# Response of slot coating flow to gap disturbances: effects of fluid properties, operating conditions, and die configurations.

Abstract Slot coating is perhaps one of the most successful film
manufacturing methods and is used to produce a wide variety of products.
It is a pre-metered method and ideal for high-precision coating: the
steady-state thickness is precisely controlled by the flow rate fed to
the die and the web speed. However, the coating method is surrounded by
small-scale disturbances that can destroy film thickness uniformity.
Among various disturbances, a periodic change of the coating gap, the
so-called gap disturbance, is considered the most dangerous in previous
studies, e.g., Romero and Carvalho (Chem Eng Sci 63(8):2161-2173, 2008),
Tsuda et al. (AIChE J 56(9):2268-2279, 2010), and Lee and Nam (AIChE J,
2015). However, the effects of fluid properties, operating conditions,
and die configurations on the slot coating flow under gap disturbances
are not examined in detail. In this study, we revisit the computer-aided
analysis of transient slot coating flow designed to predict the
thickness variations under gap disturbance. Extensive computational
predictions reveal important parameters for suppressing film
nonuniformity that can be induced by gap disturbance.

Keywords Slot coating, Frequency response, Finite element method, Uniformity, Gap disturbance

Introduction

Slot coating is a common method for producing thin film, e.g., optical films, adhesive tapes, battery separators, and etc. It is classified as a pre-metered method (1): the final wet film thickness is set by the flow rate and the web speed and is independent of coating liquid properties. Therefore, the method is suitable for highly precise film thickness control. In the slot coating process, a coating liquid is fed from the feed slot and fills the gap between the slot die lip and the moving web. There are two gas-liquid boundaries, or menisci, upstream and downstream of the slot coating flow, respectively. This liquid region bounded by the die, the web, and the menisci is called the coating bead.

For successful uniform film production, the coating flow needs to be steady-state, two-dimensional, and stable. The range of operating conditions that allows the coating bead to generate such flows is called the coating window. Many previous studies have focused on computational (2) or experimental (3) identification of such windows in steady-state coating flows.

However, coating flows are always surrounded by small-scale disturbances due to rotating elements installed in many of the processing units, such as gears, shafts, and rolls. Those periodic disturbances can cause severe nonuniformity in the produced film along the moving web direction, which significantly degrades the final product quality. Therefore, it is important to analyze the sensitivity of the coating bead under various disturbances. Once the flow response to these beads is understood, the slot coating process can be designed and optimized to minimize nonuniformity in the film thickness. (4)

Among all disturbances, a periodic change of the coating gap, i.e., the distance between the die lip and the moving substrate, is typically identified as the most dangerous. This type of disturbance is usually called the gap disturbance and is caused by substrate thickness variations, mechanical vibration of the coating die or roll, and roll run outs. According to Romero and Carvalho, (5) the amplification factor, the ratio of the relative amplitude of the film thickness oscillation to a given periodic disturbance, for the gap oscillation is almost 3-5 times stronger than other types of disturbances. Several different die lip configurations, such as underbite and overbite, were tested, and their effects on the film thickness variations were evaluated. Tsuda et al. (6) used empirical modal analysis to predict the effect of disturbances including gap oscillation. Their numerical results show that sinuous mode meniscus motion caused by gap disturbance can be explained by capillary waves. Lee and Nam (7) suggested that film thickness variation can be explained by considering the meniscus, or free surface, fluctuation relative to the oscillation of the substrate due to gap disturbances. Interplay between the amplitude and the phase lag of the fluctuation are important in determining film thickness variations.

Here, we identify important parameters such as fluid properties, operating conditions, and die configurations for the film thickness variation caused by gap oscillation using a computational frequency response analysis of slot coating flow. In this study, we use the Galerkin/finite element method to solve the transient Navier-Stokes equation under periodic disturbance. The degree of film thickness variation is quantified using the amplification factor. Its magnitude with respect to different gap disturbance frequencies is examined and compared to determine the effects of the parameters.

Governing equations and boundary conditions

The slot coating flow is a small-scale laminar flow with free surfaces and is governed by an incompressible Navier-Stokes equation:

[nabla] x n = 0 (1)

[rho]([partial derivative]u/[partial derivative]t + [nabla]h) - [nabla] x T = 0, (2)

where u is the velocity vector, p is the density of the liquid, and T = -pI + 2[micro]S is the stress tensor for Newtonian fluid. Here p and S = (1/2)([nabla]u + [nabla][u.sup.T]) are pressure and rate of strain, respectively. The boundary conditions including prescribed velocity profiles at the feed slot inlet, the kinematic condition, and the Navier slip condition for describing dynamic contact line are described in Fig. 1.

In this study, we not only considered Newtonian fluid, but also shear-thinning fluid. To describe such a fluid, we use the Carreau-Yasuda model, (8) where nonNewtonian viscosity [eta] is a function of shear rate y:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (3)

where [[eta].sub.0] is a zero shear viscosity (Pa.s), [[eta].sub.[infinity]] is an infinite shear viscosity (Pa.s), [lambda] is a relaxation time (s), n is a power-law index, and a is a parameter introduced by Yasuda et al. (8) Note that the shear rate in this two-dimensional flow can be expressed in terms of the rate of strain:

[??] = [square root of -[II.sub.s], (4)

where IIS is the second invariant of the rate of strain. In this case, the shear stress tensor is T = -pI + 2[eta]([??])S.

Boundary conditions are described in Fig. 1. Details are explained elsewhere. (5,7) To compute flow response to gap disturbances, a sinusoidal function is imposed to the moving web:

[y.sub.w](0 = [y.sub.w0] + [y.sub.wm] sin([omega]f), (5)

where [y.sub.w0] is the steady web location, [y.sub.wm] is the amplitude of the oscillation, and [omega] = 2[pi]f is the angular frequency of the oscillation with frequency f. Therefore, the gap height between the web and the slot die is a function of time:

H(t) = [H.sub.0] - [H.sub.m] sin([omega]t), (6)

with steady gap height [H.sub.0] and amplitude [H.sub.m]. As a result of the gap disturbance, film thickness variation is expected to be a sinusoidal function:

h(t) = [h.sub.0] + [h.sub.m] sin([omega]t + [PHI]), (7)

where [H.sub.0] is the steady-state wet film thickness, [h.sub.m] is the amplitude of the thickness variation, and [PHI] is the phase lag. To quantify the magnitude of film thickness variation, the amplification factor is defined as

[[alpha].sub.H] [h.sub.m]/[h.sub.o]/[H.sub.m]/[H.sub.o]. (8)

This factor is the ratio of the relative magnitude of film thickness variation to the relative magnitude of gap oscillation.

Operating conditions in this study are listed in Table 1. These parameters can be grouped as four dimensionless numbers: [N.sub.Ca] [equivalent to] [micro]Vw/[sigma] (capillary number), [N.sub.Re] [equivalent to] [rho]VwH0/[mu] (Renolds number), [R.sub.gt] [equivalent to] H0/h0 (gap-to-thickness ratio), and [P.sup.*.sub.vac] [equivalent to] [P.sub.vac]H0/[sigma] (dimensionless vacuum pressure). Other parameters for the model are slip coefficient, [beta] = 0.1 [g.sup.-1][s.sup.-1], upstream static contact angle, [[theta].sub.u] = 60[degrees], and dynamic contact angle, [[theta].sub.dyn] = 120[degrees].

