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Exact solution of the hydrodynamic focusing driven by hydrostatic pressure

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Microfluidic nanoprecipitation makes use of hydrodynamic focusing (HF) to accurately control the diffusive mixing of reactants. Both stability and precise handling of flow streams are essential for this application. However, flow stability is hardly attained when fluids are supplied by syringe pumps, due to the unavoidable fluctuations associated to the driving mechanical system. The alternative use of hydrostatic pressure is constantly increasing in microfluidic laboratories, though precise mathematical descriptions have not been reported so far. This paper presents a quantitative model for the HF driven by gravity in slit microchannels. The model analytically predicts the focusing width of the sample stream from the relative heights of the sample and sheath reservoirs. Fluids with different densities and viscosities are considered, which impact on the pressure provided by the hydrostatic columns, as well as on the flow pattern of HF. Theoretical predictions were successfully validated against experimental data. Flow-focusing experiments were carried out in hybrid PMMA/OCA chips with slit microchannels, using fluids with different physicochemical properties. Finally, a color reaction induced by pH-shift was implemented as a practical example. Different levels of diffusive mixing and reaction were attained along the focused stream by varying the relative heights of the fluid columns, precisely as predicted by calculations. The model thus provides a rational basis for the design of HF experiments using hydrostatics.

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  1. Ajdari A (2004) Steady flows in networks of microfluidic channels: building on the analogy with electrical circuits. C R Phys 5(5):539–546.

  2. Bazban-Shotorbani S, Dashtimoghadam E, Karkhaneh A, Hasani-Sadrabadi MM, Jacob KI (2016) Microfluidic directed synthesis of alginate nanogels with tunable pore size for efficient protein delivery. Langmuir 32(19):4996–5003.

  3. Berli CLA (2008) Equivalent circuit modeling of electrokinetically driven analytical microsystems. Microfluid Nanofluid 4(5):391–399.

  4. Berli CLA, Deiber JA (2004) Theoretical analysis of the gravity-driven capillary viscometers. Rev Sci Instrum 75(4):976–982.

  5. Bihi I, Vesperini D, Kaoui B, Goff AL (2019) Pressure-driven flow focusing of two miscible liquids. Phys Fluids 31(6):062001.

  6. Chen S-YC, Hung PJ, Lee PJ (2011) Microfluidic array for three-dimensional perfusion culture of human mammary epithelial cells. Biomed Microdevices 13(4):753–758.

  7. Gao W, Liu M, Chen S, Zhang C, Zhao Y (2019) Droplet microfluidics with gravity-driven overflow system. Chem Eng J 362:169–175.

  8. Giorello A, Minetti F, Nicastro A, Berli CLA (2020) The effect of gravity on microfluidic flow focusing. Sens Actuators B 307:127595.

  9. Gnyawali V, Saremi M, Kolios MC, Tsai SSH (2017) Stable microfluidic flow focusing using hydrostatics. Biomicrofluidics 11(3):034104.

  10. Groisman A, Enzelberger M, Quake SR (2003) Microfluidic memory and control devices. Science 300(5621):955–958.

  11. Gu H, Rong F, Tang B, Zhao Y, Fu D, Gu Z (2013) Photonic crystal beads from gravity-driven microfluidics. Langmuir 29(25):7576–7582.

  12. Hajian R, Hardt S (2015) Formation and lateral migration of nanodroplets via solvent shifting in a microfluidic device. Microfluid Nanofluid 19(6):1281–1296.

  13. Hu X, Cubaud T (2016) Inertial destabilization of highly viscous microfluidic stratifications. Phys Rev Fluids 1(4):044101.

  14. Huh D, Bahng JH, Ling Y, Wei H-H, Kripfgans OD, Fowlkes JB, Grotberg JB, Takayama S (2007) Gravity-driven microfluidic particle sorting device with hydrodynamic separation amplification. Anal Chem 79(4):1369–1376.

  15. Islam M, Natu R, Martinez-Duarte RA (2015) Study on the limits and advantages of using a desktop cutter plotter to fabricate microfluidic networks. Microfluid Nanofluid 19:973–985.

  16. Jacobson SC, Ramsey JM (1997) Electrokinetic focusing in microfabricated channel structures. Anal Chem 69(16):3212–3217.

