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Assisted dynamic wetting in liquid coatings
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Colloids and Surfaces A: Physicochem. Eng. Aspects 311 (2007) 55–60
Review
Assisted dynamic wetting in liquid coatings Masato Yamamura ∗ Department of Applied Chemistry, Kyushu Institute of Technology, Sensui, Tobata, Kitakyushu, Fukuoka 804-8550, Japan Available online 24 August 2007
Abstract During forced wetting above a critical speed, visible amounts of air are entrained between a liquid and a moving solid, regardless of the wettability of the surface. The present article reviews how hydrodynamic and electrostatic forces, as well as the colloidal particle motion and the surface roughness, delay the onset of air entrainment to higher surface speeds. The onset velocities of air entrainment are compared with that in a filler-free, plunging tape flow on a smooth, uncharged surface in order to access how the dynamic wetting is assisted on a macroscopic length scale. The combined effect between different wetting assistances is also discussed. © 2007 Elsevier B.V. All rights reserved. Keywords: Dynamic wetting failure; Air entrainment; Coating; Wetting
Contents 1. 2. 3. 4. 5. 6. 7.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hydrodynamic assist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electrostatic assist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Roughness-assisted wetting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Particle-assisted wetting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Combined wetting assistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1. Introduction For successful dynamic wetting, a gas in contact with a moving solid is completely displaced by an advancing liquid, or remains as invisible traces that subsequently collapse via attractive conjoining forces [1,2]. However, visible amounts of air are entrained between the liquid and the solid when the liquid is forced to advance at too high a speed. Such a dynamic wetting failure, often referred to as air entrainment, widely occurs in nature as well as in industrial applications such as coating, printing, painting, laminating, and polymer processing. On a macroscopic length scale observable by conventional optical techniques, the advancing liquid front intersects the solid surface at an apparent dynamic contact angle, θ D ([3] and references therein). At sufficiently low surface speeds, θ D remains ∗
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in equilibrium near the static angle. The contact angle increases with the increasing surface speed, and eventually approaches 180◦ at a critical speed. Beyond that limit, the dynamic contact line spontaneously breaks into two or more inclined, but still steady straight-line segments, leading to an intrusion of sawtooth-shaped air sheets [4]. With even further speeds, a three-dimensional flow instability subsequently promotes an intermittent entrainment of visible air bubbles through singular free-surface cusps [5,6]. The drastic transitions in the dynamic wetting behavior have been dealt with in earlier summaries [7] and a more recent review [8]. However, the physics of the visible air entrainment is still far from complete understanding, and what determines the transition of wetting failure still remains unresolved. The major difficulty stems from the fact that the local molecular displacement on the solid inherently couples with macroscopic liquid/gas flows in the vicinity of the singular cusp. The previous hydrodynamic theories on the scale of a few microns give only limited insights into the local events of the molecular dis-
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particles on the postponed dynamic wetting, and the combined effect between hydrodynamic and particle assistances will be addressed in Section 6. 2. Hydrodynamic assist
Fig. 1. Dynamic wetting failure in plunging tape flow.
placement, whereas the current kinetic theories on a molecular length scale [9] often ignore the global hydrodynamics. The recent progresses in the Shikhmurzaev model [10–12] and the phase-field diffuse-interface theory [13] enable us to predict the dynamic contact angle in a given external flow field, yet the computations themselves have been limited to simpler flow configurations compared to those of practical interest. One of the most common flow configurations to access the wetting failure is the plunging-tape flow, in which a continuous strand of plastic film [14–17], fiber [18,19] or rod is impinged into a stagnant pool of liquid at a constant speed (Fig. 1). The onset velocity in the plunging tape flow generally depends on the surface tension [14,16], entry angle of the solid [20], and non-Newtonian behavior of the fluids [17,18] in a complicated manner. For the specific case when a smooth, uncharged solid is impinged into a pool of Newtonian, filler-free fluids, the onset velocity of the air entrainment, vae , is inversely related to the liquid viscosity, μ, and obeys the simple power law vae ∼ μn where n denotes the negative power law index [15,16]. The power-law correlations of the dimensional variables are successful in capturing the trend of experimental data for wide range of fluid properties. The onset velocity of air entrainment, beyond which the successful dynamic wetting fails, often limits the maximum speed of industrial operations because air bubbles are detrimental to the quality of the final products. Despite formidable difficulties inherent in dynamic wetting failure, there has been a great deal of progress in the experimental analysis of the onset of air entrainment. In this article, we attempt to provide a brief guide to the recent experiments on assisted dynamic wetting. In order to access how the dynamic wetting is assisted on a macroscopic length scale, the onset velocities of the air entrainment for different flow fields are compared to that in the simplest plunging tape flow on a smooth, uncharged surface. Section 2 emphasizes the important role that the hydrodynamic force can play in impinging curtain coating flows. Section 3 assesses how the electrostatic force delays the onset of air entrainment on a charged surface. Section 4 examines how the surface roughness influences the onset of dynamic wetting failure for different viscosities of the advancing liquids. Section 5 will consider the effect of dispersed colloidal
The capillary pressure and the viscous stresses in both the advancing and receding phases generally compete with each other at the onset of wetting failure. However, another hydrodynamic force, in particular, the inertial force, also affects the onset of visible air entrainment. Typical examples include the curtain coating, in which an extruded liquid forms a free falling thin sheet before it impinges onto a moving solid. Blake et al. [21,22] demonstrated that the inertia gained during the free fall of a liquid significantly postpones the onset of dynamic wetting to higher surface speeds. Fig. 2 shows the variation in the onset velocity of air entrainment with the volumetric flow rate of the curtain per unit curtain width, Q. The test fluid was a solution of glycerol and de-ionized water with a shear viscosity of μ = 0.32 Pa s, a surface tension of σ = 65 mN/m, and a density of ρ = 1238 kg/m3 . The liquid was impinged over a distance of h = 0.178 m onto a smooth poly(ethyleneterephthalate) tape with a roughness of Rz = 0.6 m. The detailed experimental setup has been found in a previous publication [22]. In order to quantify the postponed wetting failure, the measured onset velocity was normalized as vae /vae 0 where vae and vae 0 are the onset velocities in the curtain and the plunging tape flows [16], respectively. With the increasing flow rate, the onset velocity of air entrainment initially increases, then decreases and eventually reaches the constant value of unity. The onset velocity at the peak is found to be vae /vae 0 = 5, indicating that the liquid is successfully applied onto the moving surface at a speed five times faster than the plunging-tape flow. The simple boundary layer model [23] shows that the critical Reynolds number, at which the contact line is directly located beneath the impinging liquid, is given by Rec = 2.6, where the Reynolds number is readily defined as the ratio between the inertial and viscous drag forces, i.e., Re = (ρQ/μ)(u/vae ) where u is the impinging velocity. Substituting the onset velocity of vae = 0.56 m/s at the peak, the critical flow rate corresponding to Rec was found to be Qc = 2.1 cm2 /s, showing good agreement with the peak flow rate in Fig. 2. This fact suggests that
Fig. 2. Onset velocities of air entrainment in curtain coating (Blake et al. [22]). The hydrodynamic force acts on the contact line to delay the onset of air entrainment to higher speeds near the critical flow rate Qc .
