Arjun P. Sudarsan and Victor M. Ugaz*
Department of Chemical Engineering, Texas A&M University, College Station, Texas 77843-3122, USA. E-mail: ugaz@tamu.edu; Fax: +1 (979) 845-6446; Tel: +1 (979) 458-1002
First published on 10th November 2005
Mixing of fluids at the microscale poses a variety of challenges, many of which arise from the fact that molecular diffusion is the dominant transport mechanism in the laminar flow regime. While considerable progress has been made toward developing strategies to achieve improved mixing in microfluidic systems, many of these techniques introduce additional complexity to device fabrication and/or operation processes. In this work, we explore the use of compact spiral-shaped flow geometries designed to achieve efficient mixing in a format that can be constructed using a single planar soft lithography step without the need for multilayer alignment. A series of 150 µm-wide by 29 µm-tall channels were constructed, each of which incorporated a series of spiral shaped sections arrayed along the flow path. Five spiral designs with varying channel lengths were investigated, and mixing studies were carried out at flow rates corresponding to Reynolds numbers ranging from 0.02 to 18.6. Under appropriate conditions, transverse Dean flows are induced that augment diffusive transport and promote enhanced mixing in considerably shorter downstream distances as compared with conventional planar straight channel designs. Mixing efficiency can be further enhanced by incorporating expansion vortex effects via abrupt changes in cross-sectional area along the flow path.
Typically, flow in micro-scale conduits is laminar with Reynolds numbers well below the threshold for turbulence (Re = Vd/v < 100, where V is the average flow velocity, d is the characteristic cross-sectional channel dimension and v is the kinematic viscosity of the fluid), leaving molecular diffusion as the predominant driving force for mixing to occur. Furthermore, values of the Péclet number are relatively high in microchannels (Pe = Vd/Dmol > 100, where Dmol is the molecular diffusivity), indicating that diffusive mixing occurs at a much slower rate than the timescales associated with fluid motion. These factors combine to produce characteristic mixing lengths (Δym ∼ V × (d2/D) = Pe × d) on the order of several centimeters, resulting in the need to employ cumbersomely long channels in order to achieve complete mixing. As a consequence of these limitations, considerable effort has been directed toward the development of strategies to achieve rapid laminar flow mixing in microfluidic systems.15–17
In general, mixing strategies can be classified as either active or passive, depending on the operational mechanism. Active mixers employ external forces, beyond the energy associated with the flow, in order to perform mixing. Some examples of techniques developed to accomplish this include electro-osmosis,18 magnetic stirring,19 bubble-induced acoustic actuation20 and ultrasonic effects.21 While generally effective, these designs are often not easy to integrate with other microfluidic components and typically add substantial complexity to the fabrication process. Moreover, since high electric fields, mechanical shearing, or generation of nontrivial amounts of heat are involved, they are not well suited for use with sensitive species (e.g., biological samples).
Passive mixers, on the other hand, avoid these problems by exploiting characteristics of specific flow fields to mix species without application of external electrical or mechanical forces. These designs are also often more straightforward to build and interface with other components. Passive micromixers can be broadly sub-classified into designs based on lamination and rotation techniques. Lamination-based mixers rely on the concept of repeated inter-layering of multiple parallel streams (a so-called split-and-recombine effect) in order to increase the interfacial area between species and accelerate the overall diffusive transport process. Both serial and parallel variations of these mixers have been investigated. Bessoth et al.22 demonstrated parallel lamination by splitting two species into sixteen streams on opposite sides of a silicon wafer, then bringing them together as alternating lamellae in a central zone where diffusive mixing occurred. The idea of serial lamination mixing, where the streams to be mixed are separated from a common inlet channel, was suggested in early work by Schwesinger et al.23 and Branebjerg et al.24 Usually, the streams are first directed in a horizontal conduit after which one or both of the streams are twisted vertically to initiate separation, followed by a sequence of turns to finally rejoin the fluids and create alternating lamellae. In reality however, if the fluid is subject to sudden changes in direction, distortion of the laminar profile is likely to occur making it extremely difficult to achieve multiple lamellae possessing identical cross-sectional profiles. In order to minimize these secondary flow effects, Schönfeld et al. have designed a split-and-recombine mixer incorporating minimal channel curvature capable of generating nearly ideal lamellar profiles.25 While these lamination mixers have been highly successful, the complex 3-D flow geometries associated with most of these designs require cumbersome fabrication and assembly steps. Moreover, the footprints occupied by these channels do not always make them conducive for integration into “on-chip” systems.
