Hostname: page-component-6b989bf9dc-476zt Total loading time: 0 Render date: 2024-04-14T22:10:18.498Z Has data issue: false hasContentIssue false
  • English
  • Français

Ring formation on an inclined surface

Published online by Cambridge University Press:  25 June 2015

Xiyu Du  and
R. D. Deegan
Xiyu Du
Affiliation:
Department of Physics and Center for the Study of Complex Systems, University of Michigan, Ann Arbor, MI 48109, USA
R. D. Deegan*
Affiliation:
Department of Physics and Center for the Study of Complex Systems, University of Michigan, Ann Arbor, MI 48109, USA
*
†Email address for correspondence: rddeegan@umich.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

A drop dried on a solid surface will typically leave a narrow band of solute deposited along the contact line. Here we examine variations of this deposit due to the inclination of the substrate using numerical simulations of a two-dimensional drop, equivalent to a strip-like drop. An asymptotic analysis of the contact line region predicts that the upslope deposit will grow faster at early times, but the growth of this deposit ends sooner because the upper contact line depins first. From our simulations we find that the deposit can be larger at either the upper or lower contact line depending on the initial drop volume and substrate inclination. For larger drops and steeper inclinations, the early lead in deposited mass at the upper contact line is wiped out by the earlier depinning of the upper contact line and subsequent continued growth at the lower contact line. Conversely, for smaller drops and shallower inclinations, the early lead of the upper contact line is insurmountable despite its earlier termination in growth. Our results show that it is difficult to reconstruct a posteriori the inclination of the substrate based solely on the shape of the deposit.

JFM classification

Interfacial Flows (free surface): Capillary flows Interfacial Flows (free surface): Contact lines Drops and Bubbles: Drops
Type
Rapids
Information
Journal of Fluid Mechanics , Volume 775 , 25 July 2015 , R3
Check for updates
Copyright
© 2015 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