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the hydrodynamic inertial force acts on the contact line locating beneath the impinging liquid, and hence, postpones the onset of air entrainment to higher surface speeds. The hydrodynamic assist of the dynamic wetting is consistent with other evidence that the contact angle, θ D , decreases with the increasing flow rate [24]. At lower flow rates Q Qc , the contact line is located far downstream due to the dominant viscous force. An increase in the flow rate beyond Qc tends to move the contact line further upstream, and eventually leads to the formation of an upstream liquid bank. In both cases, the liquid impact no longer acts on the contact line locating far from the impinging liquid. The resulting weaker inertial force promotes a negligible assist in the dynamic wetting, giving rise to the onset of air entrainment at the same speed as in the plunging tape flow, i.e., vae /vae 0 = 1. A similar hydrodynamic assistance has also been observed in the curtain coating with different fluid properties [12], the forward-roll coating [25], the reverse-roll coating [26] and the jet coating flow [27].
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In their experiments, the flow rate was one order of magnitude greater than the critical value, i.e., Re > Rec so that the hydrodynamic assist only slightly influences the onset of the dynamic wetting failure in Fig. 3. Fermin et al. experimentally demonstrated a similar electrostatic assistance in curtain [29] and slot [30,31] coating configurations. From a theoretical viewpoint, Blake et al. [28] combined the existing molecular kinetic theory with the electrostatic effects, and derived an expression that relates the dynamic contact angle to both the surface speed and the voltage. Although the theory fits the experimental data, they simply assumed that the hydrodynamics and electrostatics are completely decoupled. It is desirable to establish a rigorous physical model including the electrostatic effects coupled with the global flow fields that, in turn, alter the local molecular displacement on a small length scale. 4. Roughness-assisted wetting
Consider a moving solid on which a charge is applied to one side, and an equal and opposite charge is applied to the reverse side. The charge generates an electrostatic field between the solid and an approaching, conductive liquid. The resulting electrostatic force, which quadratically depends on the field strength, acts normal to the gas/liquid interface to reduce the dynamic contact angle, and eventually postpones the onset of air entrainment to higher speeds. Blake et al. [28] measured the onset velocity of air entrainment on a charged surface in curtain coatings. Fig. 3 shows the normalized onset velocity of air entrainment as a function of the applied voltage. In the test fluid, sodium chloride was added to a 60% (w/w) solution of glycerol and de-ionized water to increase the electrical conductivity. The onset velocity predicted from the power-law correlation for a plunging-tape flow [16] was found to be vae 0 = 0.17 m/s, which agrees with the measured value of 0.18 m/s in their curtain coating experiments in the absence of a voltage. The onset velocity of air entrainment monotonically increases with the increasing applied voltage, showing a noticeable electrostatic assist of dynamic wetting.
In the forced wetting on a rough, but still chemically homogenous solid surface, the dynamic contact line moves up and down the peaks and valleys, and travels a greater distance than it would over a smooth surface. Thus we can expect that the rough surface entrained air at a lower speed than the smooth surface. However, recent experiments have revealed alternative evidence that the onset velocity increases on a solid with a properly chosen surface roughness. Benkreira [32] performs plunging-tape experiments using a rough surface with the peak-to-valley height roughness of Rz = 3.3 m. Fig. 4 shows the onset velocities of the air entrainment as a function of fluid viscosity. On the smooth surface, the normalized onset velocities are close to unity and agree with those predicted from the power-law correlation [15]. The normalized onset velocity on the roughness surface becomes negative below a critical viscosity of μc = 0.4 Pa s, suggesting that the introduction of the surface roughness enhances the dynamic wetting failure. Above the critical viscosity, on the contrary, the reverse dependence was observed; the onset velocity of air entrainment becomes higher on a rough surface. This feature has been attributed to the contact line motion that does not wet the surface along the peak and the valley but rather skips from peak to peak [33]. The less contact between the liquid and the solid leads to a lower frictional force and thus the lower
Fig. 3. Onset velocities of air entrainment in curtain coating on a charged surface (Blake et al. [28]). An increase in applied voltage leads to higher onset velocities via electrostatic assist of wetting.
Fig. 4. Onset velocities of air entrainment in plunging tape flow on a rough surface (Benkreira [32]). The onset velocity becomes higher above a critical liquid viscosity for the peak-to-valley height roughness of Rz = 3.30 m, whereas it remains constant for the smooth surface of Rz = 0.60 m.
