Modelling and analysis for contact angle hysteresis on...

Modelling and analysis for contact anglehysteresis on rough surfaces

Xianmin Xu

Institute of Computational Mathematics, Chinese Academy of Sciences

Collaborators: Xiaoping Wang(HKUST)

Workshop on Modeling and Simulation of Interface Dynamics in Fluids/Solids and Applications

National University of Singapore, May 14-18, 2018

1 Background

2 Analysis by a simple phase-field model

3 Analysis by using Onsager principle as an approximation tool

4 The modified Wenzel’s and Cassie’s equations

5 Summary

1 Background




5 Summary

Wetting phenomena

Wetting describes how liquid drops stay and spread on solidsurfaces.

Young’s equation

Young’s Equation:

γLV cos θY = γSV − γSL

θY : the contact angle on a homogeneous

smooth solid surface

The static contact angle is determined by surface tensions inthe system

T. Young, An essay on the cohesion of fluids, Philos. Trans. R. Soc. London (1805)

Complicated wetting phenomena

Surface inhomogeneity or roughnesschanges largely the wetting behavior.Lotus effect

Contact angle hysteresis

Applications: self-cleaning materials,painting filtration, soil sciences,plant biology, oil industry ...

Tuteja, A., Choi W., Ma M., et al. Science 318 (2007): 1618-1622.

Wenzel’s equation

Wenzel’s Equation:

cos θW = r cos θY

roughness parameter r =|Σrough||Σ|

R.N. Wenzel Resistance of solid surfaces to wetting by water,Industrial & Engineering Chemistry, (1936).

Cassie’s equation

Cassie’s Equation:cos θC = λ cos θY1 + (1− λ) cos θY2

specially:cos θCB = λ cos θY1 − (1− λ)where λ is area fraction.

A. Cassie & S. Baxter, Wettability of porous surfaces,Transactions of the Faraday Society, (1944).

Cassie’s equation

Cassie’s Equation:cos θC = λ cos θY1 + (1− λ) cos θY2

specially:cos θCB = λ cos θY1 − (1− λ)where λ is area fraction.

A. Cassie & S. Baxter, Wettability of porous surfaces,Transactions of the Faraday Society, (1944).


Wenzel’s and Cassie’s equations are seldom supported byexperiments quantitatively

The theoretical analysis for contact angle hysteresis is difficult

R. H. Dettre, R.E. Johnson, J. Phys. Chem. 1964,J. F. Joanny, P. G. De Gennes, J Chem Phys, 1984,M Reyssat, D Quere, J. Phys. Chem. B 2009, ..., Many others

P. G. De Gennes, Wetting: statics and dynamics, Reviews of Modern Physics, 1985.

D. Bonn, J. Eggers, etc. Wetting and spreading, Reviews of Modern Physics, 2009.

H. Y. Erbil, The debate on the dependence of apparent contact angles on drop contact area or three-phasecontact line: A review, Surface Science Reports, 2014


Wenzel’s and Cassie’s equations are seldom supported byexperiments quantitatively

The theoretical analysis for contact angle hysteresis is difficult

R. H. Dettre, R.E. Johnson, J. Phys. Chem. 1964,J. F. Joanny, P. G. De Gennes, J Chem Phys, 1984,M Reyssat, D Quere, J. Phys. Chem. B 2009, ..., Many others

P. G. De Gennes, Wetting: statics and dynamics, Reviews of Modern Physics, 1985.

D. Bonn, J. Eggers, etc. Wetting and spreading, Reviews of Modern Physics, 2009.

H. Y. Erbil, The debate on the dependence of apparent contact angles on drop contact area or three-phasecontact line: A review, Surface Science Reports, 2014

A recent experiment on contact angle hysteresis

Experiments in D. Guan, et al. PRL (2016) show that there isan obvious contact angle hysteresis.

Furthermore, the contact angle hysteresis is velocitydependent and might be asymmetric.

Motivation

To understand the dynamic contact angle hysteresis bymathematical method, especially to understand theasymmetric behaviour of velocity dependence of CAH.

1 Background




5 Summary

The mathematical model for static wetting problem

Minimize the energy in the system:

I (φ) = γLV |ΣLV |+∫

ΣSL

γSL(x)ds +

∫ΣSV

γSV (x)ds

under a volume conservation constraint.

A diffuse interface model


min Iε(φε) =

∫

Ωε2 |∇φ

ε|2 + f (φε)ε dxdy +

∫∂Ων(φε)ds,

if∫

Ωφε = V0;

+∞, otherwise.

with f (φ) = (1−φ2)2

4 .

φ is the phase-field equation

ε is the interface thickness parameter.

