Unevenness of Thin Liquid Layer by Contact Angle Variation of Substrate During Coating Process

coatings

Article Unevenness of Thin Liquid Layer by Contact Angle Variation of Substrate during Coating Process

Na Kyong Kim, Dong Hee Kang and Hyun Wook Kang * Department of Mechanical Engineering, Chonnam National University, 77 Yongbong-ro, Buk-gu, Gwangju 61186, Korea; [email protected] (N.K.K.); [email protected] (D.H.K.) * Correspondence: [email protected]; Tel.: +82-62-530-1662

Received: 8 January 2019; Accepted: 15 February 2019; Published: 1 March 2019

Abstract: During a thin film application, the surface of the coating liquid applied to the substrate becomes uneven because of the geometry of the substrate, viscosity of the coating liquid, surface tension, and its contact angle with the substrate. The surface is particularly uneven at the edge corner portion of the substrate and is thicker than the average coating thickness. This study used the volume-of-fluid (VOF) method to examine the surface unevenness of the coating liquid in terms of the contact angle of the substrate surface and sides. After the coating liquid was evenly applied to the substrate, the maximum height of the uneven region of the coating liquid at the edge of the substrate increased as time passed. The point of maximum height moved away from the edge corner portion of the substrate. The coating liquid applied to the substrate with a contact angle less than 90◦ exhibited a pinning effect in which the contact point was fixed at the edge. The surface unevenness was more pronounced in the absence of the pinning effect than in its presence, due to the effects of the viscosity of the coating fluid and the surface energy of the substrate.

Keywords: thin layer coating; contact angle; edge beads; uniformity; volume of ﬂuid

1. Introduction During a thin film application, the surface of a substrate is treated with inorganic or organic material in the form of a thin film to endow the substrate surface with high functionality, such as conductivity, magnetic properties, light reflectivity, and anti-corrosion property [1,2]. There are many coating techniques such as spray processes, dip coating, sol-gel, CVD, PVD, micro-arc oxidation, and anodization [3–15]. This coating technology is employed for special-purpose lenses, automobile and ship painting, and roll-to-roll printing [16–26]. Following thin film application, the surface of the coating fluid at the edge of the substrate becomes uneven because of the effects of the viscosity of the coating fluid, surface tension, and surface contact angle with the substrate [27]. During photoresist application in lithography processes, the surface at the ends of the wafers (edge bead) is uneven, causing problems that could lead to material failure [28]. In the case of an unevenly coated lens surface, optical signals cannot be received accurately, and signal processing errors can occur [29]. To resolve this problem, the uneven parts on a coated surface can be removed using chemical or mechanical removal technologies [30–32]. However, these methods need to be performed as an additional process after the coating, and they are time-consuming and expensive. Experimental studies have shown that a uniform thin film can be attained without any additional removal processes by adjusting the ultraviolet or plasma exposure time [32,33]. However, because the unevenness of a coating fluid is affected by many factors such as its viscosity, surface tension, and surface contact angle with the substrate, it is necessary to study the various physical phenomena and the variables involved. The effects of certain physical parameters, such as the wetting properties between the ink and the surface (hydrophilic or hydrophobic), can be studied more conveniently and economically through

Coatings 2019, 9, 162; doi:10.3390/coatings9030162 www.mdpi.com/journal/coatings Coatings 2019, 9, 162 2 of 13 numerical simulations than through experiments. Numerical simulations can be used to quantitatively examine the unevenness of a coating ﬂuid at the microsecond (µs) scale, which is otherwise difﬁcult through experiments. Despite the potential advantages of numerical simulation, it has been thus far performed only on simple surfaces, coated using the spray method [34,35]. The motion of the free surface between a fluid and the surrounding air can be simulated based on the Navier–Stokes equations, via two main methods: The moving-grid method and the volume-of-fluid (VOF) method [36–38]. The moving grid method is a Lagrangian finite element method in which a grid is created to represent the free surface so that the position of the boundary surface of the moving fluid can be expressed geometrically. However, to achieve this, excessively large grid and structure remeshing (at each time step) is required [39,40]. The other approach, i.e., the VOF, is a Eulerian finite element method and can be used to express complex fluid motion in terms of the change in the fraction of volume a fluid occupies in a given volume over time. This study uses the VOF method to analyze the coating fluid behavior and surface unevenness at the edge portion of a coated surface, based on the changes in the contact angle of the substrate surface, which is an important factor influencing the surface unevenness.

2. Numerical Simulation

2.1. Volume of Fluid Numerical simulations were performed to analyze the unevenness of the coating fluid in terms of the surface contact angle of the substrate, using the VOF mode in CFD-ACE+ (ESI group) software 2017.0. In computational fluid analyses, the VOF method was used to numerically simulate abnormal flow models and to model free surfaces. It uses the fraction of the fluid volume occupied by liquids or gases in a grid to calculate the position and movement of the free surfaces, which are the boundary surfaces between the two fluids. Current numerical models include both liquid and gas phases. Therefore, the integral equation form is used to describe the motion of fluids. The mass and momentum conservation equations are as follows, respectively:

∂ Z I ρdV + ρu · ndS = 0 (1) ∂t V S

∂ Z I I Z ρudV + ρuu · ndS = τ · ndS+ (ρg + f )dV (2) ∂t V S S V where ρ is the density (kg/m3), t is time (s), u is the velocity vector (m/s), n is the unit normal vector, g is the acceleration due to gravity (m/s2), f is the surface tension force (N), τ is the stress tensor (N/m2), S is the closed surface (m2), and V is the control volume (m3). φ is defined as the volume fraction of the liquid for the VOF model. If φ is 0, the grid interior is filled with gas; if φ is 1, the grid interior is filled with liquid. If 0 < φ < 1, the value of φ in the grid specifies the liquid-to-gas ratio.  1, inside the liquid phase;  φ = > 0, < 1 at the free surface; (3)  0, inside the gas phase.

