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Regimes of thermocapillary migration of droplets under partial wetting conditions

Published online by Cambridge University Press:  18 March 2010

J. M. GOMBA  and
G. M. HOMSY
J. M. GOMBA*
Affiliation:
Department of Mechanical Engineering, University of California, Santa Barbara, California 93106-5070, USA
G. M. HOMSY
Affiliation:
Department of Mechanical Engineering, University of California, Santa Barbara, California 93106-5070, USA
*
†Present address: Instituto de Física Arroyo Seco, UNCPBA and CONICET, 7000 Tandil, Argentina. Email address for correspondence: [email protected]
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Abstract

We study the thermocapillary migration of two-dimensional droplets of partially wetting liquids on a non-uniform heated substrate. An equation for the thickness profile of the droplet is derived by employing lubrication approximations. The model includes the effect of a non-zero contact angle introduced through a disjoining–conjoining pressure term. Instead of assuming a fixed shape for the droplet, as in previous works, here we allow the droplet to change its profile with time. We identify and describe three different regimes of behaviour. For small contact angles, the droplet spreads into a long film profile with a capillary ridge near the leading edge, a behaviour that resembles the experiments on Marangoni films reported by Ludviksson & Lightfoot (Am. Inst. Chem. Eng. J., vol. 17, 1971, pp. 1166). For large contact angles, the droplet moves as a single entity, weakly distorted from its static shape. This regime is the usual one reported in experiments on thermocapillary migration of droplets. We also show some intriguing morphologies that appear in the transition between these two regimes. The occurrence of these three regimes and their dependence on various parameters is analysed.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 647: Dedicated to Professor S. H. Davis on his 70th birthday , 25 March 2010 , pp. 125 - 142
Copyright
Copyright © Cambridge University Press 2010

