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Evaporation dynamics of wavy sessile droplets

Published online by Cambridge University Press:  02 May 2025

Yaochen Mei Open the ORCID record for Yaochen Mei [Opens in a new window]  and
Xinping Zhou Open the ORCID record for Xinping Zhou [Opens in a new window]
Yaochen Mei
Affiliation:
School of Mechanical Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, PR China
Xinping Zhou*
Affiliation:
School of Mechanical Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, PR China State Key Laboratory of Intelligent Manufacturing Equipment and Technology, Huazhong University of Science and Technology, Wuhan 430074, PR China
*
Corresponding author: Xinping Zhou, [email protected]
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Abstract

We study the evaporation dynamics of non-thin non-spherical-cap (i.e. wavy) droplets. These droplets exhibit surface curvature that varies periodically with the polar angle, which profoundly influences their evaporation flux, internal flow dynamics, and the resultant deposition patterns upon complete evaporation. The droplet is considered quasi-static throughout its entire lifetime. The asymptotic expansions of the evaporation flux in the diffusion-limited model, and the induced internal inviscid flow of the droplets, are derived through asymptotic analysis. Under the assumption of small deformation amplitudes, the accuracies of these two expansions are validated numerically. Expanding upon these asymptotic results, we also investigate the surface density profile of the droplet deposition after it dries up. The results indicate that the freely moving contact line of the droplet leads to the deposited stain exhibiting a mountain-like morphology. The internal inviscid flow along with the non-spherical-cap shape eliminates the divergence of the deposited surface density profile at droplet’s centre. This work provides a theoretical basis for geometrically controlled sessile droplet evaporation, which may have practical applications in industry.

JFM classification

Drops and Bubbles: Drops Phase change: Condensation/evaporation
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 1010 , 10 May 2025 , A38
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Copyright
© The Author(s), 2025. Published by Cambridge University Press

