18.357 Interfacial Phenomena, Lecture 17: Coating, Dynamic Contact Lines

17. Coating: Dynamic Contact Lines Last time we considered the Landau-Levich-Derjaguin Problem and deduced −3 h ∼ ℓc Ca2/3 for Ca = µV σ < 10 1/3 h ∼ ℓc Ca for Ca → 1. The influence of surfactants Surfactants decrease σ which affects h slightly. But the principle effect is to generate Marangoni stresses that increase fluid emplacement: h typically doubles. Figure 17.1: The influence of surfactants on fiber coating. Gradients in Γ induce Marangoni stresses that enhance deposition. Fiber coating: ( Normal stress: p0 + σ R11 + 1 R1 1 b 1 R2 ) = p0 − ρgz. ∼ ⇒ curvature pressures dominant, can’t be balanced by gravity. If b ≪ ℓc , Thus, the interface must take the form of a catenoid: R11 + R12 = 0. e ) where h ≈ b ln(2ℓc /b). For wetting, θe = 0 ⇒ r(z) = b cosh z−h b Note: 1. gravity prevents meniscus from extending to ∞ ⇒ h deduced by cutting it off at ℓc . 2. h is just a few times b (h ≪ ℓc ) ⇒ lateral extent greatly exceeds its height. Forced wetting on fibers e.g. optical fiber coating. Figure 17.2: Etching of the microtips of Atomic Force Microscopes. As the fiber is withdrawn from the acid bath, the meniscus retreats and a sharp tip forms. Chapter 17. Coating: Dynamic Contact Lines Figure 17.3: Left: Forced wetting on a fiber. Right: The coating thickness as a function of the Reyonolds number Re. pf ilm ∼ p0 + σ b , pmeniscus ∼ p0 ⇒ Δp ∼ σ b resists entrainment. σ Force balance: µ eU2 ∼ Δp . L = bL √ e 1 Pressure match: L2 ∼ b ⇒ L ∼ be, substitute into the previous equation to find e ≈ bCa2/3 (Bretherton′ s Law) (17.1) Note: • this scaling is valid when e ≪ b, i.e. Ca2/3 ≪ 1. • At higher Ca, film is the viscous boundary layer that develops during pulling: δ ∼ Ls is the submerged length. ( µ Ls ρ U )1/2 , where Displacement of an interface in a tube E.g. air evacuating a water-filled pipette, pumping oil out of rock with water. Figure 17.4: Left: Displacing a liquid with a vapour in a tube. Right: The dependence of the film thickness left by the intruding front as a function of Ca = µU/σ. σ In the limit of h ≪ r, the pressure gradient in the meniscus ∇p ∼ rl , where l is the extent of the dynamic meniscus. σ σh 1/2 As on a fiber, pressure matching: p0 + 2σ when h ≪ r. r − r−h ∼ p0 + l2 ⇒ l ∼ (hr) 2 1/2 Force balance: µU/h ∼ σ/rl ∼ σ/r(hr) ⇒ ' -v " '-v" viscous curvature h ∼ rCa2/3 (Bretherton 1961) where Ca = µU σ . Thick films: what if h = ord(r)? For h ∼ r, Taylor (1961) found h ∼ (r − h)Ca2/3 . (17.2) 17.1. Contact Line Dynamics 17.1 Chapter 17. Coating: Dynamic Contact Lines Contact Line Dynamics Figure 17.5: The form of a moving meniscus near a wall or inside a tube for three different speeds. We consider the withdrawal of a plate from a fluid bath (Fig. 16.6) or fluid displacement within a cylindrical tube. Observations: • at low speeds, the contact line advances at the dynamic contact angle θd < θe • dynamic contact angle θd decreases progressively as U increases until U = UM . • at sufficiently high speed, the contact line cannot keep up with the imposed speed and a film is entrained onto the solid. Now consider a clean system free of hysteresis. Force of traction pulling liquid towards a dry region: F (θd ) = γSV − γSL − γ cos θd . Note: • F (θe ) = 0 in equilibrium. How does F depend on U ? What is θd (U )? • the retreating contact line (F < 0) was examined with retraction experiments e.g. plate withdrawal. • the advancing contact line (F > 0) was examined by Hoff­ mann (1975) for the case of θe = 0. • he found θd ∼ U 1/3 ∼ Ca1/3 (Tanner’s Law) Dussan (1979): drop in vicinity of contact line advances like a tractor tread Figure 17.6: Dynamic contact angle θd as a function of the differential speed U . For U > UM , the fluid wets the solid. 17.1. Contact Line Dynamics Chapter 17. Coating: Dynamic Contact Lines Figure 17.7: The advancing and retreating contact angles of a drop. Figure 17.8: A drop advancing over a solid boundary behaves like a tractor tread (Dussan 1979 ), ad­ vancing though a rolling motion. Flow near advancing contact line We now consider the flow near the contact line of a spreading liquid (θd > θe ): • consider θd ≪ 1, so that slope tan θd = • velocity gradient: dU dz ≈ z x ≈ θd ⇒ z ≈ θd x. U θd x • rate of viscous dissipation in the corner 1 1 1 ∞ 1 zmax =θd x 2 2 U dv Φ= µ dU = µ dx dz dz θd2 x2 corner 0 0 1 ∞ 2 1 U 3µU 2 ∞ dx xdx = Φ = 3µ θ d θd x θd2 x2 0 0 J ∞ dx J L dx de Gennes’ approximation: 0 x ≈ a x = ln L/a ≡ ℓD where L is the drop size and a is the molecular size. From experiments 15 < ℓD < 20. (17.3) (17.4) Energetics: 3µℓD · U2 θd rate of work done by surface forces equals the rate of viscous dissipation. Recall: FU = Φ = • F = γSV − γSL − γ cos θd = γ (cos θe − cos θd ) • in the limit θe < θd ≪ 1, cos θ ≈ 1 − θ2 2 ⇒F ≈ γ 2 e θd2 − θe2 (17.5) ) • substitute F into the energetics equation to get the contact line speed: ) U∗ e 2 U= θd θd − θe2 6ℓD where U ∗ = µγ ≈ 30m/s. (17.6) 17.1. Contact Line Dynamics Chapter 17. Coating: Dynamic Contact Lines Note: 3 1. rationalizes Hoffmann’s data (obtained for θe = 0) ⇒ U ∼ θD 2. U = 0 for θd = θe (static equilibrium) 3. U = 0 as θd → 0: dissipation in sharp wedge impedes motion. 4. U (θd ) has a maximum when dU dθd = U∗ 6ℓD e ) 3θd2 − θe2 ⇒ θd = θe √ 3 ⇒ Umax = U∗ √ θ3 9 3ℓD e Figure 17.9: Left: Schematic illustration of the flow in the vicinity of an advancing contact line. Right: The dependence of the dynamic contact angle on the speed of withdrawal. E.g. In water, U ∗ = 70m/s. With θe = 0.1 radians and ℓD = 20, Umax = 0.2mm/s ⇒ sets upper bound on extraction speed for water coating flows.