Hydrodynamics of wetting phenomena Jacco Snoeijer PHYSICS OF FLUIDS
Outline 1. Creeping flow: hydrodynamics at low Reynolds numbers (2 hrs) 2. Thin films and lubrication flows (3 hrs + problem session 1.5 hrs) 3. Static and moving contact lines (3 hrs) A. Statics (and a bit on soft substrates) B. Dynamics
to splash or not to splash Duez, Ybert, Clanet & Bocquet, Nature Physics 2007
to splash or not to splash Duez, Ybert, Clanet & Bocquet, Nature Physics 2007
to splash or not to splash Duez, Ybert, Clanet & Bocquet, Nature Physics 2007
multi-scale macroscopic Snoeijer & Andreotti, Annual Review of Fluid Mechanics 2013
multi-scale macroscopic mesoscopic (hydro) Snoeijer & Andreotti, Annual Review of Fluid Mechanics 2013
multi-scale macroscopic mesoscopic (hydro) microscopic Snoeijer & Andreotti, Annual Review of Fluid Mechanics 2013
corner flow Huh & Scriven 1971: - assume corner geometry -> straight interface - Stokes flow (no inertia, Re=0) no shear stress no slip
corner flow Huh & Scriven 1971: - assume corner geometry -> straight interface - Stokes flow (no inertia, Re=0) co-moving with contact line (receding)
corner flow Huh & Scriven 1971 streamfunction (2D, Stokes flow): 2 ( 2 ψ) = 0 r φ
corner flow Huh & Scriven 1971 streamfunction: ψ = r( Asinφ + Bcosφ + Cφ sinφ + Dφ cosφ) r φ constants A, B, C, D from boundary conditions
corner flow Huh & Scriven 1971 streamfunction: ψ = r( Asinφ + Bcosφ + Cφ sinφ + Dφ cosφ) r φ θ = 120 θ = 60
corner flow Huh & Scriven 1971 streamfunction: ψ = r( Asinφ + Bcosφ + Cφ sinφ + Dφ cosφ) r φ what happens as r 0? θ = 120 θ = 60
singularity at r=0 Huh & Scriven 1971 - velocity at r = 0 multi-valued - infinite pressure and shear stress
scaling dimensional analysis: speed U position r viscosity η F shear
scaling 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 ~ drτ(r) x r= 0
hydrodynamics fails...... when reaching molecular scales! Huh & Scriven 1971:
hydrodynamics fails...... when reaching molecular scales!
hydrodynamics fails...... when reaching molecular scales! many different theories to regularize singularity 1. slip boundary conditions 2. molecular kinetic theory 3....
1. slip length slip boundary condition: velocity at wall ~ shear stress u wall = l slip u z l slip
1. slip length slip boundary condition: velocity at wall ~ shear stress SFA, mechanical reponse (Cottin-Bizonne et al. PRL 2005)
1. slip length slip boundary condition: velocity at wall ~ shear stress l slip τ ~ ηu l slip
2. molecular kinetic theory thermally activated hopping of molecules freq ~ exp E k B T
2. 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
2. molecular kinetic theory thermally activated hopping of molecules f cl ~γ(cosθ - cosθ e ) forward/backward: freq ~ exp ± f l 2 cl micro k B T contact line speed: U ~ sinh f l 2 cl micro k B T
2. molecular kinetic theory thermally activated hopping of molecules f cl ~γ(cosθ - cosθ e ) - another source of dissipation - important when viscous dissipation is small (close to pinning-depinning)
intermediate conclusion - moving contact line: divergence viscous stress - multi-scale: coupling molecular physics and macroscopic flow - hydrodynamics, above ~ 10nm: dynamic contact angle & Cox-Voinov
corner shape? is the interface shape really a simple corner?