3. Electrostatic assist
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onset velocity of air entrainment at high viscosities. The earlier work by Buonopane et al. [34] has shown a similar roughnessassisted dynamic wetting using various liquid/solid systems, yet the substrate chemistry did not remain constant. In curtain coatings, Clarke [35] demonstrated the assisted dynamic wetting on a rough surface of Rz = 4.4 m. In his experiment, however, the momentum of the impinging liquid sheet can pressurize the contact line to some extent, because the Reynolds numbers of the curtain were not sufficiently high to ensure that the contact line locates far upstream from the impinging liquid. Thus the dynamic wetting was possibly hydrodynamically assisted in addition to the roughness effect. 5. Particle-assisted wetting The introduction of colloidal particles in liquids generally promotes shear-rate dependent rheological changes arising from complicated solid–liquid interactions [36]. Classical hydrodynamic theories show that the viscosity of a dilute suspension linearly increases with respect to the particle volume fraction. Thus the onset velocity of dynamic wetting failure is expected to monotonically decrease with the increasing particle concentration once the power law holds. In contrast, recent experiments revealed that the onset velocity first increases and then decreases with particle content at sufficiently low particle concentrations such that the suspension viscosity agrees with that of the pure fluid [37]. Fig. 5 shows the variation in the onset velocity of air entrainment with the particle number density, i.e., the number of particles per unit liquid volume. The normalized onset velocity of air entrainment increases, exhibits a peak at the critical particle number density of Σ max = 1 × 1011 m−3 and then decreases as the particle content increases. The maximum velocity at the peak was found to be vae /vae 0 = 1.6, indicating that the suspension was successfully applied onto the moving surface at a speed 60% faster that of the filler-free liquid. Such a particle-assisted dynamic wetting has been observed not only for suspensions of spherical polymeric particles but also for non-spherical inorganic fillers [38].
Fig. 5. Onset velocities of air entrainment in dilute suspension flow (Yamamura et al. [37]). The spherical poly(methyl-methacrylate) particles with diameters ranged from 8 to 50 m were dispersed in the coating liquids at weight fractions of 0.1 wt%. The normalized onset velocity of air entrainment increases, exhibits a peak, and then decreases as the particle content increases.
Yamamura et al. [37] postulated that the particle motion can split the intruding air-film into fine fingers, which subsequently shrink against the viscous drag force to assist the dynamic wetting. The key physical factors for describing the particleassistance are assumed to be the time scale required for the finger shrinkage, τ s , and the time interval of the particle contact, τ c . In the case of a low particle number density, the contact time interval becomes much longer than the time required for the finger shrinkage (τ c τ s ), and hence, the finger can completely shrink before the following particle comes into contact with the surface. The finger shrinkage in the direction opposite to the surface motion cancels the viscous drag flow, pulls the finger further upstream, and eventually causes the dynamic contact line to recede at even higher surface speeds. At high particle number densities, on the contrary, the finger does not shrink during a finite contact interval (τ c τ s ). Thus the finger formation no longer influences the onset of wetting failure, thus leading to a lower onset velocity of air entrainment. This “film-splitting” hypothesis is consistent with the measured increase and decrease in the onset velocity with the increasing particle number density. Assuming a sequential series of particle contacts and film splittings at low fluid viscosities, the one-dimensional lubrication theory yields the expression for the time-averaged onset velocity of air entrainment as [37]: vae − vae 0 1 = , 1/3 vae 0 (1/(Σ l)) + 6Σ 1/3 Cal
(1)
where Ca(= μA vae 0 /σ) and l denote the capillary number in air, and the length of the intruding air film before the particle contact. The lubrication theory has been verified by comparing the predicted and measured critical particle number densities at the peak. The critical particle number densities at the peak agree with each other in a quantitative sense for the case when the air film length l is readily determined by the direct flow visualization. Unfortunately, no rigorous physical model is currently available to predict the particle-assisted dynamic wetting speed if the air film length is unknown. For dense suspensions showing a particular shear-thinning behavior at low shear rates, Chu et al. [39] recently reported that, by adding porous or hard spherical inorganic particles in aqueous 20 wt% poly(vinyl alcohol) solutions, the onset of the “break line” defect was significantly delayed to higher surface speeds in a slot-die coating system. The particle-assisted wetting has been attributed to the fact that the polymer adsorption on the particle surface and the resultant inter-particle bridging can lead to an increase in the local surface tension, which yields a more stable gas–liquid interface that postpones the onset of wetting failure. Evidently, those physical mechanisms of wetting assistance completely differ from that in the previously mentioned dilute suspensions of simpler particle/fluid binary systems. Still to clarify is whether the film splitting or the inter-particle bridging mechanism dominates the dynamic wetting behavior for a given suspension system, and how the particle characteristics affect the wetting assistance.
M. Yamamura / Colloids and Surfaces A: Physicochem. Eng. Aspects 311 (2007) 55–60
Fig. 6. Onset velocities of air entrainment in dilute suspension jet coating flow (Yamamura et al. [38]). The introduction of particles further assists the dynamic wetting in addition to the co-existing hydrodynamic effect.
