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Modelling the effect of soluble surfactants on droplet deformation and breakup in simple shear flow

Published online by Cambridge University Press:  24 July 2024

Yan Ba ,
Haihu Liu Open the ORCID record for Haihu Liu [Opens in a new window] ,
Wenqiang Li  and
Wenjing Yang
Yan Ba
Affiliation:
School of Astronautics, Northwestern Polytechnical University, 127 West Youyi Road, Xi'an 710072, PR China
Haihu Liu*
Affiliation:
School of Energy and Power Engineering, Xi'an Jiaotong University, 28 West Xianning Road, Xi'an 710049, PR China
Wenqiang Li*
Affiliation:
School of Astronautics, Northwestern Polytechnical University, 127 West Youyi Road, Xi'an 710072, PR China
Wenjing Yang
Affiliation:
School of Astronautics, Northwestern Polytechnical University, 127 West Youyi Road, Xi'an 710072, PR China
*
†Email addresses for correspondence: haihu.liu@xjtu.edu.cn, lwq@nwpu.edu.cn
†Email addresses for correspondence: haihu.liu@xjtu.edu.cn, lwq@nwpu.edu.cn
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Abstract

A hybrid lattice Boltzmann and finite difference method is applied to study the influence of soluble surfactants on droplet deformation and breakup in simple shear flow. First, the influence of bulk surfactant parameters on droplet deformation in two-dimensional shear flow is investigated, and the surfactant solubility is found to influence droplet deformation by changing average interface surfactant concentration and non-uniform effects induced by non-uniform interfacial tension and Marangoni forces. In addition, the droplet deformation first increases and then decreases with Biot number, increases significantly with adsorption number $k$ and decreases with Péclet number or adsorption depth; and among the parameters, $k$ is the most influential one. Then, we consider three-dimensional shear flow and investigate the roles of surfactants on droplet deformation and breakup for different capillary numbers and viscosity ratios. Results show that in the soluble case with $k=0.429$, the droplet exhibits nearly the same deformation as in the insoluble case due to the balance between surfactant adsorption and desorption; upon increasing $k$ from 0.429 to 1, the average interface surfactant concentration is greatly enhanced, leading to significant increase in droplet deformation. The critical capillary number of droplet breakup $C{{a}_{cr}}$ is identified for varying viscosity ratios in clean, insoluble and soluble ($k=0.429$ and 1) systems. As the viscosity ratio increases, $C{{a}_{cr}}$ first decreases and then increases rapidly in all systems. The addition of surfactants always favours droplet breakup, and increasing solubility or $k$ could further reduce $C{{a}_{cr}}$ by increasing average interface surfactant concentration and local surfactant concentration near the neck during the necking stage.

JFM classification

Micro-/Nano-fluid dynamics: Microfluidics Multiphase and Particle-laden Flows: Multiphase flow Drops and Bubbles: Drops
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 988 , 10 June 2024 , A41
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Copyright
© The Author(s), 2024. Published by Cambridge University Press