The position of the free surface was determined using the following passive transport equation:

∂φ + u · ∇φ = 0 (4) ∂t The shape of the free surface was recreated using the second-order piecewise linear interface construction (PLIC). For the grid including the free surface, the density ρ and the dynamic viscosity µ (kg/(m·s)) are based on the volume fraction φ and can be calculated as follows: Coatings 2019, 9, 162 3 of 13

ρ = φρl + (1 − φ)ρg (5)

Coatings 2019, 9, x FOR PEER REVIEW µ = φµl + (1 − φ)µg 3 of 13 (6) where ρl and ρg are the densities of the liquid and gas, respectively, and µl and µg are the dynamic μμ=+−φφ(1 )μ viscosities of the liquid and gas, respectively.lg The surface tension force can be expressed as follows:(6)

where ρl and ρg are the densities of the liquid and gas, respectively, and μl and μg are the dynamic f = σκns, ns = ∇φ/|∇φ|, κ = −(∇s · ns) (7) viscosities of the liquid and gas, respectively. The surface tension force can be expressed as follows: −1 where σ is the surface tension coefficient==∇∇=−∇⋅κφφκ (N/m), κ is the local free surface curvature (m ), and ns is f σ nns ,/,()sss n (7) the unit normal vector to the free surface. ∇s denotes the gradient operator applied along the direction tangentwhere to theσ is surfacethe surface [41 ].tension coefficient (N/m), κ is the local free surface curvature (m−1), and ns is the unit normal vector to the free surface. ∇s denotes the gradient operator applied along the direction 2.2. Simulationtangent to the Model surface and [41]. Conditions To perform computational fluid analysis on the unevenness of the coating fluid in terms of the 2.2. Simulation Model and Conditions surface contact angle of the substrate, a computational domain was set up, as shown in Figure1. Due to theTo axisymmetric perform computational condition fluid of theanalysis numerical on the model,unevenness the simulationof the coating was fluid conducted in terms of in the 2D. The substratesurface height contact was angle set of to the 50 substrate,µm and itsa computational length to 1000 domainµm. A was wall set condition up, as shown and in a Figure no-slip 1. conditionDue to the axisymmetric condition of the numerical model, the simulation was conducted in 2D. The were applied to the substrate surface. An axis symmetric condition from the substrate centerline was substrate height was set to 50 μm and its length to 1000 μm. A wall condition and a no-slip condition ◦ set up.were The applied static to contact the substrate angle surface. between An theaxis substrate symmetric and condition the coating from the fluid substrate was setcenterline to wetting was (15 , ◦ ◦ ◦ ◦ ◦ 30 ,set 45 up., 60 The, 75 static, and contact 90 ) angle conditions, between i.e., the the substr shapeate and of thethe coating fluid fluid was front set wasto wetting only driven(15°, by surface30°, tension45°, 60°, without 75°, and any90°) additionalconditions, dynamici.e., the shape flow of effects. the coating The coatingfluid front fluid, was which only driven had an by initial heightsurface of 30 tensionµm, was without set to any be appliedadditional to dynamic the entire flow area effects. of the The substrate coating surface,fluid, which and had the forman initial ofa thin film washeight assumed. of 30 μm, The was coating set to be fluid applied was to set the as entire a Newtonian area of the fluid, substrate with surface, constant and liquid the form viscosity. of a The coatingthin fluid film was had assumed. a density The of coating 997 kg/m fluid3, was a dynamic set as a Newtonian viscosity of fluid, 0.0012 with kg/ms, constant and liquid a surface viscosity. tension 3 of 0.0236The coating N/m. Itfluid was had assumed a density that of the997 coatingkg/m , a fluiddynamic and viscosity substrate of were0.0012 surrounded kg/ms, and bya surface gas (air) at tension of 0.0236 N/m. It was assumed that the coating fluid and substrate were surrounded by gas room temperature (25 ◦C) and ambient pressure (101,325 Pa). The gas density was set to 1.1614 kg/m3, (air) at room temperature (25 °C) and ambient pressure (101,325 Pa). The gas density was set to 1.1614 and its dynamic viscosity was set to 1.846 × 10−5 kg/ms. The gravitational acceleration was set to kg/m3, and its dynamic viscosity was set to 1.846 × 10−5 kg/ms. The gravitational acceleration was set − 2 9.81to m/s−9.81 m/sin the2 in y-direction.the y-direction. Any Any evaporation, evaporation, which might might have have influenced influenced viscosity viscosity and surface and surface tension,tension, was was ignored ignored during during the the thin thin film film applicationapplication process. process. For For the the numerical numerical simulation, simulation, the time the time stepstep was was 0.1 µ0.1s andμs and the the structured structured grid grid was was used.used. The The grid grid independence independence and and convergence convergence test [42] test [42] was carriedwas carried out, out, as shownas shown in in Figure Figure S1. S1.

FigureFigure 1. Schematic 1. Schematic of of the the computational computational domain and and boundary boundary conditions conditions of the of coating the coating fluid.ﬂuid.

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3. Results 3. Results 3.1. Validation of the Numerical Model 3.1. Validation of the Numerical Model To validate the numerical model, the results of the surface contact angle simulation were comparedTo validate with the theoretical numerical model,data. The the resultstheoretical of the da surfaceta can contact be analyzed angle simulation from the were standpoint compared of withthermodynamic theoretical data. equili Thebrium, theoretical which data is attained can be analyzed when a fromliquid the is standpoint placed on of a thermodynamicsolid surface. A equilibrium,theoretical curve which was is attained plotted throug when ah liquidthe geometrical is placed onaspect a solid ratio surface. (Ho/Wo), A where theoretical Wo and curve Ho are was the plottedwidth throughand height the of geometrical the liquid drops, aspect respectively, ratio (Ho/W basedo), where on theW changeso and H ino are the thesubstrate width contact and height angle of[43]. the liquidFigure drops,2 shows respectively, the theoretical based curve on the plotted changes with in respect the substrate to the contactgeometric angle aspect [43]. ratio Figure of 2the showsliquid the drops theoretical obtained curve from plotted the simulation. with respect Theto resu thelts geometric show that aspect the simulation ratio of the is liquidin reasonable drops obtainedagreement from with the the simulation. theoretical The data. results In addition, show that validation the simulation was conducted is in reasonable experimentally agreement using with the thecontact theoretical angle data.data Inon addition, a silicon validationwafer as shown was conducted in Figure experimentally S2. Therefore, usingour numerical the contact model angle is dataconsidered on a silicon accurate wafer enough as shown to inexamine Figure S2.the Therefore, coating fluid our numericalsurface in modelterms of is consideredthe substrate accurate contact enoughangle. to examine the coating ﬂuid surface in terms of the substrate contact angle.

