## 2 Materials and Methods
### Thermodynamic Methods  
#### 2.3.1 Dispersive and Polar Energies and Lewis Acid-Base Parameters  
The determination of the free energy of adsorption \(\Delta G_{a}^{0}\) of the solvents on S-DVB/HMU was carried out from the retention volume measurements \(Vn\) using Equation (1):  
\[\Delta G_{a}^{0}(T)=-RTlnVn+C(T) \tag{1}\]  
where \(T\) is the absolute temperature, \(R\) is the perfect gas constant, and \(C(T)\) is a characteristic of interaction.  
The expression of \(\Delta G_{a}^{0}(T)\) of organic solvents is composed of the summation of the dispersive component \(\Delta G_{a}^{d}(T)\) and the polar component \(\Delta G_{a}^{sp}(T)\) of adsorption by varying the temperature:  
\[\Delta G_{a}^{0}(T)=\Delta G_{a}^{d}(T)+\Delta G_{a}^{p}(T) \tag{2}\]  
[PERSON] [37; 38] gave a new expression of \(\Delta G_{a}^{d}(T)\):  
\[\Delta G_{a}^{d}(T)=-\frac{\alpha_{0S}}{H^{6}}\left[\frac{3\mathcal{N}}{2(4 \pi\varepsilon_{0})^{2}}\bigg{(}\frac{\varepsilon_{S}\,\varepsilon_{X}}{( \varepsilon_{S}+\varepsilon_{X})}\alpha_{0X}\bigg{)}\right] \tag{3}\]  
where \(\varepsilon_{0}\) is the dielectric constant of vacuum, \(\alpha_{0}\) represents the deformation polarizability relative to the solid (S) and the solvent (\(X\)), \(\varepsilon\) is the ionization energy, and \(H\) is the separation distance between the solid substrate and the adsorbed molecule.  
The combination of Equations (1) and (3) allowed writing Equation (4):  
\[RTlnVn=\frac{\alpha_{0S}}{H^{6}}\left[\frac{3\mathcal{N}}{2(4\pi\varepsilon_{0 })^{2}}\bigg{(}\frac{\varepsilon_{S}\,\varepsilon_{X}}{(\varepsilon_{S}+ \varepsilon_{X})}\alpha_{0X}\bigg{)}\right]-\Delta G_{a}^{p}(T)+C(T) \tag{4}\]  
The interaction parameter \(\mathcal{P}_{SX}\) used as a chromatographic index was given by Equation (5):  
\[\mathcal{P}_{SX}=\frac{\varepsilon_{S}\,\varepsilon_{X}}{(\varepsilon_{S}+ \varepsilon_{X})}\alpha_{0X} \tag{5}\]  
For n-alkanes, \(RTlnVn\) can be written as:  
\[\left\{\begin{aligned} RTlnVn(n-alkane)=A(T)\bigg{[}\frac{3 \mathcal{N}}{2(4\pi\varepsilon_{0})}\mathcal{P}_{SX}(nn-alkane)\bigg{]}+C(T)\\ A(T)=\frac{\alpha_{0S}}{\left[H(T)\right]^{6}}\end{aligned}\right. \tag{6}\]  
where \(A(T)\) is a constant depending on \(\alpha_{0S}\) and \(H(T)\).  
The values of \(\Delta G_{a}^{p}(polar)\) of adsorption were obtained from Equation (7):  
\[\Delta G_{a}^{p}(T,\,polar)=RTlnVn\,\left(T,\,polar\right)-RTlnVn\,\left(T,\,X_{ n.p.}\right) \tag{7}\]  
where \(X_{n.p.}\) is the geometric point obtained from the projection of the polar point intersecting the straight line of n-alkanes.  
While the values of \(\left(-\Delta H_{a}^{p}\right)\) and \(\left(-\Delta S_{a}^{p}\right)\) of adsorbed polar solvents were deduced from Relation (8) when the linearity of \(\Delta G_{a}^{p}(T)\) was confirmed:  
\[\Delta G_{a}^{p}(T)=\Delta H_{a}^{p}-T\Delta S_{a}^{p} \tag{8}\]  
The following Equation (9) allowed obtaining the Lewis enthalpic (\(K_{A}\), \(K_{D}\)) and entropic (\(\omega_{A}\), \(\omega_{D}\)) Lewis acid-base constants:  
\[\left\{\begin{aligned} &(-\Delta H^{p})=\ K_{A}\times DN^{\prime}+K_{D} \times AN^{\prime}\\ &\left(-\Delta S_{a}^{p}\right)=\omega_{A}\times DN^{\prime}+ \omega_{D}\times AN^{\prime}\end{aligned}\right. \tag{9}\]  
where \(DN^{\prime}\) and \(AN^{\prime}\) are, respectively, the corrected electron donor and acceptor numbers of the polar molecule [56; 57].