DYNAMIC STUDY OF THE CONTACT ANGLE HYSTERESIS IN THE PRESENCE OF PERIODIC DEFECTS *

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1 11 th Natioal Cogress o Theoretical ad Applied Mechaics, 2-5 Sept. 2009, Borovets, Bulgaria DYNAMIC STUDY OF THE CONTACT ANGLE HYSTERESIS IN THE PRESENCE OF PERIODIC DEFECTS * STANIMIR ILIEV, NINA PESHEVA Istitute of Mechaics, Bulgaria Academy of Scieces, Acad. G. Bochev St. 4, 1113 Sofia, Bulgaria s: stai@imbm.bas.bg, ia@imbm.bas.bg VADIM NIKOLAYEV ESEME, Service des Basses Températures, INAC/CEA, Greoble, Frace ad ESEME, PMMH-ESPCI-P6-P7, 10, rue Vauqueli, Paris Cedex 5, Frace vadim.ikolayev@espci.fr ABSTRACT. A modificatio of the cotact lie dissipatio model is proposed to treat the cotact lie motio o chemically heterogeeous substrates with sharp border betwee the two types of domais with differet surface eergies. It is applied to the quasi-static motio of the cotact lies i the Wilhelmy plate geometry where a solid plate is withdraw/immersed vertically at a costat speed from/ito a bath of liquid. The plate surface is a 2D patter of roud spots (defects) of surface eergy differet from the rest of the plate. The spot patter is spatially periodic i both horizotal ad vertical directios. A umerical code is developed for simulatio of the motio of the 3D liquid meiscus icludig the three-phase cotact lie for differet plate velocities. Critical velocities are obtaied beyod which the liquid etraimet or dryig thresholds are attaied. The average cotact agle is obtaied from the capillary cotributio to the force actig o the plate. It is show that the chemical heterogeeity leads to a larger the zero value of the average cotact agle for the etraimet threshold ad smaller tha 180 value for the dryig threshold. KEY WORDS: relaxatio, dissipative system dyamics, dyamic cotact agles 1. Itroductio. The spreadig/recedig of liquid o o-ideal solid surfaces is still a ope problem of geeral iterest. This problem is very importat sice most real surfaces appearig i ature, i the laboratories ad i the differet techological processes, are rough ad/or heterogeeous. We study here the ect of the substrate chemical heterogeeity o the cotact lie (CL) motio ad i particular o the dyamic cotact agle (CA) i the framework of the Cotact Lie Dissipatio Approach (CLDA) [1, 2]. We cosider the motio of the cotact lie i the Wilhelmy plate geometry (a vertical flat solid * S.I. has received fiacial support from the NSF-Bulgaria uder grat umber DО /2008.

2 Iliev, Pesheva ad Nikolayev plate is withdraw/pushed from/ito a liquid bath at costat velocity u, see Fig. 1). The heterogeeous surface of the solid plate cosists of two types of domais; the surface eergy chages sharply o the borders betwee them. The ect of the substrate chemical heterogeeity o the CL motio i the framework of the CLDA is studied i a umber of previous works. I [3-5] the ifluece of the heterogeeity is modeled by icludig i the uatio for the CL motio of a additioal term: a elastic restorig force. The cocept of the elastic force is developed earlier, see e.g. [6]. Nikolayev [7] obtaied a aalytical CLDA uatio describig the CL dyamics ad solved it umerically for the CL motio o a substrate with periodic heterogeeity. However, he assumed that the slope of the liquid surface is small, which is ot always satisfied. Our goal here is to exted such a aalysis to ay iterface slope by performig a full-scale 3D dyamic umerical simulatio of the CL slow motio o a substrate with periodic heterogeeity withi the CLDA. 2. Problem formulatio. The classical Wilhelmy plate surface is described with Cartesia coordiates ( y, z ), where the y-axis is horizotal ad the z-axis is directed upward. The plate surface is ifiite ad cosists of regularly spaced idetical circular spots with surface eergy differet from that of the rest of the surface. Fig. 1. Schematic drawig of the cosidered system. The spot ceters form a quadratic lattice with uit cell of size λ. The spot radius is deoted by a. Betwee the spots the surface is homogeeous is characterized by uilibrium cotact agle θ ( 1) that it forms with the liquid meiscus. The spots are ( 2) ( 1) characterized by uilibrium cotact agle θ θ so that the surface eergy of the plate chages sharply at the borders of the spots. The liquid meiscus forms a cotact lie L with the movig plate. The dyamic cotact agle value θ ( R), where R L deotes a poit belogig to the plate surface, varies alog the CL. The distace betwee the plate ad the (distat) bath wall opposite to the plate is deoted by d x. The cotact lie Lt ( ) at time momet t is described by its height hyt (, ) relative to the meiscus level sufficietly far from the plate, where the meiscus is horizotal.

