Journal of Colloid and Interface Science

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1 Journal of Colloid and Interface Science 354 (211) Contents lists available at ScienceDirect Journal of Colloid and Interface Science Landau Levich menisci M. Maleki a,b, M. Reyssat b, F. Restagno c, D. Quéré b, C. Clanet d, a IASBS, P.O. Box , Zanjan 45195, Iran b PMMH, UMR 7636 du CNRS, ESPCI, 755 Paris, France c LPS, UMR 852 du CNRS, Université Paris Sud XI, 914 Orsay, France d LadHyX, UMR 7646 du CNRS, Ecole Polytechnique, Palaiseau, France article info abstract Article history: Received 1 May 21 Acceted 27 July 21 Available online 3 August 21 Keywords: Di coating Dynamic meniscus Lubrication Wavy film Wetting As shown by Landau, Levich and Derjaguin, a late withdrawn out of a wetting bath at low caillary numbers deforms the very to of the liquid reservoir. At this lace, a dynamic meniscus forms, whose shae and curvature select the thickness of the film entrained by the late. In this aer, we measure accurately the thickness of the entrained film by reflectometry, and characterize the dynamic meniscus, which is found to decay exonentially towards the film. We show how this shae is modified when reversing the motion: as a late enetrates the bath, the dynamic meniscus can buckle and resent a stationary wavy rofile, which we discuss. Ó 21 Elsevier Inc. All rights reserved. 1. Introduction Coating is a widesread industrial rocess that consists in deositing a liquid film of uniform thickness on a solid [9,13]. It is commonly achieved by extracting a solid from a bath, a rocess often referred to as di coating [11,2,22,26]. Di coating has been extensively studied in various geometries such as tubes, Hele- Shaw cells and fibers [2,14,16] and for various fluid roerties [3,19,17,21,8,1,18,15]. Let us describe the different henomena observed when a vertical late is moved out of a bath of wetting liquid of density q and surface tension c. At rest, first, a meniscus develos along the late [5]. Such a static meniscus can be seen in Fig. 1c. Wetting and gravity are resonsible for the deformation of the free surface: for a wetting liquid, the contact angle is zero between the solid and the liquid, while gravity imoses a horizontal surface far from the wall. Both constraints can be satisfied by curving the interface by a radius R, such that the increase of gravitational energy qg R 4 is comensated by the decrease of surface energy cr 2. This balance indicates that the meniscus size scales as the caillary length a, with a ¼ c=qg ffiffi. If the solid late is moved uwards at a velocity V >, some liquid is entrained as seen in Fig. 1d and e. The first quantitative analysis of this rocess was done by Landau, Levich and Derjaguin [1,6]. The driving force for entrainment is viscosity: molecules close to the wall follow its motion and entrain their neighbors. Corresonding author. address: clanet@ladhyx.olytechnique.fr (C. Clanet). The viscous force er unit volume scales as gv =h 2, denoting h as the entrained thickness. Two different volumetric forces resist viscous entrainment, namely gravity qg and surface tension. As emhasized above, the meniscus is curved: according to Lalace, its ressure is lower than the ambient ressure by an amount c/ a. Conversely, the ressure in the entrained flat film (far above the meniscus) is the ambient ressure, which imlies a ressure gradient oosing liquid entrainment. The corresonding Lalace force is c/ak, where k is the characteristic length of the dynamic meniscus connecting the static meniscus to the flat film. The ratio between gravity and caillary resistances reduces to k/a, that is, the ratio between the sizes of the dynamic and static menisci. If k < a, caillary resistance dominates gravity. We call it the caillary wier regime since the dynamic meniscus acts as an elastic wier on a windshield, limiting the film thickness right beyond the wier (Fig. 1d). In this regime, the balance between viscosity and surface tension leads to: h 2 =k aca, where Ca = gv /c is the caillary number. Since the static meniscus is hardly deformed by the motion of the wall [4], the curvature of the dynamic meniscus, h /k 2, matches the curvature of the static meniscus 1/a. This imlies a second relation: k 2 ah. We eventually deduce the film thickness h aca 2=3 and the length of the dynamic meniscus k aca 1=3. These exressions are exected to be valid in the small deformation regime k a, that is, for Ca 1. A rigorous asymtotic matching allowed Landau, Levich and Derjaguin to derive the numerical factors in these laws [1,6]: h :94 aca 2=3 ð1þ k :65 aca 1=3 ð2þ /$ - see front matter Ó 21 Elsevier Inc. All rights reserved. doi:1.116/j.jcis

