## 3 Retention Modeling
### Model Development
#### 3.1.1 Dual Influential Model  
The first category of models is based on the idea that modifier influences the solute retention both in the mobile and stationary phases. We assume that the ratio of the solute concentration in the mobile phase with modifier \(c_{m}\) over the corresponding concentration without modifier is proportional to the modifier concentration \(c_{mod}\) (Equation (3)). Likewise, the solute concentration in the stationary phase is increased proportional to the modifier loading in the stationary phase (Equation (4)). The loading is assumed to be independent of the presence of the solute and is a function of the pressure and temperature of the mobile phase. This class of models is referred to as dual influential models (DIM). Equations (3) and (4) are suggested in the spirit of the Setschenow equation for the salting out effect and the log-linear model for solubility in mixed solvents [20]:  
\[\lg\frac{c_{m}}{c_{m,0}}=ac_{mod} \tag{3}\]  
\[\lg\frac{c_{s}}{c_{s,0}}=bq_{mod} \tag{4}\]  
where \(c_{m,0}\) and \(c_{s,0}\) are the concentrations of a solute in mobile and stationary phases at zero modifier concentration, \(c_{mod}\) is the modifier concentrations (mg mL\({}^{-1}\)) in mobile phase, \(q_{mod}\) is the loading (mg mL\({}^{-1}\)) of modifier on stationary phase by adsorption, and both \(a\) and \(b\) are empirical parameters (mL g\({}^{-1}\)) that can be regarded as a strength of the modifier effect on the mobile and stationary phases, respectively. When the absolute values of \(a\) and \(b\) are large, the solute concentrations are more sensitive to small changes of modifier concentration. The values of \(a\) and \(b\) depend on the nature of the solute, the mobile and stationary phase, and the operational conditions (temperature, pressure).  
If we now assume the simplest case, namely a linear isotherm for the modifier adsorption equilibrium, we get Equation (5), as follows:  
\[q_{mod}=K_{eq,\ lin}c_{mod} \tag{5}\]  
where \(K_{eq,\ lin}\) is the equilibrium constant. Equations (2)-(5) can be rearranged into the following expression for the retention factors \(k\) at the modifier concentration \(c_{mod}\) and \(k_{0}\) is the retention factor at \(c_{mod}=0\):  
\[\lg k=\lg k_{0}-Ac_{mod} \tag{6}\]  
where \(A=a-bK_{eq,lin}\). The constant \(A\) is the overall strength of the modifier effect, which accounts for the change in the retention factor from its value \(k_{0}\) under modifier-free conditions.  
Another natural extension is to plug a non-linear adsorption isotherm into the retention model. Several previous works have demonstrated that the adsorption of individual organic solvents in supercritical CO\({}_{2}\) follows Langmuir-type isotherm models [13; 21; 22; 23; 24].  
In this study, we also demonstrate that Langmuir isotherm, Equation (7), provides an acceptable fit to the experimental data for methanol on silica-based materials (see Section S1 in Supplementary Materials).  
\[q_{mod}=\frac{q_{max}K_{eq,L}c_{mod}}{1+K_{eq,L}c_{mod}} \tag{7}\]  
The fractional surface coverage \(\theta\) of the stationary phase by modifier molecules is given as follows:  
\[\theta=\frac{q_{mod}}{q_{max}}=\frac{K_{eq,L}c_{mod}}{1+K_{eq,L}c_{mod}} \tag{8}\]  
Here, \(K_{eq,L}\) is the Langmuir equilibrium constant (mg mL\({}^{-1}\)) and \(q_{max}\) is the maximum modifier loading (mg mL\({}^{-1}\)) at monolayer coverage. Several \(K_{eq,L}\) values were measured as a part of our preliminary work (see Section S1 in Supplementary Materials).  
Equation (8) together with Equations (2)-(4) results in the retention model given by Equation (9).  
\[\lg k=\lg k_{0}+B\theta-ac_{mod} \tag{9}\]  
\[\lg k=\lg k_{0}+B\frac{K_{eq,~{}L}c_{mod}}{1+K_{eq,~{}L}c_{mod}}-ac_{mod} \tag{10}\]  
where \(B=bq_{max}\). Equation (10) gives a concentration-explicit form of Equation (9) and will be tested for the ability to estimate \(k_{0}\) values along with linear-DIM model, Equation (6).