Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-30T01:30:13.108Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic analysis of the evaporation dynamics of partially wetting droplets

Published online by Cambridge University Press:  06 July 2017

Nikos Savva Open the ORCID record for Nikos Savva [Opens in a new window] ,
Alexey Rednikov  and
Pierre Colinet
Nikos Savva*
Affiliation:
School of Mathematics, Cardiff University, Cardiff CF24 4AG, UK
Alexey Rednikov
Affiliation:
Université Libre de Bruxelles, TIPs-Fluid Physics Unit, CP 165/67, 1050 Brussels, Belgium
Pierre Colinet
Affiliation:
Université Libre de Bruxelles, TIPs-Fluid Physics Unit, CP 165/67, 1050 Brussels, Belgium
*
†Email address for correspondence: savvan@cf.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the dynamics of an axisymmetric, partially wetting droplet of a one-component volatile liquid. The droplet is supported on a smooth superheated substrate and evaporates into a pure vapour atmosphere. In this process, we take the liquid properties to be constant and assume that the vapour phase has poor thermal conductivity and small dynamic viscosity so that we may decouple its dynamics from the dynamics of the liquid phase. This leads to a so-called ‘one-sided’ lubrication-type model for the evolution of the droplet thickness, which accounts for the effects of evaporation, capillarity, gravity, slip and kinetic resistance to evaporation. By asymptotically matching the flow near the contact line region and the bulk of the droplet in the limit of a small slip length and commensurably small evaporation and kinetic resistance effects, we obtain coupled evolution equations for the droplet radius and volume. The predictions of our asymptotic analysis, which also include an estimate of the evaporation time, are found to be in excellent agreement with numerical simulations of the governing lubrication model for a broad range of parameter regimes.

JFM classification

Phase change: Condensation/evaporation Interfacial Flows (free surface): Contact lines Drops and Bubbles: Drops
Type
Papers
Information
Journal of Fluid Mechanics , Volume 824 , 10 August 2017 , pp. 574 - 623
Check for updates
Copyright
© 2017 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abramowitz, M. & Stegun, I. A. 1972 Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover.Google Scholar
Ajaev, V. S. 2005 Spreading of thin volatile liquid droplets on uniformly heated surfaces. J. Fluid Mech. 528, 279–296.Google Scholar
Anderson, D. M. & Davis, S. H. 1995 The spreading of volatile liquid droplets on heated surfaces. Phys. Fluids 7 (2), 248–265.Google Scholar
Bonn, D., Eggers, J., Indekeu, J., Meunier, J. & Rolley, E. 2009 Wetting and spreading. Rev. Mod. Phys. 81 (2), 739–805.Google Scholar
Bourges-Monnier, C. & Shanahan, M. E. R. 1995 Influence of evaporation on contact angle. Langmuir 11 (7), 2820–2829.Google Scholar
Boyd, J. P. 2000 Chebyshev and Fourier Spectral Methods, 2nd edn. Dover.Google Scholar
Brutin, D.(Ed.) 2015 Droplet Wetting and Evaporation: From Pure to Complex Fluids, 1st edn. Academic.Google Scholar
Burelbach, J. P., Bankoff, S. G. & Davis, S. H. 1988 Nonlinear stability of evaporating/condensing liquid films. J. Fluid Mech. 195, 463–494.CrossRefGoogle Scholar
