Modelling and analysis for contact angle hysteresis on rough surfaces

Modelling and analysis for contact angle hysteresis 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

Wetting phenomena Wetting describes how liquid drops stay and spread on solid surfaces.

Young s equation Young s Equation: γ LV cos θ Y = γ SV γ SL θ Y : the contact angle on a homogeneous smooth solid surface The static contact angle is determined by surface tensions in the system T. Young, An essay on the cohesion of fluids, Philos. Trans. R. Soc. London (1805)

Complicated wetting phenomena Surface inhomogeneity or roughness changes 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 θ W = r cos θ Y roughness parameter r = Σ rough Σ R.N. Wenzel Resistance of solid surfaces to wetting by water, Industrial & Engineering Chemistry, (1936).

Cassie s equation Cassie s Equation: cos θ C = λ cos θ Y1 + (1 λ) cos θ Y2 A. Cassie & S. Baxter, Wettability of porous surfaces, Transactions of the Faraday Society, (1944).

Cassie s equation Cassie s Equation: cos θ C = λ cos θ Y1 + (1 λ) cos θ Y2 specially: cos θ CB = λ cos θ Y1 (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 by experiments quantitatively 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-phase contact line: A review, Surface Science Reports, 2014

Contact angle hysteresis Wenzel s and Cassie s equations are seldom supported by experiments 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-phase contact 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 is an obvious contact angle hysteresis. Furthermore, the contact angle hysteresis is velocity dependent and might be asymmetric.

Motivation To understand the dynamic contact angle hysteresis by mathematical method, especially to understand the asymmetric 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) = γ LV Σ LV + γ SL (x)ds + γ SV (x)ds Σ SL Σ SV under a volume conservation constraint.

A diffuse interface model A diffuse interface model ε min I ε (φ ε Ω 2 φε 2 + f (φε ) ε dxdy + Ω ν(φε )ds, ) = if Ω φε = V 0 ; +, 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)

Cahn-Hilliard Equation with relaxed boundary condition εφ t = ( ε 2 φ + F (φ) ) in Ω (0, ), φ t + α ( ε n φ + ν (φ, x) ) = 0 on Ω (0, ), n ( ε 2 φ + F (φ)) = 0 on Ω (0, ), φ(, t) = φ 0 ( ) on Ω {0} (1) The relaxed boundary condition models the dynamics of contact 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 rough surfaces εφ t = ( ε φ + F (φ)) ε φ t + u w,τ τ φ = α ( n φ + ν (φ,x) ε in Ω (0, ), ), on Ω (0, ), n ( ε φ + F (φ) ε ) = 0 on Ω (0, ), φ(, t) = φ 0 ( ) on Ω {0} (2) Ω = (0, L) ( h(x, t), h(x, t)) with h(x, t) = h 0 + δh((x + Ut)/δ). ν(φ, x) = ν(φ, x δ ) periodic function with respect to x. The time scale is changed

Asymptotic analysis The curvature of the interface is constant The boundary condition Ṙ + ȧ cos β = α(n Γ n β cos(θ Y )) u w,τ τ n β. (3)

Dynamic contact angle on rough surface The equation: Let x ct be the contact point, we have ẋ ct = α(cos θ Y (x ct +Ut) cos θ d ) sin θ a H ct cos [ 1 H θa 1+(H ct ct cos θa ] )2 sin θ U a H ct cos θa θ a = g(θa) [( f h (θa) + cos 2 θ a )ẋct + ( f (θa) + cos2 θ a xct ct h ct 0 H ( x+ut )dx ) U ]. δ (4) where we use the notations H ct = H xct + Ut xct + Ut ( ), h ct = h 0 + δh( ), δ δ cos θ a g(θ a) = cos θ a + (θ a π 2 ) sin, f (θa) = (θ a π θa 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: θ t = [ ] ˆα cos θ cos(ˆθ Y (ˆx)) sin θ + v g(θ), ˆx t = ˆα cos θ cos(ˆθ Y (ˆx)) sin θ. (5) Here g(θ) = cos 3 (θ) cos θ+(θ π 2 ) sin θ and ˆθ Y (ˆx) = θ Y (Hˆx).

The analysis result Theorem For the chemically patterned surface with θ Y (x) = θ Y ( δ x ) 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 and contact angle hysteresis. (a). For period δ large enough, θ(t) is a periodic function with θ1 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 ˆθ 1 (ɛ) such that ˆθ 1 (δ) θ(t) θ2 after an initial transient time, and ˆθ 1 (δ) θ2 as δ 0. (c). For δ small and v < 0, there exists a ˆθ 2 (ɛ) such that θ1 θ(t) ˆθ 2 (δ) after an initial transient time, and ˆθ 2 (δ) θ1 as δ 0. The advancing and the receding processes follow different trajectories giving different advancing and receding contact angles. θ i θ Yi + v when v is small. α X. Wang, X. Xu, DCDS-A (2017)

110 105 100 95 90 85 80 75 Numerical example 1 The channel with serrated boundaries 110 105 θ a (degree) 100 95 90 Receding Advancing θ a (degree) 85 80 Receding Advancing 75 70 0.6 0.5 0.4 0.3 0.2 0.1 0 0.1 ˆxct 70 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.1 0.2 ˆxct (a) δ = 0.04 (b) δ = 0.008 Figure : Contact angle hysteresis on a rough boundary with a serrated shape.

Example 2 The channel with smooth oscillating boundary Relatively small velocity 150 140 U=0.4 U=0.2 U=0.1 θ a (degree) 130 120 Receding Advancing 110 U= 0.1 100 U= 0.2 U= 0.4 90 0.6 0.5 0.4 0.3 0.2 0.1 0 0.1 0.2 ˆxct Figure : Velocity dependence of the contact angle hysteresis(with relatively small velocity).

