# Encapsulation of Fennel Essential Oil in Calcium Alginate Microbeads via Electrostatic Extrusion
## 3 Results and Discussion
### Swelling Kinetics  
Investigating the swelling behavior of encapsulated systems is highly relevant since the essential feature of encapsulated EOs' release from initially dry hydrogels lies in the transition from immobile molecules within the dry material to their mobility upon swelling of the matrix by water [36]. This process gives rise to what can be termed as \"swelling-controlled\" drug release. In cases where the matrix exhibits sufficient hydrophilicity, the increased mobility of the molecules can reduce resistance to transport within the swollen layer, thereby regulating the rate of release primarily by the rate of water penetration [37]. Understanding which mechanism governs the release is crucial for designing encapsulates with specific release characteristics. For instance, if release is primarily controlled by water penetration into the glassy matrix, modifying the EO's diffusivity within the equilibrium-swollen gel, such as via cross-linking, may have minimal impact [37]. Conversely, if the matrix quickly undergoes penetration and plasticization but then swells slowly, focusing efforts on modifying the rate of secondary swelling would be most advantageous. The simple power law approach used in the present study offers researchers a straightforward method to correlate and assess release data, as it provides a convenient measure of release rate constancy via the parameter \(n\), where values of \(n<0.5\) indicate classical Fickian diffusion of molecules from areas of high concentration to areas of low concentration. Values of \(n\) above 1 indicate the super case II transport mechanism, often referred to as relaxation-controlled transport, as it is attributed to structural changes in polymers due to penetration of water and subsequent plasticization. The range of behavior with \(n\) values falling between 0.5 and 1 is often referred to as viscoelastic or anomalous transport [37]. The constants \(k\) and \(n\) represent parameters specific to the polymer/bioactive agent/dissolution medium system under consideration. Optimal kinetic behavior in swelling-controlled release systems is typically associated with \(n=1.0\), indicating time-independent drug release, which is highly desirable. The swelling kinetics of the freeze-dried fennel EO alginate microbeads, which had significantly higher encapsulation efficiency and loading capacity, were calculated by fitting to a simple power-law model, which, according to the high correlation coefficient R\({}^{2}\) values, showed good correlation with the experimental data. The results are shown in Figure 2, while the values of \(k\) and \(n\) obtained from data fitting are shown in Table 4.  
\(k=\) constant specific to the gel network; \(n=\) diffusion exponent; R\({}^{2}=\) correlation coefficient. \begin{table}
\begin{tabular}{c c c c c} \hline
**Alginate (\%, \(w\)/\(v\))** & **Whey Protein (\%, \(w\)/\(v\))** & \(k\) & \(n\) & \(\mathbf{R^{2}}\) \\ \hline \multirow{2}{*}{1} & 0 & 0.3743 & 0.1957 & 0.9605 \\  & 0.75 & 0.0583 & 0.5675 & 0.9188 \\  & 1.5 & \(3.22\times 10^{-5}\) & 1.9704 & 0.9358 \\ \hline \multirow{2}{*}{1.5} & 0 & 0.4129 & 0.1678 & 0.9803 \\  & 0.75 & 0.1572 & 0.3549 & 0.9715 \\  & 1.5 & \(4.36\times 10^{-5}\) & 1.9737 & 0.9776 \\ \hline \multirow{2}{*}{2} & 0 & 0.0007 & 1.425509 & 0.9914 \\  & 0.75 & 0.2918 & 0.235893 & 0.9722 \\ \cline{1-1}  & 1.5 & 0.0004 & 1.4777 & 0.9436 \\ \hline \end{tabular}
\end{table} 
Table 4: The kinetic parameters for the swelling kinetics of fennel EO alginate beads correlated using the power law model. Figure 2: Plots of fraction of fennel EO alginate beads swelling in water versus time and modeling for the correlation of kinetic parameters: (**a**) An amount of 1.0% alginate; (**b**) An amount of 1.5% alginate; (**c**) An amount of 2% alginate. Data are expressed as mean \(\pm\) standard deviation. Symbols: experimental data. Lines: modeling results. (**m**) = model.