Jacco Snoeijer

Physics of Fluids - University of Twente

contact line dynamics

flow near contact linesliding drops

static contact line

molecular scales macroscopic

γsv

γ

γ cos θe = γsv - γslYoung’s law (1805):

θe γsl

Ingbrigtsen & Toxvaerd (2007)

static vs dynamic

receding

advancing

moving contact lines

static vs dynamic

receding

advancing

moving contact lines

hydrodynamic forces down to molecular scales !!

‘multi-scale problem’

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

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)

corner flow

Huh & Scriven 1971:

- assume corner geometry -> straight interface- Stokes flow (no inertia)

no slip

no shear s

tress

corner flow

Huh & Scriven 1971:

- assume corner geometry -> straight interface- Stokes flow (no inertia)

co-moving with contact line (receding)

corner flow

Huh & Scriven 1971

φr

!

"2("2#) = 0

streamfunction (2D, Stokes flow):

corner flow

Huh & Scriven 1971

φr

!

"2("2#) = 0

streamfunction (2D, Stokes flow):

(note that for irrotational flow )

!

"2# = 0

corner flow

Huh & Scriven 1971

φr

!

"2("2#) = 0

streamfunction (2D):

biharmonic equation

corner flow

Huh & Scriven 1971

φr

!

" = r Asin# + Bcos# + C# sin# + D# cos#( )

streamfunction:

constants A, B, C, D from boundary conditions

corner flow

Huh & Scriven 1971

φr

!

" = r Asin# + Bcos# + C# sin# + D# cos#( )

streamfunction:

θ = 120° θ = 60°

corner flow

Huh & Scriven 1971

φr

!

" = r Asin# + Bcos# + C# sin# + D# cos#( )

streamfunction:

θ = 120° θ = 60°

what happens as r 0 ?

singularity at r=0Huh & Scriven 1971

- velocity at r = 0 multi-valued- infinite pressure and shear stress

exercise

dimensional analysis: speed Uposition r viscosity η

Fshear

exercise

dimensional analysis: speed Uposition r viscosity η

1. scaling shear stress τ with r ?

2. total shear force Fshear on plate?

Fshear

!

Fshear

~ dr" (r)r= 0

x

#

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 conditions2. van der Waals forces3. molecular kinetic theory4. ...

1. slip length

slip boundary condition:

velocity at wall ~ shear stress

!

uwall = lslip"u

"z

!

lslip

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

!

" ~#U

lslip

!

lslip

2. van der Waals forces

introducing disjoining pressure: π(h)

equilibrium shape: U=0

2. van der Waals forces

introducing disjoining pressure: π(h)

equilibrium shape: U=0

precursor film, molecular scale

2. van der Waals forces

introducing disjoining pressure: π(h)

dynamics:

precursor film, molecular scale

!

" ~#U

lfilm

1. & 2. are ’similar’

both provide regularization of hydrodynamic singularity:

slip

!

" ~#U

lmicro

precursor film

3. molecular kinetic theorythermally activated ‘hopping’ of molecules

!

freq ~ exp "E

kBT

#

$ %

&

' (

3. molecular kinetic theorythermally activated ‘hopping’ of molecules

fcl ~ γ(cosθ - cosθe)

Blake & Haynes 1969:

!

freq ~ exp "E ± fcl lmicro

2

kBT

#

$ %

&

' (

3. molecular kinetic theorythermally activated ‘hopping’ of molecules

forward/backward:

!

freq ~ exp ±fcl lmicro

2

kBT

"

# $

%

& '

fcl ~ γ(cosθ - cosθe)

contact line speed:

!

U ~ sinhfcl lmicro

2

kBT

"

# $

%

& '

3. molecular kinetic theoryliquid helium (Prevost et al. PRL 1999)

Fcl

spee

d

!

U ~ sinhfcl lmicro

2

kBT

"

# $

%

& '

3. molecular kinetic theoryliquid helium (Prevost et al. PRL 1999)

- another source of dissipation- important when viscous dissipation is small

Fcl

spee

d

!

U ~ sinhfcl lmicro

2

kBT

"

# $

%

& '

3. molecular kinetic theorydrop coalesence (Andrieu et al. JFM 2002)

3. molecular kinetic theorydrop coalesence (Andrieu et al. JFM 2002)

observed relaxation timescale: t ~ R / (10-6 m/s)

viscous time: t ~ R / (70 m/s)

4. ...

conclusion

- moving contact line: gives ’divergence’ viscous stress

- multi-scale: coupling molecular physics and macroscopic flow

- many different mechanisms

next lecture: hydrodynamics, above ~10nm