Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Perspective
  • Published:

Physicochemical hydrodynamics of droplets out of equilibrium

Nature Reviews Physics volume 2, pages 426–443 (2020)Cite this article

Subjects

Abstract

Droplets abound in nature and technology. In general, they are multicomponent, and, when out of equilibrium, have gradients in concentration, implying flow and mass transport. Moreover, phase transitions can occur, in the form of evaporation, solidification, dissolution or nucleation of a new phase. The droplets and their surrounding liquid can be binary, ternary or contain even more components, with several in different phases. Since the early 2000s, rapid advances in experimental and numerical fluid dynamical techniques have enabled major progress in our understanding of the physicochemical hydrodynamics of such droplets, further narrowing the gap from fluid dynamics to chemical engineering and colloid and interfacial science, arriving at a quantitative understanding of multicomponent and multiphase droplet systems far from equilibrium, and aiming towards a one-to-one comparison between experiments and theory or numerics. This Perspective discusses examples of the physicochemical hydrodynamics of droplet systems far from equilibrium and the relevance of such systems for applications.

This is a preview of subscription content, access via your institution

Access options

Buy this article

  • Purchase on Springer Link
  • Instant access to full article PDF
Buy now

Prices may be subject to local taxes which are calculated during checkout

Fig. 1: Immiscible droplets in the bulk, driven by physicochemical hydrodynamical effects.
Fig. 2: Various types of coalescence of two droplets consisting of different liquids.
Fig. 3: Collective dissolution for droplets.
Fig. 4: Multicomponent droplet evaporation.
Fig. 5: Physicochemical hydrodynamical effects evolving phase transitions.
Fig. 6: Principles of solvent exchange.
Fig. 7: Evaporating ouzo droplet.

Similar content being viewed by others

References

  1. Levich, V. G. Physicochemical Hydrodynamics (Prentice Hall, 1962).

  2. Cates, M. E. & Tjhung, E. Theories of binary fluid mixtures: from phase-separation kinetics to active emulsions. J. Fluid Mech. 836, https://doi.org/10.1017/jfm.2017.832 (2018).

  3. Lauga, E. & Powers, T. R. The hydrodynamics of swimming microorganisms. Rep. Prog. Phys. 72, 096601 (2009).

    ADS  MathSciNet  Google Scholar 

  4. Maass, C. C., Krüger, C., Herminghaus, S. & Bahr, C. Swimming droplets. Annu. Rev. Cond. Matter Phys. 7, 171–193 (2016).

    ADS  Google Scholar 

  5. Moran, J. L. & Posner, J. D. Phoretic self-propulsion. Annu. Rev. Fluid Mech. 49, 511–540 (2017).

    ADS  MathSciNet  MATH  Google Scholar 

  6. Golestanian, R.Phoretic active matter. Preprint at https://arxiv.org/abs/1909.03747 (2019).

  7. Manikantan, H. & Squires, T. M. Surfactant dynamics: hidden variables controlling fluid flows. J. Fluid Mech. 892 https://doi.org/10.1017/jfm.2020.170 (2020).

  8. Cazabat, A. M. & Guéna, G. Evaporation of macroscopic sessile droplets. Soft Matter 6, 2591–2612 (2010).

    ADS  Google Scholar 

  9. Erbil, H. Y. Evaporation of pure liquid sessile and spherical suspended drops: a review. Adv. Colloid Interface Sci. 170, 67–86 (2012).

    Google Scholar 

  10. Sefiane, K. Patterns from drying drops. Adv. Colloid Interface Sci. 206, 372–381 (2014).

    Google Scholar 

  11. Lohse, D. & Zhang, X. Surface nanobubble and surface nanodroplets. Rev. Mod. Phys. 87, 981–1035 (2015).

    ADS  MathSciNet  Google Scholar 

  12. Jain, A. & Verma, K. K. Recent advances in applications of single-drop microextraction: a review. Anal. Chim. Acta 706, 37–65 (2011).

    Google Scholar 

  13. de Wit, A. Chemo-hydrodynamic patterns and instabilities. Annu. Rev. Fluid Mech. 52, 531–555 (2020).

    ADS  MATH  Google Scholar 

  14. Lohse, D. Bubble puzzles: from fundamentals to applications. Phys. Rev. Fluids 3, 110504 (2018).

    ADS  Google Scholar 

  15. Young, N., Goldstein, J. & Block, M. J. The motion of bubbles in a vertical temperature gradient. J. Fluid Mech. 6, 350–356 (1959).

    ADS  MATH  Google Scholar 

  16. Li, Y. et al. Bouncing oil droplet in a stratified liquid and its sudden death. Phys. Rev. Lett. 122, 154502 (2019).

    ADS  Google Scholar 

  17. Anderson, J. L. Colloid transport by interfacial forces. Annu. Rev. Fluid Mech. 21, 61–99 (1989).

    ADS  MATH  Google Scholar 

  18. Anderson, J. L., Lowell, M. E. & Prieve, D. C. Motion of a particle generated by chemical gradients. Part 1. Non-electrolytes. J. Fluid Mech. 117, 107–121 (1982).

    ADS  MATH  Google Scholar 

  19. Izri, Z., Van Der Linden, M. N., Michelin, S. & Dauchot, O. Self-propulsion of pure water droplets by spontaneous Marangoni-stress-driven motion. Phys. Rev. Lett. 113, 248302 (2014).

    ADS  Google Scholar 

  20. Marbach, S., Yoshida, H. & Bocquet, L. Osmotic and diffusio-osmotic flow generation at high solute concentration. I. Mechanical approaches. J. Chem. Phys. 146, 194701 (2017).

    ADS  Google Scholar 

  21. Prieve, D. C., Malone, S. M., Khair, A. S., Stout, R. F. & Kanj, M. Y. Diffusiophoresis of charged colloidal particles in the limit of very high salinity. Proc. Natl Acad. Sci. USA 116, 18257–18262 (2019).

    ADS  Google Scholar 

  22. Yang, F., Shin, S. & Stone, H. A. Diffusiophoresis of a charged drop. J. Fluid Mech. 852, 37–59 (2018).

    ADS  MathSciNet  MATH  Google Scholar 

  23. Morozov, M. & Michelin, S. Nonlinear dynamics of a chemically-active drop: from steady to chaotic self-propulsion. J. Chem. Phys. 150, 044110 (2019).

    ADS  Google Scholar 

  24. Riegler, H. & Lazar, P. Delayed coalescence behavior of droplets with completely miscible liquids. Langmuir 24, 6395–6398 (2008).

    Google Scholar 

  25. Karpitschka, S. & Riegler, H. Quantitative experimental study on the transition between fast and delayed coalescence of sessile droplets with different but completely miscible liquids. Langmuir 26, 11823–11829 (2010).

