Piercing an interface with a brush: collaborative stiffening
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Piercing an interface with a brush : collaborativesti ff ening F. Chiodi, B. Roman & J. Bico
Physique et M´ecanique des Milieux H´et´erog`enes,UMR 7636 CNRS ESPCI-Paris6-Paris7ESPCI, 10 rue Vauquelin 75005 Paris
PACS – Pattern formation
PACS – Beams, plates, and shells
PACS – Flows in micro-electromechanical systems (MEMS) and nano-electromechanicalsystems (NEMS)
Abstract. - The hairs of a painting brush withdrawn from a wetting liquid self-assemble intoclumps whose sizes rely on a balance between liquid surface tension and hairs bending rigidity. Herewe study the situation of an immersed carpet in an evaporating liquid bath : the free extremitiesof the hairs are forced to pierce the liquid interface. The compressive capillary force on the tip offlexible hairs leads to buckling and collapse. However we find that the spontaneous association ofhairs into stronger bundles may allow them to resist capillary buckling. We explore in detail thedi ff erent structures obtained and compare them with similar patterns observed in micro-structuredsurfaces such as carbon nanotubes “forests”. Introduction. –
Everyday’s life experience teaches us that wet hairs assemble intobundles. This phenomenon is however amplified at the scale of Micro-Electro-Mechanical-Systems (MEMS) since surface forces tend to dominate over bulk forces when the scale isreduced. Indeed, if L is the typical size of a structure, surface forces are proportional to L , while elastic or gravity forces scale as L and L , respectively. Controlling ’stiction’ isthen a challenging issue in micro-engineering technologies since it often leads to the fatalcollapse of microstructures [6–8]. Nevertheless, the self-assembly of micro-structures throughcapillary forces can also be viewed as a useful tool to build complex shapes [1–5]. Beyondengineering applications, surface forces may also have a strong e ff ect on living structures.For instance, filamentous fungi living in aqueous environment have di ffi culty in growingtheir hypha through the water interface into the air. Indeed some species have to producesurfactant molecules that reduce capillary forces in order to develop the aerial structuresnecessary for dissemination [24].In the case of slender structures, the interaction between elasticity and interfacial forcescan be defined by a typical elastocapillary length scale, L EC = ! B/ γ ∼ ! Eh / γ , where E is the Young modulus of the material, h and B are the thickness and the bending sti ff nessper unit width of the structure, respectively, and γ the liquid surface tension or the solidadhesion energy [25–29]. The validity of this macroscopic length scale has recently beenconfirmed at the scale of graphene sheets through atomistic simulations [30].In this paper we study the case of a carpet-like structure immersed in a drying liq-p-1. Chiodi, B. Roman & J. Bicouid bath. This situation is important for practical situations in microtechnologies sincemicrostructures are often dried out of a solvent and brought to pierce the liquid interfaceduring the evaporation process. Recent experiments with wet carbon nanutubes [9–13], ZnO[14] or Si [15–17] nanorods ‘carpets’ and polymeric micro-pilars arrays [18–22] exhibit a sur-prising large variety of bundle structures ranging from ‘tepee’ shapes to cellular patterns.Surprising helicoidal structures have also recently been observed with soft PDMS carpets[23]. However no attempt has been made to classify the di ff erent regimes.Although macroscopic studies have shown that an isolated structure buckles upon capil-lary forces if its length is larger than a critical length of the order of L EC [29], little is knownabout the collective piercing (or collapsing) of an assembly of bristles. May the structureassemble to resist the capillary forces and pierce the interface? The aim of this paper isto present a configuration diagram that predicts the final equilibrium states as a functionof the length of the bristles, their rigidity and their lattice spacing. We will first extendthe results on the formation of bundles, study the piercing of isolated bundles, and finallydeduce a configuration diagram. Although our study is limited to macroscopic regular 1Dbrushes we believe that our results are relevant to the scale of nanotubes ‘carpets’. Experimental setup. –
