Thin film dynamics: theory and applications

  • Andrea L. Bertozzi
  • Mark Bowen
Part of the NATO Science Series book series (NAII, volume 75)

Abstract

This paper is based on a series of four lectures, by the first author, on thin films and moving contact lines. Section 1 presents an overview of the moving contact line problem and introduces the lubrication approximation. Section 2 summarizes results for positivity preserving schemes. Section 3 discusses the problem of films driven by thermal gradients with an opposing gravitational force. Such systems yield complex dynamics featuring undercompressive shocks. We conclude in Section 4 with a discussion of dewetting films.

Keywords

Entropy Manifold Verse 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    R. Abeyaratne and J. Knowles, Implications of viscosity and strain gradient effects for the kinetics of propagating phase boundaries in solids, SIAM J. Appl. Math. 51 (1991), 1205–1221.MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    D. J. Acheson, Elementary Fluid Dynamics, Oxford University Press, Oxford, 1990.MATHGoogle Scholar
  3. [3]
    R. Almgren, A. L. Bertozzi, and M. P. Brenner, Stable and unstable singularities in the unforced Hele-Shaw cell, Phys. Fluids 8 (6) (1996), 1356–1370.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    D. G. Aronson and J. Graveleau, A self-similiar solution to the focusing problem for the porous medium equation, European J. Appl. Math. 4 (1993), 65.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    F. S. Bai, C. M. Elliott, A. Gardiner, A. Spence, and A. M. Stuart, The viscous Cahn-Hilliard equation. I. Computations, Nonlinearity 8 (2) (1995), 131–160.MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    F. S. Bai, A. Spence, and A. M. Stuart, Numerical computations of coarsening in the one-dimensional Cahn-Hilliard model of phase separation, Phys. D 78 (3-4) (1994), 155–165.MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    G. I. Barenblatt, Scaling, Self-similarity, and Intermediate Asymptotics, Cambridge University Press, Cambridge, 1996.MATHGoogle Scholar
  8. [8]
    J. W. Barrett, J. F. Blowey, and H. Garcke, Finite element approximation of a fourth order degenerate parabolic equation, Numer. Math. 80 (4) (1998), 525–556.MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    P. W. Bates and P. C. Fife, Spectral comparison principles for the Cahn-Hilliard and phase-field equations, and time scales for coarsening, Phys. D 43 (2-3) (1990), 335–348.MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    R. M. Beam and R. F. Warming, Alternating direction implicit methods for parabolic equations with a mixed derivative, SIAM J. Sci. Statist. Comput. 1 (1980), 131–159.MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    E. Beretta, M. Berstch, and R. D. Passo, Nonnegative solutions of a fourth order nonlinear degenerate parabolic equation, Arch. Rational Mech. Anal. 129 (1995), 175–200.MathSciNetMATHCrossRefGoogle Scholar
  12. [12]
    F. Bernis, Change in sign of the solutions to some parabolic problems, in: Nonlinear Analysis and Applications (Arlington, Tex., 1986) (V. Lakshmikantham, ed.), Lecture Notes in Pure and Appl. Math. 109, Dekker, New York, 1987, 75–82.Google Scholar
  13. [13]
    F. Bernis, Finite speed of propagation and continuity of the interface for slow viscous flows, Adv. Differential Equations 1 (3) (1996), 337–368.MathSciNetMATHGoogle Scholar