    Google Scholar 

  26. Karpitschka, S. & Riegler, H. Noncoalescence of sessile drops from different but miscible liquids: hydrodynamic analysis of the twin drop contour as a self-stabilizing traveling wave. Phys. Rev. Lett. 109, 066103 (2012).

    ADS  Google Scholar 

  27. Koldeweij, R. B., Van Capelleveen, B. F., Lohse, D. & Visser, C. W. Marangoni-driven spreading of miscible liquids in the binary pendant drop geometry. Soft Matter 15, 8525–8531 (2019).

    ADS  Google Scholar 

  28. Yeo, Y., Basaran, O. A. & Park, K. A new process for making reservoir-type microcapsules using ink-jet technology and interfacial phase separation. J. Control. Release 93, 161–173 (2003).

    Google Scholar 

  29. Visser, C. W., Kamperman, T., Karbaat, L. P., Lohse, D. & Karperien, M. In-air microfluidics enables rapid fabrication of emulsions, suspensions, and 3D modular (bio) materials. Sci. Adv. 4, eaao1175 (2018).

    ADS  Google Scholar 

  30. Yu, H., Kant, P., Dyett, B., Lohse, D. & Zhang, X. Splitting droplet through coalescence of two different three-phase contact lines. Soft Matter 15, 6055–6061 (2019).

    ADS  Google Scholar 

  31. Berg, S. Marangoni-driven spreading along liquid–liquid interfaces. Phys. Fluids 21, 032105 (2009).

    ADS  MATH  Google Scholar 

  32. Wodlei, F., Sebilleau, J., Magnaudet, J. & Pimienta, V. Marangoni-driven flower-like patterning of an evaporating drop spreading on a liquid substrate. Nat. Commun. 9, 820 (2018).

    ADS  Google Scholar 

  33. Jehannin, M. et al. Periodic precipitation patterns during coalescence of reacting sessile droplets. Langmuir 31, 11484–11490 (2015).

    Google Scholar 

  34. Paxton, W. et al. Catalytic nanomotors: autonomous movement of striped nanorods. J. Am. Chem. Soc. 126, 13424–13431 (2004).

    Google Scholar 

  35. Golestanian, R., Liverpool, T. B. & Ajdari, A. Propulsion of a molecular machine by asymmetric distribution of reaction products. Phys. Rev. Lett. 94, 220801 (2005).

    ADS  Google Scholar 

  36. Jiang, H.-R., Yoshinaga, N. & Sano, M. Active motion of a Janus particle by self-thermophoresis in a defocused laser beam. Phys. Rev. Lett. 105, 268302 (2010).

    ADS  Google Scholar 

  37. Buttinoni, I. et al. Dynamical clustering and phase separation in suspensions of self-propelled colloidal particles. Phys. Rev. Lett. 110, 238301 (2013).

    ADS  Google Scholar 

  38. Michelin, S. & Lauga, E. Phoretic self-propulsion at finite Péclet numbers. J. Fluid Mech. 747, 572–604 (2014).

    ADS  MathSciNet  MATH  Google Scholar 

  39. Michelin, S., Lauga, E. & Bartolo, D. Spontaneous autophoretic motion of isotropic particles. Phys. Fluids 25, 061701 (2013).

    ADS  Google Scholar 

  40. Golovin, A., Gupalo, Y. P. & Ryazantsev, Y. S. Change in shape of drop moving due to the chemithermocapillary effect. J. Appl. Mech. Tech. Phys. 30, 602–609 (1989).

    ADS  Google Scholar 

  41. Rednikov, A. Y., Ryazantsev, Y. S. & Velarde, M. G. Drop motion with surfactant transfer in a homogeneous surrounding. Phys. Fluids 6, 451–468 (1994).

    ADS  MathSciNet  MATH  Google Scholar 

  42. Schmitt, M. & Stark, H. Swimming active droplet: a theoretical analysis. EPL 101, 44008 (2013).

    ADS  Google Scholar 

  43. Morozov, M. & Michelin, S. Self-propulsion near the onset of Marangoni instability of deformable active droplets. J. Fluid Mech. 860, 711–738 (2019).

    ADS  MathSciNet  MATH  Google Scholar 

  44. Abécassis, B., Cottin-Bizonne, C., Ybert, C., Ajdari, A. & Bocquet, L. Boosting migration of large particles by solute contrasts. Nat. Mater. 7, 785–789 (2008).

    ADS  Google Scholar 

  45. Palacci, J., Abecassis, B., Cottin-Bizonne, C., Ybert, C. & Bocquet, L. Colloidal motility and pattern formation under rectified diffusiophoresis. Phys. Rev. Lett. 104, 138302 (2010).

    ADS  Google Scholar 

  46. Banerjee, A., Williams, I., Azevedo, R. N., Helgeson, M. E. & Squires, T. M. Soluto-inertial phenomena: designing long-range, long-lasting, surface-specific interactions in suspensions. Proc. Natl Acad. Sci. USA 113, 8612–8617 (2016).

    ADS  Google Scholar 

  47. Banerjee, A. & Squires, T. M. Long-range, selective, on-demand suspension interactions: combining and triggering soluto-inertial beacons. Sci. Adv. 5, eaax1893 (2019).

    ADS  Google Scholar 

  48. Krüger, C., Klös, G., Bahr, C. & Maass, C. C. Curling liquid crystal microswimmers: a cascade of spontaneous symmetry breaking. Phys. Rev. Lett. 117, 048003 (2016).

    ADS  MathSciNet  Google Scholar 

  49. Suga, M., Suda, S., Ichikawa, M. & Kimura, Y. Self-propelled motion switching in nematic liquid crystal droplets in aqueous surfactant solutions. Phys. Rev. E 97, 062703 (2018).

    ADS  Google Scholar 

  50. Hu, W.-F., Lin, T.-S., Rafai, S. & Misbah, C. Chaotic swimming of phoretic particles. Phys. Rev. Lett. 123, 238004 (2019).

    ADS  Google Scholar 

  51. Thutupalli, S., Seemann, R. & Herminghaus, S. Swarming behavior of simple model squirmers. New J. Phys. 13, 073021 (2011).

    ADS  Google Scholar 

  52. Palacci, J., Sacanna, S., Steinberg, A. P., Pine, D. J. & Chaikin, P. M. Living crystals of light-activated colloidal surfers. Science 339, 936–940 (2013).

    ADS  Google Scholar 

  53. Moerman, P. G. et al. Solute-mediated interactions between active droplets. Phys. Rev. E 96, 032607 (2017).

    ADS  Google Scholar 

  54. Lippera, K., Morozov, M., Benzaquen, M. & Michelin, S. Collisions and rebounds of chemically-active droplets. J. Fluid Mech. 886, A17 (2020).

