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Axisymmetric evolution of thin films in spin coating

Published online by Cambridge University Press:  17 March 2025

Xue-Li Wang ,
Jian Qin  and
Peng Gao Open the ORCID record for Peng Gao [Opens in a new window]
Xue-Li Wang
Affiliation:
Department of Modern Mechanics, University of Science and Technology of China, Hefei, Anhui 230026, PR China
Jian Qin
Affiliation:
Institute of Applied Physics and Computational Mathematics, Beijing 100088, PR China
Peng Gao*
Affiliation:
Department of Modern Mechanics, University of Science and Technology of China, Hefei, Anhui 230026, PR China
*
Corresponding author: Peng Gao, [email protected]
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Abstract

Spin coating is the process of generating a uniform coating film on a substrate by centrifugal forces during rotation. In the framework of lubrication theory, we investigate the axisymmetric film evolution and contact-line dynamics in spin coating on a partially wetting substrate. The contact-line singularity is regularized by imposing a Navier slip model. The interface morphology and the contact-line movement are obtained by numerical solution and asymptotic analysis of the lubrication equation. The results show that the evolution of the liquid film can be classified into two modes, depending on the rotational speed. At lower speeds, the film eventually reaches an equilibrium state, and we provide a theoretical description of how the equilibrium state can be approached through matched asymptotic expansions. At higher speeds, the film exhibits two or three distinct regions: a uniform thinning film in the central region, an annular ridge near the contact line, and a possible Landau–Levich–Derjaguin-type (LLD-type) film in between that has not been reported previously. In particular, the LLD-type film occurs only at speeds slightly higher than the critical value for the existence of the equilibrium state, and leads to the decoupling of the uniform film and the ridge. It is found that the evolution of the ridge can be well described by a two-dimensional quasi-steady analysis. As a result, the ridge volume approaches a constant and cannot be neglected to predict the variation of the contact-line radius. The long-time behaviours of the film thickness and the contact radius agree with derived asymptotic solutions.

JFM classification

Interfacial Flows (free surface): Capillary flows Interfacial Flows (free surface): Contact lines Interfacial Flows (free surface): Thin films
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 1007 , 25 March 2025 , A56
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Copyright
© The Author(s), 2025. Published by Cambridge University Press

