Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-19T05:44:20.005Z Has data issue: false hasContentIssue false
  • English
  • Français

A Discrete Model For Pattern Formation In Volatile ThinFilms

Published online by Cambridge University Press:  09 July 2012

M. Malik-Garbi  and
O. Agam
Get access

Abstract

We introduce a model, similar to diffusion limited aggregation (DLA), which serves as adiscrete analog of the continuous dynamics of evaporation of thin liquid films. Withinmean field approximation the dynamics of this model, averaged over many realizations ofthe growing cluster, reduces to that of the idealized evaporation model in which surfacetension is neglected. However fluctuations beyond the mean field level play an importantrole, and we study their effect on the conserved quantities of the idealized evaporationmodel. Assuming the cluster to be a fractal, a heuristic approach is developed in order toexplain the distinctive increase of the fractal dimension with the cluster size.

Keywords

viscous fingeringdiffusion limited aggregationreaction-diffusion
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 7 , Issue 4: Modelling phenomena on micro- and nanoscale , 2012 , pp. 39 - 52
Copyright
© EDP Sciences, 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Elbaum, M., Lipson, S.G.. How does a thin wetted film dry up? Phys. Rev. Lett., 72 (1994), 3562–3565. CrossRefGoogle ScholarPubMed
Lipson, S.G.. Pattern formation in drying water films. Physica Scripta, T67 (1996), 63–66. CrossRefGoogle Scholar
Leizerson, I., Lipson, S.G., Lyushnin, A.V.. Finger instability in wetting-dewetting phenomena. Langmuir, 20 (2004), 291–194. CrossRefGoogle ScholarPubMed
Lipson, S. G.. A thickness transition in evaporating water films. Phase Transitions, 77 (2004), 677–688. CrossRefGoogle Scholar
Leizerson, I., Lipson, S.G.. How does a thin volatile film move? Langmuir, 20 (2004), 8423–8425. CrossRefGoogle ScholarPubMed
Samid-Merzel, N., Lipson, S.G., Tannhauser, D.S.. Pattern formation in drying water films. Phys. Rev., E 57 (1998), 2906–2913. Google Scholar
Taylor, G., Saffman, P.G.. A note on the motion of bubbles in a Hele-Shaw cell and porous medium. Quart. J. Mech. Appl. Math., 12 (1959), 265–279. CrossRefGoogle Scholar
Witten, T.A., Sander, L.M.. Diffusin-limited aggragation. Phys. Rev., B 27 (1983), 5686–5697. CrossRefGoogle Scholar
Mathiesen, J., Procaccia, I., Swinney, H. L., Thrasher, M.. The universality class of diffusion-limited aggregation and viscous-limited aggregation. Europhys. Lett., 76 (2006) No. 2, 257–263. CrossRefGoogle Scholar
Arneodo, A., Couder, Y., Grasseau, G., Hakim, V., Rabaud, M.. Uncovering the analytical Saffman-Taylor finger in unstable viscous fingering and diffusion-limited aggregation. Phys. Rev. Lett., 63 (1989), 984–987. CrossRefGoogle ScholarPubMed
Arneodo, A., Elezgaray, J., Tabard, M., Tallet, F.. Statistical analysis of off-lattice diffusion-limited aggregates in channeland sector geometries. Phys. Rev., E 53 (1996), 6200–6223. Google Scholar
Somfai, E., Ball, R.C., DeVita, J.P., Sander, L.M.. Diffusion-limited aggregation in channel geometry. Phys. Rev., E 68 (2003), 020401 Google Scholar
Hastings, M.B., Levitov, L.S.. Laplacian growth as one-dimensional turbulence. Physica, D 116 (1998), 244–252. CrossRefGoogle Scholar
Agam, O.. Viscous fingering in volatile thin films. Phys. Rev., E 79 (2009), 021603. Google Scholar
Diamant, H., Agam, O.. Localized Rayleigh instability in evaporation fronts. Phys. Rev. Lett., 104 (2010), 047801. CrossRefGoogle ScholarPubMed
Entov, V.M., Étingof, P.I.. Some exact sdolutions of the thin-sheet stamping problem. Fluid Dyn., 27 (1992), 169–176. CrossRefGoogle Scholar
Doi, M.. 2nd quantization representation for classical many-particle system. J. Phys., A 9 (1976), 1465–1477. Google Scholar
Peliti, L.. Path integral approach to birth-death processes on a lattice. J. Physique, 46 (1985), 1469–1483. CrossRefGoogle Scholar
Lee, B. P.. Renormalization-group calculation for the reaction kA → ∅. Phys., A 27 (1994), 2633–2652. CrossRefGoogle Scholar
Cardy, J., Tauber, U. C.. Theory of branching and annihilating random walks Phys. Rev Lett., 77 (1996), 4780–4783. CrossRefGoogle ScholarPubMed
Here the Hamiltonian which defines the evolution does not account for the constraint that A particle cannot be born on a site ocuupied by B particle. This constraint can be taken into account by replcing the term with , where Θ(x) is the haviside function and ϵ is a positive infintesimal number.