    ADS  MathSciNet  MATH  Google Scholar 

  55. Peña, A. A. & Miller, C. A. Solubilization rates of oils in surfactant solutions and their relationship to mass transport in emulsions. Adv. Colloid Interface Sci. 123, 241–257 (2006).

    Google Scholar 

  56. Herminghaus, S. et al. Interfacial mechanisms in active emulsions. Soft Matter 10, 7008–7022 (2014).

    ADS  Google Scholar 

  57. Hokmabad, B. V., Baldwin, K. A., Krüger, C., Bahr, C. & Maass, C. C. Topological stabilization and dynamics of self-propelling nematic shells. Phys. Rev. Lett. 123, 178003 (2019).

    ADS  Google Scholar 

  58. Poesio, P., Beretta, G. P. & Thorsen, T. Dissolution of a liquid microdroplet in a nonideal liquid–liquid mixture far from thermodynamic equilibrium. Phys. Rev. Lett. 103, 064501 (2009).

    ADS  Google Scholar 

  59. Lagzi, I., Soh, S., Wesson, P. J., Browne, K. P. & Grzybowski, B. A. Maze solving by chemotactic droplets. J. Am. Chem. Soc. 132, 1198–1199 (2010).

    Google Scholar 

  60. Cejkova, J., Novak, M., Stepanek, F. & Hanczyc, M. M. Dynamics of chemotactic droplets in salt concentration gradients. Langmuir 30, 11937–11944 (2014).

    Google Scholar 

  61. Jin, C., Krüger, C. & Maass, C. C. Chemotaxis and autochemotaxis of self-propelling droplet swimmers. Proc. Natl Acad. Sci. USA 114, 5089–5094 (2017).

    ADS  Google Scholar 

  62. Epstein, P. S. & Plesset, M. S. On the stability of gas bubbles in liquid–gas solutions. J. Chem. Phys. 18, 1505–1509 (1950).

    ADS  Google Scholar 

  63. Duncan, P. B. & Needham, D. Microdroplet dissolution into a second-phase solvent using a micropipet technique: test of the Epstein–Plesset model for an aniline–water system. Langmuir 22, 4190–4197 (2006).

    Google Scholar 

  64. Picknett, R. G. & Bexon, R. The evaporation of sessile or pendant drops in still air. J. Colloid Interface Sci. 61, 336–350 (1977).

    ADS  Google Scholar 

  65. Deegan, R. D. et al. Capillary flow as the cause of ring stains from dried liquid drops. Nature 389, 827–829 (1997).

    ADS  Google Scholar 

  66. Hu, H. & Larson, R. G. Evaporation of a sessile droplet on a substrate. J. Phys. Chem. B 106, 1334 (2002).

    Google Scholar 

  67. Popov, Y. O. Evaporative deposition patterns: spatial dimensions of the deposit. Phys. Rev. E 71, 036313 (2005).

    ADS  Google Scholar 

  68. Gelderblom, H. et al. How water droplets evaporate on a superhydrophobic substrate. Phys. Rev. E 83, 026306 (2011).

    ADS  Google Scholar 

  69. Stauber, J. M., Wilson, S. K., Duffy, B. R. & Sefiane, K. On the lifetimes of evaporating droplets. J. Fluid Mech. 744, R2 (2014).

    ADS  Google Scholar 

  70. Dietrich, E. et al. Role of natural convection in the dissolution of sessile droplets. J. Fluid Mech. 794, 45–67 (2016).

    ADS  MathSciNet  Google Scholar 

  71. Laghezza, G. et al. Collective and convective effects compete in patterns of dissolving surface droplets. Soft Matter 12, 5787–5796 (2016).

    ADS  Google Scholar 

  72. Carrier, O. et al. Evaporation of water: evaporation rate and collective effects. J. Fluid Mech. 798, 774–786 (2016).

    ADS  MathSciNet  MATH  Google Scholar 

  73. Zhu, X., Verzicco, R., Zhang, X. & Lohse, D. Diffusive interaction of multiple surface nanobubbles: shrinkage, growth, and coarsening. Soft Matter 14, 2006–2014 (2018).

    ADS  Google Scholar 

  74. Michelin, S., Guérin, E. & Lauga, E. Collective dissolution of microbubbles. Phys. Rev. Fluids 3, 043601 (2018).

    ADS  Google Scholar 

  75. Bao, L. et al. Flow-induced dissolution of femtoliter surface droplet arrays. Lab Chip 18, 1066–1074 (2018).

    Google Scholar 

  76. Wray, A. W., Duffy, B. R. & Wilson, S. K. Competitive evaporation of multiple sessile droplets. J. Fluid Mech. 884, A45 (2020).

    ADS  MathSciNet  MATH  Google Scholar 

  77. Chong, K. L., Li, Y., Ng, C. S., Verzicco, R. & Lohse, D. Convection-dominated dissolution for single and multiple immersed sessile droplets. J. Fluid Mech. 892, A21 (2020).

    ADS  MathSciNet  MATH  Google Scholar 

  78. Deegan, R. D. et al. Contact line deposits in an evaporating drop. Phys. Rev. E 62, 756–765 (1998).

    ADS  Google Scholar 

  79. Scriven, L. & Sternling, C. The Marangoni effects. Nature 187, 186–188 (1960).

    ADS  Google Scholar 

  80. Hu, H. & Larson, R. G. Analysis of the effects of Marangoni stresses on the microflow in an evaporating sessile droplet. Langmuir 21, 3972–3980 (2005).

    Google Scholar 

  81. Bennacer, R. & Sefiane, K. Vortices, dissipation and flow transition in volatile binary drops. J. Fluid Mech. 749, 649–665 (2014).

    ADS  Google Scholar 

  82. Tan, H. et al. Evaporation-triggered microdroplet nucleation and the four life phases of an evaporating ouzo drop. Proc. Natl Acad. Sci. USA 113, 8642–8647 (2016).

    ADS  MATH  Google Scholar 

  83. Kim, H. et al. Controlled uniform coating from the interplay of Marangoni flows and surface-adsorbed macromolecules. Phys. Rev. Lett. 116, 124501 (2016).

    ADS  Google Scholar 

  84. Dietrich, E. et al. Segregation in dissolving binary component sessile droplets. J. Fluid Mech. 812, 349–369 (2017).

    ADS  MATH  Google Scholar 

  85. Karpitschka, S., Liebig, F. & Riegler, H. Marangoni contraction of evaporating sessile droplets of binary mixtures. Langmuir 33, 4682–4687 (2017).

    Google Scholar 

  86. Diddens, C. et al. Evaporating pure, binary and ternary droplets: thermal effects and axial symmetry breaking. J. Fluid Mech. 823, 470–497 (2017).

