Elastocapillary instability under partial wetting conditions: bending versus buckling
Bruno Andreotti, Antonin Marchand, Siddhartha Das, Jacco H. Snoeijer
aa r X i v : . [ phy s i c s . f l u - dyn ] S e p Elastocapillary instability under partial wetting conditions: bending versus buckling
Bruno Andreotti , Antonin Marchand , Siddhartha Das and Jacco H. Snoeijer Physique et M´ecanique des Milieux H´et´erog`enes, UMR 7636 ESPCI -CNRS,Univ. Paris-Diderot, 10 rue Vauquelin, 75005, Paris Physics of Fluids Group and J. M. Burgers Centre for Fluid Dynamics,University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands. (Dated: November 13, 2018)The elastocapillary instability of a flexible plate plunged in a liquid bath is analysed theoretically.We show that the plate can bend due to two separate destabilizing mechanisms, when the liquid ispartially wetting the solid. For contact angles θ e > π/
2, the capillary forces acting tangential to thesurface are compressing the plate and can induce a classical buckling instability. However, a secondmechanism appears due to capillary forces normal to surface. These induce a destabilizing torquethat tends to bend the plate for any value of the contact angle θ e >
0. We denote these mechanisms as“buckling” and “bending” respectively and identify the two corresponding dimensionless parametersthat govern the elastocapillary stability. The onset of instability is determined analytically and thedifferent bifurcation scenarios are worked out for experimentally relevant conditions.
I. INTRODUCTION
Water-walking arthropods like water striders are ableto float, despite their density, thanks to surface tensionforces [1]. Their superhydrophobic legs are submittedto a repulsive force along the contact line where thethree phases (liquid, vapour and solid) meet. As the legsare long and flexible, they deform under these capillaryforces [2]. Figure 1 shows an experiment performed withextremely long artificial legs made of a soft solid, which isplunged into a liquid bath. One observes an elastocapil-lary instability that is triggered by increasing the contactangle θ e : the initially immersed solid is pushed out of theliquid to the free surface, whenever θ e is sufficiently large.In the limiting case of complete wetting, i.e. θ e = 0,a compressive force is exerted on an elastic rod initiallyimmersed in a liquid when its end pierces the liquid in-terface [3–5]. Such a rod buckles when the compressiveforce exceeds Eulers critical load. Consider the case ofa plate of thickness H much smaller than its length L and its width W . It is submitted to a capillary forceequal to the water surface tension γ LV times the perime-ter ≃ W . The critical force is equal to ( π/ BW/L ,where B is the bending stiffness, which can be expressedas B = EI/W , where E it the Young’s modulus, I themoment of inertia and W the width of the plate. There-fore, buckling occurs if the plate is longer than a criticallength L cr = ( π/ p B/ (2 γ LV ). It is proportional to theelasto-capillary length ℓ EC = (cid:18) Bγ LV (cid:19) / , (1)which is the length scale controlling a large class of elas-tocapillary problems [4–12].One may wonder if the instability observed in Fig. 1is of the same physical nature. Indeed, one can expect abuckling instability if the contact angle θ e is larger than π/
2. Namely, the total downward force that the reser-voir exerts on the solid is proportional to γ LV cos θ e , and FIG. 1. (a,b) Photographs of an elastomeric wire of radius R = 300 µ m and density ρ = 1 . · kg / m brought intocontact with a mixture of ethanol and water. From (a) to (b),the contact angle is increased continuously by decreasing theconcentration of ethyl alcohol. The wire exhibits a bucklinginstability above a critical contact angle. (c,d) Independentdetermination of the contact angle, using drops of the samemixture on a plane substrate made of the same elastomer asin (a,b). On the left, the advancing contact angle is 60 ◦ with a50% volumic solution of ethyl alcohol in water. On the right,the advancing contact angle is 95 ◦ with a 5% volumic solutionof ethyl alcohol in water. hence changes from “stretching” to “compressing” whenthe contact angle exceeds π/
2. However, there is a secondmechanism that can lead to elastic deformations. Fig-ure 2 compares the capillary energy of an extremely flexi-ble object that either remains vertical or floats on the free
FIG. 2. Capillary energy of a partially immersed plate, in theabsence of gravity and elasticity. (a) Two sides of the plateare immersed, representing an energy of 2 γ SL per immersedlength. (b) When bending towards the surface, one of thesides is no longer wetted and part of the liquid-vapor interfaceis covered. The associated energy per length is γ SV + γ SL − γ LV . The energy difference between (b) and (a) equals thespreading parameter S = γ SV − γ SL − γ LV . For the partiallywetting case S <
0, so that state (b) is energetically morefavorable than state (a). surface of the liquid. The free energy difference is propor-tional to the spreading parameter S = γ SV − γ SL − γ LV ,where γ SV , γ SL and γ LV are respectively the surface ten-sions of the solid/vapour, solid/liquid and liquid/vapourinterfaces. As a consequence, bending is favourable underpartial wetting conditions, S <