  17. Karnik R, Gu F, Basto P, Cannizzaro C, Dean L, Kyei-Manu W, Langer R, Farokhzad OC (2008) Microfluidic platform for controlled synthesis of polymeric nanoparticles. Nano Lett 8(9):2906–2912.

  18. Kestin J, Khalifa EE, Correia RJ (1981) Tables of the dynamic and kinematic viscosity of aqueous NaCl solutions in the temperature range 20–150 °C and the pressure range 0.1–35 Mpa. J Phys Chem Ref Data 10:71–87.

  19. Kim C, Hwang DH, Lee S, Kim S-J (2016) Water-head pumps provide precise and fast microfluidic pumping and switching versus syringe pumps. Microfluid Nanofluid 20(1):4.

  20. Knight JB, Vishwanath A, Brody JP, Austin RH (1998) Hydrodynamic focusing on a silicon chip: mixing nanoliters in microseconds. Phys Rev Lett 80(17):3863–3866.

  21. Komeya M, Hayashi K, Nakamura H, Yamanaka H, Sanjo H, Kojima K, Sato T, Yao M, Kimura H, Fujii T, Ogawa T (2017) Pumpless microfluidic system driven by hydrostatic pressure induces and maintains mouse spermatogenesis in vitro. Sci Rep 7(1):15459.

  22. Larsen MU, Shapley NC (2007) Stream spreading in multilayer microfluidic flows of suspensions. Anal Chem 79(5):1947–1953.

  23. Lee SH, Rasaiah JC (2011) Proton transfer and the mobilities of the H+ and OH− ions from studies of a dissociating model for water. J Chem Phys 135:124505.

  24. Lee G-B, Chang C-C, Huang S-B, Yang R-J (2006) The hydrodynamic focusing effect inside rectangular microchannels. J Micromech Microeng 16(5):1024–1032.

  25. Leung MHM, Shen AQ (2018) Microfluidic assisted nanoprecipitation of PLGA nanoparticles for curcumin delivery to leukemia jurkat cells. Langmuir 34(13):3961–3970.

  26. Li Z, Mak SY, Sauret A, Shum HC (2014) Syringe-pump-induced fluctuation in all-aqueous microfluidic system implications for flow rate accuracy. Lab Chip 14(4):744–749.

  27. Lide DR (2004) CRC handbook of chemistry and physics, 85th edn. CRC Press, Boca Raton

  28. Liu K, Xiang J, Ai Z, Zhang S, Fang Y, Chen T, Zhou Q, Li S, Wang S, Zhang N (2017) PMMA microfluidic chip fabrication using laser ablation and low temperature bonding with Oca film and Loca. Microsyst Technol 23:1937–1942.

  29. Martínez Rivas CJ, Tarhini M, Badri W, Miladi K, Greige-Gerges H, Nazari QA, Galindo Rodríguez SA, Román RÁ, Fessi H, Elaissari A (2017) Nanoprecipitation process: from encapsulation to drug delivery. Int J Pharm 532(1):66–81.

  30. Middha E, Manghnani PN, Ng DZL, Chen H, Khan SA, Liu B (2019) Direct visualization of the ouzo zone through aggregation-induced dye emission for the synthesis of highly monodispersed polymeric nanoparticles. Mater Chem Front 3(7):1375–1384.

  31. Moon B-U, Abbasi N, Jones SG, Hwang DK, Tsai SSH (2016) Water-in-water droplets by passive microfluidic flow focusing. Anal Chem 88(7):3982–3989.

  32. Robertson G, Bushell TJ, Zagnoni M (2014) Chemically induced synaptic activity between mixed primary hippocampal co-cultures in a microfluidic system. Integr Biol 6(6):636–644.

  33. Sadeghi A (2019) Analytical solutions for mass transport in hydrodynamic focusing by considering different diffusivities for sample and sheath flows. J Fluid Mech 862:517–551.

  34. Schubert S, Delaney JJT, Schubert US (2011) Nanoprecipitation and nanoformulation of polymers: from history to powerful possibilities beyond poly(lactic acid). Soft Matter 7(5):1581–1588.

  35. Seo D-b, Agca Y, Feng ZC, Critser JK (2007) Development of sorting, aligning, and orienting motile sperm using microfluidic device operated by hydrostatic pressure. Microfluid Nanofluid 3(5):561–570.

