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Viscous to inertial coalescence of liquid lenses: A lattice Boltzmann investigation

Thomas Scheel1,2,*, Qingguang Xie1,†, Marcello Sega3,1,‡, and Jens Harting1,4,§

  • 1Helmholtz Institute Erlangen-Nürnberg for Renewable Energy (IEK-11), Forschungszentrum Jülich, Cauerstr. 1, D-91058 Erlangen, Germany
  • 2Department of Physics, Friedrich-Alexander-Universität Erlangen-Nürnberg, Cauerstr. 1, D-91058 Erlangen, Germany
  • 3Department of Chemical Engineering, University College London, London WC1E 7JE, United Kingdom
  • 4Department of Chemical and Biological Engineering and Department of Physics, Friedrich-Alexander-Universität Erlangen-Nürnberg, Cauerstr. 1, D-91058 Erlangen, Germany

  • *t.scheel@fz-juelich.de
  • †q.xie@fz-juelich.de
  • ‡m.sega@ucl.ac.uk
  • §j.harting@fz-juelich.de
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Phys. Rev. Fluids 8, 074201 – Published 10 July, 2023

DOI: https://doi.org/10.1103/PhysRevFluids.8.074201

Abstract

Liquid lens coalescence is an important mechanism involved in many industrial and scientific applications. It has been investigated both theoretically and experimentally, yet it is numerically very challenging to obtain consistent results over the wide ranges of surface tension and viscosity values that are necessary to capture the asymptotic temporal behavior in the viscous and inertial limits. We report results of massively parallel simulations based on the color gradient lattice Boltzmann method, which overcome these limitations, and investigate the scaling laws of both regimes. For the two-dimensional case we find good agreement with the similarity solution of the thin-sheet equation, where in the viscous regime the connecting bridge grows linearly with time and in the inertial regime proportionally to t2/3. In three dimensions, the viscous growth of the bridge also exhibits a linear time dependence, while in the inertial regime the growth of both the bridge height and the bridge width is proportional to t1/2.

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References (62)