Solution methods

Because of the free boundaries, i.e., two menisci in the slot coating flow, the flow domain is unknown a priori. This unknown physical domain, x = (x, y), is mapped into the reference domain, [xi] = ([xi], [eta]), which is related by elliptic differential equations (9):

[nabla] * [D.sub.[xi]]([xi], [eta]) [nabla][xi] = 0, [nabla] x [D.sub.[eta]] ([xi], [eta])[nabla][eta] = 0, (9)

where [D.sub.[xi]] = and [D.sub.[eta]] are mesh diffusivities used to control the node spacing determined by the coordinate potentials [xi], and [eta], respectively. The Navier-Stokes equation is solved using the time derivatives at fixed iso-parametric coordinates:

p[v + (v - x) - [nabla] v] - [nabla] x T = 0, (10)

where v is defined by [partial derivative]v/[partial derivative]t = v - x [nabla]v, and x is the mesh velocity, the time derivative of the nodal position. Formulation of free boundary problems using the Galerkin/finite element method is explained elsewhere. (5,7)

Table 2 shows mesh conditions used in this study. The tolerance of the residual is set to [10.sup.-8], and the time step for transient computation is at least 40 per cycle of an input sine oscillation.

Results and discussion

In this study, operating conditions of the base case in terms of dimensionless numbers are [N.sub.Ca] = 0.2, [N.sub.Re] = 2 and [R.sub.gt] = 2. The base case parameters are summarized in Table 1. The imposed gap disturbances are set to 1% of the gap height, i.e., [H.sub.m] = 0.01H0. The vacuum pressure is adjusted to maintain the same upstream bead length, [L.sub.u] = 6[H.sub.0]. Therefore, although other dimensionless numbers remain equal, [P.sup.*.sub.vac] can be different. All predictions are based on Newtonian fluid except for the shear-thinning fluid. The base slot die used in this study is shown in Fig. 2. Note that in all amplification factor plots, the marked points are computed by the finite element methods. Curves that connect the points are interpolated by a cubic spline method. Therefore, the maximum peaks in the plots may be located between points.

Effects of fluid properties

Viscosity: Newtonian vs shear-thinning fluid

In this section, we examine the effect of fluid viscosity, including shear thinning, on film uniformity under gap disturbances. Model coating solution, 0.25 wt% xanthan gum/water solution, is chosen as a shear-thinning fluid in this study. The Carreau-Yasuda model parameters for this solution are [[eta].sub.0] = 13.2 Pa.s, [[eta].sub.[infinity]] = 0.00212 Pa.s, [lambda] = 60.7 s, n = 0.689 and a = 0.591. (10) The graph of viscosity of xanthan gum solution as a function of shear rate is shown in Fig. 3.

Numerous studies for shear-thinning liquid use apparent, or characteristic, viscosity at a representative shear rate estimated by the characteristics of a given flow system, e.g., Kamipli and Ryan," to determine dimensionless numbers, such as capillary and Reynolds numbers. In frequency response analysis of the slot coating flow, however, it is difficult to determine the representative shear rate. As shown in Fig. 4, shear rate is dramatically different between the regions under the downstream die lip, where the pressure gradient oscillations occur, and under the downstream free surface, where thickness variations are detected. Furthermore, the shear rate changes dramatically across the coating gap under the die lip. Especially near the die lip, where the no-slip condition dictates the fluid to stop, the shear rate locally increases significantly, when the wet film thickness is small enough to induce a strong adverse pressure gradient. In this study, however, the final wet thickness is half of the downstream gap height ([R.sub.gt] = 2) in the computations considered, which causes Couette flow in the coating gap, and the viscosity under the die lip become virtually constant at a steady state, as shown in Fig. 4.

In this study, we set the characteristic shear rate as [[??].sup.c] = H(t = 0)/[V.sub.w] for the sake of simplicity, and the corresponding capillary number becomes [N.sup.s.sub.Ca] = [eta]([[??].sup.c])[V.sub.w]/[sigma]. However, it should be emphasized that the viscosity [eta]([[??].sup.c]) is suitable for describing the effect of the pressure gradient oscillation under the die lip, not for the free surface oscillation. Therefore, we do not compare results from the shear-thinning and Newtonian fluids at the same dimensionless numbers. Instead, for the purpose of comparison, the viscosity values of Newtonian fluids are chosen to match that of xanthan gum solution at a certain shear rate: 10, 20, 50, and 100 [s.sup.-1] as shown in Fig. 3. Corresponding dimensionless numbers are summarized in Table 3.

The amplification factors [[alpha].sub.H] as a function of the frequency of the gap disturbances are shown in Fig. 5. In general, [[alpha].sub.H] increases as the frequency increases, reaches a maximum peak, and rapidly decreases. Although there are slight shifts, the maximum peak appears near 60-70 Hz for all viscosity values. The [[alpha].sub.H] of the shear-thinning solution shows the highest and the broadest curves compared with others.

Newtonian fluids with different viscosity values (different [N.sub.Ca]) are virtually the same up to f ~ 30Hz. In this low-frequency regime, the downstream meniscus oscillations are not affected by [N.sub.Ca]. This strongly suggests that the surface tension does not significantly affect the shape of the meniscus: its variation is mainly determined by the balance between pressure and viscous stress. Note that inertia is negligible in the force balance, because [N.sub.Re] is less than 0.3 for all cases. As the viscosity increases, the pressure gradient oscillations under the die lip increase proportionally to the viscosity. Although this pressure forces increases, the meniscus become less mobile due to the increased viscosity. Therefore, the meniscus oscillations remain similar at different viscosities of Newtonian fluids.

Above f > 30 Hz, the [[alpha].sub.H] curves for Newtonian fluids begin to differ: higher viscosity leads to higher [[alpha].sub.H]. As the viscosity increases, i.e., [N.sub.Ca] increases, the maximum peak increases. This trend is also observed for decreasing surface tension and will be discussed later.

The peak [[alpha].sub.H] of the shear-thinning fluid is close to that of the highest Newtonian viscosity. In the low-frequency region, however, the [[alpha].sub.H] value is higher than that of Newtonian fluids. This high [[alpha].sub.H] is responsible for the broader [[alpha].sub.H] curve of the shear-thinning fluid.

As pointed out by previous works, (5,7) the pressure gradient oscillation under the die lip is responsible for the downstream-free surface oscillation, which in turn causes the thickness variation and hence change in [[alpha].sub.H]. However, the viscosity related to the pressure gradient oscillation and the viscosity related the free surface oscillation are different in the shear-thinning fluid.

In the downstream coating gap, the fluid is subjected under relatively high shear rate, and the viscosity drops to 8.79 mPa.s, which is significantly smaller than other Newtonian viscosities considered here. However, the viscosity rapidly increases along the flow direction after the downstream die lip exit, where the downstream meniscus starts to develop. Due to the vanishing shear stress at the interface, [??] rapidly decreases along the meniscus, and the viscosity increases significantly to almost [[eta].sub.0], as shown in Fig. 4c.

Near the die lip exit, the viscosity is still low enough to generate large free surface motions. However, the fluid loses its mobility as it flows downstream; thereby, the large amplitude of the free surface oscillation is not rapidly decayed. Therefore, the [[alpha].sub.H] of the shear-thinning fluid is always higher than that of a Newtonian fluid. This phenomenon is noticeable in the frequency range 1-60 Hz. From these results, one can conclude that shear-thinning fluid is more susceptible to gap oscillation than is Newtonian fluid.