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References

Benilov, E.S. & Benilov, M.S. 2015 A thin drop sliding down an inclined plate. J. Fluid Mech. 773, 75–102.CrossRefGoogle Scholar
Bertozzi, A.L. & Brenner, M.P. 1997 Linear stability and transient growth in driven contact lines. Phys. Fluids 9 (3), 530–539.CrossRefGoogle Scholar
Boettcher, K.E.R. & Ehrhard, P. 2014 Contact-line instability of liquids spreading on top of rotating substrates. Eur. J. Mech. (B/ Fluids) 43, 33–44.CrossRefGoogle Scholar
Buckingham, R., Shearer, M. & Bertozzi, A. 2003 Thin film traveling waves and the Navier slip condition. SIAM J. Appl. Maths 63 (2), 722–744.Google Scholar
Derjaguin, B.V. 1943 Thickness of liquid layer adhering to walls of vessels on their emptying. Acta Physicochim. USSR 20, 349–352.Google Scholar
Du, H.W. & Gao, P. 2022 Asymptotic theory of gas entrainment in a two-phase Couette flow. Eur. J. Mech. (B/Fluids) 94, 148–154.CrossRefGoogle Scholar
Dussan, V.E.B. & Davis, S.H. 1974 On the motion of a fluid–fluid interface along a solid surface. J. Fluid Mech. 65 (1), 71–95.CrossRefGoogle Scholar
Eggers, J. 2005 Existence of receding and advancing contact lines. Phys. Fluids 17 (8), 082106.CrossRefGoogle Scholar
Emslie, A.G., Bonner, F.T. & Peck, L.G. 1958 Flow of a viscous liquid on a rotating disk. J. Appl. Phys. 29 (5), 858–862.CrossRefGoogle Scholar
Fraysse, N. & Homsy, G.M. 1994 An experimental study of rivulet instabilities in centrifugal spin coating of viscous Newtonian and non-Newtonian fluids. Phys. Fluids 6 (4), 1491–1504.CrossRefGoogle Scholar
Gao, P., Li, L., Feng, J.J., Ding, H. & Lu, X.Y. 2016 Film deposition and transition on a partially wetting plate in dip coating. J. Fluid Mech. 791, 358–383.CrossRefGoogle Scholar
Greenspan, H.P. 1978 On the motion of a small viscous droplet that wets a surface. J. Fluid Mech. 84 (1), 125–143.CrossRefGoogle Scholar
Hocking, L.M. 1983 The spreading of a thin drop by gravity and capillarity. Q. J. Mech. Appl. Maths 36 (1), 55–69.CrossRefGoogle Scholar
Holloway, K.E., Habdas, P., Semsarillar, N., Burfitt, K. & de Bruyn, J.R. 2007 Spreading and fingering in spin coating. Phys. Rev. E 75 (4), 046308.CrossRefGoogle ScholarPubMed
Huh, C. & Scriven, L.E. 1971 Hydrodynamic model of steady movement of a solid/liquid/fluid contact line. J. Colloid Interface Sci. 35 (1), 85–101.CrossRefGoogle Scholar
Huppert, H.E. 1982 Flow and instability of a viscous current down a slope. Nature 300 (5891), 427–429.CrossRefGoogle Scholar
Landau, L.D. & Levich, B.V. 1942 Dragging of a liquid by a moving plate. Acta Physicochim. USSR 17, 42–54.Google Scholar
Lopes, P.A., Santos, B.C., de Almeida, A.T. & Tavakoli, M. 2021 Reversible polymer-gel transition for ultra-stretchable chip-integrated circuits through self-soldering and self-coating and self-healing. Nat. Commun. 12 (1), 4666.CrossRefGoogle ScholarPubMed
Mahmoodi, S. 2023 Microstructure and resolution of etched patterns of Photoresist (AZP4620) layers spin-coated under artificially elevated gravity accelerations by two-axis spin coating technology. J. Mater. Sci. 34 (6), 550.Google Scholar
McKinley, I.S. & Wilson, S.K. 2001 The linear stability of a ridge of fluid subject to a jet of air. Phys. Fluids 13 (4), 872–883.CrossRefGoogle Scholar
McKinley, I.S. & Wilson, S.K. 2002 The linear stability of a drop of fluid during spin coating or subject to a jet of air. Phys. Fluids 14 (1), 133–142.CrossRefGoogle Scholar
McKinley, I.S., Wilson, S.K. & Duffy, B.R. 1999 Spin coating and air-jet blowing of thin viscous drops. Phys. Fluids 11 (1), 30–47.CrossRefGoogle Scholar
