Rapid and Efficient Mixing in a Slip-Driven Three-Dimensional Flow in a Rectangular Channel J. Rafael Pachecoa* Kang Ping Chena and Mark A. Hayesb aDepartment of Mechanical and Aerospace Engineering Arizona State University Tempe, AZ 85287-6106 bChemistry and Biochemistry Department & The Center for Solid State Electronics Research Arizona State University, PO Box 871604 Tempe, AZ 85287-1604 Abstract A method for generating mixing in an electro-osmotic flow of an electrolytic solution in a three-dimensional channel is proposed. When the width-to-height aspect ratio of the channel cross-section is large, mixing of a blob of a solute in a slip-driven three-dimensional flow in a rectangular channel can be used to model and assess the effectiveness of this method. It is demonstrated through numerical simulations that under certain operating conditions, rapid and efficient mixing can be achieved. Future investigation will include the solution of the exact equations and experimentation. Keywrods: Mixing; Electroosmotic flow; low Reynolds number flow; * Corresponding author. Tel. (480) 965-5604; Fax: (480) 965-1384; Email: rpacheco@ asu.edu 11. Introduction Enhanced mixing plays an important role in biological and chemical analysis in microfluidic systems. The Reynolds number, which measures the importance of the inertia force relative to the viscous force, is very small for flows in a microfluidic device due to small dimensions of the device. Conventional methods used in generating mixing in macro-scale fluid flows require sufficiently large Reynolds numbers and they become ineffective when applied to micro-scale flows. Thus, the search for effective mixing mechanisms suitable for micro-scale flows is becoming an active area of research. Current strategies proposed to enhance mixing in a microchannel can be classified as passive methods and active methods. Passive methods (also known as static mixers) do not require external forces, except for those to deliver the fluid, and the mixing process relies entirely on diffusion or chaotic advection (Stone, Strook & Ajdari 2004; Nguyen & Wu 2005). For example, oriented grooves on one wall of the channel have been used to generate transverse flow to enhance mixing. Representative work in this area includes Johnson & Locascio (2002) for electro-osmotic flows and Strook et al (2002a, 2002b) for pressure-driven flows. Nguyen & Wu (2005) provided a comprehensive classification of passive mixers. Active methods use disturbances generated by an external field for the mixing process. Active mixers are categorized according to the type of external disturbances created in the flow, such as pressure (Deshmukh et al 2000); electrohydrodynamics (El Moctar et al 2003); dielectrophoretics (Deval et al 2002); electrokinetics (Oddy et al 2001; Jacobson et al 1991; Tang et al 2002); magnetohydrodynamics (Bau et al 2001; Yi et al 2002a; 2002b); and acoustics 2(Moroney et al 1991; Zhu & Kim 1997). These active micromixers need external power sources for its operation and the structures may require complex fabrication processes. Integration into a microfluidic system can also be challenging and expensive. Some of these strategies mentioned above are general and others are useful for mixing in an electrolyte solution only. The work reported in this paper is concerned with mixing in an electrolyte solution. The method proposed here belongs to the category of active mixing based on electrokinetics, and it differs from those cited above in both the geometry employed and in the way the electric field induced transverse flow is generated. Electroosmosis has proven to be an attractive method for transporting and manipulating fluids in micro-devices. When an electrolyte solution comes into contact with a solid surface, the surface will in general acquire a surface electric charge and counterions accumulate in a thin layer adjacent to the solid surface. When an external electric field is applied, the counterions in this thin electric double-layer (EDL) are set into motion and the viscous force drags the fluid beneath into motion. In many applications, the EDL is very thin. Then electroosmotic flow in a two-dimensional channel can be modeled by specifying a “slip velocity” at the solid wall. The slip velocity is related to the strength of the electric field and the so-called zeta-potential which is the static electric potential difference across the EDL. A characteristic of a two-dimensional electroosmotic flow is that it is essentially a plug flow outside of the EDL when both the top and the bottom walls have the same electric charge. The velocity of the plug flow is independent of the dimension of the channel. This provides the advantage of easy transport of fluids. On the other hand, mixing in this plug-like electroosmotic flow is not efficient. This severely limits the application of electroosmotic micro-devices for rapid 3diagnosis, since rapid diagnosis requires rapid mixing of samples with reagents, and the reagents used in typical applications possess relatively low diffusivity. Several methods have been proposed for enhanced mixing in electroosmotic flows of electrolytc solutions in micro-channels. These include using periodic and time-dependent surface charge for a two-dimensional channel (Ajdar, 1995; Qian & Bau, 2002), and using patterned surface charge to induce more complicated multidirectional electroosmotic flows for a three-dimensional channel with infinite transverse span (Stroock et al. 2000, 2001). In Stroock et al. (2000), the external electric field is applied along the longitudinal direction of the channel, and periodic, step-like surface charge variation is imposed in either the longitudinal direction or the transverse direction. These surface charge variations generate either multidirectional flows or recirculating cellular flows. The channel discussed in Stroock et al. (2000) is unbounded in both the longitudinal and transverse directions. Qian & Bau (2002) considered a two-dimensional channel, and the surface charge varies periodically in the flow direction. An alternative method for generating efficient mixing in electroosmotic flows in a long but laterally confined three-dimensional micro-channel is proposed here. Instead of varying charges on the surface walls in space and/or time (Stroock et al. 2000; Qian & Bau 2002), which may not be easy to implement in practice, a secondary time-dependent external