Surface tension

To analyze the effect of surface tension, three different surface tension values are chosen. Corresponding dimensionless numbers are summarized in Table 4. Note that, in this surface tension test, the viscosity and the web speed are chosen as the base case (Table 1). Therefore, the frequency range of [[alpha].sub.H] peak is different from that in the previous section.

Figure 6 shows the effect of surface tension on the film thickness variation. As discussed in the previous section, the surface tension does not affect the free surface motion in the low-frequency regime. However, near the [[alpha].sub.H] peak, high surface tension helps to decrease the thickness variation.

Under the same viscosity, the amount of pressure gradient oscillation under the die lip will remain the same, as will the motion of the downstream meniscus near the die lip exit. Therefore, higher frequency gap disturbance will generate a shorter wavelength surface wave. As the surface tension increases, the leveling effect due to capillary pressure increases as well, which helps to decrease the magnitude of the surface wave. However, this effect is apparently not significant, as shown in Fig. 6: the peak decreases slightly.

Effects of operating conditions

Web speed

In slot coating, increasing web speed leads to increasing viscous stress, similar to viscosity. However, the web speed can also control the residence time of fluid particles inside the coating gap. This means that the number of gap oscillations encountered by fluid particles decreases as the web speed increases. Consequently, the effect of web speed is substantially stronger than that of viscosity or surface tension. Dimensionless numbers for testing the effect are summarized in case 1 of Table 5.

Figure 7 shows [[alpha].sub.H] as a function of the frequency for different web speeds and hence [N.sub.Ca]? As [N.sub.Ca] increases by adjusting web speed, the maximum [[alpha].sub.H] value increases. This change in maximum value is consistent with results of viscosity and surface tension. However, the peak frequency for the maximum [[alpha].sub.H] shifts toward higher frequency. Furthermore, in the low-frequency region, lower web speed produces higher [[alpha].sub.H]. This is not in agreement with the results shown in previous sections.

These two differences from viscosity and surface tension come from the fact that the web speed changes the number of gap oscillations experienced by fluid particles inside the coating gap. For the low-frequency region around O.f-50 Hz, fluid particles, which travel through the coating gap to the downstream meniscus, experience more gap oscillations when the web moves slowly, leading to higher [[alpha].sub.H]. Meanwhile, fluid particles must experience sufficient gap oscillations inside the coating gap in order to achieve the maximum [[alpha].sub.H]. Therefore, the gap oscillation needs to be faster to reach the maximum at higher web speed, and the peak frequency shifts toward higher frequency.

Flow rate (gap-to-thickness ratio)

When the web speed is fixed, the steady-state film thickness is solely controlled by the flow rate fed into the feed slot. Therefore, for a fixed gap height, this flow rate is inversely proportional to gap-to-thickness ratio [R.sub.gt]. In this study, we adjust [R.sub.gt] to analyze the effect of flow rate on the thickness variations under gap disturbance. Dimensionless numbers used for computations are summarized in case 2 of Table 5.

Amplification factor [[alpha].sub.H] increases as [R.sub.gt] increases, as shown in Fig. 8. When the flow rate is small, the downstream meniscus bends further, and the free surface is located closer to the oscillating web. Therefore, it is easier for the web to perturb the free surface, which leads to high [[alpha].sub.H] It should be mentioned that although [[alpha].sub.H], i.e., the relative magnitude of thickness variation, increases, the absolute magnitude of the thickness variation [h.sub.m] is not significantly different from different [R.sub.gt] except [R.sub.gt] = 5.00, especially near the peak.

The shape of [[alpha].sub.H] curve of noticeably different when [R.sub.gt] [greater than or equal to] 3.33. In low-frequency range (1-20 Hz), [[alpha].sub.H] increases rapidly compared with low [R.sub.gt] curves. Similar but opposite trends appear in high-frequency range (~500 Hz). As a result, bulges or bumps in [[alpha].sub.H] curves appear in both low-frequency and high-frequency ranges. We strongly suspect that this deviation from single-peak [[alpha].sub.H] curve is related to the downstream die lip vortex caused by strong adverse pressure gradient from small flow rate. Note that one can easily find by examining the rectilinear flow under the downstream die lip that [R.sub.gt] = 3 (or h0/H0 = 1/3) is associated with the onset of flow reversal in the downstream coating gap and hence the onset of the die lip vortex. The analysis of the downstream die lip vortex in slot coating flow is discussed elsewhere. (12) A downstream die lip vortex is generated near [R.sub.gt] = 3.33 and the vortex becomes larger when [R.sub.gt] = 5.00, at the same time the size of bulge in [[alpha].sub.H] curve increases more. Although the reason is not clear, one can postulate that the rotating frequency of the vortex and/or the flow topology near the die lip exit of the downstream meniscus (different flow directions above and below the separating streamline) affects the downstream meniscus oscillations.

Vacuum pressure

To coat a thin layer, a large pressure gradient must be created across the coating bead. One way to produce such a bead is by applying a vacuum pressure. (13) The upstream meniscus is pulled by the vacuum pressure, and consequently, the upstream bead length is adjusted. As pointed out in previous studies, (5,7) a large upstream bead helps to minimize the impact of flow rate disturbance on thickness variation. Case 3 of Table 5 summarizes dimensionless numbers used to determine the effect of vacuum pressure, which is essentially the same as examining the effect of upstream bead length, on [[alpha].sub.H] under gap disturbance.

Figure 9 shows [[alpha].sub.H] as a function of frequency for the three different vacuum pressure values. Unlike flow rate disturbances, the larger upstream bead size exacerbates the thickness variations: the [[alpha].sub.H] peak increases as [P.sup.*.sub.ac] increases. As the bead size grows, the area between the oscillating web and the coating liquid, the so called footprint, increases. The larger footprint leads to greater absorption of oscillations into the coating bead, which is responsible for higher [[alpha].sub.H]. Furthermore, the increased vacuum pressure causes stronger upstream pressure gradient, and the larger upstream meniscus oscillation.

Effect of die configurations

Previous studies have focused on modifying the upstream die lip geometry, e.g., underbite and overbite configurations. (5,7) In the present study, we focus on the effect of downstream die lip geometry, such as [L.sub.d] and [H.sub.d], as shown in the right upper schematic diagram of Fig. 10, while the upstream die geometry is fixed. For the effect of [H.sub.d], we test two different operating conditions, i.e., constant (or Q) or constant [R.sub.gt]. [R.sub.gt] increases as [H.sub.d] increases when [h.sub.0] is fixed, whereas [h.sub.0] needs to be increased to fix [R.sub.gt]. Dimensionless numbers used in this test are summarized in Table 6. Note that [P.sup.*.sub.vac] is adjusted to maintain the same upstream bead length for all cases.

Figure 10a shows the effect of [L.sub.d] on [[alpha].sub.H] As discussed in the previous section about the vacuum pressure effect, [L.sub.d] also affects the footprint for contacting the oscillating web. Therefore, the larger [L.sub.d] causes the higher [[alpha].sub.H] peak. However, change of the downstream bead size does not affect [[alpha].sub.H] in the low-frequency range (0.1-50 Hz). In this range, the pressure gradient under the downstream die lip has sufficient time to follow the periodic oscillation of the gap height. Consequently, the effect of the downstream die lip footprint size is negligible. As the gap oscillates faster, however, the smaller footprint size cannot produce enough pressure gradient oscillation. This leads to small [[alpha].sub.H] peaks for small [L.sub.d] cases. Therefore, comparing with the results from the vacuum pressure changes, one can conclude that the lower frequency region is mostly controlled by the upstream bead size rather than the downstream bead size. Note that, as shown in Table 6, [P.sup.*.sub.vac] is not changed by [L.sub.d]. the upstream bead size is not affected by [L.sub.d].

