FriederMugele Physics of Complex Fluids University of Twente coorganizers: JaccoSnoeier Physics of Fluids / UT Anton Darhuber MesoscopicTransport Phenomena / Tu/e speakers: JoséBico (ESPCI Paris) Daniel Bonn (UvA) MichielKreutzer (TUD) Ralph Lindken (TUD) 1 program Monday: 12:00 –13:00h registration + lunch 13:00h welcome: FriederMugele 13:15h –14:00h FriederMugele: Wetting basics (Young-Laplace equation; Young equation; examples) 14:10-15:25h JaccoSnoeijer: Wetting flows: the lubrication approximation 15:25-15:50h coffee break 15:50-16:35h JaccoSnoeijer: Coating flows: the Landau-Levichproblem and its solution using asymptotic matching 16:45-17:30h Anton Darhuber: Surface tension, capillary forces and disjoining pressure I Tuesday: 9:00h-9:45h FriederMugele: Dewetting 9:5510:40 Anton Darhuber: Surface tension, capillary forces and disjoining pressure II 10:40-11:05h coffee break 11:05h-11:50h Anton Darhuber: Surface tension-gradient-driven flows 12:00h-12:45h Daniel Bonn: Evaporating drops 12:45-14:00h lunch 14:00h-14:45h Daniel Bonn: Drop impact 15:55h-15:40h JoséBico: Elastocapillarity (I) 15:40-16:05h coffee break 16:05h –16:50h JoséBico: Elasticity & Capillarity (II) 18:30 -... joint dinner & get together 2 program Wednesday: 9:00h-9:45h MichielKreutzer: Two-phase flow in microchannels: the Brethertonproblem 9:55h-10:40h Michiel Kreutzer: Drop generation& emulsificationin microchannels 10:40h-11:05h coffeebreak 11:05h-11:50h Michiel Kreutzer: Jet instabilitiesin microchannels 12:00h-12:45h Ralph Lindken: PiVcharacterization of capillarity-driven flows 12:45-14:00h lunch 14:00h-15:00h: occasion for excercises 15:00h-17:00h lab tour (Physics of Complex Fluids / Physics of Fluids) Thursday: 9:00h-9:45h JaccoSnoeijer: Contact line dynamics(I) 9:55h-10:40h JaccoSnoeijer: Contact line dynamics (II) 10:40h-11:05h coffee break 11:05h-11:50h FriederMugele: Wetting of heterogeneous surfaces: Wenzel, Cassie-Baxter 12:00h-12:45h: JaccoSnoeijer: Contact angle hysteresis 12:45-14:00h lunch 14:00h-14:45h JoséBico: Sperhydrophobicity 14:55h-15:40h Anton Darhuber: Thermocapillaryflows 15:40h-16:05h coffee break 16:05h-16:50h Anton Darhuber: Surfactant-driven and solutocapillaryflows Friday: 9:00h-9:45h FriederMugele: Electrowetting: basic principles 9:55h-10:40h FriederMugele: Eectrowettingapplications. 10:40h-11:05h coffee break 11:05-12:00h round up –highlights / short summaries by students 12:00h closure 3 principles of wetting and capillarity æ 1 1 ö s -s ç ÷ cosq = sv sl Dp = s lv ç + ÷ = s lvk Y è R1 R2 ø s lv capillary(Laplace) equation Young equation 4 capillarity-induced instabilities drivingforce: minimizationof surfaceenergy time Rayleigh-Plateau instability 5 drops in microchannels drop generation drop dynamics Anna et al. APL 2003 6 wetting and dewettingflows coating technology dewetting of paint e.g. heating Landau-Levichfilms 7 fundamental flow properties v lubrication flows contact line motion 8 wetting & molecular interactions nanoscaledrop qY x0 disjoining pressure verticalscale: 100 nm 9 capillary forces capillary