  1. Z. Hu and R. C. Srivastava, Evolution of raindrop size distribution by coalescence, breakup, and evaporation: Theory and observations, J. Atmos. Sci. 52, 1761 (1995).
  2. A. Garaizar, J. R. Espinosa, J. A. Joseph, and R. Collepardo-Guevara, Kinetic interplay between droplet maturation and coalescence modulates shape of aged protein condensates, Sci. Rep. 12, 4390 (2022).
  3. J. I. Martínez-Herrera and J. J. Derby, Viscous sintering of spherical particles via finite element analysis, J. Am. Ceram. Soc. 78, 645 (1995).
  4. J. Eggers, J. R. Lister, and H. A. Stone, Coalescence of liquid drops, J. Fluid Mech. 401, 293 (1999).
  5. A. L. Yarin, G. G. Chase, W. Liu, S. V. Doiphode, and D. H. Reneker, Liquid drop growth on a fiber, AIChE J. 52, 217 (2006).
  6. S. Bansal, V. von Arnim, T. Stegmaier, and H. Planck, Effect of fibrous filter properties on the oil-in-water-emulsion separation and filtration performance, J. Hazard. Mater. 190, 45 (2011).
  7. H. Wijshoff, Drop dynamics in the inkjet printing process, Curr. Opin. Colloid Interface Sci. 36, 20 (2018).
  8. M. A. Hack, M. Costalonga, T. Segers, S. Karpitschka, H. Wijshoff, and J. H. Snoeijer, Printing wet-on-wet: Attraction and repulsion of drops on a viscous film, Appl. Phys. Lett. 113, 183701 (2018).
  9. J. Sun, B. Bao, M. He, H. Zhou, and Y. Song, Recent advances in controlling the depositing morphologies of inkjet droplets, ACS Appl. Mater. Interfaces 7, 28086 (2015).
  10. T. M. Eggenhuisen, Y. Galagan, A. F. K. V. Biezemans, T. M. W. L. Slaats, W. P. Voorthuijzen, S. Kommeren, S. Shanmugam, J. P. Teunissen, A. Hadipour, W. J. H. Verhees et al., High efficiency, fully inkjet printed organic solar cells with freedom of design, J. Mater. Chem. A 3, 7255 (2015).
  11. H. Eggers, F. Schackmar, T. Abzieher, Q. Sun, U. Lemmer, Y. Vaynzof, B. S. Richards, G. Hernandez-Sosa, and U. W. Paetzold, Inkjet-printed micrometer-thick perovskite solar cells with large columnar grains, Adv. Energy Mater. 10, 1903184 (2020).
  12. P. Maisch, K. C. Tam, D. Jang, M. Steinberger, F. Yang, C. J. Brabec, and H.-J. Egelhaaf, Inkjet printed organic and perovskite photovoltaics–review and perspectives, in Organic Flexible Electronics, Woodhead Publishing Series in Electronic and Optical Materials (Woodhead Publishing, 2021), pp. 305–333.
  13. O. Ronsin and J. Harting, Formation of crystalline bulk heterojunctions in organic solar cells: Insights from phase-field simulations, ACS Appl. Mater. Interfaces 14, 49785 (2022).
  14. R. Detsch, S. Blob, T. Zehnder, and A. R. Boccaccini, Evaluation of cell inkjet printing technique for biofabrication, BioNanoMaterials 17, 185 (2016).
  15. S. Ruiz-Alonso, I. Villate-Beitia, I. Gallego, M. Lafuente-Merchan, G. Puras, L. Saenz-Del-Burgo, and J. L. Pedraz, Current insights into 3D bioprinting: An advanced approach for eye tissue regeneration, Pharmaceutics 13, 308 (2021).
  16. P. Kumar, S. Ebbens, and X. Zhao, Inkjet printing of mammalian cells—Theory and applications, Bioprinting 23, e00157 (2021).
  17. D. Lohse, Fundamental fluid dynamics challenges in inkjet printing, Annu. Rev. Fluid Mech. 54, 349 (2022).
  18. M. Kuang, J. Wang, B. Bao, F. Li, L. Wang, L. Jiang, and Y. Song, Inkjet printing patterned photonic crystal domes for wide viewing-angle displays by controlling the sliding three phase contact line, Adv. Opt. Mater. 2, 34 (2014).
  19. L. Duchemin, J. Eggers, and C. Josserand, Inviscid coalescence of drops, J. Fluid Mech. 487, 167 (2003).
  20. D. G. A. L. Aarts, H. N. W. Lekkerkerker, H. Guo, G. H. Wegdam, and D. Bonn, Hydrodynamics of Droplet Coalescence, Phys. Rev. Lett. 95, 164503 (2005).