2 36 M. Maleki et al. / Journal of Colloid and Interface Science 354 (211) mm (a) (b) (c) (d) (e) 1 +1 Ca = V / Fig. 1. Deformation of a liquid bath induced by the motion of a vertical solid late at different caillary numbers Ca = gv /c. Ca > corresonds to an uward motion of the late that comes out coated with a film. Conversely, a late coated at Ca > can be lunged in the bath (Ca < ). For high caillary numbers, we observe in Fig. 1e that the dynamical meniscus becomes much larger than the static one and that the film has a non-uniform thickness. In this regime, gravity dominates surface tension which leads to the force balance gv =h 2 qg. As early noticed by Derjaguin, the entrainment law becomes h aca 1=2. Since the gravitational drainage of a film of thickness h takes lace at a velocity qgh 2 =g, we also deduce that the film falls down at the entrainment velocity V, exlaining why it cannot reach a uniform thickness. Once the late is coated, its motion can be reversed (Ca < ). At small velocity ( 1<Ca < ), the static meniscus is weakly affected by the motion of the wall (Fig. 1b), while it gets inverted at high negative caillary numbers, as seen in Fig. 1a where we observe a centimetre-size black region below the free surface. In this regime, the meniscus shae looks like the one obtained during the imact of liquid jets in a ool of the same liquid [12], and air entrainment may occur [25]. Here, we focus on the domain of small caillary numbers [Fig. 1d and b] and measure the characteristics of the dynamic meniscus for entrainment (Section 2) and immersion (Section 3). For entrainment, we comare the size and shae of the dynamic meniscus to the redictions of Landau Levich. For immersion, we show how this shae is affected by the direction of the motion, and interret our results using redictions by Wilson and Jones [25]. 2. Dragging a late out of a wetting liquid 2.1. Measurements of the film thickness The thickness of the liquid layer was measured by reflectometry. The measurement consists in analyzing the interference fringes of the light reflecting on the different layers of the late. The various indices of reflection are known, so that the thickness of the liquid layer can be deduced from the analysis of the interference attern. The sectrum of light wavelength is 4 85 nm. The incident and reflected beams are driven by an otical fiber at a distance of tyically 5 mm from the substrate. A sectrometer analyses the light reflected by the substrate: we used silicon wafers, smooth at the atomic scale and reflective like mirrors. The thicknesses accessible by this technique are in the range of 2 nm to 3 lm, with a recision of tyically ±3%. Less than 1 second is necessary for acquiring one data oint, and the error bars corresond to the data size, in Fig. 2 and others. For films thicker than 1 lm, we comlemented the otical technique by weight measurements: the late was totally extracted from the bath and its mass determined using a recise balance. We checked for three different velocities that the results obtained using the reflectometer and the balance were the same, with a comarable recision. h ( μm ) Ca = V / Fig. 2. Thickness h of the film entrained by a silicon wafer extracted at V out of a bath of silicone oil. Full and emty symbols corresond to reflectometry and weight measurements, resectively. The symbols indicate the viscosity of silicone oil (g = 5 mpa s, diamonds; g = 1 mpa s, circles; g = 3 mpa s, squares; g = 1 mpa s, asterisks), which all have the same surface tension (c = 2 mn/m), density (q = 97 kg/m 3 ) and caillary length (a = 1.5 mm). The solid line is the Landau Levich law (Eq. (1)). As liquids, we used silicone oils (PDMS) of viscosity g from 5 to 1 mpa s, and constant surface tension c = 2 mn/m and density q = 97 kg/m 3. The corresonding caillary length a is 1.5 mm. Silicone oils totally wet the wafers. The substrates were fixed, and the coating achieved by lowering the reservoir of liquid, using a steby-ste motor and a software to control its dislacement. The withdrawal velocities V could be varied between 1 lm/s and 2 cm/s. We dislay in Fig. 2 the results of the measurements. The data are comared to the Landau Levich law (Eq. (1)), and an excellent agreement is found on more than two decades in caillary number (5.1 5 < Ca <1 2 ). The film is always thick enough (h >1lm) to neglect the role of long range forces, which generally tend to thicken the film in a wetting case. The agreement is not found only for the scaling law ðh Ca 2=3 Þ, but also for the numerical coefficient. On the other hand, data for Ca >1 2 deviate from (Eq. (1)) and the deviation increases with Ca. For such caillary numbers, the Landau Levich assumtions are not satisfied anymore and gravity thins the entrained film [24,23,7]. As shown in the introduction, one exects the thickness to scale as Ca 1=2 at high caillary number, in qualitative agreement with the tendency observed at large Ca in Fig. 2. However, a comlete theoretical study of this regime remains to be done. We now discuss the actual shae of the dynamic meniscus, first recalling the classical results of the Landau Levich analysis and then comaring them to the exerimental rofiles obtained by reflectometry (see Fig. 3).

