‘Gas cushion’ model and hydrodynamic boundary conditions for superhydrophobic textures. Tatiana V. Nizkaya,1 Evgeny S. Asmolov,1,2,3 and Olga I. Vinogradova1,4,5,∗ 1A.N. Frumkin Institute of Physical Chemistry and Electrochemistry, Russian Academy of Sciences, 31 Leninsky Prospect, 119071 Moscow, Russia 2Central Aero-Hydrodynamic Institute, 140180 Zhukovsky, Moscow region, Russia 3Institute of Mechanics, M. V. Lomonosov Moscow State University, 119991 Moscow, Russia 4Department of Physics, M. V. Lomonosov Moscow State University, 119991 Moscow, Russia 5DWI - Leibniz Institute for Interactive Materials, RWTH Aachen, Forckenbeckstraße 50, 52056 Aachen, Germany 4 1 Superhydrophobic Cassie textures with trapped gas bubbles reduce drag, by generating large 0 effectiveslip, which isimportant for avarietyofapplications thatinvolveamanipulation ofliquids 2 at the small scale. Here we discuss how the dissipation in the gas phase of textures modifies their t friction properties. We propose an operator method, which allows us the mapping of the flow in c the gas subphase to a local slip boundary condition at the liquid/gas interface. The determined O uniquely local slip length depends on the viscosity contrast and underlying topography, and can be immediately used to evaluate an effective slip of the texture. Besides superlubricating Cassie 6 surfacesourapproachisvalidforroughsurfacesimpregnatedbyalow-viscosity‘lubricant’,andeven ] forWenzeltextures,wherealiquidfollows thesurface relief. Theseresultsprovideaframework for n therational design of textured surfaces for numerousapplications. y d PACSnumbers: 83.50.Rp,47.61.-k,68.03.-g - u l f I. INTRODUCTION. z . s c x i y s y Superhydrophobic (SH) textures have raised a consid- e h erable interest and motivated numerous studies during p the past decade. Such surfaces in the Cassie state, i.e., [ where the texture is filled with gas, can induce excep- L δ tional wetting properties [1] and, due to their superlu- 3 FIG. 1: Sketch of the typical 1D SH surface represented by v bricatingpotential[2–5],arealsoextremelyimportantin rectangular grooves rigorously studied here, but some of our 0 contextoffluiddynamics. Toquantifythedragreduction conclusionsaregeneralandapplyforany1Dand2Dtextured 1 associated with two-component (e.g., gas and solid) SH surfaces - pillars, holes, lamelae. 6 surfaceswithgivenareafractionsit isconvenienttocon- 4 struct the effective slip boundary condition(on the scale . 1 largerthanthepatterncharacteristiclength)fortheaver- tions 0 agedvelocityfield. Thisconditionisappliedattheimag- 4 inary smooth homogeneous surface [6, 7], which mimics ∂u ∂u 1 z =0: u=u , µ τ =µ gτ, (1) theactualoneandfullycharacterizestheflowatthereal g g : ∂z ∂z v surface and is generally a tensor [8, 9]. Once eigenval- i ues of the slip-length tensor, which depend on both the where u and µ are the velocity and the dynamic viscos- X hydrodynamicboundaryconditionatthesolid/liquidin- ity of the liquid, and u and µ are those of the gas, r g g a terfaceandviscousdissipationinthegasphase,aredeter- uτ = (ux,uy) is the tangential velocity. Although this mined, they canbe usedto solvecomplex hydrodynamic problem has been resolved numerically for rectangular problems without tedious calculations. A key difficulty grooves [13, 14], such a strategy appears rather hope- is that there is no general analytical theory that relates less in context of exact analytical results, especially for this dissipation to the relief of the texture, so that prior complex configurations, which are typical for many ap- work often neglected it, by imposing idealized shear-free plications. An elegant semi-analytical approach based boundary conditions at the gas sectors [10–12]. on an assumption of a constant shear inside the groove hasbeen proposedrecently[15]. Note that althoughthis Toaccountforadissipationwithinthegassubphaseit derivation made a significant step forward, it remains is necessaryto