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Chaotic Mixer for Microchannels

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Science  25 Jan 2002:
Vol. 295, Issue 5555, pp. 647-651
DOI: 10.1126/science.1066238

Abstract

It is difficult to mix solutions in microchannels. Under typical operating conditions, flows in these channels are laminar—the spontaneous fluctuations of velocity that tend to homogenize fluids in turbulent flows are absent, and molecular diffusion across the channels is slow. We present a passive method for mixing streams of steady pressure-driven flows in microchannels at low Reynolds number. Using this method, the length of the channel required for mixing grows only logarithmically with the Péclet number, and hydrodynamic dispersion along the channel is reduced relative to that in a simple, smooth channel. This method uses bas-relief structures on the floor of the channel that are easily fabricated with commonly used methods of planar lithography.

Microfluidic systems are now widely used in biology and biotechnology; applications include analysis (of DNA and proteins) (1), sorting (of cells) (2), high-throughput screening (3), chemical reactions (4), and transfers of small volumes (1 to 100 nl) of materials (5). Typical uses of microfluidic devices (e.g., chemical analysis in the field) require that these systems be inexpensive and simple to operate; microfluidic components that operate with pressure flow and few moving parts are desirable. Microfluidic designs should also be compatible with the planar, layer-by-layer geometries that are imposed by current, lithography-based techniques of microfabrication. Physically, flows of common liquids at practical pressures in microfluidic channels (typical cross-sectional dimension, l ∼ 100 μm) are characterized by low values of the Reynolds number (Re= Ul/ν < 100, where U is the average flow speed and ν is the kinematic viscosity of the fluid) (6); general strategies for controlling flow in microfluidic devices should not depend on inertial effects, because these only become important for Re ≫ 1.

Mixing of the fluid flowing through microchannels is important in a variety of applications: e.g., in the homogenization of solutions of reagents used in chemical reactions, and in the control of dispersion of material along the direction of Poiseuille flows (7,8). At low Re, in simple channels (i.e., with smooth walls), pressure flows are laminar and uniaxial, so the mixing of material between streams in the flow is purely diffusive. Even on the scale of a microchannel, this diffusive mixing is slow compared with the convection of material along the channel; the Péclet number is large (Pe = Ul/D > 100, where D is the molecular diffusivity) (9). For such uniaxial flows, the distance along the channel that is required for mixing to occur is Δy mU × (l 2/D) = Pe × l (10). This mixing length can be prohibitively long (≫1 cm) and grows linearly with Pe.

To reduce the mixing length there must be transverse components of flow that stretch and fold volumes of fluid over the cross section of the channel. These stirring flows will reduce the mixing length by decreasing the average distance, Δr, over which diffusion must act in the transverse direction to homogenize unmixed volumes. In a steady chaotic flow, the stretching and folding of volumes of the fluid proceed exponentially as a function of the axial distance traveled by the volume: Δr = lexp(−Δy/λ), where the initial transverse distance is taken to be l, and λ is a characteristic length determined by the geometry of trajectories in the chaotic flow (11,12). For large Pe, in a flow that is chaotic over most of its cross section, we expect an important reduction of the mixing length relative to that in an unstirred flow: Δy m ∼ λln(Pe) (13).

Several protocols for mixing based on chaotic flows have been proposed and demonstrated in macroscopic systems (typical dimension >1 cm) (12); mixing on microscales remains difficult. The group of Beebe has demonstrated chaotic stirring in a helical microchannel; in this design, stirring occurs as a result of eddies at the bends in the channel in flows of intermediateRe (i.e., Re > 1) (14). This mixer is complicated to fabricate and inefficient at lowRe (Re < 1). Active mixers require either external, variable-frequency pumps or internal moving components (15). Active mixers require variable-frequency pumps off-chip or moving components on-chip. Here, we present a general strategy for creating transverse flows in microchannels that can be used to induce chaotic stirring at low Re (0 <Re < 100).

To generate transverse flows in microchannels by using a steady axial pressure gradient, we place ridges on the floor of the channel at an oblique angle, θ, with respect to the long axis (ŷ) of the channel (Fig. 1A). This type of structure can be fabricated with two steps of photolithography. We used soft lithographic methods to make the channels in poly(dimethylsiloxane) (16, 17). These ridges—similar to rifling in a gun barrel—present an anisotropic resistance to viscous flows (flows with low Re): There is less resistance to flow in the direction parallel to the peaks and valleys of the ridges (alongŷ′) than in the orthogonal direction (along′) (18). As a result of this anisotropy, an axial pressure gradient (along ŷ) generates a mean (averaged over a period of the topography) transverse component in the flow (along ) that originates at the structured surface; the fluid circulates back across the top of the channel (along −). As indicated schematically by the two trajectories drawn in Fig. 1A, the full flow has helical streamlines. The optical micrograph in Fig. 1B shows the trajectories of two streams of fluid (red and green) in a microchannel such as the one in Fig. 1A.

