J. Non-Newtonian Fluid Mech. 118 (2004) 137–156 Low-flow limit in slot coating of dilute solutions of high molecular weight polymer O.J. Romero a , W.J. Suszynski b , L.E. Scriven b , M.S. Carvalho a,∗ a Department of Mechanical Engineering, Pontificia Universidade Catolica do Rio de Janeiro, Rua Marques de Sao Vicente 225, Gavea, Rio de Janeiro, RJ 22453-900, Brazil b Department of Chemical Engineering & Materials Science, University of Minnesota, MN, USA Accepted 14 March 2004 Abstract Slot coating is a common method in the manufacture of a wide variety of products. It belongs to a class of coating method known as premetered coating: the thickness of the coated liquid layer in principle is set by the flow rate fed to the die and the speed of the substrate moving past, and is independent of other process variables. Thus, premetered methods are ideal for high precision coating. An important operating limit of slot coating is the minimum thickness that can be coated at a given substrate speed, generally referred to as the low-flow limit. The mechanism that defines this limit balances the viscous, capillary and inertial forces in the flow. Although most of the liquids coated industrially are polymeric solutions and dispersions that are not Newtonian, previous analyses of the low-flow limit in slot coating dealt only with Newtonian liquids. In this paper, the low-flow limit in slot coating of an extensional thickening polymer solution is examined both by theory and by experiment. The continuity and momentum equations coupled with an algebraic non-Newtonian constitutive equation that relates stress to the rate-of-strain and relative-rate-of-rotation tensors were solved by the Galerkin/finite element method to model the flows. The flows themselves were visualized by video microscopy and the low-flow limit was found by observing, at given substrate speed, the feed rate at which the flow becomes unstable and breaks up. Various solutions of low molecular weight polyethylene glycol (PEG) and high molecular weight polyethylene oxide (PEO) in water were used in order to evaluate the effect of mildly viscoelastic behavior on the process. At the concentration level of the high molecular weight polymer explored here, the viscoelastic behavior of the solutions could not be accessed by oscillatory tests; the only measurable response to the addition of PEO was the rise of the apparent extensional viscosity. © 2004 Elsevier B.V. All rights reserved. Keywords: Free surface flow; Slot coating; Low-flow limit; Dilute polymer solutions 1. Introduction Slot coating is commonly used in the manufacturing of adhesive and magnetic tapes, specialty papers, imaging films, and many other products. In this process, the coating liquid is pumped to a coating die in which an elongated chamber distributes it across the width of a narrow slot through which the flow rate per unit width at the slot exit is made uniform. Exiting the slot, the liquid fills (wholly or partially) the gap between the adjacent die lips and the substrate translating rapidly past them. The liquid in the gap, bounded upstream, and downstream by gas–liquid interfaces, or menisci, forms the coating bead, as shown in Fig. 1. The competition among ∗ Corresponding author. Fax: +55-21-3114-1165. E-mail address: msc@mec.puc-rio.br (M.S. Carvalho). 0377-0257/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jnnfm.2004.03.004 viscous, capillary and pressure forces, and in some cases inertial and elastic forces, sets the range of operating parameters in which the viscous free surface flow of the liquid can be two-dimensional and steady, which is the desired state. In order to sustain the coating bead at higher substrate speeds, the gas pressure upstream of the upstream meniscus is made lower than ambient, i.e. a slight vacuum is applied to the upstream meniscus [2]. Slot coating belongs to a class of coating methods known as pre-metered coating: the thickness of the coated liquid layer is set by the flow rate fed to the coating die and the speed of the moving substrate, and is independent of other process variables. Thus, pre-metered methods are ideal for high precision coating. However, the nature of the flow in the coating bead, and therefore the uniformity of the liquid layer it delivers, can be affected by the substrate speed, the viscosity and any non-Newtonian properties of the liquid, and the configuration of the die lips 138 O.J. Romero et al. / J. Non-Newtonian Fluid Mech. 118 (2004) 137–156 FEED SLOT SLOT DIE DISTRIBUTION CHAMBER SUCTION VACUUM BOX Static contac line Static contac line UPSTREAM LIP UPSTREAM MENISCUS DOWNSTREAM LIP COATING GAP Pvac H0 COATING BEAD DOWNSTREAM MENISCUS Patm FILM THICKNESS, t SUBSTRATE VELOCITY, Vw DYNAMIC CONTACT LINE Fig. 1. Sketch of slot coating bead. immediately upstream and downstream of the slot exit. The region in the space of operating parameters of a coating process where the delivered liquid layer is adequately uniform is usually referred to as a coating window. Knowledge of coating window’s for different coating methods is needed in order to predict whether a particular method can be used to coat a given substrate at a prescribed production rate. Ruschak [24] analyzed the coating window of a slot coating bead dominated by surface tension force (capillary pressure) in the upstream and downstream menisci; Silliman [28] and Higgins and Scriven [15] took the viscous drag of the substrate and die lips into account. More refined theoretical analyses and experiments on the limits of operability and flow stability within those limits were made subsequently [4,12,25]; the results are typified in Fig. 2. The figure shows that the coating window is bounded by three modes of failure: (1) When the coated layer is thicker than the thinnest that can be produced at a fixed gap and substrate speed, i.e. t > tmin in Fig. 2, too great a vacuum at the upstream free surface causes liquid to be drawn along the die surface into the vacuum chamber. This diversion of liquid destroys premetering. (2) Too little vacuum at the upstream free surface leaves the net viscous drag force on the upstream part of the Pvac Pvac Patm (1) (3) Patm Pvac COATING Pvac WINDOW Patm Low flow limit (2) H0 /t0 H0 /tmin Pvac Patm Ca H0 /t Fig. 2. Coating window of a slot coating process in the plane of vacuum Pvac vs. gap-to-thickness ratio H0 /t, at a fixed capillary number Ca. The boundaries of the window are set by different bead break-up mechanisms. tmin represents the minimum film thickness that can be deposited onto the substrate at a given capillary number. bead unbalanced by the pressure gradient that is imposed by capillary pressure forces in the menisci upstream and downstream and the difference in external pressure on those menisci (i.e. vacuum). In consequence the upstream meniscus shifts toward the feed slot until the bead drastically rearranges into a three-dimensional form that delivers separate rivulets to the substrate. Between the rivulets are dry lanes that extend upstream through the bead. Along those lanes air is sucked into the vacuum chamber. It is in this regime that, at given vacuum (ambient pressure downstream minus air pressure exerted on upstream meniscus), there is a lower limit to the thickness of continuous liquid layer that can be coated from a downstream gap of specified clearance. As Fig. 2 shows, the limit can be lowered by applying greater vacuum and thereby shifting the upstream meniscus away from the edge of the feed slot. (3) At given substrate speed, too low a flow rate per unit width from the slot causes the downstream meniscus to curve so much that it cannot bridge the gap clearance H0 . Consequently, the meniscus becomes progressively more three-dimensional, alternate parts of it invading the gap until the bead takes a form that delivers separate rivulets or chains of droplets to the substrate moving