3 Dyamic Study of the Cotact Agle Hysteresis We study the motio of the CL i the framework of the CLDA. I the CLDA oe eglects the viscous dissipatio i the bulk of the liquid with respect to that occurrig i the viciity of the movig CL. The dissipatio is the cosidered as 2 localized at the CL. The dissipatio rate per CL legth Σ = ξv 2 is proportioal to l the squared local CL velocity v (assumed to be positive whe the liquid advace). We deote by ξ the frictio coiciet, which has the same dimesioality as the liquid viscosity. We assume here that ξ is idepedet of v, which holds, e.g., for the molecular-kietic model of the cotact lie motio [8]. For simplicity we assume here that both types of domais have the same ξ value. Accordig to the CLDA, the motio of the poit R of the CL is give by: (2.1) v ( L( R, t) ) = γ (cosθ ( R) cos θ( R )) ξ, where θ ( R ) is the uilibrium (Youg) cotact agle ual to θ ( 2) or to ( 2) θ depedig o R, γ is the surface tesio. Whe the plate is movig dowwards with velocity u, the CL velocity with respect to the solid surface is v = h + u, where h is the time derivative of the CL height. Equatio (2.1) holds whe θ ( ) R is a cos smooth fuctio i the eighborhood of the poit R. However, whe R belogs to the border betwee two domais where the surface eergy chages sharply, oe should use the coditio obtaied by Iliev ad Pesheva [9]. Accordig to it, the border poit R L remais immobile with respect to the plate while () 1 ( 2) ( 2) ( 1) ( 2) ( 1) θ( R) θ, θ if θ > θ or θ( R ) θ, θ if ( 2) ( 1) θ < θ. This pheomeo is at the origi of the stick-slip behavior of the CL.Equatio (2.1) ca be made dimesioless by expressig the legths i terms of the capillary legth l = γ / ρ g c ( ρ : liquid desity; g : gravity acceleratio) ad the time i terms of the characteristic time τ = l / 0 c ξ γ, i.e., x = x/ l, y = y / l, z = z / l, t = t / τ (ad therefore the c c c 0 dimesioless velocity is v = ξv / γ ): (2.5) v ( L( R, t )) = cosθ ( R ) cosθ. Our study here focuses o a specific system studied previously i [7] for which (1) (2) θ = 70, θ = 110, a = 0.1, ad λ = Numerical algorithm. We employ the followig umerical algorithm [2] havig three basic steps. First, for a give positio of the CL at the movig plate the statioary shape of the meiscus of the liquid with fixed volume i the domai eclosed betwee the surfaces x = 0 ad x = d is determied. We impose periodic boudary coditios i x y-directio ad we set the right cotact agle at the cotaier wall opposite to the movig plate. Next, velocity at every poit of the CL is obtaied directly from the

4 Iliev, Pesheva ad Nikolayev uality of variatios of the free eergy ad of the CL dissipatio (which is uivalet to usig Eq. (2.1) [2]). Third, the CL positio at the ext istat of time is foud explicitly from the kiematic coditio Δ R = v Δ t. The above algorithm is repeated for the successive time steps. The mai igrediets of this algorithm are the determiatio of the meiscus shape with give volume of the liquid i the tak ad give cotact lie, ad the calculatio of the velocity of the cotact lie. The determiatio of the meiscus shape is realized by a iterative miimizatio procedure based o the local variatios method ad adapted to treat the Wilhelmy plate geometry i [10]. 4. Numerical results. We study the CL motio for differet plate velocities. Our umerical simulatios show that i a certai iterval of (dimesioless) plate velocities * * u u, u r a a double periodic solutio (both i t ad y ) for hyt (, ) appears after the CL passes through several rows of defects. Outside this iterval the double periodic * solutios do ot appear. The liquid is etraied by the plate whe u u r (dyamic complete wettig; Ladau-Levich trasitio). The gas etraimet betwee the liquid * ad the plate (dyamic complete uwettig) occurs whe u u a. I Fig. 2 we show the obtaied cotact lies (i a referece system immobile with respect to the plate). The dotted lie shows the border of the circular defect where the uilibrium cotact (2) agle is θ = 110. The successive cotact lies are show with solid lies (goig from top to bottom). Oe ca see that, whe the CL reaches the upper ed of the defect, the CL accelerates sice the local CL velocity sharply icreases with cos 70 cos110. The part of the CL which lies outside the defect is also etraied. We observe also that whe the cotact lie reaches the lower ed of the defect a part of the cotact lie remais fixed to the border of the defect. These cotact lies which have part stuck to the border of the defect are show with dashed lies. Thus a stick-slip behavior of the CL i the CLDA is modeled. Here we are iterested i the calculatio of the wettig force (or else the capillary restorig force) which the liquid exerts o the solid plate. It ca be determied experimetally by tesiometric methods i Wilhelmy plate geometry ad is used to study the dyamic CA ad its depedece o the CL velocity [5]. Whe a periodic motio of the CL is attaied, the time ad space averaged force is (the time period is P= λ / u ): P λ γ (4.1) F = cosθ ( y) ( z ( y) ) dydt λp 0 0 With this force oe ca itroduce the ective dyamic CA heterogeeous surface, (4.2) cos θ = F / γ. θ for the