solveStokesequations byapplying condi- approximate and does not take into account the total dissipation in the gas subphase. To bypass this problem, it is advantageous to replace the two-phase approach,by a single-phase problem with ∗Correspondingauthor: [email protected] spatially dependent partial slip boundary condition [2, 2 16], which takes a form project both the velocity and its normal derivative on a grid{y }N at the liquid/gasinterface. Then the opera- ∂u i i=1 z =0: u −b(x,y) τ =0, (2) tor becomes a matrix Pij that relatesthe shear rate at a τ ∂z given point i with velocities in every other points of the interface j =1..N (the condition is essentiallynonlocal). where b(x,y) is the local slip length at the gas areas, Unlike the local slip length, the operator depends only which is normally assumed to conform the texture relief on the texture relief, but not on the solution outside. It according to predictions of the ‘gas cushion’ model [17] is universal and, once calculated for a given topography, µ can be applied for any geometry of the outside flow and bx,y(x,y)≃kx,y e(x,y), (3) µ any viscosity ratio. Then, in view of Eq.(1), the non- g localboundarycondition for fluid flow pasta SH surface where prefactors kx,y = 1 can reduce to 1/4 if the net reads: gas flux becomes zero (due to end walls) [18]. Such an N ∂u (y ,0) µ approach,justified for a continuous gas layer at a homo- x,y i − Px,yu (y ,0)=0, i=1..N. (5) geneous surface [17] and later for shallow grooves[18], is ∂z µg ij x,y j j=1 X by no means obvious for an arbitrary texture, where the ThisboundaryconditionallowstosolvetheStokesequa- gassubphasecanbe deepandstronglyconfined. Insuch tionsfortheliquidphaseseparatelyandtodeterminethe asituationitremainslargelyunknownifthegasflowcan local slip length by using Eq.(2): be indeed excluded from the analysis being equivalently replacedbyb(x,y),andhow(andwhether)thislocalslip bx,y(y)= µ ux,y(y,0). (6) profile is uniquely related to the relief of the texture. µ Px,y[u ] g x,y In this article, we propose a general theoretical Werecallthathelocalsliplengthmaydependnotonlyon method, which allows us to generalize the ‘gas cushion’ thetexturerelief,butalsoonthestateoftheliquidphase, model for any 1D and 2D two-phase SH textures in the whichcouldaffectthevelocityu . However,forasingle Cassie state, or rough surfaces impregnated by a low- x,y surface we consider here the local slip length is uniquely viscosity ‘lubricant’. We also show that our approach relatedto the texture reliefand the generalizationof the canbeappliedevenfortexturesintheone-phaseWenzel ‘gas cushion’ model can be constructed. For confined state, where the liquid follows the topological variations configurations (e.g. flow in a thin channel) the local slip of the texture. length will of course be a property of the whole system, but note that the Px,y-operatorswill remain exactly the same. II. THEORY. To calculate the matrices Px,y we should solve the ij problem in the gas phase and extractthe normalderiva- A. General consideration tive of the solution either analytically or numerically. In this paper we rigorously calculate them for a rectangu- To illustrate our approach,we consider 1D SH surface lar groove using a Fourier method. For an arbitrary 1D of period L, and assume the interface to be flat with no geometry Px,y can be expressed in the form of a bound- meniscus curvature (see Fig. 1). Such an idealized situ- ary integral operator involving Green’s functions for the ation, which neglects an additional mechanism for a dis- Stokes flow [29]. As a side note, we remark that it can sipation due to a meniscus [19, 20], has been considered be similarly constructed for 2D surfaces, but of course in most previous publications [6, 11, 21] and observed by using Green’s functions for 3D Stokes flow [29]. Note in recent experiments [22]. We then impose no-slip at howeverthat one does not expect the main physicalpic- the solid