Carey, V. P. 2007 Liquid–Vapor Phase-Change Phenomena: An Introduction to the Thermophysics of Vaporization and Condensation Processes in Heat Transfer Equipment, 2nd edn. CRC Press.Google Scholar
Cazabat, A.-M. & Guéna, G. 2010 Evaporation of macroscopic sessile droplets. Soft Matt. 6 (12), 2591–2612.Google Scholar
Choi, K., Ng, A. H. C., Fobel, R. & Wheeler, A. R. 2012 Digital microfluidics. Annu. Rev. Anal. Chem. 5 (1), 413–440.Google Scholar
Cioulachtjian, S., Launay, S., Boddaert, S. & Lallemand, M. 2010 Experimental investigation of water drop evaporation under moist air or saturated vapour conditions. Intl J. Therm. Sci. 49 (6), 859–866.Google Scholar
Colinet, P. & Rednikov, A. 2011 On integrable singularities and apparent contact angles within a classical paradigm. Eur. Phys. J. Spec. Topics 197 (1), 89–113.Google Scholar
Cox, R. G. 1986 The dynamics of the spreading of liquids on a solid surface. Part 1. Viscous flow. J. Fluid Mech. 168, 169–194.CrossRefGoogle Scholar
Davidovits, P., Worsnop, D. R., Jayne, J. T., Kolb, C. E., Winkler, P., Vrtala, A., Wagner, P. E., Kulmala, M., Lehtinen, K. E. J., Vesala, T. & Mozurkewich, M. 2004 Mass accommodation coefficient of water vapor on liquid water. Geophys. Res. Lett. 31 (22), L22111.CrossRefGoogle Scholar
Davis, S. H. & Hocking, L. M. 1999 Spreading and imbibition of viscous liquid on a porous base. Phys. Fluids 11 (1), 48–57.Google Scholar
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
Deegan, R. D., Bakajin, O., Dupont, T. F., Huber, G., Nagel, S. R. & Witten, T. A. 2000 Contact line deposits in an evaporating drop. Phys. Rev. E 62 (1), 756–765.Google Scholar
Dehaeck, S., Rednikov, A. & Colinet, P. 2014 Vapor-based interferometric measurement of local evaporation rate and interfacial temperature of evaporating droplets. Langmuir 30 (8), 2002–2008.Google Scholar
Dugas, V., Broutin, J. & Souteyrand, E. 2005 Droplet evaporation study applied to DNA chip manufacturing. Langmuir 21 (20), 9130–9136.Google Scholar
Dunn, G. J., Wilson, S. K., Duffy, B. R., David, S. & Sefiane, K. 2009 The strong influence of substrate conductivity on droplet evaporation. J. Fluid Mech. 623, 329–351.Google Scholar
Eames, I. W., Marr, N. J. & Sabir, H. 1997 The evaporation coefficient of water: a review. Intl J. Heat Mass Transfer 40 (12), 2963–2973.CrossRefGoogle Scholar
Eggers, J. 2005a Contact line motion for partially wetting fluids. Phys. Rev. E 72 (6), 061605.Google ScholarPubMed
Eggers, J. 2005b Existence of receding and advancing contact lines. Phys. Fluids 17 (8), 082106.Google Scholar
Eggers, J. & Pismen, L. M. 2010 Nonlocal description of evaporating drops. Phys. Fluids 22 (11), 112101.CrossRefGoogle 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 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
de Gennes, P.-G. 1985 Wetting: statics and dynamics. Rev. Mod. Phys. 57 (3), 827–863.Google Scholar
Girard, F., Antoni, M. & Sefiane, K. 2008 On the effect of Marangoni flow on evaporation rates of heated water drops. Langmuir 24 (17), 9207–9210.CrossRefGoogle ScholarPubMed
Gokhale, S. J., Plawsky, J. L. & Wayner, P. C. 2003 Experimental investigation of contact angle, curvature, and contact line motion in dropwise condensation and evaporation. J. Colloid Interface Sci. 259 (2), 354–366.Google Scholar
