Hostname: page-component-7c8c6479df-hgkh8 Total loading time: 0 Render date: 2024-03-28T16:53:21.827Z Has data issue: false hasContentIssue false

Stability of a liquid ring on a substrate

Published online by Cambridge University Press:  08 February 2013

Alejandro G. González*
Affiliation:
Instituto de Física Arroyo Seco, Universidad Nacional del Centro, de la Provincia de Buenos Aires, Pinto 399, 7000, Tandil, Argentina
Javier A. Diez
Affiliation:
Instituto de Física Arroyo Seco, Universidad Nacional del Centro, de la Provincia de Buenos Aires, Pinto 399, 7000, Tandil, Argentina
Lou Kondic
Affiliation:
Department of Mathematical Sciences, New Jersey Institute of Technology, University Heights, Newark, NJ 07102, USA
*
Email address for correspondence: aggonzal@exa.unicen.edu.ar

Abstract

We study the stability of a viscous incompressible fluid ring on a partially wetting substrate within the framework of long-wave theory. We discuss the conditions under which a static equilibrium of the ring is possible in the presence of contact angle hysteresis. A linear stability analysis (LSA) of this equilibrium solution is carried out by using a slip model to account for the contact line divergence. The LSA provides specific predictions regarding the evolution of unstable modes. In order to describe the evolution of the ring for longer times, a quasi-static approximation is implemented. This approach assumes a quasi-static evolution and takes into account the concomitant variation of the instantaneous growth rates of the modes responsible for either collapse of the ring into a single central drop or breakup into a number of droplets along the ring periphery. We compare the results of the LSA and the quasi-static model approach with those obtained from nonlinear numerical simulations using a complementary disjoining pressure model. We find remarkably good agreement between the predictions of the two models regarding the expected number of drops forming during the breakup process.

Type
Papers
Copyright
©2013 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

