## 4 Results and Discussion
### Applying Mixed-Retention Models  
According to the original MRM, Equation (12), the retention factor \(k\) varies linearly with surface coverage \(\theta\):  
\[k=(k_{c}-k_{0})\theta+k_{0} \tag{16}\]  
As seen in the \(k\) vs. \(\theta\) plot (Figure 6), the experimental data does not show linear correlations among any solute/stationary phase pairs. Only the data points of each solute/stationary phase pair at relatively lower modifier concentrations can be treated as a linear function. The reason might be that the assumption of single layer adsorption to apply Langmuir isotherm does not hold true. Studies have demonstrated the adsorption of alcohols on silica materials could exhibit multilayer behavior above a certain concentration [26; 29; 30]. If this is the case, two issues should be considered. Firstly, the calculated \(\theta\) values are only hypothetical and do not have any physical meaning. Secondly, the proposed model, Equation (16), is based on the theory that when methanol is adsorbed as a single layer, the solute molecules are attracted to the methanol molecules, which leads to \(k_{c}\). However, when multilayer adsorption appears, the affinity between the solute and adsorbed methanol changes, which leads to a different \(k_{c}^{\prime}\). In this case, \(k_{c}\) depends on the modifier concentration, and the model assumptions break down. Because of these reasons, an advanced model Equation (14) with an empirical factor \(M\) was suggested. As discussed above for the DIMs, when the experimental value of \(K_{eq,L}\) is not available, an estimated value can be taken. The results from regression by MRM-Lan-kck and MRM-Lan-kckM methods are given in Table 5.  
Figure 6: Retention factors of various solutes on SIL60 and SIL-Aerogel. SFC conditions: 200 bar, 40 \({}^{\circ}\)C, injection volume 2 \(\mu\)L, flow rate 2 mL min\({}^{-1}\), modifier (methanol) fraction at 0.1–30 vol.%. The dashed straight lines were plotted to guide the eyes at low coverages. We have observed that when Equation (14) was applied using the experimental measured \(K_{eq,L}\) (MRM-Lan-kcK), the data could not be regressed and the parameters did not converge. The suspected reason was over-parameterization, which leads to deterioration of estimation accuracy, and the resulted estimations and errors were unreliable. However, the fitting worked in all cases for phenol and toluene as solutes and the estimated \(k_{0}\) values were always comparable to those from DIM-Lan-kcK. The parameter \(M\) was found to lie in the range 100-200. Although the retention factors of toluene and phenol are very different at zero modifier concentration, they have similar \(M\) values. We may surmise that \(M\) is not solute-dependent, but modifier-dependent. This is reasonable because \(M\) was introduced to correct the strength of the modifier influence at high modifier concentration where multilayer adsorption may take place and the influence of solutes shall be minimal. Therefore, to solve the overparameterization problem, \(M\) was fixed to 150 in all cases for the next method, MRM-Lan-kcK. Unlike developing method DIM-Lan-kcK' based on the result from DIM-Lan-kc, it is not possible to treat the \(K_{eq,L}\) as a fitting parameter in the MRMs: it is not reasonable to add an extra fitting parameter to a model which is already overparameterized. We suggest a rough estimation of \(K_{eq,L}\) = 20 mL g\({}^{-1}\) as a representative value across all studied silica materials. With these estimates for \(M\) and \(K_{eq,L}\), the regressions were performed and the results show reasonable agreement (89-147%) with the method DIM-Lan-kcK (Table 5). One exception was the case of rosmarinic acid, which has an exceptionally high affinity to silica. In this case, the estimations of \(k_{0}\) by MRM are not reliable. For the second strongest retained solute, nicotinamide, the DIM-Lan and MRM-Lan methods give consistent estimations (304 \(\pm\) 16 and 277 \(\pm\) 10, respectively), even when arbitrary selected parameters \(K_{eq,L}\) and \(M\) were taken.