past. This transition from a continuous coated liquid layer is what is called here the low-flow limit: the minimum thickness of liquid that can be deposited from a gap of specified clearance at a given substrate speed. And, as Fig. 2 makes plain, it is independent of the vacuum applied, given that the vacuum is great enough to draw the upstream meniscus away from the feed slot. The outcome is the same when at a given flow rate per unit width from the slot, the substrate speed is too high. In this case, the low-flow limit is sometimes referred to as the high-speed limit (see [4]). The outcome is essentially the same when at a given flow rate per unit width from the slot and a given substrate speed, the clearance of the downstream gap is too great. In this case, the low-flow limit is refered to as the wide-gap limit: the maximum gap from which a given thickness of liquid coating can be deposited on a substrate moving at specified speed. O.J. Romero et al. / J. Non-Newtonian Fluid Mech. 118 (2004) 137–156 139 feed slot upstream downstream DCL DOWNSTREAM SLOT DIE SCL H0 R min liquid t min coa ted la yer V substrate Fig. 3. Detail of the downstream meniscus configuration. The low-flow limit occurs when the free surface is as curved as it can be and still meet the die. The threshold of the low-flow limit is well described by the viscocapillary flow model, provided viscous stress at the downstream meniscus is not excessive and the inertia of the liquid is not appreciable. In this model, the streamlines are rectilinear and parallel, or nearly so, in most of the flow; the upstream meniscus is part of a circular cylinder and the downstream meniscus nearly so but governed by the Landau–Levich relation, which is strictly valid only for flow at low capillary number, Ca ≡ µV/σ (viscosity times substrate velocity divided by surface tension) and Reynolds number, Re ≡ ρq/µ (density times flow rate per unit width of slot divided by viscosity). According to the viscocapillary model developed by Higgins and Scriven [15], given the substrate speed, the minimum flow rate per unit width—and hence the minimum liquid layer thickness—is set by the greatest adverse pressure gradient that can be created in the downstream part of the gap. That arises when the upstream meniscus is hemicylindrical and situated just upstream of the feed slot, so that the pressure jump there is the highest it can be, and the downstream meniscus is as curved as it can possibly be and still connect the free surface of the thin coated layer to the surface of the die lip that defines the downstream gap, as illustrated in Fig. 3. The maximum pressure difference is given by: Pmax = 2σ , H0 − t (1) t is the coated film thickness. Ruschak [24] extended the work of Landau and Levich [17] and showed that the pressure drop across a downstream meniscus in a slot coating bead can be related to the film thickness t, the web speed V and liquid properties: σ P = 1.34Ca2/3 . (2) t The minimum liquid layer thickness at a given capillary number Ca can be obtained by combining the two previous equations: 1 tmin = ; H0 1 + 1.49Ca−2/3 Ca 1, Re 1. (3) Above the critical gap-to-thickness ratio, two-dimensional steady flow cannot exist, according to the viscocapillary flow model. Moreover, the formula indicates that the minimum thickness of liquid that can be coated from a downstream gap of specified clearance is greater, the higher the capillary number. Thus, for example, raising the production speed makes the minimum liquid layer thicker, which might be accommodated by adding solvent, which in turn would heighten the demand on the drying operation downstream of the coater. Experiments by Sartor [25] and Carvalho and Kheshgi [4] with Newtonian liquids accorded with the viscocapillary model at capillary numbers below unity but indicated, as expected, that it becomes less and less useful as capillary number and Reynolds number rise beyond unity. High capillary number (Ca ≈ 5) and Reynolds number (Re ≈ 3) occur in higher-speed coating operation. However, the film profile equation of Kheshgi [16], which expresses the viscocapillary model augmented to take partial account of liquid inertia, represents the data to somewhat higher values of these parameters, where it, too, fails and beyond which only the full Navier-Stokes system accords with the data. In the experiments of Lee and Liu [19] as in those of Gutoff and Kendrick [14] to determine a “low-flow limit”, no vacuum was applied at the upstream meniscus. The results therefore correspond to Point t0 in Fig. 2, which is obviously a thickness greater than the low-flow limit tmin , which requires vacuum. Solving the Navier-Stokes system for two-dimensional viscous flow with free surfaces, Carvalho and Kheshgi [4] confirmed the viscocapillary model of the low-flow limit at low capillary number and low Reynolds number and, more importantly, found that the limit, instead of continuing to 140 O.J. Romero et al. / J. Non-Newtonian Fluid Mech. 118 (2004) 137–156 rise with these parameters, begins to fall when they reach higher values, in accord with their experiments. Most past analyses of slot coating flow have dealt solely with Newtonian liquids. However, the liquids coated in practice are polymer solutions or colloidal suspensions, or both. In general, polymer solutions may be shear-thinning, extension-thickening, and develop viscoelastic stresses when sheared or extended. These liquids can depart substantially from Newtonian behavior at the strain rates they suffer in the coating flow. In the particular case of the low-flow limit, such rheological effects can change the forces that operate near and at the downstream meniscus of the coating bead and thereby alter the limit. Dontula [10] analyzed experimentally the flow of polymer solutions in several coating processes. Model aqueous solutions with nearly constant viscosity and adjustable elasticity were developed to study the role of elasticity in coating and other free surface flows (see also [9]). These liquids were prepared by adding small amounts of a high molecular weight polymer (poly-ethylene oxide—PEO) to a more concentrated aqueous solution of the same polymer but of a much lower molecular weight (polyethylene glycol—PEG). The experiments showed that the free surface flow in the coating bead can be drastically changed even when only minute amounts of the high molecular polymers were added to the coating liquid. The elasticity appears to arise from the strongly extensional nature of the flows at the downstream free surface of the coating bead. However, at the low concentrations of the high molecular weight polymer, the elastic moduli of the polymeric solutions were so small that they could not be discerned from small-amplitude oscillatory tests. The only measurable manifestation of non-Newtonian behavior was the rise of the apparent extensional viscosity. Ning et al. [21] studied experimentally the effect of polymer additives on slot coating. However, like Lee and Liu [19], they used no vacuum (reduced pressure) on the upstream meniscus and so the “coatability limit” they measured is not, as claimed, the low-flow limit of the process. In this work, the low-flow limit in slot coating of certain dilute aqueous polymeric solutions was examined by both experiments and theory. With a benchtop slot coating apparatus, a coating gap and substrate speed were set. A coating bead was established by feeding liquid to the slot die at high enough volumetric flow rate per unit width being coated that the deposited layer was half the thickness of the gap and the bead was easy to establish. Then the flow rate was slowly reduced until the coated layer broke into alternating stripes of liquid and dry lanes. At each such operating condition, the positions of the downstream and upstream menisci were observed before and after the break in order to determine the breakup