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Enhanced dip coating on a soft substrate

Vincent Bertin ^1,2,3,*, Jacco H. Snoeijer ³, Elie Raphaël ², and Thomas Salez ¹

¹Université Bordeaux, CNRS, LOMA, UMR 5798, 33405 Talence, France
²UMR CNRS Gulliver 7083, ESPCI Paris, PSL Research University, 75005 Paris, France
³Physics of Fluids Group, Faculty of Science and Technology, and Mesa+ Institute, University of Twente, 7500AE Enschede, The Netherlands

^*v.l.bertin@utwente.nl

Phys. Rev. Fluids 7, L102002 – Published 24 October, 2022

DOI: https://doi.org/10.1103/PhysRevFluids.7.L102002

Abstract

A solid, withdrawn from a wetting liquid bath, entrains a thin liquid film. This simple process, first described by Landau, Levich, and Derjaguin (LLD), is commonly observed in everyday life. It also plays a central role in liquid capture by animals, and is widely used for surface-coating purposes in industry. Motivated by the emerging interest in the mechanics of very soft materials, and in particular the resulting elastocapillary coupling, we develop a dip-coating model that accounts for the additional presence of a soft solid layer atop the rigid plate. The elastic response of this soft layer is described by a Winkler's foundation. Using a combination of numerical, scaling, and asymptotic-matching methods, we find a new softness-dependent power-law regime for the thickness of entrained liquid at a small capillary number, which corresponds to a modified physics at play in the dynamic meniscus. The crossover between this regime and the classical dip-coating one occurs when the substrate's deformation is comparable to the thickness of the entrained liquid film.

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References (42)

