# Mixing Enhancement in Serpentine Micromixers with a Non-Rectangular Cross-Section

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Geometrical Design of the Micromixer

_{in}corresponding to 0.5 W for both the standard rectangular section micromixer as well as for the newly investigated design.

## 3. Numerical Model and Mixing Assessment

**u**(m∙s

^{−1}) is the velocity vector, ρ (kg∙m

^{3}) is the fluid density, η (kg∙m

^{−1}∙s

^{−1}) is the fluid viscosity, t (s) is the time, and p (Pa) is the pressure. The flow field equations are solved using a generalized minimal residual method (GMRES) iterative solver with a geometrical multigrid pre-conditioner and a Vanka algorithm for the pre- and post-smoothing. No-slip boundary conditions were set for the walls of the micromixer. A free tetrahedral mesh is used for the entire microchannel with no less than ~200,000 elements for all the geometries studied.

^{−3}) is the concentration of the species of interest, and D (m

^{2}∙s

^{−1}) is its diffusion constant, respectively. The same iterative numerical solver as for the Navier–Stokes equations is used, but the maximum element size in the mesh is constrained to less than 10 μm through the whole geometry to avoid the possible numerical errors that can be associated with this type of solution. Thus, for all the concentration simulations, the number of mesh elements in the numerical model has never been less than ~1,900,000. For all the simulations described in this work, we used the computational package COMSOL Multiphysics 5.1 (COMSOL Inc., Stockholm, Sweden) and its computational fluid dynamics/chemical engineering module. The accuracy of the numerical work employed for this type of flow has been previously validated against data obtained from other microfluidic devices [49,50]. In particular, when comparing model data with measured data on similar planar curved microchannels, as presented by Jiang et al. [50], we find mixing times to be less than 0.05 s for Dean numbers equal to or above 140, in agreement with the experimental results.

_{bins}of equal size regions. For a system with two components, the mixing index is defined as

_{1/j}and p

_{2/j}are the conditional probabilities for Components 1 and 2, respectively, to be located in bin j. They represent the fraction of Components 1 and 2, respectively, in each bin relative to the total. They are calculated as the ratio of the average bin grayscale intensity from the corresponding concentration image, normalized by the maximum intensity, i.e., 255 for grayscale image data. In this particular study, since we have two chemical species and the fluids used are incompressible the two conditional probabilities are related as p

_{2/j}= 1 − p

_{1/j}. As shown in Equation (4), the mixing index M is normalized by a factor of ln2, where 2 corresponds to the number of components. Thus, the mixing index will take the value M = 0 for completely segregated components, while it will assume the value M = 1 for the completely mixed case.

## 4. Results and Discussion

^{−3}and a viscosity η of 0.001 kg∙m

^{−1}∙s

^{−1}. The diffusion coefficient D was fixed to 1.0 × 10

^{−9}m

^{2}∙s

^{−1}corresponding to the diffusion values for most ions in aqueous solutions. For all the simulations, the working solutions were considered to be pure water introduced through one of the inlets and dyed water with a concentration c = 1 moL∙m

^{−3}introduced through the opposite inlet. The flow rates are maintained for both fluid components, with the same mean fluid velocity at both inlets. This means fluid velocity is varied from 0.0075 to 0.75 m∙s

^{−1}to span a range of Reynolds numbers corresponding to Re = 1–100.