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References

Arfken, G.B. & Weber, H.J. 2005 Mathematical Methods for Physicists. 6th edn. Elsevier.Google Scholar
Basaran, O.A., Gao, H. & Bhat, P.P. 2013 Nonstandard inkjets. Annu. Rev. Fluid Mech. 45 (1), 85–113.Google Scholar
Bodiguel, H., Doumenc, F. & Guerrier, B. 2009 Pattern formation during the drying of a colloidal suspension: pinning of a receding contact line. Eur. Phys. J. 166 (1), 29–32.Google Scholar
Bodiguel, H., Doumenc, F. & Guerrier, B. 2010 Stick-slip patterning at low capillary numbers for an evaporating colloidal suspension. Langmuir 26 (13), 10758–10763.Google Scholar
Boulogne, F., Ingremeau, F. & Stone, H.A. 2017 Coffee-stain growth dynamics on dry and wet surfaces. J. Phys.: Condens. Matter 29 (7), 074001.Google ScholarPubMed
Chen, Y.J., Suzuki, K., Mahara, H. & Yamaguchi, T. 2012 Quasi-logarithmic spacing law in dewetting patterns from the drying meniscus of a polymer solution. Chem. Phys. Lett. 529 (1), 74–78.Google Scholar
D’Ambrosio, H.-M., Wilson, S.K., Wray, A.W. & Duffy, B.R. 2023 The effect of the spatial variation of the evaporative flux on the deposition from a thin sessile droplet. J. Fluid Mech. 970, A1.Google Scholar
Deegan, R.D. 2000 Pattern formation in drying drops. Phys. Rev. E 61 (1), 475–485.Google ScholarPubMed
Deegan, R.D., Bakajin, O., Dupont, T.F., Huber, G., Nagel, S.R. & Witten, T.A. 1997 Capillary flow as the cause of ring stains from dried liquid drops. Nature 389(6653), 827–829.Google Scholar
Ding, D. & Bostwick, J.B. 2022 Pressure modes of the oscillating sessile drop. J. Fluid Mech. 944, R1.Google Scholar
Ding, H., Zhu, X., Gao, P. & Lu, X.Y. 2018 Ratchet mechanism of drops climbing a vibrated oblique plate. J. Fluid Mech. 835, R1.Google Scholar
Du, X. & Deegan, R. 2015 Ring formation on an inclined surface. J. Fluid Mech. 775, R3.Google Scholar
Eral, H.B., Augustine, D.M., Duits, M.H.G. & Mugele, F. 2011 Suppressing the coffee stain effect: how to control colloidal self-assembly in evaporating drops using electrowetting. Soft Matt. 7 (10), 4954–4958.Google Scholar
Erbil, H.Y. 2012 Evaporation of pure liquid sessile and spherical suspended drops: a review. Adv. Colloid Interface Sci. 170 (1–2), 67–86.Google ScholarPubMed
Ferrera, C., López-Herrera, J.M., Herrada, M.A., Montanero, J.M. & Acero, A.J. 2013 Dynamical behavior of electrified pendant drops. Phys. Fluids 25 (1), 012104.Google Scholar
Fischer, B.J. 2002 Particle convection in an evaporating colloidal droplet. Langmuir 18 (1), 60–67.Google Scholar
Freed-Brown, J.E. 2015 Deposition from evaporating drops: power laws and new morphologies in coffee stains. PhD thesis, University of Chicago.Google Scholar
Gelderblom, H., Bloemen, O. & Snoeijer, J.H. 2012 Stokes flow near the contact line of an evaporating drop. J. Fluid Mech. 709, 69–84.Google Scholar
Gubarenko, S.I., Chiarot, P., Ben Mrad, R. & Sullivan, P.E. 2008 Plane model of fluid interface rupture in an electric field. Phys. Fluids 20 (4), 043601.Google Scholar
Harris, D.J., Hu, H., Conrad, J.C. & Lewis, J.A. 2007 Patterning colloidal films via evaporative lithography. Phys. Rev. Lett. 98 (14), 148301.Google ScholarPubMed
Hu, H. & Larson, R.G. 2005 Analysis of the microfluid flow in an evaporating sessile droplet. Langmuir 21 (9), 3963–3971.Google Scholar
Hu, H. & Larson, R.G. 2006 Marangoni effect reverses coffee-ring depositions. J. Phys. Chem. B 110 (14), 7090–7094.Google ScholarPubMed
Jing, J. et al. 1998 Automated high resolution optical mapping using arrayed, fluid-fixed DNA molecules. Proc. Natl Acad. Sci. USA 95 (14), 8046–8051.Google ScholarPubMed
Lafuma, A. & Quéré, D. 2003 Superhydrophobic states. Nat. Mater. 2 (7), 457–460.Google ScholarPubMed
Lebedev, N.N. 1965 Special Functions and Their Applications. Prentice-Hall (from Russian, transl. R.A. Silverman).Google Scholar
Li, F. & Mugele, F. 2008 How to make sticky surfaces slippery: contact angle hysteresis in electrowetting with alternating voltage. Appl. Phys. Lett. 92 (24), 244108.Google Scholar
Lohse, D. 2022 Fundamental fluid dynamics challenges in inkjet printing. Annu. Rev. Fluid Mech. 54 (1), 349–382.Google Scholar
Mangel, R.F. & Baer, E. 1962 The evaporation of water drops from a ‘Teflon’ surface. Chem. Engng Sci. 17 (9), 705–706.Google Scholar
Masoud, H. & Felske, J.D. 2009 a Analytical solution for inviscid flow inside an evaporating sessile drop. Phys. Rev. E 79 (1), 016301.Google ScholarPubMed
Masoud, H. & Felske, J.D. 2009 b Analytical solution for Stokes flow inside an evaporating sessile drop: spherical and cylindrical cap shapes. Phys. Fluids 21 (4), 957.Google Scholar
Moore, M.R., Vella, D. & Oliver, J.M. 2021 The nascent coffee ring: how solute diffusion counters advection. J. Fluid Mech. 920, A54.Google Scholar