Figure 10b and c show the effect of [H.sub.d] with constant [h.sub.0] and [R.sub.gt], respectively. The results suggest that [R.sub.gt] does not significantly affect the thickness variation. Therefore, one can conclude that the magnitude of the steady-state pressure gradient inside the downstream coating bead does not increase or decrease the amplitude of the pressure gradient oscillations caused by the gap disturbances. According to our computational predictions, small downstream gap height can help to suppress the downstream meniscus oscillation and decrease [[alpha].sub.H], although the reason for that is not clear.

Final remarks

This work presents computational predictions on the film thickness variation in slot coating due to gap disturbances. The effects of fluid properties, operating conditions, and die configurations are examined by comparing the amplification factor [[alpha].sub.H], the relative amplitude of the film thickness oscillation, to the imposed coating gap disturbance.

According to our predictions, the fluid properties of viscosity and surface tension do not significantly affect the periodic response of the film thickness. Even the [[alpha].sub.H] curve of an extremely strong shear-thinning fluid, xanthan gum solution, does not differ considerably from Newtonian fluid results. Considering the effect of surface tension, Newtonian fluid of low viscosity is suitable for suppressing gap disturbance to benefit from the leveling effect. However, the web speed is the most influential parameter for the gap disturbances. It can change both [[alpha].sub.H] peak height and location in the amplification factor plot. In addition, the existence of a microvortex under the downstream die lip significantly changes the shape of the [[alpha].sub.H] curve, according to the results of a gap-tothickness ratio [R.sub.gt], more than 3. The larger upstream coating bead size caused by stronger vacuum leads to larger film thickness variation. This trend is opposite that of flow rate disturbance, which can be efficiently damped by large upstream coating bead. For the die lip configuration, a small downstream gap height is desirable for suppressing the thickness variations from the gap disturbances.

In steady-state flows, Reynolds and capillary numbers can characterize the coating flow. However, in periodic flows, additional dimensionless numbers that represent the characteristic time for the momentum transfer and the characteristic time for the traveling wave are needed. For example, both viscosity and web speed can increase [N.sub.Ca] in the same manner but produce very different [[alpha].sub.H] curves. This strongly suggests that another dimensionless number needs to be introduced to characterize this periodic flow response. We are currently investigating this issue.

DOI 10.1007/s11998-015-9684-1

S. Lee, J. Nam ([mail])

Department of Chemical Engineering, Sungkyunkwan

University, 2006, Seobu-ro, Jangan-gu, Suwon, Gyeonggi-do

440-746, Korea

e-mail: nam.jaewook@gmail.comjaewooknam@skku.edu

Acknowledgments This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (NRF-2013R1A1A1004986) and by the Global Ph.D Fellowship Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2014H1A2A1019789). This work was also supported by LG Chem (No. AP1400056).

References

(1.) Higgins, BG, Scriven, LE, "Capillary Pressure and Viscous Pressure Drop Set Bounds on Coating Bead Operability." Chem. Eng. Sci., 35 (3):673-682 (1980)

(2.) Nam, J, Carvalho, MS, "Flow in Tensioned-Web-Over-Slot Die Coating: Effect of Die Lip Design." Chem. Eng. Sci., 65 (13):3957-3971 (2010)

(3.) Sartor, L, "Slot Coating: Fluid Mechanics and Die Design". PhD Thesis, University of Minnesota, 1990

(4.) Perez, EB, and Carvalho, MS, "Optimization of Slot-Coating Processes: Minimizing the Amplitude of Film-Thickness Oscillation." J. Eng. Math., 71 (1):97-108 (2011)

(5.) Romero, OJ, Carvalho, MS, "Response of Slot Coating Flows to Periodic Disturbances." Chem. Eng. Sci., 63 (8):2161-2173 (2008)

(6.) Tsuda, T, de Santos, JM, Scriven, LE, "Frequency Response Analysis of Slot Coating." AlChE J., 56 (9):2268-2279, (2010)

(7.) Lee, S, Nam, J, "Analysis of Slot Coating Flow Under Tilted Die." AlChE J. (2015). doi:10.1002/aic,14752

(8.) Yasuda, H, "The Two-Center Expansion of an Exponential Function: The Reduction of Yasudas Expression to that of Briels." J. Chem. Phys., 74 (11):6531-6532, (1981)

(9.) De Santos, JM, "Two-Phase Cocurrent Downflow Through Constricted Passages". PhD Thesis, University of Minnesota, 1991

(10.) Escudier, MP, Gouldson, IW, Pereira, AS, Pinho FT, Poole, RJ. "On the Reproducibility of the Rheology of Shear-Thinning Liquids." J. NonNewtonian Fluid Mech., 97 (2):99-124, (2001)

(11.) Kamigli, F, Ryan, M E, "Perturbation Method in Gas-Assisted Power-Law Fluid Displacement in a Circular Tube and a Rectangular Channel. " Chem. Eng. J., 75 (3):167-176, (1999)

(12.) Nam, J, Scriven, LE, Cavalho, MS, "Tracking Birth of Vortex in Flows." J. Comput. Phys., 228 (12):4549-4567 (2009)

(13.) Beguin, AE, "Method of Coating Strip Material", US Patent 2,681,294, 1954

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Please note: Some tables or figures were omitted from this article.

Keywords Slot coating, Frequency response, Finite element method, Uniformity, Gap disturbance

Introduction

Slot coating is a common method for producing thin film, e.g., optical films, adhesive tapes, battery separators, and etc. It is classified as a pre-metered method (1): the final wet film thickness is set by the flow rate and the web speed and is independent of coating liquid properties. Therefore, the method is suitable for highly precise film thickness control. In the slot coating process, a coating liquid is fed from the feed slot and fills the gap between the slot die lip and the moving web. There are two gas-liquid boundaries, or menisci, upstream and downstream of the slot coating flow, respectively. This liquid region bounded by the die, the web, and the menisci is called the coating bead.

For successful uniform film production, the coating flow needs to be steady-state, two-dimensional, and stable. The range of operating conditions that allows the coating bead to generate such flows is called the coating window. Many previous studies have focused on computational (2) or experimental (3) identification of such windows in steady-state coating flows.

However, coating flows are always surrounded by small-scale disturbances due to rotating elements installed in many of the processing units, such as gears, shafts, and rolls. Those periodic disturbances can cause severe nonuniformity in the produced film along the moving web direction, which significantly degrades the final product quality. Therefore, it is important to analyze the sensitivity of the coating bead under various disturbances. Once the flow response to these beads is understood, the slot coating process can be designed and optimized to minimize nonuniformity in the film thickness. (4)

Among all disturbances, a periodic change of the coating gap, i.e., the distance between the die lip and the moving substrate, is typically identified as the most dangerous. This type of disturbance is usually called the gap disturbance and is caused by substrate thickness variations, mechanical vibration of the coating die or roll, and roll run outs. According to Romero and Carvalho, (5) the amplification factor, the ratio of the relative amplitude of the film thickness oscillation to a given periodic disturbance, for the gap oscillation is almost 3-5 times stronger than other types of disturbances. Several different die lip configurations, such as underbite and overbite, were tested, and their effects on the film thickness variations were evaluated. Tsuda et al. (6) used empirical modal analysis to predict the effect of disturbances including gap oscillation. Their numerical results show that sinuous mode meniscus motion caused by gap disturbance can be explained by capillary waves. Lee and Nam (7) suggested that film thickness variation can be explained by considering the meniscus, or free surface, fluctuation relative to the oscillation of the substrate due to gap disturbances. Interplay between the amplitude and the phase lag of the fluctuation are important in determining film thickness variations.