Prove the sharp interface limit(L. Modica, ARMA. 1987,1989, X.Xu& X. Wang, SIAP 2011)

H−1-gradient flow: the Cahn-Hillard equationMany studies on the equation: analysis, numerics and variousapplications



min Iε(φε) =

∫

Ωε2 |∇φ


∫∂Ων(φε)ds,

if∫

Ωφε = V0;

+∞, otherwise.

with f (φ) = (1−φ2)2

4 .







min Iε(φε) =

∫

Ωε2 |∇φ


∫∂Ων(φε)ds,

if∫

Ωφε = V0;

+∞, otherwise.

with f (φ) = (1−φ2)2

4 .





Cahn-Hilliard Equation with relaxed boundary condition

εφt = ∆(− ε2∆φ+ F ′(φ)

)in Ω× (0,∞),

φt + α(ε∂nφ+ ν ′(φ, x)

)= 0 on ∂Ω× (0,∞),

∂n(−ε2∆φ+ F ′(φ)) = 0 on ∂Ω× (0,∞),

φ(·, t) = φ0(·) on Ω× 0

(1)

The relaxed boundary condition models the dynamics ofcontact angle

The problem admits a unique weak solution

X. Chen, X.-P. Wang, and X., Arch. Rational Mech. Anal. 2014

Cahn-Hilliard Equation with relaxed BC on a moving roughsurfaces

εφt = ∆(− ε∆φ+ F ′(φ)

ε

)in Ω× (0,∞),

φt + uw ,τ∂τφ = −α(∂nφ+ ν′(φ,x)

ε

), on ∂Ω× (0,∞),

∂n(−ε∆φ+ F ′(φ)ε ) = 0 on ∂Ω× (0,∞),

φ(·, t) = φ0(·) on Ω× 0

(2)

Ω = (0, L)× (−h(x , t), h(x , t)) with h(x , t) = h0 + δH((x + Ut)/δ).

ν(φ, x) = ν(φ, xδ ) periodic function with respect to x .

The time scale is changed

Asymptotic analysis

The curvature of the interface is constant

The boundary condition

R + a cosβ = −α(nΓ · nβ − cos(θY ))− uw ,ττ · nβ. (3)

Dynamic contact angle on rough surface

The equation: Let xct be the contact point, we havexct = α(cos θY (xct +Ut)−cos θd )

sin θa−H′ct cos θa−[

11+(H′ct )2 −

H′ct cos θa

sin θa−H′ct cos θa

]U

θa = − g(θa)hct

[(f (θa) + cos2 θa

)xct +

(f (θa) + cos2 θa

hct

∫ xct0 H′( x+Ut

δ)dx)U].

(4)where we use the notations

H′ct = H′(xct + Ut

δ), hct = h0 + δH(

xct + Ut

δ),

g(θa) =cos θa

cos θa + (θa − π2

) sin θa, f (θa) = (θa −

π

2+ sin θa cos θa)H′(

xct + Ut

δ).

X. Xu, Y. Zhao and X.-P. Wang, submitted(2018).

Dynamic contact angle on chemically patterned surface

Consider a chemically patterned flat surface:

The previous equation is reduced to:θt =

[α cos θ−cos(θY (x))

sin θ + v]g(θ),

xt = −α cos θ−cos(θY (x))sin θ .

(5)

Here g(θ) = cos3(θ)cos θ+(θ−π

2 ) sin θ and θY (x) = θY (Hx).

The analysis result

TheoremFor the chemically patterned surface with θY (x) = θY ( x

δ) given above and assuming interface speed v small, the

solution (θ(t), x(t)) of system (5) satisfies the following properties which display the stick-slip behaviour andcontact angle hysteresis.

(a). For period δ large enough, θ(t) is a periodic function with θ∗1 ≤ θ(t) ≤ θ∗2 after an initial transient time,as the contact point x(t) moves forward (v > 0) or backward (v < 0).

(b). For δ small and v > 0, there exists a θ1(ε) such that θ1(δ) ≤ θ(t) ≤ θ∗2 after an initial transient time,

and θ1(δ)→ θ∗2 as δ → 0.

(c). For δ small and v < 0, there exists a θ2(ε) such that θ∗1 ≤ θ(t) ≤ θ2(δ) after an initial transient time,

and θ2(δ)→ θ∗1 as δ → 0.

The advancing and the receding processes follow different trajectoriesgiving different advancing and receding contact angles.

θ∗i ≈ θYi + vα

when v is small.

X. Wang, X. Xu, DCDS-A (2017)

Numerical example 1

The channel with serrated boundaries

−0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.170

75

80

85

90

95

100

105

110

x ct

θa(d

egre

e)

Receding

Advancing

(a) δ = 0.04

−0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.270

75

80

85

90

95

100

105

110

θa(d

egre

e)

x ct

Receding

Advancing

(b) δ = 0.008

Figure : Contact angle hysteresis on a rough boundary with a serratedshape.