0, whatever the value ofthe contact angle θ e . This is manifestly different from thebuckling instability, which can only occur for θ e > π/ L forthe instability in the situation of Fig. 1, where the platecan be supposed to be infinite? What are the parameterscontrolling the instability?To illustrate the two mechanisms of elastocapillary in-stability, bending and buckling, we consider a long elasticplate that is hanging freely under the influence of grav-ity. We assume the thickness to be sufficiently small toallow for a thin plate elastic description. The bottom ofthe plate is brought into contact with a liquid reservoirthat partially wets the liquid, with an equilibrium con-tact angle θ e . To reveal the effect of surface wettability,we focus on the case where both sides of the plate arewetted by the same angle. This is fundamentally differ-ent from the situation prior to piercing of a rod througha meniscus [2–5], for which one of the contact lines ispinned to the edge of the solid – in that case the con-tact angle can attain any value. Our goal is to computethe shape of the plate and to analytically determine the threshold of instability for different θ e .Our main finding is that the elastocapillary instabil-ity can occur even when θ e < π/
2, which is the regimewhere the capillary forces are pulling on the plate and no“buckling” is to be expected. Indeed, this is due to thebending mechanism described in Fig. 2, due to the par-tial wettability of the substrate. In general, the thresholdof instability depends on two dimensionless parametersthat can be associated to bending and buckling respec-tively. Interestingly, the bending parameter is not onlydetermined by the elastocapillary length ℓ EC , but alsoinvolves the characteristic size of the meniscus.The paper is organized as follows. We first formu-late the elastocapillary problem and identify the relevantdimensionless quantities in Sec. II. In Sec. III we ana-lytically determine the threshold of instability by linearanalysis and numerically compute the nonlinear bifurca-tion diagrams. The results are interpreted in experimen-tal context in Sec. IV, where we also discuss the influenceof contact angle hysteresis. The paper concludes with adiscussion on the distribution of capillary forces in Sec. V. II. ELASTOCAPILLARY FORMULATION
The strategy of the calculation is to separately treatthe portion of the plate that is outside the bath and themeniscus region near the contact line – see Fig. 3. Forsimplicity we assume that the plate and fluid are densitymatched, or equivalently, that the bottom of the platereaches only just below the surface. In practice, we findthat the characteristic length of the plate outside thereservoir is significantly larger than the capillary lengththat sets the size of the meniscus. This means that we canconsider the forces and torques induced by the meniscus(Sec. II B) as a boundary condition for the dry part ofthe plate (Sec. II A). The dimensionless equations andboundary condition are then summarized in Sec. II C.
A. The plate outside the reservoir
Given that the plate is very thin we can describe theshape by a line that we parametrize by its angle φ ( s )with respect to the vertical direction (Fig. 3). We usea curvilinear coordinate s that has s = 0 at the level ofthe bath, and the curvature is κ = dφ/ds . In cartesiancoordinates, we use the parametrisation x = χ ( z ), where z is orientated upward and z = 0 corresponds to thelevel of the liquid reservoir. The relation between thetwo representations is: dχ/ds = sin φ and dz/ds = cos φ .We consider a very long plate that, due to gravity, followsthe boundary condition φ ( ∞ ) = 0, cf. Fig. 3.Since away from the meniscus there are no forces ap-plied to the plate, the shape can be computed from theelastica equations [13, 14]: Bφ ′′ = F z sin φ. (2) FIG. 3. Left: Photograph of an elastomeric plate partiallybent by capillary forces. We show that such bending to a sta-ble, finite angle is only possible due to contact angle hystere-sis. The plate thickness is H = 0 . E = 66 kPa and its density is 1 . · kg / m . As not partic-ularly clean tap water is used, the surface tension is around γ = 0 .
05 N / m. Right: Definitions of the vertical coordinate z and curvilinear coordinate s . The deformation of the plate ischaracterized by the deflection χ ( z ) or local angle φ ( s ). Thecharacteristic curvature of the plate is ℓ − χ . Inset: sketch ofthe resultant capillary forces in the meniscus region. At thecontact lines there are pulling forces along the liquid-vaporinterface of magnitude γ . In addition, the hydrostatic pres-sure pulls or pushes on the plate depending on the level withrespect to bath. The scale of the meniscus is ℓ γ . Here, B is the bending stiffness, which reflects the inter-nal elastic torque (per unit width) due to a curvature φ ′ .For a thin plate of thickness H and elastic modulus E one finds B = EH / F z is the vertical component ofthe force (per unit plate width) on a cross-section of theplate and we will find below that F x = 0. The torquebalance then reads: T i = − Bφ ′ . (3)This internal force moment is exerted by the upper por-tion of the plate on the lower portion of the plate. Notethat for the situation in Fig. 3 the curvature is negative, φ ′ < F z on a cross-section consists of two contri-butions, due to gravity along the plate and due to surfacetension at the meniscus boundary. At a location s alongthe plate, the vertical force due to gravity is simply theweight below s , i.e. ρgHs per unit width of the plate. As we assume the plate to have the same density as theliquid, or equivalently that the bottom dips just belowthe surface, we only take into account the portion of theplate that is outside the reservoir.The capillary forces exerted in the meniscus region canbe obtained by the virtual work principle, as sketched inFig. 