mechanism. Aqueous solutions, like the ones developed by Dontula et al. [9], of low molecular weight polyethylene glycol to adjust viscosity, and high molecular weight polyethylene oxide, to adjust elasticity, were used in order to evaluate the effects of non-Newtonian behavior. The theoretical analysis consisted of solving the momentum and continuity equation system for steady two-dimensional flow with free surfaces, including an algebraic non-Newtonian constitutive equation that relates stress to the rate-of-strain and relative-rate-of-rotation tensors. Generalized Newtonian models of this class cannot represent viscoelastic stresses per se but are perhaps the simplest ones that may capture the different ways that dilute polymer molecules behave in extension-dominated and shear-dominated flow zones. The system of differential equations, boundary conditions and constitutive equation was solved by the Galerkin/finite element method. At each operating condition (set by the liquid properties as well as coating gap and web speed), a sequence of states was found by Newton iterations initialized by first-order, pseudo-arc-length continuation, i.e. a solution path in parameter space was constructed as the flow rate to the die was diminished. The theoretically determined low-flow limit was defined by the parameter values at which the angle between the downstream free surface and the downstream die lip became less than 10◦ . 2. Experimental analysis 2.1. Experimental set-up and procedure The experimental set-up is sketched in Fig. 4. The coating liquid was fed to the coating die by a progressive cavity pump (model C4015ESSQ3SAA, Moyno Industrial Products, Springfield, OH). The flow rate was controlled by the pump speed and measured with a Coriolis flow meter (model MFC 100/MFS 3000, KROHNE Americ Inc, Peabody, MA). The precision coating die, 10 cm wide, was assembled with shims such that the height of the slot was 125 m. A vacuum chamber of approximately 300 cm3 was mounted upstream the coating die. Two vacuum pumps (model VFC084P-5T, Fuji Electric Co, Tokyo, Japan) in series were used in order to achieve high enough vacuum (low enough pressure) to maintain the coating bead at the highest substrate speeds. The set-up was able to maintain a pressure of approximately 8 kPa below atmospheric. The coated film thickness was uniform across the width of the die; hence it was calculated simply by dividing the flow rate by the coating width and surface speed of the cylinder being coated. The die was mounted on an adjustable plate, such that the coating gap, i.e. the distance between the die and the cylinder surface, could be easily changed and made uniform. In most of the experiments, the coating gap was fixed at 100 m. The liquid was coated directly onto the glass cylinder, or roll. Its surface speed could be controlled from 0.2 up to 2.8 m/s. The coated liquid, except a trace residue, was removed from the roll surface by a scraper blade and recycled. A video camera (model DXC-750, Sony Inc., Tokyo, Japan) with microscope lens (Magnazoom 6000 with 0.5×adapter, Navitar, Rochester, NY) was focused on a right-angle mirror mounted inside the open end of the roll in order to visualize the flow in the coating bead through the glass wall, as shown in Fig. 5. The coating liquids used in the experiments were solutions O.J. Romero et al. / J. Non-Newtonian Fluid Mech. 118 (2004) 137–156 141 FILM THICKNESS GLASS ROLL SLOT COATER Pressure sensor SCRAPER 1 VACUUM BOX MASS FLOWMETER LIQUID PAN SCRAPER 2 PROGRESSIVE CAVITY PUMP LIQUID TRAP HAND-SET VALVE VACUUM PUMP Fig. 4. Schematic of the experimental slot coating apparatus. of polyethylene glycol (molecular weight 8 × 103 g/mol) and polyethylene oxide (molecular weight 4 × 106 g/mol). Three concentration of PEG were used, 20, 25, and 30% by weight, and to each, small amounts of PEO were added to give them slight to moderate viscoelastic character. Four different concentrations of PEO were used: 0, 0.005, 0.01, and 0.05% by weight. The rheological characterization of the solutions is described in the next subsection. The equilibrium surface tension of the solutions was measured with a Kruss Drop Analysis System DSA10, and were all in the range of 58–61 mN/m. The minimum liquid layer thickness that could be coated at each speed of the cylinder was found by the following procedure: (1) The pump speed was set to deliver a relatively thick coated layer, i.e. approximately 50 m—half of the coating gap. Enough vacuum was applied to the upstream free surface, or meniscus, to establish the coating bead; in some cases this was aided by drawing the tip of a slender wooden stick across the gap exit (a common expedient in practical coating). (2) The pump speed was lowered in small steps. After less than 10 s, steady state was reached and the positions of the dynamic contact line and the downstream static contact line were observed. If the upstream meniscus had moved toward the feed slot, the vacuum was raised in order to pull the meniscus back to about its former location. The flow rate was lowered until the downstream Fig. 5. Photograph of the experimental set-up with camera and microscope lens. 142 O.J. Romero et al. / J. Non-Newtonian Fluid Mech. 118 (2004) 137–156 meniscus invaded the coating bead. Details of how the bead broke up are described in following subsections. After this happened, one or more lanes of uncoated roll surface appeared and no manipulation with the pointed wooden stick at the gap exit would re-establish a continuous, uniform coating. This operating condition was considered to be the onset of the low-flow limit. The minimum layer thickness that could be coated was calculated from the flow rate and the surface speed of the roll. 2.2. Rheological characterization of the solutions The coating liquids used in the experiments were solutions of polyethylene glycol (molecular weight 8 × 103 g/mol) and polyethylene oxide (molecular weight 4 × 106 g/mol). These aqueous polymer solutions were proposed by Dontula et al. [9] as model liquids to study the role of elasticity in coating and other free surface flows. Similar solutions were used by Christanti and Walker [5] and Cooper-White et al. [6] to study jet break up and drop formation dynamics. The solutions were transparent and behaved similarly to Boger liquids. Their rheological properties depended on concentration. Dontula et al. [9] prepared different solutions of PEG and PEO in water in their study. The concentration of PEG they have tested varied from 16 to 43 wt.%, and that of PEO, from 0 to 0.1 wt.%. They characterized the rheological properties of the solutions in shear and in approximately extensional flows. The elastic modulus G could be reliably measured only in the more viscous solutions (larger PEG concentration and higher concentration of PEO). The apparent extensional viscosity of the solutions was measured in an opposed nozzle device. The higher concentration solutions displayed an extension-thickening behavior that presumably came from the stretching of polymer coils. In the present work, 12 different solutions with different concentrations of PEG and PEO were tested; the concentrations were lower than those reported by Dontula et al. [9]. The elastic modulus G of the solutions containing PEO could not be measured in small amplitude oscillatory tests because of the high uncertainties at small torques. The shear viscosity was measured first in a RFS-II rheometer (Rheometrics Inc., Piscataway, NJ) with a cone-and-plate fixture of diameter of 50 mm and angle of 0.04 rad. The shear viscosity of all solutions were virtually independent of shear rate. In order to determine the inherent viscosity of the solutions tested as a function of the PEO concentration with higher accuracy, the shear viscosity was also measured using glass capillary viscosimeter (Cannon-Fenske). Table 1 presents the viscosity of the solutions at different PEO concentration (from 0 to 0.1 wt.