    ADS  MathSciNet  MATH  Google Scholar 

  87. Kim, H., Muller, K., Shardt, O., Afkhami, S. & Stone, H. A. Solutal Marangoni flows of miscible liquids drive transport without surface contamination. Nat. Phys. 13, 1105 (2017).

    Google Scholar 

  88. Li, Y. et al. Evaporation-triggered segregation of sessile binary droplets. Phys. Rev. Lett. 120, 224501 (2018).

    ADS  Google Scholar 

  89. Kim, H. & Stone, H. A. Direct measurement of selective evaporation of binary mixture droplets by dissolving materials. J. Fluid Mech. 850, 769–783 (2018).

    ADS  MATH  Google Scholar 

  90. Edwards, A. et al. Density-driven flows in evaporating binary liquid droplets. Phys. Rev. Lett. 121, 184501 (2018).

    ADS  Google Scholar 

  91. Li, Y. et al. Gravitational effect in evaporating binary microdroplets. Phys. Rev. Lett. 122, 114501 (2019).

    ADS  Google Scholar 

  92. Marin, A. et al. Solutal Marangoni flow as the cause of ring stains from drying salty colloidal drops. Phys. Rev. Fluids 4, 041601 (2019).

    ADS  Google Scholar 

  93. Hosoi, A. E. & Bush, J. W. M. Evaporative instabilities in climbing films. J. Fluid Mech. 442, 217–239 (2001).

    ADS  MathSciNet  MATH  Google Scholar 

  94. Shin, S., Jacobi, I. & Stone, H. A. Bénard–Marangoni instability driven by moisture absorption. EPL 113, 24002 (2016).

    ADS  Google Scholar 

  95. Karpitschka, S. The value of a fading tracer. J. Fluid Mech. 856, 1–4 (2018).

    ADS  MathSciNet  MATH  Google Scholar 

  96. Li, Y. et al. Rayleigh–Taylor instability by segregation in an evaporating multicomponent microdroplet. J. Fluid Mech. in the press.

  97. Keiser, L., Bense, H., Colinet, P., Bico, J. & Reyssat, E. Marangoni bursting: evaporation-induced emulsification of binary mixtures on a liquid layer. Phys. Rev. Lett. 118, 074504 (2017).

    ADS  Google Scholar 

  98. Durey, G. et al. Marangoni bursting: evaporation-induced emulsification of a two-component droplet. Phys. Rev. Fluids 3, 100501 (2018).

    ADS  Google Scholar 

  99. Hernández-Sánchez, J. F., Eddi, A. & Snoeijer, J. H. Marangoni spreading due to a localized alcohol supply on a thin water film. Phys. Fluids 27, 032003 (2015).

    ADS  Google Scholar 

  100. Maheshwari, S., Van Der Hoef, M., Prosperetti, A. & Lohse, D. Molecular dynamics study of multicomponent droplet dissolution in a sparingly miscible liquid. J. Fluid Mech. 833, 54–69 (2017).

    ADS  MathSciNet  MATH  Google Scholar 

  101. Chu, S. & Prosperetti, A. Dissolution and growth of a multicomponent drop in an immiscible liquid. J. Fluid Mech. 798, 787–811 (2016).

    ADS  MathSciNet  MATH  Google Scholar 

  102. Lohse, D. Towards controlled liquid–liquid microextraction. J. Fluid Mech. 804, 1–4 (2016).

    ADS  MathSciNet  Google Scholar 

  103. Deville, S. Freezing Colloids: Observations, Principles, Control, and Use. Applications in Materials Science, Life Science, Earth Science, Food Science, and Engineering (Springer, 2017).

  104. Dedovets, D., Monteux, C. & Deville, S. Five-dimensional imaging of freezing emulsions with solute effects. Science 360, 303–306 (2018).

    Google Scholar 

  105. Vitale, S. & Katz, J. Liquid droplet dispersions formed by homogeneous liquid–liquid nucleation: ‘the ouzo effect’. Langmuir 19, 4105–4110 (2003).

    Google Scholar 

  106. Ganachaud, F. & Katz, J. Nanoparticles and nanocapsules created using the ouzo effect: spontaneous emulsification as an alternative to ultrasonic and high-shear devices. Chem. Phys. Chem 6, 209–216 (2005).

    Google Scholar 

  107. Lepeltier, E., Bourgaux, C. & Couvreur, P. Nanoprecipitation and the ‘ouzo effect’: application to drug delivery devices. Adv. Drug Deliv. Rev. 71, 86–97 (2014).

    Google Scholar 

  108. Voorhees, P. W. The theory of Ostwald ripening. J. Stat. Phys. 38, 231–252 (1985).

    ADS  Google Scholar 

  109. Solans, C., Izquierdo, P., Nolla, J., Azemar, N. & Garcia-Celma, M. J. Nano-emulsions. Curr. Opin. Colloid Interface Sci. 10, 102–110 (2005).

    Google Scholar 

  110. Zemb, T. N. et al. How to explain microemulsions formed by solvent mixtures without conventional surfactants. Proc. Natl Acad. Sci. USA 113, 4260–4265 (2016).

    ADS  Google Scholar 

  111. Zhang, X. H. et al. From transient nanodroplets to permanent nanolenses. Soft Matter 8, 4314–4317 (2012).

    ADS  Google Scholar 

  112. Joanny, J. & de Gennes, P. A model for contact angle hysteresis. J. Chem. Phys. 81, 552 (1984).

    ADS  Google Scholar 

  113. de Gennes, P. G. Wetting: statics and dynamics. Rev. Mod. Phys. 57, 827–863 (1985).

    ADS  MathSciNet  Google Scholar 

  114. Liu, Y. & Zhang, X. Evaporation dynamics of nanodroplets and their anomalous stability on rough substrates. Phys. Rev. E 88, 012404 (2013).

    ADS  Google Scholar 

  115. Lohse, D. & Zhang, X. Pinning and gas oversaturation imply stable single surface nanobubble. Phys. Rev. E 91, 031003(R) (2015).

    ADS  Google Scholar 

  116. Zhang, X. et al. Formation of surface nanodroplets under controlled flow conditions. Proc. Natl Acad. Sci. USA 112, 9253–9257 (2015).

    ADS  Google Scholar 

  117. Yu, H., Lu, Z., Lohse, D. & Zhang, X. Gravitational effect on the formation of surface nanodroplets. Langmuir 31, 12628–12634 (2015).

    Google Scholar 

  118. Yu, H., Maheshwari, S., Zhu, J., Lohse, D. & Zhang, X. Formation of surface nanodroplets facing a structured microchannel wall. Lab Chip 17, 1496–1504 (2017).

    Google Scholar 

  119. Zeng, B., Wang, Y., Zhang, X. & Lohse, D. Solvent exchange in a Hele-Shaw cell: universality of surface nanodroplet nucleation. J. Phys. Chem. C 123, 5571–5577 (2019).

