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Effect of gravity-induced shape change on the diffusion-limited evaporation of thin sessile and pendant droplets

Hannah-May D'Ambrosio*

Stephen K. Wilson†

Alexander W. Wray‡ and Brian R. Duffy

  • School of Mathematics and Statistics, University of Glasgow, University Place, Glasgow G12 8QQ, United Kingdom

  • Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, United Kingdom and Department of Mathematics and Statistics, University of Strathclyde, Livingstone Tower, 26 Richmond Street, Glasgow G1 1XH, United Kingdom

  • Department of Mathematics and Statistics, University of Strathclyde, Livingstone Tower, 26 Richmond Street, Glasgow G1 1XH, United Kingdom

  • *Contact author: hannah-may.dambrosio@glasgow.ac.uk
  • †Contact author: sw3197@bath.ac.uk
  • ‡Contact author: alexander.wray@strath.ac.uk
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Phys. Rev. E 111, 045107 – Published 25 April, 2025

DOI: https://doi.org/10.1103/PhysRevE.111.045107

Abstract

A comprehensive study of the effect of gravity-induced shape change on the diffusion-limited evaporation of thin sessile and pendant droplets on a horizontal substrate is performed. Specifically, theoretical predictions for the evolution, and hence the lifetime, of sessile and pendant droplets evaporating in four modes of evaporation, namely, the constant contact radius (CR), the constant contact angle (CA), the stick-slide (SS), and the stick-jump (SJ) modes, are obtained. In particular, it is shown that gravity-induced shape change can cause quantitative differences in the evolution of sessile and pendant droplets compared to that of a droplet in the absence of (or in the neglect of) the effect of gravity (a “zero-gravity droplet”). For example, whereas sessile and pendant droplets evaporating in the CR mode evolve in qualitatively the same manner as a zero-gravity droplet, the evolution of droplets evaporating in the CA mode is more complicated. Specifically, while a zero-gravity droplet evaporating in the CA mode evolves according to the well-known d2 and 2/3 laws, an initially large sessile droplet evolves according to qualitatively different d and 1/2 laws, and an initially large pendant droplet evolves with the contact radius and the volume (but not, of course, the contact angle) behaving as if the droplet was evaporating in the CR mode. It is also found, perhaps somewhat unexpectedly, that the maximum height of a sessile droplet evaporating in the CA mode is a nonmonotonic function of time when the initial volume of the droplet is sufficiently large. Furthermore, it is found that for all four modes of evaporation a sessile droplet always evaporates faster, and hence has a shorter lifetime, than a zero-gravity droplet with the same initial volume, which in turn always evaporates faster, and hence has a shorter lifetime, than a pendant droplet with the same initial volume. It is also shown that for all four modes of evaporation the lifetime of a droplet is a monotonically increasing function of the initial volume of the droplet, that the lifetime of a droplet evaporating in the CA mode is always longer than that of the same droplet evaporating in the CR mode, and that the lifetimes of droplets evaporating in the SS and SJ modes both always lie between the lifetimes of the same droplet evaporating in the extreme modes.

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References (89)