Atwater, H. A. & Polman, A. 2010 Plasmonics for improved photovoltaic devices. Nat. Mater. 9, 205.Google Scholar
Baroud, C. N., Gallaire, F. & Dangla, R. 2010 Dynamics of microfluidic droplets. Lab on a Chip 10, 2032.CrossRefGoogle ScholarPubMed
Beltrame, P., Knobloch, E., Hanggi, P. & Thiele, U. 2011 Rayleigh and depinning instabilities of forced liquid ridges on heterogeneous substrates. Phys. Rev. E 83, 016305.Google Scholar
Beltrame, P. & Thiele, U. 2010 Time integration and steady-state continuation method for lubrication equations. SIAM J. Appl. Dyn. Syst. 9, 484.Google Scholar
Bonn, D., Eggers, J., Indekeu, J., Meunier, J. & Rolley, E. 2009 Wetting and spreading. Rev. Mod. Phys. 81, 739.Google Scholar
Boyd, J. P. 2000 Chebyshev and Fourier Spectral Methods. Dover.Google Scholar
Chugunov, S. S., Schulz, D. L. & Akhatov, I. S. 2011 Ring-shaped sessile droplet. In Proc. ASME 2011 Int. Mech. Eng. Congress. ASME.Google Scholar
Conway, J., Korns, H. & Fisch, M. R. 1997 Evaporation kinematics of polystyrene bead suspensions. Langmuir 13, 426.CrossRefGoogle Scholar
Craster, R. V. & Matar, O. K. 2009 Dynamics and stability of thin liquid films. Rev. Mod. Phys. 81, 1131.Google Scholar
Davis, S. H. 1980 Moving contact lines and rivulet instabilities. Part I: the static rivulet. J. Fluid Mech. 98, 225.Google Scholar
Diez, J., González, A. G. & Kondic, L. 2009 On the breakup of fluid rivulets. Phys. Fluids 21, 082105.Google Scholar
Diez, J., González, A. G. & Kondic, L. 2012 Instability of a transverse liquid rivulet on an inclined plane. Phys. Fluids 24, 032104.Google Scholar
Diez, J. & Kondic, L. 2002 Computing three-dimensional thin film flows including contact lines. J. Comput. Phys. 183, 274.CrossRefGoogle Scholar
Diez, J. & Kondic, L. 2007 On the breakup of fluid films of finite and infinite extent. Phys. Fluids 19, 072107.Google Scholar
Dussan V, E. B. 1976 The moving contact line: the slip boundary condition. J. Fluid Mech. 77, 665.Google Scholar
Favazza, C., Kalyanaraman, R. & Sureshkumar, R. 2006 Robust nanopatterning by laser-induced dewetting of metal nanofilms. Nanotechnology 17, 4229.Google Scholar
Fowlkes, J. D., Kondic, L., Diez, J. & Rack, P. D. 2011 Self-assembly versus directed assembly of nanoparticles via pulsed laser induced dewetting of patterned metal films. Nano Lett. 11, 2478.CrossRefGoogle ScholarPubMed
de Gennes, P. G. 1985 Wetting: statics and dynamics. Rev. Mod. Phys. 57, 827.Google Scholar
González, A. G., Diez, J., Gomba, J., Gratton, R. & Kondic, L. 2004 Spreading of a thin two-dimensional strip of fluid on a vertical plane: experiments and modeling. Phys. Rev. E 70, 026309.CrossRefGoogle ScholarPubMed
González, A. G., Diez, J., Gratton, R. & Gomba, J. 2007 Rupture of a fluid strip under partial wetting conditions. Europhys. Lett. 77, 44001.Google Scholar
Greenspan, H. P. 1978 On the motion of a small viscous droplet that wets a surface. J. Fluid Mech. 84, 125.Google Scholar
Herminghaus, S., Jacobs, K., Mecke, K., Bischof, J., Fery, A., Ibn-Elhaj, M. & Schlagowski, S. 1998 Spinodal dewetting in liquid crystal and liquid metal films. Science 282, 916.Google Scholar
Hocking, L. M. 1990 Spreading and instability of a viscous fluid sheet. J. Fluid Mech. 221, 373.Google Scholar
Hocking, L. M. & Miksis, M. J. 1993 Stability of a ridge of fluid. J. Fluid Mech. 247, 157.Google Scholar
King, J. R., Münch, A. & Wagner, B. 2006 Linear stability of a ridge. Nonlinearity 19, 2813.Google Scholar
Koh, Y. Y., Lee, Y. C., Gaskell, P. H., Jimack, P. K. & Thompson, H. M. 2009 Droplet migration: quantitative comparisons with experiment. Eur. Phys. J. Special Topics 166, 117.Google Scholar
Kondic, L., Diez, J., Rack, P., Guan, Y. & Fowlkes, J. 2009 Nanoparticle assembly via the dewetting of patterned thin metal lines: understanding the instability mechanism. Phys. Rev. E 79, 026302.Google Scholar
Langbein, D. 1990 The shape and stability of liquid menisci at solid edges. J. Fluid Mech. 213, 251.Google Scholar
Lin, T.-S., Kondic, L. & Filippov, A. 2012 Thin films flowing down inverted substrates: Three-dimensional flow. Phys. Fluids 24, 022104.Google Scholar
McGraw, J. D., Li, J., Tran, D. L., Shi, A.-C. & Dalnoki-Veress, K. 2010 Plateau–Rayleigh instability in a torus: formation and breakup of a polymer ring. Soft Matt. 6, 1258.Google Scholar
Mitlin, V. S. 1993 Dewetting of solid surface: analogy with spinodal decomposition. J. Colloid Interface Sci. 156, 491.CrossRefGoogle Scholar
Münch, A. & Wagner, B. 2005 Contact-line instability of dewetting thin films. Physica D 209, 178.Google Scholar
Pairam, E. & Fernández-Nieves, A. 2009 Generation and stability of toroidal droplets in a viscous liquid. Phys. Rev. Lett. 102, 234501.Google Scholar
Park, J. & Moon, J. 2006 Control of colloidal particle deposit patterns within picoliter droplets ejected by ink-jet printing. Langmuir 22, 3506.Google Scholar
Plateau, J. 1849 Statique experimentale et theorique des liquides soumis aux seules forces moleculaires. Acad. Sci. Bruxelles Mem. 23, 5.Google Scholar
Rayleigh, Lord 1879 On the capillary phenomena of jets. Proc. R. Soc. Lond. 29, 71.Google Scholar
Roy, R. V. & Schwartz, L. W. 1999 On the stability of liquid ridges. J. Fluid Mech. 391, 293.Google Scholar
Savva, N. & Kalliadasis, S. 2011 Dynamics of moving contact lines: a comparison between slip and precursor film models. Europhys. Lett. 94, 64004.Google Scholar
Schafle, C., Brinkmann, M., Bechinger, C., Leiderer, P. & Lipowsky, R. 2010 Morphological wetting transitions at ring-shaped surface domains. Langmuir 26, 11878.Google Scholar
Sekimoto, K., Oguma, R. & Kawasaki, K. 1987 Morphological stability analysis of partial wetting. Ann. Phys. (N.Y.) 176, 359.Google Scholar
Squires, T. M. & Quake, S. R. 2005 Microfluidics: fluid physics at the nanoliter scale. Rev. Mod. Phys. 77, 977.Google Scholar
Stone, H. A., Stroock, A. D. & Ajdari, A. 2004 Engineering flows in small devices. Annu. Rev. Fluid Mech. 36, 381.Google Scholar
Thiele, U. 2010 Thin film evolution equations from (evaporating) dewetting liquid layers to epitaxial growth. J. Phys. Condens. Matter 22, 084019.Google Scholar
Thiele, U. & Knobloch, E. 2003 Front andback instability of a liquid film on a slightly inclined plate. Phys. Fluids 15, 892.Google Scholar
Voicu, N. E., Harkema, S. & Steiner, U. 2006 Electric-field-induced pattern morphologies in thin liquid films. Adv. Funct. Mater. 16, 926.Google Scholar
Worthington, A. M. 1879 On the spontaneous segmentation of a liquid annulus. Proc. R. Soc. Lond. 30, 49.Google Scholar
Wu, Y., Fowlkes, J. D., Rack, P. D., Diez, J. A. & Kondic, L. 2010 On the breakup of patterned nanoscale copper rings into droplets via pulsed-laser-induced dewetting: competing liquid-phase instability and transport mechanisms. Langmuir 26, 11972.Google Scholar
Wu, Y., Fowlkes, J. D., Roberts, N. A., Diez, J. A., Kondic, L., González, A. G. & Rack, P. D. 2011 Competing liquid phase instabilities during pulsed laser induced self-assembly of copper rings into ordered nanoparticle arrays on ${\mathrm{SiO} }_{2} $ . Langmuir 27, 13314.Google Scholar
Yang, L. & Homsy, G. M. 2006 Steady three-dimensional thermocapillary flows and dryout inside a V-shaped wedge. Phys. Fluids 18, 042017.Google Scholar