Passive rotation mixers that are designed to generate transverse flows across the channel cross-section have also been investigated. Johnson et al.26 and Stoock et al.27 have demonstrated this by pattering the floor of a microchannel with oblique grooves to generate lateral flows that repeatedly stretch and fold fluid segments across the channel cross-section. In these designs faster chaotic mixing is achieved at lower Re. Howell et al.28 showed a modification of this basic design, which involves patterning ridges on the top and bottom of the channel. In this mixer, combined transverse flows generated at the top and bottom of the channel contributed to mixing. A 3-D serpentine geometry with recurring “C-shaped” units has been demonstrated by Liu et al.29 The efficiency of this mixer increases with Re due to presence of eddies formed at the channel bends. Kim et al.30 demonstrated a mixer having triangular projections on the channel walls that induce stretching and folding of the fluid segments. A mixer with combined effects of splitting-and-recombining and advection has been realized in a micromixer comprised of a series of “F-shaped” mixing elements.31
Three-dimensional passive rotation using a breakup process has also been shown to be efficient for mixing.32 While this mixer design is effective for Re ranging from 1–50, its efficiency is highest at Re = 10 due to the combined influence of stretching and folding, breakup and diffusion processes. At lower flow rates (Re = 1) stretching and folding of the interface is absent and mixing is breakup and diffusion limited. At higher flow rates (Re = 50) diffusion is absent and fluids are mixed via stretching and folding and breakup processes. Other examples of mixing techniques that have recently been introduced involve self-circulation in a mixing chamber that is efficient at high flow rates (Re ≥ 50),33 an in-plane passive micromixer with modified telsa structures that employed the “Coanda effect” to improve mixing,34 and a mixer based on surface tension in a geometrical mixing chamber.35 Ismagilov et al.3,36 have developed a technique where multiple fluids are mixed by recirculation inside confined liquid droplets that are dispersed in an immiscible liquid. Günther et al.37 have shown that liquids can be mixed by recirculation that is associated with the introduction of a gas phase that forms a segmented gas–liquid slug flow.
Mixing in more complex 3-D flow networks has also been investigated. For example, micro-stereolithography has been used to construct mixing geometries mimicking conventional macroscale static mixers.38 Another example involves construction of 3-D microvascular networks incorporating arrays of square-spiral towers via direct-write assembly.39 The resulting design consisted of square-spiral towers embedded within a 3-D network and promoted fluid mixing through chaotic advection. Mixing efficiency in this case exhibited an exponential dependence on Re. A barrier embedded Kenics micro-mixer that operates using combined stretching/folding and splitting/reorientation processes has also been fabricated using micro-stereolithography.40 Although these mixing designs have proven to be effective, many of their highly intricate 3-D structures require timescales on the order of hours to days to construct, often with expensive specialized equipment.
Transverse Dean flows that arise as a result of centrifugal effects experienced by fluids traveling along a curved trajectory offer the attractive possibility of providing enhanced mixing in an easily fabricated planar 2-D format by simply introducing curvature to the flow path. The magnitude of these effects is characterized by the dimensionless Dean number (κ = δ0.5Re, where δ is the ratio of the channel hydrodynamic radius to the flow path radius of curvature), which expresses the ratio of inertial and centrifugal forces to viscous forces.41 These centrifugal effects induce a secondary flow field characterized by the presence of two counter-rotating vortices located above and below the plane of symmetry of the channel, coinciding with its plane of curvature (Fig. 1). Additional vortex pairs are known to appear at higher Dean numbers, and recent studies have shown that enhanced mixing can occur in these flows.42,43 However, the corresponding Re in these experiments is fairly large (Re ≫ 100) and outside the range of conditions that are realistically achievable in most microfluidic systems.
Fig. 1 Illustration of Dean flow effects in a curved microchannel (‘i’ and ‘o’ denote the inner and outer channel walls respectively). The transverse flow field is characterized by the presence of two counter-rotating vortices located above and below the channel midplane. At higher κ, the combined action of these vortices cause two fluid streams to almost completely switch positions within the channel. |
Dean effects have been investigated to study mixing in serpentine-like microchannels.39,44,45 As fluid flows downstream in a curved channel, centrifugal effects pull the fluid that is closer to the inner wall radially along the midplane towards the outer wall. Simultaneously, fluid that is closer to the outer wall is swept inwards along the channel walls. Ultimately, a nearly complete 180° rotation is induced above and below the midplane that causes the two fluids to almost completely switch positions within the channel (Fig. 1; velocity and concentration profiles were obtained from the analytical solution to a first order perturbation expansion of the equations of motion in a channel of circular cross-section for the ideal case of two immiscible species46). Unfortunately, in serpentine channels that are made up of opposing curved segments, this effect is reversed as the fluids flow from one bend to the other, and this cycle continues along the entire length of the channel. Consequently, the interface between the two fluids simply undulates between the channel walls without achieving appreciable mixing. One way of overcoming this flow reversal problem is to design a channel such that the transverse secondary flows are sustained over longer distances. Howell et al.47 have shown this to be possible in wide (>1 mm) spiral channels and Vanka et al.48 have studied this effect in a 793 µm-wide spiral channel at a flow rate corresponding to Re = 6.8. In such channels the fluid experiences a reduction in the channel radius of curvature as it flows downstream, accompanied by a corresponding increase in the strength of the transverse secondary flow.49
In this work, we present a study of fluid mixing in 150 µm-wide, 29 µm-tall spiral microchannels. Five different designs with varying channel lengths are investigated and flow studies are carried out for a wide range of flow conditions (0.02 ≤ Re ≤ 18.6). Further, we show that by abruptly increasing the cross-sectional area of the channel, expansion vortex effects can be harnessed to rapidly increase the extent of the mixing interface between streams.
