Modelling and analysis for contact angle hysteresis on rough surfaces
|
|
- Loreen Stewart
- 5 years ago
- Views:
Transcription
1 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
2 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
3 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
4 Wetting phenomena Wetting describes how liquid drops stay and spread on solid surfaces.
5 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)
6 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):
7 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).
8 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).
9 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).
10 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, D. Bonn, J. Eggers, etc. Wetting and spreading, Reviews of Modern Physics, 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
11 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, D. Bonn, J. Eggers, etc. Wetting and spreading, Reviews of Modern Physics, 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
12 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.
13 Motivation To understand the dynamic contact angle hysteresis by mathematical method, especially to understand the asymmetric behaviour of velocity dependence of CAH.
14 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
15 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.
16 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.
17 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)
18 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) H 1 -gradient flow: the Cahn-Hillard equation Many studies on the equation: analysis, numerics and various applications
19 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
20 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
21 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)
22 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).
23 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).
24 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)
25 Numerical example 1 The channel with serrated boundaries θ a (degree) Receding Advancing θ a (degree) Receding Advancing ˆxct ˆxct (a) δ = 0.04 (b) δ = Figure : Contact angle hysteresis on a rough boundary with a serrated shape.
26 Example 2 The channel with smooth oscillating boundary Relatively small velocity U=0.4 U=0.2 U=0.1 θ a (degree) Receding Advancing 110 U= U= 0.2 U= ˆxct Figure : Velocity dependence of the contact angle hysteresis(with relatively small velocity).
27 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= 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= x The numerical results are consistent with experiments qualitatively. (Left: numerical results, right: Experiments by Penger Tong at el.(hkust))
28 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
29 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. M. Doi, Soft matter physics, Oxford University Press, 2014
30 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 M. Doi, Soft matter physics, Oxford University Press, 2014
31 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
32 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
33 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
34 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)
35 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.
36 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.
37 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, 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
38 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, 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.
39 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) increasing velocity, Ca=0.0025,0,005,0.01, x 10 4
40 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
41 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
42 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
43 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 ( 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
44 The modified Wenzel s equation For geometric roughness, θ Y (x) is a constant function cos θ a = 1 ε ε ( 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
45 The modified Wenzel s equation For geometric roughness, θ Y (x) is a constant function cos θ a = 1 ε ε ( 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 ( y h ε + z h ε ) 2 dxdy cos(θ Y ),
