## 3 Results and Discussion
### Optimisation of the Extraction Performance

Table 3: Experimental design points and the corresponding predicted and measured Y\({}_{\text{TP}}\) values for the extractions carried out with each of the CDs used. GAE, gallic acid equivalent. Figure 2: Contour graphs presenting the effect of simultaneous variation of independent variables on the response. Assignments: \(\beta\)-CD, \(\beta\)-cyclodextrinrin; HP-\(\beta\)-CD, hydroxypropyl \(\beta\)-cyclodextrinrin; m-\(\beta\)-CD, methyl \(\beta\)-cyclodextrinrin. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Cyclodextrin** & **Maximum Predicted Response (Y\({}_{\text{TP}}\), mg GAE g\({}^{-1}\) dw)** & **Optimal Conditions** & \\ \hline  & \(\text{\bf{C}}_{\text{CD}}\) **(to/////**\%**)** & \(\text{\bf{R}}_{\text{LBS}}\) **(mL g\({}^{-1}\))** & **pH** \\ \hline \(\beta\)-CD & 47.48 \(\pm\) 2.29 & 0.88 & 93 & 3.75 \\ HP-\(\beta\)-CD & 54.40 \(\pm\) 4.42 & 1.40 & 100 & 3.90 \\ m-\(\beta\)-CD & 54.72 \(\pm\) 2.11 & 1.40 & 94 & 4.54 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Mathematical models and associated statistics derived from the experimental design. Figure 3: Desirability function for each of the CDs tested, displaying optimal conditions and maximum predicted response values. Assignments: m-\(\beta\)-CD, methyl \(\beta\)-cyclodextrintrin; HP-\(\beta\)-CD, hydroxypropyl \(\beta\)-cyclodextrintrin; \(\beta\)-CD, \(\beta\)-cyclodextrintrin. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Cyclodextrin** & **2 nd Order Polynomial Equations** & \(\text{\bf{R}}^{2}\) & \(\text{\bf{P}}\) \\ \hline \(\beta\)-CD & 43.60 \(+\) 5.48X\({}_{2}\) - 5.63X\({}_{3}\) - 3.79X\({}_{2}^{2}\) - 4.74X\({}_{3}^{2}\) & 0.98 & 0.0006 \\ HP-\(\beta\)-CD & 46.36 \(+\) 6.11X\({}_{2}\) - 4.04X\({}_{3}\) - 2.39X\({}_{2}\)X\({}_{3}\) - 4.66X\({}_{2}^{2}\) & 0.97 & 0.0025 \\ m-\(\beta\)-CD & 47.75 \(+\) 2.08X\({}_{1}\) + 6.06X\({}_{2}\) - 6.02X\({}_{3}\) + 2.92X\({}_{2}\)X\({}_{3}\) - 2.79X\({}_{2}^{2}\) - 5.48X\({}_{3}^{2}\) & 1.00 & 0.0024 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Mathematical models and associated statistics derived from the experimental design. For all CDs, optimum R\({}_{\text{L/S}}\) varied closely within 93-100 mL g\({}^{-1}\) (Table 5). This outcome showed that the influence exerted by R\({}_{\text{L/S}}\) on the extraction performance was not significantly affected by the structure of CD. The magnitude of R\({}_{\text{L/S}}\) is related with the concentration gradient between the liquid phase (extraction medium) and the surface of the solid particle, which is directly involved in mass transfer. If R\({}_{\text{L/S}}\) is below a certain limit, then the equilibria established may not favor fast diffusion of the solute during extraction, owing to non-negligible resistance to mass transfer [15]. Several examinations on polyphenol extraction from plant tissues using conventional organic solvents suggested R\({}_{\text{L/S}}\) optima between 81 [16] and 100 mL g\({}^{-1}\)[17; 18; 19]. Considering that the average R\({}_{\text{L/S}}\) value in this study was 96 mL g\({}^{-1}\), it could be argued that an aqueous medium containing any of the CDs assayed would behave as a common solvent in this regard. The statistically significant effects of pH revealed by the models for the extraction with any of the CDs assayed pointed emphatically to the role of the pH in the extraction performance. For all CDs, the optimal pH was below 5, which evidenced that extractions were favoured at acidic pH. One possible reason for this might be related to the ionisation of the phenolic hydroxyl groups, which possess weak acidity. Assuming that encapsulation of polyphenols within the CD cavity is the main effect that enhances extraction, then polyphenol dissociation would increase their polarity, leading to weaker interactions with CD cavity, which is hydrophobic. As dissociation would increase at a higher pH, it would be likely that suppression of dissociation at pH \(<\) 5 would maintain polyphenols in their molecular (non-dissociated) form, hence promoting more powerful polyphenol-CD interactions.