  21. J. D. Paulsen, J. C. Burton, and S. R. Nagel, Viscous to Inertial Crossover in Liquid Drop Coalescence, Phys. Rev. Lett. 106, 114501 (2011).
  22. J. D. Paulsen, J. C. Burton, S. R. Nagel, S. Appathurai, M. T. Harris, and O. A. Basaran, The inexorable resistance of inertia determines the initial regime of drop coalescence, Proc. Natl. Acad. Sci. USA 109, 6857 (2012).
  23. J. E. Sprittles and Y. D. Shikhmurzaev, A parametric study of the coalescence of liquid drops in a viscous gas, J. Fluid Mech. 753, 279 (2014).
  24. P. Zimmermann, A. Mawbey, and T. Zeiner, Calculation of droplet coalescence in binary Liquid–Liquid systems: An incompressible Cahn–Hilliard/Navier–Stokes approach using the non-random two-liquid model, J. Chem. Eng. Data 65, 1083 (2020).
  25. A. Menchaca-Rocha, A. Martínez-Dávalos, R. Núñez, S. Popinet, and S. Zaleski, Coalescence of liquid drops by surface tension, Phys. Rev. E 63, 046309 (2001).
  26. W. D. Ristenpart, P. M. McCalla, R. V. Roy, and H. A. Stone, Coalescence of Spreading Droplets on a Wettable Substrate, Phys. Rev. Lett. 97, 064501 (2006).
  27. R. D. Narhe, D. A. Beysens, and Y. Pomeau, Dynamic drying in the early-stage coalescence of droplets sitting on a plate, Europhys. Lett. 81, 46002 (2008).
  28. M. W. Lee, D. K. Kang, S. S. Yoon, and A. L. Yarin, Coalescence of two drops on partially wettable substrates, Langmuir 28, 3791 (2012).
  29. A. Eddi, K. G. Winkels, and J. H. Snoeijer, Influence of Droplet Geometry on the Coalescence of Low Viscosity Drops, Phys. Rev. Lett. 111, 144502 (2013).
  30. P. R. Kaneelil, A. A. Pahlavan, N. Xue, and H. A. Stone, Three-Dimensional Self-Similarity of Coalescing Viscous Drops in the Thin-Film Regime, Phys. Rev. Lett. 129, 144501 (2022).
  31. J. C. Burton and P. Taborek, Role of Dimensionality and Axisymmetry in Fluid Pinch-Off and Coalescence, Phys. Rev. Lett. 98, 224502 (2007).
  32. M. A. Hack, W. Tewes, Q. Xie, C. Datt, K. Harth, J. Harting, and J. H. Snoeijer, Self-Similar Liquid Lens Coalescence, Phys. Rev. Lett. 124, 194502 (2020).
  33. H. Chen, X. Pan, Q. Nie, Q. Ma, H. Fang, and Z. Yin, Probing the coalescence of non-Newtonian droplets on a substrate, Phys. Fluids 34, 032109 (2022).
  34. C. Klopp and A. Eremin, On droplet coalescence in quasi-two-dimensional fluids, Langmuir 36, 10615 (2020).
  35. X. Xia, C. He, and P. Zhang, Universality in the viscous-to-inertial coalescence of liquid droplets, Proc. Natl. Acad. Sci. USA 116, 23467 (2019).
  36. K. P. N. Datadien, G. Di Staso, H. M. A. Wijshoff, and F. Toschi, A quantitative comparison of physical accuracy and numerical stability of lattice Boltzmann color gradient and pseudopotential multicomponent models for microfluidic applications, Commun. Comput. Phys. 32, 450 (2022).
  37. P. Español and P. B. Warren, Perspective: Dissipative particle dynamics, J. Chem. Phys. 146, 150901 (2017).
  38. H. Liu, Q. Kang, C. R. Leonardi, S. Schmieschek, A. Narváez, B. D. Jones, J. R. Williams, A. J. Valocchi, and J. Harting, Multiphase lattice Boltzmann simulations for porous media applications, Comput. Geosci. 20, 777 (2016).
  39. P. J. A. Janssen and P. D. Anderson, Modeling film drainage and coalescence of drops in a viscous fluid, Macromol. Mater. Eng. 296, 238 (2011).
  40. J. Kamp, J. Villwock, and M. Kraume, Drop coalescence in technical liquid/liquid applications: A review on experimental techniques and modeling approaches, Rev. Chem. Eng. 33, 1 (2017).
  41. M. Latva-Kokko and D. H. Rothman, Diffusion properties of gradient-based lattice Boltzmann models of immiscible fluids, Phys. Rev. E 71, 056702 (2005).
  42. T. Reis and T. N. Phillips, Lattice Boltzmann model for simulating immiscible two-phase flows, J. Phys. A: Math. Theor. 40, 4033 (2007).
