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

Published online by Cambridge University Press:  09 June 2010

L. Fraštia ,
U. Thiele  and
L. M. Pismen
L. Fraštia*
Affiliation:
Department of Mathematical Sciences, Loughborough University Loughborough, Leicestershire, LE11 3TU, UK
U. Thiele
Affiliation:
Department of Mathematical Sciences, Loughborough University Loughborough, Leicestershire, LE11 3TU, UK
L. M. Pismen
Affiliation:
Minerva Center for Nonlinear Physics of Complex Systems Technion–Israel Institute of Technology, 32000 Haifa, Israel
*
* Corresponding author. E-mail:[email protected]
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Abstract

We determine the steady-state structures that result from liquid-liquid demixing in afree surface film of binary liquid on a solid substrate. The considered model correspondsto the static limit of the diffuse interface theory describing the phase separationprocess for a binary liquid (model-H), when supplemented by boundary conditions at thefree surface and taking the influence of the solid substrate into account. The resultingvariational problem is numerically solved employing a Finite Element Method on an adaptivegrid. The developed numerical scheme allows us to obtain the coupled steady-state filmthickness profile and the concentration profile inside the film. As an example wedetermine steady state profiles for a reflection-symmetric two-dimensional droplet forvarious surface tensions of the film and various preferential attraction strength of onecomponent to the substrate. We discuss the relation of the results of the present diffuseinterface theory to the sharp interface limit and determine the effective interfacetension of the diffuse interface by several means.

Keywords

Cahn-Hilliard theorymodel-Hphase separationdiffuse interfacephase-field modeldemixing coupled to dewettingsurface evolutionvariational methodFEM
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 6 , Issue 1: Instability and patterns. Issue dedicated to the memory of A. Golovin , 2011 , pp. 62 - 86
Copyright
© EDP Sciences, 2010

