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Dynamical equations for the contact line of an evaporating or condensing sessile drop

Published online by Cambridge University Press:  13 June 2012

Eliot Fried  and
Michel Jabbour
Eliot Fried*
Affiliation:
Department of Mechanical Engineering, McGill University, Montreal, QC H3A 0C3, Canada
Michel Jabbour
Affiliation:
Department of Mathematics, University of Kentucky, Lexington, KY 40506, USA
*
†Email address for correspondence: eliot.fried@mcgill.ca
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Abstract

The equations that govern, away from equilibrium and accounting for dissipation, the evolution of the contact line of an evaporating or condensing sessile drop on a rigid, planar substrate are derived. Aside from the normal and tangential components of the standard (Newtonian) force balance, these include a configurational force balance. At equilibrium, the normal component of the standard force balance reduces to the modified Young’s equation first mentioned by Gibbs. The remaining balances are purely dissipative and hence are vacuous in equilibrium. A complete description of contact-line dynamics generally involves all three equations. The theory is embedded in a thermodynamic framework that ensures consistency of all constitutive relations with the second law. In the linearly dissipative case, these involve six contact-line viscosities. When viscous coupling is neglected, only three viscosities remain. One is associated with stretching of the fluid along the contact line. The remaining two are related to dissipation that accompanies mass transfer between liquid and vapour phases during evaporation or condensation.

JFM classification

Phase change: Condensation/evaporation Drops and Bubbles: Drops Mathematical Foundations: General fluid mechanics
Type
Papers
Information
Journal of Fluid Mechanics , Volume 703 , 25 July 2012 , pp. 204 - 237
Copyright
Copyright © Cambridge University Press 2012