  14. [14]
    F. Bernis, Finite speed of propagation for thin viscous flows when 2 ≤ n < 3, C. R. Acad. Sci. Paris Sér. I Math. 322 (1996), 1169–1174.MathSciNetMATHGoogle Scholar
  15. [15]
    F. Bernis and A. Friedman, Higher order nonlinear degenerate parabolic equations, J. Differential Equations 83 (1990), 179–206.MathSciNetMATHCrossRefGoogle Scholar
  16. [16]
    F. Bernis, L. A. Peletier, and S. M. Williams, Source type solutions of a fourth order nonlinear degenerate parabolic equation, Nonlinear Anal.18 (3) (1992), 217–234.MathSciNetMATHCrossRefGoogle Scholar
  17. [17]
    A. L. Bertozzi, Loss and gain of regularity in a lubrication equation for thin viscous films, in: Free Boundary Problems: Theory and Applications (J. I. Díaz et al., eds.), Pitman Res. Notes Math. Ser. 323, Longman, Harlow, 1995, 72–85.Google Scholar
  18. [18]
    A. L. Bertozzi, Symmetric singularity formation in lubrication-type equations for interface motion, SIAM J. Appl. Math. 56 (3) (1996), 681–714.MathSciNetMATHCrossRefGoogle Scholar
  19. [19]
    A. L. Bertozzi, The mathematics of moving contact lines in thin liquid films, Notices Amer. Math. Soc. 45 (6) (1998), 689–697.MathSciNetMATHGoogle Scholar
  20. [20]
    A. L. Bertozzi and M. P. Brenner, Linear stability and transient growth in driven contact lines, Phys. Fluids 9 (3) (1997), 530–539.MathSciNetMATHCrossRefGoogle Scholar
  21. [21]
    A. L. Bertozzi, M. P. Brenner, T. F. Dupont, and L. P. Kadanoff, Singularities and similarities in interface flow, in: Trends and Perspectives in Applied Mathematics (L. Sirovich, ed.), Appl. Math. Sci. 100, Springer-Verlag, New York, 1994, 155–208.CrossRefGoogle Scholar
  22. [22]
    A. L. Bertozzi, G. Grün, and T. P. Witelski, Dewetting films: bifurcations and concentrations, Nonlinearity 14 (6) (2001), 1569–1592.MathSciNetMATHCrossRefGoogle Scholar
  23. [23]
    A. L. Bertozzi, A. Münch, X. Fanton, and A.-M. Cazabat, Contact line stability and ‘undercompressive shocks’ in driven thin film flow, Phys. Rev. Lett. 81 (23) (1998), 5169–5172.CrossRefGoogle Scholar
  24. [24]
    A. L. Bertozzi, A. Münch, and M. Shearer, Undercompressive shocks in thin film flows, Phys. D 134 (4) (1999), 431–464.MathSciNetMATHCrossRefGoogle Scholar
  25. [25]
    A. L. Bertozzi, A. Münch, M. Shearer, and K. Zumbrun, Stability of compressive and undercompressive thin film travelling waves, European J. Appl. Math. 12 (3) (2001), 253–291.MathSciNetMATHGoogle Scholar
  26. [26]
    A. L. Bertozzi and M. C. Pugh, The lubrication approximation for thin viscous films: regularity and long time behavior of weak solutions, Comm. Pure Appl. Math. 49 (2) (1996), 85–123.MathSciNetMATHCrossRefGoogle Scholar
  27. [27]
    A. L. Bertozzi and M. C. Pugh, Long-wave instabilities and saturation in thin film equations, Comm. Pure Appl. Math. 51 (6) (1998), 625–661.MathSciNetCrossRefGoogle Scholar
  28. [28]
    A. L. Bertozzi and M. Shearer, Existence of undercompressive travelling waves in thin film equations, SIAM J. Math. Anal. 32 (1) (2000), 194–213.MathSciNetMATHCrossRefGoogle Scholar
  29. [29]
    J. Bischof, D. Scherer, S. Herminghaus, and P. Leiderer, Dewetting modes of thin metallic films: Nucleation of holes and spinodal dewetting, Phys. Rev. Lett. 77 (8) (1996), 1536–1539.CrossRefGoogle Scholar