FigureFigure 2. Comparison2. Comparison between between the simulation the simulation results result and theoreticals and theoretical curve plotted curve using plotted the geometric using the aspectgeometric ratio ofaspect the liquid ratio of drop. the liquid drop. 3.2. Same Wetting Condition 3.2. Same Wetting Condition After the thin film is applied, the coating fluid surface becomes uneven at the edge portion of the After the thin film is applied, the coating fluid surface becomes uneven at the edge portion of substrate due to the effects of the viscosity of the coating liquid, surface tension, and contact angle the substrate due to the effects of the viscosity of the coating liquid, surface tension, and contact angle with the substrate. Because the surface is uneven at the edge of the substrate, the wetting of the with the substrate. Because the surface is uneven at the edge of the substrate, the wetting of the substrate sides is an important variable affecting the formation of a uniform thin film. If the substrate substrate sides is an important variable affecting the formation of a uniform thin film. If the substrate has undergone preprocessing, such as a sputter process or dip coating, the substrate surface and sides has undergone preprocessing, such as a sputter process or dip coating, the substrate surface and sides will be under the same wetting conditions. Accordingly, the behavior of the coating fluid was analyzed will be under the same wetting conditions. Accordingly, the behavior of the coating fluid was over time by assuming that the substrate surface and sides had the same contact angle (same wetting analyzed over time by assuming that the substrate surface and sides had the same contact angle (same condition). Table1 lists the conditions (cases) for performing the computation analysis. wetting condition). Table 1 lists the conditions (cases) for performing the computation analysis.

Table 1. Parameters used in the simulation cases under the same wetting conditions. Table 1. Parameters used in the simulation cases under the same wetting conditions. Case Contact Angle of Substrate (◦) Contact Angle of Side Wall (◦) Liquid Properties Contact Angle of Contact Angle of ◦Case Liquid Properties same_15 Substrate15 (°) Side Wall 15 (°) same_30◦ 30 30 same_15°◦ 15 15 same_45 45 45 ρ = 997 kg/m3 ◦ same_60same_30° 6030 30 60 σ = 0.0236 N/m same_75same_45°◦ 7545 45 75 ρ = 997 kg/m3 ◦ same_90same_60° 9060 60 90 σ = 0.0236 N/m same_75° 75 75 same_90° 90 90

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FigureFigure3a 3a shows shows the the analysis analysis resultsresults ofof the coating fluid fluid surface surface over over time time when when the the contact contact angle angle ofof the the substrate substrate surface surface is is equal equal to to that that of of the the sides,sides, i.e.,i.e., 1515°.◦. FigureFigure 33bb showsshows aa quantitativequantitative analysis of theof surfacethe surface profile profile of the of coatingthe coating fluid. fluid. As time As ti passes,me passes, the maximumthe maximum height height and lengthand length of the of uneven the regionuneven increase, region increase, and the pointand the of point maximum of maximum height movesheight moves away fromaway thefrom edge the corneredge corner portion portion (edges). With(edges). the applicationWith the application of the coating of thefluid, coating the fluid, surface the becomessurface becomes uneven uneven at the substrateat the substrate edges edges with the with the passage of time, and the fluid tends to reach an energy equilibrium state because of the passage of time, and the fluid tends to reach an energy equilibrium state because of the effects of the effects of the surface tension and the surface energy of the substrate. The coating fluid even covers surface tension and the surface energy of the substrate. The coating fluid even covers the sides of the the sides of the substrate because of its hydrophilic properties. The surface shape changes while the substrate because of its hydrophilic properties. The surface shape changes while the contact point is contact point is fixed at the edge corner portion of the substrate. fixed at the edge corner portion of the substrate.

FigureFigure 3. 3.( a(a)) Coating Coating ﬂuidfluid behaviorbehavior over over time time at at the the ed edgege of of the the substrate substrate under under the the same same wetting wetting conditioncondition of of 15 15°.◦.( (bb)) SurfaceSurface profile proﬁle under under the the same same wetting wetting condition condition of of15°. 15 ◦.

ThisThis phenomenon phenomenon isis calledcalled the pinning effect. effect. In In this this phenomenon, phenomenon, even even when when the the volume volume of a of a dropletdroplet attached attached to to aa solidsolid surfacesurface (sessile droplet) droplet) is is reduced, reduced, the the parallel parallel contact contact line line of the of the droplet droplet remainsremains fixed. fixed. TheThe computationalcomputational analysis analysis results results show show that that the the coated coated surface surface exhibits exhibits a pinning a pinning effecteffect for for contact contact angles angles rangingranging from 30° 30◦ toto 75° 75◦ becausebecause of of its its hydrophilic hydrophilic properties. properties. Moreover, Moreover, the the coatingcoating fluid fluid covered covered thethe sidessides ofof the substrate. Fi Figuresgures S3–S6 S3–S6 show show the the results results of ofthe the computational computational analysisanalysis for for contact contact angles angles rangingranging from 30° 30◦ toto 75°. 75◦ . FigureFigure4a 4a shows shows the the analysis analysis results results of theof the behavior behavior of the of coatingthe coating fluid fluid over over time time when when the contact the anglecontact of theangle substrate of the substrate surface is surface equal tois thatequal of to the that sides, of the i.e., sides, 90◦. Figurei.e., 90°.4b showsFigure a4b quantitative shows a quantitative analysis of the surface profile of the coating fluid. As in the case of the hydrophilic analysis of the surface profile of the coating fluid. As in the case of the hydrophilic surface, the surface, the maximum height and length of the uneven region increase as time passes. However, maximum height and length of the uneven region increase as time passes. However, unlike the unlike the substrate surface with a small contact angle, the contact point moves away from the corner substrate surface with a small contact angle, the contact point moves away from the corner portion portion (edge) over time, and the coating does not cover the edge surface of the substrate. This is (edge) over time, and the coating does not cover the edge surface of the substrate. This is because because the applied fluid tends to reduce its contact area with the substrate, and the surface shape the applied fluid tends to reduce its contact area with the substrate, and the surface shape changes changes while the contact line is not fixed because of the large contact angle between the coating fluid whileand the the substrate. contact line is not fixed because of the large contact angle between the coating fluid and the substrate. CoatingsCoatings 20192019, ,99, ,x 162 FOR PEER REVIEW 66 of of 13 13

FigureFigure 4. 4. (a(a) )Coating Coating fluid ﬂuid behavior behavior over over time time at at the the ed edgege of of the the substrate substrate under under the the same same wetting wetting ◦ ◦ conditioncondition of of 90°. 90 .((bb) )Surface Surface profile proﬁle under under the the same same wetting wetting condition condition of of 90°. 90 .

