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Substrate elastic deformation due to vertical component of liquid-vapor interfacial tension

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Abstract

Young’s equation is a fundamental equation in capillarity and wetting, which reflects the balance of the horizontal components of the three interfacial tensions with the contact angle (CA). However, it does not consider the vertical component of the liquid-vapor interfacial tension (VCLVIT). It is now well understood that the VCLVIT causes the elastic deformation of the solid substrate, which plays a significant role in the fabrication of the microfluidic devices because of the wide use of the soft materials. In this paper, the theoretical, experimental, and numerical aspects of the problem are reviewed. The effects of the VCLVIT-induced surface deformation on the wetting and spreading, the deflection of the microcantilever, and the elasto-capillarity and electroelasto-capillarity are discussed. Besides a brief review on the historical development and the recent advances, some suggestions on the future research are also provided.

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References

  1. Bonn, D., Eggers, J., Indekeu, J., Meunier, J., and Rolley, E. Wetting and spreading. Reviews of Modern Physics, 81(2), 739–805 (2009)

    Article  Google Scholar 

  2. De Gennes, P. G. Wetting: statics and dynamics. Reviews of Modern Physics, 57(3), 827–863 (1985)

    Article  Google Scholar 

  3. Adamson, A. W. and Gast, A. P. Physical Chemistry of Surfaces, 6th ed., Wiley, New York (1997)

    Google Scholar 

  4. Leger, L. and Joanny, J. F. Liquid spreading. Reports on Progress in Physics, 55(4), 431–486 (1992)

    Article  Google Scholar 

  5. Finn, R. Equilibrium Capillary Surfaces, Springer, New York (1986)

    Book  MATH  Google Scholar 

  6. Young, T. An essay on the cohesion of fluids. Philosophical Transactions of the Royal Society of London, 95, 65–87 (1805)

    Article  Google Scholar 

  7. Hondros, E. D. Dr. Thomas Young-natural philosopher. Journal of Materials Science, 40(9–10), 2119–2123 (2005)

    Article  Google Scholar 

  8. Finn, R. The contact angle in capillarity. Physics of Fluids, 18(4), 047102 (2006)

    Article  MathSciNet  Google Scholar 

  9. Maxwell, J. C. The Encyclopedia Britannica: Capillary Action, 9th ed., Encyclopedia Britannica, Inc., London, 566 (1875)

    Google Scholar 

  10. De Gennes, P. G., Brochard-Wyart, F., and Quéré, D. Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves, Springer, Berlin (2004)

    Book  MATH  Google Scholar 

  11. Hu, W. R. and Xu, S. C. Microgravity Fluid Mechanics (in Chinese), Science Press, Beijing (1999)

    Google Scholar 

  12. Lester, G. R. Contact angles of liquids at deformable solid surfaces. Journal of Colloid Science, 16(4), 315–326 (1961)

    Article  Google Scholar 

  13. Rusanov, A. I. Theory of wetting of elastically deformed bodies, 1. deformation with a finite contact-angle (in Russian). Colloid Journal of the USSR, 37(4), 614–622 (1975)

    Google Scholar 

  14. Fortes, M. A. Deformation of solid surfaces due to capillary forces. Journal of Colloid and Interface Science, 100(1), 17–26 (1984)

    Article  Google Scholar 

  15. Yu, Y. S. and Zhao, Y. P. Elastic deformation of soft membrane with finite thickness induced by a sessile liquid droplet. Journal of Colloid and Interface Science, 339(2), 489–494 (2009)

    Article  MathSciNet  Google Scholar 

  16. Liu, J. L., Nie, Z. X., and Jiang, W. G. Deformation field of the soft substrate induced by capillary force. Physica B, 404(8–11), 1195–1199 (2009)

    Article  Google Scholar 

  17. Shanahan, M. E. R. and de Gennes, P. G. Equilibrium of the triple line solid/liquid/fluid of a sessile drop. Adhesion (ed. Allen, K. W.), Elsevier Applied Science, London, 71–81 (1987)

    Google Scholar 

  18. Shanahan, M. E. R. and Carré, A. Spreading and dynamics of liquid drops involving nanometric deformations on soft substrates. Colloids and Surfaces A, 206(1–3), 115–123 (2002)

    Article  Google Scholar 

  19. Shanahan, M. E. R. and Carré, A. Nanometric solid deformation of soft materials in capillary phenomena. Nano-Surface Chemistry (ed. Rosoff, M.), Marcel Dekker Inc., New York (2002)

    Google Scholar 

  20. White, L. R. The contact angle on the elastic substrate, 1. the role of disjoining pressure in the surface mechanics. Journal of Colloid and Interface Science, 258(1), 82–96 (2003)