  30. [30]
    S. Boatto, L. Kadanoff, and P. Olla, Travelling wave solutions to thin film equations, Phys. Rev. E 48 (1993), 4423.MathSciNetCrossRefGoogle Scholar
  31. [31]
    A. Bourlioux and A. J. Majda, Theoretical and numerical structure of unstable detonations, Philos. Trans. Roy. Soc. London Ser. A 350 (1995), 29–69.MATHCrossRefGoogle Scholar
  32. [32]
    R. Buckingham, M. Shearer, and A. L. Bertozzi, Thin films traveling waves and the navier slip condition, preprint, Duke University, 2002.Google Scholar
  33. [33]
    J. P. Burelbach, S. G. Bankoff, and S. H. Davis, Nonlinear stability of evaporating/condensing liquid films, J. Fluid Mech. 195 (1988), 463–494.MATHCrossRefGoogle Scholar
  34. [34]
    P. Carles and A.-M. Cazabat, The thickness of surface-tension-gradient-driven spreading films, J. Colloid Interface Sci. 157 (1993), 196–201.CrossRefGoogle Scholar
  35. [35]
    A.-M. Cazabat, F. Heslot, S. M. Troian, and P. Carles, Finger instability of this spreading films driven by temperature gradients, Nature 346 (6287) (1990), 824–826.CrossRefGoogle Scholar
  36. [36]
    P. Constantin, T. F. Dupont, R. E. Goldstein, L. P. Kadanoff, M. J. Shelley, and S.-M. Zhou, Droplet breakup in a model of the Hele-Shaw cell, Phys. Rev. E 47 (6) (1993), 4169–4181.MathSciNetCrossRefGoogle Scholar
  37. [37]
    P. G. de Gennes, Wetting: statics and dynamics, Rev. Modern Phys. 57 (1985), 827–880.CrossRefGoogle Scholar
  38. [38]
    T. F. Dupont, R. E. Goldstein, L. P. Kadanoff, and S.-M. Zhou, Finite-time singularity formation in Hele Shaw systems, Phys. Rev. E 47 (6) (1993), 4182–4196.MathSciNetCrossRefGoogle Scholar
  39. [39]
    E. B. Dussan V and S. Davis, On the motion of a fluid-fluid interface along a solid surface, J. Fluid Mech. 65 (1974), 71–95.MATHCrossRefGoogle Scholar
  40. [40]
    J. C. Eilbeck, J. E. Furter, and M. Grinfeld, On a stationary state characterization of transition from spinodal decomposition to nucleation behaviour in the Cahn-Hilliard model of phase separation, Phys. Lett. A 135 (4-5) (1989), 272–275.MathSciNetCrossRefGoogle Scholar
  41. [41]
    M. Elbaum, S. G. Lipsom, and J. F. Wettlaufer, Evaporation pre-empts complete wetting, Europhys. Lett. 29 (6) (1995), 457–462.CrossRefGoogle Scholar
  42. [42]
    M. H. Eres, L. W. Schwartz, and R. V. Roy, Fingering phenomena for driven coating films, Phys. Fluids 12 (6) (2000), 1278–1295.MathSciNetMATHCrossRefGoogle Scholar
  43. [43]
    T. Erneux and D. Gallez, Can repulsive forces lead to stable patterns in thin liquid films?, Phys. Fluids 9 (4) (1997), 1194–1196.CrossRefGoogle Scholar
  44. [44]
    L. C. Evans, Weak Convergence Methods for Nonlinear Partial Differential Equations, CBMS Regional Conf. Ser. in Math. 74, Amer. Math. Soc., Providence, RI, 1990.MATHGoogle Scholar
  45. [45]
    L. C. Evans, Partial Differential Equations, Grad. Stud. Math. 19, Amer. Math. Soc., Providence, RI, 1998.MATHGoogle Scholar
  46. [46]
    X. Fanton, A.-M. Cazabat, and D. Quéré, Thickness and shape of films driven by a Marangoni flow, Langmuir 12 (24) (1996), 5875–5880.CrossRefGoogle Scholar
  47. [47]
    R. Ferreira and F. Bernis, Source-type solutions to thin-film equations in higher dimensions, European J. Appl. Math. 8 (5) (1997), 507–524.MathSciNetMATHCrossRefGoogle Scholar
  48. [48]
