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Scaling laws of the maximum spreading factor for impact of nanodroplets on solid surfaces

Published online by Cambridge University Press:  22 February 2022

Yi-Feng Wang ,
Yi-Bo Wang ,
Xin He ,
Ben-Xi Zhang ,
Yan-Ru Yang ,
Xiao-Dong Wang Open the ORCID record for Xiao-Dong Wang [Opens in a new window]  and
Duu-Jong Lee
Yi-Feng Wang
Affiliation:
State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources, North China Electric Power University, Beijing 102206, PR China Research Center of Engineering Thermophysics, North China Electric Power University, Beijing 102206, PR China
Yi-Bo Wang
Affiliation:
State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources, North China Electric Power University, Beijing 102206, PR China Research Center of Engineering Thermophysics, North China Electric Power University, Beijing 102206, PR China
Xin He
Affiliation:
State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources, North China Electric Power University, Beijing 102206, PR China Research Center of Engineering Thermophysics, North China Electric Power University, Beijing 102206, PR China
Ben-Xi Zhang
Affiliation:
State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources, North China Electric Power University, Beijing 102206, PR China Research Center of Engineering Thermophysics, North China Electric Power University, Beijing 102206, PR China
Yan-Ru Yang
Affiliation:
State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources, North China Electric Power University, Beijing 102206, PR China Research Center of Engineering Thermophysics, North China Electric Power University, Beijing 102206, PR China
Xiao-Dong Wang*
Affiliation:
State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources, North China Electric Power University, Beijing 102206, PR China Research Center of Engineering Thermophysics, North China Electric Power University, Beijing 102206, PR China
Duu-Jong Lee*
Affiliation:
Department of Chemical Engineering, National Taiwan University, Taipei 106, Taiwan Department of Mechanical Engineering, City University of Hong Kong, Kowloon Tang 999077, Hong Kong
*
†Email addresses for correspondence: wangxd99@gmail.com, djlee@ntu.edu.tw
†Email addresses for correspondence: wangxd99@gmail.com, djlee@ntu.edu.tw
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Abstract

This study investigates the dynamics of low-viscosity nanodroplets impacting surfaces with static contact angles from θ = 73° to 180° via molecular dynamics (MD) simulations. Two typical morphologies of impacting nanodroplets are observed at the maximum spreading state, a Hertz-ball-like in a low-Weber-number range and a thin-film-like in a high-Weber-number range. Only inertial and capillary forces dominate the impact for the former, whereas viscous force also becomes dominant for the latter. Regardless of morphologies at the maximum spreading state, the ratio of spreading time to contact time always remains constant on an ideal superhydrophobic surface with θ = 180°. With the help of different kinematic approximations of the spreading time and scaling laws of the contact time, scaling laws of the maximum spreading factor ${\beta _{max}}\sim W{e^{1/5}}$ in the low-Weber-number range (capillary regime) and ${\beta _{max}}\sim W{e^{2/3}}R{e^{ - 1/3}}$ (or ${\beta _{max}}\sim W{e^{1/2}}O{h^{1/3}}$) in the high-Weber-number range (cross-over regime) are obtained. Here, We, Re, and Oh are the Weber number, Reynolds number, and Ohnesorge number, respectively. Although the scaling laws are proposed only for the ideal superhydrophobic surface, they are tested valid for θ over 73° owing to the ignorable zero-velocity spreading effect. Furthermore, combining the two scaling laws leads to an impact number, $W{e^{3/10}}O{h^{1/3}} = 2.1$. This impact number can be used to determine whether viscous force is ignorable for impacting nanodroplets, thereby distinguishing the capillary regime from the cross-over regime.

