## 2 Background  
### Packing Model for Binary Mixtures  
The relationship between porosity (or void ratio) of the binary mixed soil and fine particles fraction exhibits a \"V-shape\" trend as shown in Figure 1. Adding fines to the mixture initially decreases the porosity as fines fill the pore space available between the large particles, leading porosity to decrease until reaching a minimum value (i.e., maximum packing density, \(\phi\), condition where \(\phi=1-n\)). The fines fraction, \(f_{f}\), at the minimum porosity is frequently called critical fine fraction, \(f_{f}^{*}\) (Figure 1a) and it refers to the amount of fine particles needed to fill the pore space completely between the larger grains [18; 42; 43]. Increasing the fraction of fine particles beyond the critical value increases porosity due to replacement of solid large particles by micro porous fines aggregates. When the fine-to-large particles diameter ratio, \(R\), is very small (<0.05), the binary mixture packing is described as \"ideal\" [28; 37; 44; 45]. In this case, the original packing of the coarse spheres is not disturbed by the fine spheres and vice versa. Consequently, the porosity of the two pure unmixed end-members (i.e., pure large sand and pure fines) remains constant after mixing. In such an ideal packing system, the porosity has been described mathematically using a linear packing model [28; 37; 44; 45]. At any fraction of fines, the porosity of the binary mixture in the case of ideal packing can be determined using only the porosity of the pure unmixed end-members. When \(R\) increases, the binary mixture packing starts to deviate from the ideal case, as the fine particles become large enough to disturb the original packing of the large particle and vice versa [15; 33; 44; 45]. In this case, the porosity of the binary mixture becomes larger than the prediction of the ideal packing model (Figure 1a). To correct for the underestimation of porosity in mixed soils, several semi-empirical models have been developed by introducing a mixing-degree coefficient (\(\lambda\)) into the ideal packing model [15; 26; 33]. The authors empirically related the mixing coefficient to \(R\), based on data set from mixtures characterized by variable \(R\), in an effort to provide an approach for predicting \(\lambda\) based on properties of end-members alone. The modified \"non-ideal\" packing model could successfully predict the porosity of sandy soil containing small and large sands as shown for example in Figure 1a. Nevertheless, accounting for the size ratio effect alone in existing models frequently fails to predict the porosity for sand-clay or sand-silt mixtures where \(R\) is very small (\(R<0.1\)) and when fine particles are less than 0.100 mm in size as pointed out by [PERSON] et al. [33] and [PERSON] et al. [35]. This is because existing \(\lambda\)-\(R\) relationship predicts almost an ideal packing for very small size ratios which does not match experimental data as demonstrated in Figure 1b for example. That is, existing models developed for binary coarse-grained soil frequently underpredict the porosity or void ratio of sand-clay and sand-fine silt mixtures. A possible reason is that very fine particles, such as clay or fine silt, tend to form aggregates that are larger than the actual particle size due to the cohesive nature of the particles [35]. Packing of very fine particles can be very different from larger particles due to the effect of inter-particle forces, such as van der Waals and electrostatic forces, that can become more important than the gravity force in governing the packing of very fine particles [39; 40]. This suggests that binary mixtures containing cohesive particles should be treated differently compared to completely non-cohesive mixtures. Outside the literature examining sand-fines mixtures, earlier studies from powder technology and chemical engineering disciplines [41], provided useful insights about the incorporation of fines cohesiveness into the packing model for glass beads powder.