contact angle hysteresis
Jacco Snoeijer
Physics of Fluids - University of Twente
small droplets can ‘stick’ to window
force on contact line
balance:
γ cos θe
γsv
γsl
force on contact line
balance:
γ cos θe
γsv
γsl
γ cos θ
γsv
force on contact line:
γsl
fcl = γ (cos θ - cos θe)
advancing - receding angles
hysteresis:
no unique value for contact angle
pinning force: H = γ (cos θr - cos θa)
’sticky’ drops
microscopic origin
geometric heterogeneity
θe
θe
small scale ‘roughness’
microscopic origin
geometric heterogeneity
θe
chemical heterogeneity:
θe(x,y)
θe
small scale ‘roughness’
large θe
smaller θe
contact line shape?
contact line shape?
heterogeneity at larger scale: macroscopic ‘wetting defects’
single defect (500µm)
many defects (10µm)
contact line shape?
heterogeneity at larger scale: macroscopic ‘wetting defects’
single defect (500µm)
‘simple’ dynamics
many defects (10µm)
collective dynamics
contact line is very ‘rough’
what will we do:
• contact line deformation of single defect
• Joanny - De Gennes model for hysteresis
• implications for dynamics...
literature:
Joanny & De Gennes,
J. Chem. Phys. 81, 552 (1984)
Bonn, Eggers, Indekeu, Meunier, Rolley,
to appear Rev. Mod. Phys. (2009)
contact line shape
z
θ0
y
contact line shape
z
θ0
z
y
x
y
contact line shape
z
y
z
y
x
η(x)
contact line position
interface
z
y
interface h(x,y) ?
h(x, y) = " 0 y + ....
!
η(x)
contact line position
interface
Δh ~
e-qy
z
y
interface h(x,y) ?
h(x, y) = " 0 y + Ae iqxe#qy
!
η(x)
contact line position
interface
z
change in local contact angle:
Δθ(x)
y
η(x)
contact line position
interface
z
Δθ(x)
y
increase of area
Δ E ~ γ q A2
contact line ‘elasticity’
Joanny & De Gennes 1984
η(x)
contact line position
sinusoidal perturbations
increase of area:
ΔE ~ γ q
A2
z
Δθ(x)
y
contact line position
sinusoidal perturbations
increase of area:
ΔE ~ γ q
z
Δθ(x)
A2
y
relaxation rate
σ
Delon et al. JFM 2008
q
single defect
θ=0
wetting defect is ‘pulling’, but....
large deformations cost energy
single defect
θ=0
η(x) contact line shape ?
single defect
θ=0
η(x) contact line shape ?
h(x, y) = " 0 y + ...
!
single defect
θ=0
η(x) contact line shape ?
"0
h(x, y) = " 0 y #
2$
!
iqx #qy
˜
dq
%
(q)
e
e
&
single defect
θ=0
η(x) contact line shape ?
"0
h(x, y) = " 0 y #
2$
iqx #qy
˜
dq
%
(q)
e
e
&
imposing θ(x) at the contact line
!
η(x)
single defect
θ=0
η(x) contact line shape ?
"0
h(x, y) = " 0 y #
2$
$h
"0
"#
= "0 +
$y
2%
!
iqx #qy
˜
dq
%
(q)
e
e
&
iqx (qy
˜
dq
&
(q)
q
e
e
'
single defect
θ=0
η(x) contact line shape ?
"0
h(x, y) = " 0 y #
2$
"0
" (x) # " 0 +
2$
!
iqx #qy
˜
dq
%
(q)
e
e
&
iqx
˜
dq
%
(q)
q
e
{
}
&
single defect
θ=0
η(x)
Nadkarni & Garoff 1992
Fdefect
"(x) ~
ln( L / x )
#
!
Fdefect ~ d" (1# cos $ e )
model for hysteresis
advancing
receding
‘jump’
asymmetry between advancing and receding
(for defects that are strong enough)
model for hysteresis
advancing
‘jump’
non-interacting defects,
density ρ
" (cos# r $ cos# a ) ~ % Fd &max
!
receding
contact line dynamics, revisited
many defects (10µm)
contact line dynamics, revisited
many defects (10µm)
if we go very slowly...
... thermally activated ‘hopping’ of molecules
contact line dynamics, revisited
thermally activated ‘hopping’ of molecules
Fcl ~ γ(cosθ - cosθe)
Rolley & Guthmann PRL 2007
speed
2
# E " f cl lmicro
&
U ~ exp%"
(
k
T
$
'
B
Fcl
!
contact line dynamics, revisited
thermally activated ‘depinning’ of contact line
Rolley & Guthmann PRL 2007
speed
2
# E " f cl lmicro
&
U ~ exp%"
(
k
T
$
'
B
Fcl
!
coupling to hydrodynamics?
for significant Ca: lubrication + slip
microscopics shows up as ‘boundary conditions’
conclusion
- microscopic description of hysteresis
- ’interacting defects’ are difficult...
depinning, contact line roughness
- coupling to hydrodynamics??