6. Combined wetting assistance Two or more wetting assistances simultaneously take place in properly designed coating systems. Understanding the feature of combined wetting assistances is of interest from both fundamental and practical viewpoints. The hydrodynamic assist of wetting has been attempted to be combined with the electrostatic [28], roughness [35] and particle assistances [38] in order to achieve a higher onset velocity of air entrainment. A typical example of the last combination is shown in Fig. 6. The Newtonian suspension jet was extruded from a round nozzle and vertically impinged onto a smooth, uncharged surface. The test fluids are silicone oils in which SiO2 particles with a mean diameter of 10 m were dispersed at the particle number densities up to 5 × 1012 m−3 . The onset velocity of air entrainment increases as the Reynolds number approaches the critical value of Rec . As described in Section 2, this is because the hydrodynamic force assists the dynamic wetting at Re = Rec . At the Reynolds number of Re = 3 near the critical value, the normalized onset velocity was (vae − vae 0 )/vae 0 = 3 in the filler-free liquid. In contrast, the suspension exhibits the normalized onset velocity of 7 at the same Reynolds number, showing that the introduction of particles further assists the dynamic wetting in addition to the co-existing hydrodynamic effect. There is a pronounced need for understanding the wetting behavior in other co-existing assistances; i.e., coating on a charged, rough surface, a particle suspension coating on a rough surface, and a suspension curtain coating on a charged surface. Unfortunately, few open publications deal with these combined assistances of practical interest. 7. Conclusions The recent progress in postponed visible air entrainment is reviewed, especially in high speed liquid coating flows. The hydrodynamic inertia forces, as well as the electrostatic forces on a charged surface, directly act on the contact line to delay the onset of dynamic wetting failure. However, the rigorous physical model to quantify the contribution of these forces is still
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incomplete. The well-defined surface roughness promotes faster coating speeds without air entrainment as the advancing liquid skips from peak to peak. It has not yet clarified whether the optimum height roughness exists in a give coating system, and how the spatial distribution of the roughness influences the dynamic wetting behavior. The particle-assisted wetting becomes pronounced in suspensions, yet it has been limited to low viscosity fluids at sufficiently low particle concentrations, such that the suspension viscosity agrees with that of the pure fluids. Understanding the wetting behavior in denser particle suspensions is challenging and the subject of future work. These wetting assistances can co-exist in order to achieve a higher onset velocity of air entrainment, yet it is not immediately clear how the different wetting assistances couple or de-couple in a given flow field. Continued research will result in a better understanding of combined assistances during the forced wetting. References [1] K. Miyamoto, On the mechanism of air entrainment, Ind. Coat. Res. 1 (1991) 71–88. [2] G.F. Teletzke, H.T. Davis, L.E. Scriven, Wetting hydrodynamics, Revue de Physique Appliquee 23 (1988) 989–1007. [3] T.D. Blake, Dynamic contact angles and wetting kinetics, Wettability, Marcel Dekker, 1993, pp. 251–308. [4] T.D. Blake, K.J. Ruschak, A maximum speed of wetting, Nature 282 (1979) 489–491. [5] J. Eggers, Air entrainment through free-surface cusps, Phys. Rev. Lett. 86 (2001) 4290–4293. [6] G.E. Innes, P.H. Gaskell, M.D. Savage, Cusps as a source of air entrainment in roll coating systems, Ind. Coat. Res. 4 (1998) 19–40. [7] S.F. Kistler, Hydrodynamics of wetting, Wettability, Marcel Dekker, 1993, pp. 311–429. [8] T.D. Blake, The physics of moving wetting lines, the physics of moving wetting lines, J. Colloid Interf. Sci. 299 (2006) 1–13. [9] M.J. de Ruijter, T.D. Blake, J. de Coninck, Dynamic wetting studies by molecular modelling simulations of droplet spreading, Langmuir 15 (1999) 7836–7847. [10] Y.D. Shikhmurzaev, Mathematical modeling of wetting hydrodynamics, Fluid Dynam. Res. 13 (1994) 45–64. [11] Y.D. Shikhmurzaev, Moving contact lines in liquid/liquid/solid systems, J. Fluid Mech. 334 (1997) 211–249. [12] T.D. Blake, Y.D. Shikhmurzaev, Dynamic wetting by liquids of different viscosity, J. Colloid Interf. Sci. 253 (2002) 196–202. [13] D. Jacqmin, Onset of wetting failure in liquid–liquid systems, J. Fluid Mech. 517 (2004) 209–228. [14] R. Burley, B.S. Kennedy, An experimental study of air entrainment at a solid/liquid/gas interface, Chem. Eng. Sci. 31 (1976) 901–911. [15] E.B. Gutoff, C.E. Kendrick, Dynamic contact angles, AIChE J. 28 (1982) 459–466. [16] R. Burley, R.P.S. Jolly, Entrainment of air into liquids by a high speed continuous solid surface, Chem. Eng. Sci. 39 (1984) 1357–1372. [17] O. Cohu, H. Benkreira, Entrainment of air by a solid surface plunging into a non-Newtonian liquid, AIChE J. 44 (1998) 2360–2368. [18] M.T. Ghannam, M.N. Esmail, Experimental study on wetting of fibers with non-Newtonian liquids, AIChE J. 43 (1997) 1579–1588. [19] P.G. Simpkins, V.J. Kuch, On air entrainment in coatings, J. Colloid Interf. Sci. 263 (2003) 562–571. [20] O. Cohu, H. Benkreira, Air entrainment in angled dip coating, Chem. Eng. Sci. 53 (1998) 533–540. [21] T.D. Blake, A. Clarke, K.J. Ruschak, Hydrodynamic assist of dynamic wetting, AIChE J. 40 (1994) 229–242. [22] T.D. Blake, R.A. Dobson, K.J. Ruschak, Wetting at high capillary numbers, J. Colloid Interf. Sci. 279 (2004) 198–205.
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[23] S.F. Kistler, The fluid mechanics of curtain coating and related viscous free surface flow with contact lines, PhD thesis, University of Minnesota, 1983, pp. 384–386. [24] T.D. Blake, M. Bracke, Y.D. Shikhmurzaev, Experimental evidence of nonlocal hydrodynamic influence on the dynamic contact angle, Phys. Fluids 11 (1999) 1995–2007. [25] H. Benkreira, Dynamic wetting in metering and pre-metered forward roll coating, Chem. Eng. Sci. 57 (2002) 3025–3032. [26] H. Benkreira, Experimental study of dynamic wetting in reverse-roll coating, AIChE J. 48 (2002) 221–226. [27] M. Yamamura, S. Suematsu, T. Kajiwara, K. Adachi, Experimental investigation of air entrainment in a vertical liquid jet flowing down onto a rotating roll, Chem. Eng. Sci. 55 (2000) 931–942. [28] T.D. Blake, A. Clarke, E.H. Stattersfield, An investigation of electrostatic assist in dynamic wetting, Langmuir 16 (2000) 2928–2935. [29] R.J. Fermin, L.E. Scriven, Electrostatically assisted coating flows, in: Proceedings of the Third European Coating Symposium, 1999, pp. 91–95. [30] R.J. Fermin, L.E. Scriven, Electrostatically assisted slot coating, in: Proceedings of the Ninth International Coating Science and Technology Symposium, 1998, pp. 149–152.
