, Sudarsan and Ugaz. 10.1073/pnas.0507976103. |
Supporting Figure 6
Supporting Figure 7
Supporting Text
Supporting Figure 8
Supporting Movie 1
Supporting Table 1
Supporting Figure 9
Fig. 6.
P-SAR micromixer incorporating two split streams. (A) Schematic of the microchannel geometry investigated (200-mm wide; 29-mm tall; 630-mm radius of curvature; Re and k computed based the 200-mm-wide segment; widths of the two split streams are chosen to maintain an equal pressure drop in each leg by compensating for their difference in length). (B) Confocal cross-sectional image taken at the entrance to the first curved channel segment (position i in A). (C) Confocal cross-sectional images of the flow inside each of the split microchannels taken at flow rates ranging from 7.3 < Re < 43.6 (2.0 < k < 12.2) at a distance of 3 mm downstream from the channel entrance (position iii in A). (D) Confocal images taken at the location where the two split streams have rejoined (position iv in A). Distinct lamellar zones are observed in the vicinity of 4.1 < k < 10.1. (E) Top view images of two aqueous streams labeled with blue and yellow food dye. At k = 2.0 the streams flow in parallel along the entire length of the channel, whereas at k = 10.1 the blue stream is transported from the inner to the outer wall, as indicated by the green color immediately upstream of the split followed by the reemergence of a blue layer at the outer wall within each individual split stream. Alternating blue, blue/green, and yellow striations are visible in the flow after the streams have rejoined, in agreement with the confocal images in D.Fig. 7.
Top view image sequence of aqueous streams (the inner stream is labeled with blue tracer dye) flowing through a curved microchannel segment (200-mm wide; 29-mm tall; 937-mm radius of curvature; i and o denote the inner and outer channel walls, respectively). The blue stream initially occupies 50% of the cross-section upon entering the curved segment, and the downstream distance where transverse Dean effects pull the blue stream outward to occupy 80% of the channel cross section (L80) is determined at each value of (and Re) by analysis of digitized images.Fig. 8.
(A) Illustration of microchannel geometry required to generate alternating lamellae of fluid species using a conventional 3D SAR micromixer design [adapted from Erbacher et al. (1)]. (B) Schematic of 2D P-SAR microchannel geometry with corresponding branching length Lb indicated. (C) Comparison of total mixing time vs. number of splits for the conventional SAR (▲) and P-SAR (<) micromixer designs shown in A and B, respectively. For the P-SAR design, dc is taken as the hydraulic diameter (54 µm) and Lb = 4 mm. For the conventional SAR design dc = 600 µm and Lb = 10 mm from the example given in ref. 1. The diffusion coefficient is taken to be D = 10–5 cm2/s.1. Erbacher, C., Bessoth, F. G., Busch, M., Verpoorte, E. & Manz, A. (1999) Mikrochim. Acta 131, 19–24.
Fig. 9.
Comparison between the operating ranges of the ASM and SHM micromixer designs for the case of an aqueous working fluid. The flow rate dependence of the downstream distance required to mix liquid streams to a level of 80% is shown for both designs (the open symbol indicates that this mixing length was estimated by extrapolation from Fig. 5A).Movie 1.
Multivortex effects in a microchannel. Parallel aqueous streams of blue and yellow dye are injected into a curved microchannel segment (29-mm tall; 630-mm radius of curvature) incorporating a sudden expansion from 100 to 500 mm in width at a distance of 1.5 mm downstream (Re computed based on the 100-mm-wide segment). As Re is increased, transverse Dean vortices (vertical plane) transport the inner stream toward the outer wall while expansion vortices (horizontal plane) simultaneously develop on either side of the entrance to the expansion. The flow direction is from the lower right to the upper left, and the video frame rate has been accelerated to 5´ actual speed.Table 1. Flow rates computed for the SHM micromixer for the case of an aqueous working fluid
Pe * | V , cm/s | Reh | V aq, cm/s | Q aq, ml/min |
2 ´ 103 | 4 ´ 10–3 | 8.52 ´ 10–5 | 7.17 ´ 10–5 | 7.32 ´ 10–7 |
2 ´ 104 | 4 ´ 10–2 | 8.52 ´ 10–4 | 7.17 ´ 10–4 | 7.32 ´ 10–6 |
2 ´ 105 | 4 ´ 10–1 | 8.52 ´ 10–3 | 7.17 ´ 10–3 | 7.32 ´ 10–5 |
9 ´ 105 | 1.8 | 3.83 ´ 10–2 | 3.23 ´ 10–2 | 3.29 ´ 10–4 |
*
Li, C. & Chen, T. (2005) Sensors Actuators B 106, 871–877.Supporting Text
Planar Split-and-Recombine (P-SAR) Mixing Time.