Here, we identify important parameters such as fluid properties, operating conditions, and die configurations for the film thickness variation caused by gap oscillation using a computational frequency response analysis of slot coating flow. In this study, we use the Galerkin/finite element method to solve the transient Navier-Stokes equation under periodic disturbance. The degree of film thickness variation is quantified using the amplification factor. Its magnitude with respect to different gap disturbance frequencies is examined and compared to determine the effects of the parameters.

Governing equations and boundary conditions

The slot coating flow is a small-scale laminar flow with free surfaces and is governed by an incompressible Navier-Stokes equation:

[nabla] x n = 0 (1)

[rho]([partial derivative]u/[partial derivative]t + [nabla]h) - [nabla] x T = 0, (2)

where u is the velocity vector, p is the density of the liquid, and T = -pI + 2[micro]S is the stress tensor for Newtonian fluid. Here p and S = (1/2)([nabla]u + [nabla][u.sup.T]) are pressure and rate of strain, respectively. The boundary conditions including prescribed velocity profiles at the feed slot inlet, the kinematic condition, and the Navier slip condition for describing dynamic contact line are described in Fig. 1.

In this study, we not only considered Newtonian fluid, but also shear-thinning fluid. To describe such a fluid, we use the Carreau-Yasuda model, (8) where nonNewtonian viscosity [eta] is a function of shear rate y:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (3)

where [[eta].sub.0] is a zero shear viscosity (Pa.s), [[eta].sub.[infinity]] is an infinite shear viscosity (Pa.s), [lambda] is a relaxation time (s), n is a power-law index, and a is a parameter introduced by Yasuda et al. (8) Note that the shear rate in this two-dimensional flow can be expressed in terms of the rate of strain:

[??] = [square root of -[II.sub.s], (4)

where IIS is the second invariant of the rate of strain. In this case, the shear stress tensor is T = -pI + 2[eta]([??])S.

Boundary conditions are described in Fig. 1. Details are explained elsewhere. (5,7) To compute flow response to gap disturbances, a sinusoidal function is imposed to the moving web:

[y.sub.w](0 = [y.sub.w0] + [y.sub.wm] sin([omega]f), (5)

where [y.sub.w0] is the steady web location, [y.sub.wm] is the amplitude of the oscillation, and [omega] = 2[pi]f is the angular frequency of the oscillation with frequency f. Therefore, the gap height between the web and the slot die is a function of time:

H(t) = [H.sub.0] - [H.sub.m] sin([omega]t), (6)

with steady gap height [H.sub.0] and amplitude [H.sub.m]. As a result of the gap disturbance, film thickness variation is expected to be a sinusoidal function:

h(t) = [h.sub.0] + [h.sub.m] sin([omega]t + [PHI]), (7)

where [H.sub.0] is the steady-state wet film thickness, [h.sub.m] is the amplitude of the thickness variation, and [PHI] is the phase lag. To quantify the magnitude of film thickness variation, the amplification factor is defined as

[[alpha].sub.H] [h.sub.m]/[h.sub.o]/[H.sub.m]/[H.sub.o]. (8)

This factor is the ratio of the relative magnitude of film thickness variation to the relative magnitude of gap oscillation.

Operating conditions in this study are listed in Table 1. These parameters can be grouped as four dimensionless numbers: [N.sub.Ca] [equivalent to] [micro]Vw/[sigma] (capillary number), [N.sub.Re] [equivalent to] [rho]VwH0/[mu] (Renolds number), [R.sub.gt] [equivalent to] H0/h0 (gap-to-thickness ratio), and [P.sup.*.sub.vac] [equivalent to] [P.sub.vac]H0/[sigma] (dimensionless vacuum pressure). Other parameters for the model are slip coefficient, [beta] = 0.1 [g.sup.-1][s.sup.-1], upstream static contact angle, [[theta].sub.u] = 60[degrees], and dynamic contact angle, [[theta].sub.dyn] = 120[degrees].

Solution methods

Because of the free boundaries, i.e., two menisci in the slot coating flow, the flow domain is unknown a priori. This unknown physical domain, x = (x, y), is mapped into the reference domain, [xi] = ([xi], [eta]), which is related by elliptic differential equations (9):

[nabla] * [D.sub.[xi]]([xi], [eta]) [nabla][xi] = 0, [nabla] x [D.sub.[eta]] ([xi], [eta])[nabla][eta] = 0, (9)

where [D.sub.[xi]] = and [D.sub.[eta]] are mesh diffusivities used to control the node spacing determined by the coordinate potentials [xi], and [eta], respectively. The Navier-Stokes equation is solved using the time derivatives at fixed iso-parametric coordinates:

p[v + (v - x) - [nabla] v] - [nabla] x T = 0, (10)

where v is defined by [partial derivative]v/[partial derivative]t = v - x [nabla]v, and x is the mesh velocity, the time derivative of the nodal position. Formulation of free boundary problems using the Galerkin/finite element method is explained elsewhere. (5,7)

Table 2 shows mesh conditions used in this study. The tolerance of the residual is set to [10.sup.-8], and the time step for transient computation is at least 40 per cycle of an input sine oscillation.

Results and discussion

In this study, operating conditions of the base case in terms of dimensionless numbers are [N.sub.Ca] = 0.2, [N.sub.Re] = 2 and [R.sub.gt] = 2. The base case parameters are summarized in Table 1. The imposed gap disturbances are set to 1% of the gap height, i.e., [H.sub.m] = 0.01H0. The vacuum pressure is adjusted to maintain the same upstream bead length, [L.sub.u] = 6[H.sub.0]. Therefore, although other dimensionless numbers remain equal, [P.sup.*.sub.vac] can be different. All predictions are based on Newtonian fluid except for the shear-thinning fluid. The base slot die used in this study is shown in Fig. 2. Note that in all amplification factor plots, the marked points are computed by the finite element methods. Curves that connect the points are interpolated by a cubic spline method. Therefore, the maximum peaks in the plots may be located between points.

Effects of fluid properties

Viscosity: Newtonian vs shear-thinning fluid

In this section, we examine the effect of fluid viscosity, including shear thinning, on film uniformity under gap disturbances. Model coating solution, 0.25 wt% xanthan gum/water solution, is chosen as a shear-thinning fluid in this study. The Carreau-Yasuda model parameters for this solution are [[eta].sub.0] = 13.2 Pa.s, [[eta].sub.[infinity]] = 0.00212 Pa.s, [lambda] = 60.7 s, n = 0.689 and a = 0.591. (10) The graph of viscosity of xanthan gum solution as a function of shear rate is shown in Fig. 3.