4. Let us consider the left side of the plate. Movingthe plate vertically by dz , one changes the horizontalposition of the contact line by dz tan φ (0), leading to anincrease of the liquid-vapor interface on the left of theplate. However, this is compensated by an equivalentdecrease of liquid-vapor interface on the right of the plate.A non-vanishing effect is that the vertical displacementincreases the length of dry plate by dz/ cos φ (0), while thewetted part is decreased by the same amount. Assumingthere is no contact angle hysteresis (∆ θ = 0), such that γ SV − γ SL = γ cos θ e , one finds the forces due to the leftand right side: F Lz = γ (sin φ (0) − cos θ e )cos φ (0) , (4) F Rz = γ ( − sin φ (0) − cos θ e )cos φ (0) . (5)Here we introduced the shorthand γ = γ LV , which willbe employed in the remainder of the paper. Again, theseforces are per unit width of the plate. Similarly, by mov-ing the plate horizontally by dx , one reduces the waterarea by dx so that: F Lx = − γ, and F Rx = γ. (6)The total horizontal force F x thus vanishes while the totalvertical force reads: F z = F Lz + F Rz = − γ cos θ e cos φ (0) . (7)These results are easily generalized to incorporate con-tact angle hysteresis, i.e. allowing for different contactangles to the left and to the right of the plate [2]. Forthis, one replaces 2 cos θ e by cos θ L + cos θ R in equation(7), where θ L and θ R denote the angles on left and right.Note that F z can be also be obtained by considering theforce contributions around each of the contact lines (cf.Sec. II B).Combining (2) with the F z induced by gravity and sur-face tension derived above, one obtains the equation forthe plate Bφ ′′ = (cid:18) ρgHs + 2 γ cos θ e cos φ (0) (cid:19) sin φ. (8)To analyze this equation it is convenient to introduce alength scale expressing the strength of gravity with re-spect to the bending stiffness: ℓ χ = (cid:18) BρgH (cid:19) / . (9) FIG. 4. The resultant vertical force on the plate can be ob-tained from the change in surface free energies due to a virtualdisplacement dz . See text for details. The analysis implicitly assumes that ℓ χ is much largerthan H .We show below that this is the typical scale over whichthe plate is curved. For the elastomeric plate used inthe experiment presented in Fig. 3, ℓ χ is around 7 mm.Scaling the curvilinear coordinate as S = sℓ χ , and Φ( S ) = φ ( s ) , (10)the torque balance (8) becomesΦ ′′ = (cid:18) S + 2 S cos θ e cos φ (0) (cid:19) sin Φ . (11)Here we introduced a dimensionless number S = γℓ χ B = (cid:18) ℓ χ ℓ EC (cid:19) , (12)The dimensionless number S expressed in (12) manifeststhe importance of the vertical surface tension forces withrespect to the bending stiffness. In the experiment pre-sented in Fig. 3, ℓ EC is around 7 . S is oforder unity ( S ≃ . S can be interpreted as the ratio of ℓ χ ,the “effective” length of the plate, and the elastocapillarylength ℓ EC . This is consistent with the general picture ofelastocapillarity, namely that surface tension can inducedeformations (such as buckling), when the plate is longerthan ℓ EC [4, 5, 10, 15, 16]. Below we will see that underpartial wetting conditions, there is another dimensionlessparameter associated to the torques induced by normalforces.Equation (11) can be solved analytically when the an-gle of deflection is small, i.e. when sin Φ(0) ≃ Φ(0) andcos Φ(0) ≃
1. This limit is relevant at large distancesfrom the meniscus, where the plate tends to a straightline (Fig. 3), as well as for describing the onset of the instability (Sec. III). The equation then becomesΦ ′′ = [ S + 2 S cos θ e ] Φ , (13)which can be solved as:Φ( S ) = Φ Ai ( S + 2 S cos θ e ) , (14)where Ai is the Airy function. The integration constantΦ determines the amplitude of the deflection and hasto be solved from the boundary condition at the menis-cus. The second Airy function Bi( S ) diverges for largearguments and thus does not comply with the boundarycondition φ ( ∞ ) = 0. Using the large S asymptotics ofAi, we find Φ( S ) ≃ Φ e − S / √ π S / . (15)The plate thus naturally tends to a vertical line. Realiz-ing that S = s/ℓ χ , we indeed find that ℓ χ sets the lengthscale over which the deflection decays along the upwarddirection. B. The meniscus region
The plate outside the liquid is described by a sec-ond order ordinary differential equation and thus requirestwo boundary conditions. A first boundary condition is φ ( ∞ ) = 0, which, for example, was used while deriv-ing (14). The second boundary condition comes fromthe torques exerted at the meniscus region. As shownin (3), the internal torque experienced by the plate isproportional to the curvature dφ/ds , which balances theexternal torque T e applied by the meniscus: T i + T e = 0 . (16)This boundary condition has to be evaluated at the posi-tion of the upper contact line. Namely, this point marksthe edge of the domain for (2), for which no normalforces were taken into account along the plate. In theparagraphs below we assume the meniscus on the left ishigher than that on the right, as in Fig. 3. The left andright contact line positions, z L and z R , are found fromthe classical meniscus solutions [17] z L = ± ℓ γ [2(1 − sin( θ L − φ ( s L ))] / , (17) z R = ± ℓ γ [2(1 − sin( θ R + φ ( s R ))] / , (18)where the sign (symbol ± ) depends on the value of thecontact angle θ e with respect to π/ θ e . Note that we now allow explicitlyfor different contact angles on both sides of the plate.The length scale of the meniscus is given by the capillarylength ℓ γ = (cid:18) γρg (cid:19) / (19)and reflects the balance between surface tension and thehydrostatic pressure (gravity). In the conditions of Fig. 3, ℓ γ is around 2 . H ≪ ℓ γ ≪ ℓ χ ∼ ℓ EC , (20)Since ℓ γ is significantly smaller than ℓ χ , it is natural touse a different scaling for the meniscus region. To avoidconfusion with the preceding paragraph, where we scaledthe curvilinear coordinate S = s/ℓ χ , we scale only thecartesian coordinate in the meniscus region: Z = zℓ γ . (21)One can assume ℓ γ ≪ ℓ EC , which suggests that thelength scale over which the capillary forces are assumedto be influential is substantially smaller than the lengthscale over which the capillary-force induced bending canbe significant. As a consequence, one can assume thatthe plate represents a negligible curvature in the menis-cus region, so that the angle can be considered constant, φ ( s L ) = φ ( s R ) = φ (0). This gives a simple relation be-tween the coordinates z = s cos φ (0).