46 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
47 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.
48 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 0 ε 0 cos(θ Y (y, z))dydz. The modified Wenzel and Cassie equations can be used to understand the contact angle hysteresis.
49 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!
For rough surface,wenzel [26] proposed the equation for the effective contact angle θ e in terms of static contact angle θ s
DERIVATION OF WENZEL S AND CASSIE S EQUATIONS FROM A PHASE FIELD MODEL FOR TWO PHASE FLOW ON ROUGH SURFACE XIANMIN XU AND XIAOPING WANG Abstract. In this paper, the equilibrium behavior of an immiscible
More informationANALYSIS FOR WETTING ON ROUGH SURFACES BY A THREE-DIMENSIONAL PHASE FIELD MODEL
ANALYSIS FOR WETTING ON ROUGH SURFACES BY A THREE-DIMENSIONAL PHASE FIELD MODEL XIANMIN XU Abstract. In this paper, we consider the derivation of the modified Wenzel s and Cassie s equations for wetting
More informationKey words. Wenzel equation, Cassie equation, wetting, rough surface, homogenization
MODIFIED WENZEL AND CASSIE EQUATIONS FOR WETTING ON ROUGH SURFACES XIANMIN XU Abstract. We study a stationary wetting problem on rough and inhomogeneous solid surfaces. We derive a new formula for the
More informationDerivation of continuum models for the moving contact line problem based on thermodynamic principles. Abstract
Derivation of continuum models for the moving contact line problem based on thermodynamic principles Weiqing Ren Courant Institute of Mathematical Sciences, New York University, New York, NY 002, USA Weinan
More informationcontact line dynamics
contact line dynamics Jacco Snoeijer Physics of Fluids - University of Twente sliding drops flow near contact line static contact line Ingbrigtsen & Toxvaerd (2007) γ γ sv θ e γ sl molecular scales macroscopic
More informationthe moving contact line
Molecular hydrodynamics of the moving contact line Tiezheng Qian Mathematics Department Hong Kong University of Science and Technology in collaboration with Ping Sheng (Physics Dept, HKUST) Xiao-Ping Wang
More informationGeneralized Cassie-Baxter equation for wetting of a spherical droplet within a smooth and heterogeneous conical cavity
Science Front Publishers Journal for Foundations and pplications of Physics, vol. 4, No. (017) (sciencefront.org) ISSN 394-3688 Generalized Cassie-axter equation for wetting of a spherical droplet within
More informationJacco Snoeijer PHYSICS OF FLUIDS
Jacco Snoeijer PHYSICS OF FLUIDS dynamics dynamics freezing dynamics freezing microscopics of capillarity Menu 1. surface tension: thermodynamics & microscopics 2. wetting (statics): thermodynamics & microscopics
More informationA phase field model for the coupling between Navier-Stokes and e
A phase field model for the coupling between Navier-Stokes and electrokinetic equations Instituto de Matemáticas, CSIC Collaborators: C. Eck, G. Grün, F. Klingbeil (Erlangen Univertsität), O. Vantzos (Bonn)
More informationcontact line dynamics
contact line dynamics part 2: hydrodynamics dynamic contact angle? lubrication: Cox-Voinov theory maximum speed for instability corner shape? dimensional analysis: speed U position r viscosity η pressure
More informationProblem Set Number 01, MIT (Winter-Spring 2018)
Problem Set Number 01, 18.377 MIT (Winter-Spring 2018) Rodolfo R. Rosales (MIT, Math. Dept., room 2-337, Cambridge, MA 02139) February 28, 2018 Due Thursday, March 8, 2018. Turn it in (by 3PM) at the Math.
More informationGeneralized Wenzel equation for contact angle of droplets on spherical rough solid substrates
Science Front Publishers Journal for Foundations and Applications of Physics, 3 (2), (2016) (sciencefront.org) ISSN 2394-3688 Generalized Wenzel equation for contact angle of droplets on spherical rough
More informationCapillarity and Wetting Phenomena
? Pierre-Gilles de Gennes Frangoise Brochard-Wyart David Quere Capillarity and Wetting Phenomena Drops, Bubbles, Pearls, Waves Translated by Axel Reisinger With 177 Figures Springer Springer New York Berlin
More informationWetting and Spreading of Drops on Rough Surfaces
Proceedings of the 6th International Congress of Chinese Mathematicians ALM 37, pp. 565 584 c Higher Education Press and International Press Beijing Boston Wetting and Spreading of Drops on Rough Surfaces
More informationModelling of interfaces and free boundaries