bridges exert mechanical forces 10 wetting of complex surfaces superhydrophobicsurfaces: the Lotus effect q 11 switching wettability voltage electrowetting& thermocapillarity 12 lecture 1: basics of wetting 13 wetting& liquid microdroplets 50 µm capillaryequation Young equation s -s Dp = p = 2ks cosq = sv sl L lv Y s H. Gauet al. Science 1999 lv 14 origin of interfacial energy O(Å) width à 0: sharp interface model (will be handled throughout this course) range of interactions (O(nm)) surface tension is excess energy w.r.t. bulk cohesive energy U ‘unhappy‘molecules at interfaces ® s » coh lv 2a2 15 interfacialtension liquid A liquid B sAB: interfacial tension interfacial tensions (of immiscible fluids) are always positive 16 interfacial tensions matter at small scales fraction of molecules close to the surface: ì -7 A× dr 3dr ï 3×10 for r=1 cm r = = í V r -3 îï 3×10 for r=1 µm à capillarity is crucial for micro-and nanofluidics 17 mechanical definition of surface tension definition A: The mechanical work d W required to create an additional surface area dA (e.g. by deforming a drop) is given by the surface tension s dW = s dA ¶F thermodynamically: s = ¶A T ,N ,V energy dimension and units: []s = ; 1J/m2 (typically: mJ/m2) area 18 mechanical definition of surface tension 2× s l soap film definition B: s is a force per unit length acting along the liquid-vapor interface aiming to shrink the interfacial area d e force f dimension and units: []s = ; 1N/m= 1J/m2 (typically: mN/m) length i n i connection to definition A work required to move the rod: t dW = 2s ldx i æ 1 dW ö force per unit length per interface: f = -ç- ÷ = s o è 2l dx ø 19 n surface tension of selected liquids material surface tension [mJ/m2] water (25°C) 73 water (100°C) 58 ethanol 23 decanol 28.5 hexane 19.4 decane 23.9 hexadecane 27.6 glycerol 63 acetone 24 mercury 485 water/oil ≈ 50 T-coefficient: (-0.07 …-0.15) mJ/ m2K 20 consequences: the Laplace pressure spherical drop R dR Pdrop Pext dVext = -dVdrop variation of internal energy: dU = - pdropdVdrop - pext dVext +s dA ! mechanical equilibrium: dU = ( pext - pdrop )dVdrop +s dA=0 dA DpL = pdrop - pext = s dVdrop 2s Laplace pressure: Dp = L R 21 generalization to arbitrary surfaces upon crossing an interface between two fluids with an interfacial tension s, the pressure increases by æ 1 1 ö ç ÷ DpL =2s k =ç + ÷s Young-Laplace law è R1 R2 ø 1 æ 1 1 ö k: mean curvature k = ç + ÷ 2 è R1 R2 ø R1, R2: principal radii of curvature (sphere: R1=R2) 22 principleradiiof curvature sign convention: air j R1 > 0 r R2 < 0 n meancurvature: r n 1 æ 1 1 ö k = ç + ÷ 2 è R1 R 2 ø liquid (k isindependent of azimuthalangle f) 23 generalization to arbitrary surfaces upon crossing an interface between two fluids with an interfacial tension s, the pressure increases by æ 1 1 ö ç ÷ DpL =2s k =ç + ÷s Young-Laplace law è R1 R2 ø 1 æ 1 1 ö k: mean curvature k = ç + ÷ 2 è R1 R2 ø R1, R2: principal radii of curvature (sphere: R1=R2) consequence: liquid surfaces in mechanical equilibrium have a constant mean curvature 50 µm (n the