Wetting morphologies on randomly oriented fibers Alban Sauret, Fran¸cois Boulogne, Beatrice Soh, Emilie Dressaire, Howard A. Stone
To cite this version: Alban Sauret, Fran¸cois Boulogne, Beatrice Soh, Emilie Dressaire, Howard A. Stone. Wetting morphologies on randomly oriented fibers. The European Physical Journal E, 2015, 38, pp.62. .
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Wetting morphologies on randomly oriented fibers Alban Sauret1 , Fran¸cois Boulogne2 , Beatrice Soh2 , Emilie Dressaire3 , and Howard A. Stone2 1
Surface du Verre et Interfaces, UMR 125, 93303 Aubervilliers, France (E-mail:
[email protected]) 2 Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA 3 Department of Mechanical and Aerospace Engineering, New York University Polytechnic School of Engineering, Brooklyn, NY 11201, USA
Abstract We characterize the different morphologies adopted by a drop of liquid placed on two randomly oriented fibers, which is a first step toward understanding the wetting of fibrous networks. The present work reviews previous modeling for parallel and touching crossed fibers and extends it to an arbitrary orientation of the fibers characterized by the tilting angle and the minimum spacing distance. Depending on the volume of liquid, the spacing distance between fibers and the angle between the fibers, we highlight that the liquid can adopt three different equilibrium morphologies: (1) a column morphology in which the liquid spreads between the fibers, (2) a mixed morphology where a drop grows at one end of the column or (3) a single drop located at the node. We capture the different morphologies observed using an analytical model that predicts the equilibrium configuration of the liquid based on the geometry of the fibers and the volume of liquid. Page 2 of 9
1
Introduction
The spreading behavior of a liquid placed on a solid substrate controls a broad range of natural and man-made processes, from the clinging of morning dew to spider webs to the coating of surfaces [1, 2, 3, 4, 5, 6, 7]. Thus, characterizing wetting phenomena offers insights into the complex physics of wet or partially wet systems. These studies also provide knowledge that can be applied to improve and develop industrial methods in which capillary forces play a key role, e.g. coating, mixing and agglomeration [8, 9, 10, 11]. When a volume of liquid is placed between two solid surfaces, a capillary bridge forms. The equilibrium shape of the liquid bridge has been studied for different configurations of the solid surfaces, e.g. flat plates and spherical grains [12, 13, 14, 15]. The shape of the liquid bridge minimizes the interfacial free energy. When the distance between the surfaces is increased, the liquid exerts an attractive force that pulls the two surfaces together [16]. This cohesive capillary force gives rise to the rich mechanical behavior of wet granular matter [8]. One configuration of solid surfaces has received less attention: the formation of capillary bridges between long cylinders, or fibers [17, 18]. It is evident that fibrous media are ubiquitous in both natural systems, such as feathers and hair [19], and engineered products, including paper and textiles [20]. Understanding the wetting of fibers is thus important for many industries. The wetting influences the dyeing of textiles, the coloring of human hair and the spreading of ink on paper. In particular,
Fig. 1. SEM pictures of drops of binder lying on glass wool. (a) Drop on1:a single (b) liquid in the state between two Figure SEM fiber, pictures of drops of drop binder lying on glass crossed fibers, (c) on anda (d) liquid in the between wool. (a) Drop single fiber, (b)column liquid state in the drop two Scaletwo bars are 10fibers, µm (pictures Saint-Gobain statefibers. between crossed (c) andfrom (d) liquid in the Research, reproduced with permission from Bintein [21]).
column state between two fibers. Scale bars are 10 µm (Pictures from Saint-Gobain Research, reproduced with permission fromforBintein [21]). pairs of fibers: parallel configurations neighboring fibers, touching crossed fibers and non-touching crossed fibers. In considering a global model for fiber arrays, we understanding the distribution of liquid array the of thus need to account for the latter case,ininanwhich fibers is arealso notcritical touching. Indeed, the closest beto the generation of fiberdistance mats used tween non-parallel fibers is anpurpose. additional that in glass wool for insulation Inparameter this situation affects thefibers equilibrium morphology of athe liquid. the glass are stuck together by binder fluid. The this paper,ofwethe study wetting morphologies on a finalInproperties fiberthe mats are in part controlled pair of fibers that are randomly oriented and spaced thus by how the wetting binder fluid is distributed among the considering a moreitsgeneral situation than 1a-d). previousInwork glass wool before solidification (figure adperformed on liquid bridges between touching or parallel dition, glass wool does not swell when in contact with fibers. In particular, we characterize the transitions beliquid the andwetting we will morphologies therefore neglect [21]. fibers tween on a this paireffect of crossed Because of challenges in visualizing the microstrucwith respect to four variables: the angle between the fibers tures, fibrous media of radius fibers athat δ, the distance betweenare thecomplex fibers h, arrays the fiber and the volume of liquid V . Thus, the new model presented in this paper describes the equilibrium wetting morpholo1 gies associated with any fiber configuration and recovers the results obtained previously for parallel and touching crossed fibers. We also highlight the understanding of the
Fig. 2. (a) rigid fibers. of radius 2 fibers, havi axis are sep
a capillary study. To con of identica
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are difficult to study experimentally. Therefore seminal work has focused on the simplest element of an array of fibers: a pair of straight parallel fibers [22, 23]. These studies have identified two liquid morphologies. A small volume of nonvolatile liquid deposited on a pair of parallel fibers can adopt a hemispherical drop shape or an extended column state. Recently, our work on crossed touching fibers has shown that, in addition to the drop and column states, the liquid can exist in a third morphology: a composite drop/column state referred to as (a) Fig. 1. SEM pictures of of drops of binder on glass wool. Fig. 1. SEM pictures of drops binder lyinglying on glass wool. (a) (a) the mixed morphology [24]. Analytical models for the on afiber, single(b) fiber, (b) liquid the drop between Drop on Drop a single liquid in theindrop statestate between twotwo crossed fibers, (c)on andparallel (d) liquidand in the column touchstate between shape crossed of the fibers, column crossed (c)state and (d) liquid in the column state between two fibers. Scale bars are 10 µm (pictures from Saint-Gobain ing fibers have been bars previously proposed two fibers. Scale are 10 µm (pictures and from compared Saint-Gobain Research, reproduced with permission from Bintein [21]). Research, reproduced with permission from Bintein [21]). with experiments. However, in most fibrous media, the fibers are ranconfigurations for neighboring pairs of fibers: parallel domlyconfigurations oriented andtouching spaced, whichfibers results innon-touching three possifor neighboring pairs fibers: parallel fibers, crossed andof crossed ble configurations for neighboring pairs of fibers: parallel fibers. In considering a global model for fiber arrays, we fibers, touching crossed fibers and non-touching crossed thus need to fibers account formodel the latter case, arrays, in which In considering a global for fiber wethe fibers,fibers. touching crossed and non-touching crossed fibers not touching. distance need to are account for model the Indeed, latter case, in which fibers.thus In considering a global for the fiberclosest arrays, wethebetween fibers is an additional parameter that fibers to are not non-parallel touching. the closest distance thus need account for theIndeed, latter case, inthe which thebeaffects the equilibrium morphology of liquid. tween non-parallel fibers is an additional parameter that (b) In this paper, we study wettingdistance morphologies fibers affects are notthetouching. Indeed, thethe closest be- on a equilibrium morphology oforiented the liquid. pair of fibers that are randomly and spaced thus Fig. 2. (a) Representation of an array of randomly oriented tween non-parallel fibers is an additional parameter that In this paper, we study the wetting morphologies onwork a considering a more general situation than previous rigid2:fibers. Schematic of the of system composed two fibers oriFigure (a) (b) Representation an array of of randomly affectspair theofequilibrium morphology of the liquid. fibers that are randomly oriented and spaced thus 2. (a) 2Representation of anthe array of randomly performed on liquid bridges between touching or parallelFig. of radius a. The z-axis defines position where theoriented two ented rigid fibers. (b) Schematic of the system composed considering awemore general situation than the previous workbe-rigid In this paper, the we wetting morphologies on fibers. (b) Schematic composed of two fibers fibers, having a tilt angleofδ,the aresystem the closest, i.e., when their fibers. Instudy particular, characterize transitions of oftwo fibers radius 2defines a. The defines posiaxis are 2separated by a distance 2 hthe +z-axis 2position a. on liquid bridges between touching parallel radius a. of The z-axis where the the two tween the wetting morphologies on a pair crossed fibers a pairperformed of fibers that are randomly oriented andoforspaced with respect to we fourcharacterize variables: the the angletransitions between thebefibers fibers, having a two tilt angle δ, having are the closest, i.e., when their fibers. In particular, tion where the fibers, a tilt angle δ, are the thus considering adistance more general situation thanfiber previous δ, the between theon fibers h, the radius a andaxis are separated by a distance 2 h + 2 a. tween the wetting morphologies a pair of crossed fibers closest, i.e., when their axis are separated by a distance work with performed on liquid bridges touching or in a capillary bridge and are thus of interest in the present the volume ofvariables: liquid V . Thus, the new modelthe presented respect to four thebetween angle between fibers 2 h +study. 