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References

REFERENCES

Anderson, D. M. & Davis, S. H. 1995 The spreading of volatile liquid droplets on heated surfaces. Phys. Fluids 7 (2), 248–265.CrossRefGoogle Scholar
Barenblatt, G. I. 1996 Scaling, Self-similarity, and Intermediate Asymptotics. Cambridge University Press.CrossRefGoogle Scholar
Beretta, E., Berstch, M. & Passo, R. D. 1995 Nonnegative solutions of a fourth order nonlinear degenerate parabolic equation. Arch. Ration. Mech. Anal. 129, 175.CrossRefGoogle Scholar
Bernis, F., Peletier, L. A. & Williams, S. M. 1992 Source type solutions of a fourth order nonlinear degenerate parabolic equation. Nonlinear Anal. Theory Methods Appl. 18, 217.CrossRefGoogle Scholar
Bouasse, H. 1924 Capillarité phénomènes superficiels. Librairie Delgrave.Google Scholar
Brochard, F. 1989 Motions of droplets on solid surfaces induced by chemical or thermal gradients. Langmuir 5 (2), 432–438.CrossRefGoogle Scholar
Brzoska, J. B., Brochard-Wyart, F. & Rondelez, F. 1993 Motions of droplets on hydrophobic model surfaces induced by thermal gradients. Langmuir 9 (8), 2220–2224.CrossRefGoogle Scholar
Cazabat, A. M., Heslot, F., Troian, S. M. & Carles, P. 1990 Fingering instability of thin spreading films driven by temperature gradients. Nature 346, 824.CrossRefGoogle Scholar
Chen, J. Z., Troian, S. M., Darhuber, A. A. & Wagner, S. 2005 Effect of contact angle hysteresis on thermocapillary droplet actuation. J. Appl. Phys. 97 (1), 014906.CrossRefGoogle Scholar
Churaev, N. V. & Sobolev, V. D. 1995 Prediction of contact angles on the basis of the Frumkin–Derjaguin approach. Adv. Colloid Interface Sci. 61, 1–16.CrossRefGoogle Scholar
Darhuber, A. A. & Troian, S. M. 2005 Principles of microfluidic actuation by modulation of surface stresses. Annu. Rev. Fluid Mech. 37, 425.CrossRefGoogle Scholar
Darhuber, A. A., Valentino, J. P., Davis, J. M., Troian, S. M. & Wagner, S. 2003 Microfluidic actuation by modulation of surface stresses. Appl. Phys. Lett. 82 (4), 657–659.CrossRefGoogle Scholar
Diez, J. A. & Kondic, L. 2001 Contact line instabilities of thin liquid films. Phys. Rev. Lett. 86, 632.CrossRefGoogle ScholarPubMed
Diez, J. A. & Kondic, L. 2007 On the breakup of fluid films of finite and infinite extent. Phys. Fluids 19 (7), 072107.CrossRefGoogle Scholar
Dussan, V. E. B. 1979 On the spreading of liquids on solid surfaces: static and dynamic contact lines. Annu. Rev. Fluid Mech. 11, 317.CrossRefGoogle Scholar
Ehrhard, P. & Davis, S. H. 1991 Non-isothermal spreading of liquid drops on horizontal plates. J. Fluid Mech. 229, 365.CrossRefGoogle Scholar
Eres, M. H., Schwartz, L. W. & Roy, R. V. 2000 Fingering phenomena for driven coating films. Phys. Fluids 12, 1278.CrossRefGoogle Scholar
Ford, M. L. & Nadim, A. 1994 Thermocapillary migration of an attached drop on a solid surface. Phys. Fluids 6, 3183.CrossRefGoogle Scholar
Gallardo, B. S., Gupta, V. K., Eagerton, F. D., Jong, L. I., Craig, V. S., Shah, R. R. & Abbott, N. L. 1999 Electrochemical principles for active control of liquids on submillimeter scales. Science 283 (5398), 57–60.CrossRefGoogle ScholarPubMed
Glasner, K. B. & Witelski, T. P. 2003 Coarsening dynamics of dewetting films. Phys. Rev. E 67, 016302.CrossRefGoogle ScholarPubMed
Gomba, J., Diez, J., González, A. G. & Gratton, R. 2005 Spreading of a micrometric fluid strip down a plane under controlled initial conditions. Phys. Rev. E 71, 016304.CrossRefGoogle Scholar
Gomba, J. M., Diez, J., Gratton, R., González, A. G. & Kondic, L. 2007 Stability study of a constant-volume thin film flow. Phys. Rev. E 76 (4), 046308.CrossRefGoogle ScholarPubMed
Gomba, J. M. & Homsy, G. M. 2009 Analytical solutions for partially wetting two-dimensional droplets. Langmuir 25 (10), 5684–5691.CrossRefGoogle ScholarPubMed