    Article  Google Scholar 

  21. Das, S., Marchand, A., Andreotti, B., and Snoeijer, J. H. Elastic deformation due to tangential capillary forces. Physics of Fluids, 23(7), 072006 (2011)

    Article  Google Scholar 

  22. Kern, R. and Müller, P. Deformation of an elastic thin solid induced by a liquid droplet. Surface Science, 264(3), 467–494 (1992)

    Article  Google Scholar 

  23. Treloar, L. R. G. The Physics of Rubber Elasticity, Clarendon, Oxford, 66 (1949)

    Google Scholar 

  24. Shanahan, M. E. R. The influence of solid micro-deformation on contact angle equilibrium. Journal of Physics D: Applied Physics, 20(7), 945–950 (1987)

    Article  Google Scholar 

  25. Shanahan, M. E. R. Statics and dynamics of wetting on thin solids. Revue de Physique Appliquée, 23(6), 1031–1037 (1988)

    Article  Google Scholar 

  26. Shanahan, M. E. R. The spreading dynamics of a liquid drop on a viscoelastic solid. Journal of Physics D: Applied Physics, 21(6), 981–985 (1988)

    Article  Google Scholar 

  27. Carré, A. and Shanahan, M. E. R. Direct evidence for viscosity-independent spreading on a soft solid. Langmuir, 11(1), 24–26 (1995)

    Article  Google Scholar 

  28. Shanahan, M. E. R. and Carré, A. Viscoelastic dissipation in wetting and adhesion phenomena. Langmuir, 11(4), 1396–1402 (1995)

    Article  Google Scholar 

  29. Carré, A., Gastel, J. C., and Shanahan, M. E. R. Viscoelastic effects in the spreading of liquids. nature, 379(6564), 432–434 (1996)

    Article  Google Scholar 

  30. Carré, A. and Shanahan, M. E. R. Effect of cross-linking on the dewetting of an elastomeric surface. Journal of Colloid and Interface Science, 191(1), 141–145 (1997)

    Article  Google Scholar 

  31. Long, D., Ajdari, A., and Leibler, L. Static and dynamic wetting properties of thin rubber films. Langmuir, 12(21), 5221–5230 (1996)

    Article  Google Scholar 

  32. Andrade, J. D., King, R. N., Gregonis, D. E., and Coleman, D. L. Surface characterization of poly (hydroxyethyl methacrylate) and related polymers, I. contact angle methods in water. Journal of Polymer Science: Polymer Symposium, 66, 313–336 (1979)

    Article  Google Scholar 

  33. Métois, J. J. Elastic straining of a thin graphite layer by a liquid droplet or a non-epitaxed Pb crystallite. Surface Science, 241(3), 279–288 (1991)

    Article  Google Scholar 

  34. Extrand, C. W. and Kumagai, Y. Contact angle and hysteresis on soft surfaces. Journal of Colloid and Interface Science, 184(1), 191–200 (1996)

    Article  Google Scholar 

  35. Saiz, E., Tomsia, A. P., and Cannon, R. M. Ridging effects on wetting and spreading of liquids on solids. Acta Materialia, 46(7), 2349–2361 (1998)

    Google Scholar 

  36. Pu, G., Guo, J. H., Gwin, L. E., and Severtson, S. J. Mechanical pinning of liquids through inelastic wetting ridge formation on thermally stripped acrylic polymers. Langmuir, 23(24), 12142–12146 (2007)

    Article  Google Scholar 

  37. Pericet-Cámara, R., Auernhammer, G. K., Koynov, K., Lorenzoni, S., Raiteri, R., and Bonaccurso, E. Solid-supported thin elastomer films deformed by microdrops. Soft Matter, 5(19), 3611–3617 (2009)

    Article  Google Scholar 

  38. Pericet-Cámara, R., Best, A., Butt, H. J., and Bonaccurso, E. Effect of capillary pressure and surface tension on the deformation of elastic surfaces by sessile liquid microdrops: an experimental investigation. Langmuir, 24(19), 10565–10568 (2008)

    Article  Google Scholar 

  39. Jerison, E. R., Xu, Y., Wilen, L. A., and Dufresne, E. R. Deformation of an elastic substrate by a three-phase contact line. Physical Review Letters, 106(18), 186103 (2011)

    Article  Google Scholar 

  40. Saiz, E., Cannon, R. M., and Tomsia, A. P. Reactive spreading: adsorption, ridging and compound formation. Acta Materialia, 48(18–19), 4449–4462 (2000)

    Article  Google Scholar 

  41. Madasu, S. and Cairncross, R. A. Static wetting on flexible substrates: a finite element formulation. International Journal for Numerical Methods in Fluids, 45(3), 301–319 (2004)

    Article  MATH  Google Scholar 

  42. Yu, Y. S., Yang, Z. Y., and Zhao, Y. P. Role of vertical component of surface tension of the droplet on the elastic deformation of PDMS membrane. Journal of Adhesion Science and Technology, 22(7), 687–698 (2008)