P. Baumeister, Optical Coating Technology (SPIE, Bellingham, WA, 2004), Vol. 137.
C. J. Brinker, Chemical Solution Deposition of Functional Oxide Thin Films (Springer, Berlin, 2013), pp. 233–261.
B. Levich and L. Landau, Dragging of liquid by a plate, Acta Physicochim. URSS 17, 141 (1942).
B. Derjaguin, Thickness of liquid layer adhering to walls of vessels on their emptying and the theory of photo-and motion-picture film coating, Acta Physicochim. URSS 20, 349 (1943).
E. Rio and F. Boulogne, Withdrawing a solid from a bath: How much liquid is coated? Adv. Colloid Interface Sci. 247, 100 (2017).
A. De Ryck and D. Quéré, Gravity and inertia effects in plate coating, J. Colloid Interface Sci. 203, 278 (1998).
B. Jin, A. Acrivos, and A. Münch, The drag-out problem in film coating, Phys. Fluids 17, 103603 (2005).
A. Q. Shen, B. Gleason, G. H. McKinley, and H. A. Stone, Fiber coating with surfactant solutions, Phys. Fluids 14, 4055 (2002).
R. Spiers, C. Subbaraman, and W. Wilkinson, Free coating of non-Newtonian liquids onto a vertical surface, Chem. Eng. Sci. 30, 379 (1975).
J. Ro and G. Homsy, Viscoelastic free surface flows: thin film hydrodynamics of Hele-Shaw and dip coating flows, J. Non-Newtonian Fluid Mech. 57, 203 (1995).
J. Ashmore, A. Q. Shen, H. Kavehpour, H. Stone, and G. McKinley, Coating flows of non-Newtonian fluids: Weakly and strongly elastic limits, J. Eng. Math. 60, 17 (2008).
W. J. Smit, C. Kusina, J.-F. Joanny, and A. Colin, Stress Field inside the Bath Determines Dip Coating with Yield-Stress Fluids in Cylindrical Geometry, Phys. Rev. Lett. 123, 148002 (2019).
M. Marchand, F. Restagno, E. Rio, and F. Boulogne, Roughness-Induced Friction on Liquid Foams, Phys. Rev. Lett. 124, 118003 (2020).
C. Datt, M. Kansal, and J. H. Snoeijer, A thin-film equation for a viscoelastic fluid, and its application to the Landau-Levich problem, J. Non-Newtonian Fluid Mech. 305, 104816 (2022).
R. Krechetnikov and G. Homsy, Experimental study of substrate roughness and surfactant effects on the Landau-Levich law, Phys. Fluids 17, 102108 (2005).
J. Seiwert, C. Clanet, and D. Quéré, Coating of a textured solid, J. Fluid Mech. 669, 55 (2011).
B. Andreotti, O. Bäumchen, F. Boulogne, K. E. Daniels, E. R. Dufresne, H. Perrin, T. Salez, J. H. Snoeijer, and R. W. Style, Solid capillarity: when and how does surface tension deform soft solids? Soft Matter 12, 2993 (2016).
R. W. Style, A. Jagota, C.-Y. Hui, and E. R. Dufresne, Elastocapillarity: Surface tension and the mechanics of soft solids, Annu. Rev. Condens. Matter Phys. 8, 99 (2017).
J. Bico, É. Reyssat, and B. Roman, Elastocapillarity: When surface tension deforms elastic solids, Annu. Rev. Fluid Mech. 50, 629 (2018).
B. Andreotti and J. H. Snoeijer, Statics and dynamics of soft wetting, Annu. Rev. Fluid Mech. 52, 285 (2020).
S. Karpitschka, A. Pandey, L. A. Lubbers, J. H. Weijs, L. Botto, S. Das, B. Andreotti, and J. H. Snoeijer, Liquid drops attract or repel by the inverted Cheerios effect, Proc. Natl. Acad. Sci. 113, 7403 (2016).
A. Hourlier-Fargette, A. Antkowiak, A. Chateauminois, and S. Neukirch, Role of uncrosslinked chains in droplets dynamics on silicone elastomers, Soft Matter 13, 3484 (2017).
M. Rivetti, V. Bertin, T. Salez, C.-Y. Hui, C. Linne, M. Arutkin, H. Wu, E. Raphaël, and O. Bäumchen, Elastocapillary levelling of thin viscous films on soft substrates, Phys. Rev. Fluids 2, 094001 (2017).
W. Kim, T. Gilet, and J. W. Bush, Optimal concentrations in nectar feeding, Proc. Natl. Acad. Sci. 108, 16618 (2011).
W. Kim and J. W. Bush, Natural drinking strategies, J. Fluid Mech. 705, 7 (2012).
F. Brau, D. Lanterbecq, L.-N. Zghikh, V. Bels, and P. Damman, Dynamics of prey prehension by chameleons through viscous adhesion, Nat. Phys. 12, 931 (2016).
A. Lechantre, A. Draux, H.-A. B. Hua, D. Michez, P. Damman, and F. Brau, Essential role of papillae flexibility in nectar capture by bees, Proc. Natl. Acad. Sci. 118, e2025513118 (2021).
S. D. Wilson, The drag-out problem in film coating theory, J. Eng. Math. 16, 209 (1982).
A. Oron, S. H. Davis, and S. G. Bankoff, Long-scale evolution of thin liquid films, Rev. Mod. Phys. 69, 931 (1997).
P.-G. De Gennes, F. Brochard-Wyart, and D. Quéré, Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves (Springer Science & Business Media, New York, 2013).
D. A. Dillard, B. Mukherjee, P. Karnal, R. C. Batra, and J. Frechette, A review of Winkler's foundation and its profound influence on adhesion and soft matter applications, Soft Matter 14, 3669 (2018).
T. G. Chandler and D. Vella, Validity of Winkler's mattress model for thin elastomeric layers: Beyond Poisson's ratio, Proc. R. Soc. London Ser. A 476, 20200551 (2020).
Assuming a flat film at $z \to \infty$ , we linearize Eq. (5) at large $z$ through $h (z) = h_{\infty} + δ h (z)$ , where $δ h (z) ≪ h_{\infty}$ . This gives $δ h^{'''} (z) = \frac{3 Ca}{h_{\infty}^{3}} [ℓ_{ec}^{2} δ h^{″} (z) - δ h (z)]$ . The solution of the latter, which is compatible with a flat film at $z \to \infty$ , takes the form $δ h (z) \propto exp (r z)$ , where $r$ is the root of the polynomial equation $r^{3} + \frac{3 Ca}{h_{\infty}^{3}} [- ℓ_{ec}^{2} r^{2} + 1] = 0$ , with a negative real part.
We note that the slope of the liquid-air interface profile at the boundary layer is $≃ \sqrt{2} ℓ_{ec} / ℓ_{cap} = \sqrt{2 L}$ . Therefore, the lubrication approximation is valid here provided that the softness parameter is small $L ≪ 1$ to ensure small slopes in the dynamical meniscus.
Assuming $K_{1}$ to be small, the linearization of Eq. (12) gives $K_{1}^{'} = 3 K_{1} (2 C + H_{\infty}) / {(C - H_{\infty})}^{4}$ . Thus, stable solutions at $ξ \to \infty$ only exist for $2 C + H_{\infty} \leq 0$ and an algebraic decay of $K_{1}$ is obtained $H_{\infty} = - 2 C$ . Combining with the equation for $C$ (in the text), we get the numerical prefactor $H_{\infty} = - 2 C = 2 / 3$ .
E. J. Hinch, Perturbation Methods, Cambridge Texts in Applied Mathematics (Cambridge University Press, Cambridge, 1991).
R. Lhermerout, H. Perrin, E. Rolley, B. Andreotti, and K. Davitt, A moving contact line as a rheometer for nanometric interfacial layers, Nat. Commun. 7, 12545 (2016).
I. Cantat, Liquid meniscus friction on a wet plate: Bubbles, lamellae, and foams, Phys. Fluids 25, 031303 (2013).
F. P. Bretherton, The motion of long bubbles in tubes, J. Fluid Mech. 10, 166 (1961).
L. Tanner, The spreading of silicone oil drops on horizontal surfaces, J. Phys. D 12, 1473 (1979).
J. R. Lister, G. G. Peng, and J. A. Neufeld, Viscous Control of Peeling an Elastic Sheet by Bending and Pulling, Phys. Rev. Lett. 111, 154501 (2013).
V. Charitatos and S. Kumar, A thin-film model for droplet spreading on soft solid substrates, Soft Matter 16, 8284 (2020).

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Enhanced dip coating on a soft substrate

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