Boundary condition induced passive chaotic mixing in straight microchannels

Microfluidics deals with control and manipulation of small volumes of fluids through channels with characteristic length scales on the order of micrometers, and it exercises precise dynamic control over the flow to study new phenomena occurring in fluids. Microfluidics has the potential to solve some of the grand engineering challenges, such as engineering better medicines, providing access to clean water, and solving energy problems due to its ability to use very small controlled volume of samples and reagents, high resolution separation, and detection with great sensitivity.^1–3 Low manufacturing cost, short timescales for analysis along with high throughput designs using device miniaturization, such as lab-on-a-chip⁴ or a reactor-on-a-chip,⁵ are other advantages. The application of microfluidics is manyfold, ranging from microelectronics, bioanalysis,^6,7 nanoparticle synthesis, organ in a chip, drug development, testing and controlling multiphase flow,^8–10 optofluidics, acoustofluidic patterning,¹¹ to detection of a single cell,^12,13 single file diffusion,¹⁴ and even a single molecule.¹⁵ 

It is clear that microfluidics offers solutions to a plethora of problems. However, growth of microfluidic technologies to the desired level has been stalled by some serious bottlenecks. Many of the applications of microfluidic chips demand homogeneous mixing of reactants. However, it is difficult to mix component fluids in microchannels spontaneously, thanks to the absence of turbulence in the viscosity dominated laminar flow regimes in the small characteristic length scales. One major bottleneck in developing an efficient microfluidic device is, thus, the difficulty in mixing miscible fluid components homogeneously and spontaneously in a low Reynolds number regime.

In a typical microchannel, cross-section length scale (l ) is $∼100$ μm or less. With small flow rates $U∼0.1$ m/s, Reynolds numbers (⁠$Re=Ulρ/μ$⁠) are typically of the order of 10 or less, where μ is the dynamic viscosity (⁠$∼10−3$ Pa S) and ρ is the density (⁠$∼103$ kg/m³ for water) of the fluid. This means that not only inertia is negligible, but also molecular diffusion is the dominant mechanism of mixing of the component fluids in this regime. However, typical diffusivity (D) of species varies from approximately 10⁻⁹ m²/s for small molecules including ions, to 10⁻¹¹ m²/s for large biomolecules. As a result, mixing timescale (⁠$l2/D$⁠) is large (>10 s), and Péclet number (⁠$Pe=Ul/D$⁠), the relative measure of diffusive timescale over advective timescale, is also large (10⁴–10⁶). The length required to mix via diffusion in a flow, thus, (⁠$lmix=lPe$⁠) typically ranges between tens of centimeters to even in meters! One of the strategies that can be employed to enhance mixing in steady laminar diffusion dominated flow regimes comes through a Smale horseshoe map and/or a baker's map.¹⁶ By effectively decreasing the striation length (⁠$lStr→0$⁠), defined as the length scale over which diffusion acts to homogenize the concentration, these maps transform the space (the flow domain) into itself through successive stretching and folding. In this case, the effective interface area across which diffusion takes place increases exponentially leading to a positive Lyapunov exponent. In other words, this is a route to chaos or a rapid growth in mixing, when the problem is translated into the language of dynamical systems.

The other pathway to achieve mixing in low Reynolds number flow goes through “chaotic advection,” where the fluid still associates itself with stretching and folding, on top of flows switching instantaneously from one streamline to another.¹⁷ This mechanism of mixing is observed in blinking flows and can be understood theoretically using a twist map.¹⁸ Although both the transformations can be analyzed rigorously using mathematical concepts of ergodicity and dynamical systems, a practical demonstration of these idealized mathematical models is rather difficult to achieve. Mixing via chaotic advection was first demonstrated in the pioneering work on the Staggered Herringbone Mixer (SHM),¹⁹ where flow transverse to the primary flow direction suitable for chaotic advection was generated by placing three dimensional (3D) patterned ridges on the floor of the channels at oblique angles with respect to the downstream direction of the flow. Mixing length varied linearly with $ln(Pe)$ confirming that the stretching and folding of the volumes grow exponentially with respect to the effective mixing length. In contrast, we take a totally new approach in our work to generate similar transverse flow by creating two-dimensional (2D) anisotropic wetting conditions on the floor of the channels experimentally and numerically. Moreover, we show that this type of enhanced mixing is not limited to just Herringbone type patterns, rather a range of computational fluid dynamics simulations reveal the effects of a range of patterns on the flow behavior and their corresponding mixing mechanisms in simple microfluidic channels.