Numerous studies for shear-thinning liquid use apparent, or characteristic, viscosity at a representative shear rate estimated by the characteristics of a given flow system, e.g., Kamipli and Ryan," to determine dimensionless numbers, such as capillary and Reynolds numbers. In frequency response analysis of the slot coating flow, however, it is difficult to determine the representative shear rate. As shown in Fig. 4, shear rate is dramatically different between the regions under the downstream die lip, where the pressure gradient oscillations occur, and under the downstream free surface, where thickness variations are detected. Furthermore, the shear rate changes dramatically across the coating gap under the die lip. Especially near the die lip, where the no-slip condition dictates the fluid to stop, the shear rate locally increases significantly, when the wet film thickness is small enough to induce a strong adverse pressure gradient. In this study, however, the final wet thickness is half of the downstream gap height ([R.sub.gt] = 2) in the computations considered, which causes Couette flow in the coating gap, and the viscosity under the die lip become virtually constant at a steady state, as shown in Fig. 4.

In this study, we set the characteristic shear rate as [[??].sup.c] = H(t = 0)/[V.sub.w] for the sake of simplicity, and the corresponding capillary number becomes [N.sup.s.sub.Ca] = [eta]([[??].sup.c])[V.sub.w]/[sigma]. However, it should be emphasized that the viscosity [eta]([[??].sup.c]) is suitable for describing the effect of the pressure gradient oscillation under the die lip, not for the free surface oscillation. Therefore, we do not compare results from the shear-thinning and Newtonian fluids at the same dimensionless numbers. Instead, for the purpose of comparison, the viscosity values of Newtonian fluids are chosen to match that of xanthan gum solution at a certain shear rate: 10, 20, 50, and 100 [s.sup.-1] as shown in Fig. 3. Corresponding dimensionless numbers are summarized in Table 3.

The amplification factors [[alpha].sub.H] as a function of the frequency of the gap disturbances are shown in Fig. 5. In general, [[alpha].sub.H] increases as the frequency increases, reaches a maximum peak, and rapidly decreases. Although there are slight shifts, the maximum peak appears near 60-70 Hz for all viscosity values. The [[alpha].sub.H] of the shear-thinning solution shows the highest and the broadest curves compared with others.

Newtonian fluids with different viscosity values (different [N.sub.Ca]) are virtually the same up to f ~ 30Hz. In this low-frequency regime, the downstream meniscus oscillations are not affected by [N.sub.Ca]. This strongly suggests that the surface tension does not significantly affect the shape of the meniscus: its variation is mainly determined by the balance between pressure and viscous stress. Note that inertia is negligible in the force balance, because [N.sub.Re] is less than 0.3 for all cases. As the viscosity increases, the pressure gradient oscillations under the die lip increase proportionally to the viscosity. Although this pressure forces increases, the meniscus become less mobile due to the increased viscosity. Therefore, the meniscus oscillations remain similar at different viscosities of Newtonian fluids.

Above f > 30 Hz, the [[alpha].sub.H] curves for Newtonian fluids begin to differ: higher viscosity leads to higher [[alpha].sub.H]. As the viscosity increases, i.e., [N.sub.Ca] increases, the maximum peak increases. This trend is also observed for decreasing surface tension and will be discussed later.

The peak [[alpha].sub.H] of the shear-thinning fluid is close to that of the highest Newtonian viscosity. In the low-frequency region, however, the [[alpha].sub.H] value is higher than that of Newtonian fluids. This high [[alpha].sub.H] is responsible for the broader [[alpha].sub.H] curve of the shear-thinning fluid.

As pointed out by previous works, (5,7) the pressure gradient oscillation under the die lip is responsible for the downstream-free surface oscillation, which in turn causes the thickness variation and hence change in [[alpha].sub.H]. However, the viscosity related to the pressure gradient oscillation and the viscosity related the free surface oscillation are different in the shear-thinning fluid.

In the downstream coating gap, the fluid is subjected under relatively high shear rate, and the viscosity drops to 8.79 mPa.s, which is significantly smaller than other Newtonian viscosities considered here. However, the viscosity rapidly increases along the flow direction after the downstream die lip exit, where the downstream meniscus starts to develop. Due to the vanishing shear stress at the interface, [??] rapidly decreases along the meniscus, and the viscosity increases significantly to almost [[eta].sub.0], as shown in Fig. 4c.

Near the die lip exit, the viscosity is still low enough to generate large free surface motions. However, the fluid loses its mobility as it flows downstream; thereby, the large amplitude of the free surface oscillation is not rapidly decayed. Therefore, the [[alpha].sub.H] of the shear-thinning fluid is always higher than that of a Newtonian fluid. This phenomenon is noticeable in the frequency range 1-60 Hz. From these results, one can conclude that shear-thinning fluid is more susceptible to gap oscillation than is Newtonian fluid.

Surface tension

To analyze the effect of surface tension, three different surface tension values are chosen. Corresponding dimensionless numbers are summarized in Table 4. Note that, in this surface tension test, the viscosity and the web speed are chosen as the base case (Table 1). Therefore, the frequency range of [[alpha].sub.H] peak is different from that in the previous section.

Figure 6 shows the effect of surface tension on the film thickness variation. As discussed in the previous section, the surface tension does not affect the free surface motion in the low-frequency regime. However, near the [[alpha].sub.H] peak, high surface tension helps to decrease the thickness variation.

Under the same viscosity, the amount of pressure gradient oscillation under the die lip will remain the same, as will the motion of the downstream meniscus near the die lip exit. Therefore, higher frequency gap disturbance will generate a shorter wavelength surface wave. As the surface tension increases, the leveling effect due to capillary pressure increases as well, which helps to decrease the magnitude of the surface wave. However, this effect is apparently not significant, as shown in Fig. 6: the peak decreases slightly.

Effects of operating conditions

Web speed

In slot coating, increasing web speed leads to increasing viscous stress, similar to viscosity. However, the web speed can also control the residence time of fluid particles inside the coating gap. This means that the number of gap oscillations encountered by fluid particles decreases as the web speed increases. Consequently, the effect of web speed is substantially stronger than that of viscosity or surface tension. Dimensionless numbers for testing the effect are summarized in case 1 of Table 5.

Figure 7 shows [[alpha].sub.H] as a function of the frequency for different web speeds and hence [N.sub.Ca]? As [N.sub.Ca] increases by adjusting web speed, the maximum [[alpha].sub.H] value increases. This change in maximum value is consistent with results of viscosity and surface tension. However, the peak frequency for the maximum [[alpha].sub.H] shifts toward higher frequency. Furthermore, in the low-frequency region, lower web speed produces higher [[alpha].sub.H]. This is not in agreement with the results shown in previous sections.

These two differences from viscosity and surface tension come from the fact that the web speed changes the number of gap oscillations experienced by fluid particles inside the coating gap. For the low-frequency region around O.f-50 Hz, fluid particles, which travel through the coating gap to the downstream meniscus, experience more gap oscillations when the web moves slowly, leading to higher [[alpha].sub.H]. Meanwhile, fluid particles must experience sufficient gap oscillations inside the coating gap in order to achieve the maximum [[alpha].sub.H]. Therefore, the gap oscillation needs to be faster to reach the maximum at higher web speed, and the peak frequency shifts toward higher frequency.

Flow rate (gap-to-thickness ratio)

When the web speed is fixed, the steady-state film thickness is solely controlled by the flow rate fed into the feed slot. Therefore, for a fixed gap height, this flow rate is inversely proportional to gap-to-thickness ratio [R.sub.gt]. In this study, we adjust [R.sub.gt] to analyze the effect of flow rate on the thickness variations under gap disturbance. Dimensionless numbers used for computations are summarized in case 2 of Table 5.