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References

Anderson, D.M., McFadden, G.B., Wheeler, A.A.. Diffuse-Interface methods in fluid mechanics . Ann. Rev. Fluid Mech., 30 (1998), 139–165.CrossRefGoogle Scholar
Antanovskii, L.K.. Microscale theory of surface tension . Phys. Rev. E, 54 (1996), 6285–6290.CrossRefGoogle ScholarPubMed
Bandyopadhyay, D., Gulabani, R., Sharma, A.. Stability and dynamics of bilayers . Ind. Eng. Chem. Res., 44 (2005), 1259–1272.CrossRefGoogle Scholar
K.-J. Bathe. Finite element procedures. Prentice-Hall, New Jersey, 2nd edition, 1995.
Binder, K.. Spinodal decomposition in confined geometry . J. Non-Equilib. Thermodyn., 23 (1998), 1–44.Google Scholar
Brusch, L., Kühne, H., Thiele, U., Bär, M.. Dewetting of thin films on heterogeneous substrates: Pinning vs. coarsening . Phys. Rev. E, 66 (2002), 011602.CrossRefGoogle Scholar
Cahn, J.W., Hilliard, J.E.. Free energy of a nonuniform System. 1. Interfacual free energy . J. Chem. Phys., 28 (1958), 258–267.CrossRefGoogle Scholar
Fischer, H.P., Maass, P., Dieterich, W.. Novel surface modes in spinodal decomposition . Phys. Rev. Lett., 79 (1997), 893–896.CrossRefGoogle Scholar
Fischer, H.P., Maass, P., Dieterich, W.. Diverging time and length scales of spinodal decomposition modes in thin films . Europhys. Lett., 42 (1998), 49–54.CrossRefGoogle Scholar
Fisher, L.S., Golovin, A.A.. Nonlinear stability analysis of a two-layer thin liquid film: Dewetting and autophobic behavior . J. Colloid Interface Sci., 291 (2005), 515–528.CrossRefGoogle ScholarPubMed
Fisher, L.S., Golovin, A.A.. Instability of a two-layer thin liquid film with surfactants: Dewetting waves . J. Colloid Interface Sci., 307 (2007), 203–214.CrossRefGoogle ScholarPubMed
Frolovskaya, O.A., Nepomnyashchy, A.A., Oron, A., Golovin, A.A.. Stability of a two-layer binary-fluid system with a diffuse interface . Phys. Fluids, 20 (2008), 112105.CrossRefGoogle Scholar
Geoghegan, M., Krausch, G.. Wetting at polymer surfaces and interfaces . Prog. Polym. Sci., 28 (2003), 261–302.CrossRefGoogle Scholar
Golovin, A.A., Davis, S.H., Nepomnyashchy, A.A.. A convective Cahn-Hilliard model for the formation of facets and corners in crystal growth . Physica D, 122 (1998), 202–230.CrossRefGoogle Scholar
Golovin, A.A., Nepomnyashchy, A.A., Davis, S.H., Zaks, M.A.. Convective Cahn-Hilliard models: From coarsening to roughening . Phys. Rev. Lett., 86 (2001), 1550–1553.CrossRefGoogle ScholarPubMed
Govor, L.V., Parisi, J., Bauer, G.H., Reiter, G.. Instability and droplet formation in evaporating thin films of a binary solution . Phys. Rev. E, 71 (2005), 051603.CrossRefGoogle ScholarPubMed
Hohenberg, P.C., Halperin, B.I.. Theory of dynamic critical phenomena . Rev. Mod. Phys., 49 (1977), 435–479.CrossRefGoogle Scholar
Jandt, K.D., Heier, J., Bates, F.S., Kramer, E.J.. Transient surface roughening of thin films of phase separating polymer mixtures . Langmuir, 12 (1996), 3716–3720.CrossRefGoogle Scholar
Jasnow, D., Viñals, J.. Coarse-grained description of thermo-capillary flow . Phys. Fluids, 8 (1996), 660–669.CrossRefGoogle Scholar
Jones, R.A.L., Norton, L.J., Kramer, E.J., Bates, F.S., Wiltzius, P.. Surface-directed spinodal decomposition . Phys. Rev. Lett., 66 (1991), 1326–1329.CrossRefGoogle ScholarPubMed
S. Kalliadasis, U. Thiele (eds.). Thin Films of Soft Matter. Springer, Wien / New York, CISM 490, 2007.
Kargupta, K., Konnur, R., Sharma, A.. Instability and pattern formation in thin liquid films on chemically heterogeneous substrates . Langmuir, 16 (2000), 10243–10253.CrossRefGoogle Scholar
Kargupta, K., Sharma, A.. Templating of thin films induced by dewetting on patterned surfaces . Phys. Rev. Lett., 86 (2001), 4536–4539.CrossRefGoogle ScholarPubMed
Karim, A., Douglas, J.F., Lee, B.P., Glotzer, S.C., Rogers, J.A., Jackman, R.J., Amis, E.J., Whitesides, G.M.. Phase separation of ultrathin polymer-blend films on patterned substrates . Phys. Rev. E, 57 (1998), R6273–R6276.CrossRefGoogle Scholar
Kenzler, R., Eurich, F., Maass, P., Rinn, B., Schropp, J., Bohl, E., Dieterich, W.. Phase separation in confined geometries: Solving the Cahn-Hilliard equation with generic boundary conditions . Comp. Phys. Comm., 133 (2001), 139–157.CrossRefGoogle Scholar
Kerle, T., Klein, J., Yerushalmi-Rozen, R.. Accelerated rupture at the liquid/liquid interface . Langmuir, 18 (2002), 10146–10154.CrossRefGoogle Scholar
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.
Lowengrub, J., Truskinovsky, L.. Quasi-incompressible Cahn-Hilliard fluids and topological transitions . Proc. R. Soc. London Ser. A-Math. Phys. Eng. Sci., 454 (1998), 2617–2654.CrossRefGoogle Scholar
Madruga, S., Thiele, U.. Decomposition driven interface evolution for layers of binary mixtures: II. Influence of convective transport on linear stability . Phys. Fluids, 21 (2009), 062104.CrossRefGoogle Scholar
Mechkov, S., Rauscher, M., Dietrich, S.. Stability of liquid ridges on chemical micro- and nanostripes . Phys. Rev. E, 77 (2008), 061605.CrossRefGoogle ScholarPubMed
Müller-Buschbaum, P., Bauer, E., Pfister, S., Roth, S.V., Burghammer, M., Riekel, C., David, C., Thiele, U.. Creation of multi-scale stripe-like patterns in thin polymer blend films . Europhys. Lett., 73 (2006), 35–41.CrossRefGoogle Scholar
Nisato, G., Ermi, B.D., Douglas, J.F., Karim, A.. Excitation of surface deformation modes of a phase-separating polymer blend on a patterned substrate . Macromolecules, 32 (1999), 2356–2364.CrossRefGoogle Scholar
Oron, A., Davis, S.H., Bankoff, S.G.. Long-scale evolution of thin liquid films . Rev. Mod. Phys., 69 (1997), 931–980.CrossRefGoogle Scholar
Pismen, L.M.. Mesoscopic hydrodynamics of contact line motion . Colloid Surf. A-Physicochem. Eng. Asp., 206 (2002), 11–30.CrossRefGoogle Scholar
Pismen, L.M., Pomeau, Y.. Disjoining potential and spreading of thin liquid layers in the diffuse interface model coupled to hydrodynamics . Phys. Rev. E, 62 (2000), 2480–2492.CrossRefGoogle ScholarPubMed
Pototsky, A., Bestehorn, M., Merkt, D., Thiele, U.. Alternative pathways of dewetting for a thin liquid two-layer film . Phys. Rev. E, 70 (2004), 025201.CrossRefGoogle ScholarPubMed
Pototsky, A., Bestehorn, M., Merkt, D., Thiele, U.. Morphology changes in the evolution of liquid two-layer films . J. Chem. Phys., 122 (2005), 224711.CrossRefGoogle ScholarPubMed
Pototsky, A., Bestehorn, M., Merkt, D., Thiele, U.. 3D Surface Patterns in liquid two-layer films . Europhys. Lett., 74 (2006), 665–671.CrossRefGoogle Scholar
Thiele, U., Brusch, L., Bestehorn, M., Bär, M.. Modelling thin-film dewetting on structured substrates and templates: Bifurcation analysis and numerical simulations . Eur. Phys. J. E, 11 (2003), 255–271.CrossRefGoogle ScholarPubMed
Thiele, U., Madruga, S., Frastia, L.. Decomposition driven interface evolution for layers of binary mixtures: I. Model derivation and stratified base states . Phys. Fluids, 19 (2007), 122106.CrossRefGoogle Scholar
Vladimirova, N., Malagoli, A., Mauri, R.. Diffusion-driven phase separation of deeply quenched mixtures . Phys. Rev. E, 58 (1998), 7691–7699.CrossRefGoogle Scholar
Vladimirova, N., Malagoli, A., Mauri, R.. Two-dimensional model of phase segregation in liquid binary mixtures . Phys. Rev. E, 60 (1999), 6968–6977.CrossRefGoogle ScholarPubMed
Wang, H., Composto, R.J.. Thin film polymer blends undergoing phase separation and wetting: Identification of early, intermediate, and late stages . J. Chem. Phys., 113 (2000), 10386–10397.CrossRefGoogle Scholar
Wang, H., Composto, R.J.. Understanding morphology evolution and roughening in phase-separating thin-film polymer blends . Europhys. Lett., 50 (2000), 622–627.CrossRefGoogle Scholar