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References

Ba, Y., Liu, H., Li, Q., Kang, Q. & Sun, J. 2016 Multiple-relaxation-time color-gradient lattice Boltzmann model for simulating two-phase flows with high density ratio. Phys. Rev. E 94, 023310.CrossRefGoogle ScholarPubMed
Ba, Y., Liu, H., Li, W. & Yang, W. 2023 A hybrid lattice Boltzmann and finite difference method for two-phase flows with soluble surfactants. Comput. Maths Applics. (submitted). arXiv:2311.15242.Google Scholar
Ba, Y., Liu, H., Sun, J. & Zheng, R. 2015 Three dimensional simulations of droplet formation in symmetric and asymmetric t-junctions using the color-gradient lattice Boltzmann model. Intl J. Heat Mass Transfer 90, 931–947.CrossRefGoogle Scholar
Baret, J.-C. 2012 Surfactants in droplet-based microfluidics. Lab on a Chip 12 (3), 422–433.CrossRefGoogle ScholarPubMed
Bazhekov, I.B., Anderson, P.D. & Meijer, H.E.H. 2006 Numerical investigation of the effect of insoluble surfactants on drop deformation and breakup in simple shear flow. J. Colloid Interface Sci. 298 (1), 369–394.CrossRefGoogle Scholar
Chen, K.-Y. & Lai, M.-C. 2014 A conservative scheme for solving coupled surface-bulk convection-diffusion equations with an application to interfacial flows with soluble surfactant. J. Comput. Phys. 257 (A), 1–18.CrossRefGoogle Scholar
Etienne, G., Kessler, M. & Amstad, E. 2017 Influence of fluorinated surfactant composition on the stability of emulsion drops. Macromol. Chem. Phys. 218, 1600365.CrossRefGoogle Scholar
Far, E.K., Gorakifard, M. & Fattahi, E. 2021 Multiphase phase-field lattice Boltzmann method for simulation of soluble surfactants. Symmetry 13 (6), 1019.Google Scholar
Feigl, K., Megias-Alguacil, D., Fischer, P. & Windhab, E.J. 2007 Simulation and experiments of droplet deformation and orientation in simple shear flow with surfactants. Chem. Engng Sci. 62 (12), 3242–3258.CrossRefGoogle Scholar
Grace, H.P. 1982 Dispersion phenomena in high-viscosity immiscible fluid systems and application of static mixers as dispersion devices in such systems. Chem. Engng Commun. 14 (3–6), 225–277.CrossRefGoogle Scholar
Guo, Z.L., Zheng, C.G. & Shi, B.C. 2002 Discrete lattice effects on the forcing term in the lattice Boltzmann method. Phys. Rev. E 65, 046308.CrossRefGoogle ScholarPubMed
Halliday, I., Hollis, A.P. & Care, C.M. 2007 Lattice Boltzmann algorithm for continuum multicomponent flow. Phys. Rev. E 76, 026708.CrossRefGoogle ScholarPubMed
Hu, W.-F., Lai, M.-C. & Misbah, C. 2018 A coupled immersed boundary and immersed interface method for interfacial flows with soluble surfactant. Comput. Fluids 168, 201–215.CrossRefGoogle Scholar
James, A.J. & Lowengrub, J. 2004 A surfactant-conserving volume-of-fluid method for interfacial flows with insoluble surfactant. J. Comput. Phys. 201 (2), 685–722.CrossRefGoogle Scholar
Janssen, P.J.A., Vananroye, A., Van Puyvelde, P., Moldenaers, P. & Anderson, P.D. 2010 Generalized behavior of the breakup of viscous drops in confinements. J. Rheol. 54 (5), 1047–1060.CrossRefGoogle Scholar
de Jesus, W.C., Roma, A.M., Pivello, M.R., Villar, M.M. & da Silveira-Neto, A. 2015 A 3D front-tracking approach for simulation of a two-phase fluid with insoluble surfactant. J. Comput. Phys. 281, 403–420.CrossRefGoogle Scholar
Kalogirou, A. & Blyth, M.G. 2020 Nonlinear dynamics of two-layer channel flow with soluble surfactant below or above the critical micelle concentration. J. Fluid Mech. 900, A7.CrossRefGoogle Scholar
Khatri, S. & Tornberg, A.-K. 2014 An embedded boundary method for soluble surfactants with interface tracking for two-phase flows. J. Comput. Phys. 256, 768–790.CrossRefGoogle Scholar
Komrakova, A.E., Shardt, O., Eskin, D. & Derksen, J.J. 2015 Effects of dispersed phase viscosity on drop deformation and breakup in inertial shear flow. Chem. Engng Sci. 126, 150–159.CrossRefGoogle Scholar
Kruijt-Stegeman, Y.W., van de Vosse, F.N. & Meijer, H.E.H. 2004 Droplet behavior in the presence of insoluble surfactants. Phys. Fluids 16 (8), 2785–2796.CrossRefGoogle Scholar