2 a. this paper describes the equilibrium wetting morpholoparallel fibers. In particular, we characterize the tranδ, the distance between the fibers h, the fiber radius a and gies associated with any fiber configuration and recovers To consider different configurations, pair sitionsthe between theliquid wetting morphologies on apresented pair of in a capillary volume of V . Thus, the new model bridgetheand are thus of interestwe in use theapresent the results obtained previously for parallel and touching of identical nylon fibers tilted with an angle δ and septhis paper describes the equilibrium wetting morpholostudy. crossed fibers with respect to four variables: the angle crossed fibers. We also highlight the understanding of the arated by a minimum separation distance h. We show a theToformation of different a capillary bridge and are thus giesthe associated fiber configuration and recovers consider the configurations, usefiber a pairof between fibers δ,with theany distance between the fibers h, in liquid morphology between randomly oriented fibers in schematic of the fiber configuration in fig. 2(b).we Each the results obtained previously for parallel and touching interest in the present study. of identical nylon fibers tilted with an angle δ and a new 3D diagram. This characterization of equilibrium is held horizontal and clamped at both ends, with one fibersepthe fiber radius a and the volume of liquid V . Thus, the crossed fibers. also highlight the understanding of the wettingWe morphologies is essential for future studies on thearated byona aminimum separation distance Weuse a rotating stage (PR01, Thorlabs)h.with ashow miToaffixed consider the different configurations, we a pair new model inwet this paper describes the equiliquid presented morphology oriented fibers properties of between fibrousrandomly media including their dryinginbe-schematic the fiber configuration in fig. 2(b). Each fiber crometerofdrive that allows for the variation of the angle nylon fibers◦ tilted with an angle δ and sepalibrium wetting morphologies associated withof any fiber of isidentical haviour. a new 3D diagram. This characterization equilibrium δ in horizontal increments and of 0.1 . The rotating stage with is mounted held clamped at both ends, one fiber by a minimum separation h.with Wea mishow configurations and recovers the results obtained previwetting morphologies is essential for future studies on the rated on a linear translation stage(PR01, (PT1,distance Thorlabs) affixed on a rotating stage Thorlabs) with a mi- a crometerof drive that allows for of of the vertithethat fiberallows configuration in Fig. 2(b). Each of wet media including theirWe drying ously properties for parallel andfibrous touching crossed fibers. alsobe- schematic crometer drive for the thevariation variation the angle cal closest distance between fibers hat in increments ofwith ◦and the Experimentalofmethods haviour. isincrements held horizontal both δ in of 0.1 . Theclamped rotating stage is ends, mounted highlight the 2understanding the liquid morphology be- fiber 5 µm. We use various fiber radii a ∈ [100; 225] µm (nyon fiber a linear translation stage (PT1, Thorlabs) a mione affixed on a rotating stage (PR01,with Thorlabs) tween randomly oriented a new oriented 3D diagram. We consider an fibers array ofin randomly fibers, whose lon fibers from Sufix Elite) and separation distances becrometer drive that allows for the variation of the vertiwithtween a micrometer drive that fibers allowsexhibit for the variation of fibers h ∈ [0, 6a]. Nylon micrometerThis characterization of size equilibrium wettingtomorpholotypical mesh is large compared the drop radius, closest the fibers h in increments of 2 Experimental methods ◦ roughness, butbetween we haveof not observed anyrotating noticeablestage i.e. an madestudies of long on fibers andproperties with a largeofporosthecalscale angle δ distance in increments 0.1 . The gies is essential forarray future the 5 µm. We use various fiber radii a ∈ [100; 225] µm (nyhysteresis with perfectly and partially wetting fluids [25]. ity, as illustrated in fig. 2(a); the notations we will use are is mounted on systematic aSufix linear translation stage distances (PT1, Thorwet fibrous media including behaviour. fibers from Elite) and separation perform experiments in the three possi-beWe consider array of their randomly oriented fibers, on whose given an in fig. 2(b). A dropdrying of liquid deposited the ar-lonWe with a configurations micrometer drivesilicone that allows for the variable fiber with oil (5 cSt, density fibers h ∈ [0, 6a]. Nylon fibers exhibit micrometerray encounters one ofcompared four possible attween typical mesh size is large tofiber the configurations: drop radius, labs) 3 ρof =roughness, 918 , but surface γ =observed 19.7 between mN/m, tion thekg/m vertical closest the fibers equilibrium liquidfibers can beand located a single fiber, scale we tension havedistance not anypuchased noticeable i.e. an array made the of long withi)a on large porosfrom Sigma-Aldrich), which is perfectly wettingfluids on the ii) on two parallel fibers (h = ̸ 0 and δ = 0), iii) at the hysteresis with perfectly and partially wetting [25]. ity, as illustrated in fig. 2(a); the notations we will use are h in increments of 5 µm. We use various fiber radii 2 Experimental methods The systematic capillary length that describes scale possiat of contact of two crossed fibers 0 andWefibers. perform experiments in Sufix thethethree given inpoint fig. 2(b). A drop of touching liquid deposited on (h the= ara ∈ [100; 225] µm (nylon fibers from Elite) gravity effects with become noticeable is cSt, defined as and δ ̸= 0), or iv)ofat thepossible point of fiber minimum distance between !the blewhich fiber configurations silicone oil (5 density ray encounters oneof four configurations: at separation distances between fibers h ∈ [0, 6 a]. Nylon We consider an array randomly oriented fibers, whose ℓ = kg/m γ/(ρg), two non-touching crossed fibers (h ̸= 0 and δ ̸= 0). Onlyρ = 3 where g is the gravitational constant and , surface tension γ = 19.7 mN/m, puchased equilibrium the liquid can be locatedresult i) oninathe single fiber, of γc 918 isexhibit the surface tension. This length is about 1.5 mmwe forhave the three latter compared configurations formation fibers micrometer-scale roughness, but typicalii)mesh size is large to the drop radius, on two parallel fibers (h ̸= 0 and δ = 0), iii) at the from Sigma-Aldrich), which is perfectly wetting on the observed noticeable with perfectly i.e. anpoint arrayofmade of long and crossed with a large fibers. The any capillary lengthhysteresis that describes the scale and at contact of twofibers touching fibersporosity, (h = 0 and not partially wetting fluids [25]. We perform systematic exas illustrated in Fig. 2(a); the notations we will use are which the gravity effects become noticeable is defined as δ ̸= 0), or iv) at the point of minimum distance between ! ℓc = γ/(ρg), is the gravitational constant and non-touching fibers (h ̸=deposited 0 and δ ̸=on 0).the Only periments in thewhere three gpossible fiber configurations with given two in Fig. 2(b). Acrossed drop of liquid 3 γ is the length is about 1.5 mm for the three latter result the formation of silicone oilsurface (5 cSt,tension. densityThis ρ = 918 kg/m , surface tension array encounters one configurations of four possible fiberinconfigurations: at equilibrium the liquid can be located (i) on a single γ = 19.7 mN/m, puchased from Sigma-Aldrich), which is fiber, (ii) on two parallel fibers (h 6= 0 and δ = 0), (iii) perfectly wetting on the fibers. The capillary length that at the point of contact of two touching crossed fibers describes the scale at which the p gravity effects become (h = 0 and δ 6= 0), or (iv) at the point of minimum noticeable is defined as `c = γ/(ρ g), where g is the distance between two non-touching crossed fibers (h 6= 0 gravitational constant and γ is the surface tension. This and δ 6= 0). Only the three latter configurations result length is about 1.5 mm for silicone oil. As this capillary 2
3
length is usually larger than the typical height H of the liquid in our experiments, we first assume that gravity effects can be neglected. We shall discuss this assumption when large volumes of fluid and/or large fibers are used, 2 as the Bond number of the system, Bo = ρ g (2 a) /γ, becomes larger than one.