JFM classification

Drops and Bubbles: Drops

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JFM Papers
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Journal of Fluid Mechanics , Volume 937 , 25 April 2022 , A12
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© The Author(s), 2022. Published by Cambridge University Press

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References

Alizadeh, A., Yamada, M., R, L.I., Shang, W., Otta, S., Zhong, S., L, G.E., Dhinojwala, A., Conway, K.R., Bahadur, V., Vinciquerra, A.J., Stephens, B. & Blohm, M.L. 2012 Dynamics of ice nucleation on water repellent surfaces. Langmuir 28, 3180–3186.CrossRefGoogle ScholarPubMed
Antonini, C., Amirfazli, A. & Marengo, M. 2012 Drop impact and wettability: from hydrophilic to superhydrophobic surfaces. Phys. Fluids 24, 102104.CrossRefGoogle Scholar
Attané, P., Girard, F. & Morin, V. 2007 An energy balance approach of the dynamics of drop impact on a solid surface. Phys. Fluids 19, 012101.CrossRefGoogle Scholar
Bartolo, D., Josserand, C. & Bonn, D. 2005 Retraction dynamics of aqueous drops upon impact on non-wetting surfaces. J. Fluid Mech. 545, 329–338.CrossRefGoogle Scholar
Benz, M., Asperger, A., Hamester, M., Welle, A., Heissler, S. & Levkin, P.A. 2020 A combined high-throughput and high-content platform for unified on-chip synthesis, characterization and biological screening. Nat. Commun. 11, 5391.CrossRefGoogle ScholarPubMed
Chandra, S. & Avedisian, C.T. 1991 On the collision of a droplet with a solid surface. Proc. R. Soc. A 432, 13–41.Google Scholar
Clanet, C., Béguin, C., Richard, D. & Quéré, D. 2004 Maximal deformation of an impacting drop. J. Fluid Mech. 517, 199–208.CrossRefGoogle Scholar
Du, J., Wang, X., Li, Y., Min, Q. & Wu, X. 2021 Analytical consideration for the maximum spreading factor of liquid droplet impact on a smooth solid surface. Langmuir 37, 7582–7590.CrossRefGoogle ScholarPubMed
Eggers, J., Fontelos, M.A., Josserand, C. & Zaleski, S. 2010 Drop dynamics after impact on a solid wall: theory and simulations. Phys. Fluids 22, 772–584.CrossRefGoogle Scholar
Galliker, P., Schneider, J., Eghlidi, H., Kress, S., Sandoghdar, V. & Poulikakos, D. 2012 Direct printing of nanostructures by electrostatic autofocussing of ink nanodroplets. Nat. Commun. 3, 890.CrossRefGoogle ScholarPubMed
Grjeu, M., Gouin, H. & Saccomandi, G. 2013 Scaling Navier-Stokes equation in nanotubes. Phys. Fluids 25, 082003.CrossRefGoogle Scholar
Glasscott, M.W., Pendergast, A.D., Goines, S., Bishop, A.R., Hoang, A.T., Renault, C. & Dick, J.E. 2019 Electrosynthesis of high-entropy metallic glass nanoparticles for designer, multi-functional electrocatalysis. Nat. Commun. 10, 2650.CrossRefGoogle ScholarPubMed
Horbach, J. & Succi, S. 2006 Lattice Boltzmann versus molecular dynamics simulation of nanoscale hydrodynamic flows. Phys. Rev. Lett. 96, 224503.CrossRefGoogle ScholarPubMed
Huebner, A., Sharma, S., Srisa-Art, M., Hollfelder, F., Edel, J.B. & de Mello, A.J. 2008 Microdroplets: a sea of applications? Lab on a Chip 8, 1244–1254.CrossRefGoogle ScholarPubMed
Jacobson, L.C., Kirby, R.M. & Molinero, V. 2014 How short is too short for the interactions of a water potential? Exploring the parameter space of a coarse-grained water model using uncertainty quantification. J. Phys. Chem. B 118, 8190–8202.CrossRefGoogle ScholarPubMed
