Dynamique d`une ligne de contact sous évaporation

GDR Films cisaillés
08 mars 2012
Dynamique d’une ligne de contact sous évaporation
Chi-Tuong PHAM 1
avec
G. BERTELOOT 2 ,
1
2
A. DAERR 2 ,
F. LEQUEUX 3
&
Laboratoire LIMSI — Université Paris-Sud 11 — France
Laboratoire MSC — Université Paris Diderot - Paris 7 — France
3
Laboratoire PPMD — ESPCI & UPMC Paris 6 — France
L. LIMAT 2
Ligne de contact en mouvement
V
Air
Eau
h(x)
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x
• Divergence des contraintes visqueuses à la ligne de contact :
V
∂U
∼η
−→ ∞
σ=η
∂z
h(x)
pour
h(x) −→ 0
(ligne de contact)
Modèle de Cox-Voinov :
θ 3 − θS3 = −9 Ca ln
L
a
avec
• En avancée (V < 0) :
θ > θS
• En reculée (V > 0) :
θ < θS
Ca = ηV /γ
(nombre capillaire)
Évaporation à la ligne de contact
• Deegan et al. (Nature 1997) : divergence du flux évaporatif près de la ligne de contact ⇐⇒
effet de pointe
− π/2−θ
π−θ
J(x) ∼ x
avec
J0 =
D csat
√g w ;
λ ρw
ρw = densité ;
≈ J0 x−1/2
Dg = constante de diffusion ; csat
w = concentration massique ;
λ = longueur caractéristique.
• Ordre de grandeur :
−→ Effet tache de café :
J0 ∼ 10−9 m3/2 s−1
Context
• We study the problem of a volatile liquid moving at speed V on a partially or totally wetting
substrate
Air
?
Θ
Liquid
h(x)
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x
V
Question: What is the effect of evaporation on the dynamics of a contact line?
Model for partial wetting
J(x) ≈
Hypothesis: No Marangoni effect, thermal gradients are neglected.
Θ
<U>
h(x)
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x
V
• Mass conservation:
∂t h+∂x [hhU i]+J(x) = 0
• Navier-Stokes + lubrication approximation:
Z
1 h
h2 ∂P
hU i =
U (x, z) dz = −
h 0
3η ∂x
hxxx
ηV
Ca =
γ
J0
√
x
with
and
h(x, t) = h(x−V t)
P = −γhxx = capillary pressure
√
3Ca 6ηJ0 x
= 2 −
h
γ h3
J0 = evaporation constant
Ansatz:
x
24ηJ0
θ (x) ≃
+
− 9Ca ln
a
γθe
{z
}
|
3
θe3
1
a
1
2
−
1
1
x2
Voinov
26900
3
Numerical simulations:
Θ (X)
27000
Analytical ansatz
26800
Voinov + evaporation
Voinov
26700
1e-6
−→ Modification of angle:
1e-3
1
X ([a. u.])
∆θ/θ ≃ 15%
• References: Berteloot et al., Europhys. Lett. 2008, 83, 14003.
1e+3
1e+6
Case of complete wetting
Air
• Navier-Stokes + lubrication approximation:
Z
1 h
h2 ∂P
hU i =
U (x, z) dz = −
h 0
3η ∂x
?
Liquid
Θ
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P = Pcapillary + Pdisjunction
• Mass conservation:
hxxx
Ca =
ηV
γ
h(x, t) = h(x − V t)
√
3Ca 6ηJ0 x
A hx
−
= 2 −
3
h
γ h
2πγ h4
J0 = evaporation constant
• Typical scalings:
2/3
|A|
x0 =
12πJ0 η
• Orders of magnitude:
and
x0 ∼ 2 µm
x
V
Hypothesis: No Marangoni effect, thermal gradients are neglected.
∂t h + ∂x [hhU i] + J(x) = 0
h(x)
A = Hamaker constant
1/2
h0 = x0
h0 ∼ 30 nm
×
|A|
2πγ
1/4
Shooting procedure
• Equation to solve on domain [ ℓmin , Lmax ]
3
1
x0
1
X2
HX
HXXX = 3Ca
−
+
.
h0
H2
H3
H4
−→ Third order ODE
=⇒
3 boundary conditions (H, H
Micro