  1. A.-M. Cazabat and G. Guéna, Evaporation of macroscopic sessile droplets, Soft Matter 6, 2591 (2010).
  2. A. F. Routh, Drying of thin colloidal films, Rep. Prog. Phys. 76, 046603 (2013).
  3. N. M. Kovalchuk, A. Trybala, and V. M. Starov, Evaporation of sessile droplets, Curr. Opin. Colloid Interface Sci. 19, 336 (2014).
  4. R. G. Larson, Transport and deposition patterns in drying sessile droplets, AIChE J. 60, 1538 (2014).
  5. Droplet Wetting and Evaporation: From Pure to Complex Fluids, edited by D. Brutin (Academic Press, New York, 2015).
  6. D. Lohse and X. Zhang, Surface nanobubbles and nanodroplets, Rev. Mod. Phys. 87, 981 (2015).
  7. D. Brutin and V. Starov, Recent advances in droplet wetting and evaporation, Chem. Soc. Rev. 47, 558 (2018).
  8. F. Giorgiutti-Dauphiné and L. Pauchard, Drying drops, Eur. Phys. J. E 41, 32 (2018).
  9. D. Zang, S. Tarafdar, Y. Yu. Tarasevich, M. D. Choudhury, and T. Dutta, Evaporation of a droplet: From physics to applications, Phys. Rep. 804, 1 (2019).
  10. Drying of Complex Fluid Drops: Fundamentals and Applications, edited by D. Brutin and K. Sefiane, Soft Matter Series No. 14 (Royal Society of Chemistry, Cambridge, 2022).
  11. H. Gelderblom, C. Diddens, and A. Marin, Evaporation-driven liquid flow in sessile droplets, Soft Matter 18, 8535 (2022).
  12. H. Y. Erbil and G. McHale, Droplet evaporation on superhydrophobic surfaces, Appl. Phys. Lett. 123, 080501 (2023).
  13. S. K. Wilson and H.-M. D'Ambrosio, Evaporation of sessile droplets, Annu. Rev. Fluid Mech. 55, 481 (2023).
  14. E. C. Tredenick, W. A. Forster, R. Pethiyagoda, R. M. van Leeuwen, and S. W. McCue, Evaporating droplets on inclined plant leaves and synthetic surfaces: Experiments and mathematical models, J. Colloid Interface Sci. 592, 329 (2021).
  15. J. L. Garcia-Cordero and Z. H. Fan, Sessile droplets for chemical and biological assays, Lab Chip 17, 2150 (2017).
  16. M. Kuang, L. Wang, and Y. Song, Controllable printing droplets for high-resolution patterns, Adv. Mater. 26, 6950 (2014).
  17. R. G. Picknett and R. Bexon, The evaporation of sessile or pendant drops in still air, J. Colloid Interface Sci. 61, 336 (1977).
  18. K. S. Birdi, D. T. Vu, and A. Winter, A study of the evaporation rates of small water drops placed on a solid surface, J. Phys. Chem. 93, 3702 (1989).
  19. H. Hu and R. G. Larson, Evaporation of a sessile droplet on a substrate, J. Phys. Chem. B 106, 1334 (2002).
  20. G. McHale, S. Aqil, N. J. Shirtcliffe, M. I. Newton, and H. Y. Erbil, Analysis of droplet evaporation on a superhydrophobic surface, Langmuir 21, 11053 (2005).
  21. T. A. H. Nguyen and A. V. Nguyen, On the lifetime of evaporating sessile droplets, Langmuir 28, 1924 (2012).
  22. J. M. Stauber, S. K. Wilson, B. R. Duffy, and K. Sefiane, On the lifetimes of evaporating droplets, J. Fluid Mech. 744, R2 (2014).
  23. J. M. Stauber, S. K. Wilson, B. R. Duffy, and K. Sefiane, On the lifetimes of evaporating droplets with related initial and receding contact angles, Phys. Fluids 27, 122101 (2015).
  24. S. K. Wilson and B. R. Duffy, Mathematical models for the evaporation of sessile droplets, in Drying of Complex Fluid Drops: Fundamentals and Applications, edited by D. Brutin and K. Sefiane, Soft Matter Series No. 14 (Royal Society of Chemistry, Cambridge, 2022), Chap. 4, pp. 47–67.
  25. R. D. Deegan, O. Bakajin, T. F. Dupont, G. Huber, S. R. Nagel, and T. A. Witten, Capillary flow as the cause of ring stains from dried liquid drops, Nature (London) 389, 827 (1997).