Analytical modeling
We consider two rigid fibers having a cylindrical crosssection, separated by a minimum distance h and tilted by an angle δ. We can define a system of coordinates Oxyz as represented schematically in Fig. 5. When needed, we use the fiber radius a to construct dimensionless parameters: the dimensionless inter-fiber distance d˜ = d/a, ˜ = h/a, the dimenthe dimensionless spacing distance h ˜ sionless wetting length L = L/a, the dimensionless crosssectional area A˜ = A/a2 and the dimensionless volume V˜ = V /a3 . Provided that the fibers are not parallel, i.e. that the tilting angle is not zero, a drop of liquid lying on two fibers will travel towards the point where the fibers are the closest, which we refer to as the “kissing point” if the fibers are in contact. We consider a liquid that has a contact angle θE and is in a column state on a pair of fibers characterized by (δ, h, a). This morphology consists of a long column of liquid, with varying cross-section and a constant height. The shape of the surface of the cross-section of the column is defined by its dimensionless radius of curvature ˜ = R/a and the angle between the line connecting the R centers of the fibers and the radius to the liquid-fiberair boundary α [Fig. 5(b)]. We define the equilibrium configuration of the general situation of two fibers that are not in contact, i.e. separated by a minimum distance h > 0 and tilted with an angle δ > 0. The inter-fiber distance, 2 d(y), varies as a function of the distance y to the point O where the two fibers are the closest [Fig. 5(a)]. ∆x is the distance between the axes of the two fibers projected in the plane (x y), and 2 h + 2 a is the closest distance between the axes of the two fibers. We have ∆x δ 2 2 . = (∆x) +4 (a+h)2 = 4 [d(y) + a] and tan 2 2y (1) Using these two expressions, we obtain 1/2 d(y) δ 2 2 2 ˜ ˜ d(y) = = y˜ tan + (1 + h) − 1, (2) a 2
In a typical experiment, we dispense a known volume V ∈ [0.5; 8] µ` of liquid using a micropipette (Eppendorf) on a pair of crossed fibers separated vertically by a separation distance h. We increase the angle between the fibers incrementally until δ ' 90◦ and then decrease δ incrementally. For each step in δ, the equilibrium state of the liquid is captured from the top and the side views with cameras (Nikon cameras D5100 and D7100 and 105 mm macro objectives) as illustrated in Figs. 3 and 4. The top view allows the measurement of the angle between the fibers and wetting length, while the side view permits the measurement of the separation distance between the fibers and the discrimination between the different states. Three possible liquid morphologies are observed: the drop state, the mixed morphology and the column state. In the drop state, the liquid collects in a single drop centered on the point where the distance between the fibers is the smallest, i.e. the node. The column morphology corresponds to the spreading of the liquid along the fibers. In this morphology, the height of the liquid remains of the same order of magnitude as the separation distance between the fibers. Finally, the mixed morphology is defined by the coexistence of a column and a drop lying at one end of a column. The position of the drop, i.e. the side of the column where it is located, is random and due to external noise when changing the tilting angle or the inter-fiber separation.
Experimentally, as the angle between the fibers is increased (Fig. 3), the length of the column of liquid decreases and the liquid switches to a mixed morphology. As the angle between the fibers is further increased, the liquid configuration becomes a drop. Then, when decreasing the angle between the fibers, the drop reverts back to the mixed morphology and eventually elongates into a column. We can also keep the tilt angle δ constant and increase the separation distance h (Fig. 4). For the particular case of parallel fibers (δ = 0), we increase the separation distance h incrementally until the drop morphology is observed. The same procedure is followed to measure the separation distance and to observe the liquid morphology. The change of morphology between the column and the drop state can be observed in the plane defined by the two fibers. Indeed, as h increases, the transition occurs when the liquid overspills the fibers. At the transition, the liquid collects in a drop. For instance, the morphology can be discriminated between figures 4(e) and 4(f).
where y˜ = y/a. ˜ > 0, we recover We observe that for δ = 0 and h the expression derived by Princen for parallel fibers [22], ˜ = 0 and δ > 0 we obtain the situation whereas for h of touching crossed fibers [24]. Using geometrical argu˜ ments, we define the radius of curvature R, ˜ ˜ = R = 1 + d − cos α , R a cos(α + θE )
(3)
and the liquid cross-sectional area, A A˜ = 2 a
=
h i ˜ 2 2 α + 2 θE − π + sin[2 (α + θE )] R
˜ sin α cos(α + θE ) − 2α + sin(2 α). +4 R (4) 3
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Fig. 3. Evolution of the morphology of a drop of volume V = 2 µℓ of silicone oil (5 cSt) on two touching crossed fibers (h = 0) Figure 3: Evolution of the morphology a dropisofvaried: volumetop V =(left) 2 µ` of silicone oil (5 cSt) on two crossedin a column of radius a = 150 µm as the angle δ between theoffibers and side (right) views. Thetouching liquid starts ◦ ◦ fibers (h = 0) of radius a = 150 µm as the angle δ between the fibers is varied: top (left) and side (right) views. state for a small tilting angle, (a) δ = 4 . As the angle is increased, a mixed morphology is observed: (b) δ = 9The , (c) δ = 15◦ , o starts state a small tilting angle, (a)◦ ,δa=drop 4 . As is increased, a mixed morphology (d) δ = 25◦ liquid and (e) δ =in35a◦column . Finally, at for larger angles, (f) δ = 45 liesthe atangle the crossing point of the fibers. Scale bars are o o o o o is observed: (b) δ = 9 , (c) δ = 15 , (d) δ = 25 and (e) δ = 35 . Finally, at larger angles, (f) δ = 45 , a drop lies 5 mm. at the crossing point of the fibers. Scale bars are 5 mm.
Fig. 3. Evolution of the morphology of a drop of volume V = 2 µℓ of silicone oil (5 cSt) on two touching crossed fibers (h = 0 of radius a = 150 µm as the angle δ between the fibers is varied: top (left) and side (right) views. The liquid starts in a column state for a small tilting angle, (a) δ = 4◦ . As the angle is increased, a mixed morphology is observed: (b) δ = 9◦ , (c) δ = 15◦ (d) δ = 25◦ and (e) δ = 35◦ . Finally, at larger angles, (f) δ = 45◦ , a drop lies at the crossing point of the fibers. Scale bars ar 5 mm.
Fig. 4. Evolution of the morphology of a drop of volume V = 2 µℓ of silicone oil (5 cSt) on two non-touching crossed fibers of radius a = 150 µm, tilted by an angle δ = 4◦ as the distance h between the fibers is increased: top (left) and side (right) views. The liquid starts in a column state at small separation distances, (a) h = 0. As the distance h is increased, the liquid adopts a drop morphology (f)-(g). Scale bars are 5 mm.
silicone oil. As this capillary length is usually larger than
the mixed morphology and the column state. In the drop
Fig. 4. Evolution ofHthe of our a drop of volume Vwe= 2state, µℓ of silicone oil (5 cSt) onintwo non-touching crossed on fibers the typical height4: ofmorphology the liquid in experiments, collects single drop centered theo Figure Evolution of the morphology of a drop of volume V = the 2 µ` liquid of silicone oil (5 cSt)a on two non-touching ◦ radius a = 150that µm, tilted by an angle δbe =neglected. 4 as the distance the fibers increased: top (left) and sideis(right) views first assume effects Weangle shallδh =between point where thehisdistance between fibers crossed gravity fibers of radius acan = 150 µm, tilted by an 4o as the distance between the fibers is the increased: top the smallThe liquid starts in a column state at small separation distances, (a) h = 0. As the distance h is increased, the liquid adopts discuss this(left) assumption when large of starts fluid and/or i.e. node. The column to a and side (right) views.volumes The liquid in a columnest, state at the small separation distances,morphology (a) h = 0. Ascorresponds the drop morphology (f)-(g). Scale bars are 5 mm. distance h is increased, the liquid adopts a drop morphology (f)-(g). Scale bars are 5 mm.
