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Evaporation-driven ring and film deposition from colloidal droplets

  • C. Nadir Kaplan (a1) and L. Mahadevan (a1) (a2) (a3) (a4)


Evaporating suspensions of colloidal particles lead to the formation of a variety of patterns, ranging from a left-over ring of a dried coffee drop to uniformly distributed solid pigments left behind wet paint. To characterize the transition between rings and uniform deposits, we investigate the dynamics of a drying droplet via a multiphase model of colloidal particles in a solvent. Our theory couples the inhomogeneous evaporation at the evolving droplet interface to the dynamics inside the drop. This includes the liquid flow, local variations of the particle concentration leading to a cross-over between dilute and dense suspensions, and the resulting propagation of the deposition front. A dimensionless parameter combining the capillary number and the droplet aspect ratio captures the formation conditions of different pattern types while correctly accounting for the transition from Stokes flow to Darcy flow at high solute concentrations.


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Abkarian, M., Nunes, J. & Stone, H. A. 2004 Colloidal crystallization and banding in a cylindrical geometry. J. Am. Chem. Soc. 126, 59785979.
Adachi, E., Dimitrov, A. S. & Nagayama, K. 1995 Stripe patterns formed on a glass surface during droplet evaporation. Langmuir 11, 10571060.
Berteloot, G., Hoang, A., Daerr, A., Kavehpour, H. P., Lequeux, F. & Limat, L. 2012 Evaporation of a sessile droplet: inside the coffee stain. J. Colloid Interface Sci. 370, 155161.
Bigioni, T. P., Lin, X. M., Nguyen, T. T., Corwin, E. I., Witten, T. A. & Jaeger, H. M. 2006 Kinetically driven self assembly of highly ordered nanoparticle monolayers. Nat. Mater. 5, 265270.
Brinkman, H. C. 1949 A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Appl. Sci. Res. A 1, 2734.
Cohen, S. I. A. & Mahadevan, L. 2013 Hydrodynamics of hemostasis in sickle-cell disease. Phys. Rev. Lett. 110, 138104.
ComsolMultiphysics 4.3a, Burlington, MA, USA.
Cook, B. P., Bertozzi, A. L. & Hosoi, A. E. 2008 Shock solutions for particle-laden thin films. SIAM J. Appl. Math. 68, 760783.
Craster, R. V., Matar, O. K. & Sefiane, K. 2009 Pinning, retraction, and terracing of evaporating droplets containing nanoparticles. Langmuir 25, 36013609.
Deegan, R. D. 2000 Pattern formation in drying drops. Phys. Rev. E 61, 475485.
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, 827829.
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, 756765.
Frastia, L., Archer, A. J. & Thiele, U. 2011 Dynamical model for the formation of patterned deposits at receding contact lines. Phys. Rev. Lett. 106, 077801.
Frastia, L., Archer, A. J. & Thiele, U. 2012 Modelling the formation of structured deposits at receding contact lines of evaporating solutions and suspensions. Soft Matt. 8, 1136311386.
de Gennes, P. G. 1985 Wetting: statics and dynamics. Rev. Mod. Phys. 57, 827863.
de Gennes, P. G., Brochard-Wyart, F. & Quéré, D. 2004 Capillarity and Wetting Phenomena. Springer.
Hu, H. & Larson, R. G. 2005 Analysis of the effects of Marangoni stresses on the microflow in an evaporating sessile droplet. Langmuir 21, 39723980.
Hu, H. & Larson, R. G. 2006 Marangoni effect reverses coffee-ring depositions. J. Phys. Chem. B 110, 70907094.
Kaplan, C. N., Wu, N., Mandre, S., Aizenberg, J. & Mahadevan, L.2015 Dynamics of evaporative colloidal patterning. arXiv:1412.1813.
Kaya, D., Belyi, V. A. & Muthukumar, M. 2010 Pattern formation in drying droplets of polyelectrolyte and salt. J. Chem. Phys. 133, 114905.