Amplification factor [[alpha].sub.H] increases as [R.sub.gt] increases, as shown in Fig. 8. When the flow rate is small, the downstream meniscus bends further, and the free surface is located closer to the oscillating web. Therefore, it is easier for the web to perturb the free surface, which leads to high [[alpha].sub.H] It should be mentioned that although [[alpha].sub.H], i.e., the relative magnitude of thickness variation, increases, the absolute magnitude of the thickness variation [h.sub.m] is not significantly different from different [R.sub.gt] except [R.sub.gt] = 5.00, especially near the peak.

The shape of [[alpha].sub.H] curve of noticeably different when [R.sub.gt] [greater than or equal to] 3.33. In low-frequency range (1-20 Hz), [[alpha].sub.H] increases rapidly compared with low [R.sub.gt] curves. Similar but opposite trends appear in high-frequency range (~500 Hz). As a result, bulges or bumps in [[alpha].sub.H] curves appear in both low-frequency and high-frequency ranges. We strongly suspect that this deviation from single-peak [[alpha].sub.H] curve is related to the downstream die lip vortex caused by strong adverse pressure gradient from small flow rate. Note that one can easily find by examining the rectilinear flow under the downstream die lip that [R.sub.gt] = 3 (or h0/H0 = 1/3) is associated with the onset of flow reversal in the downstream coating gap and hence the onset of the die lip vortex. The analysis of the downstream die lip vortex in slot coating flow is discussed elsewhere. (12) A downstream die lip vortex is generated near [R.sub.gt] = 3.33 and the vortex becomes larger when [R.sub.gt] = 5.00, at the same time the size of bulge in [[alpha].sub.H] curve increases more. Although the reason is not clear, one can postulate that the rotating frequency of the vortex and/or the flow topology near the die lip exit of the downstream meniscus (different flow directions above and below the separating streamline) affects the downstream meniscus oscillations.

Vacuum pressure

To coat a thin layer, a large pressure gradient must be created across the coating bead. One way to produce such a bead is by applying a vacuum pressure. (13) The upstream meniscus is pulled by the vacuum pressure, and consequently, the upstream bead length is adjusted. As pointed out in previous studies, (5,7) a large upstream bead helps to minimize the impact of flow rate disturbance on thickness variation. Case 3 of Table 5 summarizes dimensionless numbers used to determine the effect of vacuum pressure, which is essentially the same as examining the effect of upstream bead length, on [[alpha].sub.H] under gap disturbance.

Figure 9 shows [[alpha].sub.H] as a function of frequency for the three different vacuum pressure values. Unlike flow rate disturbances, the larger upstream bead size exacerbates the thickness variations: the [[alpha].sub.H] peak increases as [P.sup.*.sub.ac] increases. As the bead size grows, the area between the oscillating web and the coating liquid, the so called footprint, increases. The larger footprint leads to greater absorption of oscillations into the coating bead, which is responsible for higher [[alpha].sub.H]. Furthermore, the increased vacuum pressure causes stronger upstream pressure gradient, and the larger upstream meniscus oscillation.

Effect of die configurations

Previous studies have focused on modifying the upstream die lip geometry, e.g., underbite and overbite configurations. (5,7) In the present study, we focus on the effect of downstream die lip geometry, such as [L.sub.d] and [H.sub.d], as shown in the right upper schematic diagram of Fig. 10, while the upstream die geometry is fixed. For the effect of [H.sub.d], we test two different operating conditions, i.e., constant (or Q) or constant [R.sub.gt]. [R.sub.gt] increases as [H.sub.d] increases when [h.sub.0] is fixed, whereas [h.sub.0] needs to be increased to fix [R.sub.gt]. Dimensionless numbers used in this test are summarized in Table 6. Note that [P.sup.*.sub.vac] is adjusted to maintain the same upstream bead length for all cases.

Figure 10a shows the effect of [L.sub.d] on [[alpha].sub.H] As discussed in the previous section about the vacuum pressure effect, [L.sub.d] also affects the footprint for contacting the oscillating web. Therefore, the larger [L.sub.d] causes the higher [[alpha].sub.H] peak. However, change of the downstream bead size does not affect [[alpha].sub.H] in the low-frequency range (0.1-50 Hz). In this range, the pressure gradient under the downstream die lip has sufficient time to follow the periodic oscillation of the gap height. Consequently, the effect of the downstream die lip footprint size is negligible. As the gap oscillates faster, however, the smaller footprint size cannot produce enough pressure gradient oscillation. This leads to small [[alpha].sub.H] peaks for small [L.sub.d] cases. Therefore, comparing with the results from the vacuum pressure changes, one can conclude that the lower frequency region is mostly controlled by the upstream bead size rather than the downstream bead size. Note that, as shown in Table 6, [P.sup.*.sub.vac] is not changed by [L.sub.d]. the upstream bead size is not affected by [L.sub.d].

Figure 10b and c show the effect of [H.sub.d] with constant [h.sub.0] and [R.sub.gt], respectively. The results suggest that [R.sub.gt] does not significantly affect the thickness variation. Therefore, one can conclude that the magnitude of the steady-state pressure gradient inside the downstream coating bead does not increase or decrease the amplitude of the pressure gradient oscillations caused by the gap disturbances. According to our computational predictions, small downstream gap height can help to suppress the downstream meniscus oscillation and decrease [[alpha].sub.H], although the reason for that is not clear.

Final remarks

This work presents computational predictions on the film thickness variation in slot coating due to gap disturbances. The effects of fluid properties, operating conditions, and die configurations are examined by comparing the amplification factor [[alpha].sub.H], the relative amplitude of the film thickness oscillation, to the imposed coating gap disturbance.

According to our predictions, the fluid properties of viscosity and surface tension do not significantly affect the periodic response of the film thickness. Even the [[alpha].sub.H] curve of an extremely strong shear-thinning fluid, xanthan gum solution, does not differ considerably from Newtonian fluid results. Considering the effect of surface tension, Newtonian fluid of low viscosity is suitable for suppressing gap disturbance to benefit from the leveling effect. However, the web speed is the most influential parameter for the gap disturbances. It can change both [[alpha].sub.H] peak height and location in the amplification factor plot. In addition, the existence of a microvortex under the downstream die lip significantly changes the shape of the [[alpha].sub.H] curve, according to the results of a gap-tothickness ratio [R.sub.gt], more than 3. The larger upstream coating bead size caused by stronger vacuum leads to larger film thickness variation. This trend is opposite that of flow rate disturbance, which can be efficiently damped by large upstream coating bead. For the die lip configuration, a small downstream gap height is desirable for suppressing the thickness variations from the gap disturbances.

In steady-state flows, Reynolds and capillary numbers can characterize the coating flow. However, in periodic flows, additional dimensionless numbers that represent the characteristic time for the momentum transfer and the characteristic time for the traveling wave are needed. For example, both viscosity and web speed can increase [N.sub.Ca] in the same manner but produce very different [[alpha].sub.H] curves. This strongly suggests that another dimensionless number needs to be introduced to characterize this periodic flow response. We are currently investigating this issue.

DOI 10.1007/s11998-015-9684-1

S. Lee, J. Nam ([mail])

Department of Chemical Engineering, Sungkyunkwan

University, 2006, Seobu-ro, Jangan-gu, Suwon, Gyeonggi-do

440-746, Korea

e-mail: nam.jaewook@gmail.comjaewooknam@skku.edu

Acknowledgments This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (NRF-2013R1A1A1004986) and by the Global Ph.D Fellowship Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2014H1A2A1019789). This work was also supported by LG Chem (No. AP1400056).