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References

1. Adachi, E., Dimitrov, A. S. & Nagayama, K. 1995 Stripe patterns formed on a glass surface during droplet evaporation. Langmuir 11, 1057–1060.CrossRefGoogle Scholar
2. Anderson, D. M., Cermelli, P., Fried, E., Gurtin, M. E. & McFadden, G. B. 2007 General dynamical sharp-interface conditions for phase transformations in viscous heat-conducting fluids. J. Fluid Mech. 581, 323–370.CrossRefGoogle Scholar
3. Baughman, K. F., Maier, R. M., Norris, T. A., Beam, B. M., Mudalige, A., Pemberton, J. E. & Curry, J. E. 2010 Evaporative deposition patterns of bacteria from a sessile drop: effect of changes in surface wettability due to exposure to a laboratory atmosphere. Langmuir 26, 7293–7298.CrossRefGoogle ScholarPubMed
4. Bedeaux, D. 2003 Non-equilibrium thermodynamic description of the three phase contact line. J. Chem. Phys. 120, 3744–3748.CrossRefGoogle Scholar
5. Billingham, J. 2006 On a model for the motion of a contact line on a smooth solid surface. Eur. J. Appl. Maths 17, 347–382.CrossRefGoogle Scholar
6. Billingham, J. 2008 Gravity-driven thin-film flow using a new contact line model. IMA J. Appl. Maths 73, 4–36.CrossRefGoogle Scholar
7. Bonn, D., Eggers, J., Indekeu, J., Meunier, J. & Rolley, E. 2009 Wetting and spreading. Rev. Mod. Phys. 81, 739–805.CrossRefGoogle Scholar
8. Brochard-Wyart, F. & de Gennes, P.-G. 1992 Dynamics of partial wetting. Adv. Colloid Interface Sci. 39, 1–11.CrossRefGoogle Scholar
9. Burton, C. V. 1892 A theory concerning the constitution of matter. Phil. Mag. 33, 191–204.CrossRefGoogle Scholar
10. Cermelli, P. & Fried, E. 2002 The evolution equation for a disclination in a nematic liquid crystal. Proc. R. Soc. Lond. A 458, 1–20.CrossRefGoogle Scholar
11. Cermelli, P., Fried, E. & Gurtin, M. E. 2004 Sharp-interface nematic–isotropic phase transitions without flow. Arch. Rat. Mech. Anal. 174, 151–178.CrossRefGoogle Scholar
12. Cermelli, P., Fried, E. & Gurtin, M. E. 2005 Transport relations for surface integrals arising in the formulation of balance laws for evolving fluid interface. J. Fluid Mech. 544, 339–351.CrossRefGoogle Scholar
13. Cherepanov, G. P. 1967 Crack propagation in continuous media. Prikl. Mat. Mekh. 31, 503–512.Google Scholar
14. Deegan, R. D. 2000 Pattern formation in drying drops. Phys. Rev. E 61, 475–485.CrossRefGoogle ScholarPubMed
15. 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, 827–829.CrossRefGoogle Scholar
16. Eshelby, J. D. 1951 The force on an elastic singularity. Phil. Trans. R. Soc. Lond. A 244, 87–112.Google Scholar
17. Eshelby, J. D. 1953 The equation of motion of a dislocation. Phys. Rev. 90, 248–255.CrossRefGoogle Scholar
18. Eshelby, J. D. 1956 The continuum theory of lattice defects. In Progress in Solid State Physics (ed. Seitz, F. & Turnbull, D. ), vol. 3, pp. 79–144. Academic Press.Google Scholar
19. Fan, F. Q. & Stebe, K. J. 2004 Assembly of colloidal particles by evaporation on surfaces with patterned hydrophobicity. Langmuir 20, 3062–3067.CrossRefGoogle ScholarPubMed
20. Freund, J. B. 2003 The atomic detail of a wetting/de-wetting flow. Phys. Fluids 15, L33–L36.CrossRefGoogle Scholar
21. Fried, E. 2008 Sharp-interface nematic–isotropic phase transformations with flow. Arch. Rat. Mech. Anal. 190, 227–265.CrossRefGoogle Scholar
22. Fried, E., Shen, A. Q. & Gurtin, M. E. 2006 Theory for solvent, momentum, and energy transfer between a surfactant solution and a vapour atmosphere. Phys. Rev. E 73, 061601.CrossRefGoogle Scholar
23. de Gans, B.-J., Duineveld, P. C. & Schubert, U. S. 2004 Inkjet printing of polymers: state of the art and future developments. Adv. Mater. 16, 203–213.CrossRefGoogle Scholar
24. de Gennes, P.-G. 1985 Wetting: statics and dynamics. Rev. Mod. Phys. 57, 827–863.CrossRefGoogle Scholar