corner shape? dimensional analysis: speed U position r viscosity η pressure p, same dimension as shear stress τ p ~ 1/ r
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
interface shape corner is not a solution, interface curved!! θ (x)
interface shape Voinov 1976, Cox 1986: θ 3 = θ e 3 9Caln(x /l micro ) corner is not a solution, interface curved!! θ (x)
interface shape Voinov 1976, Cox 1986: θ 3 = θ e 3 9Caln(x /l micro ) - interface is curved θ (x) - angle variation ~ Ca - singularity: ln(x)
Cox-Voinov using lubrication theory, let us derive: θ 3 = θ e 3 9Caln(x /l micro )
lubrication equation h << 10-3 m h >> 10-9 m balance between viscosity η and surface tension γ Ca = Uη/γ
lubrication equation h << 10-3 m h >> 10-9 m h'''= 3Ca h 2 receding h'''= 3Ca h 2 advancing
lubrication equation h'''= 3Ca h 2 analytical solution due to Duffy & Wilson 1997:
lubrication equation h'''= 3Ca h 2 analytical solution due to Duffy & Wilson 1997: h(x) in implicit form: x(s), h(s)
lubrication equation h'''= 3Ca h 2 take a deep breath...
lubrication equation h'''= 3Ca h 2 first, verify corner solution for h(x) does not exist
lubrication equation h'''= 3Ca h 2 first, verify corner solution for h(x) does not exist h(x) = xθ h''= h'''= 0 not a solution!!
varying θ(x) h'''= 3Ca h 2 try solution with slowly varying θ(x) h(x) = xθ(x)
varying θ(x) h'''= 3Ca h 2 try solution with slowly varying θ(x) h(x) = xθ(x) h'= θ + xθ'(x)
varying θ(x) h'''= 3Ca h 2 try solution with slowly varying θ(x) h(x) = xθ(x) h'= θ + xθ'(x) h''= θ' h'''= θ''
Cox-Voinov θ 2 θ''= 3Ca x 2 approximate solution: Cox-Voinov law θ 3 (x) = θ e 3 9Caln(x /l micro )
multi-scale macroscopic Snoeijer & Andreotti, Annual Review of Fluid Mechanics 2013
multi-scale macroscopic mesoscopic (hydro) Snoeijer & Andreotti, Annual Review of Fluid Mechanics 2013
multi-scale macroscopic mesoscopic (hydro) microscopic Snoeijer & Andreotti, Annual Review of Fluid Mechanics 2013
scale dependence of angle Rame & Garoff 1996
dynamic contact angle? Le Grand et al. J. Fluid Mech. 2005
dynamic contact angle? Le Grand et al. J. Fluid Mech. 2005
dynamic contact angle Le Grand et al. J. Fluid Mech. 2005 θ ( ) Ca = Uη/γ
dynamic contact angle Le Grand et al. J. Fluid Mech. 2005 θ ( ) but attention: θ(x) depends on scale of measurement Ca = Uη/γ
application (i): spreading drop perfectly wetting θ e =0 : R(t)??
application (i): spreading drop perfectly wetting θ e =0 : outer solution: spherical cap θ app 0 R(t)??
application (i): spreading drop perfectly wetting θ e =0 : outer solution: spherical cap θ app 0 R(t)?? inner solution θ(x) 3 = 9Ca ln(x /l micro )
application (i): spreading drop perfectly wetting θ e =0 : outer solution: spherical cap θ app 0 R(t)?? inner solution θ(x) 3 = 9Ca ln(x /l micro ) see: Bonn, Eggers, Indekeu, Meunier, Rolley, Rev. Mod. Phys. (2009) 3 θ app matching: R = 9Ca ln 2e 2 l micro
application (i): spreading drop perfectly wetting θ e =0 : θ app 0 R(t) dynamics: geometry: dr dt ~ θ 3 θ ~ Vol R 3
application (i): spreading drop perfectly wetting θ e =0 : θ app 0 R(t) dynamics: geometry: dr dt ~ θ 3 θ ~ Vol R 3 what is R(t)?
application (i): spreading drop perfectly wetting θ e =0 : θ app 0 R(t) dynamics: geometry: dr dt ~ θ 3 θ ~ Vol R 3 Tanner s law: R(t) ~ t 1/10
application (ii): dewetting
application (ii): dewetting
application (ii): dewetting
application (ii): dewetting
conclusion contact line dynamics: multi-scale (micro to macro) interface curved! (dynamic contact angle?) Cox-Voinov type models effective description for spreading and wetting dynamics Snoeijer & Andreotti, Annual Review of Fluid Mechanics 2013
Hydro & Mouillage 1. Creeping flow 2. Thin films and lubrication 3. Static & moving contact lines Merci!