FriederMugele - Universiteit Twente in Enschede: High … · 2017-05-16 · FriederMugele Physics...
Transcript of FriederMugele - Universiteit Twente in Enschede: High … · 2017-05-16 · FriederMugele Physics...
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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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wetting and dewetting flows
coating technology dewetting of paint
e.g. heating
Landau-Levich films
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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
+=
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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
+==∆
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112RR
pL
κ: mean curvature
+=
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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 =
+=∆
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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