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The nascent coffee ring with arbitrary droplet contact set: an asymptotic analysis

Published online by Cambridge University Press:  12 April 2022

Madeleine Rose Moore Open the ORCID record for Madeleine Rose Moore [Opens in a new window] ,
D. Vella Open the ORCID record for D. Vella [Opens in a new window]  and
J.M. Oliver
Madeleine Rose Moore*
Affiliation:
Department of Physics & Mathematics, University of Hull, Cottingham Road, Kingston-upon-Hull HU6 7RX, UK Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK
D. Vella
Affiliation:
Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK
J.M. Oliver
Affiliation:
Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK
*
†Email address for correspondence: [email protected]
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Abstract

We consider the effect of droplet geometry on the early-stages of coffee-ring formation during the evaporation of a thin droplet with an arbitrary simple, smooth, pinned contact line. We perform a systematic matched asymptotic analysis of the small capillary number, large solutal Péclet number limit for two different evaporative models: a kinetic model, in which the evaporative flux is effectively constant across the droplet, and a diffusive model, in which the flux is singular at the contact line. For both evaporative models, solute is transported to the contact line by a capillary flow in the droplet bulk, while local to the contact line, solute diffusion counters advection. The resulting interplay leads to the formation of the nascent coffee-ring profile. By exploiting a coordinate system embedded in the contact line, we solve explicitly the local leading-order problem, deriving a similarity profile (in the form of a gamma distribution) that describes the nascent coffee ring. Notably, for an arbitrary contact line geometry, the ring characteristics change due to the concomitant asymmetry in the shape of the droplet free surface, the evaporative flux (for diffusive evaporation) and the mass flux into the contact line. We utilize the asymptotic model to determine the effects of contact line geometry on the growth of the coffee ring for a droplet with an elliptical contact set. Our results offer mechanistic insight into the effect of contact line curvature on the development of the coffee ring from deposition up to jamming of the solute; moreover, our model predicts when finite concentration effects become relevant.

JFM classification

Phase change: Condensation/evaporation Interfacial Flows (free surface): Thin films
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 940 , 10 June 2022 , A38
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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Footnotes