ToTo perform perform a a quantitative quantitative analysis analysis of of the the degr degreeee of of unevenness unevenness on on the the coated coated surface surface and and the the behaviorbehavior of of the the applied applied contact surfacesurface fluidfluid ininterms terms of of the the changes changes in in the the substrate substrate surface surface energy, energy, the maximum height (h), x-direction position of the maximum height of the surface unevenness (x ), and the maximum height (h), x-direction position of the maximum height of the surface unevennessh (xh), uneven region (l ) were measured. Each measured value was divided by the initial height of the and uneven regionuneven (luneven) were measured. Each measured value was divided by the initial height of coating fluid (h ) to express the values as dimensionless variables. Figure5a shows the analysis results the coating fluido (ho) to express the values as dimensionless variables. Figure 5a shows the analysis resultsof the maximumof the maximum height ofheight the surface of the surface unevenness unevenness based on based the contact on the anglecontact between angle between the substrate the substrateand the coating and the fluid. coating The fluid. deformation The deformation during the during initial 30theµ initials shows 30 similarμs shows trends similar in all trends the analysis in all thecases, analysis and a cases, higher and level a higher of deformation level of deformation is seen after is 30 seenµs because after 30 of μ thes because increase of inthe the increase contact in angle. the contactWith the angle. increase With in the the increase surface contact in the surface angle, the contact maximum angle, height the maximum increases height because increases the coating because liquid thetends coating to reduce liquid the tends contact to reduce area with the thecontact substrate area with surface the duesubstrate to the surfac low surfacee due to energy. the low Because surface of energy.these results, Because the of surface these unevennessresults, the wassurface more un severeevenness in the was absence more ofsevere the pinning in the effectabsence than of inthe its pinningpresence. effect Figure than5b in shows its presence. the analysis Figure results 5b shows of the x-directionthe analysis position results ofof thethe maximumx-direction height position of the of thesurface maximum unevenness. height The of the deformation surface unevenne during thess. initial The 30deformationµs shows similarduring trends the initial in all 30 the μ cases,s shows and similarthe x-direction trends in position all the ofcases, the maximumand the x-directio height isn closeposition to the of substratethe maximum edge with height the is increase close to in the the substratecontact angle. edge with This the is because increase the in appliedthe contact coating angle. fluid This at is the because substrate the cornerapplied edge coating tends fluid to reduceat the substratethe contact corner area edge with tends the substrate, to reduce and the the contact shape ar ofea thewith surface the substrate, changes, and while the the shape contact of the line surface is not changes,fixed. Figure while5c the shows contact the analysisline is not results fixed. of Figure the uneven 5c shows region. the analysis In all the results cases, theof the deformation uneven region. of the Incoating all the fluid cases, during the deformation the initial 20 ofµs the is more coating rapid, fluid showing during similar the initial trends. 20 μ Thes is deformationmore rapid, fromshowing 20 to similar30 µs is trends. gradual, The following deformation which from the amount20 to 30 ofμs changeis gradual, in the following uneven regionwhich rapidlythe amount increases of change again. in the uneven region rapidly increases again.

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FigureFigure 5. ( 5.a) ( Maximuma) Maximum height, height, (b ()b maximum) maximum position, position, andand ((cc)) unevenuneven region region analysis analysis of of the the surface surface unevennessunevenness at theat the edge edge of of the the substrate substrate with with the the same same wetting wetting conditions.conditions.

3.3.3.3. Different Different Wetting Wetting Condition Condition LikeLike silicon silicon wafers, wafers, the the sides sides of of the the substrate substrate areare sometimessometimes not processed during during substrate substrate processing.processing. Assuming Assuming this this is is the the case, case, the the behavior behavior ofof thethe coatingcoating fluid ﬂuid over over time time was was analyzed analyzed for for differentdifferent contact contact angles angles of of the the substrate substrate surface surface and and sidessides (different(different wetting co conditions).nditions). Table Table 22 lists lists thethe conditions conditions under under which which the the computational computational analysis analysis waswas executed.executed.

TableTable 2. Parameters2. Parameters used used in in the the simulation simulation cases cases underunder different wetting conditions. conditions.

Contact Angle of◦ Contact Angle of ◦ CaseCase Contact Angle of Substrate ( ) Contact Angle of Side WallLiquid ( ) Properties Liquid Properties Substrate (°) Side Wall (°) diff_15◦ 15 180 diff_15° 15 180 diff_30◦ 30 180 ◦ diff_30° 30 180 diff_45 45 180 ρ = 997 kg/m3 ◦ 3 diff_60 diff_45° 6045 180 180 ρ = 997 σkg/m= 0.0236 N/m diff_75◦ diff_60° 7560 180 180 σ = 0.0236 N/m ◦ diff_90 diff_75° 9075 180 180 diff_90° 90 180 Figure6a shows the analysis results of the coating fluid surface over time for a substrate surface contact angle of 15◦ and a side contact angle of 180◦. When the coating fluid is applied to the substrate, at 8 µs, satellite droplets are formed, which deviate away from the surface of the coating fluid at the edge portion of the substrate. Satellite droplets are created due to the high fluid inertia force that can Coatings 2019, 9, x FOR PEER REVIEW 8 of 13

Figure 6a shows the analysis results of the coating fluid surface over time for a substrate surface contact angle of 15° and a side contact angle of 180°. When the coating fluid is applied to the substrate, atCoatings 8 μs, 2019satellite, 9, 162 droplets are formed, which deviate away from the surface of the coating fluid at8 ofthe 13 edge portion of the substrate. Satellite droplets are created due to the high fluid inertia force that can break the surface tension force. These satellite droplets have average diameters of 5 μm and velocity break the surface tension force. These satellite droplets have average diameters of 5 µm and velocity magnitudes of 1.86 m/s. The satellite droplets move to the outside of the liquid film and fall down magnitudes of 1.86 m/s. The satellite droplets move to the outside of the liquid film and fall down due due to gravity. Figure 6b shows a quantitative analysis of the surface profile of the coating fluid over to gravity. Figure6b shows a quantitative analysis of the surface profile of the coating fluid over time. time. The pinning effect, in which the contact point is fixed due to the hydrophilic properties of the The pinning effect, in which the contact point is fixed due to the hydrophilic properties of the substrate, substrate, is induced at the edge corner portion, as shown in Figure 5b. However, unlike the results is induced at the edge corner portion, as shown in Figure5b. However, unlike the results shown in shown in Figure 5b, in which the coating fluid even covers the sides of the substrate, Figure 6b shows Figure5b, in which the coating fluid even covers the sides of the substrate, Figure6b shows that the that the sides are not covered owing to their non-wetting properties. When the substrate surface sides are not covered owing to their non-wetting properties. When the substrate surface contact angle contact angle ranges from 30° to 75°, the pinning effect and satellite droplets occur in all the analysis ranges from 30◦ to 75◦, the pinning effect and satellite droplets occur in all the analysis cases. The cases. The substrate’s sides are not covered because of its hydrophilic properties, similar to that substrate’s sides are not covered because of its hydrophilic properties, similar to that observed when observed when the angle is 15°. In addition, the satellite droplets are also formed with similar average the angle is 15◦. In addition, the satellite droplets are also formed with similar average diameters of diameters of 5 μm and velocity magnitudes of 1.75 m/s to that observed when the angle is 15°. Figures 5 µm and velocity magnitudes of 1.75 m/s to that observed when the angle is 15◦. Figures S7–S10 S7–S10 show the computational analysis results for contact angles ranging from 30° to 75°. show the computational analysis results for contact angles ranging from 30◦ to 75◦.

