acknowledgmentcox solution: matched asymptotics for ... coarse-grains microscopic physics into...
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Moving contact lines and enhancedslip on textured substrates
James J. FengDepartment of Chemical and Biological Engineering &Department of MathematicsUniversity of British Columbia, Vancouver, Canada
(IMA Oct 15, 2009, Minneapolis, MN)
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AcknowledgmentPengtao Yue (Dept. of Mathematics, Virginia Tech)Chunfeng Zhou (Dept. CEMS, Univ. Minnesota)Peng Gao (Dept. of Chem. & Bio. Engineering, UBC)
Yue et al., Sharp interface limit of the Cahn-Hilliard model for moving contact lines. J. Fluid Mech. (accepted 2009)
Gao & Feng, Enhanced slip on a patterned substrate due to depinning of contact line. Phys. Fluids (to appear Nov. 2009)
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OutlineMotivation: difficulties; macroscopic and microscopic models
Mesoscopic diffuse-interface theory Numerical implementation
How to produce meaningful results? Convergence to sharp-interface limit Viscoelastic effects on contact line motion
Enhanced slip on textured substrates Flow regimes: depinning of contact line Slip length
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Fluid mechanical formulation:
• Stress singularity, velocity discontinuity
• Apparent slip at the contact line• How to specify the contact line velocity?
2
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Fluid mechanical formulation:
Static contact angle Microscopic vs. macroscopicdynamic contact angle
• How to specify contact angle?
• Key: micro-macro coupling
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Previous models
Macroscopic continuum model: Navier-Stokes with slip BCMicroscopic models: hydrodynamicmodel; molecular kinetic theory
Mesoscopic models: Diffuse-interfacemodel with Cahn-Hilliard diffusion
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Macroscopic continuum model
Navier-Stokes with slip: 2 conditions- Navier condition: U=λ(dU/dy); or numerical slip- Dynamic contact angle: θD=θS? θD(U)?
Refs: Renardy et al. JCP (2001); Homsy et al. PoF (2004); Spelt, JCP 207 (2005)
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Microscopic hydrodynamic models
Cox solution: matched asymptotics forsmall Ca and small slip length lm
θD3 −θm
3 = 9Ca ln Llm
⎛⎝⎜
⎞⎠⎟
Ref: Cox, J. Fluid Mech. 168 (1986);Agreement with spreading data (Tanner’s law)
inner
outer
intermediate
3
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A mesoscopic model?
The diffuse-interface model:integrates macro- and micro-scales: Coarse-grainsmicroscopic physics into energies and adds tomacroscopic fluid dynamicsremoves singularity by Cahn-Hilliard diffusionContinuum model suitable for large-scale, high-Cacomputation in complex geometriesDoes interface capturing and complex rheology onthe fly
Refs: Jacqmin, JFM (2000); Qian et al., JFM (2006); Khatavkar et al., JFM (2007)
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Fundamental questions:
Does this phenomenology reproduce reality insome sense?
What mesoscopic model parameters to use? Interfacial thickness Interfacial energy Cahn-Hilliard diffusion
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x
φ
-1 -0.5 0 0.5 1-2
-1
0
1
2
Diffuse interfaceSharp interface
ξ ∝ ε1
The Model: Cahn-Hillard mixing energy
]4
)1()(
2
1[
2
222
εϕϕλ −+∇=mixf
Ω
1=ϕ
1−=ϕ
Gradient energy Energy ofhomogenous phase
ξ ~ 4εσ = (2 2 /3) λ / ε
• •
⎟⎠⎞⎜
⎝⎛=
εφ
2tanh
x
Takes care of fluid-fluid σ12/46
Wall potential energy: wettability
Wetting potential fw that prescribes θs(Jacqmin, J. Fluid Mech. 402, 2000):
fw (ϕ) = −σ cosθSϕ(3−ϕ 2 )
4+σ B +σ A
2
Takes care of static contact angle θs
σ B − σ A = σ cosθS
Therefore:
fw (+1) = σ A fw (−1) = σ B(in bulk A); (in bulk B)
4
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Viscoelasticity: dumbbell elastic energy
k, Boltzman constant
T , temperature
Ψ = Ψ(Q ,x ,t), distribution function
Q being the connector vector
A(Ψ) = kTΨ lnΨ + 1
2H( Q ⋅Q )
τ
p=
1+ϕ2
nH < QQ > -nkTI( )
Q
QQH
⋅=Φ 21
Takes care of bulk fluid rheology (if needed)14/46
Governing Equations:Newtonian-Oldroyd-B system
Momentum Equations:
∂u∂t
+ u ⋅∇u = −∇p +∇ ⋅ µD +τ p( ) +G∇ϕ, where D = ∇u + (∇u)T
Continuity equation:
Cahn-Hilliard equation (interface):
Viscoelastic stress (Oldroyd-B, for example)
τ p + λ1τ∇
p = 2µ p (D + λ2 D∇)
∂ϕ∂t
+ u ⋅∇ϕ = ∇ ⋅ (γ∇G); Chemical potential G = λ(−Δϕ +(ϕ 2 −1)ϕ
ε 2 )
Variation of energy
δx (virtual work principle)
δQ
δϕ
∇ ⋅u = 0
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Boundary conditions:
On solid wall (everywhere, including the contact line):
No-slip condition:
No mass flux through wall:
Rapid interfacial relaxation (natural BC from variation):
n ⋅∇G = 0
λn ⋅∇ϕ + fw′ (ϕ ) = 0
u = 0
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Implementation: physical & numerical issues
Interface must be thin: sharp interface limit toavoid “interfacial distortion”
Interface must be well resolved Generally 7-10 grids across the interface Adaptive
Meshing
4th order PDE (Cahn-Hilliard Eq.)
