## 2 Materials and Methods
### Effect of LSZ on the Aggregation Behavior of CNHox  
Figure 7 displays the temporal variations in the hydrodynamic diameter \((d_{\rm h})\) of lysozyme-coated CNHox at various mass ratios. The pH in this experiment was 5.65 \(\pm\) 0.32 without making any pH adjustment. In Figure 7, the circles filled in different colors  
Figure 6: Zeta potential of lysozyme-coated CNHox in 10 mM KCl as a function of the mass ratio of lysozyme to CNHox (g/g). stand for different KCl concentrations. The initial slopes \(\left(\mathrm{d}d_{\mathrm{h}}/\mathrm{d}t\right)_{t\to 0}\) do not make a big difference in each case in Figure 7a compared with Figure 3. This result implies that all the negative charge on surface of CNHox is neutralized by the positively charged lysozyme. The interaction between CNHox particles is dominated by van der Waals attraction rather than electric repulsive force. This interaction leads to a fast aggregation rate in a large range of electrolyte concentrations. We also cannot ignore that although the initial slope in each case is almost the same, higher electrolyte concentrations lead to particle diameter changes more quickly. This may be due to the screening effect. Figure 7b,c shows almost same trend in particle diameter changes, but apparently a higher electrolyte concentration is needed to reach the CCC at a mass ratio of 50 g/g; one  
Figure 7: Hydrodynamic diameter changes in lysozyme-coated CNHox particles with time for the mass ratios of (**a**) 0.0875 g/g, (**b**) 0.01 g/g, and (**c**) 50 g/g and various KCl concentrations. can assume this can be attributed to the relatively higher surface charge density. Details will be discussed in Figure 8.  
The different symbols in Figure 8 represent the data collected at various LSZ/CNHox mass ratios. The aggregation behaviors of the CNHox with LSZ are also DLVO-like. The addition of LSZ with a mass ratio of 0.0875 g/g decreases the CCC and dramatically induces aggregation, and low-to-high salt concentrations exist in the fast aggregation regime at the IEP. This can be attributed to the fact that the coated lysozyme neutralizes the negatively charged CNHox surface. The aggregation rates were unaffected by the salt concentration at the IEP, where the \(1/W\) was almost unified. In the case of a high dose of LSZ (50 g/g), the increment in CCC to 60.53 mM proves the larger magnitude of the zeta potential due to the charge reversal. It is possible to explain the unexpected increase in \(1/W\) at the low electrolyte concentration by the presence of a non-DLVO attractive force; during the charge neutralization and the charge reversal of CNHox, the adsorption of LSZ causes the surface charge to become heterogenous (shown in Figure 9). In the conditions of bare CNHox with trivalent counterions, patch-charge attraction was also observed [15]. This phenomenon is also usually discussed in the study of polyelectrolytes [28]. We also need to pay attention to the fact that salt dependence in the slow aggregation regime at a high mass ratio is slightly weaker than in cases at a low mass ratio and without additional LSZ. However, in the DLVO theory, the slope of stability plots should be barely changed at different the electrolyte concentrations [43]. Therefore, the slightly gentle slope observed at a high mass ratio in this study implies the existence of a non-DLVO force due to heterogeneity. [PERSON] et al. [44] have proved that the hydrophobicity and hydrophilicity can be attributed to the efficiency of colloidal stabilization. In the meanwhile, proteins tend to expose most hydrophilic areas in aqueous solution. The hydration force, therefore, plays a significant role in the aggregation at a high mass ratio of 50 g/g, contributing to the high CCC. Figure 8: Reciprocal stability ratio \(1/W\) against KCl concentration for bare CNHox (blue circles), CNHox with LSZ at the IEP (red triangles), CNHox with LSZ at low mass ratio (green squares), and CNHox with LSZ at high mass ratio (yellow circles).