1D model brushes are build by clamping lamellae of length L (centimetric) cut from bi-oriented polypropylene sheets (Innovia Films, E " h (of 15, 30, 50 or 90 µ m) on a base with a regular spacing d (ranging from millime-ters to centimeters). The elestocapillary length L EC is measured for each thickness with the‘racket’ technique described by Py et al. [1]. The brushes are first immersed into a bath ofcommercial dish-washing solution that totally wets the lamellae ( γ = 26 . Ldh
Fig. 1: Sketch of the experiment. Lamellae of length L and thickness h are clamped on an immersedbase with a regular inter-spacing d . As the liquid is progressively removed from the reservoir, thefree tips of the lamellae are forced to pierce the liquid/air interface. Forming bundles. –
We first present the sticking of wet lamellae by capillary forces,once out of the liquid bath. When a macroscopic brush with a regular lattice is withdrawn(tips down) from a liquid bath, pairs of intermediate bundles successively stick together,leading to large hierarchical bundles [26, 28] (in this situation, the lamellae do not have topierce the interface; we will consider later if these bundles are stable in the inverted case).A balance between adhesion and elastic bending energy gives the distance from the root L stick at which two hairs with an initial spacing d join . In the limit of small deformations This argument is similar to the classical argument derived by Obreimo ff in 1930 to estimate the splittingstrength of mica [31] p-2iercing an interface with a brush : collaborative sti ff ening (a)(b)(c)(d) 50 mm Fig. 2: Typical experiment: (a) immersed brush, (the upper dark line corresponds to the liquidsurface); (b) as the liquid is progessively removed, the interface reaches the tips of the lamellae,isolated lamellae buckle and eventually collapse but may also bundle together and pierce the liquidsurface; (c) final bundle ( L = 90 mm, d = 50 mm, L EC = 33 . ( d/L stick L stick = " / ( dL EC ) / . (1)This relation can be extrapolated to intermediate bundles of size N/ N , by multiplying the bending sti ff ness by factor N/ ff ective distance between these in-termediates bundles N d/
2. This leads to an e ff ective elasto-capillary length of ( N/ / L EC ,so that the joining length L stick of a pair of intermediate bundles merging into a bundle ofsize N is in this case given by [26]: L stick ( N ) = √ N / ( dL EC ) / . (2)We see that the formation of large bundle require long lamellae ( L stick increases with N ).Conversely, for a given brush with lamellae of length L , the maximum size of a bundle N max is easily derived from this last equation by taking L stick = L : N max = 2 " / " L d L EC / . (3)However smaller bundles generally are also present: when the sum of the sizes of neighboringbundles exceeds N max , they cannot stick together because they would lead to a bundleexceeding N max . When a brush is withdrawn bundles are randomly formed. The statisticalp-3. Chiodi, B. Roman & J. Bico L = 45 mm L = 40 mm L = 30 mm L = 20 mm L = 15 mm20 mm(a)(b)(c)(d)(e) Fig. 3: Final state for decreasing lamellae lengths ( d = 5 mm, L EC = 5 . distribution of the size of the bundles is found to follow a self-similar size law [32]. Thispeculiar distribution has a maximum size N max and also a minimal size, which is of theorder of N min " . N max .The condition of small deformations ( d/L stick
1) assumed for deriving the previousrelations is however not always verified in practice ( e.g. experiment displayed in fig. 1). Theequilibrium shape of the lamellae can nevertheless be described in the general situation bysolving numerically Euler’s elastica relation [33]:
B d θ ds e z + t × R = , (4)where θ is the angle made by the tangent to the lamella t with the vertical at the curvilinearcoordinate s , e z the vector perpendicular to the plane and R the constant vectorial tensionof the beam (in the present case, R only has an horizontal component). Most boundaryconditions required to solve the equation are trivial: θ = 0 at the contact point and at theclamped end, the horizontal displacement is equal to d/ √ /L EC . The non-dimensionalform of eq. 4 was solved numerically for increasing values of the non-dimensional distance d/L EC . The corresponding ratio L stick / ( dL EC ) / is displayed in fig. 4 and is found to befairly well fitted by a linear correlation: L stick / ( dL EC ) / " (9 / / (1 + 0 . d/L EC ). Wecan finally extrapolate this relation to the case of a pair of intermediate bundles of size N/ ff ective elasto-capillary length of ( N/ / L EC , and an e ff ective distance N d/
