Summer School
"Mathematical Approaches to Complex Fluids"
Cambridge, July 2013
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Uwe Thiele
Gradient dynamics formulations of thin film equations
and
Depinning transitions and deposition patterns
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Extended abstract and references
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The short course consists of 3x45min lectures. It starts with
a brief review of a number of experiments on dewetting and evaporating
thin films/drops of simple and complex liquids including suspensions
and solutions. Then the remainder of the first lecture introduces the
concept of a gradient dynamics description of the evolution of
interface-dominated films and drops on solid substrates (building on
knowledge about the equilibrium behaviour of such systems gained in
Len Pismen's lecture). We start with a formulation for a single layer
of simple non-volatile liquid, and continue with a formulation for a
two-layer film. To be able to discuss mixtures we then reformulate the
diffusion equation as a gradient dynamics and combine the
obtained elements in a gradient dynamics formulation for films of
mixtures. This will bring us well into the second lecture.
The second part of the course uses the obtained models to investigate
the depinning transitions and deposition patterns in a number of
different settings that can all be described by the introduced
evolution equations. This part starts by reviewing the techniques of
linear stability analysis, numerical continuation of steady states and
time simulation. It is explained how their combination allows one to
gain a rather complete understanding of the behaviour of a system. The
case of dewetting of a film of simple liquid serves as an example
(instability-dominated vs nucleation-dominated dewetting).
In a depinning transitions a steady structure transforms into a moving
one when a driving force passes a critical value. This results in
qualitative changes in the transport behaviour. Drops pinned by
substrate heterogeneities are a common example. They begin to slide at
a critical driving force along the substrate. A similar mechanism may
depin droplets of partially wetting liquid on a rotating
cylinder. There, gravity takes the role of the heterogeneity and the
rotation corresponds to the lateral driving. After establishing the
parallels in the underlying film profile evolution equations we
discuss the bifurcation behaviour. Next, we focus on the deposition of
line patterns at (i) at receding three-phase contact lines of
evaporating suspensions and (if time permits) in the Langmuir-Blodgett
transfer of surfactant monolayers onto a moving plate. In passing we
explain why the onset of the deposition of line patterns can be seen
as a depinning transition what allows us to understand why all the
discussed transitions are similar.
Note that all systems are only discussed in the case of one substrate
dimension. Some results for two substrate dimensions may be found
of the given literature.
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References (the ones that I co-authored can be downloaded from
www.uwethiele.de/publ.html)
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Long-wave equations
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A. Oron, S. H. Davis and S. G. Bankoff. Rev. Mod. Phys. 69, 931--980 (1997).
U Thiele, in Thin films of soft matter, eds. S. Kalliadasis and U. Thiele, Springer Wien, p.25-94 (2007).
Gradient dynamics form
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V.S. Mitlin, J. Colloid Interface Sci. 156, 491-497 (1993).
U. Thiele, J. Phys.-Cond. Mat. 22, 084019 (2010).
U. Thiele, Eur. Phys. J. Special Topics, 197, 213-220 (2011).
U. Thiele, A. J. Archer and M. Plapp, Phys. Fluids 24, 102107 (2012).
Dewetting
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U. Thiele, Eur. Phys. J. E 12, 409-416 (2003).
A Pototsky, M Bestehorn, D Merkt and U Thiele, Phys. Rev. E 70, 025201(R) (2004).
R. Seemann et al.. J. Phys.-Condes. Matter 17, S267-S290 (2005).
D. Bonn, J. Eggers, J. Indekeu, J. Meunier and E. Rolley. Rev. Mod. Phys. 81, 739-805 (2009).
Depinning transitions
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U. Thiele and E. Knobloch, Phys. Rev. Lett. 97, 204501 (2006); New J. Phys. 8, 313 (2006).
P. Beltrame, E. Knobloch, P. Hanggi and U. Thiele, Phys. Rev. E 83, 016305 (2011);
U. Thiele, J. Fluid Mech 671, 121-136 (2011).
Deposition patterns
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L. Frastia, A. J. Archer, U. Thiele, Phys. Rev. Lett. 106, 077801 (2011); Soft Matter 8, 11363-11386 (2012).
M. H. Kopf, S. V. Gurevich, R. Friedrich and U. Thiele, New J. Phys. 14, 023016 (2012).
W, Han and Z, Lin, Angew. Chem. Int. Ed. 51:1534-1546, (2012).
U. Thiele, http://arxiv.org/abs/1307.0958
General
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P.-G. de Gennes. Rev. Mod. Phys. 57, 827--863 (1985).
S. H. Strogatz. Nonlinear Dynamics and Chaos. Addison-Wesley (1994).