[31] R.J. Fermin, L.E. Scriven, Electrostatic effects in coating flows, in: Proceedings of the 10th International Coating Science and Technology Symposium, 2000, pp. 85–88. [32] H. Benkreira, The effect of substrate roughness on air entrainment in dip coating, Chem. Eng. Sci. 59 (2004) 2745–2751. [33] A. Menchaca-Rocha, The mobility of mercury drops on rough glass surfaces, J. Colloid Interf. Sci. 149 (1992) 472–480. [34] R.A. Buonopane, E.B. Gutoff, M.M.T. Rimore, Effect of plunging tape surface properties on air entrainment velocity, AIChE J. 32 (1986) 682–683. [35] A. Clarke, Coating on a rough surface, AIChE J. 48 (2002) 2149–2156. [36] R.G. Larson, The Structure and Rheology of Complex Fluids, Oxford University Press, 1999. [37] M. Yamamura, H. Miura, H. Kage, Postponed air entrainment in dilute suspension coatings, AIChE J. 51 (2005) 2171–2177. [38] M. Yamamura, A. Matsunaga, Y. Mawatari, K. Adachi, H. Kage, Particleassisted dynamic wetting in a suspension liquid jet impinged onto a moving solid at different flow rates, Chem. Eng. Sci. 61 (2006) 5421–5426. [39] W.B. Chu, J.W. Yang, Y.C. Wang, T.J. Liu, C. Tiu, J. Guo, The effect of inorganic particles on slot die coating of poly(vinyl alcohol) solutions, J. Colloid Interf. Sci. 297 (2006) 215–225.
Colloids and Surfaces A: Physicochem. Eng. Aspects 311 (2007) 55–60
Review
Assisted dynamic wetting in liquid coatings Masato Yamamura ∗ Department of Applied Chemistry, Kyushu Institute of Technology, Sensui, Tobata, Kitakyushu, Fukuoka 804-8550, Japan Available online 24 August 2007
Abstract During forced wetting above a critical speed, visible amounts of air are entrained between a liquid and a moving solid, regardless of the wettability of the surface. The present article reviews how hydrodynamic and electrostatic forces, as well as the colloidal particle motion and the surface roughness, delay the onset of air entrainment to higher surface speeds. The onset velocities of air entrainment are compared with that in a filler-free, plunging tape flow on a smooth, uncharged surface in order to access how the dynamic wetting is assisted on a macroscopic length scale. The combined effect between different wetting assistances is also discussed. © 2007 Elsevier B.V. All rights reserved. Keywords: Dynamic wetting failure; Air entrainment; Coating; Wetting
Contents 1. 2. 3. 4. 5. 6. 7.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hydrodynamic assist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electrostatic assist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Roughness-assisted wetting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Particle-assisted wetting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Combined wetting assistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1. Introduction For successful dynamic wetting, a gas in contact with a moving solid is completely displaced by an advancing liquid, or remains as invisible traces that subsequently collapse via attractive conjoining forces [1,2]. However, visible amounts of air are entrained between the liquid and the solid when the liquid is forced to advance at too high a speed. Such a dynamic wetting failure, often referred to as air entrainment, widely occurs in nature as well as in industrial applications such as coating, printing, painting, laminating, and polymer processing. On a macroscopic length scale observable by conventional optical techniques, the advancing liquid front intersects the solid surface at an apparent dynamic contact angle, θ D ([3] and references therein). At sufficiently low surface speeds, θ D remains ∗
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in equilibrium near the static angle. The contact angle increases with the increasing surface speed, and eventually approaches 180◦ at a critical speed. Beyond that limit, the dynamic contact line spontaneously breaks into two or more inclined, but still steady straight-line segments, leading to an intrusion of sawtooth-shaped air sheets [4]. With even further speeds, a three-dimensional flow instability subsequently promotes an intermittent entrainment of visible air bubbles through singular free-surface cusps [5,6]. The drastic transitions in the dynamic wetting behavior have been dealt with in earlier summaries [7] and a more recent review [8]. However, the physics of the visible air entrainment is still far from complete understanding, and what determines the transition of wetting failure still remains unresolved. The major difficulty stems from the fact that the local molecular displacement on the solid inherently couples with macroscopic liquid/gas flows in the vicinity of the singular cusp. The previous hydrodynamic theories on the scale of a few microns give only limited insights into the local events of the molecular dis-
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particles on the postponed dynamic wetting, and the combined effect between hydrodynamic and particle assistances will be addressed in Section 6. 2. Hydrodynamic assist
Fig. 1. Dynamic wetting failure in plunging tape flow.
