Frieder Mugele. Physics of Complex Fluids. University of Twente. Jacco Snoeier Physics of Fluids / UT

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1 coorganizers: Frieder Mugele Physics of Comple Fluids Jacco Snoeier Physics of Fluids / UT University of Twente Anton Darhuber Mesoscopic Transport Phenomena / Tu/e speakers: José Bico (ESPCI Paris) Daniel Bonn (UvA) Michiel Kreutzer (TUD) Ralph Lindken (TUD)

2 program Monday: :00 3:00h registration + lunch 3:00h welcome: Frieder Mugele 3:5h 4:00h Frieder Mugele: Wetting basics (Young-Laplace equation; Young equation; eamples) 4:0-5:5h Jacco Snoeijer: Wetting flows: the lubrication approimation 5:5-5:50h coffee break 5:50-6:35h Jacco Snoeijer: Coating flows: the Landau-Levich problem and its solution using asymptotic matching 6:45-7:30h Anton Darhuber: Surface tension, capillary forces and disjoining pressure I Tuesday: 9:00h-9:45h Frieder Mugele: Dewetting 9:550:40 Anton Darhuber: Surface tension, capillary forces and disjoining pressure II 0:40-:05h coffee break :05h-:50h Anton Darhuber: Surface tension-gradient-driven flows :00h-:45h Daniel Bonn: Evaporating drops :45-4:00h lunch 4:00h-4:45h Daniel Bonn: Drop impact 5:55h-5:40h José Bico: Elastocapillarity (I) 5:40-6:05h coffee break 6:05h 6:50h José Bico: Elasticity & Capillarity (II) 8: joint dinner & get together

3 program Wednesday: 9:00h-9:45h Michiel Kreutzer: Two-phase flow in microchannels: the Bretherton problem 9:55h-0:40h Michiel Kreutzer: Drop generation& emulsification in microchannels 0:40h-:05h coffee break :05h-:50h Michiel Kreutzer: Jet instabilities in microchannels :00h-:45h Ralph Lindken: PiV characterization of capillarity-driven flows :45-4:00h lunch 4:00h-5:00h: occasion for ecercises 5:00h-7:00h lab tour (Physics of Comple Fluids / Physics of Fluids) Thursday: 9:00h-9:45h Jacco Snoeijer: Contact line dynamics(i) 9:55h-0:40h Jacco Snoeijer: Contact line dynamics (II) 0:40h-:05h coffee break :05h-:50h Frieder Mugele: Wetting of heterogeneous surfaces: Wenzel, Cassie-Bater :00h-:45h: Jacco Snoeijer: Contact angle hysteresis :45-4:00h lunch 4:00h-4:45h José Bico: Sperhydrophobicity 4:55h-5:40h Anton Darhuber: Thermocapillary flows 5:40h-6:05h coffee break 6:05h-6:50h Anton Darhuber: Surfactant-driven and solutocapillary flows Friday: 9:00h-9:45h Frieder Mugele: Electrowetting: basic principles 9:55h-0:40h Frieder Mugele: Eectrowetting applications. 0:40h-:05h coffee break :05-:00h round up highlights / short summaries by students :00h closure 3

4 principles of wetting and capillarity p + R R κ cosθ Y sv sl capillary (Laplace) equation Young equation 4

5 capillarity-induced instabilities driving force: minimization of surface energy time Rayleigh-Plateau instability 5

6 drops in microchannels drop generation drop dynamics Anna et al. APL 003 6

7 wetting and dewetting flows coating technology dewetting of paint e.g. heating Landau-Levich films 7

8 fundamental flow properties v lubrication flows contact line motion 8

9 wetting & molecular interactions nanoscale drop θ Y 0 vertical scale: 00 nm disjoining pressure 9