Noblin, X., Buguin, A. & Brochard-Wyart, F. 2005 Triplon modes of puddles. Phys. Rev. Lett. 94 (16), 166102.Google ScholarPubMed
Noblin, X., Kofman, R. & Celestini, F. 2009 Ratchetlike motion of a shaken drop. Phys. Rev. Lett. 102 (19), 194504.Google ScholarPubMed
Oh, J.M., Ko, S.H. & Kang, K.H. 2008 Shape oscillation of a drop in AC electrowetting. Langmuir 24 (15), 8379–8386.Google ScholarPubMed
Ozawa, K., Nishitani, E. & Doi, M. 2005 Modeling of the drying process of liquid droplet to form thin film. Japan. J. Appl. Phys. 44 (6R), 4229.Google Scholar
Parsa, M., Harmand, S. & Sefiane, K. 2018 Mechanisms of pattern formation from dried sessile drops. Adv. Colloid Interface Sci. 254, 22–47.Google ScholarPubMed
Petsi, A.J. & Burganos, V.N. 2005 Potential flow inside an evaporating cylindrical line. Phys. Rev. E 72 (4), 047301.Google ScholarPubMed
Petsi, A.J. & Burganos, V.N. 2006 Evaporation-induced flow in an inviscid liquid line at any contact angle. Phys. Rev. E 73 (4), 041201.Google Scholar
Picknett, R.G. & Bexon, R. 1977 The evaporation of sessile or pendant drops in still air. J. Colloid Interface Sci. 61 (2), 336–350.Google Scholar
Popov, Y.O. 2005 Evaporative deposition patterns: spatial dimensions of the deposit. Phys. Rev. E 71 (3), 036313.Google ScholarPubMed
Rade, L. & Westergren, B. 1999 Mathematics Handbook for Science and Engineering. 4th edn. Springer-Verlag.Google Scholar
Rayleigh, Lord. 1879 On the capillary phenomena of jets. Proc. R. Soc. Lond. A 29 (196–199), 71–97.Google Scholar
Sáenz, P.J., Wray, A.W., Che, Z., Matar, O.K., Valluri, P., Kim, J. & Sefiane, K. 2017 Dynamics and universal scaling law in geometrically-controlled sessile drop evaporation. Nat. Commun. 8 (1), 1–9.Google Scholar
Sefiane, K. 2014 Patterns from drying drops. Adv. Colloid Interface Sci. 206, 372–381.Google ScholarPubMed
Sharp, J.S., Farmer, D.J. & Kelly, J. 2011 Contact angle dependence of the resonant frequency of sessile water droplets. Langmuir 27 (15), 9367–9371.Google ScholarPubMed
Slobozhanin, L.A. & Alexander, J.I.D. 2006 The stability of two connected drops suspended from the edges of circular holes. J. Fluid Mech. 563, 319–355.Google Scholar
Stauber, J.M., Wilson, S.K., Duffy, B.R. & Sefiane, K. 2014 On the lifetimes of evaporating droplets. J. Fluid Mech. 744, R2.Google Scholar
Stauber, J.M., Wilson, S.K., Duffy, B.R. & Sefiane, K. 2015 On the lifetimes of evaporating droplets with related initial and receding contact angles. Phys. Fluids 27 (12), 122101.Google Scholar
Talbot, E., Bain, C., De Dier, R., Sempels, W. & Vermant, J. 2016 Droplets drying on surfaces. In Fundamentals of Inkjet Printing: The Science of Inkjet and Droplets (ed. Haoth, S.D.), pp. 251–279. Wiley.Google Scholar
Tarasevich, Y.Y. 2005 Simple analytical model of capillary flow in an evaporating sessile drop. Phys. Rev. E 71 (2), 027301.Google Scholar
Thampi, S.P. & Basavaraj, M.G. 2023 Drying drops of colloidal dispersions. Ann. Rev. Chem. Biomol. Engng 14 (1), 53–83.Google ScholarPubMed
Thiele, U. 2014 Patterned deposition at moving contact lines. Adv. Colloid Interface Sci. 206, 399–413.Google ScholarPubMed
Thieret, M. 2008 Biology and Mechanics of Blood Flows, Part II: Mechanics and Medical Aspects. Springer.Google Scholar
Timm, M.L., Dehdashti, E., Darban, A.J. & Masoud, H. 2019 Evaporation of a sessile droplet on a slope. Sci. Rep. 9 (1), 19803.Google ScholarPubMed
Vukasinovic, B., Smith, M.K. & Glezer, A. 2007 Dynamics of a sessile drop in forced vibration. J. Fluid Mech. 587, 395–423.Google Scholar
Wang, Z., Orejon, D., Takata, Y. & Sefiane, K. 2022 Wetting and evaporation of multicomponent droplets. Phys. Rep. 960, 1–37.Google Scholar
Widjaja, E. & Harris, M.T. 2008 Particle deposition study during sessile drop evaporation. AIChE J. 54 (9), 2250–2260.Google Scholar
Wilson, S.K. & D’Ambrosio, H.M. 2023 Evaporation of sessile droplets. Annu. Rev. Fluid Mech. 55 (1), 481–509.Google Scholar
Wray, A.W. & Moore, M.R. 2023 Evaporation of non-circular droplets. J. Fluid Mech. 961, A11.Google Scholar
Xu, X.M., Vereecke, G., Van den Hoogen, E., Smeers, J., Armini, S., Delande, T. & Struyf, H. 2013 Wetting challenges in cleaning of high aspect ratio nano-structures. Solid State Phenom. 195, 235–238.Google Scholar
Zhang, F., Zhou, X. & Ding, H. 2023 Effects of gravity on natural oscillations of sessile drops. J. Fluid Mech. 962, A10.Google Scholar
Zheng, R. 2009 A study of the evaporative deposition process: pipes and truncated transport dynamics. Eur. Phys. J. E 29 (2), 205–218.Google ScholarPubMed
Zhou, X. & Zhang, F. 2017 Bifurcation of a partially immersed plate between two parallel plates. J. Fluid Mech. 817, 122–137.Google Scholar
Zigelman, A. & Manor, O. 2016 A model for pattern deposition from an evaporating solution subject to contact angle hysteresis and finite solubility. Soft Matt. 12 (26), 5693–5707.Google Scholar