Split into two 2nd order PDEs
5
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Decomposition of C-H Equation
s is a positive number to enhance convergence
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AMPHI: Adaptive Meshing for Phase-field (φ) Mesh locally refined around interfaces 2D and 3D codes
Finite elements with adaptive mesh
AMPHI-2D: Yue et al., J. Comput. Phys. 219, 47 (2006); 223, 1 (2007) Various applications: see pubs in http://www.math.ubc.ca/~jfengAMPHI-3D: Zhou et al., J. Comput. Phys. (to appear 2009)
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Adaptive mesh generator: GRUMMP
Unstructured Mesh Generator;Open Source software by Professor Carl Ollivier-Gooch from Mechanical Engineering Dept., UBC:http://tetra.mech.ubc.caRefinement based on user-defined length scale: weused phase-field φ;
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Finest grid covers interface
Criterion for remeshing:
6
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Adaptive meshing: retraction of torus
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Model parameters and dimensionless groups
Dimensional parameters:
λ: mixing energy density; ε: capillary width; σ=(2√2/3) λ/ ε γ: Cahn-Hilliard mobility
Dimensionless groups:
Cn = εL
(Cahn number)
S =γµL
; lD= γµ diffusion length
What values to use to produce meaningful results?
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The sharp interface limit (SIL)
1. Well established for flows without contact lines• Caginalp & Chen, Euro. J. Appl. Math. 9 (1998). • Lowengrub & Truskinovsky, P. Roy. Soc. A 454 (1998).• Practical matter: how to achieve SIL computationally?
γ ~ ε n or S ~ Cnn / 2
θD3 −θm
3 = 9Ca ln Llm
⎛⎝⎜
⎞⎠⎟
2. Does SIL exist for moving contact line?• If it does, how to achieve SIL computationally?• Is this limit “correct” w.r.t. Cox’s formula and data?
with n = -1, 1 or 2
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Consider two simple flow geometries:
fluid 1
fluid 2
V
V
D?
M?L
x
y
fluid 1 fluid 2
V
V
M?R
L
D?
x
y
Couette flow (planar) Poiseuille flow (tube)
θM
θD
θDθM
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1. Sharp interface limit achievedCouette flow, θs=90o
SIL with ε→0, γ fixed(or Cn→0, S fixed)
x
y
-0.3 -0.2 -0.1 0 0.1 0.2 0.30
0.2
0.4
0.6
0.8
1
Cn=0.01Cn=0.005Cn=0.02Cn=0.04
Ca=0.03
Ca=0.02
Ca=0.01
Flow direction
S=0.01
Threshold Cn ≤ 4S
Cn
Δ θM
0 0.01 0.02 0.03 0.04 0.050
10
20
30
40
50
60S=3.16×10-2
S=10-2
S=3.16×10-3
S=10-3
ΔθM
Ca=0.02
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2. Comparing SIL with Cox’s theoryDo we predict the correct behavior?
• Compare the apparent contact angle with Cox’s formula:
• In our diffuse-interface model, we have “diffusion length”
θD3 −θm
3 = 9Ca ln Llm
⎛⎝⎜
⎞⎠⎟
lD = µγ
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Diffusion length v.s. slip length
x
y
-0.2 -0.18 -0.16 -0.14 -0.12 -0.10
0.02
0.04
0.06
0.08
0.1
≈2.55lD
lm ≈ 2.55 lD
Ca
θM
10-4 10-3 10-2 10-180
100
120
140
160
180
Cox, δ=2.55×10-2
Cox, δ=8.06×10-3
This work, S=10-2
This work, S=3.16×10-3
θΜ
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Logarithmic divergence:
θD3 −θm
3 = 9Ca ln Llm
⎛⎝⎜
⎞⎠⎟
• Well-known prediction of Cox’s formula:
(θD
3- θm
3 )/(
9Ca
)
• Take our numerical results to S → 0:
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Reflections on lm ~ lD:• Why is sharp interface limit approached thus:
• Why numerically this amounts to Cn ≤ αS.
