Summer School "Mathematical Approaches to Complex Fluids" Cambridge, July 2013 ---------------------------------------- Uwe Thiele Gradient dynamics formulations of thin film equations and Depinning transitions and deposition patterns ---------------------------------------- Extended abstract and references -------------------------- 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. -------------------------------------------------- References (the ones that I co-authored can be downloaded from www.uwethiele.de/publ.html) -------------------------------------------------- Long-wave equations ----------------- 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 ------------------- 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 -------- 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 ----------------- 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 ---------------- 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 ------ P.-G. de Gennes. Rev. Mod. Phys. 57, 827--863 (1985). S. H. Strogatz. Nonlinear Dynamics and Chaos. Addison-Wesley (1994).