  26. R. D. Deegan, O. Bakajin, T. F. Dupont, G. Huber, S. R. Nagel, and T. A. Witten, Contact line deposits in an evaporating drop, Phys. Rev. E 62, 756 (2000).
  27. K. Sefiane, Patterns from drying drops, Adv. Colloid Interface Sci. 206, 372 (2014).
  28. X. Zhong, A. Crivoi, and F. Duan, Sessile nanofluid droplet drying, Adv. Colloid Interface Sci. 217, 13 (2015).
  29. D. Mampallil and H. B. Eral, A review on suppression and utilization of the coffee-ring effect, Adv. Colloid Interface Sci. 252, 38 (2018).
  30. M. Parsa, S. Harmand, and K. Sefiane, Mechanisms of pattern formation from dried sessile drops, Adv. Colloid Interface Sci. 254, 22 (2018).
  31. X. Yang, Z. Jiang, P. Lyu, Z. Ding, and X. Man, Deposition pattern of drying droplets, Commun. Theor. Phys. 73, 047601 (2021).
  32. H. Y. Erbil, G. McHale, and M. I. Newton, Drop evaporation on solid surfaces: Constant contact angle mode, Langmuir 18, 2636 (2002).
  33. A. K. Panwar, S. K. Barthwal, and S. Ray, Effect of evaporation on the contact angle of a sessile drop on solid substrates, J. Adhes. Sci. Technol. 17, 1321 (2003).
  34. B. Sobac and D. Brutin, Triple-line behavior and wettability controlled by nanocoated substrates: Influence on sessile drop dvaporation, Langmuir 27, 14999 (2011).
  35. E. L. Talbot, A. Berson, P. S. Brown, and C. D. Bain, Evaporation of picoliter droplets on surfaces with a range of wettabilities and thermal conductivities, Phys. Rev. E 85, 061604 (2012).
  36. K. Gleason and S. A. Putnam, Microdroplet evaporation with a forced pinned contact line, Langmuir 30, 10548 (2014).
  37. M. Gao, P. Kong, L.-X. Zhang, and J.-N. Liu, An experimental investigation of sessile droplets evaporation on hydrophilic and hydrophobic heating surface with constant heat flux, Int. Commun. Heat Mass Transfer 88, 262 (2017).
  38. S. Armstrong, G. McHale, R. Ledesma-Aguilar, and G. G. Wells, Pinning-free evaporation of sessile droplets of water from solid surfaces, Langmuir 35, 2989 (2019).
  39. S. Armstrong, G. McHale, R. Ledesma-Aguilar, and G. G. Wells, Evaporation and electrowetting of sessile droplets on slippery liquid-like surfaces and slippery liquid-infused porous surfaces (SLIPS), Langmuir 36, 11332 (2020).
  40. C. Bourgès-Monnier and M. E. R. Shanahan, Influence of evaporation on contact angle, Langmuir 11, 2820 (1995).
  41. H. Y. Erbil, G. McHale, S. M. Rowan, and M. I. Newton, Determination of the receding contact angle of sessile drops on polymer surfaces by evaporation, Langmuir 15, 7378 (1999).
  42. S. Semenov, V. M. Starov, R. G. Rubio, H. Agogo, and M. G. Velarde, Evaporation of sessile water droplets: Universal behaviour in presence of contact angle hysteresis, Colloids Surf. A 391, 135 (2011).
  43. T. A. H. Nguyen and A. V. Nguyen, Increased evaporation kinetics of sessile droplets by using nanoparticles, Langmuir 28, 16725 (2012).
  44. T. A. H. Nguyen, A. V. Nguyen, M. A. Hampton, Z. P. Xu, L. Huang, and V. Rudolph, Theoretical and experimental analysis of droplet evaporation on solid surfaces, Chem. Eng. Sci. 69, 522 (2012).
  45. S. Dash and S. V. Garimella, Droplet evaporation dynamics on a superhydrophobic surface with negligible hysteresis, Langmuir 29, 10785 (2013).
  46. M. A. Kadhim, N. Kapur, J. L. Summers, and H. Thompson, Experimental and theoretical investigation of droplet evaporation on heated hydrophilic and hydrophobic surfaces, Langmuir 35, 6256 (2019).
  47. E. Adachi, A. S. Dimitrov, and K. Nagayama, Stripe patterns formed on a glass surface during droplet evaporation, Langmuir 11, 1057 (1995).