25. de Gennes, P.-G., Brochard-Wyart, F. & Quéré, D. 2004 Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves. Springer.CrossRefGoogle Scholar
26. Ghosh, M., Fan, F. Q. & Stebe, K. J. 2007 Spontaneous pattern formation by dip coating of colloidal suspensions on homogeneous surfaces. Langmuir 23, 2180–2183.CrossRefGoogle ScholarPubMed
27. Gibbs, J. W. 1876 On the equilibrium of heterogeneous substances. In Trans. Conn. Acad. Arts Sci., 3, 108–248 (reprinted in Gibbs, J. W. 1961 The Scientific Papers, vol. 1. Dover).Google Scholar
28. Gurtin, M. E. 1988 Multiphase thermomechanics with interfacial structure 1. Heat conduction and the capillary balance law. Arch. Rat. Mech. Anal. 104, 185–221.CrossRefGoogle Scholar
29. Gurtin, M. E. 1993 Thermomechanics of Evolving Phase Boundaries in the Plane. Oxford University Press.CrossRefGoogle Scholar
30. Gurtin, M. E. 1995 The nature of configurational forces. Arch. Rat. Mech. Anal. 131, 67–100.CrossRefGoogle Scholar
31. Gurtin, M. E. 2000 Configurational Forces as Basic Concepts in Continuum Physics. Springer.Google Scholar
32. Gurtin, M. E. & Struthers, A. 1990 Multiphase thermomechanics with interfacial structure 3. Evolving phase boundaries in the presence of bulk deformation. Arch. Rat. Mech. Anal. 112, 97–160.CrossRefGoogle Scholar
33. Hu, H. & Larson, R. G. 2002 Evaporation of a sessile droplet on a substrate. J. Phys. Chem. B 106, 1334–1344.CrossRefGoogle Scholar
34. Hu, H. & Larson, R. G. 2006 Marangoni effect reverses coffee-ring depositions. J. Phys. Chem. B 110, 7090–7094.CrossRefGoogle ScholarPubMed
35. Jing, J., Reed, J., Huang, J., Hu, X., Clarke, V., Edington, J., Housman, D., Anantharaman, T. S., Huff, E. J., Mishra, B., Porter, B., Shenker, A., Wolfson, E., Hiort, C., Kantor, R., Aston, C. & Schwartz, D. C. 1998 Automated high resolution optical mapping using arrayed, fluid-fixed DNA molecules. Proc. Natl. Acad. Sci. USA 95, 8046–8051.CrossRefGoogle ScholarPubMed
36. Johannessen, E. & Bedeaux, D. 2003 The nonequilibrium van der Waals square gradient model. (II.) Local equilibrium of the Gibbs surface. Physica A 330, 354–372.CrossRefGoogle Scholar
37. Lagrange, J. L. 1788 Mécanique Analytique. Desaint, (translated as Lagrange, J. L. 1997 Analytical Mechanics. Kluwer).Google Scholar
38. Laplace, P. S. 1806 Méchanique Céleste, Supplément au   Livre. Impresse Imperiale, (Translated as Laplace, P. S. 1966 Celestial Mechanics, vol. IV. Chelsea.Google Scholar
39. Larmor, J. 1897 A dynamical theory of the electric and luminiferous medium. Part III. Relations with material media. Proc. R. Soc. Lond. 190, 205–300.Google Scholar
40. Lee, H. H., Chou, K. S. & Huang, K. C. 2005 Inkjet printing of nanosized silver colloids. Nanotechnology 16, 2436–2441.CrossRefGoogle ScholarPubMed
41. Lin, Y., Su, Z., Xiao, G., Balizan, E., Kaur, G., Niu, Z. & Wang, Q. 2011 Self-assembly of virus particles on flat surfaces via controlled evaporation. Langmuir 27, 1398–1402.CrossRefGoogle ScholarPubMed
42. Lunati, I. 2007 Young’s law and the effects of interfacial energy on the pressure at the solid–fluid interface. Phys. Fluids 19, 118105.CrossRefGoogle Scholar
43. Noether, E. 1918 Invariante variationsprobleme. Nachr. Königl. Ges. Wiss. Göttingen, Math.-phys. Kl. 2, 235–257 (translated as Noether, E. 1971 Transp. Theory Stat. Phys. 1, 186–207).Google Scholar
44. Petrov, P. G. & Petrov, J. G. 1992 A combined molecular-hydrodynamics approach to wetting kinetics. Langmuir 8, 1762–1767.CrossRefGoogle Scholar
45. Plawsky, J. L., Ojha, M., Chatterjee, A. & Wayner, P. C. 2008 Review of the effects of surface topography, surface chemistry, and fluid physics on evaporation at the contact line. Chem. Engng Commun. 196, 658–696.CrossRefGoogle Scholar
46. Pompe, T. & Herminghaus, S. 2000 Three-phase contact line energetics from nanoscale liquid surface topographies. Phys. Rev. Lett. 85, 1930–1933.CrossRefGoogle ScholarPubMed