placement, whereas the current kinetic theories on a molecular length scale [9] often ignore the global hydrodynamics. The recent progresses in the Shikhmurzaev model [10–12] and the phase-field diffuse-interface theory [13] enable us to predict the dynamic contact angle in a given external flow field, yet the computations themselves have been limited to simpler flow configurations compared to those of practical interest. One of the most common flow configurations to access the wetting failure is the plunging-tape flow, in which a continuous strand of plastic film [14–17], fiber [18,19] or rod is impinged into a stagnant pool of liquid at a constant speed (Fig. 1). The onset velocity in the plunging tape flow generally depends on the surface tension [14,16], entry angle of the solid [20], and non-Newtonian behavior of the fluids [17,18] in a complicated manner. For the specific case when a smooth, uncharged solid is impinged into a pool of Newtonian, filler-free fluids, the onset velocity of the air entrainment, vae , is inversely related to the liquid viscosity, μ, and obeys the simple power law vae ∼ μn where n denotes the negative power law index [15,16]. The power-law correlations of the dimensional variables are successful in capturing the trend of experimental data for wide range of fluid properties. The onset velocity of air entrainment, beyond which the successful dynamic wetting fails, often limits the maximum speed of industrial operations because air bubbles are detrimental to the quality of the final products. Despite formidable difficulties inherent in dynamic wetting failure, there has been a great deal of progress in the experimental analysis of the onset of air entrainment. In this article, we attempt to provide a brief guide to the recent experiments on assisted dynamic wetting. In order to access how the dynamic wetting is assisted on a macroscopic length scale, the onset velocities of the air entrainment for different flow fields are compared to that in the simplest plunging tape flow on a smooth, uncharged surface. Section 2 emphasizes the important role that the hydrodynamic force can play in impinging curtain coating flows. Section 3 assesses how the electrostatic force delays the onset of air entrainment on a charged surface. Section 4 examines how the surface roughness influences the onset of dynamic wetting failure for different viscosities of the advancing liquids. Section 5 will consider the effect of dispersed colloidal
The capillary pressure and the viscous stresses in both the advancing and receding phases generally compete with each other at the onset of wetting failure. However, another hydrodynamic force, in particular, the inertial force, also affects the onset of visible air entrainment. Typical examples include the curtain coating, in which an extruded liquid forms a free falling thin sheet before it impinges onto a moving solid. Blake et al. [21,22] demonstrated that the inertia gained during the free fall of a liquid significantly postpones the onset of dynamic wetting to higher surface speeds. Fig. 2 shows the variation in the onset velocity of air entrainment with the volumetric flow rate of the curtain per unit curtain width, Q. The test fluid was a solution of glycerol and de-ionized water with a shear viscosity of μ = 0.32 Pa s, a surface tension of σ = 65 mN/m, and a density of ρ = 1238 kg/m3 . The liquid was impinged over a distance of h = 0.178 m onto a smooth poly(ethyleneterephthalate) tape with a roughness of Rz = 0.6 m. The detailed experimental setup has been found in a previous publication [22]. In order to quantify the postponed wetting failure, the measured onset velocity was normalized as vae /vae 0 where vae and vae 0 are the onset velocities in the curtain and the plunging tape flows [16], respectively. With the increasing flow rate, the onset velocity of air entrainment initially increases, then decreases and eventually reaches the constant value of unity. The onset velocity at the peak is found to be vae /vae 0 = 5, indicating that the liquid is successfully applied onto the moving surface at a speed five times faster than the plunging-tape flow. The simple boundary layer model [23] shows that the critical Reynolds number, at which the contact line is directly located beneath the impinging liquid, is given by Rec = 2.6, where the Reynolds number is readily defined as the ratio between the inertial and viscous drag forces, i.e., Re = (ρQ/μ)(u/vae ) where u is the impinging velocity. Substituting the onset velocity of vae = 0.56 m/s at the peak, the critical flow rate corresponding to Rec was found to be Qc = 2.1 cm2 /s, showing good agreement with the peak flow rate in Fig. 2. This fact suggests that
Fig. 2. Onset velocities of air entrainment in curtain coating (Blake et al. [22]). The hydrodynamic force acts on the contact line to delay the onset of air entrainment to higher speeds near the critical flow rate Qc .
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the hydrodynamic inertial force acts on the contact line locating beneath the impinging liquid, and hence, postpones the onset of air entrainment to higher surface speeds. The hydrodynamic assist of the dynamic wetting is consistent with other evidence that the contact angle, θ D , decreases with the increasing flow rate [24]. At lower flow rates Q Qc , the contact line is located far downstream due to the dominant viscous force. An increase in the flow rate beyond Qc tends to move the contact line further upstream, and eventually leads to the formation of an upstream liquid bank. In both cases, the liquid impact no longer acts on the contact line locating far from the impinging liquid. The resulting weaker inertial force promotes a negligible assist in the dynamic wetting, giving rise to the onset of air entrainment at the same speed as in the plunging tape flow, i.e., vae /vae 0 = 1. A similar hydrodynamic assistance has also been observed in the curtain coating with different fluid properties [12], the forward-roll coating [25], the reverse-roll coating [26] and the jet coating flow [27].
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In their experiments, the flow rate was one order of magnitude greater than the critical value, i.e., Re > Rec so that the hydrodynamic assist only slightly influences the onset of the dynamic wetting failure in Fig. 3. Fermin et al. experimentally demonstrated a similar electrostatic assistance in curtain [29] and slot [30,31] coating configurations. From a theoretical viewpoint, Blake et al. [28] combined the existing molecular kinetic theory with the electrostatic effects, and derived an expression that relates the dynamic contact angle to both the surface speed and the voltage. Although the theory fits the experimental data, they simply assumed that the hydrodynamics and electrostatics are completely decoupled. It is desirable to establish a rigorous physical model including the electrostatic effects coupled with the global flow fields that, in turn, alter the local molecular displacement on a small length scale. 4. Roughness-assisted wetting
Consider a moving solid on which a charge is applied to one side, and an equal and opposite charge is applied to the reverse side. The charge generates an electrostatic field between the solid and an approaching, conductive liquid. The resulting electrostatic force, which quadratically depends on the field strength, acts normal to the gas/liquid interface to reduce the dynamic contact angle, and eventually postpones the onset of air entrainment to higher speeds. Blake et al. [28] measured the onset velocity of air entrainment on a charged surface in curtain coatings. Fig. 3 shows the normalized onset velocity of air entrainment as a function of the applied voltage. In the test fluid, sodium chloride was added to a 60% (w/w) solution of glycerol and de-ionized water to increase the electrical conductivity. The onset velocity predicted from the power-law correlation for a plunging-tape flow [16] was found to be vae 0 = 0.17 m/s, which agrees with the measured value of 0.18 m/s in their curtain coating experiments in the absence of a voltage. The onset velocity of air entrainment monotonically increases with the increasing applied voltage, showing a noticeable electrostatic assist of dynamic wetting.