2) merging into a bundle of size N : L stick ( N ) " √ N / ( dL EC ) / (1 + 0 . √ N dL EC ) . (5)p-4iercing an interface with a brush : collaborative sti ff eningSolving this relation for L stick = L then gives the maximum size N max corresponding to thenon-dimensional spacing d/L EC . Once we have characterized the size of the bundles that d / L EC L /( dL ) EC ~ stick L stick d ! t R Fig. 4: Sticking length of a pair of hairs accounting for the finite value of the spacing d . Symbols:numerical solutions of the elastica equation. Line: linear correction of the zero-order relation (eq. 5), L stick = (9 / / ( dL EC ) / (1 + 0 . d/L EC ) can spontaneously form on a given brush, we now wonder if these elastic structures maypierce the liquid interface, or buckle and collapse. Piercing an interface with an isolated bundle. –
We first consider the case ofan isolated lamella that pierces the liquid surface (insert in fig. 5). If the liquid wets thematerial, the vertical capillary force pushing the lamella downward at the liquid interfaceis given by: F cap = 2 w γ , where w is the width of the lamella . Classical Euler bucklingcriterion predicts a critical length for the slender lamella above which it buckles: L crit = π $ Bw γ w = π √ L EC , with B = Eh − ν ) , (6)where E and ν are the material Young modulus and Poisson ratio, respectively, and h the thickness of the lamellae. Although the actual postbuckling behavior is more complex[29], we will suppose, for the sake of simplicity, that lamellae with lengths exceeding L EC eventually collapse towards the base.In the case of a brush, the buckled lamellae generally hit their neighbors and mergeinto larger bundles. May these more rigid structures resist capillary loading and pierce theinterface? Since the liquid can lubricate the relative displacement between lamellae, wewould expect a bundle involving N lamellae to be N times sti ff er than a single lamella,leading to an increase of L crit by a factor √ N . The minimum piercing size N ( L ), abovewhich a bundle of a given length L is strong enough to resist piercing is given by: N ( L ) = 8 π " LL EC . (7) We suppose here that the contact angle is zero, otherwise γ should simply be replaced by γ cos θ , where θ is the contact angle of the liquid on the surface p-5. Chiodi, B. Roman & J. BicoIn order to compare this theoretical prediction with the results obtained with our brushesseparated with a spacing d , we measured the minimum size N crit ( d, L ) above which a bundleof artificially fixed size resits. N crit ( d, L ) is found significantly lower than the predicted value N ( L ) as the spacing d is increased (fig. 5). We qualitatively interpret this result by thelarger width of the base of the bundle, which increases the e ff ective sti ff ness of the structure.Experimental data obtained with lamellae of di ff erent lengths and thicknesses collapse in asingle master curve when N crit /N is plotted as a function of d/L . A fair fit of this mastercurve is (full line in fig. 5): N crit ( d, L ) = N ( L )1 + 16 d/L . (8)As the liquid bath is drained, lamellae tend to merge into bundles. Bundles containing morethan N crit lamellae pierce the interface while the other ones collapse. h = 30 µ m, L = 40 mm h = 30 µ m, L = 45 mm h = 30 µ m, L = 50 mm h = 50 µ m, L = 80 cm N ( d,L )/ N ( L ) d / L dF cap L crit Fig. 5: Main figure: minimum size of a bundle of a given length required to pierce the liquid interfaceas a function of the distance between lamellae. Continuous line: empirical fit N crit ( d, L ) /N ( L ) =1 / (1 + 16 d/L ). Inset: compressive capillary force acting on a single lamella or on a bundle. Configuration diagram. –
The fate of a given brush can be predicted by studyingthe piercing conditions of the bundles that it spontaneously develops. This is done bycomparing the critical size N crit with the maximum and minimum bundle sizes N max and N min . Indeed, if for a given length L , the size N crit (eq. 8) exceeds N max (eq. 3), the wholebrush is expected to collapse (case 1 in fig. 6). If N crit lies between N max and N min , lamellaemerge into bundles as the liquid is removed ; the largest bundles should pierce the interfacewhile the smaller ones should collapse (case 2). If N crit becomes lower than N min even thesmallest bundles are expected to pierce the surface of the liquid (case 3). A last situationarises for small values of N max . When N max is lower than 2, lamellae do not form bundlesand can either remain straight if L < L crit (case 4) or otherwise collapse (case 5, whichjoins case 1). The di ff erent cases are summarized in table 1 and sketched in fig. 6. Thesepredicted regimes are in good