Jambovane, S.R., Nune, S.K., Kelly, R.T., Mcgrail, B.P., Wang, Z., Nandasiri, M.I., Katipamula, S., Trader, C. & Schaef, H.T. 2016 Continuous, one-pot synthesis and post-synthetic modification of nanoMOFs using droplet nanoreactors. Sci. Rep. 6, 36657.CrossRefGoogle ScholarPubMed
Josserand, C. & Thoroddsen, S.T. 2016 Drop impact on a solid surface. Annu. Rev. Fluid Mech. 48, 365–391.CrossRefGoogle Scholar
Kannan, R. & Sivakumar, D. 2008 Drop impact process on a hydrophobic grooved surface. Colloids Surf. A Physicochem. Eng. Asp. 317, 694–704.CrossRefGoogle Scholar
Kawase, T., Shimoda, T., Newsome, C., Sirringhaus, H. & Friend, R.H. 2003 Inkjet printing of polymer thin film transistors. Thin Solid Films 438-439, 279–287.CrossRefGoogle Scholar
Kim, J. 2007 Spray cooling heat transfer: the state of the art. Int. J. Heat Fluid Flow 28, 753–767.CrossRefGoogle Scholar
Kim, H.Y. & Chun, J.H. 2001 The recoiling of liquid droplets upon collision with solid surfaces. Phys. Fluids 13, 643–659.CrossRefGoogle Scholar
Koishi, T., Yasuoka, K. & Zeng, X.C. 2017 Molecular dynamics simulation of water nanodroplet bounce back from flat and nanopillared surface. Langmuir 33, 10184–10192.CrossRefGoogle ScholarPubMed
Kou, J., Wang, Y., Liu, X., Zhang, X., Chen, G., Xu, X., Bao, J., Yang, K. & Yuwen, L. 2020 Continuous preparation of antimony nanocrystals with near infrared photothermal property by pulsed laser ablation in liquids. Sci. Rep. 10, 15095.CrossRefGoogle ScholarPubMed
Kreder, M.J., Alvarenga, J., Kim, P. & Aizenberg, J. 2016 Design of anti-icing surfaces: smooth, textured or slippery? Nat. Rev. Mater. 1, 15003.CrossRefGoogle Scholar
Laan, N., de Bruin, K.G., Bartolo, D., Josserand, C. & Bonn, D. 2014 Maximum diameter of impacting liquid droplets. Phys. Rev. Appl. 2, 044018.CrossRefGoogle Scholar
Landau, L.D. & Lifshits, E.M. 1965 Theory of Elasticity. Fizmatlit Publishers Russia.Google Scholar
Lee, J.B., Laan, N., de Bruin, K.G., Skantzaris, G., Shahidzadeh, N., Derome, D., Carmeliet, J. & Bonn, D. 2015 Universal rescaling of drop impact on smooth and rough surfaces. J. Fluid Mech. 786, R4.CrossRefGoogle Scholar
Li, B.X., Li, X.H. & Chen, M. 2017 Spreading and breakup of nanodroplet impinging on surface. Phys. Fluids 29, 012003.CrossRefGoogle Scholar
Li, X.H., Zhang, X.X. & Chen, M. 2015 Estimation of viscous dissipation in nanodroplet impact and spreading. Phys. Fluids 27, 052007.CrossRefGoogle Scholar
Liang, G. & Mudawar, I. 2017 Review of spray cooling – part 1: single-phase and nucleate boiling regimes, and critical heat flux. Int. J. Heat Mass Transf. 115, 1174–1205.CrossRefGoogle Scholar
Madejski, J. 1976 Solidification of droplets on a cold surface. Int. J. Heat Mass Transfer 19, 1009–1013.CrossRefGoogle Scholar
Molinero, V. & Moore, E.B. 2009 Water modeled as an intermediate element between carbon and silicon. J. Phys. Chem. B 113, 4008–4016.CrossRefGoogle ScholarPubMed
Montero de Hijes, P., Sanz, E., Joly, L., Valeriani, C. & Caupin, F. 2018 Viscosity and self-diffusion of supercooled and stretched water from molecular dynamics simulations. J. Chem. Phys. 149, 094503.CrossRefGoogle ScholarPubMed
Okumura, K., Chevy, F., Richard, D., Quéré, D. & Clanet, C. 2003 Water spring: a model for bouncing drops. Europhys. Lett. 62, 237–243.CrossRefGoogle Scholar
Padilla Espinosa, I.M., Jacobs, T.D. & Martini, A. 2021 Evaluation of force fields for molecular dynamics simulations of platinum in bulk and nanoparticle forms. J. Chem. Theory Comput. 17, 4486–4498.CrossRefGoogle ScholarPubMed