    N. Fraysse and G. M. Homsy, An experimental study of rivulet instabilities in centrifugal spin coating of viscous Newtonian and non-Newtonian fluids, Phys. Fluids 6 (4) (1994), 1491–1504.CrossRefGoogle Scholar
  49. [49]
    K. Glasner, A diffuse interface approach to Hele-Shaw flow, preprint, Duke University, 2001.Google Scholar
  50. [50]
    S. Goldstein, ed., Modern Developments in Fluid Dynamics, vol. 2, Dover Publ., New York, 1965.Google Scholar
  51. [51]
    A. A. Golovin, B. Y. Rubinstein, and L. M. Pismen, Effect of van der Waals interaction on the fingering instability of thermally driven thin wetting films, Langmuir 17 (2001), 3930–3936.CrossRefGoogle Scholar
  52. [52]
    H. P. Greenspan, On the motion of a small viscous droplet that wets a surface, J. Fluid Mech. 84 (1978), 125–143.MATHCrossRefGoogle Scholar
  53. [53]
    M. Grinfeld and A. Novick-Cohen, The viscous Cahn-Hilliard equation: Morse decomposition and structure of the global attractor, Trans. Amer. Math. Soc. 351 (6) (1999), 2375–2406.MathSciNetMATHCrossRefGoogle Scholar
  54. [54]
    G. Grün, On the convergence of entropy consistent numerical schemes for lubrication-type equations in multiple space dimensions, to appear in Math. Comp. Google Scholar
  55. [55]
    G. Grün, On the numerical simulation of wetting phenomena, in: Proc. 15th GAMM-Seminar Kiel, Numerical Methods of Composite Materials (W. Hackbusch and S. Sauter, eds.), Vieweg-Verlag, Braunschweig, to appear.Google Scholar
  56. [56]
    G. Grün and M. Rumpf, Nonnegativity preserving convergent schemes for the thin film equation, Numer. Math. 87 (1) (2000), 113–152.MathSciNetMATHCrossRefGoogle Scholar
  57. [57]
    P. J. Haley and M. J. Miksis, The effect of the contact line on droplet spreading, J. Fluid Mech. 223 (1991), 57–81.MathSciNetMATHCrossRefGoogle Scholar
  58. [58]
    S. Herminghaus, K. Jacobs, K. Mecke, J. Bischof, A. Fery, M. Ibn-Elhaj, and S. Schlagowski, Spinodal dewetting in liquid crystal and liquid metal films, Science 282 (5390) (1998), 916–919.CrossRefGoogle Scholar
  59. [59]
    C. Huh and L. E. Scriven, Hydrodynamic model of steady movement of a solid/liquid/fluid contact line, J. Colloid Interface Sci. 35 (1971), 85–101.CrossRefGoogle Scholar
  60. [60]
    H. Huppert, Flow and instability of a viscous current down a slope, Nature 300 (1982), 427–429.CrossRefGoogle Scholar
  61. [61]
    E. Isaacson, D. Marchesin, and B. Plohr, Transitional waves for conservation laws, SIAM J. Math. Anal. 21 (1990), 837–866.MathSciNetMATHCrossRefGoogle Scholar
  62. [62]
    J. N. Israelachvili, Intermolecular and Surface Forces, 2nd ed., Academic Press, New York, 1992.Google Scholar
  63. [63]
    D. E. Kataoka and S. M. Troian, A theoretical study of instabilities at the advancing front of thermally driven coating films, J. Colloid Interface Sci. 192 (1997), 350–362.CrossRefGoogle Scholar
  64. [64]
    D. E. Kataoka and S. M. Troian, Stabilizing the advancing front of thermally driven climbing films, J. Colloid Interface Sci. 203 (1998), 335–344.CrossRefGoogle Scholar
  65. [65]
    J. B. Keller and S. Antman, eds., Bifurcation Theory and Nonlinear Eigenvalue Problems, Benjamin, New York-Amsterdam, 1969.MATHGoogle Scholar
  66. [66]
    R. Konnur, K. Kargupta, and A. Sharma, Instability and morphology of thin liquid films on chemically heterogeneous substrates, Phys. Rev. Lett. 84 (5) (2000), 931–934.CrossRefGoogle Scholar
  67. [67]