Dynamic contact angle hysteresis and velocity dependence Set θ Y 1 = 3π 4, θ Y 2 = 11π 12 on a flat surface 3.2 α=3 3 2.8 2.6 θ 2.4 2.2 2 1.8 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 1.6 0.5 0 0.5 1 1.5 2 2.5 3 x The numerical results are consistent with experiments qualitatively. (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 = (x 1, x 2,, x f ) represents a set of parameters which specify the non-equilibrium state of a system. It satisfies: dx i dt = j µ ij (x) A x j, where A(x) is the free energy, µ ij is kinetic coefficient. Onsager s reciprocal relation: µ ij = µ ji There exists ζ ij (x) (friction coefficient), such that ζ ij = ζ ji and k ζ ikµ kj = δ ij. Therefore ζ ij (x)ẋ j = A x j i M. Doi, Soft matter physics, Oxford University Press, 2014

The Onsager principle Let x = (x 1, x 2,, x f ) represents a set of parameters which specify the non-equilibrium state of a system. It satisfies: dx i dt = j µ ij (x) A x j, where A(x) is the free energy, µ ij is kinetic coefficient. Onsager s reciprocal relation: µ ij = µ ji There exists ζ ij (x) (friction coefficient), such that ζ ij = ζ ji and k ζ ikµ kj = δ ij. Therefore ζ ij (x)ẋ j = A x j i The equation can be derived by minimizing the Rayleighian: R(x, ẋ) = 1 ζ ij ẋ i ẋ j + 2 i,j i A x i ẋ i with respect to ẋ i. 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) = {a 1 (t), a 2 (t),, a N (t)} be the set of the parameters which specifies the position of the boundary The motion of the system, i.e. the time derivative ȧ(t) is determined by min R(ȧ, a) = Φ(ȧ, a) + i A a i ȧ i Here A(a) is the potential energy of the system, Φ(ȧ, 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 ȧ)

Approximation for Stokesian system with free boundary The resulting system Φ ȧ i + A a i = 0. (6) The equation is a force balance of two kinds of forces: the hydrodynamic friction force Φ/ ȧ i, and the potential force A/ a i. 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 ( r + z = h(t) r 0 cos θ(t) ln r 0 cos θ(t) r 2 r 2 0 cos2 θ(t) ), (7) DeGennes,Brochard-Wyart, Quere, Capillary and wetting phenomena, 2002.

Derivation from Onsager principle Assume the shape is radial symmetric ( r + z = h(t) r 0 cos θ(t) ln r 0 cos θ(t) r 2 r 2 0 cos2 θ(t) DeGennes,Brochard-Wyart, Quere, Capillary and wetting phenomena, 2002. The derivative of the surface energy is ), (7) Ȧ 2πγr 0 (cos θ cos θ Y (z))ḣ. (8) θ Y depends only on the height position The energy dissipation is approximated by Φ = 2πηr 0 sin 2 θ θ sin θ cos θ ln ε (ḣ v)2. (9) Huh, Scriven, J. Colloid & Interface Sciences,1970.

Derivation from Onsager principle By using Onsager principle, we could derive A ODE system [ ] γ(θ sin θ cos θ) θ t = 2η ln ε sin 2 θ (cos θ cos(θ Y (z))) + v g(θ), γ(θ sin θ cos θ) z t = 2η ln ε sin 2 θ (cos θ cos(θ Y (z))). where g(θ) = ( r 0 sin θ(1 ln( 2rc r 0 cos θ )) ) 1 (10) 135 130 125 120 115 increasing velocity, Ca=0.0025,0,005,0.01,0.02 110 105 100 95 4 3 2 1 0 x 10 4

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 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 ε The equation ( ) u div ε = 0, in B u 1+ uε n S n Γ = cos θy ε, on Lε, u ε(1, y) = 0, u ε(1, y) is periodic in y with period 1, (11) The contact line L ε := {x = ψ ε (y), z = φ ε (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 θ a = k 1 + k 2 = 1 ε ε 0 1 + ( y ψ ε) 2 cos(θy ε (y) θε g (y))dy, (12) where θy ε (y) = θ Y ( y ε, φε( y )) is the Young s angle along the contact line, and ε θg ε θg ε (y) = arcsin((m L n S ) τ L ), is a geometric angle of the solid surface at the contact point y, with τ being the tangential direction of the contact line, m L is the normal of L ε p, the projection of the contact line L ε in z = 0 surface. X. Xu, SIAM J. Appl. Math., 2016

The modified Wenzel s equation For geometric roughness, θ Y (x) is a constant function cos θ a = 1 ε ε 0 1 + ( y ψ ε ) 2 cos(θ Y θ ε g (y))dy, (13) integral average of the Young s angle minus a geometric angle on contact line the classical Wenzel s equation: cos θ a = 1 ε 2 ε 0 ε 0 1 + ( y h ε + z h ε ) 2 dxdy cos(θ Y ),

The modified Cassie s equation For planar but chemically inhomogeneous solid surface, the geometrical angle is 0, and the macroscopic contact angle is given by cos θ a = 1 ε ε 0 cos θ Y (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 θ a = 1 ε 2 ε ε 0 0 cos(θ Y (y, z))dydz.

Summary Contact angle hysteresis can be qualitatively analysed by a Cahn-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 of the classical Wenzel and Cassie equation future work: Dynamic problems in 3D Stochastic homogenization Thank you very much!