## 3 Results and Discussion
### Numerical Characterization of Electrothermal Induced Convection
#### 3.1.1 Standing-Wave Electrothermal Fluid Motion
When the field frequency equals f\({}_{\text{ACET}}\) (the critical frequency for flow reversal in SWET, f\({}_{\text{ACET}}\), is \(\sqrt{11}\) times the Debye relaxation frequency f\({}_{\text{MW}}\) = \(\sigma/2\pi\epsilon\)), as shown in Figure 2D, the flow velocity originated by SWET decays dramatically, and it is well known that this serves as the critical frequency point beyond which the in-phase electrothermal streaming reverses in flow direction, and the flow velocity reaches a global minimum as indicated by the frequency-flow velocity relationship diagram in Figure 3. When the imposed signal frequency further increases and attains 10 GHz exceeding well the value of f\({}_{\text{ACET}}\), the induced free charge disappears and only the polarized bound charge exists due to a special charge relaxation process, and the electrothermal fluid motion is dominated by dielectric force acting on a permittivity gradient, wherein the fluid medium of lower polarizability (lower permittivity due to a higher temperature elevation at electrode edges) is repelled to the region of a weaker electric field strength (top of the electrode spacing), as evidenced by a high-frequency flow reversal (Figure 2E) compared to the low-frequency situation (Figure 2C). In addition, the SWET flow velocity at 10 GHz under evident charge relaxation is much smaller than that at 100 kHz. Through the previous qualitative analysis of standing-wave induced electrothermal convection, it can be seen that the SWET flow rate above the center of each electrode unit is basically zero due to the counteraction of two symmetric electrokinetic vortices in opposite streaming directions. At the same time, if the solution contains biomolecules, they will be concentrated in the region of slow flow rate on top of the electrodes under SWET convective mixing, which greatly improves the probability of the active immunoreaction between the free antigen analyte and antibody probe fixed on functionalized metal surfaces. In order to simplify the work of the parameter optimization process in the calibration experiments and to explore the influence of different experimental parameters on the SWET bulk convection and interfacial immunoreaction kinetics, it is necessary to perform a parametric scanning of the field frequency of the imposed SW signal to preliminarily determine the specific frequency range within which the optimal device performance is located. Figure 2F,G shows the flow rate and vorticity magnitude of SWET as a function of the SW signal frequency fsw under a constant driving voltage, respectively. As the frequency is scanned from 10\({}^{3}\) to 10\({}^{11}\) Hz, it can be seen from Figure 3 that the flow rate and vorticity both adopt large values in the low frequency conductivity plateau for field frequency much lower than the critical relaxation frequency f = \(\sqrt{11}\)f\({}_{\text{MW}}\). However, when the frequency increases approaches it, the flow velocity and vortex vorticity drop rapidly to a minimum value, then  
reverse in variation trend, grow with frequency, and maintain a stable permittivity plateau in the high frequency range for frequencies exceeding well f\({}_{\text{ACET}}\). Based on the above simulation results, when the frequency of the applied SW voltage signal is between 1 kHz and 10 MHz, the flow rate due to the action of SWET has a relatively high level, and the inhomogeneous bulk fluid is mainly subjected to Coulomb force acting on the induced free charge, which complies with the polarization trait of enhanced microfluidic immunoassay driven by active electrothermal convection. Figure 3: Simulation analysis of various field variables in a TWET pump device with the bottom electrode array subjected to a four-phase TW signal of a given voltage amplitude V\({}_{\text{TW}}\) = 6 V. (**A,B**) A surface plot of the zeroth-order electrostatic potential and an arrow plot of the zeroth-order electric field for real (**A**) and imaginary (**B**) parts of the TW field with a spatially varying phase independent of the applied TW field frequency.