Latva-Kokko, M. & Rothman, D.H. 2005 Diffusion properties of gradient-based lattice Boltzmann models of immiscible fluids. Phys. Rev. E 71, 056702.CrossRefGoogle ScholarPubMed
Li, J., Renardy, Y.Y. & Renardy, M. 2000 Numerical simulation of breakup of a viscous drop in simple shear flow through a volume-of-fluid method. Phys. Fluids 12 (2), 269–282.CrossRefGoogle Scholar
Lishchuk, S.V., Care, C.M. & Halliday, I. 2003 Lattice Boltzmann algorithm for surface tension with greatly reduced microcurrents. Phys. Rev. E 67, 036701.CrossRefGoogle ScholarPubMed
Liu, H., Ba, Y., Wu, L., Li, Z., Xi, G. & Zhang, Y. 2018 A hybrid lattice Boltzmann and finite difference method for droplet dynamics with insoluble surfactants. J. Fluid Mech. 837, 381–412.CrossRefGoogle Scholar
Liu, H., Valocchi, A.J. & Kang, Q. 2012 Three-dimensional lattice Boltzmann model for immiscible two-phase flow simulations. Phys. Rev. E 85, 046309.Google ScholarPubMed
Liu, H., Zhang, J., Ba, Y., Wang, N. & Wu, L. 2020 Modelling a surfactant-covered droplet on a solid surface in three-dimensional shear flow. J. Fluid Mech. 897, A33.CrossRefGoogle Scholar
Liu, H. & Zhang, Y. 2010 Phase-field modeling droplet dynamics with soluble surfactants. J. Comput. Phys. 229 (24), 9166–9187.CrossRefGoogle Scholar
Maffettone, P.L. & Minale, M. 1998 Equation of change for ellipsoidal drops in viscous flows. J. Non-Newtonian Fluid Mech. 78 (2–3), 227–241.CrossRefGoogle Scholar
McClements, D.J. 2007 Critical review of techniques and methodologies for characterization of emulsion stability. Crit. Rev. Food Sci. Nutr. 47 (7), 611–649.CrossRefGoogle ScholarPubMed
Megías-Alguacil, D., Fischer, P. & Windhab, E.J. 2006 Determination of the interfacial tension of low density difference liquid–liquid systems containing surfactants by droplet deformation. Chem. Engng Sci. 61 (5), 1386–1394.CrossRefGoogle Scholar
Milliken, W.J. & Leal, L.G. 1994 The influence of surfactant on the deformation and breakup of a viscous drop-the effect of surfactant solubility. J. Colloid Interface Sci. 166 (2), 275–285.CrossRefGoogle Scholar
Muradoglu, M. & Tryggvason, G. 2008 A front-tracking method for computation of interfacial flows with soluble surfactants. J. Comput. Phys. 227 (4), 2238–2262.CrossRefGoogle Scholar
Muradoglu, M. & Tryggvason, G. 2014 Simulations of soluble surfactants in 3D multiphase flow. J. Comput. Phys. 274, 737–757.CrossRefGoogle Scholar
Nganguia, H., Young, Y.N., Vlahovska, P.M., Blawzdziewicz, J., Zhang, J. & Lin, H. 2013 Equilibrium electro-deformation of a surfactant-laden viscous drop. Phys. Fluids 25, 092106.CrossRefGoogle Scholar
Riechers, B., Maes, F., Akoury, E., Semin, B., Gruner, P. & Baret, J.-C. 2016 Surfactant adsorption kinetics in microfluidics. Proc. Natl Acad. Sci. USA 113 (41), 11465–11470.CrossRefGoogle ScholarPubMed
Sackmann, E.K., Fulton, A.L. & Beebe, D.J. 2014 The present and future role of microfluidics in biomedical research. Nature 507 (7491), 181–189.CrossRefGoogle ScholarPubMed
Severino, M., Campana, D.M. & Giavedoni, M.D. 2005 Effects of a surfactant on the motion of a confined gas–liquid interface. The influence of the Péclet number. Latin Am. Appl. Res. 35 (3), 225–232.Google Scholar
Shang, X., Luo, Z., Bai, B., He, L. & Hu, G. 2024 Front tracking simulation of droplet displacement on solid surfaces by soluble surfactant-driven flows. Phys. Rev. Fluids 9, 014002.CrossRefGoogle Scholar
Shapira, M. & Haber, S. 1990 Low Reynolds number motion of a droplet in shear flow including wall effects. Intl J. Multiphase Flow 16 (2), 305–321.CrossRefGoogle Scholar
Sharma, D., Sharma, J., Arya, R.K., Ahuja, S. & Agnihotri, S. 2018 Surfactant enhanced drying of waterbased poly(vinyl alcohol) coatings. Prog. Org. Coat. 125, 443–452.CrossRefGoogle Scholar
Shin, S., Chergui, J., Juric, D., Kahouadji, L., Matar, O.K. & Craster, R.V. 2018 A hybrid interface tracking – level set technique for multiphase flow with soluble surfactant. J. Comput. Phys. 359, 409–435.CrossRefGoogle Scholar
van der Sman, R.G.M. & Meinders, M.B.J. 2016 Analysis of improved lattice Boltzmann phase field method for soluble surfactants. Comput. Phys. Commun. 199, 12–21.CrossRefGoogle Scholar