It is well known that surfaces play a big role in micro and nanoscales because the boundary layer essentially spans the whole cross sections of the channels. Since the ratio of solid–liquid interface area (⁠$A∼l2$⁠) in a channel and the volume of fluid (⁠$V∼l3$⁠) varies inversely to the characteristic length scale of the system (⁠$∼1/l$⁠), in small length scales, such as micro- or nanochannels (⁠$l→0$⁠), the effect of boundary conditions become increasingly dominant in this asymptotic regime. In this work, we exploit this fact and report that unlike,¹⁹ where 3D miropatterning involving rather complex fabrication process was needed to achieve chaotic advection, here we have achieved mixing at Re $≤$ 10 by creating essentially 2D hydrophobic patterns using water-repellant ink dots on the channel floor. To be precise, we have used simple hydrophobic permanent marker ink²⁰ that contains hydrophobic polymer resin to create these very thin hydrophobic patterns on one of the inner floors of the straight channel. The ratio of the thickness of the polymer films to the height of the channels is <0.02, meaning that the hydrophobic regions are essentially 2D with respect to the height of the channels. These hydrophobic dots create a discontinuous jump in the wall boundary conditions forcing the streamlines to readjust instantaneously while crossing the boundary between no-slip to hydrophobic regions. There is less resistance to flow over the water-repellant regions and the liquid flows faster over them than the no-slip regions on the floor of the channels. Moreover, the hydrophobic patterns constrain the fluid to avoid those regions from sticking to the floor, in turn potentially forcing the fluid to generate a velocity component in the transverse direction. Therefore, depending on the strength of hydrophobicity of the ink invisible ridges can be formed periodically at the locations of the hydrophobic patterns. We hypothesize that because of this anisotropic and abrupt flow resistance change due to the difference in wetting boundary conditions on the floor, the fluid generates an average transverse flow, resulting in twisting and stretching of the fluid volume over the cross section of the channel, or creating counter rotating recirculation zones leading to chaotic advection.

The whole fabrication process of microchannels with anisotropic boundary conditions is quite simple and can be done using well known soft photolithography process with polydimethylsiloxane (PDMS) elastomer. Designs of various microchannels are shown in Fig. 1. To create slip regions on the floor of the channels, we have used an unique and easy technique. Recently, it has been shown that many commercially available permanent marker inks, such as Sharpie^® inks, create elastic thin polymer films upon drying on a substrate and exhibit water-repellent hydrophobic behavior.²⁰ The thickness of these hydrophobic polymer films ranges between a few hundred nanometers to a few micrometers. Taking advantage of this hydrophobic property and easy availability of school-supply permanent markers, we have created slip dots on the floor of the channels by simply putting the dots using an ultrafine Sharpie^® marker manually. After the hydrophobic dots are created on a glass slide (Fig. 1), PDMS open channels created by standard soft photolithography process²¹ are bonded with the slide using plasma boding techniques.^22,23 The hydrophobic spots created manually on the floor of the channels are not fully symmetrical as expected and can be seen in Fig. 1(c). Several microchannels have been made with width and depth in the range of 100 to 300 μm and the length of the channels in the range of 5–10 mm. In our experiments, Reynolds number is always kept at 10 by controlling the flow speed in the syringe pumps.