    Google Scholar 

  120. Bao, L., Werbiuk, Z., Lohse, D. & Zhang, Z. Controlling the growth modes of femtoliter sessile droplets nucleating on chemically patterned surfaces. J. Phys. Chem. Lett. 7, 1055–1059 (2016).

    Google Scholar 

  121. Peng, S., Mega, T. L. & Zhang, X. Collective effects in microbubble growth by solvent exchange. Langmuir 32, 11265–11272 (2016).

    Google Scholar 

  122. Dyett, B. et al. Growth dynamics of surface nanodroplets during solvent exchange at varying flow rates. Soft Matter 14, 5197 (2018).

    ADS  Google Scholar 

  123. Peng, S., Lohse, D. & Zhang, X. Spontaneous pattern formation of surface nanodroplets from competitive growth. ACS Nano 9, 11916–11923 (2015).

    Google Scholar 

  124. Xu, C. et al. Collective interactions in the nucleation and growth of surface droplets. Soft Matter 13, 937–944 (2017).

    ADS  Google Scholar 

  125. Dyett, B., Hao, H., Lohse, D. & Zhang, X. Coalescence driven self-organization of growing nanodroplets around a microcap. Soft Matter 14, 2628–2637 (2018).

    ADS  Google Scholar 

  126. Schlichting,H.Boundary Layer Theory 7th edn (McGraw Hill, 1979).

  127. Zhang, X. et al. Mixed mode of dissolving immersed microdroplets at a solid–water interface. Soft Matter 11, 1889–1900 (2015).

    ADS  Google Scholar 

  128. Tan, H. et al. Self-wrapping of an ouzo drop induced by evaporation on a superamphiphobic surface. Soft Matter 13, 2749–2759 (2017).

    ADS  Google Scholar 

  129. Tan, H. et al. Microdroplet nucleation by dissolution of a multicomponent drop in a host liquid. J. Fluid Mech. 870, 217–246 (2019).

    ADS  MathSciNet  Google Scholar 

  130. Kirkaldy, J. S. & Brown, L. Diffusion behaviour in ternary, multiphase systems. Can. Metall. Q. 2, 89–115 (1963).

    Google Scholar 

  131. Ruschak, K. J. & Miller, C. A. Spontaneous emulsification in ternary systems with mass transfer. Ind. Eng. Chem. Fundamentals 11, 534–540 (1972).

    Google Scholar 

  132. Miller, C.A., Neogi, P. Interfacial Phenomena: Equilibrium and Dynamic Effects Vol. 139 (CRC, 2007).

  133. Otero, J., Meeker, S. & Clegg, P. S. Compositional ripening of particle-stabilized drops in a three-liquid system. Soft Matter 14, 3783–3790 (2018).

    ADS  Google Scholar 

  134. Choi, C.-H., Weitz, D. A. & Lee, C.-S. One step formation of controllable complex emulsions: from functional particles to simultaneous encapsulation of hydrophilic and hydrophobic agents into desired position. Adv. Mater. 25, 2536–2541 (2013).

    Google Scholar 

  135. Haase, M. F. & Brujic, J. Tailoring of high-order multiple emulsions by the liquid–liquid phase separation of ternary mixtures. Angew. Chemie Int. Ed. 53, 11793–11797 (2014).

    Google Scholar 

  136. Zarzar, L. D. et al. Dynamically reconfigurable complex emulsions via tunable interfacial tensions. Nature 518, 520 (2015).

    ADS  Google Scholar 

  137. Lopian, T. et al. Morphologies observed in ultraflexible microemulsions with and without the presence of a strong acid. ACS Cent. Sci. 2, 467–475 (2016).

    Google Scholar 

  138. Lu, Z. et al. Universal nanodroplet branches from confining the ouzo effect. Proc. Natl Acad. Sci. USA 114, 10332–10337 (2017).

    ADS  Google Scholar 

  139. Moerman, P. G., Hohenberg, P. C., Vanden-Eijnden, E. & Brujic, J. Emulsion patterns in the wake of a liquid–liquid phase separation front. Proc. Natl Acad. Sci. USA 115, 3599–3604 (2018).

    ADS  Google Scholar 

  140. Hajian, R. & Hardt, S. Formation and lateral migration of nanodroplets via solvent shifting in a microfluidic device. Microfluid. Nanofluidics 19, 1281–1296 (2015).

    Google Scholar 

  141. Pregl, F. Die quantitative organische Mikroanalyse (Springer, 1917).

  142. Rezaee, M. et al. Determination of organic compounds in water using dispersive liquid–liquid microextraction. J. Chromatogr. A 1116, 1–9 (2006).

    Google Scholar 

  143. Rezaee, M., Yamini, Y. & Faraji, M. Evolution of dispersive liquid–liquid microextraction method. J. Chromatogr. A 1217, 2342–2357 (2010).

    Google Scholar 

  144. Zgola-Grzeskowiak, A. & Grzeskowiak, T. Dispersive liquid–liquid microextraction. Trends Anal. Chem. 30, 1382–1399 (2011).

    Google Scholar 

  145. Guiterrez, J. M. P., Hinkley, T., Taylor, J. W., Yanev, K. & Cronin, L. Evolution of oil droplets in a chemorobotic platform. Nat. Commun. 5, 5571 (2014).

    ADS  Google Scholar 

  146. Ocaña-González, J. A., Fernández-Torres, R., Bello-López, M. Á. & Ramos-Payán, M. New developments in microextraction techniques in bioanalysis. A review. Anal. Chim. Acta 905, 8–23 (2016).

    Google Scholar 

  147. Li, M., Dyett, B., Yu, H., Bansal, V. & Zhang, X. Functional femtoliter droplets for ultrafast nanoextraction and supersensitive online microanalysis. Small 15, 1804683 (2019).

    Google Scholar 

  148. Li, M., Dyett, B. & Zhang, X. Automated femtoliter droplet-based determination of oil–water partition coefficient. Anal. Chem. 91, 10371–10375 (2019).

    Google Scholar 

  149. Qian, J. et al. One-step nanoextraction and ultrafast microanalysis based on nanodroplet formation in an evaporating ternary liquid microfilm. Adv. Mater. Technol. 5, 1900740 (2020).

    Google Scholar 

  150. Gutiérrez, J. et al. Nano-emulsions: new applications and optimization of their preparation. Curr. Opin. Colloid Interface Sci. 13, 245–251 (2008).

    Google Scholar 

  151. Zhang, C., Pansare, V. J., Prud’homme, R. K. & Priestley, R. D. Flash nanoprecipitation of polystyrene nanoparticles. Soft Matter 8, 86–93 (2012).

