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Thin liquid film morphology driven by electro-static field

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Abstract

The development of stationary patterns on a thin polymer surface subject to an electric field is studied by means of the hexagonal-planform weakly nonlinear stability analysis and numerical simulations. The time evolution of the interface between the air and the polymer film on the unbounded spatial domain is described by a thin film equation, incorporating the electric driving force and the surface diffusion. The nonlinear interfacial growth includes the amplitude equations and superposition of one-dimensional structures at regular orientations. The pattern selection is driven by the subcritical instability mechanism in which the relative thickness of the polymer film plays a critical role.

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References

  1. Chou, S. Y. and Zhuang, L. Lithographically induced self-assembly of periodic polymer micropillar arrays. Journal of Vacuum Science & Technolgy B, 17(6), 3197–3202 (1999)

    Article  Google Scholar 

  2. Schäffer, E., Thurn-Albrecht, T., Russell, T. P., and Stelner, U. Electrically induced structure formation and pattern transfer. nature, 403, 874–877 (2000)

    Article  Google Scholar 

  3. Wu, N. and Russel, W. B. Electric-field-induced patterns in thin polymer films: weakly nonlinear and fully nonlinear evolution. Langmuir, 21(26), 12290–12302 (2005)

    Article  Google Scholar 

  4. Kim, D. and Lu, W. Interface instability and nanostructure patterning. Computational Materials Science, 38, 418–425 (2006)

    Article  Google Scholar 

  5. Lin, Z. Q., Kerle, T., Baker, S. M., Hoagland, D. A., Schäffer, E., Stelner, U., and Russell, T. P. Electric field induced instabilities at liquid/liquid interfaces. Journal of Chemical Physics, 114(5), 2377–2381 (2001)

    Article  Google Scholar 

  6. Pease, L. F. III and Russel, W. B. Electrostatically induced submicron patterning of thin perfect and leaky dielectric films: a generalized linear stability analysis. Journal of Chemical Physics, 118(8), 3790–3803 (2003)

    Article  Google Scholar 

  7. Pease, L. F. III and Russel, W. B. Limitations on length scales for electrostatically induced submicrometer pillars and holes. Langmuir, 20(3), 795–804 (2004)

    Article  Google Scholar 

  8. Schäffer, E., Thurn-Albrecht, T., Russell, T. P., and Stelner, U. Electrohydrodynamic instabilities in polymer films. Europhysics Letters, 53(4), 518–524 (2001)

    Article  Google Scholar 

  9. Tian, E. M. Pattern formation induced by an electric field in thin liquid films. Journal of Mathematics, Statistics and Allied Fields, 1(1), 1–6 (2007)

    Google Scholar 

  10. Wu, N. and Russel, W. B. Micro- and nano-patterns created via electrohydrodynamic instabilities. Nano Today, 4, 180–192 (2009)

    Article  Google Scholar 

  11. Yeoh, H. K., Xu, Q., and Basaran, O. A. Equilibrium shapes and stability of a liquid film subjected to a nonuniform electric field. Physics of Fluids, 19, 114111–114132 (2007)

    Article  Google Scholar 

  12. Pease, L. F. III and Russel, W. B. Linear stability analysis of thin leaky dielectric films subjected to electric fields. Journal of Non-Newtonian Fluid Mechanics, 102, 233–250 (2002)

    Article  MATH  Google Scholar 

  13. Shanker, V. and Sharma, A. Instability of the interface between thin fluid films subjected to electric fields. Journal of Colloid and Interface Science, 274, 294–308 (2004)

    Article  Google Scholar 

  14. Verma, R., Sharma, A., Kargupta, K., and Bhaumik, J. Electric field induced instability and pattern formation in thin liquid films. Langmuir, 21(8), 3710–3721 (2005)

    Article  Google Scholar 

  15. Scanlon, J. W. and Segel, L. A. Finite amplitude cellular convection induced by surface tension. Journal of Fluid Mechanics, 30, 149–162 (1967)

    Article  MATH  Google Scholar 

  16. Segel, L. A. and Stuart, J. T. On the question of the preferred mode in cellular thermal convection. Journal of Fluid Mechanics, 13, 289–306 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  17. Segel, L. A. The nonlinear interaction of a finite number of disturbances to a layer of fluid heated from below. Journal of Fluid Mechanics, 21, 359–384 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  18. Busse, F. H. The stability of finite amplitude cellular convection and its relation to an extremum principle. Journal of Fluid Mechanics, 30, 625–649 (1967)

    Article  MATH  Google Scholar 

  19. Palm, E. Nonlinear thermal convection. Annual Reviews of Fluid Mechanics, 7, 39–61 (1975)

    Article  Google Scholar 

  20. Tian, E. M. and Wollkind, D. J. Nonlinear stability analyses of pattern formation in thin liquid films. Interfaces and Free Boundaries, 5, 1–25 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  21. Wollkind, D. J., Sriranganathan, R., and Oulton, D. B. Interfacial patterns during plane front alloy solidification. Physica D, 12(1–3), 215–240 (1984)

    Article  Google Scholar 

  22. Wollkind, D. J. and Stephenson, L. E. Chemical turing pattern formation analyses: comparison of theory with experiment. SIAM Journal of Applied Mathematics, 61(2), 387–431 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  23. Chang, H. C. Wave evolution on a falling film. Annual Reviews of Fluid Mechanics, 26, 103–136 (1994)

    Article  Google Scholar 

  24. Oron, A., Davis, S. H., and Bankoff, S. G. Long-scale evolution of thin liquid films. Reviews of Modern Physics, 69(3), 931–980 (1997)

    Article  Google Scholar 

  25. Landau, L. D. and Lifshitz, E. M. Electrodynamics of Continuous Media, Pergamon Press, New York (1960)

    MATH  Google Scholar 

  26. Saville, D. A. Electrohydrodynamics: the Taylor-Melcher leaky dielectric model. Annual Reviews of Fluid Mechanics, 29, 27–64 (1997)

    Article  MathSciNet  Google Scholar 

  27. Zaks, M. A., Auer M., and Busse, F. H. Undulating rolls and their instabilities in a Rayleigh-Benard layer. Physical Review E, 53(5), 4807–4819 (1996)

    Article  Google Scholar 

  28. Davis, S. H. Thermocapillary instabilities. Annual Reviews of Fluid Mechanics, 19, 403–435 (1987)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Wright State University, Dayton, Ohio, 45435, USA

    E. M. Tian, T. P. Svobodny & J. D. Phillips

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  1. E. M. Tian
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  2. T. P. Svobodny
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  3. J. D. Phillips
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Correspondence to E. M. Tian.

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Tian, E.M., Svobodny, T.P. & Phillips, J.D. Thin liquid film morphology driven by electro-static field. Appl. Math. Mech.-Engl. Ed. 32, 1039–1046 (2011). https://doi.org/10.1007/s10483-011-1479-9

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  • DOI: https://doi.org/10.1007/s10483-011-1479-9

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2010 Mathematics Subject Classification

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