Adachi, E., Dimitrov, A. S. & Nagayama, K. 1995 Stripe patterns formed on a glass surface during droplet evaporation. Langmuir 11 (4), 1057–1060.Google Scholar
Adam, C. D. 2012 Fundamental studies of bloodstain formation and characteristics. Forensic Sci. Intl 219 (1), 76–87.Google Scholar
Ang, K.-C. 2008 Introducing the boundary element method with Matlab. Intl J. Math. Ed. Sci. Tech. 39 (4), 505–519.Google Scholar
Attinger, D., Moore, C., Donaldson, A., Jafari, A. & Stone, H. A. 2013 Fluid dynamics topics in bloodstain pattern analysis: comparative review and research opportunities. Forensic Sci. Intl 231 (1–3), 375–396.Google Scholar
Bodiguel, H., Doumenc, F. & Guerrier, B. 2009 Pattern formation during the drying of a colloidal suspension. Eur. Phys. J. 166 (1), 29–32.Google Scholar
Bodiguel, H., Doumenc, F. & Guerrier, B. 2010 Stick–slip patterning at low capillary numbers for an evaporating colloidal suspension. Langmuir 26 (13), 10758–10763.CrossRefGoogle ScholarPubMed
de Bruin, K. G., Stoel, R. D. & Limborgh, J. C. M. 2011 Improving the point of origin determination in bloodstain pattern analysis. J. Forensic Sci. 56 (6), 1476–1482.Google Scholar
Cheng, W., Park, N., Walter, M. T., Hartman, M. R. & Luo, D. 2008 Nanopatterning self-assembled nanoparticle superlattices by moulding microdroplets. Nat. Nanotech. 3 (11), 682–690.Google Scholar
Craster, R. V., Matar, O. K. & Sefiane, K. 2009 Pinning, retraction, and terracing of evaporating droplets containing nanoparticles. Langmuir 25 (6), 3601–3609.Google Scholar
Deegan, R. D. 2000 Pattern formation in drying drops. Phys. Rev. E 61 (1), 475–485.CrossRefGoogle ScholarPubMed
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
Eral, H. B., Mampallil Augustine, D., Duits, M. H. G. & Mugele, F. 2011 Suppressing the coffee stain effect: how to control colloidal self-assembly in evaporating drops using electrowetting. Soft Matt. 7 (10), 4954–4958.Google Scholar
de Gennes, P. G. 1985 Wetting: statics and dynamics. Rev. Mod. Phys. 57 (3), 827–863.CrossRefGoogle Scholar
Grzelczak, M., Vermant, J., Furst, E M. & Liz-Marzán, L. M. 2010 Directed self-assembly of nanoparticles. ACS Nano 4 (7), 3591–3605.Google Scholar
Harris, D. J., Hu, H., Conrad, J. C. & Lewis, J. A. 2007 Patterning colloidal films via evaporative lithography. Phys. Rev. Lett. 98 (14), 148301.Google Scholar
Hu, H. & Larson, R. G. 2005 Analysis of the effects of Marangoni stresses on the microflow in an evaporating sessile droplet. Langmuir 21 (9), 3972–3980.CrossRefGoogle Scholar
Hu, H. & Larson, R. G. 2006 Marangoni effect reverses coffee-ring depositions. J. Phys. Chem. B 110 (14), 7090–7094.Google Scholar
Joksimovic, R., Watanabe, S., Riemer, S., Gradzielski, M. & Yoshikawa, K. 2014 Self-organized patterning through the dynamic segregation of DNA and silica nanoparticles. Sci. Rep. 4, 3660.Google Scholar
Kuncicky, D. M. & Velev, O. D. 2008 Surface-guided templating of particle assemblies inside drying sessile droplets. Langmuir 24 (4), 1371–1380.Google Scholar
Kusumaatmaja, H. & Yeomans, J. M. 2007 Modeling contact angle hysteresis on chemically patterned and superhydrophobic surfaces. Langmuir 23 (11), 6019–6032.Google Scholar
Larson, R. G. 2014 Transport and deposition patterns in drying sessile droplets. AIChE J. 60 (5), 1538–1571.Google Scholar
Lin, Z. Q. & Granick, S. 2005 Patterns formed by droplet evaporation from a restricted geometry. J. Am. Chem. Soc. 127 (9), 2816–2817.CrossRefGoogle ScholarPubMed
Musterd, M., van Steijn, V., Kleijn, C. R. & Kreutzer, M. T. 2014 Droplets on inclined plates: local and global hysteresis of pinned capillary surfaces. Phys. Rev. Lett. 113, 066104.Google Scholar
Orejon, D., Sefiane, K. & Shanahan, M. E. R. 2011 Stick–slip of evaporating droplets: substrate hydrophobicity and nanoparticle concentration. Langmuir 27 (21), 12834–12843.Google Scholar
Park, J. & Moon, J. 2006 Control of colloidal particle deposit patterns within picoliter droplets ejected by ink-jet printing. Langmuir 22 (8), 3506–3513.Google Scholar
Rio, E., Daerr, A., Lequeux, F. & Limat, L. 2006 Moving contact lines of a colloidal suspension in the presence of drying. Langmuir 22 (7), 3186–3191.Google Scholar
Snoeijer, J. H. & Andreotti, B. 2013 Moving contact lines: scales, regimes, and dynamical transitions. Annu. Rev. Fluid Mech. 45, 269–292.Google Scholar
Soltman, D. & Subramanian, V. 2008 Inkjet-printed line morphologies and temperature control of the coffee ring effect. Langmuir 24 (5), 2224–2231.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
Witten, T. A. 2009 Robust fadeout profile of an evaporation stain. Europhys. Lett. 86, 64002.CrossRefGoogle Scholar
Yarin, A. L., Szczech, J. B., Megaridis, C. M., Zhang, J. & Gamota, D. R. 2006 Lines of dense nanoparticle colloidal suspensions evaporating on a flat surface: formation of non-uniform dried deposits. J. Colloid Interface Sci. 294 (2), 343–354.Google Scholar
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 (7360), 308–311.CrossRefGoogle ScholarPubMed