    ADS  Google Scholar 

  152. Akbulut, M. et al. Generic method of preparing multifunctional fluorescent nanoparticles using flash nanoprecipitation. Adv. Funct. Mater. 19, 718–725 (2009).

    Google Scholar 

  153. Zeng, Z. et al. Scalable production of therapeutic protein nanoparticles using flash nanoprecipitation. Adv. Healthc. Mater. 8, 1801010 (2019).

    Google Scholar 

  154. Coquerel, G. Crystallization of molecular systems from solution: phase diagrams, supersaturation and other basic concepts. Chem. Soc. Rev. 43, 2286–2300 (2014).

    Google Scholar 

  155. Sun, X., Zhang, Y., Chen, G. & Gai, Z. Application of nanoparticles in enhanced oil recovery: a critical review of recent progress. Energies 10, 345 (2017).

    Google Scholar 

  156. Rao, F. & Liu, Q. Froth treatment in Athabasca oil sands bitumen recovery process: a review. Energy Fuels 27, 7199–7207 (2013).

    Google Scholar 

  157. He, L., Lin, F., Li, X., Sui, H. & Xu, Z. Interfacial sciences in unconventional petroleum production: from fundamentals to applications. Chem. Soc. Rev. 44, 5446–5494 (2015).

    Google Scholar 

  158. Crossley, S., Faria, J., Shen, M. & Resasco, D. E. Solid nanoparticles that catalyze biofuel upgrade reactions at the water/oil interface. Science 327, 68–72 (2010).

    ADS  Google Scholar 

  159. Jia, T. Z. et al. Membraneless polyester microdroplets as primordial compartments at the origins of life. Proc. Natl Acad. Sci. USA 116, 15830–15835 (2019).

    Google Scholar 

  160. Zwicker, D., Decker, M., Jaensch, S., Hyman, A. A. & Jülicher, F. Centrosomes are autocatalytic droplets of pericentriolar material organized by centrioles. Proc. Natl Acad. Sci. USA 111, 2636–2645 (2014).

    ADS  Google Scholar 

  161. Zwicker, D., Seyboldt, R., Weber, C. A., Hyman, A. A. & Jülicher, F. Growth and division of active droplets provides a model for protocells. Nat. Phys. 13, 408 (2017).

    Google Scholar 

  162. Golestanian, R. Division for multiplication. Nat. Phys. 13, 323–324 (2017).

    Google Scholar 

  163. Lee, J. K., Kim, S., Nam, H. G. & Zare, R. N. Microdroplet fusion mass spectrometry for fast reaction kinetics. Proc. Natl Acad. Sci. USA 112, 3898–3903 (2015).

    ADS  Google Scholar 

  164. Lee, J. K., Samanta, D., Nam, H. G. & Zare, R. N. Micrometer-sized water droplets induce spontaneous reduction. J. Am. Chem. Soc. 141, 10585–10589 (2019).

    Google Scholar 

  165. Fallah-Araghi, A. et al. Enhanced chemical synthesis at soft interfaces: a universal reaction–adsorption mechanism in microcompartments. Phys. Rev. Lett. 112, 028301 (2014).

    ADS  Google Scholar 

  166. Overbeek, J. T. G. & Voorn, M. J. Phase separation in polyelectrolyte solutions. Theory of complex coacervation. J. Cell. Comp. Physiol. 49, 7–26 (1957).

    Google Scholar 

  167. Kizilay, E., Kayitmazer, A. & Dubin, P. Complexation and coacervation of polyelectrolytes with oppositely charged colloids. Adv. Colloid Interface Sci. 167, 24–37 (2011).

    Google Scholar 

  168. Oparin, A. I. et al. The Origin of Life on the Earth (Oliver & Boyd, 1957).

  169. Shin, Y. & Brangwynne, C. P. Liquid phase condensation in cell physiology and disease. Science 357, eaaf4382 (2017).

    Google Scholar 

  170. Seo, S. et al. Microphase behavior and enhanced wet-cohesion of synthetic copolyampholytes inspired by a mussel foot protein. J. Am. Chem. Soc. 137, 9214–9217 (2015).

    Google Scholar 

  171. Chang, L.-W. et al. Sequence and entropy-based control of complex coacervates. Nat. Commun. 8, 1273 (2017).

    ADS  Google Scholar 

  172. Wijshoff, H. The dynamics of the piezo inkjet printhead operation. Phys. Rep. 491, 77–177 (2010).

    ADS  Google Scholar 

  173. Kuang, M., Wang, L. & Song, Y. Controllable printing droplets for high-resolution patterns. Adv. Mater. 26, 6950–6958 (2014).

    Google Scholar 

  174. Hoath, S. D. Fundamentals of Inkjet Printing: The Science of Inkjet and Droplets (Wiley, 2016).

  175. Dijksman, J. F. Design of Piezo Inkjet Print Heads: From Acoustics to Applications (Wiley, 2019).

  176. de Jong, J. et al. Marangoni flow on an inkjet nozzle plate. Appl. Phys. Lett. 91, 204102 (2007).

    ADS  Google Scholar 

  177. Beulen, B. et al. Flows on the nozzle plate of an inkjet printhead. Exp. Fluids 42, 217–224 (2007).

    Google Scholar 

  178. de Jong, J. et al. Entrapped air bubbles in piezo-driven inkjet printing: their effect on the droplet velocity. Phys. Fluids 18, 121511 (2006).

    ADS  MATH  Google Scholar 

  179. Fraters, A. et al. Inkjet nozzle failure by heterogeneous nucleation: bubble entrainment, cavitation, and diffusive growth. Phys. Rev. Appl. 12, 064019 (2019).

    ADS  Google Scholar 

  180. Han, W. & Lin, Z. Learning from ‘coffee rings’: ordered structures enabled by controlled evaporative self-assembly. Angew. Chemie Int. Ed. 51, 1534–1546 (2012).

    Google Scholar 

  181. Cai, Y. & Zhang-Newby, B.-m Marangoni flow-induced self-assembly of hexagonal and stripelike nanoparticle patterns. J. Am. Chem. Soc. 130, 6076–6077 (2008).

    Google Scholar 

  182. Reverchon, E., De Marco, I. & Torino, E. Nanoparticles production by supercritical antisolvent precipitation: a general interpretation. J. Supercrit. Fluids 43, 126–138 (2007).

    Google Scholar 

  183. Chan, H.-K. & Kwok, P. C. L. Production methods for nanodrug particles using the bottom-up approach. Adv. Drug Deliv. Rev. 63, 406–416 (2011).

    Google Scholar 

  184. Huang, W. & Zhang, C. Tuning the size of poly (lactic-co-glycolic acid) (PLGA) nanoparticles fabricated by nanoprecipitation. Biotechnol. J. 13, 1700203 (2018).