University of Regensburg Regensburg, March 2009 Outline 1 Introduction 2 Obstacle problems 3 Stefan problem 4 Shape optimization Introduction What is a free boundary problem? Solve a partial differential
More informationHydrodynamics of wetting phenomena. Jacco Snoeijer PHYSICS OF FLUIDS
Hydrodynamics of wetting phenomena Jacco Snoeijer PHYSICS OF FLUIDS Outline 1. Creeping flow: hydrodynamics at low Reynolds numbers (2 hrs) 2. Thin films and lubrication flows (3 hrs + problem session
More informationShear Flow of a Nematic Liquid Crystal near a Charged Surface
Physics of the Solid State, Vol. 45, No. 6, 00, pp. 9 96. Translated from Fizika Tverdogo Tela, Vol. 45, No. 6, 00, pp. 5 40. Original Russian Text Copyright 00 by Zakharov, Vakulenko. POLYMERS AND LIQUID
More informationSuperhydrophobic surfaces. José Bico PMMH-ESPCI, Paris
Superhydrophobic surfaces José Bico PMMH-ESPCI, Paris Superhydrophobic surfaces José Bico PMMH-ESPCI, Paris? Rain droplet on a window film pinning tear 180? mercury calefaction Leidenfrost point, T = 150
More informationPraktikum zur. Materialanalytik
Praktikum zur Materialanalytik Functionalized Surfaces B510 Stand: 20.10.2017 Table of contents Introduction 2 Basics 2 Surface tension 2 From wettability to the contact angle 4 The Young equation 5 Wetting
More informationLecture 6: Irreversible Processes
Materials Science & Metallurgy Master of Philosophy, Materials Modelling, Course MP4, Thermodynamics and Phase Diagrams, H. K. D. H. Bhadeshia Lecture 6: Irreversible Processes Thermodynamics generally
More informationExperimental and Theoretical Study of Motion of Drops on Horizontal Solid Surfaces with a Wettability Gradient Nadjoua Moumen
Experimental and Theoretical Study of Motion of Drops on Horizontal Solid Surfaces with a Wettability Gradient Nadjoua Moumen Department of Chemical and Biomolecular Engineering Clarkson University Outline
More informationwith deterministic and noise terms for a general non-homogeneous Cahn-Hilliard equation Modeling and Asymptotics
12-3-2009 Modeling and Asymptotics for a general non-homogeneous Cahn-Hilliard equation with deterministic and noise terms D.C. Antonopoulou (Joint with G. Karali and G. Kossioris) Department of Applied
More informationSolid-State Dewetting: Modeling & Numerics
Solid-State Dewetting: Modeling & Numerics Wei Jiang (ö ) Collaborators: Weizhu Bao (NUS, Singapore), Yan Wang (CSRC, Beijing), Quan Zhao (NUS, Singapore), Tiezheng Qian (HKUST, HongKong), David J. Srolovitz
More informationTRANSPORT IN POROUS MEDIA
1 TRANSPORT IN POROUS MEDIA G. ALLAIRE CMAP, Ecole Polytechnique 1. Introduction 2. Main result in an unbounded domain 3. Asymptotic expansions with drift 4. Two-scale convergence with drift 5. The case
More informationLine Tension Effect upon Static Wetting
Line Tension Effect upon Static Wetting Pierre SEPPECHER Université de Toulon et du Var, BP 132 La Garde Cedex seppecher@univ tln.fr Abstract. Adding simply, in the classical capillary model, a constant
More informationG : Statistical Mechanics
G25.2651: Statistical Mechanics Notes for Lecture 15 Consider Hamilton s equations in the form I. CLASSICAL LINEAR RESPONSE THEORY q i = H p i ṗ i = H q i We noted early in the course that an ensemble
More informationASPHALT WETTING DYNAMICS
ASPHALT WETTING DYNAMICS Troy Pauli, Fran Miknis, Appy Beemer, Julie Miller, Mike Farrar, and Will Wiser 5 TH Annual Pavement Performance Prediction Symposium Adhesion & Cohesion of Asphalt Pavements Cheyenne,
More informationISCST shall not be responsible for statements or opinions contained in papers or printed in its publications.
Modeling of Drop Motion on Solid Surfaces with Wettability Gradients J. B. McLaughlin, Sp. S. Saravanan, N. Moumen, and R. S. Subramanian Department of Chemical Engineering Clarkson University Potsdam,
More informationStep Bunching in Epitaxial Growth with Elasticity Effects
Step Bunching in Epitaxial Growth with Elasticity Effects Tao Luo Department of Mathematics The Hong Kong University of Science and Technology joint work with Yang Xiang, Aaron Yip 05 Jan 2017 Tao Luo
More informationMicrofluidics 2 Surface tension, contact angle, capillary flow
MT-0.6081 Microfluidics and BioMEMS Microfluidics 2 Surface tension, contact angle, capillary flow 28.1.2017 Ville Jokinen Surface tension & Surface energy Work required to create new surface = surface
