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 ... 50 mJ sv a 2 m2

How to relate wetting behavior to microscopic interaction energies ?

33 Gedankenexperiment

A A A A

d0

A B A B

dW = U final -Uinit

= 2s Av - 0 = VAA (¥) -VAA (d0 ) dW = s Av +s Bv -s AB (III)

® 2s Av = -VAA (d0 ) (I) = VAB (¥) -VAB (d0 )

(2s Bv = -VBB (d0 )) (II) = -VAB (d0 ) 34 Gedankenexperiment (II)

® 2s Av = -VAA (d0 ) (I)

2s Bv = -VBB (d0 ) (II)

s Av +s Bv -s AB = -VAB (d0 ) (III)

A: solid; B: liquid

(III)-(II) s sv - (s lv +s sl ) = S =Vll (d0 ) -Vsl (d0 )

binding energies: <0

Vll > Vsl Þ S < 0 à partial wetting

à complete wetting Vsl > Vll Þ S > 0

2 van derWaals interaction: Vsl µ a sal Vll µ al

S µ al (a s -al ) à complete wetting if solid more polarisablethan liquid 35 wetting and gravity

hydrostatic pressure

z (Dp + Dr g(h - z))×dA g 0 h0 -z

ks lv × dA

x

Young equation capillary equation

s sv -s sl cosqY = ks lv = Dp + Drg(h0 - z) s lv à now k=k(z)

36 non-dimensionalization

1 ~ s lv ~ 1 ~ dimensionless variables: z = R z ¶ x = ¶ ~x p = p ® k = k R R R

2 ~ ~ Dr g R ~ ~ ~ ~ ~ k = Dp + (h0 - z ) = Dp + Bo (h0 - z ) s lv

Bo: Bond number Bo << 1 à gravity negligible

equivalently: capillary length lc = Dr g / s lv

R << lc à gravity negligible

water in air: l ≈ 2.7mm à gravity is usually negligible in microfluidics

37 summary

n equilibrium shape of wetting structures is determined by minimum of surface energy n variation of free energy functional results in 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 n occurrence of complete vs. partial wetting is determined by relative strength of adhesive vs. cohesive forces n gravity is negligible on length scales << capillary length

lc = Dr g / s lv »O(mm)

38