10 capillary forces capillary bridges eert mechanical forces 0

11 wetting of comple surfaces superhydrophobic surfaces: the Lotus effect θ

12 switching wettability voltage electrowetting & thermocapillarity

13 lecture : basics of wetting 3

14 wetting & liquid microdroplets 50 µm capillary equation p p L κ Young equation cosθ Y sv sl H. Gau et al. Science 999 4

15 origin of interfacial energy O(Å) width 0: sharp interface model (will be handled throughout this course) range of interactions (O(nm)) surface tension is ecess energy w.r.t. bulk cohesive energy U coh unhappy molecules at interfaces a 5

16 interfacial tension liquid A liquid B AB : interfacial tension interfacial tensions (of immiscible fluids) are always positive 6

17 interfacial tensions matter at small scales fraction of molecules close to the surface: A dr V 3dr r for r cm for r µm r capillarity is crucial for micro- and nanofluidics 7

18 mechanical definition of surface tension definition A: The mechanical work δ W required to create an additional surface area da (e.g. by deforming a drop) is given by the surface tension δ W da thermodynamically: F A T, N, V energy dimension and units: [ ] ; J/m (typically: mj/m ) area 8

19 mechanical definition of surface tension l soap film d e f i n i t i o n definition B: is a force per unit length acting along the liquid-vapor interface aiming to shrink the interfacial area force dimension and units: [ ] ; N/m J/m length (typically: mn/m) connection to definition A work required to move the rod: force per unit length per interface: δw f lδ δw l δ 9

20 surface tension of selected liquids material water (5 C) water (00 C) ethanol decanol heane decane headecane glycerol acetone mercury water/oil surface tension [mj/m ] T-coefficient: ( ) mj / m K 0

21 consequences: the Laplace pressure spherical drop R δr variation of internal energy: P drop P et δ U p dv p dv + mechanical equilibrium: δu ( p p ) dv + da 0 p L p et drop drop drop p et drop drop et da dv drop et! dv da et dv drop Laplace pressure: p L R

22 generalization to arbitrary surfaces upon crossing an interface between two fluids with an interfacial tension s, the pressure increases by Young-Laplace law κ + R R p L κ: mean curvature + R R κ R, R : principal radii of curvature (sphere: R R )

23 principle radii of curvature sign convention: air ϕ n r R > 0 R < 0 n r mean curvature: κ R + R liquid (κ is independent of azimuthal angle φ) 3

24 generalization to arbitrary surfaces upon crossing an interface between two fluids with an interfacial tension s, the pressure increases by p L κ R + R Young-Laplace law κ: mean curvature κ + R R R, R : principal radii of curvature (sphere: R R ) consequence: liquid surfaces in mechanical equilibrium have a constant mean curvature (n the absence of other forces) 50 µm H. Gau et al. Science 999 4

25 variational derivation of Laplace equation equilibrium surface profile minimum of Gibbs free energy (at constant volume) G ( F p ) min V surf! pressure: Lagrange multiplier F surf : functional of surface profile A: F surf [ A] da eplicit representation of surface: z z(, y) da r da r r s s y + ( z) + ( yz) y F surf [ A] da + volume: V z (, y) d dy ( z) + ( z) d dy y s r 0 z r s y 0 y z y y 5

26 functional minimization { ( ) ( ) }! + z + z p z d min G[ z(, y)] dy y f ( z, z, z) y Euler-Lagrange equation: d d f + d dy f ( z) ( z) y f z 0 f d d ( z) S z S z % z S z z ( z z + z z) S y y / S ( z ( + ( z) + ( z) ) z( z z + z z ) S 3 y y y d d d dy ( z) ( z ( + ( z) ) z z z) f 3 S y symmetrically: ( z) y ( z ( + ( z) ) z z z) f 3 S yy y y y y f z p 6

27 Young Laplace equation mean curvature κ R + R z( + ( y z) ) ( ( + ( z)( z) y z) ( + ( y y z) z) + ) 3/ yy z( + ( z) ) p non-linear second order partial differential equation z two-dimensional version: p 3 + ( z) 7

