The Optimal Control of Geometry and Voltage Parameters on Electrokinetic Transport to Avoid Sample Leakage in Microfluidic Chips

Abstract

Computer simulation is used to investigate sample transport phenomena of cross microfluidic chips. In this study, Kirchhoff circuit theory is employed to calculate the electric field strength and approximate electroosmotic flow. It is apparent from the results that both the simulation and the theoretical data show similar trends in the electroosmosis of cross microchips. The main target in this study is to summarize the optimal controlling parameter values for avoiding sample leakage in the transport process. The effects of the applied voltage ratio, the geometry ratio and the zeta potential were simulated using a computational fluid dynamics and multiphysics solver software package (CFD-ACE+). Under our designed conditions, two major conclusions were reached: (1) for high-voltage ratios, the sample leakage can be avoided as the geometry ratio is large enough at 0.5 or greater, and (2) for small geometries, maintaining a smaller voltage ratio, 0.3 or less, is essential for avoiding sample leakage. The key is to govern the sample velocity in the upstream faster than that in the downstream. Although real experimental conditions can be further fine tuned under microscopy monitoring, these conclusions are helpful to design the proper channel geometry and set up suitable voltage parameters to avoid sample leakages in one cross-channel chip.

Appendix: The Kirchhoff Theory Model

In the cross-channel conduits in Fig. 1, the electric current in each channel follows the relationship

$$ I_{1} = 2I_{2} + I_{3} $$

(6)

where I ₁ is the current in the upper port of the vertical channel, I ₂ is the current in both side channels, and I ₃ is the current in the lower port of the vertical channel.

In a channel, the electric resistance (R ^*) is proportional to the channel length L and inversely proportional to the cross-sectional area (A) of channel. In Eq. 7, the constant ρ is the inverse of the bulk liquid conductivity κ.

$$ R^* = {\frac{\rho \times L}{A}} $$

(7)

The above equation can be rearranged to solve for the electrical potential of the downstream after the intersection. The cross biochip requires four electrodes to control the potential in the separation process. To calculate the potential at the cross part (V _cross),

$$ V_{\text{cross}} = {\frac{{\left( {V_{1} \times L_{2} \times L_{3} + 2L_{1} \times L_{3} \times R} \right)}}{{\left( {L_{2} \times L_{3} + 2L_{1} \times L_{3} \times R + L{}_{1} \times L_{2} } \right)}}} $$

(8)

$$ {\frac{\Updelta V}{{L_{3} }}} = {\frac{{V_{\text{cross}} - V_{3} }}{{L_{3} }}} $$

(9)

$$ E_{\text{steady}} = {\frac{{\left( {V_{1} - V_{\text{cross}} } \right)}}{{L_{1} }}} - {\frac{{2\left( {V_{\text{cross}} - V{}_{2}} \right) \times R}}{{L_{2} }}} $$

(10)

where L ₁ is 0.5 mm, L ₂ is 0.4 mm (left and right), L ₃ is 1.5 mm and R is the channel cross-sectional area ratio A2/A1. When channel depth is not changed, the electric resistance of each channel is dependent on the channel width.

For an infinite rectangular channel, the Helmholtz–Smolouchowski equation was obtained as follows:

$$ V_{s} = \mu_{\text{eo}} E. $$

(11)

In Eq. 5, μ _eo is the electroosmotic mobility

$$ \mu_{\text{eo}} = {\frac{{\varepsilon \varepsilon_{0} \xi }}{\eta }} $$

(12)

where ε is the electric permittivity of the solution, η is the viscosity, and ζ is the zeta potential.

In Eq. 12 under the same zeta potential the electroosmotic mobility (μ _eo) is constant. However, Eq. 10 indicates that the electric field strength changes when the cross-sectional area varies in different channel width ratios. Therefore, according to Eq. 11 the electroosmotic flow velocity (V _s) changes when the channel width ratio varies.