    N. Kopell and L. N. Howard, Bifurcations and trajectories joining critical points, Adv. Math. 18 (1975), 306–358.MathSciNetMATHCrossRefGoogle Scholar
  68. [68]
    R. S. Laugesen and M. C. Pugh, Energy levels of steady states for thin film type equations, http://xyz.lanl.gov/abs/math.AP/0003208, to appear in J. Differential Equations.Google Scholar
  69. [69]
    R. S. Laugesen and M. C. Pugh, Linear stability of steady states for thin film and Cahn-Hilliard type equations, Arch. Rational Mech. Anal. 154 (1) (2000), 3–51.MathSciNetMATHCrossRefGoogle Scholar
  70. [70]
    P. D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, CBMS-NSF Regional Conf. Ser. in Appl. Math. 11, SIAM, Philadelphia, PA, 1973.MATHCrossRefGoogle Scholar
  71. [71]
    F. Melo, J. F. Joanny, and S. Fauve, Fingering intability of spinning drops, Phys. Rev. Lett. 63 (18) (1989), 1958–1961.CrossRefGoogle Scholar
  72. [72]
    V. S. Mitlin, Dewetting of a solid surface: analogy with spinodal decomposition, J. Colloid Interface Sci. 156 (1993), 491–497.CrossRefGoogle Scholar
  73. [73]
    V. S. Mitlin and N. V. Petviashvili, Nonlinear dynamics of dewetting: kinetically stable structures, Phys. Lett. A 192 (1994), 323–326.CrossRefGoogle Scholar
  74. [74]
    A. Münch, Shock transitions in Marangoni-gravity driven thin film flow, Nonlinearity 13 (2000), 731–746.MathSciNetMATHCrossRefGoogle Scholar
  75. [75]
    A. Novick-Cohen and L. A. Peletier, Steady states of the one-dimensional Cahn-Hilliard equation, Proc. Roy. Soc. Edinburgh Sect. A 123 (6) (1993), 1071-1098.MathSciNetMATHCrossRefGoogle Scholar
  76. [76]
    A. Oron and S. G. Bankoff, Dewetting of a heated surface by an evaporating liquid film under conjoining/disjoing pressures, J. Colloid Interface Sci. 218 (1999), 152–166.CrossRefGoogle Scholar
  77. [77]
    A. Oron and S. G. Bankoff, Dynamics of a condensing liquid film under conjoining/disjoining pressures, Phys. Fluids 13 (5) (2001), 1107–1117.CrossRefGoogle Scholar
  78. [78]
    A. Oron, S. H. Davis, and S. G. Bankoff, Long-scale evolution of thin liquid films, Rev. Modern Phys. 69 (3) (1997), 931–980.CrossRefGoogle Scholar
  79. [79]
    L. A. Peletier, The porous media equation, in: Applications of Nonlinear Analysis in the Physical Sciences (H. Amann et al., eds.), Pitman, New York, 1981, 229–241.Google Scholar
  80. [80]
    G. Reiter, Dewetting of thin polymer films, Phys. Rev. Lett. 68 (1) (1992), 75–78.MathSciNetCrossRefGoogle Scholar
  81. [81]
    L. G. Reyna and M. J. Ward, Metastable internal layer dynamics for the viscous Cahn-Hilliard equation, Methods Appl. Anal. 2 (3) (1995), 285–306.MathSciNetMATHGoogle Scholar
  82. [82]
    M. Schneemilch and A.-M. Cazabat, Shock separation in wetting films driven by thermal gradients, Langmuir 16 (25) (2000), 9850–9856.CrossRefGoogle Scholar
  83. [83]
    M. Schneemilch and A.-M. Cazabat, Wetting films in thermal gradients, Langmuir 16 (23) (2000), 8796–8801.CrossRefGoogle Scholar
  84. [84]
    L. W. Schwartz, Viscous flows down an inclined plane: Instability and finger formation, Phys. Fluids A 1 (3) (1989), 443–445.MATHCrossRefGoogle Scholar
  85. [85]
    L. W. Schwartz, R. V. Roy, R. R. Eley, and S. Petrash, Dewetting patterns in a drying liquid film, J. Colloid Interface Sci. 234 (2) (2001), 363–374.CrossRefGoogle Scholar
  86. [86]
