# 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

## Access Full Article

top## Abstract

top## How to cite

topFraštia, 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

top- D.M. Anderson, G.B. McFadden, A.A. Wheeler. Diffuse-Interface methods in fluid mechanics. Ann. Rev. Fluid Mech., 30 (1998), 139–165.
- L.K. Antanovskii. Microscale theory of surface tension. Phys. Rev. E, 54 (1996), 6285–6290.
- D. Bandyopadhyay, R. Gulabani, A. Sharma. Stability and dynamics of bilayers. Ind. Eng. Chem. Res., 44 (2005), 1259–1272.
- K.-J. Bathe. Finite element procedures. Prentice-Hall, New Jersey, 2nd edition, 1995.
- K. Binder. Spinodal decomposition in confined geometry. J. Non-Equilib. Thermodyn., 23 (1998), 1–44.
- L. Brusch, H. Kühne, U. Thiele, M. Bär. Dewetting of thin films on heterogeneous substrates: Pinning vs. coarsening. Phys. Rev. E, 66 (2002), 011602.
- J.W. Cahn, J.E. Hilliard. Free energy of a nonuniform System. 1. Interfacual free energy. J. Chem. Phys., 28 (1958), 258–267.
- H.P. Fischer, P. Maass, W. Dieterich. Novel surface modes in spinodal decomposition. Phys. Rev. Lett., 79 (1997), 893–896.
- 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.
- 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.
- 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.
- 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.
- M. Geoghegan, G. Krausch. Wetting at polymer surfaces and interfaces. Prog. Polym. Sci., 28 (2003), 261–302.
- 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.
- 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.
- 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.
- P.C. Hohenberg, B.I. Halperin. Theory of dynamic critical phenomena. Rev. Mod. Phys., 49 (1977), 435–479.
- 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.
- D. Jasnow, J. Viñals. Coarse-grained description of thermo-capillary flow. Phys. Fluids, 8 (1996), 660–669.
- 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.
- S. Kalliadasis, U. Thiele (eds.). Thin Films of Soft Matter. Springer, Wien / New York, CISM 490, 2007.
- K. Kargupta, R. Konnur, A. Sharma. Instability and pattern formation in thin liquid films on chemically heterogeneous substrates. Langmuir, 16 (2000), 10243–10253.
- K. Kargupta, A. Sharma. Templating of thin films induced by dewetting on patterned surfaces. Phys. Rev. Lett., 86 (2001), 4536–4539.
- 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.
- 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.
- T. Kerle, J. Klein, R. Yerushalmi-Rozen. Accelerated rupture at the liquid/liquid interface. Langmuir, 18 (2002), 10146–10154.
- 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.
- 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.
- 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.
- S. Mechkov, M. Rauscher, S. Dietrich. Stability of liquid ridges on chemical micro- and nanostripes. Phys. Rev. E, 77 (2008), 061605.
- 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.
- 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.
- A. Oron, S.H. Davis, S.G. Bankoff. Long-scale evolution of thin liquid films. Rev. Mod. Phys., 69 (1997), 931–980.
- L.M. Pismen. Mesoscopic hydrodynamics of contact line motion. Colloid Surf. A-Physicochem. Eng. Asp., 206 (2002), 11–30.
- 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.
- 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.
- A. Pototsky, M. Bestehorn, D. Merkt, U. Thiele. Morphology changes in the evolution of liquid two-layer films. J. Chem. Phys., 122 (2005), 224711.
- A. Pototsky, M. Bestehorn, D. Merkt, U. Thiele. 3D Surface Patterns in liquid two-layer films. Europhys. Lett., 74 (2006), 665–671.
- 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.
- 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.
- N. Vladimirova, A. Malagoli, R. Mauri. Diffusion-driven phase separation of deeply quenched mixtures. Phys. Rev. E, 58 (1998), 7691–7699.
- N. Vladimirova, A. Malagoli, R. Mauri. Two-dimensional model of phase segregation in liquid binary mixtures. Phys. Rev. E, 60 (1999), 6968–6977.
- 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.
- H. Wang, R.J. Composto. Understanding morphology evolution and roughening in phase-separating thin-film polymer blends. Europhys. Lett., 50 (2000), 622–627.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.