Gotkis, Y., Ivanov, I., Murisic, N. & Kondic, L. 2006 Dynamic structure formation at the fronts of volatile liquid drops. Phys. Rev. Lett. 97 (18), 186101.CrossRefGoogle ScholarPubMed
Ho, C.-M. & Tai, Y.-C. 1998 Micro-electro-mechanical systems (MEMS) and fluid flows. Annu. Rev. Fluid Mech. 30, 579.CrossRefGoogle Scholar
Huppert, H. 1982 Flow and instability of a viscous current down a slope. Nature 300, 427.CrossRefGoogle Scholar
Kataoka, D. E. & Troian, S. M. 1998 Stabilizing the advancing front of thermally driven climbing films. J. Colloid Interface Sci. 203, 335.CrossRefGoogle ScholarPubMed
Ludviksson, V. & Lightfoot, E. N. 1971 The dynamics of thin liquid films in the presence of surface-tension gradients. Am. Inst. Chem. Eng. J. 17, 1166.CrossRefGoogle Scholar
Mitlin, V. S. & Petviashvili, N. V. 1994 Nonlinear dynamics of dewetting: kinetically stable structures. Phys. Lett. A 192, 323–326.CrossRefGoogle Scholar
Oron, A., Davis, S. H. & Bankoff, S. G. 1997 Long-scale evolution of thin liquid films. Rev. Mod. Phys. 69, 931.CrossRefGoogle Scholar
Perazzo, C. A. & Gratton, J. 2004 Navier–Stokes solutions for parallel flow in rivulets on an inclined plane. J. Fluid Mech. 507, 367–379.CrossRefGoogle Scholar
Pollack, M. G., Fair, R. B. & Shenderov, A. D. 2000 Electrowetting-based actuation of liquid droplets for microfluidic applications. Appl. Phys. Lett. 77, 1725.CrossRefGoogle Scholar
Pratap, V., Moumen, N. & Subramanian, R. S. 2008 Thermocapillary motion of a liquid drop on a horizontal solid surface. Langmuir 24 (9), 5185–5193.CrossRefGoogle ScholarPubMed
Press, W. H., Teukolsky, S. A., Vetterling, W. T. & Flannery, B. P. 1992 Numerical Recipes in Fortran 77. Cambridge University Press.Google Scholar
Sammarco, T. S. & Burns, M. A. 1999 Thermocapillary pumping of discrete drops in microfabricated analysis devices. AIChE J. 45 (2), 350–366.CrossRefGoogle Scholar
Schwartz, L. W. 1998 Hysteretic effects in droplet motions on heterogenous substrates: direct numerical simulations. Langmuir 14, 3440.CrossRefGoogle Scholar
Schwartz, L. W. 2001 On the asymptotic analysis of surface-stress-driven thin-layer flow. J. Engng Math. 39, 171.CrossRefGoogle Scholar
Schwartz, L. W. & Eley, R. R. 1998 Simulation of droplet motion of low-energy and heterogenous surfaces. J. Colloid Interface Sci. 202, 173.CrossRefGoogle Scholar
Schwartz, L. W., Roux, D. & Cooper-White, J. J. 2005 On the shapes of droplets that are sliding on a vertical wall. Physica D 209, 236–244.CrossRefGoogle Scholar
Smith, M. K. 1995 Thermocapillary migration of a two-dimensional liquid droplet on a solid surface. J. Fluid Mech. Digit. Arch. 294, 209–230.CrossRefGoogle Scholar
Stone, H. A., Stroock, A. D. & Ajdari, A. 2004 Engineering flows in small devices: microfluidics toward lab-on-a-chip. Annu. Rev. Fluid Mech. 36, 381.CrossRefGoogle Scholar
Subramanian, R. S. & Balasubramanian, R. 2001 The Motion of Bubbles and Drops in Reduced Gravity. Cambridge University Press.Google Scholar
Sur, J., Bertozzi, A. L. & Behringer, R. P. 2003 Reverse undercompressive shock structures in driven thin film flow. Phys. Rev. Lett. 90, 126105.CrossRefGoogle ScholarPubMed
Sur, J., Witelski, T. P. & Behringer, R. P. 2004 Steady-profile fingering flows in Marangoni driven thin films. Phys. Rev. Lett. 93 (24), 247803.CrossRefGoogle ScholarPubMed
Teletzke, G. F., Davis, H. T. & Scriven, L. E. 1987 How liquids spread on solids. Chem. Eng. Comm. 55, 41–82.CrossRefGoogle Scholar
Valentino, J. P., Darhuber, A. A., Troian, S. M. & Wagner, S. 2003 Thermocapillary actuation of liquids using patterned microheater arrays. Mater. Res. Soc. Symp. Proc. 773, N10.3.1–N10.3.5.CrossRefGoogle Scholar
Zhornitskaya, L. & Bertozzi, A. L. 2000 Positivity preserving numerical schemes for lubrication-type equations. SIAM J. Numer. Anal. 37, 523.CrossRefGoogle Scholar