^{−3}) and high (c = 1 moL∙m

^{−3}) concentration, respectively, left in the dyed fluid distribution map.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Sackmann, E.K.; Fulton, A.L.; Beebe, D.J. The present and future role of microfluidics in biomedical research. Nature
**2014**, 507, 181–189. [Google Scholar] [CrossRef] [PubMed] - Capretto, L.; Carugo, D.; Mazzitelli, S.; Nastruzzi, C.; Xunli, Z. Microfluidic and lab-on-a-chip preparation routes for organic nanoparticles and vesicular systems for nanomedicine applications. Adv. Drug Deliv. Rev.
**2013**, 65, 1496–1543. [Google Scholar] [CrossRef] [PubMed] - Chin, C.D.; Linder, V.; Sia, S.K. Lab-on-a-chip devices for global health: Past studies and future opportunities. Lab Chip
**2007**, 7, 41–57. [Google Scholar] [CrossRef] [PubMed] - Fair, R.B. Digital microfluidics: Is a true lab-on-a-chip possible? Microfluid. Nanofluid.
**2007**, 3, 245–281. [Google Scholar] [CrossRef] - Geong, G.S.; Chung, S.; Kim, C.B. Applications of micromixing technology. Analyst
**2010**, 135, 460–473. [Google Scholar] [CrossRef] - Chiu, D.T.; deMello, A.J.; Di Carlo, D.; Doyle, P.S.; Hansen, C.; Maceiczyk, R.M.; Wootton, R.C.R. Small but perfectly formed? Successes, challenges, and opportunities for microfluidics in the chemical and biological sciences. Chem
**2017**, 2, 201–223. [Google Scholar] [CrossRef] - Jayamohan, H.; Sant, H.J.; Gale, B.K. Applications of microfluidics for molecular diagnostics. Methods Mol. Biol.
**2013**, 949, 305–334. [Google Scholar] [CrossRef] [PubMed][Green Version] - Du, W.; Li, L.; Nichols, K.P.; Ismagilov, R.F. SlipChip. Lab Chip
**2009**, 9, 2286–2292. [Google Scholar] [CrossRef] [PubMed] - Bamford, R.A.; Smith, A.; Metz, J.; Glover, G.; Titball, R.W.; Pagliara, S. Investigating the physiology of viable but non-culturable bacteria by microfluidics and time-lapse microscopy. BMC Biol.
**2017**, 15, 121. [Google Scholar] [CrossRef] [PubMed] - Zilionis, R.; Nainys, J.; Veres, A.; Savova, V.; Zemmour, D.; Klein, A.M.; Mazutis, L. Single-cell barcoding and sequencing using droplet microfluidics. Nat. Protoc.
**2017**, 12, 44–73. [Google Scholar] [CrossRef] [PubMed] - Junkin, M.; Tay, S. Microfluidic single-cell analysis for systems immunology. Lab Chip
**2014**, 14, 1246–1260. [Google Scholar] [CrossRef] [PubMed] - Pagliara, S.; Franze, K.; McClain, C.R.; Wylde, G.W.; Fisher, C.L.; Franklin, R.J.M.; Kabla, A.J.; Keyser, U.F.; Chalut, K.J. Auxetic nuclei in embryonic stem cells exiting pluripotency. Nat. Mater.
**2014**, 13, 638–644. [Google Scholar] [CrossRef] [PubMed] - Tsao, C.-W. Polymer microfluidics: Simple, low-cost fabrication process bridging academic lab research to commercialized production. Micromachines
**2016**, 7, 225. [Google Scholar] [CrossRef] - Waheed, S.; Cabot, J.M.; Macdonald, N.P.; Lewis, T.; Guijt, R.M.; Paullab, B.; Breadmore, M.C. 3D printed microfluidic devices: Enablers and barriers. Lab Chip
**2016**, 11, 1993–2013. [Google Scholar] [CrossRef] [PubMed] - Shan, C.; Chen, F.; Yang, Q.; Jiang, Z.; Hou, X. 3D multi-microchannel helical mixer fabricated by femtosecond laser inside fused silica. Micromachines
**2018**, 9, 29. [Google Scholar] [CrossRef] - Sun, G.; Panpan, W.; Shenguang, G.; Lei, G.; Jinghua, Y.; Mei, Y. Photoelectrochemical sensor for pentachlorophenol on microfluidic paper-based analytical devicebased on the molecular imprinting technique. Biosens. Bioelectron.
**2014**, 56, 97–103. [Google Scholar] [CrossRef] [PubMed] - Lee, C.Y.; Chang, C.L.; Wang, Y.N.; Fu, L.M. Microfluidic mixing: A review. Int. J. Mol. Sci.
**2011**, 12, 3263–3287. [Google Scholar] [CrossRef] [PubMed] - Nguyen, N.-T. Micromixers: Fundamentals, Design and Fabrication, 2nd ed.; Elsevier: Oxford, UK, 2012; ISBN 978-1-43-773520-8. [Google Scholar]
- Nguyen, N.-T.; Wu, Z. Mixers—A review. J. Micromech. Microeng.