47. de Ruijter, M. J., Charlot, M., Voué, M. & De Coninck, J. 2000 Experimental evidence of several time scales in drop spreading. Langmuir 16, 2363–2368.CrossRefGoogle Scholar
48. de Ruijter, M. J., De Coninck, J., Blake, T. D., Clarke, A. & Rankin, A. 1997 Contact angle relaxation during the spreading of partially wetting drops. Langmuir 13, 7293–7298.CrossRefGoogle Scholar
49. de Ruijter, M. J., De Coninck, J. & Oshanin, G. 1999 Droplet spreading: partial wetting regime revisited. Langmuir 15, 2209–2216.CrossRefGoogle Scholar
50. Rice, J. R. 1968 A path-independent integral and the approximate analysis of strain concentrations by notches and cracks. J. Appl. Mech. 35, 379–386.CrossRefGoogle Scholar
51. Rowlinson, J. S. & Widom, B. 1982 Molecular Theory of Capillarity. Dover.Google Scholar
52. Saiz, E., Benhassine, M., De Coninck, J. & Tomsia, A. P. 2010 Early stages of dissolutive spreading. Scr. Materialia 62, 934–938.CrossRefGoogle Scholar
53. Saiz, E., Tomsia, A. P., Rauch, N., Scheu, C., Ruehle, M., Benhassine, M., Seveno, D., de Coninck, J. & Lopez-Esteban, S. 2007 Nonreactive spreading at high temperature: molten metals and oxides on molybdenum. Phys. Rev. E 76, 041602.CrossRefGoogle ScholarPubMed
54. Serrin, J. 1959 Mathematical principles of classical fluid mechanics. In Handbuch der Physik, VIII/1, pp. 125–263. Springer.Google Scholar
55. Shikhmurzaev, Y. D. 1993 The moving contact line on a smooth solid surface. Intl. J. Multiphase Flow 19, 589–610.CrossRefGoogle Scholar
56. Shikhmurzaev, Y. D. 1997 Moving contact lines in liquid/liquid/solid systems. J. Fluid Mech. 359, 313–328.CrossRefGoogle Scholar
57. Shikhmurzaev, Y. D. 2008a Capillary Flows with Forming Interfaces. Chapman & Hall/CRC.Google Scholar
58. Shikhmurzaev, Y. D. 2008b On Young’s (1805) equation and Finn’s (2006) ‘counterexample’. Phys. Lett. A 372, 707.CrossRefGoogle Scholar
59. Shklyaev, O. E. & Fried, E. 2008 Interaction between a disclination and a uniaxial–isotropic phase interface in a nematic liquid-crystal. J. Colloid Interface Sci. 317, 298–313.CrossRefGoogle Scholar
60. Shklyaev, O. E., Shen, A. Q. & Fried, E. 2009 Evolution equation for a disclination line located between the uniaxial and isotropic phases of a nematic liquid-crystal. J. Colloid Interface Sci. 329, 140–152.CrossRefGoogle ScholarPubMed
61. Shmuylovich, L., Shen, A. Q. & Stone, H. A. 2002 Surface morphology of drying latex films: multiple ring formation. Langmuir 18, 3441–3445.CrossRefGoogle Scholar
62. Smalyukh, I., Zribi, O. V., Butler, J. C., Lavrentovich, O. D. & Wong, G. C. L. 2006 Structure and dynamics of liquid crystalline pattern formation in drying droplets of DNA. Phys. Rev. Lett. 96, 177801.CrossRefGoogle ScholarPubMed
63. Steigmann, D. J. & Li, D. 1995 Energy-minimizing states of capillary systems with bulk, surface, and line phases. IMA J. Appl. Maths 55, 1–17.CrossRefGoogle Scholar
64. Truesdell, C. & Toupin, R. A. 1960 The classical field theories. In Handbuch der Physik (ed. Flügge, S. ), Vol. III/1, pp. 226–293. Springer.Google Scholar
65. Truskett, V. N. & Stebe, K. J. 2003 Influence of surfactants on an evaporating drop: fluorescence images and particle deposition patterns. Langmuir 19, 8271–8279.CrossRefGoogle Scholar
66. Voinov, O. V. 1976 Hydrodynamics of wetting. Fluid Dyn. 11, 714–721.CrossRefGoogle Scholar
67. Ward, C. A. & Sasges, M. R. 1998 Effect of gravity on contact angle: a theoretical investigation. J. Chem. Phys. 109, 3651–3660.CrossRefGoogle Scholar
68. Wheeler, D., Warren, J. A. & Boettinger, W. J. 2010 Modeling the early stages of reactive wetting. Phys. Rev. E 82, 051601.CrossRefGoogle ScholarPubMed
69. Young, T. 1805 An essay on the cohesion of fluids. Phil. Trans. R. Soc. Lond. 95, 65–87.Google Scholar