1. Capillary forces
Before addressing the torques, we first specify the var-ious capillary forces exerted by the liquid on the solidplate. The detailed spatial distribution of capillary forcesis a difficult question in itself, as addressed e.g. in [18, 19].As will be commented in Sec. V A, the resultant forcescan be represented as shown in the inset of Fig. 3. First,there is a force per length of magnitude γ that pulls alongthe liquid-vapor interface. Second, there is a contributiondue to hydrostatic pressure in the liquid, which is unbal-anced whenever the two contact lines are at a differentheight (i.e. when z L = z R ). This pressure is acting nor-mal to the solid surface and has to be integrated betweenthe two contact lines.Projecting the tangential force contributions, one findsthe resultant force along the plate F s = − γ (cos θ L + cos θ R ) , (22)taken in the positive s direction. For cos θ e <
0, equilib-rium angle θ e > π/
2, this force is compressing the plate.Similar to the classical buckling instability, such a com-pressive force has a destabilizing effect. For cos θ e > F n = γ (sin θ R − sin θ L ) + F p , (23)where F p is the unbalanced hydrostatic pressure appear-ing on the left of the plate. Assuming that z L > z R , or equivalently θ L < θ R , this hydrostatic pressure is ob-tained by integration as F p = Z s L s R ds p ( s )= − Z s L s R ds ρgs cos φ (0)= − ρg (cid:0) s L − s R (cid:1) cos φ (0)= − γ Z L − Z R cos φ (0) , (24)where we in the last step we used ρg = γ/ℓ γ and s = Zℓ γ / cos φ (0). These equation can be further workedout using (17), where we take φ ( s L ) = φ ( s R ) = φ (0).Combined with (23) this finally gives F n = − γ (cos θ L + cos θ R ) tan φ (0) . (25)Let us emphasize that these resultant force compo-nents (22,25) can be projected in the ( x, z ) directionsin order to compare to the virtual work result discussedin Sec. II A. Indeed, the projection gives F x = 0 whileone recovers the correct F z upon replacing 2 cos θ e bycos θ L + cos θ R in (7). This illustrates the importance ofthe force due to the unbalanced hydrostatic pressure F p .Its magnitude is of order γ [see (24)] and involves the ex-pressions that depend on the contact angles θ L and θ R .Most importantly, only by adding F p to γ (sin θ R − sin θ L ),one recovers the capillary forces obtained from the virtualwork principle [see section II A]. Therefore F p should beinterpreted as a capillary force.