Conventional split-and-recombine (SAR) micromixers typically rely on 3D microchannel networks to divide (split), redirect, and subsequently reassemble (recombine) liquid streams to generate lamellae of alternating species (e.g., Fig. 2A Inset). In these configurations, the total time, t, required to mix two inlet streams can be expressed by an equation of the form ,
where n is the number of split streams produced from each primary inlet stream, Lb is the length required for branching (Fig 8 A and B), dc is the diameter of the channel before and after the branching, and D is the diffusion coefficient (1). The first term on the right-hand side of the equation represents the time required for redirecting the fluid flow from the point where the primary stream is split to the point where the individual split streams recombine. The second term on the right-hand side corresponds to the time associated with diffusive interspecies transport across lamellae upon recombination. In a conventional SAR micromixer (Fig. 8A), the time associated with fluid branching increases linearly with the number of splits because of the corresponding increase in total cross-sectional area. In the P-SAR design (Fig. 8B), however, the branching time is essentially independent of the number of splits (we say "essentially" because there is some variation in path length between the outermost and innermost split streams; we have used the contour length of the outermost split stream in calculations to be conservative). Based on this analysis, a comparison of total mixing time as a function of the number of splits shows that there is an optimal number of splits in the conventional SAR design reflecting the balance between branching and diffusive timescales, whereas in the P-SAR design the mixing time continuously decreases with increasing number of splits (Fig. 8C).
Asymmetric Serpentine Mixer (ASM) Effective Flow Rate Range.
To make a direct comparison between the ASM and the staggered herringbone mixer (SHM) design of Stroock et al. (2), we must (i) determine the corresponding flow rates for the SHM in the case of an aqueous working fluid, and (ii) evaluate flow parameters (e.g., Reynolds and Péclet numbers) with respect to the channel hydraulic diameter. From the operating range of 2 ´ 103 < Pe < 9 ´ 105 given in figure 3c of Stroock et al. (2), we determined the corresponding flow velocities from V = PeD/d [where D = 2 ´ 10–8 cm2/s for a fluorescently labeled polymer in a 80% glycerol/20% water solution and d = 0.01 cm (3)]. Using these velocity values, we then computed equivalent Reynolds numbers based on the channel hydraulic diameter under the same operating conditions using Reh = dhV/v (where dh = 0.0119 cm is the hydraulic diameter and v = 0.56 cm2/s is the kinematic viscosity of the glycerol medium). Next, flow velocities corresponding to an aqueous working fluid at the same Reynolds number were determined for the SHM from Vaq = Rehµaq/dhraq (where µaq = 0.01 g/cm•s and = 0.9982 g/cm3). These velocities then were used to compute corresponding flow rates from Qaq = VaqA (where A = 0.00017 cm2 is the channel cross-sectional area). These values are summarized in Table 1, and the flow conditions needed to achieve a given mixing length using the SHM and ASM designs are compared in Fig. 9. From this analysis, it can be seen that based on fluid properties of water at room temperature, the corresponding flow rates required to achieve a level of 80% mixing in a downstream distance of »7 mm are Q ³ 10–1 ml/min for the ASM design and Q £ 10–6 ml/min for the SHM. Note that the use of tracers with different diffusion coefficients means that although the values of Pe are similar in both the ASM and SHM systems, the values of the Schmidt number (Sc = n/D) are quite different (»103 in our experiments vs. »107 in ref. 2).1. Erbacher, C., Bessoth, F. G., Busch, M., Verpoorte, E. & Manz, A. (1999) Mikrochim. Acta 131, 19
–24.2.
Stroock, A. D., Dertinger, S. K. W., Ajdari, A., Mezic, I., Stone, H. A. & Whitesides, G. M. (2002) Science 295, 647–651.3. Li, C. & Chen, T. (2005) Sensors Actuators B 106, 871–877.