%) in 20 and 30 wt.% PEG solutions. The variation of the inherent viscosity, defined as (η0 − ηs )/ηs with PEO concentration in presented in Fig. 6. It is independent of the PEG concentration and varies linearly with the high molecular weight polymer (PEO) concentration. The intrinsic viscosity, i.e. the slope of the line, is approximately [η] ≈ 0.8 m3 /kg. At equilibrium, the polymer coils start to overlap when the reduced concentration, defined as c∗ = c × [η], is approximately c∗ = 1 (see [20]). The maximum reduced concentration of the solutions tested here was below this limit and consequently all the test liquids were in the dilute regime. The relaxation time of the solutions were estimated by taking λ ≡ ηp /G, where ηp is the polymer contribution for the viscosity (ηp = η0 − ηs ) and G is the elastic modulus of the liquid. It is a function of the polymer concentration c and its molecular weight Mw (G ≡ (c/Mw )RT). The estimated relaxation time of each solution is also presented in Table 1. As expected in the dilute regime, the relaxation time is independent of the PEO concentration. The estimated relaxation time was λ ≈ 0.02 and ≈ 0.06 s for the 20 and 30 wt.% PEG solutions, respectively. The solutions’ extensional response was probed with an opposed nozzle device (RFX, Rheometrics Inc, Piscataway, NJ). Fig. 7 presents the apparent extensional viscosity of all the solutions tested. Several nozzle diameters were used to cover the range of nominal extensional rates reported, from 100 to 4000 s−1 . The breaks between sets of points associated with each solution are where the nozzle diameter was changed. This sensitivity of the measurements to the nozzle diameters has been reported before (see [8]). The apparent extensional viscosity could not be measured at lower apparent extension rate because the torques generated were below Table 1 Estimated relaxation time of the solutions at different PEO concentration PEG (wt.%) PEO (wt.%) c (kg/m3 ) ηs (mPa s) η0 (mPa s) (η0 − ηs )/ηs G (Pa) λ (s) 20 0 0.01 0.03 0.05 0.1 0 0.103 0.310 0.517 1.035 17.07 17.07 17.07 17.07 17.07 17.07 18.86 21.83 25.13 30.96 0 0.105 0.278 0.472 0.813 0.00 0.06 0.19 0.32 0.64 0.000 0.028 0.025 0.025 0.022 30 0 0.01 0.03 0.05 0.1 0 0.105 0.315 0.526 1.054 44.48 44.48 44.48 44.48 44.48 44.48 51.56 57.04 63.78 84.09 0 0.159 0.282 0.434 0.891 0.00 0.06 0.19 0.32 0.64 0.000 0.108 0.064 0.060 0.061 O.J. Romero et al. / J. Non-Newtonian Fluid Mech. 118 (2004) 137–156 143 1 Inherent viscosity, (η0−ηs)/η s 0.9 Water+PEG(20%)+PEO Water+PEG(30%)+PEO 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.2 0.4 0.6 0.8 1 1.2 3 PEO concentration, kg/m Fig. 6. Inherent viscosity (η0 − ηs )/ηs as a function of PEO concentration. The intrinsic viscosity is [η] ≈ 0.8 m3 /kg. the range of the sensor. In the range at which the measurements were made, the apparent extensional viscosity of all the solutions tested was nearly independent of apparent extension rate. Presumably the apparent extensional viscosity had reached the constant value associated with high enough extension rate that all the coils are stretched out [22]. From the estimated relaxation time of each liquid tested, the different PEO solutions on 20 wt.% PEG in water should show an extensional thickenning behavior at a deformation rate of γ̇ ≈ 1/λ ≈ 50 s−1 , which is lower than the minimum extensional rate at which reliable measurements could be made. The three solutions lacking PEO appeared to be Newtonian in that the ratio of their apparent extensional to shear viscosity was approximately 4. This ratio is often referred to as a Trouton ratio (although it is not Trouton’s). The higher the concentration of PEO, the greater was the almost constant apparent extensional viscosity. In the case of the solution with 30% PEG and 0.05% PEO, the ratio of apparent extensional to shear viscosity was approximately 15. 2.3. Results Flow visualization through the glass wall of the rotating cylinder used as substrate revealed how the upstream and downstream menisci of the coating bead and their contact lines located after each change of operating conditions, as illustrated in Fig. 8, a sequence of photographs as the flow rate falls with no vacuum applied to the upstream free meniscus. In every case shown in Fig. 8, the downstream contact line pinned at the corner (radius of about 25 m) of the downstream land but the upstream contact line located in response to the particular operating conditions. At large enough flow rates, the flow was two dimensional and both contact lines were straight—Fig. 8(a). As the flow rate fell at constant roll speed (Ca = 0.12), the upstream contact line moved toward the feed slot. At a certain operating condition, that contact line reached the feed slot and became three-dimensional; a “V” shaped air pocket formed at the dynamic contact line—Fig 8(b). At even lower flow rate, the downstream contact line invaded the gap and the bead broke, i.e. it became three-dimensional and delivered stripes rather than continuous liquid layer—Fig. 8(d). However, as noted above, this was not the way the bead failed at the low-flow limit. This type of coating failure could be avoided by raising the vacuum, i.e. reducing the pressure on the upstream meniscus, and thereby pulling it away from the feed slot. The response of the contact lines to the vacuum applied is shown in Fig. 9. At the operating conditions at which the deposited liquid film thickness was approximately half of the gap between the die and the rotating cylinder, the downstream contact line located at the corner of the downstream die lip, as shown in Fig. 8. Previous experiments [25] have shown the downstream contact line located on the die shoulder; however in those experiments the coating thicknesses were larger than the ones explored here and the substrate speed were lower. Close to the onset of the low-flow limit, in contrast, the downstream meniscus receded into the coating gap, i.e. toward the feed slot, locally at certain locations across the width of the coating bead. The sequence of photos of the coating bead in Fig. 10 shows this mode of failure. The capillary number was Ca = 0.084, and the vacuum applied to the upstream free surface was Pvac H0 /µV = 180. When the coated layer was thick (low H0 /t), the flow was two-dimensional and the contact lines were straight. As the flow rate was reduced, the forces of the two-dimensional flow evidently could not balance close to the downstream meniscus, for the flow there became three-dimensional at H0 /t = 4.5 (see Fig. 10(c)). When the flow rate was reduced even further, e.g. H0 /t = 5.45, air fingers formed at 144 O.J. Romero et al. / J. Non-Newtonian Fluid Mech. 118 (2004) 137–156 App. Extensional Viscosity, cP 1000 100 (a)- water + PEG(20%)+ PEO(0.05%) PEO(0.01%) PEO(0.005%) PEO(0%) 10 10 App. Extensional Viscosity, cP 1000 100 1000 10000 App. Extension Rate, s-1 100 PEO(0,05%) (b)- water + PEG(25%)+ PEO(0,01%) PEO(0,005%) PEO(0%) 10 10 App. Extensional Viscosity, cP 1000 100 1000 App. Extension Rate, s 10000 -1 100 PEO(0,05%) (c)- water + PEG(30%)+ PEO(0,01%) PEO(0,005%) PEO(0%) 10 10 100 1000 App.Extension Rate,s 10000 -1 Fig. 7. Apparent extensional viscosity as a function of apparent extension rate of the polymeric solutions used. (a) 20 wt.% PEG; (b) 25 wt.% PEG and (c) 30 wt.