  36. Simonnet C, Groisman A (2006) High-throughput and high-resolution flow cytometry in molded microfluidic devices. Anal Chem 78(16):5653–5663.

  37. Stiles T, Fallon R, Vestad T, Oakey J, Marr DWM, Squier J, Jimenez R (2005) Hydrodynamic focusing for vacuum-pumped microfluidics. Microfluid Nanofluid 1(3):280–283.

  38. Takayama S, McDonald JC, Ostuni E, Liang MN, Kenis PJA, Ismagilov RF, Whitesides GM (1999) Patterning cells and their environments using multiple laminar fluid flows in capillary networks. Proc Natl Acad Sci USA 96(10):5545.

  39. Takayama S, Ostuni E, LeDuc P, Naruse K, Ingber DE, Whitesides GM (2001) Laminar flows: subcellular positioning of small molecules. Nature 411:1016

  40. Thiele J, Steinhauser D, Pfohl T, Förster S (2010) Preparation of monodisperse block copolymer vesicles via flow focusing in microfluidics. Langmuir 26(9):6860–6863.

  41. Tresset G, Marculescu C, Salonen A, Ni M, Iliescu C (2013) Fine control over the size of surfactant–polyelectrolyte nanoparticles by hydrodynamic flow focusing. Anal Chem 85(12):5850–5856.

  42. Tripathi S, Chakravarty P, Agrawal A (2014) On non-monotonic variation of hydrodynamically focused width in a rectangular microchannel. Curr Sci 107(8):1260–1274

  43. Urteaga R, Elizalde E, Berli CLA (2018) Transverse solute dispersion in microfluidic paper-based analytical devices (μPADs). Analyst 143:2259–2266.

  44. Wang L, Sánchez S (2015) Self-assembly via microfluidics. Lab Chip 15(23):4383–4386.

  45. Wang X, Zhao D, Phan DTT, Liu J, Chen X, Yang B, Hughes CCW, Zhang W, Lee AP (2018) A hydrostatic pressure-driven passive micropump enhanced with siphon-based autofill function. Lab Chip 18(15):2167–2177.

  46. Wraight C (2006) Chance and design—proton transfer in water, channels and bioenergetic proteins. Biochem Biophys Acta 1757:886–912.

  47. Wu Z, Nguyen N-T (2005) Hydrodynamic focusing in microchannels under consideration of diffusive dispersion: theories and experiments. Sens Actuators B Chem 107(2):965–974.

  48. Xu J, Zhang S, Machado A, Lecommandoux S, Sandre O, Gu F, Colin A (2017) Controllable microfluidic production of drug-loaded PLGA nanoparticles using partially water-miscible mixed solvent microdroplets as a precursor. Sci Rep 7(1):4794.

  49. Yamada H, Yoshida Y, Terada N, Hagihara S, Komatsu T, Terasawa A (2008) Fabrication of gravity-driven microfluidic device. Rev Sci Instrum 79(12):124301.

  50. Zeng W, Jacobi I, Beck DJ, Li S, Stone HA (2015) Characterization of syringe-pump-driven induced pressure fluctuations in elastic microchannels. Lab Chip 15(4):1110–1115.

  51. Zhang K, Liang Q, Ma S, He T, Ai X, Hu P, Wang Y, Luo G (2010) A gravity-actuated technique for flexible and portable microfluidic droplet manipulation. Microfluid Nanofluid 9(4):995–1001.

  52. Zhu X, Yi Chu L, Chueh B-h, Shen M, Hazarika B, Phadke N, Takayama S (2004) Arrays of horizontally-oriented mini-reservoirs generate steady microfluidic flows for continuous perfusion cell culture and gradient generation. Analyst 129(11):1026–1031.

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The authors acknowledge the technical assitance of Maria Virginia Minetti with the plotter cutting. The authors thank the financial support from ASaCTeI, Santa Fe (IP-2018-049), ANPCyT (PICT-2015-1051), and Universidad Nacional del Litoral (CAI+D 504 20150100051LI), Argentina.

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Correspondence to Claudio L. A. Berli.

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Minetti, F., Giorello, A., Olivares, M.L. et al. Exact solution of the hydrodynamic focusing driven by hydrostatic pressure. Microfluid Nanofluid 24, 15 (2020).

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  • Gravity-driven flow
  • Hydrodynamic focusing
  • Hydrostatic pressure
  • Nanoprecipitation