Figure 1

Three-dimensional twisting flow in a channel with obliquely oriented ridges on one wall. (A) Schematic diagram of channel with ridges. The coordinate systems (x,y, z) and (x′, y′,z) define the principal axes of the channel and of the ridges. The angle θ defines the orientation of the ridges with respect to the channel. The amplitude of the ridges, αh, is small compared to the average height of the channel, h(α < 0.3). The width of the channel is w and principal wavevector of the ridges is q. The red and green lines represent trajectories in the flow. The streamlines of the flow in the cross section are shown below the channel. The angular displacement, Δφ, is evaluated on an outer streamline. (B) Optical micrograph showing a top view of a red stream and a green stream flowing on either side of a clear stream in a channel such as in (A) with h = 70 μm,w = 200 μm, α = 0.2, q = 2π/200 μm−1, and θ = 45°. (C) Fluorescent confocal micrographs of vertical cross sections of a microchannel such as in (A). The frames show the rotation and distortion of a stream of fluorescent solution that was injected along one side of the channel such as the stream of green solution in (B). The measured values of Δφ are indicated (29).

The frames in Fig. 1C are confocal micrographs of the vertical cross section of a channel similar to those in Fig. 1, A and B. As indicated on these images, we use the leading edge of the fluorescent region to measure the angular displacement of the fluid in the cross section, Δφ. Within the Stokes flow regime (Re < 1) and for small ridges relative to the height of the channel (relative height, α < 0.3), we have checked that the form of the flow (i.e., the shape of the trajectories) is independent of Re. We also found that the form of the flow remains qualitatively the same for Re < 100. Furthermore, the experimentally observed dependence of the average rate of rotation,dΔφ/dy, on geometrical parameters (q, h, w, and θ) can be rationalized with a simple model (19, 20).

The ability to generate transverse flows in microchannels makes it possible to design steady chaotic flows for use in microfluidic systems. A mixer based on patterns of grooves on the floor of the channel is shown schematically in Fig. 2A; we refer to this design as the staggered herringbone mixer (SHM). One way to produce a chaotic flow is to subject volumes of fluid to a repeated sequence of rotational and extensional local flows (21). This sequence of local flows is achieved in the SHM by varying the shape of the grooves as a function of axial position in the channel: The change in the orientation of the herringbones between half cycles exchanges the positions of the centers of rotation (local rotational flow, “c” in the Fig. 2A) and the up- and down-wellings (local extensional flow, “u” and “d” in Fig. 2A) in the transverse flow. Figure 2B shows the evolution of two streams through one cycle of the SHM.

Figure 2

Staggered herringbone mixer (SHM). (A) Schematic diagram of one-and-a-half cycles of the SHM. A mixing cycle is composed of two sequential regions of ridges; the direction of asymmetry of the herringbones switches with respect to the centerline of the channel from one region to the next. The streamlines of the flow in the cross section are shown schematically above the channel. The angle, Δφm, is the average angular displacement of a volume of fluid along an outer streamline over one half cycle in the flow generated by the wide arms of the herringbones. The fraction of the width of the channel occupied by the wide arms of the herringbones is p. The horizontal positions of the centers of rotation, the upwellings, and the downwellings of the cellular flows are indicated by c, u, and d, respectively. (B) Confocal micrographs of vertical cross sections of a channel as in (A). Two streams of fluorescent solution were injected on either side of a stream of clear solution (29). The frames show the distribution of fluorescence upstream of the features, after one half cycle, and after one full cycle. The fingerlike structures at the end of the fluorescent filaments on the bottom left of the second two frames are due to the weak separation of streamlines that occurs in the rectangular grooves even at low Re. In this experiment,h = 77 μm, w = 200 μm, α = 0.23, q = 2π/100 μm−1,p = 2/3, and θ = 45°, and there were 10 ridges per half cycle. Re < 10−2. Δφm ∼ 180°.

In the SHM, the efficiency of mixing is controlled by two parameters: p, a measure of the asymmetry of the herringbones; and Δφm, a measure of the amplitude of the rotation of the fluid in each half cycle (22). The angular displacement, Δφm, is controlled by the geometry of the ridges (20) and the number of herringbones per half cycle (10 in the case shown in Fig. 2). As p goes to one-half (i.e., symmetric herringbones) or Δφm goes to zero, the flow becomes nonchaotic. For p = 2/3 and Δφm > 60°, most of the cross-sectional area is involved in the chaotic flow (23). As in the twisting flow (Fig. 1), the form of flow in the SHM is independent of Rein the Stokes regime, and we have verified experimentally that it remains qualitatively the same for Re < 100.