large fibers are used, as the Bond number of the system, the spreading of the liquid along the fibers. In this morBo = ρg(2a)2 /γ, becomes larger than one. phology, the height of the liquid remains of the same order of magnitude as the separation distance between the In a typical experiment, we dispense a known volume silicone oil. As this capillary length is usually larger than the mixed morphology and the column state. In the drop V ∈ [0.5; 8] µℓ of liquid using a micropipette (Eppendorf) fibers. Finally, the mixed morphology is defined by the the typical height H of the liquid in our experiments, we state, the liquid collects in a single drop centered on th on a pair of crossed fibers separated vertically by a separa- coexistence of a column and a drop lying at one end of first assume that gravity effects can be neglected. We shall point where the distance between the fibers is the small tion distance h. We increase the angle between the fibers a column. The position of the drop, i.e. the side of the discuss this assumption when◦ large volumes of fluid and/or 4 column est, i.e. where the node. column morphology corresponds it isThe located, is random and due to exter-to incrementally until δ ≃ 90 and then decrease δ increlarge fibers are used, as the Bond number of the system, nal thenoise spreading of the liquid along the fibers. In this mor when changing the tilting angle or the inter-fiber mentally. For each step in δ, the equilibrium state of the
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2a
zz
yy
x
xx
zz y y
xx
) ( yy ) 22dd (
dd( ( y y))
2h h+ 2a 2h h+++2a 2a 2a
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Fig.5.5.Schematic Schematic and and notations notations of of the the system system composed composed of Fig. of
shape of of the the surface ofthe the cross-section thecolumn column describe the liquid morphology on parallel fibers. For The shape surface of cross-section ofofthe ˜== ˜R is defined defined bywe itsobtain dimensionless radiusofofcurvature curvature ˜ = 0,by its dimensionless radius R h the expression derived by Sauret et al. and the the angle angle between betweenthe theline lineconnecting connectingthe thecenters centers R/a for and touching crossed fibers [24]. of the the fibers fibers and andthe theradius radiusto tothe theliquid-fiber-air liquid-fiber-airboundary boundary determine of fluid (fig. To 5(b)). We define definethe themaximum equilibriumvolume configuration α (fig. 5(b)). We the equilibrium configuration ofof that can be contained inofofatwo column is a symmetric general situation two fibersmorphology thatare arenot notinthat incontact, contact, the general situation fibers that separated by minimum distancehh>> and tilted of liquid i.e. separated aa minimum distance and tilted state, weby consider the expression of00the volume with an withlying an angle angle δ> > 0. 0. on δthe fibers in this morphology: The The inter-fiber inter-fiber distance, distance,22d(y), d(y),varies variesasasaafunction functionofof Z the the distance distance yy to to the the point pointOOwhere where the twofibers fibersare arethe the ˜ two L closest between the closest (fig. (fig. 5(a)). 5(a)). ∆x ∆x isis the distance between axesofof ˜ y ) d˜ V˜thedistance = A(˜ ythe , axes (8) the two two fibers fibers projected projected in in the theplane plane (xy), y),and and22hh++2 2a a ˜(x −L is the the closest closest distance distance between betweenthe theaxes axesofofthe thetwo twofibers. fibers. We have ˜ ˜ the halfhave where A(˜ y ) is the cross-sectional !area and L ! "" δ ∆x length of the a constraint δ = ∆x. on the vol2 2 column. Imposing 2 (∆x) + 4(a+h) 4(a+h)2= =4[d(y) 4[d(y)++a]a]2 and and tan tan =2y . (∆x)2 + ume of liquid V˜ leads to a unique 2value of the wetted 2 2y (1) ˜ (1) length 2 L. Using two expressions, we obtain Using these these two expressions, we obtain ˜ In addition, solving the quadratic equation (6) for R ## !! "" $$1/2 1/2 d(y) (3), we observe 22 δsolution δ + (1 + in d(y) = y˜22tanthe ˜˜and=substituting ˜˜2relation d(y) (2) d(y) = y˜ atan (2) +h) h)2 d˜ −−1,1, 22 + (1value that= d˜aareaches maximum max when varying α
twofibers fibers radius 22a.a. (a) Top Top view view and and (b) cross-section two ofof radius (a) Figure 5: Schematic and notations of (b) thecross-section system com- for a given θE . Therefore, if the local inter-fiber distance view. where y/a. view. where˜ y˜y˜ = posed of two fibers of radius 2 a. (a) Top view and (b) d(˜ yobserve )=isy/a. larger thanδ d˜= state cannot exist. ˜column max , theh We that ˜ >> 0,0,we We observe that for for δ = 00 and and h werecover recover cross-section view. This condition defines the maximum length of a liquid the expression derived by Princen for parallel fibers the expression derived by Princen for parallel fibers[22], [22], a column. We can also keep the tilt angle δ constant and a column. We can also keep the tilt angle δ constant and whereas ˜L ˜ max ˜˜state, for h = 00 and δδ >> 00 we obtain the situation ofof) = d˜max . column since this corresponds to d( increase the separation distance h (fig. 4). For the particwhereas for h = and we obtain the situation increase the separation distance h (fig. 4). For the partic- touching crossed fibers [24]. Using geometrical arguments, Using relation write the maximum ulardetermine case of parallel fibers (δ =the 0), we increase the separaTo analytically cross-sectional shape of touching crossed fibers(2), [24]. we Using geometrical arguments,spreading ular case of parallel fibers (δ = 0), we increase the separa˜ we define the radius of curvature R, tion distance h incrementally until the drop morphology is ˜ length: the column at until the the equilibrium, we assume tion distancemorphology h incrementally drop morphology is we define the radius of curvature R, observed. The same procedure is followed to measure the ˜ that at each y˜ fromis followed the “kissing” point, observed. Thedistance same procedure to measure the the α i1/2 ˜ = RR =h11++dd˜−−cos separation distance and to observe the liquid morphology. cos α, R ˜ separation distance and to observe the liquid morphology. ˜+max ˜ 2 (3) R = a = (1 , (1 + h) (3) cross-sectional only between dependsthe oncolumn the distance cos(α θE ) 2 − + d The change ofshape morphology and the beThe change of morphology the column and the ˜ max a= cos(α + θE ) ˜ y )inbetween . (9) L tween d(˜ and the contact angle . the liquid cross-sectional dropthe statetwo canfibers, be observed the on plane defined by the two θEand drop state can be observed in the defined by thewhen two and the liquid cross-sectionalarea, area, tan(δ/2) fibers. Indeed, ason h an increases, theplane transition occurs A fibers. force balance infinitesimal volume dV = A dL Indeed, as h increases, the occursthe when A the to liquid overspills the fibers. Attransition the transition, liq˜ 2 [2α +that leads theoverspills equilibrium condition A˜˜ = We R π+ sin[2 (α length + θE )]] increases when deA =observe the the For fibers. At the[24]: transition, the can liqthe wetting E − ˜ 2 [2α +2θ uidliquid collects in a drop. instance, the morphology A = a22 = R 2θ − π + sin[2 (α + θE )]] ˜ uid collects in a drop. For instance, the morphology can a creasing the tiltingE angle δ and the separation distance h. h π figs. 4(e) and (f). i A˜ be discriminated between ˜ sin α cos(α + θE ) − 2α + sin(2α). (4) be discriminated between figs. 4(e) and (f). + 4 R ˜ ˜ particular that + forθEtouching crossed fibers 4 = 0. (5) Note in + − α − θE R − α cos θE + ˜ sin α cos(α 4R ) − 2α + sin(2α). (4) √(h = 0) ˜ 2 R and a perfectly wetting we haveshape d˜maxof= 2. To determine analytically theliquid, cross-sectional determine analytically theequilibrium, cross-sectional shape of 3 Analyticalthe modeling ˜ we the To column at the For amorphology given separation distance h andassume tilt angle δ, the expression of A˜ given by the relation 3Substituting Analytical modeling the column morphology atthe the“kissing” equilibrium, we assume that at each distance y ˜ from point, the crossmaximum wetting length defines the maximum volume (4)We in equation (5) leads to a quadratic equation for the that at each distance y˜ from the “kissing” point, the crossconsider two rigid fibers having a cylindrical cross- sectional shape only depends on the distance between the ˜ a[22]: of thed(˜ and thus the regime of existence of the We consider two rigid fibers having a cylindrical crossradius of curvature R sectional onlyon depends on the distance the ˜ column section, separated by minimum distance h and tilted by two fibers,shape y ) and the contact angle θE . Abetween force bal˜ section, by define a minimum distance hiand tilted by ance h separated column state. Indeed, the maximum volume of liquid an angle δ. We can a system of coordinates Oxyz two fibers, d(˜ y ) and on the contact angle θ . A force balE on an infinitesimal volume dV = AdL leads to the 2 ˜represented an angle system of Oxyz asR schematically in (α fig.