More informationMath 311, Partial Differential Equations, Winter 2015, Midterm
Score: Name: Math 3, Partial Differential Equations, Winter 205, Midterm Instructions. Write all solutions in the space provided, and use the back pages if you have to. 2. The test is out of 60. There
More informationWetting contact angle
Wetting contact angle Minh Do-Quang www.flow.kth.se Outline Statics; capillarity and wetting Dynamics; models describing dynamic wetting Hydrodynamics (Tanner-Cox-Voinov law) Molecular kinetics theory
More informationBoundary Conditions for the Moving Contact Line Problem. Abstract
Boundary Conditions for the Moving Contact Line Problem Weiqing Ren Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA Weinan E Department of Mathematics and PACM,
More informationPHYSICS OF FLUID SPREADING ON ROUGH SURFACES
INTERNATIONAL JOURNAL OF NUMERICAL ANALYSIS AND MODELING Volume 5, Supp, Pages 85 92 c 2008 Institute for Scientific Computing and Information PHYSICS OF FLUID SPREADING ON ROUGH SURFACES K. M. HAY AND
More informationInvestigation of energy dissipation due to contact angle hysteresis in capillary effect
Journal of Physics: Conference Series PAPER OPEN ACCESS Investigation of energy dissipation due to contact angle hysteresis in capillary effect To cite this article: Bhagya Athukorallage and Ram Iyer 216
More information(Communicated by Benedetto Piccoli)
NETWORKS AND HETEROGENEOUS MEDIA Website: http://aimsciences.org c American Institute of Mathematical Sciences Volume 2, Number 2, June 2007 pp. 211 225 A NEW MODEL FOR CONTACT ANGLE HYSTERESIS Antonio
More informationin Bounded Domains Ariane Trescases CMLA, ENS Cachan
CMLA, ENS Cachan Joint work with Yan GUO, Chanwoo KIM and Daniela TONON International Conference on Nonlinear Analysis: Boundary Phenomena for Evolutionnary PDE Academia Sinica December 21, 214 Outline
More informationdrops in motion Frieder Mugele the physics of electrowetting and its applications Physics of Complex Fluids University of Twente
drops in motion the physics of electrowetting and its applications Frieder Mugele Physics of Complex Fluids niversity of Twente 1 electrowetting: the switch on the wettability voltage outline q q q q q
More informationSupplementary Information on Thermally Enhanced Self-Propelled Droplet Motion on Gradient Surfaces
Electronic Supplementary Material (ESI) for RSC Advances. This journal is The Royal Society of Chemistry 2015 Supplementary Information on Thermally Enhanced Self-Propelled Droplet Motion on Gradient Surfaces
More informationSta$s$cal mechanics of hystere$c capillary phenomena: predic$ons of contact angle on rough surfaces and liquid reten$on in unsaturated porous media
Sta$s$cal mechanics of hystere$c capillary phenomena: predic$ons of contact angle on rough surfaces and liquid reten$on in unsaturated porous media Michel Louge h@p://grainflowresearch.mae.cornell.edu/
More informationGlobal well-posedness of the primitive equations of oceanic and atmospheric dynamics
Global well-posedness of the primitive equations of oceanic and atmospheric dynamics Jinkai Li Department of Mathematics The Chinese University of Hong Kong Dynamics of Small Scales in Fluids ICERM, Feb
More informationGeneralized Phase Field Models with Anisotropy and Non-Local Potentials
Generalized Phase Field Models with Anisotropy and Non-Local Potentials Gunduz Caginalp University of Pittsburgh caginalp@pitt.edu December 6, 2011 Gunduz Caginalp (Institute) Generalized Phase Field December
More information1. INTRODUCTION 2. PROBLEM FORMULATION ROMAI J., 6, 2(2010), 1 13
Contents 1 A product formula approach to an inverse problem governed by nonlinear phase-field transition system. Case 1D Tommaso Benincasa, Costică Moroşanu 1 v ROMAI J., 6, 2(21), 1 13 A PRODUCT FORMULA
More informationSome asymptotic properties of solutions for Burgers equation in L p (R)
ARMA manuscript No. (will be inserted by the editor) Some asymptotic properties of solutions for Burgers equation in L p (R) PAULO R. ZINGANO Abstract We discuss time asymptotic properties of solutions
More informationFrictional rheologies have a wide range of applications in engineering
A liquid-crystal model for friction C. H. A. Cheng, L. H. Kellogg, S. Shkoller, and D. L. Turcotte Departments of Mathematics and Geology, University of California, Davis, CA 95616 ; Contributed by D.