2. Capillary torques
Having established the capillary forces in the meniscusregion, we are in a position to compute the associatedtorques. Since we are interested in the boundary condi-tion for the plate outside the reservoir, we compute thetorque around the highest contact line, i.e. s L . Usingthe convention that positive torques induce a rotation inclockwise direction, the normal forces then give a torque T n = − γ ( s L − s R ) sin θ R = − γℓ γ ( Z L − Z R ) sin θ R cos φ (0) . (26)From the construction in the inset of Fig. 3 it is clearthat this torque is destabilizing. Namely, if we considera small perturbation where the plate is slightly bent tothe right, the meniscus on the left rises higher than themeniscus on the right. As a consequence the surface ten-sion force on the right has a larger moment arm thanthat its counterpart on the left. The induced torque onthe plate acts in the same direction as the initial pertur-bution, and hence, has a destabilizing effect.Similarly to (24), there is a torque induced by the hy-drostatic pressure. This is obtained by integrating overthe pressure, now including a moment arm s L − s : T p = Z s L s R ds p ( s )( s L − s )= Z s L s R ds ρgs cos φ (0)( s − s L )= 16 γℓ γ ( Z L + 2 Z R )( Z L − Z R ) cos φ (0) . (27)Interestingly, this torque scales as ( Z L − Z R ) , whichreflects the fact that both the integrated pressure andthe moment arm are proportional to Z L − Z R . For smallasymmetry we can thus neglect T p with respect to themoment induced by the force at the contact line.Finally, the torque induced by the tangential forces isstrictly zero when ∆ θ = 0, as the forces act in the samedirections. For small hysteresis, the resultant torque isof order ∼ γH ∆ θ , since the arm for the tangential forceis half the thickness of the plate. Clearly, this can beneglected with respect to T n , for which the arm is givenby ℓ γ . To summarize, we find the external torque T e = T n + T p , (28)which for small φ (0) and small hysteresis is dominatedby T n . C. Dimensionless equations
The results of the preceding paragraphs can be sum-marized as follows. We found that the plate in the regionoutside the bath is governed by the length scale ℓ χ , whichin practice is much larger than the size of the meniscus ℓ γ . To separate the regimes, we use the dimensionlesscurvilinear coordinate S = s/ℓ χ outside the bath, forwhich the shape can be solved from (11), i.e.Φ ′′ = (cid:18) S + 2 S cos θ e cos Φ(0) (cid:19) sin Φ . (29)This it to be complemented by a boundary condition at S = s L /ℓ χ ≈
0, since s L is of the order of ℓ γ ≪ ℓ χ .This boundary condition is most conveniently expressedin terms of Z = z/ℓ γ . When Z L > Z R this givesΦ ′ (0) = T (cid:18) − ( Z L − Z R ) sin θ R cos Φ(0) + ( Z L + 2 Z R )( Z L − Z R ) Φ(0) (cid:19) , (30)while for Z R > Z L one has Φ ′ (0) = T (cid:18) ( Z L − Z R ) sin θ L cos Φ(0) − (2 Z L + Z R )( Z L − Z R ) Φ(0) (cid:19) , (31) where T = ℓ χ ℓ γ ℓ EC . (32)In the experiment presented in Fig. 3, T is of order unity( T ≃ . Z L,R are determined by thecontact angles from (17), and the plate inclination at thebottom Φ(0). The latter parameter follows as a result ofthe calculation and can be used to identify the bucklinginstability.Apart from the contact angle θ e (and the hysteresis∆ θ ), the problem is governed by two dimensionless pa-rameters that can be interpreted as a ratio of lengthscales: S = (cid:18) ℓ χ ℓ EC (cid:19) , T = ℓ χ ℓ γ ℓ EC . (33)The first of these parameters can be interpreted as theability to induce buckling for the tangential capillaryforce, provided that cos θ <
0. Consistent with the stan-dard view of elastocapillarity, this effect is governed bythe elastocapillary length ℓ EC with respect to the “ef-fective” length of the plate, set by ℓ χ . The second di-mensionless parameter sets the strength of the bending induced by the torque generated in the meniscus. As itinvolves a torque, the capillary length ℓ γ intervenes asthe moment arm. III. BIFURCATIONS
It is clear that the straight plate, Φ( S ) = 0, is a so-lution of (29) that satisfies the boundary condition (30).We now analyze the stability of these solutions in termsof the parameters S and T , for different values of θ e .Throughout this section we assume no hysteresis, i.e.∆ θ = 0 or equivalently θ L = θ R = θ e . We first perform alinear analysis to identify the threshold and discuss theregimes where bending or buckling are dominant. Subse-quently, we numerically compute the bifurcation diagramby following the various solution branches in the nonlin-ear regime. A. Instability threshold
The threshold of instability of the straight plate is ob-tained by linearizing the problem for small Φ, as alreadydone in (11), yielding a solutionΦ( S ) = Φ Ai( S + 2 S cos θ e ) . (34)Similarly, the boundary condition (30) can be expandedas, Φ ′ (0) ≃ − T | cos θ e | sin θ e p − sin θ e ) Φ(0) , (35) FIG. 5. The threshold of stability T versus S for differ-ent values of the contact angle θ e = π/ , π/ , π/ , π/
2. Forcos θ e >
0, the bending threshold T increases with S sincethe vertical capillary force has a stabilizing effect. The limit-ing case cos θ e = 0 has F z = 0, for which the bending thresh-old is independent of S . where the expression for the meniscus rise (17) was used.Combining (34) and (35) one obtains the equation fora “neutral mode”, which is a solution of the deflectionprofile for arbitrary (small) perturbation amplitude Φ :2 T sin θ e | cos θ e | p − sin θ e ) + Ai ′ [2 S cos θ e ]Ai [2 S cos θ e ] = 0 . (36)Indeed, this equation provides the threshold for theinstability in terms of the parameters T , S and θ e . Thiscan be seen, e.g. by varying one of these parameterswhile keeping the other two constant. One finds thatthe internal moment T i dominates the external torque T e (stable) or vice versa (unstable), as the parameter isvaried across the neutral condition (36). B. Bending instability: θ e < π/ We now reveal the destabilizing effect of the torque inthe meniscus, associated to the parameter T , which triesto bend the plate. This mechanism is most relevant for θ < π/