Figure 2 shows the schematic of our experiments where two miscible fluids are pumped into the microfluidic reactor through the inlets by two syringe pumps (New Era Pump Systems, NE-1002X) and collected in the cuvette placed at the outlet. The fluids collected at the outlet are analyzed by UV-VIS absorption spectroscopy (Beckman Coulter, DU730) to determine the mixing index and the overall effectiveness of the hydrophobic spots in the mixing of component fluids. In our experiment, we use de-ionized (DI) distilled water (Barnstead NANOpureDiamond water purification system, specific resistivity 18.2 MΩ-cm) in one of the syringe pumps, and DI water mixed with Allura Red dye (Millipore Sigma, D = 3.23 $×10−10$ m²/s, concentration 2 $×10−5$ M) in the other. The spectrophotometer was calibrated with pure water with concentration = 0 and with inlet Allura Red solution as 1. In a simple microchannels with usual no-slip inner walls, the output confirms a fully separated flow, i.e., one of the cuvette collects red colored fluid with the same concentration of the input colored fluid, confirmed by the UV-VIS spectroscopic analysis, whereas the other output cuvette collects colorless water. The spectrophotometer is calibrated with a colored solution of known concentration. When the output collected in a cuvette is placed in a UV-VIS spectrometer absorption of light transmitted through the medium becomes directly proportional to the medium concentration. In this case, the fully separated flow through a straight microchannel is characterized by the mixing index 0.5. When a microchannel with hydrophobic marker dots on the floor is used, two input liquids start to mix. To ensure that the adsorption of Allura Red dyes by the marker ink or the dilution of marker ink in water does not affect mixing results, we flush the microchannels with clear, distilled, and de-ionized water for an hour before collecting mixing data. A more direct investigation of the amount of dye adsorbed in the marker dots, if any, can be made by dismantling the microchannels after each experiment, scraping off the marker ink and do elemental analysis of the sample. However, we have not done this analysis in this work. Each experiment is repeated at least five times to ensure reproducibility of the experimental observations. A fully mixed homogeneous fluid's concentration is analyzed by the spectrophotometer, and the mixing index is scaled as 0.

The effect of the slip dots on the degree of mixing at the outlet of the channels, measured experimentally using a spectrophotometer, is shown in Fig. 3. A transition from the laminar separated flow to a homogeneous mixing of component fluids is observed when the number of hydrophobic dots exceeds 3 in a 7 mm channel.

To understand the details of the mixing mechanism, a numerical investigation is done using the finite element simulation software package COMSOL (version 5.4). The bulk flow governed by Navier–Stokes equation $ρ[∂u→∂t+(u→·∇→)u→]=−∇→p+μ∇2u→$⁠, where $u→$ is the velocity vector, convection diffusion equations $∂c∂t+(u→·∇→)c=D∇2c$⁠, where c is the concentration of the fluid, and the continuity equation $∇→·u→=0$ is solved. Hydrophobic spots of thickness $∼2 μ$m, similar to experimental marker polymer films,²⁰ are considered as slip regions (⁠$u→·n→=0$⁠), where $n→$ is the normal vector at the solid wall boundary with a no-penetration condition. No slip conditions are imposed on the rest of the inner wall regions of the channel by using the condition $u→=0$⁠. The steady incompressible flow with laminar outflow and zero outlet static pressure is employed. In the code, Reynolds number is kept at 10 to begin with, to match it with our experimental conditions. However, we have also observed significant mixing in the straight microchannel with patterned hydrophobic regions at lower Re = 1 and Re = 0.1 also.