Fig. 2 (A) Overview of the printed-circuit board master fabrication process. (B) Soft lithography process for fabrication of planar channels using a melt-processable thermoplastic elastomer from the master molds constructed in (A). |
Microfluidic devices were then fabricated using a melt-processable thermoplastic elastomer that was synthesized by combining commercially available polystyrene–(polyethylene/polybutylene)–polystyrene (SEBS) triblock copolymers (e.g. CP-9000, Kraton-G series) in mineral oil (light mineral oil; Fisher Scientific; Hampton, NH).51 Resin and mineral oil (33 wt% copolymer) were mixed and placed under vacuum overnight at room temperature in order to allow the oil to evenly coat the resin surface. The mixture was then heated to 170 °C under vacuum for 4 h to allow the resin and oil to intermix and to remove any residual air pockets. Finally, the mixture was cooled to room temperature and the solidified gel was cut into smaller pieces and placed on top of the PC board master mold that had been preheated to 120 °C on a hot plate (Fig. 2B). Once the elastomer began to soften, a glass plate was placed on top of the slab and gentle pressure was applied by hand to ensure complete contact with the structures on the mold. After cooling and release, the solidified gel incorporates the shape of the structures on the master. Fluidic access holes were made using a syringe needle, and the molded slab was thermally bonded to a flat surface of the elastomer to form enclosed channel networks.
(1) |
Fig. 3 (A) Schematic of the spiral channel network incorporating three mixing sections. Each section consists of an inlet and outlet spiral joined by a central ‘S’-section. (B) Segment of a spiral channel depicting the trapezoidal cross-sectional profile. (C) Dean number along the spiral contour at different Re. |
Spiral design | Arcs on spiral | Max. radius of curvature | Length of inlet/outlet spiral | Length of mixing section | Footprint of mixing section |
---|---|---|---|---|---|
a All channels are 29 µm tall. | |||||
1 | 2 | 0.52 | 1.47 | 3.97 | 1.2 × 1.0 |
2 | 4 | 0.81 | 3.77 | 8.57 | 1.7 × 1.5 |
3 | 6 | 1.27 | 7.35 | 15.73 | 2.9 × 2.3 |
4 | 8 | 1.98 | 12.96 | 26.95 | 4.4 × 3.6 |
5 | 10 | 3.10 | 21.73 | 44.49 | 6.9 × 5.5 |
The first channel design we tested consisted of spiral sections made up of two arcs in each of the inlet and outlet spirals, with the longer arc having a 520 µm radius of curvature. Fig. 4 shows the mixing intensity for this channel at flow rates corresponding to Re = 0.02–18.6, as compared with straight channels of the same length. At the lowest flow rate, mixing levels of up to 80% can be achieved in under 19 mm. At the highest flow rate investigated, mixing levels of 90% can be achieved at the same distance. Unlike straight channels, the mixing length becomes shorter with increasing Re, as expected based on the fact that the Dean number (and hence the strength of the secondary flow promoting mixing) is directly proportional to Re.
Fig. 4 Mixing intensity within a two-arc, three-section spiral channel as a function of Re (filled symbols). The mixing intensity for a straight channel of equal length is also shown for comparison (open symbols). |
By increasing the length of individual spiral contours, higher levels of mixing can be achieved within the first spiral section (Fig. 5A). The increased length not only provides a longer time for diffusion at slow flow rates, but also helps in sustaining the transverse secondary flow. The strength of the transverse secondary flows is at a maximum in the arcs with the smallest radius of curvature (i.e., in the central region of the spiral), and hence at higher flow rates most mixing is expected to occur in these segments. Fig. 5B–E gives a measure of the mixing intensity along the contour of the first section of the channels (for a four-arc, six-arc, eight-arc and ten-arc designs) and its observed that most mixing occurs at the innermost region of the spiral flow path. Fig. 6 shows the mixing intensity at the end of each section for the four-arc, six-arc, eight-arc and ten-arc spiral channels. In the four-arc channel, mixing levels of 90% are achieved at the end of the second section, whereas in the eight- and ten-arc channels 90% mixing is obtained at the end of the first section.
Fig. 5 (A) Digital color images of the parallel blue and yellow streams entering the first section of an eight-arc spiral and corresponding gray-scale images depicting the evolution of the green mixed interface. Mixing intensity values as a function of Re along the contour of the section is indicated on the image. (B–E) Mixing intensity along the flow path of the first section of four-arc (B), six-arc (C), eight-arc (D) and ten-arc (E) spiral channels. |
Fig. 6 Mixing intensity as a function of Re at the end of each section for the four-arc (A), six-arc (B), eight-arc (C) and ten-arc (D) spiral channels. |
Fig. 7 Gray-scale images depicting expansion of the mixed interface with increasing Re in a two-arc spiral channel incorporating expansion vortex effects. |
This journal is © The Royal Society of Chemistry 2006 |