More informationWeek 2 Notes, Math 865, Tanveer
Week 2 Notes, Math 865, Tanveer 1. Incompressible constant density equations in different forms Recall we derived the Navier-Stokes equation for incompressible constant density, i.e. homogeneous flows:
More informationAn improved threshold dynamics method for wetting dynamics
An improved threshold dynamics method for wetting dynamics Dong Wang a, Xiao-Ping Wang b,, Xianmin Xu c,d a Department of Mathematics, University of Utah, Salt Lake City, UT, 8, USA. b Department of Mathematics,
More informationThe 2D Magnetohydrodynamic Equations with Partial Dissipation. Oklahoma State University
The 2D Magnetohydrodynamic Equations with Partial Dissipation Jiahong Wu Oklahoma State University IPAM Workshop Mathematical Analysis of Turbulence IPAM, UCLA, September 29-October 3, 2014 1 / 112 Outline
More informationTopography driven spreading. School of Biomedical & Natural Sciences, Nottingham Trent University. Clifton Lane, Nottingham NG11 8NS, UK.
Postprint Version G. McHale, N. J. Shirtcliffe, S. Aqil, C. C. Perry and M. I. Newton, Topography driven spreading, Phys. Rev. Lett. 93, Art. No. 036102 (2004); DOI: 10.1103/PhysRevLett.93.036102. The
More informationThermosolutal Convection at Infinite Prandtl Number with or without rotation: Bifurcation and Stability in Physical Space
1/29 Thermosolutal Convection at Infinite Prandtl Number with or without rotation: Bifurcation and Stability in Physical Space Jungho Park Department of Mathematics New York Institute of Technology SIAM
More informationLecture No 1 Introduction to Diffusion equations The heat equat
Lecture No 1 Introduction to Diffusion equations The heat equation Columbia University IAS summer program June, 2009 Outline of the lectures We will discuss some basic models of diffusion equations and
More informationClassical solutions for the quasi-stationary Stefan problem with surface tension
Classical solutions for the quasi-stationary Stefan problem with surface tension Joachim Escher, Gieri Simonett We show that the quasi-stationary two-phase Stefan problem with surface tension has a unique
More informationV (r,t) = i ˆ u( x, y,z,t) + ˆ j v( x, y,z,t) + k ˆ w( x, y, z,t)
IV. DIFFERENTIAL RELATIONS FOR A FLUID PARTICLE This chapter presents the development and application of the basic differential equations of fluid motion. Simplifications in the general equations and common
More informationCoupled second order singular perturbations for phase transitions
Coupled second order singular perturbations for phase transitions CMU 06/09/11 Ana Cristina Barroso, Margarida Baía, Milena Chermisi, JM Introduction Let Ω R d with Lipschitz boundary ( container ) and
More information(Communicated by Aim Sciences)
NETWORKS AND HETEROGENEOUS MEDIA c American Institute of Mathematical Sciences Volume X, Number 0X, XX 200X Website: http://aimsciences.org pp. X XX Antonio DeSimone, Natalie Grunewald, Felix Otto SISSA-International
More informationSurface and Interfacial Tensions. Lecture 1
Surface and Interfacial Tensions Lecture 1 Surface tension is a pull Surfaces and Interfaces 1 Thermodynamics for Interfacial Systems Work must be done to increase surface area just as work must be done
More informationMath 4263 Homework Set 1
Homework Set 1 1. Solve the following PDE/BVP 2. Solve the following PDE/BVP 2u t + 3u x = 0 u (x, 0) = sin (x) u x + e x u y = 0 u (0, y) = y 2 3. (a) Find the curves γ : t (x (t), y (t)) such that that
More informationSTABILITY ANALYSIS OF DYNAMIC SYSTEMS
C. Melchiorri (DEI) Automatic Control & System Theory 1 AUTOMATIC CONTROL AND SYSTEM THEORY STABILITY ANALYSIS OF DYNAMIC SYSTEMS Claudio Melchiorri Dipartimento di Ingegneria dell Energia Elettrica e
More informationChapter 4. COSMOLOGICAL PERTURBATION THEORY
Chapter 4. COSMOLOGICAL PERTURBATION THEORY 4.1. NEWTONIAN PERTURBATION THEORY Newtonian gravity is an adequate description on small scales (< H 1 ) and for non-relativistic matter (CDM + baryons after
More informationModeling the combined effect of surface roughness and shear rate on slip flow of simple fluids