    Google Scholar 

  43. Yu, Y. S. and Zhao, Y. P. Deformation of PDMS membrane and microcantilever by a water droplet: comparison between Mooney-Rivlin and linear elastic constitutive models. Journal of Colloid and Interface Science, 332(2), 467–476 (2009)

    Article  MathSciNet  Google Scholar 

  44. Wang, F. C. Boundary Slip and Contact Angle Hysteresis in the Nanoscale Liquid-Solid Interfacial Mechanics (in Chinese), Ph. D. dissertation, Graduate University of Chinese Academy of Sciences, Beijing (2012)

    Google Scholar 

  45. Rugar, D. and Hansma, P. Atomic force microscopy. Physics Today, 43(10), 23–30 (1990)

    Article  Google Scholar 

  46. Dimitriadis, E. K., Horkay, F., Maresca, J., Kachar, B., and Chadwick, R. S. Determination of elastic moduli of thin layers of soft material using the atomic force microscopy. Biophysical Journal, 82(5), 2798–2810 (2002)

    Article  Google Scholar 

  47. Magonov, S. N. and Reneker, D. H. Characterization of polymer surfaces with atomic force microscopy. Annual Review of Materials Science, 27, 175–222 (1997)

    Article  Google Scholar 

  48. Zhao, L. M., Schaefer, D., and Marten, M. R. Assessment of elasticity and topography of aspergillus nidulans spores via atomic force microscopy. Applied and Environmental Microbiology, 71(2), 955–960 (2005)

    Article  Google Scholar 

  49. Jensenius, H., Thaysen, J., Rasmussen, A. A., Veje, L. H., Hansen, O., and Boisen, A. A microcantilever-based alcohol vapor sensor-application and response model. Applied Physics Letters, 76(18), 2615–2617 (2000)

    Article  Google Scholar 

  50. Bonaccurso, E. and Butt, H. J. Microdrops on atomic force microscope cantilevers: evaporation of water and spring constant calibration. Journal of Physics and Chemistry B, 109(1), 253–263 (2005)

    Article  Google Scholar 

  51. Haschke, T., Bonaccurso, E., Butt, H. J., Lautenschlager, D., Schönfeld, F., and Wiechert, W. Sessile-drop-induced bending of atomic force microscope cantilevers: a model system for monitoring microdrop evaporation. Journal of Micromechanics and Microengineering, 16(11), 2273–2280 (2006)

    Article  Google Scholar 

  52. Butt, H. J., Golovko, D. S., and Bonaccurso, E. On the derivation of Young’s equation for sessile drops: nonequilibrium effects due to evaporation. Journal of Physics and Chemistry B, 111(19), 5277–5283 (2007)

    Article  Google Scholar 

  53. Golovko, D. S., Bonanno, P., Lorenzoni, S., Stefani, F., Raiteri, R., and Bonaccurso, E. Evaporative cooling of sessile water microdrops measured with atomic force microscope cantilevers. Journal of Micromechanics and Microengineering, 18(9), 095026 (2008)

    Article  Google Scholar 

  54. Golovko, D. S., Butt, H. J., and Bonaccurso, E. Transition in the evaporation kinetics of water microdrops on hydrophilic surfaces. Langmuir, 25(1), 75–78 (2009)

    Article  Google Scholar 

  55. Jeon, S., Desikan, R., Tian, F., and Thundat, T. Influence of nanobubbles on the bending of microcantilevers. Applied Physics Letters, 88(10), 103118 (2006)

    Article  Google Scholar 

  56. Lee, H. J., Chang, Y. S., Lee, Y. P., Jeong, K. H., and Kim, H. Y. Deflection of microcantilever by growing vapor bubble. Sensors and Actuators A, 136(2), 717–722 (2007)

    Article  Google Scholar 

  57. Zheng, X. P., Zhao, H. P., Gao, L. T., Liu, J. L., Yu, S. W., and Feng, X. Q. Elasticity-driven droplet movement on a microbeam with gradient stiffness: a biomimetic self-propelling mechanism. Journal of Colloid and Interface Science, 323(1), 133–140 (2008)

    Article  Google Scholar 

  58. Mastrangelo, C. H. and Hsu, C. H. Mechanical stability and adhesion of micro structures under capillary forces—part I: basic theory. Journal of Microelectromechanical Systems, 2(1), 33–43 (1993)

    Article  Google Scholar 

  59. Mastrangelo, C. H. and Hsu, C. H. Mechanical stability and adhesion of micro structures under capillary forces—part II: experiments. Journal of Microelectromechanical Systems, 2(1), 44–55 (1993)