5 Dyamic Study of the Cotact Agle Hysteresis I Fig 3. we plot the results for cosθ as fuctio of u ~ u* r z 0.7 cosθ y ~ u* a ~ u Fig. 2. Uit cell λxλ of the plate surface icludig oe of the surface defects. The dash-dotted lie is the border of the defect. The CLs positios (i a referece system immobile with respect to the plate) for the double periodical solutio are show with time step Δ t = 0.15 for the plate movig upwards with velocity u = 0.1. The CLs are show with solid lies except for those which are partially stuck to the border of the defect (dashed lies). Fig. 3. The cosie of the ective cotact agle as fuctio of the plate velocity u. A cotact agle hysteresis 13 is observed whe u 0. Critical velocities u * ad u * r a are obtaied above which the etraimet trasitios occur for recedig or advacig motio, respectively. For the cosidered type of heterogeeity we obtai a depedece close to liear betwee the velocity ad the cosie of the ective CA. The CA hysteresis is observed both i dyamics ad i statics. The static hysteresis (for u 0 ) is of the order of 13. The dyamic hysteresis ca be defied as a differece i θ for u of the same modules but opposite sigs. Threshold velocities u *, u * (ad correspodig critical advacig ad a r *a *r recedig dyamic cotact agles θ, θ ) are obtaied. Beyod them the etraimet trasitios occur. Oe ca see that at the etraimet threshold * * r cosθ < 1 (i.e., θ > 0 * a ad θ < 180 ). Note that, withi CLDA, the etraimet trasitios for a homogeeous plate occur whe cosθ = 1. This differece appears because, for heterogeeous plate, the etraimet threshold is attaied first for oe of the domais. Such a behavior is similar to that described by the hydrodyamic approach [11]. It shows that the experimetally observed iuality for the Ladau- Levich threshold θ * > 0 [11] ca be explaied by the iteractio with the surface heterogeeity withi the quasi-static CLDA, without usig more complicated

6 Iliev, Pesheva ad Nikolayev hydrodyamic approaches. 5. Coclusio. A quasi-static cotact lie dissipatio approach is applied to simulate the slow motio of the cotact lie o a chemically heterogeeous substrate with sharp spatial variatio of the surface eergy. A stick-slip behavior of the CL is observed. The cotact agle hysteresis is obtaied both i statics ad dyamics. Critical velocities ad critical ective cotact agles are determied beyod which the liquid or gas are etraied by the plate. The fiiteess of the dyamic cotact agle at Ladau-Levich liquid etraimet trasitio ca be explaied by the iteractio with the substrate heterogeeity. R E F E R E N C E S [1] de Gees, P. G. Wettig: statics ad dyamics. Rev. Mod. Phys., 57 (1985), [2] Iliev S., N. Pesheva, V. S. Nikolayev. Quasistatic relaxatio of arbitrarily shaped sessile drops. Phys. Rev. E, 72 (2005), [3] Raphael, E., P.G. de Gees. Dyamics of wettig with oideal surfaces. The sigle defect problem. J. Chem. Phys., 90 (1989), [4] Joay, J.-F., M. O. Robbis. Motio of a Cotact Lie o a Heterogeeous Surface. J. Chem. Phys., 92 (1990), [5] Mouliet, S., C. Guthma, E. Rolley. Dissipatio i the dyamics of a movig cotact lie: ect of the substrate disorder. Eur. Phys. J. B, 37, (2004), [6] Joay J.F., P.G de Gees. A model for cotact agle hysteresis. J. Chem. Phys., 81 (1984), [7] Nikolayev, V. S. Dyamics ad depiig of the triple cotact lie i the presece of periodic surface defects. J. Phys.: Codes. Matter., 17 (2005), [8] Blake, T. D., J. M. Hayes. Kietics of Liquid/Liquid Displacemet. J. Colloid Iterface Sci., 30 (1969), [9] Iliev, S., N. Pesheva. Wettig Properties of Well Structured Heterogeeous Substrates. Lagmuir, 19 (2003), [10] Iliev S., N. Pesheva, V. S. Nikolayev. Dyamic modellig of cotact lie deformatio: compariso with experimet. Phys. Rev. E, 78 (2008), [11] Delo G., Fermigier M., Soeijer J., Adreotti B. Relaxatio of a dewettig cotact lie. Part 2: Experimets. J. Fluid Mech. 604 (2008),

Analysis of composites with multiple rigid-line reinforcements by the BEM

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