1

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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

7

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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:

8

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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

10

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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

11

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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

12

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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