Determination of the Thickness and Composition Profiles for a Film of Binary Mixture on a Solid Substrate

L. Fraštia; U. Thiele; L. M. Pismen

Mathematical Modelling of Natural Phenomena (2010)

  • Volume: 6, Issue: 1, page 62-86
  • ISSN: 0973-5348

Abstract

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We determine the steady-state structures that result from liquid-liquid demixing in a free surface film of binary liquid on a solid substrate. The considered model corresponds to the static limit of the diffuse interface theory describing the phase separation process for a binary liquid (model-H), when supplemented by boundary conditions at the free surface and taking the influence of the solid substrate into account. The resulting variational problem is numerically solved employing a Finite Element Method on an adaptive grid. The developed numerical scheme allows us to obtain the coupled steady-state film thickness profile and the concentration profile inside the film. As an example we determine steady state profiles for a reflection-symmetric two-dimensional droplet for various surface tensions of the film and various preferential attraction strength of one component to the substrate. We discuss the relation of the results of the present diffuse interface theory to the sharp interface limit and determine the effective interface tension of the diffuse interface by several means.

How to cite

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Fraštia, L., Thiele, U., and Pismen, L. M.. "Determination of the Thickness and Composition Profiles for a Film of Binary Mixture on a Solid Substrate." Mathematical Modelling of Natural Phenomena 6.1 (2010): 62-86. <http://eudml.org/doc/197666>.

@article{Fraštia2010,
abstract = {We determine the steady-state structures that result from liquid-liquid demixing in a free surface film of binary liquid on a solid substrate. The considered model corresponds to the static limit of the diffuse interface theory describing the phase separation process for a binary liquid (model-H), when supplemented by boundary conditions at the free surface and taking the influence of the solid substrate into account. The resulting variational problem is numerically solved employing a Finite Element Method on an adaptive grid. The developed numerical scheme allows us to obtain the coupled steady-state film thickness profile and the concentration profile inside the film. As an example we determine steady state profiles for a reflection-symmetric two-dimensional droplet for various surface tensions of the film and various preferential attraction strength of one component to the substrate. We discuss the relation of the results of the present diffuse interface theory to the sharp interface limit and determine the effective interface tension of the diffuse interface by several means.},
author = {Fraštia, L., Thiele, U., Pismen, L. M.},
journal = {Mathematical Modelling of Natural Phenomena},
keywords = {Cahn-Hilliard theory; model-H; phase separation; diffuse interface; phase-field model; demixing coupled to dewetting; surface evolution; variational method; FEM},
language = {eng},
month = {6},
number = {1},
pages = {62-86},
publisher = {EDP Sciences},
title = {Determination of the Thickness and Composition Profiles for a Film of Binary Mixture on a Solid Substrate},
url = {http://eudml.org/doc/197666},
volume = {6},
year = {2010},
}

TY - JOUR
AU - Fraštia, L.
AU - Thiele, U.
AU - Pismen, L. M.
TI - Determination of the Thickness and Composition Profiles for a Film of Binary Mixture on a Solid Substrate
JO - Mathematical Modelling of Natural Phenomena
DA - 2010/6//
PB - EDP Sciences
VL - 6
IS - 1
SP - 62
EP - 86
AB - We determine the steady-state structures that result from liquid-liquid demixing in a free surface film of binary liquid on a solid substrate. The considered model corresponds to the static limit of the diffuse interface theory describing the phase separation process for a binary liquid (model-H), when supplemented by boundary conditions at the free surface and taking the influence of the solid substrate into account. The resulting variational problem is numerically solved employing a Finite Element Method on an adaptive grid. The developed numerical scheme allows us to obtain the coupled steady-state film thickness profile and the concentration profile inside the film. As an example we determine steady state profiles for a reflection-symmetric two-dimensional droplet for various surface tensions of the film and various preferential attraction strength of one component to the substrate. We discuss the relation of the results of the present diffuse interface theory to the sharp interface limit and determine the effective interface tension of the diffuse interface by several means.
LA - eng
KW - Cahn-Hilliard theory; model-H; phase separation; diffuse interface; phase-field model; demixing coupled to dewetting; surface evolution; variational method; FEM
UR - http://eudml.org/doc/197666
ER -

