## 4 Discussion
### Maximum Packing Density Prediction  
The results shown in Figures 5 and 6 highlight that minimizing porosity (i.e., maximizing packing density) at \(f_{f}^{*}\), does not only require very small \(R\) but also \(d_{l}\) has to exceed certain value. That is, packing density at any given fines fraction is function of absolute particle sizes rather than just the size ratio. This is a major difference between the current model and previous models (e.g., References [15; 33]) introduced for sand-fines mixtures. In Figure 7, we compare the measured maximum packing density (i.e., \(1-n\) at lowest porosity point) for all mixtures in Table 1, with predictions from three different approaches: (a) ideal packing (\(r=0\)), (b) incorporating only \(R\) in the model (i.e., \(x=1\) for all mixtures) and (c) the current model incorporating both \(R\) and cohesive packing (\(p=0.15\), while \(x\) varies with \(d_{l}\)). The mean absolute percentage error for the three approaches was found to be 12.6%, 10.5% and 1.0% respectively. Incorporating the effect of \(R\) alone can only slightly improve the predictions compared to the ideal packing case (Figure 7) since \(R\) is very small anyhow for sand-clay and sand-silt mixtures. The analysis also shows that ignoring fines cohesive packing behavior can result in significant overestimation of maximum packing density (Figure 7). The predictions of the three approaches are close to each other only when approaching ideal packing condition as mentioned earlier for the two mixtures in Figure 4 (i.e., the largest measured maximum packing density in Figure 7). The current model can then improve the prediction of the expected maximum packing density for mixtures composed of large and very small particles. According to the current model and observation, maximum packing density of sand-fines mixtures requires: (a) \(d_{l}>0.750\) mm, (b) \(R\) to be very small and (c) \(n_{l}\) and \(n_{f}\) to be as small as possible. Figure 6: Porosity as function of fines fraction as predicted by current model for two cases: the first one (green dotted curve) \(d_{l}=0.75\) mm and \(d_{f}=0.002\) mm, while the second case (red solid curve) \(d_{l}=0.25\) mm and \(d_{f}=0.00067\) mm. R is the same for both cases but the green curve is much closer to the ideal packing case (dashed black curve). \(n_{l}\) and \(n_{f}=0.4\) and \(0.59\) respectively for all cases. It is important to note that the current model is developed for binary mixtures characterized by very small \(R\) (\(<\)0.1) where the fine particles have particle size less than 0.150 mm. The original model and correction functions [Equations (1)-(6)] were developed for spherical glass beads which can have very different shape compared to clay and silts particles, yet the analysis shown in this study suggests the validity of such formulations. Although particle shape can impact the packing, it is possible that at very low \(R\), as it is the case for sand-clay mixtures, the impact of \(R\) and absolute particle size is dominating the packing behavior. It is also possible that the similar particle specific gravity of glass beads and sandy soil (both being predominantly composed of silica) is a major factor in having similar \(d_{cri}\) (Figure 2) and cohesive packing behavior. This is because specific gravity is a major control on inter-particle and gravity forces [36,37]. As \(R\) increases above 0.1 and the particle size of the fines increases above 0.150 mm, the role of particle shape is expected to become more important (as particle sizes become more similar), while the impact of cohesive packing diminishes and thus, the current model may not provide accurate prediction. For such mixtures containing coarser particles and characterized by large \(R\), the model developed by [PERSON] et al. [25] can be used.