2, for which it turns out the only destabilizingmechanism: the buckling parameter S is stabilizing inthis range as it multiplies with cos θ e >
0. In this contextof bending, (36) indeed provides the critical T beyondwhich the flat solution becomes unstable. Namely, forlarger T the torque in the meniscus T n becomes largerthan the internal torque of the plate T i for small pertur-bations, hence leading to instability.The result of the stability analysis is shown in Fig. 5,depicting the critical T versus S for several contact an-gles below π/
2. One observes the following trends. First,upon increasing θ e the instability is triggered at a smaller T . This occurs as the destabilizing normal forces are π /4 −π /4 −π /2 π /2 1086420 FIG. 6. Typical bifurcation diagram for bending induced in-stability, obtained from numerical integration of (29,30). So-lutions branches characterized by the angle at the bottom ofthe plate, Φ(0), upon varying the bending strength T . Thevalues of S = 10 . θ e = π/ proportional to sin θ e , and thus becomes more influentialfor larger contact angles. Second, the instability thresh-old increases with S , which represents the strength ofthe tangential capillary forces. In the regime θ e < π/ θ e >
0, these tangential forces are pulling on theplate and are indeed stabilizing; hence, one requires alarger value for T to induce the instability. In the limitof large S one can expand the Airy functions, yieldingthe asymptotics T ∼ S / . Finally, for the limiting casewhere the tangential capillary forces vanish, θ e = π/ T does not depend on S and can be com-puted analytically as T ( π/
2) = − Ai ′ (0)2Ai(0) = 3 / Γ(2 / / ≈ . · · · (37)We now analyze the non-linear behavior of the solu-tions above the threshold. We numerically solve (29,30)and characterize the various solutions with Φ(0), theplate inclination at the bottom. Figure 6 shows a typicalbifurcation diagram for the bending induced instability,by depicting the variation of Φ(0) with T . One recog-nises a supercritical pitchfork bifurcation, with a criticalexponent 1. The diagram is for a given value of S and θ e , and shows that there is a critical T beyond which thetrivial solution Φ(0) = 0 becomes unstable and bifurcatesinto two stable branches. Above threshold, the plate in-clination saturates at a finite angle Φ(0). The critical T , can be easily read off from Fig. 5 by drawing a lineparallel to T axis, passing through the corresponding S (here S = 10) and obtaining the T from the point ofintersection of this line (here θ e = π/ FIG. 7. The threshold of stability T versus S for θ e = 2 π/ F z now compresses the wire and leads tobuckling, even in the absence of bending ( T = 0). Similarto the classical buckling instability, the higher order branchescorrespond to all extrema of the Airy function. C. Buckling instability: θ e > π/ We now consider the case θ e > π/ T ≪ θ e ≈ π . According to (36), the onset of buckling is as-sociated to the rightmost maximum of the Airy function,i.e. Ai ′ ( c ) = 0, which gives S = c θ e , with c = − . · · · . (38)This value is indicated by the closed circle in Fig. 7. Itcan be seen in the figure that for T = 0, the threshold islowered due to the destabilizing nature of the torque inthe meniscus region.Beyond the onset, one observes a sequence of branchesassociated to the other maximima and minima of theAiry function, at more negative arguments. In analogyto the classical buckling, these correspond to the higherorder modes. The stability threshold is obtained from(36) which gives, for T ≪
1, Ai ′ ( c n ) = 0, so that S = c n θ e . (39)The higher order branches correspond to large argumentsof the Airy function, and can be determined accuratelyfrom asymptotics of Ai( s ): c n ≃ − (cid:20) π n + 1) (cid:21) / . (40)Similar to Fig. 6, in Fig. 8, we illustrate a typical bifur-cation diagram for the buckling induced ( θ e > π/
2) in-stability and characterize the various solution with Φ(0). π /4 −π /4 1086420 FIG. 8. Typical bifurcation diagram for the buckling inducedinstability, obtained from numerical integration of (29,30).Solutions branches characterized by the angle at the bottomof the plate, φ (0), upon varying the buckling strengths S .The values of T = 0 .
25 and θ e = 2 π/ FIG. 9. Shapes of the plate with Φ(0) = 0 for different buck-ling strengths S . From left to right: S = 2 . S = 4 . S = 5 . S = 6 . T = 0 .
25 and θ e = 2 π/ For the first branch, as we move along Φ(0) = 0, oncrossing a critical S , the solution becomes unstable andproduces two additional unstable solutions. This impliesthat there is no stable solution at a finite angle. Phys-ically, this suggests that on slight perturbation from itsequilibrium position, the wire (or plate) ends up on thesurface [as in Fig. 1], corresponding to Φ(0) = π/
2. Onmoving further along Φ(0) = 0, one encounters furtherbifurcations which correspond to higher order modes be-coming unstable. Similarly to the previous bifurcation di-agram, the critical values for S are obtained from Fig. 7,by drawing a line parallel to S axis, passing through T = 0 .
25. Typical solutions with Φ(0) = 0 are depictedin Fig. 9. As in classical buckling, the successive branchesare separated by half a wavelength.