In the forced wetting on a rough, but still chemically homogenous solid surface, the dynamic contact line moves up and down the peaks and valleys, and travels a greater distance than it would over a smooth surface. Thus we can expect that the rough surface entrained air at a lower speed than the smooth surface. However, recent experiments have revealed alternative evidence that the onset velocity increases on a solid with a properly chosen surface roughness. Benkreira [32] performs plunging-tape experiments using a rough surface with the peak-to-valley height roughness of Rz = 3.3 m. Fig. 4 shows the onset velocities of the air entrainment as a function of fluid viscosity. On the smooth surface, the normalized onset velocities are close to unity and agree with those predicted from the power-law correlation [15]. The normalized onset velocity on the roughness surface becomes negative below a critical viscosity of μc = 0.4 Pa s, suggesting that the introduction of the surface roughness enhances the dynamic wetting failure. Above the critical viscosity, on the contrary, the reverse dependence was observed; the onset velocity of air entrainment becomes higher on a rough surface. This feature has been attributed to the contact line motion that does not wet the surface along the peak and the valley but rather skips from peak to peak [33]. The less contact between the liquid and the solid leads to a lower frictional force and thus the lower
Fig. 3. Onset velocities of air entrainment in curtain coating on a charged surface (Blake et al. [28]). An increase in applied voltage leads to higher onset velocities via electrostatic assist of wetting.
Fig. 4. Onset velocities of air entrainment in plunging tape flow on a rough surface (Benkreira [32]). The onset velocity becomes higher above a critical liquid viscosity for the peak-to-valley height roughness of Rz = 3.30 m, whereas it remains constant for the smooth surface of Rz = 0.60 m.
3. Electrostatic assist
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onset velocity of air entrainment at high viscosities. The earlier work by Buonopane et al. [34] has shown a similar roughnessassisted dynamic wetting using various liquid/solid systems, yet the substrate chemistry did not remain constant. In curtain coatings, Clarke [35] demonstrated the assisted dynamic wetting on a rough surface of Rz = 4.4 m. In his experiment, however, the momentum of the impinging liquid sheet can pressurize the contact line to some extent, because the Reynolds numbers of the curtain were not sufficiently high to ensure that the contact line locates far upstream from the impinging liquid. Thus the dynamic wetting was possibly hydrodynamically assisted in addition to the roughness effect. 5. Particle-assisted wetting The introduction of colloidal particles in liquids generally promotes shear-rate dependent rheological changes arising from complicated solid–liquid interactions [36]. Classical hydrodynamic theories show that the viscosity of a dilute suspension linearly increases with respect to the particle volume fraction. Thus the onset velocity of dynamic wetting failure is expected to monotonically decrease with the increasing particle concentration once the power law holds. In contrast, recent experiments revealed that the onset velocity first increases and then decreases with particle content at sufficiently low particle concentrations such that the suspension viscosity agrees with that of the pure fluid [37]. Fig. 5 shows the variation in the onset velocity of air entrainment with the particle number density, i.e., the number of particles per unit liquid volume. The normalized onset velocity of air entrainment increases, exhibits a peak at the critical particle number density of Σ max = 1 × 1011 m−3 and then decreases as the particle content increases. The maximum velocity at the peak was found to be vae /vae 0 = 1.6, indicating that the suspension was successfully applied onto the moving surface at a speed 60% faster that of the filler-free liquid. Such a particle-assisted dynamic wetting has been observed not only for suspensions of spherical polymeric particles but also for non-spherical inorganic fillers [38].
Fig. 5. Onset velocities of air entrainment in dilute suspension flow (Yamamura et al. [37]). The spherical poly(methyl-methacrylate) particles with diameters ranged from 8 to 50 m were dispersed in the coating liquids at weight fractions of 0.1 wt%. The normalized onset velocity of air entrainment increases, exhibits a peak, and then decreases as the particle content increases.
Yamamura et al. [37] postulated that the particle motion can split the intruding air-film into fine fingers, which subsequently shrink against the viscous drag force to assist the dynamic wetting. The key physical factors for describing the particleassistance are assumed to be the time scale required for the finger shrinkage, τ s , and the time interval of the particle contact, τ c . In the case of a low particle number density, the contact time interval becomes much longer than the time required for the finger shrinkage (τ c τ s ), and hence, the finger can completely shrink before the following particle comes into contact with the surface. The finger shrinkage in the direction opposite to the surface motion cancels the viscous drag flow, pulls the finger further upstream, and eventually causes the dynamic contact line to recede at even higher surface speeds. At high particle number densities, on the contrary, the finger does not shrink during a finite contact interval (τ c τ s ). Thus the finger formation no longer influences the onset of wetting failure, thus leading to a lower onset velocity of air entrainment. This “film-splitting” hypothesis is consistent with the measured increase and decrease in the onset velocity with the increasing particle number density. Assuming a sequential series of particle contacts and film splittings at low fluid viscosities, the one-dimensional lubrication theory yields the expression for the time-averaged onset velocity of air entrainment as [37]: vae − vae 0 1 = , 1/3 vae 0 (1/(Σ l)) + 6Σ 1/3 Cal
(1)
where Ca(= μA vae 0 /σ) and l denote the capillary number in air, and the length of the intruding air film before the particle contact. The lubrication theory has been verified by comparing the predicted and measured critical particle number densities at the peak. The critical particle number densities at the peak agree with each other in a quantitative sense for the case when the air film length l is readily determined by the direct flow visualization. Unfortunately, no rigorous physical model is currently available to predict the particle-assisted dynamic wetting speed if the air film length is unknown. For dense suspensions showing a particular shear-thinning behavior at low shear rates, Chu et al. [39] recently reported that, by adding porous or hard spherical inorganic particles in aqueous 20 wt% poly(vinyl alcohol) solutions, the onset of the “break line” defect was significantly delayed to higher surface speeds in a slot-die coating system. The particle-assisted wetting has been attributed to the fact that the polymer adsorption on the particle surface and the resultant inter-particle bridging can lead to an increase in the local surface tension, which yields a more stable gas–liquid interface that postpones the onset of wetting failure. Evidently, those physical mechanisms of wetting assistance completely differ from that in the previously mentioned dilute suspensions of simpler particle/fluid binary systems. Still to clarify is whether the film splitting or the inter-particle bridging mechanism dominates the dynamic wetting behavior for a given suspension system, and how the particle characteristics affect the wetting assistance.
M. Yamamura / Colloids and Surfaces A: Physicochem. Eng. Aspects 311 (2007) 55–60
Fig. 6. Onset velocities of air entrainment in dilute suspension jet coating flow (Yamamura et al. [38]). The introduction of particles further assists the dynamic wetting in addition to the co-existing hydrodynamic effect.