area, i.e. neglect slippage of liquid [23–26] and turetobe alteredinthese(moretechnicallychallenging) gas [27] past hydrophobic surface, which is justified pro- situations, and we leave the study of these complex ge- vided the nanometric slip is small compared to param- ometries for a future work. eters of the texture. No further assumptions are made, asidefromdistinguishingbetweenlongitudinalandtrans- verse gas flow, with kx =1 and ky =1/4,to address the B. Periodic rectangular grooves. most anisotropic case. The linearity of Stokes equations implies that the For aninitial applicationof our approach,we consider boundary condition at the liquid/gas interface for lon- now periodic rectangular grooves of width δ and depth gitudinalandtransversedirectionscanbe formulatedas: e. The fraction of gas area is then φ=δ/L. In this par- ticular case the problem inside the groove can be solved ∂u z =0: gτ −Px,y[u ]=0, (4) using the Fourier method. For the longitudinal flow this gτ ∂z yields an analytical expression for the DtN matrix: where we introduced the linear operator Px,y that be- 1 longs to a general class of Dirichlet-to-Neumann (DtN) Pixj = δ FimΠmlFl−j1 , (7) ones [28]. The meaning of Eq.(4) becomes clear if we Fim =co(cid:16)s(km∗ yi/δ), Π(cid:17)ml =km∗ coth(km∗ e/δ)δml, 3 where k∗ = (2m−1)π and δ is the Kronecker delta. zerobyhavingthesameslope(whichhasnotbeentaken m ml Note that the matrix combination inside the brackets in intoaccountinrecentwork[15]). Thisslopecanbefound Eq.(7) depends only on the aspect ratio, e/δ (and the byasymptoticanalysisinthevicinityofthegroovesedge. spatial grid used). The same is true for the transverse Motivatedbyanearliersingle-phaseanalysis[31,32], we direction, although the matrix Py can be obtained only can now construct the asymptotic solution for the two- ij semi-analytically (see Appendix B). phaseflowneartheedgebyusingpolarcoordinates(r,θ) Having calculated Px,y, we can then use the Fourier (see Fig. 3(a) and Appendix C for details). ij method to solve the Stokes equations for liquid with the non-local boundary condition Eq.(5) (see Appendix A for details). We stress again that the resulting problem is not affected by the texture relief or the method used to find Px,y due to a half-space liquid domain. From ij theliquidvelocityfieldwecanextractboththelocalslip length profile b (y) (by using Eq. 6) and the effective x,y sliptensorbk,⊥ (byaveragingovertextureperiod)aswill eff be discussed below. III. RESULTS AND DISCUSSION. A. Local slip length FIG. 3: Polar coordinates (side view) used to evaluate the Figs. 2(a) and (b) show profiles of the longitudinal, flow near the edge of the groove, (a) and illustration (top bx(y), and transverse, by(y), local slip lengths at fixed view)ofthelocalsliplengthbehaviorattheedgeof2Dpillars groovewidth, δ/L=0.75, and aspect ratio, e/δ, varying (b),hollows (c) and channels (d). from 0.1 to infinity. The calculations are made using µ/µ =50,whichcorrespondstoaSHtexturefilledwith g Close to the edge, when r ≪ 1, the general solution gas. It can be seen that for shallow grooves, e/δ ≪ 1, of the Stokes equations implies a power-law dependence localsliplengthssaturatetoconstantvaluespredictedby of velocities on the distance, u ∝ rλ: u = rλasin(λθ), Eq.(3) at the central part of the gas sector, but bx,y(y) u = rλ[csin(λθ)+hcos(λθ)]. Similaxr arguments are gx vanish at the edge of the groove. Thus the local slip validforthetransverseconfiguration. Thisyieldsalinear profilescanberoughlyapproximatedbyatrapezoid[30]. dependence for the slip lengths, bx,y = rδb′ (λ). The For deeper grooves the local slip curves look more as x,y exponent λ can be found from the boundary conditions parabolic. At e/δ ≥ 1 they converge to a single curve at the solid walls and the liauid/gas interface. suggesting that bx,y(y) of deep grooves are controlled by For large µ/µ we obtain (see Appendix C): g thevalueofδonly,beingindependentonatexturedepth. This result does not support Eq.