  48. M. E. R. Shanahan, Simple theory of “stick-slip” wetting hysteresis, Langmuir 11, 1041 (1995).
  49. H. Bodiguel, F. Doumenc, and B. Guerrier, Stick-slip patterning at low capillary numbers for an evaporating colloidal suspension, Langmuir 26, 10758 (2010).
  50. D. Orejon, K. Sefiane, and M. E. R. Shanahan, Stick-slip of evaporating droplets: substrate hydrophobicity and nanoparticle concentration, Langmuir 27, 12834 (2011).
  51. A. Askounis, D. Orejon, V. Koutsos, K. Sefiane, and M. E. R. Shanahan, Nanoparticle deposits near the contact line of pinned volatile droplets: Size and shape revealed by atomic force microscopy, Soft Matter 7, 4152 (2011).
  52. A. Askounis, K. Sefiane, V. Koutsos, and M. E. R. Shanahan, Structural transitions in a ring stain created at the contact line of evaporating nanosuspension sessile drops, Phys. Rev. E 87, 012301 (2013).
  53. A. Askounis, K. Sefiane, V. Koutsos, and M. E. R. Shanahan, The effect of evaporation kinetics on nanoparticle structuring within contact line deposits of volatile drops, Colloids Surf. A 441, 855 (2014).
  54. E. Dietrich, E. S. Kooij, X. Zhang, H. J. W. Zandvliet, and D. Lohse, Stick-jump mode in surface droplet dissolution, Langmuir 31, 4696 (2015).
  55. N. Murisic and L. Kondic, On evaporation of sessile drops with moving contact lines, J. Fluid Mech. 679, 219 (2011).
  56. N. Shahidzadeh-Bonn, S. Rafaï, A. Azouni, and D. Bonn, Evaporating droplets, J. Fluid Mech. 549, 307 (2006).
  57. P. L. Kelly-Zion, C. J. Pursell, S. Vaidya, and J. Batra, Evaporation of sessile drops under combined diffusion and natural convection, Colloids Surf., A 381, 31 (2011).
  58. F. Carle, B. Sobac, and D. Brutin, Experimental evidence of the atmospheric convective transport contribution to sessile droplet evaporation, Appl. Phys. Lett. 102, 061603 (2013).
  59. O. Carrier, N. Shahidzadeh-Bonn, R. Zargar, M. Aytouna, M. Habibi, J. Eggers, and D. Bonn, Evaporation of water: Evaporation rate and collective effects, J. Fluid Mech. 798, 774 (2016).
  60. M. Moore, A. Amirfazli, and S. F. Chini, Sessile and pendant micro-liter drops evaporate at different rates: An experimental approach, J. Comput. Appl. Mech. 47, 109 (2016).
  61. B. Dollet and F. Boulogne, Natural convection above circular disks of evaporating liquids, Phys. Rev. Fluids 2, 053501 (2017).
  62. T. K. Pradhan and P. K. Panigrahi, Evaporation induced natural convection inside a droplet of aqueous solution placed on a superhydrophobic surface, Colloids Surf. A 530, 1 (2017).
  63. A. M. J. Edwards, P. S. Atkinson, C. S. Cheung, H. Liang, D. J. Fairhurst, and F. F. Ouali, Density-driven flows in evaporating binary liquid droplets, Phys. Rev. Lett. 121, 184501 (2018).
  64. Y. Li, C. Diddens, P. Lv, H. Wijshoff, M. Versluis, and D. Lohse, Gravitational effect in evaporating binary microdroplets, Phys. Rev. Lett. 122, 114501 (2019).
  65. A. P. Sommer, Limits of the impact of gravity on self-organizing nanospheres, J. Phys. Chem. B 108, 8096 (2004).
  66. I. Sandu and C. T. Fleaca, The influence of gravity on the distribution of the deposit formed onto a substrate by sessile, hanging, and sandwiched hanging drop evaporation, J. Colloid Interface Sci. 358, 621 (2011).
  67. M. A. Hampton, T. A. H. Nguyen, A. V. Nguyen, Z. P. Xu, L. Huang, and V. Rudolph, Influence of surface orientation on the organization of nanoparticles in drying nanofluid droplets, J. Colloid Interface Sci. 377, 456 (2012).
  68. N. R. Devlin, K. Loehr, and M. T. Harris, The importance of gravity in droplet evaporation: A comparison of pendant and sessile drop evaporation with particles, AIChE J. 62, 947 (2016).
