# Numerical modeling of the spray coating of spinning bodies

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## Abstract

We derive the lubrication form of the fluid mechanical equations governing the motion of a thin liquid film on an arbitrarily curved, rotating, axisymmetric substrate. The resulting equations are discretized and then solved numerically using an efficient implicit finite difference algorithm. Non-Newtonian effects are included by incorporating an Ellis viscosity model. The primary application for this work is to model the spin coating of the interior of two-piece metal beverage cans, and we consider this problem in some depth. Specifically, we show how adjusting several parameters can eliminate one possible defect in the spin coating process: the tendency for droplets to detach from the substrate when the can is spun at high rotation rates.

## Keywords

Spin coating Spray coating Numerical modeling## References

- 1.Emslie, AG, Bonner, FT, Peck, LG, “Flow of a Viscous Liquid on a Rotating Disk.”
*J. Appl. Phys.*,**29**(5) 858–862 (1958)CrossRefGoogle Scholar - 2.Rehg, TG, Higgins, BG, “Spin Coating of Colloid Suspensions.”
*AICHE J.*,**38**(4) 489–501 (1992)CrossRefGoogle Scholar - 3.Peurrung, LM, Graves, DB, “Spin Coating Over Topography.”
*Transactions on Semiconductor Manufacturing*,**6**(1) 72–76 (1993)CrossRefGoogle Scholar - 4.Wilson, SK, Hunt, R, Duffy, BR, “The Rate of Spreading in Spin Coating.”
*J. Fluid Mech.*,**413**65–88 (2000)CrossRefGoogle Scholar - 5.McKinley IS, Wilson, SK, “The Linear Stability of a Drop of Fluid During Spin Coating or Subject to a Jet of Air.”
*Phys. Fluids*,**14**(1) 133–142 (2002)CrossRefGoogle Scholar - 6.Acrivos, A, Shah, MJ, Petersen, EE, “On the Flow of a Non-Newtonian Liquid on a Rotating Disk.”
*J. Appl. Phys.*,**31**963–968 (1984)CrossRefGoogle Scholar - 7.Lawrence, CJ, “The Mechanics of Spin Coating of Polymer Films.”
*Phys. Fluids*,**31**(10) 2786–2795 (1988)CrossRefGoogle Scholar - 8.Jenekhe, SA, Schuldt, SB, “Coating Flow of Non-Newtonian Fluids on a Flat Rotating Disk.”
*Ind. Eng. Chem. Fundam.*,**23**432–436 (1984)CrossRefGoogle Scholar - 9.Lawrence, CJ, Zhou, W, “Spin Coating of Non-Newtonian Fluids.”
*J. Non-Newton. Fluid Mech.*,**39**(2) 137–187 (1991)CrossRefGoogle Scholar - 10.Britten, JA, Thomas, IM, “Non-Newtonian Flow Effects During Spin Coating Large-Area Optical Coatings with Colloidal Suspensions.”
*J. Appl. Phys.*,**71**(2) 972–979 (1992)CrossRefGoogle Scholar - 11.Borkar, AV, Tsamopoulos, JA, Guptai, SA, Gupta, RK, “Spin-Coating of Viscoelastic and Nonvolatile Fluids Over a Planer Disk.”
*Phys. Fluids*,**6**(11) 3539–3553 (1994)CrossRefGoogle Scholar - 12.Cheung, KP, Grover, RY, Wang, R, Gurkovich, RC, Wangand, G, Scheinbeim, J, “Substrate Effect on the Thickness of Spin-Coated Ultrathin Polymer Film.”
*Appl. Phys. Lett.*,**87**(21) 4103 (2005)CrossRefGoogle Scholar - 13.Meyerhofer, D, “Characteristics of Resist Films Produced by Spinning.”