    A. Sharma and R. Khanna, Pattern formation in unstable thin liquid films, Phys. Rev. Lett. 81 (16) (1998), 3463–3466.CrossRefGoogle Scholar
  87. [87]
    A. Sharma and R. Khanna, Pattern formation in unstable thin liquid films under the influence of antagonistic short- and long-range forces, J. Chem. Phys. 110 (10) (1999), 4929–4936.CrossRefGoogle Scholar
  88. [88]
    M. Shearer, D. G. Schaeffer, D. Marchesin, and P. Paes-Leme, Solution of the Riemann problem for a prototype 2 × 2 system of non-strictly hyperbolic conservation laws, Arch. Rational Mech. Anal. 97 (1987), 299–320.MathSciNetMATHCrossRefGoogle Scholar
  89. [89]
    M. Slemrod, Admissibility criteria for propagating phase boundaries in a van der Waals fluid, Arch. Rational Mech. Anal. 81 (1983), 301–315.MathSciNetMATHCrossRefGoogle Scholar
  90. [90]
    V. M. Starov, Spreading of droplets of nonvolatile liquids over a flat solid, J. Colloid Interface Sci. USSR 45 (1983), 1154.Google Scholar
  91. [91]
    S. M. Troian, E. Herbolzheimer, S. A. Safran, and J. F. Joanny, Fingering instabilities of driven spreading films, Europhys. Lett. 10 (1) (1989), 25–30.CrossRefGoogle Scholar
  92. [92]
    M. P. Valignat, S. Villette, J. Li, R. Barberi, R. Bartolino, E. Dubois-Violette, and A.-M. Cazabat, Wetting and anchoring of a nematic liquid crystal on a rough surface, Phys. Rev. Lett. 77 (10) (1996), 1994–1997.CrossRefGoogle Scholar
  93. [93]
    F. Vandenbrouck, M. P. Valignat, and A.-M. Cazabat, Thin nematic films: metastability and spinodal dewetting, Phys. Rev. Lett. 82 (13) (1999), 2693–2696.CrossRefGoogle Scholar
  94. [94]
    M. J. Ward, Metastable patterns, layer collapses, and coarsening for a one-dimensional Ginzburg-Landau equation, Stud. Appl. Math. 91 (1) (1994), 51–93.MathSciNetMATHGoogle Scholar
  95. [95]
    M. B. Williams and S. H. Davis, Nonlinear theory of film rupture, J. Colloid Interface Sci. 90 (1982), 220–228.CrossRefGoogle Scholar
  96. [96]
    T. P. Witelski and A. J. Bernoff, Stability of self-similar solutions for van der Waals driven thin film rupture, Phys. Fluids 11 (9) (1999), 2443–2445.MathSciNetMATHCrossRefGoogle Scholar
  97. [97]
    T. P. Witelski and A. J. Bernoff, Dynamics of three-dimensional thin film rupture, Phys. D 147 (2000), 155–176.MathSciNetMATHCrossRefGoogle Scholar
  98. [98]
    T. P. Witelski and M. Bowen, ADI methods for high order parabolic equations, preprint, Duke University, 2001.Google Scholar
  99. [99]
    C. C. Wu, New theory of MHD shock waves, in: Viscous Profiles and Numerical Methods for Shock Waves (M. Shearer, ed.), SIAM, Philadelphia, PA, 1991.Google Scholar
  100. [100]
    R. Xie, A. Karim, J. F. Douglas, C. C. Han, and R. A. Weiss, Spinodal dewetting of thin polymer films, Phys. Rev. Lett 81 (6) (1998), 1251–1254.CrossRefGoogle Scholar
  101. [101]
    W. W. Zhang and J. R. Lister, Similarity solutions for van der Waals rupture of a thin film on a solid substrate, Phys. Fluids 11 (9) (1999), 2454–2462.MathSciNetMATHCrossRefGoogle Scholar
  102. [102]
    L. Zhornitskaya and A. L. Bertozzi, Positivity-preserving numerical schemes for lubrication-type equations, SIAM J. Numer. Anal. 37 (2) (2000), 523–555 (electronic).MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2002

Authors and Affiliations

  • Andrea L. Bertozzi
    • 1
  • Mark Bowen
    • 1
  1. 1.Department of MathematicsDuke UniversityDurhamUSA

Personalised recommendations