% PEG. the downstream lip and penetrated all the way through the bead to the vacuum chamber, so that one or more dry stripes appeared on the cylinder (see Fig. 10(d)). It is important to notice that the upstream free surface remained well upstream of the feed slot, for that proves that mechanism that caused the low-flow limit is related to the movement of the downstream meniscus. Such a mechanism of the low-flow limit accords with the theoretical models of [24,15], the more refined one of [12], and that of [4]. It is clear that the bead breakup seen in earlier experiments in which no vacuum was applied to the upstream meniscus was not the bead breakup at the low-flow limit in slot coating. The conditions at the onset of the low-flow limit of 20% PEG solutions measured experimentally are summarized in Fig. 11, which also records the critical gap-to-thickness ratio versus capillary number at the onset of the low-flow limit that is predicted by the viscocapillary model. The critical condition found with the solution lacking PEO agrees well with the viscocapillary model based on Newtonian behavior, as expected because the low molecular weight polymer solution is a Newtonian liquid and the experiments were performed at low enough capillary number that the viscocapillary model is accurate. At a fixed capillary number, as the concentration of PEO rises and consequently the viscoelasticity of the liquid does too, the minimum thickness that can be coated rises. At Ca = 0.2, the minimum layer thickness that could be coated with the 0.05% PEO solution was close to twice of that which could be coated with the solution O.J. Romero et al. / J. Non-Newtonian Fluid Mech. 118 (2004) 137–156 145 Fig. 8. View through the glass roll of the coating bead as the flow rate is reduced. The roll speed was fixed (Ca = 0.12) and no vacuum was applied to the upstream free surface. The cylinder was moving from bottom to top of each photo. As the film thickness falls, the upstream free surface moves towards the feed slot and becomes three-dimensional. As the flow rate is decreased even further, the “V” pattern grows until the coating bead breaks (Frame d). lacking PEO. The coating window of the process was smaller the higher the extensional viscosity of the liquid used. The low-flow limits of the solutions with 25 and 30 wt.% PEG are shown in Fig. 12. The trend is similar to that in Fig. 11, i.e. the addition of small quantities of PEO made the minimum thickness larger. A minor difference is that the critical condition found with the solution lacking PEO is not as close to the predictions of the viscocapillary model with Newtonian behavior. Because the flow near the downstream meniscus is dominated by extensional deformation, the ratio of the viscous to the surface tension force is better described by a capillary number based on the apparent extensional viscosity of the coating solutions, viz. Cau ≈ ηu V/3σ. The data of Fig. 11 are reploted in Fig. 13 as a function of this modified capillary number. The apparent extensional viscosity used in the definition of the modified capillary number is the one measured at the highest extensional rate at which the data was taken, e.g. ˙ ≈ 4000 s−1 . When the extensional thickening of the liquid is taken into account in this way in the definition of the capillary number, all the data lie approximately on the same curve. The shape of the downstream meniscus as it invades the coating bead at the critical conditions of the onset of the low-flow limit depends on the forces acting on that meniscus. As the liquid becomes more viscoelastic, i.e. as the Fig. 9. Coating bead configuration at H0 /t = 2, Ca = 0.12 and increasing level of vacuum applied to the upstream free surface. With no vacuum, Pvac = 0, the bead is not continuous (Frame a). At Pvac = 98.5, an uniform coating bead is formed (Frame d). 146 O.J. Romero et al. / J. Non-Newtonian Fluid Mech. 118 (2004) 137–156 Fig. 10. Position of the downstream contact line as flow rate is reduced. The low-flow limit occurred at H0 /t = 5.4 (Frame d). The flow conditions were Ca = 0.084 and Pvac H0 /µV = 180. Capillary Number, Ca = ηοV/σ 10 PEO(0%) water + PEG(20%) + PEO(0.005%) PEO(0.01%) PEO(0.05%) 1 UNSTABLE Viscocapilllary Model 0.1 STABLE 0.01 0 2 4 6 8 10 12 Gap / Film Thickness, H0 / t Fig. 11. Onset of the low-flow limit as a function of capillary number. Twenty weight percent PEG solutions. concentration of high molecular weight PEO rises, the “V” shaped meniscus profile becomes more acute, as illustrated in Fig. 14. 3. Theoretical analysis The flow in the coating bead was described by the complete two-dimensional, steady-state mass and momentum conservation equations coupled with an algebraic constitutive equation that approximates the non-Newtonian behavior of the liquid. This section describes the formulation of the theoretical model, how its equations were solved, and the predictions solutions yielded. Carvalho and Kheshgi [4] established that at low capillary and Reynolds numbers the flow upstream of the feed slot does not affect the critical operating conditions at the onset of the low-flow limit. Because the analysis presented here focuses on the effect of non-Newtonian behavior of the liquid at low capillary number and vanishing Reynolds number, the flow domain where the governing equations are solved could be restricted to the region close to the downstream free surface, as sketched in Fig. 15. H0 is the gap between the die and the moving substrate. The synthetic inflow boundary O.J. Romero et al. / J. Non-Newtonian Fluid Mech. 118 (2004) 137–156 147 Fig. 12. Onset of the low-flow limit as a function of capillary number. (a) 25 and (b) 30 wt.% PEG solutions. 3)V 10 Capillary Number*, CaU = ( 0U PEO(0%) PEO(0.005%) water+PEG(20%)+ PEO(0.01%) PEO(0.05%) 1 Viscocapillary Model 0.1 0.01 0 2 4 6 8 10 12 Gap / Film Thickness, H0 / t Fig. 13. Low-flow limit of 20 wt.% PEG solutions as a function of the modified capillary number, defined in terms of the apparent extensional viscosity. 148 O.J. Romero et al. / J. Non-Newtonian Fluid Mech. 118 (2004) 137–156 Fig. 14. Shape of the three-dimensional downstream contact line at the onset of the low-flow limit for (a) Newtonian liquid (0 wt. % PEO) and (b) extensional-thickening solution (0.05 wt.% PEO). was placed at 5H0 upstream of the static contact line. The outflow plane was placed 15H0 downstream of the static contact line. In order to validate the predictions obtained with this restricted domain, the predicted low-flow limit for Newtonian liquids was compared with Carvalho and Kheshgi’s [4] results of solving the governing equations in a complete two-dimensional cross-section of the coating bead. 3.1. Governing equations and boundary conditions The velocity and pressure field of Stokes flow are governed by the momentum and continuity equations: ∇ · T = 0; ∇ · v = 0. (4) ¯¯ ¯ ¯ The constitutive equation that relates the stress T and the ¯¯ liquid deformation is presented in the following section. The boundary conditions that were adopted are the following: (1) Inflow: Couette–Poiseuille velocity profile. 6q y y 2 u=− ; v = 0. − H0 H0 H0 (5) Here q is the flow rate per unit width, an input parameter, and H0 is the gap between the die and the substrate. (2) Moving substrate: no-slip, no-penetration. u = Vw ; v = 0, (6) where, Vw is the substrate velocity. (3) Outflow: fully developed flow. n · ∇v = 0. ¯ ¯ (7) (4) Free surface: kinematic condition and force balance. n · v = 0; ¯ ¯ 1 dt n·T = ¯ − npamb . Ca ds ¯ ¯ ¯¯ (8) here t and n are the unit tangent and unit normal vector ¯ surface, and p along¯ the free amb is the ambient pressure. (5) Static contact line: pinned at the corner of the die lip. xscl = xcorner . ¯ ¯ (6) Die land: no-slip, no-penetration. v=0 ¯ (9) (10) 3.2. Constitutive equation Microstructured liquids can depart substantially from Newtonian liquid behavior in complex flows. Shear-thinning and extension-thickening may vary with the intensity of deformation rate, and viscoelastic stresses may arise. Moreover, polymer molecules in solution behave quite differently in flow regions where the liquid is persistently stretched in the direction in which the molecules are aligned, as compared to regions where the molecules rotate with respect to the rate-of-strain field. Some differential constitutive equations can qualitatively describe how polymer molecules