Example 2

The channel with smooth oscillating boundary

Relatively small velocity

−0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.290

100

110

120

130

140

150

x ct

θa(d

eg

ree

)

Receding

Advancing

U=0.1

U=0.2

U=−0.1

U=−0.2

U=−0.4

U=0.4

Figure : Velocity dependence of the contact angle hysteresis(withrelatively small velocity).

Dynamic contact angle hysteresis and velocity dependence

Set θY 1 = 3π4 , θY 2 = 11π

12 on a flat surface

−0.5 0 0.5 1 1.5 2 2.5 31.6

1.8

2

2.2

2.4

2.6

2.8

3

3.2

x

θ

advancing, v=2

receding, v=2

advancing, v=1.5

receding, v=1.5

advancing, v=1

receding, v=1

advancing, v=0.5

receding, v=0.5

α=3

The numerical results are consistent with experimentsqualitatively. (Left: numerical results, right: Experiments by Penger Tong at el.(HKUST))

1 Background




5 Summary

The Onsager principle

Let x = (x1, x2, · · · , xf ) represents a set of parameters which specify thenon-equilibrium state of a system. It satisfies:

dxi

dt= −

∑j

µij (x)∂A

∂xj,

where A(x) is the free energy, µij is kinetic coefficient.

Onsager’s reciprocal relation:µij = µji

There exists ζij (x) (friction coefficient), such that ζij = ζji and∑

k ζikµkj = δij .Therefore ∑

j

ζij (x)xj = −∂A

∂xi

The equation can be derived by minimizing the Rayleighian:

R(x , x) =1

2

∑i,j

ζij xi xj +∑

i

∂A

∂xixi

with respect to xi .

M. Doi, Soft matter physics, Oxford University Press, 2014



dxi

dt= −

∑j

µij (x)∂A

∂xj,





j


∂xi


R(x , x) =1

2

∑i,j

ζij xi xj +∑

i

∂A

∂xixi





dxi

dt= −

∑j

µij (x)∂A

∂xj,





j


∂xi


R(x , x) =1

2

∑i,j

ζij xi xj +∑

i

∂A

∂xixi





dxi

dt= −

∑j

µij (x)∂A

∂xj,





j


∂xi


R(x , x) =1

2

∑i,j

ζij xi xj +∑

i

∂A

∂xixi



Approximation for Stokesian system with free boundary

Stokesian hydrodynamic system with some free boundary

Suppose the boundary is evolving driven by some potential forces, e.g. gravity,

surface tension, etc.

Let a(t) = a1(t), a2(t), · · · , aN(t) be the set of theparameters which specifies the position of the boundary

The motion of the system, i.e. the time derivative a(t) isdetermined by

min R(a, a) = Φ(a, a) +∑

i

∂A

∂aiai

Here A(a) is the potential energy of the system, Φ(a, a) is the energy dissipation

function(defined as a half of the minimum of the energy dissipated per unit time

in the fluid when the boundary is changing at rate a)


Stokesian hydrodynamic system with some free boundary

Suppose the boundary is evolving driven by some potential forces, e.g. gravity,

surface tension, etc.

Let a(t) = a1(t), a2(t), · · · , aN(t) be the set of theparameters which specifies the position of the boundary

The motion of the system, i.e. the time derivative a(t) isdetermined by

min R(a, a) = Φ(a, a) +∑

i

∂A

∂aiai

Here A(a) is the potential energy of the system, Φ(a, a) is the energy dissipation

function(defined as a half of the minimum of the energy dissipated per unit time

in the fluid when the boundary is changing at rate a)


The resulting system

∂Φ

∂ai+∂A

∂ai= 0. (6)

The equation is a force balance of two kinds of forces: thehydrodynamic friction force ∂Φ/∂ai , and the potential force−∂A/∂ai .

The ODE system (6) can be solved numerically

X. Xu, Y. Di, M. Doi, Phys. Fluids, 2016.

Derivation from Onsager principle

Assume the shape is radial symmetric

z = h(t)− r0 cos θ(t) ln

(r +

√r2 − r2

0 cos2 θ(t)

r0 cos θ(t)

), (7)

DeGennes,Brochard-Wyart, Quere, Capillary and wetting phenomena, 2002.

The derivative of the surface energy is

A ≈ 2πγr0(cos θ − cos θY (z))h. (8)

θY depends only on the height position

The energy dissipation is approximated by

Φ =2πηr0 sin2 θ

θ − sin θ cos θ| ln ε|(h − v)2. (9)

Huh, Scriven, J. Colloid & Interface Sciences,1970.




