contact line dynamics

Transcription

1 contact line dynamics Jacco Snoeijer Physics of Fluids - University of Twente sliding drops flow near contact line

2 static contact line Ingbrigtsen & Toxvaerd (2007) γ γ sv θ e γ sl molecular scales macroscopic Young s law (1805): γ cos θ e = γ sv - γ sl

3 static vs dynamic receding moving contact lines advancing

4 static vs dynamic receding moving contact lines advancing hydrodynamic forces down to molecular scales!! multi-scale problem

5 outline part 1: basic ideas simple model for flow near the contact line singularity microscopic physics part 2: hydrodynamics dynamic contact angle? lubrication: Cox-Voinov theory forced wetting

6 outline part 1: basic ideas simple model for flow near the contact line singularity microscopic physics literature: Huh & Scriven, J. Colloid Interface Sci, 35, 85 (1971) Bonn, Eggers, Indekeu, Meunier, Rolley, to appear Rev. Mod. Phys. (2009)

7 corner flow Huh & Scriven 1971: - assume corner geometry -> straight interface - Stokes flow (no inertia) no shear stress no slip

8 corner flow Huh & Scriven 1971: - assume corner geometry -> straight interface - Stokes flow (no inertia) co-moving with contact line (receding)

9 corner flow Huh & Scriven 1971 streamfunction (2D, Stokes flow): " 2 (" 2 #) = 0 r φ

10 corner flow Huh & Scriven 1971 streamfunction (2D, Stokes flow): " 2 (" 2 #) = 0 r φ (note that for irrotational flow " 2 # = 0 )

11 corner flow Huh & Scriven 1971 streamfunction (2D): " 2 (" 2 #) = 0 r φ biharmonic equation

12 corner flow Huh & Scriven 1971 streamfunction: " = r( Asin# + Bcos# + C# sin# + D# cos# ) r φ constants A, B, C, D from boundary conditions

13 corner flow Huh & Scriven 1971 streamfunction: " = r( Asin# + Bcos# + C# sin# + D# cos# ) r φ θ = 120 θ = 60

14 corner flow Huh & Scriven 1971 streamfunction: " = r( Asin# + Bcos# + C# sin# + D# cos# ) r φ what happens as r 0? θ = 120 θ = 60

15 singularity at r=0 Huh & Scriven velocity at r = 0 multi-valued - infinite pressure and shear stress

16 exercise dimensional analysis: speed U position r viscosity η F shear

17 exercise dimensional analysis: speed U position r viscosity η 1. scaling shear stress τ with r? 2. total shear force F shear on plate? F shear F shear ~ dr"(r) x # r= 0

18 hydrodynamics fails when reaching molecular scales! Huh & Scriven 1971:

19 hydrodynamics fails when reaching molecular scales!

20 hydrodynamics fails when reaching molecular scales! many different theories to regularize singularity 1. slip boundary conditions 2. van der Waals forces 3. molecular kinetic theory 4....

21 1. slip length slip boundary condition: velocity at wall ~ shear stress u wall = l slip "u "z l slip

22 1. slip length slip boundary condition: velocity at wall ~ shear stress SFA, mechanical reponse (Cottin-Bizonne et al. PRL 2005)

23 1. slip length slip boundary condition: velocity at wall ~ shear stress l slip " ~ #U l slip

24 2. van der Waals forces introducing disjoining pressure: π(h) equilibrium shape: U=0

25 2. van der Waals forces introducing disjoining pressure: π(h) equilibrium shape: U=0 precursor film, molecular scale

26 2. van der Waals forces introducing disjoining pressure: π(h) precursor film, molecular scale dynamics: " ~ #U l film

27 1. & 2. are similar both provide regularization of hydrodynamic singularity: " ~ #U l micro slip precursor film

28 3. molecular kinetic theory thermally activated hopping of molecules # freq ~ exp " E & % ( $ k B T '

29 3. molecular kinetic theory thermally activated hopping of molecules f cl ~ γ(cosθ - cosθ e ) Blake & Haynes 1969: # freq ~ exp " E ± f 2 cll micro & % ( $ k B T '

30 3. molecular kinetic theory thermally activated hopping of molecules f cl ~ γ(cosθ - cosθ e ) forward/backward: " freq ~ exp ± f 2 cll micro % $ ' # k B T & contact line speed: " U ~ sinh f 2 cll micro % $ ' # k B T &

31 3. molecular kinetic theory liquid helium (Prevost et al. PRL 1999) speed " U ~ sinh f 2 cll micro % $ ' # k B T & F cl

32 3. molecular kinetic theory liquid helium (Prevost et al. PRL 1999) speed " U ~ sinh f 2 cll micro % $ ' # k B T & F cl - another source of dissipation - important when viscous dissipation is small

33 3. molecular kinetic theory drop coalesence (Andrieu et al. JFM 2002)

34 3. molecular kinetic theory drop coalesence (Andrieu et al. JFM 2002) observed relaxation timescale: t ~ R / (10-6 m/s) viscous time: t ~ R / (70 m/s)

35 4....

36 conclusion - moving contact line: gives divergence viscous stress - multi-scale: coupling molecular physics and macroscopic flow - many different mechanisms next lecture: hydrodynamics, above ~10nm