**2005**, 15, R1–R16. [Google Scholar] [CrossRef] - Cai, G.; Xue, L.; Zhang, H.; Lin, J. A review of micromixers. Micromachines
**2018**, 8, 274. [Google Scholar] [CrossRef] - Brandhoff, L.; Zirath, H.; Salas, M.; Haller, A.; Peham, J.; Wiesinger-Mayr, H.; Spittler, A.; Schnetz, G.; Lang, W.; Vellekoop, M.J. A multi-purpose ultrasonic streaming mixer for integrated magnetic bead ELISAs. J. Micromech. Microeng.
**2015**, 25, 104001. [Google Scholar] [CrossRef] - Phan, H.V.; Coskun, M.B.; Sesen, M.; Pandraud, G.; Neild, A.; Alan, T. Vibrating membrane with discontinuities for rapid and efficient microfluidic mixing. Lab Chip
**2015**, 15, 4206–4216. [Google Scholar] [CrossRef] [PubMed] - Nama, N.; Huang, P.-H.; Huang, T.J.; Constanzo, F. Investigation of micromixing by acoustically oscillated sharp-edges. Biomicrofluidics
**2016**, 10, 024124. [Google Scholar] [CrossRef] [PubMed] - Patel, M.V.; Tovar, A.R.; Lee, A.P. Lateral cavity acoustic transducer as an on-chip cell/particle microfluidic switch. Lab Chip
**2012**, 12, 139–145. [Google Scholar] [CrossRef] [PubMed] - Huang, P.-H.; Ren, L.; Nama, N.; Li, S.; Li, P.; Yao, X.; Cuento, R.A.; Wei, C.-H.; Chen, Y.; Xie, Y.; et al. An acoustofluidic sputum liquefier. Lab Chip
**2015**, 15, 3125–3131. [Google Scholar] [CrossRef] [PubMed] - Destgeer, G.; Im, S.; Ha, B.H.; Jung, J.H.; Ansari, M.A.; Sung, H.J. Adjustable, rapidly switching microfluidic gradient generation using focused travelling surface acoustic wave. Appl. Phys. Lett.
**2014**, 104, 023506. [Google Scholar] [CrossRef] - Krishnaveni, T.; Renganathan, T.; Picardo, J.R.; Pushpavanam, S. Numerical study of enhanced mixing in pressure-driven flows in microchannels using a spatially periodic electric field. Phys. Rev. E
**2017**, 96, 033117. [Google Scholar] [CrossRef] [PubMed] - Ryu, K.S.; Shaikh, K.; Goluch, E.; Fana, Z.; Liu, C. Micro magnetic stir-bar mixer integrated with parylene microfluidic channels. Lab Chip
**2004**, 6, 608–613. [Google Scholar] [CrossRef] [PubMed] - Abbas, Y.; Miwa, J.; Zengerle, R.; von Stetten, F. Active continuous-flow micromixer using an external braille pin actuator array. Micromachines
**2013**, 4, 80–89. [Google Scholar] [CrossRef] - Tofteberg, T.; Skolimowski, M.; Andreassen, E.; Geschke, O. A novel passive micromixer: Lamination in a planar channel system. Microfluid. Nanofluid.
**2010**, 8, 209–215. [Google Scholar] [CrossRef] - Stroock, A.D.; Dertinger, S.K.W.; Ajdari, A.; Mezic, I.; Stone, H.A.; Whitesides, G.M. Chaotic mixer for microchannels. Science
**2002**, 295, 647–651. [Google Scholar] [CrossRef] [PubMed] - Kee, S.P.; Gavriilidis, A. Design and characterization of the staggered herringbone mixer. Chem. Eng. J.
**2008**, 142, 109–121. [Google Scholar] [CrossRef] - Fodor, P.S.; Kaufman, M. The evolution of mixing in the staggered herring bone micromixer. Mod. Phys. Lett. B
**2011**, 25, 1111–1125. [Google Scholar] [CrossRef] - Alam, A.; Afzal, A.; Kim, K.-Y. Mixing performance of a planar micromixer with circular obstructions in a curved microchannel. Chem. Eng. Res. Des.
**2014**, 92, 423–434. [Google Scholar] [CrossRef] - Kim, D.S.; Lee, S.W.; Kwon, T.H.; Lee, S.S. A barrier embedded chaotic micromixer. J. Micromech. Microeng.
**2004**, 15, 798–805. [Google Scholar] [CrossRef] - Scherr, T.; Quitadamo, C.; Tesvich, P.; Park, D.S.; Tiersch, T.; Hayes, D.; Choi, J.W.; Nandakumar, K.; Monroe, W.T. A planar microfluidic mixer based on logarithmic spirals. J. Micromech. Microeng.