IV. EXPERIMENTAL PERSPECTIVEA. Influence of thickness
In this section we would like translate the analysis interms of dimensionless numbers S and T to an exper-imental situation. As an illustration, we consider a casewhere we fix the material properties of the liquid and theelastic solid, and vary the thickness H . The instabilityis then reached below a critical thickness, for which thebending rigidity is sufficiently weak. Alternatively, onemay perform an experiment as sketched in Fig. 1, wherethe thickness is fixed but the contact angle is varying.Retracing the steps of the analysis, one finds that S and T are constructed from the thickness of the plate H and the material parameters γ , E , ( ρg ). The bendingstiffness B by itself is not a material parameter as itdepends on the thickness as B = EH /
12. By selectingthe properties of the liquid and the elastic solid, one fixestwo length scales ℓ EG = E ρg , and ℓ γ = (cid:18) γρg (cid:19) / . (41)In the experiment presented in Fig. 3, ℓ EG is around56 cm and the capillary length is ℓ γ = 2 . ℓ EG /ℓ γ ≃ T and S in terms of thelength scales defined in (41) as: T = ( H ℓ EG ) / ℓ γ ℓ EG H = H − / ℓ EG − / ℓ γ , (42) S = ( H ℓ EG ) / ℓ γ ℓ EG H = H − / ℓ EG − / ℓ γ . (43)Interestingly, the length scales ℓ EG and ℓ γ appear in dif-ferent combinations in the two parameters. We thus an-ticipate that the threshold thickness H presents differentscaling laws, depending on whether the instability is dueto bending ( T ) or due to buckling ( S ).In Fig. 10 we show the critical thickness for differentvalues of the contact angle. We scaled the thickness byassuming T ∼ O (1), which implies a characteristic thick-ness H : H = ℓ / γ ℓ − / EG . (44)In the experiment presented in Fig. 3, H is around0 .
47 mm. The upper line in Fig. 10 represents the thresh-old thickness for instability. The main trend of the graphis that the instability is more difficult to reach, i.e. re-quires a smaller thickness of the plate, as the contactangle θ e is decreased.A key result, however, is that even for very small con-tact angles there is still an instability due to the torqueexerted by the normal forces. This contrasts the classi-cal buckling picture, since for small θ e the capillary forcesare not compressing, but are in fact pulling on the plate. π /4 /4 π /2 π FIG. 10. Experimental perspective: the threshold of instabil-ity by varying the plate thickness H and the contact angle θ e ,for ℓ EG /ℓ γ = 250. The plate thickness H is normalized by H ,defined in (44) as a threshold value for intermediate θ e . Fortypical experimental conditions, H = 0 .
47 mm.
Note that in this regime of small θ e , T → ∞ , and there-fore H , as defined in (44), is no longer the correct lengthscale for the threshold thickness. Rather, in this regimethe critical thickness is obtained from,2 (cid:18) H H (cid:19) / sin θ e | cos θ e | p − sin θ e ) +Ai ′ (cid:20) (cid:0) H H (cid:1) / (cid:16) ℓ EG ℓ γ (cid:17) / cos θ e (cid:21) Ai (cid:20) (cid:0) H H (cid:1) / (cid:16) ℓ EG ℓ γ (cid:17) / cos θ e (cid:21) = 0 . (45)For θ e → H/ H → √ (cid:18) H H (cid:19) / θ e − " (cid:18) H H (cid:19) / (cid:18) ℓ EG ℓ γ (cid:19) / / = 0 , (46)which yields: H H = (cid:18) ℓ γ ℓ EG (cid:19) / θ e / . (47)This asymptotic form is shown as the dashed line inFig. 10. Indeed, this regime involves a different com-bination of ℓ EG and ℓ γ than H .Above π/
2, one enters the usual buckling regime. Thegraph also reveals the higher order buckling modes. Fromthe previous paragraphs, a good approximation can beobtained as: H H = (cid:18) ℓ EG ℓ γ (cid:19) / (cid:20) π n + 1) (cid:21) − / ( − θ e ) / . (48)The onset is generated by the highest of the lines. Fi-nally, note that there is an optimal contact angle, slightly0before θ e = π , for which the instability is most easilyreached. The scaling near θ e = π turns out H H = (cid:18) | c | (cid:19) / (cid:18) ℓ EG ℓ γ (cid:19) / ≈ . (cid:18) ℓ EG ℓ γ (cid:19) / . (49)To summarize, the critical thickness below which theplate becomes unstable increases with the contact an-gle θ e , except very close to θ e = π , where the thick-ness displays a maximum. Intriguingly, the dependenceof the characteristic thickness on the material parame-ters is not universal, but depends on the contact angle.Three regimes can be identified, involving different com-binations of ρg , γ and E . At very small contact angles,one finds from (47): H ∝ ( ρg ) − / γ / E − / θ e / , (50)Close to θ e = π one has (49): H ∝ ( ρg ) − / γ / E − / , (51)while at intermediate contact angles one has (44) H ∝ ( ρg ) − / γ / E − / . (52) B. Influence of contact angle hysteresis
Another important experimental feature is that onecannot eliminate a substantial hysteresis of the contactangle. This means that θ L = θ R , with typical experimen-tal values ∆ θ ≈ . θ = 0). Figures 11a and 11b are both obtainedby varying θ e , for two different values of the plate thick-ness. The upper plot corresponds to H/ H = 1 .
54, forwhich a single bifurcation is observed. The lower plotcorresponds to H/ H = 0 .
77, for which the higher orderbuckling modes are crossed.The effect of contact angle hysteresis is revealed inFig. 12, comparing the bifurcation diagrams for ∆ θ = 0and ∆ θ = 0 .