respond to flow kinematics. The solution of viscoelastic free surface flows using differential constitutive models was presented recently by Pasquali and Scriven [23] and Lee et al. [18]. Algebraic models that relate stress to the rate-of-strain tensor and relative-rate-of-rotation tensor cannot account for viscoelastic stresses per se but are perhaps the simplest way of trying to capture the distinctive behavior of Fig. 15. Sketch of flow domain. O.J. Romero et al. / J. Non-Newtonian Fluid Mech. 118 (2004) 137–156 polymer molecules in extension-dominated as compared to shear-dominated flow zones. Using this type of constitutive equation can enhance understanding of the effect of soluble polymers in coating liquids. In the particular case of free surface flows of low concentration solutions of high molecular weight polymers, the shear thinning and extensional thickening effects appear to have more consequences than viscoelastic behavior [27]. The rate-of-strain and the relative-rate-of-rotation are: D ≡ 21 [∇v + (∇v)T ] and W̄ ≡ W − Ω, ¯¯ ¯¯ ¯ ¯ ¯¯ ¯¯ T where W = 1/2[∇v − (∇v) ] is the vorticity tensor, and ¯ Ω is the¯¯ tensor that¯ represents the angular velocity of the ¯ ¯ principal direction of the rate-of-strain tensor following the liquid particle motion. Astarita [1] defined a flow classification index to measure the degree to which a fluid particle avoids stretching; the index R is based on the relative rate of rotation W̄ and rate of deformation D tensors: ¯¯ ¯¯ 2 tr(W̄ ) R ≡ − ¯¯ 2 . (11) tr(D ) ¯¯ The flow classification index takes the value, R = 0, in pure extension and the value, R = 1, in shear flows. Moreover, as the flow approaches a rigid body motion, i.e. as D → 0, ¯¯ R approaches infinity. Schunk and Scriven [27] adopted the flow classification index defined by Astarita [1] and proposed a generalized Newtonian model in which the liquid viscosity is an arithmetic average of the shear viscosity and the uniaxial extensional viscosity, the latter being represented by a constitutive equation of the same functional form as Carreau’s for shear viscosity. The constitutive relation put forward by Thompson et al. [29] follows the same idea, but extends the capability of this type of model, for it is able to describe shear thinning and normal stress differences in simple shear flow and extensional thickening in pure extensional flow simultaneously. For the stress tensor it gives T = −pI + α1 D + α3 D2 + α4 [D · W̄ − W̄ · D]. (12) ¯¯ ¯¯ ¯¯ ¯¯ ¯¯ ¯¯ ¯¯ ¯¯ This is the most general way of writing a stress tensor that is symmetric as a function of the rate-of-strain and relative rate-of-rotation tensor, and can fit independently measured material functions of shear viscosity and extensional viscosity. The coefficients in (12) depend on not only those material functions of the liquid, but also the first and second normal stress coefficients. Here, a simplified version of this equation, a generalized Newtonian model, is used: T = −pI + α1 D. (13) ¯¯ ¯¯ ¯¯ The coefficient α1 is a function of the shear viscosity ηs , a viscosity in extension ηu , defined as ηu ≡ Txx − Tyy /3˙ in uniaxial extension flow, and the flow classification index R: (1−R) . α1 = 2ηR s ηu (14) 149 Each viscosity function depends on the deformation rate; in the framework of this class of models, the functions describing these dependencies are found empirically. Although Eq. (13) does not account for viscoelastic behavior per se, it is able to describe the effect of extension-thickening common to high molecular weight polymeric solutions. 3.3. Solution method Because of the free surfaces, the flow domain at each set of parameter values was unknown a priori. To solve this free boundary problem by means of standard techniques for boundary value problems, the set of differential equations and boundary conditions posed in the unknown domain Eq. (4) had to be transformed to an equivalent set defined in a known reference domain. This transformation was made by a mapping x = x(ξ ) that connects the two domains. The ¯ ¯domain ¯ unknown physical was parameterized by the position vector x, and the reference domain by ξ . The mapping ¯ the one described by De Santos ¯ [7]. The inused here was verse of the mapping that minimizes the functional is governed by a pair of elliptic differential equations identical to those encountered in diffusional transport with variable diffusion coefficients. The coordinates potentials ξ and η of the reference domain satisfied ∇ · (Dξ ∇ξ) = 0; ∇ · (Dη ∇η) = 0. (15) Dξ and Dη are diffusion-like coefficients used to control gradients in coordinate potentials, and thereby the spacing between curves of constant ξ on the one hand and of constant η on the other that make up the sides of the elements that were employed; they were quadrilateral elements. Boundary conditions were needed in order to solve the second-order partial differential Eq. (15). The solid walls and synthetic inlet and outlet planes were described by functions of the coordinates and along them stretching functions were used to distribute the terminii of the coordinate curves selected to serve as element sides. The free boundary (gas–liquid interface) required enlarging the system of governing equations with the kinematic condition, viz. Eq. (8). The discrete version of the mapping equations is generally referred to as mesh generation equations. The system of governing equations was solved by Galerkin’s method with quadrilateral finite elements. The velocity and node positions were represented in terms of biquadratic basis functions φj , and the pressure in terms of linear discontinuous basis functions χj : u = nj=1 Uj φj ; v = nj=1 Vj φj ; p = m j=1 Pj χj ; n n (16) x = j=1 Xj φj ; y = j=1 Yj φj because the stress tensor depended on the second derivative of the velocity field (through the definition of the index R) and that derivative is discontinuous across the sides of biquadractic elements, an additional variable L was introduced ¯¯ to represent the velocity gradient as a continuous function, 150 O.J. Romero et al. / J. Non-Newtonian Fluid Mech. 118 (2004) 137–156 i.e. L = ∇v. It was represented in terms of bilinear basis ¯¯ functions ψ¯j : Lux = Lvx = p LUXj ψj ; Luy = j=1 LVXj ψj ; Lvy = pj=1 p pj=1 LUY j ψj ; j=1 LVY j ψj . (17) Thus, the system of partial differential equations, and boundary conditions was reduced to a set of simultaneous algebraic equations for the coefficients of the basis functions of all the fields. This set was non-linear and sparse. It was solved by Newton’s method. In order to improve the initial guess at each new set of operating conditions a pseudo-arc-length continuation method, as described by Bolstad and Keller [3], was used. The first successful free surface flow was computed using a fixed boundary flow field with slippery surface in place of the free boundary as the initial condition for Newton’s method. The domain was divided into 256 graded quadrilateral elements, with 6400 unknowns; tests showed that using more elements produced changes in the values of the coefficients no greater than 1%. Fig. 16 shows detail of the mesh near the free surface. 3.4. Results Solutions were computed at capillary numbers from Ca = 0.11 up to 1.5; these values cover the range from where capillary pressure is dominant in at least part of the flow to where it is generally comparable in magnitude to viscous stress. The operating conditions at the low-flow limit were determined theoretically by following a solution path constructed at a fixed capillary number (e.g. fixed substrate velocity, liquid viscosity and surface tension) by lowering the flow rate, and consequently the thickness of the coated liquid layer. 