References

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  1. D.M. Anderson, G.B. McFadden, A.A. Wheeler. Diffuse-Interface methods in fluid mechanics. Ann. Rev. Fluid Mech., 30 (1998), 139–165. 
  2. L.K. Antanovskii. Microscale theory of surface tension. Phys. Rev. E, 54 (1996), 6285–6290. 
  3. D. Bandyopadhyay, R. Gulabani, A. Sharma. Stability and dynamics of bilayers. Ind. Eng. Chem. Res., 44 (2005), 1259–1272. 
  4. K.-J. Bathe. Finite element procedures. Prentice-Hall, New Jersey, 2nd edition, 1995.  
  5. K. Binder. Spinodal decomposition in confined geometry. J. Non-Equilib. Thermodyn., 23 (1998), 1–44. Zbl0941.76078
  6. L. Brusch, H. Kühne, U. Thiele, M. Bär. Dewetting of thin films on heterogeneous substrates: Pinning vs. coarsening. Phys. Rev. E, 66 (2002), 011602. 
  7. J.W. Cahn, J.E. Hilliard. Free energy of a nonuniform System. 1. Interfacual free energy. J. Chem. Phys., 28 (1958), 258–267. 
  8. H.P. Fischer, P. Maass, W. Dieterich. Novel surface modes in spinodal decomposition. Phys. Rev. Lett., 79 (1997), 893–896. 
  9. H.P. Fischer, P. Maass, W. Dieterich. Diverging time and length scales of spinodal decomposition modes in thin films. Europhys. Lett., 42 (1998), 49–54. 
  10. L.S. Fisher, A.A. Golovin. Nonlinear stability analysis of a two-layer thin liquid film: Dewetting and autophobic behavior. J. Colloid Interface Sci., 291 (2005), 515–528. 
  11. L.S. Fisher, A.A. Golovin. Instability of a two-layer thin liquid film with surfactants: Dewetting waves. J. Colloid Interface Sci., 307 (2007), 203–214. 
  12. O.A. Frolovskaya, A.A. Nepomnyashchy, A. Oron, A.A. Golovin. Stability of a two-layer binary-fluid system with a diffuse interface. Phys. Fluids, 20 (2008), 112105. Zbl1182.76256
  13. M. Geoghegan, G. Krausch. Wetting at polymer surfaces and interfaces. Prog. Polym. Sci., 28 (2003), 261–302. 
  14. A.A. Golovin, S.H. Davis, A.A. Nepomnyashchy. A convective Cahn-Hilliard model for the formation of facets and corners in crystal growth. Physica D, 122 (1998), 202–230. Zbl0952.74050
  15. A.A. Golovin, A.A. Nepomnyashchy, S.H. Davis, M.A. Zaks. Convective Cahn-Hilliard models: From coarsening to roughening. Phys. Rev. Lett., 86 (2001), 1550–1553. 
  16. L.V. Govor, J. Parisi, G.H. Bauer, G. Reiter. Instability and droplet formation in evaporating thin films of a binary solution. Phys. Rev. E, 71 (2005), 051603. 
  17. P.C. Hohenberg, B.I. Halperin. Theory of dynamic critical phenomena. Rev. Mod. Phys., 49 (1977), 435–479. 
  18. K.D. Jandt, J. Heier, F.S. Bates, E.J. Kramer. Transient surface roughening of thin films of phase separating polymer mixtures. Langmuir, 12 (1996), 3716–3720. 
  19. D. Jasnow, J. Viñals. Coarse-grained description of thermo-capillary flow. Phys. Fluids, 8 (1996), 660–669. Zbl1025.76521
  20. R.A.L. Jones, L.J. Norton, E.J. Kramer, F.S. Bates, P. Wiltzius. Surface-directed spinodal decomposition. Phys. Rev. Lett., 66 (1991), 1326–1329. 
  21. S. Kalliadasis, U. Thiele (eds.). Thin Films of Soft Matter. Springer, Wien / New York, CISM 490, 2007.  
  22. K. Kargupta, R. Konnur, A. Sharma. Instability and pattern formation in thin liquid films on chemically heterogeneous substrates. Langmuir, 16 (2000), 10243–10253. 
  23. K. Kargupta, A. Sharma. Templating of thin films induced by dewetting on patterned surfaces. Phys. Rev. Lett., 86 (2001), 4536–4539. 
  24. A. Karim, J.F. Douglas, B.P. Lee, S.C. Glotzer, J.A. Rogers, R.J. Jackman, E.J. Amis, G.M. Whitesides. Phase separation of ultrathin polymer-blend films on patterned substrates. Phys. Rev. E, 57 (1998), R6273–R6276. 