Hervet, H. & de Gennes, P.-G. 1984 The dynamics of wetting: precursor films in the wetting of ‘dry’ solids. C. R. Acad. Sci. Paris 299, 499–503.Google Scholar
Hocking, L. M. 1983 The spreading of a thin drop by gravity and capillarity. Q. J. Mech. Appl. Maths 36 (1), 55–69.Google Scholar
Hocking, L. M. 1992 Rival contact-angle models and the spreading of drops. J. Fluid Mech. 239, 671–781.Google Scholar
Hocking, L. M. 1995 On contact angles in evaporating liquids. Phys. Fluids 7 (12), 2950–2955.Google Scholar
Hu, H. & Larson, R. G. 2006 Marangoni effect reverses coffee-ring depositions. J. Phys. Chem. B 110 (14), 7090–7094.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.Google Scholar
Janeček, V. & Nikolayev, V. S. 2012 Contact line singularity at partial wetting during evaporation driven by substrate heating. Europhys. Lett. 100 (1), 14003.Google Scholar
Kelly-Zion, P. L., Pursell, C. J., Booth, R. S. & VanTilburg, A. N. 2009 Evaporation rates of pure hydrocarbon liquids under the influences of natural convection and diffusion. Intl J. Heat Mass Transfer 52 (13–14), 3305–3313.Google Scholar
Kim, J. 2007 Spray cooling heat transfer: the state of the art. Intl J. Heat Fluid Flow 28 (4), 753–767.Google Scholar
King, J. R. 2001 Thin-film flows and high-order degenerate parabolic equations. In IUTAM Symposium on Free Surface Flows (ed. King, A. C. & Shikhmurzaev, Y. D.), pp. 7–18. Springer.Google Scholar
Lacey, A. A. 1982 The motion with slip of a thin viscous droplet over a solid surface. Stud. Appl. Maths 67 (3), 217–230.Google Scholar
Lauga, E., Brenner, M. & Stone, H. 2007 Microfluidics: the no-slip boundary condition. In Springer Handbook of Experimental Fluid Mechanics (ed. Tropea, C., Yarin, A. L. & Foss, J. F.), chap. 19, pp. 1219–1240. Springer.CrossRefGoogle Scholar
Lopes, M. C., Bonaccurso, E., Gambaryan-Roisman, T. & Stephan, P. 2013 Influence of the substrate thermal properties on sessile droplet evaporation: effect of transient heat transport. Colloids Surf. A 432, 64–70.Google Scholar
Marek, R. & Straub, J. 2001 Analysis of the evaporation coefficient and the condensation coefficient of water. Intl J. Heat Mass Transfer 44 (1), 39–53.Google Scholar
Moosman, S. & Homsy, G. M. 1980 Evaporating menisci of wetting fluids. J. Colloid Interface Sci. 73 (1), 212–223.Google Scholar
Morris, S. J. S. 2001 Contact angles for evaporating liquids predicted and compared with existing experiments. J. Fluid Mech. 432, 1–30.Google Scholar
Murisic, N. & Kondic, L. 2011 On evaporation of sessile drops with moving contact lines. J. Fluid Mech. 679, 219–246.Google Scholar
Oliver, J. M., Whiteley, J. P., Saxton, M. A., Vella, D., Zubkov, V. S. & King, J. R. 2015 On contact-line dynamics with mass transfer. Eur. J. Appl. Maths 26, 1–49.Google Scholar
Paul, B. 1962 Compilation of evaporation coefficients. ARS J. 32 (9), 1321–1328.Google Scholar
Pismen, L. & Eggers, J. 2008 Solvability condition for the moving contact line. Phys. Rev. E 78 (5), 056304.Google Scholar
Plawsky, J. L., Panchamgam, S. S., Gokhale, S. J., Wayner, P. C. & DasGupta, S. 2004 A study of the oscillating corner meniscus in a vertical constrained vapor bubble system. Superlattices Microstruct. 35 (3–6), 559–572.CrossRefGoogle Scholar