+ 5.θcoordinates When we equilibrium ancethat on ancan infinitesimal volume dV in= aAdL leads state to theis defined πδ.−We 2 αcan − 2define θE + asin[2 2α + sin(2 α) condition be at [24] equilibrium column E )] −needed, asuse represented schematically in fig. dimensionless 5. When we equilibrium condition [24] the fiber radius a to construct paramh i needed, by %& ' ( ˜ use thethe fiber radius a to inter-fiber construct dimensionless param˜ = d/a, π ˜ eters: dimensionless distance d the +4 R sin α cos(α + θE ) − α cos θE = 0. (6) ˜ − α cos θE (+ A˜= 0. %& ' R 4 − α − θ (5) ˜ E A π eters: the dimensionless inter-fiber d = d/a, the ˜ = distance Z ˜ ˜ max R dimensionless spacing distance h h/a, the dimensionless 2 ˜ L 4 − α − θE R − α cos θE + = 0. (5) ˜ = h/a, the dimensionless ˜ dimensionless spacing distance h ˜ 2 ˜ y ) d˜ ˜max = R wetting length L/a, the dimensionless cross-sectional A(˜ y .rela(10) Therefore, forLa=given liquid, i.e. a specified value of Substituting the Vexpression ˜ of A given by the ˜ and 2L ˜ max ˜ = V /a3 . wetting = L/a, the dimensionless cross-sectional −˜ L the dimensionless volume V area A˜ length = A/a the contact angle θ , equation (6) can be solved to obtain E expression of A given by the relationSubstituting (4) in eq. (5)the leads to a quadratic equation for the 3 andfibers the dimensionless volume V˜ = V /a . area A˜ = A/a Provided that2 the are not parallel, i.e. that the tilt˜ as ˜ [22]:to a quadratic tion (4) in eq. (5) R leads equation for˜ the the dimensionless radius of curvature R a function of radius of curvature ˜ Provided that the fibers are not parallel, i.e. that the tiltFor a volume of liquid V larger than V ing angle is not zero, a drop of liquid lying on two fibers max , the liq˜ [22]: of curvature R α.ing not zero, drop where of liquid willangle travelistowards thea point thelying fiberson aretwo thefibers clos- radius 2 ˜ uid would not be able to spread in a column state and R [π − 2α − 2θE + sin[2(α + θE )]] − 2α + sin(2α) For the particular a perfectly wetting liquid R est,travel which we referthe to case as theof “kissing if the will towards point where the point” fibers are thefibers clos˜ 2 [π − be 2α either − 2θE +insin[2(α + θE )]] − 2α + sin(2α) could a mixed morphology or in a drop state ˜the ˜ in refer toa as the “kissing point” fibers 4R[sin α cos(α + θE ) − α cos θE ] = 0. (6) (θest, =which 0),contact. weweobtain simple expression of ifR using equa- + Eare as we shall see in the following. However, even for a We consider a liquid that has a contact angleinθEequation and ˜ are(6) in contact. + 4R[sin α cos(α + θE˜) − α cos θE ] = 0. (6) tion [26, 24]. Substituting this expression ˜ Therefore, for a given liquid, i.e. a specified value of the volume of liquid V < V , we need to compare the is We in aconsider column state on that a pair characterized by a liquid hasof afibers contact angle θE and max (3)is(δ, and with equations we obtain a direct relation angle θaE ,given eq. (6) solved tomorphologies obtain theof diTherefore, forenergies liquid, i.e. a specified value thedefined as a). This morphology consists of acharacterized long column of contact in h, a column state on a(2), pair of fibers by surface ofcan all be possible ˜ solved between y˜with and α: contact θE , − eq. (6) can be to A obtain the A di- are the radius ofγcurvature R aswhere a function of and α. liquid, varying cross-section and height. (δ, h, a). This morphology consists of aa constant long column of mensionless E =angle γA cos θ A LV E SL LV SL ˜ as a function mensionless radius of curvature R of α. liquid,s with varying cross-section and a constant height. liquid-air and liquid-fiber surface areas. These energies 2 δ ˜ y˜2 tan2 + 1+h are minimized to determine which morphology will be 2 preferentially adopted by the liquid. r −1 ! The transition between the drop state and the mixed π −1 cos α.(7) morphology is more complex as the shape of the drop = 1+ 2 α − sin(2 α) between two fibers does not have an analytical descripFor δ = 0, we recover from equation (7) the expres- tion. We assume that the surface energy associated with sion derived by Princen [22] and Proti`ere et al. [26] to the drop morphology is that of a sphere of equivalent
5
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3.0 DROP 2.5 2.0 1.5
h˜
radius [3 V˜ /(4 π)]1/3 , pierced by two fibers. Note that Proti`ere et al. [26] show that a better quantitative agreement between theoretical and experimental results can be obtained by modeling the drop with a shape close to a hemisphere with an energy equal to : " # r 2/3 Edrop 1/3 ˜ 2/3 2 ˜ ˜ ˜ = 0.6 (36 π) V −π 6 V /π − 4d , Edrop = γ a2 (11) where the pre-factor 0.6 is empirical and takes into account that the shape of the liquid in the drop state is not exactly a sphere. The corresponding surface energy associated with the column morphology is Z
h π i ˜ R − θE − α − α cos θE d˜ y 2 ˜ −L (12) Note that this formulation allows us to recover the expression previously obtained for two parallel fibers[26] as ˜ are constant along the column in this situation α and R and the energy reduces to h π i ˜col = 8 L ˜ ˜ − α cos θE . E − θE − α R (13) 2 ˜col = Ecol = 4 E γ a2
˜ L
In the present situation, we evaluate equation (12) nu˜ obtained in merically using the expressions for α and R ˜ the previous section for varying distance d(y) between the fibers similarly to the derivation by Sauret et al. [24]. The drop shape on a pair of fibers is much more complex to describe as there is no analytical expression that captures the shape of the drop. In addition, we also need to impose a constraint on the volume: the liquid can be either in a column morphology V˜col , in a drop morphology V˜drop or in a mixed state but the total volume of liquid V˜ should always sat˜ of isfy V˜ = V˜col + V˜drop . The dimensionless energy E ˜ =E ˜col + E ˜drop reaches a minimum for a the system E given volume V˜ , a given tilt angle δ and a given sepa˜ In addition, we assume that there is ration distance h. no activation barrier between the various morphologies. By doing so, we observe qualitatively the transition between the mixed morphology and the drop state but no quantitative evolution can be obtained. Therefore, this transition is captured experimentally only.
4 4.1
Morphology diagrams Parallel fibers
We first conducted experiments with drops of silicone oil on parallel nylon fibers to verify the analytical model for perfectly wetting liquids. The experimental results ˜ are reported in Fig. 6 in a morphology diagram of h as a function of V˜ (the results for dodecane, i.e, a partially wetting liquid are reported in the appendix). We
1.0 0.5 COLUMN 0.0
0
200
400
600
800
1000
V˜
˜ Fig. 6. Morphology diagram in the parameter space (V˜ , h) Figure 6: Morphology in the parameter space using silicone oil (θE = 0◦diagram ) on parallel fibers (δ = 0◦ ) of radii ◦ ˜ using a(V˜ =, [100, 150, 230] µm and V ∈ [0.5, 4] µℓ. h) silicone oila(θvolume on parallel fibers E = 0of )liquid ◦ Red circles show the drop morphology the (δ = 0 ) of radii a = [100, 150, 230]and µm blue and squares a volume column morphology. The light orange region corresponds to the of liquid V ∈ [0.5, 4] µ`. Red circles show the drop morregion where both morphologies are observed. The horizontal phology and blue squares the column morphology. The solid line corresponds to the theoretical maximum separation √ both light orange region corresponds to the region where distance where the column state is possible, d˜max = 2.