28 cylindrical coordinates surface parameterization: r r( ϕ, z) S + r + ( r) ϕ z r volume: V dv dz dr r dϕ dz dϕ r area: A da dz r dϕ S( r, ϕ) cylindrical symmetry: r 0 r r( z) ϕ p r S S r 3 zz S + ( zr) ordinary differential equation 8

29 an eample fiber immersed in water (complete wetting; no gravity) z0 radius R r 0 Sr r zz 3 S z BCs: r : κ 0 r R: r 0 S + ( r) z 0 r' ' 3 + ' r r' dz + r' r + r' d r r + r' const. R r' ( r / R) > 0 solution: z>> R r ( z) R cosh( z / R) ep( z / R) 9

30 three phase equilibrium: wetting θ π 0 < θ < π θ 0 θ sl sv non-wetting partial wetting complete wetting sl : solid-liquid interfacial energy; sv (solid-vapor); (liquid-vapor) 30

31 spreading parameter controls wetting behavior partial wetting complete wetting spreading parameter S [ ] ( + ) A F init F final sv sl S > 0 : complete wetting S < 0 : partial wetting 3

32 contact angle in partial wetting situation sv sl θ Y d cos θ θ Y d (horizontal) force balance energy minimization sv + cosθ δw { + cos θ } d 0 sl Y sl Y sv Young equation cosθ Y sv sl v : vapor or second immiscible liquid 3

33 connecting wetting behavior & surface properties S sv ( + sl ) > 0 : complete < 0 : partial wetting wetting high energy surfaces (metals, ionic crystals, covalent materials ) are usually wetted sv E coh a mj m low energy surfaces (polymers, molecular crystals) are usually partially wetted sv k B T mj a m How to relate wetting behavior to microscopic interaction energies? 33

34 Gedankeneperiment A A A A d 0 A A B B δw U final U init Av 0 VAA( ) VAA( d0) δ W + Av Bv AB (III) Av V AA ( d0) ( d ) ( ) Bv V BB 0 (I) (II) V AB V AB ( ) VAB( d0) ( d 0 ) 34

35 Gedankeneperiment (II) Av Av V AA ( d0) + Bv Bv V BB ( d0) AB V AB ( d 0 ) (I) (II) (III) A: solid; B: liquid (III)-(II) sv ( + sl ) S Vll ( d0) Vsl ( d0) binding energies: <0 V ll > V S sl < 0 partial wetting V sl > V S ll > 0 complete wetting van der Waals interaction: S α l ( α s αl ) Vsl α s α l Vll α complete wetting if solid more polarisable than liquid l 35

36 wetting and gravity hydrostatic pressure z g h 0 -z ( 0 p + ρ g( h z)) da κ da Young equation cosθ Y sv sl now κκ(z) capillary equation κ p + ρg( h 0 z) 36

37 non-dimensionalization dimensionless variables: z R ~ z R ~ p ~ p R κ κ ~ R κ ~ ~ p + ρ g R ~ ( h 0 ~ z ) ~ p + Bo ~ ( h 0 ~ z ) Bo: Bond number Bo << gravity negligible equivalently: capillary length λ c ρ g / R << λ c gravity negligible water in air: λ.7mm gravity is usually negligible in microfluidics 37

38 summary equilibrium shape of wetting structures is determined by minimum of surface energy variation of free energy functional results in p + R R κ cosθ Y sv sl capillary (Laplace) equation Young equation occurrence of complete vs. partial wetting is determined by relative strength of adhesive vs. cohesive forces gravity is negligible on length scales << capillary length λ c ρ g / O( mm) 38

Jacco Snoeijer PHYSICS OF FLUIDS

Jacco Snoeijer PHYSICS OF FLUIDS dynamics dynamics freezing dynamics freezing microscopics of capillarity Menu 1. surface tension: thermodynamics & microscopics 2. wetting (statics): thermodynamics & microscopics