FigureFigure 6. 6. (a)( aCoating) Coating fluid ﬂuid behavior behavior over over time timeat the at ed thege of edge the ofsubstrate the substrate under a under substrate a substrate surface ◦ ◦ contactsurface angle contact of angle15° and of 15a sideand contact a side angle contact of angle180°. of(b) 180 Surface.(b )profile Surface under proﬁle different under differentwetting conditions.wetting conditions.

FigureFigure 77aa shows the analysis results of the coating fluidfluid surface overover time for a substrate surface ◦ ◦ contactcontact angle angle of of 90° 90 andand a a side side contact contact angle angle of of 180°. 180 .With With the the increase increase in in the the substrate substrate surface surface contact contact angle,angle, the the size size of of the the satellite satellite droplets droplets increases. increases. These These satellite satellite droplets droplets have have average average diameters diameters of 16 of μ16mµ andm and velocity velocity magnitudes magnitudes of 1.55 of 1.55 m/s. m/s. The satellite The satellite droplets droplets are 5.3 are times 5.3 times larger larger in diameter in diameter and ◦ ◦ 1.2and times 1.2 times lower lower in velocity in velocity than the than satellite the satellite droplet droplet formed formed at the atlow the contact low contact angles angles (15°–75°). (15 At–75 50). μAts, 50theµ substrates, the substrate edge is edge not coated is not coatedbecause because of the movement of the movement of the contact of the po contactint. Figure point. 7b Figure shows7 ba quantitativeshows a quantitative analysis of analysis the surface of the profile surface of profile the coating of the fluid coating over fluid time. over In time.the coating In the fluid coating surface fluid surface profile, the maximum height and length of the uneven region increase as time passes, similar to that shown in Figure4b. However, the loss of coating fluid is due to the effect of the satellite droplets, and the maximum height of the surface unevenness is lower than that shown in Figure4b. Coatings 2019, 9, x FOR PEER REVIEW 9 of 13 profile, the maximum height and length of the uneven region increase as time passes, similar to that shown in Figure 4b. However, the loss of coating fluid is due to the effect of the satellite droplets, and theCoatings maximum2019, 9, 162 height of the surface unevenness is lower than that shown in Figure 4b. 9 of 13

FigureFigure 7. 7. (a)( aCoating) Coating fluid ﬂuid behavior behavior over over time timeat the at ed thege of edge the ofsubstrate the substrate under a under substrate a substrate surface ◦ ◦ contactsurface angle contact of angle90° and of 90a sideand contact a side angle contact of angle180°. of(b) 180 Surface.(b )profile Surface under proﬁle different under differentwetting conditions.wetting conditions.

ToTo performperform aa quantitative quantitative analysis analysis of theof the unevenness unevenness of the of coating the coating fluid surface fluid surface and the behaviorand the h behaviorof the applied of the fluid, applied the maximumfluid, the heightmaximum ( ) of height the surface (h) of unevenness, the surface theunevenness, x-direction the position x-direction of the maximum height (x ), and the region with surface unevenness (luneven) were measured. Each of the position of the maximumh height (xh), and the region with surface unevenness (luneven) were measured. h Eachmeasured of the valuesmeasured were values divided were by divided the initial by heightthe initial of theheight coating of the fluid coating ( o) fluid of the (h coatingₒ) of the liquid coating to liquidexpress to them express as dimensionless them as dimensionless variables. Figure variable8a showss. Figure the analysis8a shows results the ofanalysis the maximum results heightof the of the surface unevenness in terms of the contact angle between the substrate and the coating fluid. maximum height of the surface unevenness in◦ term◦s of the contact angle between the substrate and theWith coating the increase fluid. With in the the contact increase angle in from the contact 15 to 75angle, the from maximum 15° to 75°, height the of maximum the surface height unevenness of the increases. When the substrate surface contact angle is 90◦, the height is lower than that when the surface unevenness increases.◦ When◦ the substrate surface contact angle is 90°, the height is lower thancontact that angle when ranges the contact from 15 angleto 75 ranges. This isfrom because 15° to of 75°. the formationThis is because of large of satellite the formation droplets of and large the satelliteloss of applied droplets coating and the fluid, loss unlike of applied that whencoating the fluid, surface unlike had athat low when contact the angle surface with had the a coatinglow contact fluid angleapplied with to the the coating substrate fluid edge applied corner to portion. the substrate Figure edge8b shows corner theportion. analysis Figure results 8b shows of the the x-direction analysis µ resultsposition of ofthe the x-direction maximum position height of the maximum surface unevenness. height of the The surface deformation unevenness. during The the deformation initial 30 s duringshows athe similar initial trend 30 μ ins shows all the a cases, similar and trend the x-direction in all the cases, position and of the the x-direction maximum heightposition becomes of the closer to the edge of the substrate with the increase in the contact angle, except when the contact angle maximum◦ height becomes closer to the◦ edge of the substrate with the increase in the contact angle, exceptis 90 . when When the the contact contact angle angle is is90°. 90 When, the substrate the contact surface angle fluctuatesis 90°, the becausesubstrate of surface the contact fluctuates point becausemovement of the at thecontact edge point and themovement formation at the of satellite edge and droplets. the formation Because of ofsatellite the surface droplets. fluctuation, Because theof µ thex-direction surface fluctuation, position of thethe maximumx-direction height position moves of the closer maximum to the substrateheight moves edge, closer as observed to the substrate at 100 s. edge,Figure as8c observed shows the at analysis100 μs. Figure results 8c of shows the uneven the analysis region. results During of the the initial uneven 20 s,region. the coating During fluid the µ initialdeforms 20 s, rapidly, the coating showing fluid similar deforms trends rapidly, in all showin the cases.g similar The deformationtrends in all the is gradual cases. The from deformation 20 to 30 s, isfollowing gradual whichfrom 20 the to change30 μs, following in the uneven which region the change increases in the rapidly uneven again. region The increases formation rapidly of the uneven again. Theregion formation can be attributedof the uneven to the region small can difference be attribut in theed to contact the small angles. difference in the contact angles.