The diagrams in Fig. 3, A to C, show the experiments we used to characterize mixing. At the entrance of the channel, distinct streams of a fluorescent and a clear solution fill opposite halves of the cross section of the channel. The micrographs in Fig. 3, A and B, show that for flows with high Péclet number (Pe = 2 × 105), there is negligible mixing in a simple channel (Fig. 3A) and incomplete mixing in a channel with straight ridges (Fig. 3B) after the flow has proceeded 3 cm—the typical dimension of a microfluidic chip—down the channel. The confocal cross sections in Fig. 3C show that thorough mixing occurs at even higher Pe (9 × 105) in a channel that contains the SHM (24). The micrographs in Fig. 3C also show the rapid increase in the number of filaments and decrease in their thickness, Δr, as a function of the number of mixing cycles.

Figure 3

Performance of SHM. (A toC) (Left) Schematic diagrams of channels with no structure on the walls (A), with straight ridges as in Fig. 1 (B), and with the staggered herringbone structure as in Fig. 2 (C). In each case, equal streams of a 1 mM solution of fluoroscein-labeled polymer in water/glycerol mixtures (0 and 80% glycerol) and clear solution were injected into the channel (29, 30). (Right) Confocal micrographs that show the distribution of fluorescent molecules in the cross section of the channels at a distance of 3 cm down the channels in (A) and (B) and at distances of 0.2, 0.4, 0.6, 0.8, 1.0, and 3 cm down the channel (i.e., after 1, 2, 3, 4, 5, and 15 cycles of mixing) in (C). Experimental parameters: (A)h = 70 μm, w = 200 μm,Re ∼ 10−2, Pe = 2 × 105; (B) h = 85 μm, w= 200 μm, α = 0.18, θ = 45°, q = 2π/100 μm−1, Re ∼ 10−2,Pe = 2 × 105; (C) h = 85 μm, w = 200 μm, α = 0.18,q = 2π/100 μm−1, p = 2/3, θ = 45°, Re ∼ 10−2,Pe = 2 × 105, six ridges per half-cycle, Δφm ∼ 60°, Re ∼ 10−2, Pe = 9 × 105. (D) Plot of the standard deviation, σ, of the fluorescence intensity in confocal images such as in (A) to (C) as a function of the distance down the channel, Δy. Only the central 50% of the area of the images [indicated in the bottom frame in (C)] was used to measure σ to eliminate variations in the fluorescence intensity due to optical effects at the channel walls. The open symbols are for the mixing channel in (C): (○) forPe = 2 × 103, (□) forPe = 2 × 104, (▵) forPe = 2 × 105, and (◊) forPe = 9 × 105. 10−2Re ≤ 10 in these experiments. The points (▴) are for a smooth channel (A), and (•) are for a channel with straight ridges (B); for both, Pe = 2 × 105. Horizontal dotted line indicates the value of σ used to evaluate Δy 90, the axial distance required for 90% mixing. (E) Plot of Δy 90as a function of ln(Pe) from the curves in (D).

To quantify the degree of mixing (convection plus diffusion) as a function of the axial distance traveled in the mixer and ofPe, we measure the standard deviation of the intensity distribution in confocal images of the cross section of the flow like those in Fig. 3, A to C: σ = 〈(I − 〈I〉)21/2, whereI is the grayscale value (between 0 and 1) of a pixel, and 〈 〉 means an average over all the pixels in the image. The value of σ is 0.5 for completely segregated streams and 0 for completely mixed streams. Figure 3D shows the evolution of σ for flows of different Pe in the SHM (open symbols), in a simple channel as in Fig. 3A (▴), and in a channel with straight ridges as in Fig. 3B (•). We see that the SHM performs well over a large range inPe; as Pe increases by a factor of ∼500 (Pe = 2 × 103 to Pe = 9 × 105), the mixing length, Δy 90, required for 90% mixing (dashed line), increases by less than a factor of 3 (Δy 90 = 0.7 cm to Δy 90 = 1.7 cm). In Fig. 3E, we see that Δy 90 increases linearly with ln(Pe) for large Pe, as expected for chaotic flows. Within the limits of our simple model of mixing (13), we estimate from the linear portion of the plot in Fig. 3E that λ is on the order of a few millimeters; the average width of the filaments of unmixed fluid decreases by a factor of ∼3 as the fluid travels this axial distance. This estimate agrees qualitatively with the evolution seen in Fig. 3C.

On the basis of the results presented in Fig. 3, consider mixing a stream of protein solution in aqueous buffer (molecular weight 105, D ∼ 10−6cm2/s) with U = 1 cm/s andl = 0.01 cm. For this system, Pe = 104. The mixing length in a simple microchannel would be Δy mPe ×l = 100 cm. On the basis of Fig. 3D, the mixing length in the SHM would be, Δy m ∼ 1 cm. Furthermore, increasing the flow speed by a factor of 10 (i.e., toPe = 105) will increase the mixing length in the SHM to Δy m ∼ 1.5 cm. With the same change in flow speed, the mixing length in the absence of stirring will increase 10-fold, to Δy m ∼ 103 cm.