  25. R. Kenzler, F. Eurich, P. Maass, B. Rinn, J. Schropp, E. Bohl, W. Dieterich. Phase separation in confined geometries: Solving the Cahn-Hilliard equation with generic boundary conditions. Comp. Phys. Comm., 133 (2001), 139–157. Zbl0985.65114
  26. T. Kerle, J. Klein, R. Yerushalmi-Rozen. Accelerated rupture at the liquid/liquid interface. Langmuir, 18 (2002), 10146–10154. 
  27. J.S. Langer. An introduction to the kinetics of first-order phase transitions. in ’Solids far from Equilibrium’ (ed. by Godreche), Cambridge University Press, (1992), 297–363.  
  28. J. Lowengrub, L. Truskinovsky. Quasi-incompressible Cahn-Hilliard fluids and topological transitions. Proc. R. Soc. London Ser. A-Math. Phys. Eng. Sci., 454 (1998), 2617–2654. Zbl0927.76007
  29. S. Madruga, U. Thiele. Decomposition driven interface evolution for layers of binary mixtures: II. Influence of convective transport on linear stability. Phys. Fluids, 21 (2009), 062104. Zbl1183.76334
  30. S. Mechkov, M. Rauscher, S. Dietrich. Stability of liquid ridges on chemical micro- and nanostripes. Phys. Rev. E, 77 (2008), 061605. 
  31. P. Müller-Buschbaum, E. Bauer, S. Pfister, S.V. Roth, M. Burghammer, C. Riekel, C. David, U. Thiele. Creation of multi-scale stripe-like patterns in thin polymer blend films. Europhys. Lett., 73 (2006), 35–41. 
  32. G. Nisato, B.D. Ermi, J.F. Douglas, A. Karim. Excitation of surface deformation modes of a phase-separating polymer blend on a patterned substrate. Macromolecules, 32 (1999), 2356–2364. 
  33. A. Oron, S.H. Davis, S.G. Bankoff. Long-scale evolution of thin liquid films. Rev. Mod. Phys., 69 (1997), 931–980. 
  34. L.M. Pismen. Mesoscopic hydrodynamics of contact line motion. Colloid Surf. A-Physicochem. Eng. Asp., 206 (2002), 11–30. 
  35. L.M. Pismen, Y. Pomeau. Disjoining potential and spreading of thin liquid layers in the diffuse interface model coupled to hydrodynamics. Phys. Rev. E, 62 (2000), 2480–2492. 
  36. A. Pototsky, M. Bestehorn, D. Merkt, U. Thiele. Alternative pathways of dewetting for a thin liquid two-layer film. Phys. Rev. E, 70 (2004), 025201. 
  37. A. Pototsky, M. Bestehorn, D. Merkt, U. Thiele. Morphology changes in the evolution of liquid two-layer films. J. Chem. Phys., 122 (2005), 224711. Zbl1187.76346
  38. A. Pototsky, M. Bestehorn, D. Merkt, U. Thiele. 3D Surface Patterns in liquid two-layer films. Europhys. Lett., 74 (2006), 665–671. 
  39. U. Thiele, L. Brusch, M. Bestehorn, M. Bär. Modelling thin-film dewetting on structured substrates and templates: Bifurcation analysis and numerical simulations. Eur. Phys. J. E, 11 (2003), 255–271. 
  40. U. Thiele, S. Madruga, L. Frastia. Decomposition driven interface evolution for layers of binary mixtures: I. Model derivation and stratified base states. Phys. Fluids, 19 (2007), 122106. Zbl1182.76757
  41. N. Vladimirova, A. Malagoli, R. Mauri. Diffusion-driven phase separation of deeply quenched mixtures. Phys. Rev. E, 58 (1998), 7691–7699. 
  42. N. Vladimirova, A. Malagoli, R. Mauri. Two-dimensional model of phase segregation in liquid binary mixtures. Phys. Rev. E, 60 (1999), 6968–6977. 
  43. H. Wang, R.J. Composto. Thin film polymer blends undergoing phase separation and wetting: Identification of early, intermediate, and late stages. J. Chem. Phys., 113 (2000), 10386–10397. 
  44. H. Wang, R.J. Composto. Understanding morphology evolution and roughening in phase-separating thin-film polymer blends. Europhys. Lett., 50 (2000), 622–627. 

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