteria, or fungi. Dairy structures have mainly been studied using microscopy techniques, i.e., optical imaging [2; 3; 4], but also using scattering techniques [5; 6; 7; 8; 9], such as dynamic light scattering and small-angle X-ray and neutron scattering. Details of CM networks in dairy gels have been inferred indirectly from mechanical and optical rheological studies [10]. Real space or microscopy techniques offer, by contrast, direct visualization of structures, but the challenges of reconstructing the bulk of the properties from a sufficiently statistical description can make the practical application of such techniques problematic. Small-angle neutron scattering combines the advantages of (i) being able to penetrate bulk samples where light fails, (ii) being non-destructive, and (iii) non-invasive, allowing a (iv) separate assessment of fat-based and CM-based structures through contrast variation (hydrogen and deuterium substitution), and finally (v) requiring a minimum effort of sample preparation [11]. Small-angle neutron scattering (SANS) has been applied to probe the relevant length scales from nanometers to micrometers of milk and dairy gels. Several SANS techniques were employed, including conventional SANS instruments, which have been used to accessing length scales from approximately 10 to a few 100 nm [12]; Ultra small-angle neutron scattering (USANS) [13], which have extended the range to approximately 10 \(\upmu\)m; and the more recent Spin Echo SANS (SESANS) [14] technique, which can probe from 10 s of nm to a few 10 s of \(\upmu\)m [15]. The latter directly probes real space correlations; however, this only occurs in one dimension, such as the slit-smeared intensity of USANS, which does not pose an issue for isotropic systems such as dairy samples. Correspondingly, conventional SANS is well suited to probe the structure of casein micelles themselves [16] but reaches its limits around the largest size of casein micelles, and thus, USANS and SESANS are required to elucidate the coagulation as well as the gel formation processes and structures [8; 16]. It is these length scales which are most directly related to the important bulk properties of dairy gels [17]. In this paper, we propose and evaluate the potential of a novel technique, namely quantitative neutron dark-field contrast imaging (DFI) [18; 19; 20], which adds the promise to combine microscopic studies (in analogy to the SESANS technique) with macroscopic observations of micro-structural variations. The latter must be expected when, e.g., gravity plays a role or external stimuli or additions to the system are not homogeneous throughout the sample.