Modeling the combined effect of surface roughness and shear rate on slip flow of simple fluids Anoosheh Niavarani and Nikolai Priezjev www.egr.msu.edu/~niavaran November 2009 A. Niavarani and N.V. Priezjev,
More informationMajor Concepts Lecture #11 Rigoberto Hernandez. TST & Transport 1
Major Concepts Onsager s Regression Hypothesis Relaxation of a perturbation Regression of fluctuations Fluctuation-Dissipation Theorem Proof of FDT & relation to Onsager s Regression Hypothesis Response
More informationA generalized MBO diffusion generated motion for constrained harmonic maps
A generalized MBO diffusion generated motion for constrained harmonic maps Dong Wang Department of Mathematics, University of Utah Joint work with Braxton Osting (U. Utah) Workshop on Modeling and Simulation
More informationSimulation of T-junction using LBM and VOF ENERGY 224 Final Project Yifan Wang,
Simulation of T-junction using LBM and VOF ENERGY 224 Final Project Yifan Wang, yfwang09@stanford.edu 1. Problem setting In this project, we present a benchmark simulation for segmented flows, which contain
More informationEXISTENCE OF SOLUTIONS TO THE CAHN-HILLIARD/ALLEN-CAHN EQUATION WITH DEGENERATE MOBILITY
Electronic Journal of Differential Equations, Vol. 216 216), No. 329, pp. 1 22. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXISTENCE OF SOLUTIONS TO THE CAHN-HILLIARD/ALLEN-CAHN
More informationOn the Landau-Levich Transition
10116 Langmuir 2007, 23, 10116-10122 On the Landau-Levich Transition Maniya Maleki Institute for AdVanced Studies in Basic Sciences (IASBS), Zanjan 45195, P.O. Box 45195-1159, Iran Etienne Reyssat and
More informationThe Wilhelmy balance. How can we measure surface tension? Surface tension, contact angles and wettability. Measuring surface tension.
ow can we measure surface tension? Surface tension, contact angles and wettability www.wikihow.com/measure-surface-tension Measuring surface tension The Wilhelmy balance F Some methods: Wilhelmy plate
More informationMath 220A - Fall 2002 Homework 5 Solutions
Math 0A - Fall 00 Homework 5 Solutions. Consider the initial-value problem for the hyperbolic equation u tt + u xt 0u xx 0 < x 0 u t (x, 0) ψ(x). Use energy methods to show that the domain of dependence
More informationINTERFACIAL PHENOMENA GRADING SCHEME
18.357 INTERFACIAL PHENOMENA Professor John W. M. Bush Fall 2010 Office 2-346 MW 2-3:30 Phone: 253-4387 (office) Room 2-135 email: bush@math.mit.edu Office hours: after class, available upon request GRADING
More informationAbsorption of gas by a falling liquid film
Absorption of gas by a falling liquid film Christoph Albert Dieter Bothe Mathematical Modeling and Analysis Center of Smart Interfaces/ IRTG 1529 Darmstadt University of Technology 4th Japanese-German
More informationJ. Bico, C. Tordeux and D. Quéré Laboratoire de Physique de la Matière Condensée, URA 792 du CNRS Collège de France Paris Cedex 05, France
EUROPHYSICS LETTERS 15 July 2001 Europhys. Lett., 55 (2), pp. 214 220 (2001) Rough wetting J. Bico, C. Tordeux and D. Quéré Laboratoire de Physique de la Matière Condensée, URA 792 du CNRS Collège de France
More informationShear Thinning Near the Rough Boundary in a Viscoelastic Flow
Advanced Studies in Theoretical Physics Vol. 10, 2016, no. 8, 351-359 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/astp.2016.6624 Shear Thinning Near the Rough Boundary in a Viscoelastic Flow
More informationMathematical modelling of collective behavior
Mathematical modelling of collective behavior Young-Pil Choi Fakultät für Mathematik Technische Universität München This talk is based on joint works with José A. Carrillo, Maxime Hauray, and Samir Salem
More informationSIMULTANEOUS AND NON-SIMULTANEOUS BLOW-UP AND UNIFORM BLOW-UP PROFILES FOR REACTION-DIFFUSION SYSTEM
Electronic Journal of Differential Euations, Vol. 22 (22), No. 26, pp. 9. ISSN: 72-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu SIMULTANEOUS AND NON-SIMULTANEOUS
More information+ S/y. The wetted portion of the surface is then delimited by a certain contact line L (here a
EUROPHYSICS LETTERS Europhys. Lett., 21 (4), pp. 483-488 (1993) 1 February 1993 Contact Line Elasticity of a Completely Wetting Liquid Rising on a Wall. E. RAPHAEL(*)( ) and J. F. JoA"Y(**) (*) Institute