absence of other forces) 24 H. Gau et al. Science 1999 variationalderivation of Laplace equation equilibrium surface profile ↔ minimum of Gibbs free energy (at constant volume) ! G = (Fsurf - pV )= min pressure: Lagrange multiplier F : functional of surface profile A: F [A] = s dA surf surf ò explicit representation of surface: z = z(x, y) r r r 2 2 dA =| dA |= Dsx ´ Dsy = 1+ (¶ x z) + (¶ y z) DxDy F [A] = s dA = s 1+ ¶ z 2 + ¶ z 2 dx dy æ Dx ö æ 0 ö surf ò òò ( x ) ( y ) r ç ÷ r ç ÷ Dsx = ç 0 ÷ Dsy = ç Dy ÷ ç¶ z Dx÷ ç ÷ è x ø è¶ y z Dyø volume: V = òò z(x, y)dx dy 25 functional minimization ! G[z(x, y)] = s 1+ ¶ z 2 + ¶ z 2 - p z dx dy = min òò{ ( x ) ( y ) } f (z,¶ x z,¶ y z) d ¶f d ¶f ¶f Euler-Lagrange equation: + - = 0 dx ¶(¶ x z) dy ¶(¶ y z) ¶z ¶f 2¶ z ¶ z = x = x ¶(¶ x z) 2 % S d æ ¶ x z ö ¶ xx z× S - ¶ x z(¶ x z ¶ xx z + ¶ y z ¶ xy z)/ S -3 2 2 ç ÷ = = S (¶ xx z ×(1+ (¶ x z) + (¶ y z) )- ¶ x z(¶ x z ¶ xx z + ¶ y z ¶ xy z)) dx è S ø S 2 d ¶f -3 2 = S (¶ xx z×(1+ (¶ y z) )- ¶ x z ¶ y z ¶ xy z) dx ¶(¶ x z) symmetrically: d ¶f -3 2 ¶f = S (¶ yy z ×(1+ (¶ x z) )- ¶ x z ¶ y z ¶ xy z) = - p dy ¶(¶ y z) ¶z 26 Young Laplace equation 2x mean curvature æ 1 1 ö ¶ z(1+ (¶ z)2 ) - 2(¶ z)(¶ z)(¶ z) + ¶ z(1+ (¶ z)2 ) Dp 2k = ç + ÷ = xx y x y xy yy x = ç ÷ 2 2 3/ 2 è R1 R2 ø (1+ (¶ x z) + (¶ y z) ) s lv non-linear second order partial differential equation ¶ xx z two-dimensional version: Dp = s lv 3 2 1+ (¶ x z) 27 cylindrical coordinates 2 æ 1 ö 2 surface parameterization: r = r(j, z) S = 1+ ç ¶j r ÷ + (¶ zr) è r ø 1 volume: V = dV = dz dr r dj = dz dj r 2 area: A = ò dA =ò dzò r dj S(r,j) ò ò ò ò 2 ò ò cylindrical symmetry: ¶j r = 0 ® r = r(z) æ 1 1 ö Dp = s ç - ¶ r÷ 2 ç 3 zz ÷ S = 1+ (¶ z r) è r S S ø à ordinary differential equation 28 an example fiber immersed in water (complete wetting; no gravity) radius R 1 ¶ zz r 0 = - 3 z=0 Sr S r z 2 BCs: r à ∞: k à 0 S = 1+ (¶ z r) r à R: r’ à 0 1 r'' 1 d æ r ö 0 = - = ç ÷ 2 3 ç 2 ÷ 1+ r' r 1+ r'2 r' dz è 1+ r' ø r = const. = R r'= (r / R)2 -1 > 0 1+ r'2 z>>R solution: r(z) = R cosh(z / R) µ exp(-z / R) 29 three phase equilibrium: wetting q = p 0 < q < p q = 0 s q lv ssl ssv non-wetting partial wetting complete wetting ssl: solid-liquid interfacial energy; ssv (solid-vapor); slv (liquid-vapor) 30 spreading parameter controls wetting behavior partial wetting complete wetting 1 spreading parameter S = [F - F ]= s - (s +s ) A init final sv sl lv S > 0 : complete wetting S < 0 : partial wetting 31 contact angle in partial wetting situation dxcos q slv qY qY ssv ssl dx (horizontal) force balance energy minimization s sv = s sl +s lv cosqY dW = {s sl +s lv cosqY -s sv }dx = 0 s sv -s sl Young equation cosqY = s lv ‘v‘: vapor or second immiscible liquid 32 connecting wetting behavior & surface properties ì> 0 : complete wetting S = s sv - (s sl + s lv ) í î< 0 : partial wetting high energy surfaces (metals, ionic crystals, covalent materials…) are usually wetted E s » coh » 500 ...5000 mJ sv a 2 m2 low energy surfaces (polymers, molecular crystals) are usually partially wetted k T s » B »10 ..