## 4 Results and Discussion
### Applying Dual Influential Models
#### 4.1.2 Models Assuming Langmuir Isotherms for Modifier Adsorption  
As mentioned in Section 3.1.1, Langmuir isotherm was adopted for constructing retention models to describe the modifier adsorption in this work. Three methods based on Langmuir isotherm to analyze the retention data were established. Together with the linear-isotherm-based method discussed in the previous section, in total four methods based on DIMs were studied. Estimated \(k_{0}\) and adj. R\({}^{2}\) for the four methods are listed in Table 4. All experimental data were obtained at 40 \({}^{\circ}\)C, 200 bar. The method DIM-Lan-kck\({}^{\prime}\) employs a corrected \(K_{eq,\ L}^{corr}\) as input instead of measured \(K_{eq,\ L}\). This method was established after analyzing the results from the first three methods and is described in the following. Figure 3: Retention factors of various solutes on SIL-60 and model fittings according to DIM-Lin-kc. SFC conditions: 200 bar, 40 \({}^{\circ}\)C, injection volume 2 \(\upmu\)L, flow rate 2 mL min\({}^{-1}\). Figure 2: Retention factors of phenol on different silicas and model fittings according to DIM-Lin-kc. SFC conditions: 200 bar, 40 \({}^{\circ}\)C, injection volume 2 \(\upmu\)L, flow rate 2 mL min\({}^{-1}\). Regarding the aim of selecting the best method, three aspects were considered: (1) experimental efforts, (2) goodness of fitting, and (3) number of fitting parameters or problem of overfitting. The DIM-Lan-kcK method relies on experimentally measured \(K_{eq,~{}L}\) as input data for fitting; thus, we kept a small number of fitting parameters to avoid overfitting. However, measurements of \(K_{eq,~{}L}\) require intensive efforts considering both time and equipment upgrades [13; 22]. In the method DIM-Lan-kc, we attempted to avoid such troublesome experimental measurements by making \(K_{eq,~{}L}\) a fitting parameter. The fitted \(K_{eq,~{}L}\) values were found to be significantly larger than the measured ones. In addition, the equilibrium constants should remain nearly constant for each stationary phase at a constant temperature regardless of the used solutes. Contrary to this, the fitted values of \(K_{eq,~{}L}\) obtained for different solutes deviate largely for SIL-60 and SIL-aerogel (data not shown). We surmise that overfitting of the retention data is a possible reason. Nevertheless, we observed that the fitted \(K_{eq,~{}L}\) values obtained in the DIM-Lan-kc method for phenol correlate well with the experimentally measured \(K_{eq,~{}L}\) values. Thus, the \(K_{eq,~{}L}\) obtained from the DIM-Lan-kc method should be first corrected by multiplying by an empirical factor, \(F\). We call this value \(K_{eq,~{}L}^{corr}\):  
\[K_{eq,~{}L}^{corr}=FK_{eq,~{}L}^{\prime} \tag{15}\]  
In the DIM-Lan-kcK' method, the corrected adsorption constant from Equation (15) was employed to solve the overfitting problem of the DIM-Lan-kc method. An estimation for the correction factor \(F\) was made through a paired \(t\)-test (see details in Supplementary Materials, Section S2, Tables S2 and S3) and found to be 0.19. The value of \(k_{0}\) estimated by the DIM-Lan-kcK' method is close (55-133%, Table 4) to that from the method DIM-Lan-kcK, where experimental values of \(K_{eq,~{}L}\) are used. Compared to linear adsorption isotherm, the use of Langmuir isotherms showed a significantly better correlation, as expected, due to a greater number of fitting parameters in the latter case. However, although the DIM-Lan-kc method had one more fitting parameter than DIM-Lan-kcK and DIM-Lan-kcK', the adj. R\({}^{2}\) did not show a visible increase. Another observation regarding the goodness of fitting is that the regression for toluene always results in the smallest adj. R\({}^{2}\). Toluene was selected as it is the only solute for which \(k_{0}\) could be measured experimentally due to its low affinity to silica. However, because of the same reason, retention times of toluene are close to the hold-up time, resulting in large uncertainties of the retention factor. Therefore, the low adj.