  69. M. R. Moore and A. W. Wray, Gravitational effects on coffee-ring formation during the evaporation of sessile droplets, J. Fluid Mech. 967, A26 (2023).
  70. X. Du and R. D. Deegan, Ring formation on an inclined surface, J. Fluid Mech. 775, R3 (2015).
  71. J. Y. Kim, I. G. Hwang, and B. M. Weon, Evaporation of inclined water droplets, Sci. Rep. 7, 42848 (2017).
  72. M. L. Timm, E. Dehdashti, A. J. Darban, and H. Masoud, Evaporation of a sessile droplet on a slope, Sci. Rep. 9, 19803 (2019).
  73. P. Dhar, R. K. Dwivedi, and A. R. Harikrishnan, Surface declination governed asymmetric sessile droplet evaporation, Phys. Fluids 32, 112010 (2020).
  74. J. Cai, F. Chen, N. T. Chamakos, A. G. Papathanasiou, B. Tan, and Q. Li, Asymmetric droplet evaporation on inclined surfaces, Prog. Nucl. Energy 163, 104820 (2023).
  75. S. Tonini and G. E. Cossali, Modeling the evaporation of sessile drops deformed by gravity on hydrophilic and hydrophobic surfaces, Phys. Fluids 35, 032113 (2023).
  76. S. Tonini and G. E. Cossali, Evaporation of drops on superhydrophobic surfaces: The effect of deformation due to the gravitational field, Phys. Fluids 35, 061707 (2023).
  77. S. Tonini and G. E. Cossali, Modeling the effect of shape deformation induced by gravity on the evaporation of pendant and sessile drops, Phys. Fluids 36, 027131 (2024).
  78. G. J. Dunn, S. K. Wilson, B. R. Duffy, D. S, and S. K, A mathematical model for the evaporation of a thin sessile liquid droplet: Comparison between experiment and theory, Colloids Surf. A 323, 50 (2008).
  79. A. W. Wray, B. R. Duffy, and S. K. Wilson, Competitive evaporation of multiple sessile droplets, J. Fluid Mech. 884, A45 (2020).
  80. A. W. Wray, P. S. Wray, B. R. Duffy, and S. K. Wilson, Contact-line deposits from multiple evaporating droplets, Phys. Rev. Fluids 6, 073604 (2021).
  81. G. J. Dunn, S. K. Wilson, B. R. Duffy, S. David, and K. Sefiane, The strong influence of substrate conductivity on droplet evaporation, J. Fluid Mech. 623, 329 (2009).
  82. K. Sefiane, S. K. Wilson, S. David, G. J. Dunn, and B. R. Duffy, On the effect of the atmosphere on the evaporation of sessile droplets of water, Phys. Fluids 21, 062101 (2009).
  83. M. Ait Saada, S. Chikh, and L. Tadrist, Evaporation of a sessile drop with pinned or receding contact line on a substrate with different thermophysical properties, Int. J. Heat Mass Transf. 58, 197 (2013).
  84. F. G. H. Schofield, D. Pritchard, S. K. Wilson, and K. Sefiane, The lifetimes of evaporating sessile droplets of water can be strongly influenced by thermal effects, Fluids 6, 141 (2021).
  85. J. F. Padday, Sessile drop profiles: Corrected methods for surface tension and spreading coefficients, Proc. R. Soc. London Ser. A 330, 561 (1972).
  86. P. Aussillous and D. Quéré, Properties of liquid marbles, Proc. R. Soc. Lond. Ser. A 462, 973 (2006).
  87. R. Finn, Equilibrium Capillary Surfaces (Springer–Verlag, New York, 1986).
  88. J. F. Padday and A. R. Pitt, The stability of axisymmetric menisci, Philos. Trans. R. Soc. London Ser. A 275, 489 (1973).
  89. A. Kumar, M. R. Gunjan, K. Jakhar, A. Thakur, and R. Raj, Unified framework for mapping shape and stability of pendant drops including the effect of contact angle hysteresis, Colloids Surf. A 597, 124619 (2020).

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Phys. Rev. E 111, 045107– Published 25 April, 2025
  • Received 31 October 2023
  • Revised 6 February 2025
  • Accepted 21 February 2025
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