Potash, M. & Wayner, P. C. 1972 Evaporation from a two-dimensional extended meniscus. Intl J. Heat Mass Transfer 15 (10), 1851–1863.Google Scholar
Poulard, C., Guéna, G. & Cazabat, A. M. 2005 Diffusion-driven evaporation of sessile drops. J. Phys.: Condens. Matter 17 (49), S4213–S4227.Google Scholar
Press, W., Teukolsky, S., Vetterling, W. & Flannery, B. 1992 Numerical Recipes in C , 2nd edn. chap. 3.1, pp. 108–110. Cambridge University Press.Google Scholar
Raj, R., Kunkelmann, C., Stephan, P., Plawsky, J. & Kim, J. 2012 Contact line behavior for a highly wetting fluid under superheated conditions. Intl J. Heat Mass Transfer 55 (9–10), 2664–2675.Google Scholar
Rednikov, A. & Colinet, P. 2013 Singularity-free description of moving contact lines for volatile liquids. Phys. Rev. E 87 (1), 010401.Google ScholarPubMed
Rednikov, A. Y. & Colinet, P. 2011 Truncated versus extended microfilm at a vapor–liquid contact line on heated substrate. Langmuir 27 (5), 1758–1769.CrossRefGoogle Scholar
Rednikov, A. Y., Rossomme, S. & Colinet, P. 2009 Steady microstructure of a contact line for a liquid on a heated surface overlaid with its pure vapor: parametric study for a classical model. Multiphase Sci. Technol. 21 (3), 213–248.CrossRefGoogle Scholar
Ren, W., Trinh, P. H. & Weinan, E. 2015 On the distinguished limits of the Navier slip model of the moving contact line problem. J. Fluid Mech. 772, 107–126.CrossRefGoogle Scholar
Ristenpart, W. D., Kim, P. G., Domingues, C., Wan, J. & Stone, H. A. 2007 Influence of substrate conductivity on circulation reversal in evaporating drops. Phys. Rev. Lett. 99 (23), 234502.Google Scholar
Savva, N. & Kalliadasis, S. 2009 Two-dimensional droplet spreading over topographical substrates. Phys. Fluids 21 (9), 092102.Google Scholar
Savva, N. & Kalliadasis, S. 2011 Dynamics of moving contact lines: a comparison between slip and precursor film models. Europhys. Lett. 94 (6), 64004.Google Scholar
Savva, N. & Kalliadasis, S. 2012 Influence of gravity on the spreading of two-dimensional droplets over topographical substrates. J. Engng Maths 73 (1), 3–16.Google Scholar
Savva, N. & Kalliadasis, S. 2013 Droplet motion on inclined heterogeneous substrates. J. Fluid Mech. 725, 462–491.Google Scholar
Savva, N. & Kalliadasis, S. 2014 Low-frequency vibrations of two-dimensional droplets on heterogeneous substrates. J. Fluid Mech. 754, 515–549.Google Scholar
Savva, N., Rednikov, A. & Colinet, P. 2014 Asymptotic analysis of evaporating droplets. In Proceedings of the 4th Micro and Nano Flows Conference, University College, London, UK (ed. Karayiannis, T., König, C. S. & Balabani, S.), p. 236. Brunel University.Google Scholar
Saxton, M. A., Whiteley, J. P., Vella, D. & Oliver, J. M. 2016 On thin evaporating drops: when is the d 2 -law valid? J. Fluid Mech. 792, 134–167.Google Scholar
Schrage, R. 1953 A Theoretical Study of Interphase Mass Transfer. Columbia University Press.Google Scholar
Schwartz, L. W. & Eley, R. R. 1998 Simulation of droplet motion on low-energy and heterogeneous surfaces. J. Colloid Interface Sci. 202 (1), 173–188.Google Scholar
Sefiane, K., Moffat, J. R., Matar, O. K. & Craster, R. V. 2008 Self-excited hydrothermal waves in evaporating sessile drops. Appl. Phys. Lett. 93 (7), 074103.CrossRefGoogle Scholar