(3), which predicts that b′ ≃2µ/µ , b′ ≃µ/(2µ ). (8) x g y g bx,y(y) are growing infinitely with e, and indicates that for large e the dissipation at the edge of the grooves be- Theaboveexpressionsforb′x andb′y giveupperandlower comes crucial. bounds on slopes among all textures, which are attained when the main flow is tangent or normal to the border of the gas area. Therefore, bx is constrained by δµ/µ , g 15 (a) e/δ ≥ 1 6 (b) e/δ ≥ 1 and by by δµ/4µg (see Fig.3(b-d)), so that the local slip profilesforarbitrarytextures shouldbe similarto shown 0.5 10 4 0.5 in Fig.2, althoughthe absolute value of maximum might xb/L 0.25 yb/L differ. It is naturalnow to propose a generalizationof Eq.(3) 5 2 0.25 for a SH surface, where we scale with δ instead of L: 0.1 µ bx,y(y)=δ βx,y(e/δ,y/δ). (9) −0.5 0 0.5 −0.5 0 0.5 µ y/L y/L g Herewe ascriberescaleddimensionlesslocalsliplengths, FIG.2: Longitudinal(a)andtransverse(b)local sliplengths βx,y, which become linear in e/δ when e/δ is small, and computed with φ = 0.75, µ/µg = 50. Dashed lines show the we recover Eq.(3). At the other extreme, when e/δ is predictions of Eq.(3). large, βx,y saturate to provide an upper limit for local slip lengths. Indeed,thedatapresentedinFigs.2(a)and(b)suggest ToverifythisAnzatzinFig.2(a)and(b)weplotβx,y thatneartheedgeofthegroovebx,yalwaysaugmentfrom asafunctionofy/δ atdifferentφande/δ. Hereweusea 4 0.5 0.2 (a) e/δ ≥ 1 (b) e/δ ≥ 1 ln sec πφ 0.15 bk ≃ L 2 , 0.3 eff π 1+ L ln shec π(cid:16)φ (cid:17)+itan πφ xβ yβ 0.1 πbxc 2 2 (11) 0.1 0.1 0.05 0.25 b⊥eff ≃ 2Lπ L hlnh(cid:16)sec(cid:16)(cid:17)π2φ(cid:17)i (cid:16) (cid:17)i . 1+ ln sec πφ +tan πφ 2πby 2 2 c −0.5 0 0.5 −0.5 0 0.5 h (cid:16) (cid:17) (cid:16) (cid:17)i y/δ y/δ Let us now try to define apparent constant local slip lengths at the gas sectors. Eq.(9) suggests the following FIG.4: Rescaledlongitudinal(a)andtransverse(b)localslip- definition length fordeep andshallow grooves. Solid anddotted curves µ correspond to the Cassie, µ/µg =50, and Wenzel, µ/µg =1, bxc,y =δµ βcx,y, (12) states at φ = 0.1,0.5,0.9 (these curves coincide for shallow g grooves and are nearly overlapping for deep grooves). Dash- where dimensionless slip lengths, βx,y, depend only on dotted lines show asymptotic solutions near the edges of the c the aspect ratio of the texture, e/δ. We fitted our the- groove. oretical results for µ/µ = 5 to Eq.(11) taking βx,y as a g c fitting parameter. The obtained values are surprisingly well described by simple functions viscosityratiooftheCassiestateasinFigs.2(a)and(b). Alsoincludedareresultscalculatedforthe Wenzelstate, erf(q e/δ) erf(q e/δ) asenedtµh/aµtgfo=rr1e.laTtihveelryesdueletpsagrreosoovmese,we/hδat≥re1m,aβrxk,yabplreo.fiWlees βcx ≃ qxx , βcy ≃ 4qyy , (13) computed for different µ/µ , e/δ and even φ, practically g with q ≃ 3.1 q ≃ 2.17. These functions saturate to x y converge into a single curve [33], which coinsides with βx ≃0.32andβy ≃0.12alreadyate/δ ≥1,byimposing c c the numerical (but not semi-analytical) results reported constraints on the attainable bx,y (see Fig. 5). c before [15]. e/δFo≤r 0sh.2a5llofowragrtoroavnessve(res/eδc≤ase0).1thfeorβax,ylopnrgoitfiuledsindaelpaenndd 0.3 (a) 0.1 (b) onlyonthedepthofthegroove,andcanbeapproximated 0.2 xβc yβc0.05 by trapezoids with the central region of a constant slip 0.1 given by Eq.(3), and linear edge regions where the local slip length is described by our asymptotic model. 00 0.5 1 00 0.5 1 e/δ e/δ FIG. 5: Apparent local slip as a function of the depth of the groove for longitudinal (a) and transverse (b) directions. B. Effective slip length SymbolsshowthevaluesobtainedusingEqs.(11),solidcurves show predictions of Eqs.(13). We finally turn to the effective slip lengths, which can Assuming βx,y found for µ/µ = 5 are universal, we c g be found by averaging the obtained numerical solution canthenuseEq.