**2012**, 22, 055019. [Google Scholar] [CrossRef] [PubMed] - Chen, X.; Li, T. A novel design for passive misscromixers based on topology optimization method. Biomed. Microdevices
**2016**, 18, 57. [Google Scholar] [CrossRef] [PubMed] - Shamloo, A.; Madadelahi, M.; Akbari, A. Numerical simulation of centrifugal serpentine micromixers and analyzing mixing quality parameters. Chem. Eng. Process. Process Intensif.
**2016**, 104, 243–252. [Google Scholar] [CrossRef] - Dean, W.R. Note on the motion of a fluid in a curved pipe. Philos. Mag.
**1927**, 4, 208–223. [Google Scholar] [CrossRef] - Liu, R.H.; Stremler, M.A.; Sharp, K.V.; Olsen, M.G.; Santiago, J.G.; Adrian, R.J.; Aref, H.; Beebe, D.J. Passive mixing in a three-dimensional serpentine microchannel. J. Microelectromech. Syst.
**2000**, 9, 190–197. [Google Scholar] [CrossRef] - Araci, I.E.; Robles, M.; Quake, S.R. A reusable microfluidic device provides continuous measurement capability and improves the detection limit of digital biology. Lab Chip
**2016**, 16, 1573–1578. [Google Scholar] [CrossRef] [PubMed] - Mengeaud, V.; Josserand, J.; Girasult, H.H. Mixing processes in a zigzag microchannel: Finite element simulation and optical study. Anal. Chem.
**2002**, 74, 4279–4286. [Google Scholar] [CrossRef] [PubMed] - Alam, A.; Kim, K.Y. Analysis of mixing in a curved microchannel with rectangular grooves. Chem. Eng. J.
**2012**, 181–182, 708–716. [Google Scholar] [CrossRef] - Cook, K.J.; Fan, Y.; Hassan, I. Mixing evaluation of a passive scaled-up serpentine micromixer with slanted grooves. J. Fluids Eng.
**2013**, 135, 081102. [Google Scholar] [CrossRef] - Javaid, M.U.; Cheema, T.A.; Park, C.W. Analysis of passive mixing in a serpentine microchannel with sinusoidal side walls. Micromachines
**2018**, 9, 8. [Google Scholar] [CrossRef] - Hossain, S.; Kim, K.-Y. Mixing performance of a serpentine micromixer with non-aligned inputs. Micromachines
**2015**, 6, 842–854. [Google Scholar] [CrossRef] - Sayah, A.; Gijs, M.A.M. Understanding the mixing process in 3D microfluidic nozzle/diffuser systems: Simulations and experiments. J. Micromech. Microeng.
**2016**, 26, 115017. [Google Scholar] [CrossRef] - Fodor, P.S.; Itomlenskis, M.; Kaufman, M. Assessment of mixing in passive microchannels with fractal surface patterning. Eur. Phys. J. Appl. Phys.
**2009**, 47, 31301. [Google Scholar] [CrossRef] - D’Alessandro, J.; Fodor, P.S. Use of grooved microchannels to improve the performance of membrane-less fuel cells. Fuel Cells
**2014**, 14, 818–826. [Google Scholar] [CrossRef] - Jiang, F.; Drese, K.S.; Hardt, S.; Küpper, M.; Schönfeld, F. Helical flows and chaotic mixing in curved micro channels. AIChE J.
**2004**, 50, 2297–2305. [Google Scholar] [CrossRef] - Camesasca, M.; Kaufman, M.; Manas-Zloczower, I. Quantifying fluid mixing with the Shannon entropy. Macromol. Theory Simul.
**2006**, 15, 595–607. [Google Scholar] [CrossRef] - Fodor, P.S.; Vyhnalek, B.; Kaufman, M. Entropic Evaluation of Dean Flow Micromixers. In Proceedings of the 2013 COMSOL Conference, Boston, MA, USA, 9–11 October 2013; Available online: https://www.comsol.com/paper/entropic-evaluation-of-dean-flow-micromixers-15053 (accessed on 21 January 2018).
- Alemaskin, K.; Manas-Zloczover, I.; Kaufman, M. Entropic analysis of color homogeneity. Polym. Eng. Sci.
**2005**, 45, 1031–1038. [Google Scholar] [CrossRef] - Xia, Y.; Whitesides, G.M. Soft lithography. Annu. Rev. Mater. Sci.
**1998**, 28, 153–184. [Google Scholar] [CrossRef]