An important application of mixing in pressure flows is in the reduction of axial dispersion. Axial dispersion is important in determining performance in pressure-driven chromatography—e.g., in the transfer of fractions from a separation column to a point detector—where it leads to peak broadening. The most rapid dispersion of a band of solute takes place when its axial length is much shorter than the mixing length of the solute in the flow. During this stage, the length of the band grows at the maximum speed of the flow (i.e., more quickly than the center of the band moves along the channel). This effect is illustrated schematically in the diagram in Fig. 4A for an unstirred Poiseuille flow. Once the length of the band is greater than the mixing length, volumes of fluid have sampled both fast and slow regions of the flow, and the broadening of the band becomes diffusive, i.e. ∼Deff τr, where D eff is an effective diffusivity that again depends on the mixing length and τr is the residence time of the band in the flow (8, 25).

Figure 4

Axial dispersion with and without SHM. (A) Schematic drawing illustrating the dispersion of a plug in Poiseuille flow. (B) Unstirred Poiseuille flow in a rectangular channel: h = 70 μm, w = 200 μm, and Pe ∼ 104. (C) Stirred flow in a staggered herringbone mixer of the same design as in Fig. 3C;Pe ∼ 104. In (B) and (C), a plug of fluorescent dye was introduced into serpentine channels of the form shown in the inset in (B). The traces represent the time evolution of the total fluorescence intensity (arbitrary units) as observed experimentally with a fluorescence microscope (2.5×/0.07 numerical aperture lens that averaged over the cross section of the channel) at different axial positions along the channel: 0 cm (black), 2.0 cm (blue), 6.2 cm (orange), 10.4 cm (red), 14.6 cm (brown), and 18.8 cm (green) downstream. The black traces (0 cm) have been truncated. The dye was fluorescein and the liquid was 80% glycerol/20% water.D ∼ 10−7 cm2/s.

The experiments presented in Fig. 4 demonstrate that, by stirring the fluid in the cross section of the flow, the SHM (Fig. 4C) reduces the extent of the initial, rapid broadening of a band of material relative to that in an unstirred flow (Fig. 4B). The mixing length in the unstirred flow is Δy m ∼ 100 cm. In this case, the asymmetry of the trace of fluorescence intensity as a function of time measured near the end of the channel (green trace inFig. 4B) indicates that the band is still broadening rapidly as it reaches the end of the channel: The fluorescent fluid in the fast, uniform flow near the center of the channel is weakly dispersed and arrives at the detector first (steep initial rise of the trace); the fluorescent fluid in the shear flow near the walls is strongly dispersed and arrives at the detector later (long tail of the trace). With the SHM (Fig. 4C), the mixing length is Δy m ∼ 1 cm (estimated from the curves in Fig. 3D for Pe = 104). In this case, the band broadens rapidly in the first few centimeters of the channel as indicated by the asymmetry of the trace acquired 2 cm downstream from the inlet (blue trace in Fig. 4C). The traces acquired further downstream are noticeably more symmetrical; this change indicates a transition to diffusive broadening. Note that, by reducing the mixing length, the SHM will also reduce D eff(8, 26).

The SHM based on topography patterned on the inner surfaces of microchannels offers a general solution to the problem of mixing fluids in microfluidic systems. The simplicity of its design allows it to be integrated easily into microfluidic structures with standard microfabrication techniques. A single design will operate efficiently over a wide range of Re (we have observed good mixing for 0 < Re < 100) and Pe (a SHM that is 3 cm long will fully mix all flows with Pe < 106). This design adds a negligible resistance to flow relative to that of a simple channel of the same dimensions. More generally, topography on the walls of microchannels can be used to manipulate the position of streams in a microchannel. For example, Fig. 1 demonstrates that two streams can cross over one another in a channel with only diffusional mixing.

We note that patterned topography on surfaces such as the staggered herringbone design can be used to generate chaotic flows in contexts other than pressure-driven flows in microchannels. For example, similar structures on the walls of round pipes and capillaries will generate efficient mixing flows. Electroosmotic flows in capillaries or channels that contain the staggered herringbone structure ought to be chaotic and mix adjacent streams efficiently (20, 27). Chaotic flows will also exist in the laminar shear flow in the boundary layer of an extended flow over a surface that presents the staggered herringbone structure. This stirring of the boundary layer should enhance the rates of diffusion-limited reactions at surfaces (e.g., electrode reactions) and heat transfer from solids into bulk flows.

  • * To whom correspondence should be addressed. E-mail: stroock{at}fas.harvard.edu, gwhitesides{at}gmwgroup.harvard.edu

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