contact line dynamics

Transcription

1 contact line dynamics part 2: hydrodynamics dynamic contact angle? lubrication: Cox-Voinov theory maximum speed for instability

2 corner shape? dimensional analysis: speed U position r viscosity η pressure p, same dimension as shear stress τ p ~ 1/ r

3 corner shape? dimensional analysis: speed U position r viscosity η pressure p, same dimension as shear stress τ p ~ 1/ r corner inconsistent with Laplace pressure: p = p 0 - γh

4 interface shape corner is not a solution, interface curved!! θ (x)

5 interface shape Voinov 1976, Cox 1986: " 3 = " e 3 # 9Caln(x /l micro ) corner is not a solution, interface curved!! θ (x)

6 interface shape Voinov 1976, Cox 1986: " 3 = " e 3 # 9Caln(x /l micro ) - interface is curved θ (x) - angle variation ~ Ca - singularity: ln(x)

7 dynamic contact angle? Le Grand et al. J. Fluid Mech. 2005

8 dynamic contact angle? Le Grand et al. J. Fluid Mech. 2005

9 dynamic contact angle Le Grand et al. J. Fluid Mech θ ( ) Ca = Uη/γ

10 dynamic contact angle Le Grand et al. J. Fluid Mech θ ( ) but attention: θ(x) depends on scale of measurement Ca = Uη/γ

11 scale dependence Marsh et al. Phys. Rev. Lett. 1993

12 scale dependence Marsh et al. Phys. Rev. Lett θ x (µm)

13 Cox-Voinov using lubrication theory, let us derive: " 3 = " e 3 # 9Caln(x /l micro )

14 lubrication equation h << 10-3 m h >> 10-9 m balance between viscosity η and surface tension γ Ca = Uη/γ

15 lubrication equation h << 10-3 m h >> 10-9 m h'''= 3Ca h 2 receding h'''= " 3Ca h 2 advancing

16 lubrication equation h'''= 3Ca h 2 analytical solution due to Duffy & Wilson 1997:

17 lubrication equation h'''= 3Ca h 2 analytical solution due to Duffy & Wilson 1997: h(x) in implicit form: x(s), h(s)

18 lubrication equation h'''= 3Ca h 2 take a deep breath...

19 lubrication equation h'''= 3Ca h 2 first, verify corner solution for h(x) does not exist

20 lubrication equation h'''= 3Ca h 2 first, verify corner solution for h(x) does not exist h(x) = x" h''= h'''= 0 not a solution!!

21 varying θ(x) h'''= 3Ca h 2 try solution with slowly varying θ(x) h(x) = x"(x)

22 varying θ(x) h'''= 3Ca h 2 try solution with slowly varying θ(x) h(x) = x"(x) h'= " + x"'(x)

23 varying θ(x) h'''= 3Ca h 2 try solution with slowly varying θ(x) h(x) = x"(x) h'= " + x"'(x) h''= "' h'''= "''

24 Cox-Voinov " 2 "''= 3Ca x 2 approximate solution: Cox-Voinov law " 3 (x) = " e 3 # 9Caln(x /l micro )

25 application (i): spreading drop perfectly wetting θ e =0 : R(t)??

26 application (i): spreading drop perfectly wetting θ e =0 : outer solution: spherical cap θ app 0 R(t)??

27 application (i): spreading drop perfectly wetting θ e =0 : outer solution: spherical cap θ app 0 R(t)?? inner solution "(x) 3 = 9Ca ln(x /l micro )

28 application (i): spreading drop perfectly wetting θ e =0 : outer solution: spherical cap θ app 0 R(t)?? inner solution "(x) 3 = 9Ca ln(x /l micro ) see: Bonn, Eggers, Indekeu, Meunier, Rolley, to appear Rev. Mod. Phys. (2009) 3 " app matching: # R & = 9Ca ln% ( $ 2e 2 l micro '

29 application (i): spreading drop perfectly wetting θ e =0 : θ app 0 R(t) dynamics: geometry: dr dt ~ " 3 " ~ Vol R 3

30 application (i): spreading drop perfectly wetting θ e =0 : θ app 0 R(t) dynamics: geometry: dr dt ~ " 3 " ~ Vol R 3 exercise: what is R(t)?