3.4.1. Newtonian liquid: flow kinematics and low-flow limit The flow field as a function of the layer thickness and hence flow rate is shown in Fig. 17 at capillary number Ca = 0.3. As the thickness of the deposited liquid falls, a recirculation attached to the free surface appears, the meniscus becomes more curved, and the inclination angle of the free surface with the downstream die land diminishes. Fig. 18 displays the computed static contact angle at different capillary numbers and gap-to-thickness ratios. At a fixed flow rate, i.e. fixed gap-to-thickness ratio, the static contact angle rises as the capillary number falls, as predicted first by Saito and Scriven [26] and later by Sartor [25], Gates and Scriven [12] and Carvalho and Kheshgi [4]. At low capillary number, surface tension force (capillary pressure) is strong and the meniscus does not need to curve as much in order to provide the adverse pressure gradient needed to match the flow in the gap to the rate at which liquid is fed. The onset of the low-flow limit at a given capillary number was determined by recording the gap-to-thickness ratio at which the static contact angle reached θ = 100 , as indicated in Fig. 18. The critical conditions is graphed as a function of the capillary number in Fig. 19. The graph also shows the critical conditions predicted by the viscocapillary model and those predicted by solving the governing equations of two-dimensional flow over the entire length of the coating bead, as reported by Carvalho and Kheshgi [4]. At high capillary number (Ca > 0.05), the viscocapillary model underpredicts the critical gap-to-thickness ratio, i.e. overpredicts the minimum film thickness. Important to what follow is the fact that the critical conditions found in the present Fig. 16. Detail of the mesh near the free surface. O.J. Romero et al. / J. Non-Newtonian Fluid Mech. 118 (2004) 137–156 151 Fig. 17. Evolution of the streamlines as the flow rate, i.e. film thickness, falls. Ca = 0.3. work by solving the governing equations only in a region close to the downstream free surface of a slot coating bead are virtually the same as those predicted by Carvalho and Kheshgi [4], and agree well with the experiments reported in the previous section and those presented by Carvalho and Kheshgi [4]. Fig. 20 shows how the second invariant of the rate-of-strain tensor evolve as a function of gap-to-thickness ratio at Ca = 0.3. The deformation rate is plotted in units of web speed over gap. The magnitude of the maximum rate of strain rises as the thickness of the liquid layer falls and the recirculation region appears. In the recirculation region and 140 Newtonian Fluid Separation angle θ degrees (1) 120 Ca=0.11 Ca=0.3 Ca=0.6 Ca=1.0 Ca=1.2 Ca=1,5 100 80 (2) 60 40 20 10 (3) 0 H0 / tmin 2 3 4 5 6 7 8 9 Gap / Film Thickness, H0 / t Fig. 18. Computed static contact angle as a function of capillary number and gap-to-thickness ratio. Newtonian liquid. 152 O.J. Romero et al. / J. Non-Newtonian Fluid Mech. 118 (2004) 137–156 10 V This work Capillary Number, Ca = 0 Carvalho and Kheshgi (2000) 1 Viscocapillary Model (1980) 0.1 0.01 0.001 0 5 10 15 20 25 Gap / Film Thickness, H0 / t Fig. 19. Critical gap-to-thickness ratio at the onset of the low-flow limit as a function of capillary number. Predictions are for Newtonian liquid. in the coated liquid layer, the rate of deformation is small. The maximum rate of deformation occurs close to the web in the region underneath the downstream meniscus, where the liquid is accelerated along its streamlines, and it is on the order of 10–20 times the web speed-to-gap ratio. That ratio is the characteristic deformation rate usually used in previous analysis of slot coating flows. The type of deformation suffered by a liquid particle can be characterized Fig. 20. Evolution of the deformation rate, in units of H0 /(1000V) as the film thickness falls. Ca = 0.3. O.J. Romero et al. / J. Non-Newtonian Fluid Mech. 118 (2004) 137–156 153 Fig. 21. Flow classification index near the downstream meniscus. Ca = 0.3. with the flow classification index R, proposed by Astarita [1]. Fig. 21 shows the R field near the downstream free surface at Ca = 0.3. The vectors shown in the plot represent the direction of the highest positive linear deformation (eigenvector associated with the positive eigenvalue of the rate-of-strain tensor multiplied by the positive eigenvalue). In the coated liquid layer, where the flow approaches a plug flow, the deformation rate vanishes and R is high; in solid body motion it would be unbounded. Under the die land, R ≈ 1 and the eigenvector associated with the positive eigenvalue of the rate-of-strain tensor are 45◦ from the flow direction, indicating a shear-dominated flow. At the center of the recirculation, the liquid motion approaches a solid body rotation, and again R is high. The flow near the free surface is dominated by extensional deformation, however; the eigenvector associated with the positive eigenvalue of the rate-of-strain tensor is aligned with the flow and R is small; in pure extensional flow it would vanish. Although they have analyzed situations at much higher flow rates (film thickness to gap ratio), Pasquali and Scriven [23] showed similar behavior by defining a molecular shear and extensional rates. The high rate of strain and the extension-dominated deformation can lead to significantly higher stresses under the meniscus in flows of extensional thickening liquids [27,10]. This is the central issue in the next section. 3.4.2. Effect of rheological properties To examine the consequences of extensional thickening, the shear viscosity was taken as constant with shear rate, as in a Boger liquid: η s = η0 . In dilute polymer solutions the viscosity tends to plateau at high enough extension rate that all of the coils are stretched out (see [22]). Therefore, the dependence of the extensional viscosity on the deformation rate is well described by a Carreau-type relation, as proposed by Fuller et al. [11]: ηu = η0 − (ηu∞ − η0 )[1 + (λu γ̇)2 ](nu −1)/2 , γ̇ ≡ tr(D2 ) is the second invariant of the rate-of-strain ten¯ sor D. The¯ parameters in this equation are η0 , the low-rate ¯¯ viscosity in both shear and extension; ηu∞ , the limiting high extension rate viscosity; λu , the characteristic time that marks the onset of extensional thickening behavior; and nu , the exponent that describes the steepness of the variation of the viscosity with extension rate. The dimensionless parameters relevant to the non-Newtonian behavior of this extensionally thickening liquid are: • Carreau number: W ≡ λu Vw /H0 ; • extension-thickening exponent: nu ; • ratio of extensional viscosities at high and low deformation rates: T ≡ ηu∞ /η0 . The computed static contact angle as a function of the gap-to-thickness ratio H0 /t and Carreau number W at Ca = 0.3 and T = 10/3 is shown in Fig. 22. The effect of the high-to-low deformation viscosity ratio T is portrayed in Fig. 23. When the coated film thickness is large (low gap-to-thickness ratio), the rate of deformation is low and the effect of extensional thickening is weak. As the gap-to-thickness ratio rises, the deformation rates γ̇ become larger, as shown in Fig. 20, and the effects of extensional thickening become stronger. At a given capillary number and flow rate, the computed static contact angle falls as the extensional thickening behavior of the liquid becomes stronger, i.e. higher W and T . Similar behavior has been reported by Lee et al. [18] when using differential constitutive models. As a consequence of the change in the free surface profile, the critical gap-to-thickness ratio at the onset of the low-flow limit falls. The minimum film thickness that can be coated rises as the liquid becomes more extensional thickening. Fig. 24 shows the streamlines and free surface 154 O.J. Romero et al. / J. Non-Newtonian Fluid Mech. 118 (2004) 137–156 Separation Angle θ degrees 140 Ca=0.3; Newtonian Model 120 Ca=0.3; T=10/3; W=1 Ca=0.3; T=10/3; W=10 100 80 60 40 W 20 10 0 H0 / tmin 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 Gap / Film Thickness, H0 / t Fig. 22. Computed static contact angle as a function of the gap-to-thickness ratio and pseudo-Weissenberg number. Ca = 0.3 and T = 10/3. 