3 M. Maleki et al. / Journal of Colloid and Interface Science 354 (211) Fig. 3. Sketch of the dynamic meniscus in the withdrawal case. The dotted line shows the osition of the static meniscus and k is the length of the dynamic meniscus connecting the static meniscus to the uniform entrained film The Landau Levich rofile For small caillary numbers, the Lalace ressure can be simly exressed as cd 2 h/dx 2 and the Stokes equation takes the form: gd 2 u/dy 2 = cd 3 h/dx 3, where x and y are the directions along and erendicular to the velocity u. Integrating this equation with classical boundary conditions [no sli at the solid surface (y = ), negligible viscous stress at the free surface (y = h)], a Poiseuille rofile is found for the flow: u(y)=v c(d 3 h/dx 3 )y(y 2h)/(2g). The film rofile being steady, the flux is conserved along x, which can be written: h V ¼ hv þ c d 3 h 3g dx 3 h3 As emhasized in Ref. [1], this equation can be made dimensionless by scaling the film thickness h by h, and x by k = h / (3Ca ) 1/3. Putting H = h/h and X = x/k, Eq. (3) becomes: dx ¼ 1 H In order to satisfy the boundary condition H(X =+1) = 1, we can write close to the film: H(X)=1+(X) ( 1), which reduces Eq. (4) to: XXX =. This linear equation admits solutions of the tye (X)=ae bx with b 3 = 1. The only solution matching the uniform film is b = 1, which leads to H =1+ae X, or: hðxþ h 1 þ ae x=k ð3þ ð4þ ð5þ In this equation, a is a constant which reflects the invariance of the dynamic meniscus (4) with resect to a translation along X. Eq. (4) can be integrated numerically down to the meniscus region, starting from the asymtotic shae H =1+ae X. It is found that H XX saturates at the value.644 in the limit X = 1: at large H, the right hand side of the Eq. (4) quickly vanishes, meaning that d 2 H/dX 2 indeed tends towards a constant. This roerty was exloited by Landau and Levich: matching this constant with the curvature 2 =a at the to of the static meniscus allowed them to deduce both the thickness of the entrained film h :644 2 aca 2=3 (Eq. (1)) and the length of the dynamic meniscus k :644 2 =3 1=3 aca 1=3 (Eq. (2)) Measurements of the meniscus characteristics We measured the stationary shae of the dynamic meniscus during the film deosition by scanning the free surface along the flow with the reflectometer. Fig. 4a shows such a scan, obtained as withdrawing a silicon wafer at V =25lm/s out of a silicone oil of surface tension c = 2 mn/m and viscosity g = 1 mpa s. The corresonding caillary number is low ( ), in the range of alicability of the Landau Levich laws. We exect from Eq. (1) a thickness h = 3.5 lm for the deosited film, in excellent agreement with the value at large x in Fig. 4a. It is also observed that the film is flat, which confirms the negligible role of gravity in the limit of small caillary numbers. In the same figure, the rofile is nicely fitted by an exonential function, h(x)=h (1 + ae x/l ), as exected from Eq. (5). We deduce from the fit a value for L, which for this articular examle is found to be 47 ± 1 lm. This value is in good agreement with the length calculated by Eq. (2), which is k = 47 lm. We reeated this exeriment for different oil viscosities and withdrawal velocities, which allowed us to vary the caillary number between 1 4 and In each case, the rofile of the dynamic meniscus could be nicely fitted by an exonential law, whose characteristic length L is lotted in Fig. 4b as a function of k, the length calculated from Eq. (2) for each exeriment. There again, the fit is quite good (the data scatter around the line L = k), without any adjustable arameter. As a conclusion, reflectometry aears to be suitable for quantitatively describing the dynamic meniscus, whose shae directly reflects the hydrodynamics of the roblem. Therefore, we can think of using this technique in situations where some added comlexity should imact the flow, and h (μm) x (μm) (a) L (μm) λ(μm) (b) Fig. 4. (a) Stationary rofile of the dynamic meniscus in the region of the entrained film obtained by reflectometry, for a silicon wafer drawn at V =25lm/s out of a bath of silicone oil of surface tension c = 2 mn/m and viscosity g = 1 mpa s. The dots are the data, and the line is an exonential fit of characteristic length L =47lm. (b) Characteristic length L of the dynamic meniscus deduced from fits such as reorted in (a) as a function of the Landau Levich length k [Eq. (2)]. The data scatter around the line L = k.