(11)tocalculatetheeffectivesliplengths for longitudinal and transverse directions: forµ/µ =1and50. TheresultsareincludedinFig.6. A g generalconclusionisthatthepredictionsofEq.(11)with hu i the local slip defined by Eq.(12) are in excellent agree- bkeff,⊥ = h∂ ux,y i . (10) ment with exact theoretical results in the whole range z x,y (cid:12)z=0 of parameters, 0.1 ≤ φ ≤ 0.9 and µ/µ ≥ 1, confirm- (cid:12) g (cid:12) ing the universality of βx,y. Note that included in Fig.6 (cid:12) c The calculations are made using the viscosity ratio of effectivesliplengthsforperfect-slipstripes[10,12],prac- the Cassie and Wenzel states. For completeness we in- tically coincide with our results for µ/µ = 50. We can g clude the data for µ/µ = 5, which correspond to oil- then conclude that SH surfaces in the Cassie state pro- g impregnated textures. Fig. 6 shows longitudinal (a,b) vide the very general upper bound for effective slip of and transverse (c,d) effective slip lengths as a function textured surfaces, valid for whatever large viscosity con- of solid fraction, 1−φ, for shallow (a,c) and deep (b,d) trast (e.g. polymer melts [34]). Finally, we observe an grooves. The eigenvalues of the effective lengths of a excellentagreementof ourresults forµ/µ =1 with ear- g striped surface with a piecewise constant local slip, bx,y, lier data even for the Wenzel state obtained by using a c have been calculated analytically [16]: completely different approach [31]. 5 tures. These textures include various pillars, and holes 0.6 0.6 e/δ=0.1 (a) e/δ=1 (b) and lamellae of a complex shape. Thus, our results may guide the design of textured surfaces with superlubri- 0.4 0.4 ||b/Leff0.2 0.2 cmaetrinsgcipenotceen,taianldinmomriec.roAflnuoidthicerdefrvuicitefsu,ltdriibreocltoigoyn,cpooulyld- be to apply our method to calculations of an electro- 0 0 osmotic [35–37] and diffusio-osmotic [38] flow past tex- 0.1 0.25 0.5 0.9 0.1 0.25 0.5 0.9 1−φ 1−φ tured surfaces. 0.3 0.3 e/δ=0.25 (c) e/δ=1 (d) Appendix A: Numerical solution in liquid 0.2 0.2 L ⊥/eff b 0.1 0.1 The non-local boundary condition given by Eq. (5) is easy to implement with the Fourier method. For sim- 0 0 0.1 0.25 0.5 0.9 0.1 0.25 0.5 0.9 plicity we consider only the longitudinal flow here, the 1−φ 1−φ procedure for the transverse flow is similar. We present the flow over the superhydrophobic surface as a sum of the undisturbed flow and the correction due to the slip: FIG. 6: Longitudinal (a,b) and transverse (c,d) effective slip lengthsfortextureswithshallow(a,c)anddeep(b,d)grooves. ux(y,z)=u0(z)+u1(y,z), Fromtoptobottomµ/µg =50,5,1. Exacttheoreticalresults where u (z) = Gz is a simple shear flow and G is the areshownbycircles,analyticalresults[Eq.(11)]withlocalslip 0 undisturbed shear rate. The correction is sought in the given by Eq.(12) are plotted by solid curves. Filled squares show earlier data for perfect slip [10, 12], filled circles show formofa cosineseries,since the solutionis symmetric in earlier data for the Wentzelstate [31]. y: ∞ u (y,z)=c + c cos(k y)exp(−k z), 1 0 m m m Now,werecallthatforpillarsinthelowφlimit,δ ≃L, m=1 the average local slip was shown to scale as [21]: X where k =2πm/L. m µ e Boundary conditions for the correction read: bx,y ≃L βtanh (14) a µ Lβ ∂u g (cid:18) (cid:19) 1 −Px[u ]=−G, 0<y <δ/2, ∂z 1 (A1) with β(φ). For deep dilute pillars Eq.(14) transforms to u =0, δ/2<y <L/2. µ 1 bx,y ≃ L β (cf. Eq.(12)), and for pillars with φ = 0.9 a µ (due to the symmetry of the problem, it is sufficient to g we evaluate β ≃0.2 [21]. This value is close to the exact consider only a half of the period). We take the same onesfoundherefordeeprectangulargrooves,βx,y,soour spatial grid that is used for the DtN matrix: yi = δηi c theory provides a good sense of the possible local slip of with N nodes over the groove 2D texture. y =δ(i−1)/2N, i=1...N i andadd M nodes at the boundary in contactwith solid: IV. CONCLUSION. y =(L−δ)(i−1)/[2(M−1)]+δ/2, i=N+1...N+M. i We cut the series to N =N +M terms and obtain the We have proposedan operator method, which allowed f following linear system: usthe mapping ofthe flowinthe gassubphasetoalocal slip boundary condition at the gas area of SH surfaces. Nf Thedeterminedsliplengthisshowntobeauniquefunc- Dikcm =−G, i=1...N, tion of the viscosity contrast and topography of the un- m=1 P Nf (A2) derlyingtexture. Ourmainresults,Eqs.