140 Separation Angle θ, degrees Ca=0.3; Newtonian Model 120 Ca=0.3; W=1; T=5/3 Ca=0.3; W=1; T=10/3 100 Ca=0.3; W=1; T=50/3 80 60 40 T 20 10 0 H0 / tmin 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 Gap / Film Thickness, H0 / t Fig. 23. Computed static contact angle as a function of the gap-to-thickness ratio and the ratio of extensional viscosities at high and low deformation rates T . Ca = 0.3 and W = 1. Fig. 24. Txx component of the stress tensor, in Pascal, close to the free surface. Ca = 0.3 and H0 /t = 4.78. O.J. Romero et al. / J. Non-Newtonian Fluid Mech. 118 (2004) 137–156 155 Capillary Number, Ca = η0V/σ 10 Newtonian Model W=1; T=5/3 W=10; T=5/3 W=100; T=5/3 W=1; T=10/3 1 W=10; T=10/3 W=1; T=50/3 0,1 3 4 5 6 7 8 9 Gap / Film Thickness, H0 / t Fig. 25. Predicted critical conditions at the onset of low-flow limit for Newtonian and extensional thickening liquids. configuration at Ca = 0.3 and H0 /t = 4.78, for both Newtonian and extensional thickening behavior (W = 1 and T = 16.7). When the extensional viscosity of the liquid rises with extension rate, the meniscus has to curve more in order to balance the higher stresses that appear near the free surface. Consequently, the static contact angle at the downstream die lip falls. The mechanism responsible for the change in the static contact angle can be determined by analyzing the stress field near the free surface. The stresses rise with Carreau number due to the extensional behavior of the liquid. Fig. 24 also shows the component Txx of the stress tensor in the region close to the free surface. The rise in the extensional viscosity near the free surface makes the stress in that region larger than in the Newtonian case; a stress boundary layer is formed close to the free surface, as predicted by Pasquali and Scriven [23] and Lee et al. [18] using differential constitutive models. This similarity in the stress field reinforces the statement of Schunk and Scriven [27] that in the less concentrated polymer solutions encountered in many coating flows, the responses of shear thinning and extension thickening appear to have more consequences than viscoelastic behavior. The change in the force balance at the free surface due to the extensional thickening of the liquid causes the solution paths at W = 0 and T = 1 to cross the horizontal line of θ = 10◦ in Figs. 22 and 23 at a lower gap-to-thickness ratio than in the Newtonian flow. The onset of the low-flow limit occurs at smaller gap-to-thickness ratio, i.e. thicker coated layer. The critical conditions at the onset of the low-flow limit are mapped in Fig. 25. The onset of the low-flow limit when extensional thickening liquids are used occurs at larger film thickness, as observed in the experiments presented in the previous section. 4. Summary Slot coating is one of the preferred methods for high precision coating. There is an important operating limit, known as the low-flow limit, when thin films are coated at relatively high speeds. It is caused by the receding action of the downstream free surface as the flow rate is reduced at a fixed substrate speed. The operating parameters at which the coating bead breaks have been determined by previous researchers in the case of Newtonian liquids. However, most of the liquids coated industrially are polymeric solutions and dispersions, which are non-Newtonian. In general, polymer solutions may be shear-thinning, extension-thickening, and develop viscoelastic stresses when sheared or extended. These liquids can behave dramatically differently from Newtonian liquids in coating flows. In the particular case of very low concentrated polymer solutions of high molecular weight (less than 0.1 wt.%), the response of extension thickening appear to have more consequences than viscoelastic behavior per se (stress relaxation) (see [10,13,27]). In this work the low-flow limit of polymeric solutions was examined by both experiments and theory. A benchtop slot coating apparatus was used to visualize the coating bead at different operating conditions and observe the breakup mechanism. Aqueous solutions of low molecular weight PEG and high molecular weight PEO were used in order to evaluate the effects of extensional-thickening behavior. The flow visualization through the glass wall of the rotating cylinder used as substrate revealed that close to the onset of the low-flow limit, the downstream static contact line recedes toward the feed slot at certain locations across the width of the coating bead and the flow becomes three-dimensional. Later, the air finger formed at the downstream lip penetrates all the way through the coating bead to the vacuum chamber and one or more dry stripes appear. The experiments also showed that, at a fixed capillary number, as the concentration of PEO rises and consequently the liquid becomes more extensional-thickening, the minimum coated thickness possible increases. The operating window of the process is smaller when liquids with high extensional viscosity are used. The theoretical analysis consisted of solving the momentum and continuity equation system for steady, 156 O.J. Romero et al. / J. Non-Newtonian Fluid Mech. 118 (2004) 137–156 two-dimensional, free surface flow including a generalized Newtonian model to describe the liquid rheological behavior. The constitutive equation used relates the stress to the rate-of-strain and relative-rate-of-rotation tensors. This class of models is perhaps the simplest one that may capture the different ways that polymer molecules behave in extension-dominated and shear-dominated flow regions, although it is not able to describe viscoelastic stresses per se. The free boundary problem was solved by mapping the unknown physical domain to a fixed reference domain. The full set of differential equations, including the transformation mapping from the physical to the reference domain, was solved all together by Galerkin/finite element method. The theoretical low-flow limit was defined by the parameters at which the static contact angle between the downstream free surface and the die lip became less than 10◦ . In the case of Newtonian liquids, the quantitative agreement between the measured and the predicted parameters at the onset of the low-flow limit is excellent. The results obtained with the simple algebraic model revealed the same flow structure and stress field reported by previous work using more sophisticated and computationally expensive differential constitutive models. Moreover, the theoretical predictions of low-flow limit of extensional-thickening liquids agree qualitatively with the experimental observations of low concentration polymer solutions, which presented high extensional viscosity and no measurable viscoelastic stresses. Acknowledgements The authors would like to thank Prof. Matteo Pasquali for discussions on viscoelastic flows and for the suggestions on the rheological characterization of the low concentration PEO solutions. This work was funded by CNPq (Brazilian Research Council, Grants 300242/98-0 and 467662/00-2) and by the Industrial Partnership for Research in Interfacial and Materials Engineering (IPRIME) of the University of Minnesota. References [1] G. Astarita, Objective and generally applicable criteria for flow classification, J. Non-Newtonian Fluid Mech. 6 (1979) 69–76. [2] A.L. Beguin, Method of Coating Strip Material, US Patent no. 2,681,294 (1954). [3] J.H. Bolstad, H.B. Keller, A multigrid continuation method for elliptic problems with folds, SIAM J. Sci. Stat. Comput. 7 (1985) 1081– 1104. [4] M.S. Carvalho, H.S. Khesghi, Low-flow limit in slot coating: theory and experiments, AIChE J. 46 (2000) 1907–1917. [5] Y. Christanti, L.M. 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