Dynamique d`une ligne de contact sous évaporation
Transcription
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) 11111111111111111111111111111111111111 00000000000000000000000000000000000000 00000000000000000000000000000000000000 11111111111111111111111111111111111111 00000000000000000000000000000000000000 11111111111111111111111111111111111111 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) 1111111111111111111111111111111111111 0000000000000000000000000000000000000 0000000000000000000000000000000000000 1111111111111111111111111111111111111 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) 11111111111111111111111111111111111111 00000000000000000000000000000000000000 00000000000000000000000000000000000000 11111111111111111111111111111111111111 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 1111111111111111111111111111111111111 0000000000000000000000000000000000000 0000000000000000000000000000000000000 1111111111111111111111111111111111111 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 1111111111111111111111111111111111111 0000000000000000000000000000000000000 0000000000000000000000000000000000000 1111111111111111111111111111111111111 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