## 4 Discussion  
DNA molecules are driven by an external electric field as they translocate through the microchannel system, which is mainly influenced by electrophoretic and electro-osmotic currents. When the motion of the buffer solution and DNA molecules both remain stable, the real velocity of DNA molecules (\(v_{real}\)) is the sum of the velocity of the electro-osmotic flow (\(v_{EOF}\)) and the velocity of the DNA molecules induced by the electrophoretic force (\(v_{EP}\)), as shown in Equation [36; 37]:  
\[v_{EOF}\ =\ \epsilon_{0}\varepsilon_{\gamma}\zeta_{wall}E/\eta \tag{1}\]  
\[v_{EP}\ =\ \epsilon_{0}\varepsilon_{\gamma}\zeta_{DNA}E/\eta \tag{2}\]  
\[v_{real}\ =\ v_{EOF}+v_{EP}\ =\ \epsilon_{0}\varepsilon_{\gamma}E(\zeta_{DNA }+\zeta_{wall})/\eta \tag{3}\]  
where \(E\) is the electric field strength, \(\epsilon_{0}\) is the vacuum permittivity, \(\varepsilon_{r}\) is the relative permittivity, \(\zeta_{wall}\) is the zeta potential of the channel wall, \(\zeta_{DNA}\) is the zeta potential of the surface of the DNA molecule, and \(\eta\) is the fluid viscosity. DNA molecules are negatively charged and thus move in an opposite direction of electrophoresis under the external electric field. In this study, we detected under a bright field that the direction of solution motion in microchannels was opposite to the direction of the electric field, indicating that the direction of electro-osmotic flow was also opposite to the external electric field. That is to say, the electrophoresis of DNA molecules and electro-osmotic flow were in the same direction. From Equation (3), it can be found that the velocity of DNA molecules increases with the strength of the electric field. Considering only the effects of electrophoretic flow and electro-osmotic flow on the dynamic characteristics of DNA molecules under a DC electric field, the velocity of DNA molecules increases with the increase in the electric field, and the dynamic characteristics of DNA molecules are well explained by electric field distribution. To better reveal the dynamic characteristics of DNA molecules in the tapered microchannel system, as shown in Figure 8, the electric field intensity distribution within the tapered microchannel system was simulated by COMSOL Multiphysics software, as shown in Appendix A. Figure 8 shows the electric field distribution within the tapered  
Figure 7: The velocity of a DNA molecule within a tapered microfluidic channel: (**a**) real-time movement images of DNA molecules; (**b**) relationships of \(v_{\rm d}-\rm U\) and \(v_{\rm D}-\rm U\) of DNA molecules (the bars are standard deviation). microfluidic channel system, where \(\theta\) = 30\({}^{\circ}\), U = 20 V as it is in the fabricated chip (shown in Figure 3). From Figure 8, it can be found that (1) the electric field strength inside the large microchannels is uniformly and weak, about 430 V/m, except for a region of reduced electric field strength between the two neighboring tapered microchannels (in the gray dashed box) and (2) that inside the tapered microchannels, the electric field strength increases sharply from 430 V/m to 4.1 \(\times\) 10\({}^{3}\) V/m. Due to the surged electric field and the confinement effect of the tapered microchannel on the fluid, the velocity of DNA molecules increases rapidly inside the tapered channel. Coupled with the uniformly distributed electric field at the exit port of the tapered microchannel, all DNA molecules leave the tapered channel at the same velocity of 117 \(\upmu\)m/s (Figure 5). (3) The electric field strength is uniformly distributed and larger inside the small microchannels by about 4.1 \(\times\) 10\({}^{3}\) V/m. The electro-osmotic and electrophoretic forces remain stable. As a result, the DNA molecules maintain a stable velocity in region III (Figure 5).