(9)and(12),can c + c cos(k y )=0, i=N +1...N , 0 m m i f be thus viewed as a general ‘gas cushion’ model for tex- m=1 tured surfaces, which transforms to the standard model, P Eq.(3), in case of shallow textures. We have proventhat N besides Cassie surfaces our approach is valid for Wen- D =k cos(k y )− Pxcos(k y ). zel textures, as well as rough surfaces impregnated by a in n n i ij n j ‘lubricant’ with lower viscosity. Xj=1 We checkedthe validity ofourapproachbystudying a OncethematrixPx isknown,thelinearsystem(A2)can ij flowpastcanonicalrectangulargrooves,butourstrategy besolvedtofindthecorrection,andthecompletevelocity can be immediately applied for 1D textures with differ- fieldinliquidcanbe calculated. Asimilarprocedurecan ent cross-sections or extended to more complex 2D tex- be applied to the transverse flow (see [18] for reference). 6 Appendix B: Dirichlet-to-Neumann matrix for a The normal derivative of the velocity (using dimen- rectangular groove sional variables y,z) reads: N For rectangulargeometries the Fourier method can be ∂ugx(ηi,0) = Pxu (η ,0), implemented to calculate the DtN matrices. ∂z ij gx j j=1 Longitudinalconfiguration. Weintroducethenon- X dimensional variables in the gas domain, η = y/δ and where ζ = z/δ. Following [14], we seek for the solution of the 1 Laplace equation in gas in the form: Px = F Π F−1 , ij δ im ml lj ∞ (cid:16) (cid:17) u (η,ζ)= c sinh[k∗ (ζ+d)]cos(k∗ η), (B1) is the DtN matrix for the longitudinal flow. gx m m m Transverse configuration. We assume that the liq- m=1 X uid/gas interface is flat, so that u = 0 at ζ = 0. The gz where km∗ = (2m−1)π, d = e/δ. Each term in this se- symmetry condition implies that ugy is symmetric in η ries is a partial solution of the Laplace equation, which while ugz is anti-symmetric. We represent the solution is symmetric inη and satisfiesthe no-slipboundary con- in gas in the following form [14]: ditions at the side walls, η = ±1/2, and at the bottom ∞ cos(k∗η) wall of the groove, ζ =−d, (see Fig. 1). u (η,ζ)= n [A K′(ζ)+B G′ (ζ)] gy k∗cosh(k∗d) n n n n Thevelocityanditsnormalderivativeattheliquid/gas n=1 n n ∞ interface are P + cos(β ζ)D H (η), n n n ∞ n=1 ζ =0: u = c sinh(k∗ d)cos(k∗ η), X gx m m m m=1 (B2) ∂∂uζgx =m∞=1km∗Pcmcosh(km∗ d)cos(km∗ η). ugz(η,ζ)=n∞=1csoisnh((kkn∗n∗ηd))[AnKn(ζ)+BnGn(ζ)] P ∞ PD The relation between them can be obtained from (B2): − sin(βnζ)βnHn′(η), n=1 n X ∞ ∂u gx = (Π u∗)cos(k∗ η), ∂ζ mn n m K = sinh(k∗ζ)−ζexp[−k∗(ζ+d)]sinh(k∗d)/d, m=1 n n n n X G = k∗ζsinh[k∗(ζ+d)], n n n H = 2exp(−β /2)[cosh(β η)−2ηcoth(β /2)sinh(β η)]. u∗ =c sinh(k∗d), n n n n n n n n where k∗ = (2n−1)π and β = nπd; A , B and D n n n n n are the unknown coefficients. The conditions of non- Π =k∗ coth(k∗ e/δ)δ , m,l=1..∞. ml m m ml permeability at the side walls, u = 0 at η = ±1/2, gy and at the bottom wall and at the interface, u = 0 Here Π is the representation of the DtN operator in gz ml at ζ = 0,−d, are satisfied automatically. The no-slip the Fourier space. boundary conditions at the walls of the groove (u = 0 gz The last step is to transform this operator into phys- at η = ±1/2 and u = 0 at ζ = −d) and the con- gy ical space, so that it can be applied as a boundary tinuity condition at the interface (u = v∗ at ζ = 0) gy condition. To do so, we introduce a spatial grid η = i have to be satisfied by a proper choice of the coeffi- (i−1)/(2N), i = 1..N at the liquid/gas interface (due cients A ,B ,D . To do so, we cut the series to N n n n to the symmetry it is sufficient to consider only a half terms and introduce a grid covering the walls of the of the period) and cut the