Abstract
The coating dynamics of a drying paint film with a yield stress is studied. The liquid is modeled as a binary mixture with one volatile component, solvent, and one nonvolatile component, resin. When the solvent has a different surface tension than the resin, solvent evaporation can lead to the creation of surface tension gradients which can potentially overcome the yield stress and dramatically affect the flow history. Using the lubrication approximations to derive the flux of the liquid film parallel to the substrate, we find that the presence of the yield stress causes several distinct flow regimes. The total flux of each of these regimes is summed, and using the continuity equation we derive an evolution equation giving the height of the free surface as a function of the distance along the substrate and time. The resulting equations are discretized and solved numerically using finite differences. High order derivative is treated implicitly, allowing for large time steps and reducing the computational requirements. We find that the presence of a yield stress greatly affects the leveling behavior of the coating. Critical yield stresses exist that can cause maximal leveling of the coating film.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Abbreviations
- x :
-
Direction \(\parallel\) to substrate (cm)
- y :
-
Direction \(\perp\) to substrate (cm)
- t :
-
Time (s)
- h :
-
Coating thickness (cm)
- p :
-
Pressure in coating (dyne/cm2)
- \(\sigma\) :
-
Surface tension of coating (dyne/cm)
- \(\rho\) :
-
Density of coating (g/cm3)
- \(\tau\) :
-
Shear stress in coating (dyne/cm2)
- u :
-
Velocity in x direction (cm/s)
- \(\theta\) :
-
Inclination angle of substrate (rad)
- L :
-
Length of substrate (cm)
- E :
-
Evaporation rate (cm/s)
- Q :
-
Flux in x direction (cm2/s)
- D :
-
Diffusion of resin into the mixture (cm2/s)
- c :
-
Concentration of resin in the mixture (−)
- R :
-
Radius of curvature of surface (cm)
- \(\tau _{0}\) :
-
Yield stress of liquid (dyne/cm2)
- g :
-
Acceleration of gravity (cm/s2)
- \(P'\) :
-
\(-p_x +\rho g \sin \theta\) (dyne/cm3)
- K :
-
Viscosity (poise)
- \(Y_{L}\) :
-
y value where \(|\tau | < -\tau _0\) (cm)
- \(Y_{H}\) :
-
y value where \(|\tau | \le \tau _0\) (cm)
- \(Y_{1}\) :
-
y value where \(\tau =-\tau _0\) (cm)
- \(Y_{2}\) :
-
y value where \(\tau =\tau _0\) (cm)
- \(\varDelta h\) :
-
\(h(x=0)-h(x=L/2)\) (cm)
References
Orchard, SE, “On the Levelling in Viscous Liquids and Gels.” Appl. Sci. Res. A, 11 451–464 (1962)
Overdiep, WS, “The Levelling of Paints.” Prog. Org. Coat., 14 159–175 (1986)
Landau, L, Lifshitz, V. Fluid Mechanics. Pergamon, Oxford, 1959
Schwartz, LW, “Theoretical and Numerical Modeling of Coating Flow on Simple and Complex Substrates Including Rheology, Drying and Marangoni Effects.” In: Durst, F, Raszillier, H (eds.) Advances in Coating and Drying of Thin Films, pp. 105–128. Shaker Verlag, Aachen (1999)
Eres, HM, Weidner, DE, Schwartz, LW, “Three-Dimensional Direct Numerical Simulation of Surface-Tension-Gradient Effects on the Leveling of an Evaporating Multi-Component Fluid.” Langmuir, 4 1859–1871 (1998)
Weidner, DE, Schwartz, LW, Eres, HM, “Suppression of Drop Formation In a Model Paint Film.” Chem. Prod. Process Model., 2 (2) 1–32 (2007)
Howison, SD, Moriarty, JA, Ockendon, JR, Terrill, JA, Wilson, SK, “A Mathematical Model for Drying Paint Layers.” J. Eng. Math., 32 (4) 377–394 (1997)
Yamamura, MW, Mawatan, M, Kage, H. “Nonuniform Thinning of Polymeric Coatings Under Marangoni Stress.” J. Chem. Eng., 43 40–45 (2010)
Seeler, F, Hager, C, Tiedje, O, Schneider, M, “Simulations and Experimental Investigation of Paint Film Leveling.” J. Coat. Technol. Res., 14 (4) 767–781 (2017)
Balmforth, N, Frigaard, I, Olverez, G, “Yielding to Stress: Recent Developments in Viscoplastic Fluid Mechanics.” Ann. Rev. Fluid Mech., 46 (1) 121–146 (2014)
Eley, R, “Applied Rheology and Architectural Coating Performance.” J. Coat. Technol. Res., 16 (2) 273–305 (2019)
Farina, A., Fusi, L, Rosso, F, “Retrieving the Bingham Model from a Bi-Viscous Model: Some Explanatory Remarks.” Appl. Math. Lett., 27 611–633 (2014)
Balmforth, N, “Viscoplastic Fluids: From Theory to Application.” In: Ovalez G, and Hormozi S (eds.) Lectures on Visco-Plastic Fluid Mechanics, pp. 41–82. CISM. Springer (2019)
Herschel, WH, Bulkley R, “Measurement of Consistency as Applied to Rubber-Benzene Solutions.” Proc. Am. Soc. Test. Mater., 26 621–633 (1926)
Oron, A, Davis SH, Bankoff, SG, “Long-Scale Evolution of Thin Liquid Film.” Rev. Mod. Phys., 69 931–980 (1997)
Probstein, RF, Physicochemical Hydrodynamics, 2nd Edition. Wiley, New York (1994)
Patton, TC, Paint Flow and Pigment Dispersion. Wiley, New York (1979)
Hewitt, I, Balmforth, N, “Viscoplastic Lubrication Theory with Application to Bearings and the Washboard Instability of a Planing Plate.” J. Non-Newton. Fluid Mech., 169, 74–90 (2012)
Tanner, LH, “The Spreading of Silicone Oil Drops on Horizontal Surfaces.” J. Phys. D, 12 1473–1484 (1979)
Schwartz, LW, Cairncross, RA, Weidner, DE, “Anomalous Behavior During Leveling of Thin Coating Layers with Surfactant.” Phys. Fluids, 8 (7) 1693–1969 (1996)
Uhlherr, PH, Park, KH, Andrews JRG, “Yield Stress from Behavior on an Inclined Plane.” In: Mena, B, Garcia-Rejon, A, Rangel-Nagaile, C (eds.) Advances in Rheology, pp. 183–190 (1984)
Acknowledgments
Many thanks to L. W. Schwartz for suggesting this problem while he was my thesis advisor, and helping me through some of the initial steps.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Weidner, D.E. Leveling of a model paint film with a yield stress. J Coat Technol Res 17, 851–863 (2020). https://doi.org/10.1007/s11998-019-00260-z
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11998-019-00260-z