Soligo, G., Roccon, A. & Soldati, A. 2019 Coalescence of surfactant-laden drops by phase field method. J. Comput. Phys. 376, 1292–1311.CrossRefGoogle Scholar
Soligo, G., Roccon, A. & Soldati, A. 2020 Deformation of clean and surfactant-laden droplets in shear flow. Meccanica 55 (2, SI), 371–386.CrossRefGoogle Scholar
Stone, H.A. 1990 A simple derivation of the time-dependent convective-diffusion equation for surfactant transport along a deforming interface. Phys. Fluids A 2 (1), 111–112.CrossRefGoogle Scholar
Taylor, G.I. 1934 The formation of emulsions in definable fields of flow. Proc. R. Soc. Lond. A 146 (A858), 0501–0523.Google Scholar
Teigen, K.E., Song, P., Lowengrub, J. & Voigt, A. 2011 A diffuse-interface method for two-phase flows with soluble surfactants. J. Comput. Phys. 230 (2), 375–393.CrossRefGoogle ScholarPubMed
Van Puyvelde, P., Vananroye, A., Cardinaels, R. & Moldenaers, P. 2008 Review on morphology development of immiscible blends in confined shear flow. Polymer 49 (25), 5363–5372.CrossRefGoogle Scholar
Vananroye, A., Van Puyvelde, P. & Moldenaers, P. 2006 Effect of confinement on droplet breakup in sheared emulsions. Langmuir 22 (9), 3972–3974.CrossRefGoogle ScholarPubMed
Vananroye, A., Van Puyvelde, P. & Moldenaers, P. 2007 Effect of confinement on the steady-state behavior of single droplets during shear flow. J. Rheol. 51 (1), 139–153.CrossRefGoogle Scholar
Vananroye, A., Van Puyvelde, P. & Moldenaers, P. 2011 Deformation and orientation of single droplets during shear flow: combined effects of confinement and compatibilization. Rheol. Acta 50 (3), 231–242.CrossRefGoogle Scholar
Xu, J.J., Li, Z.L., Lowengrub, J. & Zhao, H.K. 2006 A level-set method for interfacial flows with surfactant. J. Comput. Phys. 212 (2), 590–616.CrossRefGoogle Scholar
Xu, J.-J., Shi, W. & Lai, M.-C. 2018 A level-set method for two-phase flows with soluble surfactant. J. Comput. Phys. 353, 336–355.CrossRefGoogle Scholar
Xu, J.-J., Yang, Y. & Lowengrub, J. 2012 A level-set continuum method for two-phase flows with insoluble surfactant. J. Comput. Phys. 231 (17), 5897–5909.CrossRefGoogle Scholar
Xu, J.J. & Zhao, H.K. 2003 An Eulerian formulation for solving partial differential equations along a moving interface. J. Sci. Comput. 19 (1–3), 573–594.CrossRefGoogle Scholar
Yang, J., Li, Y. & Kim, J. 2022 A correct benchmark problem of a two-dimensional droplet deformation in simple shear flow. Mathematics 10 (21), 4092.CrossRefGoogle Scholar
Zhang, J., Liu, H., Wei, B., Hou, J. & Jiang, F. 2021 a Pore-scale modeling of two-phase flows with soluble surfactants in porous media. Energy Fuels 35 (23), 19374–19388.CrossRefGoogle Scholar
Zhang, J., Shu, S., Guan, X. & Yang, N. 2021 b Regime mapping of multiple breakup of droplets in shear flow by phase-field lattice Boltzmann simulation. Chem. Engng Sci. 240 (31), 116673.CrossRefGoogle Scholar
Zhang, Y., Xu, J. & He, X. 2018 Effect of surfactants on the deformation of single droplet in shear flow studied by dissipative particle dynamics. Mol. Phys. 116 (14), 1851–1861.CrossRefGoogle Scholar
Zhao, Q., Ren, W. & Zhang, Z. 2021 A thermodynamically consistent model and its conservative numerical approximation for moving contact lines with soluble surfactants. Comput. Meth. Appl. Mech. Engng 385, 114033.CrossRefGoogle Scholar
Zhou, W., Xing, Y., Liu, X. & Yan, Y. 2023 Modeling of droplet dynamics with soluble surfactant by multi-relaxation-time phase-field lattice Boltzmann method. Phys. Fluids 35, 012109.Google Scholar
Zhu, G., Kou, J., Sun, S., Yao, J. & Li, A. 2019 Numerical approximation of a phase-field surfactant model with fluid flow. J. Sci. Comput. 80 (1), 223–247.CrossRefGoogle Scholar
Zhu, G., Kou, J., Yao, J., Li, A. & Sun, S. 2020 A phase-field moving contact line model with soluble surfactants. J. Comput. Phys. 405 (15), 109170.CrossRefGoogle Scholar
Zong, Y., Zhang, C., Liang, H., Wang, L. & Xu, J. 2020 Modeling surfactant-laden droplet dynamics by lattice Boltzmann method. Phys. Fluids 32, 122105.CrossRefGoogle Scholar