## 2 Biosensor Geometry and Theory
### Governing Equations  
The application of a non-uniform Alternating Current (AC) electric field on a fluid provokes inhomogeneities of the electrical permittivity \(\varepsilon\) and the conductivity \(\sigma\) of the solution due to the temperature gradient subsequent to the Joule effect. These inhomogeneities of \(\varepsilon\) and \(\sigma\) give rise to the electrothermal force given by this expression:  
\[\overset{\rightarrow}{F}_{E}=-\frac{1}{2}\Bigg{\{}\bigg{(}\frac{\
abla \sigma}{\sigma}-\frac{\
abla\varepsilon}{\varepsilon}\bigg{)}\cdot\overset{ \rightarrow}{E}\frac{\varepsilon\overset{\rightarrow}{E}}{1+\omega^{2} \varepsilon^{2}/\sigma^{2}}+\frac{1}{2}|\overset{\rightarrow}{E}|\,\
abla \varepsilon\Bigg{\}} \tag{1}\]  
where \(\omega=2\pi f\) represents the angular frequency of the electric field \(\overset{\rightarrow}{E}\), \(\sigma\) is the electrical conductivity and \(\varepsilon\) is permittivity of the fluid. For aqueous media at \(293\) K, \(\
abla\varepsilon/\varepsilon=-0.04\
abla T\) and \(\
abla\sigma/\sigma=0.02\
abla T\)[24] where \(T\) is temperature of the fluid. The electrothermal flow inside the microfluidic chips is assumed to be steady and laminar. The fluid is Newtonian. The flow is governed by the continuity, Navier-Stokes and the energy balance equations:  
\[\left\{ \begin{array}{l}\
abla\overset{\rightarrow}{U}=0\\ \rho\overset{\rightarrow}{U}\cdot\
abla\overset{\rightarrow}{U}=-\
abla p +\mu\
abla^{2}\overset{\rightarrow}{U}+\overset{\rightarrow}{F}_{E}\\ \rho C_{p}\overset{\rightarrow}{U}\cdot\
abla T=k\
abla^{2}T+\sigma| \overset{\rightarrow}{E}|\,^{2}\end{array} \right. \tag{2}\]  
where \(\overset{\rightarrow}{U}\), \(p\), \(\rho\), \(\mu\), \(k\), and \(C_{p}\) are the velocity vector, pressure, density, dynamic viscosity, thermal conductivity, and specific heat of the fluid, respectively. The transport of the antigen in bulk liquid phase from the left inlet is expressed by the Fick second law involving both conduction and convection:  
\[\frac{\partial C}{\partial t}+\overset{\rightarrow}{U}\cdot\
abla C=D\
abla ^{2}C \tag{3}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline \hline \multicolumn{1}{c}{Positive electrode radius \(R_{EL,int}\) (\(\mu\)m)} & 10 \\ \multicolumn{1}{c}{Internal radius of reaction surface \(R_{EL}^{int}\) (\(\mu\)m)} & 15 \\ \multicolumn{1}{c}{External radius of reaction surface \(R_{EL}^{int}\) (\(\mu\)m)} & 22 \\ \multicolumn{1}{c}{Internal radius of the negative electrode \(R_{EL}^{int}\) (\(\mu\)m)} & 25 \\ \multicolumn{1}{c}{External radius of the negative electrode \(R_{EL}^{int}\) (\(\mu\)m)} & 30 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Geometrical parameters for the second biosensor model. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline \hline \multicolumn{1}{c}{Positive electrode radius \(R_{EL,int}\) (\(\mu\)m)} & 10 \\ \multicolumn{1}{c}{Internal radius of reaction surface \(R_{EL}^{int}\) (\(\mu\)m)} & 15 \\ \multicolumn{1}{c}{External radius of reaction surface \(R_{EL}^{int}\) (\(\mu\)m)} & 22 \\ \multicolumn{1}{c}{Internal radius of the negative electrode \(R_{EL}^{int}\) (\(\mu\)m)} & 25 \\ \multicolumn{1}{c}{External radius of the negative electrode \(R_{EL}^{int}\) (\(\mu\)m)} & 30 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Geometrical parameters for type 1 of the biosensor model. Here, \(C\) is the bulk concentration (mol/m\({}^{3}\)), \(D\) is the diffusion coefficient of the antigen, and \(t\) is the time (s). Equation (3) reveals clearly that the analyte concentration in the bulk is affected by the velocity field which is itself influenced by the electrothermal force described in Equation (1). Therefore, the concentration at the biosensor surface will be affected by this force. The reaction between the antibody fixed at the reaction surface and the suspended target analyte (antigen) is modeled by the so-called first-order Langmuir adsorption equation.