31 application (i): spreading drop perfectly wetting θ e =0 : θ app 0 R(t) dynamics: geometry: dr dt ~ " 3 " ~ Vol R 3 Tanner s law: R(t) ~ t 1/10

32 application (ii): forced wetting low Ca high Ca film air liquid

33 application (ii): forced wetting low Ca high Ca film air liquid what is critical Ca?

34 contact line stability low speeds: contact line equilibrates at height z cl z cl Ca (lubrication + slip length to treat contact line)

35 contact line stability low speeds: contact line equilibrates at height z cl z cl z cl Ca Ca Snoeijer, Delon, Fermigier, Andreotti, J. Fluid Mech. 2007

36 evolution towards film dimple Snoeijer, Delon, Fermigier, Andreotti, J. Fluid Mech. 2007

37 evolution towards film z cl t U p -U cl Snoeijer, Delon, Fermigier, Andreotti, J. Fluid Mech. 2007

38 apparent contact angle maximum Ca achieved at z cl = z cl Ca

39 apparent contact angle maximum Ca achieved at z cl = z cl static bath: Ca z cl = 2(1" sin#)

40 apparent contact angle maximum Ca achieved at z cl = z cl θ app 0 static bath: Ca z cl = 2(1" sin#)

41 application (ii): forced wetting Ca θ app =0 Matching static bath to Duffy & Wilson solution Eggers, Phys. Rev. Lett Ca

42 application (ii): forced wetting Ca θ app =0 Cox-Voinov: does not describe Ca c!! Ca (due to approximations)

43 inclined plate Ca c Ca Ca c depends on inclination angle: not universal! α / θ eq Ziegler, Snoeijer, Eggers, EPJST 2009

44 drops sliding on incline U partial wetting θ e = 45

45 drops sliding on incline increasing speed Ca=ηU/γ U partial wetting θ e = 45 Podgorski et al. PRL 01 Le Grand et al. JFM 05

46 drops sliding on incline Podgorski et al. PRL 01 U some basic questions: critical speed? 3D hydrodynamics in corner? how sharp is the tip-singularity?

47 corner geometry Φ Ω geometry: h(x,y) «Cone-like»

48 corner geometry force balance: viscosity vs surface tension y φ Φ Ω high pressure low pressure x 2D cone model experiment: tracer particles Limat & Stone, Europhys. Lett. 04 Snoeijer, Rio, Le Grand, Limat, Phys. Fluids 05

49 tip? how sharp is the corner tip?

50 tip? small but finite radius of curvature R

51 recent measurements Peters, Snoeijer, Daerr, Limat, to be submitted 1/R Ca

52 distance << R: straight c.l. theory for tip

53 theory for tip distance << R: straight c.l. h'''= 3Ca h 2 $ h' 3 = " 3 e # 9Ca ln& % x l micro Cox-Voinov : divergence small length ' ) (

54 theory for tip y distance ~ R: two principal curvatures "'= 3Ca h 2 " # h''$ 2h w 2 x

55 theory for tip Peters, Snoeijer, Daerr, Limat, to be submitted 1/R Cox-Voinov like result: Ca % R ( " 3 = # 3 e $ 9Ca ln' * & lmicro ) R " l micro e # e 3 / 9Ca

56 theory for tip 1/R best fit: l micro = 8 nm Cox-Voinov like result: % R ( " 3 = # 3 e $ 9Ca ln' * & lmicro ) R " l micro e # e 3 / 9Ca 1/Ca

57 conclusion contact line dynamics: multi-scale (micro to macro) interface curved Cox-Voinov critical speed of wetting is beyond Cox-Voinov

58 open questions... rapidly advancing contact line: - 2 phase flow problem - inertial effects?

59 open questions... air entrainment U liquid rapidly advancing contact line: - 2 phase flow problem - inertial effects?