To further benchmark the code, we first created herringbone type 2D hydrophobic patters on the floor of the channels to mimic the 3D grooved channels in SHM previously reported.¹⁹ It shows similar mixing profiles (Fig. 3 of Ref. 19) in comparison with our simulation results (Fig. 4 of this paper). From the numerical simulations, we have identified two different pathways to achieve homogeneous mixing of component fluids in the low Reynolds number laminar flow regime. Preferential bending of the flow can be achieved by introducing periodic asymmetries in the boundary conditions, created by asymmetric hydrophobic patterns about the centerline of the channel shown in Fig. 5(a). Analogous to the Magnus effect,²⁴ where difference between the speeds of a fluid flowing over two opposite sides of an object gives rise to a pressure difference in the transverse direction of a flow results in bending of the fluid, here our “duck” shape patterns, for example, create a transverse pressure gradient generated through the difference of uneven flow resistances due to the inhomogeneity of the flow boundary conditions about the centerline. As can be seen from Fig. 1, the hydrophobic dots used in the experiments are not perfectly circular. This is expected as these dots were created manually using a black sharpie marker. We have done simulations with perfectly circular hydrophobic dots; however, numerical results do not show significant mixing with those perfectly symmetrical circular dots. This is why we created duck shaped hydrophobic regions to break the symmetry and then simulation showed significant mixing behavior. Hence, we can conclude that asymmetry in the hydrophobic patterns plays an important role in mixing.

We argue that the floor of the channels is no-slip everywhere except the locations where there is a hydrophobic film. Thus, the hydrophobic regions on the floor apply less resistance to the fluid flow in comparison with the corresponding no slip region on the other side of the centerline. Each of the patterns, thus, may direct one component fluid (in our case could be blue) to intrude into the other (red), eventually creating a twisting, stretching, and folding of both the components as the fluids flow downstream and leading to a Smale horseshoe type mixing pathway as seen in Fig. 5.

Another pathway of mixing is found by creating counter rotating recirculation zones with the help of “V” shaped hydrophobic patterns (Fig. 6). It is clear that the stagnation point between the recirculating fluids aligns with the position of the apex point of the V shape. This feature is evident in the herringbone pattern (Fig. 4) also, where the stagnation points can be shifted periodically in the transverse direction by moving the apex points of the asymmetric V shapes. This creates different recirculation patterns as the fluids flow downstream leading toward effective mixing through chaotic advection. To check whether this enhanced mixing is a signature of deterministic chaos or not, we investigated the power-law dependence of the mixing length on the Péclet number Pe. When mixing occurs only via diffusion, the channel length required for mixing is given by $lmix∼Utdiff∼Ul2/D∼lPe$⁠, whereas if there is a chaotic root to mixing via stretching and folding of liquid volumes over the cross section of the channel, then the mixing length would vary as $lmix∼l ln(Pe)$⁠.^19,25,26 A linear profile of l_mix with respect to $ln(Pe)$ can, thus, be considered as a signature of chaotic advection in the system. In Fig. 7, we have indeed seen this behavior in our irregular V shaped passive microfluidic mixers.

We were also curious to see if we can create a superposition of the Magnus type bending with the recirculating cells using both the pathways mentioned above, and the results are shown in Fig. 8. It shows that alternating twisting and recirculation gives rise to excellent mixing in the low Reynolds number laminar flow in a straight microchannel.

In Fig. 9, we show that even at Re = 0.1, Duck shape and symmetric V shaped hydrophobic patches can lead to significant mixing at the outlet of a 10 mm channel. Mixing index in numerical solutions is calculated using the same method as described in Ref. 19.

In conclusion, we claim that rapid mixing in low Reynolds number flow even in a straight microchannel can be achieved by manipulating twisting, stretching, and recirculation in the flow created by passive interventions through carefully chosen hydrophobic patterns on the walls/floor of the microchannels. In this easy and inexpensive method, we would not even need complex three dimensional micro-/nano-patterning to create precise grooves and ridges on the floor of the channels¹⁹ to introduce chaos, rather mixing can be manipulated by drawing hydrophobic no-slip spots on the floor of the channel by drawing the patterns with hydrophobic solutions/inks. This Letter may lead the microfluidic industry to an easy and inexpensive fabrication process to achieve efficient homogeneous mixing with a small mixing length scale.