4 362 M. Maleki et al. / Journal of Colloid and Interface Science 354 (211) h (μ m) A H = h /h W x (μm) Fig. 5. Stationary rofile of the dynamic meniscus, for a late coated by a thin film (h = 4.46 lm) of silicone oil (c = 2 mn/m and g = 35 mpa s) immersed at V =1lm/s inside a bath of the same silicone oil. The data are obtained by reflectometry. The rofile is found to be wavy, with a half-wavelength W 2 lm and amlitude A.3 lm X = x / Fig. 6. Stationary rofile of the dynamic meniscus, for a late coated at a velocity V by silicone oil (c = 2 mn/m and g = 35 mpa s) and then immersed at V = V inside a bath of the same oil. The film thickness h is scaled by the deosited thickness h, and the x coordinate by the length k = h /(3gV/c) 1/3 of the dynamic meniscus. The data corresond to V =1lm/s (triangles) and V =2lm/s (squares). The line is the numerical solution of Eq. (6), which yields a solution close to Eq. (7), with a =1. the dynamic meniscus such as coating with surfactants, with non-newtonian fluids, or immersion exeriments as done below. 3. Driving the wetted late in the ool 3.1. Observations We now discuss what haens as immersing at a velocity V a late first coated at a velocity V (Fig. 1b). There again, the motion distorts the to of the quasi-static meniscus, but the shae of the interface is found to differ from reviously. As shown in Fig. 5, we now observe stationary riles, instead of the monotonous rofile in Fig. 4a: the film (of thickness h = 4.45 lm) covering the late buckles in the dynamic meniscus (small values of x). The amlitude A of the wave is a fraction of a micrometer while the wavelength is tyically 4 lm. At smaller x, h grows quickly and meets the meniscus, a region not shown in the figure since our reflectometry method only works for small sloes and thin films. We now comment on the origin of such oscillations Wavy Landau Levich meniscus At small caillary numbers (gv/c 1), the only difference with the withdrawing case is a change in sign for the late velocity. Relacing V by V in Eq. (4) yields the following equation for the rofile [2,25]: dx ¼ H 1 where h and x are scaled by h =.94a(gV /c) 2/3 and k = h /(3gV/c) 1/3. Close to the film, the linearization of Eq. (6) [H =1+X] leads to XXX =, whose accetable solutions are ðxþ ¼a 1 e b 1X and ðxþ ¼a 2 e b2x with b 1 ¼ 1=2þi 3 =2 and b2 ¼ 1=2 i 3 =2. The film thickness thus varies as: HðXÞ ¼1 þ ae ffiffi /= 3 e X=2 cos 3 X=2 þ / ð7þ where the hase / reflects the invariance of the dynamic meniscus (6) with resect to a translation along X. This function indeed shows an oscillatory behavior, damed by an exonential decay as going to the film region. The characteristic length scale for this inverted dynamic meniscus is 2k. The corresonding wavelength is 4k= 3, which yields W (defined in Fig. 6) of2k= 3 3:6k. For the articular case of Fig. 5, we have k =55lm and we measure ð6þ W = 2 ± 5 lm, in excellent agreement with the exected value of 198 lm. The linear solution (7) can be used to integrate the non-linear Eq. (6). There again, H XX saturates in the limit X = 1, at a value deending on a. This is not a surrise because of the indeendence of the coating and immersion velocities V and V. Using the relation h xx = h /k 2 H XX together with k ¼ h =ð3caþ 1=3 ; h ¼ :94aCa 2=3 and the curvature 2 =a at the to of the static meniscus, we get: H XX ð 1Þ ¼ :64 Ca 2=3 ð8þ Ca If Ca = Ca, we recover the former limit H XX = Comarison with the exeriments Eq. (6) suggests to resent the results in a dimensionless fashion, which is done in Fig. 6 for two series of data. The data are observed to collase in a single curve, and this curve is fairly well fitted by the numerical solution of Eq. (6) (solid line), which is solved and matched with the film (this yields a solution very close to Eq. (7) with a = 1). The fit nicely catures the eriod of the wave and slightly overestimates the height of the largest bum. The numerical solution also redicts the existence of a second minimum close to X =, but, as emhasized earlier, this region could not be characterized exerimentally. In Fig. 6, re-coating and immersion were erformed at the same velocity (V = V ). This situation is the most natural one. It arises in roll coating, where a roller half immersed in a bath drags oil as it turns and re-enters the bath at the same velocity after half a turn. It also corresonds to the case described by Bretherton, where a long bubble is moved in a tube filled with liquid [2]. The front and the rear art of the bubble move at the same velocity: the front region generates a film along the wall, while riles aear at the rear of the bubble. We considered other values for the ratio V/V, and always observed riles. We show in Fig. 7 how the ratio V/V controls their amlitude and wavelength. While increasing V/V by a factor 4, the reduced amlitude A/h slowly decreases by a factor 3 while the wavelength increases by a comarable amount. In addition, we calculated the film rofile exected from Eq. (7), and deduced the theoretical values of A and W. We reort in Fig. 7 the comarison between both these quantities (solid lines) and the measured ones.