More informationMATH 131P: PRACTICE FINAL SOLUTIONS DECEMBER 12, 2012
MATH 3P: PRACTICE FINAL SOLUTIONS DECEMBER, This is a closed ook, closed notes, no calculators/computers exam. There are 6 prolems. Write your solutions to Prolems -3 in lue ook #, and your solutions to
More informationContinuum Model of Avalanches in Granular Media
Continuum Model of Avalanches in Granular Media David Chen May 13, 2010 Abstract A continuum description of avalanches in granular systems is presented. The model is based on hydrodynamic equations coupled
More informationHigh-Gain Observers in Nonlinear Feedback Control. Lecture # 2 Separation Principle
High-Gain Observers in Nonlinear Feedback Control Lecture # 2 Separation Principle High-Gain ObserversinNonlinear Feedback ControlLecture # 2Separation Principle p. 1/4 The Class of Systems ẋ = Ax + Bφ(x,
More informationNonlinear elasticity and gels
Nonlinear elasticity and gels M. Carme Calderer School of Mathematics University of Minnesota New Mexico Analysis Seminar New Mexico State University April 4-6, 2008 1 / 23 Outline Balance laws for gels
More informationSolid-State Dewetting: From Flat to Curved Substrates
Solid-State Dewetting: From Flat to Curved Substrates Yan Wang with Prof. Weizhu Bao, Prof. David J. Srolovitz, Prof. Wei Jiang and Dr. Quan Zhao IMS, 2 May, 2018 Wang Yan (CSRC) Solid-State Dewetting
More informationOn the displacement of two immiscible Stokes fluids in a 3D Hele-Shaw cell
On the displacement of two immiscible Stokes fluids in a 3D Hele-Shaw cell Gelu Paşa Abstract. In this paper we study the linear stability of the displacement of two incompressible Stokes fluids in a 3D
More informationIntroduction to Marine Hydrodynamics
1896 1920 1987 2006 Introduction to Marine Hydrodynamics (NA235) Department of Naval Architecture and Ocean Engineering School of Naval Architecture, Ocean & Civil Engineering First Assignment The first
More informationBasic Theory of Dynamical Systems
1 Basic Theory of Dynamical Systems Page 1 1.1 Introduction and Basic Examples Dynamical systems is concerned with both quantitative and qualitative properties of evolution equations, which are often ordinary
More informationDiffuse and sharp interfaces in Biology and Mechanics
Diffuse and sharp interfaces in Biology and Mechanics E. Rocca Università degli Studi di Pavia SIMAI 2016 Politecnico of Milan, September 13 16, 2016 Supported by the FP7-IDEAS-ERC-StG Grant EntroPhase
More informationAsymptotic Analysis of the Approximate Control for Parabolic Equations with Periodic Interface
Asymptotic Analysis of the Approximate Control for Parabolic Equations with Periodic Interface Patrizia Donato Université de Rouen International Workshop on Calculus of Variations and its Applications
More informationUniversity of Groningen. Wetting on rough surfaces Palasantzas, Georgios; De Hosson, J.T.M. Published in: Acta Materialia
University of Groningen Wetting on rough surfaces Palasantzas, Georgios; De Hosson, J.T.M. Published in: Acta Materialia DOI: 10.1016/S1359-6454(01)00238-5 IMPORTANT NOTE: You are advised to consult the
More informationKramers formula for chemical reactions in the context of Wasserstein gradient flows. Michael Herrmann. Mathematical Institute, University of Oxford
eport no. OxPDE-/8 Kramers formula for chemical reactions in the context of Wasserstein gradient flows by Michael Herrmann Mathematical Institute, University of Oxford & Barbara Niethammer Mathematical
More informationPiecewise Smooth Solutions to the Burgers-Hilbert Equation
Piecewise Smooth Solutions to the Burgers-Hilbert Equation Alberto Bressan and Tianyou Zhang Department of Mathematics, Penn State University, University Park, Pa 68, USA e-mails: bressan@mathpsuedu, zhang
More informationAnalysis of a non-isothermal model for nematic liquid crystals
Analysis of a non-isothermal model for nematic liquid crystals E. Rocca Università degli Studi di Milano 25th IFIP TC 7 Conference 2011 - System Modeling and Optimization Berlin, September 12-16, 2011
More informationConsider a particle in 1D at position x(t), subject to a force F (x), so that mẍ = F (x). Define the kinetic energy to be.