CoatingsCoatings 20192019,, 99,, 162x FOR PEER REVIEW 1010 ofof 1313

Figure 8.8. ((aa)) MaximumMaximum height,height, ((bb)) maximummaximum position,position, andand ((cc)) unevenuneven regionregion analysisanalysis ofof thethe surfacesurface unevenness atat thethe edgeedge ofof thethe substratesubstrate withwith differentdifferent wettingwetting conditions.conditions. 4. Conclusions 4. Conclusion This study examined coating fluid behavior and surface unevenness in terms of the surface energy This study examined coating fluid behavior and surface unevenness in terms of the surface of the substrate surface and sides during a thin film coating process. The VOF method was used energy of the substrate surface and sides during a thin film coating process. The VOF method was to simulate the free surface between the coating liquid and the surrounding air and to describe the used to simulate the free surface between the coating liquid and the surrounding air and to describe surface unevenness formation at the microsecond scale. the surface unevenness formation at the microsecond scale. • When the contact contact angle angle of of the the substrate substrate surface surface was was equal equal to that to thatof the of sides, the sides, the length the length and height and heightof the uneven of the uneven region regionincreased increased with the with increase the increase in the contact in the contact angle an angled passage and passage of time. of When time. ◦ ◦ Whenthe substrate the substrate contact contact angle ranged angle ranged from 15° from to 75°, 15 ato pinning 75 , a pinning effect was effect induced, was induced, with fixed with parallel fixed parallelcontact lines contact of the lines droplets of the on droplets the substrate on the edge substrate corner edge portion, corner and portion, the coating and thefluid coating covered fluid the coveredsides of the the substrate. sides of the Consequently, substrate. Consequently, the substrate thesides substrate were also sides coated. were also coated. ◦ • When thethe substratesubstrate contactcontact angleangle waswas 9090°,, the substratesubstrate edgeedge surfacesurface waswas notnot coated,coated, becausebecause thethe applied fluidfluid tended to reducereduce thethe contactcontact areaarea withwith thethe substrate.substrate. Moreover,Moreover, thethe surfacesurface shapeshape changedchanged while the contactcontact line at thethe cornercorner portionportion waswas notnot fixed,fixed, asas thethe contactcontact angleangle betweenbetween thethe coatingcoating fluidfluid andand thethe substratesubstrate waswas high.high. BecauseBecause ofof this,this, thethe coatingcoating fluidfluid onon thethe substratesubstrate inin thethe absence of the pinning effect effect experienced experienced more more severe severe unevenness unevenness than than in in its its presence. presence. • When thethe contactcontact angles of the substrate surface and sides were different, thethe formationformation ofof satellitesatellite droplets and thethe lossloss ofof applicationapplication solutionsolution werewere observedobserved inin allall thethe analysisanalysis cases.cases. TheThe satellitesatellite droplets couldcould havehave ledled toto problemsproblems inin thethe coatingcoating process.process. They formed as they escaped from the

Coatings 2019, 9, 162 11 of 13

cohesiveness of the coating ﬂuid due to the difference in the ﬂuid inertia, which created energy equilibrium, and the surface energy of the substrate side (non-wetting conditions). • When the contact angles ranged from 15◦ to 75◦, the pinning effect was generated at the edge corner portion, similar to that observed when the contact angle of the substrate contact surface was equal to that of the sides. However, the results were promising in cases where the substrate sides were not covered because of their high surface contact angle. Therefore, to reduce surface unevenness, it is necessary to actively control not only the wetting of the substrate surface but also of the side surface. The sides of the substrate must have a contact angle greater than 90◦, and the substrate surface must have a surface contact angle lower than 90◦.

Supplementary Materials: The following are available online at http://www.mdpi.com/2079-6412/9/3/162/ s1: Figure S1. Variation of the maximum height of the surface unevenness with the grid number; Figure S2: Comparison between the (a) static contact angle of the silicon wafer and (b) the simulation result of the contact angle; Figure S3: (a) Coating fluid behavior over time at the edge of the substrate under the same wetting condition of 30◦, (b) surface profile under the same wetting condition of 30◦, (c) maximum height, (d) maximum position, and (e) uneven region analysis of the surface unevenness at the edge of the substrate under the same wetting condition of 30◦; Figure S4: (a) Coating fluid behavior over time at the edge of the substrate under the same wetting condition of 45◦, (b) surface profile under the same wetting condition of 45◦, (c) maximum height, (d) maximum position, and (e) uneven region analysis of the surface unevenness at the edge of the substrate under the same wetting condition of 45◦; Figure S5: (a) Coating fluid behavior over time at the edge of the substrate under the same wetting condition of 60◦, (b) surface profile under the same wetting condition of 60◦, (c) maximum height, (d) maximum position, and (e) uneven region analysis of the surface unevenness at the edge of the substrate under the same wetting condition of 60◦; Figure S6: (a) Coating fluid behavior over time at the edge of the substrate under the same wetting condition of 75◦, (b) surface profile under the same wetting condition of 75◦, (c) maximum height, (d) maximum position, and (e) uneven region analysis of the surface unevenness at the edge of the substrate under the same wetting condition of 75◦; Figure S7: (a) Coating fluid behavior over time at the edge of the substrate under a substrate surface contact angle of 30◦ and a side contact angle of 180◦, (b) surface profile under different wetting conditions, (c) maximum height, (d) maximum position, and (e) uneven region analysis of the surface unevenness at the edge of the substrate under different wetting conditions; Figure S8: (a) Coating fluid behavior over time at the edge of the substrate under a substrate surface contact angle of 45◦ and a side contact angle of 180◦, (b) surface profile under different wetting conditions, (c) maximum height, (d) maximum position, and (e) uneven region analysis of the surface unevenness at the edge of the substrate under different wetting conditions; Figure S9: (a) Coating fluid behavior over time at the edge of the substrate under a substrate surface contact angle of 60◦ and a side contact angle of 180◦, (b) surface profile under different wetting conditions, (c) maximum height, (d) maximum position, and (e) uneven region analysis of the surface unevenness at the edge of the substrate under different wetting conditions; Figure S10: (a) Coating fluid behavior over time at the edge of the substrate under a substrate surface contact angle of 75◦ and a side contact angle of 180◦, (b) surface profile under different wetting conditions, (c) maximum height, (d) maximum position, and (e) uneven region analysis of surface unevenness at the edge of the substrate under different wetting conditions. Author Contributions: Conceptualization, N.K.K.; Methodology, N.K.K. and H.W.K.; Software, N.K.K. and D.H.K.; Validation, N.K.K.; Formal analysis, N.K.K.; Investigation, N.K.K., D.H.K., and H.W.K.; Data curation, N.K.K., D.H.K., and H.W.K.; Writing—original draft preparation, N.K.K.; Writing—review and editing, N.K.K., D.H.K., and H.W.K.; Visualization, N.K.K.; Supervision, H.W.K. Funding: This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (NRF-2016R1C1B2012136). Conflicts of Interest: The authors declare no conflict of interest.