morphologies are observed. The horizontal solid line corresponds to the theoretical maximum separation √ distance the columndiagrams state is possible, d˜max = 2. 4where Morphology
4.1 Parallel fibers We first conducted experiments with drops of silicone oil find that the drop-column transition for the max√ the occurs on parallel nylon fibers to verify analytical model for ˜ = d˜max imum separation h = 2, which is in agreement perfectly wetting liquids. The experimental results are rewith the analytical solution derived in the previous sec˜ as a funcported in fig. 6 in a morphology diagram of h tion and consistent with previous experimental results tion of V˜ (the results for dodecane, i.e., a partially wetobtained forare this geometry. For larger volumes liquid, ting reported in the appendix). We offind that ˜ >liquid V 400, we observe a coexistence region in which for a the drop-column transition occurs for the maximum sepa√ ˜˜ and˜V˜ the liquid given h can either be in a drop state ration h = dmax = 2, which is in agreement with the or a column morphology. This region widens analytical solution derived in coexistence the previous section and with increasing volumes. The coexistence region was also consistent with previous experimental results obtained for observed by Proti` et al.volumes [26] of liquid, V˜ > 400, we this geometry. Forere larger ˜ and observe a coexistence region in which for a given h We can understand the coexistence region by con˜ V the liquid can either be in a drop state or a column sidering the This Bond number of the widens system, with defined as morphology. coexistence region increas2 Bo = ρ g (2 a) /γ, where g is the gravitational constant, ing volumes. The coexistence region was also observed by 2 a isere a characteristic length scale associated to the sepProti` et al. [26] aration distance and γ surface tension. Bond We can understand is thethe coexistence region The by considnumber describes the relative influence of the gravitaering the Bond number of the system, defined as Bo = relative to the surface tension effects. Within ρtional g(2 a)2force /γ, where g is gravitational constant, 2 a is the coexistence region, we generally find Bo > 1, which a characteristic length scale associated to the separation is an indication effects of gravity thenumber liquid distance and γ isthat the the surface tension. The on Bond describes relative Gravity influencecan of hinder the gravitational force cannot bethe neglected. the liquid from relative to surface effects.configuration, Within the coexistence spreading into thetension more stable i.e. the region, generally find Boin>the 1, coexistence which is anregion indication columnwe state, which results obthat theHowever, effects ofcapturing gravity quantitatively on the liquid this cannot be neserved. transition glected. Gravitynumerical can hinder the liquidto from spreading into would require simulations define the shape the the column state,which which of amore dropstable and aconfiguration, column in thei.e. presence of gravity, results inthe thescope coexistence region study. observed. capis out of of the present OurHowever, studies have turing quantitatively this transition would require numerbeen performed in an horizontal plane, but we can infer ical simulations to define the shape of a drop and a column that gravitational effects on the parameters space of the in the presence of gravity, which is out of the scope of the drop could be modified slightly when this plane is tilted. 6
Fig. 7. Mo a = [100, 1 a volume show the d ogy and ye orange reg and the m sponds to V˜ , below w
present st izontal pl on the pa slightly w
4.2 Touch
We then ˜= fibers (h the analy We report morpholo rescaled v plot the a the colum represents cannot ex serve a go and the e The t drop stat the exact the rough however, diagram) gies are p ogy to th region to V˜ for V˜ which the occurs. T mixed an Bond num
Eur. Phys. J. E (2015) 38: 62
Eur. Phys. J. E (2015) 38: 62
ROP
MN 1000
˜ pace (V˜ , h) 0◦ ) of radii ∈ [0.5, 4] µℓ. squares the ponds to the e horizontal m separation √ = 2.
silicone oil l model for ults are reas a funcrtially wete find that mum sepant with the ection and btained for > 400, we ˜ and iven h r a column ith increasbserved by
by considd as Bo = tant, 2 a is separation nd number tional force coexistence indication not be neeading into tate, which wever, capuire numerd a column cope of the
Fig. 8. Morphology diagram for a drop of silicone oil places on Figure 8:crossed Morphology drop of silicone oil separated fibers ofdiagram radii a =for 150aµm in the parameter ◦ ˜ separated space using a volume V˜ =fibers 592 of of silicone (θE150 = 0µm ). places(δ,onh) crossed radii oil a = Red circles show the space drop morphology, bluea squares ˜ using in the parameter (δ, h) volumethe V˜ column = 592 morphology and yellow diamonds are the mixed morphology. of silicone oil (θ = 0 ◦ ). Red circles show the drop The thick solid lineEis the prediction of our model.
Fig. 7. Morphology diagram for touching crossed fibers of radii a = [100,7: 150,Morphology 180, 230] µm indiagram the parameter space (V˜ , crossed δ) using Figure for touching ◦ a volume V ∈ [0.5, µℓ of150, silicone (θEµm = 0in ).the Redparamcircles fibers of radii a =7][100, 180,oil230] show the drop˜ morphology, blue squares the column morpholeter space (V , δ) using a volume V ∈ [0.5, 7] µ` of siliogy and yellow diamonds are the mixed morphology. The light cone oilregion (θE = 0 ◦ ). Redtocircles show the where drop morpholorange corresponds the parameters both drop ogy, blue squares the column morphology diand the mixed morphology are observed. The and solid yellow line correamonds are the mixed morphology. The light orange responds to the theoretical maximum angle, for a given volume ˜ , below V which thetocolumn state is possible. gion corresponds the parameters where both drop and
morphology, blue squares the column morphology and yellow diamonds are the mixed morphology. The thick solid line function is the prediction model. is a weak of V˜ and of theour Bond number, but we are unable to detect it through our experiments.
the mixed morphology are observed. The solid line corresponds to the theoretical maximum angle, for a given present study. Our studies have been performed in an hor- our experiments. volume V˜ , below which the column state is possible. 4.3 Separated crossed fibers izontal plane, but we can infer that gravitational effects on the parameters space of the drop could be modified In this last configuration, the tilt angle δ and the minislightly when this plane is tilted. ˜ are both non-zero and influence mum spacing distance h 4.2 Touching crossed fibers the resulting morphology. In addition, the dimensionless 4.3 Separated crossed fiberswhich leads to a We then perform experiments with two touching crossed- volume V˜ is a parameter to consider, 4.2 Touching crossed fibers ˜ = 0 and fibers (h δ > 0) to further explore the validity of huge parameter space to investigate. To compare experiwith the our tilt analytical weminiperthe then analytical model for liquidswith withtwo zero contactcrossed angle. mental In this measurements last configuration, angle δmodel, and the We perform experiments touching formed experiments with drops of silicone oil at constant ˜ We report the observed morphologies in Fig. 7. The col˜ mum spacing distance h are both non-zero and influence fibers (h = 0 and δ > 0) to further explore the validity of volume, V˜ = 592 (corresponding to 2 µℓ on fibers of radii umnanalytical morphology is observed at with smallzero enough tilt angle. angle the resulting morphology. In addition, the dimensionthe model for liquids contact a = 150 µm) ˜and vary the tilt angle as well as the sepδ orreport rescaled volume V˜morphologies . On the morphology diagram, We the observed in fig. 7. The column aration less volume V a parameter whichis leads distance.is The results of to ourconsider, investigation prewe also plot istheobserved analytical prediction for tilt the angle transition morphology at small enough δ or sented to a huge parameter space totouching investigate. To compare in fig. 8. Similar to the crossed-fiber sys˜ rescaled V . On the and morphology we also tem between volume the column state the two diagram, other morpholoexperimental our analytical we observemeasurements three possiblewith morphologies: drop, model, mixed plot analytical prediction for theVmax transition gies, the which represents the volume beyondbetween which and we performed experiments dropsofofour silicone oil at column, each defined inwith a region parameter the columnstate statecannot and the twobetween other morphologies, a column exist crossed fiberswhich (see space ˜volume, constant V˜ report = 592our (corresponding to 2 µ` on (δ, h). We also analytical prediction in a column state represents the volume Vmax beyond equation (10)). We observe a goodwhich agreement between fibersdiagram of radii(black a = thick 150 µm) vary the tilttheangle as this line)and that captures region cannot exist between crossed (see eq. (10)). We obthe analytical prediction andfibers the experimental results. where state is observed. We should emphawell asthe the column separation distance. The results of our invesserve a good agreement between the analytical prediction ˜ = 0) corresponds to the size that the horizontal axis (h The between and thetransition experimental results.the mixed morphology and tigation is presented in Fig. 8. Similar to the touching situation of touching crossed fibers andpossible we again observe drop state is not captured by an analytical expression crossed-fiber system we observe three morpholoThe transition between the mixed morphology and the transition from and a column morphology to a mixed moras the exact shape of the drop is not known and because gies: drop, mixed