H(ℓmin ) = 0 (target)
H ′ (ℓmin ) = unknown


 ′′
H (ℓmin ) = unknown



H(ℓmin ) = 1
H ′ (ℓmin ) = 0


 ′′
H (ℓmin ) = free
′
, H ′′ ), one at least missing.
shooting
Macro
←−−−−−−−−−−−



H(Lmax ) = free
H ′ (Lmax ) = Θmax


 ′′
H (Lmax ) = 0
−−−−−−−−−−−→



H(Lmax ) = unknown
H ′ (Lmax ) = unknown


 ′′
H (Lmax ) = 0 (target)
Vanishing solution at zero
numeric
macroscopic corner
H(X) [a.u]
10000
tip of liquid
X
1
X1/2
0.0001
1e-08
1e-08
0.0001
1
10000
X [a.u]
• At leading order
√
h(x) = α x
with
α4 =
• Connecting this parabolic solution to a macroscopic corner
1
2 |A| 2
1
−→ crossover length λcross ∼ 2
θ
3π γ
See
[Poulard et al., Langmuir 2005] and
2 |A|
3π γ
hmacro (x) = θx
[Joanny & de Gennes, CRAS 1984]
• Note: these quantities are independent on J0
Numerical simulations and analytical calculations
1.6
1.4
0.045
receding Ca = 10-7
static Ca = 0
advancing Ca = 10-7
0.04
Receding
0.035
1.2
Voinov
0.03
0.6
θ
0.8
1.e3
Precursor
0.025
Advancing
3
Lmax = 10
4
Lmax = 10
analytical
analytical
0.02
Η(X)
Θ(X)
1
0.015
0.4
1
0.01
0.2
1
0
0.001
0.01
0.1
1
10
X ([a. u.])
X
100
1.e3
1000
0.005
-4e-07
10000
-2e-07
−→ Wedge connected to a precursor film H(X) = 1 + λ1 X 2 −
Length ∼ a few x0
0
2e-07
4e-07
Ca
≃ 5 µm
Thickness ∼
7
8
2
X
105
+ O(X 4 )
h0 ≃ 30 nm
Wetting laws
4
Lmacro
3
θ3 = (1 + √ ) θm
+ 1)
− 9Ca(ln
ℓmicro
2.3
with
• References: CTP et al., Europhys. Lett. 2010, 92, 54005.
3
=
θm
h0
x0
3
∼
J0 η 0
1
3
|A| 4 γ 4
∼ (10−2 )3
Dynamics of a complete wetting sessile droplet under evaporation
Air
?
Θ
h(x)
Liquid
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x
V
p
J(r) = j0 / R2 − r2
• Case of a spherical cap: evaporative flux
• Volume of a spherical cap at small contact angle θ: V =
• Mass conservation:
=⇒
θ
0
dV
=−
dt
Z
2π
0
Z
π 3
R θ
4
R
0
J(r)r dr dϕ = −2πj0 R.
3RθṘ + R2 θ̇ = −8j0 .
• Wetting law:
A
θ3 = √ + B Ṙ
R
with
A and B depending on physical constants of the problem
R
r
Comparison of the model with experiments
0.004
0.02
R(t)
θ(t)
0.0035
0.015
3.10
0.003
0.005
0.0015
2.10
-3
R(Tf - t)
θ(t) [rad]
R(t) [m]
0.002
-3
-3
0.01
0.0025
3.10
0.45
(Tf - t)
10
0
0.001
2.10
Tf - t
-3
-0.005
0.0005
tf’
0
0
5
10
15
20
25
30
35
tf
40
45
R(tf - t)
0.32
(tf - t)
-0.01
50
0.11
(tf - t)
t [s]
1
tf - t
10
−→ Fast spreading followed by slow retraction
• In agreement with experiments of Cazabat’s group [Langmuir 2005 & Soft Matter 2010]
Conclusions
• Wetting laws have been proposed for partial and complete wetting liquid under
diffusive evaporation. Cox-Voinov law has been generalized.
• For complete wetting case, the apparent contact angle scales like
3
θm
∼
J0 η0
1
3
|A| 4 γ 4
• This result is in agreement with other approaches by Cazabat et al. [Langmuir
2005]
and
Doumenc & Guerrier [EPJST 2011]
• This wetting law captures the early stages of spreading and retraction of an
evaporating droplet in complete wetting conditions