References

1. Cui, H.; Zayat, M.; Parejo, P.G.; Levy, D. Highly efficient inorganic transparent UV-protective thin-film coating by low temperature sol-gel procedure for application on heat-sensitive substrates. Adv. Mater. 2008, 20, 65–68. [CrossRef] 2. Yuan, Y.; Giri, G.; Ayzner, A.L.; Zoombelt, A.P.; Mannsfeld, S.C.; Chen, J.; Nordlund, D.; Toney, M.F.; Hunag, J.; Bao, Z. Ultra-high mobility transparent organic thin film transistors grown by an off-centre spin-coating method. Nat. Commun. 2014, 5, 3005. [CrossRef][PubMed] 3. Galeano, B.; Korff, E.; Nicholson, W.L. Inactivation of vegetative cells, but not spores, of Bacillus anthracis, B. cereus, and B. subtilis on stainless steel surfaces coated with an antimicrobial silver- and zinc-containing zeolite. Appl. Environ. Microbiol. 2003, 69, 4329–4331. [CrossRef][PubMed] Coatings 2019, 9, 162 12 of 13

4. Kendig, M.; Scully, J. Basic aspects of electrochemical impedance application for the life prediction of organic coatings on metals. Corrosion 1990, 46, 22–29. [CrossRef] 5. Feuerstein, A.; Knapp, J.; Taylor, T.; Ashary, A.; Bolcavage, A.; Hitchman, N. Technical and economical aspects of current thermal barrier coating systems for gas turbine engines by thermal spray and EBPVD: A review. J. Therm. Spray Technol. 2008, 17, 199–213. [CrossRef] 6. Pham, V.H.; Cuong, T.V.; Hur, S.H.; Shin, E.W.; Kim, J.S.; Chung, J.S.; Kim, E.J. Fast and simple fabrication of a large transparent chemically-converted graphene film by spray-coating. Carbon 2010, 48, 1945–1951. [CrossRef] 7. Li, J.; Wu, R.; Jing, Z.; Yan, L.; Zha, F.; Lei, Z. One-step spray-coating process for the fabrication of colorful superhydrophobic coatings with excellent corrosion resistance. Langmuir 2015, 31, 10702–10707. [CrossRef] [PubMed] 8. Barrows, A.T.; Pearson, A.J.; Kwak, C.K.; Dunbar, A.D.; Buckley, A.R.; Lidzey, D.G. Efficient planar heterojunction mixed-halide perovskite solar cells deposited via spray-deposition. Energy Environ. Sci. 2014, 7, 2944–2950. [CrossRef] 9. Chen, L.M.; Hong, Z.; Kwan, W.L.; Lu, C.H.; Lai, Y.F.; Lei, B.; Liu, C.P.; Yang, Y. Multi-source/component spray coating for polymer solar cells. ACS Nano 2010, 4, 4744–4752. [CrossRef][PubMed] 10. Galagan, Y.; de Vries, I.G.; Langen, A.P.; Andriessen, R.; Verhees, W.J.; Veenstra, S.C.; Kroon, J.M. Technology development for roll-to-roll production of organic photovoltaics. Chem. Eng. Process. 2011, 50, 454–461. [CrossRef] 11. Lu, Y.; Ganguli, R.; Drewien, C.A.; Anderson, M.T.; Brinker, C.J.; Gong, W.; Guo, Y.; Soyez, H.; Dunn, B.; Huang, M.H.; et al. Continuous formation of supported cubic and hexagonal mesoporous films by sol–gel dip-coating. Nature 1997, 389, 364–368. [CrossRef] 12. Brinker, C.J.; Frye, G.C.; Hurd, A.J.; Ashley, C.S. Fundamentals of sol-gel dip coating. Thin Solid Films 1991, 201, 97–108. [CrossRef] 13. Kim, D.J.; Hahn, S.H.; Oh, S.H.; Kim, E.J. Influence of calcination temperature on structural and optical