cosine series (B1) to N terms groove and the interface and containing 3N nodes (N accordingly. Considering Eq. (B2) at each grid node we at each wall/interface). Calculating the tangential ve- have: locity at each point of the groove, we obtain a sys- tem of 3N linear equations for a 3N-component vector N ugx(ηi,0)= Fimu∗m, Zk = {A1,...AN,B1...,BN,D1,...Dn}. The right- hand sides of the equations are equal to zero at groove’s m=1 X walls (no-slip) and to liquid velocity at the interface, and hence, u (0,η ) = v∗(η ). The solution satisfying the no-slip gy i i boundary conditions and taking the prescribed values at N the interface can be expressed in a matrix form: u∗ = F−1u (η ,0), m mi gx i i=1 N X Z = M v∗(η ), k kj j where F =cos(k∗ η ) is the collocation matrix. im m i Xj=0 7 where v∗(η ) is a N-component vector of velocity at the Thus the exponent depends on the viscosity ratio µ/µ j g interface gridpoints andM is a 3N×N matrix. Then only. Previousanalyticalsolutions,λ=2/3, forasingle- kj the normal derivative at the interface can be expressed phaserectangularhydrophilicgroovewithµ/µ =1 [31], g in the following way: and λ = 1/2 for a flat shear-free interface with ideal µ/µ = ∞ [10, 39] satisfy the equation obtained. When g ∂v (η ,0) N A theviscosityratioislarge,asforliquid/gascase,wecon- g j = 2 n exp(−k∗d)tanh(k∗d) ∂ζ d n n struct an asymptotic solution of (C3) in terms of series nX=1 (cid:20) in µg/µ≪1: + B k∗ cos(k∗η )=Q Z , n n n j ik k i 1 µg µ2g λ= − +O . (C4) where Qik is N ×3N matrix. 2 µπ µ2! Back in dimensional variables y,z, we obtain the fol- lowing representationfor the N ×N DtN matrix : The asymptotic solution (C4) is close to that for alter- Py =δ−1Q M . nating no-slip and perfect-slip stripes with λideal = 1/2. ij ik kj The local slip length near the edge can be defined as u (r,π) µ x r ≪1: b (r)= ≃2 rδ. x ∂ u (r,π) µ θ x g Appendix C: Asymptotic solution near the edge of the grooves Therefore, b is linear in the distance r from the corner x at the liquid/gas interface. The slope of the dependence Here we obtain a solution in the vicinity of the groove is large, of order of µ/µ . g cornerby using polarcoordinates(r,θ) [30,31], withthe Fortheflowtransversetothegrooves,werepresentthe origin at (yc,zc) = (−δ/2,0), so that y = yc +rδcosθ, solution in liquid in terms of a streamfunction ψ which z = rδsinθ (see Fig. 3(a)). Similar approach has been satisfies a biharmonic equation ∆2ψ = 0. A general so- appliedearlierforsingle-phaseflowstodescribesingular- lution can be presented in the form [31]: ities near sharpcorners. For the flow overa surface with rectangulargrooves,the shearstresshasfoundtobe sin- ψ(r,θ) =rλ[asin(λθ)+gsin((λ−2)θ) gular, i.e., proportional to r−1/3 for longitudinal and to (C5) +ccos(λθ)+hcos((λ−2)θ)]. r−0.455 for transverse configurations [31]. The edge be- tween different slipping flat interfaces has also been con- The radial and the angular components of the liquid ve- sidered, with alternating no-slipand slip stripes [32, 39], locity are trapezoidal and triangular profiles of the local slip b(y) [30]. ∂ ψ θ Forthetwo-phaseflownearthecornerofagroovewith u (r,θ)= , u (r,θ)=−∂ ψ. (C6) r θ r r flat interface, the liquid/solid, liquid/gas and gas/solid interfaces correspond to θ = 0, θ = π and θ = 3π/2 Equations similar to (C5), (C6) can be also written (if the wall of the groove is vertical). A general solution for the gas streamfunction ψ and velocity components g of the dimensionless Laplace equation is a power depen- u , u . We apply the no-slip boundary conditions at gr gθ dence on the distance r: θ = 0 and θ = 3π/2 and the continuity conditions at the gas/liquid interface, θ = π, and, similar to (C3), we r≪1: u = rλ[asin(λθ)+gcos(λθ)], (C1) x obtain the equation governing λ: u = rλ[csin(λθ)+hcos(λθ)], (C2) gx tan(λπ) µ 2λ2−4λ+1+cos(λπ) where a, g, c and h are constants which may be found = , (C7) 2(λ−1) µ 4(1−λ)cos2(λπ/2) bymatching (C1),(C2)withthe flowatr ∼1.However, g theexponentλcanbeobtainedsolelyfromtheboundary conditions. For a shear-freeinterface, µ/µg =∞, we have from(C7) The no-slip boundary condition for liquid phase at λideal = 1/2. Therefore, for large µ/µg we can again θ = 0, the no-slip boundary condition for gas phase at construct an asymptotic solution of (C7). The local slip θ = 3π/2 and the coupling conditions at the gas/liquid length, to the first order in µg/µ, reads interface, Eq.