agreement with experiments displayed fig. 2 and fig. 3.Although our model brushes are one-dimensional, we expect our results to be qualita-tively valid for two-dimensional ‘carpets’. As a main quantitative di ff erence, the scalingwith N / in eq. 3 becomes N / [28]. The direct comparison with experiments carriedwith nanorods is not precise since L EC was nor measured, but provides some qualitativeindication. For instance, cellular patterns similar to case 2 are observed in the experimentsp-6iercing an interface with a brush : collaborative sti ff ening d / L EC L / L EC (1)(2)(3) (4) (5)(a)(b)(c)(d)(e) N = N crit max N = N crit min N = 2 max N = 1 crit Fig. 6: Configuration diagram and comparison with experiments. Case 1: bundles of any possiblesize collapse. Case 2: the largest bundles resist an pierce the surface, while smaller one collapse.Case 3: Bundles of all accessible sizes resist. Case 4: lamellae do not form bundles but are sti ff enough to pierce the surface. Case 5: lamellae do not form bundles and collapse (this case joinsCase 1). Experimental parameters corresponding to the experiments illustrated in fig. 2 ( ! ) andfig. 3 ( • ). described by Chakrapani et al. [11]. In these experiments, the radius of the multi-wall nan-otubes b is assumed to be of the order of 15 nm and the material Young modulus E ∼ L EC ∼ . µ m (in the case of rods L EC = ! π Eb / γ [28]). The length of thetubes is much larger than L EC ( L ∼ µ m), which favors the collapsed cases 1 and 2,while the lattice spacing d ∼ . µ m may not be small enough to prevent collapse. Con-versely, ‘tepee’ structures reminding case 3 are formed in the experiments described by Lau et al. [10], with b ∼
25 nm, giving L EC ∼ . µ m of the same order of magnitude as thelength L ∼ µ m and the lattice spacing ( d ∼ . µ m). Experiments with Si rods exhibit thesame structure [15] with b ∼
20 nm, E ∼
130 GPa, leading to L EC ∼ . µ m for a length L ∼ µ m and a spacing d ∼ . µ m. If these nanorods were isolated they would not havepierce the interface but instead buckle and eventually collapse. This study suggests thatcollaborative piercing is possible for arbitrary flexible structures if they can merge into largeenough bundles.Note finally that other theoretical approaches have been proposed in the literature. Theformation of bundles is then interpreted in terms of lateral interactions [11,13,15,22], which isbasically equivalent to the aggregation described without the piercing problem. In addition, Table 1: Di ff erent cases described in the configuration diagram. case 1 N max < N crit the whole brush collapsescase 2 N min < N crit < N max biggest bundles pierce, while smaller collapsecase 3 N crit < N min bundles of any size piercecase 4 N max < L < L crit lamellae do not form bundles and remain straightcase 5 N max < L crit < L lamellae do not form bundles and collapsep-7. Chiodi, B. Roman & J. Bicothe finite thickness of the hairs becomes important when the distance between the hairs issmall ( h ∼ d ) and has been also considered [17]. However, to the best of our knowledge, thecombination of the size distribution of the bundles with possible buckling is original. Conclusion. –
The fate of a brush immersed in a drying liquid bath is determinedby two competitive interfacial phenomena: compressive capillary forces may induce thebuckling and eventually the collapse of the bristles, while lateral attractive capillary forceslead to collaborative sti ff ening through the formation of bundles. Di ff erent final states havebeen observed with model experiments on macroscopic brushes depending on the physicalparameters of the brush. We showed that these physical parameters can be condensed intotwo non-dimensional parameters: L/L EC and d/L EC , where L is the length of the hairs, d their spacing and L EC an elastocapillary length comparing bending sti ff ness to surface forces.Dense brushes of rigid hairs tend to resist capillary forces while floppy hairs in scarce brushescollapse. We found that arbitrary flexible hairs (that would collapse as individuals), maydevelop a collaborative sit ff ening by sticking to their close-enough neighbors and manage topierce the interface. Although our study is limited to one-dimensional brushes, we believethat our results are qualitatively valid for two-dimensional situations and may help designing‘hairy’ microstructures. ∗ ∗ ∗ This work was partially founded by the Soci´et´e des Amis de l’ESPCI. We thank DominicVella and Guillaume Batot for fruitful discutions.
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