  43. S. Leclaire, M. Reggio, and J.-Y. Trépanier, Progress and investigation on lattice Boltzmann modeling of multiple immiscible fluids or components with variable density and viscosity ratios, J. Comput. Phys. 246, 318 (2013).
  44. A. Montessori, M. Lauricella, M. La Rocca, S. Succi, E. Stolovicki, R. Ziblat, and D. Weitz, Regularized lattice Boltzmann multicomponent models for low capillary and Reynolds microfluidics flows, Comput. Fluids 167, 33 (2018).
  45. X. Shan and G. Doolen, Multicomponent lattice-Boltzmann model with interparticle interaction, J. Stat. Phys. 81, 379 (1995).
  46. Q. Xie and J. Harting, From dot to ring: the role of friction on the deposition pattern of a drying colloidal suspension droplet, Langmuir 34, 5303 (2018).
  47. D. Hessling, Q. Xie, and J. Harting, Diffusion dominated evaporation in multicomponent lattice Boltzmann simulations, J. Chem. Phys. 146, 054111 (2017).
  48. R. Benzi, S. Succi, and M. Vergassola, The lattice Boltzmann equation: Theory and applications, Phys. Rep. 222, 145 (1992).
  49. A. K. Gunstensen, D. H. Rothman, S. Zaleski, and G. Zanetti, Lattice Boltzmann model of immiscible fluids, Phys. Rev. A 43, 4320 (1991).
  50. S. Leclaire, A. Parmigiani, O. Malaspinas, B. Chopard, and J. Latt, Generalized three-dimensional lattice Boltzmann color-gradient method for immiscible two-phase pore-scale imbibition and drainage in porous media, Phys. Rev. E 95, 033306 (2017).
  51. P. L. Bhatnagar, E. P. Gross, and M. Krook, A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems, Phys. Rev. 94, 511 (1954).
  52. S. Leclaire, M. Reggio, and J.-Y. Trépanier, Isotropic color gradient for simulating very high-density ratios with a two-phase flow lattice Boltzmann model, Comput. Fluids 48, 98 (2011).
  53. G. Tryggvason, B. Bunner, A. Esmaeeli, D. Juric, N. Al-Rawahi, W. Tauber, J. Han, S. Nas, and Y.-J. Jan, A front-tracking method for the computations of multiphase flow, J. Comput. Phys. 169, 708 (2001).
  54. C. Hirt and B. Nichols, Volume of fluid (VOF) method for the dynamics of free boundaries, J. Comput. Phys. 39, 201 (1981).
  55. S. Osher and J. A. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys. 79, 12 (1988).
  56. S. Bogner, U. Rüde, and J. Harting, Curvature estimation from a volume of fluid indicator function for the simulation of surface tension and wetting with a free surface lattice Boltzmann method, Phys. Rev. E 93, 043302 (2016).
  57. X. Shan and H. Chen, Lattice Boltzmann model for simulating flows with multiple phases and components, Phys. Rev. E 47, 1815 (1993).
  58. M. R. Swift, W. R. Osborn, and J. M. Yeomans, Lattice Boltzmann Simulation of Nonideal Fluids, Phys. Rev. Lett. 75, 830 (1995).
  59. T. Erneux and S. H. Davis, Nonlinear rupture of free films, Phys. Fluids Fluid Dyn. 5, 1117 (1993).
  60. C. Coreixas, B. Chopard, and J. Latt, Comprehensive comparison of collision models in the lattice Boltzmann framework: Theoretical investigations, Phys. Rev. E 100, 033305 (2019).
  61. L. Wang and J. Sun, Lattice Boltzmann modeling for the coalescence between a free droplet in gases and a sessile droplet on wettable substrate with contact angle hysteresis, Proc. Inst. Mech. Eng. Part C: J. Mech. Eng. Sci. 232, 431 (2018).
  62. A. Montessori, M. Lauricella, N. Tirelli, and S. Succi, Mesoscale modelling of near-contact interactions for complex flowing interfaces, J. Fluid Mech. 872, 327 (2019).

Outline

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Phys. Rev. Fluids 8, 074201– Published 10 July, 2023
  • Received 13 January 2023
  • Accepted 2 June 2023
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