Melo, F., Joanny, J.F. & Fauve, S. 1989 Fingering instability of spinning drops. Phys. Rev. Lett. 63 (18), 1958–1961.CrossRefGoogle ScholarPubMed
Moriarty, J.A., Schwartz, L.W. & Tuck, E.O. 1991 Unsteady spreading of thin liquid films with small surface tension. Phys. Fluids A 3 (5), 733–742.CrossRefGoogle Scholar
Oron, A., Davis, S.H. & Bankoff, S.G. 1997 Long-scale evolution of thin liquid films. Rev. Mod. Phys. 69 (3), 931–980.CrossRefGoogle Scholar
Pan, Y., Wang, Z.B., Zhao, X.Y., Deng, W.W. & Xia, H.H. 2022 On axisymmetric dynamic spin coating with a single drop of ethanol. J. Fluid Mech. 951, A30.CrossRefGoogle Scholar
Qin, J., Lu, W. & Gao, P. 2024 A numerical method for dynamic wetting using mesoscopic contact-line models. Commu. Comput. Phys. 36, 977–995.Google Scholar
Santoro, R., Venkateswaran, S., Amadeo, F., Zhang, R., Brioschi, M., Callanan, A., Agrifoglio, M., Banfi, C., Bradley, M. & Pesce, M. 2018 Acrylate-based materials for heart valve scaffold engineering. Biomater. Sci. 6 (1), 154–167.CrossRefGoogle Scholar
Sarkın, A.S., Ekren, N. & Saglam, Ş. 2020 A review of anti-reflection and self-cleaning coatings on photovoltaic panels. Sol. Energy 199, 63–73.CrossRefGoogle Scholar
Savva, N. & Kalliadasis, S. 2009 Two-dimensional droplet spreading over topographical substrates. Phys. Fluids 21 (9), 092102.CrossRefGoogle Scholar
Savva, N., Rezgdnikov, A. & Colinet, P. 2017 Asymptotic analysis of the evaporation dynamics of partially wetting droplets. J. Fluid Mech. 824, 574–623.CrossRefGoogle Scholar
Schwartz, L.W. & Roy, R.V. 2004 Theoretical and numerical results for spin coating of viscous liquids. Phys. Fluids 16 (3), 569–584.CrossRefGoogle Scholar
Sibley, D.N., Nold, A. & Kalliadasis, S. 2015 The asymptotics of the moving contact line: cracking an old nut. J. Fluid Mech. 764, 445–462.CrossRefGoogle Scholar
Snoeijer, J.H. & Andreotti, B. 2013 Moving contact lines: scales, regimes, and dynamical transitions. Annu. Rev. Fluid Mech. 45 (1), 269–292.CrossRefGoogle Scholar
Snoeijer, J.H., Delon, G., Fermigier, M. & Andreotti, B. 2006 Avoided critical behavior in dynamically forced wetting. Phys. Rev. Lett. 96 (17), 174504.CrossRefGoogle ScholarPubMed
Spaid, M.A. & Homsy, G.M. 1996 Stability of Newtonian and viscoelastic dynamic contact lines. Phys. Fluids 8 (2), 460–478.CrossRefGoogle Scholar
Spaid, M.A. & Homsy, G.M. 1997 Stability of viscoelastic dynamic contact lines: an experimental study. Phys. Fluids 9 (4), 823–832.CrossRefGoogle Scholar
Troian, S.M., Herbolzheimer, E., Safran, S.A. & Joanny, J.F. 1989 Fingering instabilities of driven spreading films. Europhys. Lett. 10 (1), 25–30.CrossRefGoogle Scholar
Tuck, E.O. & Schwartz, L.W. 1990 A numerical and asymptotic study of some third-order ordinary differential equations relevant to draining and coating flows. SIAM Rev. 32 (3), 453–469.CrossRefGoogle Scholar
Wang, M.W. & Chou, F.C. 2001 Fingering instability and maximum radius at high rotational Bond number. J. Electrochem. Soc. 148 (5), G283.CrossRefGoogle Scholar
Wilson, S.D.R. 1982 The drag-out problem in film coating theory. J. Engng Maths 16 (3), 209–221.CrossRefGoogle Scholar
Wilson, S.K., Hunt, R. & Duffy, B.R. 2000 The rate of spreading in spin coating. J. Fluid Mech. 413, 65–88.CrossRefGoogle Scholar
Xia, Y.-T., Qin, J. & Gao, P. 2020 Dynamical wetting transition on a chemically striped incline. Phys. Fluids 32 (2), 022101.CrossRefGoogle Scholar
Zhao, Y., Song, S., Ren, X., Zhang, J., Lin, Q. & Zhao, Y. 2022 Supramolecular adhesive hydrogels for tissue engineering applications. Chem. Rev. 122 (6), 5604–5640.CrossRefGoogle ScholarPubMed