## 4 Results and Discussion
### Applying Dual Influential Models
#### 4.1.2 Models Assuming Langmuir Isotherms for Modifier Adsorption
R\({}^{2}\) for toluene should not be interpreted as a failure of the model. To summarize, the DIM-Lin-kc method is not recommended in any case because it has the worst goodness of fitting. Having the same level of goodness of fitting, the DIM-Lan \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Stationary Phase**} & \multirow{2}{*}{**Solute**} & \multirow{2}{*}{**DIM-Lan-kc**} & \multicolumn{3}{c}{**DIM-Lan-kc**} & \multicolumn{3}{c}{**DIM-Lan-kc**} & \multicolumn{3}{c}{**DIM-Lan-kc**} & \multicolumn{3}{c}{**DIM-Lan-kc**} \\ \cline{3-13}  & & & & & & & & & & & & & \\ \cline{3-13}  & \multirow{2}{*}{**Solute**} & \multirow{2}{*}{**\(k_{0}\)**} & \multirow{2}{*}{\(\mathbf{R}^{2}\)**} & \multirow{2}{*}{\(k_{0}\)**} & \multirow{2}{*}{\(K_{eq,L}\)**} & \multirow{2}{*}{\(\mathbf{R}^{2}\)**} & \multirow{2}{*}{\(k_{0}\)**} & \multirow{2}{*}{\(K_{eq,L}\)**} & \multirow{2}{*}{\(\mathbf{R}^{2}\)**} & \multirow{2}{*}{\(k_{0}\)**} & \multirow{2}{*}{\(K_{eq,L}^{corr}\)**} & \multirow{2}{*}{\(\mathbf{R}^{2}\)**} & \multirow{2}{*}{\(k_{0}^{(\text{DM-Lan-kcK})}\)**} & \multirow{2}{*}{\(\mathbf{R}^{2}\)} & \(\mathbf{k_{0}^{(\text{DM-Lan-kcK})}\)**} \\ \cline{3-13}  & & & & & & & & & & & & & \\ \hline \multirow{4}{*}{SIL-60} & \multirow{2}{*}{Phenol} & \(4.4\pm 0.3\) & 0.92 & \(6.2\pm 0.2\) & & 0.99 & \(8.1\pm 0.1\) & \(62\pm 2\) & 1.00 & \(5.9\pm 0.2\) & & 0.99 & 0.96 \\  & Naproxen & \(25\pm 5\) & 0.74 & \(36\pm 4\) & & 0.93 & \(330\pm 19\) & \(170\pm 4\) & 1.00 & \(32\pm 4\) & & 0.91 & 0.91 \\  & Nicotinamide & \(42\pm 5\) & 0.92 & \(304\pm 16\) & & 1.00 & \(103\pm 20\) & \(40\pm 3\) & 1.00 & \(219\pm 14\) & & 1.00 & 0.72 \\  & Rosmarnic & \multirow{2}{*}{\(483\pm 81\)} & \multirow{2}{*}{0.94} & \multirow{2}{*}{\(1.02\pm 0.08\)} & \multirow{2}{*}{1.00} & \multirow{2}{*}{(2.9\pm 12) \(\times\)} & \multirow{2}{*}{\(45\pm 4\) } & \multirow{2}{*}{1.00} & \multirow{2}{*}{(5.6\pm 0.4) \(\times\)} & \multirow{2}{*}{1.00} & \multirow{2}{*}{0.55} \\  & acid & & & & & & & & & & & \\  & Toluene & 0.2 & 0.89 & \(02\) & 0.97 & \(02\) & & & & & & & & \\  & Phenol & \(3.0\pm 0.2\) & 0.89 & \(4.1\pm 0.1\) & 11.2 & 0.99 & \(5.6\pm 0.1\) & \(58\pm 2\) & 1.00 & \(4.1\pm 0.1\) & 11.1 & 0.99 & 1.01 \\ SIL-300 & Phenol & \(0.9\pm 0.1\) & 0.78 & \(1.1\) & 5.3 & 0.97 & \(1.4\) & \(24\pm 3\) & 1.00 & \(1.1\) & 4.6 & 0.97 & 0.99 \\ SIL- & Phenol & \(2.0\pm 0.2\) & 0.90 & \(2.6\pm 0.1\) & & 0.98 & \(3.9\pm 0.1\) & \(63\pm 3\) & 1.00 & \(2.3\pm 0.1\) & & 0.96 & 0.90 \\  & Naproxen & \(96\pm 23\) & 0.93 & \(139\pm 8\) & 7.1 & 1.00 & \(641\pm 178\) & \(26\pm 3\) & 1.00 & \(185\pm 10\) & 11.9 & 1.00 & 1.33 \\  & Toluene & 0.1 & 0.74 & 0.1 & & 0.83 & 0.1 & \(438\pm 0.87\) & 0.1 & \(0.84\) & 1.05 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Regression results for DIMs. kcK' method shows significant advantages compared to DIM-Lan-kcK for less experimental efforts and to DIM-Lan-kc for less risk of overfitting (Figures 4 and 5). Based on the above results, several issues shall be addressed when applying the DIMs for estimating \(k_{0}\) from retention data in the presence of a modifier. First of all, when the methanol adsorption isotherm on the to-be-tested stationary phase in supercritical CO\({}_{2}\) is available at the required temperature and pressure, the method DIM-Lan-kcK can be applied with the measured \(K_{eq,L}\) for the most reliable estimation. However, in most cases, adsorption isotherms are not available. If so, phenol should be injected into the column as  
Figure 4: Retention factors of phenol and naproxen on SIL-Aerogel and model fittings according to DIM-Lan-kcK’. SFC conditions: 200 bar, 40 \({}^{\circ}\)C, injection volume 2 \(\mu\)L, flow rate 2 mL min\({}^{-1}\). Figure 5: Retention factors of toluene on SIL-60 and SIL-Aerogel and model fittings according to DIM-Lan-kcK’. SFC conditions: 200 bar, 40 \({}^{\circ}\)C, injection volume 2 \(\mu\)L, flow rate 2 mL min\({}^{-1}\). a model solute at various methanol concentration.