Article last updated 07 March 2023

References

REFERENCES

Anyfantakis, M. & Baigl, D. 2015 Manipulating the coffee-ring effect: interactions at work. ChemPhysChem 16 (13), 2726–2734.CrossRefGoogle ScholarPubMed
Boulogne, F., Ingremeau, F. & Stone, H.A. 2016 Coffee-stain growth dynamics on dry and wet surfaces. J. Phys.: Condens. Matter 29 (7), 074001.Google ScholarPubMed
Bruna, M. & Chapman, S.J. 2012 a Diffusion of multiple species with excluded-volume effects. J. Chem. Phys. 137 (20), 204116.CrossRefGoogle ScholarPubMed
Bruna, M. & Chapman, S.J. 2012 b Excluded-volume effects in the diffusion of hard spheres. Phys. Rev. E 85 (1), 011103.CrossRefGoogle ScholarPubMed
Choi, S., Stassi, S., Pisano, A.P. & Zohdi, T.I. 2010 Coffee-ring effect-based three-dimensional patterning of micro/nanoparticle assembly with a single droplet. Langmuir 26 (14), 11690–11698.CrossRefGoogle ScholarPubMed
Cui, L., Zhang, J., Zhang, X., Li, Y., Wang, Z., Gao, H., Wang, T., Zhu, S., Yu, H. & Yang, B. 2012 Avoiding coffee ring structure based on hydrophobic silicon pillar arrays during single-drop evaporation. Soft Matt. 8 (40), 10448–10456.CrossRefGoogle Scholar
De Gennes, P.-G. 1985 Wetting: statics and dynamics. Rev. Mod. Phys. 57 (3), 827–863.CrossRefGoogle Scholar
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.CrossRefGoogle 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.CrossRefGoogle Scholar
Dicuangco, M., Dash, S., Weibel, J.A. & Garimella, S.V. 2014 Effect of superhydrophobic surface morphology on evaporative deposition patterns. Appl. Phys. Lett. 104 (20), 201604.CrossRefGoogle Scholar
Freed-Brown, J.E. 2015 Deposition from evaporating drops: power laws and new morphologies in coffee stains. PhD thesis. University of Chicago.Google Scholar
Guazzelli, É. & Pouliquen, O. 2018 Rheology of dense granular suspensions. J. Fluid Mech. 852, P1.CrossRefGoogle 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.CrossRefGoogle ScholarPubMed
He, A., Yang, H., Xue, W., Sun, K. & Cao, Y. 2017 Tunable coffee-ring effect on a superhydrophobic surface. Opt. Lett. 42 (19), 3936–3939.CrossRefGoogle ScholarPubMed
Hu, H. & Larson, R.G. 2002 Evaporation of a sessile droplet on a substrate. J. Phys. Chem. B 106 (6), 1334–1344.CrossRefGoogle Scholar
Hu, H. & Larson, R.G. 2006 Marangoni effect reverses coffee-ring depositions. J. Phys. Chem. B 110 (14), 7090–7094.CrossRefGoogle ScholarPubMed
Kajiya, T., Kaneko, D. & Doi, M. 2008 Dynamical visualization of ‘coffee stain phenomenon’ in droplets of polymer solution via fluorescent microscopy. Langmuir 24, 12369–12374.CrossRefGoogle ScholarPubMed
Kang, S.J., Vandadi, V., Felske, J.D. & Masoud, H. 2016 Alternative mechanism for coffee-ring deposition based on active role of free surface. Phys. Rev. E 94 (6), 063104.CrossRefGoogle Scholar
Kaplan, C.N. & Mahadevan, L. 2015 Evaporation-driven ring and film deposition from colloidal droplets. J. Fluid Mech. 781, R2.CrossRefGoogle Scholar
Kellogg, O.D. 1929 Foundations of Potential Theory. Springer.CrossRefGoogle Scholar
Kim, H., Boulogne, F., Um, E., Jacobi, I.A., Button, E. & Stone, H.A. 2016 Controlled uniform coating from the interplay of Marangoni flows and surface-adsorbed macromolecules. Phys. Rev. Lett. 116 (12), 124501.CrossRefGoogle ScholarPubMed
Kim, S.J., Kang, K.H., Lee, J.-G., Kang, I.S. & Yoon, B.J. 2006 Control of particle-deposition pattern in a sessile droplet by using radial electroosmotic flow. Analyt. Chem. 78 (14), 5192–5197.CrossRefGoogle Scholar
Kubyshkina, V., Orejon, D., Dover, C.M. & Sefiane, K. 2020 Geometrical deposits on microstructured surfaces. J. Bionic Engng 17 (4), 851–865.CrossRefGoogle Scholar
Lacey, A.A. 1982 The motion with slip of a thin viscous droplet over a solid surface. Stud. Appl. Maths 67 (3), 217–230.CrossRefGoogle Scholar
Larson, R.G. 2014 Transport and deposition patterns in drying sessile droplets. AIChE J. 60 (5), 1538–1571.CrossRefGoogle Scholar
Layani, M., Gruchko, M., Milo, O., Balberg, I., Azulay, D. & Magdassi, S. 2009 Transparent conductive coatings by printing coffee ring arrays obtained at room temperature. ACS Nano 3 (11), 3537–3542.CrossRefGoogle ScholarPubMed