column, each defined in a region drop state is not captured by an analytical expression as phology and eventually a(δ,drop at large angles. The ver˜ of the rough estimate of equation (11). Our experimenof our parameter space h). We also report our anathe exact shape of the drop is not known and because of tical axis (δ = 0) corresponds to a pair of parallel fibers, tal results, however,ofindicate coexistence regionresults, (light lytical prediction in this diagram (black thick line) that the rough estimate eq. (11).a Our experimental and in agreement with our previous results (fig. 6) we only however, a coexistence (light orange in the captures the region where the column state is observed. orange inindicate the diagram) where region both the mixed and observe two states: column and drop. Figure 8 further con˜ = diagram) where both mixed The and transitions the drop morpholodrop morphologies arethe present. from the firms We should emphasize thethe horizontal (h 0) the possibility to that predict observedaxis morphology gies aremorphology present. Theto transitions from theregion mixedand morpholmixed the coexistence from on corresponds to the situation of touching crossed fibers two fibers randomly oriented in space. ogy to the coexistence coexistence the coexistence region region to the and dropfrom statethe appear to be and we again observe the transition from a column morregion to the of drop appear be independent of the phology to a mixed morphology and eventually a drop independent thestate V˜ for V˜ >to100. We can estimate V˜ for V˜ > 100. We can estimate the Bond number at the Bond number at which the transition between the at large angles. The vertical axis (δ = 0) corresponds which the transition between the mixed and drop states 5 Conclusions mixed and states occurs. Thetransition results suggest occurs. The drop results suggest that the betweenthat the to a pair of parallel fibers, and in agreement with our the transition between the mixed and drop morphologies previous resultswe(Fig. we only observe two states: and colthis paper, have6)investigated experimentally mixed and drop morphologies is also independent of the In theoretically the wetting morphologies on a pair of ranis also independent of the Bond number. It is possible, umn and drop. Fig. 8 further confirms the possibility to Bond number. It is possible, however, that the transition however, that the transition is a weak function of V˜ and predict the observed morphology on two fibers randomly the Bond number, but we are unable to detect it through oriented in space. 7
Fig. 9. Mor ented wetted eter space (V blue squares the mixed m the region w
domly place vious studie morphologi drop locate one drop at cal and exp morphology possible situ scribe the w oriented fib tilt angle δ ˜=h fibers h tained prev and touchin ing distance a zero and three possib article allow ˜ V˜ ) t (δ, h, randomly o
The ana tween the touching fib should be n and partiall transitions gies will he fiber array ated by a li is especially
oil places on he parameter il (θE = 0◦ ). s the column morphology. l.
, but we are
d the minind influence mensionless h leads to a pare experidel, we perat constant bers of radii as the sepation is preed-fiber sysdrop, mixed r parameter rediction in s the region uld emphaonds to the gain observe mixed mores. The verrallel fibers, g. 6) we only further conmorphology
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should be noted that the model is applicable to perfectly and partially wetting liquids. The characterization of the transitions between the three different wetting morphologies will help to better understand the behavior of a wet fiber array and describe the capillary interactions generated by a liquid bridge between two fibers. This approach is especially relevant to systems composed of flexible fibers where the capillary force can lead to the clustering of fibers [16, 19]. Our results suggest that a system of fibers can be used to manipulate liquids on a micro scale. For example, by mechanically altering the angle and/or spacing distance between the fibers, or by triggering changes in the liquid volume through condensation or evaporation,[27, 28] we can change the morphology adopted by the liquid on the fibers. Additionally, since the model that we have proposed is applicable to both perfectly and partially wetting liquids, we can consider using the transitions between wetting morphologies on fibers to estimate the Fig. 9. Morphology diagram for a pair of fibers randomly ori- contact angle of liquids on fibers. Presently, the conFigure 9: Morphology diagram for a pair of fibers ranented wetted by a drop of silicone oil (θE = 0◦ ) in the paramtact angle of a liquid on fibers can be computed using ◦ domly oriented wetted by circles a drop show of silicone oil (θ ˜ Red E =0 ) eter space (V˜ , δ, h). the drop morphology, a method proposed by Carroll [18] that involves solv˜ and inblue thesquares parameter space morphology (V˜ , δ, h). Redyellow circlesdiamonds show the the column are ing elliptic integrals. A possible alternative to such a drop morphology, blueThe squares the column the mixed morphology. light orange region morphology corresponds to cumbersome method, for example, would be to use the the region both morphologies are morphology. observed. and yellowwhere diamonds are the mixed The transition between the drop and column states of a liquid light orange region corresponds to the region where both on parallel fibers. morphologies are observed.
domly placed and oriented fibers. In agreement with prestudies, we show that in the most general case three Acknowledgements 5vious Conclusions morphologies are observed: a column morphology, a single drop located at the node, and a mixed morphology with We thank H´el`ene Lannibois-Drean and Franois Vianney In this paper, weend haveof investigated experimentally and from Saint-Gobain Research and Pierre-Brice Bintein for one drop at one a column. We report our analytitheoretically the wetting morphologies a pair of ran- providing the SEM pictures of glass wool. FB acknowlcal and experimental findings in a newon three-dimensional morphology shown fibers. in fig. 9, captures domly placeddiagram and oriented In which agreement withall edges that the research leading to these results partially possible studies, situations. parameters de- received funding from the People Programme (Marie previous we The showthree that relevant in the most general to case scribemorphologies the wetting are morphologies randomly Curie Actions) of the European Union’s Seventh Framethree observed: abetween column two morphology, fibers are the volume liquid V˜ = morpholV /a3 , the aoriented single drop located at the node,ofand a mixed work Programme (FP7/2007-2013) under REA grant tilt with angle one δ and the at minimum distance ogy drop one endseparation of a column. We between report agreement 623541. ED is supported by set-up funds ˜ = h/a. Additionally, we show that the results obfibers h our analytical and experimental findings in a new three- by the NYU Polytechnic School of Engineering. HAS tained previously for thediagram more restrictive parallel dimensional morphology shown incases Fig. of 9, which and touching crossed fibers can be recovered for zero spac- thanks the Princeton MRSEC for partial support of this captures all possible situations. The three relevant pa- research. ing distance between the fibers coupled, respectively, with rameters to describe the wetting morphologies between a zero and non-zero angle between the fibers. Thus, the two randomly oriented are have the volume of liquid three possible situationsfibers that we highlighted in this 3 V˜article = V /aallow , theustilt angle δ and the minimum separation Partially wetting liquid to define a full three-dimensional diagram A ˜ = h/a. Additionally, we show distance fibers h ˜ V˜between (δ, h, ) to predict the liquid morphology on a pair of that the results obtained randomly oriented fiberspreviously (fig. 9). for the more restric- Most experiments performed to investigate the wetting tive cases of parallel and touching crossed fibers can be morphologies on a pair of fibers use silicone oil that is The analytical describes the transition berecovered for zero model spacingthat distance between the fibers a perfectly wetting liquid (θE = 0o ). To ensure that tween the different morphologies of the liquid on noncoupled, respectively, with a zero and non-zero angle be- our model correctly captures the influence of the contouching fibers is validated by experimental results. It tween the fibers. Thus, the three possible situations that tact angle, we also performed experiments using dodeshould be noted that the model is applicable to perfectly we highlighted this article allow us to define a cane (density ρ = 748 kg/m3 , surface tension γ = 25.4 andhave partially wettinginliquids. The characterization of the ˜ ˜ full three-dimensional diagram (δ, h, Vwetting ) to predict the mN/m, purchased from Sigma-Aldrich), which is a partransitions between the three different morphololiquid morphology on a pair of randomly oriented fibers gies will help to better understand the behavior of a wet tially wetting liquid and compared these results to the (Fig. fiber 9). array and describe the capillary interactions gener- analytical prediction. The contact angle of the dodecane a liquid bridge two fibers. This approach entally and ated Theby analytical model between that describes the transition be- on nylon fibers has been measured and estimated to be is especially relevantmorphologies to systems composed of flexible fibers 13 ± 1 ◦ . The liquid-fiber contact angle was measured pair of ran- tween the different of the liquid on nontouching fibers is validated by experimental results. It using the shape of a drop on a single fiber [18].