References

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(2.) Nam, J, Carvalho, MS, "Flow in Tensioned-Web-Over-Slot Die Coating: Effect of Die Lip Design." Chem. Eng. Sci., 65 (13):3957-3971 (2010)

(3.) Sartor, L, "Slot Coating: Fluid Mechanics and Die Design". PhD Thesis, University of Minnesota, 1990

(4.) Perez, EB, and Carvalho, MS, "Optimization of Slot-Coating Processes: Minimizing the Amplitude of Film-Thickness Oscillation." J. Eng. Math., 71 (1):97-108 (2011)

(5.) Romero, OJ, Carvalho, MS, "Response of Slot Coating Flows to Periodic Disturbances." Chem. Eng. Sci., 63 (8):2161-2173 (2008)

(6.) Tsuda, T, de Santos, JM, Scriven, LE, "Frequency Response Analysis of Slot Coating." AlChE J., 56 (9):2268-2279, (2010)

(7.) Lee, S, Nam, J, "Analysis of Slot Coating Flow Under Tilted Die." AlChE J. (2015). doi:10.1002/aic,14752

(8.) Yasuda, H, "The Two-Center Expansion of an Exponential Function: The Reduction of Yasudas Expression to that of Briels." J. Chem. Phys., 74 (11):6531-6532, (1981)

(9.) De Santos, JM, "Two-Phase Cocurrent Downflow Through Constricted Passages". PhD Thesis, University of Minnesota, 1991

(10.) Escudier, MP, Gouldson, IW, Pereira, AS, Pinho FT, Poole, RJ. "On the Reproducibility of the Rheology of Shear-Thinning Liquids." J. NonNewtonian Fluid Mech., 97 (2):99-124, (2001)

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(13.) Beguin, AE, "Method of Coating Strip Material", US Patent 2,681,294, 1954

Table 1: Operating conditions and parameters of the base case flow used in this study Operating parameters Parameter Range Base case Density (g/[cm.sup.3]) [rho] 0.95-1.2 1.0 Viscosity (cP) [mu] 10-1000 20 Surface tension (dyn/cm) [sigma] 10-60 40 Web speed (m/s) [V.sub.w] 0.05-5 0.4 Gap height ([micro]m) [H.sub.o] 80-300 100 Vacuum pressure (Pa) [P.sub.vac] 0-5000 2940 Wet thickness ([micro]m) [h.sub.o] 10-100 50 Table 2: The number of elements and degrees of freedom for the computational model Number of Degrees of elements freedom Newtonian fluid 384 1661 Carreau-Yasuda 454 1961 fluid Table 3: Dimensionless numbers for the shear-thinning fluid (xanthan gum solution) and Newtonian fluids [N.sub.Ca] [N.sub.Re] [P.sup.*.sub.vac] Xanthan gum solution 0.0147 1.14 0.79 Newtonian ([mu] = 0.0577 0.29 2.34 0.0346 Pa.s) Newtonian ([mu] = 0.0903 0.18 3.58 0.0542 Pa.s) Newtonian ([mu] = 0.1657 0.10 6.33 0.0994 Pa.s) Newtonian ([mu] = 0.2627 0.06 9.66 0.1576 Pa.s) The viscosity of the Newtonian fluids are chosen based on Fig. 3. [R.sub.gt] is fixed at 2. Note that the dimensionless numbers for xanthan gum solution are evaluated using the viscosity evaluated at the characteristic shear rate [y.sup.c] = H(t = 0)/[V.sub.w] Table 4: Dimensionless numbers for different surface tension values [N.sub.Ca] [N.sub.Re] [P.sup.*.sub.vac] [sigma] = 10 dyn/cm 0.8 2 27.30 [sigma] = 20 dyn/cm 0.4 2 13.85 [sigma] = 40 dyn/cm 0.2 2 7.35 Other parameters are the same as in Table 1, and [R.sub.gt] = 2 Table 5: Dimensionless numbers for different web velocity, flow rate, and vacuum pressure [N.sub.Ca] [N.sub.Re] [P.sup.*.sub.vac] Case 1: Web speed [V.sub.w] = 0.1 m/s 0.05 0.5 2.04 [V.sub.w] = 0.2 m/s 0.10 1.0 3.91 [V.sub.w] = 0.4 m/s 0.20 2.0 7.35 [V.sub.w] = 0.6 m/s 0.30 3.0 10.46 Case 2: Flow rate Q = 8 [m.sup.2]/s 0.2 2 13.41 ([R.sub.gt] = 5.00) Q = 12 [m.sup.2]/s 0.2 2 11.13 ([R.sub.gt] = 3.33) Q = 16 [m.sup.2]/s 0.2 2 9.18 ([R.sub.gt] = 2.50) O = 20 [m.sup.2]/s 0.2 2 7.35 ([R.sub.gt] = 2.00) Q = 24 [m.sup.2]/s 0.2 2 5.59 ([R.sub.gt] = 1.67) Case 3: Vacuum pressure [P.sub.vac] = 1040 Pa 0.2 2 2.60 ([L.sub.u] = 2 [h.sub.0]) [P.sub.vac] = 1995 Pa 0.2 2 4.99 ([L.sub.u] = 4 [h.sub.0]) [P.sub.vac] = 2940 Pa 0.2 2 7.35 ([L.sub.u] = 6 [h.sub.0]) Other parameters are the same as in Table 1, and [R.sub.gt] is fixed to 2 except for case 2. To control the gap/to/ thickness ratio [R.sub.gt], the web speed is fixed at 0.4 m/s Table 6: Dimensionless numbers for different downstream die lip configurations [P.sup.*. [N.sub.Ca] [N.sub.Re] sub.vac] Case 1: Downstream bead length Ld = 5[H.sub.0]~10[H.sub.0] 0.2 2 2.94 Case 2: Downstream gap height with fixed [h.sub.0] [H.sub.d] = 0.8[H.sub.0]\sim 0.2 2 4.18 ([R.sub.gt] = 1.6) [H.sub.d] = 1.0[H.sub.0]\sim 0.2 2 7.35 ([R.sub.gt] = 2.0) [H.sub.d] = 1.2[H.sub.0]\sim 0.2 2 8.38 ([R.sub.gt] = 2.4) [H.sub.d] = 1.4[H.sub.0]\sim 0.2 2 8.70 ([R.sub.gt] = 2.8) [H.sub.d] = 1.6[H.sub.0]\sim 0.2 2 8.79 (R.sub.gt] = 3.2) Case 3: Downstream gap height with fixed [R.sub.gt] [H.sub.d] = 0.8 [H.sub.0] 0.2 2 7.50 [H.sub.d] = 1.0 [H.sub.0] 0.2 2 7.35 [H.sub.d] = 1.2 [H.sub.0] 0.2 2 7.30 [H.sub.d] = 1 4 [H.sub.] 0.2 2 7.28 [H.sub.d] = 1.6 [H.sub.] 0.2 2 7.28 [R.sub.gt] is fixed at 2 except for case 2

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Please note: Some tables or figures were omitted from this article.

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Author: | Lee, Semi; Nam, Jaewook |
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Publication: | Journal of Coatings Technology and Research |

Geographic Code: | 1USA |

Date: | Sep 1, 2015 |

Words: | 5685 |

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