Semenov, S., Trybala, A., Rubio, R. G., Kovalchuk, N., Starov, V. & Velarde, M. G. 2014 Simultaneous spreading and evaporation: recent developments. Adv. Colloid Interface Sci. 206, 382–398.Google Scholar
Shahidzadeh-Bonn, N., Rafaï, S., Azouni, A. & Bonn, D. 2006 Evaporating droplets. J. Fluid Mech. 549, 307–313.Google Scholar
Shikhmurzaev, Y. D. 2008 Capillary Flows with Forming Interfaces. Taylor & Francis.Google Scholar
Sibley, D. N., Nold, A. & Kalliadasis, S. 2015a The asymptotics of the moving contact line: cracking an old nut. J. Fluid Mech. 764, 445–462.Google Scholar
Sibley, D. N., Nold, A., Savva, N. & Kalliadasis, S. 2013 On the moving contact line singularity: asymptotics of a diffuse-interface model. Eur. Phys. J. E 36 (3), 26.Google Scholar
Sibley, D. N., Nold, A., Savva, N. & Kalliadasis, S. 2015b A comparison of slip, disjoining pressure, and interface formation models for contact line motion through asymptotic analysis of thin two-dimensional droplet spreading. J. Engng Maths 94, 19–41.Google Scholar
Sibley, D. N., Savva, N. & Kalliadasis, S. 2012 Slip or not slip? A methodical examination of the interface formation model using two-dimensional droplet spreading on a horizontal planar substrate as a prototype system. Phys. Fluids 24 (8), 082105.Google Scholar
Sobac, B. & Brutin, D. 2012 Thermocapillary instabilities in an evaporating drop deposited onto a heated substrate. Phys. Fluids 24 (3), 032103.Google Scholar
Sodtke, C., Ajaev, V. S. & Stephan, P. 2008 Dynamics of volatile liquid droplets on heated surfaces: theory versus experiment. J. Fluid Mech. 610, 343–362.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
Stephan, P. C. & Busse, C. A. 1992 Analysis of the heat transfer coefficient of grooved heat pipe evaporator walls. Intl J. Heat Mass Transfer 35 (2), 383–391.Google Scholar
Todorova, D., Thiele, U. & Pismen, L. M. 2012 The relation of steady evaporating drops fed by an influx and freely evaporating drops. J. Engng Maths 73 (1), 17–30.CrossRefGoogle Scholar
Trefethen, L. N. 2000 Spectral Methods in MATLAB. SIAM.CrossRefGoogle Scholar
Tsoumpas, Y., Dehaeck, S., Rednikov, A. & Colinet, P. 2015 Effect of Marangoni flows on the shape of thin sessile droplets evaporating into air. Langmuir 31 (49), 13334–13340.CrossRefGoogle ScholarPubMed
Vellingiri, R., Savva, N. & Kalliadasis, S. 2011 Droplet spreading on chemically heterogeneous substrates. Phys. Rev. E 84 (3), 036305.Google Scholar
Voinov, O. V. 1976 Hydrodynamics of wetting. Fluid Dyn. 11 (5), 714–721.Google Scholar
Wayner, P. C., Kao, Y. K. & LaCroix, L. V. 1976 The interline heat-transfer coefficient of an evaporating wetting film. Intl J. Heat Mass Transfer 19 (5), 487–492.Google Scholar
Wu, Q. & Wong, H. 2004 A slope-dependent disjoining pressure for non-zero contact angles. J. Fluid Mech. 506, 157–185.Google Scholar
Xu, F. & Jensen, O. E. 2016 Drop spreading with random viscosity. Proc. R. Soc. Lond. A 472 (2194), 20160270.Google Scholar
Yi, T. & Wong, H. 2007 Theory of slope-dependent disjoining pressure with application to Lennard–Jones liquid films. J. Colloid Interface Sci. 313 (2), 579–591.Google Scholar
Zheng, L., Wang, Y.-X., Plawsky, J. L. & Wayner, P. C. 2002 Effect of curvature, contact angle, and interfacial subcooling on contact line spreading in a microdrop in dropwise condensation. Langmuir 18 (13), 5170–5177.Google Scholar