8
can be used xample, by ng distance in the liqion [27, 28], e liquid on e have proally wetting etween wetntact angle gle of a liqd proposed ntegrals. A method, for en the drop s.
ianney from or providing that the reunding from he European 07-2013) unhe Princeton s supported Engineering.
he wetting oil that is re that our contact anecane (denmN/m, purally wetting ical predic-
V
Fig. 10. Morphology diagram in the parameter space (V˜ , ˜ using a partially wetting liquid (dodecane) on parallel h) fibers of radii a = [100, 150, 230] µm and a volume of liquid V ∈ [0.5, 4] µℓ. Red circles show the drop morphologies and blue squares the column morphology. The light orange region We perform systematic experiments using a pair of corresponds the coexistence and from the coexistence region to the region region where both morphology are observed. Author contribution statement parallel fibers varying the volume of liquid V˜ = V /a3 The to the drop state seem to remain independent of horizontal solid again line corresponds to the analytical predic˜ max ≃ 1.33. ˜ = h/a. The re- tion and the minimum spacing distance h V˜ . for dodecane: h
All authors contributed equally to the paper. sulting morphology diagram is shown in Fig. 10. The transition between the drop state and the column morWe thank H´el`ene Lannibois-Drean and Fran¸cois Vianney from phology is captured by the analytical calculation, which Saint-Gobain Research and Pierre-Brice for providing ˜ maxBintein predicts a maximum h ' 1.33 that for athe liquid the SEM pictures of glassseparation wool. FB acknowledges reo with contact angle of 13 . As seen in the morphology search leading to these results partially received funding from diagram, for small rescaled volumes, V˜ of ≤ the 400,European there is the People Programme (Marie Curie Actions) Union’s Seventh Framework Programme (FP7/2007-2013) a sharp transition between drop and column statesunat ˜ max der REA=grant the Princeton h 1.33.agreement As with623541. siliconeHAS oil, thanks we observe a coexMRSEC partial of this research. ED is supported istence for region (insupport orange) in which generally Bo > 1, byan set-up funds by the NYU Polytechnic School of Engineering. indication that gravity cannot be neglected in such cases. We note that the coexistence region observed for dodecane is larger than that for silicone oil, suggesting Appendix A. contact Partially wetting that the larger angle and theliquid hysteretic effect in the contact line on nylon fibers makes it more difficult to Most experiments performedEur. to investigate the wetting Phys. J. E (2015) 38: 62 spread on the fibers. morphologies on a pair of fibers use silicone oil that is a perfectly wetting liquid (θE = 0◦ ). To ensure that our 3.0 model correctly captures the influence of the contact anDROP gle, we also performed experiments using dodecane (den2.5 kg/m3 , surface tension γ = 25.4 mN/m, pursity ρ = 748 chased from Sigma-Aldrich), which is a partially wetting liquid and2.0 compared these results to the analytical prediction. The contact angle of the dodecane on nylon fibers has 1.5 been measured and estimated to be 13 ± 1◦ . The liquidfiber contact angle was measured using the shape of a drop 1.0fiber [18]. on a single We perform systematic experiments using a pair of 0.5 parallel fibers varying the volume of liquid V˜ = V /a3 ˜ = h/a. The resultand the minimum spacing distance h COLUMN 0.0 ing morphology diagram is400 shown600 in fig. 800 10. The1000 transi0 200 tion between the drop state andV˜ the column morphology is captured by the analytical calculation, which predicts a˜ Fig. 10. Morphology diagram in the parameter space (V , ˜ max maximum separation h ≃ 1.33 for a liquid with contact ˜ using10: h) partially wetting liquidin (dodecane) on parallel Figure diagram the parameter space ◦a Morphology angle of 13 . As seen in the morphology diagram, forofsmall fibers of radii a = [100, 150, 230] µm and a volume liquid ˜ ˜ ( V , h) using a partially wetting liquid (dodecane) on ˜ rescaled volumes, V ≤circles 400, there is a drop sharpmorphologies transition beV ∈ [0.5, 4] µℓ. Red show the and parallel fibers of radii a = [100, 150, 230] µm and a vol˜ tween drop and states at hmax 1.33. As with blue squares the column column morphology. The = light orange region ume of liquid V ∈ [0.5, 4] µ`. Red circles show the drop corresponds to the region where both morphology are observed. morphologies squares thetocolumn morphology. The horizontal and solidblue line corresponds the analytical predic˜ max ≃ corresponds The light orange region to the region where tion for dodecane: h 1.33. both morphology are observed. The horizontal solid line corresponds to the analytical predictions for dodecane: ˜ max ' 1.33. h h˜
ustering of
by Carroll [18] that involves solving elliptic integrals. A possible alternative to such a cumbersome method, for example, would be to use the transition between the drop and column states of a liquid on parallel fibers.
We summarize the experimental results for dodecane on touching crossed fibers in a morphology diagram of the angle between crossed fibers, δ, as a function of the dimensionless volume V˜ = V /a3 (Fig. 11). These results can be compared to the analytical model for liquids with a non-zero contact angle. We observe a good agreement between the analytical prediction (black solid line) and the experimental results for the transition between the column and either mixed morphology or drop state. We observe a coexistence region where both the mixed morphology and drop state are present, which is similar that observed for silicone crossedcrossed fibers,fibers but Fig.to11. Morphology diagram for oil theon touching ˜ larger. The transitions from the mixed morphology to (h = 0) in the parameter space (V , δ). Red circles show the drop morphology, blue squares the column morphology and yellow diamonds are the mixed morphology. The light orange region corresponds to the region where both column and mixed9 morphologies are observed. The solid black line corresponds to the analytical prediction of the maximum angle where the
Fig. 11. Morphology diagram for the touching crossed fibers (h = 0) 11: in the parameter diagram space (V˜ ,for δ).the Redtouching circles show the Figure Morphology crossed drop morphology, blue squares the column morphology and ˜ fibers (h = 0) in the parameter space (V , δ). Red circles yellow diamonds are the mixedblue morphology. Thecolumn light orange show the drop morphology, squares the morregion corresponds to the region where both column and mixed phology and yellow diamonds are the mixed morphology. morphologies are observed. The solid black line corresponds The orangeprediction region corresponds to theangle region where to thelight analytical of the maximum where the both column mixed column state isand possible formorphologies a given volumeare V˜ .observed. The
solid black line corresponds to the analytical prediction of the maximum angle where the column state is possible silicone oil, we observe for a given volume V˜ . a coexistence region (in orange) in which generally Bo > 1, an indication that gravity cannot be neglected in such cases. We note that the coexistence region observed for dodecane is larger than that for silicone oil, suggesting that the larger contact angle and the References hysteretic effect in the contact line on nylon fibers makes it [1] more difficult J. toSimao, spreadD. onThomas, the fibers. P. Contal, T. Frising, S. Call´e, We summarize the experimental results for dodecane J. C. Appert-Collin, and D. B´ emer, “Clogging of on touching crossed fibers in a morphology diagram of the fibre filters by submicron droplets. Phenomena and angle between crossed fibers, δ, as a function of the dimeninfluence of operating conditions,” J. Aerosol Sci., sionless volume V˜ = V /a3 (fig. 11). These results can be vol. 35, pp. 263–278, 2004. [2] M. Brinkmann, J. Kierfeld, and R. Lipowsky, “A general stability criterion for droplets on structured substrates,” J. Phys. A. Math. Gen., vol. 37, pp. 11547–11573, 2004. [3] C. H. Chen, Q. Cai, C. Tsai, C. L. Chen, G. Xiong, Y. Yu, and Z. Ren, “Dropwise condensation on superhydrophobic surfaces with two-tier roughness,” Appl. Phys. Lett., vol. 90, p. 173108, 2007. [4] Y. Zheng, H. Bai, Z. Huang, X. Tian, F.-Q. Nie, Y. Zhao, J. Zhai, and L. Jiang, “Directional water collection on wetted spider silk.,” Nature, vol. 463, pp. 640–643, 2010. [5] T. S. Yu, J. Park, H. Lim, and K. S. Breuer, “Fog deposition and accumulation on smooth and textured hydrophobic surfaces,” Langmuir, vol. 28, pp. 12771–12778, 2012.
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