Post on 07-Sep-2018
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
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
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).
Contact angle hysteresis
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
Contact angle hysteresis
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
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
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
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
A diffuse interface model
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
A diffuse interface model
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
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
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
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
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
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
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
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)
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)
Approximation for Stokesian system with free boundary
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.
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.
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.
Derivation from Onsager principle
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
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
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.
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.
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.
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!