**Figure 1.**3D and top views of the standard serpentine micromixer with a rectangular cross-section defined by W = 200 μm and H = 100 μm.

**Figure 2.**3D and top views of the new serpentine micromixer design employing non-rectangular cross-sections. As shown above, the orientation of the cross-section of the mixer is changed after each turn of the serpentine.

**Figure 3.**Velocity magnitude (cross-sectional maps) (

**top**) and concentration distribution (surface map) (

**bottom**) along the channel of a standard serpentine micromixer (R

_{in}= W/2 = 100 μm, and Re = 20).

**Figure 4.**Velocity magnitude (cross-sectional maps) (

**top**) and concentration distribution (surface map) (

**bottom**) along the channel of a non-rectangular cross-section serpentine micromixer (R

_{in}= W/2 = 100 μm, and Re = 20).

**Figure 5.**Transversal concentration distribution at various positions along the channel for (

**a**) a standard serpentine micromixer and (

**b**) a serpentine micromixer with a non-rectangular cross-section (Re = 20). The concentration is mapped at both the midpoint and the end of each mixing unit.

**Figure 6.**Transversal concentration distribution at various positions along the channel for (

**a**) a standard serpentine micromixer and (

**b**) a serpentine micromixer with a non-rectangular cross-section (Re = 100).

**Figure 7.**Position dependence of the mixing index along serpentine micromixers at different Reynolds numbers: (

**a**) a standard serpentine microchannel and (

**b**) a modified channel.

**Figure 8.**Arrow plots of the velocity field for the two mixers design: (

**a**) a standard micromixer and (

**b**) a non-rectangular cross-section micromixer. Both sets of data are collected for the 2nd mixing cycle at Re = 20. The left plots are for data collected at the midpoint of the first leg of the mixing cycle, while the right images are the corresponding plots for the second leg of the mixing cycle.

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Clark, J.; Kaufman, M.; Fodor, P.S. Mixing Enhancement in Serpentine Micromixers with a Non-Rectangular Cross-Section. *Micromachines* **2018**, *9*, 107.
https://doi.org/10.3390/mi9030107

**AMA Style**

Clark J, Kaufman M, Fodor PS. Mixing Enhancement in Serpentine Micromixers with a Non-Rectangular Cross-Section. *Micromachines*. 2018; 9(3):107.
https://doi.org/10.3390/mi9030107

**Chicago/Turabian Style**

Clark, Joshua, Miron Kaufman, and Petru S. Fodor. 2018. "Mixing Enhancement in Serpentine Micromixers with a Non-Rectangular Cross-Section" *Micromachines* 9, no. 3: 107.
https://doi.org/10.3390/mi9030107