    Google Scholar 

  185. Zhang, L.-f Indirect methods of detecting and evaluating inclusions in steel: a review. J. Iron Steel Res. Int. 13, 1–8 (2006).

    Google Scholar 

  186. Leenaars, A., Huethorst, J. & Van Oekel, J. Marangoni drying: a new extremely clean drying process. Langmuir 6, 1701–1703 (1990).

    Google Scholar 

  187. Marra, J. & Huethorst, J. Physical principles of Marangoni drying. Langmuir 7, 2748–2755 (1991).

    Google Scholar 

  188. O’Brien, S. On marangoni drying: nonlinear kinematic waves in a thin film. J. Fluid Mech. 254, 649–670 (1993).

    ADS  MATH  Google Scholar 

  189. Thess, A. & Boos, W. A model for Marangoni drying. Phys. Fluids 11, 3852–3855 (1999).

    ADS  MATH  Google Scholar 

  190. Matar, O. & Craster, R. Models for Marangoni drying. Phys. Fluids 13, 1869–1883 (2001).

    ADS  MathSciNet  MATH  Google Scholar 

  191. Okorn-Schmidt, H. F. et al. Particle cleaning technologies to meet advanced semiconductor device process requirements. ECS J. Solid State Sci. Technol. 3, N3069–N3080 (2014).

    Google Scholar 

  192. Bico, J., Roman, B., Moulin, L. & Boudaoud, A. Adhesion: elastocapillary coalescence in wet hair. Nature 432, 690 (2004).

    ADS  Google Scholar 

  193. Duprat, C., Protiere, S., Beebe, A. & Stone, H. A. Wetting of flexible fibre arrays. Nature 482, 510 (2012).

    ADS  Google Scholar 

  194. Bico, J., Reyssat, É. & Roman, B. Elastocapillarity: when surface tension deforms elastic solids. Annu. Rev. Fluid Mech. 50, 629–659 (2018).

    ADS  MathSciNet  MATH  Google Scholar 

  195. Vazquez, G., Alvarez, E. & Navaza, J. M. Surface tension of alcohol water. Water from 20 to 50 degrees C. J. Chem. Eng. Data 40, 611–614 (1995).

    Google Scholar 

  196. Hell, S. W. Microscopy and its focal switch. Nat. Meth. 6, 24–32 (2009).

    Google Scholar 

  197. Webb, R. H. Confocal optical microscopy. Rep. Prog. Phys. 59, 427–471 (1996).

    ADS  Google Scholar 

  198. Marquet, P. et al. Digital holographic microscopy: a noninvasive contrast imaging technique allowing quantitative visualization of living cells with subwavelength axial accuracy. Optics Lett. 30, 468–470 (2005).

    ADS  Google Scholar 

  199. Garcia-Sucerquia, J. et al. Digital in-line holographic microscopy. Appl. Opt. 45, 836–850 (2006).

    ADS  Google Scholar 

  200. Merola, F. et al. Driving and analysis of micro-objects by digital holographic microscope in microfluidics. Opt. Lett. 36, 3079–3081 (2011).

    ADS  Google Scholar 

  201. Wereley, S. T. & Meinhart, C. D. Recent advances in micro-particle image velocimetry. Annu. Rev. Fluid Mech. 42, 557–576 (2010).

    ADS  Google Scholar 

  202. Versluis, M. High-speed imaging in fluids. Exp. Fluids 54, 1458 (2013).

    Google Scholar 

  203. van der Bos, A., Zijlstra, A., Gelderblom, E. & Versluis, M. iLIF: illumination by laser-induced fluorescence for single flash imaging on a nanoseconds timescale. Exp. Fluids 51, 1283–1289 (2011).

    Google Scholar 

  204. van der Bos, A. et al. Velocity profile inside piezoacoustic inkjet droplets in flight: comparison between experiment and numerical simulation. Phys. Rev. Appl. 1, 014004 (2014).

    ADS  Google Scholar 

  205. Switkes, M. & Ruberti, J. W. Rapid cryofixation/freeze fracture for the study of nanobubbles at solid–liquid interfaces. Appl. Phys. Lett. 84, 4759–4761 (2004).

    ADS  Google Scholar 

  206. Schlücker, S. Surface-enhanced Raman spectroscopy: concepts and chemical applications. Angew. Chemie Int. Ed. 53, 4756–4795 (2014).

    Google Scholar 

  207. Whitesides, G. M. The origins and the future of microfluidics. Nature 442, 368–373 (2006).

    ADS  Google Scholar 

  208. Squires, T. & Quake, S. Microfluidics: fluid physics at the nanoliter scale. Rev. Mod. Phys. 77, 977–1026 (2005).

    ADS  Google Scholar 

  209. Garstecki, P., Fuerstman, M. J., Stone, H. A. & Whitesides, G. M. Formation of droplets and bubbles in a microfluidic T-junction: scaling and mechanism of break-up. Lab Chip 6, 437–446 (2006).

    Google Scholar 

  210. Eijkel, J. C. T. & van den Berg, A. Nanofluidics: what is it and what can we expect from it? Microfluid. Nanofluidics 1, 249–267 (2005).

    Google Scholar 

  211. deMello, A. J. Control and detection of chemical reactions in microfluidic systems. Nature 442, 394–402 (2006).

    ADS  Google Scholar 

  212. Lach, S., Yoon, S. M. & Grzybowski, B. A. Tactic, reactive, and functional droplets outside of equilibrium. Chem. Soc. Rev. 45, 4766–4796 (2016).

    Google Scholar 

  213. Spandan, V., Lohse, D., de Tullio, M. D. & Verzicco, R. A fast moving least squares approximation with adaptive Lagrangian mesh refinement for large scale immersed boundary simulations. J. Comput. Phys. 375, 228–239 (2018).

    ADS  MathSciNet  MATH  Google Scholar 

  214. Diddens, C. Detailed finite element method modeling of evaporating multi-component droplets. J. Comput. Phys. 340, 670–687 (2017).

    ADS  MathSciNet  Google Scholar 

  215. Heil, M. & Hazel, A. L. in Fluid–Structure Interaction (eds Schafer, M. & Bungartz, H.-J.) 19–49 (Springer, 2006).

  216. Heil, M. & Hazel, A. L. Fluid–structure interaction in internal physiological flows. Annu. Rev. Fluid Mech. 43, 141–162 (2011).

    ADS  MathSciNet  MATH  Google Scholar 

  217. Seo, J. H. & Mittal, R. A sharp-interface immersed boundary method with improved mass conservation and reduced spurious pressure oscillations. J. Comput. Phys. 230, 7347–7363 (2011).