properties of TiO2 thin films prepared by sol–gel dip coating. Mater. Lett. 2002, 57, 355–360. [CrossRef] 14. Asri, R.I.M.; Harun, W.S.W.; Hassan, M.A.; Ghani, S.A.C.; Buyong, Z. A review of hydroxyapatite-based coating techniques: Sol–gel and electrochemical depositions on biocompatible metals. J. Mech. Behav. Biomed. Mater. 2016, 57, 95–108. [CrossRef][PubMed] 15. Dehghanghadikolaei, A.; Ansary, J.; Ghoreishi, R. Sol-gel process applications: A mini-review. Proc. Nat. Res. Soc. 2018, 2, 02008. [CrossRef] 16. Mehta, P.; Al-Kinani, A.A.; Haj-Ahmad, R.; Arshad, M.S.; Chang, M.W.; Alany, R.G.; Ahmad, Z. Electrically atomised formulations of timolol maleate for direct and on-demand ocular lens coatings. Eur. J. Pharm. Biopharm. 2017, 119, 170–184. [CrossRef][PubMed] 17. Sedransk, K.L.; Tenhaeff, W.E.; Gleason, K.K. Grafting CVD of poly(vinyl pyrrolidone) for durable scleral lens coatings. Chem. Vapor Depos. 2010, 16, 23–28. [CrossRef] 18. Li, B.; Blaschke, H.; Ristau, D. Combined laser calorimetry and photothermal technique for absorption measurement of optical coatings. Appl. Opt. 2006, 45, 5827–5831. [CrossRef][PubMed] 19. Malherbe, L.; Mandin, C. VOC emissions during outdoor ship painting and health-risk assessment. Atmos. Environ. 2007, 41, 6322–6330. [CrossRef] 20. Dik, J.; Janssens, K.; Van Der Snickt, G.; van der Loeff, L.; Rickers, K.; Cotte, M. Visualization of a lost painting by Vincent van Gogh using synchrotron radiation based X-ray fluorescence elemental mapping. Anal. Chem. 2008, 80, 6436–6442. [CrossRef][PubMed] 21. Osticioli, I.; Zoppi, A.; Castellucci, E.M. Fluorescence and Raman spectra on painting materials: Reconstruction of spectra with mathematical methods. J. Raman Spectrosc. 2006, 37, 974–980. [CrossRef] 22. Celebi, U.B.; Vardar, N. Investigation of VOC emissions from indoor and outdoor painting processes in shipyards. Atmos. Environ. 2008, 42, 5685–5695. [CrossRef] 23. Krebs, F.C.; Fyenbo, J.; Jørgensen, M. Product integration of compact roll-to-roll processed polymer solar cell modules: methods and manufacture using flexographic printing, slot-die coating and rotary screen printing. J. Mater. Chem. 2010, 20, 8994–9001. [CrossRef] 24. Krebs, F.C. Polymer solar cell modules prepared using roll-to-roll methods: knife-over-edge coating, slot-die coating and screen printing. Sol. Energy Mater. Sol. Cells 2009, 93, 465–475. [CrossRef] Coatings 2019, 9, 162 13 of 13

25. Ahn, S.H.; Guo, L.J. High-speed roll-to-roll nanoimprint lithography on ﬂexible plastic substrates. Adv. Mater. 2008, 20, 2044–2049. [CrossRef]

26. Sonawane, R.S.; Kale, B.B.; Dongare, M.K. Preparation and photo-catalytic activity of Fe–TiO2 thin films prepared by sol–gel dip coating. Mater. Chem. Phys. 2004, 85, 52–57. [CrossRef] 27. Smith, S.; Stolle, D. Nonisothermal two-dimensional film casting of a viscous polymer. Polym. Eng. Sci. 2000, 40, 1870–1877. [CrossRef] 28. Pollentier, I.; Somanchi, A.; Burkeen, F.; Vedula, S. Influence of immersion lithography on wafer edge defectivity. Solid State Technol. 2008, 51, 38–42. 29. Zhang, X.; Li, B. Calibration optimization of laser-induced deflection signal for measuring absorptance of laser components. Appl. Opt. 2015, 54, 1861–1869. [CrossRef][PubMed] 30. Lee, H.; Lee, K.; Ahn, B.; Xu, J.; Xu, L.; Oh, K.W. A new fabrication process for uniform SU-8 thick photoresist structures by simultaneously removing edge bead and air bubbles. J. Micromech. Microeng. 2011, 21, 125006. [CrossRef] 31. Park, H.W.; Kim, H.J.; Roh, J.H.; Choi, J.K.; Cha, K.R. Simple and cost-effective method for edge bead removal by using a taping method. J. Korean Phys. Soc. 2018, 73, 1473–1478. [CrossRef] 32. Uddin, M.A.; Chan, H.P.; Chow, C.K.; Chan, Y.C. Effect of spin coating on the curing rate of epoxy adhesive for the fabrication of a polymer optical waveguide. J. Electron. Mater. 2004, 33, 224–228. [CrossRef] 33. Sharma, A.; Verheijen, M.A.; Wu, L.; Karwal, S.; Vandalon, V.; Knoops, H.C.; Sundaram, R.S.; Hofmann, J.P.;

Kessels, W.M.M.; Bol, A.A. Low-temperature plasma-enhanced atomic layer deposition of 2-D MoS2: large area, thickness control and tuneable morphology. Nanoscale 2018, 10, 8615–8627. [CrossRef][PubMed] 34. Ye, Q.; Pulli, K. Numerical and experimental investigation on the spray coating process using a pneumatic atomizer: influences of operating conditions and target geometries. Coatings 2017, 7, 13. [CrossRef] 35. Joshi, A.; James, S. Molecular dynamics simulation study of cold spray process. J. Manuf. Process. 2018, 33, 136–143. [CrossRef] 36. de Lamotte, A.; Delafosse, A.; Calvo, S.; Toye, D. Identifying dominant spatial and time characteristics of flow dynamics within free-surface baffled stirred-tanks from CFD simulations. Chem. Eng. Sci. 2018, 192, 128–142. [CrossRef] 37. Wu, L.; Gong, M.; Wang, J. Development of a DEM–VOF model for the turbulent free-surface flows with particles and its application to stirred mixing system. Ind. Eng. Chem. Res. 2018, 57, 1714–1725. [CrossRef] 38. Li, L.; Liu, Z.; Cao, M.; Li, B. Large eddy simulation of bubbly flow and slag layer behavior in ladle with discrete phase model (DPM)–Volume of fluid (VOF) coupled model. JOM 2015, 67, 1459–1467. [CrossRef] 39. Muzaferija, S.; Peri´ c, M. Computation of free-surface flows using the finite-volume method and moving grids. Numer. Heat Transf. Part B 1997, 32, 369–384. [CrossRef] 40. Laadhari, A.; Székely, G. Fully implicit finite element method for the modeling of free surface flows with surface tension effect. Int. J. Numer. Methods Eng. 2017, 111, 1047–1074. [CrossRef] 41. Brackbill, J.U.; Kothe, D.B.; Zemach, C. A continuum method for modeling surface tension. J. Comput. Phys. 1992, 100, 335–354. [CrossRef] 42. Hirsch, C. Consistency, stability and error analysis of numerical schemes. In Numerical Computation of Internal and External Flows, 2nd ed.; John Wiley & Sons: Oxford, UK, 2007; pp. 291–292. 43. Kang, H.W.; Sung, H.J.; Lee, T.M.; Kim, D.S.; Kim, C.J. Liquid transfer between two separating plates for micro-gravure-offset printing. J. Micromech. Microeng. 2008, 19, 015025. [CrossRef]

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