1. Clearly, the left-right symmetry of theproblem is broken by the hysteresis, as reflected by thesplitting of the branches for Φ(0) > < π /4 π /4 −π /4 /4 π /2 −π /2 π /2 π π /4 π /4 −π /4 /4 π /2 −π /2 π /2 π FIG. 11. Bifurcation diagram, Φ(0) versus θ e for ℓ EG /ℓ γ =250. The two panels correspond to different cross-sections ofFig. 10, namely (a) H/ H = 1 .
54, and (b) H/ H = 0 . V. DISCUSSIONA. Spatial distribution of capillary forces
The total force exerted by the liquid on the solid is γ cos θ e per unit contact line, as follows from a thermo-dynamic argument based on virtual work (Sec. II). Inthis paper, we have treated these capillary forces as ifthey were perfectly localized at the contact line – see theinset of Fig. 3. However, this is not necessarily a repre-sentation of the real distribution of capillary forces, sincethe solid can be submitted to a Laplace pressure wher-ever the solid surface is curved [18, 20, 21]. In the case ofthe plate, for example, this curvature is localized at thebottom edges and could induce an upward force due toa Laplace pressure on the solid. To restore the thermo-dynamic resultant force, this necessarily means that ad-ditional downward forces must be present at the contactline to counteract this effect. Here we will not discuss theorigin and nature of this Laplace pressure and we referthe interested reader to [18, 21] ; assuming that it exists,how are the results derived in this paper affected?We consider the two-dimensional situation depicted in1 π /4 π /4 −π /4 /4 π /2 −π /2 π /2 π FIG. 12. Effect on hysteresis on the bifurcation diagram,Φ(0) versus θ e , in the case H/ H = 1 .
71 and ℓ EG /ℓ γ = 250.∆ θ = 0 . Fig. 13, where the bottom of the solid can take an arbi-trary shape. The immersed part of the solid is submittedto a distribution of pressure γ s κ proportional to the cur-vature κ and to the relevant surface tension coefficient γ s .The total force exerted on the solid due to this curvatureeffect is written as a contour integral: ~F κ = Z RL γ s κ~n dl = Z RL γ s d~tdl dl = (cid:2) γ s ~t (cid:3) RL (53)where l is the curvilinear coordinate, ~t is the local tan-gent vector and ~n the local normal vector. Elementarygeometry gives d~t/dl = κ~n , where κ is the local curva-ture of the surface. Hence, (53) shows that the Laplacecontribution to the total capillary force depends only onthe tangent vectors at the contact line and is indepen-dent of the shape of the immersed solid. As mentionedabove, this must be compensated by an additional forceat the contact line to restore the thermodynamic result.Similarly, the total moment of the Laplace force can bewritten as a contour integral: ~τ κ = Z RL γ s ~r ∧ κ~n dl = Z RL γ s d~r ∧ ~tdl dl = (cid:2) γ s ~r ∧ ~t (cid:3) RL (54)where we have used the property d~r/dl = ~t . As for theresulting force, the moment also does not depend on theshape of the object and is equal to the moment of a force γ s ~t that would be localized at the contact line. In otherwords, neither the total force nor the total moment ex-erted on the upper part of the plate depend on the dis-tribution of capillary forces below the surface.We thus conclude that all results presented in this pa-per are perfectly insensitive to the true spatial distribu-tion of the capillary forces. Reciprocally, it also impliesthat the Laplace pressure on a solid cannot be charac-terised using bending or buckling experiments. Instead, FIG. 13. Sketch of the quantities required to compute theLaplace pressure exerted on a solid of arbitrary shape. Seetext for details. one must measure the deformations of the solid surface[20].
B. Conclusion
In this paper we identified two separate mechanismsthat can lead to elastocapillary instability of a flexibleplate partially immersed in a liquid. The tangential com-ponents of the capillary forces can induce buckling when-ever the contact angle θ e > π/
2. By contrast, the nor-mal components of the capillary forces, which are propor-tional to γ sin θ e , have a destabilizing effect for arbitrary θ e >
0. The underlying physical mechanism can be in-ferred from the inset of Fig. 3: a small perturbation ofthe plate inclination induces a longer moment arm oneone side of the plate, such that the resultant torque isdestabilizing. Alternatively, one may consider a free en-ergy argument, showing that the immersed state of thewire is energetically unfavorable whenever the liquid ispartially wetting (Fig. 2). We found that the dimension-less number associated to this bending mechanism doesnot only involve the elastocapillary length ℓ EC , but alsothe capillary length ℓ γ . The capillary length appears asit sets the typical moment arm for the torque. Changingthe solid from a plate to a thin wire of radius R ≪ ℓ c , asin Fig. 1, the length scale for the moment arm becomes R . By estimating the physical parameters for the wire inFig. 1, we conclude that in this particular example theinstability is not triggered by bending or buckling indi-vidually, but by a combination of the two mechanisms.The analysis of the present paper focussed on the tran-sition of an immersed plate, partially wetted on bothsides, to a state where the plate is pushed to free surface.This can be considered as the inverse of the “piercing”problem that is relevant e.g. for water striders [2]. In thecase of piercing, the contact line remains pinned on theedge of the solid before entering the liquid, such that one2side of the solid remains completely dry. It would be in-teresting to see if the equivalent of the stability threshold(36) could be derived for piercing as well. Acknowledgements:
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