(r +

√r2 − r2

0 cos2 θ(t)

r0 cos θ(t)

), (7)






Φ =2πηr0 sin2 θ






(r +

√r2 − r2

0 cos2 θ(t)

r0 cos θ(t)

), (7)






Φ =2πηr0 sin2 θ




By using Onsager principle, we could derive

A ODE systemθt =

[− γ(θ−sin θ cos θ)

2η| ln ε| sin2 θ(cos θ − cos(θY (z))) + v

]g(θ),

zt = −γ(θ−sin θ cos θ)

2η| ln ε| sin2 θ(cos θ − cos(θY (z))).

(10)

where g(θ) =(r0 sin θ(1− ln( 2rc

r0 cos θ )))−1

−4 −3 −2 −1 0

x 10−4

95

100

105

110

115

120

125

130

135

increasing velocity,

Ca=0.0025,0,005,0.01,0.02

1 Background




5 Summary

The simplified sharp-interface model in 3D

The domain with a rough surface, with period ε

The equationdiv(

∇uε√1+|∇uε|

)= 0, in Bu

nS · nΓ = cos θεY , on Lε,uε(1, y) = 0,uε(1, y) is periodic in y with period 1,

(11)

The contact line Lε := x = ψε(y), z = φε(y).

G. De Philippis, F. Maggi, Regularity of free boundaries in anisotropic capillary problems and the validity of

Young’s law, Arch. Rational Mech. Anal. 2014.

The simplified sharp-interface model in 3D

The domain with a rough surface, with period εThe equation

div(

∇uε√1+|∇uε|

)= 0, in Bu

nS · nΓ = cos θεY , on Lε,uε(1, y) = 0,uε(1, y) is periodic in y with period 1,

(11)

The contact line Lε := x = ψε(y), z = φε(y).G. De Philippis, F. Maggi, Regularity of free boundaries in anisotropic capillary problems and the validity of

Young’s law, Arch. Rational Mech. Anal. 2014.

Homogenization

Asymptotic expansions, in the leading order:

The homogenized surface is given by z = k(1− x).

the apparent contact angle

cos θa =−k

√1 + k2

=1

ε

∫ ε

0

√1 + (∂yψε)2 cos(θεY (y)− θεg (y))dy , (12)

where θεY (y) = θY ( yε, φε( y

ε)) is the Young’s angle along the contact line, and

θεgθεg (y) = arcsin((mL × nS ) · τL),

is a geometric angle of the solid surface at the contact point y , with τ being thetangential direction of the contact line, mL is the normal of Lεp , the projection ofthe contact line Lε in z = 0 surface.

X. Xu, SIAM J. Appl. Math., 2016

The modified Wenzel’s equation

For geometric roughness, θY (x) is a constant function

cos θa =1

ε

∫ ε

0

√1 + (∂yψε)2 cos(θY − θεg (y))dy , (13)

integral average of the Young’s angleminus a geometric angle on contact line

the classical Wenzel’s equation:

cos θa =1

ε2

∫ ε

0

∫ ε

0

√1 + (∂yhε + ∂zhε)2dxdy cos(θY ),

The modified Wenzel’s equation

For geometric roughness, θY (x) is a constant function

cos θa =1

ε

∫ ε

0

√1 + (∂yψε)2 cos(θY − θεg (y))dy , (13)

integral average of the Young’s angleminus a geometric angle on contact line

the classical Wenzel’s equation:

cos θa =1

ε2

∫ ε

0

∫ ε

0

√1 + (∂yhε + ∂zhε)2dxdy cos(θY ),

The modified Cassie’s equation

For planar but chemically inhomogeneous solid surface, thegeometrical angle is 0, and the macroscopic contact angle isgiven by

cos θa =1

ε

∫ ε

0cos θY (y , z)

∣∣z=φε(y)

dy . (14)

integral average of the Young’s angle on the contact line

The classical Cassie’s equation: the area integral average

cos θa =1

ε2

∫ ε

0

∫ ε

0cos(θY (y , z))dydz .

The modified Wenzel and Cassie equations can be used tounderstand the contact angle hysteresis.



cos θa =1

ε

∫ ε

0cos θY (y , z)

∣∣z=φε(y)

dy . (14)



cos θa =1

ε2

∫ ε

0

∫ ε





cos θa =1

ε

∫ ε

0cos θY (y , z)

∣∣z=φε(y)

dy . (14)



cos θa =1

ε2

∫ ε

0

∫ ε



Summary

Contact angle hysteresis can be qualitatively analysed by aCahn-Hilliard equation with relaxed boundary condition

Onsager principle is a useful approximation tool for studying CAH

A modified Wenzel and Cassie equation should be used instead ofthe classical Wenzel and Cassie equation

future work:

Dynamic problems in 3D

Stochastic homogenization

Thank you very much!

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