(1), lead to a linear system on a, g, c, h which yields the following equation on λ: ur(r,π) 1 µ r≪1: b (r)= ≃ rδ. y ∂ u (r,π) 2µ θ r g 3λπ (µ−µ )tan(λπ) g tan = . (C3) 2 µ+µ tan2(λπ) (cid:18) (cid:19) g 8 [1] D.Quere, Annu.Rev.Mater. Res. 38, 71 (2008). (2013). [2] L. Bocquet and J. L. Barrat, Soft Matter 3, 685 (2007). [23] O.I.VinogradovaandG.E.Yakubov,Langmuir19,1227 [3] J. P. Rothstein, Annu.Rev.Fluid Mech. 42, 89 (2010). (2003). [4] O.I.VinogradovaandA.L.Dubov,MendeleevCommun. [24] C. Cottin-Bizonne, B. Cross, A. Steinberger, and 19, 229 (2012). E. Charlaix, Phys.Rev.Lett. 94, 056102 (2005). [5] R. S. Voronov, D. V. Papavassiliou, and L. L. Lee, Ind. [25] O. I. Vinogradova, K. Koynov, A. Best, and F. Feuille- Eng. Chem. Res. 47, 2455 (2008). bois, Phys. Rev.Lett. 102, 118302 (2009). [6] O.I.VinogradovaandA.V.Belyaev,J.Phys.: Condens. [26] L. Joly, C. Ybert, and L. Bocquet, Phys. Rev. Lett. 96, Matter 23, 184104 (2011). 046101 (2006). [7] K. Kamrin, M. Z. Bazant, and H. A. Stone, J. Fluid [27] D.Seoand W.A.Ducker,Phys.Rev.Lett.111, 174502 Mech. 658, 409 (2010). (2013). [8] H. A. Stone, A. D. Stroock, and A. Ajdari, Annual Re- [28] A. Quarteroni and A. Valli, Domain Decomposition view of Fluid Mechanics 36, 381 (2004). Methods for Partial Differential Equations (Oxford Sci- [9] M.Z.BazantandO.I.Vinogradova,J.FluidMech.613, ence Publications, 1999). 125 (2008). [29] C. Pozrikidis, Boundary Integral and Singularity Meth- [10] J. R. Philip, J. Appl. Math. Phys.23, 353 (1972). ods for Linearised Viscous Flow (Cambridge University [11] N.V.Priezjev,A.A.Darhuber,andS.M.Troian,Phys. Press, 1992). Rev.E 71, 041608 (2005). [30] J. Zhou, E. S. Asmolov, F. Schmid, and O. I. Vino- [12] E.LaugaandH.A.Stone,J.FluidMech.489,55(2003). gradova, J. Chem. Phys. 139, 174708 (2013). [13] D.Maynes,K.Jeffs,B.Woolford,andB.W.Webb,Phys. [31] C. Y.Wang, Phys. Fluids 15, 1114 (2003). Fluids 19, 093603 (2007). [32] E. S. Asmolov, J. Zhou, F. Schmid, and O. I. Vino- [14] C. Ng, H. Chu, and C. Wang, Phys. Fluids 22, 102002 gradova, Phys.Rev.E 88, 023004 (2013). (2010). [33] We remark that this curve can be very well fitted by a [15] C. Sch¨onecker, T. Baier, and S. Hardt, J. Fluid Mech. symmetric fourth-order polynomial with slopes defnied 740, 168 (2014). by Eq.(8). [16] A. V. Belyaev and O. I. Vinogradova, J. Fluid Mech. [34] P. G. de Gennes, C. R. Acad. Sci. Paris 288 B, 219 652, 489 (2010). (1979). [17] O.I. Vinogradova, Langmuir 11, 2213 (1995). [35] A. V. Belyaev and O. I. Vinogradova, Phys. Rev. Lett. [18] T. V. Nizkaya, E. S. Asmolov, and O. I. Vinogradova, 107, 098301 (2011). Soft Matter 9, 11671 (2013). [36] T. M. Squires, Phys.Fluids 20, 092105 (2008). [19] M. Sbragaglia and A. Prosperetti, Phys. Fluids 19, [37] S. S. Bahga, O. I. Vinogradova, and M. Z. Bazant, 043603 (2007). J. Fluid Mech. 644, 245 (2010). [20] J. Hyv¨aluoma and J. Harting, Phys. Rev. Lett. 100, [38] D. M. Huang, C. Cottin-Bizzone, C. Ybert,and L. Boc- 246001 (2008). quet, Phys.Rev.Lett. 20, 092105 (2008). [21] C.Ybert,C.Barentin,C.Cottin-Bizonne,P.Joseph,and [39] E. S. Asmolov and O. I. Vinogradova, J. Fluid Mech. L. Bocquet, Phys.Fluids 19, 123601 (2007). 706, 108 (2012). [22] E.Karatay,A.S.Haase,C.W.Visser,C.Sun,D.Lohse, P. A. Tsai, and R. G. H. Lammertink, PNAS 110, 8422 0.6 0.6 e/δ=0.1 (a) e/δ=1 (b) 0.4 0.4 L / f f ||be 0.2 0.2 0 0 0.1 0.25 0.5 0.9 0.1 0.25 0.5 0.9 φ φ 1− 1− 0.3 0.3 e/δ=0.25 (c) e/δ=1 (d) 0.2 0.2 L / f ⊥ ef b 0.1 0.1 0 0 0.1 0.25 0.5 0.9 0.1 0.25 0.5 0.9 φ φ 1− 1−