*J. Appl. Phys.*,**49**(7) 3993–3997 (1978)CrossRefGoogle Scholar - 14.Bornside, DE, Macosko, CW, Scriven, LE, “Spin Coating: One-Dimensional Model.”
*J. Appl. Phys.*,**66**(11) 5185–5193 (1989)CrossRefGoogle Scholar - 15.Sukanek, PC, “A Model for Spin Coating with Topography.”
*J. Electrochem. Soc.*,**136**(10) 3019–3026 (1989)CrossRefGoogle Scholar - 16.Overdiep, WS, “Levelling of Paints.” In: Spalding, DB (ed.)
*Physicochemical Hydrodynamics*, pp. 683–697, Advance Publications, London, 1977Google Scholar - 17.Sukanek, PC, “Dependence of Film Thickness on Speed in Spin Coating.”
*J. Electrochem. Soc.*,**138**(6) 1712–1719 (1991)CrossRefGoogle Scholar - 18.Levinson, WA, Arnold, A, DeHodgins, O, “Spin-Coating Behavior of Polymide Precursor Solutions.”
*Polym. Eng. Sci.*,**33**(15) 980–988 (1993)CrossRefGoogle Scholar - 19.Spaid, MA, Homsy, GM, “Viscolastic Free-surface Flows—Spin-coating and Dynamic Contact Lines.”
*J. Nonnewton. Fluid Mech.*,**55**249–281 (1994)CrossRefGoogle Scholar - 20.Desouza, MN, Leaver, KD, Eskiyerli, MH, “The Characterization Method Applied to Spin-coating in One-dimension.”
*Comput. Mater. Sci.*,**4**(3) 233–240 (1995)CrossRefGoogle Scholar - 21.Gu, J, Bullwinkel, MD, Campbell, GA, “Spin-Coating on Substrate with Topography.”
*J. Electrochem. Soc.*,**142**(3) 907–913 (1995)CrossRefGoogle Scholar - 22.Chang, Y,Chen, WC, “Theoretical Analysis on Spin Coating of Polyimide Precursor Solutions.”
*J. Electrochem. Soc.*,**148**(4) 77–81 (2001)CrossRefGoogle Scholar - 23.Cregan, V, O’Brien, SBG, “Extended Asymptotic Solutions to the Spin-coating Model with Small Evaporation.”
*Appl. Math. Comput.*,**223**76–87 (2013)Google Scholar - 24.Li, W, Carvalho, MS, Kumar, S, “Liquid-Film Coating on Topographically Patterned Rotating Cylinders.”
*Phys. Rev. Fluids*,**2**(2) 024001 (2017)CrossRefGoogle Scholar - 25.Mahmoodi, S, Guoqing, M, Khajavi, MN, “Two-Dimensional Spin Coating with Vertical Centrifugal Force and the Effect of Artificial Gravity on Surface Leveling.”
*J. Coat. Technol. Res.*,**13**(6) 1123–1137 (2016)CrossRefGoogle Scholar - 26.Greenberg, MD,
*Foundations of Applied Mathematics*. Prentice-Hall, Englewood Cliffs, 1978Google Scholar - 27.Schwartz, LW, Weidner, DE, Eley, RR, “An Analysis of the Effect of Surfactant on the Leveling Behavior of a Thin Liquid Coating Layer.”
*Langmuir*,**11**(10) 3690–3693 (1995)CrossRefGoogle Scholar - 28.Byrd, RB, Stewart, WE, Lightfoot EN,
*Transport Phenomena*. Wiley, New York, 1960Google Scholar - 29.Gerek, G, Coucher, RG,
*Coated Container*, 1976. US Patent 3,947,617Google Scholar - 30.Gerek, G, Coucher, RG,
*Container Coating Method*, 1977. US Patent 4,025,664Google Scholar - 31.Coucher, RG, Gerek, G,
*Container Coating Method*, 1980. US Patent 4,183,974Google Scholar - 32.Dervy, L, Gogola, M, Walsh, W,
*Bottom Profile for Drawn and Ironed Can Body*, 2007. US Patent App. 11/890,107Google Scholar