## 2 Materials and Methods
### Comparison between Experimental Data and DLVO Prediction  
[PERSON] et al. [46] have demonstrated the following relationship between critical coagulation ionic strength (CCIS) and surface charge density (\(\sigma\)) from the DLVO theory:  
\[CCIS=\left(\frac{9}{8\pi\mathrm{exp}(2)}\right)^{\frac{1}{3}}\frac{1}{\lambda _{\mathrm{B}}}\bigg{(}\frac{\sigma^{2}}{\epsilon_{0}\epsilon_{\mathrm{r}}H} \bigg{)}^{\frac{2}{3}}, \tag{5}\]  
where \(H\) is the Hamaker constant, which characterizes the magnitude of van der Waals interaction potential between particles; \(\epsilon_{0}\epsilon_{r}\) is the dielectric constant; and \(\lambda_{\mathrm{B}}\) is the Bjerrum length, which can be calculated with Equation (6):  
\[\lambda_{\mathrm{B}}=\frac{e^{2}}{4\pi k_{\mathrm{B}}T\epsilon_{0}\epsilon_{ \mathrm{r}}}, \tag{6}\]  
where \(e\) is the elementary charge. Within the Debye-Huckel approximation, the surface charge density \(\sigma\) can be calculated with Equation (7):  
\[\sigma=\epsilon_{0}\epsilon_{r}\kappa\psi_{\mathrm{d}\mathrm{l}}, \tag{7}\]  
Figure 9: Sketch map of patch-charge attraction between two LSZ-saturated CNHox particles. where the diffuse layer potential \(\psi_{dl}\) is roughly equivalent to the zeta potential, and \(\kappa^{-1}\) is the Debye length, which corresponds to the thickness of the diffuse double layer. \[\kappa^{-1}=\left(\frac{\varepsilon_{0}\varepsilon_{r}k_{\mathrm{B}}T}{2N_{ \mathrm{A}}e^{2}I}\right)^{\frac{1}{2}}, \tag{8}\]  
where \(I\) is the ionic strength, which can be calculated with Equation (9):  
\[I=\frac{1}{2}{\sum_{\mathrm{i}}}z_{\mathrm{i}}^{2}c_{\mathrm{i}}, \tag{9}\]  
where \(z_{\mathrm{i}}\) is the i-th ion valence, and \(c_{\mathrm{i}}\) is the concentration of i-th ion. The correlation between the surface charge density and CCIS of CNHoxs in the presence of LSZ is depicted in Figure 10. The experimental data of the CNHox in the presence of LSZ are represented by the red filled circles. The broken dotted line, dashed line, solid line, and broken line correspond to the DLVO predictions with Hamaker constants of 4 \(\times\) 10\({}^{-21}\) J, 1 \(\times\) 10\({}^{-20}\) J, 2 \(\times\) 10\({}^{-20}\) J, and 7 \(\times\) 10\({}^{-20}\) J, respectively. The experimental result from [PERSON] et al. (2015) (open triangle) and the experimental result of this research (red filled circles) are well aligned with the theoretical DLVO prediction. The DLVO theory works very well with bare CNHox and CNHox in the presence of LSZ. These four chosen Hamaker constants cover the results from [PERSON] et al. (2015) and this research. In Figure 10, the result of the CNHox at a high mass ratio is located at the calculated line with a relatively lower Hamaker constant (4 \(\times\) 10\({}^{-21}\) J), which is well-aligned with the result calculated by [PERSON] et al. (2017) for a 3.3 \(\times\) 10\({}^{-21}\) J Hamaker constant of LSZ-water-LSZ. This phenomenon indicates that the attractive interaction between the LSZ-water-LSZ is smaller than that of CNHox-water-CNHox. The zeta potentials at different mass ratios as a function of electrolyte concentration are depicted in Figure 11. In the case of bare CNHox and a low mass ratio, the results are not much different. The adsorption of a small amount of LSZ slightly reduces the net  
Figure 10: Comparison between surface charge density (\(\sigma\)) of LSZ-coated CNHox (red filled circles), bare CNHox (2015) (white open triangle), and critical coagulation ionic strength (CCIS) with DLVO prediction with different Hmaker constant \(H\) (dash-dot line, solid line, dotted line, and broken line). charge compared with bare CNHox; at a high mass ratio, CCC shifts to a relatively higher electrolyte concentration and a lower magnitude of zeta potential, and this agrees with the outcome depicted in Figure 11. The magnitude of zeta potential, in this case, reaches 40 mV, which is the largest without an additional electrolyte compared to the other cases.