    Article  Google Scholar 

  60. Syms, R. R. A., Yeatman, E. M., Bright, V. M., and Whitesides, G. M. Surface tension-powered self-assembly of microstructures—the state-of-the-art. Journal of Microelectromechanical Systems, 12(4), 387–417 (2003)

    Article  Google Scholar 

  61. Syms, R. R. A. and Yeatman, E. M. Self-assembly of three-dimensional microstructures using rotation by surface tension forces. Electronics Letters, 29(8), 662–664 (1993)

    Article  Google Scholar 

  62. Green, P. W., Syms, R. R. A., and Yeatman, E. M. Demonstration of three-dimensional microstructure self-assembly. Journal of Microelectromechanical Systems, 4(4), 170–176 (1995)

    Article  Google Scholar 

  63. Syms, R. R. A. Equilibrium of hinged and hingeless structures rotated using surface tension forces. Journal of Microelectromechanical Systems, 4(4), 177–184 (1995)

    Article  Google Scholar 

  64. Bico, J., Roman, B., Moulin, L., and Boudaoud, A. Adhesion: elastocapillary coalescence in wet hair. nature, 432(7018), 690 (2004)

    Article  Google Scholar 

  65. Py, C., Reverdy, P., Doppler, L., Bico, J., Roman, B., and Baroud, C. N. Capillary origami: spontaneous wrapping of a droplet with an elastic sheet. Physical Review Letters, 98(15), 156103 (2007)

    Article  Google Scholar 

  66. Py, C., Reverdy, P., Doppler, L., Bico, J., Roman, B., and Baroud, C. N. Capillary origami. Physics of Fluids, 19(9), 091104 (2007)

    Article  Google Scholar 

  67. Guo, X. Y., Li, H., Ahn, B. Y., Duoss, E. B., Hsia, K. J., Lewis, J. A., and Nuzzo, R. Two- and three-dimensional folding of thin film single-crystalline silicon for photovoltaic power applications. Proceedings of the National Academy of Sciences of the United States of America, 106(48), 20149–20154 (2009)

    Article  Google Scholar 

  68. Li, H., Guo, X. Y., Nuzzo, R. G., and Hsia, K. J. Capillary induced self-assembly of thin foils into 3D structures. Journal of the Mechanics and Physics of Solids, 58(12), 2033–2042 (2010)

    Article  MATH  Google Scholar 

  69. Patra, N., Wang, B. Y., and Král, P. Nanodroplet activated and guided folding of graphene nanostructures. Nano Letters, 9(11), 3766–3771 (2009)

    Article  Google Scholar 

  70. Yuan, Q. Z. and Zhao, Y. P. Precursor film in dynamic wetting, electrowetting and electro-elastocapillarity. Physical Review Letters, 104(24), 246101 (2010)

    Article  Google Scholar 

  71. Wang, Z. Q., Wang, F. C., and Zhao, Y. P. Tap dance of water droplet. Proceedings of the Royal Society A, 468(2145), 2485–2495 (2012)

    Article  Google Scholar 

  72. Yu, Y. S., Wang, Z. Q., and Zhao, Y. P. Experimental and theoretical investigations of evaporation of sessile water droplet on hydrophobic surfaces. Journal of Colloid and Interface Science, 365(1), 254–259 (2012)

    Article  MathSciNet  Google Scholar 

  73. Deegan, R. D., Bakajin, O., Dupont, T. F., Huber, G., Nagel, S. R., and Witten, T. A. Capillary flow as the cause of ring stains from dried liquid drops. nature, 389(6653), 827–829 (1997)

    Article  Google Scholar 

  74. Yunker, P. J., Still, T., Lohr, M. A., and Yodh, A. G. Suppression of the coffee-ring effect by shape-dependent capillary interactions. nature, 476(7360), 308–311 (2011)

    Article  Google Scholar 

  75. Wang, Y. and Zhao, Y. P. Electrowetting on curved surfaces. Soft Matter, 8(9), 2599–2606 (2012)

    Article  Google Scholar 

  76. Roman, B. and Bico, J. Elasto-capillarity: deforming an elastic structure with a liquid droplet. Journal of Physics: Condensed Matter, 22(49), 493101 (2010)

    Article  Google Scholar 

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  1. Department of Engineering Mechanics, School of Civil Engineering and Architecture, Hubei University of Technology, Wuhan, 430068, P. R. China

    Ying-song Yu  (余迎松)

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Correspondence to Ying-song Yu  (余迎松).

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Communicated by Ya-pu ZHAO

Project supported by the National Natural Science Foundation of China (No. 11002051)

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Yu, Ys. Substrate elastic deformation due to vertical component of liquid-vapor interfacial tension. Appl. Math. Mech.-Engl. Ed. 33, 1095–1114 (2012). https://doi.org/10.1007/s10483-012-1608-x

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