Kobayashi, M., Makino, M., Okuzono, T. & Doi, M. 2010 Interference effects in the drying of polymer droplets on substrate. J. Phys. Soc. Japan 79, 044802.
Landau, L. D. & Lifshitz, E. M. 2004 Fluid Mechanics, 2nd edn. Elsevier.
Lebovka, N. I., Gigiberiya, V. A., Lytvyn, O. S., Tarasevich, Y. Y., Vodolazskaya, I. V. & Bondarenko, O. P. 2014 Drying of sessile droplets of laponite-based aqueous nanofluids. Colloids Surf. A 462, 5263.
Lin, X. M., Jaeger, H. M., Sorensen, C. M. & Klabunde, K. J. 2001 Formation of long-range-ordered nanocrystal superlattices on silicon nitride substrates. J. Phys. Chem. B 105, 33533357.
Maheshwari, S., Zhang, L., Zhu, Y. & Chang, H. C. 2008 Coupling between precipitation and contact-line dynamics: multiring stains and stick-slip motion. Phys. Rev. Lett. 100, 044503.
Marín, Á. G., Gelderblom, H., Lohse, D. & Snoeijer, J. H. 2011 Order-to-disorder transition in ring-shaped colloidal stains. Phys. Rev. Lett. 107, 085502.
Marín, Á. G., Gelderblom, H., Susarrey-Arce, A., van Houselt, A., Lefferts, L., Gardeniers, J. G. E., Lohse, D. & Snoeijer, J. H. 2012 Building microscopic soccer balls with evaporating colloidal fakir drops. Proc. Natl Acad. Sci. USA 109, 1645516458.
Narayanan, S., Wang, J. & Lin, X. M. 2004 Dynamical self-assembly of nanocrystal superlattices during colloidal droplet evaporation by in situ small angle x-ray scattering. Phys. Rev. Lett. 93, 135503.
Okuzono, T., Kobayashi, M. & Doi, M. 2009 Final shape of a drying thin film. Phys. Rev. E 80, 021603.
Oron, A., Davis, S. H. & Bankoff, S. G. 1997 Long-scale evolution of thin liquid films. Rev. Mod. Phys. 69, 931980.
Parisse, F. & Allain, C. 1996 Shape changes of colloidal suspension droplets during drying. J. Phys. II 6, 11111119.
Parisse, F. & Allain, C. 1997 Drying of colloidal suspension droplets: experimental study and profile renormalization. Langmuir 13, 35983602.
Popov, Y. O. 2005 Evaporative deposition patterns: spatial dimensions of the deposit. Phys. Rev. E 71, 036313.
Shmuylovich, L., Shen, A. Q. & Stone, H. A. 2002 Surface morphology of drying latex films: multiple ring formation. Langmuir 18, 34413445.
Stickel, J. J. & Powell, R. L. 2005 Fluid mechanics and rheology of dense suspensions. Annu. Rev. Fluid Mech. 37, 129149.
Tarasevich, Y. Y., Vodolazskaya, I. V. & Bondarenko, O. P. 2013 Modeling of spatial-temporal distribution of the components in the drying sessile droplet of biological fluid. Colloids Surf. A 432, 99103.
Tarasevich, Y. Y., Vodolazskaya, I. V. & Isakova, O. P. 2011 Desiccating colloidal sessile drop: dynamics of shape and concentration. Colloid Polym. Sci. 289, 10151023.
Thiele, U. 2014 Patterned deposition at moving contact lines. Adv. Colloid Interface Sci. 206, 399413.
Witten, T. A. 2009 Robust fadeout profile of an evaporation stain. Euro. Phys. Lett. 86, 64002.
Yang, X., Li, C. Y. & Sun, Y. 2014 From multi-ring to spider web and radial spoke: competition between the receding contact line and particle deposition in a drying colloidal drop. Soft Matt. 10, 44584463.
Yunker, P. J., Still, T., Lohr, M. A. & Yodh, A. G. 2011 Suppression of the coffee-ring effect by shape-dependent capillary interactions. Nature 476, 308311.
Zhang, L., Maheshwari, S., Chang, H. C. & Zhu, Y. 2008 Evaporative self-assembly from complex DNA-colloid suspensions. Langmuir 24, 39113917.
Zheng, R. 2009 A study of the evaporative deposition process: pipes and truncated transport dynamics. Eur. Phys. J. E 29, 205218.
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Evaporation-driven ring and film deposition from colloidal droplets

  • C. Nadir Kaplan (a1) and L. Mahadevan (a1) (a2) (a3) (a4)


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