    ADS  MathSciNet  MATH  Google Scholar 

  218. Kim, J. Phase-field models for multi-component fluid flows. Commun. Comput. Phys. 12, 613–661 (2012).

    MathSciNet  MATH  Google Scholar 

  219. Sussman, M., Smereka, P. & Osher, S. A level set approach for computing solutions to incompressible two-phase flow. J. Comput. Phys. 114, 146–159 (1994).

    ADS  MATH  Google Scholar 

  220. Sethian, J. A. & Smereka, P. Level set methods for fluid interfaces. Annu. Rev. Fluid Mech. 35, 341–372 (2003).

    ADS  MathSciNet  MATH  Google Scholar 

  221. de Langavant, C. C., Guittet, A., Theillard, M., Temprano-Coleto, F. & Gibou, F. Level-set simulations of soluble surfactant driven flows. J. Comput. Phys. 348, 271–297 (2017).

    ADS  MathSciNet  MATH  Google Scholar 

  222. Gibou, F., Fedkiw, R. & Osher, S. A review of level-set methods and some recent applications. J. Comput. Phys. 353, 82–109 (2018).

    ADS  MathSciNet  MATH  Google Scholar 

  223. Chen, S. & Doolen, G. D. Lattice Boltzmann method for fluid flows. Annu. Rev. Fluid Mech. 30, 329–364 (1998).

    ADS  MathSciNet  MATH  Google Scholar 

  224. Aidun, C. K. & Clausen, J. R. Lattice-Boltzmann method for complex flows. Annu. Rev. Fluid Mech. 42, 439–472 (2010).

    ADS  MathSciNet  MATH  Google Scholar 

  225. Perlekar, P., Benzi, R., Clercx, H. J., Nelson, D. R. & Toschi, F. Spinodal decomposition in homogeneous and isotropic turbulence. Phys. Rev. Lett. 112, 014502 (2014).

    ADS  Google Scholar 

  226. Hessling, D., Xie, Q. & Harting, J. Diffusion dominated evaporation in multicomponent lattice Boltzmann simulations. J. Chem. Phys. 146, 054111 (2017).

    ADS  Google Scholar 

  227. Koplik, J. & Banavar, J. R. Continuum deductions from molecular hydrodynamics. Annu. Rev. Fluid Mech. 27, 257–292 (1995).

    ADS  Google Scholar 

  228. Frenkel, D. & Smit, B. Understanding Molecular Simulations: From Algorithm to Applications (Academic, 1996).

  229. Lauga, E., Brenner, M. P. & Stone, H. A. in Handbook of Experimental Fluid Dynamics (eds Tropea, C. et al.) 1219–1240 (Springer, 2007).

  230. Bocquet, L. & Charlaix, E. Nanofluidics, from bulk to interfaces. Chem. Soc. Rev. 39, 1073–1095 (2010).

    Google Scholar 

  231. Maheshwari, S., van der Hoef, M., Zhang, X. & Lohse, D. Stability of surface nanobubbles: a molecular dynamics study. Langmuir 32, 11116–11122 (2016).

    Google Scholar 

  232. Maheshwari, S., van der Hoef, M., Rodriguez Rodriguez, J. & Lohse, D. Leakiness of pinned neighboring surface nanobubbles induced by strong gas–surface interaction. ACS Nano 12, 2603–2609 (2018).

    Google Scholar 

  233. Diddens, C., Li, Y. & Lohse, D. Competing Marangoni and Rayleigh convection in evaporating binary droplets. Preprint at arXiv https://arxiv.org/abs/2005.14138 (2020).

  234. Taylor, G. Dispersion of soluble matter in solvent flowing slowly through a tube. Proc. R. Soc. Lond. A 219, 186–203 (1953).

    ADS  Google Scholar 

  235. Aris, R. On the dispersion of a solute in a fluid flowing through a tube. Proc. R. Soc. Lond. A 235, 67–77 (1956).

    ADS  Google Scholar 

  236. Aris, R. On the dispersion of a solute by diffusion, convection and exchange between phases. Proc. R. Soc. Lond. A 252, 538–550 (1959).

    ADS  MathSciNet  MATH  Google Scholar 

  237. Anna, S. L., Bontoux, N. & Stone, H. A. Formation of dispersions using ‘flow focusing’ in microchannels. Appl. Phys. Lett. 82, 364–366 (2003).

    ADS  Google Scholar 

  238. Utada, A. et al. Monodisperse double emulsions generated from a microcapillary device. Science 308, 537–541 (2005).

    ADS  Google Scholar 

  239. Solans, C., Morales, D. & Homs, M. Spontaneous emulsification. Curr. Opin. Colloid Interface Sci. 22, 88–93 (2016).

    Google Scholar 

Download references

Acknowledgements

We thank our colleagues, postdocs, PhD students and other students for their contributions to our understanding of physicochemical hydrodynamics of droplets and for the intellectual stimulation we have enjoyed when doing physics together. In the context of the subjects covered here, we thank L. Bao, K. Leong Chong, P. Kant, Z. Lu, A. Prosperetti, V. Spandan, M. Versluis, C.-W. Visser, H. Wijshoff, H. Yu and in particular C. Diddens, Yanshen Li, Yaxing Li and H. Tan. We also thank A. Juel, S. Karpitschka, C. Maas, A. Prosperetti, J. Snoeijer and H. Stone for comments on the manuscript. D.L. thanks D. van Gils for drawing many figures and acknowledges support from the Dutch Research Council (NWO) under several projects, from the ERC Advanced Grant “DDD” under the project number 740479, and from the ERC Proof-of-Concept Grant “NanoEX” under the project number 862032. X.H.Z. acknowledges support from the Natural Science and Engineering Council of Canada (NSERC) and from the Canada Research Chairs programme.

Author information

Authors and Affiliations

  1. Physics of Fluids Group, Max Planck Center for Complex Fluid Dynamics, MESA+ Research Institute and J. M. Burgers Centre for Fluid Dynamics, University of Twente, Enschede, The Netherlands

    Detlef Lohse & Xuehua Zhang

  2. Max Planck Institute for Dynamics and Self-Organization, Göttingen, Germany

    Detlef Lohse

  3. Department of Chemical and Materials Engineering, University of Alberta, Edmonton, Alberta, Canada

    Xuehua Zhang

Authors
  1. Detlef Lohse
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xuehua Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Both authors contributed to all aspects of manuscript preparation, revision and editing.

Corresponding authors

Correspondence to Detlef Lohse or Xuehua Zhang.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Reviewer information

Nature Reviews Physics thanks C. Misbah and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lohse, D., Zhang, X. Physicochemical hydrodynamics of droplets out of equilibrium. Nat Rev Phys 2, 426–443 (2020). https://doi.org/10.1038/s42254-020-0199-z

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s42254-020-0199-z

This article is cited by

Search

Advanced search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing