## 3 Results and Discussion
### Analysis and Discussion of Release Results  
An attempt to quantify the drug release curves and to associate them with the dominant physical mechanisms will be presented here. Let us start from the simplest case, which is the direct release of the drug. The process in this case can be characterized as dissolution. The final concentration of the appreitant is simply its solubility in the liquid C\({}_{\rm eq}\). The drug concentration evolution can be described from the solution of the following simple conservation equation for the drug concentration in liquid:  
\[\rm V\frac{dC}{dt}=AK(C-C_{eq}) \tag{1}\]  
where V is the volume of the fluid, A the sample-fluid interfacial area, and K the overall dissolution rate constant. In general, the dissolution is a two-step process: the first step is the removal of the drug from the solid to the dissolved state, and the second the mass transfer from the solid-liquid interface to the bulk liquid. In principle, the constant K is related to the partial dissolution rate constant k and to the mass transfer coefficient h as K = 1/(1/k + 1/h). However, it is not possible to assess experimentally the relative contributions of mass transfer and dissolution on the release rate. Considering the experimental characteristic release time of thirty minutes and the high mixing rate employed (100 rpm), it can be argued that the dominant step is the dissolution, so K coincides to k. Assuming the initial condition C = 0, the above equation can be integrated to give:  
\[\rm C=C_{eq}(1-exp(\frac{Ak}{V}t)) \tag{2}\]  
The fractional release curve can simply be computed by the ratio CV/m, where m is the mass of the drug, which is the same for all experiments performed in the present work. The above equation is fitted to the experimental data and leads to the following parameter value: AK/V = 8 \(\times\) 10\({}^{-4}\) s\({}^{-1}\). The next system to analyze is the PEO-drug system. In general, PEO undergoes swelling and erosion (dissolution) in the liquid phase. The relative rate of the two processes depends on the PEO chain length (i.e., on the molecular weight of the PEO). In particular, large chains inhibit the erosion of the polymer matrix, so considerable swelling occurs prior to erosion. The behavior of PEO influences the release profile of the drug. In the case of erosion domination, the drug is released from the eroded matrix, so a zero-order release kinetic profile is expected. In the case of swelling domination, the drug has to diffuse through the gel layer created by polymer swelling, so the undergoing release mechanism is diffusion, which can be described (employing an approximating technique like the linear driving force formula [69]) as a first-order process. The experimental release data for the PEO-appreitant system are fitted using both zero- and first-order kinetic models, and it was found that the first-order model can perfectly characterize the data. This means that swelling dominates erosion, which is quite compatible to the high-molecular-weight (2 \(\times\) 10\({}^{6}\)) of the PEO of the present work [70; 71]. The integrated first-order equation for evolution of drug concentration is given as C = C\({}_{\rm F}\)(1 \(-\) exp(\(-\)K\({}_{\rm D}\)t)), where C\({}_{\rm F}\) is the final drug concentration in the liquid and KD is the first-order rate coefficient, which is related to the drug diffusivity in a gel slab of thickness L as K\({}_{\rm D}\) = 3D/L\({}^{2}\). The comparison between the experimental and model release curves are shown in Figure 16. It appears that the kinetic constant does not depend on the drug content in the PEO matrix and it takes the value K\({}_{\text{D}}\) = 2 \(\times\) 10\({}^{-3}\) s\({}^{-1}\). The next step is the system modeling in the presence of clay. A first attempt is to try a first-order kinetic model (diffusion through swollen gel layer), found to work in the absence of clay. The fitting procedure was fairly successful (deviation smaller than 2% from the experimental data points) for the case of 10 wt % clay.