• What value to use for γ or lD or S:
Reduce interfacial thickness ε, butkeep mobility γ or lD =(µγ)1/2 constant
There’s no escaping some degree ofphenomenology (not surprisingly)
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3. Viscoelastic effects
Wei et al., Dynamic wetting of Boger fluidsJ. Colloid Interface Sci., 313, p. 274 (2007)
Boger fluid
Experimental observations:
Viscoelasticity enhances“viscous bending” at theclose vicinity of contact line
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Experimental data (Wei et al. 2007)
Newtonian fit
VE enhances viscous bending near contact line
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Numerical Results
x
y
-0.2 -0.15 -0.1 -0.05 0 0.05 0.10
0.2
0.4
0.6
0.8
1
NewtonianWi=0.0.0784Wi=0.784Wi=3.92
Oldroyd-B
Newtonian
x
y
-0.2 -0.18 -0.16 -0.14 -0.12 -0.10
0.05
0.1
0.15
0.2
NewtonianWi=0.0.0784Wi=0.784Wi=3.92
More bending for VE systems
Displacement in pipe flow:µ*=1, β=0.5, θs=90o, Ca=0.02, S=0.01, Cn=0.01
D
H
l
VWi
55.2
λ=Effective
Weissenburg #
9
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Explanation for the extra bending
Color map: polymer normal stress τxxBlack lines: φ=-0.9 and 0.9Arrows: velocity
The extra polymerstress acts on one sideof the interface causesa larger interfacecurvature
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Newtonian fluids displacing Oldroyd-B fluids
x
y
-0.2 -0.15 -0.1 -0.05 0 0.05 0.10
0.2
0.4
0.6
0.8
1
NewtonianWi=0.0.0784Wi=0.784Wi=3.92
x
y
-0.2 -0.18 -0.16 -0.14 -0.12 -0.10
0.05
0.1
0.15
0.2
NewtonianWi=0.0.0784Wi=0.784Wi=3.92
VE hinders bending of the interface
Newtonian
Oldroyd-B
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Enhanced Slip on Patterned Surface
No slip:
slip length b
V
wall= 0 Partial slip:
V
wall= b γ
wall
• As an application of diffuse-interface model to simulating moving contact lines
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Slip Length
On superhydrophobic or microtextured surfaces:b ~ 20 µm, 40% drag reduction (Rothstein et al. PoF 2004)
Possible explanation: microbubbles entrapped in cavities and grooves
On smooth surfaces: b ~ 10 nm Insignificant for typical fluid dynamics problems
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Shear-rate dependence?Shear independence Most microchannel & capillaries (Choi et al., PoF 2006; Truesdell
et al., PRL 2006)
Increase with shear rate Small microchannel (d~1 µm, ~105 s-1; Choi et al., PoF 2003) Surface force apparatus (Zhu & Granick, PRL 2001) Atomic force microscope (Craig et al., PRL 2001)
γ
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Negative slip length? Curvature effect
Steinberger et al. Nature Mat. (2007) Hyväluoma & Harting PRL (2008)
Bubbles mayincrease frictionContact line alwayspinned
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Outstanding questions:
How the meniscus shape and motion, ifcontact line depins, affect slip?
How the slip length depends on the shearrate?
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Physical Model & MethodologyAssumptions Creeping flow Incompressible fluids Periodic BC on left & right
Dimensionless parameters Ca: ratio of viscous force to
surface tension W=ls/lg: width of the ridge
Diffuse-interface model with finite elementmethod: much more stringent Cn ≤ 0.2S=0.002
liquid
gas
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Four Flow Regimes
Ca
W
0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
2
Ca
W
0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
2
I II
III
IV
Regime I
Regime II
Regime III
Regime IV
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Discontinuous transition: II III
Regime I Regime II Regime IIIcontinuous discontinuous
Increasing shear rate
shear force surface tension
Fs1 + Fs2 + Fp = σ (1+ cosθs )
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Ca
b/lg
0 0.2 0.4 0.6 0.8 1
10-1
100
101
IIIRegime I II
How does b depend on flow rate?
Shear independence in regimes I and IIIThe slip length increase with the shear rate in regime IIGreatly enhanced slip in regime IIITrend consistent with experiment
W=0.5
Ca
b/lg
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
IVRegime I II
W=1.1
b/lgb/lg
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Effect of W
W
b/lg
0 0.5 1 1.5 20
0.05
0.1
0.15
0.2
Ca = 0.1Ca = 0.6double viscosityscaling law
Regime I
Regime II
W
b/lg
0 0.5 1 1.5 20
2
4
6Ca = 1.0double viscosityscaling law
Regime III
Regime IV
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Experimental considerations
102-103105-106shear rate (/s)
6472surface tension(dyn/cm)
1.51x10-3Viscosity (Pa s)
glycerinwaterFor l = 10 µm:
• Trend: no shear dependence at low Ca, followed by increase at Ca~0.5 due to contact line depinning
• How to reach Ca~0.5 in experiments?
• Can we directly observe depinning in experiments?46/46
SummarySharp interface limit of diffuse-interface computation:convergence to sharp-interface limitViscoelasticity enhances viscous bending whendisplacing Newtonian fluidContact line depinning greatly enhances apparent slipIncrease of the slip length with shear rate at high CaAdvantages to the diffuse-interface model for movingcontact lines