Pasandideh-Fard, M., Qiao, Y.M., Chandra, S. & Mostaghimi, J. 1996 Capillary effects during droplet impact on a solid surface. Phys. Fluids 8, 650–659.CrossRefGoogle Scholar
Richard, D., Clanet, C. & Quéré, D. 2002 Contact time of a bouncing drop. Nature 417, 811.CrossRefGoogle ScholarPubMed
Roisman, I.V., Rioboo, R. & Tropea, C. 2002 Normal impact of a liquid drop on a dry surface: model for spreading and receding. Proc. R. Soc. A 458, 1411–1430.CrossRefGoogle Scholar
Teare, D.O.H., Spanos, C.G., Ridley, P., Kinmond, E.J., Roucoules, V. & Badyal, J.P.S. 2002 Pulsed plasma deposition of super-hydrophobic nanospheres. Chem. Mater. 14, 4566–4571.CrossRefGoogle Scholar
Ukiwe, C. & Kwok, D.Y. 2005 On the maximum spreading diameter of impacting droplets on well-prepared solid surfaces. Langmuir 21, 666–673.CrossRefGoogle ScholarPubMed
Vaikuntanathan, V., Kannan, R. & Sivakumar, D. 2010 Impact of water drops onto the junction of a hydrophobic texture and a hydrophilic smooth surface. Colloids Surf. A Physicochem. Eng. Asp. 369, 65–74.CrossRefGoogle Scholar
Wang, Y.B., Wang, Y.F., Gao, S.R., Yang, Y.R., Wang, X.D. & Chen, M. 2020 a Universal model for the maximum spreading factor of impacting nanodroplets: from hydrophilic to hydrophobic surfaces. Langmuir 36, 9306–9316.CrossRefGoogle ScholarPubMed
Wang, Y.-B., Wang, Y.-F., Wang, X., Zhang, B.-X., Yang, Y.-R., Wang, X.-D. & Chen, M. 2021 a Splash of impacting nanodroplets on solid surfaces. Phys. Rev. Fluids 6, 094201.CrossRefGoogle Scholar
Wang, Y.-F., Wang, Y.-B., Xie, F.-F., Liu, J.-Y., Wang, S.-L., Yang, Y.-R., Gao, S.-R. & Wang, X.-D. 2020 b Spreading and retraction kinetics for impact of nanodroplets on hydrophobic surfaces. Phys. Fluids 32, 092005.CrossRefGoogle Scholar
Wang, Y.B., Wang, X.D., Yang, Y.R. & Chen, M. 2019 The maximum spreading factor for polymer nanodroplets impacting a hydrophobic solid surface. J. Phys. Chem. C 123, 12841–12850.CrossRefGoogle Scholar
Wang, Y.-B., Wang, Y.-F., Yang, Y.-R., Wang, X.-D. & Chen, M. 2021 b Spreading time of impacting nanodroplets. J. Phys. Chem. B 125, 5630–5635.CrossRefGoogle ScholarPubMed
Wildeman, S., Visser, C.W., Sun, C. & Lohse, D. 2016 On the spreading of impacting drops. J. Fluid Mech. 805, 636–655.CrossRefGoogle Scholar
Worthington, A.M. 1876 On the forms assumed by drops of liquids falling vertically on a horizontal plate. Pro. R. Soc. London 25, 261–271.Google Scholar
Xie, F.F., Lu, G., Wang, X.D. & Wang, D.Q. 2018 Enhancement of coalescence-induced nanodroplet jumping on superhydrophobic surfaces. Langmuir 34, 11195–11203.CrossRefGoogle ScholarPubMed
Xie, F.F., Lv, S.H., Yang, Y.R. & Wang, X.D. 2020 Contact time of a bouncing nanodroplet. J Phys Chem Lett. 11, 2818–2823.CrossRefGoogle ScholarPubMed
Yaguchi, H., Yano, T. & Fujikawa, S. 2010 Molecular dynamics study of vapor-liquid equilibrium state of an argon nanodroplet and its vapor. J. Fluid. Sci. Technol. 5, 180–191.CrossRefGoogle Scholar
Yarin, A.L. 2006 Drop impact dynamics: splashing, spreading, receding, bouncing…. Annu. Rev. Fluid Mech. 38, 159–192.CrossRefGoogle Scholar
Zhu, Y., Piehowski, P.D., Zhao, R., Chen, J., Shen, Y., Moore, R.J., Shukla, A.K., Petyuk, V.A., Campbell-Thompson, M., Mathews, C.E., Smith, R.D., Qian, W.-J. & Kelly, R.T. 2018 Nanodroplet processing platform for deep and quantitative proteome profiling of 10-100 mammalian cells. Nat. Commun. 9, 882.CrossRefGoogle ScholarPubMed