Chapter 4 Energy and Stability 4.1 Energy in 1D Consider a particle in 1D at position x(t), subject to a force F (x), so that mẍ = F (x). Define the kinetic energy to be T = 1 2 mẋ2 and the potential energy
More informationemulsions, and foams March 21 22, 2009
Wetting and adhesion Dispersions in liquids: suspensions, emulsions, and foams ACS National Meeting March 21 22, 2009 Salt Lake City Ian Morrison 2009 Ian Morrison 2009 Lecure 2 - Wetting and adhesion
More informationHomogenization and error estimates of free boundary velocities in periodic media
Homogenization and error estimates of free boundary velocities in periodic media Inwon C. Kim October 7, 2011 Abstract In this note I describe a recent result ([14]-[15]) on homogenization and error estimates
More informationFrieder Mugele. Physics of Complex Fluids. University of Twente. Jacco Snoeier Physics of Fluids / UT
coorganizers: Frieder Mugele Physics of Comple Fluids Jacco Snoeier Physics of Fluids / UT University of Twente Anton Darhuber Mesoscopic Transport Phenomena / Tu/e speakers: José Bico (ESPCI Paris) Daniel
More informationParameter Dependent Quasi-Linear Parabolic Equations
CADERNOS DE MATEMÁTICA 4, 39 33 October (23) ARTIGO NÚMERO SMA#79 Parameter Dependent Quasi-Linear Parabolic Equations Cláudia Buttarello Gentile Departamento de Matemática, Universidade Federal de São
More informationDLVO interaction between the spheres
DLVO interaction between the spheres DL-interaction energy for two spheres: D w ( x) 64c π ktrϕ e λ DL 2 x λ 2 0 0 D DLVO interaction w ( x) 64πkTRϕ e λ DLVO AR /12x 2 x λd 2 0 D Lecture 11 Contact angle
More informationOPTIMAL CONVERGENCE RATES FOR THE COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH POTENTIAL FORCES
OPTIMAL CONVERGENCE RATES FOR THE COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH POTENTIAL FORCES RENJUN DUAN Department of Mathematics, City University of Hong Kong 83 Tat Chee Avenue, Kowloon, Hong Kong,
More informationContinuum Modeling of Transportation Networks with Differential Equations
with Differential Equations King Abdullah University of Science and Technology Thuwal, KSA Examples of transportation networks The Silk Road Examples of transportation networks Painting by Latifa Echakhch
More informationLattice Bhatnagar Gross Krook model for the Lorenz attractor
Physica D 154 (2001) 43 50 Lattice Bhatnagar Gross Krook model for the Lorenz attractor Guangwu Yan a,b,,liyuan a a LSEC, Institute of Computational Mathematics, Academy of Mathematics and System Sciences,
More informationCapillary surfaces and complex analysis: new opportunities to study menisci singularities. Mars Alimov, Kazan Federal University, Russia
Capillary surfaces and complex analysis: new opportunities to study menisci singularities Mars limov Kazan Federal University Russia Kostya Kornev Clemson University SC Outline Intro to wetting and capillarity
More informationSliding Friction in the Frenkel-Kontorova Model
arxiv:cond-mat/9510058v1 12 Oct 1995 to appear in "The Physics of Sliding Friction", Ed. B.N.J. Persson (Kluwer Academic Publishers), 1995 Sliding Friction in the Frenkel-Kontorova Model E. Granato I.N.P.E.,
More informationA PRACTICALLY UNCONDITIONALLY GRADIENT STABLE SCHEME FOR THE N-COMPONENT CAHN HILLIARD SYSTEM
A PRACTICALLY UNCONDITIONALLY GRADIENT STABLE SCHEME FOR THE N-COMPONENT CAHN HILLIARD SYSTEM Hyun Geun LEE 1, Jeong-Whan CHOI 1 and Junseok KIM 1 1) Department of Mathematics, Korea University, Seoul
More information