6. Combined wetting assistance Two or more wetting assistances simultaneously take place in properly designed coating systems. Understanding the feature of combined wetting assistances is of interest from both fundamental and practical viewpoints. The hydrodynamic assist of wetting has been attempted to be combined with the electrostatic [28], roughness [35] and particle assistances [38] in order to achieve a higher onset velocity of air entrainment. A typical example of the last combination is shown in Fig. 6. The Newtonian suspension jet was extruded from a round nozzle and vertically impinged onto a smooth, uncharged surface. The test fluids are silicone oils in which SiO2 particles with a mean diameter of 10 m were dispersed at the particle number densities up to 5 × 1012 m−3 . The onset velocity of air entrainment increases as the Reynolds number approaches the critical value of Rec . As described in Section 2, this is because the hydrodynamic force assists the dynamic wetting at Re = Rec . At the Reynolds number of Re = 3 near the critical value, the normalized onset velocity was (vae − vae 0 )/vae 0 = 3 in the filler-free liquid. In contrast, the suspension exhibits the normalized onset velocity of 7 at the same Reynolds number, showing that the introduction of particles further assists the dynamic wetting in addition to the co-existing hydrodynamic effect. There is a pronounced need for understanding the wetting behavior in other co-existing assistances; i.e., coating on a charged, rough surface, a particle suspension coating on a rough surface, and a suspension curtain coating on a charged surface. Unfortunately, few open publications deal with these combined assistances of practical interest. 7. Conclusions The recent progress in postponed visible air entrainment is reviewed, especially in high speed liquid coating flows. The hydrodynamic inertia forces, as well as the electrostatic forces on a charged surface, directly act on the contact line to delay the onset of dynamic wetting failure. However, the rigorous physical model to quantify the contribution of these forces is still
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incomplete. The well-defined surface roughness promotes faster coating speeds without air entrainment as the advancing liquid skips from peak to peak. It has not yet clarified whether the optimum height roughness exists in a give coating system, and how the spatial distribution of the roughness influences the dynamic wetting behavior. The particle-assisted wetting becomes pronounced in suspensions, yet it has been limited to low viscosity fluids at sufficiently low particle concentrations, such that the suspension viscosity agrees with that of the pure fluids. Understanding the wetting behavior in denser particle suspensions is challenging and the subject of future work. These wetting assistances can co-exist in order to achieve a higher onset velocity of air entrainment, yet it is not immediately clear how the different wetting assistances couple or de-couple in a given flow field. Continued research will result in a better understanding of combined assistances during the forced wetting. References [1] K. Miyamoto, On the mechanism of air entrainment, Ind. Coat. Res. 1 (1991) 71–88. [2] G.F. Teletzke, H.T. Davis, L.E. Scriven, Wetting hydrodynamics, Revue de Physique Appliquee 23 (1988) 989–1007. [3] T.D. Blake, Dynamic contact angles and wetting kinetics, Wettability, Marcel Dekker, 1993, pp. 251–308. [4] T.D. Blake, K.J. Ruschak, A maximum speed of wetting, Nature 282 (1979) 489–491. [5] J. Eggers, Air entrainment through free-surface cusps, Phys. Rev. Lett. 86 (2001) 4290–4293. [6] G.E. Innes, P.H. Gaskell, M.D. Savage, Cusps as a source of air entrainment in roll coating systems, Ind. Coat. Res. 4 (1998) 19–40. [7] S.F. Kistler, Hydrodynamics of wetting, Wettability, Marcel Dekker, 1993, pp. 311–429. [8] T.D. Blake, The physics of moving wetting lines, the physics of moving wetting lines, J. Colloid Interf. Sci. 299 (2006) 1–13. [9] M.J. de Ruijter, T.D. Blake, J. de Coninck, Dynamic wetting studies by molecular modelling simulations of droplet spreading, Langmuir 15 (1999) 7836–7847. [10] Y.D. Shikhmurzaev, Mathematical modeling of wetting hydrodynamics, Fluid Dynam. Res. 13 (1994) 45–64. [11] Y.D. Shikhmurzaev, Moving contact lines in liquid/liquid/solid systems, J. Fluid Mech. 334 (1997) 211–249. [12] T.D. Blake, Y.D. Shikhmurzaev, Dynamic wetting by liquids of different viscosity, J. Colloid Interf. Sci. 253 (2002) 196–202. [13] D. Jacqmin, Onset of wetting failure in liquid–liquid systems, J. Fluid Mech. 517 (2004) 209–228. [14] R. Burley, B.S. Kennedy, An experimental study of air entrainment at a solid/liquid/gas interface, Chem. Eng. Sci. 31 (1976) 901–911. [15] E.B. Gutoff, C.E. Kendrick, Dynamic contact angles, AIChE J. 28 (1982) 459–466. [16] R. Burley, R.P.S. Jolly, Entrainment of air into liquids by a high speed continuous solid surface, Chem. Eng. Sci. 39 (1984) 1357–1372. [17] O. Cohu, H. Benkreira, Entrainment of air by a solid surface plunging into a non-Newtonian liquid, AIChE J. 44 (1998) 2360–2368. [18] M.T. Ghannam, M.N. Esmail, Experimental study on wetting of fibers with non-Newtonian liquids, AIChE J. 43 (1997) 1579–1588. [19] P.G. Simpkins, V.J. Kuch, On air entrainment in coatings, J. Colloid Interf. Sci. 263 (2003) 562–571. [20] O. Cohu, H. Benkreira, Air entrainment in angled dip coating, Chem. Eng. Sci. 53 (1998) 533–540. [21] T.D. Blake, A. Clarke, K.J. Ruschak, Hydrodynamic assist of dynamic wetting, AIChE J. 40 (1994) 229–242. [22] T.D. Blake, R.A. Dobson, K.J. Ruschak, Wetting at high capillary numbers, J. Colloid Interf. Sci. 279 (2004) 198–205.
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