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Frieder MugelePhysics of Complex Fluids

University of Twente

coorganizers:

Jacco SnoeierPhysics of Fluids / UT

Anton DarhuberMesoscopic Transport Phenomena / Tu/e

speakers:José Bico (ESPCI Paris)Daniel Bonn (UvA)Michiel Kreutzer (TUD)Ralph Lindken (TUD)

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program

Monday:12:00 – 13:00h registration + lunch13:00h welcome: Frieder Mugele13:15h – 14:00h Frieder Mugele: Wetting basics (Young-Laplace equation; Young equation; examples)14:10-15:25h Jacco Snoeijer: Wetting flows: the lubrication approximation15:25-15:50h coffee break15:50-16:35h Jacco Snoeijer: Coating flows: the Landau-Levich problem and its solution using asymptotic matching16:45-17:30h Anton Darhuber: Surface tension, capillary forces and disjoining pressure I

Tuesday: 9:00h-9:45h Frieder Mugele: Dewetting9:5510:40 Anton Darhuber: Surface tension, capillary forces and disjoining pressure II10:40-11:05h coffee break11:05h-11:50h Anton Darhuber: Surface tension-gradient-driven flows12:00h-12:45h Daniel Bonn: Evaporating drops12:45-14:00h lunch14:00h-14:45h Daniel Bonn: Drop impact15:55h-15:40h José Bico: Elastocapillarity (I)15:40-16:05h coffee break16:05h – 16:50h José Bico: Elasticity & Capillarity (II)18:30 - ... joint dinner & get together

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programWednesday:9:00h-9:45h Michiel Kreutzer: Two-phase flow in microchannels: the Bretherton problem9:55h-10:40h Michiel Kreutzer: Drop generation& emulsification in microchannels10:40h-11:05h coffee break11:05h-11:50h Michiel Kreutzer: Jet instabilities in microchannels12:00h-12:45h Ralph Lindken: PiV characterization of capillarity-driven flows12:45-14:00h lunch14:00h-15:00h: occasion for excercises15:00h-17:00h lab tour (Physics of Complex Fluids / Physics of Fluids)

Thursday:9:00h-9:45h Jacco Snoeijer: Contact line dynamics(I)9:55h-10:40h Jacco Snoeijer: Contact line dynamics (II)10:40h-11:05h coffee break11:05h-11:50h Frieder Mugele: Wetting of heterogeneous surfaces: Wenzel, Cassie-Baxter12:00h-12:45h: Jacco Snoeijer: Contact angle hysteresis12:45-14:00h lunch14:00h-14:45h José Bico: Sperhydrophobicity14:55h-15:40h Anton Darhuber: Thermocapillary flows15:40h-16:05h coffee break16:05h-16:50h Anton Darhuber: Surfactant-driven and solutocapillary flows

Friday:9:00h-9:45h Frieder Mugele: Electrowetting: basic principles9:55h-10:40h Frieder Mugele: Eectrowetting applications. 10:40h-11:05h coffee break11:05-12:00h round up – highlights / short summaries by students 12:00h closure

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principles of wetting and capillarity

lv

slsvY σ

σσθ

−=cos

Young equationcapillary (Laplace) equation

κσσ lvlv RRp =

+=∆

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capillarity-induced instabilities

driving force:minimization of surface energy

time

Rayleigh-Plateau instability

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drops in microchannels

Anna et al. APL 2003

drop generation drop dynamics

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wetting and dewetting flows

coating technology dewetting of paint

e.g. heating

Landau-Levich films

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fundamental flow properties

lubrication flows contact line motion

v

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wetting & molecular interactions

vertical scale: 100 nm

nanoscale drop

θY

x0

disjoining pressure

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capillary forces

capillary bridges exert mechanical forces

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wetting of complex surfaces

superhydrophobic surfaces: the Lotus effect

θ

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switching wettability

voltage

electrowetting & thermocapillarity

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lecture 1:

basics of wetting

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wetting & liquid microdroplets

50 µm

H. Gau et al. Science 1999lvLpp κσ2==∆lv

slsvY σ

σσθ

−=cos

capillary equation Young equation

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origin of interfacial energy

range of interactions (O(nm))

‘unhappy‘ molecules at interfaces 22aU coh

lv ≈→ σ

O(Å)

surface tension is excess energy w.r.t. bulk cohesive energy

width à 0: sharp interface model(will be handled throughout this course)

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interfacial tension

interfacial tensions (of immiscible fluids) are always positive

liquid A

liquid B

σAB: interfacial tension

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interfacial tensions matter at small scales

fraction of molecules close to the surface:

⋅

⋅==

⋅−

−

3

7

103

1033rdr

VdrA for r=1 cm

for r=1 µm

r

à capillarity is crucial for micro- and nanofluidics

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mechanical definition of surface tension

dAW σδ =

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 σ

thermodynamically:VNTA

F

,,∂∂

=σ

[ ] ;area

energy=σdimension and units: 1J/m2 (typically: mJ/m2)

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mechanical definition of surface tension

[ ] ;lengthforce

=σdimension and units: 1N/m = 1J/m2 (typically: mN/m)

definition

soap film

lσ⋅2

definition B:σ is a force per unit length acting along the liquid-vapor interface aiming to shrink the interfacial area

connection to definition A xlW δσδ 2=work required to move the rod:

force per unit length per interface: σδ

δ=

−−=

xW

lf

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surface tension of selected liquids

≈ 50water/oil485mercury24acetone63glycerol27.6hexadecane23.9decane19.4hexane

2328.5

ethanoldecanol

58water (100°C)73water (25°C)surface tension [mJ/m2]material

T-coefficient: (-0.07 … -0.15) mJ / m2K

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consequences: the Laplace pressure

spherical dropR

δRPdrop

Pext

dAdVpdVpU extextdropdrop σδ +−−=variation of internal energy:

mechanical equilibrium: 0)(!=+−= dAdVppU dropdropext σδ

dropextdropL dV

dAppp σ=−=∆

dropext dVdV −=

Laplace pressure:R

pLσ2

=∆

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generalization to arbitrary surfaces

upon crossing an interface between two fluids with an interfacial tension s, the pressure increases by

Young-Laplace lawσκσ

+==∆

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112RR

pL

κ: mean curvature

+=

21

1121

RRκ

R1, R2: principal radii of curvature (sphere: R1=R2)

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principle radii of curvature

nr

R1 > 0R2 < 0

liquid

air

+=

21

1121

RRκ

mean curvature:

nrϕ

(κ is independent of azimuthal angle φ)

sign convention:

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generalization to arbitrary surfaces

upon crossing an interface between two fluids with an interfacial tension s, the pressure increases by

Young-Laplace lawσκσ

+==∆

21

112RR

pL

κ: mean curvature

+=

21

1121

RRκ

R1, R2: principal radii of curvature (sphere: R1=R2)

consequence: liquid surfaces in mechanical equilibrium have a constant mean curvature(n the absence of other forces)

50 µm

H. Gau et al. Science 1999

50 µm

H. Gau et al. Science 1999

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variational derivation of Laplace equation

equilibrium surface profile ↔ minimum of Gibbs free energy (at constant volume)

( ) min!=−= VpFG surf

Fsurf: functional of surface profile A: ∫= dAAFsurf σ][

explicit representation of surface: ),( yxzz =

yx ssAddA rrr∆×∆== || ( ) ( ) yxzz yx ∆∆∂+∂+= 221

pressure: Lagrange multiplier

( ) ( ) dydxzzdAAF yxsurf ∫∫∫ ∂+∂+== 221][ σσ

volume: dydxyxzV ∫∫= ),(

∆∂

∆=∆

xz

xs

x

x 0r

∆∂∆=∆

yzys

y

y

0r

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functional minimization

( ) ( ){ } min!

1)],([ 22 =−∂+∂+= ∫∫ dydxzpzzyxzG yxσ

),,( zzzf yx ∂∂

Euler-Lagrange equation: ( ) ( ) 0=∂∂

−∂∂∂

+∂∂∂

zf

zf

dyd

zf

dxd

yx

( ) Szz

zf xx

x

∂=

∂=

∂∂∂

%22

( )2

/S

SzzzzzSzS

zdxd xyyxxxxxxx ∂∂+∂∂∂−⋅∂

=

∂ ( ) ( )( ) ( )( )zzzzzzzzS xyyxxxxyxxx ∂∂+∂∂∂−∂+∂+⋅∂= − 223 1

( ) ( )( )( )zzzzzSz

fdxd

xyyxyxxx

∂∂∂−∂+⋅∂=∂∂∂ − 23 1

pzf

−=∂∂

( ) ( )( )( )zzzzzSz

fdyd

xyyxxyyy

∂∂∂−∂+⋅∂=∂∂∂ − 23 1

symmetrically:

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Young Laplace equation

lvyx

xyyxyyxyxx pzz

zzzzzzzσ∆

=∂+∂+

∂+∂+∂∂∂−∂+∂2/322

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))()(1())(1()()()(2))(1(

non-linear second order partial differential equation

=

+=

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112RR

κ

2x mean curvature

two-dimensional version: 32)(1 z

zpx

xxlv

∂+

∂=∆ σ

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cylindrical coordinates

surface parameterization: ),( zrr ϕ= ( )2211 rr

rS z∂+

∂+= ϕ

area: ∫ ∫∫ == ),( ϕϕ rSdrdzdAAvolume: ∫ ∫ ∫ ∫∫∫ === 2

21 rddzdrdrdzdVV ϕϕ

cylindrical symmetry: )(0 zrrr =→=∂ϕ

∂−=∆ r

SSrp zz3

11σ 2)(1 rS z∂+=

à ordinary differential equation

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an examplefiber immersed in water (complete wetting; no gravity)

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10S

rSr

zz∂−=

2)(1 rS z∂+=

322 '1

'''1

10r

rrr +

−+

=

+=

2'1'1

rr

dzd

r

Rconstr

r==

+.

'1 2( ) 01/' 2 >−= Rrr

radius R

rz

z=0

BCs: r à ∞: κ à 0r à R: r’à 0

)/cosh()( RzRzr = )/exp( RzRz

−∝>>

solution:

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three phase equilibrium: wetting

σsl: solid-liquid interfacial energy; σsv (solid-vapor); σlv (liquid-vapor)

σlv

σsvσslθ

non-wetting partial wetting complete wetting

θ = π 0 < θ < π θ = 0

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spreading parameter controls wetting behavior

partial wetting complete wetting

spreading parameter [ ] )(1lvslsvfinalinit FF

AS σσσ +−=−=

S > 0 : complete wettingS < 0 : partial wetting

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contact angle in partial wetting situation

(horizontal) force balance

lv

slsvY σ

σσθ

−=cosYoung equation

‘v‘: vapor or second immiscible liquid

σlv

σsv σslθY

energy minimization

θY

dx cos θ

dx

Ylvslsv θσσσ cos+= { } 0cos =−+= dxW svYlvsl σθσσδ

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connecting wetting behavior & surface properties

high energy surfaces (metals, ionic crystals, covalent materials…) are usually wetted

22 5000...500 mmJ

aEcoh

sv ≈≈σ

low energy surfaces (polymers, molecular crystals) are usually partially wetted

22 50...10 mmJ

aTkB

sv ≈≈σ

How to relate wetting behavior to microscopic interaction energies ?

<>

+−=wettingpartialwettingcomplete

S lvslsv :0:0

)( σσσ

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Gedankenexperiment

A

d0

A

A

A

initfinal UUW −=δ

)()(02 0dVV AAAAAv −∞=−= σ

B

A

B

A

)(2 0dVAAAv −=→ σ (I)

( ))(2 0dVBBBv −=σ (II)

)()( 0dVV

W

ABAB

ABBvAv

−∞=

−+= σσσδ (III)

)( 0dVAB−=

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Gedankenexperiment (II))(2 0dVAAAv −=→ σ (I)

)(2 0dVBBBv −=σ (II)

)( 0dVABABBvAv −=−+ σσσ (III)

A: solid; B: liquid

(III)-(II) )()()( 00 dVdVS slllsllvsv −==+− σσσbinding energies: <0

0<⇒> SVV slll

0>⇒> SVV llsl

à partial wetting

à complete wetting

van der Waals interaction: lsslV αα∝ 2lllV α∝

)( lslS ααα −∝ à complete wetting if solid more polarisable than liquid

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wetting and gravity

)( 0 zhgplv −∆+∆= ρκσlv

slsvY σ

σσθ

−=cos

dAzhgp ⋅−∆+∆ ))(( 0ρ

dAlv ⋅κσ

h0 - zg

x

z

hydrostatic pressure

capillary equationYoung equation

à now κ=κ(z)

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non-dimensionalization

dimensionless variables: zRz ~= xx R~

1∂=∂ κκ ~1

R=→

)~~(~)~~(~~00

2

zhBopzhRgplv

−+∆=−∆

+∆=σ

ρκ

pR

p lv ~σ=

Bo: Bond number Bo << 1 à gravity negligible

equivalently: capillary length lvc g σρλ /∆=

R << λc à gravity negligible

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

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summary

n equilibrium shape of wetting structures is determined by minimum of surface energy

n variation of free energy functional results in

lv

slsvY σ

σσθ

−=cos

Young equationcapillary (Laplace) equation

κσσ lvlv RRp =

+=∆

21

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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

)(/ mmOg lvc ≈∆= σρλ