Li, F. & Mugele, F. 2008 How to make sticky surfaces slippery: contact angle hysteresis in electrowetting with alternating voltage. Appl. Phys. Lett. 92 (24), 244108.CrossRefGoogle Scholar
Li, Y., Diddens, C., Segers, T., Wijshoff, H., Versluis, M. & Lohse, D. 2020 Evaporating droplets on oil-wetted surfaces: suppression of the coffee-stain effect. Proc. Natl Acad. Sci. USA 117 (29), 16756–16763.CrossRefGoogle ScholarPubMed
Li, Y., Lv, C., Li, Z., Quéré, D. & Zheng, Q. 2015 From coffee rings to coffee eyes. Soft Matt. 11 (23), 4669–4673.CrossRefGoogle ScholarPubMed
Li, Y., Lv, P., Diddens, C., Tan, H., Wijshoff, H., Versluis, M. & Lohse, D. 2018 Evaporation-triggered segregation of sessile binary droplets. Phys. Rev. Lett. 120 (22), 224501.CrossRefGoogle ScholarPubMed
Mampallil, D. & Eral, H.B. 2018 A review on suppression and utilization of the coffee-ring effect. Adv. Colloid Interface Sci. 252, 38–54.CrossRefGoogle ScholarPubMed
Masoud, H. & Felske, J.D. 2009 Analytical solution for Stokes flow inside an evaporating sessile drop: spherical and cylindrical cap shapes. Phys. Fluids 21 (4), 042102.CrossRefGoogle Scholar
Moore, M.R., Vella, D. & Oliver, J.M. 2021 The nascent coffee ring: how solute diffusion counters advection. J. Fluid Mech. 920, A54.CrossRefGoogle Scholar
Murisic, N. & Kondic, L. 2011 On evaporation of sessile drops with moving contact lines. J. Fluid Mech. 679, 219–246.CrossRefGoogle Scholar
Oliver, J.M., Whiteley, J.P., Saxton, M.A., Vella, D., Zubkov, V.S. & King, J.R. 2015 On contact-line dynamics with mass transfer. Eur. J. Appl. Maths 26 (5), 671–719.CrossRefGoogle Scholar
Olver, F.W.J., Lozier, D.W., Boisvert, R.F. & Clark, C.W. 2010 NIST Handbook of Mathematical Functions. Cambridge University Press.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.CrossRefGoogle ScholarPubMed
Pahlavan, A.A., Yang, L., Bain, C.D. & Stone, H.A. 2021 Evaporation of binary-mixture liquid droplets: the formation of picoliter pancakelike shapes. Phys. Rev. Lett. 127, 024501.CrossRefGoogle ScholarPubMed
Pham, T. & Kumar, S. 2017 Drying of droplets of colloidal suspensions on rough substrates. Langmuir 33 (38), 10061–10076.CrossRefGoogle ScholarPubMed
Popov, Y.O. 2005 Evaporative deposition patterns: spatial dimensions of the deposit. Phys. Rev. E 71, 036313.CrossRefGoogle ScholarPubMed
Popov, Y.O. & Witten, T.A. 2003 Characteristic angles in the wetting of an angular region: deposit growth. Phys. Rev. E 68 (3), 036306.CrossRefGoogle ScholarPubMed
Ristenpart, W.D., Kim, P.G., Domingues, C., Wan, J. & Stone, H.A. 2007 Influence of substrate conductivity on circulation reversal in evaporating drops. Phys. Rev. Lett. 99 (23), 234502.CrossRefGoogle ScholarPubMed
Sáenz, P.J., Wray, A.W., Che, Z., Matar, O.K., Valluri, P., Kim, J. & Sefiane, K. 2017 Dynamics and universal scaling law in geometrically-controlled sessile drop evaporation. Nat. Commun. 8, 14783.CrossRefGoogle Scholar
Saxton, M.A. 2016 Modelling the contact-line dynamics of an evaporating drop. DPhil thesis, University of Oxford.Google Scholar
Swinehart, D.F. 1962 The Beer–Lambert law. J. Chem. Educ. 39 (7), 333.CrossRefGoogle Scholar
Timm, M.L., Dehdashti, E., Darban, A.J. & Masoud, H. 2019 Evaporation of a sessile droplet on a slope. Scientific Rep. 9 (1), 19803.CrossRefGoogle ScholarPubMed
Tredenick, E.C., Forster, W.A., Pethiyagoda, R., van Leeuwen, R.M. & McCue, S.W. 2021 Evaporating droplets on inclined plant leaves and synthetic surfaces: experiments and mathematical models. J. Colloid Interface Sci. 592, 329–341.CrossRefGoogle ScholarPubMed
Van Dyke, M. 1964 Perturbation Methods in Fluid Mechanics. Academic Press.Google Scholar
Weon, B.M. & Je, J.H. 2013 Self-pinning by colloids confined at a contact line. Phys. Rev. Lett. 110 (2), 028303.CrossRefGoogle Scholar
Witten, T.A. 2009 Robust fadeout profile of an evaporation stain. Europhys. Lett. 86 (6), 64002.CrossRefGoogle Scholar
Wray, A.W., Papageorgiou, D.T., Craster, R.V., Sefiane, K. & Matar, O.K. 2014 Electrostatic suppression of the ‘coffee stain effect’. Langmuir 30 (20), 5849–5858.CrossRefGoogle ScholarPubMed
Zhong, X. & Duan, F. 2016 Flow regime and deposition pattern of evaporating binary mixture droplet suspended with particles. Eur. Phys. J. E 39 (2), 18.CrossRefGoogle ScholarPubMed