## 3 Results and Discussion
### Effect of the Applied Voltage  
Figure 5 illustrates the effect of electrodes angles (\(\theta\), \(\alpha\)) on the transient evolution of CRP complex concentration for V\({}_{\rm rms}\) = 15 V and u\({}_{\rm ave}\) = 100 \(\upmu\)m/s. Four values of the angle \(\theta\) = \(\alpha\) (namely, 40, 80, 120, and 160\({}^{\circ}\)) were considered in order to study the effect of these angles on the biosensor response. The results are also compared to those obtained in the case without the electrothermal effect (V\({}_{\rm rms}\) = 0 V). As expected, this figure reveals that when the electrodes angle increases, the electrothermal effect becomes more important and subsequently there is an improvement of the reaction rate. Table 5 illustrates the initial slopes for association and dissociation binding reaction corresponding to four values of the electrode aperture angles \(\theta\) and \(\alpha\). The initial slope was calculated by choosing few points near the initial time of each phase (i.e., near \(t\) = 0 s for the association phase and near \(t\) = 800 s for the dissociation phase). Then, a linear interpolation based on the least square method was used to compute the slope. The slope of the curves gives the binding rate and is related to the response time of the biosensor. The enhancement factors of these initial slopes are also presented. It is clear that the improvement of the binding reaction is significant during both the association phase and the dissociation phase. The most significant enhancement factor for the adsorption and desorption of CRP protein pair under 15 V was obtained for \(\alpha\) = \(\theta\) = 160\({}^{\circ}\). Figure 5: Temporal evolution of the average surface complex concentration for different values of the electrode aperture angles with applied voltage V\({}_{\rm rms}\)= 15 V. Four values of the angle \(\theta\) = \(\alpha\) (namely, 40, 80, 120, and 160\({}^{\circ}\)) are considered. The average inlet velocity is fixed at 100 \(\upmu\)m/s. The initial slopes depend not only on the electrode aperture angles but also on the applied voltage. Figure 6 presents the three-dimensional surface plot of the slope at the origin of time related to the association and dissociation stages versus the applied voltage and the electrode aperture angle for a fixed value of the average inlet velocity \(\mathrm{u_{ave}}=100~{}\upmu\)m/s. The large value of the initial slope was achieved for the largest applied voltage and for the largest electrode aperture angle. Figure 7 illustrates the temperature rise versus the applied voltage and electrode aperture angle for \(\mathrm{u_{ave}}=100~{}\upmu\)m/s. From this figure, it is noted that the temperature rise is strongly influenced by the applied voltage. However, the electrode aperture angle has a small and linear effect on the temperature rise. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \begin{tabular}{c} **Initial Slope** \\ **(Association) \(\times\) 10\({}^{-11}\)** \\ \end{tabular} & \begin{tabular}{c} **Initial Slope** \\ **(Dissociation) \(\times\) 10\({}^{-11}\)** \\ \end{tabular} & \
\begin{tabular}{c} **Enhancement Factor** \\ **(Association)** \\ \end{tabular} &
\begin{tabular}{c} **Enhancement Factor** \\ **(Dissociation)** \\ \end{tabular} \\ \hline \(\alpha=\theta=40^{\circ}\) & 4.37 & 3.37 & 1.22 & 1.14 \\ \(\alpha=\theta=80^{\circ}\) & 4.85 & 3.66 & 1.36 & 1.24 \\ \(\alpha=\theta=120^{\circ}\) & 4.43 & 3.98 & 1.52 & 1.35 \\ \(\alpha=\theta=160^{\circ}\) & 5.90 & 4.25 & 1.65 & 1.44 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Initial slopes and enhancement factors of the binding reactions under 15 V for different values of the electrode aperture angles. Figure 6: (**a**) Surface plot of initial slope related to the association and (**b**) dissociation stages versus applied voltage \(\mathrm{V_{rms}}\) and the electrode aperture angle \(\alpha=0\). The average inlet velocity is fixed at 100 \(\upmu\)m/s. The initial slope for the both phases varies linearly with the applied voltage.