## 3 Results and Discussion
### Extraction Kinetics and Temperature Effects  
To appraise the temperature effects on polyphenol recovery, kinetics was traced using all three CDs over a range of 40 to 80 \({}^{\circ}\)C (Figure 4). The model that could effectively describe the patterns recorded was second-order kinetics [26]:  
\[\mathrm{Y_{TP(t)}}\ =\ \frac{\mathrm{Y_{TP(s)}^{2}}\mathit{kt}}{1+\mathrm{Y_{TP(s) }}\mathit{kt}} \tag{4}\]  
where \(\mathrm{Y_{TP(t)}}\) and \(\mathrm{Y_{TP(s)}}\) represent the TP yield at any time \(t\) and at equilibrium (saturation), respectively, \(k\) is the second-order extraction rate constant. When \(t\) approaches 0, the initial extraction rate, \(h\), given as \(\mathrm{Y_{TP(t)}}\mathit{t}\), is defined as follows:  
\[h\ =\ k\mathrm{Y_{TP(s)}^{2}} \tag{5}\]  
In Table 7, the values determined for \(k\), \(\mathrm{Y_{TP(s)}}\) and \(h\), using SigmaPlot(tm) 12.5, can be seen. For all CDs, the three kinetic parameters exhibited an increase as a response to raising the temperature up to 80 \({}^{\circ}\)C.  
The highest \(\mathrm{Y_{TP(s)}}\) was achieved with m-\(\beta\)-CD (98.39 mg GAE g\({}^{-1}\) dm) at 80 \({}^{\circ}\)C, and it was only 5.3% lower than that achieved with 60% methanol (Figure 5). Both \(\beta\)-CD and HP-\(\beta\)-CD were significantly less effective, giving \(\mathrm{Y_{TP(s)}}\) 75.85 and 81.93 mg GAE g\({}^{-1}\) dm, respectively. To further assess the impact of temperature, samples obtained at the end of each treatment (180 min) were also assayed for antioxidant activity (Figure 6). In line with \(\mathrm{Y_{TP(s)}}\), extracts obtained with m-\(\beta\)-CD exhibited the highest \(\mathrm{A_{AR}}\) (1112.51 \(\upmu\)mol DPPH g\({}^{-1}\) dm), followed by HP-\(\beta\)-CD (836.17 \(\upmu\)mol DPPH g\({}^{-1}\) dm) and \(\beta\)-CD (824.08 \(\upmu\)mol DPPH g\({}^{-1}\) dm). The results for \(\mathrm{P_{R}}\) were in concurrence, giving corresponding  
Figure 4: Kinetics of polyphenol extraction from _S. fruticosa_, with each of the CDs tested, within a range from 40 to 80 \({}^{\circ}\)C. Extractions were performed under optimized conditions. Assignments: m-\(\beta\)-CD, methyl \(\beta\)-cyclodextrinrin; HP-\(\beta\)-CD, hydroxypropyl \(\beta\)-cyclodextrinrin; \(\beta\)-CD, \(\beta\)-cyclodextrinrin. values of 241.88, 210.72, and 185.74 umol AAE g\({}^{-1}\) dm. This outcome suggested that, using m-\(\beta\)-CD, polyphenol-enriched extracts with improved antioxidant characteristics may be produced at 80 \({}^{\circ}\)C.  
The effect of temperature on \(k\) was better illustrated by establishing correlations between \(k\) and \(T\) (Figure 7). These correlations could be very effectively described using an exponential model [27]: \(k=k_{0}\)  
\[k\ \ =\ \ k_{0}+\mathrm{a}e^{-\mathrm{b}T} \tag{6}\]  
where \(k\) corresponds to the second-order extraction rate and \(k_{0}\) to a pre-exponential factor. In Table 8, the parameters \(k_{0}\), \(\mathrm{a}\), and \(\mathrm{b}\), calculated by SigmaPlot\({}^{\mathrm{TM}}\) 12.5, are given analytically. Extraction with m-\(\beta\)-CD displayed the lowest \(\mathrm{b}\) value, which suggested that it was the least affected by temperature, as opposed to the extraction with HP-\(\beta\)-CD. This finding evidenced that m-\(\beta\)-CD provided the most effective and the least energy-demanding extraction of polyphenols. To ascertain this and obtain a tentative estimation of the barriers required for the extraction with each CD tested, the activation energy was determined as follows [28]:  
\[ln(\frac{k_{ref}}{k})\ =\ \left(-\frac{E_{a}}{R}\right)\left(\frac{1}{T}-\frac{1 }{T_{ref}}\right) \tag{7}\]  
where \(T_{\mathrm{ref}}\) was chosen as the mean temperature of testing (60 \({}^{\circ}\)C) and \(T\) = 40 \({}^{\circ}\)C. \(k_{\mathrm{ref}}\) and \(k\) were the corresponding second-order extraction rate constants. \(E_{\mathrm{a}}\) is the activation energy (J mol\({}^{-1}\)) and \(R\) the universal gas constant (8.314 J K\({}^{-1}\) mol\({}^{-1}\)).