5 M. Maleki et al. / Journal of Colloid and Interface Science 354 (211) A/h 1-1 W/λ V/V V/V Fig. 7. Amlitude A and half-wavelength W of the wavy meniscus defined in Fig. 5, as a function of the ratio between immersion and withdrawal velocities. A is scaled by h and W by k = h /(3gV/c) 1/3, the length of the dynamic meniscus. The oil viscosity g is either 1 mpa s (circles) or 35 mpa s (triangles). The solid lines show the result of the numerical integration of Eq. (6). Deviations are observed at small velocity V, which is interreted in the text as resulting from gravitational drainage. The agreement is quite good rovided that the ratio V/V is large enough. We see in articular that the dimensionless half-wavelength W/k tends towards a constant 3.5, in excellent agreement with the calculated value W = 3.6k. Below V/V.1, both the amlitude and the wavelength deviate from the exected values. In this limit, the drainage of the film should erturb the analysis: for V = (fixed wall), liquid still enters the ool, yet by a different mechanism (gravitational drainage). Then, as theoretically shown by Wilson and Jones [25], riles should also exist in the region matching the film and the meniscus. The transition between Landau Levich Bretherton and Wilson Jones riles is exected to occur when the drainage velocity V d of the film exceeds the wall velocity. Since, V d qgh 2 =g, this criterion can be written V/V <(gv /c) 1/3, indeendent of g! For caillary numbers of the order of 1 3, the transition is exected for V/V <.1, as indeed seen in Fig. 7. In this fixed wall limit, gravity can no longer be neglected in the dynamic meniscus and the stationary condition imlies: qgh 3 ¼ qgh3 ch 3 d 3 h dx 3 Following [25], this equation can be made dimensionless by scaling h by h and x by a new characteristic length k g =(a 2 h ) 1/3. For our micrometric films, we exect k g to be in the range of 1 3 lm, as observed exerimentally. Eq. (9) then becomes: dx ¼ H 3 1 ð9þ ð1þ Close to the film (H =1+), we obtain XXX =3, which imlies oscillations of half-wavelength W =2k g /3 5/6, i.e. aroximately 2.5k g, indeendent of V. This exlains why in Fig. 7b the quantity W/k quickly decreases: W tends towards a constant value, while k diverges at small V. The study of the evolution of the amlitude A in this small velocity regime remains to be done. 4. Conclusion We studied by reflectometry the characteristics of Landau Levich dynamic menisci, that is, the transition region between a static meniscus and a film entrained by a moving solid. If the solid is extracted from the bath, the shae of the stretched dynamic meniscus was found to be exonential (close to the film), with a characteristic length slowly increasing as a function of the caillary number (as Ca 1=3 ), in excellent agreement with the classical Landau Levich icture. In addition, we could describe in a very wide range of caillary numbers the law of deosition, which rovides recise data on late coating, even at large caillary numbers for which gravity also contributes to thin the entrained film. Once coated, the late can re-enter the bath. Looking at the common case where the deosition and coating velocities are of the same order, we reorted that the dynamic meniscus has a stationary wavy shae. We studied the amlitude, wavelength and enveloe of this shae, and found that all these characteristics are quantitatively catured in the Landau Levich Bretherton frame. Strong deviations were also observed at small immersion velocities, which were interreted as resulting from gravitational drainage that also rovokes interface undulations, yet of different wavelength as roosed by Wilson and Jones. Acknowledgments This work has been suorted by The Center for International Research and Collaborations (ISMO). References [1] K. Afanasiev, A. Münch, B. 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